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Spectroscopy and Excited States

Numerically Precise Benchmark of Many-Body Self-Energies on Spherical Atoms
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  • S. Vacondio*
    S. Vacondio
    Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via G. Campi 213/a, Modena 41121, Italy
    Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy
    *E-mail: [email protected]
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    Orcidhttps://orcid.org/0000-0001-9494-7518
  • D. Varsano
    D. Varsano
    Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy
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    Orcidhttps://orcid.org/0000-0001-7675-7374
  • A. Ruini
    A. Ruini
    Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via G. Campi 213/a, Modena 41121, Italy
    Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy
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  • A. Ferretti
    A. Ferretti
    Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy
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Journal of Chemical Theory and Computation

Cite this: J. Chem. Theory Comput. 2022, 18, 6, 3703–3717
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https://doi.org/10.1021/acs.jctc.2c00048
Published May 13, 2022

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Abstract

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We investigate the performance of beyond-GW approaches in many-body perturbation theory by addressing atoms described within the spherical approximation via a dedicated numerical treatment based on B-splines and spherical harmonics. We consider the GW, second Born (2B), and GW + second order screened exchange (GW+SOSEX) self-energies and use them to obtain ionization potentials from the quasi-particle equation (QPE) solved perturbatively on top of independent-particle calculations. We also solve the linearized Sham–Schlüter equation (LSSE) and compare the resulting xc potentials against exact data. We find that the LSSE provides consistent starting points for the QPE but does not present any practical advantage in the present context. Still, the features of the xc potentials obtained with it shed light on possible strategies for the inclusion of beyond-GW diagrams in the many-body self-energy. Our findings show that solving the QPE with the GW+SOSEX self-energy on top of a PBE or PBE0 solution is a viable scheme to go beyond GW in finite systems, even in the atomic limit. However, GW shows a comparable performance if one agrees to use a hybrid starting point. We also obtain promising results with the 2B self-energy on top of Hartree–Fock, suggesting that the full time-dependent Hartree–Fock vertex may be another viable beyond-GW scheme for finite systems.

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1. Introduction

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The GW approximation (1−3) of many-body perturbation theory (MBPT) has been the state-of-the-art method for the prediction of excitation spectra for the last 40 years. (4−7) It has been applied to the band structure calculation of a variety of condensed matter systems, with the band gap being the property best predicted. Over the last 15 years, it has also found a successful application in molecules. (8−14) GW, however, still fails at predicting a number of properties and phenomena, such as occupied bandwidths, (15−23) satellites, (24) strong correlations, (25) and orbital energy ordering and spacing. (26−28) It being framed within Hedin’s iterative scheme, a self-consistent approach to GW should be implemented in principle. In practice, this is often detrimental for the prediction of band structures and energy levels, (29,30) so different flavors of self-consistency are pursued instead, (21,31,32) including no self-consistency at all, that is, single shot G0W0 calculations. This gives rise to a starting point dependence of GW, which is also numbered among its drawbacks.
In Hedin’s iterative scheme, GW corresponds to the neglect of vertex corrections. In order to fix the shortcomings of GW, there has been a recent resurgence of interest in the field of vertex corrections. While it has been suggested that vertex corrections may have little effect on the band gap prediction, (33) they have been found to improve other observable quantities such as ionization potentials of extended systems (34,35) and molecules. (27,36,37) The ambiguity concerning self-consistency has also been addressed: By now it has been accepted that vertex corrections and self-consistency effects tend, at least partially, to cancel out, (23,38,39) justifying non-self-consistent GW calculations on the one hand and suggesting that full self-consistency should be pursued only in the presence of vertex corrections on the other. There is no unambiguous way to go beyond GW. Following Hedin’s scheme, the most natural approach would be to simply take the first self-energy diagram of higher order than GW in the screened interaction. (33,34,38) Alternative approaches include those targeting positive-definiteness of the spectral function (40−42) or compliance with the Ward identity. (20,35) For computational simplicity, local 2-point vertices have also been proposed. (35,43−45)
In the present work, we benchmark low-order expansions of self-energies obtained by employing nontrivial vertex functions (see Appendix A). In particular, besides the GW approximation, we consider the second Born (2B) self-energy (46) and the GW plus second order screened exchange (GW+SOSEX) self-energy introduced in refs (27and47). We use spherical atoms as our test bed, reducing the numerical problem to one spatial dimension and treating it with a dedicated all-electron implementation based on a B-spline basis and spherical harmonics. The simplicity of the problem allows us to avoid the use of a number of approximate techniques commonly adopted in general-purpose codes, including pseudopotentials, frequency representations, (48,49) and extrapolations to the complete basis set limit. The numerical error stemming from these approximations may hinder the assessment of method accuracy and makes the comparison of results from different codes difficult. Moreover, assuming a negligible contribution from relativistic effects, the numerical precision we achieve allows for a direct comparison with experiments. To our knowledge, in the literature there are few MBPT studies focused on atoms, (8,9,50−53) while some general MBPT benchmarks include atomic systems as a part of larger test sets. (54,55)
Another important aspect to be assessed when using self-energy-based methods is the study of the optimal starting point and possibly even of the effect of self-consistency. If on the one hand 2B is naturally framed within the time-dependent Hartree–Fock (TDHF) approach (8,36,37) and is therefore fit for a HF reference, on the other hand, GW and GW+SOSEX have shown more ambiguity. In molecules, GW seems to work best on average with a hybrid DFT starting point. (11,14,26) This can be explained by the fact that HF constitutes a decent guess for orbitals and eigenvalues to start with but offers a poor description of screening in self-energies containing a random phase approximation (RPA) screened interaction such as GW. As already witnessed in solid state systems, the RPA screening tends to perform much better when built on the KS system, (56) which however constitutes an especially bad guess for energy levels in molecules, where local and semilocal functionals miss the 1/r decay of the xc potential. (57,58) Therefore, hybrid functionals such as PBE0 should in principle bring in the best of both worlds, but this is not always true: In fact, the HF reference and different flavors of self-consistent schemes have also been considered in specific cases and have proven better than a hybrid reference. (52,54,59−61) GW+SOSEX has so far shown less ambiguity, performing much better with PBE and PBE0 (27,62) rather than with HF; (28) self-consistent schemes have only been marginally addressed with this self-energy. (63)
In this work, other than considering the usual HF, PBE, and PBE0 references in perturbative (or “one-shot”) many-body calculations, we also address self-consistency in a less traditional way by employing the self-consistent solution of the linearized Sham–Schlüter equation (LSSE) (64) as a starting point. As shown by our results, in atoms this constitutes a privileged starting point whose highest occupied molecular orbital (HOMO) is left mostly unchanged by a single iteration of the Dyson equation, provided that the self-energy used in the Dyson equation is the same as that used in the LSSE. (65,66) This is a strong hint that, as is expected, (67) the density of the starting point is largely preserved, given the relation that HOMO energy levels have with the decay of the density. (68,69) Band gaps similar to those of self-consistent GW have been found in condensed matter by adopting this procedure, (70) and a good agreement with experiment has also been found for selected systems. (71)
Given the availability of very accurate xc potentials for selected atoms, (58,72) we also propose the solution of the LSSE as a further benchmark for self-energies, as is also done with total energy functionals. (73) In this respect, the LSSE can be considered an alternative, if not an approximation, to the self-consistent Dyson equation of MBPT. In fact, both approaches are based on the stationarity of the Klein energy functional, the latter with an unconstrained Green’s function (GF) and the former under the restriction of the GF being noninteracting and subject to a local potential. (67) The accuracy of the LSSE as an approximation to the self-consistent Dyson equation has not been thoroughly investigated yet. Solving the LSSE with the HF self-energy has been found to yield good approximations to the HF HOMOs, and the upper valence thus obtained is also in closer agreement with HF than one would get with local and semilocal KS-DFT functionals. (74,75) In this case the static, spatially nonlocal Fock exchange self-energy is approximated with a static, spatially local xc potential. In general, however, the xc potential is required to approximate dynamical, spatially nonlocal self-energies obtained in the self-consistent solution of the Dyson equation, which may be too demanding. As of now, an argument involving the adiabatic connection between the self-consistent Dyson equation and the self-consistent KS equations has been proposed in order to justify the LSSE solution as a starting point for many-body calculations (see Appendix B of ref (65)): solving the Dyson quasi-particle equation on top of it (without renormalization factor) provides a first-order approximation to the self-consistent quasi-particle energies.
In general, our results suggest that despite the consistency that the LSSE starting points display for many-body calculations these are not a good approximation to the self-consistent Dyson equation. The numerical effort required for the solution of the LSSE is not even justified by an increased accuracy with respect to traditional perturbative schemes starting from (generalized) KS-DFT solutions. In fact, when the GW and GW+SOSEX self-energies are used, we find it beneficial to use such perturbative schemes, which are known to generally render the RPA screened interaction W reasonably accurate as discussed earlier in this section. Confirming the common knowledge on finite systems, (11,14,26) we find that even in atoms the GW self-energy needs a fraction of nonlocal exchange in the KS-DFT starting point to achieve good accuracy, which we obtain with the PBE0 functional. Instead, the GW+SOSEX self-energy displays no such need, also showing a reduced starting point dependence in going from PBE to PBE0. (27) The 2B self-energy is also capable of yielding good, if not excellent, accuracy in many atoms at a reduced computational cost, provided that the HF starting point is used. This draws our interest to the full TDHF vertex.
This article is organized as follows: In section 2, we present the two ways in which we deal with self-energies in this work: by plugging them either in the many-body QPE or in the LSSE. Then we introduce the self-energies under investigation and briefly detail how we perform frequency integrations. We present and discuss the results in section 3, starting from the exposition of the main numerical aspects of our treatment (section 3.1). Then, given the specificity of the approach, we devote section 3.2 to elucidating the features of the xc potentials obtained from the self-consistent solution of the LSSE. Next, in section 3.3, we assess the performance of the selected self-energies by looking at ionization potentials (IPs) as obtained from the negative of the HOMO energies. Finally, we draw our conclusions in section 4 and elaborate on the possible forthcoming work.

2. Theoretical Framework

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2.1. The Green’s Function and the Dyson Equation

The mathematical key object of MBPT is the one-particle Green’s function (GF), from which the charged excitation spectrum of a system can be extracted. Starting from the Hartree GF, GH, the GF of the system of interacting electrons G can be obtained via the Dyson equation, (76)
(1)
involving the self-energy operator Σ. The self-energy can be cast as a functional of the interacting GF, that is, Σ = Σ[G], in order to write approximations to Σ as sums of a few physically meaningful Feynman diagrams.
Having Σ = Σ[G] implies that the Dyson equation should in principle be solved self-consistently. In practical implementations, the calculation of G is often perturbative (also termed “one-shot”), consisting of a single iteration of the Dyson equation upon an independent-particle solution of the electronic problem. Here we follow this path starting from either a HF or a KS-DFT GF, whereas in the next section we will tackle self-consistency by letting Σ = Σ[G] in the Sham–Schlüter equation (although we will quickly approximate G with GKS). Therefore, the Dyson equation reads either
(2)
when starting from a HF solution or
(3)
when starting from a KS-DFT solution. In the latter case, the xc potential, vxc, is removed and the nonlocal Fock exchange, Σx, is added; in the former case, the Fock exchange is already present in GHF, and only the correlation self-energy, Σc = Σ – Σx, is needed in going from the noninteracting GHF to the interacting G. From the GF G thus obtained, the spectral function
(4)
can be computed, and its peaks can be identified with the charged excitations of the system. Here ϵF is the Fermi energy. The operator A is meant to reproduce the spectra from angle-resolved photoemission experiments, its trace representing the density of states. In the absence of strong correlation, the main peaks of the spectral function are found in correspondence of the so-called quasi-particle (QP) energies, ϵQP. For numerical convenience, a diagonal approximation of the GF, self-energy, and spectral function operators is often adopted, yielding the quasi-particle equations (QPEs),
(5)
and
(6)
for the computation of QP energies with the HF and the KS-DFT starting points, respectively. The i index runs over the independent-particle states of the system, to which QP states correspond one by one. We refer to the difference between the QP energy and the independent-particle energy as QP shift.

2.2. The xc Potential and the Sham–Schlüter Equation

Both MBPT and KS-DFT are capable of yielding the exact ground-state electron density of a system, provided that the exact self-energy Σ and the exact xc potential vxc are used. Applying to both sides of eq 3 (now with Σ = Σ[G]) a linear density operator , which maps G to the ground-state density of the system, yields
(7)
which is known as the Sham–Schlüter equation (SSE). (64) The density condition employed for deriving it implies that GKS produces the same density as G. The equation is in the form Ax = b, and can be solved as a linear system in the unknown x, which is the xc potential, vxc(r). Since the unknown itself is needed to compute GKS, which enters the kernel A and the inhomogeneous term b, the problem must, in general, be solved iteratively. Most often, it is linearized by performing the substitution G → GKS, which amounts to retaining the lowest order in GKS in eq 3. By doing so one obtains the linearized Sham–Schlüter equation (LSSE):
(8)
where χ0KS is the KS independent-particle static polarizability. We emphasize that the dependence Σ = Σ[G] in the full SSE implies that the full Dyson equation for G, eq 1, is obeyed self-consistently, while this is not the case for the LSSE, where Σ = Σ[GKS] and there is no reference to G any longer. This implies that the density condition used to derive the SSE no longer holds exactly, and it has to be verified how close the densities produced by the fully self-consistent G and by the LSSE-self-consistent GKS are. The KS scheme differs significantly from the MBPT scheme, in that it does not account for frequency dependence and is affected by the KS gap problem. (77−79) Nonetheless, one iteration of the Dyson equation is expected to preserve the density of GKS as obtained from the self-consistent LSSE. (65−67)
Alternatively, the LSSE can also be derived via an optimized effective potential (OEP) strategy starting from total energy functionals. (67,80,81) In the framework of MBPT, it is possible to build total energy functionals that depend directly on the GF. (82,83) In this case, the LSSE can be obtained (67,84) by minimizing the Klein functional with the constraint that the GF be given by a static and spatially local potential. The Klein functional reads (83,85−88)
(9)
and its (unconstrained) optimization implies that the Dyson equation is obeyed self-consistently at stationarity. In this expression, Trω{} stands for , EH is the Hartree energy and G0 is the free-particle GF corresponding to H0 = T + vext. The Φxc[G] functional, which carries electron interaction contributions beyond Hartree, is also related to the self-energy by
(10)
Constraining the Klein functional to GF Gs obtained from a static and spatially local potential vs(r) yields
(11)
where Ts is the kinetic energy of noninteracting particles (subject to a local potential) of density ρ. Imposing the variational condition yields
(12)
where vxcoep(r) is obtained as the solution of the LSSE, eq 8. Importantly, as in the unconstrained optimization, the self-energy in the LSSE/OEP case is obtained from the Φ functional via eq 10. Nevertheless, Σ is no longer a functional of G but rather of GKS (see eqs 7 and 8).

2.3. Self-Energies

In the present work, we focus on the second Born (2B), GW, and GW plus second-order screened exchange (GW+SOSEX) self-energies. As seen in the first line of Figure 1, besides the Fock exchange, the 2B self-energy includes a second order direct (2Bd) and a second order exchange (2Bx) diagram. These are all possible second order irreducible diagrams. As long as occupied states are concerned, they balance each other with respect to the self-screening error, meaning that solving the Dyson equation with this self-energy in a one-electron system yields a vanishing QP shift of the occupied level. On the other hand, in 2B, screening is accounted for through the use of one bubble in the direct diagram, making this self-energy unsuitable for extended systems (especially metals) and possibly large polarizable molecules.

Figure 1

Figure 1. Self-energy diagrams benchmarked in this work. Solid lines represent the fermion propagators, dashed lines the bare Coulomb interaction, and wiggly lines the dressed Coulomb interaction.

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Screening is better accounted for within the GW approximation (second line of Figure 1), where it is introduced at the random phase approximation (RPA) level. This comes with the price of having only a direct diagram in the self-energy, giving rise to a self-screening error. (88−90) Ameliorating the situation with the inclusion of the bare 2Bx diagram, balancing the 2Bd diagram naturally contained in GW, is unsuitable in extended systems and has proven unsuccessful even in molecules. (26) A partial inclusion of RPA screening at second order via the SOSEX diagram (third line of Figure 1) has led to more promising results, (27,62,63) and the diagram itself can be justified by symmetry arguments (27) originally made for RPA total energies. (91,92) These arguments aim at restoring the antisymmetry of the two-particle GF upon odd permutation of its space–time arguments (also referred to as crossing symmetry), as the fermionic nature of electrons requires. A justification of the SOSEX diagram within the T-matrix formalism is also possible. (47) It must be noted that even with the SOSEX diagram full cancellation of the self-screening error occurs only up to second order in the bare Coulomb interaction, that is, at the 2B level of theory. It is also unknown whether the partial screening in this diagram can actually make it suitable for metals.
We now give the analytical expressions for the self-energies, in a real-space/frequency representation. The 2B self-energy reads
(13)
with Σx being the Fock self-energy,
(14)
and
(15)
v is the Coulomb interaction. The GW self-energy reads:
(16)
Finally, the SOSEX contribution to the GW+SOSEX self-energy reads
(17)
with the GW+SOSEX self-energy obviously given by ΣGW+SOSEX = ΣGW + ΣSOSEX. All the screened interactions W are to be intended in the RPA, that is, the Dyson equation,
(18)
is obeyed with χ0 being the independent-particle irreducible polarizability, that is,
(19)
We conclude this section by linking the self-energies to their respective total energy functionals, when available. Approximate KS-DFT total energies can be obtained via the constrained Klein functional by making approximations on the Φ functional (see eq 11). We remark that even with the exact Φ functional, the exact energy cannot be obtained as long as KS GFs are fed to the Klein functional (therefore making it the constrained Klein functional). An approximation on the Φ functional in turn entails an approximation on the self-energy (see eq 10). In particular, the Φ functional yielding the HF self-energy upon differentiation (first diagram of either line in Figure 2) leads to the exact-exchange (EXX) energy upon insertion in eq 11. Following this reasoning, the 2B self-energy corresponds to the MP2 energy (first line in Figure 2), and the GW self-energy corresponds to the RPA energy (second line in Figure 2). On the other hand, the GW+SOSEX self-energy (at least in the current formulation with W treated at the RPA level) is not Φ-derivable and is therefore not rigorously suitable for the variational argument presented in section 2.2. The total energy expressions termed RPA+SOSEX introduced in the frameworks of coupled cluster theory (91) and of the adiabatic connection formula (92) are not formally linked to the GW+SOSEX self-energy. We nonetheless apply the LSSE treatment to the GW+SOSEX self-energy for benchmarking purposes.

Figure 2

Figure 2. MP2 and RPA Φ functionals, from which the 2B and GW self-energies, respectively, are obtained upon differentiation with respect to the GF.

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For coherence with this picture and the literature, we adopt the following nomenclature: we will refer to the potential obtained from the HF self-energy through the solution of the LSSE as the EXX potential, the potential obtained from the 2B self-energy will be the MP2 potential, and the one obtained from the GW self-energy will be the RPA potential. On the other hand, the GW+SOSEX self-energy will simply correspond to the GW+SOSEX potential.

2.4. Frequency Integrations

Implementing the solution of the Dyson equation in a one-shot fashion greatly simplifies the numerical problem at hand. In this case, all the GFs appearing in eqs 13–19 can be taken to be noninteracting and cast in a spectral representation by using the orbitals ψα and eigenvalues ϵα of the underlying independent particle Hamiltonian, according to
(20)
where η is a positive, vanishingly small real value ensuring the time-ordering expression of the GF; we use Greek letters to indicate suitable multi-indices. Frequency integrals can then be easily performed by means of the residue theorem, when the explicit knowledge of the poles of W is not required. In fact, while it is in principle possible to cast W as a sum over poles, for example, by solving an eigenvalue problem, (93) here we solve eq 18 via a matrix inversion at given frequencies in the complex plane, so we do not have access to the explicit position of the poles of W. Before proceeding, we introduce the following notation for partially and fully saturated Coulomb integrals, respectively:
(21)
A similar notation is also used for W(ω).
Next, we consider frequency integrations for each self-energy and see how the frequency dependence of W is dealt with, when present. In the case of the 2B self-energy, there is no W and the solution of the frequency integrals using the residue theorem simply yields
(22)
(23)
with
(24)
We have also set θμ = θ(ϵμ – ϵF) and θ̅μ = 1 – θμ = θ(ϵF – ϵμ). We place 0 in the subscripts of self-energies to indicate that they are computed with a one-shot procedure.
In the case of GW, the frequency integration involving W can be dealt with using a contour deformation technique. (94,95) The same can be done with the SOSEX diagram after carrying out the integral in ω2 of eq 17; by doing so using the residue theorem, one gets
(25)
with
(26)
The fact that Iμν has the same pole structure as W allows for the application of the contour deformation technique.
When Σ enters the LSSE (eq 8), the frequency integral in the RHS, that is, the inhomogeneous term, can be again performed using the residue theorem, and the integrals involving W that were treated using the contour deformation simply reduce to integrals over the imaginary axis. (96,97) As an example, we consider the frequency integral in the correlation contribution to the RPA inhomogeneous term, obtained when the GW self-energy is used:
(27)
with Wp = W – vc, and we stress that is obtained from GKS in this last expression. Expressing each GF in spectral form according to eq 20 (also dropping the KS label for ease of notation), and integrating with respect to ω the three GF of eq 27, one obtains
(28)
The fact that this expression has the same pole structure as W allows once again for integration with respect to ω′ in eq 27 via the contour deformation technique, which in this case trivially reduces to an integration over the imaginary frequency axis. The result in eq 28 can also be obtained for the MP2 and SOSEX inhomogeneous terms. While for the latter a continuation to the imaginary axis is again applied to compute the leftover frequency integral, a fully analytical expression can be worked out for the former using the residue theorem.

3. Results and Discussion

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3.1. Numerical Details

The numerical problem for spherical atoms can be reduced to one dimension, namely, the radial one. The leftover angular degrees of freedom can be treated analytically by means of the spherical harmonics machinery. In other words, matrices are made block diagonal in the spherical harmonics representation. To give a flavor of how the angular problem is treated, we mention that the coupling of spherical harmonics in direct self-energy diagrams such as 2Bd and GW yields the Wigner 3j symbols, whereas in exchange diagrams such as 2Bx and SOSEX, the Wigner 6j symbols appear. The basics of this machinery can be found in ref (98).
The numerical radial problem is treated with a B-spline basis set in analogy with the method presented in ref (99). A cubic grid extending over a 30 Bohr-radius rmax is employed, with vanishing functions at the boundaries as imposed by the B-splines. Matrix diagonalizations and inversions are performed in the B-spline basis, ensuring smoothness and faithful representation of functions and operators. We checked that 300 B-splines ensure convergence within 1 mRy of the energy values presented throughout. We had to sometimes resort to a smaller basis set of 100 B-splines for more demanding computations, such as those producing the GW+SOSEX potentials. We verified that this does not heavily impact the quality of the results anyway, which remain converged well within a threshold of 10 mRy even with this smaller basis.
Self-energies are calculated using the techniques presented in section 2.4 over a frequency grid to solve eqs 5 and 6. The calculation of LSSE inhomogeneous terms is also presented in section 2.4, but the numerical solution of the LSSE requires more care. The LSSE linear system is rank-deficient, (66) due to the degree of freedom given by the possibility of incorporating inside the xc potential an arbitrary energy shift. In the spherical problem, this degree of freedom can be saturated by imposing the known theoretical asymptotic behavior
(29)
with v̅xc → 0 faster than 1/r for r → ∞. In our implementation, we solve for rv̅xc with a singular value decomposition (SVD) treatment for the linear system inversion. The SVD allows us to impose the vanishing of vxc for r → ∞. Thanks to the B-spline basis set, the unphysical oscillations in the potentials found when solving the problem on a radial grid are avoided. (98,99) However, we found numerical noise in the matching of vxc with the long-range solution −1/r. Although the SVD showed some efficacy in removing such noise, we found that it still impacted the quality of the KS HOMOs. We managed to improve them by adopting the following procedure: At each LSSE iteration, we discard vxc in the interval [rcut, rmax], with rcut being the radial coordinate at which the noise onset is found, and replace it with −1/r; then we shift vxc in the leftover interval [0, rcut] in order to match −1/r at rcut. Unfortunately rcut is system-dependent and has to be evaluated for every atom and spin polarization. Nevertheless, this procedure allows us to obtain EXX HOMOs in very good agreement with the HF ones and MP2 and RPA HOMOs in excellent agreement with those in the literature. (98,99)

3.2. EXX, MP2, RPA, and GW+SOSEX xc Potentials

The xc potentials for atoms as obtained from the self-consistent solution of eq 8 using the EXX, MP2, and RPA functionals have already been documented and characterized in the literature for a number of atoms, mostly up to Ar. (74,75,98−104) Here we also consider the solution of eq 8 with the GW+SOSEX self-energy and present calculations for an extended set of spherical atoms, including those from the fourth period (i.e., K, Ca, Mn, Zn, As, and Kr), in which the 3d shell comes into play starting from Mn. Moreover, we extend the RPA calculations previously performed on closed-shell spherical atoms (99) to spin-polarized spherical atoms having half-filled shells.
In Figure 3, we show the potentials for two of the heavier atoms we present in this work, Ca and Zn. The shell structure of the electronic charge in atoms is reflected on the potential through the presence of bumps. These bumps are found between two regions in space where the charge density originates from different orbitals. (79,105,106) Local functionals used in KS-DFT such as local-density approximation (LDA) cannot describe the charge localization due to the atomic shell structure, displaying smoother behavior in the xc potential. PBE does better in this regard but still yields xc potentials far away from the LSSE solutions, which properly account for exchange. Moreover, neither LDA nor PBE capture the correct 1/r decay of the xc potential (insets of Figure 3), which is due to the predominance of exchange in the long-range. (57,58) Finally, it can also be noticed how filling the shells progressively shortens the range of the potentials, in analogy with the density.

Figure 3

Figure 3. EXX, RPA, LDA, and PBE xc potentials for Ca (left) and Zn (right); (inset) xc potentials multiplied by the radial coordinate.

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The inclusion of correlation of any kind generally has the effect of damping the bumps of the xc potential. In order to study correlation with a suitable scale, it is customary to resort to the correlation potential, which is obtained by removing the exchange contribution from the xc potential. Here we adopt the definition of ref (99), relying on the difference between two potentials evaluated at the respective self-consistent densities:
(30)
that is, the potential to be subtracted is the EXX one, vx evaluated at its self-consistent density ρx. Some correlation potentials calculated by us are plotted in Figures 4, 5, and 6. For He, Be, and Ne a comparison is made with the exact results obtained by Umrigar and Gonze. (58) We note that we compare potentials obtained at self-consistency using the respective functionals: this is not the approach adopted by all authors, some preferring to evaluate the potentials with one iteration of the LSSE starting from accurate densities. (98)

Figure 4

Figure 4. Correlation potentials for helium, beryllium, and neon. The MP2 correlation potential for beryllium is computed on top of EXX orbitals and eigenvalues. Exact correlation potentials are from ref (58).

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Figure 5

Figure 5. Alternative beyond-RPA correlation potentials for beryllium. Self-consistency has little impact on the GW+SOSEX HOMO: we have ϵHOMOscGW+SOSEX = −0.733 Ry and ϵHOMOGW+SOSEX@EXX = −0.735 Ry. Exact correlation potential is from ref (58).

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Figure 6

Figure 6. Correlation potentials for sodium and potassium. Thick lines are for the spin majority channel; thin lines are for the spin minority channel.

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In the next section we discuss the IPs as obtained from the KS HOMOs, which can be rationalized observing the behavior of the respective xc potential as displayed in Figure 4. The MP2 functional is known to yield the main features of the exact correlation potential, although often missing the correct scale. (98,103) For instance, in He the MP2 potential is close to the exact one in the region near the nucleus but poorly matches it in the medium and long range, consistently attaining higher values. As a result, the MP2 HOMO level of He is too shallow and shows little improvement as compared with the EXX one. In contrast, the RPA potential offers a poorer description in the short-range radial region but more closely resembles the exact potential in the middle and long range. This leads to a markedly improved KS eigenvalue, almost exactly matching the one from experiments. In evaluating these results, one should not be fooled by the logarithmic scale commonly adopted for plotting the potentials, which has the effect of extending over a considerable length very small radial regions.
Moving on to Be, we recall that for this atom no self-consistent xc potential can be obtained from the MP2 functional. (98) The MP2 energy functional is divergent in the presence of a vanishing gap, and the self-consistent procedure leads indeed to the closure of the KS gap. This instability is likely to occur in systems with small KS gaps, as is the case for Be. Ca also presents a small KS gap, and our calculations confirm the presence of the instability also for this atom. As a consequence, the MP2 correlation potentials for Be and Ca are computed by iterating the LSSE solver only once starting from EXX orbitals and eigenvalues. Switching to the RPA functional restores the possibility of having a self-consistent xc potential. It must be noted, however, that the improvement toward the exact potential in Be is modest, as is the improvement of the KS HOMO. Finally, for Ne the MP2 functional produces an especially bad result, whereas the RPA functional performs much better. As we shall see in the next section, the 2B self-energy also performs poorly with Ne. There we discuss why it is likely that it is necessary to fully include screening at least at the RPA level in atoms such as Ne.
We now consider the GW+SOSEX potentials. As a first remark, we observe that the tail of the correlation potential is strikingly off in He, with the zero being approached from below. The issue becomes less relevant in heavier atoms but is still present as one can see in Ne. As a consequence, the GW+SOSEX HOMO levels are generally deeper (with the exception of the group 1 atoms) than the RPA ones, which are already too low as compared with experiment. Even in the short-range region, the GW+SOSEX potential is not always satisfactory: while there is an improvement with respect to RPA in He and Ne, in Be there is a visible worsening. It should be noted, however, that the inclusion of a screened exchange diagram allows for a self-consistent solution in Be, whereas the bare second-order diagram in MP2 did not. If one computes the GW+SOSEX potential for Be with a single iteration of the LSSE on an EXX self-consistent solution, as we did in the MP2 case, a short-range description in line with the ones of MP2 and RPA is recovered (see Figure 5). In this case, self-consistency appears to exacerbate the bad features of the GW+SOSEX potential, as already seen with the MP2 functional. (98) The KS HOMO is subject to little influence by self-consistency anyway, since it mainly affects the potential in the short-range.
In Figure 5, we also briefly explore the “RPA plus MP2 exchange” (RPA+MP2x) functional in the case of Be, yielding a self-energy consisting of the GW plus 2Bx diagram in the SSE. The resulting xc potential presents features in opposition with those of the exact potential, suggesting that the unscreened 2Bx diagram is not properly balanced by the GW diagram. In He, we even find that the peak in the correlation potential disappears. Overall, all our calculations show that the RPA+MP2x functional performs badly with atoms and should probably not be employed in general. The related GW+2Bx self-energy has also been found to perform poorly in the past. (26) This suggests that in order to build upon the RPA/GW level of theory some amount of screening should always be included in the higher-order diagrams. Our results show that doing it with the SOSEX diagram restores the features of the exact potential, although retaining a suboptimal asymptotic behavior. The GW+SOSEX diagram can then be viewed more positively and considered as a starting point for more sophisticated self-energies, which could possibly yield better results. For instance, it is an ingredient of the full second-order (in the screened interaction) diagram, which would also restore variationality if employed in the LSSE, it being a Ψ-derivable self-energy. (85,107) Additionally, a set of diagrams enforcing the positive-definiteness of the spectral function could be obtained starting from the SOSEX diagram, following the recipe provided in ref (40).
Another positive aspect of the GW+SOSEX potential is that it does circumstantially improve on the RPA one. As we discuss in section 3.3, the GW+SOSEX self-energy leads to a better prediction of the HOMO of group 1 atoms (save for H) with either the self-consistent LSSE solution discussed in this section or the one-shot QPE starting from HF (see Figure 8). Therefore, we present in Figure 6 the correlation potentials of two atoms from group 1 (Na and K). We can only speculate on what the improvements of the GW+SOSEX potential are on the RPA one, since no exact potential is currently available for these atoms. Looking at the majority spin-channel potentials, we see that the GW+SOSEX potential is found between the MP2 and the RPA potentials in the medium and long range. From the results of the next section, we can see that for atoms in group 1 the best prediction of the HOMO is given by the MP2 functional. Thus, including a screened exchange diagram, which tends to bring the RPA potential closer to the MP2 one, has the effect of improving the ionization potential, although not making it as accurate as the MP2 one.

3.3. Ionization Potentials

In this section, we address the performance of the different self-energies described in Figure 1. We do this by evaluating the mean absolute error (MAE) with respect to experiment of the IPs computed with each self-energy approximation (Figure 7). These are obtained as the negative of the HOMO energies, that is, IP = −ϵHOMO. In Table 1, we also provide the maximum and minimum errors found on the test set, and in Figure 8, we provide the error contributions resolved on each atom. The LSSE results are considered at the scf level without an iteration of the QPE. In fact, as we discuss at the end of this section, the LSSE does provide a consistent starting point for the QPE, if the same self-energy approximation is used in both steps, so that the QPE produces a mostly negligible QP shift on the HOMO.

Figure 7

Figure 7. Mean absolute error (MAE) on the ionization potentials for each method and starting point. In white, we indicate the independent-particle self-consistent field (scf) methods.

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Table 1. Negative of HOMO Energies (Ry) as Computed with Different Self-Energies and Starting Points: Mean Absolute Error (MAE), Maximum Absolute Error (Max AE), and Minimum Absolute Error (Min AE)
 @HF@PBE@PBE0LSSE
–ϵHOMO2B
MAE0.0150.0720.0400.051
Max AE0.0880.4910.2500.273
Min AE0.0000.0010.0000.000
–ϵHOMOGW
MAE0.0220.0300.0170.027
Max AE0.0480.0860.0420.060
Min AE0.0010.0010.0010.003
–ϵHOMOGW+SOSEX
MAE0.0260.0140.0170.032
Max AE0.0540.0390.0310.047
Min AE0.0010.0010.0010.000

Figure 8

Figure 8. Deviation from experiment of the ionization potentials of the neutral atoms as computed with the selected self-energies and starting points. The LSSE-MP2 IPs of Be and Ca and the LSSE-GW+SOSEX IP of Mn are computed starting from EXX orbitals and eigenvalues. Experimental IPs are from ref (108).

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The first notable result is that on average the 2B self-energy performs remarkably well on the studied atoms on top of the HF starting point. However, not considering alkali metals, it falls short of this result if the PBE or PBE0 functionals are used instead, also displaying a strong starting point dependence. We note that the 2B self-energy is generated by one iteration through Hedin’s scheme with the TDHF vertex, and subsequent truncation to second order in the Coulomb interaction. Therefore, employing the HF starting point implies having a consistent functional derivative of the self-energy with respect to the GF in the vertex equation (i.e., a physical polarizability). (36) This, together with other results from the literature, (23,36,104) points to a possible relevant role of a consistent inclusion of the vertex corrections, which should be further investigated. This may also explain why, as seen in the previous section, the MP2 functional treated at the LSSE level produces underwhelming results, along with perturbative QPE schemes using starting points different from HF.
We note, however, that with some atoms, such as Ne, the 2B self-energy fails even with the HF starting point. This can be explained with the findings of Bruneval, (52) who pointed out the sizable difference that including all the infinite GW self-energy diagrams can sometimes make as compared with including just the first single-ring one, that is, the 2Bd diagram in this work. This difference suggests that screening plays an important role also in finite systems, as already emphasized by Shirley and Martin in ref (8). Here, we find it to be especially important in Ne in order to obtain an accurate result. However, this does not imply that the RPA screening on top of a HF solution (RPA@HF), which we are using in GW@HF, is necessarily the best choice. In fact, according to ref (52), the QP shift in N is also sensitive to the full inclusion of the RPA screening beyond the 2Bd diagram via GW. However, 2B performs quite well with N, whereas GW does considerably worse. Including the SOSEX diagram further worsens the result for N, suggesting that adding more diagrams with the RPA@HF screening can do little to improve the description.
The behavior described for N is confirmed for many of the selected atoms: in most cases, GW@HF overbinds the HOMO and GW+SOSEX@HF increases the error. A similar trend has also been recently found using the GW100 test set. (28) Therefore, the introduction of the SOSEX diagram does not seem justified with the HF starting point, owing to the suboptimal RPA@HF screening. A few notable exceptions are alkali metals, Li, Na, and K: it could be argued that the self-screening of the lone s electron is particularly relevant in these atoms and is thus the main source of error for GW, which can be ameliorated by GW+SOSEX. However, this explanation is not confirmed by the H atom, the archetypical system for self-screening detection: in this case the error of GW@HF is already small to start with and GW+SOSEX@HF does worse. As we shall see, the picture is puzzlingly the opposite when using (generalized) KS-DFT solutions from the PBE and PBE0 functionals as starting points.
Interestingly, the results for GW@HF and GW+SOSEX@HF are generally mirrored in the LSSE framework: GW+SOSEX consistently improves on GW in atoms of group 1 and consistently does worse in almost all the others. This may be explained by the fact that the EXX contribution is dominant in determining the LSSE occupied orbitals and eigenvalues, which end up being similar to the HF ones, especially in the upper valence. (75) We therefore conclude that the KS spectrum as obtained from the LSSE is not sufficient to provide a better RPA screening than the HF one. The remaining discrepancies between the QPE@HF and the LSSE methods may be explained by the following two major facts: (i) the solution of the LSSE has a different unoccupied spectrum, with the LUMO being a KS bound state, whereas it is an unbound state in QPE@HF; (ii) the LSSE also involves a form of KS self-consistency. As compared with GW@HF and GW+SOSEX@HF, these two differences provide some advantage to the LSSE in noble gases, in atoms of the N family, and in Zn (the LSSE-GW+SOSEX IP of Zn surprisingly matching the experiment) but end up worsening the results in group 1 and 2 atoms.
If solutions using density functionals in the (generalized) KS scheme are used as a starting point, the picture drastically changes. GW@PBE is seen to do worse than GW@HF, now strongly underbinding the HOMO in many lighter atoms. Adding the SOSEX diagram greatly improves the HOMO energies, reducing by half the MAE. The GW+SOSEX self-energy is now seen to perform worst with the atoms with which it performed best using the HF starting point, that is the group 1 atoms; and conversely, GW performs better. Notably, the rationale for the SOSEX diagram as a cure for the self-screening problem of GW seems now justified if H is considered. This confirms early observations that the GW self-screening problem in H can be exacerbated by a starting point that suffers from a self-interaction/delocalization error and is instead small when an accurate starting point is adopted. (50,51)
Going from the PBE to the PBE0 starting point greatly affects the performance of GW, which is now comparable with GW+SOSEX@PBE and scarcely the one of GW+SOSEX. On the one hand, this confirms a reduced starting point dependence of the GW+SOSEX self-energy, (27,28) also seen in self-consistent schemes. (63) On the other hand, a fraction of exchange in the starting point might as well be sufficient for obtaining accurate IPs with GW. (14) Of course one should consider larger test sets for a more reliable assessment: the IPs of ref (27) obtained with the larger G2 test set seem to confirm this picture. The more recent results of ref (28) obtained with the GW100 test set do show a marked improvement with the GW+SOSEX self-energy instead, even with the PBE0 reference. Moreover, other observables such as deeper binding energies may also be considered for a more comprehensive assessment.
We conclude this section by checking the validity of the LSSE starting points as approximate self-consistent solutions. In Table 2, we report the ionization potentials of all atoms as obtained from self-consistent solutions of the LSSE and from the solution of the QPE (eq 6) on top of the LSSE solutions. As expected on the basis of the LSSE construction (preserving the charge density, and therefore the IP, to first order in converting the KS equations into the Dyson equation), (65−67) it can be seen that 2B@LSSE-MP2 and GW@LSSE-RPA calculations exhibit almost no QP shift in the IPs, the discrepancy between the KS HOMO and the QP HOMO being at most 1 mRy (with the exception of the 2B@LSSE-MP2 calculation for Zn). The discrepancies between the GW+SOSEX HOMOs are more sizable instead but still within 10 mRy (with the exception of Li). The reason why this occurs may be that the GW+SOSEX self-energy is not suitable for the variational argument presented in section 2.2. Moreover, the suboptimal long-range behavior of the GW+SOSEX xc potential may also hint at some underlying numerical issue in the solution of the LSSE. In fact, these potentials are more sensitive to the rcut parameter chosen for matching with the long-range solution (see section 3 of the Supporting Information); therefore, a larger error bar is to be expected on the IPs as obtained from the GW+SOSEX HOMO energies than one gets with the MP2 and RPA potentials, however not undermining the assessment presented in this section. One may conservatively estimate this error from the QP shift obtained from the Dyson equation, which, as already mentioned, can be up to the order of 10 mRy.
Table 2. IPs (Ry) from the Self-Consistent Solution of the LSSE vs IPs from the One-Shot Solution of the QPE upon the Self-Consistent LSSEa
 LSSEQPE@LSSE 
atomsMP2RPAGW+SOSEX2BGWGW+SOSEXexpt (108)
H1.0000.9971.0110.9990.9971.0170.999467
He1.7851.8021.8331.7851.8031.8411.80714
Li0.3960.4550.4370.3960.4550.4490.39628
Be 0.7100.733 0.7090.7410.68521
N1.0001.0721.1061.0011.0711.1131.06824
Ne1.3161.5941.6291.3161.5941.6391.58496
Na0.3810.4260.4030.3810.4260.4110.37772
Mg0.6040.5960.5950.6030.5960.6010.56199
P0.7700.7920.7990.7710.7930.8070.7707575
Ar1.1171.1831.1961.1171.1831.2061.15831
K0.3330.3630.3400.3330.3640.3450.31904
Ca 0.4830.495 0.4830.5030.44931
Mn0.6230.570 0.6230.570 0.546390
Zn0.8000.7070.6890.7950.7070.6960.6904609
As0.7380.7570.7590.7370.7580.7660.71945
Kr1.0241.0691.0751.0251.0701.0821.02895
a

No self-consistent MP2 xc potentials can be computed for Be and Ca owing to an instability due to the closure of the KS gap in the self-consistency procedure. No self-consistent GW+SOSEX potential for Mn is present either due to numerical issues. Basis set size = 100 B-splines.

In order to check whether the LSSE could be an approximate pathway to self-consistency, it would also be useful to compare the MP2 and RPA IPs with the self-consistent 2B and GW ones, respectively. In Table 3, we compare our IPs from the solution of the LSSE with those from the self-consistent calculations of refs (9and46) (see also section 1 of the Supporting Information). We find a generally poor agreement of the 2B/MP2 IPs and a partial agreement of the GW/RPA ones, that is, only for He and Ne do we have similar GW/RPA IPs. While some aspects of our numerical treatment may differ from those of ref (46), this does not seem sufficient to explain so-large a discrepancy between the 2B and MP2 results. Therefore, it can be argued that the LSSE does not provide a good approximation to the self-consistent Dyson equation when the 2B self-energy is employed. The agreement between the two methods appears to be sometimes better with the GW self-energy but still system-dependent. Further precise self-consistent calculations are needed to assess when and whether the two methods yield similar results. Other observable quantities should also be targeted in the LSSE approximate self-consistency, such as the HOMO–LUMO gap upon inclusion of the derivative discontinuity correction. (70)
Table 3. Self-Consistent LSSE RPA and MP2 IPs (Ry) of Selected Atoms Compared to Self-Consistent GW and 2B Ionization Potentials from the Literature
atomsLSSE-RPA (B-spline)sc-GW (9) (Slater orb.)LSSE-MP2 (B-spline)sc-2B (46) (Slater orb.)expt (108)
He1.8041.8051.7861.8111.80714
Be0.7110.636 0.6600.68521
Ne1.5931.6001.3121.4971.58496
Mg0.5970.5350.6050.5530.56199

4. Conclusions and Perspectives

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In this work, we have presented a benchmark of three self-energies derived within MBPT, that is, 2B, GW, and GW+SOSEX. We implemented the resolution of the many-body QPE and of the self-consistent LSSE with these self-energies for a set of spherical atoms, aided by an easily converged B-spline basis set. We chose to also consider the solution of the LSSE for two reasons: to possibly have a consistent starting point for the QPE and, as a further benchmark, to compare the xc potentials we get from it with the exact ones available in the literature.
The EXX, MP2, and RPA xc potentials computed by solving the LSSE reproduce the ones present in the literature. We extended these calculations to atoms belonging to the fourth period and to spin-polarized cases, the latter kind not yet considered for the RPA potential. Furthermore, we explored the GW+SOSEX self-energy in the LSSE framework: we find that it brings the correlation potentials closer to the exact ones in the short-range for He and Ne but not for Be; in the long-range, the description is even worse, with these potentials reaching zero from below. This is especially noticeable in lighter atoms, whose HOMO is affected worst by this bad feature. In contrast, the GW+SOSEX potential seems to improve on GW in group 1 atoms by bringing the description closer to the MP2 one.
The bottom line of this part of the work is that, as is known in MBPT, also in the LSSE framework “more diagrams” does not necessarily mean “better results”: the GW+2Bx/RPA+MP2x calculations we briefly presented are explanatory in this regard. At least we find the GW+SOSEX potentials to restore the correct main features of the exact potentials by building upon the GW+2Bx/RPA+MP2x level of theory. Therefore, we believe that improving on the GW+SOSEX level of theory with a sensibly chosen criterion could fix the issue of the long-range behavior. Among these criteria, we include variationality (e.g., considering the self-energy diagram of second order in the screened interaction) or positive definiteness of the spectral function (following ref (40)).
Concerning the IPs, we find the solutions of the LSSE to provide a consistent starting point for the QPE, yielding small to vanishing QP shifts in the HOMO and therefore hinting at conservation of the starting density to a large degree. (67) However, the comparison with the existing literature, though limited, suggests that this does not necessarily mean that the IPs we obtain from the self-consistent LSSE are a good approximation to the ones obtained from the self-consistent Dyson equation. We also find that the LSSE generally performs poorly as compared with the resolution of the QPE starting from independent-particle solutions given by either the HF method or semilocal/hybrid functionals in (generalized) KS-DFT. In this regard, we get the best performance from GW+SOSEX@PBE/PBE0, 2B@HF, and GW@PBE0. This confirms the common knowledge that hybrid functionals provide good starting points for GW applied to finite systems, (11) even in the atomic limit.
The justification of the SOSEX diagram as a cure for the self-screening error of GW seems to be correct only with the PBE and PBE0 starting points. On the contrary, adding the SOSEX diagram to GW with the HF starting point appears to be inaccurate, likely due to the poor description given by the RPA screening computed on top of a HF solution. These performances are in line with those found with larger test sets involving small molecules. (27,28) However, we must point out that, as in ref (27), we find that the GW+SOSEX self-energy does not bring a marked improvement to the IPs, on average, as compared to GW, when the PBE0 reference is used; analogous results have been recently found by Bruneval et al. in ref (14) by using another hybrid reference, leading them to claim that GW is still the best choice within MBPT for computing IPs of molecules. Almost as recently, and by using the same test set, the work by Wang et al. presented in ref (28) has instead shown an improvement on GW with the GW+SOSEX self-energy even starting from the PBE0 reference. As we already stated in the introduction, the arbitrariness of the starting point adds a layer of complexity to the assessment of a self-energy-based method, and the possibility of tuning a parameter in hybrid functionals seems to allow GW to be most often up to the task, (14,109,110) perhaps at the expense of conceptual clarity.
Finally, we are positively impressed by the good performance of the 2B self-energy. This motivates us to build upon this level of theory by including all the self-energy diagrams generated by the TDHF vertex, which has recently been shown to be a promising route to electronic excitations in molecules. (36,37)

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.2c00048.

  • Comparison of results with existing literature, convergence of the xc potentials, and perturbative versus self-consistent solution of the LSSE (PDF)

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Author Information

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  • Corresponding Author
    • S. Vacondio - Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via G. Campi 213/a, Modena 41121, Italy;  Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy;  Orcidhttps://orcid.org/0000-0001-9494-7518;  Email: [email protected]
  • Authors
    • D. Varsano - Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy;  Orcidhttps://orcid.org/0000-0001-7675-7374
    • A. Ruini - Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via G. Campi 213/a, Modena 41121, Italy;  Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy
    • A. Ferretti - Centro S3, CNR−Istituto Nanoscienze, 41125 Modena, Italy
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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The authors acknowledge stimulating discussions with Maria Hellgren, Nicola Marzari, and Tommaso Chiarotti. This work was partially supported by the MaX – MAterials design at the eXascale – Centre of Excellence, funded by the European Union program H2020-INFRAEDI-2018-1 (Grant No. 824143).

Appendix A: Self-Energies from Vertex Corrections

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Iterating through Hedin’s scheme with a nontrivial vertex function would in principle be an ideal and systematic way to devise beyond-GW approximations. However, doing so can quickly lead to very complex methods. This procedure allows for the inclusion of self-energy diagrams to infinite order. Nevertheless, since the vertex function includes the derivative of the self-energy, Σ, with respect to the Green’s function G, new diagrams are added at each iteration, making a rigorous implementation of Hedin’s scheme numerically unfeasible. In order to overcome this complexity, approximations and truncations in the iteration process must be made. In the GW approximation, vertex corrections are neglected altogether, making the vertex function trivial and allowing for a self-consistent solution independent of the starting point.
The simplest vertex correction is the time-dependent Hartree–Fock (TDHF) scheme, obtained by plugging the Fock self-energy into the vertex equation (see Figure 9, top). This vertex has been studied in a perturbative fashion, that is, with a single or a few iterations through Hedin’s scheme, starting from a HF reference. (8,36,37) By plugging the GW self-energy into the vertex function, the time-dependent GW (TDGW) method is obtained instead (see Figure 9, middle). For simplicity, the dependence of the screened interaction on the Green’s function is generally ignored in the differentiation contained in the vertex function. By doing so, a TDGW vertex is obtained, leading to the same skeleton self-energy diagrams generated by the TDHF vertex, but framed in terms of the screened interaction.

Figure 9

Figure 9. Vertex equations in the TDHF, TDGW, and GW+SOSEX approximations.

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A way to further simplify the scheme is to truncate the infinite summation of self-energy diagrams generated by a vertex function, retaining only lower orders. This allows one to avoid the often cumbersome solution of the vertex equation and simply compute a number of selected self-energy diagrams. For instance, with truncation to second order in the Coulomb interaction, the summation generated by the TDHF vertex yields the second Born (2B) self-energy. With truncation to second order in the screened Coulomb interaction, the summation generated by the TDGW vertex yields the GW self-energy plus a second-order exchange contribution, the latter corresponding to the 2B exchange (2Bx) diagram expressed in terms of the dressed interaction W (see Figure 10). When this diagram is implemented, the two screened interactions have most often been considered statically (34) or with a plasmon pole approximation; (33,38) only recently has their frequency dependence been fully taken into account. (23,28) Moreover, a treatment for this diagram enforcing positive definite spectral functions has also been proposed. (40−42)

Figure 10

Figure 10. Self-energy diagram of order 2 in the screened interaction W.

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A simpler second-order approximation, devoid of double frequency integrals, can be devised by retaining only the zeroth and first-order terms in the TDHF vertex (see Figure 9, bottom) and then substituting them into the self-energy. This produces a second-order diagram with one screened Coulomb interaction, leaving the other at the bare level. This diagram is termed the second-order screened exchange (SOSEX) diagram and the self-energy approximation is referred to as GW+SOSEX. This has been studied so far for molecules (27,28,62) and model systems. (63)

References

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This article references 110 other publications.

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    The GW compendium: a practical guide to theoretical photoemission spectroscopy
    Golze, Dorothea; Dvorak, Marc; Rinke, Patrick
    Frontiers in Chemistry (Lausanne, Switzerland) (2019), 7 (), 377CODEN: FCLSAA; ISSN:2296-2646. (Frontiers Media S.A.)
    A review. The GW approxn. in electronic structure theory has become a widespread tool for predicting electronic excitations in chem. compds. and materials. In the realm of theor. spectroscopy, the GW method provides access to charged excitations as measured in direct or inverse photoemission spectroscopy. The no. of GW calcns. in the past two decades has exploded with increased computing power and modern codes. The success of GW can be attributed to many factors: favorable scaling with respect to system size, a formal interpretation for charged excitation energies, the importance of dynamical screening in real systems, and its practical combination with other theories. In this review, we provide an overview of these formal and practical considerations. We expand, in detail, on the choices presented to the scientist performing GWcalcns. for the first time. We also give an introduction to the many-body theory behind GW a review of modern applications like mols. and surfaces, and a perspective on methods which go beyond conventional GW calcns. This review addresses chemists, physicists and material scientists with an interest in theor. spectroscopy. It is intended for newcomers to GW calcns. but can also serve as an alternative perspective for experts and an up-to-date source of computational techniques.
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    Strinati, G. Dynamical Shift and Broadening of Core Excitons in Semiconductors. Phys. Rev. Lett. 1982, 49, 1519– 1522,  DOI: 10.1103/PhysRevLett.49.1519
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    Dynamical shift and broadening of core excitons in semiconductors
    Strinati, G.
    Physical Review Letters (1982), 49 (20), 1519-22CODEN: PRLTAO; ISSN:0031-9007.
    A non-Hermitian eigenvalue equation is proposed to det. binding energies and widths of core excitons in semiconductors, taking into account the time dependence of screening effects through the dielec. matrix ε-1(.vector.r,.vector.r';ω). Deviations from static screening contribute both an increase of the binding energy and a narrowing of the Auger width. Numerical ests. of both effects for the Si 2p transition give qual. agreement with exptl. data when the exciton size is reduced by band-structure effects.
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    Hybertsen, M. S.; Louie, S. G. First-Principles Theory of Quasiparticles: Calculation of Band Gaps in Semiconductor and Insulators. Phys. Rev. Lett. 1985, 55, 1418– 1421,  DOI: 10.1103/PhysRevLett.55.1418
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    First-principles theory of quasiparticles: calculation of band gaps in semiconductors and insulators
    Hybertsen, Mark S.; Louie, Steven G.
    Physical Review Letters (1985), 55 (13), 1418-21CODEN: PRLTAO; ISSN:0031-9007.
    A first-principles theory is given for the quasiparticle energies of semiconductors and insulators. The full dielec. matrix is used to evaluate the nonlocal, energy-dependent electron self-energy operator. Both local-field effects and dynamical screening are essential for understanding quasiparticle energies. Results for the band gaps, optical transitions, and band dispersions for Si and diamond are in excellent agreement with existing exptl. data.
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    Godby, R. W.; Schlüter, M.; Sham, L. J. Accurate Exchange-Correlation Potential for Silicon and Its Discontinuity on Addition of an Electron. Phys. Rev. Lett. 1986, 56, 2415– 2418,  DOI: 10.1103/PhysRevLett.56.2415
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    Accurate exchange-correlation potential for silicon and its discontinuity on addition of an electron
    Godby, R. W.; Schlueter, M.; Sham, L. J.
    Physical Review Letters (1986), 56 (22), 2415-18CODEN: PRLTAO; ISSN:0031-9007.
    An accurate d.-functional exchange-correlation potential Vxc(r) was detd. for Si, from calcns. of the self-energy Σ(r,r',ω). No local-d. approxn. (LDA) is used for Vxc. The band structure with this Vxc is in remarkably close agreement with that obtained with the LDA, while both differ significantly from the quasiparticle spectrum of Σ. The 50% band-gap error found in LDA calcns. is therefore not caused by the LDA but by the discontinuity, Δ, in the exact Vxc on addn. of an electron.
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    Marzari, N.; Ferretti, A.; Wolverton, C. Electronic-structure methods for materials design. Nat. Mater. 2021, 20, 736– 749,  DOI: 10.1038/s41563-021-01013-3
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    Electronic-structure methods for materials design
    Marzari, Nicola; Ferretti, Andrea; Wolverton, Chris
    Nature Materials (2021), 20 (6), 736-749CODEN: NMAACR; ISSN:1476-1122. (Nature Portfolio)
    A review. The accuracy and efficiency of electronic-structure methods to understand, predict and design the properties of materials has driven a new paradigm in research. Simulations can greatly accelerate the identification, characterization and optimization of materials, with this acceleration driven by continuous progress in theory, algorithms and hardware, and by adaptation of concepts and tools from computer science. Nevertheless, the capability to identify and characterize materials relies on the predictive accuracy of the underlying phys. descriptions, and on the ability to capture the complexity of realistic systems. We provide here an overview of electronic-structure methods, of their application to the prediction of materials properties, and of the different strategies employed towards the broader goals of materials design and discovery.
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    Shirley, E. L.; Martin, R. M. GW quasiparticle calculations in atoms. Phys. Rev. B 1993, 47, 15404– 15412,  DOI: 10.1103/PhysRevB.47.15404
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    GW-quasiparticle calculations in atoms
    Shirley, Eric L.; Martin, Richard M.
    Physical Review B: Condensed Matter and Materials Physics (1993), 47 (23), 15404-12CODEN: PRBMDO; ISSN:0163-1829.
    The authors present quasiparticle calcns. done in atoms using various versions of Hedin's GW approxn. The performance of GW in major-group elements is illustrated by calcns. of first and second ionization energies from the same initial configuration, along with various promotion energies. The results significantly improve Hartree-Fock eigenvalues and are comparable, in accuracy, to Hartree-Fock, total-energy differences. In the 3d (iron-series) transition metals, there are also significant improvements in the 4s- and 3d-state electron addn. energies in the first half of the series. In the second half of the series, the authors find that strong correlations among 3d electrons of opposite spin are not accounted for by GW. They also study the screening effects by cores on valence-electron addn. and removal energies. In particular, the "alkalilike" problem of one valence electron bound to a closed-shell core was examd.. The generalized GW approxn. describes quant. important effects of core-relaxation and core-valence correlation on the valence electrons. These core-relaxation and core-valence correlation effects can be incorporated into an effective potential suitable for use in calcns. of valence properties of atoms, mols., and solids (following paper).
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    Stan, A.; Dahlen, N. E.; Leeuwen, R. v. Fully self-consistent GW calculations for atoms and molecules. EPL 2006, 76, 298,  DOI: 10.1209/epl/i2006-10266-6
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    Rostgaard, C.; Jacobsen, K. W.; Thygesen, K. S. Fully self-consistent GW calculations for molecules. Phys. Rev. B 2010, 81, 085103,  DOI: 10.1103/PhysRevB.81.085103
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    Fully self-consistent GW calculations for molecules
    Rostgaard, C.; Jacobsen, K. W.; Thygesen, K. S.
    Physical Review B: Condensed Matter and Materials Physics (2010), 81 (8), 085103/1-085103/10CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We calc. single-particle excitation energies for a series of 34 mols. using fully self-consistent GW, one-shot G0W0, Hartree-Fock (HF), and hybrid d.-functional theory (DFT). All calcns. are performed within the projector-augmented wave method using a basis set of Wannier functions augmented by numerical AOs. The GW self-energy is calcd. on the real frequency axis including its full frequency dependence and off-diagonal matrix elements. The mean abs. error of the ionization potential (IP) with respect to expt. is found to be 4.4, 2.6, 0.8, 0.4, and 0.5 eV for DFT-PBE, DFT-PBE0, HF, G0W0[HF], and self-consistent GW, resp. This shows that although electronic screening is weak in mol. systems, its inclusion at the GW level reduces the error in the IP by up to 50% relative to unscreened HF. In general GW overscreens the HF energies leading to underestimation of the IPs. The best IPs are obtained from one-shot G0W0 calcns. based on HF since this reduces the overscreening. Finally, we find that the inclusion of core-valence exchange is important and can affect the excitation energies by as much as 1 eV.
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    Bruneval, F.; Marques, M. A. L. Benchmarking the Starting Points of the GW Approximation for Molecules. J. Chem. Theory Comput. 2013, 9, 324– 329,  DOI: 10.1021/ct300835h
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    Benchmarking the Starting Points of the GW Approximation for Molecules
    Bruneval, Fabien; Marques, Miguel A. L.
    Journal of Chemical Theory and Computation (2013), 9 (1), 324-329CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    The GW approxn. is nowadays being used to obtain accurate quasiparticle energies of atoms and mols. In practice, the GW approxn. is generally evaluated perturbatively, based on a prior self-consistent calcn. within a simpler approxn. The final result thus depends on the choice of the self-consistent mean-field chosen as a starting point. Using a recently developed GW code based on Gaussian basis functions, we benchmark a wide range of starting points for perturbative GW, including Hartree-Fock, LDA, PBE, PBE0, B3LYP, HSE06, BH and HLYP, CAM-B3LYP, and tuned CAM-B3LYP. In the evaluation of the ionization energy, the hybrid functionals are clearly superior results starting points when compared to Hartree-Fock, to LDA, or to the semilocal approxns. Furthermore, among the hybrid functionals, the ones with the highest proportion of exact-exchange usually perform best. Finally, the reliability of the frozen-core approxn., that allows for a considerable speed-up of the calcns., is demonstrated.
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    Kaplan, F.; Harding, M. E.; Seiler, C.; Weigend, F.; Evers, F.; van Setten, M. J. Quasi-Particle Self-Consistent GW for Molecules. J. Chem. Theory Comput. 2016, 12, 2528– 2541,  DOI: 10.1021/acs.jctc.5b01238
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    12
    Quasi-Particle Self-Consistent GW for Molecules
    Kaplan, F.; Harding, M. E.; Seiler, C.; Weigend, F.; Evers, F.; van Setten, M. J.
    Journal of Chemical Theory and Computation (2016), 12 (6), 2528-2541CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present the formalism and implementation of quasi-particle self-consistent GW (qsGW) and eigenvalue only quasi-particle self-consistent GW (evGW) adapted to std. quantum chem. packages. Our implementation is benchmarked against high-level quantum chem. computations (coupled-cluster theory) and exptl. results using a representative set of mols. Furthermore, we compare the qsGW approach for five mols. relevant for org. photovoltaics to self-consistent GW results (scGW) and analyze the effects of the self-consistency on the ground state d. by comparing calcd. dipole moments to their exptl. values. We show that qsGW makes a significant improvement over conventional G0W0 and that partially self-consistent flavors (in particular evGW) can be excellent alternatives.
  13. 13
    Hung, L.; da Jornada, F. H.; Souto-Casares, J.; Chelikowsky, J. R.; Louie, S. G.; Öğüt, S. Excitation spectra of aromatic molecules within a real-space GW -BSE formalism: Role of self-consistency and vertex corrections. Phys. Rev. B 2016, 94, 085125,  DOI: 10.1103/PhysRevB.94.085125
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    Excitation spectra of aromatic molecules within a real-space GW-BSE formalism: Role of self-consistency and vertex corrections
    Hung, Linda; da Jornada, Felipe H.; Souto-Casares, Jaime; Chelikowsky, James R.; Louie, Steven G.; Ogut, Serdar
    Physical Review B (2016), 94 (8), 085125/1-085125/13CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
    We present first-principles calcns. on the vertical ionization potentials (IPs), electron affinities (EAs), and singlet excitation energies on an arom.-mol. test set (benzene, thiophene, 1,2,5-thiadiazole, naphthalene, benzothiazole, and tetrathiafulvalene) within the GW and Bethe-Salpeter equation (BSE) formalisms. Our computational framework, which employs a real-space basis for ground-state and a transition-space basis for excited-state calcns., is well suited for high-accuracy calcns. on mols., as we show by comparing against G0W0 calcns. within a plane-wave-basis formalism. We then generalize our framework to test variants of the GW approxn. that include a local d. approxn. (LDA) -derived vertex function (ΓLDA) and quasiparticle-self-consistent (QS) iterations. We find that ΓLDA and quasiparticle self-consistency shift IPs and EAs by roughly the same magnitude, but with opposite sign for IPs and the same sign for EAs. G0W0 and QSGWΓLDA are more accurate for IPs, while G0W0ΓLDA and QSGW are best for EAs. For optical excitations, we find that perturbative GW-BSE underestimates the singlet excitation energy, while self-consistent GW-BSE results in good agreement with previous best-est. values for both valence and Rydberg excitations. Finally, our work suggests that a hybrid approach, in which G0W0 energies are used for occupied orbitals and G0W0ΓLDA for unoccupied orbitals, also yields optical excitation energies in good agreement with expt. but at a smaller computational cost.
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    Bruneval, F.; Dattani, N.; van Setten, M. J. The GW Miracle in Many-Body Perturbation Theory for the Ionization Potential of Molecules. Frontiers in Chemistry 2021, 9, 749779,  DOI: 10.3389/fchem.2021.749779
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    14
    The GW miracle in many-body perturbation theory for the ionization potential of molecules
    Bruneval, Fabien; Dattani, Nike; van Setten, Michiel J.
    Frontiers in Chemistry (Lausanne, Switzerland) (2021), 9 (), 749779CODEN: FCLSAA; ISSN:2296-2646. (Frontiers Media S.A.)
    We use the GW100 benchmark set to systematically judge the quality of several perturbation theories against high-level quantum chem. methods. First of all, we revisit the ref. CCSD(T) ionization potentials for this popular benchmark set and establish a revised set of CCSD(T) results. Then, for all of these 100 mols., we calc. the HOMO energy within second and third-order perturbation theory (PT2 and PT3), and, GW as post-Hartree-Fock methods. We found GW to be the most accurate of these three approxns. for the ionization potential, by far. Going beyond GW by adding more diagrams is a tedious and dangerous activity: We tried to complement GW with second-order exchange (SOX), with second-order screened exchange (SOSEX), with interacting electron-hole pairs (WTDHF), and with a GW d.-matrix (γGW). Only the γGW result has a pos. impact. Finally using an improved hybrid functional for the non-interacting Green's function, considering it as a cheap way to approx. self-consistency, the accuracy of the simplest GW approxn. improves even more. We conclude that GW is a miracle: Its subtle balance makes GW both accurate and fast.
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    Shung, K. W. K.; Mahan, G. D. Calculated Photoemission Spectra of Na. Phys. Rev. Lett. 1986, 57, 1076– 1079,  DOI: 10.1103/PhysRevLett.57.1076
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    Calculated photoemission spectra of sodium
    Shung, Kenneth W. K.; Mahan, G. D.
    Physical Review Letters (1986), 57 (8), 1076-9CODEN: PRLTAO; ISSN:0031-9007.
    A detailed calcn. was made of the photoemission spectra of Na for electrons emitted normal to the (110) surface. The exptl. results of E. Jensen and E. W. Plummer (1985) were taken as an indication that the conduction band of Na is severely distorted and hence that a charge-d. wave exists in the system. The calcn. confirms just the opposite, i.e., that the band is not strongly distorted from a free-electron band. The calcn. is able to reproduce the exptl. data closely; it also predicts an angle dependence of the spectra, a feature which needs to be examd. exptl.
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    Shung, K. W. K.; Sernelius, B. E.; Mahan, G. D. Self-energy corrections in photoemission of Na. Phys. Rev. B 1987, 36, 4499– 4502,  DOI: 10.1103/PhysRevB.36.4499
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    Self-energy corrections in photoemission of sodium
    Shung, Kenneth W. K.; Sernelius, Bo E.; Mahan, G. D.
    Physical Review B: Condensed Matter and Materials Physics (1987), 36 (8), 4499-502CODEN: PRBMDO; ISSN:0163-1829.
    The photoemission spectra were calcd. of Na, where self-energy corrections and surface effects were fully incorporated. The self-energy was calcd. with the Rayleigh-Schroedinger perturbation method. The energy shifts in both initial and final states were demonstrated by comparing the calcn. with the measured results of E. W. Plummer and coworkers (1984, 1985). Both the imaginary part and the real part of the self-energy contribute to the obsd. large bandwidth redn. in Na. These effects are believed to be important, in general, in analyzing photoemission spectra.
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    Northrup, J. E.; Hybertsen, M. S.; Louie, S. G. Theory of quasiparticle energies in alkali metals. Phys. Rev. Lett. 1987, 59, 819– 822,  DOI: 10.1103/PhysRevLett.59.819
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    Theory of quasiparticle energies in alkali metals
    Northrup, John E.; Hybertsen, Mark S.; Louie, Steven G.
    Physical Review Letters (1987), 59 (7), 819-22CODEN: PRLTAO; ISSN:0031-9007.
    The quasiparticle band structures of Na and Li are calcd. by evaluation of the electron self-energy to first order in the dynamically screened Coulomb interaction. For Na, results explain quant. the large bandwidth redn. obsd. in recent angle-resolved photoemission expts. The inclusion of exchange-correlation effects in the dielec. screening and the requirement that the energy dependence of the electron Green's function be treated self-consistently are important in obtaining the correct bandwidths.
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    Lyo, I.-W.; Plummer, E. W. Quasiparticle band structure of Na and simple metals. Phys. Rev. Lett. 1988, 60, 1558– 1561,  DOI: 10.1103/PhysRevLett.60.1558
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    Quasiparticle band structure of sodium and simple metals
    Lyo, In Whan; Plummer, E. W.
    Physical Review Letters (1988), 60 (15), 1558-61CODEN: PRLTAO; ISSN:0031-9007.
    An abs. measurement of the quasiparticle band structure of Na(110) was made by using angle-resolved photoemission. The occupied band is parabolic, but with an 18% narrowing in band width compared to self-consistent band calcns. A simple model calcn. of the self-energy correction for a homogeneous electron gas produced good agreement with the data, if a dielec. response beyond the random-phase approxn. was included. Expt. and theory are compared for other simple metals.
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    Yasuhara, H.; Yoshinaga, S.; Higuchi, M. Why is the Bandwidth of Sodium Observed to be Narrower in Photoemission Experiments?. Phys. Rev. Lett. 1999, 83, 3250– 3253,  DOI: 10.1103/PhysRevLett.83.3250
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    Why is the Bandwidth of Sodium Observed to be Narrower in Photoemission Experiments?
    Yasuhara, H.; Yoshinaga, S.; Higuchi, M.
    Physical Review Letters (1999), 83 (16), 3250-3253CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    The exptl. predicted narrowing in the bandwidth of Na is interpreted in terms of the nonlocal self-energy effect on quasiparticle energies of the electron liq. The calcd. self-energy correction is an increasing function of the wave no. variable. The usual anal. of angle-resolved photoemission expts. assumes the final-state energies on the nearly free-electron-like model and hence incorrectly ascribes the nonlocal self-energy correction to the final-state energies to occupied-state energies, seemingly leading to a narrowing in the bandwidth.
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    Takada, Y. Inclusion of Vertex Corrections in the Self-Consistent Calculation of Quasiparticles in Metals. Phys. Rev. Lett. 2001, 87, 226402,  DOI: 10.1103/PhysRevLett.87.226402
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    Inclusion of Vertex Corrections in the Self-Consistent Calculation of Quasiparticles in Metals
    Takada, Yasutami
    Physical Review Letters (2001), 87 (22), 226402/1-226402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Based on the Bethe-Salpeter equation and the Ward identity derived from it, we provide a scheme for constructing the vertex function in the self-consistent iteration loop to det. the electron self-energy. The scheme is implemented in the homogeneous electron gas at the sodium d.
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    van Schilfgaarde, M.; Kotani, T.; Faleev, S. Quasiparticle Self-Consistent GW Theory. Phys. Rev. Lett. 2006, 96, 226402,  DOI: 10.1103/PhysRevLett.96.226402
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    Quasiparticle Self-Consistent GW Theory
    Van Schilfgaarde, M.; Kotani, Takao; Faleev, S.
    Physical Review Letters (2006), 96 (22), 226402/1-226402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    In past decades the scientific community has been looking for a reliable first-principles method to predict the electronic structure of solids with high accuracy. Here we present an approach which we call the quasiparticle self-consistent GW approxn. It is based on a kind of self-consistent perturbation theory, where the self-consistency is constructed to minimize the perturbation. We apply it to selections from different classes of materials, including alkali metals, semiconductors, wide band gap insulators, transition metals, transition metal oxides, magnetic insulators, and rare earth compds. Apart from some mild exceptions, the properties are very well described, particularly in weakly correlated cases. Self-consistency dramatically improves agreement with expt., and is sometimes essential. Discrepancies with expt. are systematic, and can be explained in terms of approxns. made.
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    Morris, A. J.; Stankovski, M.; Delaney, K. T.; Rinke, P.; García-González, P.; Godby, R. W. Vertex corrections in localized and extended systems. Phys. Rev. B 2007, 76, 155106,  DOI: 10.1103/PhysRevB.76.155106
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    Vertex corrections in localized and extended systems
    Morris, Andrew J.; Stankovski, Martin; Delaney, Kris T.; Rinke, Patrick; Garcia-Gonzalez, P.; Godby, R. W.
    Physical Review B: Condensed Matter and Materials Physics (2007), 76 (15), 155106/1-155106/9CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    Within many-body perturbation theory, we apply vertex corrections to various closed-shell atoms and to jellium, using a local approxn. for the vertex consistent with starting the many-body perturbation theory from a Kohn-Sham Green's function constructed from d.-functional theory in the local-d. approxn. The vertex appears in two places - in the screened Coulomb interaction W and in the self-energy Σ - and we obtain a systematic discrimination of these two effects by turning the vertex in Σ on and off. We also make comparisons to std. GW results within the usual RPA, which omits the vertex from both. When a vertex is included for closed-shell atoms, both ground-state and excited-state properties demonstrate little improvement over std. GW. For jellium, we observe marked improvement in the quasiparticle bandwidth when the vertex is included only in W, whereas turning on the vertex in Σ leads to an unphys. quasiparticle dispersion and work function. A simple anal. suggests why implementation of the vertex only in W is a valid way to improve quasiparticle energy calcns., while the vertex in Σ is unphys., and points the way to the development of improved vertices for ab initio electronic structure calcns.
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    Kutepov, A. L. Electronic structure of Na, K, Si, and LiF from self-consistent solution of Hedin’s equations including vertex corrections. Phys. Rev. B 2016, 94, 155101,  DOI: 10.1103/PhysRevB.94.155101
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    Electronic structure of Na, K, Si, and LiF from self-consistent solution of Hedin's equations including vertex corrections [Erratum to document cited in CA169:150966]
    Kutepov, Andrey L.
    Physical Review B (2016), 94 (15), 155101/1-155101/23CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
    A few self-consistent schemes to solve the Hedin equations are presented. They include vertex corrections of different complexity. Commonly used quasiparticle approxn. for the Green's function and static approxn. for the screened interaction are avoided altogether. Using alkali metals Na and K as well as semiconductor Si and wide-gap insulator LiF as examples, it is shown that both the vertex corrections in the polarizability P and in the self-energy Σ are important. Particularly, vertex corrections in Σ with proper treatment of frequency dependence of the screened interaction always reduce calcd. bandwidths/band gaps, improving the agreement with expt. The complexity of the vertex included in P and in Σ can be different. Whereas in the case of polarizability one generally has to solve the Bethe-Salpeter equation for the corresponding vertex function, it is enough (for the materials in this study) to include the vertex of the first order in the self-energy. The calcns. with appropriate vertices show remarkable improvement in the calcd. bandwidths and band gaps as compared to the self-consistent GW approxn. as well as to the self-consistent quasiparticle GW approxn.
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    Guzzo, M.; Lani, G.; Sottile, F.; Romaniello, P.; Gatti, M.; Kas, J. J.; Rehr, J. J.; Silly, M. G.; Sirotti, F.; Reining, L. Valence Electron Photoemission Spectrum of Semiconductors: Ab Initio Description of Multiple Satellites. Phys. Rev. Lett. 2011, 107, 166401,  DOI: 10.1103/PhysRevLett.107.166401
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    Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites
    Guzzo, Matteo; Lani, Giovanna; Sottile, Francesco; Romaniello, Pina; Gatti, Matteo; Kas, Joshua J.; Rehr, John J.; Silly, Mathieu G.; Sirotti, Fausto; Reining, Lucia
    Physical Review Letters (2011), 107 (16), 166401/1-166401/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    The exptl. valence band photoemission spectrum of semiconductors exhibits multiple satellites that cannot be described by the GW approxn. for the self-energy in the framework of many-body perturbation theory. Taking silicon as a prototypical example, we compare exptl. high energy photoemission spectra with GW calcns. and analyze the origin of the GW failure. We then propose an approxn. to the functional differential equation that dets. the exact one-body Green's function, whose soln. has an exponential form. This yields a calcd. spectrum, including cross sections, secondary electrons, and an est. for extrinsic and interference effects, in excellent agreement with expt. Our result can be recast as a dynamical vertex correction beyond GW, giving hints for further developments.
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    Choi, S.; Kutepov, A.; Haule, K.; van Schilfgaarde, M.; Kotliar, G. First-principles treatment of Mott insulators: linearized QSGW+DMFT approach. npj Quantum Materials 2016, 1, 16001,  DOI: 10.1038/npjquantmats.2016.1
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    Marom, N.; Caruso, F.; Ren, X.; Hofmann, O. T.; Körzdörfer, T.; Chelikowsky, J. R.; Rubio, A.; Scheffler, M.; Rinke, P. Benchmark of GW methods for azabenzenes. Phys. Rev. B 2012, 86, 245127,  DOI: 10.1103/PhysRevB.86.245127
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    Benchmark of GW methods for azabenzenes
    Marom, Noa; Caruso, Fabio; Ren, Xinguo; Hofmann, Oliver T.; Korzdorfer, Thomas; Chelikowsky, James R.; Rubio, Angel; Scheffler, Matthias; Rinke, Patrick
    Physical Review B: Condensed Matter and Materials Physics (2012), 86 (24), 245127/1-245127/16CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    Many-body perturbation theory in the GW approxn. is a useful method for describing electronic properties assocd. with charged excitations. A hierarchy of GW methods exists, starting from non-self-consistent G0W0, through partial self-consistency in the eigenvalues and in the Green's function (scGW0), to fully self-consistent GW (scGW). Here, we assess the performance of these methods for benzene, pyridine, and the diazines. The quasiparticle spectra are compared to photoemission spectroscopy (PES) expts. with respect to all measured particle removal energies and the ordering of the frontier orbitals. We find that the accuracy of the calcd. spectra does not match the expectations based on their level of self-consistency. In particular, for certain starting points G0W0 and scGW0 provide spectra in better agreement with the PES than scGW.
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    Ren, X.; Marom, N.; Caruso, F.; Scheffler, M.; Rinke, P. Beyond the GW approximation: A second-order screened exchange correction. Phys. Rev. B 2015, 92, 081104,  DOI: 10.1103/PhysRevB.92.081104
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    Beyond the GW approximation: a second-order screened exchange correction
    Ren, Xinguo; Marom, Noa; Caruso, Fabio; Scheffler, Matthias; Rinke, Patrick
    Physical Review B: Condensed Matter and Materials Physics (2015), 92 (8), 081104/1-081104/6CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    Motivated by the recently developed renormalized second-order perturbation theory for ground-state energy calcns., we propose a second-order screened exchange correction (SOSEX) to the GW self-energy. This correction follows the spirit of the SOSEX correction to the RPA for the electron correlation energy and can be clearly represented in terms of Feynman diagrams. We benchmark the performance of the perturbative G0W0 + SOSEX scheme for a set of mol. systems, including the G2 test set from quantum chem. as well as benzene and tetracyanoethylene. We find that G0W0 + SOSEX improves over G0W0 for the energy levels of the highest occupied and lowest unoccupied MOs. In addn., it can resolve some of the difficulties encountered by the GW method for relative energy positions as exemplified by benzene where the energy spacing between certain valence orbitals is severely underestimated.
  28. 28
    Wang, Y.; Rinke, P.; Ren, X. Assessing the G0W0Γ0(1) Approach: Beyond G0W0 with Hedin’s Full Second-Order Self-Energy Contribution. J. Chem. Theory Comput 2021, 17, 5140– 5154,  DOI: 10.1021/acs.jctc.1c00488
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    28
    Assessing the G0W0Γ0(1) Approach: Beyond G0W0 with Hedin's Full Second-Order Self-Energy Contribution
    Wang, Yanyong; Rinke, Patrick; Ren, Xinguo
    Journal of Chemical Theory and Computation (2021), 17 (8), 5140-5154CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present and benchmark a self-energy approach for quasiparticle energy calcns. that goes beyond Hedin's GW approxn. by adding the full second-order self-energy (FSOS-W) contribution. The FSOS-W diagram involves two screened Coulomb interaction (W) lines, and adding the FSOS-W to the GW self-energy can be interpreted as first-order vertex correction to GW (GWΓ(1)). Our FSOS-W implementation is based on the resoln.-of-identity technique and exhibits better than O(N5) scaling with system size for small- to medium-sized mols. We then present one-shot GWΓ(1) (G0W0Γ0(1)) benchmarks for the GW100 test set and a set of 24 acceptor mols. For semilocal or hybrid d. functional theory starting points, G0W0Γ0(1) systematically outperforms G0W0 for the first vertical ionization potentials and electron affinities of both test sets. Finally, we demonstrate that a static FSOS-W self-energy significantly underestimates the quasiparticle energies.
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    von Barth, U.; Holm, B. Self-consistent GW0 results for the electron gas: Fixed screened potential W0 within the random-phase approximation. Phys. Rev. B 1996, 54, 8411– 8419,  DOI: 10.1103/PhysRevB.54.8411
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    Self-consistent GW0 results for the electron gas: fixed screened potential W0 within the random-phase approximation
    von Barth, Ulf; Holm, Bengt
    Physical Review B: Condensed Matter (1996), 54 (12), 8411-8419CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    With the aim of properly understanding the basis for and the utility of many-body perturbation theory as applied extended metallic systems, we have calcd. the electronic self-energy of the homogeneous electron gas within the GW approxn. The calcn. had been carried out in a self-consistent way; i.e., the one-electron Green function (G) obtained from Dyson's equation is the same as that used to calc. the self-energy. The self-consistency is restricted in the sense that the screened interaction W is kept fixed and equal to that of the random-phase approxn. for the gas. We have found that the final results are marginally affected by the broadening of the quasiparticles, and that their self-consistent energies are still close to their free-electron counterparts as they are in non-self-consistent calcns. The redn. in strength of the quasiparticles and the development of satellite structure (plasmons) gives, however, a markedly smaller dynamic self-energy leading to, e.g., a smaller redn. in the quasiparticle strength as compared to non-self-consistent results. The relatively bad description of plasmon structure within the non-self-consistent GW approxn. is marginally improved. A first attempt at including W in the self-consistency cycle leads to an even broader and structureless satellite spectrum in disagreement with expt.
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    Holm, B.; von Barth, U. Fully self-consistent GW self-energy of the electron gas. Phys. Rev. B 1998, 57, 2108– 2117,  DOI: 10.1103/PhysRevB.57.2108
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    Fully self-consistent GW self-energy of the electron gas
    Holm, B.; von Barth, U.
    Physical Review B: Condensed Matter and Materials Physics (1998), 57 (4), 2108-2117CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    We present fully self-consistent results for the self-energy of the electron gas within the GW approxn. This means that the self-consistent Green's function G, as obtained from Dyson's equation, is used not only for obtaining the self-energy but also for constructing the screened interaction W within the RPA. Such a theory is particle and energy conserving in the sense of Kadanoff and Baym. We find an increase in the wt. of the quasiparticle as compared to ordinary non-self-consistent calcns. but also to calcns. with partial self-consistency using a fixed W. The quasiparticle bandwidth is larger than that of free electrons and the satellite structure is broad and featureless; both results clearly contradict the exptl. evidence. The total energy, though, is as accurate as that from quantum Monte Carlo calcns., and its deriv. with respect to particle no. agrees with the Fermi energy as obtained directly from the pole of the Green's function at the Fermi level. Our results indicate that, unless vertex corrections are included, non-self-consistent results are to be preferred for most properties except for the total energy.
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    Schindlmayr, A.; García-González, P.; Godby, R. W. Diagrammatic self-energy approximations and the total particle number. Phys. Rev. B 2001, 64, 235106,  DOI: 10.1103/PhysRevB.64.235106
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    Diagrammatic self-energy approximations and the total particle number
    Schindlmayr, Arno; Garcia-Gonzalez, P.; Godby, R. W.
    Physical Review B: Condensed Matter and Materials Physics (2001), 64 (23), 235106/1-235106/6CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    There is increasing interest in many-body perturbation theory as a practical tool for the calcn. of ground-state properties. As a consequence, unambiguous sum rules such as the conservation of particle no. under the influence of the Coulomb interaction have acquired an importance that did not exist for calcns. of excited-state properties. In this paper we obtain a rigorous, simple relation whose fulfillment guarantees particle-no. conservation in a given diagrammatic self-energy approxn. Hedin's G0W0 approxn. does not satisfy this relation and hence violates the particle-no. sum rule. Very precise calcns. for the homogeneous electron gas and a model inhomogeneous electron system allow the extent of the nonconservation to be estd.
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    Stan, A.; Dahlen, N. E.; van Leeuwen, R. Levels of self-consistency in the GW approximation. J. Chem. Phys. 2009, 130, 114105,  DOI: 10.1063/1.3089567
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    Levels of self-consistency in the GW approximation
    Stan, Adrian; Dahlen, Nils Erik; van Leeuwen, Robert
    Journal of Chemical Physics (2009), 130 (11), 114105/1-114105/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We perform GW calcns. on atoms and diat. mols. at different levels of self-consistency and investigate the effects of self-consistency on total energies, ionization potentials, and particle no. conservation. We further propose a partially self-consistent GW scheme in which we keep the correlation part of the self-energy fixed within the self-consistency cycle. This approxn. is compared to the fully self-consistent GW results and to the GW0 and the G0W0 approxns. Total energies, ionization potentials, and two-electron removal energies obtained with our partially self-consistent GW approxn. are in excellent agreement with fully self-consistent GW results while requiring only a fraction of the computational effort. We also find that self-consistent and partially self-consistent schemes provide ionization energies of similar quality as the G0W0 values but yield better total energies and energy differences. (c) 2009 American Institute of Physics.
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    Bobbert, P. A.; van Haeringen, W. Lowest-order vertex-correction contribution to the direct gap of silicon. Phys. Rev. B 1994, 49, 10326– 10331,  DOI: 10.1103/PhysRevB.49.10326
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    Lowest-order vertex-correction contribution to the direct gap of silicon
    Bobbert, P. A.; van Haeringen, W.
    Physical Review B: Condensed Matter and Materials Physics (1994), 49 (15), 10326-31CODEN: PRBMDO; ISSN:0163-1829.
    The authors have calcd. the contribution of the lowest-order vertex-correction diagram to the direct gap of silicon at the Γ-point, taking into account the dynamic screening of the electron-electron interaction. The authors' best calcn. yields a contribution of 0.12 eV. This result supports the assumption of the GW approxn. that vertex corrections can be neglected. The authors do not find a significant shift of the abs. energies.
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    Grüneis, A.; Kresse, G.; Hinuma, Y.; Oba, F. Ionization Potentials of Solids: The Importance of Vertex Corrections. Phys. Rev. Lett. 2014, 112, 096401,  DOI: 10.1103/PhysRevLett.112.096401
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    Ionization potentials of solids: the importance of vertex corrections
    Grueneis, Andreas; Kresse, Georg; Hinuma, Yoyo; Oba, Fumiyasu
    Physical Review Letters (2014), 112 (9), 096401/1-096401/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    The ionization potential is a fundamental key quantity with great relevance to diverse material properties. We find that state of the art methods based on d. functional theory and simple diagrammatic approaches as commonly taken in the GW approxn. predict the ionization potentials of semiconductors and insulators unsatisfactorily. Good agreement between theory and expt. is obtained only when diagrams resulting from the antisymmetry of the many-electron wave function are taken into account via vertex corrections in the self-energy. The present approach describes both localized and delocalized states accurately, making it ideally suited for a wide class of materials and processes.
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    Tal, A.; Chen, W.; Pasquarello, A. Vertex function compliant with the Ward identity for quasiparticle self-consistent calculations beyond GW. Phys. Rev. B 2021, 103, L161104,  DOI: 10.1103/PhysRevB.103.L161104
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    Vertex function compliant with the Ward identity for quasiparticle self-consistent calculations beyond GW
    Tal, Alexey; Chen, Wei; Pasquarello, Alfredo
    Physical Review B (2021), 103 (16), L161104CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    We extend the quasiparticle self-consistent approach beyond the GW approxn. by using a range-sepd. vertex function. The developed approach yields band gaps, dielec. consts., and band positions with an accuracy similar to highest-level electronic-structure calcns. without exceeding the cost of regular quasiparticle self-consistent GW. We introduce an exchange-correlation kernel that accounts for the vertex over the full spatial range. In the long range it complies with the Ward identity, while it is approximated through the adiabatic local d. functional in the short range. In this approach, the renormalization factor is balanced and the higher-order diagrams are effectively taken into account.
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    Maggio, E.; Kresse, G. GW Vertex Corrected Calculations for Molecular Systems. J. Chem. Theory Comput. 2017, 13, 4765– 4778,  DOI: 10.1021/acs.jctc.7b00586
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    GW Vertex Corrected Calculations for Molecular Systems
    Maggio, Emanuele; Kresse, Georg
    Journal of Chemical Theory and Computation (2017), 13 (10), 4765-4778CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Hedin's scheme is solved with the inclusion of the vertex function (GWΓ) for a set of small mols. The computational scheme allows for the consistent inclusion of the vertex both at the polarizability level and in the self-energy. A diagrammatic anal. shows that the self-energy formed with this four-point vertex does not lead to double counting of diagrams, that can be classified as direct "bubbles" and exchange diagrams. By removing the exchange diagrams from the self-energy, a simpler approxn. is obtained, called GWtc-tc. Very good agreement with expensive wave function-based methods is obtained for both approxns.
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    Mejuto-Zaera, C.; Weng, G.; Romanova, M.; Cotton, S. J.; Whaley, K. B.; Tubman, N. M.; Vlček, V. Are multi-quasiparticle interactions important in molecular ionization?. J. Chem. Phys. 2021, 154, 121101,  DOI: 10.1063/5.0044060
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    Are multi-quasiparticle interactions important in molecular ionization?
    Mejuto-Zaera, Carlos; Weng, Guorong; Romanova, Mariya; Cotton, Stephen J.; Whaley, K. Birgitta; Tubman, Norm M.; Vlcek, Vojtech
    Journal of Chemical Physics (2021), 154 (12), 121101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Photoemission spectroscopy directly probes individual electronic states, ranging from single excitations to high-energy satellites, which simultaneously represent multiple quasiparticles (QPs) and encode information about electronic correlation. The 1st-principles description of the spectra requires an efficient and accurate treatment of all many-body effects. This is esp. challenging for inner valence excitations where the single QP picture breaks down. Here, the authors provide the full valence spectra of small closed-shell mols., exploring the independent and interacting quasiparticle regimes, computed with the fully correlated adaptive sampling CI method. The authors critically compare these results to calcns. with the many-body perturbation theory, based on the GW and vertex cor. GWΓ approaches. The latter explicitly accounts for 2-QP quantum interactions, which have often been neglected. For mol. systems, the vertex correction universally improves the theor. spectra, and it is crucial for the accurate prediction of QPs as well as capturing the rich satellite structures of high-energy excitations. GWΓ offers a unified description across all relevant energy scales. Probably the multi-QP regime corresponds to dynamical correlations, which can be described via perturbation theory. (c) 2021 American Institute of Physics.
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    Shirley, E. L. Self-consistent GW and higher-order calculations of electron states in metals. Phys. Rev. B 1996, 54, 7758– 7764,  DOI: 10.1103/PhysRevB.54.7758
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    Self-consistent GW and higher-order calculations of electron states in metals
    Shirley, Eric L.
    Physical Review B: Condensed Matter (1996), 54 (11), 7758-7764CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    Past work, treating simple metals in the GW [Green function (G) with dynamically screened Coulomb interaction (W)] approxn., has largely neglected effects of self-consistency and higher-order vertex corrections on occupied bandwidths. This work presents self-consistent GW results, plus nearly self-consistent higher-order results, for jellium, illustrating that both effects are large, yet largely canceling (e.g., 0.65-eV effects on the sodium bandwidth, but a combined effect of only 0.13 eV). This supports findings that many-body effects substantially reduce such bandwidths.
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    Kutepov, A. L. Vertex corrections in self-consistent GWΓ calculations: ground state properties of vanadium. arXiv 2018, 1809.06654 [cond-mat], https://arxiv.org/abs/1809.06654.
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    Stefanucci, G.; Pavlyukh, Y.; Uimonen, A.-M.; van Leeuwen, R. Diagrammatic expansion for positive spectral functions beyond GW: Application to vertex corrections in the electron gas. Phys. Rev. B 2014, 90, 115134,  DOI: 10.1103/PhysRevB.90.115134
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    Diagrammatic expansion for positive spectral functions beyond GW: application to vertex corrections in the electron gas
    Stefanucci, G.; Pavlyukh, Y.; Uimonen, A.-M.; van Leeuwen, R.
    Physical Review B: Condensed Matter and Materials Physics (2014), 90 (11), 115134/1-115134/17, 17 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We present a diagrammatic approach to construct self-energy approxns. within many-body perturbation theory with pos. spectral properties. The method cures the problem of neg. spectral functions which arises from a straightforward inclusion of vertex diagrams beyond the GW approxn. Our approach consists of a two-step procedure: We first express the approx. many-body self-energy as a product of half-diagrams and then identify the minimal no. of half-diagrams to add in order to form a perfect square. The resulting self-energy is an unconventional sum of self-energy diagrams in which the internal lines of half a diagram are time-ordered Green's functions, whereas those of the other half are anti-time-ordered Green's functions, and the lines joining the two halves are either lesser or greater Green's functions. The theory is developed using noninteracting Green's functions and subsequently extended to self-consistent Green's functions. Issues related to the conserving properties of diagrammatic approxns. with pos. spectral functions are also addressed. As a major application of the formalism we derive the minimal set of addnl. diagrams to make pos. the spectral function of the GW approxn. with lowest-order vertex corrections and screened interactions. The method is then applied to vertex corrections in the three-dimensional homogeneous electron gas by using a combination of anal. frequency integrations and numerical Monte Carlo momentum integrations to evaluate the diagrams.
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    Pavlyukh, Y.; Uimonen, A.-M.; Stefanucci, G.; van Leeuwen, R. Vertex Corrections for Positive-Definite Spectral Functions of Simple Metals. Phys. Rev. Lett. 2016, 117, 206402,  DOI: 10.1103/PhysRevLett.117.206402
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    Vertex corrections for positive-definite spectral functions of simple metals
    Pavlyukh, Y.; Uimonen, A.-M.; Stefanucci, G.; van Leeuwen, R.
    Physical Review Letters (2016), 117 (20), 206402/1-206402/6CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    We present a systematic study of vertex corrections in a homogeneous electron gas at metallic densities. The vertex diagrams are built using a recently proposed pos.-definite diagrammatic expansion for the spectral function. The vertex function not only provides corrections to the well known plasmon and particle-hole scatterings, but also gives rise to new phys. processes such as the generation of two plasmon excitations or the decay of the one-particle state into a two-particle-one-hole state. By an efficient Monte Carlo momentum integration we are able to show that the addnl. scattering channels are responsible for a redn. of the bandwidth, the appearance of a secondary plasmon satellite below the Fermi level, and a substantial redistribution of spectral wts. The feasibility of the approach for first-principles band-structure calcns. is also discussed.
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    Pavlyukh, Y.; Stefanucci, G.; van Leeuwen, R. Dynamically screened vertex correction to GW. Phys. Rev. B 2020, 102, 045121,  DOI: 10.1103/PhysRevB.102.045121
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    Dynamically screened vertex correction to GW
    Pavlyukh, Y.; Stefanucci, G.; van Leeuwen, R.
    Physical Review B (2020), 102 (4), 045121CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    Diagrammatic perturbation theory is a powerful tool for the investigation of interacting many-body systems, the self-energy operator Σ encoding all the variety of scattering processes. In the simplest scenario of correlated electrons described by the GW approxn. for the electron self-energy, a particle transfers a part of its energy to neutral excitations. Higher-order (in screened Coulomb interaction W) self-energy diagrams lead to improved electron spectral functions (SFs) by taking more complicated scattering channels into account and by adding corrections to lower order self-energy terms. However, they also may lead to unphys. neg. spectral functions. The resoln. of this difficulty has been demonstrated in our previous works. The main idea is to represent the self-energy operator in a Fermi golden rule form which leads to a manifestly pos. definite SF and allows for a very efficient numerical algorithm. So far, the method has only been applied to the three-dimensional electron gas, which is a paradigmatic system, but a rather simple one. Here we systematically extend the method to two dimensions including realistic systems such as monolayer and bilayer graphene. We focus on one of the most important vertex function effects involving the exchange of two particles in the final state. We demonstrate that it should be evaluated with the proper screening and discuss its influence on the quasiparticle properties.
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    Del Sole, R.; Reining, L.; Godby, R. W. GWΓ approximation for electron self-energies in semiconductors and insulators. Phys. Rev. B 1994, 49, 8024– 8028,  DOI: 10.1103/PhysRevB.49.8024
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    GWΓ approximation for electron self-energies in semiconductors and insulators
    Del Sole, R.; Reining, Lucia; Godby, R. W.
    Physical Review B: Condensed Matter (1994), 49 (12), 8024-8CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    The widely used GW approxn. for the self-energy operator of a system of interacting electrons may, in principle, be improved using an approx. vertex correction Γ. The authors est. Γ using the local-d. approxn. The authors report the results of a comparable series of GW calcns. the band structure of Si, in which such a vertex correction is (i) excluded entirely, (ii) included only in the screened Coulomb interaction W, and (iii) included in both W and the expression for the self-energy. The authors also discuss the symmetry properties of the exact vertex correction and how they may be retained in further improvements.
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    Shishkin, M.; Marsman, M.; Kresse, G. Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections. Phys. Rev. Lett. 2007, 99, 246403,  DOI: 10.1103/PhysRevLett.99.246403
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    Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections
    Shishkin, M.; Marsman, M.; Kresse, G.
    Physical Review Letters (2007), 99 (24), 246403/1-246403/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Self-consistent GW calcns., maintaining only the quasiparticle part of the Green's function G, are reported for a wide class of materials, including small gap semiconductors and large gap insulators. We show that the inclusion of the attractive electron-hole interaction via an effective nonlocal exchange correlation kernel is required to obtain accurate band gaps in the framework of self-consistent GW calcns. If these are accounted for via vertex corrections in W, the band gaps are found to be within a few percent of the exptl. values.
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    Chen, W.; Pasquarello, A. Accurate band gaps of extended systems via efficient vertex corrections in GW. Phys. Rev. B 2015, 92, 041115,  DOI: 10.1103/PhysRevB.92.041115
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    Accurate band gaps of extended systems via efficient vertex corrections in GW
    Chen, Wei; Pasquarello, Alfredo
    Physical Review B: Condensed Matter and Materials Physics (2015), 92 (4), 041115/1-041115/5CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We propose the use of an approx. bootstrap exchange-correlation kernel to account for vertex corrections in self-consistent GW calcns. We show that the approx. kernel gives accurate band gaps for a variety of extended systems, including simple sp semiconductors, wide band-gap insulators, and transition-metal compds. with either closed or open d shells. The accuracy is comparable with that obtained via the soln. of the Bethe-Salpeter equation but only at a fraction of the computational cost.
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    Dahlen, N. E.; van Leeuwen, R. Self-consistent solution of the Dyson equation for atoms and molecules within a conserving approximation. J. Chem. Phys. 2005, 122, 164102,  DOI: 10.1063/1.1884965
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    Self-consistent solution of the Dyson equation for atoms and molecules within a conserving approximation
    Dahlen, Nils Erik; van Leeuwen, Robert
    Journal of Chemical Physics (2005), 122 (16), 164102/1-164102/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We have calcd. the self-consistent Green's function for a no. of atoms and diat. mols. This Green's function is obtained from a conserving self-energy approxn., which implies that the observables calcd. from the Green's functions agree with the macroscopic conservation laws for particle no., momentum, and energy. As a further consequence, the kinetic and potential energies agree with the virial theorem, and the many possible methods for calcg. the total energy all give the same result. In these calcns. we use the finite temp. formalism and calc. the Green's function on the imaginary time axis. This allows for a simple extension to nonequil. systems. We have compared the energies from self-consistent Green's functions to those of non-self-consistent schemes and also calcd. ionization potentials from the Green's functions by using the extended Koopmans' theorem.
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    Romaniello, P.; Bechstedt, F.; Reining, L. Beyond the GW approximation: Combining correlation channels. Phys. Rev. B 2012, 85, 155131,  DOI: 10.1103/PhysRevB.85.155131
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    Beyond the GW approximation: combining correlation channels
    Romaniello, Pina; Bechstedt, Friedhelm; Reining, Lucia
    Physical Review B: Condensed Matter and Materials Physics (2012), 85 (15), 155131/1-155131/15CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    In many-body perturbation theory (MBPT) the self-energy Σ = iGWΓ plays a key role since it contains all the many-body effects of the system. The exact self-energy is not known; as a first approxn. one can set the vertex function Γ to unity which leads to the GW approxn. The latter properly describes the high-d. regime, where screening is important; in the low-d. regime, instead, other approxns. are proposed, such as the T matrix, which describes multiple scattering between two particles. Here we combine the two approaches. Starting from the fundamental equations of MBPT, we show how one can derive the T-matrix approxn. to the self-energy in a common framework with GW. This allows us to elucidate several aspects of this formulation, including the origin of, and link between, the electron-hole and the particle-particle T matrix, the derivation of a screened T matrix, and the conversion of the T matrix into a vertex correction. The exactly solvable Hubbard mol. is used for illustration.
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    Leon, D. A.; Cardoso, C.; Chiarotti, T.; Varsano, D.; Molinari, E.; Ferretti, A. Frequency dependence in GW made simple using a multipole approximation. Phys. Rev. B 2021, 104, 115157,  DOI: 10.1103/PhysRevB.104.115157
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    Frequency dependence in GW made simple using a multipole approximation
    Leon, Dario A.; Cardoso, Claudia; Chiarotti, Tommaso; Varsano, Daniele; Molinari, Elisa; Ferretti, Andrea
    Physical Review B (2021), 104 (11), 115157CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    In the GW approxn., the screened interaction W is a nonlocal and dynamical potential that usually has a complex frequency dependence. A full description of such a dependence is possible but often computationally demanding. For this reason, it is still common practice to approx. W(ω) using a plasmon pole (PP) model. Such an approach, however, may deliver an accuracy limited by its simplistic description of the frequency dependence of the polarizability, i.e., of W. In this work, we explore a multipole approach (MPA) and develop an effective representation of the frequency dependence of W. We show that an appropriate sampling of the polarizability in the frequency complex plane and a multipole interpolation can lead to a level of accuracy comparable with full-frequency methods at a much lower computational cost. Moreover, both accuracy and cost are controllable by the no. of poles used in MPA. Eventually, we validate the MPA approach in selected prototype systems, showing that full-frequency quality results can be obtained with a limited no. of poles.
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    Chiarotti, T.; Marzari, N.; Ferretti, A. Unified Green’s function approach for spectral and thermodynamic properties from algorithmic inversion of dynamical potentials. Phys. Rev. Research 2022, 4, 013242,  DOI: 10.1103/PhysRevResearch.4.013242
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    Unified Green's function approach for spectral and thermodynamic properties from algorithmic inversion of dynamical potentials
    Chiarotti, Tommaso; Marzari, Nicola; Ferretti, Andrea
    Physical Review Research (2022), 4 (1), 013242CODEN: PRRHAI; ISSN:2643-1564. (American Physical Society)
    Dynamical potentials appear in many advanced electronic-structure methods, including self-energies from many-body perturbation theory, dynamical mean-field theory, electronic-transport formulations, and many embedding approaches. Here, we propose a novel treatment for the frequency dependence, introducing an algorithmic inversion method that can be applied to dynamical potentials expanded as sum over poles. This approach allows for an exact soln. of Dyson-like equations at all frequencies via a mapping to a matrix diagonalization, and provides simultaneously frequency-dependent (spectral) and frequency-integrated (thermodn.) properties of the Dyson-inverted propagators. The transformation to a sum over poles is performed introducing nth order generalized Lorentzians as an improved basis set to represent the spectral function of a propagator. Numerical results for the homogeneous electron gas at the G0W0 level are provided to argue for the accuracy and efficiency of such unified approach.
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    Nelson, W.; Bokes, P.; Rinke, P.; Godby, R. W. Self-interaction in Green’s-function theory of the hydrogen atom. Phys. Rev. A 2007, 75, 032505,  DOI: 10.1103/PhysRevA.75.032505
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    Self-interaction in Green's-function theory of the hydrogen atom
    Nelson, W.; Bokes, P.; Rinke, Patrick; Godby, R. W.
    Physical Review A: Atomic, Molecular, and Optical Physics (2007), 75 (3, Pt. A), 032505/1-032505/4CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    At. hydrogen provides a unique test case for computational electronic structure methods, since its electronic excitation energies are known anal. With only one electron, hydrogen contains no electronic correlation and is therefore particularly susceptible to spurious self-interaction errors introduced by certain computational methods. In this paper we focus on many-body perturbation theory (MBPT) in Hedin's GW approxn. While the Hartree-Fock and the exact MBPT self-energy are free of self-interaction, the correlation part of the GW self-energy does not have this property. Here we use at. hydrogen as a benchmark system for GW and show that the self-interaction part of the GW self-energy, while nonzero, is small. The effect of calcg. the GW self-energy from exact wave functions and eigenvalues, as distinct from those from the local-d. approxn., is also illuminating.
  51. 51
    Sakuma, R.; Aryasetiawan, F. Self-energy calculation of the hydrogen atom: Importance of the unbound states. Phys. Rev. A 2012, 85, 042509,  DOI: 10.1103/PhysRevA.85.042509
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    Self-energy calculation of the hydrogen atom: importance of the unbound states
    Sakuma, R.; Aryasetiawan, F.
    Physical Review A: Atomic, Molecular, and Optical Physics (2012), 85 (4-A), 042509/1-042509/6CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    We present the calcn. of the self-energy of the isolated hydrogen atom within the GW approxn. starting from the noninteracting Green's function constructed from the exact wave functions of the hydrogen atom. The error in the electron removal energy of the 1s state is found to be about 0.02 eV, which is much smaller than what one would expect. This small error is explained by the cancellation of the self-screening errors between different l contributions of the self-energy. The unbound continuum states are found to be crucial to get the correct self-energy.
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    Bruneval, F. Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies. J. Chem. Phys. 2012, 136, 194107,  DOI: 10.1063/1.4718428
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    Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies
    Bruneval, Fabien
    Journal of Chemical Physics (2012), 136 (19), 194107/1-194107/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A systematic evaluation of the ionization energy within the GW approxn. is carried out for the first row atoms, from H to Ar. We describe a Gaussian basis implementation of the GW approxn., which does not resort to any further tech. approxn., besides the choice of the basis set for the electronic wavefunctions. Different approaches to the GW approxn. have been implemented and tested, for example, the std. perturbative approach based on a prior mean-field calcn. (Hartree-Fock GW@HF or d.-functional theory GW@DFT) or the recently developed quasiparticle self-consistent method (QSGW). The HOMO energies of atoms obtained from both GW@HF and QSGW are in excellent agreement with the exptl. ionization energy. The LUMO energies of the singly charged cation yield a noticeably worse est. of the ionization energy. The best agreement with respect to expt. is obtained from the total energy differences within the RPA functional, which is the total energy corresponding to the GW self-energy. We conclude with a discussion about the slight concave behavior upon no. electron change of the GW approxn. and its consequences upon the quality of the orbital energies. (c) 2012 American Institute of Physics.
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    Li, J.; Holzmann, M.; Duchemin, I.; Blase, X.; Olevano, V. Helium Atom Excitations by the GW and Bethe-Salpeter Many-Body Formalism. Phys. Rev. Lett. 2017, 118, 163001,  DOI: 10.1103/PhysRevLett.118.163001
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    Helium atom excitations by the GW and bethe-salpeter many-body formalism
    Li, Jing; Holzmann, Markus; Duchemin, Ivan; Blase, Xavier; Olevano, Valerio
    Physical Review Letters (2017), 118 (16), 163001/1-163001/6CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
    A review. The helium atom is the simplest many-body electronic system provided by nature. The exact soln. to the Schr.ovrddot.odinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodol. Here, we check the ab initio many-body GW approxn. and the Bethe-Salpeter equation (BSE) against the exact soln. for helium. Starting from the Hartree-Fock method, we show that the GW and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and GW approxns. is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent d.-functional theory.
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    Koval, P.; Foerster, D.; Sánchez-Portal, D. Fully self-consistent GW and quasiparticle self-consistent GW for molecules. Phys. Rev. B 2014, 89, 155417,  DOI: 10.1103/PhysRevB.89.155417
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    Fully self-consistent GW and quasiparticle self-consistent GW for molecules
    Koval, P.; Foerster, D.; Sanchez-Portal, D.
    Physical Review B: Condensed Matter and Materials Physics (2014), 89 (15), 155417/1-155417/19CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    Two self-consistent schemes involving Hedin's GW approxn. are studied for a set of sixteen different atoms and small mols. We compare results from the fully self-consistent GW approxn. (SCGW) and the quasiparticle self-consistent GW approxn. (QSGW) within the same numerical framework. Core and valence electrons are treated on an equal footing in all the steps of the calcn. We use basis sets of localized functions to handle the space dependence of quantities and spectral functions to deal with their frequency dependence. We compare SCGW and QSGW on a qual. level by comparing the computed densities of states (DOS). To judge their relative merit on a quant. level, we compare their vertical ionization potentials (IPs) with those obtained from coupled-cluster calcns. CCSD(T). Our results are futher compared with "one-shot" G0W0 calcns. starting from Hartree-Fock solns. (G0W0-HF). Both self-consistent GW approaches behave quite similarly. Averaging over all the studied mols., both methods show only a small improvement (somewhat larger for SCGW) of the calcd. IPs with respect to G0W0-HF results. Interestingly, SCGW and QSGW calcns. tend to deviate in opposite directions with respect to CCSD(T) results. SCGW systematically underestimates the IPs, while QSGW tends to overestimate them. G0W0-HF produces results which are surprisingly close to QSGW calcns. both for the DOS and for the numerical values of the IPs.
  55. 55
    van Setten, M. J.; Caruso, F.; Sharifzadeh, S.; Ren, X.; Scheffler, M.; Liu, F.; Lischner, J.; Lin, L.; Deslippe, J. R.; Louie, S. G.; Yang, C.; Weigend, F.; Neaton, J. B.; Evers, F.; Rinke, P. GW100: Benchmarking G0W0 for Molecular Systems. J. Chem. Theory Comput. 2015, 11, 5665– 5687,  DOI: 10.1021/acs.jctc.5b00453
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    GW100: Benchmarking G0W0 for Molecular Systems
    van Setten, Michiel J.; Caruso, Fabio; Sharifzadeh, Sahar; Ren, Xinguo; Scheffler, Matthias; Liu, Fang; Lischner, Johannes; Lin, Lin; Deslippe, Jack R.; Louie, Steven G.; Yang, Chao; Weigend, Florian; Neaton, Jeffrey B.; Evers, Ferdinand; Rinke, Patrick
    Journal of Chemical Theory and Computation (2015), 11 (12), 5665-5687CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present the GW100 set. GW100 is a benchmark set of the ionization potentials and electron affinities of 100 mols. computed with the GW method using three independent GW codes and different GW methodologies. The quasi-particle energies of the highest-occupied MOs (HOMO) and lowest-unoccupied MOs (LUMO) are calcd. for the GW100 set at the G0W0@PBE level using the software packages TURBOMOLE, FHI-aims, and BerkeleyGW. The use of these three codes allows for a quant. comparison of the type of basis set (plane wave or local orbital) and handling of unoccupied states, the treatment of core and valence electrons (all electron or pseudopotentials), the treatment of the frequency dependence of the self-energy (full frequency or more approx. plasmon-pole models), and the algorithm for solving the quasi-particle equation. Primary results include ref. values for future benchmarks, best practices for convergence within a particular approach, and av. error bars for the most common approxns.
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    van Loon, E. G. C. P.; Rösner, M.; Katsnelson, M. I.; Wehling, T. O. Random phase approximation for gapped systems: Role of vertex corrections and applicability of the constrained random phase approximation. Phys. Rev. B 2021, 104, 045134,  DOI: 10.1103/PhysRevB.104.045134
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    Random phase approximation for gapped systems: Role of vertex corrections and applicability of the constrained random phase approximation
    van Loon, Erik G. C. P.; Roesner, Malte; Katsnelson, Mikhail I.; Wehling, Tim O.
    Physical Review B (2021), 104 (4), 045134CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    The many-body theory of interacting electrons poses an intrinsically difficult problem that requires simplifying assumptions. For the detn. of electronic screening properties of the Coulomb interaction, the RPA (RPA) provides such a simplification. Here we explicitly show that this approxn. is justified for band structures with sizable band gaps. This is when the electronic states responsible for the screening are energetically far away from the Fermi level, which is equiv. to a short electronic propagation length of these states. The RPA contains exactly those diagrams in which the classical Coulomb interaction covers all distances, whereas neglected vertex corrections involve quantum tunneling through the barrier formed by the band gap. Our anal. of electron-electron interactions provides a real-space analogy to Migdal's theorem on the smallness of vertex corrections in electron-phonon problems. An important application is the increasing use of constrained RPA calcns. of effective interactions. We find that their usage of Kohn-Sham energies accounts for the leading local (excitonic) vertex correction in insulators.
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    Almbladh, C.-O.; von Barth, U. Exact results for the charge and spin densities, exchange-correlations potentials, and density-functional eigenvalues. Phys. Rev. B 1985, 31, 3231– 3244,  DOI: 10.1103/PhysRevB.31.3231
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    Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues
    Almbladh, C. O.; Von Barth, U.
    Physical Review B: Condensed Matter and Materials Physics (1985), 31 (6), 3231-44CODEN: PRBMDO; ISSN:0163-1829.
    The authors derive asymptotically exact results for the charge and spin densities far away from finite systems (atoms and mols.) and far outside solid surfaces. These results are then used to obtain the correct asymptotic form of the exchange-correlation potential of d.-functional (DF) theory and to prove that, for all systems, the eigenvalue of the uppermost occupied DF orbital equals the exact ionization potential. For spin-polarized finite systems, the uppermost DF eigenvalue in each spin channel is also given by exact excitation energies.
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    Umrigar, C. J.; Gonze, X. Accurate exchange-correlation potentials and total-energy components for the helium isoelectronic series. Phys. Rev. A 1994, 50, 3827– 3837,  DOI: 10.1103/PhysRevA.50.3827
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    Accurate exchange-correlation potentials and total-energy components for the helium isoelectronic series
    Umrigar, C. J.; Gonze, Xavier
    Physical Review A: Atomic, Molecular, and Optical Physics (1994), 50 (5), 3827-37CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    Starting from very accurate many-body wave functions, we have constructed essentially exact electron, correlation-energy, and exchange-energy densities, exchange-correlation potentials, and components of the total energies for helium and two-electron ions (H-, Be2+, Ne8+, Hg78+). These d.-functional results are compared to the corresponding quantities obtained from a variety of commonly used approx. d. functionals, namely, the local-d. approxn. and various generalized-gradient approxns., in order to test the accuracy of the approx. functionals. Although the generalized-gradient approxns. yield improved energies compared to the local-d. approxn., the exchange and correlation potentials (esp. the latter) obtained from the generalized-gradient approxns. are in poor agreement with the corresponding exact potentials. The large-distance asymptotic behavior of the exact exchange-correlation potential to O(1/r4) is found to agree with theor. predictions. The short-range behavior of the exchange-correlation potential is very close to quadratic. The prospects for improved generalized-gradient approxns. are discussed.
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    Blase, X.; Attaccalite, C.; Olevano, V. First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications. Phys. Rev. B 2011, 83, 115103,  DOI: 10.1103/PhysRevB.83.115103
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    First-principles GW calculations for fullerenes, porphyrins, phthalocyanine, and other molecules of interest for organic photovoltaic applications
    Blase, X.; Attaccalite, C.; Olevano, V.
    Physical Review B: Condensed Matter and Materials Physics (2011), 83 (11), 115103/1-115103/9CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We evaluate the performances of ab initio GW calcns. for the ionization energies and HOMO-LUMO gaps of 13 gas phase mols. of interest for org. electronic and photovoltaic applications, including the C60 fullerene, pentacene, free-base porphyrins and phthalocyanine, PTCDA, and std. monomers such as thiophene, fluorene, benzothiazole, or thiadiazole. Std. G0W0 calcns., i.e., starting from eigenstates obtained with local or semilocal functionals, significantly improve the ionization energy and band gap as compared to d. functional theory Kohn-Sham results, but the calcd. quasiparticle values remain too small as a result of overscreening. Starting from Hartree-Fock-like eigenvalues provides much better results and is equiv. to performing self-consistency on the eigenvalues, with a resulting accuracy of 2%-4% as compared to expt. Our calcns. are based on an efficient Gaussian-basis implementation of GW with explicit treatment of the dynamical screening through contour deformation techniques.
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    Faber, C.; Attaccalite, C.; Olevano, V.; Runge, E.; Blase, X. First-principles GW calculations for DNA and RNA nucleobases. Phys. Rev. B 2011, 83, 115123,  DOI: 10.1103/PhysRevB.83.115123
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    First-principles GW calculations for DNA and RNA nucleobases
    Faber, Carina; Attaccalite, Claudio; Olevano, V.; Runge, E.; Blase, X.
    Physical Review B: Condensed Matter and Materials Physics (2011), 83 (11), 115123/1-115123/5CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    On the basis of first-principles GW calcns., we study the quasiparticle properties of the guanine, adenine, cytosine, thymine, and uracil DNA and RNA nucleobases. Beyond std. G0W0 calcns., starting from Kohn-Sham eigenstates obtained with (semi)local functionals, a simple self-consistency on the eigenvalues allows us to obtain vertical ionization energies and electron affinities within an av. 0.11 and 0.18 eV error, resp., as compared to state-of-the-art coupled-cluster and multiconfigurational perturbative quantum chem. approaches. Further, GW calcns. predict the correct π-character of the highest occupied state, due to several level crossings between d. functional and GW calcns. Our study is based on a recent Gaussian-basis implementation of GW calcns. with explicit treatment of dynamical screening through contour deformation techniques.
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    Blase, X.; Boulanger, P.; Bruneval, F.; Fernandez-Serra, M.; Duchemin, I. GW and Bethe-Salpeter study of small water clusters. J. Chem. Phys. 2016, 144, 034109,  DOI: 10.1063/1.4940139
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    GW and Bethe-Salpeter study of small water clusters
    Blase, Xavier; Boulanger, Paul; Bruneval, Fabien; Fernandez-Serra, Marivi; Duchemin, Ivan
    Journal of Chemical Physics (2016), 144 (3), 034109/1-034109/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We study within the GW and Bethe-Salpeter many-body perturbation theories the electronic and optical properties of small (H2O)n water clusters (n = 1-6). Comparison with high-level CCSD(T) Coupled-Cluster at the Single Double (Triple) levels and ADC(3) Green's function third order algebraic diagrammatic construction calcns. indicates that the std. non-self-consistent G0W0@PBE or G0W0@PBE0 approaches significantly underestimate the ionization energy by about 1.1 eV and 0.5 eV, resp. Consequently, the related Bethe-Salpeter lowest optical excitations are found to be located much too low in energy when building transitions from a non-self-consistent G0W0 description of the quasiparticle spectrum. Simple self-consistent schemes, with update of the eigenvalues only, are shown to provide a weak dependence on the Kohn-Sham starting point and a much better agreement with ref. calcns. The present findings rationalize the theory to expt. possible discrepancies obsd. in previous G0W0 and Bethe-Salpeter studies of bulk water. The increase of the optical gap with increasing cluster size is consistent with the evolution from gas to dense ice or water phases and results from an enhanced screening of the electron-hole interaction. (c) 2016 American Institute of Physics.
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    Knight, J. W.; Wang, X.; Gallandi, L.; Dolgounitcheva, O.; Ren, X.; Ortiz, J. V.; Rinke, P.; Körzdörfer, T.; Marom, N. Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods. J. Chem. Theory Comput. 2016, 12, 615– 626,  DOI: 10.1021/acs.jctc.5b00871
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    Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods
    Knight, Joseph W.; Wang, Xiaopeng; Gallandi, Lukas; Dolgounitcheva, Olga; Ren, Xinguo; Ortiz, J. Vincent; Rinke, Patrick; Korzdorfer, Thomas; Marom, Noa
    Journal of Chemical Theory and Computation (2016), 12 (2), 615-626CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    The performance of different GW methods is assessed for a set of 24 org. acceptors. Errors are evaluated with respect to coupled cluster singles, doubles, and perturbative triples [CCSD(T)] ref. data for the vertical ionization potentials (IPs) and electron affinities (EAs), extrapolated to the complete basis set limit. Addnl. comparisons are made to exptl. data, where available. We consider fully self-consistent GW (scGW), partial self-consistency in the Green's function (scGW0), non-self-consistent G0W0 based on several mean-field starting points, and a "beyond GW" second-order screened exchange (SOSEX) correction to G0W0. We also describe the implementation of the self-consistent Coulomb hole with screened exchange method (COHSEX), which serves as one of the mean-field starting points. The best performers overall are G0W0 + SOSEX and G0W0 based on an IP-tuned long-range cor. hybrid functional with the former being more accurate for EAs and the latter for IPs. Both provide a balanced treatment of localized vs delocalized states and valence spectra in good agreement with photoemission spectroscopy (PES) expts.
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    Loos, P.-F.; Romaniello, P.; Berger, J. A. Green Functions and Self-Consistency: Insights From the Spherium Model. J. Chem. Theory Comput. 2018, 14, 3071– 3082,  DOI: 10.1021/acs.jctc.8b00260
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    Green Functions and Self-Consistency: Insights From the Spherium Model
    Loos, Pierre-Francois; Romaniello, Pina; Berger, J. A.
    Journal of Chemical Theory and Computation (2018), 14 (6), 3071-3082CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We report an exhaustive study of the performance of different variants of Green function methods for the spherium model in which two electrons are confined to the surface of a sphere and interact via a genuine long-range Coulomb operator. We show that the spherium model provides a unique paradigm to study electronic correlation effects from the weakly correlated regime to the strongly correlated regime, since the mathematics are simple while the physics is rich. We compare perturbative GW, partially self-consistent GW and second-order Green function (GF2) methods for the computation of ionization potentials, electron affinities, energy gaps, correlation energies as well as singlet and triplet neutral excitations by solving the Bethe-Salpeter equation (BSE). We discuss the problem of self-screening in GW and show that it can be partially solved with a second-order screened exchange correction (SOSEX). We find that, in general, self-consistency deteriorates the results with respect to those obtained within perturbative approaches with a Hartree-Fock starting point. Finally, we unveil an important problem of partial self-consistency in GW: in the weakly correlated regime, it can produce artificial discontinuities in the self-energy caused by satellite resonances with large wts.
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    Niquet, Y. M.; Gonze, X. Band-gap energy in the random-phase approximation to density-functional theory. Phys. Rev. B 2004, 70, 245115,  DOI: 10.1103/PhysRevB.70.245115
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    Band-gap energy in the random-phase approximation to density-functional theory
    Niquet, Y. M.; Gonze, X.
    Physical Review B: Condensed Matter and Materials Physics (2004), 70 (24), 245115/1-245115/12CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    We calc. the interacting bandgap energy of a solid within the RPA to d. functional theory (DFT). The interacting bandgap energy is defined as Eg = ERPA(N + 1) + ERPA(N-1)-2ERPA(N), where ERPA(N) is the total DFT-RPA energy of the N-electron system. We compare the interacting bandgap energy with the Kohn-Sham bandgap energy, which is the difference between the conduction and valence band edges in the Kohn-Sham band structure. We show that they differ by an unrenormalized "G0W0" self-energy correction (i.e., a GW self-energy correction computed using Kohn-Sham orbitals and energies as input). This provides a well-defined and meaningful interpretation to G0W0 quasiparticle bandgap calcns., but questions the physics behind the renormalization factors in the expression of the bandgap energy. We also sep. the kinetic from the Coulomb contributions to the DFT-RPA bandgap energy, and discuss the related problem of the deriv. discontinuity in the DFT-RPA functional. Last we discuss the applicability of our results to other functionals based on many-body perturbation theory.
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    Niquet, Y. M.; Fuchs, M.; Gonze, X. Asymptotic behavior of the exchange-correlation potentials from the linear-response Sham–Schlüter equation. J. Chem. Phys. 2003, 118, 9504– 9518,  DOI: 10.1063/1.1566739
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    Asymptotic behavior of the exchange-correlation potentials from the linear-response Sham-Schluter equation
    Niquet, Y. M.; Fuchs, M.; Gonze, X.
    Journal of Chemical Physics (2003), 118 (21), 9504-9518CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The linear-response Sham-Schluter equation can be used to calc. an exchange-correlation potential starting from a given approxn. for the self-energy. The asymptotic behavior of these potentials is, however, much debated, a recent work suggesting that they could blow up in finite systems. Here we investigate the asymptotic behavior of the linear-response Sham-Schluter potentials in the GW and second-order approxns. for the self-energy. We show that these potentials do not diverge, and that the correlation potential itself has a -α/(2r4) tail (under appropriate conditions), where α depends on the self-energy. We also provide further justification for the quasiparticle approxn. to the linear-response Sham-Schluter equation, that is much simpler to solve while likely being of comparable accuracy. Calcns. for real mols. or solids using this approxn. should be within the reach of present computers.
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    Casida, M. E. Generalization of the optimized-effective-potential model to include electron correlation: A variational derivation of the Sham-Schlüter equation for the exact exchange-correlation potential. Phys. Rev. A 1995, 51, 2005– 2013,  DOI: 10.1103/PhysRevA.51.2005
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    Generalization of the optimized-effective-potential model to include electron correlation: a variational derivation of the Sham-Schlueter equation for the exact exchange-correlation potential
    Casida, Mark E.
    Physical Review A: Atomic, Molecular, and Optical Physics (1995), 51 (3), 2005-13CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    The now classic optimized-effective-potential (OEP) approach of Sharp and Horton, [Phys, Rev. 90, 317 (1953)] and Talman and Shadwick [Phys. Rev. A 14, 36 (1976)] seeks the local potential that is variationally optimized to best approx. the Hartree-Fock exchange operator. The resulting OEP can be identified as the exchange potential of Kohn-Sham d.-functional theory. The present work generalizes this OEP approach to treat the correlated case, and shows that the Kohn-Sham exchange-correlation potential is the variationally best local approxn. to the exchange-correlation self-energy. This provides a variational derivation of the equation for the exact exchange-correlation potential that was derived by Sham and Schlueter using a d. condition. Implications for an approx. phys. interpretation of the Kohn-Sham orbitals are discussed. A correlated generalization of the Sharp-Horton-Krieger-Li-Isfrate [Phys. Lett. A 146, 256 (1990)] approxn. of the exchange potential is introduced in the quasiparticle limit.
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    Perdew, J. P.; Levy, M. Comment on “Significance of the highest occupied Kohn-Sham eigenvalue. Phys. Rev. B 1997, 56, 16021– 16028,  DOI: 10.1103/PhysRevB.56.16021
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    Perdew, John P.; Levy, Mel
    Physical Review B: Condensed Matter (1997), 56 (24), 16021-16028CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    With more explanation than usual and without appeal to Janak's theorem, we discuss the statement and proof of the ionization potential theorems for the exact Kohn-Sham d.-functional theory of a many-electron system: (1) For any av. electron no. N between the integers Z - 1 and Z, and thus for N → Z from below, the highest occupied or partly occupied Kohn-Sham orbital energy is minus the ionization energy of the Z-electron system. (2) For Z - 1 < N < Z, the exact Kohn-Sham effective potential vs(r) tends to zero as |r| → ∞. We then argue that an objection to these theorems [L. Kleinman, Phys. Rev. B 56, 12042 (1997)] overlooks a crucial step in the proof of theorem (2): The asymptotic exponential decay of the exact electron d. of the Z-electron system is controlled by the exact ionization energy, but the decay of an approx. d. is not controlled by the approx. ionization energy. We discuss relevant evidence from the numerical construction of the exact Kohn-Sham potential. In particular, we point out a model two-electron problem for which the ionization potential theorems are exactly confirmed. Finally, we comment on related issues: the self-interaction correction, the discontinuity of the exact Kohn-Sham potential as N passes through the integer Z, and the generalized sum rule on the exchange-correlation hole.
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    Harbola, M. K. Relationship between the highest occupied Kohn-Sham orbital eigenvalue and ionization energy. Phys. Rev. B 1999, 60, 4545– 4550,  DOI: 10.1103/PhysRevB.60.4545
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    Relationship between the highest occupied Kohn-Sham orbital eigenvalue and ionization energy
    Harbola, Manoj K.
    Physical Review B: Condensed Matter and Materials Physics (1999), 60 (7), 4545-4550CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    Recent arguments for and against the equivalence of the highest occupied orbital eigenvalue of the Kohn-Sham theory and ionization energy are discussed. For phys. realistic systems with a non-integral no. of electrons, which are described by the thermal av. of two systems, each with an integer no. of electrons, an equiv. Kohn-Sham system exists. This is shown by writing explicit expressions for the exchange-correlation potential constructed to give the mixed-state d., and then relating it to the mixed-state exchange-correlation energy functional by employing the virial theorem sum rule of Levy and Perdew [Phys. Rev. A 32, 2010 (1985)]. Further, the functional deriv. of the mixed-state exchange-correlation energy functional is obtained in terms of this potential. This is then used to show, without recourse to Janak's theorem [Phys. Rev. B 18, 7165 (1978)], that .vepsiln.max(N)=-I(Z), where Z is an integer and (Z-1)<N<Z. Thus the original arguments about the equivalence of the highest occupied Kohn-Sham orbital eigenenergy and the ionization energy which were based on Janak's theorem are valid, and the two quantities are equal.
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    Klimeš, J.; Kresse, G. Kohn-Sham band gaps and potentials of solids from the optimized effective potential method within the random phase approximation. J. Chem. Phys. 2014, 140, 054516,  DOI: 10.1063/1.4863502
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    Kohn-Sham band gaps and potentials of solids from the optimised effective potential method within the random phase approximation
    Klimes, Jiri; Kresse, Georg
    Journal of Chemical Physics (2014), 140 (5), 054516/1-054516/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We present an implementation of the optimized effective potential (OEP) scheme for the exact-exchange (EXX) and RPA energy functionals and apply these methods to a range of bulk materials. We calc. the Kohn-Sham (KS) potentials and the corresponding band gaps and compare them to the potentials obtained by std. local d. approxn. (LDA) calcns. The KS gaps increase upon going from the LDA to the OEP in the RPA and finally to the OEP for EXX. This can be explained by the different depth of the potentials in the bonding and interstitial regions. To obtain the true quasi-particle gaps the deriv. discontinuities or G0W0 corrections need to be added to the RPA-OEP KS gaps. The predicted G0W0@RPA-OEP quasi-particle gaps are about 5% too large compared to the exptl. values. However, compared to G0W0 calcns. based on local or semi-local functionals, where the errors vary between different materials, we obtain a rather consistent description among all the materials. (c) 2014 American Institute of Physics.
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    Hellgren, M.; Baguet, L.; Calandra, M.; Mauri, F.; Wirtz, L. Electronic structure of TiSe2 from a quasi-self-consistent G0W0 approach. Phys. Rev. B 2021, 103, 075101,  DOI: 10.1103/PhysRevB.103.075101
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    Electronic structure of TiSe2 from a quasi-self-consistent G0W0 approach
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    Physical Review B (2021), 103 (7), 075101CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
    In a previous work, it was shown that the inclusion of exact exchange is essential for a first-principles description of both the electronic and the vibrational properties of TiSe2, M.Hellgren et al. [Phys.Rev.Lett.119, 176401 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.176401]. The GW approxn. provides a parameter-free description of screened exchange but is usually employed perturbatively (G0W0), making results more or less dependent on the starting point. In this work, we develop a quasi-self-consistent extension of G0W0 based on the RPA (RPA) and the optimized effective potential of hybrid d. functional theory. This approach generates an optimal G0W0 starting point and a hybrid exchange parameter consistent with the RPA. While self-consistency plays a minor role for systems such as Ar, BN, and ScN, it is shown to be crucial for TiS2 and TiSe2. We find the high-temp. phase of TiSe2 to be a semimetal with a band structure in good agreement with expt. Furthermore, the optimized hybrid functional agrees well with our previous est. and therefore accurately reproduces the low-temp. charge-d.-wave phase.
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    Varsano, D.; Barborini, M.; Guidoni, L. Kohn-Sham orbitals and potentials from quantum Monte Carlo molecular densities. J. Chem. Phys. 2014, 140, 054102,  DOI: 10.1063/1.4863213
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    We show the possibility to ext. Kohn-Sham orbitals, orbital energies, and exchange correlation potentials from accurate quantum Monte Carlo (QMC) densities for atoms (He, Be, Ne) and mols. (H2, Be2, H2O, and C2H4). The variational Monte Carlo (VMC) densities based on accurate Jastrow Antisymmetrised Geminal Power wave functions are calcd. through different estimators. Using these ref. densities, we ext. the Kohn-Sham quantities with the method developed by Zhao, Morrison, and Parr (ZMP). We compare these extd. quantities with those obtained form CISD densities and with other data reported in the literature, finding a good agreement between VMC and other high-level quantum chem. methods. Our results demonstrate the applicability of the ZMP procedure to QMC mol. densities, that can be used for the testing and development of improved functionals and for the implementation of embedding schemes based on QMC and d. functional theory. (c) 2014 American Institute of Physics.
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    A self-consistent set of equations was derived for an at. central potential such that the energy given by the orbitals for the potential is minimized. This effective potential behaves like -e2/r for large electron-nucleus distances (r) [e = electronic charge]. The equations were solved for C, Ne, and Al; the resulting total energies exceed the published Hartree-Fock total energies by <0.005%. The theory provides an effective, local, central exchange potential analogous to the Xα-statistical-exchange potential (J. C. Slater, (1974).
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    The authors present accurate optimized-potential-model (OPM) results for spherical spin-polarized atoms emphasizing the precise construction of the OPM exchange potential from the numerical soln. of the OPM integral equation, esp. for large r. The results are used to discuss the quality of the local spin-d. approxn. (LSDA) and a generalized-gradient expansion (GGA) A. D. Becke (1988) for describing these atoms. The LSDA can produce substantial errors (beyond what is known from unpolarized atoms) for quantities which are directly related to the spin polarization of these systems. In particular, the LSDA overestimates the magnetization d. in the interior of Cu by a factor of 2. While the GGA improves integral quantities like total ground-state and exchange energies, remarkably it is less successful for energy differences like Ex↑ - Ex↓. Most important, however, it is not able to reduce the LSDA's errors for local quantities like the difference between spin-up and spin-down exchange potentials and magnetization densities significantly nor does it reverse the LSDA's incorrect ordering of the two highest occupied majority-spin eigenvalues of Cr and Cu.
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    The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn-Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS d.-functional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approx. functional if the GKS potential operator is continuous and the d. change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from meta-generalized gradient approxns. (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen mols., solid aluminum arsenide, and solid argon provide numerical illustrations.
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    The Hohenberg-Kohn theorem was extended to fractional electron no. N, for an isolated open system described by a statistical mixt. The curve of lowest av. energy EN vs. N is a series of straight line segments with slope discontinuities at integral N. As N increases through an integer M, the chem. potential and the highest occupied Kohn-Sham orbital energy both jump from EM-EM-1 to EM+1-EM.
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    Kraisler, E.; Hodgson, M. J. P.; Gross, E. K. U. From Kohn–Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential. J. Chem. Theory Comput. 2021, 17, 1390– 1407,  DOI: 10.1021/acs.jctc.0c01093
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    From Kohn-Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential
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    Accurately describing excited states within Kohn-Sham (KS) d. functional theory (DFT), particularly those which induce ionization and charge transfer, remains a great challenge. Common exchange-correlation (xc) approxns. are unreliable for excited states owing, in part, to the absence of a deriv. discontinuity in the xc energy (Δ), which relates a many-electron energy difference to the corresponding KS energy difference. We demonstrate, anal. and numerically, how the relationship between KS and many-electron energies leads to the step structures obsd. in the exact xc potential in four scenarios: electron addn., mol. dissocn., excitation of a finite system, and charge transfer. We further show that steps in the potential can be obtained also with common xc approxns., as simple as the LDA, when addressed from the ensemble perspective. The article therefore highlights how capturing the relationship between KS and many-electron energies with advanced xc approxns. is crucial for accurately calcg. excitations, as well as the ground-state d. and energy of systems which consist of distinct subsystems.
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    A review with 93 refs. is given on the concept of orbital- and eigenvalue-dependent exchange-correlation (xc) energy functionals. We show how such functionals can be derived in a systematic fashion via a perturbation expansion, utilizing the Kohn-Sham system as a noninteracting ref. system. We demonstrate that the second-order contribution to this expansion of the xc-energy functional includes the leading term of the van der Waals interaction. The optimized-potential method (OPM), which allows the calcn. of the multiplicative xc-potential corresponding to an orbital- and eigenvalue-dependent xc-energy functional via an integral equation, is discussed in detail. We examine an approx. anal. soln. of the OPM integral equation, pointing out that, for eigenvalue-dependent functionals, the three paths used in the literature for the derivation of this approxn. yield different results. Finally, a no. of illustrative results, both for the exchange-only limit and for the combination of the exact exchange with various correlation functionals, are given.
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    This review provides a perspective on the use of orbital-dependent functionals, which is currently considered one of the most promising avenues in modern d.-functional theory. The focus here is on four major themes: the motivation for orbital-dependent functionals in terms of limitations of semilocal functionals; the optimized effective potential as a rigorous approach to incorporating orbital-dependent functionals within the Kohn-Sham framework; the rationale behind and advantages and limitations of four popular classes of orbital-dependent functionals; and the use of orbital-dependent functionals for predicting excited-state properties. For each of these issues, both formal and practical aspects are assessed.
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    Dahlen, N. E.; van Leeuwen, R.; von Barth, U. Variational energy functionals of the Green function and of the density tested on molecules. Phys. Rev. A 2006, 73, 012511,  DOI: 10.1103/PhysRevA.73.012511
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    Variational energy functionals of the Green function and of the density tested on molecules
    Dahlen, Nils Erik; van Leeuwen, Robert; von Barth, Ulf
    Physical Review A: Atomic, Molecular, and Optical Physics (2006), 73 (1, Pt. A), 012511/1-012511/13CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    We have calcd. total energies of atoms and diat. mols. from the Luttinger-Ward functional, using self-energy approxns. to second order as well as the GW approxn. In order to assess the variational quality of this functional, we have also solved the Dyson equation self-consistently. The Luttinger-Ward functional is compared to the variational functional due to Klein, and we demonstrate that the variational property of the latter functional is inferior to that of the Luttinger-Ward functional. We also show how to obtain variational d. functionals from the functionals of the Green function. These orbital functional schemes are important for systems where d.-functional theory using local functionals of the d. necessarily fails. We derive an optimized effective potential (OEP) scheme that is based on the Luttinger-Ward functional and, unlike the conventional OEP schemes, produces energies in good agreement with the values obtained from the self-consistent Green function. Our calcns. show that, when applied to mols., the Luttinger-Ward functional is more sensitive to the quality of the input Green function than when applied to atoms, but the energies are remarkably close to the self-consistent values when the Hartree-Fock Green function is used as input. This Luttinger-Ward functional is therefore a simple and efficient method for studying the merits of various self-energy approxns. while avoiding the computationally demanding task of solving the Dyson equation self-consistently.
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    von Barth, U.; Dahlen, N. E.; van Leeuwen, R.; Stefanucci, G. Conserving approximations in time-dependent density functional theory. Phys. Rev. B 2005, 72, 235109,  DOI: 10.1103/PhysRevB.72.235109
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    Conserving approximations in time-dependent density functional theory
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    Physical Review B: Condensed Matter and Materials Physics (2005), 72 (23), 235109/1-235109/10CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
    In the present work, we propose a theory for obtaining successively better approxns. to the linear response functions of time-dependent d. or current-d. functional theory. The new technique is based on the variational approach to many-body perturbation theory (MBPT) as developed during the sixties and later expanded by us in the mid-nineties. Due to this feature, the resulting response functions obey a large no. of conservation laws such as particle and momentum conservation and sum rules. The quality of the obtained results is governed by the phys. processes built in through MBPT but also by the choice of variational expressions. We here present several conserving response functions of different sophistication to be used in the calcn. of the optical response of solids and nanoscale systems.
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    In principle, the Luttinger-Ward Green's-function formalism allows one to compute simultaneously the total energy and the quasiparticle band structure of a many-body electronic system from first principles. We present approx. and exact expressions for the correlation energy within the GW-RPA that are more amenable to computation and allow for developing efficient approxns. to the self-energy operator and correlation energy. The exact form is a sum over differences between plasmon and interband energies. The approx. forms are based on summing over screened interband transitions. We also demonstrate that blind extremization of such functionals leads to unphys. results: imposing phys. constraints on the allowed solns. (Green's functions) is necessary. Finally, we present some relevant numerical results for at. systems.
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    Physical Review A: Atomic, Molecular, and Optical Physics (2007), 75 (3, Pt. A), 032505/1-032505/4CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
    At. hydrogen provides a unique test case for computational electronic structure methods, since its electronic excitation energies are known anal. With only one electron, hydrogen contains no electronic correlation and is therefore particularly susceptible to spurious self-interaction errors introduced by certain computational methods. In this paper we focus on many-body perturbation theory (MBPT) in Hedin's GW approxn. While the Hartree-Fock and the exact MBPT self-energy are free of self-interaction, the correlation part of the GW self-energy does not have this property. Here we use at. hydrogen as a benchmark system for GW and show that the self-interaction part of the GW self-energy, while nonzero, is small. The effect of calcg. the GW self-energy from exact wave functions and eigenvalues, as distinct from those from the local-d. approxn., is also illuminating.
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    Romaniello, P.; Guyot, S.; Reining, L. The self-energy beyond GW: Local and nonlocal vertex corrections. J. Chem. Phys. 2009, 131, 154111,  DOI: 10.1063/1.3249965
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    It is commonly accepted that the GW approxn. for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approxn.: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the at. limit, in which, instead, correlation is directly responsible for the obsd. problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be cor. by a two-point vertex contribution to the self-energy derived from time-dependent d. functional theory (TDDFT). We explain why the addn. of a particle, instead, requires the use of a different approx. vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the at. limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qual. We discuss the effects of the self-screening error as well as the at. limit using GW for the exactly solvable two-site Hubbard model. (c) 2009 American Institute of Physics.
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    Grüneis, A.; Marsman, M.; Harl, J.; Schimka, L.; Kresse, G. Making the random phase approximation to electronic correlation accurate. J. Chem. Phys. 2009, 131, 154115,  DOI: 10.1063/1.3250347
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    Making the random phase approximation to electronic correlation accurate
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    We show that the inclusion of second-order screened exchange to the RPA allows for an accurate description of electronic correlation in atoms and solids clearly surpassing the random phase approxn., but not yet approaching chem. accuracy. From a fundamental point of view, the method is self-correlation free for one-electron systems. From a practical point of view,the approach yields correlation energies for atoms, as well as for the jellium electron gas within a few kcal/mol of exact values, atomization energies within typically 2-3 kcal/mol of expt., and excellent lattice consts. for ionic and covalently bonded solids (0.2% error). The computational complexity is only O(N5), comparable to canonical second-order Moller-Plesset perturbation theory, which should allow for routine calcns. on many systems. (c) 2009 American Institute of Physics.
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    Ren, X.; Rinke, P.; Scuseria, G. E.; Scheffler, M. Renormalized second-order perturbation theory for the electron correlation energy: Concept, implementation, and benchmarks. Phys. Rev. B 2013, 88, 035120,  DOI: 10.1103/PhysRevB.88.035120
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    We present a renormalized second-order perturbation theory (rPT2), based on a Kohn-Sham (KS) ref. state, for the electron correlation energy that includes the RPA (RPA), second-order screened exchange (SOSEX), and renormalized single excitations (rSE). These three terms all involve a summation of certain types of diagrams to infinite order, and can be viewed as "renormalization" of the second-order direct, exchange, and single-excitation (SE) terms of Rayleigh-Schroedinger perturbation theory based on a KS ref. In this work, we establish the concept of rPT2 and present the numerical details of our SOSEX and rSE implementations. A preliminary version of rPT2, in which the renormalized SE (rSE) contribution was treated approx., has already been benchmarked for mol. atomization energies and chem. reaction barrier heights and shows a well-balanced performance. In this work, we present a refined version of rPT2, in which we evaluate the rSE series of diagrams rigorously. We then extend the benchmark studies to noncovalent interactions, including the rare-gas dimers, and the S22 and S66 test sets, as well as the cohesive energy of small copper clusters, and the equil. geometry of 10 diat. mols. Despite some remaining shortcomings, we conclude that rPT2 gives an overall satisfactory performance across different electronic situations, and is a promising step towards a generally applicable electronic-structure approach.
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    Lebègue, S.; Arnaud, B.; Alouani, M.; Bloechl, P. E. Implementation of an all-electron GW approximation based on the projector augmented wave method without plasmon pole approximation: Application to Si, SiC, AlAs, InAs, NaH, and KH. Phys. Rev. B 2003, 67, 155208,  DOI: 10.1103/PhysRevB.67.155208
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    Implementation of an all-electron GW approximation based on the projector augmented wave method without plasmon pole approximation: application to Si, SiC, AlAs, InAs, NaH, and KH
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    Physical Review B: Condensed Matter and Materials Physics (2003), 67 (15), 155208/1-155208/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
    An implementation of the GW approxn. (GWA) based on the all-electron projector-augmented-wave (PAW) method is presented, where the screened Coulomb interaction is computed within the RPA (RPA) instead of the plasmon-pole model. Two different ways of computing the self-energy are reported. The method is used successfully to det. the quasiparticle energies of six semiconducting or insulating materials: Si, SiC, AlAs, InAs, NaH, and KH. To illustrate the method the real and imaginary part of the frequency-dependent self-energy together with the spectral function of silicon are computed. Finally, the GWA results are compared with other calcns., highlighting that all-electron GWA results can differ markedly from those based on pseudopotential approaches.
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    The authors present a new method for efficient, accurate calcns. of many-body properties of periodic systems. The main features are (i) use of a real-space/imaginary-time representation, (ii) avoidance of any model form for the screened interaction W, (iii) exact sepn. of W and the self-energy Σ into short- and long-ranged parts, and (iv) the use of novel anal. continuation techniques in the energy domain. The computer time scales approx. linearly with system size. The authors give results for jellium and Si, including the spectral function of Si obtained from the Dyson equation.
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    Second-order perturbation theory based on the Kohn-Sham Hamiltonian leads to an implicit d. functional for the correlation energy EcMP2, which is explicitly dependent on both occupied and unoccupied Kohn-Sham single-particle orbitals and energies. The corresponding correlation potential vcMP2, which has to be evaluated by the optimized potential method, was found to be divergent in the asymptotic region of atoms, if pos.-energy continuum states are included in the calcn. [Facco Bonetti et al., Phys. Rev. Lett. 86, 2241 (2001)]. On the other hand, Niquet et al., [J. Chem. Phys. 118, 9504 (2003)] showed that vcMP2 has the same asymptotic -α/(2r4) behavior as the exact correlation potential, if the system under study has a discrete spectrum only. We study vMP2c for atoms in a spherical cavity within a basis-set-free finite differences approach, ensuring a completely discrete spectrum by requiring hard-wall boundary conditions at the cavity radius. Choosing this radius sufficiently large, one can devise a numerical continuation procedure which allows to normalize vMP2c consistent with the std. choice vc(r ∞) = 0 for free atoms, without modifying the potential in the chem. relevant region. An important prerequisite for the success of this scheme is the inclusion of very high-energy virtual states. Using this technique, we have calcd. vMP2c for all closed-shell and spherical open-shell atoms up to argon. One finds that vMP2c reproduces the shell structure of the exact correlation potential very well but consistently overestimates the corresponding shell oscillations. In the case of spin-polarized atoms one observes a strong interrelation between the correlation potentials of the two spin channels, which is completely absent for std. d. functionals. However, our results also demonstrate that EMP2c can only serve as a first step towards the construction of a suitable implicit correlation functional: The fundamental variational instability of this functional is recovered for beryllium, for which a breakdown of the self-consistent Kohn-Sham iteration is obsd. Moreover, even for those atoms for which the self-consistent iteration is stable, the results indicate that the inclusion of vMP2c in the total Kohn-Sham potential does not lead to an improvement compared to the complete neglect of the correlation potential.
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    As part of a project to obtain better optical response functions for nanomaterials and other systems with strong excitonic effects, we here calc. the exchange-correlation (XC) potential of d. functional theory (DFT) at a level of approxn. which corresponds to the dynamically screened exchange or GW approxn. In this process, we have designed a numerical method based on cubic splines, which appears to be superior to other techniques previously applied to the "inverse engineering problem" of DFT, i.e., the problem of finding an XC potential from a known particle d. The potentials we obtain do not suffer from unphys. ripple and have, to within a reasonable accuracy, the correct asymptotic tails outside localized systems. The XC potential is an important ingredient in finding the particle-conserving excitation energies in atoms and mols., and our potentials perform better in this regard as compared to the local-d. approxn. potential, potentials from generalized gradient approxns., and a DFT potential based on MP2 theory.
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    Perturbation theory on the basis of the Kohn-Sham Hamiltonian leads to an implicit d. functional for the correlation energy Ec. In this contribution we investigate the corresponding correlation potential vc. It is shown that for finite systems the vc obtained by direct application of the optimized potential method diverges in the asymptotic region. The presence of unoccupied states, inherent in any perturbative form of Ec, is identified as the origin of this unphys. behavior. An approx. variational procedure is developed in order to avoid this difficulty. The potential resulting from this method qual. reproduces the shell structure of the exact at. vc.
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    In this work we studied a new functional for the correlation energy obtained from the exact-exchange (EXX) approxn. within time-dependent d. functional theory. Correlation energies have been calcd. for a no. of different atoms showing excellent agreement with results from more sophisticated methods. These results lose little accuracy by approximating the EXX kernel by its static value, a procedure which enormously simplifies the calcns. The correlation potential, obtained by taking the functional deriv. with respect to the d., turns out to be remarkably accurate for all atoms studied. This potential has been used to calc. ionization potentials, static polarizabilities, and van der Waals coeffs. with results in close agreement with expt. (c) 2010 American Institute of Physics.
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    Golze, D.; Keller, L.; Rinke, P. Accurate Absolute and Relative Core-Level Binding Energies from GW. J. Phys. Chem. Lett. 2020, 11, 1840– 1847,  DOI: 10.1021/acs.jpclett.9b03423
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    Accurate Absolute and Relative Core-Level Binding Energies from GW
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    Journal of Physical Chemistry Letters (2020), 11 (5), 1840-1847CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    We present an accurate approach to compute X-ray photoelectron spectra based on the GW Green's function method that overcomes the shortcomings of common d. functional theory approaches. GW has become a popular tool to compute valence excitations for a wide range of materials. However, core-level spectroscopy is thus far almost uncharted in GW. We show that single-shot perturbation calcns. in the G0W0 approxn., which are routinely used for valence states, cannot be applied for core levels and suffer from an extreme, erroneous transfer of spectral wt. to the satellite spectrum. The correct behavior can be restored by partial self-consistent GW schemes or by using hybrid functionals with almost 50% of exact exchange as a starting point for G0W0. We also include relativistic corrections and present a benchmark study for 65 mol. 1s excitations. Our abs. and relative GW core-level binding energies agree within 0.3 and 0.2 eV with expt., resp.

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  • Abstract

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    Figure 1

    Figure 1. Self-energy diagrams benchmarked in this work. Solid lines represent the fermion propagators, dashed lines the bare Coulomb interaction, and wiggly lines the dressed Coulomb interaction.

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    Figure 2

    Figure 2. MP2 and RPA Φ functionals, from which the 2B and GW self-energies, respectively, are obtained upon differentiation with respect to the GF.

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    Figure 3

    Figure 3. EXX, RPA, LDA, and PBE xc potentials for Ca (left) and Zn (right); (inset) xc potentials multiplied by the radial coordinate.

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    Figure 4

    Figure 4. Correlation potentials for helium, beryllium, and neon. The MP2 correlation potential for beryllium is computed on top of EXX orbitals and eigenvalues. Exact correlation potentials are from ref (58).

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    Figure 5

    Figure 5. Alternative beyond-RPA correlation potentials for beryllium. Self-consistency has little impact on the GW+SOSEX HOMO: we have ϵHOMOscGW+SOSEX = −0.733 Ry and ϵHOMOGW+SOSEX@EXX = −0.735 Ry. Exact correlation potential is from ref (58).

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    Figure 6

    Figure 6. Correlation potentials for sodium and potassium. Thick lines are for the spin majority channel; thin lines are for the spin minority channel.

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    Figure 7

    Figure 7. Mean absolute error (MAE) on the ionization potentials for each method and starting point. In white, we indicate the independent-particle self-consistent field (scf) methods.

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    Figure 8

    Figure 8. Deviation from experiment of the ionization potentials of the neutral atoms as computed with the selected self-energies and starting points. The LSSE-MP2 IPs of Be and Ca and the LSSE-GW+SOSEX IP of Mn are computed starting from EXX orbitals and eigenvalues. Experimental IPs are from ref (108).

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    Figure 9

    Figure 9. Vertex equations in the TDHF, TDGW, and GW+SOSEX approximations.

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    Figure 10

    Figure 10. Self-energy diagram of order 2 in the screened interaction W.

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  • References

    This article references 110 other publications.

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      Physical Review (1965), 139 (3A), 796-823CODEN: PHRVAO; ISSN:0031-899X.
      A set of successively more accurate self-consistent equations for the 1-electron Green's function were derived. They correspond to an expansion in a screened potential rather than the bare Coulomb potential. The 1st equation is adequate for many purposes. Each equation follows from the demand that a corresponding expression for the total energy be stationary with respect to variations in the Green's function. The main information to be obtained, besides the total energy, is 1-particle-like excitation spectra, i.e., spectra characterized by the quantum nos. of a single particle. This includes the low-excitation spectra in metals as well as configurations in atoms, mols., and solids with one electron outside or one electron missing from a closed-shell structure. In the latter cases, an approx. description is obtained by a modified Hartree-Fock equation involving a "Coulomb hole" and a static screened potential in the exchange term. As an example, spectra of some atoms are discussed. To investigate the convergence of successive approxn. for the Green's function, extensive calcns. were made for the electron gas at a range of metallic ds. The results are expressed in terms of quasiparticle energies Ε(k) and quasiparticle interactions f(k,k'). The very 1st approxn. gives a good value for the magnitude of Ε(k.). To est. the deriv. of Ε(k), both the 1st- and the 2nd-order terms are needed. The derivative, and thus the sp. heat, differs from the free-particle value by only a few percent. The correction to the sp. heat keeps the same sign down to the lowest alkalimetal ds., and is smaller than those obtained recently by Silverstein (CA 59, 144d) and by Rice (CA 62, 7218f). The results for the paramagnetic susceptibility are unreliable in the alkalimetal-d.-region owing to poor convergence of the expansion for f. Besides the proof of a modified Luttinger-Ward-Klein variational principle and a related self-consistency idea, there is not much new in principle but emphasis is on the development of a numerically manageable approxn. scheme.
    2. 2
      Reining, L. The GW approximation: content, successes and limitations. Wiley Interdisciplinary Reviews: Computational Molecular Science 2018, 8, e1344  DOI: 10.1002/wcms.1344
      There is no corresponding record for this reference.
    3. 3
      Golze, D.; Dvorak, M.; Rinke, P. The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy. Front. Chem. 2019, 7, 377,  DOI: 10.3389/fchem.2019.00377
      3
      The GW compendium: a practical guide to theoretical photoemission spectroscopy
      Golze, Dorothea; Dvorak, Marc; Rinke, Patrick
      Frontiers in Chemistry (Lausanne, Switzerland) (2019), 7 (), 377CODEN: FCLSAA; ISSN:2296-2646. (Frontiers Media S.A.)
      A review. The GW approxn. in electronic structure theory has become a widespread tool for predicting electronic excitations in chem. compds. and materials. In the realm of theor. spectroscopy, the GW method provides access to charged excitations as measured in direct or inverse photoemission spectroscopy. The no. of GW calcns. in the past two decades has exploded with increased computing power and modern codes. The success of GW can be attributed to many factors: favorable scaling with respect to system size, a formal interpretation for charged excitation energies, the importance of dynamical screening in real systems, and its practical combination with other theories. In this review, we provide an overview of these formal and practical considerations. We expand, in detail, on the choices presented to the scientist performing GWcalcns. for the first time. We also give an introduction to the many-body theory behind GW a review of modern applications like mols. and surfaces, and a perspective on methods which go beyond conventional GW calcns. This review addresses chemists, physicists and material scientists with an interest in theor. spectroscopy. It is intended for newcomers to GW calcns. but can also serve as an alternative perspective for experts and an up-to-date source of computational techniques.
    4. 4
      Strinati, G. Dynamical Shift and Broadening of Core Excitons in Semiconductors. Phys. Rev. Lett. 1982, 49, 1519– 1522,  DOI: 10.1103/PhysRevLett.49.1519
      4
      Dynamical shift and broadening of core excitons in semiconductors
      Strinati, G.
      Physical Review Letters (1982), 49 (20), 1519-22CODEN: PRLTAO; ISSN:0031-9007.
      A non-Hermitian eigenvalue equation is proposed to det. binding energies and widths of core excitons in semiconductors, taking into account the time dependence of screening effects through the dielec. matrix ε-1(.vector.r,.vector.r';ω). Deviations from static screening contribute both an increase of the binding energy and a narrowing of the Auger width. Numerical ests. of both effects for the Si 2p transition give qual. agreement with exptl. data when the exciton size is reduced by band-structure effects.
    5. 5
      Hybertsen, M. S.; Louie, S. G. First-Principles Theory of Quasiparticles: Calculation of Band Gaps in Semiconductor and Insulators. Phys. Rev. Lett. 1985, 55, 1418– 1421,  DOI: 10.1103/PhysRevLett.55.1418
      5
      First-principles theory of quasiparticles: calculation of band gaps in semiconductors and insulators
      Hybertsen, Mark S.; Louie, Steven G.
      Physical Review Letters (1985), 55 (13), 1418-21CODEN: PRLTAO; ISSN:0031-9007.
      A first-principles theory is given for the quasiparticle energies of semiconductors and insulators. The full dielec. matrix is used to evaluate the nonlocal, energy-dependent electron self-energy operator. Both local-field effects and dynamical screening are essential for understanding quasiparticle energies. Results for the band gaps, optical transitions, and band dispersions for Si and diamond are in excellent agreement with existing exptl. data.
    6. 6
      Godby, R. W.; Schlüter, M.; Sham, L. J. Accurate Exchange-Correlation Potential for Silicon and Its Discontinuity on Addition of an Electron. Phys. Rev. Lett. 1986, 56, 2415– 2418,  DOI: 10.1103/PhysRevLett.56.2415
      6
      Accurate exchange-correlation potential for silicon and its discontinuity on addition of an electron
      Godby, R. W.; Schlueter, M.; Sham, L. J.
      Physical Review Letters (1986), 56 (22), 2415-18CODEN: PRLTAO; ISSN:0031-9007.
      An accurate d.-functional exchange-correlation potential Vxc(r) was detd. for Si, from calcns. of the self-energy Σ(r,r',ω). No local-d. approxn. (LDA) is used for Vxc. The band structure with this Vxc is in remarkably close agreement with that obtained with the LDA, while both differ significantly from the quasiparticle spectrum of Σ. The 50% band-gap error found in LDA calcns. is therefore not caused by the LDA but by the discontinuity, Δ, in the exact Vxc on addn. of an electron.
    7. 7
      Marzari, N.; Ferretti, A.; Wolverton, C. Electronic-structure methods for materials design. Nat. Mater. 2021, 20, 736– 749,  DOI: 10.1038/s41563-021-01013-3
      7
      Electronic-structure methods for materials design
      Marzari, Nicola; Ferretti, Andrea; Wolverton, Chris
      Nature Materials (2021), 20 (6), 736-749CODEN: NMAACR; ISSN:1476-1122. (Nature Portfolio)
      A review. The accuracy and efficiency of electronic-structure methods to understand, predict and design the properties of materials has driven a new paradigm in research. Simulations can greatly accelerate the identification, characterization and optimization of materials, with this acceleration driven by continuous progress in theory, algorithms and hardware, and by adaptation of concepts and tools from computer science. Nevertheless, the capability to identify and characterize materials relies on the predictive accuracy of the underlying phys. descriptions, and on the ability to capture the complexity of realistic systems. We provide here an overview of electronic-structure methods, of their application to the prediction of materials properties, and of the different strategies employed towards the broader goals of materials design and discovery.
    8. 8
      Shirley, E. L.; Martin, R. M. GW quasiparticle calculations in atoms. Phys. Rev. B 1993, 47, 15404– 15412,  DOI: 10.1103/PhysRevB.47.15404
      8
      GW-quasiparticle calculations in atoms
      Shirley, Eric L.; Martin, Richard M.
      Physical Review B: Condensed Matter and Materials Physics (1993), 47 (23), 15404-12CODEN: PRBMDO; ISSN:0163-1829.
      The authors present quasiparticle calcns. done in atoms using various versions of Hedin's GW approxn. The performance of GW in major-group elements is illustrated by calcns. of first and second ionization energies from the same initial configuration, along with various promotion energies. The results significantly improve Hartree-Fock eigenvalues and are comparable, in accuracy, to Hartree-Fock, total-energy differences. In the 3d (iron-series) transition metals, there are also significant improvements in the 4s- and 3d-state electron addn. energies in the first half of the series. In the second half of the series, the authors find that strong correlations among 3d electrons of opposite spin are not accounted for by GW. They also study the screening effects by cores on valence-electron addn. and removal energies. In particular, the "alkalilike" problem of one valence electron bound to a closed-shell core was examd.. The generalized GW approxn. describes quant. important effects of core-relaxation and core-valence correlation on the valence electrons. These core-relaxation and core-valence correlation effects can be incorporated into an effective potential suitable for use in calcns. of valence properties of atoms, mols., and solids (following paper).
    9. 9
      Stan, A.; Dahlen, N. E.; Leeuwen, R. v. Fully self-consistent GW calculations for atoms and molecules. EPL 2006, 76, 298,  DOI: 10.1209/epl/i2006-10266-6
      There is no corresponding record for this reference.
    10. 10
      Rostgaard, C.; Jacobsen, K. W.; Thygesen, K. S. Fully self-consistent GW calculations for molecules. Phys. Rev. B 2010, 81, 085103,  DOI: 10.1103/PhysRevB.81.085103
      10
      Fully self-consistent GW calculations for molecules
      Rostgaard, C.; Jacobsen, K. W.; Thygesen, K. S.
      Physical Review B: Condensed Matter and Materials Physics (2010), 81 (8), 085103/1-085103/10CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We calc. single-particle excitation energies for a series of 34 mols. using fully self-consistent GW, one-shot G0W0, Hartree-Fock (HF), and hybrid d.-functional theory (DFT). All calcns. are performed within the projector-augmented wave method using a basis set of Wannier functions augmented by numerical AOs. The GW self-energy is calcd. on the real frequency axis including its full frequency dependence and off-diagonal matrix elements. The mean abs. error of the ionization potential (IP) with respect to expt. is found to be 4.4, 2.6, 0.8, 0.4, and 0.5 eV for DFT-PBE, DFT-PBE0, HF, G0W0[HF], and self-consistent GW, resp. This shows that although electronic screening is weak in mol. systems, its inclusion at the GW level reduces the error in the IP by up to 50% relative to unscreened HF. In general GW overscreens the HF energies leading to underestimation of the IPs. The best IPs are obtained from one-shot G0W0 calcns. based on HF since this reduces the overscreening. Finally, we find that the inclusion of core-valence exchange is important and can affect the excitation energies by as much as 1 eV.
    11. 11
      Bruneval, F.; Marques, M. A. L. Benchmarking the Starting Points of the GW Approximation for Molecules. J. Chem. Theory Comput. 2013, 9, 324– 329,  DOI: 10.1021/ct300835h
      11
      Benchmarking the Starting Points of the GW Approximation for Molecules
      Bruneval, Fabien; Marques, Miguel A. L.
      Journal of Chemical Theory and Computation (2013), 9 (1), 324-329CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      The GW approxn. is nowadays being used to obtain accurate quasiparticle energies of atoms and mols. In practice, the GW approxn. is generally evaluated perturbatively, based on a prior self-consistent calcn. within a simpler approxn. The final result thus depends on the choice of the self-consistent mean-field chosen as a starting point. Using a recently developed GW code based on Gaussian basis functions, we benchmark a wide range of starting points for perturbative GW, including Hartree-Fock, LDA, PBE, PBE0, B3LYP, HSE06, BH and HLYP, CAM-B3LYP, and tuned CAM-B3LYP. In the evaluation of the ionization energy, the hybrid functionals are clearly superior results starting points when compared to Hartree-Fock, to LDA, or to the semilocal approxns. Furthermore, among the hybrid functionals, the ones with the highest proportion of exact-exchange usually perform best. Finally, the reliability of the frozen-core approxn., that allows for a considerable speed-up of the calcns., is demonstrated.
    12. 12
      Kaplan, F.; Harding, M. E.; Seiler, C.; Weigend, F.; Evers, F.; van Setten, M. J. Quasi-Particle Self-Consistent GW for Molecules. J. Chem. Theory Comput. 2016, 12, 2528– 2541,  DOI: 10.1021/acs.jctc.5b01238
      12
      Quasi-Particle Self-Consistent GW for Molecules
      Kaplan, F.; Harding, M. E.; Seiler, C.; Weigend, F.; Evers, F.; van Setten, M. J.
      Journal of Chemical Theory and Computation (2016), 12 (6), 2528-2541CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present the formalism and implementation of quasi-particle self-consistent GW (qsGW) and eigenvalue only quasi-particle self-consistent GW (evGW) adapted to std. quantum chem. packages. Our implementation is benchmarked against high-level quantum chem. computations (coupled-cluster theory) and exptl. results using a representative set of mols. Furthermore, we compare the qsGW approach for five mols. relevant for org. photovoltaics to self-consistent GW results (scGW) and analyze the effects of the self-consistency on the ground state d. by comparing calcd. dipole moments to their exptl. values. We show that qsGW makes a significant improvement over conventional G0W0 and that partially self-consistent flavors (in particular evGW) can be excellent alternatives.
    13. 13
      Hung, L.; da Jornada, F. H.; Souto-Casares, J.; Chelikowsky, J. R.; Louie, S. G.; Öğüt, S. Excitation spectra of aromatic molecules within a real-space GW -BSE formalism: Role of self-consistency and vertex corrections. Phys. Rev. B 2016, 94, 085125,  DOI: 10.1103/PhysRevB.94.085125
      13
      Excitation spectra of aromatic molecules within a real-space GW-BSE formalism: Role of self-consistency and vertex corrections
      Hung, Linda; da Jornada, Felipe H.; Souto-Casares, Jaime; Chelikowsky, James R.; Louie, Steven G.; Ogut, Serdar
      Physical Review B (2016), 94 (8), 085125/1-085125/13CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
      We present first-principles calcns. on the vertical ionization potentials (IPs), electron affinities (EAs), and singlet excitation energies on an arom.-mol. test set (benzene, thiophene, 1,2,5-thiadiazole, naphthalene, benzothiazole, and tetrathiafulvalene) within the GW and Bethe-Salpeter equation (BSE) formalisms. Our computational framework, which employs a real-space basis for ground-state and a transition-space basis for excited-state calcns., is well suited for high-accuracy calcns. on mols., as we show by comparing against G0W0 calcns. within a plane-wave-basis formalism. We then generalize our framework to test variants of the GW approxn. that include a local d. approxn. (LDA) -derived vertex function (ΓLDA) and quasiparticle-self-consistent (QS) iterations. We find that ΓLDA and quasiparticle self-consistency shift IPs and EAs by roughly the same magnitude, but with opposite sign for IPs and the same sign for EAs. G0W0 and QSGWΓLDA are more accurate for IPs, while G0W0ΓLDA and QSGW are best for EAs. For optical excitations, we find that perturbative GW-BSE underestimates the singlet excitation energy, while self-consistent GW-BSE results in good agreement with previous best-est. values for both valence and Rydberg excitations. Finally, our work suggests that a hybrid approach, in which G0W0 energies are used for occupied orbitals and G0W0ΓLDA for unoccupied orbitals, also yields optical excitation energies in good agreement with expt. but at a smaller computational cost.
    14. 14
      Bruneval, F.; Dattani, N.; van Setten, M. J. The GW Miracle in Many-Body Perturbation Theory for the Ionization Potential of Molecules. Frontiers in Chemistry 2021, 9, 749779,  DOI: 10.3389/fchem.2021.749779
      14
      The GW miracle in many-body perturbation theory for the ionization potential of molecules
      Bruneval, Fabien; Dattani, Nike; van Setten, Michiel J.
      Frontiers in Chemistry (Lausanne, Switzerland) (2021), 9 (), 749779CODEN: FCLSAA; ISSN:2296-2646. (Frontiers Media S.A.)
      We use the GW100 benchmark set to systematically judge the quality of several perturbation theories against high-level quantum chem. methods. First of all, we revisit the ref. CCSD(T) ionization potentials for this popular benchmark set and establish a revised set of CCSD(T) results. Then, for all of these 100 mols., we calc. the HOMO energy within second and third-order perturbation theory (PT2 and PT3), and, GW as post-Hartree-Fock methods. We found GW to be the most accurate of these three approxns. for the ionization potential, by far. Going beyond GW by adding more diagrams is a tedious and dangerous activity: We tried to complement GW with second-order exchange (SOX), with second-order screened exchange (SOSEX), with interacting electron-hole pairs (WTDHF), and with a GW d.-matrix (γGW). Only the γGW result has a pos. impact. Finally using an improved hybrid functional for the non-interacting Green's function, considering it as a cheap way to approx. self-consistency, the accuracy of the simplest GW approxn. improves even more. We conclude that GW is a miracle: Its subtle balance makes GW both accurate and fast.
    15. 15
      Shung, K. W. K.; Mahan, G. D. Calculated Photoemission Spectra of Na. Phys. Rev. Lett. 1986, 57, 1076– 1079,  DOI: 10.1103/PhysRevLett.57.1076
      15
      Calculated photoemission spectra of sodium
      Shung, Kenneth W. K.; Mahan, G. D.
      Physical Review Letters (1986), 57 (8), 1076-9CODEN: PRLTAO; ISSN:0031-9007.
      A detailed calcn. was made of the photoemission spectra of Na for electrons emitted normal to the (110) surface. The exptl. results of E. Jensen and E. W. Plummer (1985) were taken as an indication that the conduction band of Na is severely distorted and hence that a charge-d. wave exists in the system. The calcn. confirms just the opposite, i.e., that the band is not strongly distorted from a free-electron band. The calcn. is able to reproduce the exptl. data closely; it also predicts an angle dependence of the spectra, a feature which needs to be examd. exptl.
    16. 16
      Shung, K. W. K.; Sernelius, B. E.; Mahan, G. D. Self-energy corrections in photoemission of Na. Phys. Rev. B 1987, 36, 4499– 4502,  DOI: 10.1103/PhysRevB.36.4499
      16
      Self-energy corrections in photoemission of sodium
      Shung, Kenneth W. K.; Sernelius, Bo E.; Mahan, G. D.
      Physical Review B: Condensed Matter and Materials Physics (1987), 36 (8), 4499-502CODEN: PRBMDO; ISSN:0163-1829.
      The photoemission spectra were calcd. of Na, where self-energy corrections and surface effects were fully incorporated. The self-energy was calcd. with the Rayleigh-Schroedinger perturbation method. The energy shifts in both initial and final states were demonstrated by comparing the calcn. with the measured results of E. W. Plummer and coworkers (1984, 1985). Both the imaginary part and the real part of the self-energy contribute to the obsd. large bandwidth redn. in Na. These effects are believed to be important, in general, in analyzing photoemission spectra.
    17. 17
      Northrup, J. E.; Hybertsen, M. S.; Louie, S. G. Theory of quasiparticle energies in alkali metals. Phys. Rev. Lett. 1987, 59, 819– 822,  DOI: 10.1103/PhysRevLett.59.819
      17
      Theory of quasiparticle energies in alkali metals
      Northrup, John E.; Hybertsen, Mark S.; Louie, Steven G.
      Physical Review Letters (1987), 59 (7), 819-22CODEN: PRLTAO; ISSN:0031-9007.
      The quasiparticle band structures of Na and Li are calcd. by evaluation of the electron self-energy to first order in the dynamically screened Coulomb interaction. For Na, results explain quant. the large bandwidth redn. obsd. in recent angle-resolved photoemission expts. The inclusion of exchange-correlation effects in the dielec. screening and the requirement that the energy dependence of the electron Green's function be treated self-consistently are important in obtaining the correct bandwidths.
    18. 18
      Lyo, I.-W.; Plummer, E. W. Quasiparticle band structure of Na and simple metals. Phys. Rev. Lett. 1988, 60, 1558– 1561,  DOI: 10.1103/PhysRevLett.60.1558
      18
      Quasiparticle band structure of sodium and simple metals
      Lyo, In Whan; Plummer, E. W.
      Physical Review Letters (1988), 60 (15), 1558-61CODEN: PRLTAO; ISSN:0031-9007.
      An abs. measurement of the quasiparticle band structure of Na(110) was made by using angle-resolved photoemission. The occupied band is parabolic, but with an 18% narrowing in band width compared to self-consistent band calcns. A simple model calcn. of the self-energy correction for a homogeneous electron gas produced good agreement with the data, if a dielec. response beyond the random-phase approxn. was included. Expt. and theory are compared for other simple metals.
    19. 19
      Yasuhara, H.; Yoshinaga, S.; Higuchi, M. Why is the Bandwidth of Sodium Observed to be Narrower in Photoemission Experiments?. Phys. Rev. Lett. 1999, 83, 3250– 3253,  DOI: 10.1103/PhysRevLett.83.3250
      19
      Why is the Bandwidth of Sodium Observed to be Narrower in Photoemission Experiments?
      Yasuhara, H.; Yoshinaga, S.; Higuchi, M.
      Physical Review Letters (1999), 83 (16), 3250-3253CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The exptl. predicted narrowing in the bandwidth of Na is interpreted in terms of the nonlocal self-energy effect on quasiparticle energies of the electron liq. The calcd. self-energy correction is an increasing function of the wave no. variable. The usual anal. of angle-resolved photoemission expts. assumes the final-state energies on the nearly free-electron-like model and hence incorrectly ascribes the nonlocal self-energy correction to the final-state energies to occupied-state energies, seemingly leading to a narrowing in the bandwidth.
    20. 20
      Takada, Y. Inclusion of Vertex Corrections in the Self-Consistent Calculation of Quasiparticles in Metals. Phys. Rev. Lett. 2001, 87, 226402,  DOI: 10.1103/PhysRevLett.87.226402
      20
      Inclusion of Vertex Corrections in the Self-Consistent Calculation of Quasiparticles in Metals
      Takada, Yasutami
      Physical Review Letters (2001), 87 (22), 226402/1-226402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Based on the Bethe-Salpeter equation and the Ward identity derived from it, we provide a scheme for constructing the vertex function in the self-consistent iteration loop to det. the electron self-energy. The scheme is implemented in the homogeneous electron gas at the sodium d.
    21. 21
      van Schilfgaarde, M.; Kotani, T.; Faleev, S. Quasiparticle Self-Consistent GW Theory. Phys. Rev. Lett. 2006, 96, 226402,  DOI: 10.1103/PhysRevLett.96.226402
      21
      Quasiparticle Self-Consistent GW Theory
      Van Schilfgaarde, M.; Kotani, Takao; Faleev, S.
      Physical Review Letters (2006), 96 (22), 226402/1-226402/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      In past decades the scientific community has been looking for a reliable first-principles method to predict the electronic structure of solids with high accuracy. Here we present an approach which we call the quasiparticle self-consistent GW approxn. It is based on a kind of self-consistent perturbation theory, where the self-consistency is constructed to minimize the perturbation. We apply it to selections from different classes of materials, including alkali metals, semiconductors, wide band gap insulators, transition metals, transition metal oxides, magnetic insulators, and rare earth compds. Apart from some mild exceptions, the properties are very well described, particularly in weakly correlated cases. Self-consistency dramatically improves agreement with expt., and is sometimes essential. Discrepancies with expt. are systematic, and can be explained in terms of approxns. made.
    22. 22
      Morris, A. J.; Stankovski, M.; Delaney, K. T.; Rinke, P.; García-González, P.; Godby, R. W. Vertex corrections in localized and extended systems. Phys. Rev. B 2007, 76, 155106,  DOI: 10.1103/PhysRevB.76.155106
      22
      Vertex corrections in localized and extended systems
      Morris, Andrew J.; Stankovski, Martin; Delaney, Kris T.; Rinke, Patrick; Garcia-Gonzalez, P.; Godby, R. W.
      Physical Review B: Condensed Matter and Materials Physics (2007), 76 (15), 155106/1-155106/9CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      Within many-body perturbation theory, we apply vertex corrections to various closed-shell atoms and to jellium, using a local approxn. for the vertex consistent with starting the many-body perturbation theory from a Kohn-Sham Green's function constructed from d.-functional theory in the local-d. approxn. The vertex appears in two places - in the screened Coulomb interaction W and in the self-energy Σ - and we obtain a systematic discrimination of these two effects by turning the vertex in Σ on and off. We also make comparisons to std. GW results within the usual RPA, which omits the vertex from both. When a vertex is included for closed-shell atoms, both ground-state and excited-state properties demonstrate little improvement over std. GW. For jellium, we observe marked improvement in the quasiparticle bandwidth when the vertex is included only in W, whereas turning on the vertex in Σ leads to an unphys. quasiparticle dispersion and work function. A simple anal. suggests why implementation of the vertex only in W is a valid way to improve quasiparticle energy calcns., while the vertex in Σ is unphys., and points the way to the development of improved vertices for ab initio electronic structure calcns.
    23. 23
      Kutepov, A. L. Electronic structure of Na, K, Si, and LiF from self-consistent solution of Hedin’s equations including vertex corrections. Phys. Rev. B 2016, 94, 155101,  DOI: 10.1103/PhysRevB.94.155101
      23
      Electronic structure of Na, K, Si, and LiF from self-consistent solution of Hedin's equations including vertex corrections [Erratum to document cited in CA169:150966]
      Kutepov, Andrey L.
      Physical Review B (2016), 94 (15), 155101/1-155101/23CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
      A few self-consistent schemes to solve the Hedin equations are presented. They include vertex corrections of different complexity. Commonly used quasiparticle approxn. for the Green's function and static approxn. for the screened interaction are avoided altogether. Using alkali metals Na and K as well as semiconductor Si and wide-gap insulator LiF as examples, it is shown that both the vertex corrections in the polarizability P and in the self-energy Σ are important. Particularly, vertex corrections in Σ with proper treatment of frequency dependence of the screened interaction always reduce calcd. bandwidths/band gaps, improving the agreement with expt. The complexity of the vertex included in P and in Σ can be different. Whereas in the case of polarizability one generally has to solve the Bethe-Salpeter equation for the corresponding vertex function, it is enough (for the materials in this study) to include the vertex of the first order in the self-energy. The calcns. with appropriate vertices show remarkable improvement in the calcd. bandwidths and band gaps as compared to the self-consistent GW approxn. as well as to the self-consistent quasiparticle GW approxn.
    24. 24
      Guzzo, M.; Lani, G.; Sottile, F.; Romaniello, P.; Gatti, M.; Kas, J. J.; Rehr, J. J.; Silly, M. G.; Sirotti, F.; Reining, L. Valence Electron Photoemission Spectrum of Semiconductors: Ab Initio Description of Multiple Satellites. Phys. Rev. Lett. 2011, 107, 166401,  DOI: 10.1103/PhysRevLett.107.166401
      24
      Valence electron photoemission spectrum of semiconductors: ab initio description of multiple satellites
      Guzzo, Matteo; Lani, Giovanna; Sottile, Francesco; Romaniello, Pina; Gatti, Matteo; Kas, Joshua J.; Rehr, John J.; Silly, Mathieu G.; Sirotti, Fausto; Reining, Lucia
      Physical Review Letters (2011), 107 (16), 166401/1-166401/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The exptl. valence band photoemission spectrum of semiconductors exhibits multiple satellites that cannot be described by the GW approxn. for the self-energy in the framework of many-body perturbation theory. Taking silicon as a prototypical example, we compare exptl. high energy photoemission spectra with GW calcns. and analyze the origin of the GW failure. We then propose an approxn. to the functional differential equation that dets. the exact one-body Green's function, whose soln. has an exponential form. This yields a calcd. spectrum, including cross sections, secondary electrons, and an est. for extrinsic and interference effects, in excellent agreement with expt. Our result can be recast as a dynamical vertex correction beyond GW, giving hints for further developments.
    25. 25
      Choi, S.; Kutepov, A.; Haule, K.; van Schilfgaarde, M.; Kotliar, G. First-principles treatment of Mott insulators: linearized QSGW+DMFT approach. npj Quantum Materials 2016, 1, 16001,  DOI: 10.1038/npjquantmats.2016.1
      There is no corresponding record for this reference.
    26. 26
      Marom, N.; Caruso, F.; Ren, X.; Hofmann, O. T.; Körzdörfer, T.; Chelikowsky, J. R.; Rubio, A.; Scheffler, M.; Rinke, P. Benchmark of GW methods for azabenzenes. Phys. Rev. B 2012, 86, 245127,  DOI: 10.1103/PhysRevB.86.245127
      26
      Benchmark of GW methods for azabenzenes
      Marom, Noa; Caruso, Fabio; Ren, Xinguo; Hofmann, Oliver T.; Korzdorfer, Thomas; Chelikowsky, James R.; Rubio, Angel; Scheffler, Matthias; Rinke, Patrick
      Physical Review B: Condensed Matter and Materials Physics (2012), 86 (24), 245127/1-245127/16CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      Many-body perturbation theory in the GW approxn. is a useful method for describing electronic properties assocd. with charged excitations. A hierarchy of GW methods exists, starting from non-self-consistent G0W0, through partial self-consistency in the eigenvalues and in the Green's function (scGW0), to fully self-consistent GW (scGW). Here, we assess the performance of these methods for benzene, pyridine, and the diazines. The quasiparticle spectra are compared to photoemission spectroscopy (PES) expts. with respect to all measured particle removal energies and the ordering of the frontier orbitals. We find that the accuracy of the calcd. spectra does not match the expectations based on their level of self-consistency. In particular, for certain starting points G0W0 and scGW0 provide spectra in better agreement with the PES than scGW.
    27. 27
      Ren, X.; Marom, N.; Caruso, F.; Scheffler, M.; Rinke, P. Beyond the GW approximation: A second-order screened exchange correction. Phys. Rev. B 2015, 92, 081104,  DOI: 10.1103/PhysRevB.92.081104
      27
      Beyond the GW approximation: a second-order screened exchange correction
      Ren, Xinguo; Marom, Noa; Caruso, Fabio; Scheffler, Matthias; Rinke, Patrick
      Physical Review B: Condensed Matter and Materials Physics (2015), 92 (8), 081104/1-081104/6CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      Motivated by the recently developed renormalized second-order perturbation theory for ground-state energy calcns., we propose a second-order screened exchange correction (SOSEX) to the GW self-energy. This correction follows the spirit of the SOSEX correction to the RPA for the electron correlation energy and can be clearly represented in terms of Feynman diagrams. We benchmark the performance of the perturbative G0W0 + SOSEX scheme for a set of mol. systems, including the G2 test set from quantum chem. as well as benzene and tetracyanoethylene. We find that G0W0 + SOSEX improves over G0W0 for the energy levels of the highest occupied and lowest unoccupied MOs. In addn., it can resolve some of the difficulties encountered by the GW method for relative energy positions as exemplified by benzene where the energy spacing between certain valence orbitals is severely underestimated.
    28. 28
      Wang, Y.; Rinke, P.; Ren, X. Assessing the G0W0Γ0(1) Approach: Beyond G0W0 with Hedin’s Full Second-Order Self-Energy Contribution. J. Chem. Theory Comput 2021, 17, 5140– 5154,  DOI: 10.1021/acs.jctc.1c00488
      28
      Assessing the G0W0Γ0(1) Approach: Beyond G0W0 with Hedin's Full Second-Order Self-Energy Contribution
      Wang, Yanyong; Rinke, Patrick; Ren, Xinguo
      Journal of Chemical Theory and Computation (2021), 17 (8), 5140-5154CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present and benchmark a self-energy approach for quasiparticle energy calcns. that goes beyond Hedin's GW approxn. by adding the full second-order self-energy (FSOS-W) contribution. The FSOS-W diagram involves two screened Coulomb interaction (W) lines, and adding the FSOS-W to the GW self-energy can be interpreted as first-order vertex correction to GW (GWΓ(1)). Our FSOS-W implementation is based on the resoln.-of-identity technique and exhibits better than O(N5) scaling with system size for small- to medium-sized mols. We then present one-shot GWΓ(1) (G0W0Γ0(1)) benchmarks for the GW100 test set and a set of 24 acceptor mols. For semilocal or hybrid d. functional theory starting points, G0W0Γ0(1) systematically outperforms G0W0 for the first vertical ionization potentials and electron affinities of both test sets. Finally, we demonstrate that a static FSOS-W self-energy significantly underestimates the quasiparticle energies.
    29. 29
      von Barth, U.; Holm, B. Self-consistent GW0 results for the electron gas: Fixed screened potential W0 within the random-phase approximation. Phys. Rev. B 1996, 54, 8411– 8419,  DOI: 10.1103/PhysRevB.54.8411
      29
      Self-consistent GW0 results for the electron gas: fixed screened potential W0 within the random-phase approximation
      von Barth, Ulf; Holm, Bengt
      Physical Review B: Condensed Matter (1996), 54 (12), 8411-8419CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      With the aim of properly understanding the basis for and the utility of many-body perturbation theory as applied extended metallic systems, we have calcd. the electronic self-energy of the homogeneous electron gas within the GW approxn. The calcn. had been carried out in a self-consistent way; i.e., the one-electron Green function (G) obtained from Dyson's equation is the same as that used to calc. the self-energy. The self-consistency is restricted in the sense that the screened interaction W is kept fixed and equal to that of the random-phase approxn. for the gas. We have found that the final results are marginally affected by the broadening of the quasiparticles, and that their self-consistent energies are still close to their free-electron counterparts as they are in non-self-consistent calcns. The redn. in strength of the quasiparticles and the development of satellite structure (plasmons) gives, however, a markedly smaller dynamic self-energy leading to, e.g., a smaller redn. in the quasiparticle strength as compared to non-self-consistent results. The relatively bad description of plasmon structure within the non-self-consistent GW approxn. is marginally improved. A first attempt at including W in the self-consistency cycle leads to an even broader and structureless satellite spectrum in disagreement with expt.
    30. 30
      Holm, B.; von Barth, U. Fully self-consistent GW self-energy of the electron gas. Phys. Rev. B 1998, 57, 2108– 2117,  DOI: 10.1103/PhysRevB.57.2108
      30
      Fully self-consistent GW self-energy of the electron gas
      Holm, B.; von Barth, U.
      Physical Review B: Condensed Matter and Materials Physics (1998), 57 (4), 2108-2117CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      We present fully self-consistent results for the self-energy of the electron gas within the GW approxn. This means that the self-consistent Green's function G, as obtained from Dyson's equation, is used not only for obtaining the self-energy but also for constructing the screened interaction W within the RPA. Such a theory is particle and energy conserving in the sense of Kadanoff and Baym. We find an increase in the wt. of the quasiparticle as compared to ordinary non-self-consistent calcns. but also to calcns. with partial self-consistency using a fixed W. The quasiparticle bandwidth is larger than that of free electrons and the satellite structure is broad and featureless; both results clearly contradict the exptl. evidence. The total energy, though, is as accurate as that from quantum Monte Carlo calcns., and its deriv. with respect to particle no. agrees with the Fermi energy as obtained directly from the pole of the Green's function at the Fermi level. Our results indicate that, unless vertex corrections are included, non-self-consistent results are to be preferred for most properties except for the total energy.
    31. 31
      Schindlmayr, A.; García-González, P.; Godby, R. W. Diagrammatic self-energy approximations and the total particle number. Phys. Rev. B 2001, 64, 235106,  DOI: 10.1103/PhysRevB.64.235106
      31
      Diagrammatic self-energy approximations and the total particle number
      Schindlmayr, Arno; Garcia-Gonzalez, P.; Godby, R. W.
      Physical Review B: Condensed Matter and Materials Physics (2001), 64 (23), 235106/1-235106/6CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      There is increasing interest in many-body perturbation theory as a practical tool for the calcn. of ground-state properties. As a consequence, unambiguous sum rules such as the conservation of particle no. under the influence of the Coulomb interaction have acquired an importance that did not exist for calcns. of excited-state properties. In this paper we obtain a rigorous, simple relation whose fulfillment guarantees particle-no. conservation in a given diagrammatic self-energy approxn. Hedin's G0W0 approxn. does not satisfy this relation and hence violates the particle-no. sum rule. Very precise calcns. for the homogeneous electron gas and a model inhomogeneous electron system allow the extent of the nonconservation to be estd.
    32. 32
      Stan, A.; Dahlen, N. E.; van Leeuwen, R. Levels of self-consistency in the GW approximation. J. Chem. Phys. 2009, 130, 114105,  DOI: 10.1063/1.3089567
      32
      Levels of self-consistency in the GW approximation
      Stan, Adrian; Dahlen, Nils Erik; van Leeuwen, Robert
      Journal of Chemical Physics (2009), 130 (11), 114105/1-114105/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We perform GW calcns. on atoms and diat. mols. at different levels of self-consistency and investigate the effects of self-consistency on total energies, ionization potentials, and particle no. conservation. We further propose a partially self-consistent GW scheme in which we keep the correlation part of the self-energy fixed within the self-consistency cycle. This approxn. is compared to the fully self-consistent GW results and to the GW0 and the G0W0 approxns. Total energies, ionization potentials, and two-electron removal energies obtained with our partially self-consistent GW approxn. are in excellent agreement with fully self-consistent GW results while requiring only a fraction of the computational effort. We also find that self-consistent and partially self-consistent schemes provide ionization energies of similar quality as the G0W0 values but yield better total energies and energy differences. (c) 2009 American Institute of Physics.
    33. 33
      Bobbert, P. A.; van Haeringen, W. Lowest-order vertex-correction contribution to the direct gap of silicon. Phys. Rev. B 1994, 49, 10326– 10331,  DOI: 10.1103/PhysRevB.49.10326
      33
      Lowest-order vertex-correction contribution to the direct gap of silicon
      Bobbert, P. A.; van Haeringen, W.
      Physical Review B: Condensed Matter and Materials Physics (1994), 49 (15), 10326-31CODEN: PRBMDO; ISSN:0163-1829.
      The authors have calcd. the contribution of the lowest-order vertex-correction diagram to the direct gap of silicon at the Γ-point, taking into account the dynamic screening of the electron-electron interaction. The authors' best calcn. yields a contribution of 0.12 eV. This result supports the assumption of the GW approxn. that vertex corrections can be neglected. The authors do not find a significant shift of the abs. energies.
    34. 34
      Grüneis, A.; Kresse, G.; Hinuma, Y.; Oba, F. Ionization Potentials of Solids: The Importance of Vertex Corrections. Phys. Rev. Lett. 2014, 112, 096401,  DOI: 10.1103/PhysRevLett.112.096401
      34
      Ionization potentials of solids: the importance of vertex corrections
      Grueneis, Andreas; Kresse, Georg; Hinuma, Yoyo; Oba, Fumiyasu
      Physical Review Letters (2014), 112 (9), 096401/1-096401/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The ionization potential is a fundamental key quantity with great relevance to diverse material properties. We find that state of the art methods based on d. functional theory and simple diagrammatic approaches as commonly taken in the GW approxn. predict the ionization potentials of semiconductors and insulators unsatisfactorily. Good agreement between theory and expt. is obtained only when diagrams resulting from the antisymmetry of the many-electron wave function are taken into account via vertex corrections in the self-energy. The present approach describes both localized and delocalized states accurately, making it ideally suited for a wide class of materials and processes.
    35. 35
      Tal, A.; Chen, W.; Pasquarello, A. Vertex function compliant with the Ward identity for quasiparticle self-consistent calculations beyond GW. Phys. Rev. B 2021, 103, L161104,  DOI: 10.1103/PhysRevB.103.L161104
      35
      Vertex function compliant with the Ward identity for quasiparticle self-consistent calculations beyond GW
      Tal, Alexey; Chen, Wei; Pasquarello, Alfredo
      Physical Review B (2021), 103 (16), L161104CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      We extend the quasiparticle self-consistent approach beyond the GW approxn. by using a range-sepd. vertex function. The developed approach yields band gaps, dielec. consts., and band positions with an accuracy similar to highest-level electronic-structure calcns. without exceeding the cost of regular quasiparticle self-consistent GW. We introduce an exchange-correlation kernel that accounts for the vertex over the full spatial range. In the long range it complies with the Ward identity, while it is approximated through the adiabatic local d. functional in the short range. In this approach, the renormalization factor is balanced and the higher-order diagrams are effectively taken into account.
    36. 36
      Maggio, E.; Kresse, G. GW Vertex Corrected Calculations for Molecular Systems. J. Chem. Theory Comput. 2017, 13, 4765– 4778,  DOI: 10.1021/acs.jctc.7b00586
      36
      GW Vertex Corrected Calculations for Molecular Systems
      Maggio, Emanuele; Kresse, Georg
      Journal of Chemical Theory and Computation (2017), 13 (10), 4765-4778CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Hedin's scheme is solved with the inclusion of the vertex function (GWΓ) for a set of small mols. The computational scheme allows for the consistent inclusion of the vertex both at the polarizability level and in the self-energy. A diagrammatic anal. shows that the self-energy formed with this four-point vertex does not lead to double counting of diagrams, that can be classified as direct "bubbles" and exchange diagrams. By removing the exchange diagrams from the self-energy, a simpler approxn. is obtained, called GWtc-tc. Very good agreement with expensive wave function-based methods is obtained for both approxns.
    37. 37
      Mejuto-Zaera, C.; Weng, G.; Romanova, M.; Cotton, S. J.; Whaley, K. B.; Tubman, N. M.; Vlček, V. Are multi-quasiparticle interactions important in molecular ionization?. J. Chem. Phys. 2021, 154, 121101,  DOI: 10.1063/5.0044060
      37
      Are multi-quasiparticle interactions important in molecular ionization?
      Mejuto-Zaera, Carlos; Weng, Guorong; Romanova, Mariya; Cotton, Stephen J.; Whaley, K. Birgitta; Tubman, Norm M.; Vlcek, Vojtech
      Journal of Chemical Physics (2021), 154 (12), 121101CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Photoemission spectroscopy directly probes individual electronic states, ranging from single excitations to high-energy satellites, which simultaneously represent multiple quasiparticles (QPs) and encode information about electronic correlation. The 1st-principles description of the spectra requires an efficient and accurate treatment of all many-body effects. This is esp. challenging for inner valence excitations where the single QP picture breaks down. Here, the authors provide the full valence spectra of small closed-shell mols., exploring the independent and interacting quasiparticle regimes, computed with the fully correlated adaptive sampling CI method. The authors critically compare these results to calcns. with the many-body perturbation theory, based on the GW and vertex cor. GWΓ approaches. The latter explicitly accounts for 2-QP quantum interactions, which have often been neglected. For mol. systems, the vertex correction universally improves the theor. spectra, and it is crucial for the accurate prediction of QPs as well as capturing the rich satellite structures of high-energy excitations. GWΓ offers a unified description across all relevant energy scales. Probably the multi-QP regime corresponds to dynamical correlations, which can be described via perturbation theory. (c) 2021 American Institute of Physics.
    38. 38
      Shirley, E. L. Self-consistent GW and higher-order calculations of electron states in metals. Phys. Rev. B 1996, 54, 7758– 7764,  DOI: 10.1103/PhysRevB.54.7758
      38
      Self-consistent GW and higher-order calculations of electron states in metals
      Shirley, Eric L.
      Physical Review B: Condensed Matter (1996), 54 (11), 7758-7764CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      Past work, treating simple metals in the GW [Green function (G) with dynamically screened Coulomb interaction (W)] approxn., has largely neglected effects of self-consistency and higher-order vertex corrections on occupied bandwidths. This work presents self-consistent GW results, plus nearly self-consistent higher-order results, for jellium, illustrating that both effects are large, yet largely canceling (e.g., 0.65-eV effects on the sodium bandwidth, but a combined effect of only 0.13 eV). This supports findings that many-body effects substantially reduce such bandwidths.
    39. 39
      Kutepov, A. L. Vertex corrections in self-consistent GWΓ calculations: ground state properties of vanadium. arXiv 2018, 1809.06654 [cond-mat], https://arxiv.org/abs/1809.06654.
      There is no corresponding record for this reference.
    40. 40
      Stefanucci, G.; Pavlyukh, Y.; Uimonen, A.-M.; van Leeuwen, R. Diagrammatic expansion for positive spectral functions beyond GW: Application to vertex corrections in the electron gas. Phys. Rev. B 2014, 90, 115134,  DOI: 10.1103/PhysRevB.90.115134
      40
      Diagrammatic expansion for positive spectral functions beyond GW: application to vertex corrections in the electron gas
      Stefanucci, G.; Pavlyukh, Y.; Uimonen, A.-M.; van Leeuwen, R.
      Physical Review B: Condensed Matter and Materials Physics (2014), 90 (11), 115134/1-115134/17, 17 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We present a diagrammatic approach to construct self-energy approxns. within many-body perturbation theory with pos. spectral properties. The method cures the problem of neg. spectral functions which arises from a straightforward inclusion of vertex diagrams beyond the GW approxn. Our approach consists of a two-step procedure: We first express the approx. many-body self-energy as a product of half-diagrams and then identify the minimal no. of half-diagrams to add in order to form a perfect square. The resulting self-energy is an unconventional sum of self-energy diagrams in which the internal lines of half a diagram are time-ordered Green's functions, whereas those of the other half are anti-time-ordered Green's functions, and the lines joining the two halves are either lesser or greater Green's functions. The theory is developed using noninteracting Green's functions and subsequently extended to self-consistent Green's functions. Issues related to the conserving properties of diagrammatic approxns. with pos. spectral functions are also addressed. As a major application of the formalism we derive the minimal set of addnl. diagrams to make pos. the spectral function of the GW approxn. with lowest-order vertex corrections and screened interactions. The method is then applied to vertex corrections in the three-dimensional homogeneous electron gas by using a combination of anal. frequency integrations and numerical Monte Carlo momentum integrations to evaluate the diagrams.
    41. 41
      Pavlyukh, Y.; Uimonen, A.-M.; Stefanucci, G.; van Leeuwen, R. Vertex Corrections for Positive-Definite Spectral Functions of Simple Metals. Phys. Rev. Lett. 2016, 117, 206402,  DOI: 10.1103/PhysRevLett.117.206402
      41
      Vertex corrections for positive-definite spectral functions of simple metals
      Pavlyukh, Y.; Uimonen, A.-M.; Stefanucci, G.; van Leeuwen, R.
      Physical Review Letters (2016), 117 (20), 206402/1-206402/6CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      We present a systematic study of vertex corrections in a homogeneous electron gas at metallic densities. The vertex diagrams are built using a recently proposed pos.-definite diagrammatic expansion for the spectral function. The vertex function not only provides corrections to the well known plasmon and particle-hole scatterings, but also gives rise to new phys. processes such as the generation of two plasmon excitations or the decay of the one-particle state into a two-particle-one-hole state. By an efficient Monte Carlo momentum integration we are able to show that the addnl. scattering channels are responsible for a redn. of the bandwidth, the appearance of a secondary plasmon satellite below the Fermi level, and a substantial redistribution of spectral wts. The feasibility of the approach for first-principles band-structure calcns. is also discussed.
    42. 42
      Pavlyukh, Y.; Stefanucci, G.; van Leeuwen, R. Dynamically screened vertex correction to GW. Phys. Rev. B 2020, 102, 045121,  DOI: 10.1103/PhysRevB.102.045121
      42
      Dynamically screened vertex correction to GW
      Pavlyukh, Y.; Stefanucci, G.; van Leeuwen, R.
      Physical Review B (2020), 102 (4), 045121CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      Diagrammatic perturbation theory is a powerful tool for the investigation of interacting many-body systems, the self-energy operator Σ encoding all the variety of scattering processes. In the simplest scenario of correlated electrons described by the GW approxn. for the electron self-energy, a particle transfers a part of its energy to neutral excitations. Higher-order (in screened Coulomb interaction W) self-energy diagrams lead to improved electron spectral functions (SFs) by taking more complicated scattering channels into account and by adding corrections to lower order self-energy terms. However, they also may lead to unphys. neg. spectral functions. The resoln. of this difficulty has been demonstrated in our previous works. The main idea is to represent the self-energy operator in a Fermi golden rule form which leads to a manifestly pos. definite SF and allows for a very efficient numerical algorithm. So far, the method has only been applied to the three-dimensional electron gas, which is a paradigmatic system, but a rather simple one. Here we systematically extend the method to two dimensions including realistic systems such as monolayer and bilayer graphene. We focus on one of the most important vertex function effects involving the exchange of two particles in the final state. We demonstrate that it should be evaluated with the proper screening and discuss its influence on the quasiparticle properties.
    43. 43
      Del Sole, R.; Reining, L.; Godby, R. W. GWΓ approximation for electron self-energies in semiconductors and insulators. Phys. Rev. B 1994, 49, 8024– 8028,  DOI: 10.1103/PhysRevB.49.8024
      43
      GWΓ approximation for electron self-energies in semiconductors and insulators
      Del Sole, R.; Reining, Lucia; Godby, R. W.
      Physical Review B: Condensed Matter (1994), 49 (12), 8024-8CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      The widely used GW approxn. for the self-energy operator of a system of interacting electrons may, in principle, be improved using an approx. vertex correction Γ. The authors est. Γ using the local-d. approxn. The authors report the results of a comparable series of GW calcns. the band structure of Si, in which such a vertex correction is (i) excluded entirely, (ii) included only in the screened Coulomb interaction W, and (iii) included in both W and the expression for the self-energy. The authors also discuss the symmetry properties of the exact vertex correction and how they may be retained in further improvements.
    44. 44
      Shishkin, M.; Marsman, M.; Kresse, G. Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections. Phys. Rev. Lett. 2007, 99, 246403,  DOI: 10.1103/PhysRevLett.99.246403
      44
      Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections
      Shishkin, M.; Marsman, M.; Kresse, G.
      Physical Review Letters (2007), 99 (24), 246403/1-246403/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Self-consistent GW calcns., maintaining only the quasiparticle part of the Green's function G, are reported for a wide class of materials, including small gap semiconductors and large gap insulators. We show that the inclusion of the attractive electron-hole interaction via an effective nonlocal exchange correlation kernel is required to obtain accurate band gaps in the framework of self-consistent GW calcns. If these are accounted for via vertex corrections in W, the band gaps are found to be within a few percent of the exptl. values.
    45. 45
      Chen, W.; Pasquarello, A. Accurate band gaps of extended systems via efficient vertex corrections in GW. Phys. Rev. B 2015, 92, 041115,  DOI: 10.1103/PhysRevB.92.041115
      45
      Accurate band gaps of extended systems via efficient vertex corrections in GW
      Chen, Wei; Pasquarello, Alfredo
      Physical Review B: Condensed Matter and Materials Physics (2015), 92 (4), 041115/1-041115/5CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We propose the use of an approx. bootstrap exchange-correlation kernel to account for vertex corrections in self-consistent GW calcns. We show that the approx. kernel gives accurate band gaps for a variety of extended systems, including simple sp semiconductors, wide band-gap insulators, and transition-metal compds. with either closed or open d shells. The accuracy is comparable with that obtained via the soln. of the Bethe-Salpeter equation but only at a fraction of the computational cost.
    46. 46
      Dahlen, N. E.; van Leeuwen, R. Self-consistent solution of the Dyson equation for atoms and molecules within a conserving approximation. J. Chem. Phys. 2005, 122, 164102,  DOI: 10.1063/1.1884965
      46
      Self-consistent solution of the Dyson equation for atoms and molecules within a conserving approximation
      Dahlen, Nils Erik; van Leeuwen, Robert
      Journal of Chemical Physics (2005), 122 (16), 164102/1-164102/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We have calcd. the self-consistent Green's function for a no. of atoms and diat. mols. This Green's function is obtained from a conserving self-energy approxn., which implies that the observables calcd. from the Green's functions agree with the macroscopic conservation laws for particle no., momentum, and energy. As a further consequence, the kinetic and potential energies agree with the virial theorem, and the many possible methods for calcg. the total energy all give the same result. In these calcns. we use the finite temp. formalism and calc. the Green's function on the imaginary time axis. This allows for a simple extension to nonequil. systems. We have compared the energies from self-consistent Green's functions to those of non-self-consistent schemes and also calcd. ionization potentials from the Green's functions by using the extended Koopmans' theorem.
    47. 47
      Romaniello, P.; Bechstedt, F.; Reining, L. Beyond the GW approximation: Combining correlation channels. Phys. Rev. B 2012, 85, 155131,  DOI: 10.1103/PhysRevB.85.155131
      47
      Beyond the GW approximation: combining correlation channels
      Romaniello, Pina; Bechstedt, Friedhelm; Reining, Lucia
      Physical Review B: Condensed Matter and Materials Physics (2012), 85 (15), 155131/1-155131/15CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      In many-body perturbation theory (MBPT) the self-energy Σ = iGWΓ plays a key role since it contains all the many-body effects of the system. The exact self-energy is not known; as a first approxn. one can set the vertex function Γ to unity which leads to the GW approxn. The latter properly describes the high-d. regime, where screening is important; in the low-d. regime, instead, other approxns. are proposed, such as the T matrix, which describes multiple scattering between two particles. Here we combine the two approaches. Starting from the fundamental equations of MBPT, we show how one can derive the T-matrix approxn. to the self-energy in a common framework with GW. This allows us to elucidate several aspects of this formulation, including the origin of, and link between, the electron-hole and the particle-particle T matrix, the derivation of a screened T matrix, and the conversion of the T matrix into a vertex correction. The exactly solvable Hubbard mol. is used for illustration.
    48. 48
      Leon, D. A.; Cardoso, C.; Chiarotti, T.; Varsano, D.; Molinari, E.; Ferretti, A. Frequency dependence in GW made simple using a multipole approximation. Phys. Rev. B 2021, 104, 115157,  DOI: 10.1103/PhysRevB.104.115157
      48
      Frequency dependence in GW made simple using a multipole approximation
      Leon, Dario A.; Cardoso, Claudia; Chiarotti, Tommaso; Varsano, Daniele; Molinari, Elisa; Ferretti, Andrea
      Physical Review B (2021), 104 (11), 115157CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      In the GW approxn., the screened interaction W is a nonlocal and dynamical potential that usually has a complex frequency dependence. A full description of such a dependence is possible but often computationally demanding. For this reason, it is still common practice to approx. W(ω) using a plasmon pole (PP) model. Such an approach, however, may deliver an accuracy limited by its simplistic description of the frequency dependence of the polarizability, i.e., of W. In this work, we explore a multipole approach (MPA) and develop an effective representation of the frequency dependence of W. We show that an appropriate sampling of the polarizability in the frequency complex plane and a multipole interpolation can lead to a level of accuracy comparable with full-frequency methods at a much lower computational cost. Moreover, both accuracy and cost are controllable by the no. of poles used in MPA. Eventually, we validate the MPA approach in selected prototype systems, showing that full-frequency quality results can be obtained with a limited no. of poles.
    49. 49
      Chiarotti, T.; Marzari, N.; Ferretti, A. Unified Green’s function approach for spectral and thermodynamic properties from algorithmic inversion of dynamical potentials. Phys. Rev. Research 2022, 4, 013242,  DOI: 10.1103/PhysRevResearch.4.013242
      49
      Unified Green's function approach for spectral and thermodynamic properties from algorithmic inversion of dynamical potentials
      Chiarotti, Tommaso; Marzari, Nicola; Ferretti, Andrea
      Physical Review Research (2022), 4 (1), 013242CODEN: PRRHAI; ISSN:2643-1564. (American Physical Society)
      Dynamical potentials appear in many advanced electronic-structure methods, including self-energies from many-body perturbation theory, dynamical mean-field theory, electronic-transport formulations, and many embedding approaches. Here, we propose a novel treatment for the frequency dependence, introducing an algorithmic inversion method that can be applied to dynamical potentials expanded as sum over poles. This approach allows for an exact soln. of Dyson-like equations at all frequencies via a mapping to a matrix diagonalization, and provides simultaneously frequency-dependent (spectral) and frequency-integrated (thermodn.) properties of the Dyson-inverted propagators. The transformation to a sum over poles is performed introducing nth order generalized Lorentzians as an improved basis set to represent the spectral function of a propagator. Numerical results for the homogeneous electron gas at the G0W0 level are provided to argue for the accuracy and efficiency of such unified approach.
    50. 50
      Nelson, W.; Bokes, P.; Rinke, P.; Godby, R. W. Self-interaction in Green’s-function theory of the hydrogen atom. Phys. Rev. A 2007, 75, 032505,  DOI: 10.1103/PhysRevA.75.032505
      50
      Self-interaction in Green's-function theory of the hydrogen atom
      Nelson, W.; Bokes, P.; Rinke, Patrick; Godby, R. W.
      Physical Review A: Atomic, Molecular, and Optical Physics (2007), 75 (3, Pt. A), 032505/1-032505/4CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      At. hydrogen provides a unique test case for computational electronic structure methods, since its electronic excitation energies are known anal. With only one electron, hydrogen contains no electronic correlation and is therefore particularly susceptible to spurious self-interaction errors introduced by certain computational methods. In this paper we focus on many-body perturbation theory (MBPT) in Hedin's GW approxn. While the Hartree-Fock and the exact MBPT self-energy are free of self-interaction, the correlation part of the GW self-energy does not have this property. Here we use at. hydrogen as a benchmark system for GW and show that the self-interaction part of the GW self-energy, while nonzero, is small. The effect of calcg. the GW self-energy from exact wave functions and eigenvalues, as distinct from those from the local-d. approxn., is also illuminating.
    51. 51
      Sakuma, R.; Aryasetiawan, F. Self-energy calculation of the hydrogen atom: Importance of the unbound states. Phys. Rev. A 2012, 85, 042509,  DOI: 10.1103/PhysRevA.85.042509
      51
      Self-energy calculation of the hydrogen atom: importance of the unbound states
      Sakuma, R.; Aryasetiawan, F.
      Physical Review A: Atomic, Molecular, and Optical Physics (2012), 85 (4-A), 042509/1-042509/6CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      We present the calcn. of the self-energy of the isolated hydrogen atom within the GW approxn. starting from the noninteracting Green's function constructed from the exact wave functions of the hydrogen atom. The error in the electron removal energy of the 1s state is found to be about 0.02 eV, which is much smaller than what one would expect. This small error is explained by the cancellation of the self-screening errors between different l contributions of the self-energy. The unbound continuum states are found to be crucial to get the correct self-energy.
    52. 52
      Bruneval, F. Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies. J. Chem. Phys. 2012, 136, 194107,  DOI: 10.1063/1.4718428
      52
      Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies
      Bruneval, Fabien
      Journal of Chemical Physics (2012), 136 (19), 194107/1-194107/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A systematic evaluation of the ionization energy within the GW approxn. is carried out for the first row atoms, from H to Ar. We describe a Gaussian basis implementation of the GW approxn., which does not resort to any further tech. approxn., besides the choice of the basis set for the electronic wavefunctions. Different approaches to the GW approxn. have been implemented and tested, for example, the std. perturbative approach based on a prior mean-field calcn. (Hartree-Fock GW@HF or d.-functional theory GW@DFT) or the recently developed quasiparticle self-consistent method (QSGW). The HOMO energies of atoms obtained from both GW@HF and QSGW are in excellent agreement with the exptl. ionization energy. The LUMO energies of the singly charged cation yield a noticeably worse est. of the ionization energy. The best agreement with respect to expt. is obtained from the total energy differences within the RPA functional, which is the total energy corresponding to the GW self-energy. We conclude with a discussion about the slight concave behavior upon no. electron change of the GW approxn. and its consequences upon the quality of the orbital energies. (c) 2012 American Institute of Physics.
    53. 53
      Li, J.; Holzmann, M.; Duchemin, I.; Blase, X.; Olevano, V. Helium Atom Excitations by the GW and Bethe-Salpeter Many-Body Formalism. Phys. Rev. Lett. 2017, 118, 163001,  DOI: 10.1103/PhysRevLett.118.163001
      53
      Helium atom excitations by the GW and bethe-salpeter many-body formalism
      Li, Jing; Holzmann, Markus; Duchemin, Ivan; Blase, Xavier; Olevano, Valerio
      Physical Review Letters (2017), 118 (16), 163001/1-163001/6CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
      A review. The helium atom is the simplest many-body electronic system provided by nature. The exact soln. to the Schr.ovrddot.odinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodol. Here, we check the ab initio many-body GW approxn. and the Bethe-Salpeter equation (BSE) against the exact soln. for helium. Starting from the Hartree-Fock method, we show that the GW and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and GW approxns. is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent d.-functional theory.
    54. 54
      Koval, P.; Foerster, D.; Sánchez-Portal, D. Fully self-consistent GW and quasiparticle self-consistent GW for molecules. Phys. Rev. B 2014, 89, 155417,  DOI: 10.1103/PhysRevB.89.155417
      54
      Fully self-consistent GW and quasiparticle self-consistent GW for molecules
      Koval, P.; Foerster, D.; Sanchez-Portal, D.
      Physical Review B: Condensed Matter and Materials Physics (2014), 89 (15), 155417/1-155417/19CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      Two self-consistent schemes involving Hedin's GW approxn. are studied for a set of sixteen different atoms and small mols. We compare results from the fully self-consistent GW approxn. (SCGW) and the quasiparticle self-consistent GW approxn. (QSGW) within the same numerical framework. Core and valence electrons are treated on an equal footing in all the steps of the calcn. We use basis sets of localized functions to handle the space dependence of quantities and spectral functions to deal with their frequency dependence. We compare SCGW and QSGW on a qual. level by comparing the computed densities of states (DOS). To judge their relative merit on a quant. level, we compare their vertical ionization potentials (IPs) with those obtained from coupled-cluster calcns. CCSD(T). Our results are futher compared with "one-shot" G0W0 calcns. starting from Hartree-Fock solns. (G0W0-HF). Both self-consistent GW approaches behave quite similarly. Averaging over all the studied mols., both methods show only a small improvement (somewhat larger for SCGW) of the calcd. IPs with respect to G0W0-HF results. Interestingly, SCGW and QSGW calcns. tend to deviate in opposite directions with respect to CCSD(T) results. SCGW systematically underestimates the IPs, while QSGW tends to overestimate them. G0W0-HF produces results which are surprisingly close to QSGW calcns. both for the DOS and for the numerical values of the IPs.
    55. 55
      van Setten, M. J.; Caruso, F.; Sharifzadeh, S.; Ren, X.; Scheffler, M.; Liu, F.; Lischner, J.; Lin, L.; Deslippe, J. R.; Louie, S. G.; Yang, C.; Weigend, F.; Neaton, J. B.; Evers, F.; Rinke, P. GW100: Benchmarking G0W0 for Molecular Systems. J. Chem. Theory Comput. 2015, 11, 5665– 5687,  DOI: 10.1021/acs.jctc.5b00453
      55
      GW100: Benchmarking G0W0 for Molecular Systems
      van Setten, Michiel J.; Caruso, Fabio; Sharifzadeh, Sahar; Ren, Xinguo; Scheffler, Matthias; Liu, Fang; Lischner, Johannes; Lin, Lin; Deslippe, Jack R.; Louie, Steven G.; Yang, Chao; Weigend, Florian; Neaton, Jeffrey B.; Evers, Ferdinand; Rinke, Patrick
      Journal of Chemical Theory and Computation (2015), 11 (12), 5665-5687CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present the GW100 set. GW100 is a benchmark set of the ionization potentials and electron affinities of 100 mols. computed with the GW method using three independent GW codes and different GW methodologies. The quasi-particle energies of the highest-occupied MOs (HOMO) and lowest-unoccupied MOs (LUMO) are calcd. for the GW100 set at the G0W0@PBE level using the software packages TURBOMOLE, FHI-aims, and BerkeleyGW. The use of these three codes allows for a quant. comparison of the type of basis set (plane wave or local orbital) and handling of unoccupied states, the treatment of core and valence electrons (all electron or pseudopotentials), the treatment of the frequency dependence of the self-energy (full frequency or more approx. plasmon-pole models), and the algorithm for solving the quasi-particle equation. Primary results include ref. values for future benchmarks, best practices for convergence within a particular approach, and av. error bars for the most common approxns.
    56. 56
      van Loon, E. G. C. P.; Rösner, M.; Katsnelson, M. I.; Wehling, T. O. Random phase approximation for gapped systems: Role of vertex corrections and applicability of the constrained random phase approximation. Phys. Rev. B 2021, 104, 045134,  DOI: 10.1103/PhysRevB.104.045134
      56
      Random phase approximation for gapped systems: Role of vertex corrections and applicability of the constrained random phase approximation
      van Loon, Erik G. C. P.; Roesner, Malte; Katsnelson, Mikhail I.; Wehling, Tim O.
      Physical Review B (2021), 104 (4), 045134CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      The many-body theory of interacting electrons poses an intrinsically difficult problem that requires simplifying assumptions. For the detn. of electronic screening properties of the Coulomb interaction, the RPA (RPA) provides such a simplification. Here we explicitly show that this approxn. is justified for band structures with sizable band gaps. This is when the electronic states responsible for the screening are energetically far away from the Fermi level, which is equiv. to a short electronic propagation length of these states. The RPA contains exactly those diagrams in which the classical Coulomb interaction covers all distances, whereas neglected vertex corrections involve quantum tunneling through the barrier formed by the band gap. Our anal. of electron-electron interactions provides a real-space analogy to Migdal's theorem on the smallness of vertex corrections in electron-phonon problems. An important application is the increasing use of constrained RPA calcns. of effective interactions. We find that their usage of Kohn-Sham energies accounts for the leading local (excitonic) vertex correction in insulators.
    57. 57
      Almbladh, C.-O.; von Barth, U. Exact results for the charge and spin densities, exchange-correlations potentials, and density-functional eigenvalues. Phys. Rev. B 1985, 31, 3231– 3244,  DOI: 10.1103/PhysRevB.31.3231
      57
      Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues
      Almbladh, C. O.; Von Barth, U.
      Physical Review B: Condensed Matter and Materials Physics (1985), 31 (6), 3231-44CODEN: PRBMDO; ISSN:0163-1829.
      The authors derive asymptotically exact results for the charge and spin densities far away from finite systems (atoms and mols.) and far outside solid surfaces. These results are then used to obtain the correct asymptotic form of the exchange-correlation potential of d.-functional (DF) theory and to prove that, for all systems, the eigenvalue of the uppermost occupied DF orbital equals the exact ionization potential. For spin-polarized finite systems, the uppermost DF eigenvalue in each spin channel is also given by exact excitation energies.
    58. 58
      Umrigar, C. J.; Gonze, X. Accurate exchange-correlation potentials and total-energy components for the helium isoelectronic series. Phys. Rev. A 1994, 50, 3827– 3837,  DOI: 10.1103/PhysRevA.50.3827
      58
      Accurate exchange-correlation potentials and total-energy components for the helium isoelectronic series
      Umrigar, C. J.; Gonze, Xavier
      Physical Review A: Atomic, Molecular, and Optical Physics (1994), 50 (5), 3827-37CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      Starting from very accurate many-body wave functions, we have constructed essentially exact electron, correlation-energy, and exchange-energy densities, exchange-correlation potentials, and components of the total energies for helium and two-electron ions (H-, Be2+, Ne8+, Hg78+). These d.-functional results are compared to the corresponding quantities obtained from a variety of commonly used approx. d. functionals, namely, the local-d. approxn. and various generalized-gradient approxns., in order to test the accuracy of the approx. functionals. Although the generalized-gradient approxns. yield improved energies compared to the local-d. approxn., the exchange and correlation potentials (esp. the latter) obtained from the generalized-gradient approxns. are in poor agreement with the corresponding exact potentials. The large-distance asymptotic behavior of the exact exchange-correlation potential to O(1/r4) is found to agree with theor. predictions. The short-range behavior of the exchange-correlation potential is very close to quadratic. The prospects for improved generalized-gradient approxns. are discussed.
    59. 59
      Blase, X.; Attaccalite, C.; Olevano, V. First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications. Phys. Rev. B 2011, 83, 115103,  DOI: 10.1103/PhysRevB.83.115103
      59
      First-principles GW calculations for fullerenes, porphyrins, phthalocyanine, and other molecules of interest for organic photovoltaic applications
      Blase, X.; Attaccalite, C.; Olevano, V.
      Physical Review B: Condensed Matter and Materials Physics (2011), 83 (11), 115103/1-115103/9CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We evaluate the performances of ab initio GW calcns. for the ionization energies and HOMO-LUMO gaps of 13 gas phase mols. of interest for org. electronic and photovoltaic applications, including the C60 fullerene, pentacene, free-base porphyrins and phthalocyanine, PTCDA, and std. monomers such as thiophene, fluorene, benzothiazole, or thiadiazole. Std. G0W0 calcns., i.e., starting from eigenstates obtained with local or semilocal functionals, significantly improve the ionization energy and band gap as compared to d. functional theory Kohn-Sham results, but the calcd. quasiparticle values remain too small as a result of overscreening. Starting from Hartree-Fock-like eigenvalues provides much better results and is equiv. to performing self-consistency on the eigenvalues, with a resulting accuracy of 2%-4% as compared to expt. Our calcns. are based on an efficient Gaussian-basis implementation of GW with explicit treatment of the dynamical screening through contour deformation techniques.
    60. 60
      Faber, C.; Attaccalite, C.; Olevano, V.; Runge, E.; Blase, X. First-principles GW calculations for DNA and RNA nucleobases. Phys. Rev. B 2011, 83, 115123,  DOI: 10.1103/PhysRevB.83.115123
      60
      First-principles GW calculations for DNA and RNA nucleobases
      Faber, Carina; Attaccalite, Claudio; Olevano, V.; Runge, E.; Blase, X.
      Physical Review B: Condensed Matter and Materials Physics (2011), 83 (11), 115123/1-115123/5CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      On the basis of first-principles GW calcns., we study the quasiparticle properties of the guanine, adenine, cytosine, thymine, and uracil DNA and RNA nucleobases. Beyond std. G0W0 calcns., starting from Kohn-Sham eigenstates obtained with (semi)local functionals, a simple self-consistency on the eigenvalues allows us to obtain vertical ionization energies and electron affinities within an av. 0.11 and 0.18 eV error, resp., as compared to state-of-the-art coupled-cluster and multiconfigurational perturbative quantum chem. approaches. Further, GW calcns. predict the correct π-character of the highest occupied state, due to several level crossings between d. functional and GW calcns. Our study is based on a recent Gaussian-basis implementation of GW calcns. with explicit treatment of dynamical screening through contour deformation techniques.
    61. 61
      Blase, X.; Boulanger, P.; Bruneval, F.; Fernandez-Serra, M.; Duchemin, I. GW and Bethe-Salpeter study of small water clusters. J. Chem. Phys. 2016, 144, 034109,  DOI: 10.1063/1.4940139
      61
      GW and Bethe-Salpeter study of small water clusters
      Blase, Xavier; Boulanger, Paul; Bruneval, Fabien; Fernandez-Serra, Marivi; Duchemin, Ivan
      Journal of Chemical Physics (2016), 144 (3), 034109/1-034109/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We study within the GW and Bethe-Salpeter many-body perturbation theories the electronic and optical properties of small (H2O)n water clusters (n = 1-6). Comparison with high-level CCSD(T) Coupled-Cluster at the Single Double (Triple) levels and ADC(3) Green's function third order algebraic diagrammatic construction calcns. indicates that the std. non-self-consistent G0W0@PBE or G0W0@PBE0 approaches significantly underestimate the ionization energy by about 1.1 eV and 0.5 eV, resp. Consequently, the related Bethe-Salpeter lowest optical excitations are found to be located much too low in energy when building transitions from a non-self-consistent G0W0 description of the quasiparticle spectrum. Simple self-consistent schemes, with update of the eigenvalues only, are shown to provide a weak dependence on the Kohn-Sham starting point and a much better agreement with ref. calcns. The present findings rationalize the theory to expt. possible discrepancies obsd. in previous G0W0 and Bethe-Salpeter studies of bulk water. The increase of the optical gap with increasing cluster size is consistent with the evolution from gas to dense ice or water phases and results from an enhanced screening of the electron-hole interaction. (c) 2016 American Institute of Physics.
    62. 62
      Knight, J. W.; Wang, X.; Gallandi, L.; Dolgounitcheva, O.; Ren, X.; Ortiz, J. V.; Rinke, P.; Körzdörfer, T.; Marom, N. Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods. J. Chem. Theory Comput. 2016, 12, 615– 626,  DOI: 10.1021/acs.jctc.5b00871
      62
      Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods
      Knight, Joseph W.; Wang, Xiaopeng; Gallandi, Lukas; Dolgounitcheva, Olga; Ren, Xinguo; Ortiz, J. Vincent; Rinke, Patrick; Korzdorfer, Thomas; Marom, Noa
      Journal of Chemical Theory and Computation (2016), 12 (2), 615-626CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      The performance of different GW methods is assessed for a set of 24 org. acceptors. Errors are evaluated with respect to coupled cluster singles, doubles, and perturbative triples [CCSD(T)] ref. data for the vertical ionization potentials (IPs) and electron affinities (EAs), extrapolated to the complete basis set limit. Addnl. comparisons are made to exptl. data, where available. We consider fully self-consistent GW (scGW), partial self-consistency in the Green's function (scGW0), non-self-consistent G0W0 based on several mean-field starting points, and a "beyond GW" second-order screened exchange (SOSEX) correction to G0W0. We also describe the implementation of the self-consistent Coulomb hole with screened exchange method (COHSEX), which serves as one of the mean-field starting points. The best performers overall are G0W0 + SOSEX and G0W0 based on an IP-tuned long-range cor. hybrid functional with the former being more accurate for EAs and the latter for IPs. Both provide a balanced treatment of localized vs delocalized states and valence spectra in good agreement with photoemission spectroscopy (PES) expts.
    63. 63
      Loos, P.-F.; Romaniello, P.; Berger, J. A. Green Functions and Self-Consistency: Insights From the Spherium Model. J. Chem. Theory Comput. 2018, 14, 3071– 3082,  DOI: 10.1021/acs.jctc.8b00260
      63
      Green Functions and Self-Consistency: Insights From the Spherium Model
      Loos, Pierre-Francois; Romaniello, Pina; Berger, J. A.
      Journal of Chemical Theory and Computation (2018), 14 (6), 3071-3082CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We report an exhaustive study of the performance of different variants of Green function methods for the spherium model in which two electrons are confined to the surface of a sphere and interact via a genuine long-range Coulomb operator. We show that the spherium model provides a unique paradigm to study electronic correlation effects from the weakly correlated regime to the strongly correlated regime, since the mathematics are simple while the physics is rich. We compare perturbative GW, partially self-consistent GW and second-order Green function (GF2) methods for the computation of ionization potentials, electron affinities, energy gaps, correlation energies as well as singlet and triplet neutral excitations by solving the Bethe-Salpeter equation (BSE). We discuss the problem of self-screening in GW and show that it can be partially solved with a second-order screened exchange correction (SOSEX). We find that, in general, self-consistency deteriorates the results with respect to those obtained within perturbative approaches with a Hartree-Fock starting point. Finally, we unveil an important problem of partial self-consistency in GW: in the weakly correlated regime, it can produce artificial discontinuities in the self-energy caused by satellite resonances with large wts.
    64. 64
      Sham, L. J.; Schlüter, M. Density-Functional Theory of the Energy Gap. Phys. Rev. Lett. 1983, 51, 1888– 1891,  DOI: 10.1103/PhysRevLett.51.1888
      There is no corresponding record for this reference.
    65. 65
      Niquet, Y. M.; Gonze, X. Band-gap energy in the random-phase approximation to density-functional theory. Phys. Rev. B 2004, 70, 245115,  DOI: 10.1103/PhysRevB.70.245115
      65
      Band-gap energy in the random-phase approximation to density-functional theory
      Niquet, Y. M.; Gonze, X.
      Physical Review B: Condensed Matter and Materials Physics (2004), 70 (24), 245115/1-245115/12CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We calc. the interacting bandgap energy of a solid within the RPA to d. functional theory (DFT). The interacting bandgap energy is defined as Eg = ERPA(N + 1) + ERPA(N-1)-2ERPA(N), where ERPA(N) is the total DFT-RPA energy of the N-electron system. We compare the interacting bandgap energy with the Kohn-Sham bandgap energy, which is the difference between the conduction and valence band edges in the Kohn-Sham band structure. We show that they differ by an unrenormalized "G0W0" self-energy correction (i.e., a GW self-energy correction computed using Kohn-Sham orbitals and energies as input). This provides a well-defined and meaningful interpretation to G0W0 quasiparticle bandgap calcns., but questions the physics behind the renormalization factors in the expression of the bandgap energy. We also sep. the kinetic from the Coulomb contributions to the DFT-RPA bandgap energy, and discuss the related problem of the deriv. discontinuity in the DFT-RPA functional. Last we discuss the applicability of our results to other functionals based on many-body perturbation theory.
    66. 66
      Niquet, Y. M.; Fuchs, M.; Gonze, X. Asymptotic behavior of the exchange-correlation potentials from the linear-response Sham–Schlüter equation. J. Chem. Phys. 2003, 118, 9504– 9518,  DOI: 10.1063/1.1566739
      66
      Asymptotic behavior of the exchange-correlation potentials from the linear-response Sham-Schluter equation
      Niquet, Y. M.; Fuchs, M.; Gonze, X.
      Journal of Chemical Physics (2003), 118 (21), 9504-9518CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The linear-response Sham-Schluter equation can be used to calc. an exchange-correlation potential starting from a given approxn. for the self-energy. The asymptotic behavior of these potentials is, however, much debated, a recent work suggesting that they could blow up in finite systems. Here we investigate the asymptotic behavior of the linear-response Sham-Schluter potentials in the GW and second-order approxns. for the self-energy. We show that these potentials do not diverge, and that the correlation potential itself has a -α/(2r4) tail (under appropriate conditions), where α depends on the self-energy. We also provide further justification for the quasiparticle approxn. to the linear-response Sham-Schluter equation, that is much simpler to solve while likely being of comparable accuracy. Calcns. for real mols. or solids using this approxn. should be within the reach of present computers.
    67. 67
      Casida, M. E. Generalization of the optimized-effective-potential model to include electron correlation: A variational derivation of the Sham-Schlüter equation for the exact exchange-correlation potential. Phys. Rev. A 1995, 51, 2005– 2013,  DOI: 10.1103/PhysRevA.51.2005
      67
      Generalization of the optimized-effective-potential model to include electron correlation: a variational derivation of the Sham-Schlueter equation for the exact exchange-correlation potential
      Casida, Mark E.
      Physical Review A: Atomic, Molecular, and Optical Physics (1995), 51 (3), 2005-13CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      The now classic optimized-effective-potential (OEP) approach of Sharp and Horton, [Phys, Rev. 90, 317 (1953)] and Talman and Shadwick [Phys. Rev. A 14, 36 (1976)] seeks the local potential that is variationally optimized to best approx. the Hartree-Fock exchange operator. The resulting OEP can be identified as the exchange potential of Kohn-Sham d.-functional theory. The present work generalizes this OEP approach to treat the correlated case, and shows that the Kohn-Sham exchange-correlation potential is the variationally best local approxn. to the exchange-correlation self-energy. This provides a variational derivation of the equation for the exact exchange-correlation potential that was derived by Sham and Schlueter using a d. condition. Implications for an approx. phys. interpretation of the Kohn-Sham orbitals are discussed. A correlated generalization of the Sharp-Horton-Krieger-Li-Isfrate [Phys. Lett. A 146, 256 (1990)] approxn. of the exchange potential is introduced in the quasiparticle limit.
    68. 68
      Perdew, J. P.; Levy, M. Comment on “Significance of the highest occupied Kohn-Sham eigenvalue. Phys. Rev. B 1997, 56, 16021– 16028,  DOI: 10.1103/PhysRevB.56.16021
      68
      Comment on "Significance of the highest occupied Kohn-Sham eigenvalue"
      Perdew, John P.; Levy, Mel
      Physical Review B: Condensed Matter (1997), 56 (24), 16021-16028CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      With more explanation than usual and without appeal to Janak's theorem, we discuss the statement and proof of the ionization potential theorems for the exact Kohn-Sham d.-functional theory of a many-electron system: (1) For any av. electron no. N between the integers Z - 1 and Z, and thus for N → Z from below, the highest occupied or partly occupied Kohn-Sham orbital energy is minus the ionization energy of the Z-electron system. (2) For Z - 1 < N < Z, the exact Kohn-Sham effective potential vs(r) tends to zero as |r| → ∞. We then argue that an objection to these theorems [L. Kleinman, Phys. Rev. B 56, 12042 (1997)] overlooks a crucial step in the proof of theorem (2): The asymptotic exponential decay of the exact electron d. of the Z-electron system is controlled by the exact ionization energy, but the decay of an approx. d. is not controlled by the approx. ionization energy. We discuss relevant evidence from the numerical construction of the exact Kohn-Sham potential. In particular, we point out a model two-electron problem for which the ionization potential theorems are exactly confirmed. Finally, we comment on related issues: the self-interaction correction, the discontinuity of the exact Kohn-Sham potential as N passes through the integer Z, and the generalized sum rule on the exchange-correlation hole.
    69. 69
      Harbola, M. K. Relationship between the highest occupied Kohn-Sham orbital eigenvalue and ionization energy. Phys. Rev. B 1999, 60, 4545– 4550,  DOI: 10.1103/PhysRevB.60.4545
      69
      Relationship between the highest occupied Kohn-Sham orbital eigenvalue and ionization energy
      Harbola, Manoj K.
      Physical Review B: Condensed Matter and Materials Physics (1999), 60 (7), 4545-4550CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      Recent arguments for and against the equivalence of the highest occupied orbital eigenvalue of the Kohn-Sham theory and ionization energy are discussed. For phys. realistic systems with a non-integral no. of electrons, which are described by the thermal av. of two systems, each with an integer no. of electrons, an equiv. Kohn-Sham system exists. This is shown by writing explicit expressions for the exchange-correlation potential constructed to give the mixed-state d., and then relating it to the mixed-state exchange-correlation energy functional by employing the virial theorem sum rule of Levy and Perdew [Phys. Rev. A 32, 2010 (1985)]. Further, the functional deriv. of the mixed-state exchange-correlation energy functional is obtained in terms of this potential. This is then used to show, without recourse to Janak's theorem [Phys. Rev. B 18, 7165 (1978)], that .vepsiln.max(N)=-I(Z), where Z is an integer and (Z-1)<N<Z. Thus the original arguments about the equivalence of the highest occupied Kohn-Sham orbital eigenenergy and the ionization energy which were based on Janak's theorem are valid, and the two quantities are equal.
    70. 70
      Klimeš, J.; Kresse, G. Kohn-Sham band gaps and potentials of solids from the optimized effective potential method within the random phase approximation. J. Chem. Phys. 2014, 140, 054516,  DOI: 10.1063/1.4863502
      70
      Kohn-Sham band gaps and potentials of solids from the optimised effective potential method within the random phase approximation
      Klimes, Jiri; Kresse, Georg
      Journal of Chemical Physics (2014), 140 (5), 054516/1-054516/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present an implementation of the optimized effective potential (OEP) scheme for the exact-exchange (EXX) and RPA energy functionals and apply these methods to a range of bulk materials. We calc. the Kohn-Sham (KS) potentials and the corresponding band gaps and compare them to the potentials obtained by std. local d. approxn. (LDA) calcns. The KS gaps increase upon going from the LDA to the OEP in the RPA and finally to the OEP for EXX. This can be explained by the different depth of the potentials in the bonding and interstitial regions. To obtain the true quasi-particle gaps the deriv. discontinuities or G0W0 corrections need to be added to the RPA-OEP KS gaps. The predicted G0W0@RPA-OEP quasi-particle gaps are about 5% too large compared to the exptl. values. However, compared to G0W0 calcns. based on local or semi-local functionals, where the errors vary between different materials, we obtain a rather consistent description among all the materials. (c) 2014 American Institute of Physics.
    71. 71
      Hellgren, M.; Baguet, L.; Calandra, M.; Mauri, F.; Wirtz, L. Electronic structure of TiSe2 from a quasi-self-consistent G0W0 approach. Phys. Rev. B 2021, 103, 075101,  DOI: 10.1103/PhysRevB.103.075101
      71
      Electronic structure of TiSe2 from a quasi-self-consistent G0W0 approach
      Hellgren, Maria; Baguet, Lucas; Calandra, Matteo; Mauri, Francesco; Wirtz, Ludger
      Physical Review B (2021), 103 (7), 075101CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
      In a previous work, it was shown that the inclusion of exact exchange is essential for a first-principles description of both the electronic and the vibrational properties of TiSe2, M.Hellgren et al. [Phys.Rev.Lett.119, 176401 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.176401]. The GW approxn. provides a parameter-free description of screened exchange but is usually employed perturbatively (G0W0), making results more or less dependent on the starting point. In this work, we develop a quasi-self-consistent extension of G0W0 based on the RPA (RPA) and the optimized effective potential of hybrid d. functional theory. This approach generates an optimal G0W0 starting point and a hybrid exchange parameter consistent with the RPA. While self-consistency plays a minor role for systems such as Ar, BN, and ScN, it is shown to be crucial for TiS2 and TiSe2. We find the high-temp. phase of TiSe2 to be a semimetal with a band structure in good agreement with expt. Furthermore, the optimized hybrid functional agrees well with our previous est. and therefore accurately reproduces the low-temp. charge-d.-wave phase.
    72. 72
      Varsano, D.; Barborini, M.; Guidoni, L. Kohn-Sham orbitals and potentials from quantum Monte Carlo molecular densities. J. Chem. Phys. 2014, 140, 054102,  DOI: 10.1063/1.4863213
      72
      Kohn-Sham orbitals and potentials from quantum Monte Carlo molecular densities
      Varsano, Daniele; Barborini, Matteo; Guidoni, Leonardo
      Journal of Chemical Physics (2014), 140 (5), 054102/1-054102/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We show the possibility to ext. Kohn-Sham orbitals, orbital energies, and exchange correlation potentials from accurate quantum Monte Carlo (QMC) densities for atoms (He, Be, Ne) and mols. (H2, Be2, H2O, and C2H4). The variational Monte Carlo (VMC) densities based on accurate Jastrow Antisymmetrised Geminal Power wave functions are calcd. through different estimators. Using these ref. densities, we ext. the Kohn-Sham quantities with the method developed by Zhao, Morrison, and Parr (ZMP). We compare these extd. quantities with those obtained form CISD densities and with other data reported in the literature, finding a good agreement between VMC and other high-level quantum chem. methods. Our results demonstrate the applicability of the ZMP procedure to QMC mol. densities, that can be used for the testing and development of improved functionals and for the implementation of embedding schemes based on QMC and d. functional theory. (c) 2014 American Institute of Physics.
    73. 73
      Fabiano, E.; Śmiga, S.; Giarrusso, S.; Daas, T. J.; Della Sala, F.; Grabowski, I.; Gori-Giorgi, P. Investigation of the Exchange-Correlation Potentials of Functionals Based on the Adiabatic Connection Interpolation. J. Chem. Theory Comput. 2019, 15, 1006– 1015,  DOI: 10.1021/acs.jctc.8b01037
      There is no corresponding record for this reference.
    74. 74
      Talman, J. D.; Shadwick, W. F. Optimized effective atomic central potential. Phys. Rev. A 1976, 14, 36– 40,  DOI: 10.1103/PhysRevA.14.36
      74
      Optimized effective atomic central potential
      Talman, James D.; Shadwick, William F.
      Physical Review A: Atomic, Molecular, and Optical Physics (1976), 14 (1), 36-40CODEN: PLRAAN; ISSN:1050-2947.
      A self-consistent set of equations was derived for an at. central potential such that the energy given by the orbitals for the potential is minimized. This effective potential behaves like -e2/r for large electron-nucleus distances (r) [e = electronic charge]. The equations were solved for C, Ne, and Al; the resulting total energies exceed the published Hartree-Fock total energies by <0.005%. The theory provides an effective, local, central exchange potential analogous to the Xα-statistical-exchange potential (J. C. Slater, (1974).
    75. 75
      Engel, E.; Vosko, S. H. Accurate optimized-potential-model solutions for spherical spin-polarized atoms: Evidence for limitations of the exchange-only local spin-density and generalized-gradient approximations. Phys. Rev. A 1993, 47, 2800– 2811,  DOI: 10.1103/PhysRevA.47.2800
      75
      Accurate optimized-potential-model solutions for spherical spin-polarized atoms: evidence for limitations of the exchange-only local spin-density and generalized-gradient approximations
      Engel, E.; Vosko, S. H.
      Physical Review A: Atomic, Molecular, and Optical Physics (1993), 47 (4-A), 2800-11CODEN: PLRAAN; ISSN:0556-2791.
      The authors present accurate optimized-potential-model (OPM) results for spherical spin-polarized atoms emphasizing the precise construction of the OPM exchange potential from the numerical soln. of the OPM integral equation, esp. for large r. The results are used to discuss the quality of the local spin-d. approxn. (LSDA) and a generalized-gradient expansion (GGA) A. D. Becke (1988) for describing these atoms. The LSDA can produce substantial errors (beyond what is known from unpolarized atoms) for quantities which are directly related to the spin polarization of these systems. In particular, the LSDA overestimates the magnetization d. in the interior of Cu by a factor of 2. While the GGA improves integral quantities like total ground-state and exchange energies, remarkably it is less successful for energy differences like Ex↑ - Ex↓. Most important, however, it is not able to reduce the LSDA's errors for local quantities like the difference between spin-up and spin-down exchange potentials and magnetization densities significantly nor does it reverse the LSDA's incorrect ordering of the two highest occupied majority-spin eigenvalues of Cr and Cu.
    76. 76
      Fetter, A. L.; Walecka, J. D. Quantum theory of many-particle systems; McGraw-Hill: New York, 1971.
      There is no corresponding record for this reference.
    77. 77
      Perdew, J. P.; Yang, W.; Burke, K.; Yang, Z.; Gross, E. K. U.; Scheffler, M.; Scuseria, G. E.; Henderson, T. M.; Zhang, I. Y.; Ruzsinszky, A.; Peng, H.; Sun, J.; Trushin, E.; Görling, A. Understanding band gaps of solids in generalized Kohn–Sham theory. Proc. Natl. Acad. Sci. U.S.A. 2017, 114, 2801– 2806,  DOI: 10.1073/pnas.1621352114
      77
      Understanding band gaps of solids in generalized Kohn-Sham theory
      Perdew, John P.; Yang, Weitao; Burke, Kieron; Yang, Zenghui; Gross, Eberhard K. U.; Scheffler, Matthias; Scuseria, Gustavo E.; Henderson, Thomas M.; Zhang, Igor Ying; Ruzsinszky, Adrienn; Peng, Haowei; Sun, Jianwei; Trushin, Egor; Gorling, Andreas
      Proceedings of the National Academy of Sciences of the United States of America (2017), 114 (11), 2801-2806CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)
      The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, the gap in the band structure of the exact multiplicative Kohn-Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS d.-functional theory. Here, we give a simple proof of a theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approx. functional if the GKS potential operator is continuous and the d. change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from meta-generalized gradient approxns. (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen mols., solid aluminum arsenide, and solid argon provide numerical illustrations.
    78. 78
      Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L. Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy. Phys. Rev. Lett. 1982, 49, 1691– 1694,  DOI: 10.1103/PhysRevLett.49.1691
      78
      Density-functional theory for fractional particle number: derivative discontinuities of the energy
      Perdew, John P.; Parr, Robert G.; Levy, Mel; Balduz, Jose L., Jr.
      Physical Review Letters (1982), 49 (23), 1691-4CODEN: PRLTAO; ISSN:0031-9007.
      The Hohenberg-Kohn theorem was extended to fractional electron no. N, for an isolated open system described by a statistical mixt. The curve of lowest av. energy EN vs. N is a series of straight line segments with slope discontinuities at integral N. As N increases through an integer M, the chem. potential and the highest occupied Kohn-Sham orbital energy both jump from EM-EM-1 to EM+1-EM.
    79. 79
      Kraisler, E.; Hodgson, M. J. P.; Gross, E. K. U. From Kohn–Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential. J. Chem. Theory Comput. 2021, 17, 1390– 1407,  DOI: 10.1021/acs.jctc.0c01093
      79
      From Kohn-Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential
      Kraisler, Eli; Hodgson, M. J. P.; Gross, E. K. U.
      Journal of Chemical Theory and Computation (2021), 17 (3), 1390-1407CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Accurately describing excited states within Kohn-Sham (KS) d. functional theory (DFT), particularly those which induce ionization and charge transfer, remains a great challenge. Common exchange-correlation (xc) approxns. are unreliable for excited states owing, in part, to the absence of a deriv. discontinuity in the xc energy (Δ), which relates a many-electron energy difference to the corresponding KS energy difference. We demonstrate, anal. and numerically, how the relationship between KS and many-electron energies leads to the step structures obsd. in the exact xc potential in four scenarios: electron addn., mol. dissocn., excitation of a finite system, and charge transfer. We further show that steps in the potential can be obtained also with common xc approxns., as simple as the LDA, when addressed from the ensemble perspective. The article therefore highlights how capturing the relationship between KS and many-electron energies with advanced xc approxns. is crucial for accurately calcg. excitations, as well as the ground-state d. and energy of systems which consist of distinct subsystems.
    80. 80
      Engel, E.; Dreizler, R. M. From Explicit to Implicit Density Functionals. J. Comput. Chem. 1999, 20, 31,  DOI: 10.1002/(SICI)1096-987X(19990115)20:1<31::AID-JCC6>3.0.CO;2-P
      80
      From explicit to implicit density functionals
      Engel, E.; Dreizler, R. M.
      Journal of Computational Chemistry (1999), 20 (1), 31-50CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      A review with 93 refs. is given on the concept of orbital- and eigenvalue-dependent exchange-correlation (xc) energy functionals. We show how such functionals can be derived in a systematic fashion via a perturbation expansion, utilizing the Kohn-Sham system as a noninteracting ref. system. We demonstrate that the second-order contribution to this expansion of the xc-energy functional includes the leading term of the van der Waals interaction. The optimized-potential method (OPM), which allows the calcn. of the multiplicative xc-potential corresponding to an orbital- and eigenvalue-dependent xc-energy functional via an integral equation, is discussed in detail. We examine an approx. anal. soln. of the OPM integral equation, pointing out that, for eigenvalue-dependent functionals, the three paths used in the literature for the derivation of this approxn. yield different results. Finally, a no. of illustrative results, both for the exchange-only limit and for the combination of the exact exchange with various correlation functionals, are given.
    81. 81
      Kümmel, S.; Kronik, L. Orbital-dependent density functionals: Theory and applications. Rev. Mod. Phys. 2008, 80, 3– 60,  DOI: 10.1103/RevModPhys.80.3
      81
      Orbital-dependent density functionals: Theory and applications
      Kuemmel, Stephan; Kronik, Leeor
      Reviews of Modern Physics (2008), 80 (1), 3-60CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)
      This review provides a perspective on the use of orbital-dependent functionals, which is currently considered one of the most promising avenues in modern d.-functional theory. The focus here is on four major themes: the motivation for orbital-dependent functionals in terms of limitations of semilocal functionals; the optimized effective potential as a rigorous approach to incorporating orbital-dependent functionals within the Kohn-Sham framework; the rationale behind and advantages and limitations of four popular classes of orbital-dependent functionals; and the use of orbital-dependent functionals for predicting excited-state properties. For each of these issues, both formal and practical aspects are assessed.
    82. 82
      Luttinger, J. M.; Ward, J. C. Ground-State Energy of a Many-Fermion System. II. Phys. Rev. 1960, 118, 1417– 1427,  DOI: 10.1103/PhysRev.118.1417
      There is no corresponding record for this reference.
    83. 83
      Klein, A. Perturbation Theory for an Infinite Medium of Fermions. II. Phys. Rev. 1961, 121, 950– 956,  DOI: 10.1103/PhysRev.121.950
      There is no corresponding record for this reference.
    84. 84
      Dahlen, N. E.; van Leeuwen, R.; von Barth, U. Variational energy functionals of the Green function and of the density tested on molecules. Phys. Rev. A 2006, 73, 012511,  DOI: 10.1103/PhysRevA.73.012511
      84
      Variational energy functionals of the Green function and of the density tested on molecules
      Dahlen, Nils Erik; van Leeuwen, Robert; von Barth, Ulf
      Physical Review A: Atomic, Molecular, and Optical Physics (2006), 73 (1, Pt. A), 012511/1-012511/13CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      We have calcd. total energies of atoms and diat. mols. from the Luttinger-Ward functional, using self-energy approxns. to second order as well as the GW approxn. In order to assess the variational quality of this functional, we have also solved the Dyson equation self-consistently. The Luttinger-Ward functional is compared to the variational functional due to Klein, and we demonstrate that the variational property of the latter functional is inferior to that of the Luttinger-Ward functional. We also show how to obtain variational d. functionals from the functionals of the Green function. These orbital functional schemes are important for systems where d.-functional theory using local functionals of the d. necessarily fails. We derive an optimized effective potential (OEP) scheme that is based on the Luttinger-Ward functional and, unlike the conventional OEP schemes, produces energies in good agreement with the values obtained from the self-consistent Green function. Our calcns. show that, when applied to mols., the Luttinger-Ward functional is more sensitive to the quality of the input Green function than when applied to atoms, but the energies are remarkably close to the self-consistent values when the Hartree-Fock Green function is used as input. This Luttinger-Ward functional is therefore a simple and efficient method for studying the merits of various self-energy approxns. while avoiding the computationally demanding task of solving the Dyson equation self-consistently.
    85. 85
      von Barth, U.; Dahlen, N. E.; van Leeuwen, R.; Stefanucci, G. Conserving approximations in time-dependent density functional theory. Phys. Rev. B 2005, 72, 235109,  DOI: 10.1103/PhysRevB.72.235109
      85
      Conserving approximations in time-dependent density functional theory
      von Barth, Ulf; Dahlen, Nils Erik; van Leeuwen, Robert; Stefanucci, Gianluca
      Physical Review B: Condensed Matter and Materials Physics (2005), 72 (23), 235109/1-235109/10CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      In the present work, we propose a theory for obtaining successively better approxns. to the linear response functions of time-dependent d. or current-d. functional theory. The new technique is based on the variational approach to many-body perturbation theory (MBPT) as developed during the sixties and later expanded by us in the mid-nineties. Due to this feature, the resulting response functions obey a large no. of conservation laws such as particle and momentum conservation and sum rules. The quality of the obtained results is governed by the phys. processes built in through MBPT but also by the choice of variational expressions. We here present several conserving response functions of different sophistication to be used in the calcn. of the optical response of solids and nanoscale systems.
    86. 86
      Ismail-Beigi, S. Correlation energy functional within the GW-RPA: Exact forms, approximate forms, and challenges. Phys. Rev. B 2010, 81, 195126,  DOI: 10.1103/PhysRevB.81.195126
      86
      Correlation energy functional within the GW-RPA: Exact forms, approximate forms, and challenges
      Ismail-Beigi, Sohrab
      Physical Review B: Condensed Matter and Materials Physics (2010), 81 (19), 195126/1-195126/21CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      In principle, the Luttinger-Ward Green's-function formalism allows one to compute simultaneously the total energy and the quasiparticle band structure of a many-body electronic system from first principles. We present approx. and exact expressions for the correlation energy within the GW-RPA that are more amenable to computation and allow for developing efficient approxns. to the self-energy operator and correlation energy. The exact form is a sum over differences between plasmon and interband energies. The approx. forms are based on summing over screened interband transitions. We also demonstrate that blind extremization of such functionals leads to unphys. results: imposing phys. constraints on the allowed solns. (Green's functions) is necessary. Finally, we present some relevant numerical results for at. systems.
    87. 87
      Stefanucci, G.; van Leeuwen, R. Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction; Cambridge University Press, 2013.
      There is no corresponding record for this reference.
    88. 88
      Martin, R. M.; Reining, L.; Ceperley, D. M. Interacting Electrons; Cambridge University Press: Cambridge, 2016.
      There is no corresponding record for this reference.
    89. 89
      Nelson, W.; Bokes, P.; Rinke, P.; Godby, R. W. Self-interaction in Green’s-function theory of the hydrogen atom. Phys. Rev. A 2007, 75, 032505,  DOI: 10.1103/PhysRevA.75.032505
      89
      Self-interaction in Green's-function theory of the hydrogen atom
      Nelson, W.; Bokes, P.; Rinke, Patrick; Godby, R. W.
      Physical Review A: Atomic, Molecular, and Optical Physics (2007), 75 (3, Pt. A), 032505/1-032505/4CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)
      At. hydrogen provides a unique test case for computational electronic structure methods, since its electronic excitation energies are known anal. With only one electron, hydrogen contains no electronic correlation and is therefore particularly susceptible to spurious self-interaction errors introduced by certain computational methods. In this paper we focus on many-body perturbation theory (MBPT) in Hedin's GW approxn. While the Hartree-Fock and the exact MBPT self-energy are free of self-interaction, the correlation part of the GW self-energy does not have this property. Here we use at. hydrogen as a benchmark system for GW and show that the self-interaction part of the GW self-energy, while nonzero, is small. The effect of calcg. the GW self-energy from exact wave functions and eigenvalues, as distinct from those from the local-d. approxn., is also illuminating.
    90. 90
      Romaniello, P.; Guyot, S.; Reining, L. The self-energy beyond GW: Local and nonlocal vertex corrections. J. Chem. Phys. 2009, 131, 154111,  DOI: 10.1063/1.3249965
      90
      The self-energy beyond GW: Local and nonlocal vertex corrections
      Romaniello, P.; Guyot, S.; Reining, L.
      Journal of Chemical Physics (2009), 131 (15), 154111/1-154111/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      It is commonly accepted that the GW approxn. for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approxn.: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the at. limit, in which, instead, correlation is directly responsible for the obsd. problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be cor. by a two-point vertex contribution to the self-energy derived from time-dependent d. functional theory (TDDFT). We explain why the addn. of a particle, instead, requires the use of a different approx. vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the at. limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qual. We discuss the effects of the self-screening error as well as the at. limit using GW for the exactly solvable two-site Hubbard model. (c) 2009 American Institute of Physics.
    91. 91
      Grüneis, A.; Marsman, M.; Harl, J.; Schimka, L.; Kresse, G. Making the random phase approximation to electronic correlation accurate. J. Chem. Phys. 2009, 131, 154115,  DOI: 10.1063/1.3250347
      91
      Making the random phase approximation to electronic correlation accurate
      Grueneis, Andreas; Marsman, Martijn; Harl, Judith; Schimka, Laurids; Kresse, Georg
      Journal of Chemical Physics (2009), 131 (15), 154115/1-154115/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We show that the inclusion of second-order screened exchange to the RPA allows for an accurate description of electronic correlation in atoms and solids clearly surpassing the random phase approxn., but not yet approaching chem. accuracy. From a fundamental point of view, the method is self-correlation free for one-electron systems. From a practical point of view,the approach yields correlation energies for atoms, as well as for the jellium electron gas within a few kcal/mol of exact values, atomization energies within typically 2-3 kcal/mol of expt., and excellent lattice consts. for ionic and covalently bonded solids (0.2% error). The computational complexity is only O(N5), comparable to canonical second-order Moller-Plesset perturbation theory, which should allow for routine calcns. on many systems. (c) 2009 American Institute of Physics.
    92. 92
      Ren, X.; Rinke, P.; Scuseria, G. E.; Scheffler, M. Renormalized second-order perturbation theory for the electron correlation energy: Concept, implementation, and benchmarks. Phys. Rev. B 2013, 88, 035120,  DOI: 10.1103/PhysRevB.88.035120
      92
      Renormalized second-order perturbation theory for the electron correlation energy: concept, implementation, and benchmarks
      Ren, Xinguo; Rinke, Patrick; Scuseria, Gustavo E.; Scheffler, Matthias
      Physical Review B: Condensed Matter and Materials Physics (2013), 88 (3), 035120/1-035120/15CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      We present a renormalized second-order perturbation theory (rPT2), based on a Kohn-Sham (KS) ref. state, for the electron correlation energy that includes the RPA (RPA), second-order screened exchange (SOSEX), and renormalized single excitations (rSE). These three terms all involve a summation of certain types of diagrams to infinite order, and can be viewed as "renormalization" of the second-order direct, exchange, and single-excitation (SE) terms of Rayleigh-Schroedinger perturbation theory based on a KS ref. In this work, we establish the concept of rPT2 and present the numerical details of our SOSEX and rSE implementations. A preliminary version of rPT2, in which the renormalized SE (rSE) contribution was treated approx., has already been benchmarked for mol. atomization energies and chem. reaction barrier heights and shows a well-balanced performance. In this work, we present a refined version of rPT2, in which we evaluate the rSE series of diagrams rigorously. We then extend the benchmark studies to noncovalent interactions, including the rare-gas dimers, and the S22 and S66 test sets, as well as the cohesive energy of small copper clusters, and the equil. geometry of 10 diat. mols. Despite some remaining shortcomings, we conclude that rPT2 gives an overall satisfactory performance across different electronic situations, and is a promising step towards a generally applicable electronic-structure approach.
    93. 93
      Albrecht, S.; Reining, L.; Del Sole, R.; Onida, G. Ab Initio Calculation of Excitonic Effects in the Optical Spectra of Semiconductors. Phys. Rev. Lett. 1998, 80, 4510– 4513,  DOI: 10.1103/PhysRevLett.80.4510
      93
      Ab Initio Calculation of Excitonic Effects in the Optical Spectra of Semiconductors
      Albrecht, Stefan; Reining, Lucia; Del Sole, Rodolfo; Onida, Giovanni
      Physical Review Letters (1998), 80 (20), 4510-4513CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      An ab initio approach to the calcn. of excitonic effects in the optical absorption spectra of semiconductors and insulators is formulated. It starts from a quasiparticle band structure calcn. and is based on the relevant Bethe-Salpeter equation. An application to bulk Si shows a substantial improvement with respect to previous calcns. in the description of the exptl. spectrum, for both peak positions and line shape.
    94. 94
      Farid, B.; Daling, R.; Lenstra, D.; van Haeringen, W. GW approach to the calculation of electron self-energies in semiconductors. Phys. Rev. B 1988, 38, 7530– 7534,  DOI: 10.1103/PhysRevB.38.7530
      94
      GW approach to the calculation of electron self-energies in semiconductors
      Farid; Daling; Lenstra; van Haeringen W
      Physical review. B, Condensed matter (1988), 38 (11), 7530-7534 ISSN:0163-1829.
      There is no expanded citation for this reference.
    95. 95
      Lebègue, S.; Arnaud, B.; Alouani, M.; Bloechl, P. E. Implementation of an all-electron GW approximation based on the projector augmented wave method without plasmon pole approximation: Application to Si, SiC, AlAs, InAs, NaH, and KH. Phys. Rev. B 2003, 67, 155208,  DOI: 10.1103/PhysRevB.67.155208
      95
      Implementation of an all-electron GW approximation based on the projector augmented wave method without plasmon pole approximation: application to Si, SiC, AlAs, InAs, NaH, and KH
      Lebegue, S.; Arnaud, B.; Alouani, M.; Bloechl, P. E.
      Physical Review B: Condensed Matter and Materials Physics (2003), 67 (15), 155208/1-155208/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
      An implementation of the GW approxn. (GWA) based on the all-electron projector-augmented-wave (PAW) method is presented, where the screened Coulomb interaction is computed within the RPA (RPA) instead of the plasmon-pole model. Two different ways of computing the self-energy are reported. The method is used successfully to det. the quasiparticle energies of six semiconducting or insulating materials: Si, SiC, AlAs, InAs, NaH, and KH. To illustrate the method the real and imaginary part of the frequency-dependent self-energy together with the spectral function of silicon are computed. Finally, the GWA results are compared with other calcns., highlighting that all-electron GWA results can differ markedly from those based on pseudopotential approaches.
    96. 96
      Rojas, H. N.; Godby, R. W.; Needs, R. J. Space-Time Method for Ab Initio Calculations of Self-Energies and Dielectric Response Functions of Solids. Phys. Rev. Lett. 1995, 74, 1827– 1830,  DOI: 10.1103/PhysRevLett.74.1827
      96
      Space-time method for ab initio calculations of self-energies and dielectric response functions of solids
      Rojas, H. N.; Godby, R. W.; Needs, R. J.
      Physical Review Letters (1995), 74 (10), 1827-30CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      The authors present a new method for efficient, accurate calcns. of many-body properties of periodic systems. The main features are (i) use of a real-space/imaginary-time representation, (ii) avoidance of any model form for the screened interaction W, (iii) exact sepn. of W and the self-energy Σ into short- and long-ranged parts, and (iv) the use of novel anal. continuation techniques in the energy domain. The computer time scales approx. linearly with system size. The authors give results for jellium and Si, including the spectral function of Si obtained from the Dyson equation.
    97. 97
      Ren, X.; Rinke, P.; Blum, V.; Wieferink, J.; Tkatchenko, A.; Sanfilippo, A.; Reuter, K.; Scheffler, M. Resolution-of-identity approach to Hartree–Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions. New J. Phys. 2012, 14, 053020,  DOI: 10.1088/1367-2630/14/5/053020
      97
      Resolution-of-identity approach to Hartree-Fock, hybrid density functionals, RPA, MP2 and GW with numeric atom-centered orbital basis functions
      Ren, Xinguo; Rinke, Patrick; Blum, Volker; Wieferink, Juergen; Tkatchenko, Alexandre; Sanfilippo, Andrea; Reuter, Karsten; Scheffler, Matthias
      New Journal of Physics (2012), 14 (May), 053020/1-053020/55CODEN: NJOPFM; ISSN:1367-2630. (Institute of Physics Publishing)
      A review. The efficient implementation of electronic structure methods is essential for first principles modeling of mols. and solids. We present here a particularly efficient common framework for methods beyond semilocal d.-functional theory (DFT), including Hartree-Fock (HF), hybrid d. functionals, RPA (RPA), second-order Moller-Plesset perturbation theory (MP2) and the GW method. This computational framework allows us to use compact and accurate numeric atom-centered orbitals (NAOs), popular in many implementations of semilocal DFT, as basis functions. The essence of our framework is to employ the 'resoln. of identity (RI)' technique to facilitate the treatment of both the two-electron Coulomb repulsion integrals (required in all these approaches) and the linear d.-response function (required for RPA and GW). This is possible because these quantities can be expressed in terms of the products of single-particle basis functions, which can in turn be expanded in a set of auxiliary basis functions (ABFs). The construction of ABFs lies at the heart of the RI technique, and we propose here a simple prescription for constructing ABFs which can be applied regardless of whether the underlying radial functions have a specific anal. shape (e.g. Gaussian) or are numerically tabulated. We demonstrate the accuracy of our RI implementation for Gaussian and NAO basis functions, as well as the convergence behavior of our NAO basis sets for the above-mentioned methods. Benchmark results are presented for the ionization energies of 50 selected atoms and mols. from the G2 ion test set obtained with the GW and MP2 self-energy methods, and the G2-I atomization energies as well as the S22 mol. interaction energies obtained with the RPA method.
    98. 98
      Jiang, H.; Engel, E. Second-order Kohn-Sham perturbation theory: Correlation potential for atoms in a cavity. J. Chem. Phys. 2005, 123, 224102,  DOI: 10.1063/1.2128674
      98
      Second-order Kohn-Sham perturbation theory: Correlation potential for atoms in a cavity
      Jiang, Hong; Engel, Eberhard
      Journal of Chemical Physics (2005), 123 (22), 224102/1-224102/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Second-order perturbation theory based on the Kohn-Sham Hamiltonian leads to an implicit d. functional for the correlation energy EcMP2, which is explicitly dependent on both occupied and unoccupied Kohn-Sham single-particle orbitals and energies. The corresponding correlation potential vcMP2, which has to be evaluated by the optimized potential method, was found to be divergent in the asymptotic region of atoms, if pos.-energy continuum states are included in the calcn. [Facco Bonetti et al., Phys. Rev. Lett. 86, 2241 (2001)]. On the other hand, Niquet et al., [J. Chem. Phys. 118, 9504 (2003)] showed that vcMP2 has the same asymptotic -α/(2r4) behavior as the exact correlation potential, if the system under study has a discrete spectrum only. We study vMP2c for atoms in a spherical cavity within a basis-set-free finite differences approach, ensuring a completely discrete spectrum by requiring hard-wall boundary conditions at the cavity radius. Choosing this radius sufficiently large, one can devise a numerical continuation procedure which allows to normalize vMP2c consistent with the std. choice vc(r ∞) = 0 for free atoms, without modifying the potential in the chem. relevant region. An important prerequisite for the success of this scheme is the inclusion of very high-energy virtual states. Using this technique, we have calcd. vMP2c for all closed-shell and spherical open-shell atoms up to argon. One finds that vMP2c reproduces the shell structure of the exact correlation potential very well but consistently overestimates the corresponding shell oscillations. In the case of spin-polarized atoms one observes a strong interrelation between the correlation potentials of the two spin channels, which is completely absent for std. d. functionals. However, our results also demonstrate that EMP2c can only serve as a first step towards the construction of a suitable implicit correlation functional: The fundamental variational instability of this functional is recovered for beryllium, for which a breakdown of the self-consistent Kohn-Sham iteration is obsd. Moreover, even for those atoms for which the self-consistent iteration is stable, the results indicate that the inclusion of vMP2c in the total Kohn-Sham potential does not lead to an improvement compared to the complete neglect of the correlation potential.
    99. 99
      Hellgren, M.; von Barth, U. Correlation potential in density functional theory at the GWA level: Spherical atoms. Phys. Rev. B 2007, 76, 075107,  DOI: 10.1103/PhysRevB.76.075107
      99
      Correlation potential in density functional theory at the GWA level: Spherical atoms
      Hellgren, Maria; von Barth, Ulf
      Physical Review B: Condensed Matter and Materials Physics (2007), 76 (7), 075107/1-075107/12CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      As part of a project to obtain better optical response functions for nanomaterials and other systems with strong excitonic effects, we here calc. the exchange-correlation (XC) potential of d. functional theory (DFT) at a level of approxn. which corresponds to the dynamically screened exchange or GW approxn. In this process, we have designed a numerical method based on cubic splines, which appears to be superior to other techniques previously applied to the "inverse engineering problem" of DFT, i.e., the problem of finding an XC potential from a known particle d. The potentials we obtain do not suffer from unphys. ripple and have, to within a reasonable accuracy, the correct asymptotic tails outside localized systems. The XC potential is an important ingredient in finding the particle-conserving excitation energies in atoms and mols., and our potentials perform better in this regard as compared to the local-d. approxn. potential, potentials from generalized gradient approxns., and a DFT potential based on MP2 theory.
    100. 100
      Wang, Y.; Perdew, J. P.; Chevary, J. A.; Macdonald, L. D.; Vosko, S. H. Exchange potentials in density-functional theory. Phys. Rev. A 1990, 41, 78– 86,  DOI: 10.1103/PhysRevA.41.78
      100
      Exchange potentials in density-functional theory
      Wang, Yue; Perdew, John P.; Chevary, J. A.; Macdonald, L. D.; Vosko, S. H.
      Physical Review A: Atomic, Molecular, and Optical Physics (1990), 41 (1), 78-86CODEN: PLRAAN; ISSN:0556-2791.
      The Harbola-Sahni exchange potential is the work needed to move an electron against the elec. field of its hole charge distribution. We prove that it is not the exact exchange potential of d.-functional theory, by showing that it yields the wrong second-order gradient expansion in the slowly varying limit. But we also discover that it yields the correct local-d. approxn. Thus the Harbola-Sahni potential is a more phys. correct version of the Slater potential, one that is better suited for mol. and solid-state applications. As a step in our derivation, we present the third-order gradient expansion of the exchange hole d., and discuss its structure. We also describe a new version of the Harbola-Sahni potential which corrects its path dependence. The exact exchange potential for an atom is given by the optimized potential model (OPM) of Talman and Shadwick. By using enhanced numerics, we confirm that the OPM potential satisfies the Levy-Perdew virial relation and exhibits correct -1/r behavior at large r. Numerical calcns. also show that the intershell max. in the exact exchange potential are needed to lower the total energy. These "bumps" are missing from the Harbola-Sahni and Slater potentials.
    101. 101
      Krieger, J. B.; Li, Y.; Iafrate, G. J. Derivation and application of an accurate Kohn-Sham potential with integer discontinuity. Phys. Lett. A 1990, 146, 256– 260,  DOI: 10.1016/0375-9601(90)90975-T
      101
      Derivation and application of an accurate Kohn-Sham potential with integer discontinuity
      Krieger, J. B.; Li, Yan; Iafrate, G. J.
      Physics Letters A (1990), 146 (5), 256-60CODEN: PYLAAG; ISSN:0375-9601.
      A new expression for the Kohn-Sham nKS) spin polarized exchange-only potential is derived and is shown to closely approx. the exact numerical results for atoms as well as exhibit the necessary integer discontinuity. The generalization of the results for any assumed exchange-correlation functional is also presented.
    102. 102
      Krieger, J. B.; Li, Y.; Iafrate, G. J. Construction and application of an accurate local spin-polarized Kohn-Sham potential with integer discontinuity: Exchange-only theory. Phys. Rev. A 1992, 45, 101– 126,  DOI: 10.1103/PhysRevA.45.101
      102
      Construction and application of an accurate local spin-polarized Kohn-Sham potential with integer discontinuity: exchange-only theory
      Krieger, J. B.; Li, Yan; Iafrate, G. J.
      Physical Review A: Atomic, Molecular, and Optical Physics (1992), 45 (1), 101-26CODEN: PLRAAN; ISSN:0556-2791.
      An accurate spin-polarized exchange-only Kohn-Sham (KS) (W. Khon and L. J. Sham, 1965) potential is constructed from a consideration of the optimized-effective-potential (OEP) method. A detailed anal. of the OEP integral equation for the exchange-only case results in a set of conditions which are manifestly satisfied by the exact OEP; these conditions are employed to construct an approx. OEP, Vxσ, and therefore an approx. KS exchange-only potential as a functional of KS orbitals. This Vxσ can be derived anal. based on a simple approxn. of the Green's functions in the OEP integral equation. The constructed potential, although approx., contains many of the key analytic features of the exact KS potential: it reduces to the exact KS result in the homogeneous-electron-gas limit, approaches -1/r as r → ∞, yields highest occupied-orbital energy eigenvalues εmσ that satisfy Koopman's theorem, and exhibits an integer discontinuity when considered as a function of fractional occupancy of the highest-energy occupied single-particle state of a given spin projection σ. In addn. εmσ nearly exactly satisfies J. F. Janak's (1978) theorem. The approx. OEP is a simple but remarkably accurate representation of the exact, numerically derived exchange-only OEP. Detailed numerical results obtained by employing Vxσ as the exchange-only potential for the atoms with closed subshells yield total energies, Hartree potentials, single-particle expectation values, and εm which are in excellent agreement with both exact OEP and Hartree-Fock (HF) results and represent a significant improvement over the results obtained by employing other exchange-only potentials. The properties of alkali-metal atoms are calcd. including the sep. spin-up and spin-down densities to obtain results in excellent agreement with those of spin-unrestricted OEP and HF methods. The accuracy of Vxσ by calcg. the total energy, εm↑, and εm↓ as a function of fractional filling f, of the highest occupied single-particle orbital for the magnesium atom (Z = 12) from N = 9-12 electrons and find excellent agreement with both spin-unrestricted OEP and HF results even when εmσ is strongly dependent on f. The authors display the integer discontinuity in Vxσ when the highest-energy spin subshell begins to be filled.
    103. 103
      Facco Bonetti, A.; Engel, E.; Schmid, R. N.; Dreizler, R. M. Investigation of the Correlation Potential from Kohn-Sham Perturbation Theory. Phys. Rev. Lett. 2001, 86, 2241– 2244,  DOI: 10.1103/PhysRevLett.86.2241
      103
      Investigation of the Correlation Potential from Kohn-Sham Perturbation Theory
      Facco Bonetti, A.; Engel, E.; Schmid, R. N.; Dreizler, R. M.
      Physical Review Letters (2001), 86 (11), 2241-2244CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Perturbation theory on the basis of the Kohn-Sham Hamiltonian leads to an implicit d. functional for the correlation energy Ec. In this contribution we investigate the corresponding correlation potential vc. It is shown that for finite systems the vc obtained by direct application of the optimized potential method diverges in the asymptotic region. The presence of unoccupied states, inherent in any perturbative form of Ec, is identified as the origin of this unphys. behavior. An approx. variational procedure is developed in order to avoid this difficulty. The potential resulting from this method qual. reproduces the shell structure of the exact at. vc.
    104. 104
      Hellgren, M.; von Barth, U. Correlation energy functional and potential from time-dependent exact-exchange theory. J. Chem. Phys. 2010, 132, 044101,  DOI: 10.1063/1.3290947
      104
      Correlation energy functional and potential from time-dependent exact-exchange theory
      Hellgren, Maria; von Barth, Ulf
      Journal of Chemical Physics (2010), 132 (4), 044101/1-044101/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      In this work we studied a new functional for the correlation energy obtained from the exact-exchange (EXX) approxn. within time-dependent d. functional theory. Correlation energies have been calcd. for a no. of different atoms showing excellent agreement with results from more sophisticated methods. These results lose little accuracy by approximating the EXX kernel by its static value, a procedure which enormously simplifies the calcns. The correlation potential, obtained by taking the functional deriv. with respect to the d., turns out to be remarkably accurate for all atoms studied. This potential has been used to calc. ionization potentials, static polarizabilities, and van der Waals coeffs. with results in close agreement with expt. (c) 2010 American Institute of Physics.
    105. 105
      van Leeuwen, R.; Gritsenko, O.; Baerends, E. J. Step structure in the atomic Kohn-Sham potential. Z. Phys. D - Atoms, Molecules and Clusters 1995, 33, 229– 238,  DOI: 10.1007/BF01437503
      105
      Step structure in the atomic Kohn-Sham potential
      van Leeuwen, Robert; Gritsenko, Oleg; Baerends, Evert Jan
      Zeitschrift fuer Physik D: Atoms, Molecules and Clusters (1995), 33 (4), 229-38CODEN: ZDACE2; ISSN:0178-7683. (Springer)
      The authors analyze the exchange-correlation potential within the Kohn-Sham approach to d. functional theory for the case of at. systems. The exchange-correlation potential is written as a sum of two potentials. One of these potentials is the long-range Coulombic potential of the coupling const. integrated exchange-correlation hole which represents the screening of the two-particle interactions due to exchange-correlation effects. The other potential contains the functional deriv. with respect to the electron d. of the coupling const. integrated pair-correlation function representing the sensitivity of this exchange-correlation screening to d. variations. An explicit expression of the exchange-part of this functional deriv. is derived using an approxn. for the Green function of the Kohn-Sham system and is shown to display a distinct at. shell structure. The corresponding potential has a clear step structure and is const. within the at. shells and changes rapidly at the at. shell boundaries. Numerical examples are presented for Be and Kr atoms using the optimized potential model (OPM).
    106. 106
      Ferretti, A.; Dabo, I.; Cococcioni, M.; Marzari, N. Bridging density-functional and many-body perturbation theory: Orbital-density dependence in electronic-structure functionals. Phys. Rev. B 2014, 89, 195134,  DOI: 10.1103/PhysRevB.89.195134
      106
      Bridging density-functional and many-body perturbation theory: orbital-density dependence in electronic-structure functionals
      Ferretti, Andrea; Dabo, Ismaila; Cococcioni, Matteo; Marzari, Nicola
      Physical Review B: Condensed Matter and Materials Physics (2014), 89 (19), 195134/1-195134/8, 8 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
      Energy functionals which depend explicitly on orbital densities, rather than on the total charge d., appear when applying self-interaction corrections to d.-functional theory; this is, e.g., the case for Perdew-Zunger and Koopmans-compliant functionals. In these formulations the total energy is not invariant under unitary rotations of the orbitals, and local, orbital-dependent potentials emerge. We argue that this is not a shortcoming, and that instead these potentials can provide, in a functional form, a simplified quasiparticle approxn. to the spectral potential, i.e., the local, frequency-dependent contraction of the many-body self-energy that is sufficient to describe exactly the spectral function. As such, orbital-d.-dependent functionals have the flexibility to accurately describe both total energies and quasiparticle excitations in the electronic-structure problem. In addn., and at variance with the Kohn-Sham case, orbital-dependent potentials do not require nonanalytic deriv. discontinuities. We present numerical solns. based on the frequency-dependent Sham-Schluter equation to support this view, and examine some of the existing functionals in this perspective, highlighting the very close agreement between exact and approx. orbital-dependent potentials.
    107. 107
      Almbladh, C.-O.; Barth, U. V.; Leeuwen, R. V. Variational total energies from Φ- and Ψ- derivable theories. Int. J. Mod. Phys. B 1999, 13, 535– 541,  DOI: 10.1142/S0217979299000436
      107
      Variational total energies from Φ- and ψ- derivable theories
      Almbladh, C. -O.; Von Barth, U.; Van Leeuwen, R.
      International Journal of Modern Physics B (1999), 13 (5 & 6), 535-541CODEN: IJPBEV; ISSN:0217-9792. (World Scientific Publishing Co. Pte. Ltd.)
      Starting from many-body perturbation theory we have constructed a new variational expression for the total energy of many-electron systems. This expression is a functional of two independent variables, the one-electron Green function and the screened Coulomb interaction. The new functional as well as a much older variational expression by Luttinger and Ward (LW) are tested on the interacting electron gas. Both functionals yield extraordinary accurate total energies although the new functional requires a much cruder input and is therefore easier to apply to more realistic systems.
    108. 108
      Rumble, J. R., Ed. CRC Handbook of Chemistry and Physics; CRC Press, 2010.
      There is no corresponding record for this reference.
    109. 109
      Atalla, V.; Zhang, I. Y.; Hofmann, O. T.; Ren, X.; Rinke, P.; Scheffler, M. Enforcing the linear behavior of the total energy with hybrid functionals: Implications for charge transfer, interaction energies, and the random-phase approximation. Phys. Rev. B 2016, 94, 035140,  DOI: 10.1103/PhysRevB.94.035140
      109
      Enforcing the linear behavior of the total energy with hybrid functionals: implications for charge transfer, interaction energies, and the random-phase approximation
      Atalla, Viktor; Ying, Zhang Igor; Hofmann, Oliver T.; Ren, Xinguo; Rinke, Patrick; Scheffler, Matthias
      Physical Review B (2016), 94 (3), 035140/1-035140/17CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
      We obtain the exchange parameter of hybrid functionals by imposing the fundamental condition of a piecewise linear total energy with respect to electron no. For the Perdew-Burke-Ernzerhof (PBE) hybrid family of exchange-correlation functionals (i.e., for an approx. generalized Kohn-Sham theory) this implies that (i) the HOMO corresponds to the ionization potential (I), (ii) the energy of the LUMO corresponds to the electron affinity (A), and (iii) the energies of the frontier orbitals are const. as a function of their occupation. In agreement with a previous study, we find that these conditions are met for high values of the exact exchange admixt. α and illustrate their importance for the tetrathiafulvalene-tetracyanoquinodimethane complex for which std. d. functional theory functionals predict artificial electron transfer. We further assess the performance for atomization energies and weak interaction energies. We find that atomization energies are significantly underestimated compared to PBE or PBE0, whereas the description of weak interaction energies improves significantly if a 1/R6 van der Waals correction scheme is employed.
    110. 110
      Golze, D.; Keller, L.; Rinke, P. Accurate Absolute and Relative Core-Level Binding Energies from GW. J. Phys. Chem. Lett. 2020, 11, 1840– 1847,  DOI: 10.1021/acs.jpclett.9b03423
      110
      Accurate Absolute and Relative Core-Level Binding Energies from GW
      Golze, Dorothea; Keller, Levi; Rinke, Patrick
      Journal of Physical Chemistry Letters (2020), 11 (5), 1840-1847CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      We present an accurate approach to compute X-ray photoelectron spectra based on the GW Green's function method that overcomes the shortcomings of common d. functional theory approaches. GW has become a popular tool to compute valence excitations for a wide range of materials. However, core-level spectroscopy is thus far almost uncharted in GW. We show that single-shot perturbation calcns. in the G0W0 approxn., which are routinely used for valence states, cannot be applied for core levels and suffer from an extreme, erroneous transfer of spectral wt. to the satellite spectrum. The correct behavior can be restored by partial self-consistent GW schemes or by using hybrid functionals with almost 50% of exact exchange as a starting point for G0W0. We also include relativistic corrections and present a benchmark study for 65 mol. 1s excitations. Our abs. and relative GW core-level binding energies agree within 0.3 and 0.2 eV with expt., resp.

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    • Comparison of results with existing literature, convergence of the xc potentials, and perturbative versus self-consistent solution of the LSSE (PDF)


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