The Structure of Density-Potential Mapping. Part I: Standard Density-Functional TheoryClick to copy article linkArticle link copied!
- Markus Penz
- Erik I. TellgrenErik I. TellgrenHylleraas Centre for Quantum Molecular Sciences, University of Oslo, Oslo 0315, NorwayMore by Erik I. Tellgren
- Mihály A. CsirikMihály A. CsirikHylleraas Centre for Quantum Molecular Sciences, University of Oslo, Oslo 0315, NorwayDepartment of Computer Science, Oslo Metropolitan University, Oslo 0130, NorwayMore by Mihály A. Csirik
- Michael RuggenthalerMichael RuggenthalerMax Planck Institute for the Structure and Dynamics of Matter, Hamburg 22761, GermanyMore by Michael Ruggenthaler
- Andre Laestadius*Andre Laestadius*E-mail: [email protected]Department of Computer Science, Oslo Metropolitan University, Oslo 0130, NorwayHylleraas Centre for Quantum Molecular Sciences, University of Oslo, Oslo 0315, NorwayMore by Andre Laestadius
Abstract
The Hohenberg–Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. Part I of this review aims at clarifying the status of the Hohenberg–Kohn theorem within DFT and Part II at different extensions of the theory that include magnetic fields. We collect evidence that the Hohenberg–Kohn theorem does not so much form the basis of DFT, but is rather the consequence of a more comprehensive mathematical framework. Such results are especially useful when it comes to the construction of generalized DFTs.
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License Summary*
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Attribution (BY): Credit must be given to the creator.
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License Summary*
You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters below:
Creative Commons (CC): This is a Creative Commons license.
Attribution (BY): Credit must be given to the creator.
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1. Introduction
(HK1) If two potentials share a common ground-state density, then they also share a common ground-state wave function or density matrix.
(HK2) If two potentials share any common eigenstate and if that eigenstate is nonzero almost everywhere (a property that is guaranteed if the unique-continuation property (UCP) holds; see Section 5), then they are equal up to a constant.
2. Preliminaries

3. Representability of Densities
4. The Hohenberg–Kohn Theorem






5. The Unique-Continuation Property
6. Hierarchy of Density Functionals





F• | convex | domain | convex | ||
---|---|---|---|---|---|
FHK1,pure | no | v-reppure | no | ||
↓ cl | |||||
FHK1,ens | ? | v-repens | ? | ||
FCS,pure | no | N-rep | yes | ||
↓ conv | |||||
FCS,ens | yes | N-rep | yes | ||
F | yes | L1 ∩ L3 | yes |
7. Density-Potential Mappings from Differentials
Figure 1
Figure 1. Example of a convex and lower-semicontinuous function with a discontinuity at ρ0 and some elements from the subdifferential displayed as linear continuous tangent functionals at ρ0, represented by dashed lines.
8. Linking to a Reference System: The Kohn–Sham Scheme


9. Density-Potential Mixing and Regularized DFT




10. Abstract Density-Potential Mapping

11. Summary
Figure 2
Figure 2. Logical implications between the different statements relating to a “unique density-potential mapping” and the HK theorem in standard DFT.
12. Outlook

Acknowledgments
E.I.T., M.A.C., and A.L. thank the Research Council of Norway (RCN) under CoE (Hylleraas Centre) Grant No. 262695, for A.L. and M.A.C. also CCerror Grant No. 287906 and for EIT also “Magnetic Chemistry” Grant No. 287950, and M.R. acknowledges the Cluster of Excellence “CUI: Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG), EXC 2056, project ID 390715994. A.L. and M.A.C. were also supported by the ERC through StG REGAL under agreement No. 101041487. The authors thank Centre for Advanced Studies (CAS) in Oslo, since this work includes insights gathered at the YoungCAS workshop “Do Electron Current Densities Determine All There Is to Know?”, held July 9–13, 2018, in Oslo, Norway.
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- 34Pino, R.; Bokanowski, O.; Ludeña, E. V.; Boada, R. L. A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces. Theor. Chem. Acc. 2007, 118, 557– 561, DOI: 10.1007/s00214-007-0367-6Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXpvVKnt78%253D&md5=77be97a81821b41bb628fcfe2ffcbe22A re-statement of the Hohenberg-Kohn theorem and its extension to finite subspacesPino, Ramiro; Bokanowski, Olivier; Ludena, Eduardo V.; Boada, Roberto LopezTheoretical Chemistry Accounts (2007), 118 (3), 557-561CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH)Bearing in mind the insight into the Hohenberg-Kohn theorem for Coulomb systems provided recently by Kryachko (Int J Quantum Chem 103:818, 2005), we present a re-statement of this theorem through an elaboration on Lieb's proof as well as an extension of this theorem to finite subspaces.
- 35Garrigue, L. Unique Continuation for Many-Body Schrödinger Operators and the Hohenberg-Kohn Theorem. Math. Phys. Anal. Geom. 2018, 21, 27, DOI: 10.1007/s11040-018-9287-zGoogle ScholarThere is no corresponding record for this reference.
- 36Tellgren, E. I.; Laestadius, A.; Helgaker, T.; Kvaal, S.; Teale, A. M. Uniform magnetic fields in density-functional theory. J. Chem. Phys. 2018, 148, 024101, DOI: 10.1063/1.5007300Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXltlCisA%253D%253D&md5=8f4caf48483a5e3e1a8801a1ddd9738eUniform magnetic fields in density-functional theoryTellgren, Erik I.; Laestadius, Andre; Helgaker, Trygve; Kvaal, Simen; Teale, Andrew M.Journal of Chemical Physics (2018), 148 (2), 024101/1-024101/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We construct a d.-functional formalism adapted to uniform external magnetic fields that is intermediate between conventional d. functional theory and Current-D. Functional Theory (CDFT). In the intermediate theory, which we term linear vector potential-DFT (LDFT), the basic variables are the d., the canonical momentum, and the paramagnetic contribution to the magnetic moment. Both a constrained-search formulation and a convex formulation in terms of Legendre-Fenchel transformations are constructed. Many theor. issues in CDFT find simplified analogs in LDFT. We prove results concerning N-representability, Hohenberg-Kohn-like mappings, existence of minimizers in the constrained-search expression, and a restricted analog to gauge invariance. The issue of additivity of the energy over non-interacting subsystems, which is qual. different in LDFT and CDFT, is also discussed. (c) 2018 American Institute of Physics.
- 37Kohn, W.; Savin, A.; Ullrich, C. A. Hohenberg–Kohn theory including spin magnetism and magnetic fields. Int. J. Quantum Chem. 2004, 100, 20– 21, DOI: 10.1002/qua.20163Google Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXntlCrsLY%253D&md5=8a84555294a27a7b1cd1660ee0fcf8dcHohenberg-Kohn theory including spin magnetism and magnetic fieldsKohn, Walter; Savin, Andreas; Ullrich, Carsten A.International Journal of Quantum Chemistry (2004), 100 (1), 20-21CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)Beginning with work by U. von Barth and L. Hedin in 1972 and continuing with recent papers in 2001 by K. Capelle and G. Vignale and by H. Eschrig and W. E. Pickett, questions have been raised about the meaning and validity of Hohenberg-Kohn theory when spin magnetism and/or magnetic fields are present. This article offers clarifications of some of these questions. In particular, it also concludes that these questions do not affect spin-d. functional theory for nondegenerate ground states as currently practiced.
- 38Kohn, W. Density Functional Theory: Fundamentals and Applications. In Highlights of condensed-matter theory; Bassani, F., Fumi, F., Tosi, M. P., Eds.; Proceedings of the international school of physics “Enrico Fermi”, Course LXXXIX at Varenna on Lake Como, 28 June-16 July 1983; North-Holland: Amsterdam, 1985; pp 1– 15.Google ScholarThere is no corresponding record for this reference.
- 39Regbaoui, R. Unique Continuation from Sets of Positive Measure. In Carleman Estimates and Applications to Uniqueness and Control Theory; Colombini, F., Zuily, C., Eds.; Progress in Nonlinear Differential Equations and Their Applications; Birkhäuser: Basel, 2001; Vol. 46; pp 179– 190.Google ScholarThere is no corresponding record for this reference.
- 40de Figueiredo, D. G.; Gossez, J.-P. Strict monotonicity of eigenvalues and unique continuation. Comm. Par. Diff. Eq. 1992, 17, 339– 346, DOI: 10.1080/03605309208820844Google ScholarThere is no corresponding record for this reference.
- 41Ladyzenskaya, O. A.; Ural’tzeva, N. N. Linear and Quasilinear Elliptic Equations; Academic Press: New York, 1968.Google ScholarThere is no corresponding record for this reference.
- 43Lammert, P. E. In search of the Hohenberg-Kohn theorem. J. Math. Phys. 2018, 59, 042110, DOI: 10.1063/1.5034215Google Scholar43https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXptVCnsrk%253D&md5=160d1fff304ab2f36fd0a5b520b47ddcIn search of the Hohenberg-Kohn theoremLammert, Paul E.Journal of Mathematical Physics (Melville, NY, United States) (2018), 59 (4), 042110/1-042110/19CODEN: JMAPAQ; ISSN:0022-2488. (American Institute of Physics)The Hohenberg-Kohn theorem, a cornerstone of electronic d. functional theory, concerns uniqueness of external potentials yielding given ground densities of an N-body system. The problem is rigorously explored in a universe of three-dimensional Kato-class potentials, with emphasis on trade-offs between conditions on the d. and conditions on the potential sufficient to ensure uniqueness. Sufficient conditions range from none on potentials coupled with everywhere strict positivity of the d. to none on the d. coupled with something a little weaker than local 3N/2-power integrability of the potential on a connected full-measure set. A second theme is localizability, i.e., the possibility of uniqueness over subsets of R3 under less stringent conditions. (c) 2018 American Institute of Physics.
- 44Jerison, D.; Kenig, C. E. Unique continuation and absence of positive eigenvalues for Schrodinger operators. Ann. Math. 1985, 121, 463– 488, DOI: 10.2307/1971205Google ScholarThere is no corresponding record for this reference.
- 45Garrigue, L. Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian. Doc. Math. 2020, 25, 869– 898, DOI: 10.4171/dm/765Google ScholarThere is no corresponding record for this reference.
- 46Garrigue, L. Hohenberg–Kohn Theorems for Interactions, Spin and Temperature. J. Stat. Phys. 2019, 177, 415– 437, DOI: 10.1007/s10955-019-02365-6Google ScholarThere is no corresponding record for this reference.
- 47Kvaal, S.; Helgaker, T. Ground-state densities from the Rayleigh–Ritz variation principle and from density-functional theory. J. Chem. Phys. 2015, 143, 184106, DOI: 10.1063/1.4934797Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvVWmtrfE&md5=c155dc8a001b3819e1ebe1aecaa1b3d9Ground-state densities from the Rayleigh-Ritz variation principle and from density-functional theoryKvaal, Simen; Helgaker, TrygveJournal of Chemical Physics (2015), 143 (18), 184106/1-184106/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The relationship between the densities of ground-state wave functions (i.e., the minimizers of the Rayleigh-Ritz variation principle) and the ground-state densities in d.-functional theory (i.e., the minimizers of the Hohenberg-Kohn variation principle) is studied within the framework of convex conjugation, in a generic setting covering mol. systems, solid-state systems, and more. Having introduced admissible d. functionals as functionals that produce the exact ground-state energy for a given external potential by minimizing over densities in the Hohenberg-Kohn variation principle, necessary and sufficient conditions on such functionals are established to ensure that the Rayleigh-Ritz ground-state densities and the Hohenberg-Kohn ground-state densities are identical. We apply the results to mol. systems in the Born-Oppenheimer approxn. For any given potential v ∈ L3/2(R3) + L∞(R3), we establish a one-to-one correspondence between the mixed ground-state densities of the Rayleigh-Ritz variation principle and the mixed ground-state densities of the Hohenberg-Kohn variation principle when the Lieb d.-matrix constrained-search universal d. functional is taken as the admissible functional. A similar one-to-one correspondence is established between the pure ground-state densities of the Rayleigh-Ritz variation principle and the pure ground-state densities obtained using the Hohenberg-Kohn variation principle with the Levy-Lieb pure-state constrained-search functional. In other words, all phys. ground-state densities (pure or mixed) are recovered with these functionals and no false densities (i.e., minimizing densities that are not phys.) exist. The importance of topol. (i.e., choice of Banach space of densities and potentials) is emphasized and illustrated. The relevance of these results for current-d.-functional theory is examd. (c) 2015 American Institute of Physics.
- 48Lewin, M.; Lieb, E. H.; Seiringer, R. Universal Functionals in Density Functional Theory. arXiv (Mathematical Physics), September 13, 2022, ver. 4, 1912.10424 (accessed 2023-01-31).Google ScholarThere is no corresponding record for this reference.
- 54Tsuneda, T. Density functional theory in quantum chemistry; Springer, 2014.Google ScholarThere is no corresponding record for this reference.
- 56Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133, DOI: 10.1103/PhysRev.140.A1133Google ScholarThere is no corresponding record for this reference.
- 67van Leeuwen, R. Density functional approach to the many-body problem: key concepts and exact functionals. Adv. Quantum Chem. 2003, 43, 25, DOI: 10.1016/S0065-3276(03)43002-5Google ScholarThere is no corresponding record for this reference.
- 68Lammert, P. E. Well-behaved coarse-grained model of density-functional theory. Phys. Rev. A 2010, 82, 012109, DOI: 10.1103/PhysRevA.82.012109Google Scholar68https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXpsF2hs78%253D&md5=93121d5e4d4e67a301a625c4eded0464Well-behaved coarse-grained model of density-functional theoryLammert, Paul E.Physical Review A: Atomic, Molecular, and Optical Physics (2010), 82 (1, Pt. A), 012109/1-012109/14CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)A coarse-grained version of quantum d.-functional theory featuring a limited spatial resoln. is shown to model formal d.-functional theory (DFT). This means that all densities are ensemble-V-representable (i.e., by mixed states), the intrinsic energy functional F is a continuous function of the d., and the representing external potential is the functional deriv. of the intrinsic energy, in the sense of directional derivs. within the domain of F. The representing potential v[ρ] also has a quasicontinuity property, specifically, v[ρ]ρ is continuous as a function of ρ. Convergence of the intrinsic energy, coarse-grained densities, and representing potentials in the limit of coarse-graining scale going to zero are studied vis-a-vis Lieb's L1 ∩ L3 theory. The intrinsic energy converges monotonically to its fine-grained (continuum) value, and coarse-grainings of a d. ρ converge strongly to ρ. If a sequence of coarse-grained densities converges strongly to ρ and their representing potentials converge weak-*, the limit is the representing potential for ρ. Conversely, L3/2 + L∞ representability implies the existence of such a coarse-grained sequence.
- 69Kümmel, S.; Perdew, J. P. Optimized effective potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potential. Phys. Rev. B 2003, 68, 035103, DOI: 10.1103/PhysRevB.68.035103Google Scholar69https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXnsVGru7w%253D&md5=3620b9847bdfc9aa6301c90903b8f369Optimized effective potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potentialKuemmel, Stephan; Perdew, John P.Physical Review B: Condensed Matter and Materials Physics (2003), 68 (3), 035103/1-035103/15CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)The optimized effective potential (OEP) is the exact Kohn-Sham potential for explicitly orbital-dependent energy functionals, e.g., the exact exchange energy. We give a proof for the OEP equation which does not depend on the chain rule for functional derivs. and directly yields the equation in its simplest form: a certain first-order d. shift must vanish. This condition explains why the highest-occupied orbital energies of Hartree-Fock and exact exchange OEP are so close. More importantly, we show that the exact OEP can be constructed iteratively from the first-order shifts of the Kohn-Sham orbitals and that these can be calcd. easily. The exact exchange potential υx(r) for spherical atoms and three-dimensional sodium clusters is calcd. Its long-range asymptotic behavior is investigated, including the approach of νx(r) to a nonvanishing const. in particular spatial directions. We calc. total and orbital energies and static elec. dipole polarizabilities for the sodium clusters employing the exact exchange functional. Exact OEP results are compared to the Krieger-Li-Iafrate and local d. approxns.
- 70Gonis, A. Functionals and Functional Derivatives of Wave Functions and Densities. World J. Condens. Matter Phys. 2014, 4, 179, DOI: 10.4236/wjcmp.2014.43022Google Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhs1ylu77M&md5=bd966da7ce2767eff3cc7a270613ff25Functionals and functional derivatives of wave functions and densitiesGonis, A.World Journal of Condensed Matter Physics (2014), 4 (3), 179-199, 22 pp.CODEN: WJCMAD; ISSN:2160-6927. (Scientific Research Publishing, Inc.)It is shown that the process of conventional functional differentiation does not apply to functionals whose domain (and possibly range) is subject to the condition of integral normalization, as is the case with respect to a domain defined by wave functions or densities, in which there exists no neighborhood about a given element in the domain defined by arbitrary variations that also lie in the domain. This is remedied through the generalization of the domain of a functional to include distributions in the form of εδ(r-r'), where δ(r-r') is the Dirac delta function and ε is a real no. This allows the detn. of the rate of change of a functional with respect to changes of the independent variable detd. at each point of the domain, with no ref. needed to the values of the functional at different functions in its domain. One feature of the formalism is the detn. of rates of change of general expectation values (that may not necessarily be functionals of the d.) with respect to the wave functions or the densities detd. by the wave functions forming the expectation value. It is also shown that ignoring the conditions of conventional functional differentiation can lead to false proofs, illustrated through a flaw in the proof that all densities defined on a lattice are ν-representable. In a companion paper, the math. integrity of a no. of long-standing concepts in d. functional theory are studied in terms of the formalism developed here.
- 71Gonis, A.; Zhang, X.-G.; Däne, M.; Stocks, G. M.; Nicholson, D. M. Reformulation of density functional theory for N-representable densities and the resolution of the v-representability problem. J. Phys. Chem. Solids 2016, 89, 23– 31, DOI: 10.1016/j.jpcs.2015.10.006Google Scholar71https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhslGktrbM&md5=1508ec66f702b96837da2ed58fa3cd19Reformulation of density functional theory for N-representable densities and the resolution of the v-representability problemGonis, A.; Zhang, X.-G.; Dane, M.; Stocks, G. M.; Nicholson, D. M.Journal of Physics and Chemistry of Solids (2016), 89 (), 23-31CODEN: JPCSAW; ISSN:0022-3697. (Elsevier Ltd.)A review. D. functional theory for the case of general, N-representable densities is reformulated in terms of d. functional derivs. of expectation values of operators evaluated with wave functions leading to a d., making no ref. to the concept of potential. The developments provide proof of existence of a math. procedure that dets. whether a d. is v-representable and in the case of an affirmative answer dets. the potential (within an additive const.) as a deriv. with respect to the d. of a constrained search functional. It also establishes the existence of an energy functional of the d. that, for v-representable densities, assumes its min. value at the d. describing the ground state of an interacting many-particle system. The theorems of Hohenberg and Kohn emerge as special cases of the formalism. Numerical results for one-dimensional non-interacting systems illustrate the formalism. Some direct formal and practical implications of the present reformulation of DFT are also discussed.
- 72Gonis, A. Is an interacting ground state (pure state) v-representable density also non-interacting ground state v-representable by a Slater determinant? In the absence of degeneracy, yes. Phys. Lett. A 2019, 383, 2772– 2776, DOI: 10.1016/j.physleta.2019.03.007Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXkvVektLk%253D&md5=070944978ef97ec9019cded329b0dd55Is an interacting ground state (pure state) v-representable density also non-interacting ground state v-representable by a Slater determinant? In the absence of degeneracy, yes!Gonis, A.Physics Letters A (2019), 383 (23), 2772-2776CODEN: PYLAAG; ISSN:0375-9601. (Elsevier B.V.)It is shown that in the absence of degeneracy a d., n(r), describing the probability of finding a particle in a small vol. at position, r, that is known to be pure-state v-representable in terms of the ground state of an interacting system of particles evolving under an external potential, v(r), is also pure state v-representable in terms of a single Slater determinant describing the ground state of a system of non-interacting particles under a potential, vs(r). This establishes the validity of the Kohn-Sham formalism of d. functional theory. An explicit form of vs(r) is derived. We also derive the exact form of the correlation functional and the corresponding potential, μc(r), that lead to the exact d. and energy of an interacting system's ground state. Finally, we demonstrate that practical implementations of the Kohn-Sham formalism can generate neither the exact d. nor energy of an interacting system's ground state, a feature that is particularly true in the case of the so-called exact exchange, in which the correlation functional and potential are set equal to zero.
- 73Tchenkoue, M.-L. M.; Penz, M.; Theophilou, I.; Ruggenthaler, M.; Rubio, A. Force balance approach for advanced approximations in density functional theories. J. Chem. Phys. 2019, 151, 154107, DOI: 10.1063/1.5123608Google Scholar73https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvF2gsb%252FP&md5=0f7767b34093b2334595a6934dafb220Force balance approach for advanced approximations in density functional theoriesTchenkoue, Mary-Leena M.; Penz, Markus; Theophilou, Iris; Ruggenthaler, Michael; Rubio, AngelJournal of Chemical Physics (2019), 151 (15), 154107/1-154107/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We propose a systematic and constructive way to det. the exchange-correlation potentials of d.-functional theories including vector potentials. The approach does not rely on energy or action functionals. Instead, it is based on equations of motion of current quantities (force balance equations) and is feasible both in the ground-state and the time-dependent settings. This avoids, besides differentiability and causality issues, the optimized-effective-potential procedure of orbital-dependent functionals. We provide straightforward exchange-type approxns. for different d. functional theories that for a homogeneous system and no external vector potential reduce to the exchange-only local-d. and Slater Xα approxns. (c) 2019 American Institute of Physics.
- 74Stefanucci, G.; van Leeuwen, R. Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction; Cambridge University Press, 2013.Google ScholarThere is no corresponding record for this reference.
- 75Ruggenthaler, M.; Tancogne-Dejean, N.; Laestadius, A.; Penz, M.; Csirik, M. A.; Rubio, A. Exchange energies with forces in density-functional theory. arXiv (Chemical Physics), March 31, 2022, ver. 1, 2203.16980 (accessed 2023-01-31).Google ScholarThere is no corresponding record for this reference.
- 76Harbola, M. K.; Slamet, M.; Sahni, V. Local exchange-correlation potential from the force field of the Fermi-Coulomb hole charge for non-symmetric systems. Phys. Lett. A 1991, 157, 60– 64, DOI: 10.1016/0375-9601(91)90409-2Google ScholarThere is no corresponding record for this reference.
- 77Slamet, M.; Sahni, V.; Harbola, M. K. Force field and potential due to the Fermi-Coulomb hole charge for nonspherical-density atoms. Phys. Rev. A 1994, 49, 809, DOI: 10.1103/PhysRevA.49.809Google Scholar77https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXitVSkurk%253D&md5=6992ef793987372299ac6aef4fdc76abForce field and potential due to the Fermi-Coulomb hole charge for nonspherical-density atomsSlamet, Marlina; Sahni, Viraht; Harbola, Manoj K.Physical Review A: Atomic, Molecular, and Optical Physics (1994), 49 (2), 809-17CODEN: PLRAAN; ISSN:0556-2791.In the work formalism for the detn. of electronic structure, the exchange-correlation energy and (local) potential of the electrons both arise via Coulomb's law from the same source, viz., the quantum-mech. Fermi-Coulomb hole charge. The potential is the work Wxc(r) done to move an electron in the field of its Fermi-Coulomb hole and the energy is the interaction energy between the electronic and hole charge densities. For nonsym. electronic d. systems for which the curl of the field may not vanish, a local effective exchange-correlation potential Wxceff(r) is detd. from the irrotational component of the field, the solenoidal component being neglected. In this paper the authors investigate this approxn. further to better understand its accuracy for nonspherical d. systems by application to a degenerate ground state of the carbon atom within the Pauli-correlated approxn. The results of the non-self-consistent calcns. indicate that the solenoidal component of the force field due to the Fermi hole is negligible and two orders of magnitude smaller than the irrotational component. Therefore, essentially all the effects of Pauli correlation are accounted for by the latter and the effective exchange potential Wxeff(r) is thereby an accurate representation of the local exchange potential in the atom. Further, the solenoidal component of the field vanishes at the nucleus, in the classically forbidden region and along certain axes of symmetry. Thus the work Wx(r) in the field of the Fermi hole for such nonspherical atoms is path independent over substantial regions of configuration space. Finally, the authors discuss the structure of the local many-body potential of nonspherical-d. atoms when both Pauli and Coulomb correlations are present.
- 78Gaiduk, A. P.; Staroverov, V. N. How to tell when a model Kohn–Sham potential is not a functional derivative. J. Chem. Phys. 2009, 131, 044107, DOI: 10.1063/1.3176515Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXptVOltL8%253D&md5=95fa0853b3bc3b542a9d4491510011c0How to tell when a model Kohn-Sham potential is not a functional derivativeGaiduk, Alex P.; Staroverov, Viktor N.Journal of Chemical Physics (2009), 131 (4), 044107/1-044107/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A model exchange-correlation potential constructed with Kohn-Sham orbitals should be a functional deriv. of some d. functional. Several necessary conditions for a functional deriv. are discussed including: (i) minimization of the total-energy expression by the ground-state soln. of the Kohn-Sham equations, (ii) path independence of the van Leeuwen-Baerends line integral, and (iii) net zero force and zero torque on the d. A no. of existing model potentials are checked for these properties and it is found that most of the potentials tested are not functional derivs. Phys. properties obtained from potentials that have no parent functionals are ambiguous and, therefore, should be interpreted with caution. (c) 2009 American Institute of Physics.
- 79Pulay, P. Convergence acceleration of iterative sequences. The case of SCF iteration. Chem. Phys. Lett. 1980, 73, 393– 398, DOI: 10.1016/0009-2614(80)80396-4Google Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3cXlt1SltLs%253D&md5=223e2e3ae6c788b5c2dab99c0feb0b70Convergence acceleration of iterative sequences. The case of SCF iterationPulay, PeterChemical Physics Letters (1980), 73 (2), 393-8CODEN: CHPLBC; ISSN:0009-2614.Based on a recent method of J. A. Pople et al. for the soln. of large systems of linear equations, a procedure is given for accelerating the convergence of slowly converging quasi-Newton-Raphson type algorithms. This procedure is particularly advantageous if the no. of parameters is so large that the calcn. and storage of the hessian is no longer practical. Application to the SCF problem is treated.
- 80Cancès, E.; Le Bris, C. Can we outperform the DIIS approach for electronic structure calculations?. Int. J. Quantum Chem. 2000, 79, 82– 90, DOI: 10.1002/1097-461X(2000)79:2<82::AID-QUA3>3.0.CO;2-IGoogle Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXksFyltb0%253D&md5=7ae38ddfc9819e04f71e6f3cf6082275Can we outperform the DIIS approach for electronic structure calculations?Cances, Eric; Le Bris, ClaudeInternational Journal of Quantum Chemistry (2000), 79 (2), 82-90CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)This article regroups various results on SCF algorithms for computing electronic structures of mol. systems. The first part deals with the convergence properties of the "conventional" Roothaan algorithm and of the level-shifting algorithm. In the second part, a new class of algorithms is introduced, for which convergence is guaranteed by math. arguments. Computational performance on various test problems is reported; advantages of this new approach are demonstrated.
- 81Cancès, E.; Kemlin, G.; Levitt, A. Convergence analysis of direct minimization and self-consistent iterations. SIAM Journal on Matrix Analysis and Applications 2021, 42, 243– 274, DOI: 10.1137/20M1332864Google ScholarThere is no corresponding record for this reference.
- 82Penz, M.; Laestadius, A. Convergence of the regularized Kohn–Sham iteration in Banach spaces. arXiv (Mathematical Physics), October 1, 2020, ver. 3, 2003.05389 (accessed 2023-01-31).Google ScholarThere is no corresponding record for this reference.
- 83Flamant, C.; Kolesov, G.; Manousakis, E.; Kaxiras, E. Imaginary-time time-dependent density functional theory and its application for robust convergence of electronic states. J. Chem. Theory Comput. 2019, 15, 6036– 6045, DOI: 10.1021/acs.jctc.9b00617Google Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvVKltr%252FN&md5=23436f2c1caae703b7069a40f70438dcImaginary-Time Time-Dependent Density Functional Theory and Its Application for Robust Convergence of Electronic StatesFlamant, Cedric; Kolesov, Grigory; Manousakis, Efstratios; Kaxiras, EfthimiosJournal of Chemical Theory and Computation (2019), 15 (11), 6036-6045CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Reliable and robust convergence to the electronic ground state within d. functional theory (DFT) Kohn-Sham (KS) calcns. remains a thorny issue in many systems of interest. In such cases, charge sloshing can delay or completely hinder the convergence. Here, we use an approach based on transforming the time-dependent DFT equations to imaginary time, followed by imaginary-time evolution, as a reliable alternative to the SCF procedure for detg. the KS ground state. We discuss the theor. and tech. aspects of this approach and show that the KS ground state should be expected to be the long-imaginary-time output of the evolution, independent of the exchange-correlation functional or the level of theory used to simulate the system. By maintaining self-consistency between the single-particle wavefunctions (orbitals) and the electronic d. throughout the detn. of the stationary state, our method avoids the typical difficulties encountered in SCF. To demonstrate dependability of our approach, we apply it to selected systems which struggle to converge with SCF schemes. In addn., through the van Leeuwen theorem, we affirm the phys. meaningfulness of imaginary time TDDFT, justifying its use in certain topics of statistical mechanics such as in computing imaginary time path integrals.
- 85Penz, M.; Csirik, M. A.; Laestadius, A. Density-potential inversion from Moreau–Yosida regularization. arXiv (Chemical Physics), December 27, 2022, ver. 2, 2212.12727 (accessed 2023-01-31).Google ScholarThere is no corresponding record for this reference.
- 86Schönhammer, K.; Gunnarsson, O.; Noack, R. M. Density-functional theory on a lattice: Comparison with exact numerical results for a model with strongly correlated electrons. Phys. Rev. B 1995, 52, 2504– 2510, DOI: 10.1103/PhysRevB.52.2504Google Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXnt1Ghu7o%253D&md5=a81804af5bd37653afcb12a59de94dbcDensity-functional theory on a lattice: comparison with exact numerical results for a model with strongly correlated electronsSchonhammer, K.; Gunnarsson, O.; Noack, R. M.Physical Review B: Condensed Matter (1995), 52 (4), 2504-10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)Most of the basic ideas of the d.-functional theory (DFT) are shown to be unrelated to the fact that the particle d. is used as the basic variable. After presenting the general formalism and a simple example unrelated to densities, the authors discuss various approaches to d.-functional theory on a lattice. One has proven useful for the understanding of various fundamental issues of the DFT. The exact Kohn-Sham band gap and the band gap in the local-d. approxn. (LDA) are compared to the exact result for a one-dimensional model. As the interacting homogeneous system can be solved exactly no further approxn. is needed to formulate the LDA.
- 87Higuchi, M.; Higuchi, K. Arbitrary choice of basic variables in density functional theory: Formalism. Phys. Rev. B 2004, 69, 035113, DOI: 10.1103/PhysRevB.69.035113Google Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXht1Ortbo%253D&md5=230b920487a80e5274cacc2ec6030d64Arbitrary choice of basic variables in density functional theory: FormalismHiguchi, Masahiko; Higuchi, KatsuhikoPhysical Review B: Condensed Matter and Materials Physics (2004), 69 (3), 035113/1-035113/9CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)The Hohenberg-Kohn theorem of the d. functional theory (DFT) is extended by modifying the Levy constrained-search formulation. The theorem allows us to choose arbitrary phys. quantities as basic variables which det. the ground-state properties of the system. Moreover, the theorem establishes a min. principle with respect to variations in chosen basic variables as well as with respect to variations in the d. By using this theorem, self-consistent single-particle equations are derived. N single-particle orbitals introduced reproduce not only the electron d. but also arbitrary phys. quantities which are chosen as basic variables. The validity of the theory is confirmed by examples where the spin d. or paramagnetic c.d. is chosen as one of basic variables. The resulting single-particle equations coincide with the Kohn-Sham equations of the spin-d. functional theory or current-d. functional theory, resp. By choosing basic variables appropriate to the system, the present theory can describe the ground-state properties more efficiently than the conventional DFT.
- 88Higuchi, K.; Higuchi, M. Arbitrary choice of basic variables in density functional theory. II. Illustrative applications. Phys. Rev. B 2004, 69, 165118, DOI: 10.1103/PhysRevB.69.165118Google Scholar88https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXktVKit7s%253D&md5=87b390d17291e24b9f1a9325548b243cArbitrary choice of basic variables in density functional theory. II. Illustrative applicationsHiguchi, Katsuhiko; Higuchi, MasahikoPhysical Review B: Condensed Matter and Materials Physics (2004), 69 (16), 165118/1-165118/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)Our recent extended constrained-search theory [M. Higuchi and K. Higuchi, Phys. Rev. B 69, 035113 (2004)] enables us to choose arbitrary quantities as the basic variables of d. functional theory. In this paper, we apply it to several cases. In the case in which the occupation matrix of localized orbitals is chosen as a basic variable, we can obtain the single-particle equation which is equiv. to that of the LDA+U method. The theory also leads to the Hartree-Fock-Kohn-Sham equation by letting the exchange energy be a basic variable. Furthermore, if the quantity assocd. with the d. of states near the Fermi level is chosen as a basic variable, the resulting single-particle equation includes the addnl. potential which could mainly modify the energy-band structures near the Fermi level.
- 89Xu, L.; Mao, J.; Gao, X.; Liu, Z. Extensibility of Hohenberg–Kohn Theorem to General Quantum Systems. Adv. Quantum Techn. 2022, 5, 2200041, DOI: 10.1002/qute.202200041Google ScholarThere is no corresponding record for this reference.
- 91Vignale, G.; Rasolt, M. Density-functional theory in strong magnetic fields. Phys. Rev. Lett. 1987, 59, 2360– 2363, DOI: 10.1103/PhysRevLett.59.2360Google Scholar91https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXislSqtA%253D%253D&md5=6c0439dbbbddf191b1da1f73d69ed733Density-functional theory in strong magnetic fieldsVignale, G.; Rasolt, MarkPhysical Review Letters (1987), 59 (20), 2360-3CODEN: PRLTAO; ISSN:0031-9007.The current-d.-functional theory is formulated for systems in arbitrarily strong magnetic fields. A set of self-consistent equations comparable to the Kohn-Sham equations for ordinary d.-functional theory is derived, and proved to be gage invariant and to satisfy the continuity equation. The exchange-correlation energy functional Exc[n,jp] [n(r) is the d. and jp(r) is the "paramagnetic" c.d.] depends on the current via the combination v(r) = V × [jp(r)/n(r)]. An explicit formula for Exc is derived, which is local in v(r).
- 92Kümmel, S.; Kronik, L. Orbital-dependent density functionals: Theory and applications. Rev. Mod. Phys. 2008, 80, 3, DOI: 10.1103/RevModPhys.80.3Google Scholar92https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXlslKms74%253D&md5=867b579a9dfef4dfa661a6af35d6d21dOrbital-dependent density functionals: Theory and applicationsKuemmel, Stephan; Kronik, LeeorReviews 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.
- 93Vignale, G.; Rasolt, M.; Geldart, D. Magnetic fields and density functional theory. Adv. Quantum Chem. 1990, 21, 235– 253, DOI: 10.1016/S0065-3276(08)60599-7Google ScholarThere is no corresponding record for this reference.
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Abstract
Figure 1
Figure 1. Example of a convex and lower-semicontinuous function with a discontinuity at ρ0 and some elements from the subdifferential displayed as linear continuous tangent functionals at ρ0, represented by dashed lines.
Figure 2
Figure 2. Logical implications between the different statements relating to a “unique density-potential mapping” and the HK theorem in standard DFT.
References
This article references 93 other publications.
- 1Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864– B871, DOI: 10.1103/PhysRev.136.B864There is no corresponding record for this reference.
- 2Lieb, E. H. Density Functionals for Coulomb-Systems. Int. J. Quantum Chem. 1983, 24, 243– 277, DOI: 10.1002/qua.5602403022https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXltFSqsL0%253D&md5=b7fbcf2dcd41abb459c278b3f7da5976Density functionals for Coulomb systemsLieb, Elliott H.International Journal of Quantum Chemistry (1983), 24 (3), 243-77CODEN: IJQCB2; ISSN:0020-7608.Some of the math. connections between N-particle wave functions and their single-particle densities ρ are discussed. The math. underpinnings of "universal d. functional" theory are given for the ground state energy. The Hohenberg-Kohn functional is not defined for all ρ. Several ways around this difficulty are given. Since the functional mentioned above is not computable, examples of explicit functionals that have the virtue of yielding rigorous bounds to the energy are reviewed.
- 3Lammert, P. E. Differentiability of Lieb functional in electronic density functional theory. Int. J. Quantum Chem. 2007, 107, 1943– 1953, DOI: 10.1002/qua.213423https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXmsFegtrc%253D&md5=0c6a93837694922655e3581026618f95Differentiability of Lieb functional in electronic density functional theoryLammert, Paul E.International Journal of Quantum Chemistry (2007), 107 (10), 1943-1953CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)A solid understanding of the Lieb functional FL is important because of its centrality in the foundations of electronic d. functional theory. A basic question is whether directional derivs. of FL at an ensemble-V-representable d. are given by (minus) the potential. A widely accepted purported proof that FL is Gateaux differentiable at EV-representable densities would say, "yes.". But that proof is fallacious, as shown here. FL is not Gateaux differentiable in the normal sense, nor is it continuous. By means of a constructive approach, however, we are able to show that the deriv. of FL at an EV-representable d. ρ0 in the direction of ρ1 is given by the potential if ρ0 and ρ1 are everywhere strictly greater than zero, and they and the ground state wave function have square integrable derivs. through second order.
- 4Kvaal, S.; Ekström, U.; Teale, A. M.; Helgaker, T. Differentiable but exact formulation of density-functional theory. J. Chem. Phys. 2014, 140, 18A518, DOI: 10.1063/1.4867005There is no corresponding record for this reference.
- 5Laestadius, A.; Penz, M.; Tellgren, E. I.; Ruggenthaler, M.; Kvaal, S.; Helgaker, T. Generalized Kohn–Sham iteration on Banach spaces. J. Chem. Phys. 2018, 149, 164103, DOI: 10.1063/1.50377905https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhvF2lsLjK&md5=98d584bc5b7535d865fd689a4fa7ef6cGeneralized Kohn-Sham iteration on Banach spacesLaestadius, Andre; Penz, Markus; Tellgren, Erik I.; Ruggenthaler, Michael; Kvaal, Simen; Helgaker, TrygveJournal of Chemical Physics (2018), 149 (16), 164103/1-164103/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A detailed account of the Kohn-Sham (KS) algorithm from quantum chem., formulated rigorously in the very general setting of convex anal. on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows us to rigorously introduce, in contrast to the common unregularized approach, a well-defined KS iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different d.-functional theories and possibly even to models outside of quantum mechanics. (c) 2018 American Institute of Physics.
- 6Laestadius, A.; Penz, M.; Tellgren, E. I.; Ruggenthaler, M.; Kvaal, S.; Helgaker, T. Kohn–Sham theory with paramagnetic currents: compatibility and functional differentiability. J. Chem. Theory Comput. 2019, 15, 4003– 4020, DOI: 10.1021/acs.jctc.9b001416https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXptlegsLk%253D&md5=50e91248c86fa01b2d737447aeff1bc1Kohn-Sham Theory with Paramagnetic Currents: Compatibility and Functional DifferentiabilityLaestadius, Andre; Tellgren, Erik I.; Penz, Markus; Ruggenthaler, Michael; Kvaal, Simen; Helgaker, TrygveJournal of Chemical Theory and Computation (2019), 15 (7), 4003-4020CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Recent work has established Moreau-Yosida regularization as a math. tool to achieve rigorous functional differentiability in d.-functional theory. In this article, we extend this tool to paramagnetic current-d.-functional theory, the most common d.-functional framework for magnetic field effects. The extension includes a well-defined Kohn-Sham iteration scheme with a partial convergence result. To this end, we rely on a formulation of Moreau-Yosida regularization for reflexive and strictly convex function spaces. The optimal Lp-characterization of the paramagnetic c.d. is derived from the N-representability condition. A crucial prerequisite for the convex formulation of paramagnetic current-d.-functional theory, termed compatibility between function spaces for the particle d. and the c.d., is pointed out and analyzed. Several results about compatible function spaces are given, including their recursive construction. The regularized, exact functionals are calcd. numerically for a Kohn-Sham iteration on a quantum ring, illustrating their performance for different regularization parameters.
- 7Capelle, K.; Vignale, G. Nonuniqueness and derivative discontinuities in density-functional theories for current-carrying and superconducting systems. Phys. Rev. B 2002, 65, 113106, DOI: 10.1103/PhysRevB.65.1131067https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XitVaqsbk%253D&md5=c8c650ba66797015f95f875b4f4eef6fNonuniqueness and derivative discontinuities in density-functional theories for current-carrying and superconducting systemsCapelle, K.; Vignale, G.Physical Review B: Condensed Matter and Materials Physics (2002), 65 (11), 113106/1-113106/4CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)Current-carrying and superconducting systems can be treated within d.-functional theory if suitable addnl. d. variables (the c.d. and the superconducting order parameter, resp.) are included in the d.-functional formalism. Here we show that the corresponding conjugate potentials (vector and pair potentials, resp.) are not uniquely detd. by the densities. The Hohenberg-Kohn theorem of these generalized d.-functional theories is thus weaker than the original one. We give explicit examples and explore some consequences.
- 8Percus, J. The role of model systems in the few-body reduction of the N-fermion problem. Int. J. Quantum Chem. 1978, 13, 89– 124, DOI: 10.1002/qua.5601301088https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE1cXhvFOqurw%253D&md5=577a618b8a3817e030987db05e3d84ceThe role of model systems in the few-body reduction of the N-fermion problemPercus, J. K.International Journal of Quantum Chemistry (1978), 13 (1), 89-124CODEN: IJQCB2; ISSN:0020-7608.The problem of upper and lower ground state energy bounds for many-fermion systems is considered from the viewpoint of reduced d. matrices. Model d. matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both math. and phys. models are constructed for setting up universally valid inequalities on d. matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.
- 9Levy, M. Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 6062– 6065, DOI: 10.1073/pnas.76.12.60629https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3cXmsFSrtQ%253D%253D&md5=141936a35c228be0c68b1263da05a33fUniversal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problemLevy, MelProceedings of the National Academy of Sciences of the United States of America (1979), 76 (12), 6062-5CODEN: PNASA6; ISSN:0027-8424.Universal variational functionals of densities, 1st-order d. matrixes, and natural spin-orbitals are explicitly displayed for variational calcns. of ground states of interacting electrons is atoms, mols., and solids. In all cases, the functionals search for constrained min. In particular, following Percus Q[ρ] = min 〈Ψρ|T + Vee|Ψρ〉 is identified as the universal functional of Hohenberg and Kohn for the sum of the kinetic and electron-electron repulsion energies of an N-representable trial electron d. ρ. Q[ρ] Searches all antisym. wavefunctions Ψρ which yield the fixed ρ. Q[ρ] Then delivers that expectation value which is a min. Similarly, W[γ] = min 〈Ψγ]Vee|Ψγ〉 is a universal functional for the electron-electron repulsion energy of an N-representable trial 1st-order d. matrix γ, where the actual external potential may be nonlocal as well as local. These universal functions do not require that a trial function for a variational calcn. be assocd. with a ground state of some external potential. Thus, the v-representability problem, which is esp. severe for trial 1st-order d. matrixes, was solved. Universal variational functionals in Hartree-Fock and other restricted wavefunction theories are also presented. Finally, natural spin-orbital functional theory is compared with traditional orbital formulations in d. functional theory.
- 10Penz, M.; Laestadius, A.; Tellgren, E. I.; Ruggenthaler, M. Guaranteed Convergence of a Regularized Kohn–Sham Iteration in Finite Dimensions. Phys. Rev. Lett. 2019, 123, 037401, DOI: 10.1103/PhysRevLett.123.03740110https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvFCgs7bJ&md5=2e9c58ed26595b1532eedbde84966390Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite DimensionsPenz, Markus; Laestadius, Andre; Tellgren, Erik I.; Ruggenthaler, MichaelPhysical Review Letters (2019), 123 (3), 037401CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)A review. The exact Kohn-Sham iteration of generalized d.-functional theory in finite dimensions with a Moreau-Yosida regularized universal Lieb functional and an adaptive damping step is shown to converge to the correct ground-state d.
- 11Penz, M.; Laestadius, A.; Tellgren, E. I.; Ruggenthaler, M.; Lammert, P. E. Erratum: Guaranteed Convergence of a Regularized Kohn–Sham Iteration in Finite Dimensions. Phys. Rev. Lett. 2020, 125, 249902, DOI: 10.1103/PhysRevLett.125.24990211https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BB3svmvVOltg%253D%253D&md5=e4445dbdddd6e1116f17a736a6737af6Erratum: Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions [Phys. Rev. Lett. 123, 037401 (2019)]Penz Markus; Laestadius Andre; Tellgren Erik I; Ruggenthaler Michael; Lammert Paul EPhysical review letters (2020), 125 (24), 249902 ISSN:.This corrects the article DOI: 10.1103/PhysRevLett.123.037401.
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- 17Eschrig, H. The Fundamentals of Density Functional Theory, 2nd ed.; Springer, 2003.There is no corresponding record for this reference.
- 18Parr, R.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press, 1989.There is no corresponding record for this reference.
- 19Teale, A. M.; Helgaker, T.; Savin, A.; Adamo, C.; Aradi, B.; Arbuznikov, A. V.; Ayers, P. W.; Baerends, E. J.; Barone, V.; Calaminici, P. DFT exchange: Sharing perspectives on the workhorse of quantum chemistry and materials science. Phys. Chem. Chem. Phys. 2022, 24, 28700– 28781, DOI: 10.1039/D2CP02827A19https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38Xis12isb3O&md5=ad832216838456d8d93b2c286cdc90d1DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials scienceTeale, Andrew M.; Helgaker, Trygve; Savin, Andreas; Adamo, Carlo; Aradi, Balint; Arbuznikov, Alexei V.; Ayers, Paul W.; Baerends, Evert Jan; Barone, Vincenzo; Calaminici, Patrizia; Cances, Eric; Carter, Emily A.; Chattaraj, Pratim Kumar; Chermette, Henry; Ciofini, Ilaria; Crawford, T. Daniel; De Proft, Frank; Dobson, John F.; Draxl, Claudia; Frauenheim, Thomas; Fromager, Emmanuel; Fuentealba, Patricio; Gagliardi, Laura; Galli, Giulia; Gao, Jiali; Geerlings, Paul; Gidopoulos, Nikitas; Gill, Peter M. W.; Gori-Giorgi, Paola; Gorling, Andreas; Gould, Tim; Grimme, Stefan; Gritsenko, Oleg; Jensen, Hans Joergen Aagaard; Johnson, Erin R.; Jones, Robert O.; Kaupp, Martin; Koster, Andreas M.; Kronik, Leeor; Krylov, Anna I.; Kvaal, Simen; Laestadius, Andre; Levy, Mel; Lewin, Mathieu; Liu, Shubin; Loos, Pierre-Francois; Maitra, Neepa T.; Neese, Frank; Perdew, John P.; Pernal, Katarzyna; Pernot, Pascal; Piecuch, Piotr; Rebolini, Elisa; Reining, Lucia; Romaniello, Pina; Ruzsinszky, Adrienn; Salahub, Dennis R.; Scheffler, Matthias; Schwerdtfeger, Peter; Staroverov, Viktor N.; Sun, Jianwei; Tellgren, Erik; Tozer, David J.; Trickey, Samuel B.; Ullrich, Carsten A.; Vela, Alberto; Vignale, Giovanni; Wesolowski, Tomasz A.; Xu, Xin; Yang, WeitaoPhysical Chemistry Chemical Physics (2022), 24 (47), 28700-28781CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)In this paper, the history, present status, and future of d.-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including mol. scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliog. of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.
- 20Gilbert, T. L. Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B 1975, 12, 2111, DOI: 10.1103/PhysRevB.12.2111There is no corresponding record for this reference.
- 21Harriman, J. E. Orthonormal orbitals for the representation of an arbitrary density. Phys. Rev. A 1981, 24, 680, DOI: 10.1103/PhysRevA.24.68021https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3MXkvVKlt70%253D&md5=43a901825aac4f07378eeee48691e0f2Orthonormal orbitals for the representation of an arbitrary densityHarriman, John E.Physical Review A: Atomic, Molecular, and Optical Physics (1981), 24 (2), 680-2CODEN: PLRAAN; ISSN:0556-2791.An alternative to the Gilbert construction is presented. For any nonneg., normalized d., an arbitrary no. of orthonormal orbitals can be constructed with squares which sum, with minimal restrictions on the occupation nos., to the given d. In 3 dimensions, substantial freedom remains in the choice of orbitals within the basic scheme. The kinetic-energy d. obtained includes the Weizsaecker term and another term which, in simple cases, is proportional to the cube of the d.
- 23Englisch, H.; Englisch, R. Hohenberg-Kohn theorem and non-V-representable densities. Physica A Stat. Mech. Appl. 1983, 121, 253– 268, DOI: 10.1016/0378-4371(83)90254-623https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3sXmtVSltb8%253D&md5=eedaef6df576b8643a696e4fa4f0da7bHohenberg-Kohn theorem and non-V-representable densitiesEnglisch, H.; Englisch, R.Physica A: Statistical Mechanics and Its Applications (Amsterdam, Netherlands) (1983), 121A (1-2), 253-68CODEN: PHYADX; ISSN:0378-4371.In the d.-functional formalism of P. Hohenberg and W. Kohn (1964), the variation is only allowed over the 1-particle densities that are pure-state-V-representable [V = potential function] (PS-V-representable). M. Levy (1982) and E. H. Lieb (1982) proved that not every ensemble-V-representable (E-V-representable) d. is PS-V-representable. Since the Hohenberg-Kohn formalism can be extended to a variation over E-V-representable densities for degenerate ground states, the result of M. L. and E. H. L. is not a counterexample to the universality of the Hohenberg-Kohn theorem. Not every N-representable d. [N = no. of electrons] is E-V-representable, as shown by examples of non-E-V-representable densities. The functional of M. L. (1979) is of value in calcg. ground-state energies, since this functional only requires the N-representability of the densities. Therefore, 2 approaches for the calcn. of excited-state energies were transferred into the framework of the formalism of M.L.
- 24Levy, M. Electron densities in search of Hamiltonians. Phys. Rev. A 1982, 26, 1200, DOI: 10.1103/PhysRevA.26.120024https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38Xltleku7w%253D&md5=033f6460f0d912815e8c7000a0b896d5Electron densities in search of HamiltoniansLevy, MelPhysical Review A: Atomic, Molecular, and Optical Physics (1982), 26 (3), 1200-8CODEN: PLRAAN; ISSN:0556-2791.By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state d., fundamental connections between densities and Hamiltonians are revealed. A tight rigorous bound connects a d. to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. With the exception of the occurrence of certain linear dependencies, a d. will not generally be assocd. with any ground-state wave function (is not wave function v representable) if arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to d.-functional theory is emphasized for non-v-representable as well as for v-representable ds. The possible appearance of noninteger occupation nos. is discussed in connection with the existence of non-v representability for some Kohn-Sham noninteracting systems.
- 25Penz, M.; van Leeuwen, R. Density-functional theory on graphs. J. Chem. Phys. 2021, 155, 244111, DOI: 10.1063/5.007424925https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BB38XjtFyr&md5=a6e0d7d71dda9db2d189f1458cc669f3Density-functional theory on graphsPenz, Markus; van Leeuwen, RobertJournal of Chemical Physics (2021), 155 (24), 244111CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The principles of d.-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void, in general, while many insights into the topol. structure of the d.-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional. (c) 2021 American Institute of Physics.
- 26Garrigue, L. Some properties of the potential-to-ground state map in quantum mechanics. Comm. Math. Phys. 2021, 386, 1803– 1844, DOI: 10.1007/s00220-021-04140-9There is no corresponding record for this reference.
- 28Barbu, V.; Precupanu, T. Convexity and Optimization in Banach Spaces, 4th ed.; Springer, 2012.There is no corresponding record for this reference.
- 30Garrigue, L. Building Kohn–Sham Potentials for Ground and Excited States. Arch. Ration. Mech. Anal. 2022, 245, 949– 1003, DOI: 10.1007/s00205-022-01804-1There is no corresponding record for this reference.
- 31Reed, M.; Simon, B. II: Fourier Analysis, Self-Adjointness; Elsevier, 1975; Vol. 2.There is no corresponding record for this reference.
- 32Lieb, E. H.; Loss, M. Analysis; American Mathematical Society: Providence, RI, 2001.There is no corresponding record for this reference.
- 34Pino, R.; Bokanowski, O.; Ludeña, E. V.; Boada, R. L. A re-statement of the Hohenberg–Kohn theorem and its extension to finite subspaces. Theor. Chem. Acc. 2007, 118, 557– 561, DOI: 10.1007/s00214-007-0367-634https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXpvVKnt78%253D&md5=77be97a81821b41bb628fcfe2ffcbe22A re-statement of the Hohenberg-Kohn theorem and its extension to finite subspacesPino, Ramiro; Bokanowski, Olivier; Ludena, Eduardo V.; Boada, Roberto LopezTheoretical Chemistry Accounts (2007), 118 (3), 557-561CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH)Bearing in mind the insight into the Hohenberg-Kohn theorem for Coulomb systems provided recently by Kryachko (Int J Quantum Chem 103:818, 2005), we present a re-statement of this theorem through an elaboration on Lieb's proof as well as an extension of this theorem to finite subspaces.
- 35Garrigue, L. Unique Continuation for Many-Body Schrödinger Operators and the Hohenberg-Kohn Theorem. Math. Phys. Anal. Geom. 2018, 21, 27, DOI: 10.1007/s11040-018-9287-zThere is no corresponding record for this reference.
- 36Tellgren, E. I.; Laestadius, A.; Helgaker, T.; Kvaal, S.; Teale, A. M. Uniform magnetic fields in density-functional theory. J. Chem. Phys. 2018, 148, 024101, DOI: 10.1063/1.500730036https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXltlCisA%253D%253D&md5=8f4caf48483a5e3e1a8801a1ddd9738eUniform magnetic fields in density-functional theoryTellgren, Erik I.; Laestadius, Andre; Helgaker, Trygve; Kvaal, Simen; Teale, Andrew M.Journal of Chemical Physics (2018), 148 (2), 024101/1-024101/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We construct a d.-functional formalism adapted to uniform external magnetic fields that is intermediate between conventional d. functional theory and Current-D. Functional Theory (CDFT). In the intermediate theory, which we term linear vector potential-DFT (LDFT), the basic variables are the d., the canonical momentum, and the paramagnetic contribution to the magnetic moment. Both a constrained-search formulation and a convex formulation in terms of Legendre-Fenchel transformations are constructed. Many theor. issues in CDFT find simplified analogs in LDFT. We prove results concerning N-representability, Hohenberg-Kohn-like mappings, existence of minimizers in the constrained-search expression, and a restricted analog to gauge invariance. The issue of additivity of the energy over non-interacting subsystems, which is qual. different in LDFT and CDFT, is also discussed. (c) 2018 American Institute of Physics.
- 37Kohn, W.; Savin, A.; Ullrich, C. A. Hohenberg–Kohn theory including spin magnetism and magnetic fields. Int. J. Quantum Chem. 2004, 100, 20– 21, DOI: 10.1002/qua.2016337https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXntlCrsLY%253D&md5=8a84555294a27a7b1cd1660ee0fcf8dcHohenberg-Kohn theory including spin magnetism and magnetic fieldsKohn, Walter; Savin, Andreas; Ullrich, Carsten A.International Journal of Quantum Chemistry (2004), 100 (1), 20-21CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)Beginning with work by U. von Barth and L. Hedin in 1972 and continuing with recent papers in 2001 by K. Capelle and G. Vignale and by H. Eschrig and W. E. Pickett, questions have been raised about the meaning and validity of Hohenberg-Kohn theory when spin magnetism and/or magnetic fields are present. This article offers clarifications of some of these questions. In particular, it also concludes that these questions do not affect spin-d. functional theory for nondegenerate ground states as currently practiced.
- 38Kohn, W. Density Functional Theory: Fundamentals and Applications. In Highlights of condensed-matter theory; Bassani, F., Fumi, F., Tosi, M. P., Eds.; Proceedings of the international school of physics “Enrico Fermi”, Course LXXXIX at Varenna on Lake Como, 28 June-16 July 1983; North-Holland: Amsterdam, 1985; pp 1– 15.There is no corresponding record for this reference.
- 39Regbaoui, R. Unique Continuation from Sets of Positive Measure. In Carleman Estimates and Applications to Uniqueness and Control Theory; Colombini, F., Zuily, C., Eds.; Progress in Nonlinear Differential Equations and Their Applications; Birkhäuser: Basel, 2001; Vol. 46; pp 179– 190.There is no corresponding record for this reference.
- 40de Figueiredo, D. G.; Gossez, J.-P. Strict monotonicity of eigenvalues and unique continuation. Comm. Par. Diff. Eq. 1992, 17, 339– 346, DOI: 10.1080/03605309208820844There is no corresponding record for this reference.
- 41Ladyzenskaya, O. A.; Ural’tzeva, N. N. Linear and Quasilinear Elliptic Equations; Academic Press: New York, 1968.There is no corresponding record for this reference.
- 43Lammert, P. E. In search of the Hohenberg-Kohn theorem. J. Math. Phys. 2018, 59, 042110, DOI: 10.1063/1.503421543https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXptVCnsrk%253D&md5=160d1fff304ab2f36fd0a5b520b47ddcIn search of the Hohenberg-Kohn theoremLammert, Paul E.Journal of Mathematical Physics (Melville, NY, United States) (2018), 59 (4), 042110/1-042110/19CODEN: JMAPAQ; ISSN:0022-2488. (American Institute of Physics)The Hohenberg-Kohn theorem, a cornerstone of electronic d. functional theory, concerns uniqueness of external potentials yielding given ground densities of an N-body system. The problem is rigorously explored in a universe of three-dimensional Kato-class potentials, with emphasis on trade-offs between conditions on the d. and conditions on the potential sufficient to ensure uniqueness. Sufficient conditions range from none on potentials coupled with everywhere strict positivity of the d. to none on the d. coupled with something a little weaker than local 3N/2-power integrability of the potential on a connected full-measure set. A second theme is localizability, i.e., the possibility of uniqueness over subsets of R3 under less stringent conditions. (c) 2018 American Institute of Physics.
- 44Jerison, D.; Kenig, C. E. Unique continuation and absence of positive eigenvalues for Schrodinger operators. Ann. Math. 1985, 121, 463– 488, DOI: 10.2307/1971205There is no corresponding record for this reference.
- 45Garrigue, L. Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian. Doc. Math. 2020, 25, 869– 898, DOI: 10.4171/dm/765There is no corresponding record for this reference.
- 46Garrigue, L. Hohenberg–Kohn Theorems for Interactions, Spin and Temperature. J. Stat. Phys. 2019, 177, 415– 437, DOI: 10.1007/s10955-019-02365-6There is no corresponding record for this reference.
- 47Kvaal, S.; Helgaker, T. Ground-state densities from the Rayleigh–Ritz variation principle and from density-functional theory. J. Chem. Phys. 2015, 143, 184106, DOI: 10.1063/1.493479747https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvVWmtrfE&md5=c155dc8a001b3819e1ebe1aecaa1b3d9Ground-state densities from the Rayleigh-Ritz variation principle and from density-functional theoryKvaal, Simen; Helgaker, TrygveJournal of Chemical Physics (2015), 143 (18), 184106/1-184106/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The relationship between the densities of ground-state wave functions (i.e., the minimizers of the Rayleigh-Ritz variation principle) and the ground-state densities in d.-functional theory (i.e., the minimizers of the Hohenberg-Kohn variation principle) is studied within the framework of convex conjugation, in a generic setting covering mol. systems, solid-state systems, and more. Having introduced admissible d. functionals as functionals that produce the exact ground-state energy for a given external potential by minimizing over densities in the Hohenberg-Kohn variation principle, necessary and sufficient conditions on such functionals are established to ensure that the Rayleigh-Ritz ground-state densities and the Hohenberg-Kohn ground-state densities are identical. We apply the results to mol. systems in the Born-Oppenheimer approxn. For any given potential v ∈ L3/2(R3) + L∞(R3), we establish a one-to-one correspondence between the mixed ground-state densities of the Rayleigh-Ritz variation principle and the mixed ground-state densities of the Hohenberg-Kohn variation principle when the Lieb d.-matrix constrained-search universal d. functional is taken as the admissible functional. A similar one-to-one correspondence is established between the pure ground-state densities of the Rayleigh-Ritz variation principle and the pure ground-state densities obtained using the Hohenberg-Kohn variation principle with the Levy-Lieb pure-state constrained-search functional. In other words, all phys. ground-state densities (pure or mixed) are recovered with these functionals and no false densities (i.e., minimizing densities that are not phys.) exist. The importance of topol. (i.e., choice of Banach space of densities and potentials) is emphasized and illustrated. The relevance of these results for current-d.-functional theory is examd. (c) 2015 American Institute of Physics.
- 48Lewin, M.; Lieb, E. H.; Seiringer, R. Universal Functionals in Density Functional Theory. arXiv (Mathematical Physics), September 13, 2022, ver. 4, 1912.10424 (accessed 2023-01-31).There is no corresponding record for this reference.
- 54Tsuneda, T. Density functional theory in quantum chemistry; Springer, 2014.There is no corresponding record for this reference.
- 56Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133, DOI: 10.1103/PhysRev.140.A1133There is no corresponding record for this reference.
- 67van Leeuwen, R. Density functional approach to the many-body problem: key concepts and exact functionals. Adv. Quantum Chem. 2003, 43, 25, DOI: 10.1016/S0065-3276(03)43002-5There is no corresponding record for this reference.
- 68Lammert, P. E. Well-behaved coarse-grained model of density-functional theory. Phys. Rev. A 2010, 82, 012109, DOI: 10.1103/PhysRevA.82.01210968https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXpsF2hs78%253D&md5=93121d5e4d4e67a301a625c4eded0464Well-behaved coarse-grained model of density-functional theoryLammert, Paul E.Physical Review A: Atomic, Molecular, and Optical Physics (2010), 82 (1, Pt. A), 012109/1-012109/14CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)A coarse-grained version of quantum d.-functional theory featuring a limited spatial resoln. is shown to model formal d.-functional theory (DFT). This means that all densities are ensemble-V-representable (i.e., by mixed states), the intrinsic energy functional F is a continuous function of the d., and the representing external potential is the functional deriv. of the intrinsic energy, in the sense of directional derivs. within the domain of F. The representing potential v[ρ] also has a quasicontinuity property, specifically, v[ρ]ρ is continuous as a function of ρ. Convergence of the intrinsic energy, coarse-grained densities, and representing potentials in the limit of coarse-graining scale going to zero are studied vis-a-vis Lieb's L1 ∩ L3 theory. The intrinsic energy converges monotonically to its fine-grained (continuum) value, and coarse-grainings of a d. ρ converge strongly to ρ. If a sequence of coarse-grained densities converges strongly to ρ and their representing potentials converge weak-*, the limit is the representing potential for ρ. Conversely, L3/2 + L∞ representability implies the existence of such a coarse-grained sequence.
- 69Kümmel, S.; Perdew, J. P. Optimized effective potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potential. Phys. Rev. B 2003, 68, 035103, DOI: 10.1103/PhysRevB.68.03510369https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXnsVGru7w%253D&md5=3620b9847bdfc9aa6301c90903b8f369Optimized effective potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potentialKuemmel, Stephan; Perdew, John P.Physical Review B: Condensed Matter and Materials Physics (2003), 68 (3), 035103/1-035103/15CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)The optimized effective potential (OEP) is the exact Kohn-Sham potential for explicitly orbital-dependent energy functionals, e.g., the exact exchange energy. We give a proof for the OEP equation which does not depend on the chain rule for functional derivs. and directly yields the equation in its simplest form: a certain first-order d. shift must vanish. This condition explains why the highest-occupied orbital energies of Hartree-Fock and exact exchange OEP are so close. More importantly, we show that the exact OEP can be constructed iteratively from the first-order shifts of the Kohn-Sham orbitals and that these can be calcd. easily. The exact exchange potential υx(r) for spherical atoms and three-dimensional sodium clusters is calcd. Its long-range asymptotic behavior is investigated, including the approach of νx(r) to a nonvanishing const. in particular spatial directions. We calc. total and orbital energies and static elec. dipole polarizabilities for the sodium clusters employing the exact exchange functional. Exact OEP results are compared to the Krieger-Li-Iafrate and local d. approxns.
- 70Gonis, A. Functionals and Functional Derivatives of Wave Functions and Densities. World J. Condens. Matter Phys. 2014, 4, 179, DOI: 10.4236/wjcmp.2014.4302270https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXhs1ylu77M&md5=bd966da7ce2767eff3cc7a270613ff25Functionals and functional derivatives of wave functions and densitiesGonis, A.World Journal of Condensed Matter Physics (2014), 4 (3), 179-199, 22 pp.CODEN: WJCMAD; ISSN:2160-6927. (Scientific Research Publishing, Inc.)It is shown that the process of conventional functional differentiation does not apply to functionals whose domain (and possibly range) is subject to the condition of integral normalization, as is the case with respect to a domain defined by wave functions or densities, in which there exists no neighborhood about a given element in the domain defined by arbitrary variations that also lie in the domain. This is remedied through the generalization of the domain of a functional to include distributions in the form of εδ(r-r'), where δ(r-r') is the Dirac delta function and ε is a real no. This allows the detn. of the rate of change of a functional with respect to changes of the independent variable detd. at each point of the domain, with no ref. needed to the values of the functional at different functions in its domain. One feature of the formalism is the detn. of rates of change of general expectation values (that may not necessarily be functionals of the d.) with respect to the wave functions or the densities detd. by the wave functions forming the expectation value. It is also shown that ignoring the conditions of conventional functional differentiation can lead to false proofs, illustrated through a flaw in the proof that all densities defined on a lattice are ν-representable. In a companion paper, the math. integrity of a no. of long-standing concepts in d. functional theory are studied in terms of the formalism developed here.
- 71Gonis, A.; Zhang, X.-G.; Däne, M.; Stocks, G. M.; Nicholson, D. M. Reformulation of density functional theory for N-representable densities and the resolution of the v-representability problem. J. Phys. Chem. Solids 2016, 89, 23– 31, DOI: 10.1016/j.jpcs.2015.10.00671https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhslGktrbM&md5=1508ec66f702b96837da2ed58fa3cd19Reformulation of density functional theory for N-representable densities and the resolution of the v-representability problemGonis, A.; Zhang, X.-G.; Dane, M.; Stocks, G. M.; Nicholson, D. M.Journal of Physics and Chemistry of Solids (2016), 89 (), 23-31CODEN: JPCSAW; ISSN:0022-3697. (Elsevier Ltd.)A review. D. functional theory for the case of general, N-representable densities is reformulated in terms of d. functional derivs. of expectation values of operators evaluated with wave functions leading to a d., making no ref. to the concept of potential. The developments provide proof of existence of a math. procedure that dets. whether a d. is v-representable and in the case of an affirmative answer dets. the potential (within an additive const.) as a deriv. with respect to the d. of a constrained search functional. It also establishes the existence of an energy functional of the d. that, for v-representable densities, assumes its min. value at the d. describing the ground state of an interacting many-particle system. The theorems of Hohenberg and Kohn emerge as special cases of the formalism. Numerical results for one-dimensional non-interacting systems illustrate the formalism. Some direct formal and practical implications of the present reformulation of DFT are also discussed.
- 72Gonis, A. Is an interacting ground state (pure state) v-representable density also non-interacting ground state v-representable by a Slater determinant? In the absence of degeneracy, yes. Phys. Lett. A 2019, 383, 2772– 2776, DOI: 10.1016/j.physleta.2019.03.00772https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXkvVektLk%253D&md5=070944978ef97ec9019cded329b0dd55Is an interacting ground state (pure state) v-representable density also non-interacting ground state v-representable by a Slater determinant? In the absence of degeneracy, yes!Gonis, A.Physics Letters A (2019), 383 (23), 2772-2776CODEN: PYLAAG; ISSN:0375-9601. (Elsevier B.V.)It is shown that in the absence of degeneracy a d., n(r), describing the probability of finding a particle in a small vol. at position, r, that is known to be pure-state v-representable in terms of the ground state of an interacting system of particles evolving under an external potential, v(r), is also pure state v-representable in terms of a single Slater determinant describing the ground state of a system of non-interacting particles under a potential, vs(r). This establishes the validity of the Kohn-Sham formalism of d. functional theory. An explicit form of vs(r) is derived. We also derive the exact form of the correlation functional and the corresponding potential, μc(r), that lead to the exact d. and energy of an interacting system's ground state. Finally, we demonstrate that practical implementations of the Kohn-Sham formalism can generate neither the exact d. nor energy of an interacting system's ground state, a feature that is particularly true in the case of the so-called exact exchange, in which the correlation functional and potential are set equal to zero.
- 73Tchenkoue, M.-L. M.; Penz, M.; Theophilou, I.; Ruggenthaler, M.; Rubio, A. Force balance approach for advanced approximations in density functional theories. J. Chem. Phys. 2019, 151, 154107, DOI: 10.1063/1.512360873https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvF2gsb%252FP&md5=0f7767b34093b2334595a6934dafb220Force balance approach for advanced approximations in density functional theoriesTchenkoue, Mary-Leena M.; Penz, Markus; Theophilou, Iris; Ruggenthaler, Michael; Rubio, AngelJournal of Chemical Physics (2019), 151 (15), 154107/1-154107/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We propose a systematic and constructive way to det. the exchange-correlation potentials of d.-functional theories including vector potentials. The approach does not rely on energy or action functionals. Instead, it is based on equations of motion of current quantities (force balance equations) and is feasible both in the ground-state and the time-dependent settings. This avoids, besides differentiability and causality issues, the optimized-effective-potential procedure of orbital-dependent functionals. We provide straightforward exchange-type approxns. for different d. functional theories that for a homogeneous system and no external vector potential reduce to the exchange-only local-d. and Slater Xα approxns. (c) 2019 American Institute of Physics.
- 74Stefanucci, 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.
- 75Ruggenthaler, M.; Tancogne-Dejean, N.; Laestadius, A.; Penz, M.; Csirik, M. A.; Rubio, A. Exchange energies with forces in density-functional theory. arXiv (Chemical Physics), March 31, 2022, ver. 1, 2203.16980 (accessed 2023-01-31).There is no corresponding record for this reference.
- 76Harbola, M. K.; Slamet, M.; Sahni, V. Local exchange-correlation potential from the force field of the Fermi-Coulomb hole charge for non-symmetric systems. Phys. Lett. A 1991, 157, 60– 64, DOI: 10.1016/0375-9601(91)90409-2There is no corresponding record for this reference.
- 77Slamet, M.; Sahni, V.; Harbola, M. K. Force field and potential due to the Fermi-Coulomb hole charge for nonspherical-density atoms. Phys. Rev. A 1994, 49, 809, DOI: 10.1103/PhysRevA.49.80977https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXitVSkurk%253D&md5=6992ef793987372299ac6aef4fdc76abForce field and potential due to the Fermi-Coulomb hole charge for nonspherical-density atomsSlamet, Marlina; Sahni, Viraht; Harbola, Manoj K.Physical Review A: Atomic, Molecular, and Optical Physics (1994), 49 (2), 809-17CODEN: PLRAAN; ISSN:0556-2791.In the work formalism for the detn. of electronic structure, the exchange-correlation energy and (local) potential of the electrons both arise via Coulomb's law from the same source, viz., the quantum-mech. Fermi-Coulomb hole charge. The potential is the work Wxc(r) done to move an electron in the field of its Fermi-Coulomb hole and the energy is the interaction energy between the electronic and hole charge densities. For nonsym. electronic d. systems for which the curl of the field may not vanish, a local effective exchange-correlation potential Wxceff(r) is detd. from the irrotational component of the field, the solenoidal component being neglected. In this paper the authors investigate this approxn. further to better understand its accuracy for nonspherical d. systems by application to a degenerate ground state of the carbon atom within the Pauli-correlated approxn. The results of the non-self-consistent calcns. indicate that the solenoidal component of the force field due to the Fermi hole is negligible and two orders of magnitude smaller than the irrotational component. Therefore, essentially all the effects of Pauli correlation are accounted for by the latter and the effective exchange potential Wxeff(r) is thereby an accurate representation of the local exchange potential in the atom. Further, the solenoidal component of the field vanishes at the nucleus, in the classically forbidden region and along certain axes of symmetry. Thus the work Wx(r) in the field of the Fermi hole for such nonspherical atoms is path independent over substantial regions of configuration space. Finally, the authors discuss the structure of the local many-body potential of nonspherical-d. atoms when both Pauli and Coulomb correlations are present.
- 78Gaiduk, A. P.; Staroverov, V. N. How to tell when a model Kohn–Sham potential is not a functional derivative. J. Chem. Phys. 2009, 131, 044107, DOI: 10.1063/1.317651578https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXptVOltL8%253D&md5=95fa0853b3bc3b542a9d4491510011c0How to tell when a model Kohn-Sham potential is not a functional derivativeGaiduk, Alex P.; Staroverov, Viktor N.Journal of Chemical Physics (2009), 131 (4), 044107/1-044107/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A model exchange-correlation potential constructed with Kohn-Sham orbitals should be a functional deriv. of some d. functional. Several necessary conditions for a functional deriv. are discussed including: (i) minimization of the total-energy expression by the ground-state soln. of the Kohn-Sham equations, (ii) path independence of the van Leeuwen-Baerends line integral, and (iii) net zero force and zero torque on the d. A no. of existing model potentials are checked for these properties and it is found that most of the potentials tested are not functional derivs. Phys. properties obtained from potentials that have no parent functionals are ambiguous and, therefore, should be interpreted with caution. (c) 2009 American Institute of Physics.
- 79Pulay, P. Convergence acceleration of iterative sequences. The case of SCF iteration. Chem. Phys. Lett. 1980, 73, 393– 398, DOI: 10.1016/0009-2614(80)80396-479https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3cXlt1SltLs%253D&md5=223e2e3ae6c788b5c2dab99c0feb0b70Convergence acceleration of iterative sequences. The case of SCF iterationPulay, PeterChemical Physics Letters (1980), 73 (2), 393-8CODEN: CHPLBC; ISSN:0009-2614.Based on a recent method of J. A. Pople et al. for the soln. of large systems of linear equations, a procedure is given for accelerating the convergence of slowly converging quasi-Newton-Raphson type algorithms. This procedure is particularly advantageous if the no. of parameters is so large that the calcn. and storage of the hessian is no longer practical. Application to the SCF problem is treated.
- 80Cancès, E.; Le Bris, C. Can we outperform the DIIS approach for electronic structure calculations?. Int. J. Quantum Chem. 2000, 79, 82– 90, DOI: 10.1002/1097-461X(2000)79:2<82::AID-QUA3>3.0.CO;2-I80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXksFyltb0%253D&md5=7ae38ddfc9819e04f71e6f3cf6082275Can we outperform the DIIS approach for electronic structure calculations?Cances, Eric; Le Bris, ClaudeInternational Journal of Quantum Chemistry (2000), 79 (2), 82-90CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)This article regroups various results on SCF algorithms for computing electronic structures of mol. systems. The first part deals with the convergence properties of the "conventional" Roothaan algorithm and of the level-shifting algorithm. In the second part, a new class of algorithms is introduced, for which convergence is guaranteed by math. arguments. Computational performance on various test problems is reported; advantages of this new approach are demonstrated.
- 81Cancès, E.; Kemlin, G.; Levitt, A. Convergence analysis of direct minimization and self-consistent iterations. SIAM Journal on Matrix Analysis and Applications 2021, 42, 243– 274, DOI: 10.1137/20M1332864There is no corresponding record for this reference.
- 82Penz, M.; Laestadius, A. Convergence of the regularized Kohn–Sham iteration in Banach spaces. arXiv (Mathematical Physics), October 1, 2020, ver. 3, 2003.05389 (accessed 2023-01-31).There is no corresponding record for this reference.
- 83Flamant, C.; Kolesov, G.; Manousakis, E.; Kaxiras, E. Imaginary-time time-dependent density functional theory and its application for robust convergence of electronic states. J. Chem. Theory Comput. 2019, 15, 6036– 6045, DOI: 10.1021/acs.jctc.9b0061783https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhvVKltr%252FN&md5=23436f2c1caae703b7069a40f70438dcImaginary-Time Time-Dependent Density Functional Theory and Its Application for Robust Convergence of Electronic StatesFlamant, Cedric; Kolesov, Grigory; Manousakis, Efstratios; Kaxiras, EfthimiosJournal of Chemical Theory and Computation (2019), 15 (11), 6036-6045CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)Reliable and robust convergence to the electronic ground state within d. functional theory (DFT) Kohn-Sham (KS) calcns. remains a thorny issue in many systems of interest. In such cases, charge sloshing can delay or completely hinder the convergence. Here, we use an approach based on transforming the time-dependent DFT equations to imaginary time, followed by imaginary-time evolution, as a reliable alternative to the SCF procedure for detg. the KS ground state. We discuss the theor. and tech. aspects of this approach and show that the KS ground state should be expected to be the long-imaginary-time output of the evolution, independent of the exchange-correlation functional or the level of theory used to simulate the system. By maintaining self-consistency between the single-particle wavefunctions (orbitals) and the electronic d. throughout the detn. of the stationary state, our method avoids the typical difficulties encountered in SCF. To demonstrate dependability of our approach, we apply it to selected systems which struggle to converge with SCF schemes. In addn., through the van Leeuwen theorem, we affirm the phys. meaningfulness of imaginary time TDDFT, justifying its use in certain topics of statistical mechanics such as in computing imaginary time path integrals.
- 85Penz, M.; Csirik, M. A.; Laestadius, A. Density-potential inversion from Moreau–Yosida regularization. arXiv (Chemical Physics), December 27, 2022, ver. 2, 2212.12727 (accessed 2023-01-31).There is no corresponding record for this reference.
- 86Schönhammer, K.; Gunnarsson, O.; Noack, R. M. Density-functional theory on a lattice: Comparison with exact numerical results for a model with strongly correlated electrons. Phys. Rev. B 1995, 52, 2504– 2510, DOI: 10.1103/PhysRevB.52.250486https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXnt1Ghu7o%253D&md5=a81804af5bd37653afcb12a59de94dbcDensity-functional theory on a lattice: comparison with exact numerical results for a model with strongly correlated electronsSchonhammer, K.; Gunnarsson, O.; Noack, R. M.Physical Review B: Condensed Matter (1995), 52 (4), 2504-10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)Most of the basic ideas of the d.-functional theory (DFT) are shown to be unrelated to the fact that the particle d. is used as the basic variable. After presenting the general formalism and a simple example unrelated to densities, the authors discuss various approaches to d.-functional theory on a lattice. One has proven useful for the understanding of various fundamental issues of the DFT. The exact Kohn-Sham band gap and the band gap in the local-d. approxn. (LDA) are compared to the exact result for a one-dimensional model. As the interacting homogeneous system can be solved exactly no further approxn. is needed to formulate the LDA.
- 87Higuchi, M.; Higuchi, K. Arbitrary choice of basic variables in density functional theory: Formalism. Phys. Rev. B 2004, 69, 035113, DOI: 10.1103/PhysRevB.69.03511387https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXht1Ortbo%253D&md5=230b920487a80e5274cacc2ec6030d64Arbitrary choice of basic variables in density functional theory: FormalismHiguchi, Masahiko; Higuchi, KatsuhikoPhysical Review B: Condensed Matter and Materials Physics (2004), 69 (3), 035113/1-035113/9CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)The Hohenberg-Kohn theorem of the d. functional theory (DFT) is extended by modifying the Levy constrained-search formulation. The theorem allows us to choose arbitrary phys. quantities as basic variables which det. the ground-state properties of the system. Moreover, the theorem establishes a min. principle with respect to variations in chosen basic variables as well as with respect to variations in the d. By using this theorem, self-consistent single-particle equations are derived. N single-particle orbitals introduced reproduce not only the electron d. but also arbitrary phys. quantities which are chosen as basic variables. The validity of the theory is confirmed by examples where the spin d. or paramagnetic c.d. is chosen as one of basic variables. The resulting single-particle equations coincide with the Kohn-Sham equations of the spin-d. functional theory or current-d. functional theory, resp. By choosing basic variables appropriate to the system, the present theory can describe the ground-state properties more efficiently than the conventional DFT.
- 88Higuchi, K.; Higuchi, M. Arbitrary choice of basic variables in density functional theory. II. Illustrative applications. Phys. Rev. B 2004, 69, 165118, DOI: 10.1103/PhysRevB.69.16511888https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXktVKit7s%253D&md5=87b390d17291e24b9f1a9325548b243cArbitrary choice of basic variables in density functional theory. II. Illustrative applicationsHiguchi, Katsuhiko; Higuchi, MasahikoPhysical Review B: Condensed Matter and Materials Physics (2004), 69 (16), 165118/1-165118/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)Our recent extended constrained-search theory [M. Higuchi and K. Higuchi, Phys. Rev. B 69, 035113 (2004)] enables us to choose arbitrary quantities as the basic variables of d. functional theory. In this paper, we apply it to several cases. In the case in which the occupation matrix of localized orbitals is chosen as a basic variable, we can obtain the single-particle equation which is equiv. to that of the LDA+U method. The theory also leads to the Hartree-Fock-Kohn-Sham equation by letting the exchange energy be a basic variable. Furthermore, if the quantity assocd. with the d. of states near the Fermi level is chosen as a basic variable, the resulting single-particle equation includes the addnl. potential which could mainly modify the energy-band structures near the Fermi level.
- 89Xu, L.; Mao, J.; Gao, X.; Liu, Z. Extensibility of Hohenberg–Kohn Theorem to General Quantum Systems. Adv. Quantum Techn. 2022, 5, 2200041, DOI: 10.1002/qute.202200041There is no corresponding record for this reference.
- 91Vignale, G.; Rasolt, M. Density-functional theory in strong magnetic fields. Phys. Rev. Lett. 1987, 59, 2360– 2363, DOI: 10.1103/PhysRevLett.59.236091https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1cXislSqtA%253D%253D&md5=6c0439dbbbddf191b1da1f73d69ed733Density-functional theory in strong magnetic fieldsVignale, G.; Rasolt, MarkPhysical Review Letters (1987), 59 (20), 2360-3CODEN: PRLTAO; ISSN:0031-9007.The current-d.-functional theory is formulated for systems in arbitrarily strong magnetic fields. A set of self-consistent equations comparable to the Kohn-Sham equations for ordinary d.-functional theory is derived, and proved to be gage invariant and to satisfy the continuity equation. The exchange-correlation energy functional Exc[n,jp] [n(r) is the d. and jp(r) is the "paramagnetic" c.d.] depends on the current via the combination v(r) = V × [jp(r)/n(r)]. An explicit formula for Exc is derived, which is local in v(r).
- 92Kümmel, S.; Kronik, L. Orbital-dependent density functionals: Theory and applications. Rev. Mod. Phys. 2008, 80, 3, DOI: 10.1103/RevModPhys.80.392https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXlslKms74%253D&md5=867b579a9dfef4dfa661a6af35d6d21dOrbital-dependent density functionals: Theory and applicationsKuemmel, Stephan; Kronik, LeeorReviews 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.
- 93Vignale, G.; Rasolt, M.; Geldart, D. Magnetic fields and density functional theory. Adv. Quantum Chem. 1990, 21, 235– 253, DOI: 10.1016/S0065-3276(08)60599-7There is no corresponding record for this reference.