**Cite This:**

*J. Chem. Theory Comput.*2019, 15, 4, 2206-2220

# Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer

- Fabian M. Faulstich
*****Fabian M. FaulstichHylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway*****E-mail: [email protected]More by Fabian M. Faulstich - ,
- Mihály MátéMihály MátéStrongly Correlated Systems “Lendület” Research Group, Wigner Research Center for Physics, H-1525, P.O. Box 49, Budapest, HungaryDepartment of Physics of Complex Systems, Eötvös Loránd University, Pf. 32, H-1518 Budapest, HungaryMore by Mihály Máté
- ,
- Andre LaestadiusAndre LaestadiusHylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, NorwayMore by Andre Laestadius
- ,
- Mihály András CsirikMihály András CsirikStrongly Correlated Systems “Lendület” Research Group, Wigner Research Center for Physics, H-1525, P.O. Box 49, Budapest, HungaryMore by Mihály András Csirik
- ,
- Libor VeisLibor VeisJ. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech RepublicMore by Libor Veis
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- Andrej AntalikAndrej AntalikJ. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech RepublicFaculty of Mathematics and Physics, Charles University, 11636 Prague, Czech RepublicMore by Andrej Antalik
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- Jiří BrabecJiří BrabecJ. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i., Dolejškova 3, 18223 Prague 8, Czech RepublicMore by Jiří Brabec
- ,
- Reinhold SchneiderReinhold SchneiderModeling, Simulation and Optimization in Science, Department of Mathematics, Technische Universität Berlin, Sekretariat MA 5-3, Straße des 17. Juni 136, 10623 Berlin, GermanyMore by Reinhold Schneider
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- Jiří PittnerJiří PittnerMore by Jiří Pittner
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- Simen KvaalSimen KvaalHylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, NorwayMore by Simen Kvaal
- , and
- Örs LegezaÖrs LegezaStrongly Correlated Systems “Lendület” Research Group, Wigner Research Center for Physics, H-1525, P.O. Box 49, Budapest, HungaryMore by Örs Legeza

## Abstract

In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO–LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the density matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the density matrix renormalization group-tailored coupled-cluster error.

## I. Introduction

*E*

_{h}) via the CCSD correction on the external part compared to the single reference CCSD method on the full space. Nonetheless, the simplistic approach of the DMRG-TCC method to the multireference problem comes with a price. The DMRG-TCC, as a CAS method, does not correlate external amplitudes with the CAS amplitudes, i.e., contributions from the external part to excited determinants within the CAS are not present. Furthermore, in situations where the choice of a reference determinant becomes unclear, e.g., strong open-shell systems, the DMRG-TCC method could run into potential problems since it is based on a single reference formulation. Although the total spin can be fixed for the CAS part in the DMRG calculations (spin-adapted DMRG (4−7)), for the full orbital space it cannot be controlled through the external CC corrections presented in this work.

## II. DMRG-TCC Method

*Ŝ*, and an external (ext) part

*T̂*, i.e., the wave function is parametrized as

*Ŝ*,

*T̂*commute since this separation is merely a partition of the overall cluster operator. In this formulation the linked CC equations are given by

_{CAS}⟩ =

*e*

^{Ŝ}|Ψ

_{HF}⟩ first and keeping it fixed for the dynamical correction via the CCSD method restricts the above equations to |Ψ

_{μ}⟩ not in the CAS, i.e., ⟨Ψ|Ψ

_{μ}⟩ = 0 for all |Ψ⟩ in the CAS (we say that |Ψ

_{μ}⟩ is in the

*L*

^{2}-orthogonal complement of the CAS). We emphasize that this includes mixed states, e.g., |Ψ

_{IJ}

^{AB}⟩ where |Ψ

_{I}

^{A}⟩ is an element of the CAS but |Ψ

_{J}

^{B}⟩ is not. We consider a CAS of

*N*-electron Slater-determinants formed from the set of spin-orbitals . This is, in the mathematical sense, a subspace of the full configuration interaction (FCI) space, i.e., the space of all

*N*-electron Slater-determinants formed from the entire set of spin-orbitals . We here assume the spin-orbitals to be eigenfunctions of the system’s Fock operator. Note that the following analysis can be applied to any single-particle operator fulfilling the properties used in ref (76)—not only the Fock operator. This mathematical analysis, among other things, rests on the structure of a one-particle operator with a distinct (and furthermore steerable) CAS-ext gap. As described below (in connection to Assumption A), choosing the Fock operator might lead to the inclusion of diffuse functions in the CAS.

*N*, i.e., (for more details we refer the reader to ref (70)), an efficient approximation scheme for strongly correlated systems is indispensable for the TCC method to have practical significance. One of the most efficient schemes for static correlation is the DMRG method. (80) Going back to the physicists White and Martin, (3) it was introduced to quantum chemistry as an alternative to the CI or CC approach. However, the major disadvantage of the DMRG is that in order to compute dynamical correlation high bond dimensions (tensor ranks) may be necessary, making the DMRG a potentially costly method. (2,80) Nevertheless, as a TNS-TCC method, the DMRG-TCC approach is an efficient method since the CAS is supposed to cover the statically correlated orbitals of the system. This avoids the DMRG method’s weak point and allows to employ a much larger CAS compared to the traditional CI-TCC method. We remark here that some terminology has different meaning in mathematics, physics, and chemistry. The number of legs of a tensor is called the order of the tensor in mathematics, while it is called the rank of the tensor in physics. The rank of the matrix corresponds to the number of nonzero singular values after matricization in mathematics, i.e., the Schmidt number in physics.

_{HF}⟩, which implies that separating the cluster operator corresponds to a partition of excitation operators. Hence,

*Ŝ*and

*T̂*commute. This makes the DMRG-TCC method’s analysis much more accessible than internally contracted MRCC methods and therewith facilitates establishing sound mathematical results. (76) We remark, however, that the computationally most demanding step of the DMRG-TCC calculation is the DMRG part, and its cost increases rapidly with

*k*. Alternative to the dynamical correction via the CC approach, the DMRG-MRCI method in ref (81) utilizes an internally contracted CI algorithm different from a conventional CI calculation.

## III. Formal Properties of the DMRG-TCC Method

*n*we find that all excitation amplitudes need to be zero but for the

*n*th one. This is somewhat surprising as the equivalence of linked and unlinked CC equations holds for rank complete truncations of the single-reference CC method.

**T**with elements

*T*

_{μ,ν}= ⟨Ψ

_{μ}|

*e*

^{T̂}|Ψ

_{ν}⟩ for μ, ν ∉ CAS. Note that, as

*T̂*increases the excitation rank,

**T**is an atomic lower triangular matrix and therefore not singular. Assuming that the linked CC equations hold, the nonsingularity of

**T**yields

*E*

_{0}⟨Ψ

_{μ}|

*e*

^{T̂}|Ψ

_{HF}⟩, describe the unlinked CC equations. To analyze the remaining terms on the r.h.s. in eq 2 we expand the inner products, i.e.,

*T̂*enters to the power of two or higher since an excitation of order two or higher acting on an at least singly excited Slater-determinant |Ψ

_{γ}⟩ yields an at least 3-fold excited Slater-determinant. However, as the external space contains mixed states, we find that ⟨Ψ

_{μ}|

*T̂*|Ψ

_{γ}⟩ is not necessarily zero, namely, for ⟨Ψ

_{μ}| = ⟨Ψ

_{α}| ∧ ⟨Ψ

_{β}| and |Ψ

_{γ}⟩ = |Ψ

_{β}⟩ with α ∈ ext and β ∈ CAS. This proves the claim.

*A*and

*B*be

*Ĥ*

_{AB}=

*Ĥ*

_{A}+

*Ĥ*

_{B}. Since the TCC approach corresponds to a partitioning of the cluster amplitudes we note that for

_{HF}

^{(AB)}⟩ = |Ψ

_{HF}

^{(A)}⟩ ∧ |Ψ

_{HF}

^{(B)}⟩, the energy of the compound systems can be written as

_{μ}

^{(AB)}|, it holds that . Splitting the argument into three cases, we note that

^{(A)}Ψ

^{(B)}| = ⟨Ψ

^{(A)}| ∧ ⟨Ψ

^{(B)}|. This proves the energy size consistency for the untruncated TCC method. From this we conclude the energy size consistency for the DMRG-TCCSD scheme, because the truncation only affects the product states ⟨Ψ

_{μ}

^{(A)}Ψ

_{μ}

^{(B)}| and these are zero in the above projection.

*Ŝ*

_{n}for

*n*> 3. More precisely, due to the fact that in the TCCSD case external space amplitudes can at most contain one virtual orbital in the CAS, the TCCSD amplitude expressions become independent of

*Ŝ*

_{4}, i.e.,

*a*′,

*b*′,

*c*′,

*d*′,

*e*′ describe orbitals in the CAS, the nonprimed variable

*a*describes an orbital in the external part and

*i*,

*j*,

*k*,

*l*,

*m*,

*n*are occupied orbitals. Note, this does not imply that we can restrict the CAS computation to a manifold characterizing excitations with rank less or equal to three as for strongly correlated systems these can still be relevant. However, it reduces the number of terms entering the DMRG-TCCSD energy computations significantly.

*Ŝ*

_{n}for

*n*= 1, 2 into account. (28) We emphasize that the additional consideration of

*Ŝ*

_{3}corresponds to an exact treatment of the CAS contributions to the energy. Furthermore, this consideration does not change the TCC method’s complexity, if the

*Ŝ*

_{3}amplitudes are available. This is due to the fact that including the CAS triple excitation amplitudes will not exceed the dominating complexities of the CCSD approach (82) nor of the DMRG method. However, the extraction of the CI-triples from the DMRG wave function is costly and a corresponding efficiency investigation is left for future work.

## IV. Analysis of the DMRG-TCC Method

### IV.A. Complete Active Space Choice

*well-chosen*CAS, i.e., a large enough CAS that covers the system’s static correlation. Consequently, we require a quantitative measurement for the quality of the CAS, which presents the first obstacle for creating a nonempirical model since the chemical concept of correlation is not well-defined. (83) In the DMRG-TCC method, we use a quantum information theory approach to classify the spin-orbital correlation. This classification is based on the

*mutual-information*

*von Neumann entropy S*(ρ) = −Tr(ρ ln ρ) of the reduced density operators ρ

_{{X}}. (84) Note that the mutual-information describes two-particle correlations. For a more general connection between multiparticle correlations and ξ-correlations, we refer the reader to the work of Szalay et al. (84) We emphasize that in practice this is a basis dependent quantity, which is in agreement with the chemical definition of correlation concepts. (83) We identify pairs of spin-orbitals contributing to a high mutual information value as strongly correlated, the pairs contributing to the plateau region, i.e., a region in which the mutual information profile is constant, as nondynamically correlated and the pairs contributing to the mutual information tail as dynamically correlated (see Figure 3). The mutual-information profile can be well approximated from a prior DMRG computation on the full system. Due to the size of the full system we only compute a DMRG solution of low

*bond dimension*(also called

*tensor rank*). These low-accuracy calculations, however, already provide a good qualitative entropy profile, i.e., the shapes of profiles obtained for low bond dimension,

*M*, agree well with the ones obtained in the FCI limit. Here, we refer to Figures 2 and 3 showing the single orbital entropy and mutual information profiles, respectively, for various

*M*values and for three different geometries of the N

_{2}molecule. The orbitals with large entropies can be identified from the low-

*M*calculations providing a routine procedure to form the CAS including the strongly correlated orbitals. (85−87) In practice this is achieved by using the following dimension reduction protocol: We start with a very low bond dimension calculation carried out on the full orbital space. Based on the corresponding entropy profile and an

*a priori*defined numerical threshold, a smaller set of orbitals is selected. In a subsequent step the same procedure is repeated on the reduced orbital set but with a larger bond dimension. This iterative dimension reduction protocol is a typical renormalization group based approach to refine the entropy spectrum that is also used in condensed matter physics.

*k*=

*N*), the DMRG-TCCSD becomes the CCSD method and, for (i.e.,

*k*=

*K*), it is the DMRG method. We recall that the CCSD method can not resolve static correlation and the DMRG method needs high tensor ranks for dynamically correlated systems. This suggests that the error obtains a minimum for some

*k*with

*N*≤

*k*≤

*K*, i.e., there exists an optimal choice of

*k*determining the basis splitting and therewith the choice of the CAS. Note that this feature becomes important for large systems since high bond dimensions become simply impossible to compute with available methods.

### IV.B. Local Analysis of the DMRG-TCC Method

*nonlinear Galerkin scheme*, (70) which is a well-established framework in numerical analysis to convert the continuous Schrödinger equation to a discrete problem. For the DMRG-TCC method a first local analysis was performed in ref (76). There, a quantitative error estimate with respect to the basis truncation was established. Faulstich et al. showed under certain assumptions (Assumption A and B in the sequel) that the DMRG-TCC method possesses a locally unique and

*quasi-optimal*solution (cf. section 4.1 in ref (76)). In case of the DMRG-TCC method the latter means: On a fixed CAS, the CC method tailored by a DMRG solution provides a truncation hierarchy that converges to the best possible dynamical correction to the given CAS. For a fixed basis set the CC solution tailored by a DMRG solution on a fixed CAS is up to a multiplicative constant the best possible solution in the approximation space defined by the basis set. In other words, the CC method provides the best possible dynamical correction for a given CAS solution such as a DMRG solution.

*f*(

*t*;

*s*) = ⟨Ψ

_{μ}|

*e*

^{–Ŝ}

*e*

^{–T̂}

*Ĥe*

^{T̂}

*e*

^{Ŝ}|Ψ

_{HF}⟩, for |Ψ

_{μ}⟩ not in the CAS. Note that we use the convention where small letters

*s*,

*t*correspond to cluster amplitudes, whereas capital letters

*Ŝ*,

*T̂*describe cluster operators. The corresponding TCC energy expression is given by

*F̂*is

*bounded*and satisfies a so-called

*. Note that spectral gap assumptions (cf. HOMO–LUMO gap) are standard in the analysis of dynamically correlated systems, and for a more detailed description of these properties in this context, we refer readers to ref (77). Second, in the theoretical framework (76) it is assumed that there exists a CAS-ext gap in the spectrum of the Fock operator, i.e., there is a gap between the*

*G*årding inequality*k*th and the

*k*+ 1st orbital energies. The CAS-ext gap (although in practice possibly very small) was sufficient for the analysis since the main purpose was to remove the HOMO–LUMO gap assumption and allow for quasi-degeneracy, which makes the general TCC approach applicable to multiconfiguration systems. Intuitively, this gap assumption means that the CAS captures the static correlation of the system.

_{k+1}– ε

_{k}, nor to the HOMO–LUMO gap ε

_{N+1}– ε

_{N}. Due to the frozen CAS-amplitudes this stability constant becomes much larger and is roughly estimated by ε

_{k+1}– ε

_{N}. This improved stability provides accurate CC amplitudes and the improved gap is not destroyed e.g. by the existence of many diffuse functions around the LUMO-level (Fermi level). In this case, the CAS includes the diffuse functions. This might not be optimal but is the simplest choice and most importantly fulfills the stability condition. The issue of basis set optimization is discussed briefly in the conclusion but a more detailed discussion is left for future work.

*Ŵ*=

*Ĥ*–

*F̂*. This operator describes the difference of the Hamiltonian and a single particle operator, here chosen to be the Fock operator. Using the similarity transformed

*Ŵ*and fixing the CAS amplitudes

*s*, the mapis assumed to have a

*small enough*Lipschitz-continuity constant (see eq 20 in ref (76)). The physical interpretation of this Lipschitz condition is at the moment unclear.

### IV.C. Error Estimates for the DMRG-TCC Method

*Ĥ*|Ψ*⟩ =

*E*|Ψ*⟩. Using the exponential parametrization and the above introduced separation of the cluster operator, we have

_{FCI}

^{CAS}⟩ = exp(

*Ŝ*

_{FCI})|Ψ

_{HF}⟩ is an approximation to the projection of |Ψ*⟩ onto the CAS

*P̂*= ∑

_{μ∈CAS}|Ψ

_{μ}⟩⟨Ψ

_{μ}| is the

*L*

^{2}-orthogonal projection onto the CAS. For a reasonably sized CAS the FCI solution |Ψ

_{FCI}

^{CAS}⟩ is rarely computationally accessible and we introduce the DMRG solution on the CAS as an approximation of |Ψ

_{FCI}

^{CAS}⟩

_{FCI}

^{CAS}⟩, the TCC method yields the best possible solution with respect to the chosen CAS, i.e.,

*f*(

*t*

_{CC}

^{*};

*s*

_{FCI}) = 0. This solution is in general different from

*t*

_{CC}fulfilling

*f*(

*t*

_{CC};

*s*

_{DMRG}) = 0 and its truncated version

*t*

_{CCSD}satisfying

*P*

_{Gal}

*f*(

*t*

_{CCSD};

*s*

_{DMRG}) = 0, where

*P*

_{Gal}denotes the

*l*

^{2}-orthogonal projection onto the corresponding Galerkin space. In the context of the DMRG-TCC theory, the Galerkin space represents a truncation in the excitation rank of the cluster operator, e.g., DMRG-TCCD, DMRG-TCCSD, etc.

*Δ*

*E*can be estimated as (76)

*l*

^{2}- or

*L*

^{2}-norm, respectively, but also measure the kinetic energy. It should be clear from context which Hilbert space is in question and we refer to ref (71) for formal definitions. The first term is defined as

_{DMRG}

^{CAS}⟩. We emphasize that the dynamical corrections via the CCSD and the untruncated CC method are here tailored by the same CAS solution. Hence, the energy error

*Δε*corresponds to a single reference CC energy error, which suggests an analysis similar to that of refs (70and72). Indeed, the

*Aubin–Nitsche duality method*(88−90) yields a quadratic

*a priori*error estimate in ∥

*t*

_{CCSD}–

*t*

_{CC}∥ (and in terms of the Lagrange mulitpliers; see Theorem 29 in ref (76)).

*Δε*

_{CAS}is connected with the error

_{μ}= ε

_{I1...In}

^{A1···An}= ∑

_{j=1}

^{n}(λ

_{Aj}– λ

_{Ij}), for 1 ≤

*n*≤

*k*, where λ

_{i}are the orbital energies. The ε

_{μ}are the (translated) Fock energies, more precisely,

*F̂*|Ψ

_{μ}⟩ = (Λ

_{0}+ ε

_{μ})|Ψ

_{μ}⟩, with Λ

_{0}= ∑

_{i=1}

^{N}λ

_{i}. Note that the wave function |Ψ

_{FCI}

^{CAS}⟩ is in general not an eigenfunction of

*Ĥ*; however, it is an eigenfunction of the projected Hamiltonian

*P̂ĤP̂*. Equation 5 involves the exponential parametrization. This can be estimated by the energy error of the DMRG wave function, denoted , namely

*δε*

_{Tr}, i.e., . Hence, for well chosen CAS the difference ∥|Ψ

_{DMRG}

^{CAS}⟩ – |Ψ

_{FCI}

^{CAS}⟩∥

_{L2}is sufficiently small such that holds. This again shows the importance of a well-chosen CAS. Furthermore, the last term in eq 6 can be eliminated via orbital rotations, as it is a sum of single excitation amplitudes.

*t**,

*s**) is a stationary point of we have . A calculation involving Taylor expanding around (

*t**,

*s**) (see Lemma 26 in ref (76)) yields

*t*

_{CC}

^{*},

*s*

_{FCI}) differs in general from the FCI solution (

*t**,

*s**).

*a priori*energy error estimate for the DMRG-TCC method. The interested reader is referred to ref (76) for a more detailed treatment of the above analysis.

### IV.D. On the *k*-Dependence of the Error Estimates

*k*-dependence of the error

*Δ*

*E*. However, the above derived error bound has a highly complicated

*k*-dependence since not only the amplitudes but also the implicit constants (in ≲) and norms depend on

*k*. Therefore, the analysis in ref (76) is not directly applicable to take the full

*k*-dependence into account.

*s*

_{DMRG}→

*s*

_{FCI}we obtain that

*t*

_{CC}→

*t*

_{CC}

^{*}since the TCC method is numerically stable, i.e., a small perturbation in

*s*corresponds to a small perturbation in the solution

*t*. Furthermore, if we assume that

*t*

_{CCSD}≈

*t*

_{CC}, which is reasonable for the equilibrium bond length of N

_{2}, the error can be bound as

*k*on

*Δ*

*E*

_{k}and

*C*

_{k}highlights the

*k*-dependence. We remark that we here used the less accurate

*l*

^{2}-structure on the amplitude space compared to the

*H*

^{1}-structure in eq 9. This yields

*k*-independent vectors (

*t*

_{CCSD},

*s*

_{DMRG}) and (

*t**,

*s**), as well as an

*k*-independent

*l*

^{2}-norm. The

*k*-depenence of

*C*

_{k}will be investigated numerically in more detail in section B.5.

## V. Splitting Error for N_{2}

*k*-dependence in the above performed error analysis explicitly is a highly nontrivial task involving many mathematical obstacles and is part of our current research. Therefore, we here extend the mathematical results from section IV with a numerical investigation on this

*k*-dependence. Our study is presented for the N

_{2}molecule using the cc-pVDZ basis, which is a common basis for benchmark computations developed by Dunning and co-workers. (91) Here we remark that in our calculations all electrons are correlated as opposed to the typical frozen-core calculation, where the two 1s orbitals are omitted from the full orbital space. We investigate three different geometries of the nitrogen dimer by stretching the molecule, thus the performance of DMRG-TCCSD method is assessed against DMRG and single reference CC methods for bond lengths

*r*= 2.118

*a*

_{0}, 2.700

*a*

_{0}, and 3.600

*a*

_{0}. In the equilibrium geometry the system is weakly correlated implying that single reference CC methods yield reliable results. For increasing bond length

*r*the system shows multireference character, i.e., static correlations become more dominant. For

*r*> 3.5

*a*

_{0}this results in the breakdown of single reference CC methods. (92) This breakdown can be overcome with the DMRG-TCCSD method once a large and well chosen CAS is formed, we therefore refer to the DMRG-TCCSD method as numerically stable with respect to the bond length along the potential energy surface (PES).

### V.A. Computational Details

*I*

_{dist}= ∑

_{ij}

*I*

_{i|j}|

*i*–

*j*|

^{2}. In order to speed up the convergence of the DMRG procedure the configuration interaction based dynamically extended active space (CI-DEAS) method is applied. (93,99) In the course of these optimization steps, the single orbital entropy (

*S*

_{i}=

*S*(ρ

_{{i}})) and the two-orbital mutual information (

*I*

_{i|j}) are calculated iteratively until convergence is reached. The size of the active space is systematically increased by including orbitals with the largest single site entropy values, which at the same time correspond to orbitals contributing to the largest matrix elements of the mutual information. Thus, the decreasingly ordered values of

*S*

_{i}define the so-called CAS vector, which provides a guide in what order to extend the CAS by including additional orbitals. The bond dimensions

*M*(tensor rank) in the DMRG method can be kept fixed or adapted dynamically (dynamic block state selection (DBSS) approach) in order to fulfill an

*a priori*defined error margin. (103,104) Accurate extrapolation to the truncation free limit is possible as a function of the truncation error

*δε*

_{Tr}. (103,105)

*d*= 4 dimensional. In this -representation an orbital can be empty, singly occupied with either a spin up or spin down electron, or doubly occupied with opposite spins. Note, in contrast to section IV we need

*N*/2 spatial orbitals to describe an

*N*-electron wave function and similar changes apply to the size of the basis set so that we use

*K*≡

*K*/2 from here on. The single orbital entropy therefore varies between 0 and ln

*d*= ln 4, while the two-orbital mutual information varies between 0 and ln

*d*

^{2}= ln 16.

*k*for the following sections.

(1) | First the CAS is formed from the full orbital space by setting | ||||

(2) | Using a given | ||||

(3) | Using the matrix product state representation of |Ψ | ||||

(4) | In the following step the cluster amplitudes for the external part, i.e., | ||||

(5) | As we discus in the next section, finding the optimal CAS, i.e., |

*k*, thus here the

*k*dependence is also omitted.

*k*split, the accuracy of |Ψ

_{DMRG}

^{CAS}⟩ depends on the DMRG truncation error,

*δε*

_{Tr}. As has been shown in refs (103and105), the relative error,

*Δ*

*E*

_{rel}= (

*E*

_{DMRG(δεTr)}

^{CAS}–

*E*

_{FCI}

^{CAS})/

*E*

_{FCI}

^{CAS}is a linear function of

*δε*

_{Tr}on a logarithmic scale. Therefore, extrapolation to the FCI limit can be carried out as a function of

*δε*

_{Tr}. In addition, the error term

_{DMRG}

^{CAS}⟩ or |Ψ

_{FCI}

^{CAS}⟩. Note that calculating all CI-coefficients scales exponentially with the size of the CAS. However, since the system is dynamically correlated zeroth order, single and double excitation coefficients are sufficient. Hence, the error terms ∥|Ψ

_{DMRG}

^{CAS}⟩ – |Ψ

_{FCI}

^{CAS}⟩∥

_{L2}and ∥(

*Ŝ*

_{FCI}–

*Ŝ*

_{DMRG(δεTr)})|Ψ

_{HF}⟩∥ in eqs 6 and 7, respectively, can be well approximated. We remark that this exponential scaling with the CAS size also effects the computational costs of the CAS CI-triples, which are needed for an exact treatment of the TCCSD energy equation. However, investigations of the influence of the CAS CI-triples on the computed energies are left for future work.

### V.B. Results and Discussion

*k*and as a function of the DMRG-truncation error

*δε*

_{Tr}. For our numerical error study we perform steps 1–4 discussed in section V.A for each

*N*/2 <

*k*<

*K*. For each geometry

*r*= 2.118

*a*

_{0}, 2.700

*a*

_{0}, and 3.600

*a*

_{0}, we also carry out very high accuracy DMRG calculations on the full orbital space, i.e., by setting the truncation error to

*δε*

_{Tr}= 10

^{–8}and

*k*=

*K*. This data is used as a reference for the FCI solution.

#### B.1. Entropy Study on the Full Orbital Space

*k*=

*K*= 28 orbitals, and for various fixed

*M*values and for

*δε*

_{Tr}= 10

^{–8}. In the latter case the maximum bond dimension was set to

*M*= 10 000. In Figure 1 a, we show the relative error of the ground-state energy as a function of the DMRG-truncation error on a logarithmic scale. For the FCI energy,

*E*

_{FCI}, the CCSDTQPH reference energy is used given in ref (107). It is visible that the relative error is a linear function of the truncation error on a logarithmic scale, thus extrapolation to the truncation free solution can be carried out according to refs (103and105).

*M*= 64, 256, 512 and with

*δε*

_{Tr}= 10

^{–8}for the three geometries. As can be seen in the figures, the entropy profiles obtained with low-rank DMRG calculations already resemble the main characteristics of the exact profile (

*M*≃ 10000). Therefore, orbitals with large single orbital entropies, also contributing to large matrix elements of

*I*

_{i|j}, can easily be identified from a low-rank computation. The ordered orbital indices define the CAS vector, and the CAS for the DMRG-TCCSD can be formed accordingly as discussed in section V.A.

*S*

_{i}shifts upward for increasing

*r*indicating the higher contribution of static correlations for the stretched geometries. Similarly the first 50–100 matrix elements of

*I*

_{i|j}also take larger values for larger

*r*while the exponential tail, corresponding to dynamic correlations, is less effected. The gap between large and small values of the orbital entropies gets larger and its position shifts rightward for larger

*r*. Thus, for the stretched geometries more orbitals must be included in the CAS during the TCC scheme in order to determine the static correlations accurately. We remark here that the orbitals contributing to the high values of the single orbital entropy and mutual information matrix elements change for the different geometries according to chemical bond forming and breaking processes. (108)

#### B.2. Numerical Investigation of the Error’s *k*-Dependence

*δε*

_{Tr}= 10

^{–8}. The CAS was formed by including all Hartee–Fock orbitals and its size was increased systematically by including orbitals with the largest entropies according to the CAS vector. Orbitals with degenerate single orbital entropies, due to symmetry considerations, are added to the CAS at the same time. Thus, there are some missing

*k*points in the following figures. For each restricted CAS we carry our the usual optimization steps of a DMRG scheme as discussed in section V.A, with low bond dimension followed by a high-accuracy calculation with

*δε*

_{Tr}= 10

^{–8}using eight sweeps. (93) Our DMRG ground-state energies for 7 <

*k*< 28 together with the CCSD (corresponding to a DMRG-TCCSD calculation where

*k*=

*N*/2 = 7) and CCSDTQ reference energies, are shown in Figure 4 near the equilibrium bond length,

*r*= 2.118

*a*

_{0}. The single-reference coupled cluster calculations were performed in NWChem, (109) we employed the cc-pVDZ basis set in the spherical representation. For

*k*=

*K*= 28 the CCSDTQPH energy was taken as a reference for the FCI energy. (107)

*k*= 7 and decreases monotonically with increasing

*k*until the full orbital solution with

*k*= 28 is reached. It is remarkable, however, that the DMRG-TCCSD energy is significantly below the CCSD energy for all CAS choices, even for a very small

*k*= 9. The error, however, shows an irregular behavior taking small values for several different

*k*values. This is due to the fact that the DMRG-TCCSD approach suffers from a methodological error, i.e., certain fraction of the correlations are lost, since the CAS is frozen in the CCSD correction. This supports the hypothesis of a

*k*-dependent constant as discussed in section IV.D. Therefore, whether orbital

*k*is part of the CAS or external part provides a different methodological error. This is clearly seen as the error increases between

*k*= 10 and 15 although the CAS covers more of the system’s static correlation with increasing

*k*. This is investigated in more detail in section B.4.

*k*-splits lead to small DMRG-TCCSD errors, the optimal

*k*value from the computational point of view, is determined not only by the error minimum but also by the minimal computational time, i.e., we need to take the computational requirements of the DMRG into account. Note that the size of the DMRG block states contributes significantly to the computational cost of the DMRG calculation. The connection of the block size to the CAS choice is shown in Figure 1b, where the maximal number of DMRG block states is depicted as a function of

*k*for the

*a priori*defined truncation error margin

*δε*

_{Tr}= 10

^{–8}. Note that max(

*M*) increases rapidly for 10 <

*k*< 20. The optimal CAS is therefore chosen such that the DMRG block states are not too large and the DMRG-TCCSD provides a low error, i.e., is a local minimum in the residual with respect to

*k*.

*k*values. Since close to the equilibrium geometry the wave function is dominated by a single reference character, it is expected that DMRG-TCCSD leads to even more robust improvements for the stretched geometries, i.e., when the multireference character of the wave function is more pronounced. Our results for the stretched geometries,

*r*= 2.700

*a*

_{0}and 3.600

*a*

_{0}, are shown in Figures 2, 3, 5, and 6. As mentioned in section B.1, for larger

*r*values static correlations gain importance signaled by the increase in the single orbital entropy in Figure 2. Thus, the multireference character of the wave function becomes apparent through the entropy profiles. According to Figure 5 the DMRG-TCCSD energy for all

*k*> 7 values is again below the CCSD computation and for

*k*> 15 it is even below the CCSDT reference energy. For

*r*= 3.600

*a*

_{0}the CC computation fluctuates with increasing excitation ranks and CCSDT is even far below the FCI reference energy, revealing the variational breakdown of the single-reference CC method for multireference problems. In contrast to this, the DMRG-TCCSD energy is again below the CCSD energy for all

*k*> 7, but above the CCSDT energy. The error furthermore shows a local minimum around

*k*= 19. For the stretched geometries static correlations are more pronounced, there are more orbitals with large entropies, thus the maximum number of DMRG block states increases more rapidly with

*k*compared to the situation near the equilibrium geometry (see Figure 1b). Thus, obtaining an error margin within 1 μ

*E*

_{h}for

*k*= 19 ≪ 28 leads to a significant save in computational time and resources. Here we remark that DMRG-TCCSD is a single-reference multireference method thus the choice of the reference determinant can effect its performance. In the our current study, however, we have verified that for

*d*≤ 4 and for all

*k*values the weight of the Hartree–Fock determinant was significantly larger than all other determinants.

#### B.3. Effect of *δε*_{Tr} on the DMRG-TCCSD

^{–4}and 10

^{–7}. Our results are shown in Figures 4, 5, and 6. For small

*k*the DMRG solution basically provides the Full-CI limit since the

*a priori*set minimum number of block states

*M*

_{min}≃ 64 already leads to a very low truncation error. Therefore, the error of the DMRG-TCCSD is dominated by the methodological error. For

*k*> 15 the effect of the DMRG truncation error becomes visible and for large

*k*the overall error is basically determined by the DMRG solution. For larger

*δε*

_{Tr}between 10

^{–4}and 10

^{–5}the DMRG-TCCSD error shows a minimum with respect to

*k*. This is exactly the expected trend, since the CCSD method fails to capture static correlation while DMRG requires large bond dimension to recover dynamic correlations, i.e., a low truncation error threshold. In addition, the error minima for different truncation error thresholds

*δε*

_{Tr}happen to be around the same

*k*values. This has an important practical consequence: the optimal

*k*-split can be determined by performing cheap DMRG-TCCSD calculations using large DMRG truncation error threshold as a function of

*k*.

*Δ*

*E*

_{GS}has a high peak for 9 <

*k*< 16. This can be explained by the splitting of the FCI space since this yields that the correlation from external orbitals with CAS orbitals is ignored. Thus, we also performed calculations for

*δε*

_{Tr}= 10

^{–5}using a CAS formed by taking

*k*orbitals according to increasing values of the single orbital entropy values in order to demonstrate the importance of the CAS extension. The corresponding error profile as a function of

*k*near the equilibrium geometry is shown in Figure 4 labeled by CAS

^{↑}. As expected, the improvement of DMRG-TCCSD is marginal compared to CCSD up to a very large

*k*≃ 23 split since ψ

_{DMRG}

^{CAS}differs only marginally from ψ

_{HF}.

#### B.4. Numerical Investigation on CAS-ext Correlations

*k*values the most important orbitals, i.e., those with the largest entropies, are included in the CAS. In Figure 7, the sorted values of the mutual information obtained by DMRG(

*k*) for 9 ≤

*k*≤ 28 is shown on a semilogarithmic scale. It is apparent from the figure that the largest values of

*I*

_{i|j}change only slightly with increasing

*k*, thus static correlations are basically included for all restricted CAS. The exponential tail of

*I*

_{i|j}corresponding to dynamic correlations, however, becomes more visible only for larger

*k*values. We conclude, for a given

*k*split the DMRG method computes the static correlations efficiently and the missing tail of the mutual information with respect to the full orbital space (

*k*= 28) calculation is captured by the TCC scheme.

*S*(ρ

_{CAS(k)}) as a function of

*k*. Here ρ

_{CAS(k)}is formed by a partial trace on the external part of |Ψ

_{DMRG}

^{FCI}⟩. The block entropy is shown in Figure 8a. The block entropy decays monotonically for

*k*> 7, i.e, the correlations between the CAS and the external part vanish with increasing

*k*. In contrast to this, when an ordering according to CAS

^{↑}is used the correlation between CAS and external part remains always strong, i.e., some of the highly correlated orbitals are distributed among the CAS and the external part. Nevertheless, both curves are smooth and they cannot explain the error profile shown in Figure 4.

#### B.5. Numerical Values for the Amplitude Error Analysis

*N*/2 ≤

*k*≤

*K*but in terms of the CC amplitudes. Therefore, we also present a more detailed description of eq 10 in section IV which includes the following terms:

*valid index-pairs*are μ = (

**,**

*i***), with**

*a***= (**

*i**i*

_{1}, ...,

*i*

_{n}) ∈ {1, ...,

*N*/2}

^{n}, and

**= (**

*a**a*

_{1}, ...,

*a*

_{n}) ∈ {

*N*/2 + 1, ...,

*K*}

^{n}. The excitation rank is given by |μ| =

*n*where

*n*= 1 stands for singles,

*n*= 2 for doubles, and so on. The μ values are the labels of excitation operators τ̂

_{i}

^{a}≔

*â*

_{a}

^{†}

*â*

_{i}, and τ̂

_{i1,...,in}

^{a1,...,an}≔ τ̂

_{in}

^{an}... τ̂

_{i1}

^{a1}. The corresponding amplitudes are given as

*t*

_{i1,...,in}

^{a1,...,an}. For invalid index-pairs, i.e., index-pairs that are out of range, the amplitudes are always zero. The various amplitudes appearing in eq 11 are calculated according to the following rules:

(1) | The tensor (12) i, i_{1}, i_{2} ∈ {1, ..., N/2} and a, a_{1}, a_{2} ∈ {N/2 + 1, ..., k}. | ||||

(2) | The tensor | ||||

(3) | The tensor | ||||

(4) | The tensor |

*e*(

*k*, δ

_{εTr}) as a function of

*k*of the nitrogen dimer near the equilibrium bond length. Note that the quantitative behavior is quite robust with respect to the bond dimension since the values only differ marginally. We emphasize that the error contribution in Figure 8 is dominated by second term in eq 11 since this is an order of magnitude larger than the contribution from the first and third terms in eq 11, respectively. The first term in eq 11 is furthermore related to the usual T1 diagnostic in CC, (110) so it is not a surprise that a small value, ∼10

^{–3}, was found. Comparing this error profile to the one shown in Figure 4 we can understand the irregular behavior and the peak in the error in

*Δ*

*E*

_{GS}between

*k*= 9 and 17, and the other peaks for

*k*> 17 but the error minimum found for

*k*= 19 remains unexplained. Furthermore, we can conclude from Figure 8b that the quotient

*Δ*

*E*

_{GS}(

*k*)/

*e*(

*k*, δ

_{εTr}) is not constant. This indicates that the constants involved in section IV in particular the constant in eq 10 in section IV.D is indeed

*k*-dependent.

## VI. Conclusion

*a priori*error estimate for the DMRG-TCC method, which aligns the error behavior of the DMRG-TCC method with variational methods except for the upper bound condition. We emphasize that the DMRG-TCC solution depends strongly on the CAS choice. Throughout the analysis we neglected this dependence as we assumed an optimal CAS choice as indicated in section IV.A. The explicit consideration of this dependence in the performed error analysis carries many mathematical challenges, which are part of our current research. Therefore, we extended this work with a numerical study of the

*k*-dependence of the DMRG-TCCSD error which showed also that the constants involved in the error estimation are most likely

*k*-dependent. This stresses the importance of further mathematical work to include this dependence explicitly in the analysis.

*k*-dependence of the DMRG-TCCSD error revealed that the predicted trend in section IV.A is correct. We demonstrated that the error indeed first decays (7 ≤

*k*≤ 9) and then increases again (25 ≤

*k*≤ 28) for low-rank approximations, i.e., 10

^{–4}respectively 10

^{–5}. This aligns with the theoretical prediction based on the properties of the DMRG and single reference CC method. Additional to this general trend, the error shows oscillations. A first hypothesis is that this behavior is related to the ignored correlations in the transition

*k*→

*k*+ 1. However, this was not able to be proven so far using entropy based measures but a similar irregular behavior can be detected by a cluster amplitude error analysis. Furthermore, such oscillations can be related to a bad reference function. Nonetheless, this scenario has here been ruled out since the Hartree–Fock determinant was found to be dominant in the CAS solution, i.e., the weight of the Hartree–Fock had largest weight in the CAS solution. The irregular behavior of the error minimum found for the DMRG-TCCSD method, therefore, could not be explained within this article and is left for future work. Despite the unknown reason for this behavior, we note that the error minima are fairly robust with respect to the bond dimension. Hence, the DMRG-TCCSD method can be extended with a screening process using low bond-dimension approximations to detect possible error minima.

*k*= 9) yields a significant improvement of the energy and that the energies for all three geometries and all CAS choices outrun the single-reference CC method. In addition, the DMRG-TCCSD method avoids the breakdown of the CC approach even for multireference (strongly correlated) systems and, using concepts of quantum information theory, allows an efficient routine application of the method. Since the numerical error study showed a significant improvement for small CAS, we suspect the DMRG-TCCSD method to be of great use for larger systems with many strongly correlated orbitals as well as a many dynamically correlated orbitals. (1,2)

## Acknowledgments

This work has received funding from the *Research Council of Norway* (RCN) under CoE Grant No. 262695 (Hylleraas Centre for Quantum Molecular Sciences), from ERC-STG-2014 under grant No. 639508, from the *Hungarian National Research, Development and Innovation Office* (NKFIH) through Grant No. K120569, from the *Hungarian Quantum Technology National Excellence Program* (Project No. 2017-1.2.1-NKP-2017-00001), from the *Czech Science Foundation* (grants no. 16-12052S, 18-18940Y, and 18-24563S), and the *Czech Ministry of Education, Youth and Sports* (project no. LTAUSA17033). Ö.L. also acknowledges financial support form the Alexander von Humboldt Foundation. F.M.F, A.L., Ö.L., and M.A.C. are grateful for the mutual hospitality received during their visits at the Wigner Research Center for Physics in Budapest and the Hylleraas Centre for Quantum Molecular Sciences. Ö.L. and J.P. acknowledge useful discussions with Marcel Nooijen.

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The following topics are discussed: Exponential era of electron correlation theory; Genuine MR CC theory in Hilbert space and in Fock space; Alternative MR CC methods. Numerical illustrations are presented.**10**Köhn, A.; Hanauer, M.; Mück, L. A.; Jagau, T.-C.; Gauss, J. State-specific multireference coupled-cluster theory.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2013,*3*, 176– 197, DOI: 10.1002/wcms.1120Google ScholarThere is no corresponding record for this reference.**11**Jeziorski, B.; Monkhorst, H. J. Coupled-cluster method for multideterminantal reference states.*Phys. Rev. A: At., Mol., Opt. Phys.*1981,*24*, 1668– 1681, DOI: 10.1103/PhysRevA.24.1668Google Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3MXmtV2qtrs%253D&md5=0434d85ec80902962638bf1a465dbc62Coupled-cluster method for multideterminantal reference statesJeziorski, Bogumil; Monkhorst, Hendrik J.Physical Review A: Atomic, Molecular, and Optical Physics (1981), 24 (4), 1668-81CODEN: PLRAAN; ISSN:0556-2791.A general coupled-cluster method valid for arbitrary multideterminantal ref. states is formulated. The resulting cluster expansion for the wave function is a generalization of that introduced by Silverstone and Sinanoglu (1966). The connected nature of the cluster operators, and the effective interaction is proven in the case when the ref. space is complete, i.e., is invariant under unitary transformations of partly occupied orbitals. For incomplete ref. spaces the disconnected terms appearing in the effective interaction are properly generated by the coupled-cluster theory. Approx. schemes for solving coupled-cluster equations are proposed and their relation with perturbation theory is briefly discussed.**12**Lee, J.; Small, D. W.; Epifanovsky, E.; Head-Gordon, M. Coupled-cluster valence-bond singles and doubles for strongly correlated systems: Block-tensor based implementation and application to oligoacenes.*J. Chem. Theory Comput.*2017,*13*, 602– 615, DOI: 10.1021/acs.jctc.6b01092Google Scholar12https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXmslSktw%253D%253D&md5=f12e7012e633ea5551d1c5ab3c7ac617Coupled-Cluster Valence-Bond Singles and Doubles for Strongly Correlated Systems: Block-Tensor Based Implementation and Application to OligoacenesLee, Joonho; Small, David W.; Epifanovsky, Evgeny; Head-Gordon, MartinJournal of Chemical Theory and Computation (2017), 13 (2), 602-615CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We demonstrate a block-tensor based implementation of coupled-cluster valence-bond singles and doubles (CCVB-SD) [Small, D. W.; Head-Gordon M. J. Chem. Phys.2012, 137, 114103] which is a simple modification to restricted CCSD (RCCSD) that provides a qual. correct description of valence correlations even in strongly correlated systems. We derive the Λ-equation of CCVB-SD and the corresponding unrelaxed d. matrixes. The resulting prodn.-level implementation is applied to oligoacenes, correlating up to 318 electrons in 318 orbitals. CCVB-SD shows a qual. agreement with exact methods for short acenes and reaches the bulk limit of oligoacenes in terms of natural orbital occupation nos., whereas RCCSD shows nonvariational behavior even for relatively short acenes. A significant redn. in polyradicaloid character is found when correlating all valence electrons instead of only the π-electrons.**13**Lindgren, I.; Mukherjee, D. On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces.*Phys. Rep.*1987,*151*, 93– 127, DOI: 10.1016/0370-1573(87)90073-1Google Scholar13https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2sXlsVWht7k%253D&md5=35a97c30d3e61cef5fce8536dc6e45f3On the connectivity criteria in the open-shell coupled-cluster theory for general model spacesLindgren, Ingvar; Mukherjee, DebashisPhysics Reports (1987), 151 (2), 93-127CODEN: PRPLCM; ISSN:0370-1573.A review with 60 refs. includes discussion of current theor. status of linked-cluster theorem, connectivity of the cluster amplitudes, and the effective Hamiltonians.**14**Lindgren, I. Linked-Diagram and Coupled-Cluster Expansions for Multi-Configurational, Complete and Incomplete Model Spaces.*Phys. Scr.*1985,*32*, 291, DOI: 10.1088/0031-8949/32/4/009Google ScholarThere is no corresponding record for this reference.**15**Mukherjee, D.; Moitra, R. K.; Mukhopadhyay, A. Applications of a non-perturbative many-body formalism to general open-shell atomic and molecular problems: calculation of the ground and the lowest π-π* singlet and triplet energies and the first ionization potential of trans-butadiene.*Mol. Phys.*1977,*33*, 955– 969, DOI: 10.1080/00268977700100871Google Scholar15https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2sXlsVWnu7w%253D&md5=087024d694b5d2045c65d015d676afe1Applications of a nonperturbative many-body formalism to general open-shell atomic and molecular problems: calculation of the ground and the lowest π-π* singlet and triplet energies and the first ionization potential of trans-butadieneMukherjee, Debashis; Moitra, Raj Kumar; Mukhopadhyay, AtriMolecular Physics (1977), 33 (4), 955-69CODEN: MOPHAM; ISSN:0026-8976.The definition of the cluster expansion operator in the nonperturbative open-shell many-body formalism (1975) in generalized to allow incorporation into the model space of determinants differing widely in energy and in their no. of electrons. This is useful for calcg. difference energies in at. and mol. systems. The generalized scheme is used to calc. the energies of the ground, lowest π-π* singlet, lowest π-π* triplet, and singly ionized states and the 1st ionization potential of trans-butadiene. A single cluster expansion operator is employed for all states. Results are given for a general and several restricted model spaces. The calcns. agree with CI results complete in the chosen basis.**16**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method with optimized reference state.*Int. J. Quantum Chem.*1984,*26*, 267– 291, DOI: 10.1002/qua.560260827Google ScholarThere is no corresponding record for this reference.**17**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. I. General formalism.*Phys. Rev. A: At., Mol., Opt. Phys.*1985,*32*, 725– 742, DOI: 10.1103/PhysRevA.32.725Google Scholar17https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2sjotlKisA%253D%253D&md5=6944b99b1a4fdd6f97d622e5ad54d847Coupled-cluster method in Fock space. I. General formalismStolarczyk; MonkhorstPhysical review. A, General physics (1985), 32 (2), 725-742 ISSN:0556-2791.There is no expanded citation for this reference.**18**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. II. Brueckner-Hartree-Fock method.*Phys. Rev. A: At., Mol., Opt. Phys.*1985,*32*, 743, DOI: 10.1103/PhysRevA.32.743Google ScholarThere is no corresponding record for this reference.**19**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. III. On similarity transformation of operators in Fock space.*Phys. Rev. A: At., Mol., Opt. Phys.*1988,*37*, 1908, DOI: 10.1103/PhysRevA.37.1908Google ScholarThere is no corresponding record for this reference.**20**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. IV. Calculation of expectation values and transition moments.*Phys. Rev. A: At., Mol., Opt. Phys.*1988,*37*, 1926, DOI: 10.1103/PhysRevA.37.1926Google ScholarThere is no corresponding record for this reference.**21**Stolarczyk, L. Z.; Monkhorst, H. J. Quasiparticle Fock-space coupled-cluster theory.*Mol. Phys.*2010,*108*, 3067– 3089, DOI: 10.1080/00268976.2010.518981Google Scholar21https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhsVyhsbfK&md5=50f59a083c979897f1e8ead8a339125bQuasiparticle Fock-space coupled-cluster theoryStolarczyk, Leszek Z.; Monkhorst, Hendrik J.Molecular Physics (2010), 108 (21-23), 3067-3089CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)The quasiparticle Fock-space coupled-cluster (QFSCC) theory, introduced by us in 1985, is described. This is a theory of many-electron systems which uses the second-quantisation formalism based on the algebraic approxn.: one chooses a finite spin-orbital basis, and builds a fermionic Fock space to represent all possible antisym. electronic states of a given system. The algebraic machinery is provided by the algebra of linear operators acting in the Fock space, generated by the fermion (creation and annihilation) operators. The Fock-space Hamiltonian operator then dets. the system's stationary states and their energies. Within the QFSCC theory, the Fock space and its operator algebra are subject to a unitary transformation which effectively changes electrons into some fermionic quasiparticles. A generalisation of the coupled-cluster method is achieved by enforcing the principle of quasiparticle-no. conservation. The emerging quasiparticle model of many-electron systems offers useful phys. insights and computational effectiveness. The QFSCC theory requires a substantial reformulation of the traditional second-quantisation language, by making full use of the algebraic properties of the Fock space and its operator algebra. In particular, the role of operators not conserving the no. of electrons (or quasiparticles) is identified.**22**Jeziorski, B.; Monkhorst, H. J. Coupled-cluster method for multideterminantal reference states.*Phys. Rev. A: At., Mol., Opt. Phys.*1981,*24*, 1668, DOI: 10.1103/PhysRevA.24.1668Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3MXmtV2qtrs%253D&md5=0434d85ec80902962638bf1a465dbc62Coupled-cluster method for multideterminantal reference statesJeziorski, Bogumil; Monkhorst, Hendrik J.Physical Review A: Atomic, Molecular, and Optical Physics (1981), 24 (4), 1668-81CODEN: PLRAAN; ISSN:0556-2791.A general coupled-cluster method valid for arbitrary multideterminantal ref. states is formulated. The resulting cluster expansion for the wave function is a generalization of that introduced by Silverstone and Sinanoglu (1966). The connected nature of the cluster operators, and the effective interaction is proven in the case when the ref. space is complete, i.e., is invariant under unitary transformations of partly occupied orbitals. For incomplete ref. spaces the disconnected terms appearing in the effective interaction are properly generated by the coupled-cluster theory. Approx. schemes for solving coupled-cluster equations are proposed and their relation with perturbation theory is briefly discussed.**23**Datta, D.; Mukherjee, D. An explicitly spin-free compact open-shell coupled cluster theory using a multireference combinatoric exponential ansatz: Formal development and pilot applications.*J. Chem. Phys.*2009,*131*, 044124, DOI: 10.1063/1.3185356Google Scholar23https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtVCnsbfJ&md5=737a2ddde25dfccaf06391516d1c5794An explicitly spin-free compact open-shell coupled cluster theory using a multireference combinatoric exponential ansatz: Formal development and pilot applicationsDatta, Dipayan; Mukherjee, DebashisJournal of Chemical Physics (2009), 131 (4), 044124/1-044124/30CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a comprehensive account of an explicitly spin-free compact state-universal multireference coupled cluster (CC) formalism for computing the state energies of simple open-shell systems, e.g., doublets and biradicals, where the target open-shell states can be described by a few configuration state functions spanning a model space. The cluster operators in this formalism are defined in terms of the spin-free unitary generators with respect to the common closed-shell component of all model functions (core) as vacuum. The spin-free cluster operators are either closed-shell-like n hole-n particle excitations (denoted by Tμ) or involve excitations from the doubly occupied (nonvalence) orbitals to the singly occupied (valence) orbitals (denoted by Seμ). In addn., there are cluster operators with exchange spectator scatterings involving the valence orbitals (denoted by Sreμ). We propose a new multireference cluster expansion ansatz for the wave operator with the above generally noncommuting cluster operators which essentially has the same phys. content as the Jeziorski-Monkhorst ansatz with the commuting cluster operators defined in the spin-orbital basis. The Tμ operators in our ansatz are taken to commute with all other operators, while the Seμ and Sreμ operators are allowed to contract among themselves through the spectator valence orbitals. An important innovation of this ansatz is the choice of an appropriate automorphic factor accompanying each contracted composite of cluster operators in order to ensure that each distinct excitation generated by this composite appears only once in the wave operator. The resulting CC equations consist of two types of terms: A "direct" term and a "normalization" term contg. the effective Hamiltonian operator. It is emphasized that the direct term is almost quartic in the cluster amplitudes, barring only a handful of terms and termination of the normalization term depends on the valence rank of the effective Hamiltonian operator and the excitation rank of the cluster operators at which the theory is truncated. Illustrative applications are presented by computing the state energies of neutral doublet radicals and doublet mol. cations and ionization energies of neutral mols. and comparing our results with the other open-shell CC theories, benchmark full CI results (when available) in the same basis, and the exptl. results. Highly encouraging results show the efficacy of the method. (c) 2009 American Institute of Physics.**24**Evangelista, F. A.; Allen, W. D.; Schaefer, H. F., III High-order excitations in state-universal and state-specific multireference coupled cluster theories: Model systems.*J. Chem. Phys.*2006,*125*, 154113, DOI: 10.1063/1.2357923Google Scholar24https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtFeksLrI&md5=1ce0b9045da90febeaced130d082bd0cHigh-order excitations in state-universal and state-specific multireference coupled cluster theories: Model systemsEvangelista, Francesco A.; Allen, Wesley D.; Schaefer, Henry F., IIIJournal of Chemical Physics (2006), 125 (15), 154113/1-154113/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)For the first time high-order excitations (n > 2) have been studied in three multireference couple cluster (MRCC) theories built on the wave operator formalism: (1) the state-universal (SU) method of Jeziorski and Monkhorst (JM) (2) the state-specific Brillouin-Wigner (BW) coupled cluster method, and (3) the state-specific MRCC approach of Mukherjee (Mk). For the H4, P4, BeH2, and H8 models, multireference coupled cluster wave functions, with complete excitations ranging from doubles to hextuples, have been computed with a new arbitrary-order string-based code. Comparison is then made to corresponding single-ref. coupled cluster and full CI (FCI) results. For the ground states the BW and Mk methods are found, in general, to provide more accurate results than the SU approach at all levels of truncation of the cluster operator. The inclusion of connected triple excitations reduces the nonparallelism error in singles and doubles MRCC energies by a factor of 2-10. In the BeH2 and H8 models, the inclusion of all quadruple excitations yields abs. energies within 1 kcal mol-1 of the FCI limit. While the MRCC methods are very effective in multireference regions of the potential energy surfaces, they are outperformed by single-ref. CC when one electronic configuration dominates.**25**Piecuch, P.; Paldus, J. Orthogonally spin-adapted multi-reference Hilbert space coupled-cluster formalism: Diagrammatic formulation.*Theor. Chim. Acta*1992,*83*, 69– 103, DOI: 10.1007/BF01113244Google ScholarThere is no corresponding record for this reference.**26**Kucharski, S.; Balková, A.; Szalay, P.; Bartlett, R. J. Hilbert space multireference coupled-cluster methods. II. A model study on H8.*J. Chem. Phys.*1992,*97*, 4289– 4300, DOI: 10.1063/1.463931Google Scholar26https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38XlvFylur0%253D&md5=144b60da7b81e1756bba0d1c88b90c7eHilbert-space-multireference-coupled-cluster methods. II. A model study on the octaatomic hydrogen clusterKucharski, S. A.; Balkova, A.; Szalay, P. G.; Bartlett, Rodney J.Journal of Chemical Physics (1992), 97 (6), 4289-300CODEN: JCPSA6; ISSN:0021-9606.The performance of various CC approaches using both single and multideterminantal refs. is investigated for the (quasi-)degenerate states of mol. systems, where inclusion of higher excitations (or equivalently nondynamic correlation) proves to be needed. The prototype system H8 represents an adequate model for our study, where we can vary the degree of degeneracy from a completely degenerate situation to a nondegenerate one in a continuous way. To obtain a reliable benchmark for CC results, the full CI (FCI) and large-scale complete active space CI (CAS CI) calcns., resp., are performed for a variety of geometries and states. The convergence of the approx. single ref. CC approaches is extremely sensitive to the level of degeneracies involved. In the nondegenerate case, the std. CC method with single and double excitations is found to be quite satisfactory; in the (quasi-)degenerate situations, however, the inclusion of triple excitations and noniterative quadruple excitations is needed to furnish semiquant. values of correlation energies. The alternative treatment of nondynamic correlation using a multideterminantal Hilbert space coupled-cluster (MRCC) method demonstrates the power of this approach, which provides a balanced description of both dynamic and nondynamic correlation in the degenerate region for all the investigated states of H8. Its convergence for nondegenerate situations, however, is less satisfactory, being affected by an intruder state problem.**27**Balková, A.; Kucharski, S.; Meissner, L.; Bartlett, R. J. A Hilbert space multi-reference coupled-cluster study of the H 4 model system.*Theor. Chim. Acta*1991,*80*, 335– 348, DOI: 10.1007/BF01117417Google Scholar27https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXmvVOgu74%253D&md5=7290c8e9f0e02182cda0277fbb514a9eA Hilbert space multi-reference coupled-cluster study of the hydrogen tetraatomic molecule model systemBalkova, A.; Kucharski, S. A.; Meissner, L.; Bartlett, Rodney J.Theoretica Chimica Acta (1991), 80 (4-5), 335-48CODEN: TCHAAM; ISSN:0040-5744.Employing the Hilbert space ansatz, a fully quadratic coupled-cluster method with a multidimensional ref. space is applied to a DZP basis study of the model system, H4. The ref. space is described by two to four configurations at the level of single and double excitations, and single and double excitation operations are included in the expansions for the cluster and wave operator through quadratic terms. The performance of quadratic MRCCSD is investigated for the ground and three excited states of the H4 system consisting of two stretched hydrogen mols. in a trapezoidal configuration where the degree of quasidegeneracy is varied from a nondegenerate situation to a completely degenerate one. Compared to full CI, in the highly degenerate region, the MRCCSD works quite well. In less degenerate regions, the accuracy is less satisfactory.**28**Kinoshita, T.; Hino, O.; Bartlett, R. J. Coupled-cluster method tailored by configuration interaction.*J. Chem. Phys.*2005,*123*, 074106, DOI: 10.1063/1.2000251Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpsleht7w%253D&md5=7c005bedda722c576125ea4cec764785Coupled-cluster method tailored by configuration interactionKinoshita, Tomoko; Hino, Osamu; Bartlett, Rodney J.Journal of Chemical Physics (2005), 123 (7), 074106/1-074106/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A method is presented which combines coupled cluster (CC) and CI to describe accurately potential-energy surfaces (PESs). We use the cluster amplitudes extd. from the complete active space CI calcn. to manipulate nondynamic correlation to tailor a single ref. CC theory (TCC). The dynamic correlation is then incorporated through the framework of the CC method. We illustrate the method by describing the PESs for HF, H2O, and N2 mols. which involve single, double, and triple bond-breaking processes. To the dissocn. limit, this approach yields far more accurate PESs than those obtained from the conventional CC method and the addnl. computational cost is negligible compared with the CC calcn. steps. We anticipate that TCC offers an effective and generally applicable approach for many problems.**29**Fang, T.; Shen, J.; Li, S. Block correlated coupled cluster method with a complete-active-space self-consistent-field reference function: The formula for general active spaces and its applications for multibond breaking systems.*J. Chem. Phys.*2008,*128*, 224107, DOI: 10.1063/1.2939014Google Scholar29https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXntFyntbc%253D&md5=3f5bd5d601d324cb38268b7aac738127Block correlated coupled cluster method with a complete-active-space self-consistent-field reference function: The formula for general active spaces and its applications for multibond breaking systemsFang, Tao; Shen, Jun; Li, ShuhuaJournal of Chemical Physics (2008), 128 (22), 224107/1-224107/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The block correlated coupled cluster (BCCC) theory is developed for a general complete-active-space (CAS) self-consistent-field ref. function. By truncating the cluster operator up to the four-block correlation level, we derive the spin orbital formulation of the CAS-BCCC4 approach. The CAS-BCCC4 approach is invariant to sep. unitary transformation within active, occupied, and virtual orbitals. We have implemented the approach and applied this approach to describe the potential energy surfaces for bond breaking processes in C2 and N2 and for a simultaneous double bond dissocn. in H2O. Numerical results show that the CAS-BCCC4 approach provides quite accurate descriptions for the entire dissocn. process in each of the studied systems. The overall performance of the present approach is found to be better than that of the internally contracted multireference CI singles and doubles or complete-active-space second-order perturbation theory. The size-extensivity error is found to be relatively small for N2. (c) 2008 American Institute of Physics.**30**Datta, D.; Kong, L.; Nooijen, M. A state-specific partially internally contracted multireference coupled cluster approach.*J. Chem. Phys.*2011,*134*, 214116, DOI: 10.1063/1.3592494Google Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXntV2ktro%253D&md5=2d2bda10ce239ab6181bd8b886edbe8aA state-specific partially internally contracted multireference coupled cluster approachDatta, Dipayan; Kong, Liguo; Nooijen, MarcelJournal of Chemical Physics (2011), 134 (21), 214116/1-214116/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A state-specific partially internally contracted multireference coupled cluster approach is presented for general complete active spaces with arbitrary no. of active electrons. The dominant dynamical correlation is included via an exponential parametrization of internally contracted cluster operators (T) which excite electrons from a multideterminantal ref. function. The remaining dynamical correlation and relaxation effects are included via a diagonalization of the transformed Hamiltonian ‾H = e-THeT in the multireference CI singles space in an uncontracted fashion. A new set of residual equations for detg. the internally contracted cluster amplitudes is proposed. The second quantized matrix elements of ‾H, expressed using the extended normal ordering of Kutzelnigg and Mukherjee, are used as the residual equations without projection onto the excited configurations. These residual equations, referred to as the many-body residuals, do not have any near-singularity and thus, should allow one to solve all the amplitudes without discarding any. There are some relatively minor remaining convergence issues that may arise from an attempt to solve all the amplitudes and an initial anal. is provided in this paper. Applications to the bond-stretching potential energy surfaces for N2, CO, and the low-lying electronic states of C2 indicate clear improvements of the results using the many-body residuals over the conventional projected residual equations. (c) 2011 American Institute of Physics.**31**Hanauer, M.; Köhn, A. Pilot applications of internally contracted multireference coupled cluster theory, and how to choose the cluster operator properly.*J. Chem. Phys.*2011,*134*, 204111, DOI: 10.1063/1.3592786Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXms1SktbY%253D&md5=9697e08b7b9e13a092dc4536a41f600aPilot applications of internally contracted multireference coupled cluster theory, and how to choose the cluster operator properlyHanauer, Matthias; Koehn, AndreasJournal of Chemical Physics (2011), 134 (20), 204111/1-204111/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The internally contracted multireference coupled cluster (icMRCC) method allows a highly accurate description of both static and dynamic correlation with a computational scaling similar to single ref. coupled cluster theory. The authors show that the method can lose its orbital invariance and size consistency when no special care is taken in the elimination of redundant excitations. Using the BeH2 model system, four schemes are compared which differ in their treatment of linear dependencies between excitations of different rank (such as between singles and doubles). While the energy curves agree within tens of μEh when truncating the cluster operator at double excitations (icMRCCSD), inclusion of triple excitations (icMRCCSDT) leads to significant differences of more than 1 mEh. One scheme clearly yields the best results, while the others even turn out to be not size consistent. The former procedure uses genuine single and double excitations and discards those linear combinations of (spectator) double and triple excitations which have the same effect on the ref. function. With this approach, the equil. structure and harmonic vibrational frequencies of ozone obtained with icMRCCSDT are in excellent agreement with CCSDTQ. The authors further apply icMRCC methods to potential energy surfaces of HF, LiF, N2, and to the singlet-triplet splitting of benzynes. In particular, the latter calcns. have been made possible by implementing the method with the proper formal scaling using automated techniques. (c) 2011 American Institute of Physics.**32**Evangelista, F. A.; Gauss, J. An orbital-invariant internally contracted multireference coupled cluster approach.*J. Chem. Phys.*2011,*134*, 114102, DOI: 10.1063/1.3559149Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXjt1WnsLg%253D&md5=192193b11256b4ec9766accd14f98683An orbital-invariant internally contracted multireference coupled cluster approachEvangelista, Francesco A.; Gauss, JuergenJournal of Chemical Physics (2011), 134 (11), 114102/1-114102/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have formulated and implemented an internally contracted multireference coupled cluster (ic-MRCC) approach aimed at solving two of the problems encountered in methods based on the Jeziorski-Monkhorst ansatz: (i) the scaling of the computational and memory costs with respect to the no. of refs., and (ii) the lack of invariance of the energy with respect to rotations among active orbitals. The ic-MRCC approach is based on a straightforward generalization of the single-ref. coupled cluster ansatz in which an exponential operator is applied to a multiconfigurational wave function. The ic-MRCC method truncated to single and double excitations (ic-MRCCSD) yields very accurate potential energy curves in benchmark computations on the Be + H2 insertion reaction, the dissocn. of hydrogen fluoride, and the sym. double dissocn. of water. Approxns. of the ic-MRCC theory in which the Baker-Campbell-Hausdorff expansion is truncated up to a given no. of commutators are found to converge quickly to the full theory. In our tests, two commutators are sufficient to recover a total energy within 0.5 mEh of the full ic-MRCCSD method along the entire potential energy curve. A formal anal. shows that the ic-MRCC method is invariant with respect to rotation among active orbitals, and that the orthogonalization procedure used to produce the set of linearly independent excitation operators plays a crucial role in guaranteeing the invariance properties. The orbital invariance was confirmed in numerical tests. Moreover, approximated versions of the ic-MRCC theory based on a truncated Baker-Campbell-Hausdorff expansion, preserve the orbital invariance properties of the full theory. (c) 2011 American Institute of Physics.**33**Lyakh, D. I.; Ivanov, V. V.; Adamowicz, L. Automated generation of coupled-cluster diagrams: Implementation in the multireference state-specific coupled-cluster approach with the complete-active-space reference.*J. Chem. Phys.*2005,*122*, 024108, DOI: 10.1063/1.1824897Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXislaiug%253D%253D&md5=7d69842cc09bb81a4a3dc980d5cbd4dbAutomated generation of coupled-cluster diagrams: Implementation in the multireference state-specific coupled-cluster approach with the complete-active-space referenceLyakh, Dmitry I.; Ivanov, Vladimir V.; Adamowicz, LudwikJournal of Chemical Physics (2005), 122 (2), 024108/1-024108/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An algorithm for generation of the spin-orbital diagrammatic representation, the corresponding algebraical formulas, and the computer code of the coupled-cluster (CC) method with an arbitrary level of the electronic excitations has been developed. The method was implemented in the general case as well as for specific application in the state-specific multireference coupled-cluster theory (SSMRCC) based on the concept of a "formal ref. state." The algorithm was tested in SSMRCC calcns. describing dissocn. of a single bond and in calcns. describing simultaneous dissocn. of two single bonds-the problem requiring up to six-particle excitations in the CC operator.**34**Hanrath, M. An exponential multireference wave-function Ansatz.*J. Chem. Phys.*2005,*123*, 084102, DOI: 10.1063/1.1953407Google Scholar34https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpvFClu7w%253D&md5=2ea2eb79a0daeaef1b48ff34c888dfd8An exponential multireference wave-function AnsatzHanrath, MichaelJournal of Chemical Physics (2005), 123 (8), 084102/1-084102/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An exponential multireference wave-function Ansatz is formulated. In accordance with the state universal coupled-cluster Ansatz of Jeziorski and Monkhorst [Phys. Rev. A 24, 1668 (1981)] the approach uses a ref. specific cluster operator. In order to achieve state selectiveness the excitation- and ref.-related amplitude indexing of the state universal Ansatz is replaced by an indexing which is based on excited determinants. There is no ref. determinant playing a particular role. The approach is size consistent, coincides with traditional single-ref. coupled cluster if applied to a single-ref., and converges to full CI with an increasing cluster operator excitation level. Initial applications on BeH2, CH2, Li2, and nH2 are reported.**35**Pittner, J.; Nachtigall, P.; Čársky, P.; Hubač, I. State-Specific Brillouin- Wigner Multireference Coupled Cluster Study of the Singlet- Triplet Separation in the Tetramethyleneethane Diradical.*J. Phys. Chem. A*2001,*105*, 1354– 1356, DOI: 10.1021/jp0032199Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXnvVOkug%253D%253D&md5=c653f767b0aba2538b8720b0dc52a5b1State-Specific Brillouin-Wigner Multireference Coupled Cluster Study of the Singlet-Triplet Separation in the Tetramethyleneethane DiradicalPittner, Jiri; Nachtigall, Petr; Carsky, Petr; Hubac, IvanJournal of Physical Chemistry A (2001), 105 (8), 1354-1356CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The potential energy curves for the twisting of tetramethyleneethane (I) in its lowest singlet and triplet states were calcd. by the state-specific two-ref. Brillouin-Wigner coupled-cluster method with single and double excitations. The calcd. potential energy curves are essentially the same as those obtained by the two-determinant CCSD method, and they are also in agreement with the previously reported d. functional theory results. Our data bring support for the previously suggested interpretation of exptl. data on I in the gas phase and in the matrix.**36**Hubač, I.; Wilson, S. On the use of Brillouin-Wigner perturbation theory for many-body systems.*J. Phys. B: At., Mol. Opt. Phys.*2000,*33*, 365, DOI: 10.1088/0953-4075/33/3/306Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXhsVOkt7o%253D&md5=fe17ef2812fd715a55747ac599265353On the use of Brillouin-Wigner perturbation theory for many-body systemsHubac, I.; Wilson, S.Journal of Physics B: Atomic, Molecular and Optical Physics (2000), 33 (3), 365-374CODEN: JPAPEH; ISSN:0953-4075. (Institute of Physics Publishing)The use of Brillouin-Wigner perturbation theory in describing many-body systems is critically re-examd.**37**Hubač, I.; Pittner, J.; Čársky, P. Size-extensivity correction for the state-specific multireference Brillouin-Wigner coupled-cluster theory.*J. Chem. Phys.*2000,*112*, 8779– 8784, DOI: 10.1063/1.481493Google Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXivFClsLk%253D&md5=346ef344a80a5066f6242bc9bb0093a4Size-extensivity correction for the state-specific multireference Brillouin-Wigner coupled-cluster theoryHubac, Ivan; Pittner, Jiri; Carsky, PetrJournal of Chemical Physics (2000), 112 (20), 8779-8784CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a simple a posteriori correction for the state-specific multireference Brillouin-Wigner coupled-cluster (MR BWCCSD) theory, which eliminates its size-extensivity error. In the converged amplitudes we drop terms that were identified to be responsible for the lack of size extensivity. We performed MR BWCCSD calcns. with this correction on CH2, SiH2, twisted ethylene, F2, and ozone that are all, from the computational point of view, typical representatives of two-ref. problems. Comparison with rigorously size-extensive calcns. and expt. shows that the size-extensivity error of the cor. MR BWCCSD is only a few tenths of kcal/mol.**38**Pittner, J.; Šmydke, J.; Čársky, P.; Hubač, I. State-specific Brillouin-Wigner multireference coupled cluster study of the F2 molecule: assessment of the a posteriori size-extensivity correction.*J. Mol. Struct.: THEOCHEM*2001,*547*, 239– 244, DOI: 10.1016/S0166-1280(01)00473-0Google Scholar38https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXltVajs7c%253D&md5=96a9e262e118ed0912e96d3b60473ce6State-specific Brillouin-Wigner multireference coupled cluster study of the F2 molecule: assessment of the a posteriori size-extensivity correctionPittner, J. V.; Smydke, J.; Carsky, P.; Hubac, I.Journal of Molecular Structure: THEOCHEM (2001), 547 (), 239-244CODEN: THEODJ; ISSN:0166-1280. (Elsevier Science B.V.)We tested a posteriori correction suggested previously for the state-specific multireference Brillouin-Wigner coupled-cluster singles and doubles (MR BW CCSD) theory, to eliminate its size-extensivity error. The correction was applied to a two-ref. BW CCSD model by using the cc-pVXZ basis sets (X = 2,3,4) and it was tested by calcg. the spectroscopic consts. of the F2 mol.**39**Fang, T.; Li, S. Block correlated coupled cluster theory with a complete active-space self-consistent-field reference function: The formulation and test applications for single bond breaking.*J. Chem. Phys.*2007,*127*, 204108, DOI: 10.1063/1.2800027Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXhsVWhu7zE&md5=c3af725bc9dd81b5d29a583075fc321fBlock correlated coupled cluster theory with a complete active-space self-consistent-field reference function: The formulation and test applications for single bond breakingFang, Tao; Li, ShuhuaJournal of Chemical Physics (2007), 127 (20), 204108/1-204108/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Block correlated coupled cluster (BCCC) theory with a complete active-space self-consistent-field (CASSCF) ref. function is presented. This theory provides an alternative multireference coupled cluster framework to describe the multireference characters of the ground-state wave functions. In this approach, a multireference block is defined to incorporate the nondynamic correlation, and all other blocks involve just a single spin orbital. The cluster operators are truncated up to the four-block correlation level, leading to the BCCC4 scheme. For a single bond breaking problem, the present CAS-BCCC4 approach with a CASSCF(2,2) ref. function computationally scales as the traditional single-ref. coupled cluster singles and doubles. We have applied the present approach to investigate the electronic structures of several model systems including H4, P4, and BeH2, and the single bond breaking processes in small systems such as F2, HF, BH, and CH4. A comparison of our results with those from full CI calcns. shows that the present approach can provide quant. descriptions for all the studied systems. The size-consistency error is found to be quite small in the dissocn. limit of diat. mols. F2, HF, and BH.**40**Chattopadhyay, S.; Mahapatra, U. S.; Mukherjee, D. Development of a linear response theory based on a state-specific multireference coupled cluster formalism.*J. Chem. Phys.*2000,*112*, 7939– 7952, DOI: 10.1063/1.481395Google Scholar40https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXivVOisro%253D&md5=0e6b4748b217a0e9eeff6bf0e3c4f63fDevelopment of a linear response theory based on a state-specific multireference coupled cluster formalismChattopadhyay, Sudip; Mahapatra, Uttam Sinha; Mukherjee, DebashisJournal of Chemical Physics (2000), 112 (18), 7939-7952CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present in this paper a linear response theory based on our recently developed state-specific multireference coupled cluster (SS-MRCC) method to compute excited state energies for systems whose ground state has a pronounced multireference character. The SS-MRCC method is built on complete active space ref. functions, and is designed to treat quasidegeneracy of varying degrees while bypassing the intruder problem. The linear response theory based on such a function [multireference coupled cluster based linear response theory (MR-CCLRT)] offers a very convenient access to computation of excited states and, in particular, to generation of potential energy surfaces (PES) for excited states where a traditional response formulation based on a single ref. theory will fail due to the quasidegeneracy at some regions of the PES and the effective Hamiltonian-based multireference response methods would be plagued by intruders. An attractive feature of the MR-CCLRT is that the computed excitation energies are size intensive in the sense that they become asymptotically equal to the sum of fragment excitation energies in the limit of noninteracting fragments. Illustrative numerical results are presented for the excited state PES of the rectangular H4 (P4) model, the trapezoidal H4 (H4) model, for Li2, and for some sample points on the excited states PES of the BeH2 complex. The ground states of all the three examples possess quasidegeneracy at some point on the PES, and there are potential intruders at some other points in the PES, and hence are appropriate to test the efficacy of the MR-CCLRT. A comparison with the (CI) full CI and MR-CCLRT results in the same basis for all the mols. shows very good performance of the theory in general, and indicates the efficacy of the method.**41**Kong, L. Connection between a few Jeziorski-Monkhorst ansatz-based methods.*Int. J. Quantum Chem.*2009,*109*, 441– 447, DOI: 10.1002/qua.21822Google Scholar41https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhsVeltA%253D%253D&md5=9e3b2d29476cca1c77d9a4b5e75791deConnection between a few Jeziorski-Monkhorst ansatz-based methodsKong, LiguoInternational Journal of Quantum Chemistry (2008), 109 (3), 441-447CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)Different Jeziorski-Monkhorst ansatz-based methods are unified according to how to group terms to eliminate the redundancy problem. It is found that some seemingly different methods used to do MRCC are equiv. It is argued that the various defining equations are not entirely proper, in the sense that the proper residual condition is not satisfied. This may partially rationalize the unsatisfactory performance of the various methods for single ref. systems. In contrast, the MRexpT method satisfies the proper residual condition and it is expected that it will outperform other JM ansatz-based methods in single-ref. cases.**42**Chattopadhyay, S.; Mahapatra, U. S.; Mukherjee, D. Property calculations using perturbed orbitals via state-specific multireference coupled-cluster and perturbation theories.*J. Chem. Phys.*1999,*111*, 3820– 3831, DOI: 10.1063/1.479685Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXltFGkurc%253D&md5=927dde8df8263b311b25e85ccc6b6674Property calculations using perturbed orbitals via state-specific multireference coupled-cluster and perturbation theoriesChattopadhyay, Sudip; Mahapatra, Uttam Sinha; Mukherjee, DebashisJournal of Chemical Physics (1999), 111 (9), 3820-3831CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper we apply the recently developed state-specific multireference coupled-cluster and perturbation theories to calc. elec. properties such as dipole moment and static polarizability using perturbed orbitals in finite fields. The theories are built on complete active space ref. functions, and are designed to treat quasidegeneracy of varying degrees while bypassing the intruder problem. Numerical results are presented for the model systems H4 with trapezoidal geometry and the lowest two singlet states of CH2. Both the systems require a multireference formulation due to quasidegeneracy. In the field-free situation, the former encounters intruders at an intermediate trapezoidal geometry in the traditional treatment using effective Hamiltonians, while the latter shows a pronounced multireference character in the two singlet states. This affects the response properties in the presence of a perturbing field. A comparison with the full CI results in the same basis indicates the efficacy of the state-specific methods in wide ranges of geometries, even when the traditional effective Hamiltonian based methods fail due to intruders.**43**Pittner, J. Continuous transition between Brillouin-Wigner and Rayleigh-Schrödinger perturbation theory, generalized Bloch equation, and Hilbert space multireference coupled cluster.*J. Chem. Phys.*2003,*118*, 10876– 10889, DOI: 10.1063/1.1574785Google Scholar43https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXksFyns78%253D&md5=9fb8dd8eb2dd496426241146ba4fe4a7Continuous transition between Brillouin-Wigner and Rayleigh-Schroedinger perturbation theory, generalized Bloch equation, and Hilbert space multireference coupled clusterPittner, JiriJournal of Chemical Physics (2003), 118 (24), 10876-10889CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A continuous transition between the Rayleigh-Schrodinger and Brillouin-Wigner perturbation theories was constructed and the Bloch equation for the corresponding wave operator was derived. Subsequently it was applied to the Hilbert space multireference coupled cluster theory and used to investigate relationships between several versions of multireference coupled cluster methods. Finally, based on those continuous transitions, new size extensivity corrections for the Brillouin-Wigner coupled cluster method were suggested. Numerical tests of size-extensivity and separability of a supermol. to closed- and open-shell fragments are also presented. Equivalence of some of the multireference coupled cluster methods with single and double excitations to full CI for two-electron systems was investigated, both theor. and numerically.**44**Mahapatra, U. S.; Datta, B.; Mukherjee, D. A size-consistent state-specific multireference coupled cluster theory: Formal developments and molecular applications.*J. Chem. Phys.*1999,*110*, 6171– 6188, DOI: 10.1063/1.478523Google Scholar44https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXitVCmt7c%253D&md5=f98011e4719a6b12ab0fe42bb71ed504A size-consistent state-specific multireference coupled cluster theory: formal developments and molecular applicationsMahapatra, Uttam Sinha; Datta, Barnali; Mukherjee, DebashisJournal of Chemical Physics (1999), 110 (13), 6171-6188CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper, we present a comprehensive account of a manifestly size-consistent coupled cluster formalism for a specific state, which is based on a ref. function composed of determinants spanning a complete active space (CAS). The method treats all the ref. determinants on the same footing and is hence expected to provide uniform description over a wide range of mol. geometry. The combining coeffs. are detd. by diagonalizing an effective operator in the CAS and are thus completely flexible, not constrained to preassigned values. A sep. exponential-type excitation operator is invoked to induce excitations to all the virtual functions from each ref. determinant. The linear dependence inherent in this choice of cluster operators is eliminated by invoking suitable sufficiency conditions, which in a transparent manner leads to manifest size extensivity. The use of a CAS also guarantees size consistency. We also discuss the relation of our method with the extant state-specific formalisms. Illustrative applications are presented for systems such as H4 in rectangular and trapezoidal geometries, the Be-H2 C2v insertion reaction path, the potential energy curves of Li2 and F2, and certain electronic states of CH2 and C2 mols. with pronounced multireference character. The results indicate the efficacy of the method for obviating the intruders and of providing accuracy.**45**Mášik, J.; Hubač, I.; Mach, P. Single-root multireference Brillouin-Wigner coupled-cluster theory: Applicability to the F 2 molecule.*J. Chem. Phys.*1998,*108*, 6571– 6579, DOI: 10.1063/1.476071Google Scholar45https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXisVart74%253D&md5=a48ba5056620f0d4437d442eddc820dbSingle-root multireference Brillouin-Wigner coupled-cluster theory: Applicability to the F2 moleculeMasik, Jozef; Hubac, Ivan; Mach, PavelJournal of Chemical Physics (1998), 108 (16), 6571-6579CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Recently developed single-root multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory, which deals with one state at a time while employing a multiconfigurational ref. wave function, is applied to the ground state of the F2 mol. using a two-determinant ref. space at the level of the CCSD approxn. The method represents a brand-new coupled-cluster (CC) approach to quasidegenerate problems which combines merits of two theories: the single-ref. CC method in a nondegenerate case and the Hilbert space MR CC method in quasidegenerate case. The method is able to switch itself from a nondegenerate to a fully degenerate case in a continuous manner, providing thus smooth potential energy surfaces. Moreover, in contrast to the Hilbert space MR CC approaches, it does not contain the so-called coupling terms and completely reduces to the std. single-ref. CC method in a highly nondegenerate region. Using a [4s,3p,1d] and [4s,3p,2d,1f ] basis sets, the calcd. potential energy curves are smooth, dissoc. correctly and the results are compared with other available multireference techniques as well as expt.**46**Hubač, I.; Neogrády, P. Size-consistent Brillouin-Wigner perturbation theory with an exponentially parametrized wave function: Brillouin-Wigner coupled-cluster theory.*Phys. Rev. A: At., Mol., Opt. Phys.*1994,*50*, 4558– 4564, DOI: 10.1103/PhysRevA.50.4558Google Scholar46https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXis1Krs70%253D&md5=10030999f110ac3516ebf1536a9630f1Size-consistent Brillouin-Wigner perturbation theory with an exponentially parametrized wave function: Brillouin-Wigner coupled-cluster theoryHubac, Ivan; Neogrady, PavelPhysical Review A: Atomic, Molecular, and Optical Physics (1994), 50 (6), 4558-64CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The size consistency of the Brillouin-Wigner perturbation theory is studied by using the Lippmann-Schwinger equation and an exponential ansatz for the wave function. The relation of this theory to the coupled-cluster method is studied, and a comparison through the effective-Hamiltonian method is also provided.**47**Adamowicz, L.; Malrieu, J.-P.; Ivanov, V. V. New approach to the state-specific multireference coupled-cluster formalism.*J. Chem. Phys.*2000,*112*, 10075– 10084, DOI: 10.1063/1.481649Google Scholar47https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXjvVOlu7s%253D&md5=fa287ceeaca8599e6dbc5670f9bdbd69New approach to the state-specific multireference coupled-cluster formalismAdamowicz, Ludwik; Malrieu, Jean-Paul; Ivanov, Vladimir V.Journal of Chemical Physics (2000), 112 (23), 10075-10084CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new development is presented in the framework of the state-specific multireference (MR) coupled-cluster (CC) theory (MRCC). The method is based on the CASSCF (complete active space SCF) wave function and it is designed specifically for calcg. excited electronic states. In the proposed approach, the cluster structure of the CC wave operator and the method to det. this operator are the key features. Since the general formulation of the CASCC method is uncontracted, i.e., allows the interaction between the nondynamic and dynamic correlation effects to affect both the CAS ref. function and the CC correlation wave operator, the method is expected to perform better than contracted perturbative approaches such as the CASPT2 (second-order perturbation theory based on the CAS wave function) method. Also, the CASCC method is not a perturbative approach and is not based on selection of an unperturbed Hamiltonian, which in the case of the CASPT2 method often leads to the "intruder state" problem. CASCC calcns. of the lowest totally sym. excited state of the H8 model system using the internally contracted and uncontracted approaches reveal some interesting features of the methodol.**48**Kállay, M.; Szalay, P. G.; Surján, P. R. A general state-selective multireference coupled-cluster algorithm.*J. Chem. Phys.*2002,*117*, 980– 990, DOI: 10.1063/1.1483856Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XltVOhs74%253D&md5=314fba7fa8a56ef47a58638e1b3d8542A general state-selective multireference coupled-cluster algorithmKallay, Mihaly; Szalay, Peter G.; Surjan, Peter R.Journal of Chemical Physics (2002), 117 (3), 980-990CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A state-selective multireference coupled-cluster algorithm is presented which is capable of describing single, double (or higher) excitations from an arbitrary complete model space. One of the active space determinants is chosen as a formal Fermi-vacuum and single, double (or higher) excitations from the other ref. functions are considered as higher excitations from this determinant as it has been previously proposed by Oliphant and Adamowicz [J. Chem. Phys. 94, 1229 (1991)]. Coupled-cluster equations are generated in terms of antisymmetrized diagrams and restrictions are imposed on these diagrams to eliminate those cluster amplitudes which carry undesirable no. of inactive indexes. The corresponding algebraic expressions are factorized and contractions between cluster amplitudes and intermediates are evaluated by our recent string-based algorithm [J. Chem. Phys. 115, 2945 (2001)]. The method can be easily modified to solve multireference CI problems. The performance of the method is demonstrated by several test calcns. on systems which require a multireference description. The problem related to the choice of the Fermi-vacuum has also been investigated.**49**Piecuch, P.; Kowalski, K. The state-universal multi-reference coupled-cluster theory: An overview of some recent advances.*Int. J. Mol. Sci.*2002,*3*, 676– 709, DOI: 10.3390/i3060676Google Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xnsl2hsL0%253D&md5=2f527cacb01aafe273c1ba320a494013The state-universal multi-reference coupled-cluster theory: an overview of some recent advancesPiecuch, Piotr; Kowalski, KarolInternational Journal of Molecular Sciences [online computer file] (2002), 3 (6), 676-709CODEN: IJMCFK; ISSN:1422-0067. (Molecular Diversity Preservation International)A review. Some recent advances in the area of multi-ref. coupled-cluster theory of the state-universal type are discussed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-ref. coupled-cluster singles and doubles method (SUMRCCSD) with the multi-ref. many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUMRCCSD formalism that carry only core and virtual orbital indexes are replaced by the first-order MRMBPT ests.; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. The new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indexes by the first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.**50**Schucan, T.; Weidenmüller, H. The effective interaction in nuclei and its perturbation expansion: An algebraic approach.*Ann. Phys.*1972,*73*, 108– 135, DOI: 10.1016/0003-4916(72)90315-6Google Scholar50https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE38XlsVKhu70%253D&md5=fe7aeddd54b795b7ae8e43663c654d06Effective interaction in nuclei and its perturbation expansion. Algebraic approachSchucan, T. H.; Weidenmueller, H. A.Annals of Physics (San Diego, CA, United States) (1972), 73 (1), 108-35CODEN: APNYA6; ISSN:0003-4916.A finite-dimensional model is considered for the Hilbert space of the A-N problem. In the frame of this model the energy-independent effective interaction W first introduced by Des Cloiseaux and Brandow is constructed. The connection is demonstrated between this explicit form and the implicit equation for W given by these authors. By using the explicit form for W, the perturbation expansion is investigated of W in powers of the interaction. (When used in the nuclear problem, this expansion leads to the folded diagrams.) This expansion is likely to diverge in most cases of practical interest. Several methods are given which can yield a convergent expansion for W. The implications of these results for paractical calcns. are discussed.**51**Kaldor, U. Intruder states and incomplete model spaces in multireference coupled-cluster theory: The 2*p*^{2}states of Be.*Phys. Rev. A: At., Mol., Opt. Phys.*1988,*38*, 6013, DOI: 10.1103/PhysRevA.38.6013Google Scholar51https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXht1Wktbg%253D&md5=97b84de48b67076e7d8c9eec4e474850Intruder states and incomplete model spaces in multireference coupled-cluster theory: the 2p2 states of berylliumKaldor, UziPhysical Review A: Atomic, Molecular, and Optical Physics (1988), 38 (12), 6013-16CODEN: PLRAAN; ISSN:0556-2791.The open-shell coupled-cluster (CC) method is applied to the excited 2p2 states of Be. With the 2s2 and 2p2 configurations included in the model (P) space, the CC equations prove very difficult to converge. When they do converge, very large (>5) excitation amplitudes are obsd., and the second 1S corresponds to the intruder 2s3s configuration, rather than the desired 2p2. The inclusion of 2s3s (but not 3s2) in the model space, which thereby becomes incomplete, improves convergence significantly, and gives energies in very good agreement with values known from other sources.**52**Malrieu, J.; Durand, P.; Daudey, J. Intermediate Hamiltonians as a new class of effective Hamiltonians.*J. Phys. A: Math. Gen.*1985,*18*, 809, DOI: 10.1088/0305-4470/18/5/014Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2MXhvFWgsb0%253D&md5=b98f331eaccb371286295249909bb1b7Intermediate Hamiltonians as a new class of effective HamiltoniansMalrieu, J. P.; Durand, P.; Daudey, J. P.Journal of Physics A: Mathematical and General (1985), 18 (5), 809-26CODEN: JPHAC5; ISSN:0305-4470.A new class of effective Hamiltonians (called intermediate Hamiltonians) is presented; only one part of their roots are exact eigenenergies of the full Hamiltonian. The theory of these intermediate Hamiltonians is presented by means of a new wave-operator R which is the analog of the wave-operated Ω in the theory of effective Hamiltonians. Solns. are obtained by a generalized degenerate perturbation theory (GDPT) and by iterative procedures. Two model systems are numerically solved which demonstrate the good convergence properties of GDPT with respect to std. degenerate perturbation theory. Continuity of the solns. is also checked in the presence of an intruder state.**53**Jankowski, K.; Malinowski, P. A valence-universal coupled-cluster single-and double-excitations method for atoms. III. Solvability problems in the presence of intruder states.*J. Phys. B: At., Mol. Opt. Phys.*1994,*27*, 1287, DOI: 10.1088/0953-4075/27/7/004Google Scholar53https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXjtVOksrg%253D&md5=c350f67988d697cf44431b3d18b049c9A valence-universal coupled-cluster single- and double-excitations method for atoms: III. Solvability problems in the presence of intruder statesJankowski, K.; Malinowski, P.Journal of Physics B: Atomic, Molecular and Optical Physics (1994), 27 (7), 1287-98CODEN: JPAPEH; ISSN:0953-4075.To better understand the problems met when solving the equations of valence-universal coupled-cluster (VU-CC) approaches in the presence of intruder states, the authors are concerned with the following aspects of the solvability problem for sets of non-linear equations: the existence and properties of multiple solns. and the attainability of these solns. by means of various numerical methods. This study is concd. on the equations obtained for Be within the framework of the recently formulated at. oriented form of the VU-CC accounting for one- and two-electron excitations (VU-CCSD/R) and based on the complete model space (2s2, 2p2). Six pairs of multiple solns. representing four 1S states are found and discussed. Three of these solns. provide amplitudes describing the 2p2 1S state for which the intruder state problem has been considered as extremely serious. Several known numerical methods have been applied to solve the same set of non-linear equations for the two-valence cluster amplitudes. It is shown that these methods perform quite differently in the presence of intruder states, which seems to indicate that the intruder state problem for VU-CC methods is partly caused by the commonly used methods of solving the non-linear equations.**54**Sharma, S.; Alavi, A. Multireference linearized coupled cluster theory for strongly correlated systems using matrix product states.*J. Chem. Phys.*2015,*143*, 102815, DOI: 10.1063/1.4928643Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhtlyns7vM&md5=0afcce08382e94e43b86e765e5b17811Multireference linearized coupled cluster theory for strongly correlated systems using matrix product statesSharma, Sandeep; Alavi, AliJournal of Chemical Physics (2015), 143 (10), 102815/1-102815/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We propose a multireference linearized coupled cluster theory using matrix product states (MPSs-LCC) which provides remarkably accurate ground-state energies, at a computational cost that has the same scaling as multireference CI singles and doubles, for a wide variety of electronic Hamiltonians. These range from first-row dimers at equil. and stretched geometries to highly multireference systems such as the chromium dimer and lattice models such as periodic two-dimensional 1-band and 3-band Hubbard models. The MPS-LCC theory shows a speed up of several orders of magnitude over the usual D. Matrix Renormalization Group (DMRG) algorithm while delivering energies in excellent agreement with converged DMRG calcns. Also, in all the benchmark calcns. presented here, MPS-LCC outperformed the commonly used multi-ref. quantum chem. methods in some cases giving energies in excess of an order of magnitude more accurate. As a size-extensive method that can treat large active spaces, MPS-LCC opens up the use of multireference quantum chem. techniques in strongly correlated ab initio Hamiltonians, including two- and three-dimensional solids. (c) 2015 American Institute of Physics.**55**Henderson, T. M.; Bulik, I. W.; Stein, T.; Scuseria, G. E. Seniority-based coupled cluster theory.*J. Chem. Phys.*2014,*141*, 244104, DOI: 10.1063/1.4904384Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXitFOjsrnK&md5=e3ebcce5c47052da1a339a27b5db275dSeniority-based coupled cluster theoryHenderson, Thomas M.; Bulik, Ireneusz W.; Stein, Tamar; Scuseria, Gustavo E.Journal of Chemical Physics (2014), 141 (24), 244104/1-244104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Doubly occupied CI (DOCI) with optimized orbitals often accurately describes strong correlations while working in a Hilbert space much smaller than that needed for full CI. However, the scaling of such calcns. remains combinatorial with system size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N3, disregarding the two-electron integral transformation from at. to MOs). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correlations, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems. (c) 2014 American Institute of Physics.**56**Lehtola, S.; Parkhill, J.; Head-Gordon, M. Cost-effective description of strong correlation: Efficient implementations of the perfect quadruples and perfect hextuples models.*J. Chem. Phys.*2016,*145*, 134110, DOI: 10.1063/1.4964317Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhs1ClsbjJ&md5=cba359a527854cc102fb61be47a44713Cost-effective description of strong correlation: Efficient implementations of the perfect quadruples and perfect hextuples modelsLehtola, Susi; Parkhill, John; Head-Gordon, MartinJournal of Chemical Physics (2016), 145 (13), 134110/1-134110/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Novel implementations based on dense tensor storage are presented for the singlet-ref. perfect quadruples (PQ) [J. A. Parkhill et al., J. Chem. Phys. 130, 084101 (2009)] and perfect hextuples (PH) [J. A. Parkhill and M. Head-Gordon, J. Chem. Phys. 133, 024103 (2010)] models. The methods are obtained as block decompns. of conventional coupled-cluster theory that are exact for four electrons in four orbitals (PQ) and six electrons in six orbitals (PH), but that can also be applied to much larger systems. PQ and PH have storage requirements that scale as the square, and as the cube of the no. of active electrons, resp., and exhibit quartic scaling of the computational effort for large systems. Applications of the new implementations are presented for full-valence calcns. on linear polyenes (CnHn+2), which highlight the excellent computational scaling of the present implementations that can routinely handle active spaces of hundreds of electrons. The accuracy of the models is studied in the π space of the polyenes, in hydrogen chains (H50), and in the π space of polyacene mols. In all cases, the results compare favorably to d. matrix renormalization group values. With the novel implementation of PQ, active spaces of 140 electrons in 140 orbitals can be solved in a matter of minutes on a single core workstation, and the relatively low polynomial scaling means that very large systems are also accessible using parallel computing. (c) 2016 American Institute of Physics.**57**Lehtola, S.; Parkhill, J.; Head-Gordon, M. Orbital optimization in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes.*Mol. Phys.*2018,*116*, 547– 560, DOI: 10.1080/00268976.2017.1342009Google Scholar57https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtVyqtr7M&md5=4788914aaa7d0a92eb8c474502445ad3Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenesLehtola, Susi; Parkhill, John; Head-Gordon, MartinMolecular Physics (2018), 116 (5-6), 547-560CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)We describe the implementation of orbital optimization for the models in the perfect pairing hierarchy. Orbital optimization, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local min. and σ-π symmetry breaking solns. were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calcns. at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimized orbitals are used. PH is found to yield good agreement with previously published d. matrix renormalization group data in the π space, capturing over 95% of the correlation energy. Full-valence calcns. made possible by our novel, efficient code reveal that strong correlations are weaker when larger basis sets or active spaces are employed than in previous calcns. The largest full-valence PH calcns. presented correspond to a (192e,192o) problem.**58**Cullen, J. Generalized valence bond solutions from a constrained coupled cluster method.*Chem. Phys.*1996,*202*, 217– 229, DOI: 10.1016/0301-0104(95)00321-5Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xpt1antw%253D%253D&md5=b7ee308cea7c5dbf21df43709ba3fd5fGeneralized valence bond solutions from a constrained coupled cluster methodCullen, JohnChemical Physics (1996), 202 (2,3), 217-29CODEN: CMPHC2; ISSN:0301-0104. (Elsevier)The GVB-PP wave function is cast into a coupled cluster form with the coupled cluster operator constrained to intra-bond double excitations. Following the coupled cluster ansatz, where the trial wave function is assumed to satisfy the Schroedinger equation, projections onto the ref. and doubly excited configurations are used to det. the energy and coeffs. resp. A decoupling of these equations results, allowing anal. solns. The active orbital space is simultaneously optimized to produce the lowest energy. This is carried out efficiently using a procedure previously developed by Head-Gordon and Pople for the direct optimization of a Hartree-Fock wave function. Preliminary calcns. for singlet/triplet states and bond dissocns. show that although this method is strictly nonvariational, local min. are found which lie within a few tenths of a millihartree of the true GVB-PP energy.**59**Goddard, W. A., III; Harding, L. B. The description of chemical bonding from ab initio calculations.*Annu. Rev. Phys. Chem.*1978,*29*, 363– 396, DOI: 10.1146/annurev.pc.29.100178.002051Google Scholar59https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE1MXht1Cjuw%253D%253D&md5=9007c9aa07bfe0efea2d94d84f1f2034The description of chemical bonding from ab initio calculationsGoddard, William A., III; Harding, Lawrence B.Annual Review of Physical Chemistry (1978), 29 (), 363-96CODEN: ARPLAP; ISSN:0066-426X.A review with 38 refs.**60**Ukrainskii, I. New variational function in the theory of quasi-one-dimensional metals.*Theor. Math. Phys.*1977,*32*, 816– 822, DOI: 10.1007/BF01089566Google ScholarThere is no corresponding record for this reference.**61**Hunt, W.; Hay, P.; Goddard, W., III Self-Consistent Procedures for Generalized Valence Bond Wavefunctions. Applications H_{3}, BH, H_{2}O, C_{2}H_{6}, and O_{2}.*J. Chem. Phys.*1972,*57*, 738– 748, DOI: 10.1063/1.1678308Google ScholarThere is no corresponding record for this reference.**62**Hurley, A.; Lennard-Jones, J. E.; Pople, J. A. The molecular orbital theory of chemical valency XVI. A theory of paired-electrons in polyatomic molecules.*Proc. R. Soc. London. Series A. Math. Phys. Sci.*1953,*220*, 446– 455Google ScholarThere is no corresponding record for this reference.**63**Živković, T. P. Existence and reality of solutions of the coupled-cluster equations.*Int. J. Quantum Chem.*1977,*12*, 413– 420, DOI: 10.1002/qua.560120849Google ScholarThere is no corresponding record for this reference.**64**Piecuch, P.; Zarrabian, S.; Paldus, J.; Čížek, J. Coupled-cluster approaches with an approximate account of triexcitations and the optimized-inner-projection technique. II. Coupled-cluster results for cyclic-polyene model systems.*Phys. Rev. B: Condens. Matter Mater. Phys.*1990,*42*, 3351, DOI: 10.1103/PhysRevB.42.3351Google ScholarThere is no corresponding record for this reference.**65**Atkinson, K. E.*An introduction to numerical analysis*; John Wiley & Sons, 2008.Google ScholarThere is no corresponding record for this reference.**66**Živković, T. P.; Monkhorst, H. J. Analytic connection between configuration-interaction and coupled-cluster solutions.*J. Math. Phys.*1978,*19*, 1007– 1022, DOI: 10.1063/1.523761Google ScholarThere is no corresponding record for this reference.**67**Kowalski, K.; Jankowski, K. Towards complete solutions to systems of nonlinear equations of many-electron theories.*Phys. Rev. Lett.*1998,*81*, 1195, DOI: 10.1103/PhysRevLett.81.1195Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXltVSrt7w%253D&md5=0ea6cd8d6989fddf2ced3e6be183ff76Towards Complete Solutions to Systems of Nonlinear Equations of Many-Electron TheoriesKowalski, Karol; Jankowski, KarolPhysical Review Letters (1998), 81 (6), 1195-1198CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Employing the homotopy method we have obtained the complete set of real solns. to the equations of the RHF method as well as the full set of solns. to the equations of the coupled-cluster-with-doubles method for the H4 and P4 models broadly applied in various many-electron studies. These are the first global results obtained so far for any formulations of the Hartree-Fock and coupled-cluster methods when applied to realistic models.**68**Piecuch, P.; Kowalski, K. In*Computational Chemistry: Reviews of Current Trends*; Leszczynski, J., Ed.; World Scientific, Singapore, 2000; Vol. 5.Google ScholarThere is no corresponding record for this reference.**69**Jeziorski, B.; Paldus, J. Valence universal exponential ansatz and the cluster structure of multireference configuration interaction wave function.*J. Chem. Phys.*1989,*90*, 2714– 2731, DOI: 10.1063/1.455919Google ScholarThere is no corresponding record for this reference.**70**Schneider, R. Analysis of the Projected Coupled Cluster Method in Electronic Structure Calculation.*Numer. Math.*2009,*113*, 433– 471, DOI: 10.1007/s00211-009-0237-3Google ScholarThere is no corresponding record for this reference.**71**Rohwedder, T. The Continuous Coupled Cluster Formulation for the Electronic Schrödinger Equation.*ESAIM: Math. Modell. Numer. Anal.*2013,*47*, 421– 447, DOI: 10.1051/m2an/2012035Google ScholarThere is no corresponding record for this reference.**72**Rohwedder, T.; Schneider, R. Error Estimates for the Coupled Cluster Method.*ESAIM: Math. Modell. Numer. Anal.*2013,*47*, 1553– 1582, DOI: 10.1051/m2an/2013075Google ScholarThere is no corresponding record for this reference.**73**Laestadius, A.; Kvaal, S. Analysis of the extended coupled-cluster method in quantum chemistry.*SIAM J. on Numer. Anal.*2018,*56*, 660– 683, DOI: 10.1137/17M1116611Google ScholarThere is no corresponding record for this reference.**74**Löwdin, P.-O. On the stability problem of a pair of adjoint operators.*J. Math. Phys.*1983,*24*, 70– 87, DOI: 10.1063/1.525604Google ScholarThere is no corresponding record for this reference.**75**Arponen, J. Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems.*Ann. Phys.*1983,*151*, 311– 382, DOI: 10.1016/0003-4916(83)90284-1Google Scholar75https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2cXis1OqtA%253D%253D&md5=7b010ab3d1ce0aba10ed5fb00da58f3dVariational principles and linked-cluster exp S expansions for static and dynamic many-body problemsArponen, JoukoAnnals of Physics (San Diego, CA, United States) (1983), 151 (2), 311-82CODEN: APNYA6; ISSN:0003-4916.The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed by using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using 2 different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the av. values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes.**76**Faulstich, F. M.; Laestadius, A.; Kvaal, S.; Legeza, Ö.; Schneider, R. Analysis of The Coupled-Cluster Method Tailored by Tensor-Network States in Quantum Chemistry.*arXiv.org*2018, 1802.05699Google ScholarThere is no corresponding record for this reference.**77**Laestadius, A.; Faulstich, F. M. The coupled-cluster formalism-a mathematical perspective.*Mol. Phys.*2019, 1– 12, DOI: 10.1080/00268976.2018.1564848Google ScholarThere is no corresponding record for this reference.**78**Piecuch, P.; Oliphant, N.; Adamowicz, L. A state-selective multireference coupled-cluster theory employing the single-reference formalism.*J. Chem. Phys.*1993,*99*, 1875– 1900, DOI: 10.1063/1.466179Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtFWjsrc%253D&md5=85bd8b4c37a2e1763e9032287669e1e5A state-selective multireference coupled-cluster theory employing the single-reference formalismPiecuch, Piotr; Oliphant, Nevin; Adamowicz, LudwikJournal of Chemical Physics (1993), 99 (3), 1875-900CODEN: JCPSA6; ISSN:0021-9606.A new state-selective multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism is described. It is an extension of a simple two-ref. formalism, which the authors presented in the authors' earlier paper [N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991)], and a rigorous formulation of another method of ours, which the authors obtained as an approxn. of the SRCC approach truncated at triple excitations (SRCCSDT) [N. Oliphant and L. Adamowicz, J. Chem. Phys. 96, 3739 (1992)]. The size extensivity of the resulting correlation energies is achieved by employing a SRCC-like ansatz for the multideterminantal wave function. General considerations are supplemented by suggesting a hierarchy of approx. schemes, with the MRCCSD approach (MRCC approach truncated at double excitations from the ref. determinants) representing the most important one. The authors' state-selective MRCCSD theory emerges through a suitable selection of the most essential cluster components appearing in the full SRCCSDTQ method (SRCC method truncated at quadruple excitations), when the latter is applied to quasidegenerate states. The complete set of equations describing the authors' MRCCSD formalism is presented and the possibility of the recursive intermediate factorization [S. A. Kucharski and R. J. Bartlett, Theor. Chim. Acta 80, 387 (1991)] of the authors' approach, leading to an efficient computer algorithm, is discussed.**79**Piecuch, P.; Adamowicz, L. State-selective multireference coupled-cluster theory employing the single-reference formalism: Implementation and application to the H8 model system.*J. Chem. Phys.*1994,*100*, 5792– 5809, DOI: 10.1063/1.467143Google Scholar79https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXjtVOksrY%253D&md5=b8df2b4213996bf687d716f91ccaa240State-selective multireference coupled-cluster theory employing the single-reference formalism: implementation and application to the H8 model systemPiecuch, Piotr; Adamowicz, LudwikJournal of Chemical Physics (1994), 100 (8), 5792-809CODEN: JCPSA6; ISSN:0021-9606.The new state-selective (SS) multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism, which was introduced previously (P. Piecuch, et al., 1993), was implemented; results are presented of pilot calcns. for the min. basis-set (MBS) model composed of eight hydrogen atoms in various geometrical arrangements. This model enables a continuous transition between degenerate and nondegenerate regimes. Comparison is made with the results of SR CC calcns. involving double (CCD), single and double (CCSD), single, double, and triple (CCSDT), and single, double, triple, and quadruple (CCSDTQ) excitations. The authors' SS CC energies are also compared with the results of Hilbert space, state-universal (SU) MR CC(S)D calcns., as well as with MR-CI results (with and without Davidson-type corrections), and with exact correlation energies obtained using the full-CI (FCI) method. Along with the ground-state energies, the authors also analyzed the resulting wave functions by examg. some selected cluster components. This anal. enabled the authors to assess the quality of the resulting wave functions. The authors' SS CC theory truncated at double excitations, which emerges through selection of the most essential clusters appearing in the full SR CCSDTQ formalism [SS CCSD (TQ) method], provided equally good results in the nondegenerate and quasidegenerate regions. The difference between the ground-state energy obtained with the SS CCSD(TQ) approach and the FCI energy did not exceed 1.1 milli-hartree over all the geometries considered. This value compares favorably with the max. difference of 2.8 milli-hartrees between the SU CCSD energies and the FCI energies obtained for the same range of geometries. The SS CCSD(T) method, emerging from the SR CCSDT theory through selection of the most essential clusters, was less stable, since it neglected very important semi-internal quadruple excitations. Unlike the genuine multideterminantal SU CC formalism, the authors' SS CC approach was not affected by the intruder-state problem, and its convergence remained satisfactory the in nondegenerate and quasidegenerate regimes.**80**Chan, G. K.-L.; Sharma, S. The density matrix renormalization group in quantum chemistry.*Annu. Rev. Phys. Chem.*2011,*62*, 465– 481, DOI: 10.1146/annurev-physchem-032210-103338Google Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmsVWmt7k%253D&md5=99fca86a8b3932bf6d9f73defd9ee37eThe density matrix renormalization group in quantum chemistryChan, Garnet Kin-Lic; Sharma, SandeepAnnual Review of Physical Chemistry (2011), 62 (), 465-481CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)A review. The d. matrix renormalization group is a method that is useful for describing mols. that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the d. matrix renormalization group works, a survey of its existing applications to mol. problems, and some thoughts on the future of the method.**81**Saitow, M.; Kurashige, Y.; Yanai, T. Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave function.*J. Chem. Phys.*2013,*139*, 044118, DOI: 10.1063/1.4816627Google Scholar81https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhtFygsbnL&md5=a30f97f373ba1a069bbce1c23a48e178Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave functionSaitow, Masaaki; Kurashige, Yuki; Yanai, TakeshiJournal of Chemical Physics (2013), 139 (4), 044118/1-044118/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We report development of the multireference CI (MRCI) method that can use active space scalable to much larger size refs. than has previously been possible. The recent development of the d. matrix renormalization group (DMRG) method in multireference quantum chem. offers the ability to describe static correlation in a large active space. The present MRCI method provides a crit. correction to the DMRG ref. by including high-level dynamic correlation through the CI treatment. When the DMRG and MRCI theories are combined (DMRG-MRCI), the full internal contraction of the ref. in the MRCI ansatz, including contraction of semi-internal states, plays a central role. However, it is thought to involve formidable complexity because of the presence of the five-particle rank reduced-d. matrix (RDM) in the Hamiltonian matrix elements. To address this complexity, we express the Hamiltonian matrix using commutators, which allows the five-particle rank RDM to be canceled out without any approxn. Then we introduce an approxn. to the four-particle rank RDM by using a cumulant reconstruction from lower-particle rank RDMs. A computer-aided approach is employed to derive the exceedingly complex equations of the MRCI in tensor-contracted form and to implement them into an efficient parallel computer code. This approach extends to the size-consistency-cor. variants of MRCI, such as the MRCI+Q, MR-ACPF, and MR-AQCC methods. We demonstrate the capability of the DMRG-MRCI method in several benchmark applications, including the evaluation of single-triplet gap of free-base porphyrin using 24 active orbitals. (c) 2013 American Institute of Physics.**82**Myhre, R. H.; Koch, H. The multilevel CC3 coupled cluster model.*J. Chem. Phys.*2016,*145*, 044111, DOI: 10.1063/1.4959373Google Scholar82https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1Kmur7E&md5=6f62a1c36755e3209a5cbbf86d3502c8The multilevel CC3 coupled cluster modelMyhre, Rolf H.; Koch, HenrikJournal of Chemical Physics (2016), 145 (4), 044111/1-044111/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an efficient implementation of the closed shell multilevel coupled cluster method where coupled cluster singles and doubles (CCSD) is used for the inactive orbital space and CCSD with perturbative triples (CC3) is employed for the smaller active orbital space. Using Cholesky orbitals, the active space can be spatially localized and the computational cost is greatly reduced compared to full CC3 while retaining the accuracy of CC3 excitation energies. For the small org. mols. considered we achieve up to two orders of magnitude redn. in the computational requirements. (c) 2016 American Institute of Physics.**83**Lyakh, D. I.; Musiał, M.; Lotrich, V. F.; Bartlett, R. J. Multireference nature of chemistry: The coupled-cluster view.*Chem. Rev.*2012,*112*, 182– 243, DOI: 10.1021/cr2001417Google Scholar83https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs12hsbjF&md5=26080054ce24a172826517cb7d772f62Multireference Nature of Chemistry: The Coupled-Cluster ViewLyakh, Dmitry I.; Musial, Monika; Lotrich, Victor F.; Bartlett, Rodney J.Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 182-243CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. The following topics are discussed: Exponential era of electron correlation theory; Genuine MR CC theory in Hilbert space and in Fock space; Alternative MR CC methods. Numerical illustrations are presented.**84**Szalay, S.; Barcza, G.; Szilvási, T.; Veis, L.; Legeza, Ö. The correlation theory of the chemical bond.*Sci. Rep.*2017,*7*, 2237, DOI: 10.1038/s41598-017-02447-zGoogle Scholar84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1cngtF2msg%253D%253D&md5=e14838c28811e3e816ab44d05ab62906The correlation theory of the chemical bondSzalay Szilard; Barcza Gergely; Legeza Ors; Szilvasi Tibor; Szilvasi Tibor; Veis LiborScientific reports (2017), 7 (1), 2237 ISSN:.The quantum mechanical description of the chemical bond is generally given in terms of delocalized bonding orbitals, or, alternatively, in terms of correlations of occupations of localised orbitals. However, in the latter case, multiorbital correlations were treated only in terms of two-orbital correlations, although the structure of multiorbital correlations is far richer; and, in the case of bonds established by more than two electrons, multiorbital correlations represent a more natural point of view. Here, for the first time, we introduce the true multiorbital correlation theory, consisting of a framework for handling the structure of multiorbital correlations, a toolbox of true multiorbital correlation measures, and the formulation of the multiorbital correlation clustering, together with an algorithm for obtaining that. These make it possible to characterise quantitatively, how well a bonding picture describes the chemical system. As proof of concept, we apply the theory for the investigation of the bond structures of several molecules. We show that the non-existence of well-defined multiorbital correlation clustering provides a reason for debated bonding picture.**85**Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy.*Phys. Rev. B: Condens. Matter Mater. Phys.*2003,*68*, 195116, DOI: 10.1103/PhysRevB.68.195116Google Scholar85https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXpvVegs7k%253D&md5=215685d20c465a36d96e9adf4bbb0ea3Optimizing the density-matrix renormalization group method using quantum information entropyLegeza, O.; Solyom, J.Physical Review B: Condensed Matter and Materials Physics (2003), 68 (19), 195116/1-195116/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)In order to optimize the ordering of the lattice sites in the momentum space and quantum chem. versions of the d.-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various mols. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chem. DMRG to a great extent. The effect of lattice site ordering on the no. of block states to be kept during the RG procedure is also investigated.**86**Barcza, G.; Legeza, Ö.; Marti, K. H.; Reiher, M. Quantum-information analysis of electronic states of different molecular structures.*Phys. Rev. A: At., Mol., Opt. Phys.*2011,*83*, 012508, DOI: 10.1103/PhysRevA.83.012508Google Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsFegsr0%253D&md5=4a41eebdc1e8bed2c9112fcaf74fa90fQuantum-information analysis of electronic states of different molecular structuresBarcza, G.; Legeza, O.; Marti, K. H.; Reiher, M.Physical Review A: Atomic, Molecular, and Optical Physics (2011), 83 (1, Pt. A), 012508/1-012508/15CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We have studied transition metal clusters from a quantum information theory perspective using the d.-matrix renormalization group (DMRG) method. We demonstrate the competition between entanglement and interaction localization and discuss the application of the CI-based dynamically extended active space procedure, which significantly reduces the effective system size and accelerates the speed of convergence for complicated mol. electronic structures. Our results indicate the importance of taking entanglement among MOs into account in order to devise an optimal DMRG orbital ordering and carry out efficient calcns. on transition metal clusters. Apart from these algorithmic observations, which lead to a recipe for black-box DMRG calcns., our work provides phys. understanding of electron correlation in mol. and cluster structures in terms of entropy measures of relevance also to recent work on tensor-network representations of electronic states. We also identify those MOs which are highly entangled and discuss the consequences for chem. bonding and for the structural transition from an dioxygen binding copper cluster to an bis-oxygen-bridged system with broken O-O bond.**87**Stein, C. J.; Reiher, M. Automated selection of active orbital spaces.*J. Chem. Theory Comput.*2016,*12*, 1760– 1771, DOI: 10.1021/acs.jctc.6b00156Google Scholar87https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XjvFyltLs%253D&md5=c46ae44d10c10dfa409cf8807a779308Automated Selection of Active Orbital SpacesStein, Christopher J.; Reiher, MarkusJournal of Chemical Theory and Computation (2016), 12 (4), 1760-1771CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)One of the key challenges of quantum-chem. multi-configuration methods is the necessity to manually select orbitals for the active space. This selection requires both expertise and experience and can therefore impose severe limitations on the applicability of this most general class of ab initio methods. A poor choice of the active orbital space may yield even qual. wrong results. This is obviously a severe problem, esp. for wave function methods that are designed to be systematically improvable. Here, we show how the iterative nature of the d. matrix renormalization group combined with its capability to include up to about 100 orbitals in the active space can be exploited for a systematic assessment and selection of active orbitals. These benefits allow us to implement an automated approach for active orbital space selection, which can turn multi-configuration models into black box approaches.**88**Aubin, J. P. Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods.*Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)*1967,*21*, 599– 637Google ScholarThere is no corresponding record for this reference.**89**Nitsche, J. Ein kriterium für die quasi-optimalität des ritzschen verfahrens.*Numer. Math.*1968,*11*, 346– 348, DOI: 10.1007/BF02166687Google ScholarThere is no corresponding record for this reference.**90**Oganesyan, L. A.; Rukhovets, L. A. Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary.*USSR Comput. Math. Math. Phys.*1969,*9*, 158– 183, DOI: 10.1016/0041-5553(69)90159-1Google ScholarThere is no corresponding record for this reference.**91**Dunning, T. H., Jr Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.*J. Chem. Phys.*1989,*90*, 1007– 1023, DOI: 10.1063/1.456153Google Scholar91https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXksVGmtrk%253D&md5=c6cd67a3748dc61692a9cb622d2694a0Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogenDunning, Thom H., Jr.Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.**92**Kowalski, K.; Piecuch, P. Renormalized CCSD (T) and CCSD (TQ) approaches: Dissociation of the N_{2}triple bond.*J. Chem. Phys.*2000,*113*, 5644– 5652, DOI: 10.1063/1.1290609Google Scholar92https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXmvFOlsrs%253D&md5=c1c67154a810a07f46da394e62b1a0bcRenormalized CCSD(T) and CCSD(TQ) approaches: Dissociation of the N2 triple bondKowalski, Karol; Piecuch, PiotrJournal of Chemical Physics (2000), 113 (14), 5644-5652CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The recently proposed renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods, which can be viewed as generalizations of the noniterative perturbative CCSD(T) and CCSD(TQf) schemes and which result from the more general method of moments of coupled-cluster equations, are applied to the dissocn. of the ground-state N2 mol. It is shown that the renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods provide significantly better results for large N-N sepns. than their unrenormalized CCSD(T) and CCSD(TQf) counterparts.**93**Szalay, S.; Pfeffer, M.; Murg, V.; Barcza, G.; Verstraete, F.; Schneider, R.; Legeza, Ö. Tensor product methods and entanglement optimization for ab initio quantum chemistry.*Int. J. Quantum Chem.*2015,*115*, 1342– 1391, DOI: 10.1002/qua.24898Google Scholar93https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXovVags7c%253D&md5=d1f7db2c2c73a2d907e9a290ccb7f22bTensor product methods and entanglement optimization for ab initio quantum chemistrySzalay, Szilard; Pfeffer, Max; Murg, Valentin; Barcza, Gergely; Verstraete, Frank; Schneider, Reinhold; Legeza, OersInternational Journal of Quantum Chemistry (2015), 115 (19), 1342-1391CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The treatment of high-dimensional problems such as the Schroedinger equation can be approached by concepts of tensor product approxn. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods-developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools-can be combined to attack highly challenging problems in quantum chem. The aim of the present paper is to give a pedagogical introduction to the theor. background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum chem." at the Workshop on Theor. Chem., Feb. 18-21, 2014, Mariapfarr, Austria. © 2015 Wiley Periodicals, Inc.**94**Murg, V.; Verstraete, F.; Legeza, Ö.; Noack, R.-h. M. Simulating strongly correlated quantum systems with tree tensor networks.*Phys. Rev. B: Condens. Matter Mater. Phys.*2010,*82*, 205105, DOI: 10.1103/PhysRevB.82.205105Google Scholar94https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhsFWmtLrF&md5=c554e73a8f20b3a417a0e2381c3ff02fSimulating strongly correlated quantum systems with tree tensor networksMurg, V.; Verstraete, F.; Legeza, O.; Noack, R. M.Physical Review B: Condensed Matter and Materials Physics (2010), 82 (20), 205105/1-205105/11CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chem. versions of the d.-matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topol. of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following [Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006)], we treat a treelike network ansatz with arbitrary coordination no. z, where the z = 2 case corresponds to the one-dimensional (1D) scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addn., we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1D interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calcns. carried out on small quantum chem. systems support our approach.**95**Nakatani, N.; Chan, G. K.-L. Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm.*J. Chem. Phys.*2013,*138*, 134113, DOI: 10.1063/1.4798639Google Scholar95https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXlt1Gks7c%253D&md5=2e926e48567120d16676ed7329a235c2Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithmNakatani, Naoki; Chan, Garnet Kin-LicJournal of Chemical Physics (2013), 138 (13), 134113/1-134113/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We investigate tree tensor network states for quantum chem. Tree tensor network states represent one of the simplest generalizations of matrix product states and the d. matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general mols. We describe an optimal tree tensor network state algorithm for quantum chem. We introduce the concept of half-renormalization which greatly improves the efficiency of the calcns. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states vs. matrix product states. We carry out benchmark calcns. both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree mols. (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree mols., whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling assocd. with tree algorithms. In tree like mols., tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calcn. with tree tensor network states correlates 110 electrons in 110 active orbitals. (c) 2013 American Institute of Physics.**96**Murg, V.; Verstraete, F.; Schneider, R.; Nagy, P. R.; Legeza, Ö. Tree tensor network state with variable tensor order: an efficient multireference method for strongly correlated systems.*J. Chem. Theory Comput.*2015,*11*, 1027– 1036, DOI: 10.1021/ct501187jGoogle Scholar96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXitVersLc%253D&md5=4287825e29765d130d6ac6b08c17d344Tree Tensor Network State with Variable Tensor Order: An Efficient Multireference Method for Strongly Correlated SystemsMurg, V.; Verstraete, F.; Schneider, R.; Nagy, P. R.; Legeza, Oe.Journal of Chemical Theory and Computation (2015), 11 (3), 1027-1036CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We study the tree-tensor-network-state (TTNS) method with variable tensor orders for quantum chem. TTNS is a variational method to efficiently approx. complete active space (CAS) CI wave functions in a tensor product form. TTNS can be considered as a higher order generalization of the matrix product state (MPS) method. The MPS wave function is formulated as products of matrixes in a multiparticle basis spanning a truncated Hilbert space of the original CAS-CI problem. These matrixes belong to active orbitals organized in a one-dimensional array, while tensors in TTNS are defined upon a tree-like arrangement of the same orbitals. The tree-structure is advantageous since the distance between two arbitrary orbitals in the tree scales only logarithmically with the no. of orbitals N, whereas the scaling is linear in the MPS array. It is found to be beneficial from the computational costs point of view to keep strongly correlated orbitals in close vicinity in both arrangements; therefore, the TTNS ansatz is better suited for multireference problems with numerous highly correlated orbitals. To exploit the advantages of TTNS a novel algorithm is designed to optimize the tree tensor network topol. based on quantum information theory and entanglement. The superior performance of the TTNS method is illustrated on the ionic-neutral avoided crossing of LiF. It is also shown that the avoided crossing of LiF can be localized using only ground state properties, namely one-orbital entanglement.**97**Gunst, K.; Verstraete, F.; Wouters, S.; Legeza, Ö.; Van Neck, D. T3NS: Three-Legged Tree Tensor Network States.*J. Chem. Theory Comput.*2018,*14*, 2026– 2033, DOI: 10.1021/acs.jctc.8b00098Google Scholar97https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXjtlOltLc%253D&md5=b3058f1bfd3cdc7de8f4728d955f21e6T3NS: Three-Legged Tree Tensor Network StatesGunst, Klaas; Verstraete, Frank; Wouters, Sebastian; Legeza, Ors; Van Neck, DimitriJournal of Chemical Theory and Computation (2018), 14 (4), 2026-2033CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a new variational tree tensor network state (TTNS) ansatz, the three-legged tree tensor network state (T3NS). Phys. tensors are interspersed with branching tensors. Phys. tensors have one phys. index and at most two virtual indexes, as in the matrix product state (MPS) ansatz of the d. matrix renormalization group (DMRG). Branching tensors have no phys. index, but up to three virtual indexes. In this way, advantages of DMRG, in particular a low computational cost and a simple implementation of symmetries, are combined with advantages of TTNS, namely incorporating more entanglement. Our code is capable of simulating quantum chem. Hamiltonians, and we present several proof-of-principle calcns. on LiF, N2, and the bis(μ-oxo) and μ-η2:η2 peroxo isomers of [Cu2O2]2+.**98**Chan, G. K.-L.; Head-Gordon, M. Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group.*J. Chem. Phys.*2002,*116*, 4462– 4476, DOI: 10.1063/1.1449459Google Scholar98https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhslCjsrs%253D&md5=f1249212b5c6414a901ff5c8a4a64542Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization groupChan, Garnet Kin-Lic; Head-Gordon, MartinJournal of Chemical Physics (2002), 116 (11), 4462-4476CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the recently developed D. Matrix Renormalization Group (DMRG) algorithm in the context of quantum chem. In contrast to traditional approaches, this algorithm is believed to yield arbitrarily high accuracy in the energy with only polynomial computational effort. We describe in some detail how this is achieved. We begin by introducing the principles of the renormalization procedure, and how one formulates an algorithm for use in quantum chem. The renormalization group algorithm is then interpreted in terms of familiar quantum chem. concepts, and its numerical behavior, including its convergence and computational cost, are studied using both model and real systems. The asymptotic convergence of the algorithm is derived. Finally, we examine the performance of the DMRG on widely studied chem. problems, such as the water mol., the twisting barrier of ethene, and the dissocn. of nitrogen. In all cases, the results are favorably comparable with the best existing quantum chem. methods, and particularly so when the nondynamical correlation is strong. Some perspectives for future development are given.**99**Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy.*Phys. Rev. B: Condens. Matter Mater. Phys.*2003,*68*, 195116, DOI: 10.1103/PhysRevB.68.195116Google Scholar99https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXpvVegs7k%253D&md5=215685d20c465a36d96e9adf4bbb0ea3Optimizing the density-matrix renormalization group method using quantum information entropyLegeza, O.; Solyom, J.Physical Review B: Condensed Matter and Materials Physics (2003), 68 (19), 195116/1-195116/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)In order to optimize the ordering of the lattice sites in the momentum space and quantum chem. versions of the d.-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various mols. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chem. DMRG to a great extent. The effect of lattice site ordering on the no. of block states to be kept during the RG procedure is also investigated.**100**Barcza, G.; Legeza, Ö.; Marti, K. H.; Reiher, M. Quantum-information analysis of electronic states of different molecular structures.*Phys. Rev. A: At., Mol., Opt. Phys.*2011,*83*, 012508, DOI: 10.1103/PhysRevA.83.012508Google Scholar100https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsFegsr0%253D&md5=4a41eebdc1e8bed2c9112fcaf74fa90fQuantum-information analysis of electronic states of different molecular structuresBarcza, G.; Legeza, O.; Marti, K. H.; Reiher, M.Physical Review A: Atomic, Molecular, and Optical Physics (2011), 83 (1, Pt. A), 012508/1-012508/15CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We have studied transition metal clusters from a quantum information theory perspective using the d.-matrix renormalization group (DMRG) method. We demonstrate the competition between entanglement and interaction localization and discuss the application of the CI-based dynamically extended active space procedure, which significantly reduces the effective system size and accelerates the speed of convergence for complicated mol. electronic structures. Our results indicate the importance of taking entanglement among MOs into account in order to devise an optimal DMRG orbital ordering and carry out efficient calcns. on transition metal clusters. Apart from these algorithmic observations, which lead to a recipe for black-box DMRG calcns., our work provides phys. understanding of electron correlation in mol. and cluster structures in terms of entropy measures of relevance also to recent work on tensor-network representations of electronic states. We also identify those MOs which are highly entangled and discuss the consequences for chem. bonding and for the structural transition from an dioxygen binding copper cluster to an bis-oxygen-bridged system with broken O-O bond.**101**Fertitta, E.; Paulus, B.; Barcza, G.; Legeza, Ö. Investigation of metal-insulator-like transition through the*ab initio*density matrix renormalization group approach.*Phys. Rev. B: Condens. Matter Mater. Phys.*2014,*90*, 245129, DOI: 10.1103/PhysRevB.90.245129Google Scholar101https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjtlGrs7Y%253D&md5=6d2f8b048038703aa3bea6393cdaf65bInvestigation of metal-insulator-like transition through the ab initio density matrix renormalization group approachFertitta, E.; Paulus, B.; Barcza, G.; Legeza, Oe.Physical Review B: Condensed Matter and Materials Physics (2014), 90 (24), 245129/1-245129/11, 11 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)We have studied the metal-insulator-like transition in pseudo-one-dimensional systems, i.e., lithium and beryllium rings, through the ab initio d. matrix renormalization group (DMRG) method. Performing accurate calcns. for different interat. distances and using quantum information theory, we investigated the changes occurring in the wave function between a metallic-like state and an insulating state built from free atoms. We also discuss entanglement and relevant excitations among the MOs in the Li and Be rings and show that the transition bond length can be detected using orbital entropy functions. Also, the effect of different orbital bases on the effectiveness of the DMRG procedure is analyzed comparing the convergence behavior.**102**Rissler, J.; Noack, R. M.; White, S. R. Measuring orbital interaction using quantum information theory.*Chem. Phys.*2006,*323*, 519– 531, DOI: 10.1016/j.chemphys.2005.10.018Google Scholar102https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XjvVanu74%253D&md5=ca82193fb0d3c9b3dbcd392adcdd9757Measuring orbital interaction using quantum information theoryRissler, Joerg; Noack, Reinhard M.; White, Steven R.Chemical Physics (2006), 323 (2-3), 519-531CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)Quantum information theory gives rise to a straightforward definition of the interaction of electrons Ip,q in two orbitals p,q for a given many-body wave function. A convenient way to calc. the von Neumann entropies needed is presented in this work, and the orbital interaction Ip,q is successfully tested for different types of chem. bonds. As an example of an application of Ip,q beyond the interpretation of wave functions, Ip,q is then used to investigate the ordering problem in the d.-matrix renormalization group.**103**Legeza, Ö.; Röder, J.; Hess, B. A. Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach.*Phys. Rev. B: Condens. Matter Mater. Phys.*2003,*67*, 125114, DOI: 10.1103/PhysRevB.67.125114Google Scholar103https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXivFaqsbY%253D&md5=0f5650c8557f55974dea14ef3e8a41d3Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approachLegeza, O.; Roder, J.; Hess, B. A.Physical Review B: Condensed Matter and Materials Physics (2003), 67 (12), 125114/1-125114/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)We have applied the momentum space version of the d.-matrix renormalization-group method (k-DMRG) in quantum chem. in order to study the accuracy of the algorithm in this new context. We have shown numerically that it is possible to det. the desired accuracy of the method in advance of the calcns. by dynamically controlling the truncation error and the no. of block states using a novel protocol that we dubbed dynamical block state selection protocol. The relationship between the real error and truncation error has been studied as a function of the no. of orbitals and the fraction of filled orbitals. We have calcd. the ground state of the mols. CH2, H2O, and F2 as well as the first excited state of CH2. Our largest calcns. were carried out with 57 orbitals, the largest no. of block states was 1500-2000, and the largest dimensions of the Hilbert space of the superblock configuration was 800 000-1 200 000.**104**Legeza, Ö.; Sólyom, J. Quantum data compression, quantum information generation, and the density-matrix renormalization-group method.*Phys. Rev. B: Condens. Matter Mater. Phys.*2004,*70*, 205118, DOI: 10.1103/PhysRevB.70.205118Google Scholar104https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXhtVGgsbvN&md5=81d2d092f4fcd276feafa092663763caQuantum data compression, quantum information generation, and the density-matrix renormalization-group methodLegeza, O.; Solyom, J.Physical Review B: Condensed Matter and Materials Physics (2004), 70 (20), 205118/1-205118/7CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)We have studied quantum data compression for finite quantum systems where the site d. matrixes are not independent, i.e., the d. matrix cannot be given as direct product of site d. matrixes and the von Neumann entropy is not equal to the sum of site entropies. Using the d.-matrix renormalization-group (DMRG) method for the one-dimensional Hubbard model, we have shown that a simple relationship exists between the entropy of the left or right block and dimension of the Hilbert space of that block as well as of the superblock for any fixed accuracy. The information loss during the RG procedure has been investigated and a more rigorous control of the relative error has been proposed based on Kholevo's theory. Our results are also supported by the quantum chem. version of DMRG applied to various mols. with system lengths up to 60 lattice sites. A sum rule that relates site entropies and the total information generated by the renormalization procedure has also been given, which serves as an alternative test of convergence of the DMRG method.**105**Legeza, Ö.; Fáth, G. Accuracy of the density-matrix renormalization-group method.*Phys. Rev. B: Condens. Matter Mater. Phys.*1996,*53*, 14349– 14358, DOI: 10.1103/PhysRevB.53.14349Google Scholar105https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XjsVCrsrg%253D&md5=d37b67aaa40eee4a8aee061f416120c5Accuracy of the density-matrix renormalization-group methodLegeza, Ors; Fath, GaborPhysical Review B: Condensed Matter (1996), 53 (21), 14349-14358CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)White's d.-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact soly. of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm paramters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.**106**Legeza, Ö., Veis, L., Mosoni, T.*QC-DMRG-Budapest, a program for quantum chemical DMRG calculations*; HAS RISSPO: Budapest, 2018.Google ScholarThere is no corresponding record for this reference.**107**Chan, G. K.-L.; Kállay, M.; Gauss, J. State-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curve.*J. Chem. Phys.*2004,*121*, 6110– 6116, DOI: 10.1063/1.1783212Google Scholar107https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXnvFeqs7g%253D&md5=409eb74e4f9716d5bab8ddf7de1e4f8bState-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curveChan, Garnet Kin-Lic; Kallay, Mihaly; Gauss, JurgenJournal of Chemical Physics (2004), 121 (13), 6110-6116CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the nitrogen binding curve with the d. matrix renormalization group (DMRG) and single-ref. and multireference coupled cluster (CC) theory. Our DMRG calcns. use up to 4000 states and our single-ref. CC calcns. include up to full connected hextuple excitations. Using the DMRG, we compute an all-electron benchmark nitrogen binding curve, at the polarized, valence double-zeta level (28 basis functions), with an estd. accuracy of 0.03 mEh. We also assess the performance of more approx. DMRG and CC theories across the nitrogen curve. We provide an anal. of the relative strengths and merits of the DMRG and CC theory under different correlation conditions.**108**Boguslawski, K.; Tecmer, P.; Barcza, G.; Legeza, Ö.; Reiher, M. Orbital Entanglement in Bond-Formation Processes.*J. Chem. Theory Comput.*2013,*9*, 2959– 2973, DOI: 10.1021/ct400247pGoogle Scholar108https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXnsFCltLY%253D&md5=88e637ee401cc68cc86ed61bd5659616Orbital Entanglement in Bond-Formation ProcessesBoguslawski, Katharina; Tecmer, Pawel; Barcza, Gergely; Legeza, Ors; Reiher, MarkusJournal of Chemical Theory and Computation (2013), 9 (7), 2959-2973CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The accurate calcn. of the (differential) correlation energy is central to the quantum chem. description of bond-formation and bond-dissocn. processes. In order to est. the quality of single- and multireference approaches for this purpose, various diagnostic tools have been developed. In this work, we elaborate on our previous observation that one- and two-orbital-based entanglement measures provide quant. means for the assessment and classification of electron correlation effects among MOs. The dissocn. behavior of some prototypical diat. mols. features all types of correlation effects relevant for chem. bonding. We demonstrate that our entanglement anal. is convenient to dissect these electron correlation effects and to provide a conceptual understanding of bond-forming and bond-breaking processes from the point of view of quantum information theory.**109**Valiev, M.; Bylaska, E.; Govind, N.; Kowalski, K.; Straatsma, T.; Dam, H. V.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T.; de Jong, W. NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations.*Comput. Phys. Commun.*2010,*181*, 1477– 1489, DOI: 10.1016/j.cpc.2010.04.018Google Scholar109https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXos1Cjur8%253D&md5=19100f255a4e6088076fb69421a9a0acNWChem: A comprehensive and scalable open-source solution for large scale molecular simulationsValiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; de Jong, W. A.Computer Physics Communications (2010), 181 (9), 1477-1489CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)A review. The latest release of NWChem delivers an open-source computational chem. package with extensive capabilities for large scale simulations of chem. and biol. systems. Utilizing a common computational framework, diverse theor. descriptions can be used to provide the best soln. for a given scientific problem. Scalable parallel implementations and modular software design enable efficient utilization of current computational architectures. This paper provides an overview of NWChem focusing primarily on the core theor. modules provided by the code and their parallel performance.**110**Lee, T. J.; Taylor, P. R. A diagnostic for determining the quality of singlereference electron correlation methods.*Int. J. Quantum Chem.*1989,*36*, 199– 207, DOI: 10.1002/qua.560360824Google ScholarThere is no corresponding record for this reference.**111**Krumnow, C.; Veis, L.; Legeza, Ö.; Eisert, J. Fermionic orbital optimization in tensor network states.*Phys. Rev. Lett.*2016,*117*, 210402, DOI: 10.1103/PhysRevLett.117.210402Google Scholar111https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsVSkt77N&md5=425ce4b4a69386812f4ed7ef6045a42cFermionie orbital optimization in tensor network statesKrumnow, C.; Veis, L.; Legeza, O.; Eisert, J.Physical Review Letters (2016), 117 (21), 210402/1-210402/6CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chem. A new freedom arising in such noniocal fcrmionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this Letter, we propose a way to overcome this challenge. We suggest a method intertwining the optimization over matrix product states with suitable fcrmionic Gaussian mode trans- formations. The described algorithm generalizes basis changes in the spirit of the Hartree-Fock method to matrix-product states, and provides a black box tool for basis optimization in tensor network methods.

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**1**Veis, L.; Antalík, A.; Brabec, J.; Neese, F.; Legeza, Ö.; Pittner, J. Coupled Cluster Method with Single and Double Excitations Tailored by Matrix Product State Wave Functions.*J. Phys. Chem. Lett.*2016,*7*, 4072– 4078, DOI: 10.1021/acs.jpclett.6b019081https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsFOktrbL&md5=6f0a8d79fff88c257aef0327abd74296Coupled Cluster Method with Single and Double Excitations Tailored by Matrix Product State Wave FunctionsVeis, Libor; Antalik, Andrej; Brabec, Jiri; Neese, Frank; Legeza, Ors; Pittner, JiriJournal of Physical Chemistry Letters (2016), 7 (20), 4072-4078CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)In the past decade, the quantum chem. version of the d. matrix renormalization group (DMRG) method has established itself as the method of choice for calcns. of strongly correlated mol. systems. Despite its favorable scaling, it is in practice not suitable for computations of dynamic correlation. We present a novel method for accurate "post-DMRG" treatment of dynamic correlation based on the tailored coupled cluster (CC) theory in which the DMRG method is responsible for the proper description of nondynamic correlation, whereas dynamic correlation is incorporated through the framework of the CC theory. We illustrate the potential of this method on prominent multireference systems, in particular, N2 and Cr2 mols. and also oxo-Mn(Salen), for which we have performed the first post-DMRG computations in order to shed light on the energy ordering of the lowest spin states.**2**Veis, L.; Antalík, A.; Legeza, Ö.; Alavi, A.; Pittner, J. The Intricate Case of Tetramethyleneethane: A Full Configuration Interaction Quantum Monte Carlo Benchmark and Multireference Coupled Cluster Studies.*J. Chem. Theory Comput.*2018,*14*, 2439– 2445, DOI: 10.1021/acs.jctc.8b000222https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXlvVKltb8%253D&md5=4509b1d7b8f169311482f9724950b512The Intricate Case of Tetramethyleneethane: A Full Configuration Interaction Quantum Monte Carlo Benchmark and Multireference Coupled Cluster StudiesVeis, Libor; Antalik, Andrej; Legeza, Ors; Alavi, Ali; Pittner, JiriJournal of Chemical Theory and Computation (2018), 14 (5), 2439-2445CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We have performed a full CI (FCI) quality benchmark calcn. for the tetramethyleneethane mol. in the cc-pVTZ basis set employing a subset of complete active space second order perturbation theory, CASPT2(6,6), natural orbitals for the FCI quantum Monte Carlo calcn. The results are in an excellent agreement with the previous large scale diffusion Monte Carlo calcns. by Pozun et al. and available exptl. results. Our computations verified that there is a max. on the potential energy surface (PES) of the ground singlet state (1A) 45° torsional angle, and the corresponding vertical singlet-triplet energy gap is 0.01 eV. We have employed this benchmark for the assessment of the accuracy of Mukherjee's coupled clusters with up to triple excitations (MkCCSDT) and CCSD tailored by the d. matrix renormalization group method (DMRG). Multireference MkCCSDT with CAS(2,2) model space, though giving good values for the singlet-triplet energy gap, is not able to properly describe the shape of the multireference singlet PES. Similarly, DMRG(24,25) is not able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method describes the shape of the ground singlet state with excellent accuracy but for the correct ordering requires computation of the zero-spin-projection component of the triplet state (3B1).**3**White, S. R.; Martin, R. L. Ab Initio Quantum Chemistry Using The Density Matrix Renormalization Group.*J. Chem. Phys.*1999,*110*, 4127– 4130, DOI: 10.1063/1.4782953https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXhtF2gtbw%253D&md5=ae0c47542a0ddc08171b93f29693e51fAb initio quantum chemistry using the density matrix renormalization groupWhite, Steven R.; Martin, Richard L.Journal of Chemical Physics (1999), 110 (9), 4127-4130CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper we describe how the d. matrix renormalization group can be used for quantum chem. calcns. for mols., as an alternative to traditional methods, such as CI or coupled cluster approaches. As a demonstration of the potential of this approach, we present results for the H2O mol. in a std. Gaussian basis. Results for the total energy of the system compare favorably with the best traditional quantum chem. methods.**4**McCulloch, I. P.; Gulácsi, M. The non-Abelian density matrix renormalization group algorithm.*Europhys. Lett.*2002,*57*, 852, DOI: 10.1209/epl/i2002-00393-04https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XivFSntr4%253D&md5=6051302fd2230ed8b19bc768a156da8bThe non-Abelian density matrix renormalization group algorithmMcCulloch, I. P.; Gulacsi, M.Europhysics Letters (2002), 57 (6), 852-858CODEN: EULEEJ; ISSN:0295-5075. (EDP Sciences)We describe here the extension of the d. matrix renormalization group algorithm to the case where the Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original d. matrix renormalization group basis. We demonstrate the usefulness of the construction via the one-dimensional Hubbard model as the symmetry group is enlarged from U(1) × U(1), up to SU(2) × SU(2).**5**Tóth, A.; Moca, C.; Legeza, Ö.; Zaránd, G. Density matrix numerical renormalization group for non-Abelian symmetries.*Phys. Rev. B: Condens. Matter Mater. Phys.*2008,*78*, 245109, DOI: 10.1103/PhysRevB.78.2451095https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXitlWjtw%253D%253D&md5=c2db4225058b3dfca6ad1b6bff6b1a1fDensity matrix numerical renormalization group for non-Abelian symmetriesToth, A. I.; Moca, C. P.; Legeza, O.; Zarand, G.Physical Review B: Condensed Matter and Materials Physics (2008), 78 (24), 245109/1-245109/11CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)We generalize the spectral sum rule preserving d. matrix numerical renormalization group (DM-NRG) method in such a way that it can make use of an arbitrary no. of not necessarily Abelian local symmetries present in the quantum impurity system. We illustrate the benefits of using non-Abelian symmetries by the example of calcns. for the T matrix of the two-channel Kondo model in the presence of magnetic field, for which conventional NRG methods produce large errors and/or take a long run-time.**6**Sharma, S.; Chan, G. K.-L. Spin-adapted density matrix renormalization group algorithms for quantum chemistry.*J. Chem. Phys.*2012,*136*, 124121, DOI: 10.1063/1.36956426https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XkvVOlur4%253D&md5=28c8a6d0e213d0040834e00b484d37fcSpin-adapted density matrix renormalization group algorithms for quantum chemistrySharma, Sandeep; Chan, Garnet Kin-LicJournal of Chemical Physics (2012), 136 (12), 124121/1-124121/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We extend the spin-adapted d. matrix renormalization group (DMRG) algorithm of to quantum chem. Hamiltonians. This involves using a quasi-d. matrix, to ensure that the renormalized DMRG states are eigenfunctions of S2 operator, and the Wigner-Eckart theorem, to reduce overall storage and computational costs. We argue that the spin-adapted DMRG algorithm is most advantageous for low spin states. Consequently, we also implement a singlet-embedding strategy due to where we target high spin states as a component of a larger fictitious singlet system. Finally, we present an efficient algorithm to calc. one- and two-body reduced d. matrixes from the spin-adapted wavefunctions. We evaluate our developments with benchmark calcns. on transition metal system active space models. These include the Fe2S2, Fe2S2(SCH3)42-, and Cr2 systems. In the case of Fe2S2, the spin-ladder spacing is on the micro-Hartree scale, and here we show that we can target such very closely spaced states. In Fe2S2(SCH3)42-, we calc. particle and spin correlation functions, to examine the role of sulfur bridging orbitals in the electronic structure. In Cr2 we demonstrate that spin-adaptation with the Wigner-Eckart theorem and using singlet embedding can yield up to an order of magnitude increase in computational efficiency. Overall, these calcns. demonstrate the potential of using spin-adaptation to extend the range of DMRG calcns. in complex transition metal problems. (c) 2012 American Institute of Physics.**7**Keller, S.; Reiher, M. Spin-adapted matrix product states and operators.*J. Chem. Phys.*2016,*144*, 134101, DOI: 10.1063/1.49449217https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XlsVWgsLs%253D&md5=b49f76ffe87ac609bebe9008dce38cabSpin-adapted matrix product states and operatorsKeller, Sebastian; Reiher, MarkusJournal of Chemical Physics (2016), 144 (13), 134101/1-134101/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Matrix product states (MPSs) and matrix product operators (MPOs) allow an alternative formulation of the d. matrix renormalization group algorithm introduced by White. Here, we describe how non-Abelian spin symmetry can be exploited in MPSs and MPOs by virtue of the Wigner-Eckart theorem at the example of the spin-adapted quantum chem. Hamiltonian operator. (c) 2016 American Institute of Physics.**8**Bartlett, R. J.; Musiał, M. Coupled-cluster theory in quantum chemistry.*Rev. Mod. Phys.*2007,*79*, 291– 352, DOI: 10.1103/RevModPhys.79.2918https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXmt1Cqtbw%253D&md5=59fd2f595def41752de72a92c8ac510cCoupled-cluster theory in quantum chemistryBartlett, Rodney J.; Musial, MonikaReviews of Modern Physics (2007), 79 (1), 291-352CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. Today, coupled-cluster theory offers the most accurate results among the practical ab initio electronic-structure theories applicable to moderate-sized mols. Though it was originally proposed for problems in physics, it has seen its greatest development in chem., enabling an extensive range of applications to mol. structure, excited states, properties, and all kinds of spectroscopy. In this review, the essential aspects of the theory are explained and illustrated with informative numerical results.**9**Lyakh, D. I.; Musiał, M.; Lotrich, V. F.; Bartlett, R. J. Multireference Nature of Chemistry: The Coupled-Cluster View.*Chem. Rev.*2012,*112*, 182– 243, DOI: 10.1021/cr20014179https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs12hsbjF&md5=26080054ce24a172826517cb7d772f62Multireference Nature of Chemistry: The Coupled-Cluster ViewLyakh, Dmitry I.; Musial, Monika; Lotrich, Victor F.; Bartlett, Rodney J.Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 182-243CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review. The following topics are discussed: Exponential era of electron correlation theory; Genuine MR CC theory in Hilbert space and in Fock space; Alternative MR CC methods. Numerical illustrations are presented.**10**Köhn, A.; Hanauer, M.; Mück, L. A.; Jagau, T.-C.; Gauss, J. State-specific multireference coupled-cluster theory.*Wiley Interdiscip. Rev.: Comput. Mol. Sci.*2013,*3*, 176– 197, DOI: 10.1002/wcms.1120There is no corresponding record for this reference.**11**Jeziorski, B.; Monkhorst, H. J. Coupled-cluster method for multideterminantal reference states.*Phys. Rev. A: At., Mol., Opt. Phys.*1981,*24*, 1668– 1681, DOI: 10.1103/PhysRevA.24.166811https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3MXmtV2qtrs%253D&md5=0434d85ec80902962638bf1a465dbc62Coupled-cluster method for multideterminantal reference statesJeziorski, Bogumil; Monkhorst, Hendrik J.Physical Review A: Atomic, Molecular, and Optical Physics (1981), 24 (4), 1668-81CODEN: PLRAAN; ISSN:0556-2791.A general coupled-cluster method valid for arbitrary multideterminantal ref. states is formulated. The resulting cluster expansion for the wave function is a generalization of that introduced by Silverstone and Sinanoglu (1966). The connected nature of the cluster operators, and the effective interaction is proven in the case when the ref. space is complete, i.e., is invariant under unitary transformations of partly occupied orbitals. For incomplete ref. spaces the disconnected terms appearing in the effective interaction are properly generated by the coupled-cluster theory. Approx. schemes for solving coupled-cluster equations are proposed and their relation with perturbation theory is briefly discussed.**12**Lee, J.; Small, D. W.; Epifanovsky, E.; Head-Gordon, M. Coupled-cluster valence-bond singles and doubles for strongly correlated systems: Block-tensor based implementation and application to oligoacenes.*J. Chem. Theory Comput.*2017,*13*, 602– 615, DOI: 10.1021/acs.jctc.6b0109212https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXmslSktw%253D%253D&md5=f12e7012e633ea5551d1c5ab3c7ac617Coupled-Cluster Valence-Bond Singles and Doubles for Strongly Correlated Systems: Block-Tensor Based Implementation and Application to OligoacenesLee, Joonho; Small, David W.; Epifanovsky, Evgeny; Head-Gordon, MartinJournal of Chemical Theory and Computation (2017), 13 (2), 602-615CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We demonstrate a block-tensor based implementation of coupled-cluster valence-bond singles and doubles (CCVB-SD) [Small, D. W.; Head-Gordon M. J. Chem. Phys.2012, 137, 114103] which is a simple modification to restricted CCSD (RCCSD) that provides a qual. correct description of valence correlations even in strongly correlated systems. We derive the Λ-equation of CCVB-SD and the corresponding unrelaxed d. matrixes. The resulting prodn.-level implementation is applied to oligoacenes, correlating up to 318 electrons in 318 orbitals. CCVB-SD shows a qual. agreement with exact methods for short acenes and reaches the bulk limit of oligoacenes in terms of natural orbital occupation nos., whereas RCCSD shows nonvariational behavior even for relatively short acenes. A significant redn. in polyradicaloid character is found when correlating all valence electrons instead of only the π-electrons.**13**Lindgren, I.; Mukherjee, D. On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces.*Phys. Rep.*1987,*151*, 93– 127, DOI: 10.1016/0370-1573(87)90073-113https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2sXlsVWht7k%253D&md5=35a97c30d3e61cef5fce8536dc6e45f3On the connectivity criteria in the open-shell coupled-cluster theory for general model spacesLindgren, Ingvar; Mukherjee, DebashisPhysics Reports (1987), 151 (2), 93-127CODEN: PRPLCM; ISSN:0370-1573.A review with 60 refs. includes discussion of current theor. status of linked-cluster theorem, connectivity of the cluster amplitudes, and the effective Hamiltonians.**14**Lindgren, I. Linked-Diagram and Coupled-Cluster Expansions for Multi-Configurational, Complete and Incomplete Model Spaces.*Phys. Scr.*1985,*32*, 291, DOI: 10.1088/0031-8949/32/4/009There is no corresponding record for this reference.**15**Mukherjee, D.; Moitra, R. K.; Mukhopadhyay, A. Applications of a non-perturbative many-body formalism to general open-shell atomic and molecular problems: calculation of the ground and the lowest π-π* singlet and triplet energies and the first ionization potential of trans-butadiene.*Mol. Phys.*1977,*33*, 955– 969, DOI: 10.1080/0026897770010087115https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE2sXlsVWnu7w%253D&md5=087024d694b5d2045c65d015d676afe1Applications of a nonperturbative many-body formalism to general open-shell atomic and molecular problems: calculation of the ground and the lowest π-π* singlet and triplet energies and the first ionization potential of trans-butadieneMukherjee, Debashis; Moitra, Raj Kumar; Mukhopadhyay, AtriMolecular Physics (1977), 33 (4), 955-69CODEN: MOPHAM; ISSN:0026-8976.The definition of the cluster expansion operator in the nonperturbative open-shell many-body formalism (1975) in generalized to allow incorporation into the model space of determinants differing widely in energy and in their no. of electrons. This is useful for calcg. difference energies in at. and mol. systems. The generalized scheme is used to calc. the energies of the ground, lowest π-π* singlet, lowest π-π* triplet, and singly ionized states and the 1st ionization potential of trans-butadiene. A single cluster expansion operator is employed for all states. Results are given for a general and several restricted model spaces. The calcns. agree with CI results complete in the chosen basis.**16**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method with optimized reference state.*Int. J. Quantum Chem.*1984,*26*, 267– 291, DOI: 10.1002/qua.560260827There is no corresponding record for this reference.**17**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. I. General formalism.*Phys. Rev. A: At., Mol., Opt. Phys.*1985,*32*, 725– 742, DOI: 10.1103/PhysRevA.32.72517https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC2sjotlKisA%253D%253D&md5=6944b99b1a4fdd6f97d622e5ad54d847Coupled-cluster method in Fock space. I. General formalismStolarczyk; MonkhorstPhysical review. A, General physics (1985), 32 (2), 725-742 ISSN:0556-2791.There is no expanded citation for this reference.**18**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. II. Brueckner-Hartree-Fock method.*Phys. Rev. A: At., Mol., Opt. Phys.*1985,*32*, 743, DOI: 10.1103/PhysRevA.32.743There is no corresponding record for this reference.**19**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. III. On similarity transformation of operators in Fock space.*Phys. Rev. A: At., Mol., Opt. Phys.*1988,*37*, 1908, DOI: 10.1103/PhysRevA.37.1908There is no corresponding record for this reference.**20**Stolarczyk, L. Z.; Monkhorst, H. J. Coupled-cluster method in Fock space. IV. Calculation of expectation values and transition moments.*Phys. Rev. A: At., Mol., Opt. Phys.*1988,*37*, 1926, DOI: 10.1103/PhysRevA.37.1926There is no corresponding record for this reference.**21**Stolarczyk, L. Z.; Monkhorst, H. J. Quasiparticle Fock-space coupled-cluster theory.*Mol. Phys.*2010,*108*, 3067– 3089, DOI: 10.1080/00268976.2010.51898121https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhsVyhsbfK&md5=50f59a083c979897f1e8ead8a339125bQuasiparticle Fock-space coupled-cluster theoryStolarczyk, Leszek Z.; Monkhorst, Hendrik J.Molecular Physics (2010), 108 (21-23), 3067-3089CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)The quasiparticle Fock-space coupled-cluster (QFSCC) theory, introduced by us in 1985, is described. This is a theory of many-electron systems which uses the second-quantisation formalism based on the algebraic approxn.: one chooses a finite spin-orbital basis, and builds a fermionic Fock space to represent all possible antisym. electronic states of a given system. The algebraic machinery is provided by the algebra of linear operators acting in the Fock space, generated by the fermion (creation and annihilation) operators. The Fock-space Hamiltonian operator then dets. the system's stationary states and their energies. Within the QFSCC theory, the Fock space and its operator algebra are subject to a unitary transformation which effectively changes electrons into some fermionic quasiparticles. A generalisation of the coupled-cluster method is achieved by enforcing the principle of quasiparticle-no. conservation. The emerging quasiparticle model of many-electron systems offers useful phys. insights and computational effectiveness. The QFSCC theory requires a substantial reformulation of the traditional second-quantisation language, by making full use of the algebraic properties of the Fock space and its operator algebra. In particular, the role of operators not conserving the no. of electrons (or quasiparticles) is identified.**22**Jeziorski, B.; Monkhorst, H. J. Coupled-cluster method for multideterminantal reference states.*Phys. Rev. A: At., Mol., Opt. Phys.*1981,*24*, 1668, DOI: 10.1103/PhysRevA.24.166822https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3MXmtV2qtrs%253D&md5=0434d85ec80902962638bf1a465dbc62Coupled-cluster method for multideterminantal reference statesJeziorski, Bogumil; Monkhorst, Hendrik J.**23**Datta, D.; Mukherjee, D. An explicitly spin-free compact open-shell coupled cluster theory using a multireference combinatoric exponential ansatz: Formal development and pilot applications.*J. Chem. Phys.*2009,*131*, 044124, DOI: 10.1063/1.318535623https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhtVCnsbfJ&md5=737a2ddde25dfccaf06391516d1c5794An explicitly spin-free compact open-shell coupled cluster theory using a multireference combinatoric exponential ansatz: Formal development and pilot applicationsDatta, Dipayan; Mukherjee, DebashisJournal of Chemical Physics (2009), 131 (4), 044124/1-044124/30CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a comprehensive account of an explicitly spin-free compact state-universal multireference coupled cluster (CC) formalism for computing the state energies of simple open-shell systems, e.g., doublets and biradicals, where the target open-shell states can be described by a few configuration state functions spanning a model space. The cluster operators in this formalism are defined in terms of the spin-free unitary generators with respect to the common closed-shell component of all model functions (core) as vacuum. The spin-free cluster operators are either closed-shell-like n hole-n particle excitations (denoted by Tμ) or involve excitations from the doubly occupied (nonvalence) orbitals to the singly occupied (valence) orbitals (denoted by Seμ). In addn., there are cluster operators with exchange spectator scatterings involving the valence orbitals (denoted by Sreμ). We propose a new multireference cluster expansion ansatz for the wave operator with the above generally noncommuting cluster operators which essentially has the same phys. content as the Jeziorski-Monkhorst ansatz with the commuting cluster operators defined in the spin-orbital basis. The Tμ operators in our ansatz are taken to commute with all other operators, while the Seμ and Sreμ operators are allowed to contract among themselves through the spectator valence orbitals. An important innovation of this ansatz is the choice of an appropriate automorphic factor accompanying each contracted composite of cluster operators in order to ensure that each distinct excitation generated by this composite appears only once in the wave operator. The resulting CC equations consist of two types of terms: A "direct" term and a "normalization" term contg. the effective Hamiltonian operator. It is emphasized that the direct term is almost quartic in the cluster amplitudes, barring only a handful of terms and termination of the normalization term depends on the valence rank of the effective Hamiltonian operator and the excitation rank of the cluster operators at which the theory is truncated. Illustrative applications are presented by computing the state energies of neutral doublet radicals and doublet mol. cations and ionization energies of neutral mols. and comparing our results with the other open-shell CC theories, benchmark full CI results (when available) in the same basis, and the exptl. results. Highly encouraging results show the efficacy of the method. (c) 2009 American Institute of Physics.**24**Evangelista, F. A.; Allen, W. D.; Schaefer, H. F., III High-order excitations in state-universal and state-specific multireference coupled cluster theories: Model systems.*J. Chem. Phys.*2006,*125*, 154113, DOI: 10.1063/1.235792324https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XhtFeksLrI&md5=1ce0b9045da90febeaced130d082bd0cHigh-order excitations in state-universal and state-specific multireference coupled cluster theories: Model systemsEvangelista, Francesco A.; Allen, Wesley D.; Schaefer, Henry F., IIIJournal of Chemical Physics (2006), 125 (15), 154113/1-154113/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)For the first time high-order excitations (n > 2) have been studied in three multireference couple cluster (MRCC) theories built on the wave operator formalism: (1) the state-universal (SU) method of Jeziorski and Monkhorst (JM) (2) the state-specific Brillouin-Wigner (BW) coupled cluster method, and (3) the state-specific MRCC approach of Mukherjee (Mk). For the H4, P4, BeH2, and H8 models, multireference coupled cluster wave functions, with complete excitations ranging from doubles to hextuples, have been computed with a new arbitrary-order string-based code. Comparison is then made to corresponding single-ref. coupled cluster and full CI (FCI) results. For the ground states the BW and Mk methods are found, in general, to provide more accurate results than the SU approach at all levels of truncation of the cluster operator. The inclusion of connected triple excitations reduces the nonparallelism error in singles and doubles MRCC energies by a factor of 2-10. In the BeH2 and H8 models, the inclusion of all quadruple excitations yields abs. energies within 1 kcal mol-1 of the FCI limit. While the MRCC methods are very effective in multireference regions of the potential energy surfaces, they are outperformed by single-ref. CC when one electronic configuration dominates.**25**Piecuch, P.; Paldus, J. Orthogonally spin-adapted multi-reference Hilbert space coupled-cluster formalism: Diagrammatic formulation.*Theor. Chim. Acta*1992,*83*, 69– 103, DOI: 10.1007/BF01113244There is no corresponding record for this reference.**26**Kucharski, S.; Balková, A.; Szalay, P.; Bartlett, R. J. Hilbert space multireference coupled-cluster methods. II. A model study on H8.*J. Chem. Phys.*1992,*97*, 4289– 4300, DOI: 10.1063/1.46393126https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK38XlvFylur0%253D&md5=144b60da7b81e1756bba0d1c88b90c7eHilbert-space-multireference-coupled-cluster methods. II. A model study on the octaatomic hydrogen clusterKucharski, S. A.; Balkova, A.; Szalay, P. G.; Bartlett, Rodney J.Journal of Chemical Physics (1992), 97 (6), 4289-300CODEN: JCPSA6; ISSN:0021-9606.The performance of various CC approaches using both single and multideterminantal refs. is investigated for the (quasi-)degenerate states of mol. systems, where inclusion of higher excitations (or equivalently nondynamic correlation) proves to be needed. The prototype system H8 represents an adequate model for our study, where we can vary the degree of degeneracy from a completely degenerate situation to a nondegenerate one in a continuous way. To obtain a reliable benchmark for CC results, the full CI (FCI) and large-scale complete active space CI (CAS CI) calcns., resp., are performed for a variety of geometries and states. The convergence of the approx. single ref. CC approaches is extremely sensitive to the level of degeneracies involved. In the nondegenerate case, the std. CC method with single and double excitations is found to be quite satisfactory; in the (quasi-)degenerate situations, however, the inclusion of triple excitations and noniterative quadruple excitations is needed to furnish semiquant. values of correlation energies. The alternative treatment of nondynamic correlation using a multideterminantal Hilbert space coupled-cluster (MRCC) method demonstrates the power of this approach, which provides a balanced description of both dynamic and nondynamic correlation in the degenerate region for all the investigated states of H8. Its convergence for nondegenerate situations, however, is less satisfactory, being affected by an intruder state problem.**27**Balková, A.; Kucharski, S.; Meissner, L.; Bartlett, R. J. A Hilbert space multi-reference coupled-cluster study of the H 4 model system.*Theor. Chim. Acta*1991,*80*, 335– 348, DOI: 10.1007/BF0111741727https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3MXmvVOgu74%253D&md5=7290c8e9f0e02182cda0277fbb514a9eA Hilbert space multi-reference coupled-cluster study of the hydrogen tetraatomic molecule model systemBalkova, A.; Kucharski, S. A.; Meissner, L.; Bartlett, Rodney J.Theoretica Chimica Acta (1991), 80 (4-5), 335-48CODEN: TCHAAM; ISSN:0040-5744.Employing the Hilbert space ansatz, a fully quadratic coupled-cluster method with a multidimensional ref. space is applied to a DZP basis study of the model system, H4. The ref. space is described by two to four configurations at the level of single and double excitations, and single and double excitation operations are included in the expansions for the cluster and wave operator through quadratic terms. The performance of quadratic MRCCSD is investigated for the ground and three excited states of the H4 system consisting of two stretched hydrogen mols. in a trapezoidal configuration where the degree of quasidegeneracy is varied from a nondegenerate situation to a completely degenerate one. Compared to full CI, in the highly degenerate region, the MRCCSD works quite well. In less degenerate regions, the accuracy is less satisfactory.**28**Kinoshita, T.; Hino, O.; Bartlett, R. J. Coupled-cluster method tailored by configuration interaction.*J. Chem. Phys.*2005,*123*, 074106, DOI: 10.1063/1.200025128https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpsleht7w%253D&md5=7c005bedda722c576125ea4cec764785Coupled-cluster method tailored by configuration interactionKinoshita, Tomoko; Hino, Osamu; Bartlett, Rodney J.Journal of Chemical Physics (2005), 123 (7), 074106/1-074106/6CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A method is presented which combines coupled cluster (CC) and CI to describe accurately potential-energy surfaces (PESs). We use the cluster amplitudes extd. from the complete active space CI calcn. to manipulate nondynamic correlation to tailor a single ref. CC theory (TCC). The dynamic correlation is then incorporated through the framework of the CC method. We illustrate the method by describing the PESs for HF, H2O, and N2 mols. which involve single, double, and triple bond-breaking processes. To the dissocn. limit, this approach yields far more accurate PESs than those obtained from the conventional CC method and the addnl. computational cost is negligible compared with the CC calcn. steps. We anticipate that TCC offers an effective and generally applicable approach for many problems.**29**Fang, T.; Shen, J.; Li, S. Block correlated coupled cluster method with a complete-active-space self-consistent-field reference function: The formula for general active spaces and its applications for multibond breaking systems.*J. Chem. Phys.*2008,*128*, 224107, DOI: 10.1063/1.293901429https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1cXntFyntbc%253D&md5=3f5bd5d601d324cb38268b7aac738127Block correlated coupled cluster method with a complete-active-space self-consistent-field reference function: The formula for general active spaces and its applications for multibond breaking systemsFang, Tao; Shen, Jun; Li, ShuhuaJournal of Chemical Physics (2008), 128 (22), 224107/1-224107/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The block correlated coupled cluster (BCCC) theory is developed for a general complete-active-space (CAS) self-consistent-field ref. function. By truncating the cluster operator up to the four-block correlation level, we derive the spin orbital formulation of the CAS-BCCC4 approach. The CAS-BCCC4 approach is invariant to sep. unitary transformation within active, occupied, and virtual orbitals. We have implemented the approach and applied this approach to describe the potential energy surfaces for bond breaking processes in C2 and N2 and for a simultaneous double bond dissocn. in H2O. Numerical results show that the CAS-BCCC4 approach provides quite accurate descriptions for the entire dissocn. process in each of the studied systems. The overall performance of the present approach is found to be better than that of the internally contracted multireference CI singles and doubles or complete-active-space second-order perturbation theory. The size-extensivity error is found to be relatively small for N2. (c) 2008 American Institute of Physics.**30**Datta, D.; Kong, L.; Nooijen, M. A state-specific partially internally contracted multireference coupled cluster approach.*J. Chem. Phys.*2011,*134*, 214116, DOI: 10.1063/1.359249430https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXntV2ktro%253D&md5=2d2bda10ce239ab6181bd8b886edbe8aA state-specific partially internally contracted multireference coupled cluster approachDatta, Dipayan; Kong, Liguo; Nooijen, MarcelJournal of Chemical Physics (2011), 134 (21), 214116/1-214116/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A state-specific partially internally contracted multireference coupled cluster approach is presented for general complete active spaces with arbitrary no. of active electrons. The dominant dynamical correlation is included via an exponential parametrization of internally contracted cluster operators (T) which excite electrons from a multideterminantal ref. function. The remaining dynamical correlation and relaxation effects are included via a diagonalization of the transformed Hamiltonian ‾H = e-THeT in the multireference CI singles space in an uncontracted fashion. A new set of residual equations for detg. the internally contracted cluster amplitudes is proposed. The second quantized matrix elements of ‾H, expressed using the extended normal ordering of Kutzelnigg and Mukherjee, are used as the residual equations without projection onto the excited configurations. These residual equations, referred to as the many-body residuals, do not have any near-singularity and thus, should allow one to solve all the amplitudes without discarding any. There are some relatively minor remaining convergence issues that may arise from an attempt to solve all the amplitudes and an initial anal. is provided in this paper. Applications to the bond-stretching potential energy surfaces for N2, CO, and the low-lying electronic states of C2 indicate clear improvements of the results using the many-body residuals over the conventional projected residual equations. (c) 2011 American Institute of Physics.**31**Hanauer, M.; Köhn, A. Pilot applications of internally contracted multireference coupled cluster theory, and how to choose the cluster operator properly.*J. Chem. Phys.*2011,*134*, 204111, DOI: 10.1063/1.359278631https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXms1SktbY%253D&md5=9697e08b7b9e13a092dc4536a41f600aPilot applications of internally contracted multireference coupled cluster theory, and how to choose the cluster operator properlyHanauer, Matthias; Koehn, AndreasJournal of Chemical Physics (2011), 134 (20), 204111/1-204111/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The internally contracted multireference coupled cluster (icMRCC) method allows a highly accurate description of both static and dynamic correlation with a computational scaling similar to single ref. coupled cluster theory. The authors show that the method can lose its orbital invariance and size consistency when no special care is taken in the elimination of redundant excitations. Using the BeH2 model system, four schemes are compared which differ in their treatment of linear dependencies between excitations of different rank (such as between singles and doubles). While the energy curves agree within tens of μEh when truncating the cluster operator at double excitations (icMRCCSD), inclusion of triple excitations (icMRCCSDT) leads to significant differences of more than 1 mEh. One scheme clearly yields the best results, while the others even turn out to be not size consistent. The former procedure uses genuine single and double excitations and discards those linear combinations of (spectator) double and triple excitations which have the same effect on the ref. function. With this approach, the equil. structure and harmonic vibrational frequencies of ozone obtained with icMRCCSDT are in excellent agreement with CCSDTQ. The authors further apply icMRCC methods to potential energy surfaces of HF, LiF, N2, and to the singlet-triplet splitting of benzynes. In particular, the latter calcns. have been made possible by implementing the method with the proper formal scaling using automated techniques. (c) 2011 American Institute of Physics.**32**Evangelista, F. A.; Gauss, J. An orbital-invariant internally contracted multireference coupled cluster approach.*J. Chem. Phys.*2011,*134*, 114102, DOI: 10.1063/1.355914932https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXjt1WnsLg%253D&md5=192193b11256b4ec9766accd14f98683An orbital-invariant internally contracted multireference coupled cluster approachEvangelista, Francesco A.; Gauss, JuergenJournal of Chemical Physics (2011), 134 (11), 114102/1-114102/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We have formulated and implemented an internally contracted multireference coupled cluster (ic-MRCC) approach aimed at solving two of the problems encountered in methods based on the Jeziorski-Monkhorst ansatz: (i) the scaling of the computational and memory costs with respect to the no. of refs., and (ii) the lack of invariance of the energy with respect to rotations among active orbitals. The ic-MRCC approach is based on a straightforward generalization of the single-ref. coupled cluster ansatz in which an exponential operator is applied to a multiconfigurational wave function. The ic-MRCC method truncated to single and double excitations (ic-MRCCSD) yields very accurate potential energy curves in benchmark computations on the Be + H2 insertion reaction, the dissocn. of hydrogen fluoride, and the sym. double dissocn. of water. Approxns. of the ic-MRCC theory in which the Baker-Campbell-Hausdorff expansion is truncated up to a given no. of commutators are found to converge quickly to the full theory. In our tests, two commutators are sufficient to recover a total energy within 0.5 mEh of the full ic-MRCCSD method along the entire potential energy curve. A formal anal. shows that the ic-MRCC method is invariant with respect to rotation among active orbitals, and that the orthogonalization procedure used to produce the set of linearly independent excitation operators plays a crucial role in guaranteeing the invariance properties. The orbital invariance was confirmed in numerical tests. Moreover, approximated versions of the ic-MRCC theory based on a truncated Baker-Campbell-Hausdorff expansion, preserve the orbital invariance properties of the full theory. (c) 2011 American Institute of Physics.**33**Lyakh, D. I.; Ivanov, V. V.; Adamowicz, L. Automated generation of coupled-cluster diagrams: Implementation in the multireference state-specific coupled-cluster approach with the complete-active-space reference.*J. Chem. Phys.*2005,*122*, 024108, DOI: 10.1063/1.182489733https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXislaiug%253D%253D&md5=7d69842cc09bb81a4a3dc980d5cbd4dbAutomated generation of coupled-cluster diagrams: Implementation in the multireference state-specific coupled-cluster approach with the complete-active-space referenceLyakh, Dmitry I.; Ivanov, Vladimir V.; Adamowicz, LudwikJournal of Chemical Physics (2005), 122 (2), 024108/1-024108/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An algorithm for generation of the spin-orbital diagrammatic representation, the corresponding algebraical formulas, and the computer code of the coupled-cluster (CC) method with an arbitrary level of the electronic excitations has been developed. The method was implemented in the general case as well as for specific application in the state-specific multireference coupled-cluster theory (SSMRCC) based on the concept of a "formal ref. state." The algorithm was tested in SSMRCC calcns. describing dissocn. of a single bond and in calcns. describing simultaneous dissocn. of two single bonds-the problem requiring up to six-particle excitations in the CC operator.**34**Hanrath, M. An exponential multireference wave-function Ansatz.*J. Chem. Phys.*2005,*123*, 084102, DOI: 10.1063/1.195340734https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpvFClu7w%253D&md5=2ea2eb79a0daeaef1b48ff34c888dfd8An exponential multireference wave-function AnsatzHanrath, MichaelJournal of Chemical Physics (2005), 123 (8), 084102/1-084102/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)An exponential multireference wave-function Ansatz is formulated. In accordance with the state universal coupled-cluster Ansatz of Jeziorski and Monkhorst [Phys. Rev. A 24, 1668 (1981)] the approach uses a ref. specific cluster operator. In order to achieve state selectiveness the excitation- and ref.-related amplitude indexing of the state universal Ansatz is replaced by an indexing which is based on excited determinants. There is no ref. determinant playing a particular role. The approach is size consistent, coincides with traditional single-ref. coupled cluster if applied to a single-ref., and converges to full CI with an increasing cluster operator excitation level. Initial applications on BeH2, CH2, Li2, and nH2 are reported.**35**Pittner, J.; Nachtigall, P.; Čársky, P.; Hubač, I. State-Specific Brillouin- Wigner Multireference Coupled Cluster Study of the Singlet- Triplet Separation in the Tetramethyleneethane Diradical.*J. Phys. Chem. A*2001,*105*, 1354– 1356, DOI: 10.1021/jp003219935https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXnvVOkug%253D%253D&md5=c653f767b0aba2538b8720b0dc52a5b1State-Specific Brillouin-Wigner Multireference Coupled Cluster Study of the Singlet-Triplet Separation in the Tetramethyleneethane DiradicalPittner, Jiri; Nachtigall, Petr; Carsky, Petr; Hubac, IvanJournal of Physical Chemistry A (2001), 105 (8), 1354-1356CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)The potential energy curves for the twisting of tetramethyleneethane (I) in its lowest singlet and triplet states were calcd. by the state-specific two-ref. Brillouin-Wigner coupled-cluster method with single and double excitations. The calcd. potential energy curves are essentially the same as those obtained by the two-determinant CCSD method, and they are also in agreement with the previously reported d. functional theory results. Our data bring support for the previously suggested interpretation of exptl. data on I in the gas phase and in the matrix.**36**Hubač, I.; Wilson, S. On the use of Brillouin-Wigner perturbation theory for many-body systems.*J. Phys. B: At., Mol. Opt. Phys.*2000,*33*, 365, DOI: 10.1088/0953-4075/33/3/30636https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXhsVOkt7o%253D&md5=fe17ef2812fd715a55747ac599265353On the use of Brillouin-Wigner perturbation theory for many-body systemsHubac, I.; Wilson, S.Journal of Physics B: Atomic, Molecular and Optical Physics (2000), 33 (3), 365-374CODEN: JPAPEH; ISSN:0953-4075. (Institute of Physics Publishing)The use of Brillouin-Wigner perturbation theory in describing many-body systems is critically re-examd.**37**Hubač, I.; Pittner, J.; Čársky, P. Size-extensivity correction for the state-specific multireference Brillouin-Wigner coupled-cluster theory.*J. Chem. Phys.*2000,*112*, 8779– 8784, DOI: 10.1063/1.48149337https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXivFClsLk%253D&md5=346ef344a80a5066f6242bc9bb0093a4Size-extensivity correction for the state-specific multireference Brillouin-Wigner coupled-cluster theoryHubac, Ivan; Pittner, Jiri; Carsky, PetrJournal of Chemical Physics (2000), 112 (20), 8779-8784CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present a simple a posteriori correction for the state-specific multireference Brillouin-Wigner coupled-cluster (MR BWCCSD) theory, which eliminates its size-extensivity error. In the converged amplitudes we drop terms that were identified to be responsible for the lack of size extensivity. We performed MR BWCCSD calcns. with this correction on CH2, SiH2, twisted ethylene, F2, and ozone that are all, from the computational point of view, typical representatives of two-ref. problems. Comparison with rigorously size-extensive calcns. and expt. shows that the size-extensivity error of the cor. MR BWCCSD is only a few tenths of kcal/mol.**38**Pittner, J.; Šmydke, J.; Čársky, P.; Hubač, I. State-specific Brillouin-Wigner multireference coupled cluster study of the F2 molecule: assessment of the a posteriori size-extensivity correction.*J. Mol. Struct.: THEOCHEM*2001,*547*, 239– 244, DOI: 10.1016/S0166-1280(01)00473-038https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3MXltVajs7c%253D&md5=96a9e262e118ed0912e96d3b60473ce6State-specific Brillouin-Wigner multireference coupled cluster study of the F2 molecule: assessment of the a posteriori size-extensivity correctionPittner, J. V.; Smydke, J.; Carsky, P.; Hubac, I.Journal of Molecular Structure: THEOCHEM (2001), 547 (), 239-244CODEN: THEODJ; ISSN:0166-1280. (Elsevier Science B.V.)We tested a posteriori correction suggested previously for the state-specific multireference Brillouin-Wigner coupled-cluster singles and doubles (MR BW CCSD) theory, to eliminate its size-extensivity error. The correction was applied to a two-ref. BW CCSD model by using the cc-pVXZ basis sets (X = 2,3,4) and it was tested by calcg. the spectroscopic consts. of the F2 mol.**39**Fang, T.; Li, S. Block correlated coupled cluster theory with a complete active-space self-consistent-field reference function: The formulation and test applications for single bond breaking.*J. Chem. Phys.*2007,*127*, 204108, DOI: 10.1063/1.280002739https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2sXhsVWhu7zE&md5=c3af725bc9dd81b5d29a583075fc321fBlock correlated coupled cluster theory with a complete active-space self-consistent-field reference function: The formulation and test applications for single bond breakingFang, Tao; Li, ShuhuaJournal of Chemical Physics (2007), 127 (20), 204108/1-204108/12CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Block correlated coupled cluster (BCCC) theory with a complete active-space self-consistent-field (CASSCF) ref. function is presented. This theory provides an alternative multireference coupled cluster framework to describe the multireference characters of the ground-state wave functions. In this approach, a multireference block is defined to incorporate the nondynamic correlation, and all other blocks involve just a single spin orbital. The cluster operators are truncated up to the four-block correlation level, leading to the BCCC4 scheme. For a single bond breaking problem, the present CAS-BCCC4 approach with a CASSCF(2,2) ref. function computationally scales as the traditional single-ref. coupled cluster singles and doubles. We have applied the present approach to investigate the electronic structures of several model systems including H4, P4, and BeH2, and the single bond breaking processes in small systems such as F2, HF, BH, and CH4. A comparison of our results with those from full CI calcns. shows that the present approach can provide quant. descriptions for all the studied systems. The size-consistency error is found to be quite small in the dissocn. limit of diat. mols. F2, HF, and BH.**40**Chattopadhyay, S.; Mahapatra, U. S.; Mukherjee, D. Development of a linear response theory based on a state-specific multireference coupled cluster formalism.*J. Chem. Phys.*2000,*112*, 7939– 7952, DOI: 10.1063/1.48139540https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXivVOisro%253D&md5=0e6b4748b217a0e9eeff6bf0e3c4f63fDevelopment of a linear response theory based on a state-specific multireference coupled cluster formalismChattopadhyay, Sudip; Mahapatra, Uttam Sinha; Mukherjee, DebashisJournal of Chemical Physics (2000), 112 (18), 7939-7952CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present in this paper a linear response theory based on our recently developed state-specific multireference coupled cluster (SS-MRCC) method to compute excited state energies for systems whose ground state has a pronounced multireference character. The SS-MRCC method is built on complete active space ref. functions, and is designed to treat quasidegeneracy of varying degrees while bypassing the intruder problem. The linear response theory based on such a function [multireference coupled cluster based linear response theory (MR-CCLRT)] offers a very convenient access to computation of excited states and, in particular, to generation of potential energy surfaces (PES) for excited states where a traditional response formulation based on a single ref. theory will fail due to the quasidegeneracy at some regions of the PES and the effective Hamiltonian-based multireference response methods would be plagued by intruders. An attractive feature of the MR-CCLRT is that the computed excitation energies are size intensive in the sense that they become asymptotically equal to the sum of fragment excitation energies in the limit of noninteracting fragments. Illustrative numerical results are presented for the excited state PES of the rectangular H4 (P4) model, the trapezoidal H4 (H4) model, for Li2, and for some sample points on the excited states PES of the BeH2 complex. The ground states of all the three examples possess quasidegeneracy at some point on the PES, and there are potential intruders at some other points in the PES, and hence are appropriate to test the efficacy of the MR-CCLRT. A comparison with the (CI) full CI and MR-CCLRT results in the same basis for all the mols. shows very good performance of the theory in general, and indicates the efficacy of the method.**41**Kong, L. Connection between a few Jeziorski-Monkhorst ansatz-based methods.*Int. J. Quantum Chem.*2009,*109*, 441– 447, DOI: 10.1002/qua.2182241https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXhsVeltA%253D%253D&md5=9e3b2d29476cca1c77d9a4b5e75791deConnection between a few Jeziorski-Monkhorst ansatz-based methodsKong, LiguoInternational Journal of Quantum Chemistry (2008), 109 (3), 441-447CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)Different Jeziorski-Monkhorst ansatz-based methods are unified according to how to group terms to eliminate the redundancy problem. It is found that some seemingly different methods used to do MRCC are equiv. It is argued that the various defining equations are not entirely proper, in the sense that the proper residual condition is not satisfied. This may partially rationalize the unsatisfactory performance of the various methods for single ref. systems. In contrast, the MRexpT method satisfies the proper residual condition and it is expected that it will outperform other JM ansatz-based methods in single-ref. cases.**42**Chattopadhyay, S.; Mahapatra, U. S.; Mukherjee, D. Property calculations using perturbed orbitals via state-specific multireference coupled-cluster and perturbation theories.*J. Chem. Phys.*1999,*111*, 3820– 3831, DOI: 10.1063/1.47968542https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXltFGkurc%253D&md5=927dde8df8263b311b25e85ccc6b6674Property calculations using perturbed orbitals via state-specific multireference coupled-cluster and perturbation theoriesChattopadhyay, Sudip; Mahapatra, Uttam Sinha; Mukherjee, DebashisJournal of Chemical Physics (1999), 111 (9), 3820-3831CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper we apply the recently developed state-specific multireference coupled-cluster and perturbation theories to calc. elec. properties such as dipole moment and static polarizability using perturbed orbitals in finite fields. The theories are built on complete active space ref. functions, and are designed to treat quasidegeneracy of varying degrees while bypassing the intruder problem. Numerical results are presented for the model systems H4 with trapezoidal geometry and the lowest two singlet states of CH2. Both the systems require a multireference formulation due to quasidegeneracy. In the field-free situation, the former encounters intruders at an intermediate trapezoidal geometry in the traditional treatment using effective Hamiltonians, while the latter shows a pronounced multireference character in the two singlet states. This affects the response properties in the presence of a perturbing field. A comparison with the full CI results in the same basis indicates the efficacy of the state-specific methods in wide ranges of geometries, even when the traditional effective Hamiltonian based methods fail due to intruders.**43**Pittner, J. Continuous transition between Brillouin-Wigner and Rayleigh-Schrödinger perturbation theory, generalized Bloch equation, and Hilbert space multireference coupled cluster.*J. Chem. Phys.*2003,*118*, 10876– 10889, DOI: 10.1063/1.157478543https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXksFyns78%253D&md5=9fb8dd8eb2dd496426241146ba4fe4a7Continuous transition between Brillouin-Wigner and Rayleigh-Schroedinger perturbation theory, generalized Bloch equation, and Hilbert space multireference coupled clusterPittner, JiriJournal of Chemical Physics (2003), 118 (24), 10876-10889CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A continuous transition between the Rayleigh-Schrodinger and Brillouin-Wigner perturbation theories was constructed and the Bloch equation for the corresponding wave operator was derived. Subsequently it was applied to the Hilbert space multireference coupled cluster theory and used to investigate relationships between several versions of multireference coupled cluster methods. Finally, based on those continuous transitions, new size extensivity corrections for the Brillouin-Wigner coupled cluster method were suggested. Numerical tests of size-extensivity and separability of a supermol. to closed- and open-shell fragments are also presented. Equivalence of some of the multireference coupled cluster methods with single and double excitations to full CI for two-electron systems was investigated, both theor. and numerically.**44**Mahapatra, U. S.; Datta, B.; Mukherjee, D. A size-consistent state-specific multireference coupled cluster theory: Formal developments and molecular applications.*J. Chem. Phys.*1999,*110*, 6171– 6188, DOI: 10.1063/1.47852344https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1MXitVCmt7c%253D&md5=f98011e4719a6b12ab0fe42bb71ed504A size-consistent state-specific multireference coupled cluster theory: formal developments and molecular applicationsMahapatra, Uttam Sinha; Datta, Barnali; Mukherjee, DebashisJournal of Chemical Physics (1999), 110 (13), 6171-6188CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In this paper, we present a comprehensive account of a manifestly size-consistent coupled cluster formalism for a specific state, which is based on a ref. function composed of determinants spanning a complete active space (CAS). The method treats all the ref. determinants on the same footing and is hence expected to provide uniform description over a wide range of mol. geometry. The combining coeffs. are detd. by diagonalizing an effective operator in the CAS and are thus completely flexible, not constrained to preassigned values. A sep. exponential-type excitation operator is invoked to induce excitations to all the virtual functions from each ref. determinant. The linear dependence inherent in this choice of cluster operators is eliminated by invoking suitable sufficiency conditions, which in a transparent manner leads to manifest size extensivity. The use of a CAS also guarantees size consistency. We also discuss the relation of our method with the extant state-specific formalisms. Illustrative applications are presented for systems such as H4 in rectangular and trapezoidal geometries, the Be-H2 C2v insertion reaction path, the potential energy curves of Li2 and F2, and certain electronic states of CH2 and C2 mols. with pronounced multireference character. The results indicate the efficacy of the method for obviating the intruders and of providing accuracy.**45**Mášik, J.; Hubač, I.; Mach, P. Single-root multireference Brillouin-Wigner coupled-cluster theory: Applicability to the F 2 molecule.*J. Chem. Phys.*1998,*108*, 6571– 6579, DOI: 10.1063/1.47607145https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXisVart74%253D&md5=a48ba5056620f0d4437d442eddc820dbSingle-root multireference Brillouin-Wigner coupled-cluster theory: Applicability to the F2 moleculeMasik, Jozef; Hubac, Ivan; Mach, PavelJournal of Chemical Physics (1998), 108 (16), 6571-6579CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Recently developed single-root multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory, which deals with one state at a time while employing a multiconfigurational ref. wave function, is applied to the ground state of the F2 mol. using a two-determinant ref. space at the level of the CCSD approxn. The method represents a brand-new coupled-cluster (CC) approach to quasidegenerate problems which combines merits of two theories: the single-ref. CC method in a nondegenerate case and the Hilbert space MR CC method in quasidegenerate case. The method is able to switch itself from a nondegenerate to a fully degenerate case in a continuous manner, providing thus smooth potential energy surfaces. Moreover, in contrast to the Hilbert space MR CC approaches, it does not contain the so-called coupling terms and completely reduces to the std. single-ref. CC method in a highly nondegenerate region. Using a [4s,3p,1d] and [4s,3p,2d,1f ] basis sets, the calcd. potential energy curves are smooth, dissoc. correctly and the results are compared with other available multireference techniques as well as expt.**46**Hubač, I.; Neogrády, P. Size-consistent Brillouin-Wigner perturbation theory with an exponentially parametrized wave function: Brillouin-Wigner coupled-cluster theory.*Phys. Rev. A: At., Mol., Opt. Phys.*1994,*50*, 4558– 4564, DOI: 10.1103/PhysRevA.50.455846https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2MXis1Krs70%253D&md5=10030999f110ac3516ebf1536a9630f1Size-consistent Brillouin-Wigner perturbation theory with an exponentially parametrized wave function: Brillouin-Wigner coupled-cluster theoryHubac, Ivan; Neogrady, PavelPhysical Review A: Atomic, Molecular, and Optical Physics (1994), 50 (6), 4558-64CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)The size consistency of the Brillouin-Wigner perturbation theory is studied by using the Lippmann-Schwinger equation and an exponential ansatz for the wave function. The relation of this theory to the coupled-cluster method is studied, and a comparison through the effective-Hamiltonian method is also provided.**47**Adamowicz, L.; Malrieu, J.-P.; Ivanov, V. V. New approach to the state-specific multireference coupled-cluster formalism.*J. Chem. Phys.*2000,*112*, 10075– 10084, DOI: 10.1063/1.48164947https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXjvVOlu7s%253D&md5=fa287ceeaca8599e6dbc5670f9bdbd69New approach to the state-specific multireference coupled-cluster formalismAdamowicz, Ludwik; Malrieu, Jean-Paul; Ivanov, Vladimir V.Journal of Chemical Physics (2000), 112 (23), 10075-10084CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A new development is presented in the framework of the state-specific multireference (MR) coupled-cluster (CC) theory (MRCC). The method is based on the CASSCF (complete active space SCF) wave function and it is designed specifically for calcg. excited electronic states. In the proposed approach, the cluster structure of the CC wave operator and the method to det. this operator are the key features. Since the general formulation of the CASCC method is uncontracted, i.e., allows the interaction between the nondynamic and dynamic correlation effects to affect both the CAS ref. function and the CC correlation wave operator, the method is expected to perform better than contracted perturbative approaches such as the CASPT2 (second-order perturbation theory based on the CAS wave function) method. Also, the CASCC method is not a perturbative approach and is not based on selection of an unperturbed Hamiltonian, which in the case of the CASPT2 method often leads to the "intruder state" problem. CASCC calcns. of the lowest totally sym. excited state of the H8 model system using the internally contracted and uncontracted approaches reveal some interesting features of the methodol.**48**Kállay, M.; Szalay, P. G.; Surján, P. R. A general state-selective multireference coupled-cluster algorithm.*J. Chem. Phys.*2002,*117*, 980– 990, DOI: 10.1063/1.148385648https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XltVOhs74%253D&md5=314fba7fa8a56ef47a58638e1b3d8542A general state-selective multireference coupled-cluster algorithmKallay, Mihaly; Szalay, Peter G.; Surjan, Peter R.Journal of Chemical Physics (2002), 117 (3), 980-990CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)A state-selective multireference coupled-cluster algorithm is presented which is capable of describing single, double (or higher) excitations from an arbitrary complete model space. One of the active space determinants is chosen as a formal Fermi-vacuum and single, double (or higher) excitations from the other ref. functions are considered as higher excitations from this determinant as it has been previously proposed by Oliphant and Adamowicz [J. Chem. Phys. 94, 1229 (1991)]. Coupled-cluster equations are generated in terms of antisymmetrized diagrams and restrictions are imposed on these diagrams to eliminate those cluster amplitudes which carry undesirable no. of inactive indexes. The corresponding algebraic expressions are factorized and contractions between cluster amplitudes and intermediates are evaluated by our recent string-based algorithm [J. Chem. Phys. 115, 2945 (2001)]. The method can be easily modified to solve multireference CI problems. The performance of the method is demonstrated by several test calcns. on systems which require a multireference description. The problem related to the choice of the Fermi-vacuum has also been investigated.**49**Piecuch, P.; Kowalski, K. The state-universal multi-reference coupled-cluster theory: An overview of some recent advances.*Int. J. Mol. Sci.*2002,*3*, 676– 709, DOI: 10.3390/i306067649https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xnsl2hsL0%253D&md5=2f527cacb01aafe273c1ba320a494013The state-universal multi-reference coupled-cluster theory: an overview of some recent advancesPiecuch, Piotr; Kowalski, KarolInternational Journal of Molecular Sciences [online computer file] (2002), 3 (6), 676-709CODEN: IJMCFK; ISSN:1422-0067. (Molecular Diversity Preservation International)A review. Some recent advances in the area of multi-ref. coupled-cluster theory of the state-universal type are discussed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-ref. coupled-cluster singles and doubles method (SUMRCCSD) with the multi-ref. many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUMRCCSD formalism that carry only core and virtual orbital indexes are replaced by the first-order MRMBPT ests.; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. The new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indexes by the first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.**50**Schucan, T.; Weidenmüller, H. The effective interaction in nuclei and its perturbation expansion: An algebraic approach.*Ann. Phys.*1972,*73*, 108– 135, DOI: 10.1016/0003-4916(72)90315-650https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE38XlsVKhu70%253D&md5=fe7aeddd54b795b7ae8e43663c654d06Effective interaction in nuclei and its perturbation expansion. Algebraic approachSchucan, T. H.; Weidenmueller, H. A.Annals of Physics (San Diego, CA, United States) (1972), 73 (1), 108-35CODEN: APNYA6; ISSN:0003-4916.A finite-dimensional model is considered for the Hilbert space of the A-N problem. In the frame of this model the energy-independent effective interaction W first introduced by Des Cloiseaux and Brandow is constructed. The connection is demonstrated between this explicit form and the implicit equation for W given by these authors. By using the explicit form for W, the perturbation expansion is investigated of W in powers of the interaction. (When used in the nuclear problem, this expansion leads to the folded diagrams.) This expansion is likely to diverge in most cases of practical interest. Several methods are given which can yield a convergent expansion for W. The implications of these results for paractical calcns. are discussed.**51**Kaldor, U. Intruder states and incomplete model spaces in multireference coupled-cluster theory: The 2*p*^{2}states of Be.*Phys. Rev. A: At., Mol., Opt. Phys.*1988,*38*, 6013, DOI: 10.1103/PhysRevA.38.601351https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXht1Wktbg%253D&md5=97b84de48b67076e7d8c9eec4e474850Intruder states and incomplete model spaces in multireference coupled-cluster theory: the 2p2 states of berylliumKaldor, UziPhysical Review A: Atomic, Molecular, and Optical Physics (1988), 38 (12), 6013-16CODEN: PLRAAN; ISSN:0556-2791.The open-shell coupled-cluster (CC) method is applied to the excited 2p2 states of Be. With the 2s2 and 2p2 configurations included in the model (P) space, the CC equations prove very difficult to converge. When they do converge, very large (>5) excitation amplitudes are obsd., and the second 1S corresponds to the intruder 2s3s configuration, rather than the desired 2p2. The inclusion of 2s3s (but not 3s2) in the model space, which thereby becomes incomplete, improves convergence significantly, and gives energies in very good agreement with values known from other sources.**52**Malrieu, J.; Durand, P.; Daudey, J. Intermediate Hamiltonians as a new class of effective Hamiltonians.*J. Phys. A: Math. Gen.*1985,*18*, 809, DOI: 10.1088/0305-4470/18/5/01452https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2MXhvFWgsb0%253D&md5=b98f331eaccb371286295249909bb1b7Intermediate Hamiltonians as a new class of effective HamiltoniansMalrieu, J. P.; Durand, P.; Daudey, J. P.Journal of Physics A: Mathematical and General (1985), 18 (5), 809-26CODEN: JPHAC5; ISSN:0305-4470.A new class of effective Hamiltonians (called intermediate Hamiltonians) is presented; only one part of their roots are exact eigenenergies of the full Hamiltonian. The theory of these intermediate Hamiltonians is presented by means of a new wave-operator R which is the analog of the wave-operated Ω in the theory of effective Hamiltonians. Solns. are obtained by a generalized degenerate perturbation theory (GDPT) and by iterative procedures. Two model systems are numerically solved which demonstrate the good convergence properties of GDPT with respect to std. degenerate perturbation theory. Continuity of the solns. is also checked in the presence of an intruder state.**53**Jankowski, K.; Malinowski, P. A valence-universal coupled-cluster single-and double-excitations method for atoms. III. Solvability problems in the presence of intruder states.*J. Phys. B: At., Mol. Opt. Phys.*1994,*27*, 1287, DOI: 10.1088/0953-4075/27/7/00453https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXjtVOksrg%253D&md5=c350f67988d697cf44431b3d18b049c9A valence-universal coupled-cluster single- and double-excitations method for atoms: III. Solvability problems in the presence of intruder statesJankowski, K.; Malinowski, P.Journal of Physics B: Atomic, Molecular and Optical Physics (1994), 27 (7), 1287-98CODEN: JPAPEH; ISSN:0953-4075.To better understand the problems met when solving the equations of valence-universal coupled-cluster (VU-CC) approaches in the presence of intruder states, the authors are concerned with the following aspects of the solvability problem for sets of non-linear equations: the existence and properties of multiple solns. and the attainability of these solns. by means of various numerical methods. This study is concd. on the equations obtained for Be within the framework of the recently formulated at. oriented form of the VU-CC accounting for one- and two-electron excitations (VU-CCSD/R) and based on the complete model space (2s2, 2p2). Six pairs of multiple solns. representing four 1S states are found and discussed. Three of these solns. provide amplitudes describing the 2p2 1S state for which the intruder state problem has been considered as extremely serious. Several known numerical methods have been applied to solve the same set of non-linear equations for the two-valence cluster amplitudes. It is shown that these methods perform quite differently in the presence of intruder states, which seems to indicate that the intruder state problem for VU-CC methods is partly caused by the commonly used methods of solving the non-linear equations.**54**Sharma, S.; Alavi, A. Multireference linearized coupled cluster theory for strongly correlated systems using matrix product states.*J. Chem. Phys.*2015,*143*, 102815, DOI: 10.1063/1.492864354https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhtlyns7vM&md5=0afcce08382e94e43b86e765e5b17811Multireference linearized coupled cluster theory for strongly correlated systems using matrix product statesSharma, Sandeep; Alavi, AliJournal of Chemical Physics (2015), 143 (10), 102815/1-102815/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We propose a multireference linearized coupled cluster theory using matrix product states (MPSs-LCC) which provides remarkably accurate ground-state energies, at a computational cost that has the same scaling as multireference CI singles and doubles, for a wide variety of electronic Hamiltonians. These range from first-row dimers at equil. and stretched geometries to highly multireference systems such as the chromium dimer and lattice models such as periodic two-dimensional 1-band and 3-band Hubbard models. The MPS-LCC theory shows a speed up of several orders of magnitude over the usual D. Matrix Renormalization Group (DMRG) algorithm while delivering energies in excellent agreement with converged DMRG calcns. Also, in all the benchmark calcns. presented here, MPS-LCC outperformed the commonly used multi-ref. quantum chem. methods in some cases giving energies in excess of an order of magnitude more accurate. As a size-extensive method that can treat large active spaces, MPS-LCC opens up the use of multireference quantum chem. techniques in strongly correlated ab initio Hamiltonians, including two- and three-dimensional solids. (c) 2015 American Institute of Physics.**55**Henderson, T. M.; Bulik, I. W.; Stein, T.; Scuseria, G. E. Seniority-based coupled cluster theory.*J. Chem. Phys.*2014,*141*, 244104, DOI: 10.1063/1.490438455https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXitFOjsrnK&md5=e3ebcce5c47052da1a339a27b5db275dSeniority-based coupled cluster theoryHenderson, Thomas M.; Bulik, Ireneusz W.; Stein, Tamar; Scuseria, Gustavo E.Journal of Chemical Physics (2014), 141 (24), 244104/1-244104/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Doubly occupied CI (DOCI) with optimized orbitals often accurately describes strong correlations while working in a Hilbert space much smaller than that needed for full CI. However, the scaling of such calcns. remains combinatorial with system size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N3, disregarding the two-electron integral transformation from at. to MOs). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correlations, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems. (c) 2014 American Institute of Physics.**56**Lehtola, S.; Parkhill, J.; Head-Gordon, M. Cost-effective description of strong correlation: Efficient implementations of the perfect quadruples and perfect hextuples models.*J. Chem. Phys.*2016,*145*, 134110, DOI: 10.1063/1.496431756https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xhs1ClsbjJ&md5=cba359a527854cc102fb61be47a44713Cost-effective description of strong correlation: Efficient implementations of the perfect quadruples and perfect hextuples modelsLehtola, Susi; Parkhill, John; Head-Gordon, MartinJournal of Chemical Physics (2016), 145 (13), 134110/1-134110/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Novel implementations based on dense tensor storage are presented for the singlet-ref. perfect quadruples (PQ) [J. A. Parkhill et al., J. Chem. Phys. 130, 084101 (2009)] and perfect hextuples (PH) [J. A. Parkhill and M. Head-Gordon, J. Chem. Phys. 133, 024103 (2010)] models. The methods are obtained as block decompns. of conventional coupled-cluster theory that are exact for four electrons in four orbitals (PQ) and six electrons in six orbitals (PH), but that can also be applied to much larger systems. PQ and PH have storage requirements that scale as the square, and as the cube of the no. of active electrons, resp., and exhibit quartic scaling of the computational effort for large systems. Applications of the new implementations are presented for full-valence calcns. on linear polyenes (CnHn+2), which highlight the excellent computational scaling of the present implementations that can routinely handle active spaces of hundreds of electrons. The accuracy of the models is studied in the π space of the polyenes, in hydrogen chains (H50), and in the π space of polyacene mols. In all cases, the results compare favorably to d. matrix renormalization group values. With the novel implementation of PQ, active spaces of 140 electrons in 140 orbitals can be solved in a matter of minutes on a single core workstation, and the relatively low polynomial scaling means that very large systems are also accessible using parallel computing. (c) 2016 American Institute of Physics.**57**Lehtola, S.; Parkhill, J.; Head-Gordon, M. Orbital optimization in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes.*Mol. Phys.*2018,*116*, 547– 560, DOI: 10.1080/00268976.2017.134200957https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhtVyqtr7M&md5=4788914aaa7d0a92eb8c474502445ad3Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenesLehtola, Susi; Parkhill, John; Head-Gordon, MartinMolecular Physics (2018), 116 (5-6), 547-560CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)We describe the implementation of orbital optimization for the models in the perfect pairing hierarchy. Orbital optimization, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local min. and σ-π symmetry breaking solns. were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calcns. at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimized orbitals are used. PH is found to yield good agreement with previously published d. matrix renormalization group data in the π space, capturing over 95% of the correlation energy. Full-valence calcns. made possible by our novel, efficient code reveal that strong correlations are weaker when larger basis sets or active spaces are employed than in previous calcns. The largest full-valence PH calcns. presented correspond to a (192e,192o) problem.**58**Cullen, J. Generalized valence bond solutions from a constrained coupled cluster method.*Chem. Phys.*1996,*202*, 217– 229, DOI: 10.1016/0301-0104(95)00321-558https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28Xpt1antw%253D%253D&md5=b7ee308cea7c5dbf21df43709ba3fd5fGeneralized valence bond solutions from a constrained coupled cluster methodCullen, JohnChemical Physics (1996), 202 (2,3), 217-29CODEN: CMPHC2; ISSN:0301-0104. (Elsevier)The GVB-PP wave function is cast into a coupled cluster form with the coupled cluster operator constrained to intra-bond double excitations. Following the coupled cluster ansatz, where the trial wave function is assumed to satisfy the Schroedinger equation, projections onto the ref. and doubly excited configurations are used to det. the energy and coeffs. resp. A decoupling of these equations results, allowing anal. solns. The active orbital space is simultaneously optimized to produce the lowest energy. This is carried out efficiently using a procedure previously developed by Head-Gordon and Pople for the direct optimization of a Hartree-Fock wave function. Preliminary calcns. for singlet/triplet states and bond dissocns. show that although this method is strictly nonvariational, local min. are found which lie within a few tenths of a millihartree of the true GVB-PP energy.**59**Goddard, W. A., III; Harding, L. B. The description of chemical bonding from ab initio calculations.*Annu. Rev. Phys. Chem.*1978,*29*, 363– 396, DOI: 10.1146/annurev.pc.29.100178.00205159https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaE1MXht1Cjuw%253D%253D&md5=9007c9aa07bfe0efea2d94d84f1f2034The description of chemical bonding from ab initio calculationsGoddard, William A., III; Harding, Lawrence B.Annual Review of Physical Chemistry (1978), 29 (), 363-96CODEN: ARPLAP; ISSN:0066-426X.A review with 38 refs.**60**Ukrainskii, I. New variational function in the theory of quasi-one-dimensional metals.*Theor. Math. Phys.*1977,*32*, 816– 822, DOI: 10.1007/BF01089566There is no corresponding record for this reference.**61**Hunt, W.; Hay, P.; Goddard, W., III Self-Consistent Procedures for Generalized Valence Bond Wavefunctions. Applications H_{3}, BH, H_{2}O, C_{2}H_{6}, and O_{2}.*J. Chem. Phys.*1972,*57*, 738– 748, DOI: 10.1063/1.1678308There is no corresponding record for this reference.**62**Hurley, A.; Lennard-Jones, J. E.; Pople, J. A. The molecular orbital theory of chemical valency XVI. A theory of paired-electrons in polyatomic molecules.*Proc. R. Soc. London. Series A. Math. Phys. Sci.*1953,*220*, 446– 455There is no corresponding record for this reference.**63**Živković, T. P. Existence and reality of solutions of the coupled-cluster equations.*Int. J. Quantum Chem.*1977,*12*, 413– 420, DOI: 10.1002/qua.560120849There is no corresponding record for this reference.**64**Piecuch, P.; Zarrabian, S.; Paldus, J.; Čížek, J. Coupled-cluster approaches with an approximate account of triexcitations and the optimized-inner-projection technique. II. Coupled-cluster results for cyclic-polyene model systems.*Phys. Rev. B: Condens. Matter Mater. Phys.*1990,*42*, 3351, DOI: 10.1103/PhysRevB.42.3351There is no corresponding record for this reference.**65**Atkinson, K. E.*An introduction to numerical analysis*; John Wiley & Sons, 2008.There is no corresponding record for this reference.**66**Živković, T. P.; Monkhorst, H. J. Analytic connection between configuration-interaction and coupled-cluster solutions.*J. Math. Phys.*1978,*19*, 1007– 1022, DOI: 10.1063/1.523761There is no corresponding record for this reference.**67**Kowalski, K.; Jankowski, K. Towards complete solutions to systems of nonlinear equations of many-electron theories.*Phys. Rev. Lett.*1998,*81*, 1195, DOI: 10.1103/PhysRevLett.81.119567https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXltVSrt7w%253D&md5=0ea6cd8d6989fddf2ced3e6be183ff76Towards Complete Solutions to Systems of Nonlinear Equations of Many-Electron TheoriesKowalski, Karol; Jankowski, KarolPhysical Review Letters (1998), 81 (6), 1195-1198CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Employing the homotopy method we have obtained the complete set of real solns. to the equations of the RHF method as well as the full set of solns. to the equations of the coupled-cluster-with-doubles method for the H4 and P4 models broadly applied in various many-electron studies. These are the first global results obtained so far for any formulations of the Hartree-Fock and coupled-cluster methods when applied to realistic models.**68**Piecuch, P.; Kowalski, K. In*Computational Chemistry: Reviews of Current Trends*; Leszczynski, J., Ed.; World Scientific, Singapore, 2000; Vol. 5.There is no corresponding record for this reference.**69**Jeziorski, B.; Paldus, J. Valence universal exponential ansatz and the cluster structure of multireference configuration interaction wave function.*J. Chem. Phys.*1989,*90*, 2714– 2731, DOI: 10.1063/1.455919There is no corresponding record for this reference.**70**Schneider, R. Analysis of the Projected Coupled Cluster Method in Electronic Structure Calculation.*Numer. Math.*2009,*113*, 433– 471, DOI: 10.1007/s00211-009-0237-3There is no corresponding record for this reference.**71**Rohwedder, T. The Continuous Coupled Cluster Formulation for the Electronic Schrödinger Equation.*ESAIM: Math. Modell. Numer. Anal.*2013,*47*, 421– 447, DOI: 10.1051/m2an/2012035There is no corresponding record for this reference.**72**Rohwedder, T.; Schneider, R. Error Estimates for the Coupled Cluster Method.*ESAIM: Math. Modell. Numer. Anal.*2013,*47*, 1553– 1582, DOI: 10.1051/m2an/2013075There is no corresponding record for this reference.**73**Laestadius, A.; Kvaal, S. Analysis of the extended coupled-cluster method in quantum chemistry.*SIAM J. on Numer. Anal.*2018,*56*, 660– 683, DOI: 10.1137/17M1116611There is no corresponding record for this reference.**74**Löwdin, P.-O. On the stability problem of a pair of adjoint operators.*J. Math. Phys.*1983,*24*, 70– 87, DOI: 10.1063/1.525604There is no corresponding record for this reference.**75**Arponen, J. Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems.*Ann. Phys.*1983,*151*, 311– 382, DOI: 10.1016/0003-4916(83)90284-175https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL2cXis1OqtA%253D%253D&md5=7b010ab3d1ce0aba10ed5fb00da58f3dVariational principles and linked-cluster exp S expansions for static and dynamic many-body problemsArponen, JoukoAnnals of Physics (San Diego, CA, United States) (1983), 151 (2), 311-82CODEN: APNYA6; ISSN:0003-4916.The exp S formalism for the ground state of a many-body system is derived from a variational principle. An energy functional is constructed by using certain n-body linked-cluster amplitudes with respect to which the functional is required to be stationary. By using 2 different sets of amplitudes one either recovers the normal exp S method or obtains a new scheme called the extended exp S method. The same functional can be used also to obtain the av. values of any operators as well as the linear response to static perturbations. The theory is extended to treat dynamical phenomena by introducing time dependence to the cluster amplitudes.**76**Faulstich, F. M.; Laestadius, A.; Kvaal, S.; Legeza, Ö.; Schneider, R. Analysis of The Coupled-Cluster Method Tailored by Tensor-Network States in Quantum Chemistry.*arXiv.org*2018, 1802.05699There is no corresponding record for this reference.**77**Laestadius, A.; Faulstich, F. M. The coupled-cluster formalism-a mathematical perspective.*Mol. Phys.*2019, 1– 12, DOI: 10.1080/00268976.2018.1564848There is no corresponding record for this reference.**78**Piecuch, P.; Oliphant, N.; Adamowicz, L. A state-selective multireference coupled-cluster theory employing the single-reference formalism.*J. Chem. Phys.*1993,*99*, 1875– 1900, DOI: 10.1063/1.46617978https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK3sXmtFWjsrc%253D&md5=85bd8b4c37a2e1763e9032287669e1e5A state-selective multireference coupled-cluster theory employing the single-reference formalismPiecuch, Piotr; Oliphant, Nevin; Adamowicz, LudwikJournal of Chemical Physics (1993), 99 (3), 1875-900CODEN: JCPSA6; ISSN:0021-9606.A new state-selective multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism is described. It is an extension of a simple two-ref. formalism, which the authors presented in the authors' earlier paper [N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991)], and a rigorous formulation of another method of ours, which the authors obtained as an approxn. of the SRCC approach truncated at triple excitations (SRCCSDT) [N. Oliphant and L. Adamowicz, J. Chem. Phys. 96, 3739 (1992)]. The size extensivity of the resulting correlation energies is achieved by employing a SRCC-like ansatz for the multideterminantal wave function. General considerations are supplemented by suggesting a hierarchy of approx. schemes, with the MRCCSD approach (MRCC approach truncated at double excitations from the ref. determinants) representing the most important one. The authors' state-selective MRCCSD theory emerges through a suitable selection of the most essential cluster components appearing in the full SRCCSDTQ method (SRCC method truncated at quadruple excitations), when the latter is applied to quasidegenerate states. The complete set of equations describing the authors' MRCCSD formalism is presented and the possibility of the recursive intermediate factorization [S. A. Kucharski and R. J. Bartlett, Theor. Chim. Acta 80, 387 (1991)] of the authors' approach, leading to an efficient computer algorithm, is discussed.**79**Piecuch, P.; Adamowicz, L. State-selective multireference coupled-cluster theory employing the single-reference formalism: Implementation and application to the H8 model system.*J. Chem. Phys.*1994,*100*, 5792– 5809, DOI: 10.1063/1.46714379https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK2cXjtVOksrY%253D&md5=b8df2b4213996bf687d716f91ccaa240State-selective multireference coupled-cluster theory employing the single-reference formalism: implementation and application to the H8 model systemPiecuch, Piotr; Adamowicz, LudwikJournal of Chemical Physics (1994), 100 (8), 5792-809CODEN: JCPSA6; ISSN:0021-9606.The new state-selective (SS) multireference (MR) coupled-cluster (CC) method exploiting the single-ref. (SR) particle-hole formalism, which was introduced previously (P. Piecuch, et al., 1993), was implemented; results are presented of pilot calcns. for the min. basis-set (MBS) model composed of eight hydrogen atoms in various geometrical arrangements. This model enables a continuous transition between degenerate and nondegenerate regimes. Comparison is made with the results of SR CC calcns. involving double (CCD), single and double (CCSD), single, double, and triple (CCSDT), and single, double, triple, and quadruple (CCSDTQ) excitations. The authors' SS CC energies are also compared with the results of Hilbert space, state-universal (SU) MR CC(S)D calcns., as well as with MR-CI results (with and without Davidson-type corrections), and with exact correlation energies obtained using the full-CI (FCI) method. Along with the ground-state energies, the authors also analyzed the resulting wave functions by examg. some selected cluster components. This anal. enabled the authors to assess the quality of the resulting wave functions. The authors' SS CC theory truncated at double excitations, which emerges through selection of the most essential clusters appearing in the full SR CCSDTQ formalism [SS CCSD (TQ) method], provided equally good results in the nondegenerate and quasidegenerate regions. The difference between the ground-state energy obtained with the SS CCSD(TQ) approach and the FCI energy did not exceed 1.1 milli-hartree over all the geometries considered. This value compares favorably with the max. difference of 2.8 milli-hartrees between the SU CCSD energies and the FCI energies obtained for the same range of geometries. The SS CCSD(T) method, emerging from the SR CCSDT theory through selection of the most essential clusters, was less stable, since it neglected very important semi-internal quadruple excitations. Unlike the genuine multideterminantal SU CC formalism, the authors' SS CC approach was not affected by the intruder-state problem, and its convergence remained satisfactory the in nondegenerate and quasidegenerate regimes.**80**Chan, G. K.-L.; Sharma, S. The density matrix renormalization group in quantum chemistry.*Annu. Rev. Phys. Chem.*2011,*62*, 465– 481, DOI: 10.1146/annurev-physchem-032210-10333880https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXmsVWmt7k%253D&md5=99fca86a8b3932bf6d9f73defd9ee37eThe density matrix renormalization group in quantum chemistryChan, Garnet Kin-Lic; Sharma, SandeepAnnual Review of Physical Chemistry (2011), 62 (), 465-481CODEN: ARPLAP; ISSN:0066-426X. (Annual Reviews Inc.)A review. The d. matrix renormalization group is a method that is useful for describing mols. that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the d. matrix renormalization group works, a survey of its existing applications to mol. problems, and some thoughts on the future of the method.**81**Saitow, M.; Kurashige, Y.; Yanai, T. Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave function.*J. Chem. Phys.*2013,*139*, 044118, DOI: 10.1063/1.481662781https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhtFygsbnL&md5=a30f97f373ba1a069bbce1c23a48e178Multireference configuration interaction theory using cumulant reconstruction with internal contraction of density matrix renormalization group wave functionSaitow, Masaaki; Kurashige, Yuki; Yanai, TakeshiJournal of Chemical Physics (2013), 139 (4), 044118/1-044118/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We report development of the multireference CI (MRCI) method that can use active space scalable to much larger size refs. than has previously been possible. The recent development of the d. matrix renormalization group (DMRG) method in multireference quantum chem. offers the ability to describe static correlation in a large active space. The present MRCI method provides a crit. correction to the DMRG ref. by including high-level dynamic correlation through the CI treatment. When the DMRG and MRCI theories are combined (DMRG-MRCI), the full internal contraction of the ref. in the MRCI ansatz, including contraction of semi-internal states, plays a central role. However, it is thought to involve formidable complexity because of the presence of the five-particle rank reduced-d. matrix (RDM) in the Hamiltonian matrix elements. To address this complexity, we express the Hamiltonian matrix using commutators, which allows the five-particle rank RDM to be canceled out without any approxn. Then we introduce an approxn. to the four-particle rank RDM by using a cumulant reconstruction from lower-particle rank RDMs. A computer-aided approach is employed to derive the exceedingly complex equations of the MRCI in tensor-contracted form and to implement them into an efficient parallel computer code. This approach extends to the size-consistency-cor. variants of MRCI, such as the MRCI+Q, MR-ACPF, and MR-AQCC methods. We demonstrate the capability of the DMRG-MRCI method in several benchmark applications, including the evaluation of single-triplet gap of free-base porphyrin using 24 active orbitals. (c) 2013 American Institute of Physics.**82**Myhre, R. H.; Koch, H. The multilevel CC3 coupled cluster model.*J. Chem. Phys.*2016,*145*, 044111, DOI: 10.1063/1.495937382https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xht1Kmur7E&md5=6f62a1c36755e3209a5cbbf86d3502c8The multilevel CC3 coupled cluster modelMyhre, Rolf H.; Koch, HenrikJournal of Chemical Physics (2016), 145 (4), 044111/1-044111/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We present an efficient implementation of the closed shell multilevel coupled cluster method where coupled cluster singles and doubles (CCSD) is used for the inactive orbital space and CCSD with perturbative triples (CC3) is employed for the smaller active orbital space. Using Cholesky orbitals, the active space can be spatially localized and the computational cost is greatly reduced compared to full CC3 while retaining the accuracy of CC3 excitation energies. For the small org. mols. considered we achieve up to two orders of magnitude redn. in the computational requirements. (c) 2016 American Institute of Physics.**83**Lyakh, D. I.; Musiał, M.; Lotrich, V. F.; Bartlett, R. J. Multireference nature of chemistry: The coupled-cluster view.*Chem. Rev.*2012,*112*, 182– 243, DOI: 10.1021/cr200141783https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs12hsbjF&md5=26080054ce24a172826517cb7d772f62Multireference Nature of Chemistry: The Coupled-Cluster ViewLyakh, Dmitry I.; Musial, Monika; Lotrich, Victor F.; Bartlett, Rodney J.**84**Szalay, S.; Barcza, G.; Szilvási, T.; Veis, L.; Legeza, Ö. The correlation theory of the chemical bond.*Sci. Rep.*2017,*7*, 2237, DOI: 10.1038/s41598-017-02447-z84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1cngtF2msg%253D%253D&md5=e14838c28811e3e816ab44d05ab62906The correlation theory of the chemical bondSzalay Szilard; Barcza Gergely; Legeza Ors; Szilvasi Tibor; Szilvasi Tibor; Veis LiborScientific reports (2017), 7 (1), 2237 ISSN:.The quantum mechanical description of the chemical bond is generally given in terms of delocalized bonding orbitals, or, alternatively, in terms of correlations of occupations of localised orbitals. However, in the latter case, multiorbital correlations were treated only in terms of two-orbital correlations, although the structure of multiorbital correlations is far richer; and, in the case of bonds established by more than two electrons, multiorbital correlations represent a more natural point of view. Here, for the first time, we introduce the true multiorbital correlation theory, consisting of a framework for handling the structure of multiorbital correlations, a toolbox of true multiorbital correlation measures, and the formulation of the multiorbital correlation clustering, together with an algorithm for obtaining that. These make it possible to characterise quantitatively, how well a bonding picture describes the chemical system. As proof of concept, we apply the theory for the investigation of the bond structures of several molecules. We show that the non-existence of well-defined multiorbital correlation clustering provides a reason for debated bonding picture.**85**Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy.*Phys. Rev. B: Condens. Matter Mater. Phys.*2003,*68*, 195116, DOI: 10.1103/PhysRevB.68.19511685https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXpvVegs7k%253D&md5=215685d20c465a36d96e9adf4bbb0ea3Optimizing the density-matrix renormalization group method using quantum information entropyLegeza, O.; Solyom, J.Physical Review B: Condensed Matter and Materials Physics (2003), 68 (19), 195116/1-195116/19CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)In order to optimize the ordering of the lattice sites in the momentum space and quantum chem. versions of the d.-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various mols. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chem. DMRG to a great extent. The effect of lattice site ordering on the no. of block states to be kept during the RG procedure is also investigated.**86**Barcza, G.; Legeza, Ö.; Marti, K. H.; Reiher, M. Quantum-information analysis of electronic states of different molecular structures.*Phys. Rev. A: At., Mol., Opt. Phys.*2011,*83*, 012508, DOI: 10.1103/PhysRevA.83.01250886https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsFegsr0%253D&md5=4a41eebdc1e8bed2c9112fcaf74fa90fQuantum-information analysis of electronic states of different molecular structuresBarcza, G.; Legeza, O.; Marti, K. H.; Reiher, M.Physical Review A: Atomic, Molecular, and Optical Physics (2011), 83 (1, Pt. A), 012508/1-012508/15CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We have studied transition metal clusters from a quantum information theory perspective using the d.-matrix renormalization group (DMRG) method. We demonstrate the competition between entanglement and interaction localization and discuss the application of the CI-based dynamically extended active space procedure, which significantly reduces the effective system size and accelerates the speed of convergence for complicated mol. electronic structures. Our results indicate the importance of taking entanglement among MOs into account in order to devise an optimal DMRG orbital ordering and carry out efficient calcns. on transition metal clusters. Apart from these algorithmic observations, which lead to a recipe for black-box DMRG calcns., our work provides phys. understanding of electron correlation in mol. and cluster structures in terms of entropy measures of relevance also to recent work on tensor-network representations of electronic states. We also identify those MOs which are highly entangled and discuss the consequences for chem. bonding and for the structural transition from an dioxygen binding copper cluster to an bis-oxygen-bridged system with broken O-O bond.**87**Stein, C. J.; Reiher, M. Automated selection of active orbital spaces.*J. Chem. Theory Comput.*2016,*12*, 1760– 1771, DOI: 10.1021/acs.jctc.6b0015687https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XjvFyltLs%253D&md5=c46ae44d10c10dfa409cf8807a779308Automated Selection of Active Orbital SpacesStein, Christopher J.; Reiher, MarkusJournal of Chemical Theory and Computation (2016), 12 (4), 1760-1771CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)One of the key challenges of quantum-chem. multi-configuration methods is the necessity to manually select orbitals for the active space. This selection requires both expertise and experience and can therefore impose severe limitations on the applicability of this most general class of ab initio methods. A poor choice of the active orbital space may yield even qual. wrong results. This is obviously a severe problem, esp. for wave function methods that are designed to be systematically improvable. Here, we show how the iterative nature of the d. matrix renormalization group combined with its capability to include up to about 100 orbitals in the active space can be exploited for a systematic assessment and selection of active orbitals. These benefits allow us to implement an automated approach for active orbital space selection, which can turn multi-configuration models into black box approaches.**88**Aubin, J. P. Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods.*Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)*1967,*21*, 599– 637There is no corresponding record for this reference.**89**Nitsche, J. Ein kriterium für die quasi-optimalität des ritzschen verfahrens.*Numer. Math.*1968,*11*, 346– 348, DOI: 10.1007/BF02166687There is no corresponding record for this reference.**90**Oganesyan, L. A.; Rukhovets, L. A. Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary.*USSR Comput. Math. Math. Phys.*1969,*9*, 158– 183, DOI: 10.1016/0041-5553(69)90159-1There is no corresponding record for this reference.**91**Dunning, T. H., Jr Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen.*J. Chem. Phys.*1989,*90*, 1007– 1023, DOI: 10.1063/1.45615391https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXksVGmtrk%253D&md5=c6cd67a3748dc61692a9cb622d2694a0Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogenDunning, Thom H., Jr.Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606.Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron.**92**Kowalski, K.; Piecuch, P. Renormalized CCSD (T) and CCSD (TQ) approaches: Dissociation of the N_{2}triple bond.*J. Chem. Phys.*2000,*113*, 5644– 5652, DOI: 10.1063/1.129060992https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3cXmvFOlsrs%253D&md5=c1c67154a810a07f46da394e62b1a0bcRenormalized CCSD(T) and CCSD(TQ) approaches: Dissociation of the N2 triple bondKowalski, Karol; Piecuch, PiotrJournal of Chemical Physics (2000), 113 (14), 5644-5652CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)The recently proposed renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods, which can be viewed as generalizations of the noniterative perturbative CCSD(T) and CCSD(TQf) schemes and which result from the more general method of moments of coupled-cluster equations, are applied to the dissocn. of the ground-state N2 mol. It is shown that the renormalized and completely renormalized CCSD(T) and CCSD(TQ) methods provide significantly better results for large N-N sepns. than their unrenormalized CCSD(T) and CCSD(TQf) counterparts.**93**Szalay, S.; Pfeffer, M.; Murg, V.; Barcza, G.; Verstraete, F.; Schneider, R.; Legeza, Ö. Tensor product methods and entanglement optimization for ab initio quantum chemistry.*Int. J. Quantum Chem.*2015,*115*, 1342– 1391, DOI: 10.1002/qua.2489893https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXovVags7c%253D&md5=d1f7db2c2c73a2d907e9a290ccb7f22bTensor product methods and entanglement optimization for ab initio quantum chemistrySzalay, Szilard; Pfeffer, Max; Murg, Valentin; Barcza, Gergely; Verstraete, Frank; Schneider, Reinhold; Legeza, OersInternational Journal of Quantum Chemistry (2015), 115 (19), 1342-1391CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)The treatment of high-dimensional problems such as the Schroedinger equation can be approached by concepts of tensor product approxn. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods-developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools-can be combined to attack highly challenging problems in quantum chem. The aim of the present paper is to give a pedagogical introduction to the theor. background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum chem." at the Workshop on Theor. Chem., Feb. 18-21, 2014, Mariapfarr, Austria. © 2015 Wiley Periodicals, Inc.**94**Murg, V.; Verstraete, F.; Legeza, Ö.; Noack, R.-h. M. Simulating strongly correlated quantum systems with tree tensor networks.*Phys. Rev. B: Condens. Matter Mater. Phys.*2010,*82*, 205105, DOI: 10.1103/PhysRevB.82.20510594https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXhsFWmtLrF&md5=c554e73a8f20b3a417a0e2381c3ff02fSimulating strongly correlated quantum systems with tree tensor networksMurg, V.; Verstraete, F.; Legeza, O.; Noack, R. M.Physical Review B: Condensed Matter and Materials Physics (2010), 82 (20), 205105/1-205105/11CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chem. versions of the d.-matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topol. of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following [Y. Shi, L. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006)], we treat a treelike network ansatz with arbitrary coordination no. z, where the z = 2 case corresponds to the one-dimensional (1D) scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addn., we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1D interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calcns. carried out on small quantum chem. systems support our approach.**95**Nakatani, N.; Chan, G. K.-L. Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm.*J. Chem. Phys.*2013,*138*, 134113, DOI: 10.1063/1.479863995https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXlt1Gks7c%253D&md5=2e926e48567120d16676ed7329a235c2Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithmNakatani, Naoki; Chan, Garnet Kin-LicJournal of Chemical Physics (2013), 138 (13), 134113/1-134113/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We investigate tree tensor network states for quantum chem. Tree tensor network states represent one of the simplest generalizations of matrix product states and the d. matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general mols. We describe an optimal tree tensor network state algorithm for quantum chem. We introduce the concept of half-renormalization which greatly improves the efficiency of the calcns. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states vs. matrix product states. We carry out benchmark calcns. both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree mols. (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree mols., whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling assocd. with tree algorithms. In tree like mols., tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calcn. with tree tensor network states correlates 110 electrons in 110 active orbitals. (c) 2013 American Institute of Physics.**96**Murg, V.; Verstraete, F.; Schneider, R.; Nagy, P. R.; Legeza, Ö. Tree tensor network state with variable tensor order: an efficient multireference method for strongly correlated systems.*J. Chem. Theory Comput.*2015,*11*, 1027– 1036, DOI: 10.1021/ct501187j96https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXitVersLc%253D&md5=4287825e29765d130d6ac6b08c17d344Tree Tensor Network State with Variable Tensor Order: An Efficient Multireference Method for Strongly Correlated SystemsMurg, V.; Verstraete, F.; Schneider, R.; Nagy, P. R.; Legeza, Oe.Journal of Chemical Theory and Computation (2015), 11 (3), 1027-1036CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We study the tree-tensor-network-state (TTNS) method with variable tensor orders for quantum chem. TTNS is a variational method to efficiently approx. complete active space (CAS) CI wave functions in a tensor product form. TTNS can be considered as a higher order generalization of the matrix product state (MPS) method. The MPS wave function is formulated as products of matrixes in a multiparticle basis spanning a truncated Hilbert space of the original CAS-CI problem. These matrixes belong to active orbitals organized in a one-dimensional array, while tensors in TTNS are defined upon a tree-like arrangement of the same orbitals. The tree-structure is advantageous since the distance between two arbitrary orbitals in the tree scales only logarithmically with the no. of orbitals N, whereas the scaling is linear in the MPS array. It is found to be beneficial from the computational costs point of view to keep strongly correlated orbitals in close vicinity in both arrangements; therefore, the TTNS ansatz is better suited for multireference problems with numerous highly correlated orbitals. To exploit the advantages of TTNS a novel algorithm is designed to optimize the tree tensor network topol. based on quantum information theory and entanglement. The superior performance of the TTNS method is illustrated on the ionic-neutral avoided crossing of LiF. It is also shown that the avoided crossing of LiF can be localized using only ground state properties, namely one-orbital entanglement.**97**Gunst, K.; Verstraete, F.; Wouters, S.; Legeza, Ö.; Van Neck, D. T3NS: Three-Legged Tree Tensor Network States.*J. Chem. Theory Comput.*2018,*14*, 2026– 2033, DOI: 10.1021/acs.jctc.8b0009897https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXjtlOltLc%253D&md5=b3058f1bfd3cdc7de8f4728d955f21e6T3NS: Three-Legged Tree Tensor Network StatesGunst, Klaas; Verstraete, Frank; Wouters, Sebastian; Legeza, Ors; Van Neck, DimitriJournal of Chemical Theory and Computation (2018), 14 (4), 2026-2033CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a new variational tree tensor network state (TTNS) ansatz, the three-legged tree tensor network state (T3NS). Phys. tensors are interspersed with branching tensors. Phys. tensors have one phys. index and at most two virtual indexes, as in the matrix product state (MPS) ansatz of the d. matrix renormalization group (DMRG). Branching tensors have no phys. index, but up to three virtual indexes. In this way, advantages of DMRG, in particular a low computational cost and a simple implementation of symmetries, are combined with advantages of TTNS, namely incorporating more entanglement. Our code is capable of simulating quantum chem. Hamiltonians, and we present several proof-of-principle calcns. on LiF, N2, and the bis(μ-oxo) and μ-η2:η2 peroxo isomers of [Cu2O2]2+.**98**Chan, G. K.-L.; Head-Gordon, M. Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group.*J. Chem. Phys.*2002,*116*, 4462– 4476, DOI: 10.1063/1.144945998https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XhslCjsrs%253D&md5=f1249212b5c6414a901ff5c8a4a64542Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization groupChan, Garnet Kin-Lic; Head-Gordon, MartinJournal of Chemical Physics (2002), 116 (11), 4462-4476CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the recently developed D. Matrix Renormalization Group (DMRG) algorithm in the context of quantum chem. In contrast to traditional approaches, this algorithm is believed to yield arbitrarily high accuracy in the energy with only polynomial computational effort. We describe in some detail how this is achieved. We begin by introducing the principles of the renormalization procedure, and how one formulates an algorithm for use in quantum chem. The renormalization group algorithm is then interpreted in terms of familiar quantum chem. concepts, and its numerical behavior, including its convergence and computational cost, are studied using both model and real systems. The asymptotic convergence of the algorithm is derived. Finally, we examine the performance of the DMRG on widely studied chem. problems, such as the water mol., the twisting barrier of ethene, and the dissocn. of nitrogen. In all cases, the results are favorably comparable with the best existing quantum chem. methods, and particularly so when the nondynamical correlation is strong. Some perspectives for future development are given.**99**Legeza, Ö.; Sólyom, J. Optimizing the density-matrix renormalization group method using quantum information entropy.*Phys. Rev. B: Condens. Matter Mater. Phys.*2003,*68*, 195116, DOI: 10.1103/PhysRevB.68.19511699https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXpvVegs7k%253D&md5=215685d20c465a36d96e9adf4bbb0ea3Optimizing the density-matrix renormalization group method using quantum information entropyLegeza, O.; Solyom, J.**100**Barcza, G.; Legeza, Ö.; Marti, K. H.; Reiher, M. Quantum-information analysis of electronic states of different molecular structures.*Phys. Rev. A: At., Mol., Opt. Phys.*2011,*83*, 012508, DOI: 10.1103/PhysRevA.83.012508100https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsFegsr0%253D&md5=4a41eebdc1e8bed2c9112fcaf74fa90fQuantum-information analysis of electronic states of different molecular structuresBarcza, G.; Legeza, O.; Marti, K. H.; Reiher, M.**101**Fertitta, E.; Paulus, B.; Barcza, G.; Legeza, Ö. Investigation of metal-insulator-like transition through the*ab initio*density matrix renormalization group approach.*Phys. Rev. B: Condens. Matter Mater. Phys.*2014,*90*, 245129, DOI: 10.1103/PhysRevB.90.245129101https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjtlGrs7Y%253D&md5=6d2f8b048038703aa3bea6393cdaf65bInvestigation of metal-insulator-like transition through the ab initio density matrix renormalization group approachFertitta, E.; Paulus, B.; Barcza, G.; Legeza, Oe.Physical Review B: Condensed Matter and Materials Physics (2014), 90 (24), 245129/1-245129/11, 11 pp.CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)We have studied the metal-insulator-like transition in pseudo-one-dimensional systems, i.e., lithium and beryllium rings, through the ab initio d. matrix renormalization group (DMRG) method. Performing accurate calcns. for different interat. distances and using quantum information theory, we investigated the changes occurring in the wave function between a metallic-like state and an insulating state built from free atoms. We also discuss entanglement and relevant excitations among the MOs in the Li and Be rings and show that the transition bond length can be detected using orbital entropy functions. Also, the effect of different orbital bases on the effectiveness of the DMRG procedure is analyzed comparing the convergence behavior.**102**Rissler, J.; Noack, R. M.; White, S. R. Measuring orbital interaction using quantum information theory.*Chem. Phys.*2006,*323*, 519– 531, DOI: 10.1016/j.chemphys.2005.10.018102https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD28XjvVanu74%253D&md5=ca82193fb0d3c9b3dbcd392adcdd9757Measuring orbital interaction using quantum information theoryRissler, Joerg; Noack, Reinhard M.; White, Steven R.Chemical Physics (2006), 323 (2-3), 519-531CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.)Quantum information theory gives rise to a straightforward definition of the interaction of electrons Ip,q in two orbitals p,q for a given many-body wave function. A convenient way to calc. the von Neumann entropies needed is presented in this work, and the orbital interaction Ip,q is successfully tested for different types of chem. bonds. As an example of an application of Ip,q beyond the interpretation of wave functions, Ip,q is then used to investigate the ordering problem in the d.-matrix renormalization group.**103**Legeza, Ö.; Röder, J.; Hess, B. A. Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach.*Phys. Rev. B: Condens. Matter Mater. Phys.*2003,*67*, 125114, DOI: 10.1103/PhysRevB.67.125114103https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD3sXivFaqsbY%253D&md5=0f5650c8557f55974dea14ef3e8a41d3Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approachLegeza, O.; Roder, J.; Hess, B. A.Physical Review B: Condensed Matter and Materials Physics (2003), 67 (12), 125114/1-125114/10CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)We have applied the momentum space version of the d.-matrix renormalization-group method (k-DMRG) in quantum chem. in order to study the accuracy of the algorithm in this new context. We have shown numerically that it is possible to det. the desired accuracy of the method in advance of the calcns. by dynamically controlling the truncation error and the no. of block states using a novel protocol that we dubbed dynamical block state selection protocol. The relationship between the real error and truncation error has been studied as a function of the no. of orbitals and the fraction of filled orbitals. We have calcd. the ground state of the mols. CH2, H2O, and F2 as well as the first excited state of CH2. Our largest calcns. were carried out with 57 orbitals, the largest no. of block states was 1500-2000, and the largest dimensions of the Hilbert space of the superblock configuration was 800 000-1 200 000.**104**Legeza, Ö.; Sólyom, J. Quantum data compression, quantum information generation, and the density-matrix renormalization-group method.*Phys. Rev. B: Condens. Matter Mater. Phys.*2004,*70*, 205118, DOI: 10.1103/PhysRevB.70.205118104https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXhtVGgsbvN&md5=81d2d092f4fcd276feafa092663763caQuantum data compression, quantum information generation, and the density-matrix renormalization-group methodLegeza, O.; Solyom, J.Physical Review B: Condensed Matter and Materials Physics (2004), 70 (20), 205118/1-205118/7CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)We have studied quantum data compression for finite quantum systems where the site d. matrixes are not independent, i.e., the d. matrix cannot be given as direct product of site d. matrixes and the von Neumann entropy is not equal to the sum of site entropies. Using the d.-matrix renormalization-group (DMRG) method for the one-dimensional Hubbard model, we have shown that a simple relationship exists between the entropy of the left or right block and dimension of the Hilbert space of that block as well as of the superblock for any fixed accuracy. The information loss during the RG procedure has been investigated and a more rigorous control of the relative error has been proposed based on Kholevo's theory. Our results are also supported by the quantum chem. version of DMRG applied to various mols. with system lengths up to 60 lattice sites. A sum rule that relates site entropies and the total information generated by the renormalization procedure has also been given, which serves as an alternative test of convergence of the DMRG method.**105**Legeza, Ö.; Fáth, G. Accuracy of the density-matrix renormalization-group method.*Phys. Rev. B: Condens. Matter Mater. Phys.*1996,*53*, 14349– 14358, DOI: 10.1103/PhysRevB.53.14349105https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK28XjsVCrsrg%253D&md5=d37b67aaa40eee4a8aee061f416120c5Accuracy of the density-matrix renormalization-group methodLegeza, Ors; Fath, GaborPhysical Review B: Condensed Matter (1996), 53 (21), 14349-14358CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)White's d.-matrix renormalization-group (DMRG) method has been applied to the one-dimensional Ising model in a transverse field (ITF), in order to study the accuracy of the numerical algorithm. Due to the exact soly. of the ITF for any finite chain length, the errors introduced by the basis truncation procedure could have been directly analyzed. By computing different properties, like the energies of the low-lying levels or the ground-state one- and two-point correlation functions, we obtained a detailed picture of how these errors behave as functions of the various model and algorithm paramters. Our experience with the ITF contributes to a better understanding of the DMRG method, and may facilitate its optimization in other applications.**106**Legeza, Ö., Veis, L., Mosoni, T.*QC-DMRG-Budapest, a program for quantum chemical DMRG calculations*; HAS RISSPO: Budapest, 2018.There is no corresponding record for this reference.**107**Chan, G. K.-L.; Kállay, M.; Gauss, J. State-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curve.*J. Chem. Phys.*2004,*121*, 6110– 6116, DOI: 10.1063/1.1783212107https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXnvFeqs7g%253D&md5=409eb74e4f9716d5bab8ddf7de1e4f8bState-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curveChan, Garnet Kin-Lic; Kallay, Mihaly; Gauss, JurgenJournal of Chemical Physics (2004), 121 (13), 6110-6116CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)We study the nitrogen binding curve with the d. matrix renormalization group (DMRG) and single-ref. and multireference coupled cluster (CC) theory. Our DMRG calcns. use up to 4000 states and our single-ref. CC calcns. include up to full connected hextuple excitations. Using the DMRG, we compute an all-electron benchmark nitrogen binding curve, at the polarized, valence double-zeta level (28 basis functions), with an estd. accuracy of 0.03 mEh. We also assess the performance of more approx. DMRG and CC theories across the nitrogen curve. We provide an anal. of the relative strengths and merits of the DMRG and CC theory under different correlation conditions.**108**Boguslawski, K.; Tecmer, P.; Barcza, G.; Legeza, Ö.; Reiher, M. Orbital Entanglement in Bond-Formation Processes.*J. Chem. Theory Comput.*2013,*9*, 2959– 2973, DOI: 10.1021/ct400247p108https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXnsFCltLY%253D&md5=88e637ee401cc68cc86ed61bd5659616Orbital Entanglement in Bond-Formation ProcessesBoguslawski, Katharina; Tecmer, Pawel; Barcza, Gergely; Legeza, Ors; Reiher, MarkusJournal of Chemical Theory and Computation (2013), 9 (7), 2959-2973CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)The accurate calcn. of the (differential) correlation energy is central to the quantum chem. description of bond-formation and bond-dissocn. processes. In order to est. the quality of single- and multireference approaches for this purpose, various diagnostic tools have been developed. In this work, we elaborate on our previous observation that one- and two-orbital-based entanglement measures provide quant. means for the assessment and classification of electron correlation effects among MOs. The dissocn. behavior of some prototypical diat. mols. features all types of correlation effects relevant for chem. bonding. We demonstrate that our entanglement anal. is convenient to dissect these electron correlation effects and to provide a conceptual understanding of bond-forming and bond-breaking processes from the point of view of quantum information theory.**109**Valiev, M.; Bylaska, E.; Govind, N.; Kowalski, K.; Straatsma, T.; Dam, H. V.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T.; de Jong, W. NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations.*Comput. Phys. Commun.*2010,*181*, 1477– 1489, DOI: 10.1016/j.cpc.2010.04.018109https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXos1Cjur8%253D&md5=19100f255a4e6088076fb69421a9a0acNWChem: A comprehensive and scalable open-source solution for large scale molecular simulationsValiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; de Jong, W. A.Computer Physics Communications (2010), 181 (9), 1477-1489CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)A review. The latest release of NWChem delivers an open-source computational chem. package with extensive capabilities for large scale simulations of chem. and biol. systems. Utilizing a common computational framework, diverse theor. descriptions can be used to provide the best soln. for a given scientific problem. Scalable parallel implementations and modular software design enable efficient utilization of current computational architectures. This paper provides an overview of NWChem focusing primarily on the core theor. modules provided by the code and their parallel performance.**110**Lee, T. J.; Taylor, P. R. A diagnostic for determining the quality of singlereference electron correlation methods.*Int. J. Quantum Chem.*1989,*36*, 199– 207, DOI: 10.1002/qua.560360824There is no corresponding record for this reference.**111**Krumnow, C.; Veis, L.; Legeza, Ö.; Eisert, J. Fermionic orbital optimization in tensor network states.*Phys. Rev. Lett.*2016,*117*, 210402, DOI: 10.1103/PhysRevLett.117.210402111https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsVSkt77N&md5=425ce4b4a69386812f4ed7ef6045a42cFermionie orbital optimization in tensor network statesKrumnow, C.; Veis, L.; Legeza, O.; Eisert, J.Physical Review Letters (2016), 117 (21), 210402/1-210402/6CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chem. A new freedom arising in such noniocal fcrmionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this Letter, we propose a way to overcome this challenge. We suggest a method intertwining the optimization over matrix product states with suitable fcrmionic Gaussian mode trans- formations. The described algorithm generalizes basis changes in the spirit of the Hartree-Fock method to matrix-product states, and provides a black box tool for basis optimization in tensor network methods.