Valence-bond theory and the evaluation of electronic energy matrix elements between nonorthogonal Slater determinants

Leasure, Steven C.; Balint‐Kurti, Gabriel G.

doi:10.1103/physreva.31.2107

Cited by 28 publications

(12 citation statements)

References 26 publications

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“…To deal with spin orthogonality properly, this structure must be recognized, because the overlap-integrals usually refer to orbital, not to spin orbitals. This has led other workers [6-91 (see also [13]) to formulate rules that also occur in our set (cf. section 2.6).…”

Section: Sand') and S ! L '~~ ' )mentioning

confidence: 95%

The generalized Slater–Condon rules

Verbeek

Lenthe

1991

Int J of Quantum Chemistry

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By relating the blocking structure of the relevant matrix of overlap-integrals to its cofactors, the Slater-Condon rules for the evaluation of an element of a matrix representation of an electronic Hamiltonian in a Slater determinant basis are generalized to the case where not all orbitals are orthogonal. This yields a set of 33 rules, which allows for an efficient implementation of the valence bond theory.

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Section: Sand') and S ! L '~~ ' )mentioning

confidence: 95%

The generalized Slater–Condon rules

Verbeek

Lenthe

1991

Int J of Quantum Chemistry

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“…Löwdin first presented a formula for the matrix elements of the Hamiltonian between Slater determinants expressed in terms of non-orthogonal spin-orbitals [31][32][33][34] , and after him different methods have been suggested to avoid the need of calculating the inverse, allowing for a rapid evaluation of the cofactors [35][36][37] , but in general they do not explicitly consider that the determinants from the set employed in a calculation may differ only in few spin-orbitals from a reference determinant. Hayes and Stone are the first who emphasized the advantages coming from prop-1-15 | 5 erly taking this peculiarity into account 38 , followed by Figari and Magnasco, who generalized the Hayes' results 39 .…”

Section: Implementation In the Code Ceres: Cofactors Versus Inverse Mmentioning

confidence: 99%

Ab Initio Non-Covalent Crystal Field Theory for Lanthanide Complexes: A Multiconfigurational Non-Orthogonal Group Functions Approach

Soncini¹,

PICCARDO²

2020

Preprint

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We present a non-orthogonal fragment ab initio methodology for the calculation of crystal field energy levels and magnetic properties in lanthanide complexes, implementing a systematic description of non-covalent contributions to metal-ligand bonding. The approach has two steps. In the first step, appropriate ab initio wavefunctions for the various ionic fragments (lanthanide ion and coordinating ligands) are separately optimized, accounting for the electrostatic influence of the surrounding environment, within various approximations. In the second and final step, the scalar relativistic (DKH2) electrostatic Hamiltonian of the whole molecule is represented on the basis of the optimized metal-ligand multiconfigurational non-orthogonal group functions (MC-NOGF), and reduced to an effective (2J+1)-dimensional non-orthogonal Configuration Interaction (CI) problem via L{\"o}wdin-partitioning. Within the proposed formalism, the projected Hamiltonian can be implemented to any desired order of perturbation theory in the fragment-localised excitations out of the degenerate space, and its eigenvalues and eigenfunctions are systematic approximations to the crystal field energies and wavefunctions. We present a preliminary implementation of the proposed MC-NOGF method to first-order degenerate perturbation theory within our own ab initio code CERES, and compare its performance both with the simpler non-covalent orthogonal ab initio approach Fragment Ab Initio Model Potential (FAIMP) approximation, and with the full CAHF/CASCI-SO method, accounting for metal-ligand covalency in a mean-field manner. We find that energies and magnetic properties for 44 complexes obtained via an iteratively optimized version of our MC-NOGF first-order non-covalent method, compare remarkably well to the full CAHF/CASCI-SO method including metal-ligand covalency, and are superior to the best purely electrostatic results achieved via an iteratively optimized version of the FAIMP approach.<br>

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“…A variety of different approaches have been developed for the efficient evaluation of nonorthogonal matrix elements, [26][27][28][29][30] which are predominantly derived from Löwdin's general formula. 31 The most popular framework in quantum chemistry is the generalized Slater-Condon rules, 27,32 where biorthogonal occupied orbitals are constructed 33,34 and a modified form of the Slater-Condon rules 35 is applied depending on the number of zero-overlap orbital pairs in the biorthogonal basis.…”

Section: Introductionmentioning

confidence: 99%

Generalized nonorthogonal matrix elements: Unifying Wick’s theorem and the Slater–Condon rules

Burton

2021

The Journal of Chemical Physics

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Matrix elements between nonorthogonal Slater determinants represent an essential component of many emerging electronic structure methods. However, evaluating nonorthogonal matrix elements is conceptually and computationally harder than their orthogonal counterparts. While several different approaches have been developed, these are predominantly derived from the first-quantized generalized Slater-Condon rules and usually require biorthogonal occupied orbitals to be computed for each matrix element. For coupling terms between nonorthogonal excited configurations, a second-quantized approach such as the nonorthogonal Wick's theorem is more desirable, but this fails when the two reference determinants have a zero many-body overlap. In this contribution, we derive an entirely generalized extension to the nonorthogonal Wick's theorem that is applicable to all pairs of determinants with nonorthogonal orbitals. Our approach creates a universal methodology for evaluating any nonorthogonal matrix element and allows Wick's theorem and the generalized Slater-Condon rules to be unified for the first time. Furthermore, we present a simple well-defined protocol for deriving arbitrary coupling terms between nonorthogonal excited configurations. In the case of overlap and one-body operators, this protocol recovers efficient formulas with reduced scaling, promising significant computational acceleration for methods that rely on such terms.

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Valence-bond theory and the evaluation of electronic energy matrix elements between nonorthogonal Slater determinants

Cited by 28 publications

References 26 publications

The generalized Slater–Condon rules

The generalized Slater–Condon rules

Ab Initio Non-Covalent Crystal Field Theory for Lanthanide Complexes: A Multiconfigurational Non-Orthogonal Group Functions Approach

Generalized nonorthogonal matrix elements: Unifying Wick’s theorem and the Slater–Condon rules

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