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The Journal of Chemical Physics

Diagrammatical Method for Geminals. I. Theory

Josef Paldus

Published Jul 15, 1972• 28 citations
Publisher

Abstract

A diagrammatical method, based on the second quantization formalism, time-independent Wick's theorem and Feynman-like diagrams, is developed for the calculation of arbitrary matrix elements of n-particle spin-independent operators between the states represented by antisymmetrized products of geminals of an arbitrary kind (i.e., strongly or weakly orthogonal, interacting, nonorthogonal or identical geminals). The method greatly facilitates any algebraic manipulation with geminal type functions, particularly when strong orthogonality is not assumed, and enables one to write down directly the algebraic expressions for the matrix elements of an arbitrary spin-independent operator between the single antisymmetrized products of arbitrary geminals, solely on the basis of simple graphs, which are easily constructed.

DOI
10.1063/1.1678296

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A diagrammatical method, based on the second quantization formalism, time-independent Wick's theorem and Feynman-like diagrams, is developed for the calculation of arbitrary matrix elements of n-particle spin-independent operators between the states represented by antisymmetrized products of geminals of an arbitrary kind (i.e., strongly or weakly orthogonal, interacting, nonorthogonal or identical geminals). The method greatly facilitates any algebraic manipulation with geminal type functions, particularly when strong orthogonality is not assumed, and enables one to write down directly the algebraic expressions for the matrix elements of an arbitrary spin-independent operator between the single antisymmetrized products of arbitrary geminals, solely on the basis of simple graphs, which are easily constructed.
The graphical technique of evaluation in the second quantization representation of matrix elements of operators associated with the interactions in atoms is developed. The diagrams corresponding to operators and many-electron states given in terms of irreducible tensorial product of creation and annihilation operators are considered. The general rules for the construction of the diagrams for operators and matrix elements are presented. The procedures of reduction of the diagrams to obtain the algebraic expression are discussed.
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.
The Lewis electron-pair picture pervades all of chemistry and a substantialdomain of materials science. “Geminals” are electron-pair quantum states whichare used to express approximate solutions of the Schrödinger equation of polyelec-tronic systems. In quantum chemistry, geminal-based methods are potentially moreeffective than the traditional ones based on the orbital picture: electrons occupyorbitals and experience only the average effect of each other. This is especially truefor strongly correlated-systems. However, the computational cost of using arbitrarygeminal wave functions is non-polynomial in the system size.It is therefore an important challenge to find additional constraints on geminalmodels which preserve the accuracy of the wavefunction description while reducingthe calculations to polynomial cost. The Ayers’ group at Mc Master has developedone such successful geminal model called AP1roG because the wavefunction is anantisymmetric product of geminals with one distinct reference orbital occupied ineach geminal [1]. This ansatz relaxes the so-called “strong-orthogonality” constraintbetween geminals. The latter makes wavefunction matrix element computationsvery easy but appears too drastic from the physical point of view, since the geminalelectron pairs are then distinguishable.We have followed a different path to relax the strong-orthogonality constraint. Wehave defined the geometrical concept of n-orthogonality [2] which can be viewed asa graded indistinguishability measure for electronic states: 1-orthogonality coincideswith strong orthogonality, the larger n the less distinguishable the n-orthogonal elec-tronic states will be. We have studied geminal models constrained by n-orthogonalityrelations for different values of n, imposed to different combinations of geminal prod-uct states [3]. However, in this work the scaling of matrix element computations wasstill unsatisfactory. We will present new ideas developed since then to remedy thisproblem.References[1] P. A Limacher, P. W. Ayers, P. A. Johnson, S. De Baerdemacker, D. Van Neck and P. Bultinck, J. of Chem. Theory and Computation 9, 1394-1401 (2013).[2] P. Cassam-Chenaı̈, Phys. Rev. A77, 032103 (2008).[3] P. Cassam-Chenaı̈, V. Rassolov, Chem. Phys. Lett. 487, 147-152 (2010).
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of particles.
In Section 14 we considered many nucleon states as the antisymmetrized products of single nucleon wave functions. Due to the antisymmetrization, the wave function is completely determined by the single nucleon states which are occupied. This suggests that a direct way to deal with many nucleon states is by using a representation in which occupied single nucleon states, rather than nucleon coordinates, are specified. To do this we have to define a fixed set of single nucleon states. The results ( 14.6 ), ( 14.7 ) indicate that we should provide matrix elements of relevant single nucleon operators, calculated in the basis of these single nucleon wave functions. The results ( 14.11 ), ( 14.12 ) and ( 14.13 ) show that we should also calculate matrix elements of two-body operators in the basis of single nucleon wave functions. These calculations should be performed in the way shown in preceding sections. They are carried out in configuration space by using wave functions of space coordinates as well as spin (and isospin) variables. Equipped with these matrix elements we can introduce the representation in terms of occupied states by using the formalism of second quantization. This formalism will now be briefly described without entering into serious discussions and rigorous proofs.