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arXiv:physics/0412079

Arrow diagram method based on overlapping electronic groups: corrections to the linked AD theorem

Wang, Yu, Lev N Kantorovich

Published Dec 14, 2004• 0 citations
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Abstract

Arrow diagram (AD) method (L. Kantorovich and B. Zapol, J. Chem. Phys. \textbf{96}, 8420 (1992); \emph{ibid}, 8427) provides a convenient means of systematic calculation of arbitrary matrix elements, $$, of symmetrical operators, $\hat{O}$, in quantum chemistry when the total system wavefunction $Ψ$ is represented as an antisymmetrised product of overlapping many-electron group functions, $Φ_{A}$, corresponding to each part (group) $A$ of the system: $Ψ=\hat{A}\prod_{A}Φ_{A}$. For extended (e.g. infinite) systems the calculation is somewhat difficult, however, as mean values of the operators require that each term of the diagram expansion is to be divided by the normalisation integral $S=$, which is given by an AD expansion as well. A linked AD theorem suggested previously (L. Kantorovich, Int. J. Quant. Chem. \textbf{76}, 511 (2000)) to deal with this problem is reexamined in this paper using a simple Hartree-Fock problem of a one-dimensional ring of infinite size which is found to be analytically solvable. We find that corrections to the linked AD theorem are necessary in a general case of a finite overlap between different electronic groups. A general method of constructing these corrections in a form of a power series expansion with respect to overlap is suggested. It is illustrated on the ring model system.

DOI
10.48550/arxiv.physics/0412079

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Arrow diagram (AD) method (L. Kantorovich and B. Zapol, J. Chem. Phys. \textbf{96}, 8420 (1992); \emph{ibid}, 8427) provides a convenient means of systematic calculation of arbitrary matrix elements, $$, of symmetrical operators, O^\hat{O}O^, in quantum chemistry when the total system wavefunction ΨΨΨ is represented as an antisymmetrised product of overlapping many-electron group functions, ΦAΦ_{A}ΦA​, corresponding to each part (group) AAA of the system: Ψ=A^∏AΦAΨ=\hat{A}\prod_{A}Φ_{A}Ψ=A^∏A​ΦA​. For extended (e.g. infinite) systems the calculation is somewhat difficult, however, as mean values of the operators require that each term of the diagram expansion is to be divided by the normalisation integral S=S=S=, which is given by an AD expansion as well. A linked AD theorem suggested previously (L. Kantorovich, Int. J. Quant. Chem. \textbf{76}, 511 (2000)) to deal with this problem is reexamined in this paper using a simple Hartree-Fock problem of a one-dimensional ring of infinite size which is found to be analytically solvable. We find that corrections to the linked AD theorem are necessary in a general case of a finite overlap between different electronic groups. A general method of constructing these corrections in a form of a power series expansion with respect to overlap is suggested. It is illustrated on the ring model system.
For nonorthogonal electron group functions, the arrow diagram (AD) method (Kantorovich and Zapol 1992 J. Chem. Phys. 96 8420; 1992 J. Chem. Phys. 96 8427) provides a convenient procedure for calculating matrix elements of arbitrary symmetrical operators . The total wavefunction of the system is represented as an antisymmetrized product of nonorthogonal many-electron group functions Φ I of each group I in the system. For extended (e.g. infinite) systems the calculation of the mean value of an operator is ill defined, however, as it requires that each term of the diagram expansion be divided by the normalization integral which is given by an AD expansion as well. In this work, we cast the mean value of a symmetrical operator in a form of an AD expansion which is a linear combination of linked ADs. By analysing an exactly solvable one-dimensional Hartree–Fock problem, we find that pre-factors, attached to every linked AD in the linear combination, can be expanded in a power series with respect to overlap. A general method of calculating these pre-factors in a form of a power series expansion with respect to overlap is suggested. This advance makes the AD theory applicable to extended systems, and allows one to calculate the mean value of an arbitrary symmetrical operator correct up to the desired order of overlap within the group function theory. In particular, we derive the effective Hamiltonian of a quantum cluster surrounded by overlapping group functions (e.g. bonds) in the environment region which is correct up to the second order with respect to overlap (an embedding problem).
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.
In this (first) paper we attempt to generalize the notion of tensor connectivity, subsequently studying how this property is affected in different tensorial operations. We show that the often implied corollary of the linked diagram theorem, namely individual size-extensivity of arbitrary connected closed diagrams, can be violated in Coulomb systems. In particular, the assumption of the existence of localized Hartree–Fock orbitals is generally incompatible with the individual size-extensivity of connected closed diagrams when the interaction tensor is generated by the true two-body part of the electronic Hamiltonian. Thus, in general, size-extensivity of a many-body method may originate in specific cancellations of super-extensive quantities, breaking the convenient one-to-one correspondence between the connectivity of arbitrary many-body equations and the size-extensivity of the expectation values evaluated by those equations (for example, when certain diagrams are discarded from the method). Nevertheless, assuming that many-body equations are evaluated for a stable many-particle system, it is possible to introduce a workaround, called the ε-approximation, which restores the individual size-extensivity of an arbitrary connected closed diagram, without qualitatively affecting the asymptotic behavior of the computed expectation values. No assumptions concerning the periodicity of the system and its strict electrical neutrality are made.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
Adopting the so-called "genealogical construction," one can express the eigenstates of collective operators corresponding to a specified mode for an NNN-atom system in terms of those for an ($N\ensuremath{-}1$)-atom system. Using these Dicke states as bases and using the Wigner-Eckart theorem, a matrix element of a collective operator of an arbitrary mode can be written as the product of an mmm-dependent factor and an mmm-independent reduced matrix element (RME). A set of recursion formulas for the RME is obtained. A graphical representation of the RME on the branching diagram for binary irreducible representations of permutation groups is then introduced. This gives a simple and systematic way of calculating the RME. This method is especially useful when the cooperation number r\mathcal{r}r is close to N2\frac{N}{2}2N​, where almost exact asymptotic expressions can be obtained easily. The result shows explicitly the geometry dependence of superradiance and the relative importance of r\mathcal{r}r-conserving and r\mathcal{r}r-nonconserving processes. This clears up the chief difficulty encountered in the Dicke-Schwendimann approach to the problem of NNN two-level atoms, spread over large regions, interacting with a multimode radiation field.
In treating a system of NNN antisymmetric particles, it is shown that, if the total Hamiltonian Hop{\mathcal{H}}_{\mathrm{op}}Hop​ is degenerate, the eigenstates of the operator used for classifying the corresponding degenerate states may be selected by means of a "projection operator" O\mathcal{O}O. If the total wave function is approximated by such a projection of a single determinant, the description of the system may be reduced to the ordinary Hartree-Fock scheme treating this determinant, if the original Hamiltonian is replaced by a complete Hamiltonian ${\ensuremath{\Omega}}_{\mathrm{op}}={\mathcal{O}}^{\ifmmode\dagger\else\textdagger\fi{}}{\mathcal{H}}_{\mathrm{op}}\mathcal{O}$ containing also many-particle interactions. This approach corresponds to a "fixed" configurational interaction, but the scheme has preserved the physical simplicity and visuality of the Hartree-Fock approximation. The idea of "doubly filled" orbitals is abandoned, and the orbitals associated with different spins will automatically try to arrange themselves in such a way that particles having antiparallel spins will tend to avoid each other due to their mutual repulsion.