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Physical Review C

Multiphonon theory: Wick’s theorem and recursion formulas for coupled phonons

Konstantin V. Protasov, BERNARD SILVESTRE-BRAC, R. Piepenbring, Maurice Grinberg

Published Apr 1, 1996• 9 citations
Publisher

Abstract

The introduction of an adequate auxiliary space allows us to replace the problem of the treatment of coupled phonons, which is very complicated and voluminous, with the problem of the treatment of usual matrices. As applications of this approach, the formulation of a Wick's theorem for coupled phonons and recursion formulas which can be used to calculate the overlap matrix elements of multiphonon states built with n coupled phonons of different types and collectivities are given. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevc.53.1646

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The introduction of an adequate auxiliary space allows us to replace the problem of the treatment of coupled phonons, which is very complicated and voluminous, with the problem of the treatment of usual matrices. As applications of this approach, the formulation of a Wick's theorem for coupled phonons and recursion formulas which can be used to calculate the overlap matrix elements of multiphonon states built with n coupled phonons of different types and collectivities are given. \textcopyright{} 1996 The American Physical Society.
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.
Wick's theorem for the expectation value of an operator between many-particle states is generalized to the case of a homogeneous linear transformation operator Oˆ\^OOˆ, which leaves invariant the 2N2N2N-dimensional linear space composed of the annihilation and the creation operators for the N fermion or boson single-particle basis. Such a transformation generally belongs to the group SO(2N,C)\mathrm{SO}(2N,C)SO(2N,C) for the case of fermion, or the group Sp(2N,C)\mathrm{Sp}(2N,C)Sp(2N,C) for the case of boson. This theorem is applied to derive a reduction theorem for the matrix element of the type $〈\ensuremath{\alpha}|{\ensuremath{\alpha}}_{1}{\ensuremath{\alpha}}_{2}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\alpha}}_{m}\^O{\ensuremath{\beta}}_{m+1}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\beta}}_{m+2}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\beta}}_{2n}^{\ifmmode\dagger\else\textdagger\fi{}}|\ensuremath{\beta}〉,$ whose many-quasiparticle states and vacua appearing in both sides are represented in different quasiparticle pictures as indicated by $\ensuremath{\alpha}$ and $\ensuremath{\beta}.$ The reduction formula is useful in calculating any nuclear transition matrix element for the quantum-number projection calculation associated with the constrained Hartree-Fock-Bogoliubov solutions, and also the generator coordinate method with quantum number projection.
By using the extension of the statistical Wick's theorem (Gaudin's theorem) to deal with generalized statistical density operators (those which can be expressed as the product of and operator carrying out a canonical transformations times a density operator) and using the appropriate limits we are able to rederive in a very simple way the standard generalized Wick's theorem for overlaps of mean-field wave functions. Due to the simplicity of the derivation it is now straightforward to consider more involved cases and some of them are discussed. The present derivation also allows us to obtain general and compact formulas for other particular cases of the generalized Wick theorem involving overlaps of multiquasiparticle excitations of product wave functions. The new expressions allow us to reduce the combinatorial complexity of the standard calculation of the above overlaps.