Theoretical and applicative properties of the correlation and G‐particle‐hole matrices

Abstract

We have recently (Valdemoro et al., Sixth International Congress of theInternational Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) reported the form of the G-particle-hole hypervirial equation, which can be identified with the anti-Hermitian part of the correlation contracted Schrödinger equation (Alcoba, Phys Rev A, 2002, 65, 032519), as a tool to obtain the second-order reduced density matrix of an N-electron system without previous knowledge of the wave-function. The results which have been obtained when solving the G-particle-hole hypervirial equation with an iterative method also described in (Valdemoro et al., Sixth International Congress of the International Society for Theoretical Chemical Physics, 2008; Alcoba et al., Int J Quantum Chem, in press) have been highly accurate. The convergence of these test calculations has been very smooth, though rather slow. One of the factors which determines the performance of the method is the accuracy with which the 3-order correlation matrices (3-CM) involved in the calculations are approximated. It is, therefore, necessary to optimize to the utmost the construction algorithms of these 3-order matrices in terms of the 2-CM. In this article, the main theoretical features of the p-CM are described. Also, some aspects of the correlation contracted Schrödinger equation and of the G-particle-hole hypervirial equation are revisited. A new theorem, concerning the sufficiency of the hypervirial of the 3-order correlation operator to guarantee a correspondence between its solution and that of the Schrödinger equation, and some preliminary results concerning the constructing algorithms of the 3-CM in terms of the 2-CM, are reported in the second part of this article.

Some theoretical questions about the G‐particle‐hole hypervirial equation

“…Thus, and Note that no Hermitian positive semidefinite G ‐particle‐hole matrix exists in this last case, which shows that the correlation matrices family is much larger than the family of the Hermitian, positive semidefinite G ‐particle‐hole family. The properties of this type of high‐order correlation matrices, which are very relevant in the study of the many‐body problem, have been widely studied21, 22, 24, 43–52.…”

“…A similar sufficiency theorem for the 2‐CCSE has also been reported by Alcoba21 but until now, no such sufficiency theorems have been proved for the ACSE and GHV cases. Note, however, that a sufficiency theorem has been proved by Valdemoro et al for the third‐order GHV equation24.…”

“…When comparing the performance of the two hypervirial equations with that obtained in a full configuration interaction (FCI) calculation, both the ACSE and the GHV equations have been found to yield highly accurate results19, 23, 25, 29–35. On the other hand, it has been shown that while fulfilling the GHV equation implies that both the 1‐CSE and the ACSE are also fulfilled23–25, the reverse is not true. As no sufficiency theorem exists for the second‐order GHV, it is pertinent to examine whether an ACSE solution, which has been shown to be spurious23 (E. R. Davidson, 2008, Private Communication) also constitutes a counter‐example for the GHV sufficiency.…”

“…There are two other families of contracted equations, which are generated by the correlation‐MCM where the correlation, or equivalently, the G ‐particle‐hole operators play a similar role as the density operator in the previously mentioned MCM, whose p ‐order members are the p ‐order correlation contracted Schrödinger equation ( p ‐CCSE)21–25 and the p ‐order G ‐particle‐hole hypervirial equation ( p ‐order GHV)22–27.…”

Extension of the cumulant definition for low particle number systems

“…The concept of cumulant 1–22 has received much attention in quantum chemistry since the last decade. Cumulant has been primarily used as the element to reconstruct density matrices (DMs), whether in wavefunction‐based theories or directly density matrix–based theories, where high‐rank DMs are approximated by low‐rank cumulants 12, 23–48. The hope is that the magnitude of high‐rank cumulants decreases rapidly so that neglecting them in the approximation of DMs only incurs tolerable errors.…”