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3,5‐contracted Schrödinger equation: Determining quantum energies and reduced density matrices without wave functions
Author(s) -
Mazziotti David A.
Publication year - 1998
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/(sici)1097-461x(1998)70:4/5<557::aid-qua2>3.0.co;2-u
Subject(s) - antisymmetric relation , perturbation theory (quantum mechanics) , physics , quantum , wave function , cumulant , generalization , quantum mechanics , mathematical physics , mathematics , mathematical analysis , statistics
Through the 3,5‐contracted Schrödinger equation (3,5‐CSchE) quantum energies and 3‐particle reduced density matrices (3‐RDMs) are determined directly without wave functions. Since the 3,5‐CSchE involves the 5‐RDM, its solution is indeterminate without N ‐representability conditions. However, the indeterminacy of the 3,5‐CSchE may be removed through a reconstruction strategy for building the 4‐ and 5‐RDMs from the 3‐RDM. We present a systematic procedure for obtaining corrections for Valdemoro's reconstruction functionals from two complementary approaches, the particle–hole duality and the theory of cumulants. With the cumulants we are able to demonstrate that we have obtained all terms in the reconstruction functionals which may be written as antisymmetric products of the lower rdms. The cumulants allow us to understand the reconstruction functionals in terms of a renormalized many‐body perturbation theory. The reconstruction functionals also lead to a natural generalization of Wick's theorem for evaluating expectation values of fermionic annihilation and creation operators with respect to correlated reference states. Previous work [Phys. Rev. A 57 , 4219 (1998)] has explored the determination of correlation energy and 2‐RDMs through the 2,4‐CSchE, also known as the density equation. Because the reconstruction functionals employed with the 3,5‐CSchE depend only on the antisymmetric products of lower RDMs in constrast to those used with the 2,4‐CSchE, the 3,5‐CSchE method presented here does not require the solution of systems of linear equations during reconstruction or the storage of the reconstructed RDMs. Application of the 3,5‐CSchE technique to a quasi‐spin model generates ground‐state energies and 2‐RDMs similar in accuracy to single–double configuration interaction (SDCI). We employ a simple iterative procedure for the solution of the 3,5‐CSchE without traditional diagonalization. The CSchE techniques offer an approximate solution of the N ‐representability problem and a new approach to electron correlation. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 557–570, 1998