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The Journal of Chemical Physics

Improved perturbative treatment of electronic energies from a minimal-norm approach to many-body perturbation theory

Tamás Juhász, David A. Mazziotti

Published Mar 22, 2005• 4 citations
Publisher

Abstract

We propose a zeroth-order Hamiltonian for many-body perturbation theory based on the unitary decomposition of the two-particle reduced Hamiltonian. For the zeroth-order Hamiltonian constrained to be diagonal in the Hartree-Fock basis set, the two-particle reduced perturbation matrix is chosen to have a minimal Frobenius norm. When compared with the Møller-Plesset partitioning, the method yields more accurate second-order energies.

DOI
10.1063/1.1862232

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We propose a zeroth-order Hamiltonian for many-body perturbation theory based on the unitary decomposition of the two-particle reduced Hamiltonian. For the zeroth-order Hamiltonian constrained to be diagonal in the Hartree-Fock basis set, the two-particle reduced perturbation matrix is chosen to have a minimal Frobenius norm. When compared with the Møller-Plesset partitioning, the method yields more accurate second-order energies.
In many cases, density-functional theory (DFT) with current standard approximate functionals offers a relatively accurate and computationally cheap description of the short-range dynamic electron correlation effects. However, in general, standard DFT does not treat the dispersion interaction effects adequately which, on the other hand, can be described by many-body perturbation theory (MBPT). It is therefore of interest to develop a hybrid model which combines the best of both the MBPT and DFT approaches. This can be achieved by splitting the two-electron interaction into long-range and short-range parts; the long-range part is then treated by MBPT and the short-range part by DFT. This work deals with the formulation of a general MBPT-DFT model (i.e., valid for any type of zeroth-order Hamiltonian) based on such a range separation. Applying the Rayleigh-Schr\"odinger formalism in this context, one finds that the generalized Bloch equation becomes self-consistent at each order of perturbation. This complication arises because the short-range part of the energy is a functional of the exact electron density, which is expanded in a perturbation series. In order to address this ``self-consistency problem'' and provide computable orbital-based expressions for any order of perturbation, a general one-electron reduced-density-matrix-based formalism is proposed. Two applications of our general formalism are presented: The derivation of a hybrid second-order M\o{}ller-Plesset-DFT model and the formulation of a range-separated optimized effective potential method based on a hybrid G\"orling-Levy-type MBPT-DFT model.
We present a generalization of the Hartree-Fock (HF) method incorporating two-body correlations. The correlations are introduced by means of a unitary operator eiF{e}^{\mathrm{iF}}eiF, where FFF is a Hermitian one- and two-body operator to be determined by the variational principle. Assuming that the matrix elements of FFF are small so that powers of FFF higher than the second may be neglected, a set of linear equations determining FFF is obtained. The relation between our method and methods used to treat singular interactions is discussed. The method is applied to two recently suggested models and is compared with the HF method, second-order perturbation theory, and with the random-phase approximation. It is found that our method yields very good results within the range of validity of our approximations, which seems to call forth applications to more realistic problems. Possible applications to the study of collective phenomena in nuclei are indicated.
A framework for developing new approximate electronic structure methods is presented, in which the correlation energy of a many-electron system in the ground state is computed as in the single-reference second-order many-body perturbation theory but with the reference one-body Hamiltonian modified by additional (generally non-local) potentials which are universal functionals of the Hartree-Fock density. The existence of such functionals that reproduce the exact correlation energy justifies the search for approximate models which may overcome some important deficiencies of the traditional density-functional methods and also perform better than the usual low-order wavefunction methods.
The second-order multireference perturbation theory employing multiple partitioning of the many-electron Hamiltonian into a zero-order part and a perturbation is formulated in terms of many-body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two-particle terms in zero-order operators. The novel formulation naturally gives rise to a “sum-over-orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one-electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999
In this paper, the second in a series of two, we complete the derivation of the lowest-order wave function of a dimensional perturbation theory (DPT) treatment for the N-body quantum-confined system. Taking advantage of the symmetry of the zeroth-order configuration, we use group theoretic techniques and the FG matrix method from quantum chemistry to obtain analytic results for frequencies and normal modes. This method directly accounts for each two-body interaction, rather than an average interaction so that even lowest-order results include beyond-mean-field effects. It is thus appropriate for the study of both weakly and strongly interacting systems and the transition between them. While previous work has focused on energies, lowest-order wave functions yield important information such as the nature of excitations and expectation values of physical observables at low orders including density profiles. Higher orders in DPT also require as input the zeroth-order wave functions. In the earlier paper we presented a program for calculating the analytic normal-mode coordinates of the large-D system and illustrated the procedure by deriving the two simplest normal modes. In this paper we complete this analysis by deriving the remaining, and more complex, normal coordinates of the system.