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Following the theoretical development of a spin-adapted state-specific multi-reference second-order perturbation theory (SA-SSMRPT2) as expounded in the preceding publication, we discuss here its implementation and the results of its applications to potential energy curves (PECs) of various electronic states of small molecules. In particular, we illustrate its efficacy in states of various spin multiplicities and varying multi-reference character. Both Mo\ller-Plesset (MP) and Epstein-Nesbet (EN) type of partitions have been explored. Also, a straightforward Rayleigh-Schrödinger (RS) and Brillouin-Wigner (BW) version of the SA-SSMRPT2 have been studied. Ground state PECs were computed for singlet states of HF, BH, and H2O molecules as well as the doublet state of NH2 and BeH radicals and compared to corresponding full configuration interaction numbers, which serve as benchmark results. As an extensive application on a production level, the ground state PECs of N2, a classic example of multiple-bond breaking, were calculated using cc-pVXZ (X = 3,4,5) basis and then extrapolated to obtain estimates of the complete basis set limit. Vibrational energy levels were extracted from these N2 PECs, which compare favorably to the experimental values. In addition, extensive studies were also carried out on PECs of the seven low-lying excited states of the N2 molecule. Finally, it is shown that the flexibility to relax configuration coefficients in SA-SSMRPT2 helps to provide good descriptions for the avoided crossing between the two lowest 1Σ states of the LiF molecule. Our results indicate (1) that more studies are needed to draw firm conclusions about the relative efficacies of the MP and EN results and (2) that the RS version works so well as compared to the BW version that the extra computational expenses needed in the later formalism is not warranted. [ABSTRACT FROM AUTHOR]

We present in this paper a comprehensive formulation of a spin-adapted size-extensive state-specific multi-reference second-order perturbation theory (SA-SSMRPT2) as a tool for applications to molecular states of arbitrary complexity and generality. The perturbative theory emerges in the development as a result of a physically appealing quasi-linearization of a rigorously size-extensive state-specific multi-reference coupled cluster (SSMRCC) formalism [U. S. Mahapatra, B. Datta, and D. Mukherjee, J. Chem. Phys. 110, 6171 (1999)]. The formulation is intruder-free as long as the state-energy is energetically well-separated from the virtual functions. SA-SSMRPT2 works with a complete active space (CAS), and treats each of the model space functions on the same footing. This thus has the twin advantages of being capable of handling varying degrees of quasi-degeneracy and of ensuring size-extensivity. This strategy is attractive in terms of the applicability to bigger systems. A very desirable property of the parent SSMRCC theory is the explicit maintenance of size-extensivity under a variety of approximations of the working equations. We show how to generate both the Rayleigh-Schrödinger (RS) and the Brillouin-Wigner (BW) versions of SA-SSMRPT2. Unlike the traditional naive formulations, both the RS and the BW variants are manifestly size-extensive and both share the avoidance of intruders in the same manner as the parent SSMRCC. We discuss the various features of the RS as well as the BW version using several partitioning strategies of the hamiltonian. Unlike the other CAS based MRPTs, the SA-SSMRPT2 is intrinsically flexible in the sense that it is constructed in a manner that it can relax the coefficients of the reference function, or keep the coefficients frozen if we so desire. We delineate the issues pertaining to the spin-adaptation of the working equations of the SA-SSMRPT2, starting from SSMRCC, which would allow us to incorporate essentially any type open-shell configuration-state functions (CSF) within the CAS. The formalisms presented here will be applied extensively in a companion paper to assess their efficacy. [ABSTRACT FROM AUTHOR]

We analyze the structure and the solutions of the irreducible k-particle Brillouin conditions (IBCk) and the irreducible contracted Schrödinger equations (ICSEk) for an n-electron system without electron interaction. This exercise is very instructive in that it gives one both the perspective and the strategies to be followed in applying the IBC and ICSE to physically realistic systems with electron interaction. The IBC1 leads to a Liouville equation for the one-particle density matrix γ1=γ, consistent with our earlier analysis that the IBC1 holds both for a pure and an ensemble state. The IBC1 or the ICSE1 must be solved subject to the constraints imposed by the n-representability condition, which is particularly simple for γ. For a closed-shell state γ is idempotent, i.e., all natural spin orbitals (NSO’s) have occupation numbers 0 or 1, and all cumulants λk with k>=2 vanish. For open-shell states there are NSO’s with fractional occupation number, and at the same time nonvanishing elements of λ2, which are related to spin and symmetry coupling. It is often useful to describe an open-shell state by a totally symmetric ensemble state. If one wants to treat a one-particle perturbation by means of perturbation theory, this mainly as a run-up for the study of a two-particle perturbation, one is faced with the problem that the perturbation expansion of the Liouville equation gives information only on the nondiagonal elements (in a basis of the unperturbed states) of γ. There are essentially three possibilities to construct the diagonal elements of γ: (i) to consider the perturbation expansion of the characteristic polynomial of γ, especially the idempotency for closed-shell states, (ii) to rely on the ICSE1, which (at variance with the IBC1) also gives information on the diagonal elements, though not in a very efficient manner, and (iii) to formulate the perturbation theory in terms of a unitary transformation in Fock space. The latter is particularly powerful, especially, when one wishes to study realistic Hamiltonians with a two-body interaction. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The k-particle irreducible Brillouin conditions IBCk and the k-particle irreducible contracted Schrödinger equations ICSEk for a closed-shell state are analyzed in terms of a Møller–Plesset–type perturbation expansion. The zeroth order is Hartree–Fock. From the IBC2(1), i.e., from the two-particle IBC to first order in the perturbation parameter μ, one gets the leading correction λ2(1) to the two-particle cumulant λ2 correctly. However, in order to construct the second-order energy E2, one also needs the second-order diagonal correction γD(2) to the one-particle density matrix γ. This can be obtained: (i) from the idempotency of the n-particle density matrix, i.e., essentially from the requirement of n-representability; (ii) from the ICSE1(2); or (iii) by means of perturbation theory via a unitary transformation in Fock space. Method (ii) is very unsatisfactory, because one must first solve the ICSE3(2) to get λ3(2), which is needed in the ICSE2(2) to get λ2(2), which, in turn, is needed in the ICSE1(2) to get γ(2). Generally the (k+1)-particle approximation is needed to obtain Ek correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSEk, ignoring λk+1 and λk+2, by a renormalized hierarchy, in which only λk+2 is ignored, and λk+1 is expressed in terms of the λp of lower particle rank via the partial trace relation for λk+2. Then the k-particle approximation is needed to obtain Ek correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields Ek+1. We also study the possibility to use some simple necessary n-representability conditions, based on the non-negativity of γ(2) and two related matrices, in order to get estimates for γD(2) in terms of λ2(1). In general these estimates are rather weak, but they can become close to the best possible bounds in special situations characterized by a very sparse structure of λ2 in terms of a localized representation. The perturbative analysis does not encourage the use of a k-particle hierarchy based on the ICSEk (or on their reducible counterparts, the CSEk), it rather favors the approach in terms of the unitary transformation, where the k-particle approximation yields the energy correct up to E2k-1. The problems that arise are related to the unavoidable appearance of exclusion-principle violating cumulants. The good experience with perturbation theory in terms of a unitary transformation suggests that one should abandon a linearly convergent iteration scheme based on the ICSEk hierarchy, in favor of a quadratically convergent one based on successive unitary transformations. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Presents corrections to the article 'Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. I. The equations satisfied by the density cumulants' which appeared in the 'Journal of Chemical Physics, 'volume 114, page 2047, year 2001 issue.

Two alternative conditions for the stationarity of the energy expectation value with respect to k-particle excitations are the k-particle Brillouin conditions BC[sub k] and the k-particle contracted Schro¨dinger equations, CSE[sub k]. These conditions express the k-particle density matrices γ[sub k] in terms of density matrices of higher particle rank. The latter can be eliminated if one expresses the γ[sub k] in terms of their cumulants λ[sub k], but this is not sufficient to make the BC[sub k] or CSE[sub k] separable (extensive), i.e., they are not expressible in terms of only connected diagrams. However, in a formulation based on the recently introduced general normal ordering with respect to arbitrary wave functions, the irreducible counterparts IBC[sub k] and ICSE[sub k] of the BC[sub k] and CSE[sub k] can be defined. They are easily evaluated explicitly in terms of the generalized Wick theorem for arbitrary wave functions, and they lead to equations for the direct construction of the cumulants λ[sub k], which are additively separable quantities and which scale linearly with the system size. The IBC[sub k] or the ICSE[sub k] are necessary conditions for γ and the λ[sub k] to represent an exact n-fermionic eigenstate of the given Hamiltonian. To specify the desired state, additional conditions must be satisfied as well, e.g., the partial trace relations which relate λ[sub 2] to γ and γ[sup 2]. The particle number and the total spin must be specified and n-representability conditions enter implicitly. While the nondiagonal elements of γ and the λ[sub k] are determined by the IBC[sub k] or the ICSE[sub k], the additional conditions mainly serve to fix the diagonal elements. A hierarchy of k-particle approximations is defined. It is based on the fact that the expansion in terms of cumulants λ[sub k] can be truncated at any particle rank, which would not be possible for the d... [ABSTRACT FROM AUTHOR]