Josef Paldus

Stability Conditions for the Solutions of the Hartree—Fock Equations for Atomic and Molecular Systems. Application to the Pi‐Electron Model of Cyclic Polyenes

The stability conditions which ensure that the Hartree—Fock determinant minimizes the energy expectation value are rederived using the language familiar in quantum chemistry. These stability conditions are then specified for the case of closed‐shell electronic systems which allow additional simplification of the conditions as well as a certain classification of the instabilities.Examples of the instabilities of different types are presented and the case of the so‐called singlet instabilities—most interesting from the physical point of view—is studied in detail for the pi‐electron model of cyclic polyenes.

Time-Independent Diagrammatic Approach to Perturbation Theory of Fermion Systems

Publisher Summary The chapter discusses the time-independent diagrammatic approach to perturbation theory of fermion systems. The chapter explores the perturbation theory for a non-degenerate level. The formulas derived serves as a starting point for the subsequent consideration of the excitation and ionization energies. The advantages of the direct calculation of excitation energies, compared with the approach in which the total energies of the pertinent electronic states are calculated separately for each state and then the excitation energies are obtained by subtracting the appropriate state energies, are quite obvious. The Rayleigh-Schrodinger (RS) perturbation theory (PT) for the case of a non-degenerate level of some Hamiltonian operator is discussed. The chapter discusses that even the Rayleigh-Schrodinger perturbation expressions for the direct calculation of the excitation energies may be obtained in a rather simple way without the involvement of the Green function formalism. On the contrary, our simple approach using the ordinary perturbation theory for separate levels presents certain desirable features of the Green function formalism. The chapter explains the diagrammatic representation of Wicks theorem and resulting diagrams. General explicit formulas for the second- and third-order excitation energy contributions are given in the chapter.

Group theoretical approach to the configuration interaction and perturbation theory calculations for atomic and molecular systems

A formalism for an efficient generation of spin‐symmetry adapted configuration interaction (CI) matrices of the N‐electron atomic or molecular systems, described by nonrelativistic spin‐independent Hamiltonians, is presented. The Gelfand and Tsetlin canonical basis for the finite dimensional irreducible representations of the unitary groups is used as an N‐electron CI basis. A simplified Gelfand‐type pattern pertaining to the N‐electron problem is introduced, which considerably simplifies the canonical basis generation and, more importantly, the calculation of representation matrices of the (infinitesimal) generators of the pertinent unitary group in this basis. The calculation of the CI matrices for the above mentioned systems is then straightforward, since any particle number conserving operator may be written as a sum of n‐degree forms in the unitary group generators. The computation of CI matrices for various Hamiltonians as well as the problems of the space‐symmetry adaptation of the Gelfand‐Tsetlin ...

Reduced multireference CCSD method: An effective approach to quasidegenerate states

Standard multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple solution problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that concentrate on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) The construction of a variational configuration interaction (CI) wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster analysis of this CI wave...

Spin‐adapted multireference coupled‐cluster approach: Linear approximation for two closed‐shell‐type reference configurations

An explicit form of the spin‐adapted multireference coupled‐cluster formalism in the linear approximation is developed for the special case of a two‐dimensional model space involving only closed‐shell‐type configurations. The formalism is applicable to a number of quasidegenerate systems with two valence orbitals of distinct spatial symmetry and should serve as a convenient testing ground for the suitability of the multireference coupled‐cluster theory. General problems of the multireference coupled‐cluster approach and its relationship with the corresponding configuration interaction formalism are discussed as well as the problems pertaining to a practical implementation of this formalism.

Correlation problems in atomic and molecular systems. V. Spin‐adapted coupled cluster many‐electron theory

The graphical techniques of spin algebras are combined with a diagrammatic approach based on the time independent Wick theorem to yield the spin‐adapted form of the coupled cluster theory. The general rules for the implementation of this formalism are formulated and illustrated on the basic coupled‐pair many‐electron theory pertaining to closed shell ground states, for which case the explicit spin‐adapted equations are derived. The advantages of the spin‐adapted form of the theory are discussed along with the new insights it affords.

Valence universal exponential ansatz and the cluster structure of multireference configuration interaction wave function

A rigorous algebraic formulation of open‐shell coupled‐cluster theory is presented. This formulation yields explicit formulas exhibiting the relationship between open‐shell cluster amplitudes and linear coefficients of multireference CI wave functions. When the valence‐universal exponential ansatz is used, the CI coefficients of states with n valence electrons contribute to the n‐body and higher‐order cluster operators. The implications of cluster conditions, requiring that the four‐body cluster amplitudes be small, are investigated. It is shown that for valence‐universal theories these conditions lead to approximate relations involving CI coefficients for states of systems differing in the number of electrons. For Lindgren’s ansatz these relations are linear in the CI coefficients corresponding to states with the largest electron number. For the valence‐nonuniversal exponential ansatz of Jeziorski and Monkhorst, the cluster conditions do not mix wave functions for systems which differ in the number of el...

Stability Conditions for the Solutions of the Hartree–Fock Equations for Atomic and Molecular Systems. II. Simple Open‐Shell Case

The stability conditions for the solutions of the Hartree–Fock equations for the simple open‐shell case, i.e., closed shell with one extra electron, are derived. It is shown that only “doublet stability” is relevant is this simple open‐shell case, the solutions being always “nondoublet unstable.” The doublet stability conditions are then derived using the mathematical methods of quantum field theory, namely, occupation number representation, Wicks theorem, and Feynman‐like diagrams. In order to familiarize the reader with the use of these concepts they are first used to rederive the singlet and nonsinglet stability conditions for the closed‐shell case. A general method of finding new Hartree–Fock solutions, in the case that the symmetry adapted Hartree–Fock solutions are unstable, is briefly discussed. The implications of the instability on the ground‐state correlation energy calculations and on the excitation energy calculations using time‐dependent Hartree–Fock theory are considered.

Reduced multireference couple cluster method. II. Application to potential energy surfaces of HF, F2, and H2O

The so-called reduced multireference (RMR) coupled cluster method restricted to singly and doubly excited clusters (CCSD) [see X. Li and J. Paldus, J. Chem. Phys. 107, 6257 (1997)] is employed to compute potential energy surfaces for the HF, F2 and H2O molecules over a wide range of geometries using basis sets of a double zeta (DZ) and DZ plus polarization (DZP) quality. The RMR-CCSD method belongs to a class of externally corrected CCSD approaches, which rely on a suitable non-CC wave function that is flexible enough to describe the dissociation process at hand and is used as a source of 3- and 4-body cluster amplitudes. These amplitudes are in turn used to achieve a more appropriate decoupling of the full CC chain of equations than that leading to the standard CCSD equations. The RMR-CCSD method employs for this purpose a MR-CISD wave function obtained with a relatively small active or model space. To illustrate the capabilities of this approach, the computed potential energy curves for the HF, F2 and H...

AUTOMATION OF THE IMPLEMENTATION OF SPIN-ADAPTED OPEN-SHELL COUPLED-CLUSTER THEORIES RELYING ON THE UNITARY GROUP FORMALISM

A new implementation of the orthogonally spin‐adapted open‐shell (OS) coupled‐cluster (CC) formalism that is based on the unitary group approach to many‐electron correlation problem is described. Although the emphasis is on the so‐called state specific single‐reference but multiconfigurational OS CC approach, the developed algorithms as well as the actual codes are also amenable to multireference CC applications of the state‐universal type. A special attention is given to simple OS doublets and OS singlet and triplet cases, the former being applicable to the ground states of radicals and the latter to the excited states of closed shell systems. The encoding of the underlying formalism is fully automated and is based on a convenient decomposition of the Hamiltonian into the effective zero‐, one‐, and two‐orbital contributions as well as on the general strategy that focuses on the excitation operator driven evaluation of individual absolute, linear, quadratic, etc., coupled cluster coefficients, rather than...