Many-electron Correlation Problem Research Articles

Recent Advances in the Study of Negative-Ion Resonances Using Multiconfigurational Propagator and a Complex Absorbing Potential.

The transient resonances are a challenge to bound state quantum mechanics. These states lie in the continuum part of the spectrum of the Hamiltonian. For this, one has to treat a continuum problem due to electron-molecule scattering and the many-electron correlation problem simultaneously. Moreover, the description of a resonance requires a wavefunction that bridges the part that resembles a bound state with another that resembles a continuum state such that the continuity of the wavefunction and its first derivative with respect to the distance between the incoming projectile and the target is maintained. A review of the recent advances in the theoretical investigation of the negative-ion resonances (NIR) is presented. The NIRs are ubiquitous in nature. They result from the scattering of electrons off of an atomic or molecular target. They are important for numerous chemical processes in upper atmosphere, space and even biological systems. A contextual background of the existing theoretical methods as well as the newly-developed multiconfigurational propagator tools based on a complex absorbing potential are discussed.

Matrix elements of unitary group generators in many-fermion correlation problem. III. Green-Gould approach

The third part of our survey series concerning the evaluation of matrix elements (MEs) of the unitary group generators and of their products in the electronic Gel’fand–Tsetlin basis of the two-column irreps of U(n)—which are essential in the unitary group approach (UGA) to the many-electron correlation problem as handled by the configuration interaction (CI) and the coupled cluster (CC) approaches—relies on what we refer to as the Green-Gould (G-G) approach. In addition to the CI and CC methods, the G-G formalism proved to be very helpful in a number of other tasks, particularly in handling of the spin-dependent operators, the density matrices, or partitioned basis sets adapted to a chosen group chain.

Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.

Matrix elements of unitary group generators in many-fermion correlation problem. II. Graphical methods of spin algebras

This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital U(2n) generators, in the basis of the electronic Gel’fand–Tsetlin (G–T) [or Gel’fand–Paldus (G–P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the U(n), $${\mathcal {S}}_N$$ , and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group $${\mathcal {S}}_N$$ . The key is a close relationship between the P-states of UGA and the Yamanouchi–Kotani (Y–K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent U(2n) generators in the UGA P-basis.

Analytic-continuation approach to the resummation of divergent series in Rayleigh-Schrödinger perturbation theory

The method of analytic continuation is applied to estimate eigenvalues of linear operators from finite order results of perturbation theory even in cases when the latter is divergent. Given a finite number of terms ${E}^{(k)},k=1,2,\ensuremath{\cdots}M$ resulting from a Rayleigh-Schr\odinger perturbation calculation, scaling these numbers by ${\ensuremath{\mu}}^{k}$ ($\ensuremath{\mu}$ being the perturbation parameter) we form the sum $E(\ensuremath{\mu})={\ensuremath{\sum}}_{k}{\ensuremath{\mu}}^{k}{E}^{(k)}$ for small $\ensuremath{\mu}$ values for which the finite series is convergent to a certain numerical accuracy. Extrapolating the function $E(\ensuremath{\mu})$ to $\ensuremath{\mu}=1$ yields an estimation of the exact solution of the problem. For divergent series, this procedure may serve as resummation tool provided the perturbation problem has a nonzero radius of convergence. As illustrations, we treat the anharmonic (quartic) oscillator and an example from the many-electron correlation problem.

Unitary group approach to the many-electron correlation problem: spin-dependent operators

Following a brief overview of the unitary group approach (UGA) to the many-electron correlation problem, focusing in particular on Shavitt’s contribution via his graphical unitary group approach, we present a short review of our earlier results for the evaluation of matrix elements (MEs) of unitary group generators or products of generators in the electronic Gel’fand–Tsetlin basis with the help of spin-adapted second-quantization-like creation and annihilation vector operators at the unitary group level. This formalism is then extended to a spin-dependent case that is required when accounting for relativistic effects by developing explicit expressions for MEs of spin-orbital creation and annihilation operators in terms of the standard spin-adapted UGA basis. This leads naturally to a segmentation of these MEs and enables the evaluation of spin-dependent one-body operators while relying largely on the segment values of the standard spin-independent UGA.

QCI and related CC approaches: a retrospection

The relationship between standard single-reference coupled-cluster (CC) and quadratic configurations interaction (QCI) methods is reviewed from the viewpoint of both past and recent developments in the methodology of post-Hartree–Fock approaches to the many-electron correlation problem, emphasising their advantages and shortcomings on the basis of many-body perturbation theory (MBPT) analysis, the required computational effort, and their practical exploitations.

Graph theoretical concepts for the unitary group approach to the many-electron correlation problem

The simplified unitary group formalism for electronic states introduced by Paldus was recast in a form based on a relatively small table of ''distinct rows'' of the array (or tableau) representatives of the appropriate canonical basis functions. This table can be represented effectively and compactly in the form of a two-rooted directed graph. Each member (configuration function) of the canonical basis is represented by a walk connecting the head and tail of the graph, and its canonical index (sequence number) can be determined easily and directly from information contained in the distinct row table or the graph. The calculation of the matrix representation of a quantum mechanical operator in the canonical basis can be organized in terms of various subgraphs of the distinct row graph, and in terms of certain ''loops'' contained in them. A relatively small number of basic subgraphs can be used to generate most of the information needed in the calculation. Certain problems associated with matrix elements of products of generators of the unitary group remain to be solved before an effective formalism for large-scale electronic structure calculations is obtained, but it is hoped that the concepts introduced here can provide insight which will be helpful inmore » solving these problems.« less

Recent advances in electronic structure theory: Method of moments of coupled-cluster equations and renormalized coupled-cluster approaches

The recently developed new approach to the many-electron correlation problem in atoms and molecules, termed the method of moments of coupled-cluster (CC) equations (MMCC), is reviewed. The ground-state MMCC formalism and its extension to excited electronic states via the equation-of-motion coupled-cluster (EOMCC) approach are discussed. The main principle of all MMCC methods is that of the non-iterative energy corrections which, when added to the ground- and excited-state energies obtained in the standard CC calculations, such as CCSD or EOMCCSD, recover the exact, full configuration interaction (CI) energies. Three types of the MMCC approximations are reviewed in detail: (i) the CI-corrected MMCC methods, which can be applied to ground and excited states; (ii) the renormalized and completely renormalized CC methods for ground states; and (iii) the quasi-variational MMCC approaches for the ground-state problem, including the quadratic MMCC models. It is demonstrated that the MMCC formalism provides a new theoretical framework for designing 'black-box' CC approaches that lead to an excellent description of entire potential energy surfaces of ground- and excited-state molecular systems with an ease of use of the standard single-reference methods. The completely renormalized (CR) CCSD(T) and CCSD(TQ) methods and their quadratic and excited-state MMCC analogues remove the failing of the standard CCSD, CCSD(T), EOMCCSD and similar methods at larger internuclear separations and for states that normally require a genuine multireference description. All theoretical ideas are illustrated by numerical examples involving bond breaking, excited vibrational states, reactive potential energy surfaces and difficult cases of excited electronic states. The description of the existing and well-established variants of the MMCC theory, such as CR-CCSD(T), is augmented by the discussion of future prospects and potentially useful recent developments, including the extension of the black-box CR-CCSD(T) method to excited states.

Method of Moments of Coupled-Cluster Equations: Externally Corrected Approaches Employing Configuration Interaction Wave Functions

A new approach to the many-electron correlation problem, termed the method of moments of coupled-cluster equations (MMCC), is further developed and tested. The main idea of the MMCC theory is that of the noniterative energy corrections which, when added to the energies obtained in the standard coupled-cluster calculations, recover the exact (full configuration interaction) energy. The MMCC approximations require that a guess is provided for the electronic wave function of interest. The idea of using simple estimates of the wave function, provided by the inexpensive configuration interaction (CI) methods employing small sets of active orbitals to define higher–than–double excitations, is tested in this work. The CI-corrected MMCC methods are used to study the single bond breaking in HF and the simultaneous breaking of both O–H bonds in H2O.

The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches

This paper is the first in a series of papers on the new approach to the many-electron correlation problem, termed the method of moments of coupled-cluster equations (MMCC). A hierarchy of MMCC approximations, including the renormalized and completely renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) methods, which can be viewed as generalizations of the well-known perturbative coupled-cluster CCSD[T], CCSD(T), CCSD(TQf), and CCSDT(Qf) schemes, is introduced. In this initial study, an emphasis is placed on the ability of the MMCC approach to describe bond breaking and large effects due to connected triples and quadruples by modifying the standard noniterative CC approaches, such as the popular CCSD(T) method. The performance of selected MMCC approaches, including the renormalized and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ) schemes, is illustrated by the results of pilot calculations for the HF and H2O molecules.

On the structure of the Clifford algebra unitary group adapted states in the many-electron correlation problem

An attempt has been made to understand the structure of the Clifford algebra unitary group adapted many-particle states from the conventional symmetric group point of view. Emphasizing the symmetric group result that the consideration of the spin-independent Hamiltonian matrix over the many-particle configuration functions (CFs) entails a particular subspace of their spatial parts only, attention is confined entirely in this subspace. Question of adapting the functions therein to the unitary group subduction chain is then shown to bring out an interesting lead to the Clifford algebra unitary group approach (CAUGA) states, thus underlining the motive and the essential gains of the CAUGA formulation. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 607–614, 2000

Matrix elements ofU(2n) generators in a multishell spin-orbit basis. I. The del-operator MEs in a two-shell composite Gelfand-Paldus basis

This is the first in a series of three articles which aimed to derive the matrix elements of the U(2n) generators in a multishell spin-orbit basis. This is a basis appropriate to many-electron systems which have a natural partitioning of the orbital space and where also spin-dependent terms are included in the Hamiltonian. The method is based on a new spin-dependent unitary group approach to the many-electron correlation problem due to Gould and Paldus [M. D. Gould and J. Paldus, J. Chem. Phys. 92, 7394, (1990)]. In this approach, the matrix elements of the U(2n) generators in the U(n) x U(2)-adapted electronic Gelfand basis are determined by the matrix elements of a single Ll(n) adjoint tensor operator called the del-operator, denoted by Delta(j)(i) (1 less than or equal to i, j less than or equal to n). Delta or del is a polynomial of degree two in the U(n) matrix E = [E-j(i)]. The approach of Gould and Paldus is based on the transformation properties of the U(2n) generators as an adjoint tensor operator of U(n) x U(2) and application of the Wigner-Eckart theorem. Hence, to generalize this approach, we need to obtain formulas for the complete set of adjoint coupling coefficients for the two-shell composite Gelfand-Paldus basis. The nonzero shift coefficients are uniquely determined and may he evaluated by the methods of Gould et al. [see the above reference]. In this article, we define zero-shift adjoint coupling coefficients for the two-shell composite Gelfand-Paldus basis which are appropriate to the many-electron problem. By definition, these are proportional to the corresponding two-shell del-operator matrix elements, and it is shown that the Racah factorization lemma applies. Formulas for these coefficients are then obtained by application of the Racah factorization lemma. The zero-shift adjoint reduced Wigner coefficients required for this procedure are evaluated first. All these coefficients are needed later for the multishell case, which leads directly to the two-shell del-operator matrix elements. Finally, we discuss an application to charge and spin densities in a two-shell molecular system. (C) 1998 John Wiley & Sons.

On a general system-partitioning in the many-electron correlation problem

On a general system-partitioning in the many-electron correlation problem

Calculation of static molecular properties in the framework of the unitary group based coupled cluster approach

The recently developed and implemented state selective, fully spin-adapted coupled cluster (CC) method that employs a single, yet effectively multiconfigurational, spin-free reference and the formalism of the unitary group approach (UGA) to the many-electron correlation problem, has been employed to calculate static electric properties of various open-shell (OS) systems using the finite field (FF) technique. Starting with the lithium atom, the method was applied at the first-order interacting space single and double excitation level (CCSD(is)) to several first- and second-row hydrides having OS ground state, namely to the CH, NH, OH, SiH, PH, and SH radicals. In the case of NH we also considered three OS excited states. In all cases the dipole moment and polarizability were determined using a high quality basis set and compared with the experiment, whenever available, as well as with various configuration interaction results and other theoretical results that are available from the literature. The agreement of our CCSD(is) values with experiment is very satisfactory except for the 3Σ− ground state of the NH radical, where the experimentally determined dipole moment is too small. No experimental data are available for the corresponding polarizabilities. It is also shown that the FF technique is not suitable for calculations of higher order static properties, such as the hyperpolarizability β and γ tensors. For this reason we formulate the linear response version of our UGA-based CCSD approach and discuss the aspects of its future implementation. Key words: static molecular properties, dipole moments, polarizabilities, free radicals, unitary group based coupled cluster method, linear response theory, finite field technique.

Unitary group based state specific open-shell-singlet coupled-cluster method: Application to ozone and comparison with Hilbert and Fock space theories

Recently introduced state-specific coupled-cluster method, which exploits the formalism of the unitary group approach to the many-electron correlation problem and enables a properly spin-adapted treatment of open-shell states, is applied to several low lying singlet and triplet electronic states of ozone at the double-zeta plus polarization level of approximation. The method employs a nonstandard cluster Ansatz, based on a single spin-free reference built from either the ground state restricted Hartree–Fock orbitals or from the restricted open-shell Hartree–Fock orbitals, specific for each excited state. The results are compared with available experimental data and with other ab initio calculations, particularly with those employing spin-orbital based, spin nonadapted multireference coupled-cluster approaches of both state universal and valence universal types, as well as equation-of-motion coupled-cluster method. The general agreement is satisfactory except for the B2 states, where the difference between the multireference spin nonadapted and our state specific, but spin adapted, approaches amounts to as much as 0.64 eV. It is shown that this difference arises due to the spin contamination that is present in the spin-orbital based multireference approaches.

Study of localized molecular orbitals using group theory methods and its approach to the many‐electron correlation problem. IV. The symmetry‐adaptation of many‐center integrals and hamiltonian matrix elements in MCSCF calculations

AbstractGroup theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self‐consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many‐center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.

Unitary group approach to reduced density matrices

The structure of the algebra generated by one-electron charge and spin density operators with an appropriately defined convolution product is investigated in the context of the unitary group approach (UGA) to the many-electron correlation problem. The new idempotent density operators, defined via the unitary groupU(n) projection operators, are introduced and employed to investigate the behavior of charge and spin density operators under the convolution product.

A study of localized molecular orbitais by the group theory method and its approach to the many-electron correlation problem: Part 2. The point group symmetry and calculation program of spin-free Hamiltonian matrix elements

The use of group symmetric localized molecular orbitals as configuration generating orbitals of many-electron valence bond functions, namely bonded tableaux (BTs ), greatly simplifies the calculation of the Hamiltonian matrix elements. A practical algorithm and program for such calculations was developed, which includes the calculation of the overlap integral between the BTs and the action of the generators and of the generator products of the unitary group on the BT states, and the Hamiltonian matrix elements are expressed in terms of a linear combination of nonequivalent two-centre and four-centre integrals and geometric factors based on the Wigner-Eikart law.

A study of localized molecular orbitals by the group theory method and its approach to the many-electron correlation problem: Part 1. Group symmetric localized molecular orbitals

It is proved that any subspace invariant under the operators of a molecular symmetry group G must be related to a definite coset decomposition according to a subgroup of G. Using this correlation the group symmetric localized molecular orbitals (SLMOs) which can simultaneously satisfy the molecular symmetry, equivalence, transferability and orthogonality were obtained. It is appropriate that SLMOs constitute configuration generating orbitais (CGOs ) of N-electron functions by virtue of the simplification of the symmetry adaptation. A corresponding program for the solutions of SLMOs was constructed.