Ab initio correlation corrections to the Hartree-Fock quasi band-structure of periodic systems employing Wannier-type orbitals

A size-consistent ab initio formalism to calculate correlation corrections to ionization potentials as well as electron affinities of periodic systems is presented. Our approach is based on a Hartree-Fock scheme which directly yields local orbitals without any a posteriori localization step. The use of local orbitals implies non-zero off-diagonal matrix elements of the Fock operator, which are treated as an additional perturbation and give rise to localization diagrams. Based on the obtained local orbitals, an effective Bloch Hamiltonian is constructed to second order of perturbation theory with all third-order localization diagrams included. In addition, the summation of certain classes of diagrams up to infinite order in the off-diagonal Fock elements as well as the Epstein-Nesbet partitioning of the full Hamiltonian are discussed. The problem of intruder states, frequently encountered in many-body perturbation theory, is dealt with by employing the theory of intermediate Hamiltonians. As model systems we have chosen cyclic periodic structures up to an oligoethylene ring in double-zeta basis; however, the theory presented here straightforwardly carries over to infinite periodic systems. Received: 30 April 1998 / Accepted: 27 July 1998 /  Published online: 7 October 1998     

Bond length alternation in cyclic polyenes. V. Local finite-order perturbation theory approach

The finite-order many-body perturbation theory using the localized Wannier orbital basis is applied to the problem of bond length alternation in the Pariser–Parr–Pople model of cyclic polyenes C_N H_N, N = 4v + 2, which may be regarded as a simplified model of polyacetylene. Both the Møller–Plesset and the Epstein–Nesbet-type partitionings of the model Hamiltonian are employed. The localized orbital basis enables an efficient truncation of the perturbation theory summations over the intermediate states as well as an elimination of energetically unimportant diagrams, thus enabling one to obtain the fourth-order Møller–Plesset-perturbation energies with a relatively small computational effort even for large polyenes. The results obtained with the second-, third-, and fourth-order Møller–Plesset and with the third-order Epstein–Nesbet perturbation theories yield very similar bond length distortions (about 0.05 Å) and stabilization energies per site (about 0.04 eV) as obtained earlier with the RHF , one-parameter AMO , and delocalized orbital perturbation theories. The effects of truncation and diagram elimination in the fourth-order Møller–Plesset perturbation theory and the abnormal behavior of the second-order Epstein–Nesbet perturbation theory results in the localized Wannier basis near the instability threshold of the RHF solutions are discussed.     

Unitary group approach to general system partitioning. II. U(n) matrix element evaluation in a composite basis

An explicit segment level formalism is derived for the matrix elements of the U(n) generators in an arbitrary (multishell) composite basis. The results of this paper, which contain the usual (spin-independent) unitary calculus approach as a limiting case, yield a more powerful and versatile algorithm than the traditional (spin-independent) unitary group formalism.     

New algorithm for high-order time-dependent hartree–fock theory for nonlinear optical properties

A new formalism and algorithm is developed for solving the general-order time-dependent Hartree–Fock (TDHF) problem. It is shown that for any order a generalization of the TDHF equations can be derived where all lower-order solutions constitute a constant term. This makes it very easy to obtain high-order solutions. As the space required for the mapping of density matrices to Fock matrices in a problem of a giver order is largely reduced, we can perform the most time-consuming steps within the core memory of the machine and easily manipulate vector products via optimum routines. The second hyperpolarizability γ is obtained from the secondorder TDHF solution via a 2n rule. The formalism also allows for expressing all terms in the equation diagrammatically, which provides additional physical insight and a more systematic evaluation of terms. To illustrate the method, TDHF results are presented for trans-butadiene and carbon monoxide for several optical processes, including correlation corrections to their static hyperpolarizabilities obtained via coupled cluster (CCSD) and many-body perturbation theory. The hybrid TDHF/CCSD method provides excellent agreement with the DC–SHG experiments (χ||⁽²⁾ = 11.4 x 10^?32 esu/mol compared to 12.9 ± 11.4 × 10^?32 esu/mol and χ||⁽³⁾ = 149 compared to 144 ± 4 × 10^?39 esu/mol).     

Perturbative calculation of intermolecular interactions in orthogonalized or biorthogonal basis sets

Two versions of a many-body perturbation theory for the computation of molecular interaction energies are investigated. The methods are based on the partitioning of the second-quantized form of the dimer Hamiltonian written either in the orthogonalized basis of the monomer MOs, or, alternatively, in the original non-orthogonal dimer basis set handling the overlap by the biorthogonal formalism. The zeroth-order Hamiltonian H ⁰ is the sum of effective monomer Fockians and the zeroth-order wave functions are exact eigenfunctions of H ⁰. Full antisymmetry is ensured by the second-quantized formalism. Basis set superposition error is accounted for by the counterpoise correction recipe. Results for He₂, (H₂)₂ and (H₂O)₂ indicate the reliability of the biorthogonal technique.     

Application of the unitary group approach to evaluate spin density for configuration interaction calculations in a basis of S2 eigenfunctions

We present an implementation of the spin‐dependent unitary group approach to calculate spin densities for configuration interaction calculations in a basis of spin symmetry‐adapted functions. Using S² eigenfunctions helps to reduce the size of configuration space and is beneficial in studies of the systems where selection of states of specific spin symmetry is crucial. To achieve this, we combine the method to calculate U(n) generator matrix elements developed by Downward and Robb (Theor. Chim. Acta 1977, 46, 129) with the approach of Battle and Gould to calculate U(2n) generator matrix elements (Chem. Phys. Lett. 1993, 201, 284). We also compare and contrast the spin density formulated in terms of the spin‐independent unitary generators arising from the group theory formalism and equivalent formulation of the spin density representation in terms of the one‐ and two‐electron charge densities.     

Electron correlation effects on magnetic properties of BH

The nuclear magnetic shielding and magnetizability tensors for the BH molecule are calculated by the coupled-Hartree–Fock method and many-body perturbation theory. As in the case of H₂, HF, and F₂, the second-order non-CHF diagrams make an inappreciable contribution.     

On construction of diatomic potential energy functions

The Clinton function α_n( R ) corresponding to the model potential energy function U_n( R ), is proposed as a test of applicability of U_n( R ). Such an analysis is given for some model potential energy functions.     

Localized orbitals for the description of molecular interaction

The intermolecular interaction between the molecules CH₂O and NH₃ was investigated by the supermolecule method. The interaction energies were first calculated at the ab initio SCF level, and the electron correlation was included via second-order Møller-Plesset perturbation theory (MP 2). The basis set superposition error (BSSE ) was taken into account by the counter-poise (CP ) method. The occupied and the virtual canonical molecular orbitals (CMOS ) of the supermolecule were separately localized by the Boys' procedure. The correlation correction was calculated by the many-body perturbation theory (MBPT ) in the localized representation. Contributions of the third- and fourth-order localized diagrams were added to those of the second-order canonical diagram. This procedure gives a correction nearly equivalent to that of MP 2. The possibility to separate LMO contributions responsible for the dispersion interaction was investigated.     

GW-BSE Calculations of Electronic Band Gap and Optical Spectrum of ZnFe2O4: Effect of Cation Distribution and Spin Configuration

The G0W0, evGW0, evGW, and scGW0 approximations to many-body perturbation theory combined with the Bethe-Salpeter approach (BSE) are applied to calculate electronic and optical properties of the open-shell spinel ferrite ZnFe₂O₄. The effect of the various degrees of self-consistency is assessed by comparison to recent experimental results. Furthermore, the influence of the method for obtaining the ground-state wavefunction is studied, including the GGA functional PBE with and without an intermediate step using the COHSEX approximation, as well as PBE+U, where we try to minimize the influence of the Hubbard potential U. Best agreement for the optical band gap and the first maxima of the excitation spectrum is obtained with the evGW method based on a PBE+U wavefunction. This method is chosen and converged carefully to yield quantitative results for the optical spectra of four different magnetic structures and cation distributions of ZnFe₂O₄. With the results we provide a possible explanation for inconsistency in experimental results.     

Restricted Hartree-Fock and unrestricted Hartree-Fock as reference states in many-body perturbation theory: a critical comparison of the two approaches

Summary The paper deals with two topics related to the problem which reference state is better for many-body perturbation theory: restricted Hartree-Fock (RHF) or unrestricted Hartree-Fock (UHF)? The first topic concerns the potential surfaces. Several examples are presented to show shortcomings of the two approaches and a simple way is presented which seems to give a useful potential curve in the whole range of interatomic distances by a composition of RHF and UHF potential curves. The second topic concerns the many-body perturbation theory for open-shell systems in the RHF formalism. The method is critically examined and compared with the ordinary many-body perturbation theory using UHF as the reference. This examination of many-body techniques provides also some insight into the problems inherent of the SCF theory: spin contamination from higher multiplets, localization of orbitals, and self-consistency effects.     

Orbital correspondence analysis in maximum symmetry: Formulation and conceptual framework

A recently proposed method for the analysis of the course of chemical reactions, based on the maximal use of available symmetry, is formulated as a set of procedural rules. The application of these rules is illustrated with a simple prototype reaction: CH₂+C₂H₄ fcyclo-C₃H₆. They are then derived, using the formalism of time-dependent perturbation theory within the Born-Oppenheimer approximation, thus bringing out the method's underlying assumptions and its relation to the widely used Woodward-Hoffmann procedure.     

Many-body perturbation theory for open-shell systems. Expansion through fourth-order

All of the diagrams which arise in the many-body perturbation theory of open-shell systems using a restricted Hartree-Fock reference function are given through fourth-order in the energy. New effects which arise in fourthorder are discussed.S.E.R.C. Advanced Fellow     

A many-body diagrammatic theory of rotation–vibration spectra

A many-body diagrammatic perturbation theory of rotation–vibration spectra is elaborated. The present approach is based on two many-body techniques, namely on the second quantization formalism (a rotating–vibrating molecule is formally treated here as a system of interacting vibrons, obeying the Bose–Einstein statistics) and the many-body diagrammatic theory of a model Hamiltonian, initially suggested in the microscopic theory of nuclei and in the last decade very frequently exploited in the accounting for the correlation effects in many electron systems. In the framework of this theory, the rotation–vibration energies are determined as the eigenvalues of a finite-dimensional model eigenproblem.     

The shifted scheme in the general-model-space diagrammatic perturbation theory

The shifted scheme of many-body perturbation theory is applied to open-shell states within the framework of the general-model-space theory. Rules for shifting the denominators of folded diagrams. which appear in open-shell perturbation expansions, are given. The finite-order energies in the shifted scheme obtained in two equivalent representations may differ. This happens, for instance, in the case of triplet states. For ³Σ_u⁺ states of the He₂, differences up to 0.07 mhartree have been found in third order. A similar phenomenon is the size inconsistency of the shifted scheme observed by Silver in the ground state of He₂. A possible advantage of the shifted scheme is its faster convergence for excited states.     

A unitary group formulation of many-body theory: The spin-shift formalism

This paper lays the algebraic foundation of a unitary group approach to many-body theory. We define a set of second quantized spin-shift operators which are used to construct spin-adapted many-electron configuration functions. We investigate the particle-hole transformation, normal ordering, and contraction of spin shifts. This gives an orbital Wick's theorem reflecting the permutational structure of the states spanning the irreducible representations of the spatial unitary group. We study the spinless Hamiltonian, and the relationship between spin shifts and unitary group generators.     

Representation theory of so(4,2) for the perturbation treatment of hydrogenic-type hamiltonians by algebraic methods

The representations of so(4,2) which are applicable to the perturbation treatment of one-electron Hamiltonians of the form H = H₀ + λV are discussed, where H₀ is a hydrogenic Hamiltonian. A unified construction of the representations of so(2,1) and so(3) is outlined and the representations of so(4) [and also so(3,1)] are then obtained using both the vector operator method and angular momentum recoupling techniques. The merging of so(2,1) and so(4) then leads in a natural way to so(4,2). An outline of perturbation theory applications such as the Stark and Zeeman effects is also given.     

Explicit representation of Gel'fand–Tsetlin states in clifford algebra unitary group approach

A explicit expression for the unitary group Clebsch–Gordan coefficients, which couple two fully antisymmetric single-column states into the two-column Gel'fand–Tsetlin states, is given in terms of isoscalar factors for the canonical subgroup chain U(n) ? U(n – 1) ? …? ? U(1). The isoscalar factors are expressed through the step numbers labeling canonical basis states and enable a straightforward construction of Gel'fand–Tsetlin states in the Clifford algebra unitary group approach, without the use of the tables for the symmetric group outer-product reduction coefficients.