Unrestricted Hartree Fock perturbation theory

The coupled and uncoupled Hartree Fock perturbation theories are generalised using unrestricted Hartree Fock theory. The variational approach to coupled theory is discussed and various corrections to the uncoupled theory are considered. Trial calculations are carried out to obtain atom-atom polarizabilities for Allyl and the first excited triplet state of cis-Butadiene, and to obtain estimates of spin and charge densities for various heterocyclics.

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Zusammenfassung Die gekoppelte and die ungekoppelte Hartree-Fock Störungstheorie werden mit Hilfe der uneingeschränkten Hartree-Fock-Theorie verallgemeinert. Das Variationsverfahren für die gekoppelte Theorie wird diskutiert und verschiedene Korrekturen für die ungekoppelte Theorie werden untersucht. Es werden Testrechnungen durchgeführt, um die Atom-Atom-Polarisierbarkeiten des Allylradikals und des ersten angeregten Tripletzustandes von cis-Butadien, und um die Spin- und Ladungsdichten verschiedener Heterozyklen abzuschätzen.

     

Multireference coupled cluster theory in Fock space

Summary A Fock space multireference coupled cluster method based on incomplete model spaces is described. Some of the essential computational aspects of the theory are discussed with the aid of the diagrammatic representation of the equations. An application to the calculation of ionization potentials and excitation energies ofs-tetrazine is presented along with comparisons with conventionalab initio calculations and experimental results.     

The Fock space coupled cluster method: theory and application

Summary The Fock space coupled cluster method and its application to atomic and molecular systems are described. The importance of conserving size extensivity is demonstrated by the electron affinities of the alkali atoms. Two types of intruder states are discussed, one attributable to the orbital energy spectrum and the other caused by two-electron interactions. They are illustrated by the excited states of Li₂ and by¹ S states of Be, respectively. It is shown how both problems may be solved using incomplete model spaces. The selection of the model space in a moderately dense spectrum is discussed in connection with N₂ excited states.Supported in part by the U.S.-Israel Binational Science Foundation     

Improved uncoupled Hartree–Fock (IUCHF) perturbation methods and bounds for the second-order energy in coupled Hartree–Fock perturbation theory

Different kinds of improved uncoupled Hartree–Fock methods are proposed for the calculation of second-order perturbation energies. Using these methods inequalities are derived for the error of the uncoupled procedure with geometric approximation.     

Fock space multireference coupled cluster theory: Study of shape resonance

The complex absorbing potential along with correlated independent particle potential (CIP) Fock space multireference coupled cluster method is used for the study of resonances. We have studied shape resonance of e^?‐ F₂, e^?‐ N₂O and e^?‐CO molecules. In particular, we have studied e^?‐ F₂ scattering at different bond lengths to know whether is bound at the equilibrium bond length of F₂. © 2013 Wiley Periodicals, Inc.     

The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

Application of second-order SCF perturbation theory to the solution of time-dependent Hartree—Fock equations

The self-consistent field based theory of frequency-independent hyperpolarizabilities is adapted to the treatment of the corresponding frequency-dependent quantities. Explicit expressions are given for the double and null frequency polarizations of a molecule in a long-established oscillating electric field.     

On the single-root approach within the framework of coupled-cluster theory in Fock space

A thorough formulation of Fock Space Brillouin–Wigner Coupled Cluster method is presented following previous developments [N.D.K. Petraco, Ľ. Horný, H.F. Schaefer, I. Hubač, J. Chem. Phys. 117 (2002) 9580]. The new method is designed to avoid the intruder states problem, and introduces the single-root solution feature which has not been considered yet within valence-universal methods. The explicit equations for the (0,1) sector of the Fock space are introduced.     

Electronic Fock space as associative superalgebra

A new algebraic structure on a finite dimensional electronic Fock space is studied in detail. This structure is defined in terms of a certain multiplication of many electron wave functions and has close interrelation with coupled cluster and similar approaches. Its study simplifies the mathematical backgrounds of these approaches. Even more, it leads to relations that would be very difficult to derive using conventional technique. Formulas for action of the creation‐annihilation operators on products of state vectors are derived. Explicit expressions for action of simplest particle‐conserving products of the creation‐annihilation operators on powers of state vectors are given. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Fock space multireference coupled cluster theory: noniterative inclusion of triples for excitation energies

In this paper we propose and numerically implement a specific scheme for calculating the excitation energies (EEs) within the Fock space multireference coupled cluster framework, which includes the contributions from noniterative triples cluster amplitudes. These contribute to the EEs at the third order. We present results for CH⁺ and N2, and study the effects of these noniterative triples on EEs. Received: 28 July 1997 / Accepted: 8 December 1997     

Influence of molecular geometry on valence space for quasidegenerate many-body perturbation theory

The Be-H₂ insertion reaction is used as a model to study the application of quasidegenerate many-body perturbation theory (QDMBPT) to polyatomic molecules where the pattern of quasidegenerate orbitals varies greatly with geometry. Full active valence space QDMBPT calculations are compared with the exact solutions within the basis and with previous QDMBPT computations that retain only a pair of quasidegenerate valence orbitals.     

Fock transforms in reciprocal-space quantum theory

In Fock's reciprocal-space treatment of the hydrogen atom,k-space is mapped onto the surface of a 4-dimensional hypersphere, and the solutions (apart from an invariant factor) are 4-dimensional hyperspherical harmonics. Fock's method can be generalized to provide solutions for the Schrödinger equation of a charged particle moving in a many-center Coulomb potential, and in this case the solutions are found by diagonalizing an overlap matrix involving products of hyperspherical harmonics. The present paper discusses a transform which can conveniently be used to evaluate the elements of the overlap matrix.     

Stationary perturbation theory

Summary After a short recapitulation of the basic concepts of stationary perturbation theory, this is applied to a many-electron Hamiltonian, with or without an external field, given in a Fock space formulation in terms of a finite basis, the exact eigenfunctions of which are the full-CI wave functions. The Lie algebra _c ⁿ of the variational group corresponding to this problem is presented. It has an important subalgebra _c ⁽¹⁾ of one-particle transformations. Hartree-Fock and coupled Hartree-Fock (also uncoupled Hartree-Fock) as well as MC-SCF and coupled MC-SCF are outlined in this framework. Many-body perturbation theory and Møller-Plesset perturbation theory are derived from the same kind of stationarity condition and a new non-perturbative iteration construction of the full-CI wave function is proposed, the first Newton-Raphson iteration cycle of which is CEPA-0. For the treatment of electron correlation for properties two variants of Møller-Plesset theory referred to as coupled (CMP) and uncoupled (UCMP) are defined, neither of which is fully satisfactory. While CMP satisfies a Brillouin condition, which implies that first order correlation corrections to first- and second-order properties vanish, it does not satisfy a Hellmann-Feynman theorem, i.e. a first order property isnot the expectation value of the operator associated with the property. Conversely UCMP satisfies a Hellmann-Feynman theorem but no Brillouin theorem. The incompatibility of the two theorems is related to an unbalanced treatment of one-particle- and higher excitations in MP theory. CMP, which is based on coupled Hartree-Fock as uncorrelated reference, appears to have slight advantages over UCMP, but neither variant looks very promising for the evaluation of 2nd order correlation corrections to 2nd-order properties. Then four variants of the perturbation theory of properties with a nonperturbative treatment of electron correlation on CEPA-0 level (but extendable to a higher level) are discussed. While those variants which are the direct counterpart of UCMP and CMP must be discarded, the perturbative CEPA-0 derived from a perturbative treatment on full-CI level appears to satisfy all important criteria, in particular it satisfies a Brillouin-Brueckner condition and a Hellmann-Feynman theorem. A simplified version, the coupled Brillouin-Brueckner CEPA-0 appears to have essentially the same qualities. It is important to replace the Brillouin condition of MP theory by the Brillouin-Brueckner condition in non-perturbative approaches, especially if one is interested in properties.     

Large-Order perturbation theory

The original motivation for studying the asymptotic behavior of the coefficients of perturbation series came from quantum field theory. An overview is given of some of the attempts to understand quantum field theory beyond finite-order perturbation series. At least in the case of the Thirring model and probably in general, the full content of a relativistic quantum field theory cannot be recovered from its perturbation series. This difficulty, however, does not occur in quantum mechanics, and the anharmonic oscillator is used to illustrate the methods used in large-order perturbation theory. Two completely different methods are discussed, the first one using the WKB approximation, and a second one involving the statistical analysis of Feynman diagrams. The first one is well developed and gives detailed information about the desired asymptotic behavior, while the second one is still in its infancy and gives instead information about the distribution of vertices of the Feynman diagrams.     

Quasi-degenerate second-order perturbation theory for occupation restricted multiple active space self-consistent field reference functions

A multi-configuration quasi-degenerate second-order perturbation method based on the occupation restricted multiple active space (ORMAS-PT/ORMAS) reference wavefunction is presented. ORMAS gives one the ability to approximate a complete active space self-consistent field (CASSCF) wavefunction using only a subset of the configurations from the CASSCF space. The essential idea behind ORMAS-PT is to use the multi-reference M?ller-Plesset formalism to correct the ORMAS reference energy. A computational scheme employing direct CI methodology is presented. Several tests are presented to demonstrate the performance of the ORMAS-PT method.     

On perturbation graph theory

A perturbation theoretical formalism is developed which enables the calculation of the (topological) resonance energy of arbitrary heteroconjugated π-electron systems. The previous method of Herndon and Párkányi is thus generalized.     

Auxiliary density perturbation theory

A new approach, named auxiliary density perturbation theory, for the calculation of second energy derivatives is presented. It is based on auxiliary density functional theory in which the Coulomb and exchange-correlation potentials are expressed by auxiliary function densities. Different to conventional coupled perturbed Kohn-Sham equations the perturbed density matrix is obtained noniteratively by solving an inhomogeneous equation system with the dimension of the auxiliary function set used to expand the auxiliary function density. A prototype implementation for the analytic calculation of molecular polarizabilities is presented. It is shown that the polarizabilities obtained with the newly developed auxiliary density perturbation approach match quantitative with the ones from standard density functional theory if augmented auxiliary function sets are used. The computational advantages of auxiliary density perturbation theory are discussed, too.