An extended PCILO method

The perturbative configuration interaction using localized orbitals (the PCILO method) was extended in the way that current limitations to the two-centre bond approach were overcome. The localized molecular orbitals contain an arbitrary number of the basis set components; this follows from the a priori stated localized bonding model of a molecule. The extended PCILO method was formulated for the CNDO, INDO and NDDO Hamiltonian approximations. The configuration interaction was performed using the Rayleigh-Schrödinger many-body perturbation theory with the Møller-Plesset type of Hamiltonian partitioning, similar to that used in the so-called modified PCILO method. Applications to molecules with semi-localized and/or semi-delocalized bonds, as benzene or diborane, are presented.     

A CSF-based multireference coupled pair approximation

Summary MRCPA (multireference coupled pair approximation) is formulated by the use of the wave operator formalism and Rayleigh-Schrödiner perturbation theory with special selection of the unperturbed part of the electronic Hamiltonian. By considering super molecule, it is shown that the theory is size consistent with the help of the new formalism. The method has been tested for three simple systems, H₂O, FH, and O₂.     

Charge vs. spin density waves in the fullerene polymer

Using an effective model Hamiltonian, we performed bond-length-optimized RHF and UHF band structure calculations to describe the electronic structure of the fullerene polymer (C₆₀)_x. For the doped system, the RHF calculations predict a gap on the 100 meV scale, in good agreement with experimental data, and show a special kind of charge-density-wave solution. At certain geometries, when the intercage conjugation is hindered, the RHF calculations are triplet instable: A carefully chosen initial guess to UHF leads to a small-amplitude spin-density-wave solution. © 1997 John Wiley & Sons, Inc.     

Communication: ROHF theory made simple

Restricted open-shell Hartree-Fock (ROHF) theory is formulated as a projected self-consistent unrestricted HF (UHF) model by mathematically constraining spin density eigenvalues. This constrained UHF (CUHF) wave function is identical to that obtained from Roothaan's effective Fock operator. The α and β CUHF Fock operators are parameter-free and have eigenvalues (orbital energies) that are physically meaningful as in UHF, except for eliminating spin contamination. This new way of solving ROHF leads to orbitals that turn out to be identical to semicanonical orbitals. The present approach removes ambiguities in ROHF orbital energies.     

Classifying variational functions for quasi-one-dimensional systems and incorporating electron correlation

A classification is proposed for the variation functions in the variable localized geminals (VLG) approximation based on the types of variational function in the unrestricted Hartree-Fock (UHF) method for one-dimensional systems. It has been found that functions of VLG type provide better incorporation of electron-correlation effects than do UHF ones. A study has been made on the phase diagram for the electronic states of polyacetylene in relation to the electron-electron parameters in the extrended Hubbard Hamiltonian containing certain types of VLG function.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 23, No. 6, pp. 665–672, November–December, 1987.We are indebted to V. Ya. Krivnov and A. Yu. Kon for a valuable discussion and to V. M. Pak for providing the optimization program.     

Peierls- and Mott-type instabilities in one-dimensional chains—coexistence or contradiction

Two possible effects—Peierls instability (bond alternation) and Mott-type correlation spin ordering—leading to metal–insulator transition (energy gap formation) in one–dimensional chains, have been considered using UHF treatment of the Hubbard-type Hamiltonian. It has been found that in the framework of approximations applied these two instabilities are mutually exclusive.     

Development of spin‐dependent relativistic open‐shell Hartree–Fock theory with time‐reversal symmetry (I): The unrestricted approach

An open‐shell Hartree–Fock (HF) theory for spin‐dependent, two‐component relativistic calculations, termed the Kramers‐unrestricted HF (KUHF) method, is developed. The present KUHF method, which is formulated as a relativistic counterpart of nonrelativistic UHF, is based on quaternion algebra and partly uses time‐reversal symmetry. The fundamental characteristics of KUHF are discussed in this study. From numerical assessments, it was revealed that KUHF gives a corresponding solution to nonrelativistic UHF; furthermore, KUHF properly describes spin‐orbit interactions. In addition, KUHF can improve the self‐consistent field convergence behavior in spin‐dependent calculations, for example, for f‐block elements.     

Electronic structure of long polyene chains with an impurity atom

The influence of an impurity atom on the -electronic structure of long polyenes is considered using the unrestricted Hartree-Fock (UHF) method. It is shown that the substitution of a carbon atom in a long polyene chain is a local perturbation in spite of the nonlinearity of the UHF Hamiltonian. The conditions under which the local states appear in the forbidden zone of long polyenes are stated. Some experiments are proposed to elucidate the nature of the forbidden zone in the -electron spectra of long polyene chains.

---

Zusammenfassung Der Einfluß eines Fremdatoms auf die -Elektronenstruktur langer Polyene wird mit Hilfe der uneingeschränkten Hartree-Fock-Methode (UHF) untersucht. Es wird gezeigt, daß die Substitution eines C-Atoms in einer langen Polyenkette eine lokale Störung darstellt, obwohl der UHF-Hamiltonoperator nicht linear ist. Die Bedingungen, unter denen die lokalen Zustände in der verbotenen Zone für die langen Polyene erscheinen, werden dargestellt. Einige Experimente zur Aufklärung der Natur der verbotenen Zonen in den -Elektronen-Spektren langer Polyenketten werden vorgeschlagen.

---

Résumé La méthode Hartree-Fock sans restrictions (UHF) est utilisée pour étudier l'influence d'une impureté atomique sur la structure électronique des grands polyénes. On montre que la substitution d'un atome de carbone dans une longue chaîne polyénique est une perturbation locale en dépit de la non linéarité de l'hamiltonien UHF. On énonce les conditions d'apparition des états locaux dans la zone interdite des longs polyènes. Certaines expériences sont proposées pour élucider la nature de la zone interdite dans le spectre d'électrons des longues chaînes polyéniques.

     

Spin alignment in organic high-spin molecules as studied by ESR and the generalized Hubbard Hamiltonian approach

In order to examine the role of topological symmetry in spin alignment of organic high-spin molecules, a novel hydrocarbon, 3,4′-diphenylmethylenebis(phenylmethylene) (3,4′-DPBPM), has beeen synthesized and characterized by single-crystal ESR. This molecule has a triplet ground state with a closely located quintet excited state in contrast to its topological isomer, 3,3′-DPBPM, with a non-degenerate septet ground state. UHF SCF calculation of the generalized Hubbard Hamiltonian satisfactorily predicts these results.     

Bond length alternation in cyclic polyenes. III. Alternant molecular orbital method

The bond length alternation problem in cyclic polyene models as described by the Pariser–Parr–Pople Hamiltonian and an empirical quasiharmonic π-core potential is investigated using the one-parameter alternant molecular orbital (AMO) method. It is shown that in contrast to the unrestricted Hartree–Fock (UHF) results, which lead to symmetric equidistant structures, the one-parameter AMO results yield bond length alternating structures similar to those obtained with the restricted HF approach. The correlation energy recovered by the AMO method is examined for the symmetric polyenes in the whole range of coupling constants for both the Pariser–Parr–Pople and Hubbard Hamiltonians and compared with exact full configuration interaction (FCI) results. For the first member of the cyclic polyene series we also compared the FCI and AMO correlation energies for different nuclear framework distortions. This comparison indicates that in contrast to the UHF results the fraction of the correlation energy recovered by the AMO approach is very uniform over the range of nuclear distortions considered. The AMO results thus strongly indicate the dimerization in the polyenic chains.     

Symplectic Methods of Fifth Order for the Numerical Solution of the Radial Shrödinger Equation

In this paper the numerical solution of the radial Shrödinger equation via new proposed symplectic-schemes is investigated. In particular, the radial Schödinger equation is transformed into Hamiltonian canonical form and is solved via symplectic integrators. Based on this approach, fifth-order methods are proposed. We compare these methods with well-known existing symplectic methods. The numerical results show the efficiency of the proposed method.     

Møller–Plesset theory for atomic ground state energies

Mø–Plesset theory, in which electron correlation energy is calculated by perturbation techniques, is used in second order to calculate energies of the ground states of atoms up to neon. The unrestricted Hartree–Fock (UHF) Hamiltonian is used as the unperturbed system and the technique is then described as unrestricted Mø–Plesset to second order (UMP2). Use of large Gaussian basis sets suggests that the limiting UMP2 energies with a complete basis of s, p, and d functions account for 75–84% of the correlation energy. Preliminary estimates of the contributions of basis functions with higher angular quantum numbers indicate that full UMP2 limits give even more accurate total energies.     

The electronic structure of the NF2 radical

Ab initio unrestricted Hartree-Fock (UHF) calculations (with and without single annihilation) have been performed on the radical NF₂ using four different basis sets; namely, a minimal basis Slater set, two minimal basis Gaussian sets and a Gaussian set of approximately double zeta accuracy.Several one-electron charge dependent properties have been calculated with each basis set and it is most apparent that near double zeta accuracy 2p functions are necessary to produce reliable values.Single annihilation of the UHF wavefunctions calculated with the two basis sets containing near double zeta 2p functions, was found to give an accurate representation of the anisotropic coupling constants at both the nitrogen and fluorine atoms. Less satisfactory agreement with the experimental isotropic coupling constants was found with all calculations.This investigation of the NF₂ radical indicates that, providing a good quality basis set is used, the single annihilated UHF method can provide accurate values for most charge and spin dependent observable properties of open shell molecules.

---

Zusammenfassung Ab intio-Rechnungen nach der uneingeschränkten Hartree-Fock-Methode (UHF-Methode) (mit und ohne einfache Auslöschung) wurden für das Radikal NF₂ mit verschiedenen Basissätzen durchgeführt. Die vier Basissätze sind: ein minimaler Basissatz von Slaterfunktionen, zwei minimale Basissätze von Gaußfunktionen und ein größerer Satz von Gaußfunktionen von angenäherter Genauigkeit einer Doppelzeta-Basis. Der Vergleich der Basissätze bei der Berechnung von ladungsabhängigen Einelektron-Eigenschaften zeigt, daß der angenäherte Doppelzeta-Basissatz zur Berechnung geeigneter Werte notwendig ist. Eine einfache Auslöschung in den UHF-Wellenfunktionen, die mit den beiden Basissätzen mit genäherten Doppelzeta-2p-Funktionen berechnet wurden, führte auf eine genaue Darstellung der anisotropen Kopplungskonstanten am Stickstoff und an Fluor. Die Übereinstimmung der berechneten isotropen Kopplungskonstanten war bei allen Rechnungen weniger gut. Diese Ergebnisse für das NF₂-Radikal zeigen, daß bei Verwendung einer geeigneten Basis mit Hilfe der UHF-Methode mit einfacher Auslöschung genaue Werte für die meisten von der Ladung oder vom Spin abhängigen Eigenschaften von Molekülen mit offenen Schalen gewonnen werden können.

     

Symplectic Methods for the Numerical Solution of the Radial Shrödinger Equation

In this paper new symplectic-schemes for the numerical solution of the radial Shrödinger equation are proposed. In particular, symplectic integrators for Hamiltonian systems have been developed. Based on this approach, second- and third-order methods are proposed. These methods are more accurate than the existing ones. We compare these methods not only with the existing symplectic methods, but also with a classical Runge–Kutta–Nyström method.     

Quasi-bound states of coupled Morse oscillators

Quasi-bound (resonance) states are present in the continuous spectrum of the Hamiltonian of two coupled Morse oscillators. Two different methods for approximating these as localized states are compared. The algebraic approach is shown to be in very good accord with the other method which is formulated in coordinate space and hence is differential in character. For these highly excited states an intermultiplet mixing term is included in the algebraic Hamiltonian.     

A numerical basis for the accurate representation of the continuum spectrum of atomic Hamiltonians

We present a method for the accurate calculation of the complete spectrum of the Schrödinger equation in terms of B-splines polynomial basis. The method is capable to represent numerically the bound and continuum spectrum of complex atomic systems. The theoretical method is discussed, and an application to hydrogenic Hamiltonian is given.AMS subject classification: 65705, 34L40     

Collinear collision of an atom with a homonuclear diatomic molecule

The problem of collinear scattering of an atom from a homonuclear diatomic molecule is formulated in terms of a first-order nonlinear matrix differential equation for the variable coefficient of reflection. For a homonuclear molecule when the target Hamiltonian is invariant under the parity transformation, only transitions between even states or odd states are possible. This selection rule reduces the number of open or closed channels that contribute to the reflection and transmission coefficients. But for numerical calculation, under the conditions of the problem, one can approximate the target Hamiltonian by the Hamiltonian of a displaced harmonic oscillator. In this approximation, the reflectional symmetry of the Hamiltonian is not preserved and transitions between any two levels of the target are possible. To simplify the problem further, the interaction between the projectile and the target is assumed to be a sum of two Gaussian terms. For this combination of the potentials the many-channel interaction can be expressed analytically. By fitting the Lennard–Jones potential with a sum of two Gaussian potentials and solving the matrix differential equation, transition probabilities are obtained for the He? H₂ collision. The numerical results are compared with the results found by Secrest and Johnson, and by Clark and Dickinson.     

Diagrammatic formulation of the second‐order many‐body multipartitioning perturbation theory

The second‐order multireference perturbation theory employing multiple partitioning of the many‐electron Hamiltonian into a zero‐order part and a perturbation is formulated in terms of many‐body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two‐particle terms in zero‐order operators. The novel formulation naturally gives rise to a “sum‐over‐orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one‐electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999     

Site-directed deep electronic tunneling through a molecular network

Electronic tunneling in a complex molecular network of N(>2) donor/acceptor sites, connected by molecular bridges, is analyzed. The "deep" tunneling dynamics is formulated using a recursive perturbation expansion, yielding a McConnell-type reduced N-level model Hamiltonian. Applications to models of molecular junctions demonstrate that the donor-bridge contact parameters can be tuned in order to control the tunneling dynamics and particularly to direct the tunneling pathway to either one of the various acceptors.     

Expressing kinetic‐energy for vibration–rotation motions in general rotating systems of axes and introducing quasirectilinear vibrational coordinates to simplify Hamiltonian forms

To understand how the internal and rotational motions of a polyatomic system depend on which rotating system of axes is selected, we derived the explicit form of the atomic velocities determined by an observer stationed on the general rotating system of axes. Using the derived velocities, we formulated the kinetic energy expression for vibration–rotation motions with respect to the rotating system of axes. From this expression, we clarified covariant metric tensors under zero angular momentum, which have been confused with an erroneous expression even in the professional literature, and the relationship between the kinetic energy expression and the rotating system of axes. Furthermore, to simplify the Hamiltonian form, we introduced quasirectilinear vibrational coordinates to describe the Hamiltonian. The resulting Hamiltonian form is superior to those of the previous studies in that the kinetic and potential energy expressions are simple and the vibrational frequencies are independent of the original internal coordinates used. In fact, we show that its application for three examples is useful. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 22–29, 2001