A direct relativistic four-component multiconfiguration self-consistent-field method for molecules

A new direct relativistic four-component Kramers-restricted multiconfiguration self-consistent-field (KR-MCSCF) code for molecules has been implemented. The program is based upon Kramers-paired spinors and a full implementation of the binary double groups (D(2h)(*) and subgroups). The underlying quaternion algebra for one-electron operators was extended to treat two-electron integrals and density matrices in an efficient and nonredundant way. The iterative procedure is direct with respect to both configurational and spinor variational parameters; this permits the use of large configuration expansions and many basis functions. The relativistic minimum-maximum principle is implemented in a second-order restricted-step optimization algorithm, which provides sharp and well-controlled convergence. This paper focuses on the necessary modifications of nonrelativistic MCSCF methodology to obtain a fully variational KR-MCSCF implementation. The general implementation also allows for the use of molecular integrals from a two-component relativistic Hamiltonian as, for example, the Douglas-Kroll-Hess variants. Several sample applications concern the determination of spectroscopic properties of heavy-element atoms and molecules, demonstrating the influence of spin-orbit coupling in MCSCF approaches to such systems and showing the potential of the new method.     

Convergence of the Rayleigh—Ritz method in self-consistent-field and multiconfiguration self-consistent-field calculations

We investigate the analytical convergence of SCF and MCSCF calculations, when the dimension of the subspaces to which the orbitals are restricted tends to infinity. We show that the completeness only inL ²(R ³;C ²) of the orbital bases does not ensure the convergence of the Ritz-energy, neither in SCF nor in MCSCF calculations, but that this convergence — as well as the convergence of the Ritz-orbitals in SCF calculations — is on the contrary guaranteed if the orbital bases are complete in the Sobolev spaceW ^1,2(R ³;C ²). Some consequences on the choice of the orbital exponents of Slater and Gauss functions are also discussed.     

Polarization Green's function with multiconfiguration self-consistent-field reference states

The polarization Green's-function formalism in the superoperator notation of Goscinski and Lukman is re-derived using a multiconfiguration self-consistent-field (MC -SCF ) reference state to establish the superoperator metric. The potential advantages of employing this more general reference state in Green's-function theories and certain inherent weaknesses associated with the traditional Hartree–Fock or Rayleigh–Schrödinger perturbation theory reference state choices are briefly discussed. The Hermiticity of the superoperators is analyzed within the framework of the MC –SCF reference state. Using a concept of order appropriate for this reference state choice, explicit formulas and computational procedures for the implementation of this Green's-function theory are presented and specialized to include terms consistent through second order.     

Multiconfiguration self-consistent-field theory based upon the fragment molecular orbital method

The fragment molecular orbital (FMO) method was combined with the multiconfiguration self-consistent-field (MCSCF) theory. One- and two-layer approaches were developed, the former involving all dimer MCSCF calculations and the latter limiting MCSCF calculations to a small part of the system. The accuracy of the two methods was tested using the six electrons in six orbitals complete active space type of MCSCF and singlet spin state for phenol+(H(2)O)(n), n=16,32,64 (6-31G( *) and 6-311G( *) basis sets); alpha helices and beta strands of phenylalanine-(alanine)(n), n=4,8,16 (6-31G( *)). Both double-zeta and triple-zeta quality basis sets with polarization were found to have very similar accuracy. The error in the correlation energy was at most 0.000 88 a.u., the error in the gradient of the correlation energy was at most 6.x10(-5) a.u./bohr and the error in the correlation correction to the dipole moment was at most 0.018 D. In addition, vertical singlet-triplet electron excitation energies were computed for phenol+(H(2)O)(n), (n=16,32,64), 6-31G( *), and the errors were found to be at most 0.02 eV. Approximately linear scaling was observed for the FMO-based MCSCF methods. As an example, an FMO-based MCSCF calculation with 1262 basis functions took 98 min on one 3.0 GHz Pentium4 node with 1 Gbyte RAM.     

Non-relativistic self-consistent-field theory. III

An exact self-consistent-field formalism, which obviates the need of evaluating interelectronic repulsion integrals, is developed. This formulation claims general applicability and can lead either to Hartree-Fock functions or to the exact solutions of a non-relativistic Hamiltonian, depending on the trial functions used.

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Zusammenfassung Es wird ein exakter seIf-consistent-field-Formalismus entwickelt, der die Berechnung von Elektronenwechselwirkungsintegralen unnötig macht. Diese Formulierung beansprucht allgemeine Anwendbarkeit und kann, je nach Art der verwendeten Funktionen, entweder zu Hartree-Fock-Funktionen oder zu exakten Lösungen eines nichtrelativistischen Hamiltonoperators führen.

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Résumé On a développé un formalisme exact de champ «gauto-cohérent», où il n'y en a pas besoin de calculer les intégrales concernant la répulsion interélectronique. Ce formalisme est complètement général et il peut Être employé pour déterminer bien des fonctions de Hartree-Fock ou les fonctions propres d'un hamiltonien non-relativistique, dépendant des caractéristiques des fonctions employées.

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This work has been supported in part by the National Research Council of Canada.     

Non-relativistic self-consistent-field theory. IV

The general, exact self-consistent-field formalism developed in III of this series has been applied to the determination of the Hartree-Fock wave function for the ground state of Li. Practical details of the calculations are given.

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Zusammenfassung Der allgemeine, exakte self-consistent-field-Formalismus, der in III dieser Serie entwickelt wurde, ist für die Bestimmung der Hartree-Fock-Funktion für den Grundzustand von Li verwendet worden. Praktische Einzelheiten dieser Berechnungen sind angeführt.

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Résumé Le formalisme exact de champ «gauto-cohérent», qui a été développé dans III de cette série, est employé pour déterminer la fonction de Hartree-Fock pour l'état fondamental de Li. On présente aussi les détails pratiques des calculs.

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This work has been supported in part by the National Research Council of Canada.     

Non-relativistic self-consistent-field theory. I

A reformulation of the general self-consistent-field formalism is presented.

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Zusammenfassung Es wird eine Umformulierung des allgemeinen self-consistent-field-Formalismus angegeben.

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Résumé Le formalisme de la méthode du champ «gauto-cohérent» est reformulé.

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This work has been supported in part by the National Research Council of Canada.     

Non-relativistic self-consistent-field theory. II

The condition to be satisfied by the orbitals which minimize the orbital energies is determined. This condition provides a link with the Thomas-Fermi approximation.

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Zusammenfassung Es wird eine Bedingung bestimmt, die von den Orbitalen, die die Orbital-Energien auf ein Minimum herabsetzen, erfüllt wird. Diese Bedingung bringt eine Verbindung zu der Thomas-Fermi-Methode.

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Résumé On a déterminé la condition qui doit Être satisfaite par les orbitales qui produisent un minimum pour les énergies des orbitales. Cette condition permet d'établir un rapport avec la théorie de Thomas-Fermi.

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This work has been supported in part by the National Research Council of Canada and has been presented in part at the 47th Annual Canadian Chemical Conference, Kingston, Ontario, Canada, June 1–3, 1964, and at the 2nd Gordon Research Conference on Theoretical Chemistry, New Hampton, N. H., U. S. A., June 29–July 3, 1964.     

Perturbation theory for multiconfiguration reference states

Vector method procedures are adapted to evaluate Rayleigh-Schrödinger perturbation corrections to a multiconfiguration zeroth order function. If this function is sufficiently flexible, this perturbation theory can be applied to low lying excited states. The effectiveness of our theory is demonstrated on the ground state of F₂ and the low lying excited states of Mg₂. Energies calculated through fourth order are compared with appropriate CI results.     

Spin component scaling in multiconfiguration perturbation theory

We investigate a term-by-term scaling of the second-order energy correction obtained by perturbation theory (PT) starting from a multiconfiguration wave function. The total second-order correction is decomposed into several terms, based on the level and the spin pattern of the excitations. To define individual terms, we extend the same spin/different spin categorization of spin component scaling in various ways. When needed, identification of the excitation level is facilitated by the pivot determinant underlying the multiconfiguration PT framework. Scaling factors are determined from the stationary condition of the total energy calculated up to order 3. The decomposition schemes are tested numerically on the example of bond dissociation profiles and energy differences. We conclude that Grimme's parameters determined for single-reference M?ller-Plesset theory may give a modest error reduction along the entire potential surface, if adopting a multireference based PT formulation. Scaling factors obtained from the stationary condition show relatively large variation with molecular geometry, at the same time they are more efficient in reducing the error when following a bond dissociation process.     

Pseudopotential method for the direct construction of localized orbitals by multiconfiguration self-consistent field theory

The orbital equations for the direct construction of localized fixed orbitals by multiconfiguration self-consistent field theory (MCSCF-FXO) are transformed without approximation into pseudopotential form by a two-step process. First the utilization of a particular family of localization is shown to separate the set of orbital equations into two sets of coupled equations, one describing “valence” orbitals and one describing “core” orbitals. In addition we obtain by appropriate choice of localization potential three different sets of MCSCF-FXO orbitals, namely: maximally screened, “one-center” and “intermediate” orbitals. In the second step the orbital equations are transformed into pseudopotential form and explicit non-local pseudopotentials yielding and core orbitals are obtained. Finally, several different physically motivated approximations to the exact pseudopotentials, and the frozen-core approximation are discussed.     

Evaluation of the Coulomb energy in relativistic self-consistent-field theory

 Computational schemes are presented with which to evaluate the electrostatic Coulomb energy in relativistic molecular electronic structure calculations using a basis of four-component Dirac spinor amplitudes. We demonstrate that algorithms may be constructed and implemented which differ only in minor details from those in common use in nonrelativistic quantum chemistry, and that the four-component formalism is neither as complicated nor as expensive as has been suggested recently in the literature. Spherically symmetrical atomic basis sets are presented which indicate that accurate representations of the Coulomb energy may be obtained using modest expansions of the electronic density in a scalar auxiliary basis set of spherical harmonic Gaussian-type functions. Received: 15 April 2002 / Accepted: 15 May 2002 / Published online: 29 July 2002     

Analytic techniques for the coupled multiconfiguration SCF perturbation theory

The perturbation theory based on the paired excitation multiconfiguration self-consistent field approach of Clementi and Veillard is considered. The coupled first-order perturbed orbital equations are analysed and an appropriate computational scheme for their solution is discussed. The proposed computational scheme is analogous to the technique employed for the solution of the coupled Hartree–Fock equations in the one-configuration approximation. However, because of the presence ofnondiagonal Lagrangian multipliers and the use of different one-electron operators for different orbitals, the present scheme raises some new computational problems. In this context a new technique for the solution of the unperturbed multiconfiguration self-consistent field equations is proposed. A simple illustration of the superiority of the multiconfiguration perturbation approach with respect to the ordinary coupled Hartree–Fock scheme is given. Also the validity of the variation formulation of the presented scheme and its relation to the finite-field approach are discussed.     

On a new self-consistent-field theory for the canonical ensemble

A significant amount of many-body problems of quantum or classical equilibrium statistical mechanics are conveniently treated at fixed temperature and system size. In this paper, we present a new functional integral approach for solving canonical ensemble problems over the entire coupling range, relying on the method of Gaussian equivalent representation of Efimov and Ganbold. We demonstrate its suitability and competitiveness for performing approximate calculations of thermodynamic and structural quantities on the example of a repulsive potential model, widely used in soft matter theory.     

Generalized brillouin theorem multiconfiguration method for excited states

The Generalized Brillouin Theorem Multiconfiguration Method (GBT-MC) of Grein and Chang is extended and applied to the calculation of excited states. Orthogonality constraints to lower states as well as second-order interaction effects of states lying close together have been taken into account. In this way quadratic convergence can be guaranteed. Difficulties with coupling coefficients and Lagrangian multipliers of SCF methods can be circumvented. Test calculations have been performed on valence electron excited states of C, H₂O, and CH₂O, and on core excited states of Li.     

The coupled-cluster method with a multiconfiguration reference state

The coupled-cluster approach to obtaining the bond-state wave functions of many-electron systems is extended, with a set of physically reasonable approximations, to admit a multiconfiguration reference state. This extension permits electronic structure calculations to be performed on correlated closed- or open-shell systems with potentially uniform precision for all molecular geometries. Explicit coupled cluster working equations are derived using a multiconfiguration reference state for the case in which the so-called cluster operator is approximated by its one- and two-particle components. The evaluation of the requisite matrix elements is facilitated by use of the unitary group generators which have recently received wide attention and use in the quantum chemistry community.     

Coupled multiconfiguration self-consistent field (MC SCF) perturbation theory

The analytical form of the perturbation theory for the MC SCF method of Veillard and Clementi is presented. The appropriate second-order energy functional which takes into account the self-consistency requirements, leads to a set of coupled first-order perturbed equations determining the perturbed configuration coefficients and orbitals. The second-order energy formula derived from this functional can be given a clear physical interpretation. The present analytical approach is compared with the finite perturbation MC SCF scheme. The possibility of the approximate solution of the coupled MC SCF perturbation equations is also discussed and the so-called uncoupled procedures are devised. In the limit of the single determinant wave function the present formulae are shown to be equivalent to the appropriate Hartree-Fock perturbation results. The differences between the one-configuration SCF and the MC SCF approach are illustrated by the calculation of the electric dipole polarizability of. HZ in the CNDO/2 approximation. It is shown that the one-configuration SCF approaches cannot account for the correct asymptotic properties of the second-order energy for large internuclear distances. This feature of the SCF perturbation theories does not depend on the specific approximations of the CNDO/2 scheme and is corrected by using the MC SCF perturbation theory.