Abstract: A 'generalised' Coulson-Fischer wave function, in which the orbitals are approximated by means of a linear expansion in distributed Gaussian s-type basis functions, is used to determine a complete potential energy curve for the ground state of the hydrogen molecule by ab initio methods. A distributed basis set is developed in which both the positions and the exponents of Gaussian s-type functions are determined by invoking the variation principle. This basis set supports a sub-hartree level of accuracy and provides a reference with respect to which calculations can be assessed, where the exponents are generated using an even-tempered prescription and an anharmonic model is employed to distribute the basis functions along the internuclear axis. This basis set also supports energy expectation values of a sub-hartree level of accuracy.
Abstract: The structure and stability of the helio hydrogen cyanide molecular ion, HeCNH2+, is investigated by standard quantum chemical methods. Single reference calculations are carried out using second-order perturbation theory (MP2), the coupled cluster expansion in the CCSD approximation, and the hybrid approach using a perturbative estimate of the triple excitation energy component designated CCSD(T). Multireference calculations using a complete active space (CASSCF) and a second-order perturbation theory estimate of correlation effects (CASPT2) are reported.
Abstract: Within the Hartree-Fock approximation, distributed basis sets of s-type Gaussian functions are used to compute those portions of the ground-state potential energy curves of the H-2, LiH and BH molecules for which the model is appropriate. The exponents are generated using the even-tempered prescription and an anharmonic model is employed to distribute the basis functions. For few-electron diatomic systems, this approach is known to support sub-mu Hartree accuracy for electronic energies. For the molecules studied in the present work, matrix Hartree-Fock energy expectation values are compared with finite difference Hartree-Fock calculations carried out at the same geometries and using grids designed to support an accuracy of at least 1 nanoHartree. The distributed basis sets developed in this work support matrix Hartree-Fock energies which differ from the corresponding finite difference energies by 0.41, 0.58 and 0.68 mu Hartree, respectively, for the ground states of the H2, LiH and BH molecules at their equilibrium geometries. For each of these systems, a sub-mu Hartree level of accuracy is supported for a range of geometries.
Abstract: The multireference, state specific, second-order, Brillouin-Wigner perturbation theory is applied to the autoaromatisation of hex-3-ene-1,5-diyne, the Bergman reaction. Calculations are reported for the reactant (hex-3-ene-1,5-diyne), the transition state and the product (1,4-didehydrobenzene). A posteriori modifications are made which, in the case of a single reference function, recover the well-known formula of second-order many-body perturbation theory, i.e. Mller-Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state-specfic multireference Rayleigh-Schrdinger-like perturbation theory. Calculations are performed for a sequence of correlation consistent basis sets and, by extrapolation, complete basis set limits of the energetics of the Bergman reaction are estimated.
Abstract: The most widely used sequence of approximations in describing the structure and properties of molecules is first to assume the Born-Oppenheimer approximation to separate the electronic and the nuclear motions, then to decouple the electronic problem by assuming an independent particle model, and then to correct the mean field description by taking account of the instantaneous interactions of the electrons. (Nuclear degrees of freedom are often treated classically.) Many-body perturbation theory (MBPT) in its second-order form (designated MP2) remains the most widely used practical technique for describing the effects of electron correlation. MBPT is also invaluable in understanding the relation between different approaches to the electron correlation problem such as configuration interaction and various cluster expansions. Recent years have seen a growing interest in the simultaneous description of electronic and nuclear motion. The nonadiabatic coupling between the electronic and nuclear motion manifest itself in numerous and rather diverse phenomena. An independent particle model can be formulated in which the averaged interactions between the electrons, between the electrons and the nuclei and between the nuclei are described quantum mechanically. Multicomponent MBPT can then be used to formulate the corresponding correlation problem accounting for electron-electron interactions, electron-nucleus interactions and nucleus-nucleus interactions in either algebraic or diagrammatic terms. The practical realization of multicomponent MBPT rests on the development of efficient algorithms and the associated computer code. In recent work, we have advocated the use literate programming techniques in the development and publication of computer code for molecular structure calculations. We briefly discuss the application of these methods to the multicomponent many-body perturbation expansion.
Abstract: The application of literate programming methods in many-body perturbation theory is illustrated by considering the computation of the third-order "ring" diagram in the correlation energy expansion for a closed-shell, singlet system. An a posteriori application of literate programming techniques to a major component of the first published computer program for performing many-body perturbation theory calculations within the algebraic approximation is given.
Abstract: We investigate the accuracy with which the electric dipole polarizability, alpha(zz), and the hyperpolarizability, beta(zzz), can be calculated by using the algebraic approximation, i.e. finite basis set expansions, and by means of the finite difference method in calculations for the ground states of the 14 electron systems N-2, CO and BF within the Hartree-Fock model at their respective experimental equilibrium geometries. For a well-chosen grid, the finite difference technique can provide Hartree-Fock energy and dipole moment expectation values approaching machine precision which can be used to assess the accuracy of corresponding calculations carried out within the algebraic approximation. The finite field approximation is used to determine polarizabilities and hyperpolarizabilities from finite difference Hartree-Fock dipole moment expectation values. The results are compared with finite basis set calculations of the corresponding quantities which are carried out analytically using coupled perturbed Hartree-Fock theory. For the N-2 molecule, the Hartree-Fock polarizability is found to be 14.9512 au within the finite basis set approximation and 14.945 au within the finite difference approach. For the CO molecule, the corresponding results are 14.4668 au and 14.4668 au, whilst for the BF molecule the values are 16.6450 au and 16.6450 au, respectively. The Hartree-Fock hyperpolarizability of the CO molecule is found to be 31.4081 au and 31.411 au within the finite basis set and finite difference approximations, respectively. The corresponding hyperpolarizability values for the BF molecule are 63.9687 au and 63.969 au, respectively.
Abstract: An anharmonic model for systematically distributing Gaussian basis sets is presented. Illustrative calculations demonstrate the new approach for the H-2(+), HeH2+ and H-2 diatomic systems. The total ground state energies for the one-electron systems and the ground state Hartree-Fock energy for the H-2 molecule supported by distributed basis sets constructed by means of the anharmonic model are compared with the corresponding energies obtained by invoking the variation principle to determine optimal exponents and positions. The use of a series of anharmonic distributions of s-type Gaussian functions centred on regularly spaced points is also investigated. Calculated energy expectation values supported by our largest distributed basis sets differ from the corresponding exact values by 0.70 mu Hartree for the H-2(+) ion and by 0.89 mu Hartree for the HeH2+ ion. For the ground state Hartree-Fock energy of the H-2 molecule, our distributed basis set yields a value which differs from the finite difference Hartree-Fock value by 0.67 mu Hartree. (c) 2007 Wiley Periodicals, Inc.
Abstract: The multireference, state-specific, second-order, Brillouin-Wigner perturbation theory is presented as a robust approach to the electron correlation problem for systems demanding the use of a multireference function. A posteriori modifications are made which, in the case of a single reference function, recover the well known formula of second-order many-body perturbation theory (MBPT2), i.e. Moller-Plesset (MP2) theory, and in the multireference case can be shown to be equivalent to state-specific multireference Rayleigh-Schrodinger-like perturbation theory. It is shown that multireference many-body Brillouin-Wigner perturbation theory when restricted to second-order is a true many-body theory, that is, it is a theory, which has the property of extensivity. (c) 2007 Wiley Periodicals, Inc.
Abstract: The multireference, state specific, second-order, Brillouin-Wigner perturbation theory is presented. A posteriori corrections are made which in the case of a single reference function recover the well-known formula of second-order many-body perturbation theory, i.e. Moller-Plesset 'MP2' theory, and in the multireference case yields a state specific multireference second-order Rayleigh-Schrodinger perturbation theory. Applications to the bond breaking processes in the ground states of the BH and FH molecules are described using basis sets for which the corresponding full configuration interaction calculations are possible. Multireference Brillouin-Wigner coupled cluster calculations, in the CCSD approximation, are presented for these diatomic molecules using the same basis sets.
Abstract: A practical Hartree-Fock theory of atomic and molecular electronic structure is developed for individual electronically excited states that does not involve off-diagonal Lagrange multipliers. An. easily implemented method for taking the orthogonality constraints into account, which has been proposed earlier by one of us, is used to impose the orthogonality of the Hartree-Fock excited state wave function of interest to states of lower energy. The applicability of systematic sequence of even-tempered basis sets with the orbital exponents, zeta(p), defined by the geometric series zeta(p) = alpha beta(p) is examined in Hartree-Fock energy calculations for excited states which have the same spatial and spin symmetry as the ground state. It is shown that a simple reoptimization of the alpha and beta parameters leads to a sequence of even-tempered basis sets capable of supporting high accuracy for excited state energies of some simple atoms.
Abstract: In this paper, we advocate the use of literate programming techniques in molecular physics and quantum chemistry. With a suitable choice of publication medium, literate programming allows both a theory and corresponding computer code to be placed in the public domain and subject to the usual "open criticism and constructive use" which form an essential ingredient of the scientific method. The use of literate programming methods leads naturally to structure and standardization in computer code. In turn, this structure leads to subroutine libraries and we describe the specification of a basic tensor algebra subroutine library, which we have recently developed, and which we expect to prove useful in a range of applications. We briefly consider the use of literate programming techniques in enhancing collaborative virtual environments, which facilitate developed cooperation between geographically distributed sites.
Abstract: Literate programming methods have not been widely used in computational molecular structure theory. We argue that significant advantages would result from the use of literate programming methods in the computational molecular sciences and, indeed, in computational science, in general. Our arguments are illustrated by a simple example of literate programming methods in ab initio electronic structure theory is described. We distinguish text-embedded code, or literate programs, from code-embedded text and suggest that the latter may form a more useful vehicle for publication than literate programs. (c) 2005 Wiley Periodicals, Inc.
Abstract: Literate programming has not so far found widespread application in quantum chemistry. Here we suggest that literate programming would do much to enhance the communication of the methods and algorithms of computational quantum chemistry. We argue that literate programming can foster a collaborative approach to the development of theory and code in quantum chemistry. We consider a collaborative approach to computational quantum chemistry via a collaborative virtual environment involving literate programming methods and contrast this with the more traditional approaches, such as the UK's Collaborative Computational Project 1. A sample literate program for the evaluation of the incomplete gamma function is presented using C and the literate programming conventions introduced by Knuth. This demonstrates the application of literate programming methodology to the heart of the molecular integral problem when Gaussian basis sets are employed. We briefly indicate how literate programming techniques may prove useful more generally in other computational sciences.
Abstract: Distributed universal even-tempered basis sets have been developed over recent years that are capable of supporting Hartree-Fock energies to an accuracy approaching the sub-muHartree level. These basis sets have also been exploited in correlation studies, in applications to polyatomic molecules, and in the calculation of electric properties, such as multipole moments, polarizabilities, and hyperpolarizabilities. Jorge and coworkers have also developed universal basis sets and have recently reported applications to diatomic molecular systems. In this article, we compare the molecular calculations reported by Jorge and coworkers with our previous studies. Particular attention is given to the degree of computational linear dependence associated with the various basis sets employed and the consequential effects of the accuracy of the calculated energies. (C) 2004 Wiley Periodicals, Inc.
Abstract: Multireference Brillouin-Wigner coupled cluster theory is applied to the H8 model, a model in which the degree of quasidegeneracy is a function of a single geometrical parameter a. Using a 6-31G basis set, multireference Brillouin-Wigner coupled cluster theory (MR-BWCC) is applied to the H8 model as a function of a both with and without an a posteriori correction for the lack of extensivity. The resulting potential energy curve is compared with the corresponding curve obtained by the application of single reference CCSD theory ("coupled cluster singles and doubles"), and with the curve obtained after adding a perturbative estimate of the triple excitation energy component, CCSD(T). The MR-BWCCSD, CCSD, and CCSD(T) curves are also compared with the full configuration interaction (FCI) curve defining the exact solution within the chosen basis set. (c) 2005 Wiley Periodicals, Inc.
Abstract: The multireference, state-specific, second-order, Brillouin-Wigner perturbation theory (MR-BWPT2) is presented. A posteriori corrections are made which, in the case of a single reference function, recover the well-known formula of second-order many-body perturbation theory, i.e. Moller-Plesset 'MP2' theory, and in the multireference case suppress terms which scale non-linearly with the number of electrons in the system. Prototype calculations are reported for the (H-2)(4) model in which the degree of quasi-degeneracy can be varied by changing a single geometric parameter. The calculated total energies obtained by a second-order Brillouin-Wigner treatment are compared with those supported by CAS-MP2 ( complete active space self-consistent field followed by second-order Moller-Plesset-like treatment of dynamic correlation effects), by MR-BWCC ( multireference Brillouin-Wigner coupled cluster expansion), and by full configuration interaction. MR-BWPT2 provides a theoretical apparatus comparable to the widely used MP2 theory, but which can be applied to quantum chemical problems requiring a multireference formalism.
Abstract: Distributed basis sets of s-type Gaussian functions are determined by invoking the variation principle for the restricted open-shell matrix Hartree-Fock ground states of the open-shell molecular systems HeH and BeH for nuclear separations of 1.500 bohr and 2.500 bohr, respectively. The calculated energy expectation values supported by these distributed basis sets are compared with the energies obtained from finite difference open-shell Hartree-Fock calculations. The restricted open-shell matrix Hartree-Fock calculations are performed by means of the asymptotic method. The accuracy of the approximations to the energy expectation values supported by the distributed basis sets of s-type Gaussian functions is comparable with that attained in previous studies of closed-shell systems. The parameters, that is, the exponents and positions defining the variationally optimized distributed basis sets, are presented and discussed. (C) 2004 Wiley Periodicals, Inc.
Abstract: Using large component basis sets of distributed s-type Gaussian functions with positions and exponents optimized so as to support Hartree-Fock total energies with an accuracy approaching the sub-muhartree level, Dirac-Hartree-Fock-Coulomb calculations are reported for the ground states of the open-shell molecular systems HeH and BeH. The small component basis sets are obtained by applying the (strict) kinetic balance condition. Explicit expressions are given for the electron repulsion integrals required for relativistic atomic and molecular electronic structure calculation, using basis sets of distributed G-spinors. (C) 2004 Wiley Periodicals, Inc.
Abstract: In a previous paper, we have made a comparison of the accuracy with which the electric dipole polarizability alpha(zz) and hyperpolarizability beta(zzz) can be calculated by using either the finite basis set approach (the algebraic approximation) or the finite difference method in calculations for the ground states of the H-2, LiH, BH and FH molecules, at their respective experimental equilibrium geometries, within the Hartree-Fock model. A re-examination of the hyperpolarizability of the BH molecule shows it to be very sensitive both to the choice of grid employed in the finite difference Hartree-Fock calculation and the construction of the basis set used in the matrix Hartree-Fock study. A new comparison of finite difference and finite basis set hyperpolarizabilities for the 13H molecule is made, together with new calculations for the LiH and FH ground states.
Abstract: The development of robust many-body methods for the molecular electronic structure problem with respect to multireference functions has attracted much attention over the past two decades. In recent years, multireference methods based on the Brillouin-Wigner expansion have been shown to overcome the intruder state problems which have plagued similar approaches based on the Rayleigh-Schrodinger series. However, there are some problems in molecular electronic structure theory, which may be handled by means of multireference methods, but can be treated by methods that avoid the use of multireference functions. We consider two examples of alternatives to multireference methods for the molecular electronic structure problem. The study of excited states having the same symmetry as the ground state or some lower lying excited state often involves the use of a multireference function. We describe an alternative procedure based on the generalized Rayleigh-Ritz variation principle. Gidopoulos, Glushkov, and Wilson have termed this the optimized trace method. The generalized Rayleigh-Ritz principle for the relativistic formulation of the molecular electronic structure problem is briefly also considered. Molecular dissociative processes are frequently described by quantum chemical methods based on multireference functions. Single-reference Hartree-Fock methods usually provide a qualitatively incorrect description of such processes even in the simplest of molecules, H, However, approximate single configuration wave functions constructed from nonorthogonal orbitals can often afford a useful approximation to bond-breaking processes. Each occupied nonorthogonal orbital is then an eigenfunction of a different Fock-like operator. Each of these Fock-like operators supports a spectrum of M single particle states, where M is the size of the basis set, and only the lowest of these is occupied. We briefly consider the relativistic formulation of the molecular electronic structure problem based on approximations involving products of nonorthogonal functions within the Furry bound state interaction picture of quantum electrodynamics. (C) 2004 Wiley Periodicals, Inc.
Abstract: Some empirical formulae for generating systematic sequences of even-tempered basis sets are investigated and compared with Schmidt-Ruedenberg ansatz. The new formulae ensure the asymptotic completeness of the sequence of even-tempered basis sets but encompass greater flexibility than the Schmidt-Ruedenberg scheme. (C) 2003 Wiley Periodicals, Inc.
Abstract: An a posteriori Brillouin-Wigner correction is applied to a limited multireference configuration interaction study of a simple model problem consisting of four H-2 molecules arranged in a distorted octagonal conformation in which the degree of quasidegeneracy depends on a single parameter designated a . Double reference configuration interaction calculations are performed as a function of the parameter a. A posteriori Brillouin-Wigner corrections are applied and compared with a generalization of the Davidson correction for multireference applications. The calculated total energies for the ground state are compared with full configuration interaction studies carried out within the same basis set of Gaussian-type functions. The basis set dependence of the problem is explored by performing calculations in two small Gaussian basis sets: one of minimum basis set quality and the other of 'double zeta' type.
Abstract: Some recent progress on the application of Brillouin-Wigner expansions in quantum chemistry is reviewed. Brillouin-Wigner perturbation theory is used to obtain both a Bloch-like equation and a Lippmann-Schwinger-like equation in Brillouin-Wigner form. Both the single reference and the multireference formulation of the expansion are considered. The Bloch-like equation together with the Lippmann-Schwinger-like equation can be used to derive a Brillouin-Wigner coupled cluster (BWCC) expansion whilst the Lippmann-Schwinger-like equation can be used twice to cast the configuration interaction expansion in Brillouin-Wigner form (BWCI). By exploiting an identity relating the Brillouin-Wigner denominator to the Rayleigh-Schrodinger denominator, a posteriori corrections to the multireference Brillouin-Wigner coupled cluster theory and to the limited multireference configuration method are defined. A brief survey of applications is made.
Abstract: Using basis sets of distributed s-type Gaussian functions with positions and exponents optimized so as to support Hartree-Fock total energies with an accuracy approaching the sub-muHartree level, Dirac-Hartree-Fock-Coulomb calculations are reported for the ground states of the H-2, LiH, and BH molecules at their respective equilibrium geometries. The calculated energies are compared with those obtained by Kullie et al. using the finite element technique for the Dirac-Hartree-Fock-Coulomb problem: Finite basis set calculations using the Dirac-Hartree-Fock-Breit approximation are also reported and compared with a first-order treatment of the frequency-independent Breit interaction. (C) 2002 Wiley Periodicals, Inc.
Abstract: A posteriori corrections to state specific multireference configuration interaction are proposed which restore linear scaling with particle number. The corrections are based on an analysis of the Brillouin-Wigner perturbation theory, using a Lippmann-Schwinger-like equation.
Abstract: A new a posteriori correction to the method of limited configuration interaction is described that attempts to restore a linear scaling with particle number. The corrections are based on an analysis of the limited configuration interaction method in terms of the Brillouin-Wigner perturbation theory using a Lippmann-Schwinger-like equation. The new correction procedure is general and, in this work, the application to the limited multireference configuration interaction approximation is considered in some detail. An illustrative application to the rigid rotation of the diimine molecule is presented and the results are compared with those obtained by employing Davidson-like corrections and the corresponding full configuration interaction energies. (C) 2002 Wiley Periodicals, Inc.
Abstract: Distributed basis sets of s-type Gaussian functions are determined by invoking the variation principle for the Hartree-Fock ground states of the H-2, LiH, and BH molecules at their respective experimental equilibrium geometries. The calculated energy expectation values supported by these finite basis sets are compared with finite difference Hartree-Fock energies reported by Kobus et al. A distributed basis set of 54 s-type Gaussian functions distributed along the internuclear axis is shown to support an accuracy of 0.05 muHartree for the Hartree-Fock ground-state energy of the H-2 molecule while a similar set containing 50 functions leads to an accuracy of 0.8 muHartree for the ground-state energy of the LiH molecule. For the BH ground state, a Hartree-Fock energy in error by 1.7 muHartree is supported by a variationally optimized distributed basis set of 65 s-type Gaussian functions distributed along the internuclear axis. The parameters, that is, the exponents and positions, defining the variationally optimized distributed basis sets are presented and discussed. (C) 2002 Wiley Periodicals, Inc.
Abstract: A comparison is made of the accuracy with which the total electronic energy can be calculated by using the finite basis set approach (the algebraic approximation) and the finite difference method in calculations employing the Hartree-Fock model for the open shell ground. (X(2)Sigma(+)) states of the fluorides BaF and YbF. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The difference between the finite basis set and finite difference Hartree-Fock energies is 2.6 muE(h) for BaF and 2.8 muE(h) for YbF. Dipole moments determined within the algebraic approximation are also compared with the corresponding finite difference expectation values.
Abstract: Some comments are made on the basis sets employed in recent studies of electron correlation energies for small molecules with particular reference to calculations for the ground states of the nitrogen and water molecules. For diatomic systems, the use of finite difference and finite basis set approximations in generating the Hartree-Fock reference function is compared. The distributed, universal, even-tempered basis sets (for which we introduce the acronym duet) and correlation consistent basis sets are compared for both the Hartree-Fock model and in treatments of correlation effects. The use of correlation treatments based on many-body perturbation theory and on coupled cluster expansions are discussed. The systematic approximation of the molecular integral supermatrix corresponding to duet basis sets is addressed, as are applications to molecular systems containing heavy atoms.
Abstract: A comparison is made of the accuracy by which the electric dipole polalizability alpha(zz) and hyperpolarizability beta(zzz) can be calculated by using the finite basis set approach (the algebraic approximation) and finite difference method in calculations employing the Hartree-Fock model. The numerical and algebraic methods were tested on the ground states of H-2, LiH, BH and FH molecules at their respective experimental equilibrium geometries. For the FH molecule at its experimental equilibrium geometry, a sequence of distributed universal even-tempered basis sets have been used to explore the convergence pattern of the total energy, dipole moment and polarizabilities. The comparison of finite difference and finite basis set methods is extended to geometries for which the nuclear separation, R-FH, lies in the range 1.5-2.2 b. The methods give consistent results to within 1% or better. In the case of the FH molecule the dependence of truncation errors of the total energy, dipole moment and polarizabilities on the geometry have been studied and are shown to be negligible.
Abstract: A comparison is made of the finite-difference method and finite basis set approach (the algebraic approximation) in calculations using the Hartree-Fock model for the ground (X (1) Sigma (+)(g)) state of the nitrogen molecule with finite nuclei. Finite difference Hartree-Fock energies are determined for the Gaussian nuclear model and the Fermi nuclear model as well as the usual point nucleus. Finite basis set Hartree-Fock energies are calculated for a Gaussian nuclear model using part of a sequence of even-tempered basis sets of Gaussian-type functions which in a previously reported study with a point nucleus model have been shown to support an accuracy approaching 1 mu Hartree in the total Hartree-Fock energy. It is demonstrated that the two methods are capable of accurately accounting for the finite nuclear charge distribution. In the case of the nitrogen molecule the effect can be very accurately estimated from atomic calculations.
Abstract: Sequences of basis sets can be constructed so as to approach the 'infinite basis set limit' in molecular electronic structure calculations. Extrapolation procedures can also be devized to provide accurate values of this limit. However, some care is required since the 'infinite basis set limit' may not correspond to the limit supported by a complete basis set. Extrapolation procedures can then lead to limiting values, which are not estimates of the exact solutions.
Abstract: The local orbit al energy function is used to assess the quality of approximate Hartree-Fock orbitals obtained by invoking the algebraic approximation and using a finite basis set expansion. Systematic sequences of distributed universal even-tempered basis sets of spherical-harmonic Gaussian-type functions are used to generate orbitals for which the corresponding tot al Hartree-Fock energy approaches the 1 mE(h) level of accuracy. A pilot study of the behaviour of the local energy function is made for the hydrogenic atom described by a sequence of even-tempered Gaussian basis sets. The results of prototype calculations for the Hartree-Fock ground state of the BF molecule at its equilibrium geometry are presented. Sequences of calculations which use atom-centred basis sets are investigated as well as sequences which also include bond centred functions. The effects of the bond centred functions on the local orbital energy function are analysed. The local orbital energy function is seen as a measure of the quality of calculations carried out wit hin the matrix Hartree-Fock approximation which can be employed in cases where the corresponding finite difference Hartree-Fock results are not available.
Abstract: The application of Brillouin-Wigner perturbation theory to a hydrogenic model problem is described using both a relativistic and a nonrelativistic formalism. The nonrelativistic study is carried out by employing a basis set of Coulomb Sturmian functions whereas calculations within the relativistic formalism use basis sets of L-spinors. A two-state hydrogenic model problem provides a dramatic example of the dependence of the convergence behaviour of both the Brillouin-Wigner and the Rayleigh-Schrodinger perturbation expansions on the choice of basis set. Two types of Brillouin-Wigner perturbation expansion are compared in both the relativistic and the nonrelativistic formulations, the first using the exact energy in the denominator factors and the second employing total energies determined by a self-consistent procedure applied at finite order. Extrapolation procedures are also investigated. For the relativistic formulation the second-order energy is divided into two components, one arising from a sum over positive-energy states and the other corresponding to a sum over the negative-energy states. The application of correction terms, based on the known relation between the denominator factors occuring in Brillouin-Wigner perturbation theory and those in the Rayleigh-Schrodinger formulation, which restore linear scaling with particle number in many-body formulations, is investigated for the hydrogenic model problem.
Abstract: The generalized multireference Brillouin-Wigner coupled cluster theory is presented for the case of a single root function. The Bloch-like equation and the Lippmann-Schwinger-like equation for the generalized multireference Brillouin-Wigner perturbation theory are obtained. These equations are then applied to the solution of the multireference coupled cluster equations for the case in which a single root function is required.
Abstract: The use of Moeller-Plesset perturbation theory, frequently designated MPn, in describing electron correlation in atomic and molecular systems is critically examined. Some recently published studies have examined the higher terms in the Moeller-Plesset perturbation expansion and the convergence behaviour of the perturbation series. From these investigations conclusions about the applicability of the low order theory have been drawn which, in view of the widespread use of MP2, demand more thorough analysis. Combining the Rayleigh-Schrodinger and the Brillouin-Wigner perturbation expansions avoids an infinite order expansion and yields a closed expression which consists of the MP2 energy components together with a remainder term. The applicablity of MP2 theory then rests upon the magnitude of this remainder term rather than the behaviour of the higher order terms in the perturbation series.
Abstract: The effects of intramolecular basis set superposition errors are less well documented than the corresponding intermolecular effects. The intramolecular basis set superposition errors are examined, using the approach of Jensen, for several basis sets developed by Pople and his co-workers, which are widely used in studies of larger molecules. Prototype calculations are reported for the ground state of the water molecule using both the matrix Hartree-Fock method and the many-body perturbation expansion for the correlation energy taken through second order. A similar investigation is carried out for some of the correlation consistent basis sets published by Dunning and his collaborators. Specifically, the following aspects are investigated: (i) the magnitude of the intramolecular basis set superposition error, (ii) the nonadditivity of intramolecular counterpoise corrections when applied in a pairwise fashion, and (iii) the use of multiple "ghost" centers. (C) 2001 John Wiley & Sons, Inc.
Abstract: Systematic sequences of distributed universal even-tempered basis sets of Gaussian functions have been shown to support an accuracy approaching the sub-mu Hartree level for the total Hartree-Fock energies for diatomic molecules containing first row atoms. They have also been shown to support high precision correlation treatments. Furthermore, the use of a similar approach for systems containing heavy atoms and for polyatomic molecules has been demonstrated. In this paper, systematic truncation of basis sets developed in this fashion is explored. An application to the Hartree-Fock ground state of the BF molecule at its equilibrium geometry is described. The parent distributed universal basis set, which contains a total of 623 primitive Gaussian functions, is truncated by systematically removing those basis functions for which the magnitude of the elements of the orbital expansion coefficient vector are less than some small tau for all occupied orbitals. The effects of truncation on the description of electron correlation effects using second order many-body perturbation theory is also explored.
Abstract: New a posteriori corrections to the method of limited configuration interaction are proposed which restore a linear scaling with particle number. The corrections are based on an analysis of the Brillouin-Wigner perturbation theory, using a Lippmann-Schwinger-like equation, and its connection with the limited configuration-interaction method. The new corrections are general and can be applied to any limited configuration-interaction approximation. An illustrative application to the ground state of the water molecule is presented and the results are compared with those obtained by employing other approaches to the correlation problem.
Abstract: A comparison is made of the accuracy with which the total electronic energy can be calculated by using either the finite basis set approach (the algebraic approximation) or finite difference methods in calculations employing the restricted Hartree-Fock model for the open-shell ground (X(2)Sigma+) states of the Group IIa fluorides BeF, MgF, CaF and SrF. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. By using two different grids, the accuracy of the finite difference calculations has been estimated to be of the order of 10(-2) mu E-h The average difference between the finite basis set and finite difference total Hartree-Fock energies is 2.75 mu E-h. Dipole moments determined within the algebraic approximation are also compared with the corresponding finite difference expectation values.
Abstract: A comparison is made of the accuracy with which the electric moments mu, Theta, Omega, and Phi, can be calculated by using the finite basis set approach (the algebraic approximation) and finite-difference method in calculations employing the Hartree-Fock model for the ground states of 16 diatomic molecules at their experimental equilibrium geometries. Specifically, the 2(n)-pole moments n = 1,2,3,4, for the N-2, CO, BF, CN-, NO+, BeF, BO, CN, N-2(+), A1F, GaF, InF, TIF, MgF, CaF, and SrF molecules are determined using basis sets and grids that have been employed in previous studies of the Hartree-Fock energy.
Abstract: Many-body treatments of electron correlation effects in small molecules, when implemented in a high performance computing environment and exploiting large and flexible basis sets, can lead to results of high precision. The results of such calculations are usually assessed by comparison with empirical estimates of the correlation energy and it is, therefore, important that these estimates be reliable. The empirical correlation energies for the N-2, CO and BF molecules and the NO+ ion are re-evaluated by making use of recent Hartree-Fock results of mu hartree accuracy and new estimates of the relativistic energy components which are based on four component atomic and molecular relativistic electronic structure calculations and take account of the non-additivity of correlation and relativistic effects. A definition of the post-Hartree-Fock energy is proposed.
Abstract: A comparison is made of the accuracy with which the total electronic energy can be calculated by using either the finite basis set approach (the algebraic approximation) or finite difference methods in calculations employing the Hartree-Fock model for the open shell ground (X(2)Sigma(+)) states of the 13-electron systems BeF, BO, CN and N-2(+). The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The accuracy of the finite difference calculations is assessed by employing two grids of different sizes, one of 32617 points and the other of 132385 points. The total Hartree-Fock energies supported by these two grids differ by less than 0.05 mu E-h. The average difference between the finite basis set and finite difference Hartree-Fock energies is 1.3 mu E-h. Dipole moments determined within the algebraic approximation are compared with the corresponding finite difference expectation values for the neutral species.
Abstract: A previously reported study of the CN- anion at the matrix Hartree-Fock level failed to match the accuracy achieved in studies of isoelectronic neutral diatomic systems. In this paper, we examine the CN- ion again at both the Hartree-Fock level and including the effects of electron correlation by means of finite-order many-body perturbation theory. A basis set which supports a matrix Hartree-Fock energy 2.0 mu Hartree above the corresponding finite difference energy is constructed. Over 98.5% of an estimate of the exact second-order correlation energy component is recovered by a basis set containing atom-centred and bond-centred functions of s, p, d, f, g and h symmetry. The calculated energies are compared with those supported by other published basis sets.
Abstract: Distributed basis sets of s-type Gaussian function for diatomic molecules are developed in which the disposition of the off-nucleus expansion centers are determined by a stochastic variational technique. The utility of this approach is investigated by means of prototype matrix Hartree-Fock calculations for the ground state of the nitrogen molecule. (C) 1999 John Wiley & Sons, Inc.
Abstract: The use of hierarchical orthogonality restrictions in handling the non orthogonality problem which arises when spin-coupled wave functions are employed to describe the electronic structure of extended molecules is investigated. (C) 1999 John Wiley & Sons, Inc.
Abstract: High-precision many-body perturbation theory calculations are reported for the ground state of the neon atom within the algebraic approximation, i.e. by using a finite basis set expansion. The second-order many-body perturbation theory correlation energy component is calculated using an even-tempered universal basis set sequence of Gaussian-type functions of increasing angular quantum number. The basis subsets for each symmetry are systematically truncated so as to attempt to maintain an accuracy of similar to 1 mu Hartree. The resulting energy components are compared with other recent results, including those obtained by employing correlation-consistent basis sets and the recent finite-element study by Flores and Kolb. A sub-mu Hartree level of accuracy was achieved for the E-2(l = 1) energy component.
Abstract: A comparison is made of the finite basis set approach (the algebraic approximation) and finite difference methods in calculations using the Hartree-Fock model for the ground (X-1 Sigma(+)) states of the heaviest Group IIIb fluorides: indium fluoride and thallium fluoride molecules. The XF molecules, X = In, Tl, are considered as a prototype for systems containing increasingly heavy atoms and numbers of electrons. New finite difference Hartree-Fock calculations for the GaF ground state improve upon earlier results. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The dependence of the finite difference calculations on the numerical grid employed is discussed. For the InF molecule, the total Hartree-Fock energy obtained with a grid of 595 x 595 points lies 15 mu E-h below the finite basis set energy. For the ground state of TlF the finite basis set energy lies about 61 mu E-h above the converged finite difference result obtained on a grid of 595 x 595 points. For the TlF molecule, the matrix element required to evaluate the electric dipole moment volume interaction for the elementary particles in the Tl nucleus is evaluated from the finite difference Hartree-Fock wavefunction.
Abstract: High-precision many-body perturbation theory calculations are reported for a series of 14-electron diatomic systems in their ground electronic states at their respective equilibrium geometries using a universal basis set of even-tempered Gaussian primitive functions. Using a systematic approach previously developed for the N-2 molecule, calculations have been carried out for the CO and BF molecules and for the NO+ ion using the matrix Hartree-Fock approximation supplemented by a second-order many-body perturbation theory description of correlation effects. The basis set is distributed across the nuclear centres and the bond centre. Functions of s, p, d, f, g and h symmetry are included on each centre. Extrapolation procedures are employed to estimate the contribution of higher symmetry types to the electron correlation energy. The calculated and extrapolated correlation energies obtained are compared with the corresponding results obtained with some of the larger correlation consistent basis sets, with calculations employing explicitly correlated functions, and with empirical estimates of the correlation energy.
Abstract: The orbital amplitude difference function is used to assess the quality of Hartree-Fock orbitals obtained by invoking the algebraic approximation for the BF ground-state. Systematic sequences of even-tempered, spherical-harmonic Gaussian-type basis functions are used to generate orbitals for which the corresponding total Hartree-Fock energy approaches the 1 mu E-h level of accuracy. Exact orbitals are obtained from finite difference calculations using a grid based on spheroidal coordinates. The finite basis set approximations for the orbital are discretized. The accuracy of the discretization is assessed. For each occupied orbital a discretized representation of the orbital amplitude difference function is generated and analysed.
Abstract: A particular formulation of the distributed Gaussian basis-set approach, the extended Gaussian cell model, is applied to the simplest polycentric molecule, the linear H-3(2+) ion. Calculations of the total energy using two extensions of the original Gaussian cell model are described and results are reported for the ground state and the first excited state. A comparison with recently reported finite element calculations is made for a number of nuclear geometries. (C) 1996 John Wiley & Sons, Inc.
Abstract: Many-body perturbation theory, formulated within the algebraic approximation, is used to determine the correlation energy of the ground state of the nitrogen molecule. Systematically constructed even-tempered basis sets of Gaussian-type functions, which have been shown to achieve an accuracy approaching similar to 1 mu Hartree in matrix Hartree-Fock calculations, have been developed to afford basis sets suitable for electron correlation studies using the second-order many-body perturbation expansion. Over 98% of an empirical estimate of the ground-state correlation energy of the N-2 molecule at its equilibrium geometry is recovered by using a basis set constructed from even-tempered sets centred on the atoms and on the bond midpoint and containing functions of s, p, d, f, g and h symmetry. The importance of functions with i symmetry is also assessed. It is estimated that the calculated correlation energy corresponds to 99.1% of the exact second-order energy. The correlation energy obtained in the present study is compared with second-order energies obtained by using previously reported basis sets, which are shown to recover, at best, about 10% less of the empirical correlation energy estimate.
Abstract: The use of distributed Gaussian basis sets in reducing the total basis set truncation error in matrix Hartree-Fock and second-order many-body perturbation theory calculations is investigated for the ground state of the water molecule at its equilibrium geometry. A distributed basis set of even-tempered Gaussian functions centred not only on the atomic nuclei bur also on the O-H bond centres and at the midpoint of the line H-H is shown to give a matrix Hartree-Fock energy of -76.067 488 mu Hartree. For diatomic molecules, distributed basis sets of this type have been shown to yield matrix Hartree-Fock energies which approach an accuracy of similar to 1 mu Hartree. The present distributed basis set, which includes functions of s, p, d and f symmetry, is employed in a second-order many-body perturbation theory study of correlation effects recovering 97.6% of an estimate of the exact second-order correlation energy given by Klopper. The effects of higher harmonics in the basis set are investigated and a basis set, which includes functions of s, p, d, f and g symmetry, is shown to be capable of recovering 98.6% of the exact second-order energy. The reliability of simple extrapolation techniques to estimate the effects of basis functions of h symmetry and higher is investigated and shown to support 99.8% of the estimate of the exact second-order correlation energy component.
Abstract: A comparison is made of the accuracy with which the total electronic energy can be calculated by using either the finite basis set approach (the algebraic approximation) or finite difference methods in calculations using the Hartree-Fock model for the ground (X(1) Sigma(+)) states of the Group IIIb fluorides: boron fluoride, aluminium fluoride and gallium fluoride molecules. The XF molecules, X = B, Al, Ga, are considered as a prototype for systems containing increasingly heavy atoms and numbers of electrons. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The dependence of the finite difference calculations on the numerical grid employed is studied and it is found to be necessary to employ a multiple grid for the heaviest system, GaF, in order to achieve the target accuracy of 1 mu E(h) for the total energy.
Abstract: The ground state Hartree-Fock electronic energy for the nitrogen molecule is determined by using basis sets of distributed s- and p-type Gaussian functions and compared with the results of fully numerical (finite difference) Hartree-Fock calculations. Calculations are also reported for the ground states of the CO and BF molecules and a universal distributed Gaussian basis set established. A potential energy curve is calculated for the CO molecule and the calculation of energy derivatives is considered. The use of distributed basis sets of s and p Gaussian functions in the study of electron correlation effects using many-body perturbation theory is demonstrated.
Abstract: The properties of the expansion of the elliptical function, exp(-alpha lambda)exp(-beta mu), in an even-tempered basis set of Gaussian functions, exp(-eta r(2)), are examined analytically. A Laplace transform is used to obtain an integral representation of the elliptical function in terms of Gaussian functions. Numerical quadrature procedures afford a method for obtaining a finite-series expansion of elliptical functions in terms of an arbitrary number of Gaussian-type functions. The Laplace transform provides a prescription for the generation of a systematic sequence of distributed basis sets of Gaussian functions from a sequence of atomic even-tempered sets. illustrative calculations are reported for the hydrogen molecular ion.
Abstract: The accuracy of the total energies obtained for the series of 14 electron molecules N-2, CO, BF, NO+ and CN- by means of matrix Hartree-Fock calculations using a universal basis set of Gaussian-type functions is reaccessed in the light of improved finite difference calculations.
Abstract: Matrix Hartree-Fock calculations for a polyatomic molecule, the ground state of the carbon dioxide molecule, are carried out using basis sets which afford an accuracy approaching the sub-mu Hartree level for the energy of the carbon monoxide molecule. Using a 28s14p14d14f atom centred and a 24s10p11d bond-centred Gaussian basis set, an upper bound for the Hartree-Fock energy of the CO2 ground state at the experimental geometry is computed as -187.725 408 Hartree. Based on our experience with diatomic molecules, we believe this value to be accurate to 56 mu Hartree. The results of the present calculations are used to access the accuracy of many of the basis sets currently used in molecular self-consistent field studies including those designated STO-3G, 4-31G, 6-31G, 6-31G(3d), 6-311G, 6-311+G(3df), D95 and D95V+(3d) as well as the basis sets cc-pVDZ, aug-cc-pVDZ, cc-pVTZ and aug-cc-pVTZ, which have been proposed for electron correlation studies.
Abstract: A comparison is made of the accuracy with which the total electronic energy can be calculated by using either the finite ba-sis set approach (the algebraic approximation) or finite difference methods in calculations using the Hartree-Fock model for the ground (x 1SIGMA+) state of the carbon monosulphide molecule, The CS molecule is considered as a prototype for systems containing atoms from different rows of the periodic table. The convergence of the calculations carried out within the algebraic approximation is monitored by employing systematically constructed basis sets of increasing size. The dependence of the finite difference calculations on the numerical grid employed is studied.
Abstract: The fifth-order Hugenholtz diagrams in the many-body perturbation expansion for the correlation energy of a closed-shell system described in zero-order by a single determinant constructed from canonical one-electron state functions are automatically generated, presented in their entirety for the first time, and analyzed.
Abstract: High-precision matrix Hartree-Fock calculations are reported for a series of five isoelectronic diatomic systems in their ground states. A universal even-tempered basis set of Gaussian-type functions is developed for the nitrogen molecule at its experimentally derived equilibrium nuclear separation and then used to parameterize the single-particle state functions in the matrix Hartree-Fock description of the neutral molecules CO and BF, and in the ions NO+ and CN- at their respective experimentally determined equilibrium geometries. The calculated energies are compared with the results of previously reported fully numerical calculations which were performed by using finite difference and/or finite element techniques and which are taken to define the Hartree-Fock limit. Energies obtained using the universal even-tempered finite basis set expansion are well within what is usually regarded as 'chemical accuracy' of similar to 1 mHartree and are only in error by similar to 2.3, similar to 1.5, similar to 90, similar to 40 and similar to 140 mu Hartree for Nz, CO, BF, NO+ and CN-, respectively.
Abstract: The use of distributed basis sets of s-type Gaussian functions is investigated for the ground states of the one-electron, homonuclear, diatomic ions H-2+, He-2(3+), Li-2(5+) and Be2(7+) and for the ground states of the corresponding neutral species H-2, He-2, Li-2 and Be2. Empirical schemes are employed both to generate the exponents defining the basis functions and to define their distribution in space. For the one-electron systems it is shown by comparing with the exact solutions of the Schrodinger equation that sub-muHartree accuracy can be achieved. For the many-electron systems the accuracy achieved decreases with increasing number of electrons but remains below approximately 0.15 mHartree even for the largest of the molecular systems considered.
Abstract: The Hartree-Fock ground state potential energy curve for the carbon monoxide molecule is calculated using both the finite difference and the finite basis set methods. The results of calculations performed at ten internuclear separations are reported and a comparision is made of the calculated potential energy curves over the range of internuclear separations, 1.970-2.330 a(0) around the experimental equilibrium value. Potential energy curve constants and spectroscopic constants are determined.
Abstract: The finite basis set Hartree-Fock method is compared with the finite difference and finite element counterparts. A comparison of the total and orbital energies for two diatomic systems demonstrates that the finite basis set approach can afford an accuracy approaching that achieved in finite difference and finite element methods provided that the large and flexible basis sets are systematically constructed. Some advantages of the finite basis set technique (the algebraic approximation) are emphasized.
Abstract: The performance of the ccMBPT program for the calculation of electron correlation effects in atoms and molecules on the IBM Shared Memory System (POWER/4) is reported.
Abstract: A comparison of matrix Hartree-Fock calculations using basis sets of Gaussian-type functions with numerical Hartree-Fock studies is made for the ground state of the nitrogen molecule. The use of both atom- and bond-centred functions is investigated. It is shown that by employing systematic sequences of even-tempered basis sets the accuracy achieved in the numerical Hartree-Fock calculations can be approached when basis sets of atom-centred functions are employed and can be matched when bond-centred functions are included. An energy of -108.993 823 Hartree is obtained within the algebraic approximation which should be compared with -108,993 808 Hartree and -108.9939 Hartree from previously reported fully numerical and semi-numerical calculations. The advantages of the matrix Hartree-Fock method are emphasized.
Abstract: The performance of the ccMBPT program for the calculation of electron correlation effects on the NEC SX-3/44 computer is described. Rates of execution observed are compared with data obtained from other machines including the Cray Y-MP C-90.
Abstract: A program for calculating the correlation energy components associated with each of the fourth-order diagrams involving triply excited intermediate states in the many-body perturbation theory expansion is presented. The program exhibits a high level of vectorization and parallelism and execution rates of 2.284 Gflops have been reported on the CRAY Y-MP/8128 computer.
Abstract: The use of universal basis of elliptical functions in molecular electronic structure calculations for diatomic systems is investigated. By considering both the unitedatom and the separated-atom limits, an empirical relation between screening constants is introduced which leads to sub-mu-Hartree accuracy over a wide range of nuclear separations. Calculated energies for the homonuclear one-electron systems X2n+ (X = XH, He, Li, Be, B, C, N, O and F) obtained by using the same set of integrals over a universal basis set of elliptical functions are compared with the exact energies. Heteronuclear one-electron systems are also studied.
Abstract: A formulation of the diagrammatic many-body perturbation theory suitable for effective implementation on multi-processor computers is described. This concurrent computation many-body perturbation theory (ccMBPT) exploits the additive separability of the correlation energy components resulting from the linked diagram expansion to devise algorithms suitable for a parallel processing environment. A dynamic load balancing technique is employed to exploit the parallel processing capabilities of computers such as the CRAY Y-MP. The performance of the technique is demonstrated in calculations of the most computationally demanding of the fourth order energy components, those involving triply excited intermediate states. Execution rates in excess of 2.28 x 10(9) floating point operations per second are observed on an eight-processor machine.
Abstract: The description of electron correlation effects is of central importance in molecular physics and theoretical chemistry. When formulated within the algebraic approximation, the many-body perturbation theory of electron correlation forms the basis of an algorithm which is well suited to implementation on multi-processor vectorcomputers. We describe the performance of the ccMBPT program on the Cray Y-MP and IBM 3090 machines.
Abstract: The efficient evaluation of the fourth-order energy terms corresponding to linked diagrams involving triply excited intermediate states which arise in the many-body perturbation expansion for the electron correlation energy of closed-shell atoms and molecules on vector processing and parallel processing computers is discussed. The linked-diagram theorem decouples the many-electron problem allowing efficient implementation on parallel processing machines. Furthermore, the computation associated with each of the resulting subproblems is very well suited to vector processing machines. A "dynamic load balancing" technique is employed in both a dedicated and a multi-user environment. Timing tests are reported for the CRAY X-MP/416 computer. Execution rates in excess of 828 Mflops are observed.
Abstract: A prototype collaborative virtual environment is described which has been specifically designed for the development of theory, the associated algorithms and computer code in the study of the molecular electronic structure problem. The environment was constructed as part of a project aimed at the further development of Brillouin-Wigner many-body methods for the handling of the electron correlation problem for systems where the use of a multireference formalism is required; a formalism which can often be plagued by the 'intruder state' problem in practical applications. The collaborative virtual environment is web-based and contains three main elements:- (i) details of the participants, (ii) a knowledge base, (iii) tools for collaboration. By developing the prototype virtual environment whilst carrying out collaborative research on a specific problem, these three elements have been carefully tailored for the needs of the molecular physics/quantum chemistry researcher. The further development of a collaborative virtual environment for molecular electronic structure theory is briefly considered.
Abstract: The problem of the orthogonality of the closed- and open-shell orbitals in self-constistent field (SCF) theory are reviewed. An easily implemented asymptotic projection method for talking orthogonality constraints into account is used to develop an alternative open-shell SCF approach for ground and excited states. Variational derivation of the open-shell Hartree-Fock equations is described which does not involve off-diagonal Lagrangian multipliers. The possibilities of designing a well-defined, open-shell many-body perturbation theory using the orbitals of the asymptotic projection method is demonstrated by means of calculations of ground and excited state energies using the different levels of approximation. We also focus on the development of alternative computational strategies capable of optimizing relatively small distributed basis sets with respect to the positions and exponents of the basis functions.
Abstract: An alternative approach to problems in quantum chemistry which can be written as an eigenvalue equation with orthogonality restrictions imposed oil eigenvectors is reviewed. The basic tenets of a simply implemented asymptotic projection method for taking the necessary orthogonality constraints into account are presented. The eigenvalue equation for a modified operator is derived and the equivalence of the original and modified problem is rigorously demonstrated. The asymptotic projection method is compared with the conventional approach to Constrained variational problems based on the elimination of off-diagonal Lagrange multipliers and with other methods. A general procedure for application of the method to excited state problems is demonstrated by means of calculations of excited state energies and excitation energies for the one-electron molecular systems, H-2(+) and H-3(++).
Abstract: The interaction between electrons and phonons in molecules is investigated using a formulation in which the total molecular Hamiltonian is subjected to two canonical transformations. The first of these transformations, which we term the normal coordinate transformation, passes from a crude representation, which we term the clamped nuclei representation, in terms of basis functions defined with respect to some fixed geometry of the nuclei, R-0, to a representation in terms of basis functions depending on R. This transformation mixes electronic and vibrational motions and leads to a simple formulae for adiabatic corrections. The second transformation, termed the momentum transformation, leads to a non-adiabatic (or diabatic) representation. Each representation supports a different partition of the total molecular Hamiltonian into a zero-order (or reference) Hamiltonian and a perturbation. For each representation, quasiparticles, i.e. renormalized electrons and phonons are defined will provide the theoretical apparatus required for the description of the electron-phonon interaction.
