Sequences Of Even-tempered Basis Sets Research Articles

On the accuracy of the algebraic approximation in molecular electronic structure calculations. III. Comparison of matrix Hartree-Fock and numerical Hartree-Fock calculations for the ground state of the nitrogen molecule

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 from previously reported fully numerical and semi-numerical calculations. The advantages of the matrix Hartree-Fock method are emphasized.

On the accuracy of the algebraic approximation in electronic structure calculations for the ground states of H32+ and H43+ using basis sets of atom-centred Gaussian-type functions

Sequences of even-tempered basis sets of atom-centered Gaussian-type functions are employed to explore the accuracy that can be achieved for the ground state energies of the H32+ and H43+ species in calculations using finite basis sets. These systems are considered as prototypes for the treatment of polyatomic molecules of arbitrary symmetry within the algebraic approximation. Keywords: algebraic approximation, basis sets, even-tempered, universal.

Universal basis sets and Cholesky decomposition of the two-electron integral matrix

It is shown that, for a one-centre universal even-tempered basis set containing N functions, only N of the N( N + 1)/2 eigenvalues of the two-electron integral matrix are larger than some small δ as N becomes large and thus only N of the rows or columns of the matrix are linearly independent in this limit. It is demonstrated that the same degree of linear dependence exists in the case of a many-centre universal basis set. This approximate numerical linear dependence can be effectively exploited in atomic and molecular electronic structure calculations using universal systematic sequences of even-tempered basis sets by means of a Cholesky decomposition of the two-electron integral matrix. This opens up the possibility of using large, flexible basis sets in high precision calculations on small molecules, in studies of extended molecular systems, and in relativistic electronic structure calculations.

Van der Waals interaction potentials

The ground state argon-argon interaction is calculated using the supermolecular approach. Electron correlation effects are described by means of the many-body perturbation theory. A systematic sequence of even-tempered basis sets is used in conjunction with the function counterpoise technique used as a test, rather than a correction, for basis set superposition. 65 per cent of the interaction energy associated with electron correlation effects is recovered. 47 per cent of the van der Waals minimum is calculated. The magnitude of remaining errors is discussed.

Second-order correlation energy of the argon atom using basis sets of Gaussian-type functions

The second-order correlation energy of the ground state of the argon atoms is calculated using basis sets of Gaussian-type functions. It is demonstrated that accurate second-order correlation energies can be obtained by employing systematic sequences of even-tempered basis sets of Gaussian-type functions. The pair correlation energies are compared with those obtained by Jankowski et al. (1979) using basis sets of exponential-type functions.

The orbital amplitude difference function for systematic sequences of even-tempered basis sets of gaussian-type functions

The orbital amplitude difference function is shown to provide a useful method for displaying basis set deficiencies, particularly in calculations using systematic sequences of even-tempered basis functions. An application to the beryllium atom is presented.

Van der Waals interaction potentials

The ground state argon-argon interaction is calculated using the supermolecular approach. Electron correlation effects are described by means of the many-body perturbation theory. A systematic sequence of even-tempered basis sets is used in conjunction with the function counterpoise technique used as a test, rather than a correction, for basis set superposition. 65 per cent of the interaction energy associated with electron correlation effects is recovered. 47 per cent of the van der Waals minimum is calculated. The magnitude of remaining errors is discussed.

Diagrammatic perturbation theory: a comparison of numerical methods with basis set expansion techniques for a model problem

A comparison has been made of numerical methods with basis set expansion techniques in the evaluation of the sum-over-states expressions which arise in the diagrammatic perturbation expansion in second and higher orders. The model problem of a ground-state hydrogenic atom with charge Z perturbed by the potential -Z'/r is used. The energy corresponding to each diagrammatic component can be evaluated explicitly for this mode. Universal systematic sequences of even-tempered basis sets of exponential-type functions and Gaussian-type functions are employed and the convergence of the individual components of the energy expansion with increasing size of basis set is examined.

Molecular multipole moment calculations using universal systematic sequences of even-tempered basis sets

Higher order multipole moments for the ground states of the HF, CO and N2 molecules are calculated using the matrix Hartree-Fock ansatz. Calculations are reported for a universal systematic sequence of even-tempered basis sets of cartesian gaussian-type functions and the behaviour of the calculated higher order multipole moments with increasing size of basis set is examined. Distributed multipole analyses are also reported.

The Dirac equation in the algebraic approximation. II. Extended basis set calculations for hydrogenic atoms

For pt.I see ibid., vol.17, p.L45 (1984b). The solution of the Dirac equation for hydrogen-like atoms within the algebraic approximation, that is, by using a finite basis set, is considered. It is shown that by making an appropriate choice of basis functions the problem which has been termed 'variational collapse' can be avoided. Applications using a systematic sequence of even-tempered basis sets are presented and convergence of the calculated energies within the algebraic approximation to the exact energies with increasing size of basis set is investigated.

Van der Waals interaction potentials

The ground state argon-argon interaction is calculated using the supermolecular approach. Electron correlation effects are described by means of the many-body perturbation theory. A systematic sequence of even-tempered basis sets is used in conjunction with the function counterpoise technique used as a test, rather than a correction, for basis set superposition. 65 per cent of the interaction energy associated with electron correlation effects is recovered. 47 per cent of the van der Waals minimum is calculated. The magnitude of remaining errors is discussed.

The use of universal even-tempered basis sets in spin-coupled wavefunctions; the electron affinity of Li(2S)

Universal systematic sequences of even-tempered basis sets of Slater orbitals have been used in theoretical studies of the series of atoms and atomic ions, Be(1S), B+(1S) and Li-(1S). The wavefunctions used were of the spin-coupled valence bond type which provide a very compact representation of the fully correlated 1S states. It was found that the same ((9, 1)s/(6, 1)p) basis set provided an excellent description of the series; the authors show that they are essentially at the basis set limit, even for Li-(1S). A useful consequence of the work is that one can obtain an accurate value for the electron affinity of Li(2S) using only 3s-, 2p- and 3p-type virtual orbitals and just twenty symmetry structures. The same basis set applied to the 22P degrees and 32S states of Li using just the relevant spin-coupled virtual orbitals of the 22S state also yields values for the term splittings that are in good agreement with experiment. The efficacy of universal even-tempered basis sets for molecular systems is illustrated by results for the LiHe+, CH4+ and BH3+ ions.

Universal systematic sequences of even-tempered basis functions in electronic structure studies of negative ions

Universal systematic sequences of even-tempered basis sets of exponential-type functions are developed for the treatment of negative ions. Calculations are reported for Li−, Be−, B−, C−, N−, O−, and F− within the self-consistent field approximation. Extrapolation procedures are employed to obtain estimates of the basis set limit.

Ab initio calculation of atomic spin-orbit coupling constants using a universal systematic sequence of even-tempered exponential-type basis sets

Ab initio calculations of spin-orbit coupling constants are reported for a number of first-row atoms and ions using a universal systematic sequence of even-tempered basis sets of exponential-type functions. The convergence of the calculations with respect to size of basis set is examined and extrapolation to the basis set limited made. The importance of the present atomic calculations to the determination of the fine structure in molecules is discussed.

Kinetic energy, potential energy, and virial ratio from calculations using systematic sequences of even-tempered basis sets

The use of systematic sequences of even-tempered basis sets in calculations of the kinetic energy, potential energy and virial ratio within the self-consistent-field molecular-orbital approach is discussed. The components of the total energy are shown to converge smoothly with increasing size of basis set. Systematic sequences of universal even-tempered basis sets are also considered.

Scaling in second-order electron correlation calculations using systematic sequences of even-tempered basis sets

Improved results can often be obtained from second-order Rayleigh-Schrodinger perturbation calculations of electron correlation energies using large basis sets by introducing a scaling factor in the zero-order Hamiltonian. The scaling parameter may be determined from full third-order calculations using a smaller basis set. This scaling procedure can be applied in a systematic fashion by employing a sequence of even-tempered basis sets. Calculations illustrating this approach for the beryllium atom and the neon atom are presented. The scaling procedure is also employed in conjunction with a universal systematic sequence of basis functions. Calculations illustrating this Correlation energy — Mang-body perturbation theory.