Gaussian-type Basis Functions Research Articles

Newton–Raphson optimization of the many-body nonadiabatic wave function expressed in terms of explicitly correlated Gaussian functions

A nonadiabatic many-body wave function is represented in terms of explicitly correlated Gaussian-type basis functions. Motions of all particles (nuclei and electrons) are treated equally and particles are distinguished via permutational symmetry. The nonadiabatic wave function is determined in a variational calculation with the use of the method proposed recently [P. M. Kozlowski and L. Adamowicz, J. Chem. Phys. 95, 6681 (1991)]. In this approach no direct separation of the center-of-mass motion from the internal motion is required. The theory of analytical first and second derivatives of the variational functional with respect to the Gaussian exponents and its computational implementation in conjunction with the Newton–Raphson optimization technique is described. Finally, some numerical examples are shown.

Implementation of analytical first derivatives for evaluation of the many-body nonadiabatic wave function with explicitly correlated Gaussian functions

A nonadiabatic many-particle wave function is generated using an expansion in terms of explicitly correlated Gaussian-type basis functions. In this approach, motions of all particles are correlated at the same time, and electrons and nuclei are distinguished via permutational symmetry. We utilize our newly proposed nonadiabatic variational approach [P. M. Kozlowski and L. Adamowicz, J. Chem. Phys. 95, 6681 (1991)], which does not require the separation of the internal and external motions. The analytical first derivative of the variational functional with respect to the nonlinear parameters appearing in the basis functions are derived and implemented to find the minimum. Numerical examples for the ground state of the hydrogen molecule are presented.

Representation of electronic wavefunctions by Lobatto shape functions: application to the photoionization cross section of H +2

It is shown that a convenient and flexible representation of molecular electronic continuum wavefunctions may be constructed from Lobatto shape functions, especially in combination with conventional Gaussian-type basis functions. When used in the logarithmic derivative version of the Kohn variational principle, an efficient procedure for calculating molecular photoionization cross sections is obtained. As a first example, the method is applied to the benchmark system H + 2. Excellent results are found with rather moderate effort for the full investigated range of photoelectron energies (up to 200 eV).

An effective method for generating nonadiabatic many-body wave function using explicitly correlated Gaussian-type functions

General formalism for the application of explicitly correlated Gaussian-type basis functions for nonadiabatic calculations on many-body systems is presented. In this approach the motions of all particles are correlated in the same time. The energy associated with the external degrees of freedom, i.e., the motion of the center of mass, is eliminated in an effective way from the total energy of the system. In order to achieve this, methodology for construction of the many-body nonadiabatic wave function and algorithms for evaluation of the multicenter and multiparticle integrals involving explicitly correlated Gaussian cluster functions are derived. Next the computational implementation of the method is discussed. Finally, variational calculations for a model three-body system are presented.

Hydrogenlike atoms in uniformly charged sphere model of atomic nucleus. II. Application of basis-set expansion method

The basis-set expansion method has been applied to the nonrelativistic and relativistic ground states of one-electron atoms H, Ar17+, Xe53+, Hg79+, and U91+ in the uniformly charged sphere (UCS) and the point-change (PC) models of atomic nucleus, using the Gaussian-type basis functions. The energies and the radial expectation values, especially 1/r2, converges faster in the UCS model than in the PC model. In the PC model, larger values of the exponent parameters of the basis functions are required both in the nonrelativistic and the relativistic calculations. Even in the UCS model, the larger values of the exponent parameters are needed in the relativistic calculations.

Relativistic perturbation theory: II. One-electron variational perturbation calculations

For pt.I see ibid., vol.19, p.149 (1986). The effect of the approximate character of the non-relativistic wavefunction on the second-order relativistic energy correction for the ground state of hydrogen-like atoms is illustrated by means of prototype calculations using Gaussian-type basis functions. Generalisation of the concept of even-tempered basis sets for use in relativistic calculations is proposed. Applications using Gaussian-type and exponential-type functions are presented and the convergence of the second-order relativistic correction with increasing size of basis set is examined. Some attention is paid to the possibility of computation of the third-order energy correction using the first-order correction to the wavefunction.

Fourth-order many-body perturbation-theory study of the electron-correlation contribution to polarizabilities of Ne.

Different electric polarizabilities of Ne are calculated by using the complete fourth-order many-body perturbation-theory (MBPT) method and a partly optimized set of Gaussian-type orbital and contracted Gaussian-type orbital basis functions. The results obtained by the MBPT method (dipole polarizability \ensuremath{\alpha}=2.712 a.u., dipole hyperpolarizability \ensuremath{\gamma}=104.6 a.u., quadrupole polarizability C=3.85 a.u., dipole-quadrupole polarizability B=-17.75 a.u.) are considered as the best currently available theoretical estimates. The importance of the electron-correlation contribution is demonstrated by the comparison with the corresponding self-consistent-field Hartree-Fock results (\ensuremath{\alpha}=2.374 a.u., \ensuremath{\gamma}=63.9 a.u., C=3.20 a.u., B=-12.75 a.u.).

Basis sets for molecular calculations

A general review is presented on the subject of basis sets used in ab initio molecular calculations. Both Slater-type and Gaussian-type basis functions are discussed from various view-points and their relative merits are appraised. A particular emphasis is placed on the problem of the basis set superposition errors (BSSE). This review is prepared particularly for those who are content with the convenience of using built-in basis sets in well-designed computer programs. A bibliography is also included to supplement and update that of Dunning and Hay published in 1977.

Complex Gaussian-type basis functions in the calculation of resonance energies and widths

Basis set calculations of the energies and widths of several resonances in electron scattering by a one-dimensional model potential are carried out using real basis sets that are augmented with various choices for a long-range complex Gaussian-type function. The results show that neither the form of the complex basis function nor the behavior of the trial wave function near the origin are important for obtaining accurate results. In fact, excellent results are obtained with a modest basis set of real bound functions augmented with a single complex Gaussian orbital when the complex exponential parameter is chosen by a stationary condition and the calculation is stabilized with respect to the bound function exponential parameter.

Automatic computation of Bessel function integrals

A numerical method for the calculation of spherical Bessel transforms via Gaussian quadrature in the complex plane is presented. The method is evaluated by transforming Slater and Gaussian-type spherical coordinate basis functions, used in electronic structure calculations, from position space to momentum space. The feasibility and efficiency of the method is explored for different regions of momenta and different orders of the spherical Bessel transform. The results illustrate that in general the method performs very well in the large p regions when applied to the Slater-type functions. On the other hand, a parameter has to be introduced for the application to Gaussian-type functions in order to avoid cancellation. In this case, the method performs well but accuracy is lost at larger p. Use of the parameter in application to Slater functions yields accurate results for all regions of p.

Conformational analysis of primary ethylene ozonide by gradient and multiconfigurational scf calculations

Four conformers of ethylene primary ozonide have been studied by ab initio gradient and MC SCF calculations, using gaussian-type basis functions. The MC SCF results indicate that the conformers are not as close in energy as suggested from single-determinant SCF calculations. The oxygen-oxygen and carbon-oxygen half-chair structures are much lower in energy than the carbon-carbon half-chair.

An AB initio study of the potential surface for the reaction H2CN+H2 → HCH+H + H2

The potential surface for the reactions H2CN+H2 → HCH+H + H2 has been studied by ab initio SCF calculations, using gaussian-type basis functions. A saddle point on the surface has been found, and a reaction path is proposed to explain the observed release of kinetic energy.

An ab initio study of the potential surface for the reaction H 2CO +H → HCO + + H 2

The potential surface for the reaction H 2CO +H → HCO + + H 2 has been studied by ab initio SCF calculations, using gaussian-type basis functions. A saddle point on the surface has been found, and a reaction path is proposed to explain the observed release of kinetic energy. The energy of activation and Δ E for the reaction have been estimated.

Molecular orbital studies of conformers and the barrier to internal rotation in 1,3-butadiene

The potential curve for rotation around the central bond in 1,3-butadiene has been estimated by ab initio calculations using Gaussian-type basis functions. The calculations, which also include limited geometry variation during rotation, suggest that in the SCF approximation the second stable form of the molecule is a gauche conformation rather than a cis. The predicted energy difference between the planar trans ground state and the stable gauche form is 2.7 kcal/mole and the barrier to internal rotation is found to be 6.0 kcal/mole using a (9,5) basis for carbon and 4 s functions on hydrogen.

Electron Correlation in Closed Shell Systems. I. Perturbation Theory Using Gaussian-Type Geminals

The calculation of electron correlation energy in closed-shell atoms and molecules is approached using Rayleigh-Schroedinger perturbation theory with the symmetric sum of Hartree-Fock operators for H0. The alleged advantages of using a VN−1 potential are questioned. Variational equations for first-order pair correlation functions are computed for He, Be, B+ and Ne by expansion in linear combinations of correlated Gaussian-type geminal basis functions containing r122 in the exponent. Such functions form mathematically complete sets, have convenient symmetry properties, and are integrable in closed form. An extensive search for optimum exponential parameters yielded trial functions for each of the ss orbital pairs giving better than 99% of the limiting second-order pair energy using only five basis functions per pair. Similar but less thorough studies of sp and pp pairs in neon are also reported. Careful attention is paid to computational accuracy. An upper bound of −0.3428 a.u. is established on the second-order contribution to the correlation energy in neon.

A theoretical study of the electronic structure and conformation of glyoxal

Ground states of trans-, perpendicular- and cis-glyoxal are studied by ab initio calculations using Gaussian-type basis functions. trans-Glyoxal is found to be the stable form of the molecule. Theoretical values of first ionization potentials and dipole moments are derived, and a population analysis is made.

The electronic structure of boron trifluoride

The electronic structure of BF3 has been calculated by a self-consistent field method using an extended set of Gaussian-type basis functions.

Gaussian-Type Functions for Polyatomic Systems. I

In view of rapid progress of computer capability, it is very desirable to have a reliable assessment of the usefulness of Gaussian-type orbitals as basis functions for large-scale molecular calculations. In the present paper several attempts are made to answer this question mainly at the level of atomic Hartree—Fock calculations. The necessary number of terms of Gaussian-type basis functions in the analytical Hartree—Fock expansion calculation is apparently more than twice as much as the number of terms needed in the expansion with Slater-type basis functions. However, this fact does not necessarily suggest a definite choice of Slater-type orbitals. Discussions pertinent to this point are presented in the latter part of the present paper.