Variational solution of the dirac equation within a multicentre basis set of gaussian functions

The Dirac equation for one-electron polynuclear systems is solved variationally within an analytical multicentre basis set. Each of the four components of the molecular spinor is expanded separately into a scalar basis set. The expansion coefficients are obtained from the variational principle. Calculations using gaussian functions are performed for hydrogen-like atoms and for hydrogen-ion-like molecules with constituents of nuclear charge Z = 1, 50 and 90.     

Polarization functions for gaussian basis sets for the first row atoms

Exponent optimization was performed for a single set ofd-type Gaussians on the first row atoms C, N, and O in fifteen small molecules. The hydrogenp-exponents were kept at the fixed value of 1.0. For the underlying valence shell basis sets, Dunning's double zeta basis sets were used. Standard exponents of polarization functions are suggested for the most common valence states of the C, N, and O atoms.     

A mixed basis set of slater and gaussian functions for molecular calculations

A procedure for using simultaneously Slater and Gaussian basis functions in molecular calculations is presented here. The analytic expressions of the integral prototypes involving both Slater and Gaussian functions are explicitly derived.     

Geometry optimizations with a small contracted gaussian basis set

The STO-2G basis set has been used to calculate equilibrium geometries of 27 molecules. A comparison with STO-3G and experimental results indicates that the exploration of potential energy surfaces of large organic molecules can be efficiently carried out using the STO-2G basis set.     

A general nonlinear expansion form for electronic wave functions

A new expansion form is presented for electronic wave functions. The wave function is a linear combination of product basis functions, and each product basis function in turn is formally equivalent to a linear combination of configuration state functions that comprise an underlying linear expansion space. The expansion coefficients that define the basis functions are nonlinear functions of a smaller number of variables. The expansion form is appropriate for both ground and excited states and to both closed and open shell molecules. The method is formulated in terms of spin-eigenfunctions using the graphical unitary group approach (GUGA), and consequently it does not suffer from spin contamination.     

The ellipsoidal gaussian basis in molecular orbital theory

New analytic integral formulas are presented for the potential energy integrals over ellipsoidal Gaussian basis functions [ exp (-x² - y² - z²)] that enter into solving the conventional expansion self-consistent field equations. Near minimal atomic orbital bases combined from large nuclear-centered primitive Gaussian sets are used in test calculations on the HF and CO molecules. The ellipsoidal exponential parameters for the valence atomic orbitals are fully optimized using a single scale factor for each atomic orbital and nuclear coordinate. The results are compared with those obtained using an unoptimized nuclear centered double-zeta spherical Gaussian basis.     

Hartree–Fock wave functions with a modified GTO basis for atoms

We have solved the atomic Hartree–Fock equations by using the algebraic approach, expanding the single-particle radial wave function in terms of a modified Gaussian type orbitals (GTOs) basis. Several atomic properties such as Kato's cusp condition for the electron density or the correct asymptotic behavior of the electron momentum density distribution are accurately verified. Additionally the energy of the atomic ground state can be obtained by using a smaller number of basis functions than in standard GTO expansions. This study has been performed for several atoms of the first three rows. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 59–64, 1997     

Calculations using a set of evenly spaced S-type gaussian functions

A basis set of evenly spaced S-type Gaussian functions with common exponents is examined. Formulas for common one- and two-electron integrals are derived. Because of thesymmetry of this basis set, a very compact two-electron integral list is produced. The number of two-electron integrals that must be stored is approximately eight times the number of basis functions. Use of this basis set in an SCF calculation is examined. Numerical results show that this approach works well for molecules containing only small atoms such as hydrogen, helium, or lithium, but that the method has problems with the core orbitals of heavier atoms. Procedures for augementing this basis set in calculations involving heavier atoms are examined.     

Continuum wave functions by least-squares scheme in a B-spline basis: Multicenter and multielectron formulations

Application of a purely algebraic least-squares approach in a spline basis to the determination of continuum orbitals in the single-electron molecular case illustrates the excellent numerical properties of the method. Initial work based on a single-center expansion is extended to an LCAO formulation within a single-particle Hamiltonian and to the full multielectron problem in the one-center scheme. Preliminary results on H and He photoionization are reported. © 1994 John Wiley & Sons, Inc.     

Nonempirical calculations of dipole moments of molecules in semifloating gaussian basis sets

For the example of the calculation of the dipole moments of the HF, HCl, H₂O, NH₃, CO, H₂CO, CH₃F molecules in two-exponent and three-exponent Gaussian basis sets, we have studied the effect of including floating functions in the basis, directly giving the effect of polarization of the electron shell of the atom in the molecule. We have established a weak dependence of the calculated dipole moment on the dimensionality of the basis, the number of floating functions, and also the orbital exponents of the hydrogen atoms. The correction introduced by the floating functions in molecules with polar bonds is considerably greater than the correlation correction. The proposed approach allows us to decrease the dimensionality of the orbital basis by a factor of 1.5–2 without making the agreement with experiment worse.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 26, No. 4, pp. 481–485, July–August, 1990.     

Exact wave functions for few-particle systems: The choice of expansion for coulomb potentials

We discuss a procedure for calculating numerical Hartree–Fock orbitals that can be applied to polyatomic systems. This approach is formulated in momentum space to avoid Coulomb singularities and uses fast Fourier transforms to solve integral convolutions. Results for a number of simple systems are presented.     

Calculating vibrational energies and wave functions of vinylidene using a contracted basis with a locally reorthogonalized coupled two-term Lanczos eigensolver

We use a contracted basis+Lanczos eigensolver approach to compute vinylidene-like vibrational states of the acetylene-vinylidene system. To overcome problems caused by loss of orthogonality of the Lanczos vectors we reorthogonalize Lanczos vector and use a coupled two-term approach. The calculations are done in CC-HH diatom-diatom Jacobi coordinates which make it easy to compute states one irreducible representation at a time. The most costly parts of the calculation are parallelized and scale well. We estimate that the vinylidene energies we compute are converged to approximately 1 cm(-1).     

Analysis of molecular orbital wave functions in terms of valence bond functions for molecular fragments. I. Theory

A method of expansion of molecular orbital wave functions into valence bond (VB ) functions is extended to molecular fragments. The wave function is projected onto a basis of mixed determinants, involving molecular orbitals as well as fragment atomic orbitals, and is further expressed as a linear combination of VB functions, characteristic of structural formulas of the fragment but whose remaining bonds are frozen. Structural weights for the fragment are deduced from this expression. Delocalized molecular orbitals are used as a startpoint, as they are after an ordinary SCF calculation. Wave functions of medium-sized molecules may be analyzed with reasonable storage requirements in a computer.     

Total energies of molecules with the local density functional approximation and gaussian basis sets

Total energies of small molecules were calculated with a local density functional (LDF) approximation within the LCAO MO SCF scheme. The local spin density functional (LSD) of Gunnarsson and Lundqvist was used. The basis sets used are of contracted gaussian type which allow comparison of LSD with Hartree-Fock (HF) results. The program for calculation of the LSD term was incorporated into the standard ab initio package. The LSD binding energies were in better agreement with experiment than those from HF.     

A hybrid of exponential and gaussian functions as a simple model of asymmetric chromatographic peaks

A hybrid of exponential and Gaussian functions is developed as a model of asymmetric peak profiles. This exponential-Gaussian hybrid function (EGH) is mathematically simple, numerically stable, and its parameters are readily determined by making graphical measurements and applying simple equations. Furthermore, the statistical moments of the EGH function can be accurately approximated (within+/-0.15% error) at any level of asymmetry using formulae that are easily programmed into a computer. These features of the EGH make it very easy to implement by most chromatographers. The EGH serves as a useful alternative to the exponentially modified Gaussian (EMG) for modeling slightly asymmetric peaks since the two models produce nearly the same profile at relatively low asymmetries. The EGH also serves as an addition to the extensive list of alternative models that are sometimes better than the EMG at describing highly asymmetric peaks. A comparison between EMG and EGH curves at various asymmetries is made by analysis of toluene, phenylalanine, and pyridine on a reversed-phase liquid chromatographic system.