Stanley A. Hagstrom

ONE-CENTER r/sub ij/ INTEGRALS OVER SLATER-TYPE ORBITALS.

Expressions are derived for the integrals which arise in calculations of atomic wavefunctions of four or more electrons, utilizing the method of explicit introduction of interelectronic distances rij into a configuration interaction wavefunction (subject to the restriction of at most one rij per configuration). The approach followed in the derivations is similar to one followed by Ohrn and Nordling for three electron integrals; however, in extending the Ohrn and Nordling scheme to four electrons for a basis of s, p, and d Slater‐type orbitals, numerical snags are encountered which have been obviated by an Euler transformation on certain of the auxiliary integrals.

High precision variational calculations for the Born-Oppenheimer energies of the ground state of the hydrogen molecule

Born-Oppenheimer approximation Hylleraas variational calculations with up to 7034 expansion terms are reported for the 1sigma(g)+ ground state of neutral hydrogen at various internuclear distances. The nonrelativistic energy is calculated to be -1.174 475 714 220(1) hartree at R = 1.4 bohr, which is four orders of magnitude better than the best previous Hylleraas calculation, that of Wolniewicz [J. Chem. Phys. 103, 1792 (1995)]. This result agrees well with the best previous variational energy, -1.174 475 714 216 hartree, of Cencek (personal communication), obtained using explicitly correlated Gaussians (ECGs) [Cencek and Rychlewski, J. Chem. Phys. 98, 1252 (1993); Cencek et al., ibid. 95, 2572 (1995); Rychlewski, Adv. Quantum Chem. 31, 173 (1998)]. The uncertainty in our result is also discussed. The nonrelativistic energy is calculated to be -1.174 475 931 399(1) hartree at the equilibrium R = 1.4011 bohr distance. This result also agrees well with the best previous variational energy, -1.174 475 931 389 hartree, of Cencek and Rychlewski [Rychlewski, Handbook of Molecular Physics and Quantum Chemistry, edited by S. Wilson (Wiley, New York, 2003), Vol. 2, pp. 199-218; Rychlewski, Explicitly Correlated Wave Functions in Chemistry and Physics Theory and Applications, edited by J. Rychlewski (Kluwer Academic, Dordrecht, 2003), pp. 91-147.], obtained using ECGs.

Single‐Center Wave Function for the Hydrogen Molecule

The ground state of the hydrogen molecule is studied using an expansion based on a single center, the molecular midpoint, with basis orbitals constructed from associated Laguerre functions with a single orbital exponent. The convergence of the expansion is studied by systematic addition of terms and is found to be slow. The best wave functions attained have energies of —1.15086 (38 axially symmetric terms), and —1.16141 (44 terms). The results are shown to be very similar to those obtained by using Slater orbitals with nonintegral principal quantum number both from the energy increments observed and from a natural spin orbital occupation number analysis. It is concluded that the slow convergence probably results from failure to represent adequately the singularities at the nuclei, and that further use of single‐center expansions in diatomic problems (except at very small internuclear distances) seems unprofitable, irrespective of what set of orbitals is used as a basis.

Mathematical and computational science issues in high precision Hylleraas-configuration interaction variational calculations: I. Three-electron integrals

The most difficult integral arising in Hylleraas-configuration interaction (Hy-CI) calculations, the three-electron triangle integral, is discussed. We focus on recursive techniques at both the double precision and quadruple precision level of accuracy while trying to minimize the use of higher precision arithmetic. Also, we investigate the use of series acceleration to overcome problems of slow convergence of certain integrals defined by infinite series. We find that a direct + tail Levin u-transformation convergence acceleration overcomes problems that arise when using other convergence acceleration techniques, and is the best method for overcoming the slow convergence of the triangle integral. The question of calibrating an acceleration method is also discussed, as well as ways to improve our work.

Transferability of electron pairs between H2O and H2O2

The transferability of strongly orthogonal geminals between H2O and H2O2 is investigated. Wavefunctions for H2O are constructed from the appropriate geminals in H2O2. Likewise, except for optimization of the oxygen‐oxygen bond geminal (ΛOO), wavefunctions for H2O2 are constructed from the appropriate geminals in H2O. The results are encouraging. In all cases, the energies of the wavefunctions constructed from the transferred geminals are actually lower than the energies of the optimum MO wavefunctions and are close to the energies of the optimum geminal wavefunctions. The optimum geminals of H2O are compared with those of H2O2. The corresponding geminals are found to be almost identical. Some factors which influence the success of geminal transferability are examined. In particular, a view of geminal localization is presented, and shielding and inductive effects are discussed. Owing to substantial shielding of the secondary oxygen nucleus in H2O2, the effective fields seen by the OH geminals turn out to b...

Nonorthogonal configuration interaction study of the ground state of BeH2

A series of configuration interaction (CI) calculations are carried out on the 1Σg+ ground state of the BeH2 molecule using a nonorthogonal one‐electron basis of elementary symmetry orbitals formed from real Slater‐type orbitals. A number of different atomic‐orbital basis sets are used. For the largest basis, a calculated energy of −15.8478 hartree at an equilibrium Be–H bond distance of 2.52 bohr is obtained with an optimized 80‐term wavefunction. Based on previous estimates of the Hartree‐Fock energy and the present results, this figure represents approximately 55% of the total correlation energy of the molecule, which is estimated to be −0.140 hartree. Using potential curves obtained from this wavefunction, the fundamental symmetric and asymmetric vibration frequencies are calculated to be 2083 and 2497 cm−1, respectively. The related force constants are also given. There are no experimental data available for comparison. The computed wavefunctions are discussed and comparisons made with elementary val...

Mathematical and computational science issues in high precision Hylleraas-configuration interaction variational calculations: II. Kinetic energy and electron-nucleus interaction integrals

Three-electron kinetic energy and electron?nucleus interaction integrals arising in Hylleraas?configuration interaction (Hy?CI) calculations are discussed. We show that the electron?nucleus interaction operators introduce no new integral complications and that the kinetic energy operators lead to integrals which have much in common with electron interaction integrals treated in the first paper of this series. We discuss the efficient evaluation of the kinetic energy integrals and give selected values of the integrals to 30 digit accuracy. The one new W auxiliary function which arises in treating the kinetic energy operator is discussed in appendix B. We also give a few details on accurate evaluations of the Condon and Shortley cks which arise in Hy?CI calculations in appendix A.

Hylleraas-configuration-interaction study of the 1 S ground state of neutral beryllium

Hylleraas-configuration-interaction (Hy-CI) method variational calculations are reported for the {sup 1}S ground state of neutral beryllium. The best nonrelativistic energy obtained was -14.667 356 4 hartree, which is estimated to be accurate to a tenth of a microhartree.

Transferability of electron pair wavefunctions. III. A minimal basis strongly orthogonal geminal calculation of the hydrogen peroxide hindered rotation potential

Preliminary to wavefunction transfer studies, we present an ab initio strongly orthogonal geminal (SOG) calculation of the H2O2 hindered rotation potential. With geometry optimized, a cis barrier of 8.6 kcal/mole and a zero trans barrier are computed. The optimum OO bond length (2.811 bohr) and OOH angle (97.0°) agree well with experiment, but the OH bond length (1.950 bohr) is longer than experimental. The trans barrier is discussed and attributed to an eclipsed OH bond–lone pair interaction which stabilizes the experimental equilibrium conformation. It is argued on the basis of (a) electron distribution differences, (b) correlation effects in the OO bond, (c) differences between SCF calculations and experiment, and (d) SOG improvement of SCF minimal basis results that correlation cannot be ruled out as a factor in the cis barrier.

Transferability of electron pair wavefunctions. IV. Recreation of the hydrogen peroxide hindered rotation potential

Transferability of strongly orthogonal geminals is used to reproduce the hindered rotation potential of hydrogen peroxide. Geminals optimized at one conformation are transferred via a truncation–rotation–reorthogonalization process to a variety of other conformations about the curve. In almost every case, the optimized cis barrier is reproduced to within 10% and the spectrum of torsional levels within the curve constructed by transfer closely resembles that within the optimized curve. Slight errors arising from restricted geometry and from truncation and reorthogonalization of the wavefunction are discussed and assessed.