This paper is concerned with the efficient computation of the ubiquitous electron repulsion integral in molecular quantum mechanics. Differences and similarities in organization of existing Gaussian integral programs are discussed, and a new strategy is developed. An analysis based on the theory of orthogonal polynomials yields a general formula for basis functions of arbitrarily high angular momentum. (ηiηj∥ηkηl) = Σα=1,nIx(uα) Iy(uα) I*z(uα) By computing a large block of integrals concurrently, the same I factors may be used for many different integrals. This method is computationally simple and numerically well behaved. It has been incorporated into a new molecular SCF program HONDO. Preliminary tests indicate that it is competitive with existing methods especially for highly angularly dependent functions.

Following an earlier proposal to evaluate electron repulsion integrals over Gaussian basis functions by a numerical quadrature based on a set of orthogonal polynomials (Rys polynomials),a computational procedure is outlined for efficient evaluation of the two-dimensional integrals Zx, Zy , and I,. Compact recurrence formulas for the integrals make the method particularly fitted to handle highangular-momentum basis functions. The technique has been implemented in the HONDO molecular orbital program.

Given any two sets of spin orbitals ai and bj , there exist equivalent sets Iii and bj such that their overlap matrix is diagonal, i.e., (Iii I bi) =diiOij. This is the basis of the corresponding orbital transformation of Amos and Hall. Their transformation is shown to have widespread application to quantum chemistry. It leads to a simple generalization of the Slater-Condon rules for the expectation value of an operator between two determinantal wavefunctions when the spin orbitals of one function have no simple orthogonality relationship to those of the other function. In the case of single,determinantal wavefunctions, use of the corresponding orbital transformation and the integral Hellmann-Feynman formula leads to a very simple expression for the energy difference associated with two similar configurations of a molecular system. Extensions to limited configuration interaction expansions are discussed. Given single-determinantal wavefunctions for two related molecular systems, it is shown that the corresponding orbitals are those which are most nearly molecularly invariant in the sense of maximum overlap. A comparison of the Pitzer-Lipscomb wavefunctions for the staggered and eclipsed forms of ethane reveals that six of the nine corresponding orbitals have an overlap of no less than 0.999998 in the two configurations. Use of the corresponding orbital transformation overcomes various computational difficulties encountered with LOwdin's cofactor method for treating the nonorthogonality problem.

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Static polarizability and second hyperpolarizability tensors are computed for a series of polyenes, polyynes, and cumulenes by ab initio SCF theory. Numerically stable finite field (FF) calculations can be achieved by using polynomial fits of either energy or induced dipole moment as a function of field strength. The nonlinear expansion coefficients from these fits correspond to the microscopic nonlinear optical property. Our results from fully coupled (FF) ab initio calculations for polarizability are in good agreement with those derived from uncoupled (sum-over-states) ab initio methods. The hyperpolarizabilities do not compare as well. A qualitative description of the chain length dependence of polarizability and hyperpolarizability for moderately long chains is discussed in terms of an empirical function. Diffuse orbital basis functions are required for qualitatively correct hyperpolarizabilities of small conjugated systems or for that matter any small molecule. For example, the average second hyperpolarizability, 7, of ethylene is computed to be -13, 1.7, and 726 au with STO-3G, 6-31G, and augmented 3-21G basis sets, respectively. The value computed with the augmented basis set agrees, within a factor of 2, with the experimental value of 1500 au. The valence set for the carbon atoms is augmented with diffuse s, p, and two Cartesian d sets, or subsets of these. The inclusion of diffuse polarization functions drastically alters the computed second hyperpolarizabilities. The results are nearly insensitive to the choice of valence set but are highly dependent on the basis set quality. We also describe the use of a corresponding orbital analysis to aid in the interpretation of ab initio results obtained by either FF or analytic derivative methods. The computed polarizability and hyperpolarizability and the and components indicate that the contribution due to the orbitals is much more significant than the orbitals. Both contributions change sign in going from the ethylenic and acetylenic chains to the cumulenic systems. The polarization of -electron density by the field is illustrated by contour surfaces of derivative -electron-density functions. Contour maps of the first derivative of charge density with respect to the field (a) for an acetylenic chain are nearly periodic, corresponding to localized polarization, whereas the third derivative density (7) corresponds to longer range charge shifts.

Efficient methods are developed for the computation of spin-orbit coupling constants in polyatomic molecules using complete active space multiconfiguration self-consistent field wave functions. All electron–nuclear and electron–electron spin-orbit interactions in the Breit–Pauli Hamiltonian are retained without storing or transforming spin-orbit integrals. This technique is applied to the calculation of spin-orbit coupling constants between singlet and triplet electronic states. Allowing nonorthogonality of the singlet and triplet molecular orbitals in the active space improves the quality of the wave functions and presents no serious computational difficulties. To test the method, spin-orbit coupling constants are computed for the diatomic molecules NH, OH+, PH, and O2 and compared with similar calculations reported in the literature. Calculations are also carried out for the organic biradical trimethylene (ĊH2CH2ĊH2). The coupling constant is found to vary from 0 to 2.5 cm−1 depending upon geometry. It is very sensitive to rotation of the terminal methylenes but relatively insensitive to the CCC angle. These results contribute to our understanding of the role of the triplet state in biradical reactions.

A new computer program for post-processing analysis of quantum-chemical electron densities is described. The code can work with Slater- and Gaussian-type basis functions of arbitrary angular momentum. It has been applied to explore the basis-set dependence of the electron density and its Laplacian in terms of local and integrated topological properties. Our analysis, including Gaussian/Slater basis sets up to sextuple/quadruple-zeta order, shows that these properties considerably depend on the choice of type and number of primitives utilized in the wavefunction expansion. Basis sets with high angular momentum (l = 5 or l = 6) are necessary to achieve convergence for local properties of the density and the Laplacian. In agreement with previous studies, atomic charges defined within Bader's Quantum Theory of Atoms in Molecules appear to be much more basis-set dependent than the Hirshfeld's stockholder charges. The former ones converge only at the quadruple-zeta/higher level with Gaussian/Slater functions.

Abstractk ovkr to understand the hindered rotatiomd and vibrational dynamics of methane trapped in Cm i.ntcrstices and to determine the structure around the interstitial site, we have carried out inelastic neutxcm scattering studies of the methane/C@ system. At temperatures of 20K and below, we observe inelastic peaks from rotational transitions of the C& These transitions allow unambiguous assignment of the hindered rc-tational energy levels and a determination of the inkradcm potential. The appearance of two peaks for.one of the J = 0+3 transitions implies the existence of wo distinct kinds of interstitial sires and the maured transition energies suggest a rotational barrier of about 26 and 16 meV for these sites. Time,dependent changes in peak heights indicate slow (tln ZY 2.6 hrs) triplet+quimet nuclear spin conversion that necessarily accompapks the J = 1+0 rotational rehtxatiom We aiso have observed a sharp inektic peak at 9.3 meV, which corresponds to a local vlhatkmal mode of (XQ rattling in its cage at -2,2 TETk.C)ther peaks involving higher-enagy vibrational excitations in CD~C@ correspond in energy to assigned peaks in the inelastic neutron scattering spectra of Ca, albeit sometimes with different intensities.