Self consistent molecular orbital calculations on polyatomic molecules: Gaussian approximation for two-electron integrals

A MO –LCAO –SCF treatment is performed on water, ammonia and methane using a recently proposed approximate method. The procedure is found capable of predicting total and orbital energies in close agreement with the results of accurate computations using double ζ basis sets of Slater type orbitals. A comparison is made with the results of similar approximate ab initio procedures.     

Symmetry properties of one- and two-electron molecular integrals

The maximum numbers of distinct one- and two-electron integrals that arise in calculating the electronic energy of a molecule are discussed. It is shown that these may be calculated easily using the character table of the symmetry group of the set of basis functions used to express the wave function. Complications arising from complex group representations and from a conflict of symmetry between the basis set and the nuclear configuration are considered and illustrated by examples.     

Gaussian expansions of orbital products for the evaluation of two-electron integrals

A method is described for evaluating multicenter integrals over contracted gaussian-type orbitals by the use of gaussian expansion of orbital products. The expansions are determined by the method of non-linear least squares with constraints. There is no restriction upon the symmetry of the orbital product and the method is applicable to all products arising from s, p and d-type orbitals. Results are given to indicate the accuracy of the method.     

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm ⁵-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.     

Multithreaded parallelization of the energy and analytic gradient in the fragment molecular orbital method

The fragment molecular orbital method in GAMESS is parallelized in a multithreaded OpenMP implementation combined with the MPI version of the two-level generalized distributed data interface. The energy and analytic gradient in gas phase and the polarizable continuum model of solvation are parallelized in this hybrid three-level scheme, achieving a large memory footprint reduction and a high parallel efficiency on Intel Xeon Phi processors. The parallel efficiency is demonstrated on the Stampede2 and Theta supercomputers using up to 2048 nodes (262 144 threads).     

Computation of some new two-electron Gaussian integrals

Summary The evaluation of a new form of two-electron integrals is required if the interelectronic distancer ₁₂ is used as a variable in then-electron functions of electron correlation methods. The McMurchie-Davidson algorithm for the generation of molecular integrals over Gaussian-type functions is ideally suited to this. The new Gaussian integrals are formed from Hermite integrals overr ₁₂ (rather than 1/r ₁₂) by standard techniques. The Hermite integrals overr ₁₂ itself are generated by a simple procedure with negligible computational effort. The key results are discussed in the context of general recursion formulas. On leave from: Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, W-4630 Bochum, Germany     

Semi-asymptotic evaluation of certain three-center two-electron integrals

Two-electron repulsion integrals between a two-center charge distribution and a charge distribution about a third center, which do not appreciably interpenetrate, are shown to be given to useful accuracy by numerical differentiation of certain three-center one-electron integrals. This method also may be used to evaluate integrals of this type for which the Mulliken or Sklar approximations are inapplicable.Supported by Contract SD-102 with the Advanced Research Projects Agency.     

STM studies of single molecules: molecular orbital aspects

As a fundamental and frequently referred concept in modern chemistry, the molecular orbital plays a vital role in the science of single molecules, which has become an active field in recent years. For the study of single molecules, scanning tunneling microscopy (STM) has been proven to be a powerful scientific technique. Utilizing specific distribution of the molecular orbitals at spatial, energy, and spin scales, STM can explore many properties of single molecule systems, such as geometrical configuration, electronic structure, magnetic polarization, and so on. Various interactions between the substrate and adsorbed molecules are also understood in terms of the molecular orbitals. Molecular engineering methods, such as mode-selective chemistry based on the molecular orbitals, and resonance tunneling between the molecular orbitals of the molecular sample and STM tip, have stimulated new advances of single molecule science.     

Analytical second derivatives of molecular electronic energy integrals obtained by spherical gaussian orbital

In this research, the complete general formulas for the analytical second derivative of the molecular integrals for spherical gaussian orbitals of electronic energy are presented. Formulas were given for the second derivative for orbital exponent, orbital and nuclear cartesian coordinates and coefficients of contracted gaussians. In order to save computational time, the formulas for the second derivative are written in terms of the original integrals. Although the formulas were presented in general for any type of application, the Floating Spherical Gaussian Orbital (FSGO) method is applied to some molecules such as LiH, H₂O and CH₂ (singlet) to check the formulas. The results were compared with the results of the finite difference method. Besides the accuracy of the analytical derivative, the saving in computational time is significant.     

Analytic derivatives for the Cholesky representation of the two-electron integrals

We propose a formalism for calculating analytic derivatives of the electronic energy with respect to nuclear coordinates using Cholesky decomposition of the two-electron integrals. The formalism is derived by exploiting the equivalence of Cholesky decomposition and density fitting when a suitable auxiliary basis set is used for expanding atomic orbital product densities in the latter. An implementation of gradients at the nonhybrid density functional theory level is presented, and sample calculations demonstrate that the errors in equilibrium geometries due to the Cholesky representation of the integrals can be controlled by adjusting the decomposition threshold.     

Pre-processing two-electron integrals for efficient utilization in many-electron self-consistent field calculations

A procedure for pre-processing a random list of two-electron integrals is described. Construction of Fock-matrices with the resulting PK-file is shown to be three times faster than use of a file containing the random two-electron integrals, indices, and a branching code.     

Many-Electron,multicenter integrals in superposition of correlated configurations method. I. one- and two-electron integrals

The calculations by means of the superposition of correlated configurations method (Hylleraas-CI ), that is, the combination of configuration interaction with the Hylleraas-type correlation factors, needs the effective evaluation of some nontrivial integrals. This series of papers gives the formulas for all types of integrals needed for molecular calculations when Gaussian lobe functions are used as a basis set. The formulas for two-electron integrals are given in the present paper. The preliminary results for two-electron systems are presented.     

Careful neglect of small two-electron integrals applied to borazane and diazirine

A formulation is given to systematically neglect two-electron integrals in a gaussian lobe basis set smaller than a prescribed threshold. With a threshold of 10^?4 au the energies are unchanged to seven significant figures and the dipole moments to four for borazane and diazirine. This technique offers a 15-20% saving in computation time for these integrals with a negligible loss of accuracy.     

Multipole-based integral estimates for the rigorous description of distance dependence in two-electron integrals

We derive multipole-based integral estimates (MBIE) as rigorous and tight upper bounds to four-center two-electron integrals in order to account for the 1/R distance decay between the charge distributions, which is missing in the Schwarz screening commonly used in ab initio methods. Our screening criteria are valid for all angular momenta and can be formulated for any order of multipoles. We have found the expansion limited to dipoles to be sufficiently tight for estimating the integrals in Hartree-Fock and density-functional theories, while the screening effort is negligible. For, e.g., a DNA fragment with 1052 atoms and 10,674 basis functions (6-31G*) the exchange part is faster by a factor of 2.1 as compared to the Schwarz screening both within our linear exchange scheme, whereas a smaller factor of 1.3 is gained for the Coulomb part within the continuous fast multipole method. Most importantly, our new MBIE screening is perfectly suited to exploit the strong distance decay of electron-correlation effects of at least 1/R4 in atomic-orbital-based formulations of correlation methods.     

An economic storage and processing method for two-electron integrals in LCAO-MO calculations

An economic algorithm for storing and processing two-electron integrals arising in LCAO-MO calculations is presented. The integrals are sorted prior to the SCF iterative scheme, classified according to equivalences in the orbital indices and finally stored on separate files that contain only integrals of one type. The novel approach of physically separating the integrals according to category is shown to be more efficient than random storage. Actual computing times for the new technique are tabulated for a representative number of molecular systems and compared with times obtained using previously reported methods.