Coupled-cluster theory in a projected atomic orbital basis

We present a biorthogonal formulation of coupled-cluster (CC) theory using a redundant projected atomic orbital (PAO) basis. The biorthogonal formulation provides simple equations, where the projectors involved in the definition of the PAO basis are absorbed in the integrals. Explicit expressions for the coupled-cluster singles and doubles equations are derived in the PAO basis. The PAO CC equations can be written in a form identical to the standard molecular orbital CC equations, only with integrals that are related to the atomic orbital integrals through different transformation matrices. The dependence of cluster amplitudes, integrals, and correlation energy contributions on the distance between the participating atomic centers and on the number of involved atomic centers is illustrated in numerical case studies. It is also discussed how the present reformulation of the CC equations opens new possibilities for reducing the number of involved parameters and thereby the computational cost.     

Numerical integration techniques for quantum chemistry

Korobov theory for multidimensional numerical integration is used to evaluate electronic integrals. This paper shows the important role played by periodization techniques. Singularity (r ₁₂ ^?1 ) in the bielectronic six-dimensional integrals is removed through a twofold three-dimensional integration. Results are presented for atomic integrals involving Slater type atomic orbitals.     

Note on an integral of A. F. Saturno

A simple derivation is given of an integral discussed by Saturno in connection with certain atomic inter-electron repulsion integrals. This derivation extends the method to any type of atomic orbital, and it can be used to derive the values of similar Coulomb integrals when the two atomic centres are a distance R apart.     

Highly accurate evaluation of atomic three-electron integrals of lowest orders

Calculations of three-electron atomic systems in Hylleraas coordinates require integrals involving all the interparticle distances r(ij), which have usually been evaluated by introducing series expansions. For integrals with the smallest powers of r(ij) these expansions do not converge at a satisfactory rate, leading some investigators to introduce convergence-acceleration procedures. This paper recommends the alternative of evaluating these integrals in closed form and presents stable explicit formulas for so doing. Some of the formulas are more compact versions of those in the literature; others have not been previously reported. It is also shown that finite-difference methods can be used with advantage to obtain additional low-order integrals. Sample integral values have been provided for test purposes.     

Unified treatment of complex and real rotation-angular functions for two-center overlap integrals over arbitrary atomic orbitals

The new combined formulas have been established for the complex and real rotation-angular functions arising in the evaluation of two-center overlap integrals over arbitrary atomic orbitals in molecular coordinate system. These formulas can be useful in the study of different quantum mechanical problems in both the theory and practice of calculations dealing with atoms, molecules, nuclei and solids when the integer and noninteger n complex and real atomic orbitals basis sets are emploed. This work presented the development of our previous paper (I.I. Guseinov in Phys Rev A 32:1864, 1985).     

Valence-bond calculations on ZNO and HGO using integrals computed through the semiempirical AM1 method

Valence-bond calculations have been carried out on ZnO and HgO using a basis set of Slatertype atomic orbitals and the one- and two-electron integrals as computed in the semiempirical AM 1 molecular orbital method. The zero differential overlap approximation has been used to calculate integrals between atomic orbital Slater determinants using the rules for matrix elements between determinants formed by orthogonal orbitals. Diabatic and adiabatic curves have been analyzed for the two systems, and results compared with molecular orbital AM 1 results. © 1992 John Wiley & Sons, Inc.     

Study of localized molecular orbitals using group theory methods and its approach to the many-electron correlation problem. IV. The symmetry-adaptation of many-center integrals and hamiltonian matrix elements in MCSCF calculations

Group theoretic methods are presented for the transformations of integrals and the evaluation of matrix elements encountered in multiconfigurational self-consistent field (MCSCF) and configuration interaction (CI) calculations. The method has the advantages of needing only to deal with a symmetry unique set of atomic orbitals (AO) integrals and transformation from unique atomic integrals to unique molecular integrals rather than with all of them. Hamiltonian matrix element is expressed by a linear combination of product terms of many-center unique integrals and geometric factors. The group symmetry localized orbitals as atomic and molecular orbitals are a key feature of this algorithm. The method provides an alternative to traditional method that requires a table of coupling coefficients for products of the irreducible representations of the molecular point group. Geometric factors effectively eliminate these coupling coefficients. The saving of time and space in integral computations and transformations is analyzed. © 1994 by John Wiley & Sons, Inc.     

Intracule functional models. II. Analytically integrable kernels

We present, within the framework of intracule functional theory (IFT), a class of kernels whose correlation integrals can be found in closed form. This approach affords three major advantages over other kernels that we have considered previously; ease of implementation, computational efficiency, and numerical stability. We show that even the simplest member of the class yields reasonable estimates of the correlation energies of 18 atomic and 56 molecular systems and we conclude that this kernel class will prove useful in the development of future IFT models.     

Analytical evaluation of three-center nuclear-attraction integrals over s-Slater orbitals for asymmetrical conformations of the centers

Analytical formulas for three-center nuclear-attraction integrals over Slater orbitals are given for any location of the three atomic centers. In the mathematical derivations the Neumann expansion has been used and new general auxiliary integrals which depend on the elliptical coordinates of one of the centers are defined. The orbital exponents within the integrals may be different.     

Use of green's coulomb function in perturbation theory for atoms

Green's Coulomb function for negative values of energy is applied to the calculation of sums and integrals arising in second-order field form of atomic perturbation theory.     

A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians

Utilizing the fact that solid-harmonic combinations of Cartesian and Hermite Gaussian atomic orbitals are identical, a new scheme for the evaluation of molecular integrals over solid-harmonic atomic orbitals is presented, where the integration is carried out over Hermite rather than Cartesian atomic orbitals. Since Hermite Gaussians are defined as derivatives of spherical Gaussians, the corresponding molecular integrals become the derivatives of integrals over spherical Gaussians, whose transformation to the solid-harmonic basis is performed in the same manner as for integrals over Cartesian Gaussians, using the same expansion coefficients. The presented solid-harmonic Hermite scheme simplifies the evaluation of derivative molecular integrals, since differentiation by nuclear coordinates merely increments the Hermite quantum numbers, thereby providing a unified scheme for undifferentiated and differentiated four-center molecular integrals. For two- and three-center two-electron integrals, the solid-harmonic Hermite scheme is particularly efficient, significantly reducing the cost relative to the Cartesian scheme.     

Unrestricted Hartree-Fock calculations of spin densities with orthogonalized atomic orbitals: Proton and 13C hyperfine splittings in some pi hydrocarbon radicals

The spin density distribution in a few hydrocarbon radicals has been calculated using orthogonalized atomic orbitals in the Unrestricted Hartree-Fock formalism of Amos and Snyder and including certain more important two-electron hybrid and exchange integrals and all the core-resonance integrals. Our calculated spin densities for the cation and anion radicals of alternant hydrocarbons, which are now different due to the breakdown of the pairing theorem, are, in general, of the right relative order so that even the simple McConnell type of relation can account partly for the observed differences in the proton splittings between cations and anions. The proton splittings for position 2 of naphthalene and anthracene radical ions are correctly predicted, thus clearing up the well-known cation-anion anomaly for this position. Comparative calculations have been made to show that the spin density results are worsened with the neglect of the integrals of the type mentioned before. An empirical analysis correlating the observed ¹³C splittings and the spin density results over a non-orthogonal basis set shows that the available ¹³C splittings in alternant hydrocarbon radical ions can be explained with a set of sigma-pi parameters which are consistent with the theory. It is shown that even though the spin densities in cations and anions may be different, these can lead to similar ¹³C splittings.     

Spin-orbit (core) and core potential integrals

Molecular integral formulas and corresponding computational algorithms are developed for the relativistic spin-orbit and core potential operators that are obtained from atomic relativistic calculations by means of the effective core potential procedure. Much use is made of earlier work on core potential integrals by McMurchie and Davidson. The resulting computer code has been made part of the ARGOS (Argonne, Ohio State) program from the C?OLUMBUS suite of programs, which computes the needed integrals over symmetry-adapted combinations of generally contacted Gaussian atomic orbitals.     

Electric and magnetic multipole integrals in STO basis sets: an analytical approach

A new method for the evaluation of one- and two-centre magnetic and electric multipole integrals for Slater-type functions is presented. The method is strictly analytical in that no approximations of any kind are involved. Two simple functions, ℐ₁ ^aug and ℐ₂ ^aug, are introduced, which employ only functions that are well known in electronic structure theory. With the use of augmentation exponents these functions apply to multipole integrals as well as other one-electron integrals, such as nuclear attraction integrals. The proposed method includes the analytic determination of derivatives of the integrals with respect to atomic displacements. Some illustrative test calculations are presented and compared to results from the literature. Received: 20 April 1998 / Accepted: 13 October 1998 / Published online: 1 February 1999     

Formulation of the LCAS MS SCF method within the Gaussian basis set

Within the presented LCAS MS (linear combination of atomic spinors–molecular spinors) SCF formalism both large and small components of the spinor radial parts have been expanded within the Gaussian basis set. The respective expressions for matrix elements as well as for one- and two-electron integrals are given.     

Spin-orbit interaction in polyatomic molecules: Ab initio computations with Gaussian orbitals

Standard sets of Gaussian atomic orbitals (STO -3G , STO -4.31G ) are used to evaluate spin-orbit coupling constants in linear molecules (CO, NNN) and spin-orbit effects on singlet–triplet transition intensities in formaldehyde. All spin-other orbit effects have been included. In all cases spin-other orbit interactions form a large fraction of the matrix elements. Simple formulae to evaluate spin-orbit one- and two-electron integrals over atomic orbitals are presented. Standard molecular integral programs can be used for the computation of spin-orbit integrals.     

The general theory of the extended method of calulation of atomic structures

A general theory of the extended method of calculation of atomic structures having complex configurations is presented. It is based on the correspondence between the radial integrals of the ordinary method and the linear combinations of those of the extended method. The rules for going over from the ordinary method to the extended one are given.     

Orbital-invariant formulation and second-order gradient evaluation in Møller-Plesset perturbation theory

Based on the Hylleraas functional form, the second and third orders of Møller-Plesset perturbation theory are reformulated in terms of arbitrary (e.g., localized) internal orbitals, and atomic orbitals in the virtual space. The results are strictly equivalent to the canonical formulation if no further approximations are introduced. The new formalism permits the extension of the local correlation method to Møller-Plesset theory. It also facilitates the treatment of weak pairs at a lower (e.g., second order) level of theory in CI and coupled cluster methods. Based on our formalism, an MP2 gradient algorithm is outlined which does not require the storage of derivative integrals, integrals with three external MO indices, and, using the method of Handy and Schaefer, the repeated solution of the coupled-perturbed SCF equations.     

Generalized hybrid-orbital method for combining density functional theory with molecular mechanicals.

The generalized hybrid orbital (GHO) method has previously been formulated for combining molecular mechanics with various levels of quantum mechanics, in particular semiempirical neglect of diatomic differential overlap theory, ab initio Hartree-Fock theory, and self-consistent charge density functional tight-binding theory. To include electron-correlation effects accurately and efficiently in GHO calculations, we extend the GHO method to density functional theory in the generalized-gradient approximation and hybrid density functional theory (denoted by GHO-DFT and GHO-HDFT, respectively) using Gaussian-type orbitals as basis functions. In the proposed GHO-(H)DFT formalism, charge densities in auxiliary hybrid orbitals are included to calculate the total electron density. The orthonormality constraints involving the auxiliary Kohn-Sham orbitals are satisfied by carrying out the hybridization in terms of a set of L?wdin symmetrically orthogonalized atomic basis functions. Analytical gradients are formulated for GHO-(H)DFT by incorporating additional forces associated with GHO basis transformations. Scaling parameters are introduced for some of the one-electron integrals and are optimized to obtain the correct charges and geometry near the QM/MM boundary region. The GHO-(H)DFT method based on the generalized gradient approach (GGA) (BLYP and mPWPW91) and HDFT methods (B3 LYP, mPW1PW91, and MPW1 K) is tested-for geometries and atomic charges-against a set of small molecules. The following quantities are tested: 1) the C--C stretch potential in ethane, 2) the torsional barrier for internal rotation around the central C--C bond in n-butane, 3) proton affinities for a set of alcohols, amines, thiols, and acids, 4) the conformational energies of alanine dipeptide, and 5) the barrier height of the hydrogen-atom transfer between n-C4H10 and n-C4H9, where the reaction center is described at the MPW1 K/6-31G(d) level of theory.     

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.