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.     

The hydrogen molecule in strong magnetic fields: optimizations of anisotropic Gaussian basis sets

The anisotropic Gaussian basis sets were optimized for the H atom and the hydrogen molecule in strong magnetic fields of 0-1000 a.u. We used five-parameter fit functions to generate anisotropic Gaussian exponents of hydrogenic atomic orbitals. These functions provided errors of energy that were comparable to the independent optimization of all the exponents. The optimal exponents were used to calculate the Hartree-Fock energies of H2 at arbitrary orientations, with respect to the magnetic field. Furthermore, the double-exponential transformation was applied to calculate highly anisotropic Coulomb integrals. Between magnetic field strengths of 1 a.u. and 100 a.u., a molecule in a triplet ground state continuously changed its stable orientation from the perpendicular geometry to the parallel geometry.     

Non-integer Slater orbital calculations

In order to calculate the one- and two-electron, two-center integrals over non-integer n Slater type orbitals, use is made of elliptical coordinates for the monoelectronic, hybrid, and Coulomb integrals. For the exchange integrals, the atomic orbitals are translated to a common center. The final integration is performed by Gaussian quadrature.As an example, an SCF ab initio calculation is performed for the LiH molecule, both with integer and non-integer principal quantum number.     

Fourier transform general formula for systematic potentials

For calculating molecular integrals of systematic potentials, a three‐dimensional (3D) Fourier transform general formula can be derived, by the use of the Abel summation method. The present general formula contains all 3D Fourier transform formulas which are well known as Bethe–Salpeter formulas (Bethe and Salpeter, Handbuch der Physik, Bd. XXXV, 1957) as special cases. It is shown that, in several of the Bethe–Salpeter formulas, the integral does not converge in the meaning of the Riemann integral but converges in the meaning of a hyper function as the Schwartz distribution. For showing an effectiveness of the present general formula, the convergence condition of molecular integrals is derived generally for all of the present potentials. It is found that molecular integrals can be converged in the meaning of the Riemann integral for the present potentials, except for those for extra super singular potentials. It is also found that the convergence condition of molecular integrals over the Slater‐type orbitals is exactly the same as that of the corresponding integrals over the Gaussian‐type orbitals for the present systematic potentials. For showing more effectiveness, the molecular integral over the gauge‐including atomic orbitals is derived for the magnetic dipole‐same‐dipole interaction. © 2012 Wiley Periodicals, Inc.     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.     

Gauge invariant gaussian orbitals and the ab initio calculation of diamagnetic susceptibility for molecules

It is shown that gauge terms can be introduced into the Gaussian functions used as the basis functions for an ab initio calculation of the energy of a molecule in the presence of a uniform magnetic field so that all the integrals become independent of the origin of the vector potential. The perturbation treatment of the diamagnetic susceptibility is considered in the molecular orbital approximation. The results show that the susceptibility can be calculated using only the unperturbed orbitals and their first-order corrections. All the integrals that arise can be expressed in terms of known functions.     

Implementation of atomic basis set composed of 1s Gaussian and 1s Slater-type orbitals to carry out quantum mechanics molecular calculations

A mixed atomic basis set formed with ls Slater-type orbitals and 1s floating spherical Gaussian orbitals is implemented. Evaluation of multicenter integrals is carried out using a method based on expansion of binary products of atomic basis functions in terms of a complete basis set, and a systematic analysis is performed. The proposed algorithm is very stable and furnishes fairly good results for total energy and geometry. An LCAO-SCF test calculation is carried out on LiH. The trends observed show that there are some combinations of mixed orbitals that are appropriate to describe the system. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 604–609, 1999     

Computation of many-center exchange integrals over Slater orbitals up to 4d by means of optimized Gaussian expansions

The previously described optimization of small Gaussian expansions, to be employed for the computation of many-center two-electron integrals over Slater basis sets, has been extended, with some refinements, up 4d orbitals. © 1993 John Wiley & Sons, Inc.     

Molecular integrals in the generalized hylleraas–CI method

In the generalized Hylleraas–CI method, the original correlation factor r^v_ij is multiplied by a Gaussian geminal. Using the approach of generating functions, the general formulas of molecular integrals in this method are derived over Cartesian Gaussian orbitals. From differentiations of the generating functions, the expanding length in the incomplete Gamma functions is reduced, and some cancellations presented in other approaches are avoided. Preliminary calculations for H₂ and H₂—H₂ systems are carried out over STO -3G basis. The results are encouraging.     

Localized orbitals based on the fermi hole

The Fermi hole provides a direct (non-iterative) method for tansforming canonical SCF molecular orbitals into localized orbitals. Except for simple overlap integrals required to maintain orthogonality, this method requires no integrals over orbitals or basis functions. This method is demonstrated by application to a furanone (C₄H₄O₂), methylacetylene, and boron trifluoride. The results of these calculations are compared to those determined by the orbital centroid criterion of localization.     

Gaussian expansion method for molecular integrals of molecular properties

The finite Gaussian Expansion method for molecular integrals proposed by Taketa, O-ohata and Huzinaga has been extended to the integrals of molecular properties. The integral formulas of so-called moment, field and field gradient integrals have been derived. It has been numerically shown that in order to evaluate the field and the field gradient integrals based on Slater type orbitals, eight- or ten-term Gaussian expansions are sufficient but this method fails to attain sufficient effective numbers for the moment integrals.     

Symmetry and equivalence restrictions in electronic structure calculations

A simple method for obtaining MCSCF orbitals and CI natural orbitals adapted to degenerate point groups, with full symmetry and equivalence restrictions, is described. Among several advantages accruing from this method are the ability to perform atomic SCF calculations on states for which the SCF energy expression cannot be written in terms of Coulomb and exchange integrals over real orbitals, and the generation of symmetry-adapted atomic natural orbitals for use in a recently proposed method for basis set contraction.     

The ab initio calculation of molecular electric, magnetic and geometric properties

We give an account of some recent advances in the development of ab initio methods for the calculation of molecular response properties, involving electric, magnetic, and geometric perturbations. Particular attention is given to properties in which the basis functions depend explicitly both on time and on the applied perturbations such as perturbations involving nuclear displacements or external magnetic fields when London atomic orbitals are used. We summarize a general framework based on the quasienergy for the calculation of arbitrary-order molecular properties using the elements of the density matrix in the atomic-orbital basis as the basic variables. We demonstrate that the necessary perturbed density matrices of arbitrary order can be determined from a set of linear equations that have the same formal structure as the set of linear equations encountered when determining the linear response equations (or time-dependent self-consistent-field equations). Additional components needed to calculate properties involving perturbation-dependent basis sets are flexible one- and two-electron integral techniques for geometric or magnetic-field differentiated integrals; in Kohn-Sham density-functional theory (KS-DFT), we also need to calculate derivatives of the exchange-correlation functional. We describe a recent proposal for evaluating these contributions based on automatic differentiation. Within this framework, it is now possible to calculate any molecular property for an arbitrary self-consistent-field reference state, including two- and four-component relativistic self-consistent-field wave functions. Examples of calculations that can be performed with this formulation are presented.     

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.     

Ab initio calculation of the C/D ratio of magnetic circular dichroism

A procedure for calculating the magnetic circular dichroism C/D ratio from density functional theory calculations is discussed. The method is simplified considerably through the application of group theory and the irreducible-tensor method and only requires integrals of the magnetic dipole moment operator over a few orbitals and published tables of symmetry factors. The implementation of the method is tested through application to several small and medium-sized molecules.     

From linear combinations to integrals: A new approach to the basis function problem

A new formalism is presented, based upon the finite element method, that permits a dual representation of orbitals in terms of exponential or Gaussian functions as both an integral over the space of exponential parameters and as a linear combination of basis functions. The method has been implemented for the atomic Hartree–Fock problem using exponential functions and test calculations made for atoms ranging from B to Cl. Accurate and consistent results can be obtained for a variety of atoms in a simple way using computational schemes that are systematic and hierarchic in nature. The new formalism is promising for any method where the calculation of integrals is not a major problem, such as some approaches of the density functional method and the pseudospectral formulation of ab initio methods. © 1992 by John Wiley & Sons, Inc.     

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.     

On numerical solution of the integral Hartree-Fock equations for molecules by the method of MO LCAO in the momentum space

To develop a numerical solution of mentioned equations the method of factorized projection of integral operator kernel is applied. All matrix elements of the method are calculated analytically, being expressed in terms of two types of standard integrals: the overlap integrals and one-electron Coulomb integrals. To calculate the integrals we used the O(4)-symmetry of hydrogen-like atomic orbitals as well as operational technique of differentiation with respect to scalar and vector parameters.     

Simplified box orbitals: A spatially restricted alternative to the slater‐type orbitals

We introduce a new type of spatially restricted basis function (zero beyond a characteristic r₀ value of the radial coordinate) that makes it possible to obtain, in nonconfined systems, similar results to STO functions. This is important because the use of this kind of functions enables the exact application a sort of zero differential overlap approximation to calculate properties of large systems. Our functions are a modification of the BO‐xZ box orbitals introduced by Lepetit et al. First, we replaced these orbitals by a function that is easier to obtain and generalize, that we named “simplified box orbital” (SBO), and we have shown some advantages of the SBO over BO and standard STO functions. Second, we obtained Gaussian developments for both the original BO orbitals and the new SBO orbitals. In this way, it becomes possible to manage our SBO orbitals with standard quantum chemistry software like GAUSSIAN or similar programs. © 2014 Wiley Periodicals, Inc.     

Gaussian and finite-element Coulomb method for the fast evaluation of Coulomb integrals

The authors propose a new linear-scaling method for the fast evaluation of Coulomb integrals with Gaussian basis functions called the Gaussian and finite-element Coulomb (GFC) method. In this method, the Coulomb potential is expanded in a basis of mixed Gaussian and finite-element auxiliary functions that express the core and smooth Coulomb potentials, respectively. Coulomb integrals can be evaluated by three-center one-electron overlap integrals among two Gaussian basis functions and one mixed auxiliary function. Thus, the computational cost and scaling for large molecules are drastically reduced. Several applications to molecular systems show that the GFC method is more efficient than the analytical integration approach that requires four-center two-electron repulsion integrals. The GFC method realizes a near linear scaling for both one-dimensional alanine alpha-helix chains and three-dimensional diamond pieces.