Evaluation of Hylleraas-CI atomic integrals by integration over the coordinates of one electron. I. Three-electron integrals

A method to evaluate the nonrelativistic electron-repulsion, nuclear attraction and kinetic energy three-electron integrals over Slater orbitals appearing in Hylleraas-CI (Hy-CI) electron structure calculations on atoms is shown. It consists on the direct integration over the interelectronic coordinate r _ij and the sucessive integration over the coordinates of one of the electrons. All the integrals are expressed as linear combinations of basic two-electron integrals. These last are solved in terms of auxiliary two-electron integrals which are easy to compute and have high accuracy. The use of auxiliary three-electron ones is avoided, with great saving of storage memory. Therefore this method can be used for Hy-CI calculations on atoms with number of electrons N ≥ 5. It has been possible to calculate the kinetic energy also in terms of basic two-electron integrals by using the Hamiltonian in Hylleraas coordinates, for this purpose some mathematical aspects like derivatives of the spherical harmonics with respect to the polar angles and recursion relations are treated and some new relations are given.     

The Hylleraas-CI method in molecular calculations: Two-electron integrals

In the Hylleraas-CI method, first proposed by Sims and Hagstrom, correlation factors of the type r are included into the configurations of a CI expansion. The computation of the matrix elements requires the evaluation of different two-, three-, and four-electron integrals. In this article we present formulas for the two-electron integrals over Cartesian Gaussian functions, the most used basis functions in molecular calculations. Most of the integrals have been calculated analytically in closed form (some of them in terms of the incomplete Gamma function), but in one case a numerical integration is required, although the interval for the integration is finite and the integrand well-behaved. We have also reported on partial and preliminary computations for the H₂ molecule using our four-center general formulas; a basis set of s- and p-type functions yielded at R = 1.4001 Å an energy of - 1.174380 a.u. to be compared with Kolos and Wolniewicz value of - 1.174475.     

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.     

Application of modified Neumann expansion for analytical evaluation of two-center,two-electron integrals over slater-type orbitals

A modified form of the Neumann expansion in terms of products of orthogonal polynomials for the inverse interelectronic distance r¹₁₂ is proposed. This expansion has been applied in order to derive a unified analytical formula for two-center and two-electron integrals over Slater-type orbitals. The results are equivalent to those given recently by Yasui and Saika, but the expansion itself can be used for building up a realistic algorithm for evaluation of three- and four-electron integrals determined by using correlated variational wave functions.     

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.     

Evaluation of Hylleraas-CI atomic integrals. III. Two-electron kinetic energy integrals

This paper is the part III of a series about the evaluation of Hylleraas-Configuration Interaction (Hy-CI) integrals by the method of direct integration over the interelectronic coordinates. The two-electron kinetic-energy integrals have been derived using the Hamiltonian in Hylleraas coordinates. We have improved the algorithm used in part II of this series and obtained general expressions. The method used for the two-electron integrals can be used in the same fashion for the evaluation of the three-electron ones. The formulas shown here have been tested in actual Hy-CI calculations of two-electron systems. The two-electron kinetic energy integrals values agree with the ones obtained using the Kolos and Roothaan transformation. The effectiveness of the different methods is discussed.     

Atomic integrals in Hylleraas–CI calculations with double-linked correlation terms

Atomic Hylleraas-CI calculations with linked correlation terms of the form r r are discussed. Formulas for the integration of the radial part and the arising auxiliary integrals are deduced and convergence proofs are given. © 1995 John Wiley & Sons, Inc.     

A comparison of efficiency and accuracy of two-electron integrals calculation between two methods in multi-configuration time-dependent hartree fock frame

An approach to the evaluation of the two-electron repulsion integrals exactly in sine finite basis representation is proposed. The two-electron coulomb potential integrals are calculated respectively in sine finite basis representation by using two-fold Gaussian quadrature rules and in discrete variable representation by using the natural potential expansion of coulomb potential $r_{12} $ . The efficiency and accuracy of two methods to calculate the two-electron repulsion integrals are compared. Some demonstrative calculations indicate that both the two ways are effective methods to do two-electron integrals calculations in the multi-configuration time-dependent hartree fock (MCTDHF) frame. By using the method to calculate the two-electron integrals in sine FBR, the working equations of MCTDHF are propagated in imaginary time. The ground state energy of helium atom obtained in the imaginary propagation is close to the Full Configuration interaction energy calculated by Molpro.     

Recurrence relations for the evaluation of electron repulsion integrals over spherical Gaussian functions

Recurrence relations are derived for the evaluation of two-electron repulsion integrals (ERIs) over Hermite and spherical Gaussian functions. Through such relations, a generic ERI or ERI derivative may be reduced to “basic” integrals, i.e., true and auxiliary integrals involving only zero angular momentum functions. Extensive use is made of differential operators, in particular, of the spherical tensor gradient ??(?). Spherical Gaussians, being nonseparable in the x, y, and z coordinates, were not included in previous formulations. The advantages of using spherical Gaussians instead of Cartesian or Hermite Gaussians are briefly discussed. © 1993 John Wiley & Sons, Inc.     

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     

Nonlinear transformation methods for accelerating the convergence of Coulomb integrals over exponential type functions

It is well known that in any ab initio molecular orbital (MO) calculation, the major task involves the computation of molecular integrals, among which the computation of Coulomb integrals are the most frequently encountered. As the molecular system gets larger, computation of these integrals becomes one of the most laborious and time consuming steps in molecular systems calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. The atomic orbital basis functions chosen in the present work are Slater type functions. These functions can be expressed as finite linear combinations of B functions which are suitable to apply the Fourier transform method. The difficulties of the numerical evaluation of the analytic expressions of the integrals of interest arise mainly from the presence of highly oscillatory semi-infinite integrals. In this work, we present a generalized algorithm based on the nonlinear transformation of Sidi, for a precise and fast numerical evaluation of molecular integrals over Slater type functions and over B functions. Numerical results obtained for the three-center two-electron Coulomb and hybrid integrals over B functions and over Slater type functions. Comparisons with numerical results obtained using alternatives approaches and an existing code are listed.     

Finite jellium models. I. Restricted Hartree-Fock calculations

Restricted Hartree-Fock calculations have been performed on the Fermi configurations of n electrons confined within a cube. The self-consistent-field orbitals have been expanded in a basis of N particle-in-a-box wave functions. The difficult one- and two-electron integrals have been reduced to a small set of canonical integrals that are calculated accurately using quadrature. The total energy and exchange energy per particle converge smoothly toward their limiting values as n increases; the highest occupied molecular orbital-lowest unoccupied molecular orbital gap and Dirac coefficient converge erratically. However, the convergence in all cases is slow.     

Slater orbital molecular integrals with numerical fourier transform methods. I. (coplanar) multicenter exchange integrals over 1s orbitals

Slater orbital r₁₂^?1 integrals are calculated with a numerical Fourier-transform method based on a formulation first given by Bonham, Peacher and Cox. Spherical wave expansions are introduced that decouple the Feynman integrations for the charge distribution Fourier transforms. The Feynman integrals are evaluated semianalytically, and their properties are analyzed in detail. The final computational step involves a numerical integration over charge distribution quantities. Results for (coplanar) multicenter exchange integrals over 1s orbitals are given. As long as the charge distributions are overlapping considerably, the method gives good results, even when these distributions are highly asymmetric. The method as presently implemented fails when highly disconnected charge distributions are involved.     

Two-center molecular repulsion integrals over slater functions

For the general two-electron two-center integral over Slater functions, use of the Neumann expansion for the electron-electron interaction term yields the standard auxiliary functions. These are expanded and integrated explicitly by two independent methods. The resulting simple analytic formula for the total integral is completely general, requiring only the Slater function quantum numbers and exponents and the internuclear separation. Hence all two-electron hydrid, coulomb, exchange, and one-center integrals are considered. The efficiency of calculation of this expression is compared with those of other methods, indicating an order of magnitude improvement in speed over recursion for the exchange integral.     

Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal‐coordinate and zeta‐function expansions

One‐electron integrals over three centers and two‐electron integrals over two centers, involving Slater‐type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r₁₂ within an ellipsoidal‐coordinate system or by employing a one‐center expansion in spherical‐harmonic and zeta‐function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta‐function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 1–13, 1999     

Molecular correlation integrals. Two center one electron integrals containing a function of interparticle distance in one center expansion approximation

Two-center one-electron integrals needed in certain molecular correlated wave function calculations, using one-center expansion approximation, have been studied. The form of the basic correlated function used in this study is The parent integral is expressed in terms of an angular integral, and an auxiliary radial integral depending upon the variables r₁, r₂, and r₁₂. Several analytical formulas, and a recursive formula are derived for the auxiliary integral, and other related integrals. All these formulas are given in computationally useful forms. Logical flow charts and FORTRAN programs were constructed for computing the basic integrals discussed in the paper. Numerical values of some integrals, thus obtained, are tabulated for comparisons.     

Auxiliary functions for Slater molecular integrals

Summary Some types of recurrence relations are modified to overcome the cases in which their conventional application is unstable in both the forward and backward directions. The original recurrence relations — connecting adjacent elements — are replaced by more general ones, where the non-adjacent elements are connected by coefficients obtained by new sets of relations derived from the original ones. This modification can be helpful for the calculation of the complicated molecular integrals with Slater Type Orbitals (STOs).As a simple test we prove that some auxiliary functions — previously evaluated by expensive expansions — appearing in two-center two-electron integrals can be thus calculated with very low computer cost and high accuracy.     

Resolutions of the Coulomb operator. VI. Computation of auxiliary integrals

We discuss the efficient computation of the auxiliary integrals that arise when resolutions of two-electron operators (specifically, the Coulomb operator [T. Limpanuparb, A. T. B. Gilbert, and P. M. W. Gill, J. Chem. Theory Comput. 7, 830 (2011)] and the long-range Ewald operator [T. Limpanuparb and P. M. W. Gill, J. Chem. Theory Comput. 7, 2353 (2011)]) are employed in quantum chemical calculations. We derive a recurrence relation that facilitates the generation of auxiliary integrals for Gaussian basis functions of arbitrary angular momentum and propose a near-optimal algorithm for its use.     

Three-center expansion of electron repulsion integrals with linear combination of atomic electron distributions

A new systematic way of constructing auxiliary basis functions for approximating the evaluation of electron repulsion integrals is proposed and applied to SCF and MCSCF wavefunction calculations. In the approximation, the one-electron density is expanded in terms of a linear combination of atomic electron distributions (LCAD), and the four-center two-electron repulsion integrals are reduced to the three- and two-center quantities. This results in a high-accuracy approximation as well as a large reduction in disk storage and input/output requirement, proportional to N³ rather than N⁴, N being the number of basis functions. Numerical results indicate that the error from the present approximation decreases as the size of molecular basis functions increases and that the LCAD version of MCSCF calculations requires only a fractional amount of the CPU time required in the conventional procedure without loss of accuracy.     

Orbit-orbit molecular integrals over Gaussian orbitals

Analytical formulas are derived for integrals of the orbit-orbit operator in the Breit-Pauli Hamiltonian. The present method differs from an earlier one in the introduction of a new auxiliary function G_n(a) which is an integral of Shavitt's F_m(t) function. A method for its evaluation is discussed and some numerical examples are given.