Multicenter and multiparticle integrals for explicitly correlated cartesian gaussian-type functions

An analytical derivation of multicenter and multiparticle integrals for explicitly correlated Cartesian Gaussian-type cluster functions is demonstrated. The evaluation method is based on the application of raising operators that transform spherical cluster Gaussian functions into Cartesian Gaussian functions.     

Analytic first derivatives for explicitly correlated,multicenter, Gaussian geminals

Variational calculations utilizing the analytic gradient of explicitly correlated Gaussian molecular integrals are presented for the ground state of the hydrogen molecule. Preliminary results serve to motivate the need for general formulas for analytic first derivatives of molecular integrals involving multicenter, explicitly correlated Gaussian geminals with respect to Gaussian exponents and coordinates of the orbital centers. Explicit formulas for analytic first derivatives of Gaussian functions containing correlation factors of the form exp(-βr_ij²) are derived and discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 991–999, 1997     

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.     

Evaluation of one-electron integrals for arbitrary operators V(r) over Cartesian Gaussians: Application to inverse-square distance and Yukawa operators

A general procedure is presented for generating one-electron integrals over any arbitrary potential operator that is a function of radial distance only. The procedure outlines that for a nucleus centered at point C integrals over Cartesian Gaussians can be written as linear combinations of 1-D integrals. These Cartesian Gaussian functions are expressed in a compact form involving easily computed auxiliary functions. It is well known that integrals over the Coulomb operator can be expressed in terms of F_n(T) integrals, where By means of a substitution for F_n(T) by other simple functions, algorithms that form integrals over an arbitrary function can be generated. Formation of such integrals is accomplished with minor editing of existing code based on the McMurchie–Davidson formalism. Further, the method is applied using the inverse-square distance and Yukawa potential operators V(r) over Cartesian Gaussian functions. Thus, the proposed methodology covers a large class of one-electron integrals necessary for theoretical studies of molecular systems by ab initio calculations. Finally, by virtue of the procedure's recursive nature it provides us with an efficient scheme of computing the proposed class of one-electron integrals. © 1993 John Wiley & Sons, Inc.     

Analytical gradients for Singer's multicenter n‐electron explicitly correlated Gaussians

Analytical gradients for Singer's basis of n‐electron multicenter explicitly correlated Gaussian functions are derived and implemented to variationally optimize the energy and wave function of molecular systems within the Born–Oppenheimer approximation. Wave functions are optimized with respect to (½n(n+1)+3n) nonlinear variational parameters and one linear coefficient per term in the basis set. Preliminary results for the ground states of H₃⁺ and H₃ suggest that the method can be more flexible and can achieve lower energies than previously reported calculations. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 151–159, 2001     

Molecular integrals for Gaussian and exponential‐type functions: Shift operators

Basis functions with arbitrary quantum numbers can be attained from those with the lowest numbers by applying shift operators. We derive the general expressions and the recurrence relations of these operators for Cartesian basis sets with Gaussian and exponential radial factors. In correspondence, the expressions of molecular integrals involving functions with arbitrary quantum numbers can be obtained by applying these operators on the integrals with the lowest quantum numbers. Since the original form of the shift operators is not appropriate to deal with integrals, we give their representation in terms of derivatives with respect to the parameters on which these integrals explicitly depend. Moreover, we translate the recurrence relations to the new representation and, finally, we analyze the general expressions ot the molecular integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 137–145, 2000     

Comment on “Symbolic Calculation of Two‐Center Overlap Integrals Over Slater‐type Orbitals”

Published by Gümü?. and Özdo?an formulas for the evaluation of two‐center overlap integrals (Gümü?, S.; Özdo?an, T. J. Chin. Chem. Soc. 2004, 51, 243) are critically analyzed. It is demonstrated that the formulas presented in this work are not original and they can easily be derived from the relationships contained in our papers (Guseinov, I. I. J. Phys. B 1970 , 3, 1399; Phys. Rev. A 1985 , 32, 1864; J. Mol Struct. (Theochem) 1995 , 336, 17) by changing the summation indices and application of a simple algebra. It should be noted that the symbolic results of overlap integrals between different combinations of quantum numbers given in Table 1 and 2 can also be obtained from the use of established in above mentioned our papers general formulas or presented in the literature relations for overlap integrals in terms of the products of molecular auxiliary functions A_n(p) and B_n(pt) (see, e.g., Lofthus, A. Mol. Phys. 1962 , 5, 105).     

Maxwell–Cartesian spherical harmonics in multipole potentials and atomic orbitals

 The nature of the Maxwell–Cartesian spherical harmonics S ⁽ⁿ⁾ _K and their relation to tesseral harmonics Y _nm is examined with the help of “tricorn arrays” that display the components of a totally symmetric Cartesian tensor of any rank in a systematic way. The arrays show the symmetries of the Maxwell–Cartesian harmonic tensors with respect to permutation of axes, the traceless properties of the tensors, the linearly independent subsets, the nonorthogonal subsets, and the subsets whose linear combinations produce the tesseral harmonics. The two families of harmonics are related by their connection with the gradients of 1/r, and explicit formulas for the transformation coefficients are derived. The rotational transformation of S ⁽ⁿ⁾ _K functions is described by a relatively simple Cartesian tensor method. The utility of the Maxwell–Cartesian harmonics in the theory of multipole potentials, where these functions originated in the work of Maxwell, is illustrated with some newer applications which employ a detracer exchange theorem and make use of the partial linear independence of the functions. The properties of atomic orbitals whose angular part is described by Maxwell–Cartesian harmonics are explored, including their angular momenta, adherence to an Uns?ld-type spherical symmetry relation, and potential for eliminating an angular momentum “contamination” problem in Cartesian Gaussian basis sets. Received: 9 July 2001 / Accepted: 7 September 2001 / Published online: 19 December 2001     

Comparative study of unconventional 1s basis functions for the 1Σ ground state of H2 and He

We investigated various nonstandard 1s basis functions (generalized Slater-Gaussian, ellipsoidal Gaussian, floating spherical and ellipsoidal Gaussian, rational function, Hulthén approximation, two-Slater-type orbital, generalized Guillemin–Zener function, and various noninteger-n elliptical orbitals) for approximating the ¹Σ ground state of H₂ and He₂⁺⁺. A CI trial wave-function including Σ_g-type MO's is adopted and molecular integrals are evaluated numerically. The energy improvement on the 1s STO is small except for noninteger-n orbitals which closely approach the “SCF limit”.     

Molecular symmetry and ab initio calculations. II. Symmetry-matrix and symmetry-supermatrix in the Dirac-Fock method

The concepts of symmetry-matrix and symmetry-supermatrix introduced in article I [J. Comput. Chem., 10, 957 (1989)] can be generalized to the Dirac-Fock method. By using the semidirect product decomposition of O_h and the linear vector space theory, the irreducible representation basis of O_h for any molecular system (O_h or its subgroups) can be deduced analytically in the nonorthonormal Cartesian Gaussian basis. This method is extended to discuss the double-valued representations of O_h* in the complex Cartesian Gaussian spinor basis. In the double-valued irreducible representation basis of D₂*, the matrix of kinetic operator c(OVERLINE)σ(/OVERLINE)·(OVERLINE)p(/OVELINE) in the Dirac-Fock equation can be reduced into a real symmetric and can be grouped into classes under the operations in D_3d. Therefore, the symmetry-matrix and symmetry-supermatrix can also be used in the Dirac-Fock method to reduce the storage of two electron integrals and calculations of Fock matrix during iterations by a factor of ca. g² (g is the order of the molecular symmetry group). In addition, a method to deal with the nonorthonormal space is presented. © 1996 by John Wiley & Sons, Inc.     

Use of one-range addition theorems in terms of ?�� for ?��-ETO and Coulomb- Yukawa like correlated interaction potentials with integer and noninteger indices in evaluation of multicenter multielectron integrals

Multicenter multielectron integrals appearing in the study of multielectron properties of atomic and molecular systems are evaluated using one-range addition theorems in terms of complete orthonormal sets of ?^??-exponential type orbitals (?^??-ETO, ?? = 1, 0, ?1, ?2, ??) for ?^??-ETO and Coulomb-Yukawa like correlated interaction potentials (CIP) introduced by the author. The final results are especially useful for the computation of arbitrary multicenter multielectron integrals that arise in the Hartree-Fock- Roothaan (HFR) approximation and also in the explicitly correlated methods based upon the use of ?^??-ETO as basis functions.     

Evaluation of molecular integral of Cartesian Gaussian type basis function with complex‐valued center coordinates and exponent via the McMurchie–Davidson recursion formula,and its application to electron dynamics

McMurchie–Davidson recursion formula is extended to derive the ab initio molecular integrals with higher angular quantum number complex Gaussian type basis function which has complex‐valued center coordinates and a complex‐valued exponent. Using the analytical recursion formulae, some calculations of electronic dynamics after beta decay of tritium hydride molecular ion HT⁺ are performed by a quantum wave packet method with thawed Gaussian basis functions of s‐ and p‐type. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Modified Cartesian Gaussian functions and their use in quantum chemistry

The general formula for the electron-electron repulsion integral in the modified Cartesian Gaussian basis set derived in Ref. [1] is simplified. A general relation between the standard and modified CG functions is given. A possible use of the modified CG functions to quantum chemical calculations which include the correlation factor r_ij² is indicated.     

Molecular symmetry and ab initio calculations: IV. Symmetry-matrix and symmetry-supermatrix in calculations of two-electron repulsion integrals

The product of two Gaussians having different centers is itself a one-center Gaussian, thus multicenter integrals with a Cartesian Gaussian basis can be reduced to one-center integrals. Recurrence relations for overlap integrals and electron repulsion integrals (ERIs) are derived at these centers. The calculations of overlap integrals and ERIs are carried out step by step from the highest symmetry case (one center) to required cases (different centers) by using the translation of Cartesian Gaussians. Full exploitation of symmetry in calculation processes can result in optimal use of these recurrence relations. Compared with the recently published algorithms, based on the recurrence relations derived by Obara and Saika [J. Chem. Phys., 84 , 3963 (1986)], the floating point operations (FLOPs) for ERI calculations (having four different centers) can be reduced by a factor of ca. 2. A significant extra saving in calculations and storage can be obtained if atoms, linear, or planar molecules are discussed. © 1997 John Wiley & Sons, Inc.     

Implementation of gradient formulas for correlated gaussians: He, ∞He,Ps2, 9Be,and ∞Be test results

New formulas in the basis of explicitly correlated Gaussian basis functions, derived in a previous article using powerful matrix calculus, are implemented and applied to find variational upper bounds for nonrelativistic ground states of ⁴He, ^∞He, Ps₂, ⁹Be, and ^∞Be. Analytic gradients of the energy are included to speed optimization of the exponential variational parameters. Five different nonlinear optimization subroutines (algorithms) are compared: TN, truncated Newton; DUMING, quasi-Newton; DUMIDH, modified Newton; DUMCGG, conjugate gradient; and POWELL, direction set (nongradient). The new analytic gradient formulas are found to significantly accelerate optimizations that require gradients. We found that the truncated Newton algorithm out-performs the other optimizers for the selected test cases. Computer timings and energy bounds are reported. © 1997 John Wiley & Sons, Inc.     

Synthesis,Characterization and Magnetic Studies of μ‐Oxamido Copper (II) ‐ Chromium (III) Heterodinuclear Complexes

Three new μ‐oxamido‐bridged heterodinuclear copper (II)‐chromium (III) complexes formulated [Cu(Me₂oxpn)Cr‐(L)₂](NO₃)₃, where Me₂oxpn denotes N,N'‐bis(3‐amino‐2, 2‐dimethylpropyl)oxamido dianion and L represents 5‐methyl‐1,10‐phenanthroline (Mephen), 4,7‐diphenyl‐1,10‐phenanthroline (Ph₂phen) or 2,2′‐bipyridine (bpy), have been synthesized and characterized by elemental analyses, IR and electronic spectral studies, magnetic moments of room‐temperature and molar conductivity measurements. It is proposed that these complexes have oxamido‐bridged structures consisting of planar copper (II) and octahedral chromium (III) ions. The variable temperature magnetic susceptibilities (4.2–300 K) of complexes [Cu(Me₂oxpn)Cr(Ph₂phen)₂](NO₃)₃ (1) and [Cu(Me₂oxpn)Cr(Mephen)₂] (NO₃)₃ (2) were further measured and studied, demonstrating the ferromagnetic interaction between the adjacent chromium (III) and copper (II) ions through the oxamido‐bridge in both complexes 1 and 2. Based on the spin Hamiltonian, ? = ‐ 2J?₁ · ?₂, the exchange integrals J were evaluated as + 21.5 an^?1 for 1 and + 22.8 cm^?1 for 2.     

Evaluation of two-center,two- and three-electron integrals involving correlation factors over Slater-type orbitals. II. Kinetic and potential energy integrals and examples of numerical results

This article is concerned with the construction of the general algorithm for evaluating two-center, two- and three-electron integrals occurring in matrix elements of one-electron operators in the basis of variational correlated functions. This problem has been solved here in prolate spherical coordinates, using the modified and extended form of the Neumann expansion of the interelectronic distance function r^k_ij derived in Part I of this series for k = ?1, 0, 1, 2. This work expands the method proposed by one of us in the preceding paper for integrals of the types mentioned above. The results of numerical calculations for different types of the two- and three-electron integrals are presented. The problem of convergence of the proposed procedures used is also discussed.     

Completeness criteria for explicitly correlated Gaussian geminal bases of axial symmetry

The completeness criteria for the basis set of explicitly correlated Gaussian-type geminals adapted to C_∞v symmetry are given. Specifically, we show that any pair function of Σ⁺ symmetry can be expanded in terms of products involving two spherical Gaussian orbitals located on the internuclear axis and a Gaussian correlating factor with a positive exponent. Pair functions corresponding to other irreducible representations of C_∞v can be expressed as linear combinations of products of a σ⁺ function and an angular factor depending on the azimuthal angles. The minimal set of the angular factors needed for completeness is given. These factors are relevant also for other explicitly correlated bases. © 1997 John Wiley & Sons, Inc.     

Variation of the ℏω with the particle number and the appearance of “kinks” for atomic clusters

The dependence of the harmonic oscillator (HO) energy level spacing ?ω on the particle number N is studied analytically for atomic (metal) clusters on the basis of their electronic densities, parametrizing Ekardt's results (for sodium clusters) by means of a Fermi distribution. An interesting feature of such an approach is that it leads, under the assumptions made, to “kinks,” that is, to “marked discontinuities in the slope” of ?ω at the closed shells. These discontinuities diminish as N increases. For large N, ?ω becomes simply: ?ω?c₁N^?1/3+c₂N^?1. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

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.