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1.
María Belén Ruiz 《Journal of mathematical chemistry》2009,46(4):1322-1355
An alternative procedure to the classical method for evaluating the four-electron Hylleraas-CI integrals is given. The method
consists of direct integration over the r
12 coordinate and integration over the coordinates of one of the electrons, reducing the integrals to lower order. The method
based on the earlier work of Calais and L?wdin and of Perkins is extended to the general angular case. In this way it is possible
to solve all of the four-electron integrals appearing in the Hylleraas-CI method. The four-electron integrals are expanded
in three-electron ones which are in turn expanded in two-electron integrals. Finally the two-electron integrals are expanded
into two-electron auxiliary integrals which usually have one negative power. The use of three- and four-electron electron
auxiliary integrals is avoided. Some special cases lead to one- and two-electron auxiliary integrals with negative powers
which do not converge individually but do converge in combination with others. These relations and their solutions are presented,
together with results of various kinds of integrals. 相似文献
2.
Jan Budziski 《International journal of quantum chemistry》2004,97(4):832-843
An algorithm for evaluation of two‐center, three‐electron integrals with the correlation factors of the type rr and rrr as well as four‐electron integrals with the correlation factors rrr and rrr in the Slater basis is presented. This problem has been solved here in elliptical coordinates, using the generalized and modified form of the Neumann expansion of the interelectronic distance function r for k ≥ ?1. Some numerical results are also included. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004 相似文献
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4.
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 相似文献
5.
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. 相似文献
6.
This paper presents a computationally efficient formula in terms of basic overlap integrals over Slater type orbitals (STOs)
for the evaluation of auxiliary function which plays a central role in calculations of multicenter molecular integrals. The basic overlap integrals are calculated
with the help of recurrence relations. The resulting simple analytical formula for the auxiliary function is completely general for p
a
≤ 1.2 and arbitrary values of parameters p and pt. The efficiency of calculation of auxiliary function is compared with other method. 相似文献
7.
Frank E. Harris 《International journal of quantum chemistry》2002,88(6):701-734
Extant analytic methods for evaluating two‐center electron repulsion integrals in a Slater‐type orbital (STO) basis using ellipsoidal coordinates and the Neumann expansion of 1/r12 have problems of numerical stability that are analyzed in detail using computer‐assisted algebraic techniques. Some of these problems can be eliminated by use of procedures known in this field 40 years ago but seemingly forgotten now. Others can be removed by use of a formulation suitable for small values of the STO screening parameter. A recent attempt at such a formulation is corrected and extended in a way permitting its practical use. The main functions encountered in the integrations over the ellipsoidal coordinate of the range 1 … ∞ are Bessel functions or generalizations thereof, as pointed out here for the first time. This fact is used to motivate the derivation of recurrence relations additional to those previously known. Novel techniques were devised for using these recurrence relations, thereby providing new ways of calculating the quantities that enter the ellipsoidal expansion. The convergence rate of this expansion and the numerical characteristics of several computational strategies are reported in enough detail to identify the ranges where various schemes can be used. This information shows that recent discussions of the “convergence characteristics of [the] ellipsoidal coordinate expansion” are in fact not that, but are instead discussions of an inability to make accurate calculations of the individual terms of the expansion. It is also seen that the parameter range suitable for use of Kotani's well‐known recursive scheme is more limited than seems generally believed. The procedures discussed in this work are capable of yielding accurate two‐center electron repulsion integrals by the ellipsoidal expansion method for all reasonable STO screening parameters, and have been implemented in illustrative public‐domain computer programs. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002 相似文献
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J. E. Prez J. C. Cesco O. E. Taurian F. S. Ortiz A. E. Rosso C. C. Denner G. O. Giubergia 《International journal of quantum chemistry》2005,102(6):1056-1060
A new approach for evaluating the four‐center bielectronic integrals (12|34), involving 1s Slater‐type orbitals, is presented. The method uses the multiplication theorem for Bessel functions. The bielectronic integral is expressed in terms of a finite sum of functions, and a scaling parameter is introduced. In the present work, the scaling parameter used is an average. The results show that the first term in the sum is always the principal contribution, and the remainder has a corrective character. The whole scheme and its numerical trend are understood on the basis of a theorem recently proved. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 相似文献
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11.
B. A. Mamedov 《International journal of quantum chemistry》2000,78(3):146-152
The multicenter charge‐density expansion coefficients [I. I. Guseinov, J Mol Struct (Theochem) 417 , 117 (1997)] appearing in the molecular integrals with an arbitrary multielectron operator were calculated for extremely large quantum numbers of Slater‐type orbitals (STOs). As an example, using computer programs written for these coefficients, with the help of single‐center expansion method, some of two‐electron two‐center Coulomb and four‐center exchange electron repulsion integrals of Hartree–Fock–Roothaan (HFR) equations for molecules were also calculated. Accuracy of the results is quite high for the quantum numbers, screening constants, and location of STOs. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 146–152, 2000 相似文献
12.
A general formula has been established for the expansion of the product of two normalized associated Legendre functions centered on the nuclei a and b. This formula has been utilized for the evaluation of two‐center overlap and nuclear attraction integrals over Slater‐type orbitals (STOs) with integer and noninteger principal quantum numbers. The formulas given in this study for the evaluation of two‐center overlap and nuclear attraction integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, orbital exponents, and internuclear distances. © 2001 Wiley Periodicals, Inc. Int J Quantum Chem, 2001 相似文献
13.
Efficiency of the algorithms for the calculation of Slater molecular integrals in polyatomic molecules 总被引:1,自引:0,他引:1
The performances of the algorithms employed in a previously reported program for the calculation of integrals with Slater-type orbitals are examined. The integrals are classified in types and the efficiency (in terms of the ratio accuracy/cost) of the algorithm selected for each type is analyzed. These algorithms yield all the one- and two-center integrals (both one- and two-electron) with an accuracy of at least 12 decimal places and an average computational time of very few microseconds per integral. The algorithms for three- and four-center electron repulsion integrals, based on the discrete Gauss transform, have a computational cost that depends on the local symmetry of the molecule and the accuracy of the integrals, standard efficiency being in the range of eight decimal places in hundreds of microseconds. 相似文献
14.
F. G. Pashaev 《Journal of mathematical chemistry》2009,45(4):891-890
An efficient method for computing overlap integral over Slater type orbitals based on the B Filter-Steinborn and Guseinov auxiliary functions is presented. The final results are expressed through the binomial coefficients with the help of which
the overlap integrals can be evaluated efficiently and accurately. The results of calculation are in good agreement with those
obtained by other method for arbitrary principal quantum numbers and different screening constants.
An erratum to this article can be found at 相似文献
15.
In this study, using complete orthonormal sets of exponential type orbitals (ETOs), a single closed analytical relation is derived for a large number of different expansions of overlap integrals over Slater type orbitals (STOs) with the same screening parameters in terms of Gegenbauer coefficients. The general formula obtained for the overlap integrals is utilized for the evaluation of multicenter nuclear attraction and electron repulsion integrals appearing in the Hartree–Fock–Roothaan equations for molecules. The formulas given in this study for the evaluation of these multicenter integrals show good rate of convergence and great numerical stability under wide range of quantum numbers, scaling parameters of STOs and internuclear distances. 相似文献
16.
Ishida K 《Journal of computational chemistry》2003,24(15):1874-1890
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. 相似文献
17.
In this study, we shall suggest analytical expressions for two-center nuclear attraction integrals over STO’s with a one-center
charge distribution by using Fourier transform method. The derivation is based on partial-fraction decompositions and Taylor
expansions of rational functions. Analytical expressions obtained by this method are expressed in terms of Gegenbauer, and
binomial coefficients and linear combinations of STO’s. Finally, it is relatively easy to express the Fourier integral representations
of two-center nuclear attraction integrals with a one-center charge distribution mentioned above as finite and infinite of
series of STO’s and irregular solid harmonics which may be considered to be limiting cases of STO’s. 相似文献
18.
T.
zdoan 《International journal of quantum chemistry》2003,92(5):419-427
The expansion formula has been presented for Slater‐type orbitals with noninteger principal quantum numbers (noninteger n‐STOs), which involves conventional STOs (integer n‐STOs) with the same center. By the use of this expansion formula, arbitrary multielectron multicenter molecular integrals over noninteger n‐STOs are expressed in terms of counterpart integrals over integer n‐STOs with a combined infinite series formula. The convergence of the method is tested for two‐center overlap, nuclear attraction, and two‐electron one‐center integrals, due to the scarcity of the literature, and fair uniform convergence and great numerical stability under wide changes in molecular parameters is achieved. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003 相似文献
19.
María Belén Ruiz 《Journal of mathematical chemistry》2011,49(10):2457-2485
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. 相似文献
20.
Three‐center electric multipole moment integrals over Slater‐type orbitals (STOs) can be evaluated by translating the orbitals on one center to the other and reducing the system to an expansion of two‐center integrals. These are then evaluated using Fourier transforms. The resulting expression depends on the overlap integrals that can be evaluated with the greatest ease. They involve expressions for STO with different screening parameters that are known analytically. This work gives the overall expressions analytically in a compact form, based on Gegenbauer polynomials. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012 相似文献