Many-center integrals

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

There are some benefits in using STO instead of GTO fiinctions, the problem consists on how to solve the many center integral computation bottleneck. A possible way will be discussed now, using CETO s. 

As it is shown above, CETO functions can be considered the bviilding blocks of cartesian STO s. In this sense, solving the many center integral evaluation problem over CETO functions will be the same as to move another step into the path to solve the integral STO problem. 

The optimization of basis set non-linear parameters, appearing in equation (5.2), constitute one of the main steps in the preliminary work before many center integral evaluation. There will be described only a step by step procedure in order to optimize non-linear parameters of the involved fimctions one by one. 

Two center density expansions into separate centers have been employed successfully in order to overcome the many center integral problem. Here it is also proved how such a naive but elegant algorithm, based essentially on a recursive Cholesky decomposition, am be well adapted to modem computational hardware architectures. CETO functions appear in this manner as a plausible alternative to the present GTO quantum chemical computational flood, constituting the foundation of another step signaling the path towards STO integral calculation. 

Monkhorst HJ, Jeziorski B (1979) No linear dependence or many-center integral problems in momentum space quantum chemistry. J Chem Phys 71(12) 5268-5269... 

B2u, B1u, and. Elu) were separated by electron repulsion both from one another and from the corresponding triplets, and that these energy differences could be interpreted in terms of reasonable values for the Coulomb repulsion integrals between atomic orbitals. They did not find it possible to evaluate all the many-center integrals required, and errors crept into their numerical calculations, but subsequent work11 12-58 68 has left little doubt that the 1800 A band of benzene has an Elu upper state and that the upper states of the 2600 A and 2100 A bands are Biu and Bltt, respectively. There was, therefore, even at that time clear evidence that electron repulsion must be included in any final theory, though aromatic molecules with less symmetry than benzene clearly presented a much more difficult problem. 

Although most molecular calculations are carried out using GTOs, in some cases (in particular for atoms and diatoms), Slater-type orbitals (STOs) are used instead. The STOs have a different radial form than the GTOs, proportional to exp(— rA) rather than to exp(—arA). The GTOs are used in preference to the STOs because the evaluation of many-center integrals is much easier for GTOs than for STOs. 

P. O. Lowdin, J. Chem. Phys., 21, 374 (1953). Approximate Formulas for Many-Center Integrals in the Theory of Molecules and Crystals. 

Exponential functions used as basis functions in actual calculations on atoms and molecules are called Slater type orbitals (STO), or simply exponential type orbitals (ETO), in quantum chemistry. In the case of molecules, exponential orbitals lead to very difficult and time-consuming many-center integrals, though there are some successful attempts to get beyond the bottleneck (J.D. Talman). 

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

Thus the many-center one-electron problem is easily solved, provided that the integrals shown in equation (65) can be evaluated. The reciprocals of the parameters can then be identified with the roots of the secular equation (63). 

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

The two-electron integrals pq kl] are < p(l)0fc(2) e2/ri2 0,(l)0j(2) > and may involve as many as four orbitals. The models of interest are restricted to one and two-center terms. Two electrons in the same orbital, [pp pp], is 7 in Pariser-Parr-Pople (PPP) theory[4] or U in Hubbard models[5], while pp qq are the two-center integrals kept in PPP. The zero-differential-overlap (ZDO) approximation[3] can be invoked to rationalize such simplification. In modern applications, however, and especially in the solid state, models are introduced phenomenologically. Particularly successful models are apt to be derived subsequently and their parameters computed separately. 

MANY CENTER AO INTEGRAL EVALUATION USING CARTESIAN... 

Keywords Many Center AO Integrals. Molecular Basis Sets, ETO. STO. CETO. 

Because of the many center nature of the fourth integral case, a detailed analysis of three center nuclear attraction integral problem is given. Using the ideas developed in Sections 4 and 5, it is described how the three center integrals become expressible in terms of one and tv/o center ones. An example involving s-type WO-CETO functions is presented as a test of the developed theory of the preceding chapters. 

This section also includes at the end an analysis of the electron spin spin contact integrals, a well known and simple relativistic correction term, which are also studied in this paper due to their close relationship with Quantum Similarity Measures [66c]. The form of sudi integrals corresponds to some kind of many function, many center overlap. 

Up to now we have assumed in this chapter the use of Slater-type orbitals. Actually, use may be made of any type of functions which form a complete set in Hilbert space. Since for practical reasons the expansion (2,1) must be always truncated, it is preferable to choose functions with a fast convergence. This requirement is probably best satisfied just for Slater-type functions. Nevertheless there is another aspect which must be taken into account. It is the rapidity with which we are able to evaluate the integrals over the basis set functions. This is particularly topical for many-center two-electron integrals. In this respect the use of the STO basis set is rather cumbersome. The only widely used alternative is a set of Gaus-slan-type functions (GTF). The properties of Gaussian-type functions are just the opposite - integrals are computed simply and, in comparison to the STO basis set, rather rapidly, but the convergence is slow. 

The above equations are sufficient to evaluate the gradients of all integrals that are met in electronic structure calculations. In actual practice, however, many tricks are used to reduce the computational work. The evaluation of the gradients when Slater-type functions are used (see Sections 3.5 and 3.6) is more difficult, but could proceed along the same lines as for Gaussian functions, at least for the two-center integrals that usually appear in semiempirical models. 

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