Cartesian exponential type orbitals

As a foreword it must be said, perhaps constructing a too late homage to the brilliant contribution of professor Boys to Quantum Chemistry, that the first description of cartesian exponential type orbitals (CETO) was made thirty years ago by Boys and Cook [1], One can probably think this fact as a consequence of the evolution of Boys s thought on the basis set problem and to the incipient ETO-GTO dilemma, which Boys has himself stated ten years earlier [2a]. 

From now on let us define a real normalized Cartesian Exponential Type Orbital (CETO) centered at the point A as ... 

In fact, our interest in the present formulation, the use of NSS s andLKD s, has been aroused when studying the integrals over Cartesian Exponential Type Orbitals [la,b] and Generalized Perturbation Theory [ld,e]. The use of both symbols in this case has been extensively studied in the above references, so we will not repeat here the already published arguments. Instead we will show the interest of using nested sums in a wide set of Quantum Chemical areas, which in some way or another had been included in our research interests [Ic]. 

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. 

A short introduction on Exponential Type Orbitals (ETO) functions is given here, in order to prepare the integral evaluation over cartesian ETO s. 

Another kind of ETO functions, which also can be formed having a cartesian term as angular component, are the functions which may be called Laplace Exponential Type Orbitals LETO. An unnormalized LETO can be defined as follows ... 

In the case of d-type orbitals, there are six Cartesian GTOs with pre-exponential factors of x, xy, y, xz, yz and z - Only five are linearly independent, e combi nation... 

We use an alternative to this method, that enables a fast and accurate evaluation of the two-center integrals. Analytical integration is possible when linear combinations of Gaussian Type Orbitals are used to describe the atomic states [1,9, 10]. Imperfect behavior of such gaussian functions at large distances does not affect the results, since the two-center matrix-elements (7) have an exponential decay for increasing intemuclear distances. For example, for integrals and expressed in cartesian coordinates, one has to evaluate expressions such as... 

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