Exponential-type orbitals

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. 

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]. 

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

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

Within the framework of spherical coordinates rA>6A>9A) define an unnormalized Spherical Exponential Type Orbital (SETO) function as ... 

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 ... 

Keywords Exponential-type orbitals Generalized Sturmians Hyperspherical harmonics Interelectron repulsion integrals Isoenergetic configurations Momentum space Quantum theory Sturmians... 

Gaussian basis functions in molecular electronic structure calculations was first suggested by McWeeny115,116 and independently by Boys117 in 1950. However, the supposedly more physical Slater (exponential) Type Orbitals (STO) remained the basis function of choice for many years because they correctly describe not only the cusp associated with the nucleus on which they are centred but also afford a suitable representation of the long range behaviour. Shavitt118 records that... 

R374 P. E. Hoggan, How Specific Exponential Type Orbitals Recently Became a Viable Basis Set Choice in Nuclear Magnetic Resonance Shielding Tensor Calculation , arXivv.org, e-Print Archive, Quantum Physics, 26 Oct 2010, 1-23, arxiv 1010.5425vl [quant-ph], Avail. URL http //aps.arxiv.org/PS cache/arxi v/ pdf/1010/1010.5425vl. pdf. 

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). 

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]. 

Atoms and Molecules with Exponential-Type Orbitals... 

Abstract A new compact two-range addition theorem for Coulomb Sturmians is presented. This theorem has been derived by breaking up the exponential-type orbitals into convenient elementary functions the Yukawa potential (e /r) and evenly-loaded solid harmonics, for which translation formulas are... 

Exponential type orbitals (ETO) are the natural choice for the basis set in atomic and molecular electronic structure calculations. The resulting multicenter integrals (for molecules) are notoriously difficult to accurately evaluate. One of the fundamental reasons is the lack of a compact and rapidly convergent addition theorem (ADT) for ETOs. 

The present work describes a breakthrough in two-electron integral calculations, as a result of Coulomb operator resolutions. This is particularly significant in that it eliminates the arduous orbital translations which were necessary until now for exponential type orbitals. The bottleneck has been eliminated from evaluation of three- and four- center integrals over Slater type orbitals and related basis functions. 

Although the majority of electronic quantum chemistry uses gaussian expansions of atomic orbitals [10,11], the present work uses exponential type orbital (ETO) basis sets which satisfy Kato s conditions for atomic orbitals they possess a cusp at the nucleus and decay exponentially at long distances from it [39]. It updates work since 1970 and detailed elsewhere [3,6,15,18,28,31,42,51,52,57,60,62]. 

The plane wave basis used as the basis for 2-D periodic solid wave-functions is approximated by localised B-splines, that can also describe the exponential type orbitals of the molecules. 

It is a remarkable gain in simplicity that the Coulomb operator resolution [66] now enables the exponential type orbital translations to be completely avoided, although some mathematical structure has been emerging in the BCLFs used to translate Slater type orbitals [74] and even more in the Shibuya-Wulfman matrix used to translate Coulomb Sturmians. 

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