Molecular function basis expansion

The simple orbital basis expansion method which is used in the implementation of most models of molecular electronic structure consists of expanding each R as a linear combination of determinants of a set of (usually) atom-centred functions of one or two standard forms. In particular most qualitative and semi-quantitative theories restrict the terms in this expansion to consist of the (approximate) occupied atomic orbitals of the constituent atoms of the molecule. There are two types of symmetry constraint implicit in this technique. 

Molecular expansion variational method Korobov [7] developed a variational method using the molecular-type basis functions involving excited... 

We begin our discussion of wave function based quantum chemistry by introducing the concepts of -electron and one-electron expansions. First, in Sec. 2.1, we consider the expansion of the approximate wave function in Slater determinants of spin orbitals. Next, we introduce in Sec. 2.2 the one-electron Gaussian functions (basis functions) in terms of which the molecular spin orbitals are usually constructed the standard basis sets of Gaussian functions are finally briefly reviewed in Sec. 2.3. 

In our four-component molecular approach, thus, we use spin-coupled, kinetically balanced, generally contracted Gaussian-type spinors (GTSs) as basis functions. The basis expansion is... 

The choice and generation of basis sets has been addressed by many authors [190,192,528,554-563]. While we consider here only the basic principles of basis-set construction, we should note that this is a delicate issue as it determines the accuracy of a calculation. Therefore, we refer the reader to the references just given and to the review in Ref. [564]. In Ref. [559] it is stressed that the selection of the number of basis functions used for the representation of a shell riiKi should not be made on the grounds of the nonrelativistic shell classification nj/j but on the natural basis of j quantum numbers resulting in basis sets of similar size for, e.g., Si/2 and pi/2 shells, while the p /2 basis may be chosen to be smaller. As a consequence, if, for instance, pi/2 and p /2 shells are treated on the tijli footing, the number of contracted basis functions may be doubled (at least in principle). The ansatz which has been used most frequently for the representation of molecular one-electron spinors is a basis expansion into Gauss-type spinors. 

Recently, an elegant approach to the bonding in clusters has been developed by Stone [35], whose Tensor Surface Harmonic Theory derives cluster skeletal molecular orbitals as expansions of vector surface harmonic functions. The skeletal molecular orbitals are generated from basis sets with 0(o) and l(jt) nodes with respect to a radial vector passing through the atoms. 

Without going into the mathematical details we shall qualitatively overview and compare the possibilities for defining polyatomic fragments in quantum chemical calculations. Since the present-day techniques always use finite basis expansions for the development of the molecular wave function, the crucial question is how to partition this basis set in the most effective way. 

In the case of atoms and molecules with central symmetry, in which case a one-center expansion is feasible, these equations are solved numerically. In the case of molecules with no central symmetry and in the case of crystals, this procedure is not possible and one must expand the molecular orbitals (MOs) or crystal orbitals (COs) as a linear combination of some basis functions (LCAO expansion). This was first conducted for molecules by Malli and Oreg< and applications can be found for diatomic or linear molecules, the only exception being the HjCO molecule, which was treated by Aoyama et... 

We have determined expansions of wave functions of several atomic and molecular systems in their ground states, at FCI level. These wave functions have been expressed in the three mentioned molecular basis sets CMO, NO, and in order to study their compacmess in different molecular orbital basis sets. Our aim is to analyze the structure and compactness of those expansions by means of the entropic indices proposed in Eqs. (5), (7), and (8) according to the seniority numbers. We have mainly chosen the systems of... 

The results reported in Table 1 also allow one to compare, in terms of the values of the indices ( and lyy, the expansions of the wave fnnctions of these systems according to the molecular orbital basis sets in which they are expressed. As can be seen from that table, the values of both indices are considerably lower in the NO and basis sets than in their CMO counterparts (except for the Be atom in the STO-3G basis set) the Be atom recovers the improvement in the and NO molecular basis sets when the larger cc-pVDZ basis set is used. These results again confirm that the NO and molecular basis sets lead to more compact wave functions, as has been reported in Refs. [6, 9, 11]. These valnes also point out that the Ic and % indices constitute suitable devices to describe quantitatively the compactness of a wave function. The high values found for the indices in the Be and Mg atoms in the three molecular basis sets can be interpreted in terms of the strong correlation exhibited by those systems. The appropriate ground-state wave functions for these atoms reqnire several dominant Slater determinants. The values reported in Table 2 reflect that seniority levels with very low contribution to the wave functions can present a broad determinantal distribution, i.e., the Li2 molecule exhibits 7 2=4 > 5 values because its W =4 = 10 " weight is expanded on 7560 Slater determinants in the STO-3G basis set [6]. Moreover, the Ia=o index values reported in that table indicate that aU systems possess a narrower... 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

The interpretation of IETS is helpful in understanding molecular junctions. Several workers have developed techniques for doing so [97-102], some based on quite complex analyses of the full Green s function [99-101], others based on a much simpler analysis in which the fact that the response is so weak is used as the basis for perturbative expansion[98]. The results of these analyses fit the spectra well. 

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