Molecular orbitals basis function expansion

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. 

From this wave function, one sees how even in the early beginning of molecular quantum mechanics, atomic orbitals were used to construct molecular wave functions. This explains why one of the first AIM definitions relied on atomic orbitals. Nowadays, molecular ab initio calculations are usually carried out using basis sets consisting of basis functions that mimic atomic orbitals. Expanding the electron density in the set of natural orbitals and introducing the basis function expansion leads to [15]... 

The key is that a single-center expansion of the transition density, implicit in a multipolar expansion of the Coulombic interaction potential, cannot capture the complicated spatial patterns of phased electron density that arise because molecules have shape. The reason is obvious if one considers that, according to the LCAO method, the basis set for calculating molecular wavefunctions is the set of atomic orbital basis functions localized at atomic centers a set of basis functions localized at one point in a molecule is unsatisfactory. 

In the calculation of molecular electronic structure by the basis function expansion method it is necessary to calculate the molecular orbital repulsion integrals by calculating the corresponding repulsion integrals involving the basis functions... 

This is a convenient result if we substitute the basis function expansion (3.133) for the molecular orbitals into this expression, we obtain a formula for the energy, which is readily evaluated from quantities available at any... 

For electronic wave function calculations one would prefer to use the Slater functions. They more correctly describe the qualitative features of the molecular orbitals than do Gaussian functions, and fewer Slater basis functions than Gaussian basis functions would be needed in the basis function expansion of for comparable results. It is possible to show, for example, that at large distances molecular orbitals decay as which is of the Slater rather than the Gaussian form. In particular, the exact solution for the Is orbital of the hydrogen atom is the Slater function... 

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. 

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. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

We have presented the extension of the SCF-MI algorithm ((jianinetti, Raimondi and Tomaghi, 1996) to the more general case of the intermolecular interactions between closed- and open-shell systems. As in our previous work, the BSSE is excluded a priori, by allowing the expansion of molecular orbitals of each fragment only in the basis functions centred on the fragment itself... 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

Which properties are least well determined by the variational method The basis functions in the LCAO expansion are either Slater orbitals with an exponential factor e r or gaussians, e ar2 r appears explicitly only as a denominator in the SCF equations thus matrix elements are of the form < fc/r 0i> these have the largest values as r->0. Thus the parts of the wavefunction closest to the nuclei are the best determined, and the largest errors are in the outer regions. This corresponds to the physical observation that the inner-shell orbitals contribute most to the molecular energy. It is unfortunate in this respect that the bonding properties depend on the outer shells. 

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