LCAO approximation calculations

One of the things illustrated by this calculation is that a surprisingly good approximation to the eigenvalue can often be obtained from a combination of approximate functions that does not represent the exact eigenfunction very closely. Eigenvalues are not vei y sensitive to the eigenfunctions. This is one reason why the LCAO approximation and Huckel theory in particular work as well as they do. 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

Any set of one-electron functions can be a basis set in the LCAO approximation. However, a well-defined basis set will predict electronic properties using fewer terms than a poorly-defined basis set. So, choosing a proper basis set in ab initio calculations is critical to the reliability and accuracy of the calculated results. 

A special aspect of this description appears if one starts the orbital optimisation process with orbitals obtained by linear combinations ofRHF orbitals of the isolated atoms (LCAO approximation s.str.). Let Pn.opt and be the starting and final orbitals of such a calculation. Then the difference between c n.opi and Papt in the vicinity of each atom merely consists in a distortion of the atomic orbitals of each atom. This distortion just compensates the contribution of the orbitals of the other atoms to Pn.ctpt in order to restore the proportionality between the partial waves of ipopi and the appropriate atomic orbital. 

Results of LCAO-MO calculations, although qualitative only, are in agreement with this interpretation. They show that hydration of both cations and anions involves a transfer of charge between the central ion and the water molecules 7—10). The decrease in the net charge of the central cation may be considerable, e.g. in [Li(OH2)ie] tbe net charge of the lithium ion is approximately -fO.4 (10). 

The 5-position of the nonprotonated 1,2,4-thiadiazole system was calculated to be the most reactive in nucleophilic substitution reactions using a simple molecular orbital method with LCAO approximation (84CHEC-I(6)463>. 

Jtfany authors refer to the HF-LCAO procedure, when discussing HF calculations made within the LCAO approximation. 

The electronic charge density in an MO extends over the whole molecule, or at least over a volume containing two or more atoms, and therefore the MOs must form bases for the symmetry point group of the molecule. Useful deductions about bonding can often be made without doing any quantum chemical calculations at all by finding these symmetry-adapted MOs expressed as linear combinations of AOs (the LCAO approximation). So we seek the LCAO MOs... 

Sometimes the estimation of the electronic structures of polymer chains necessitates the inclusion of long-range interactions and intermolecular interactions in the chemical shift calculations. To do so, it is necessary to use a sophisticated theoretical method which can take account of the characteristics of polymers. In this context, the tight-binding molecular orbital(TB MO) theory from the field of solid state physics is used, in the same sense in which it is employed in the LCAO approximation in molecular quantum chemistry to describe the electronic structures of infinite polymers with a periodical structure -11,36). In a polymer chain with linearly bonded monomer units, the potential energy if an electron varies periodically along the chain. In such a system, the wave function vj/ (k) for electrons at a position r can be obtained from Bloch s theorem as follows(36,37) ... 

O. Vahtras et al., Integral approximations for LCAO-SCF calculations. Chem. Phys. Lett. 213, 514-518 (1993)... 

Not even the SCF procedure can overcome this problem. In the case of atoms, the central field remains a valid and good approximation. Assuming a rigid linear structure in the molecular case is clearly not good enough, although it contains an element of truth. This inherent problem plagues all LCAO-SCF calculations to an even more serious extent. 

However, in a quantum chemical context there is often one overwhelming difficulty that is common to both Newton-like and variable-metric methods, and that is the difficulty of storing the hessian or an approximation to its inverse. This problem is not so acute if one is using such a method in optimizing orbital exponents or internuclear distances, but in optimizing linear coefficients in LCAO type calculations it can soon become impossible. In modern calculations a basis of say fifty AOs to construct ten occupied molecular spin-orbitals would be considered a modest size, and that would, even in a closed-shell case, give one a hessian of side 500. In a Newton-like method the problem of inverting a matrix of such a size is a considerable... 

In order to perform a qualitative analysis of the /1-decay-induced redistribution of electron density it is sufficient to calculate the molecular electron states in the MO LCAO approximation, i.e., not taking into account the correlation of electrons. Below we present the calculation data for a number of molecules, which we have obtained using the Gaussian-70 program with the basis of s and p functions. We have used the extended atomic basis 4-31G, which contains about twice as many atomic functions as the minimal one (Ditchfield et al, 1971). 

The electron density distribution was calculated according to Mulliken (1955 see also Herzberg, 1966). In the MO LCAO approximation the ith molecular orbital is... 

We have calculated the data presented in the table in collaboration with G. V. Smeloy (Kaplan et al., 1983, 1985). In the MO LCAO approximation we have used the same bases of atomic functions as in calculations of the excitation probabilities of the corresponding molecules (see Section III,B,1). Allowing for electron correlation, calculations of the number of Cl configurations and the atomic bases were the same as those given in Section III,B,2. 

In view of the large size of the valine molecule, our calculations (Kaplan et al., 1983) were carried out on a minimal basis of 51 Slater orbitals in the MO LCAO approximation using the method described in Sections II, C, and III, B, 1. We have taken into account 608 singly excited states of the ion (valine-He)+. The results of calculations for valine on a minimal basis were corrected with regard to the results of calculation of the influence of the basis length on the excitation probabilities of the fragments that are shown enclosed in boxes in Fig. 9 (the rest of the molecule was replaced by a hydrogen atom). 

The molecular integrals [Eqs. (4.2)-(4.4)] are calculated by these molecular orbitals of LCAO approximation. In Roothaan molecular orbital theory, the developing coefficients are determined as the electronic energy (E) becomes minimal (51MI1). Then the developing coefficients (Crj) and the molecular orbital energies (e,) are obtained by solving the Fock equa-... 

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