Approximate SCF-MO Methods

At the beginning of this chapter it was stated that ab initio calculations require exact calculation of all integrals contributing to the elements of the Fock matrix, but we have seen that, as we encounter systems with more and more electrons and nuclei, the number of three- and four-center two-electron integrals becomes enormous, driving the cost of the calculation out of the reach of most researchers. This has led to efforts to find sensible and systematic simplifications to the LCAO-MO-SCF method—simplifications that remain within the general theoretical SCF framework but shorten computation of the Fock matrix. 

A number of variants of a systematic approach meeting the above criteria have been developed by Pople and co-workers, and these are now widely used. The approximations 

Differential overlap dS between two AOs, Xa and x. is the product of these functions in the differential volume element dv  

The only way for the differential overlap to be zero in Ju is for Xa or xb, or both, to be identically zero in dv. Zero differential overlap (ZDO) between Xa and xb in all volume elements requires that Xa and Xb can never be finite in the same region, that is, the functions do not touch. It is easy to see that, if there is ZDO between Xa and xb (understood to apply in all dv), then the familiar overlap integral S must vanish too. The converse is not tme, however. S is zero for any two orthogonal functions even if they touch. An example is provided by an s and a p function on the same center. 

It is a much stronger statement to say that Xa and xb have ZDO than it is to say they are orthogonal. Indeed, it is easy to think of examples of orthogonal AOs but impossible to think of any pair of AOs separated by a finite or zero distance and having ZDO. Because AOs decay exponentially, there is always some interpenetration. 

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If the full molecular symmetry is assumed, the ground states of the cation radical of fulvalene and the anion radical of heptafulvalene are both predicted to be of symmetry by using the semiempirical open-shell SCF MO method The lowest excited states of both radicals are of symmetry and are predicted to be very close to the ground states in the framework of the Hiickel approximation these states are degenerate in both cases (Fig. 4). Therefore, it is expected that in both these radicals the ground state interacts strongly with the lowest excited state through the nuclear deformation of symmetry ( — with the result that the initially assumed molecular... 

D. N. Natida and K. Jug, Tbeor. Chim. AcUi, 57, 95 (19S0). SINDOl. A Semiempirical SCF MO Method for Molecular Binding Energy and Geometry. 1.. Approximations and Parameterira-tions. 

In (16.83), vai is the number of valence electrons in the molecule, V(i) is the potential energy of valence electron i in the field of the nuclei and the inner-shell (core) electrons, and Hvai (0 is the one-electron part of Hyai-TTie CNDO and INDO methods are SCF MO methods that iteratively solve the Roothaan equations using approximations for the integrals in the Fock matrix elements. 

Ab initio calculations are based on iterative procedures and provide the basis for self-consistent field-molecular orbital SCF-MO) methods. Electron-electron repulsion is specifically taken into account. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved in the Roothaan method by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved. 

A quantum-chemistry approximation method is variational if the energy calculated by the method is never less than the true energy of the state being calculated. Since an SCF MO wave-function energy is equal to the variational integral (8.1), the SCF MO method is variational. Although being variational is a desirable property, we shall see that many of the calculation methods currently used (such as MP, CC, DFT) are not variational. 

The condition (17.38) determining the tt-MO coefficients for benzene was derived solely from symmetry considerations, without use of the Hiickel approximations. Thus the MOs (17.41) are (except for normalization constants) the correct minimal-basis-set SCF TT-electron MOs for benzene. (The Hiickel energies ei,..., e are, however, not the true SCF orbital energies. The Hiickel method ignores electron repulsions and takes the total TT-electron energy as the sum of orbital energies. The SCF MO method takes electron... 

Semiempirical calculations have been carried out by an unparameterized SCF-MO method with integral approximations [5], various versions of the CNDO [37 to 42] and INDO [6, 38, 43 to 45] methods, the MNDO [46, 47] and MINDO [48] methods, the extended Hiickel method [3, 4, 49, 50] (presumably also [51 ]), a Pariser-Parr-Pople-type open-shell method [49] (presumably also [51]), and a simple MO approach [52]. Besides some other molecular properties, the charge distribution (atomic charges and/or overlap populations) [5, 38,40,41,43,49 to 51] and the spin density distribution (and thus, the hyperfine coupling constants, compare above and p. 241) [3 to 6, 46, 48] have been the subjects of many of these studies. 

Semiempirical calculations for j or AEf were carried out by the SCF-MO method with integral approximations [23], by various versions of the CNDO [24 to 27] and INDO [1 to 4, 28] methods, by the MNDO method [29], by the extended Huckel and Pariser-Parr-Pople methods [30], and by a simple [31] MO method [32]. 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

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