Overlap Mulliken approximation

Most methods of this type are based on the so-called zero-differential overlap (ZDO) approximation. Their development begins by using an approximation to the atomic-orbital-based two-electron integrals introduced by Mulliken ... 

The off-diagonal elements of Extended Hiickel theory, (fi v) represent the effects of bonding between the atoms and are assumed to be proportional to the overlap, Sj y. An approximation for differential overlap referred to as the Mulliken approximation... 

The pairwise overlap, symmetrical orthogonalisation and Mulliken approximation together validate the NDO approximation — the orbital product in the orthogonalised GHO basis vanishes to the extent that the Mulliken approximation is realistic. This conclusion obviously has enormous consequences for any NDO approximation schemes. 

In their treatment of N3 and HN3 Yonezawa and co-workers abandoned the zero differential overlap approximation and considered all valence electrons. The Mulliken approximation... 

It is shown that the LCAO molecular Hartree-Fock equations for a closed-shell configuration can be reduced to a form identical with that of the Hoffmann extended Hiickel approximation if (i) we accept the Mulliken approximation for overlap charge distributions and (ii) we assume a uniform charge distribution in calculating two-electron integrals over molecular orbitals. Numerical comparisons indicate that this approximation leads to results which, while unsuitable for high accuracy calculations, should be reasonably satisfactory for molecules that cannot at present be handled with facility by standard LCAO molecular Hartree-Fock methods. 

Formally, these equations are similar to the SCF LCAO MO equations. Thus, a good definition of //effective may yield satisfactory solutions, without the tedious iteration procedure of the SCF method. In practice, two calculation procedures have evolved from these equations, the Hiickel approximation and the Wheland-Mulliken approximation. Neither of them specifies the analytical form of H, but instead treats some of the matrix elements H,. as adjustable parameters. Their main difference resides in the neglect (Hiickel) or non-neglect (Wheland-Mulliken) of overlap. Their common feature is the tight-binding approximation, namely the neglect of all matrix elements which involve non-bonded atoms. 

What leads to a large Fy value The simplest and most powerful way to think about F j interactions is in terms of orbital overlap, making implicit use of the Mulliken approximation, i.e.. 

Population analysis with semi-empirical methods requires a special comment. These methods normally employ the ZDO approximation, i.e. the overlap S is a unit matrix. The population analysis can therefore be performed directly on the density matrix. In some cases, however, a Mulliken population analysis is performed with DS, which requires an explicit calculation of the S matrix. 

The strength of the Fajb interaction and its variations with distance and orientation can be conveniently visualized in terms of the overlap of 7ra and 7tb NBOs, on the basis of a Mulliken-type approximation (cf. Eq. (1.34)). As an example, the top two panels of Fig. 3.38 compare the overlapping 7ta-7tb orbital contours for trails 1 and cis 2 isomers of butadiene. As shown in Fig. 3.38, the overlap in the cis isomer 2 (S = 0.2054) is slightly weaker than that in the trans isomer 1 (S = 0.2209), due to the unfavorable orientation of the 7ta across the nodal plane of the 7tb in the latter case. Consistently with the weaker 7ta-7tb overlap, the JtA F nh ) interaction is less, namely 0.0608 in 1 versus 0.0564 in 2. The delocalization tail of the 7fa NEMO is correspondingly less than its value in the trans isomer... 

These interactions are second order in PAB, so they are weaker than those occurring between degenerate orbitals (first order in PAB). The more stabilized a, the easier is the reaction between A and B. To maximize this stabilization, the numerator PAB2 must be increased and/or the denominator (EA° — EB°) decreased. Since Mulliken s approximation (p. 13) takes PAB proportional to the overlap between VPA° and we can see that ... 

The bonding to antibonding orbital separation is a function of the resonance integral, which, in turn, is proportional to the overlap of the two interacting orbitals (Mulliken s approximation, p. 13). 

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

In contrast to the nice, neat two-center-two-electron bond model, it is not so easy to determine the overall bonding character from MO orbital drawings alone. We need another measure. This comes from the Mulliken overlap population which is a numerical indicator of bonding (positive) and antibonding (negative) character between a pair of atoms within a molecule. For N2 in an approximate calculation the overall overlap populations are +0.68 for the three a filled MOs and +0.54 each for the it MOs. If one considers each it interaction of bond order one then the overall bond order is clearly three. 

Tables 1.2-1.6 contain the results of an approximate MO calculation (Fenske-Hall) on the BF molecule. From this output (a) construct a MO diagram showing MO energy levels and qualitative AO compositions in MO drawings (b) examine the HOMO and LUMO relative to Lewis acid/base behavior and compare it with CO. Would BF be suitable for coordination to, e.g., a Cr center (c) use the Mulliken charges to predict the direction of the dipole moment (d) examine the Mulliken overlap populations and decide whether it is proper to describe the B-F bond as a single, double or triple bond. 

Since H j is proportional to Sijf one expects which is an MO approximation to the bond energy, to depend directly on the Mulliken overlap population as given in Equation 9. It should not be expected to include effects caused by the bond polarity, however. 

The simplified and convenient self-consistent-charge (SCC) - scheme is used to obtained an approximate SCF potential. By the use of Mulliken population analysis , we estimate the effective charge on each atom in the cluster from the orbital population and also estimate the strength of the covalent bonding between atoms from the overlap population. 

Thus this model maps density over atoms rather than spatial coordinates. If overlap is included some other definition of charge density such as Mulliken s17 may be employed. Eq. (30) and (31) are then used with this wave function to calculate the hyperfine constants as a function of the pn s. If symmetry is high enough, there will be enough hyperfine constants to determine all the p s, otherwise additional approximations may be necessary. For transition metal complexes, where spin-orbit effects are appreciable, it is necessary to include admixtures of excited-state configurations that are mixed with the ground state by the spin-orbit operator. To determine the extent of admixture, we must know the value of the spin-orbit constant X and the energy of the excited states. 

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