Mulliken-type approximations

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... 

We started by generating a data base of inner-shell correlation contributions for some 130 molecules that cover the first two rows of the periodic table. In order to reduce the number of parameters in the model to be fitted, we introduced a Mulliken-type approximation for the parameters Dab (Da+Db)/2. Furthermore we did retain different parameters for single and multiple bonds, but assumed Da=b (3/2)Da=b-... 

It is clear that such a Hamiltonian can only rationalize a Huckel-type Hamiltonian (eventually in a non-minimal basis set) and that it essentially leads to a pure Mulliken-type approximation for the bicentric integrals (which play the key role in the construction of the bond through electron delocalization). It is known that such approximations usually fail an empirical k parameter is used in the proportionality of the hopping integrals to the overlap and atomic energies (Wolfsberg-Helmholtz approximation). 

Equation (11.7) implies that orbital products can well be approximated in the average by some Mulliken-type expansion [206,218]. 

The interconnections of the two approximations have been extensively discussed. From a theoretical point of view, calculations without overlap (Hiickel type) can be considered as a way to reproduce with less labor the results of the calculations with overlap (Wheland-Mulliken type) provided that the quantities are interpreted as bearing over an orthogonalized basis. Thus, in practice, the Hiickel approximation is used most frequently with parameters H = a, and = f rs along each pair of adjacent atoms. These parameters are all expressed in terms of the corresponding values a and p for a standard carbon compound which need not be explicited until the end of the calculation. The individual energies of the electrons are finally obtained in the form... 

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 ... 

Called by Mulliken the LCAO ( linear combination of atomic orbitals ) form. For a critique of this type of approximation see R. S. Mulliken, J. Chem. Phys. 3, 375 (1935). It appears to have been first suggested by Len-nard-Jones. 

Condensed versions for the Fukui ftinction based upon the idea of approximate integration of the electron density in atomic regions (the simplest model being a Mulliken population type analysis [26]) were proposed by Mortier and Yang [27], For a given atom A one then writes... 

Since the work of McMurry, Mulliken, Sidman, and Pople, it has been fairly well established that the 290 m/i transition in saturated ketones is the n - n transition involving orbitals roughly of the type depicted in Figure 2. For a first approximation, these orbitals may be represented as... 

We are using, with Mulliken, lower case letters for denoting the quantum numbers of orbitals and capital letters for quantum numbers of the more-electron systems. The type of these quantum numbers depends on the symmetry of the nuclear distribution of the system considered. The case of atomic spectroscopy is thus approximated by the highest symmetry possible—the spherical symmetry around a single, very, very small, nucleus. 

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