Mulliken approximation

The constant k can be adjusted to give best agreement with exper-iinen t. It is I onn d th at a good value to n se is soin ewh at larger th an would be indicated by the Mulliken approximation and a very standard value used by many groups is ... 

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 derivation of this result introduces a generalization of 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 the spirit of the Huckel-Wheland theory, the Mulliken approximation for electron repulsion integrals 53) is used in the maximization of the localization sum (Eq. (28)). The details of the calculations are reported elsewhere. S4)... 

Further enlarging the set of Coulomb integrals has been done in the Fenske-Hall method practiced more or less widely in the 1970s and 1980s. It takes into account all possible two-electron integrals, but calculates them using the Mulliken approximation eq. (2.29). Nevertheless no decisive success has been achieved in this direction. 

This equation is the exact expression arising from the solution of a 2 x 2 determinant. In order to understand qualitatively how this expression varies as a function of eav, Epiotis has used the Mulliken approximation, Hy = kSy ). Further assuming that Sy < 1, Eq. (1) simplifies to Eq. (2) ... 

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... 

In the work of Larsson and Pyykko [51] the energy dependence of the atomic orbitals is defined by a polynomial fit of the charge dependence of the energy of any one AO on the population of the other AOs, as derived from Dirac-Fock calculations [49], Orbital populations are obtained by the Mulliken approximation. The orbital energies are corrected for the Coulomb interaction of the total charge on other atoms. The wavefunctions are of double-zeta variety, the radial parameters being optimised separately for the different spinors. 

Here L is a correction of the Mulliken approximation for the kinetic energy and All is entirely empirical and contains adjustable bond parameters. These are optimized in order to minimize the deviation from experiment for a set of reference compounds. In a way similar to INDO/S [Eq. (45)], two sets of Slater orbital exponents are used one K(0)] for intra-atomic integrals and the other (0 for molecular integrals. For comparison with experimental heats of formation, the calculated binding energies Eb [Eq. (4)] are corrected by the zero-point energies obtained from vibration analyses. Later, a substantially modified version of SINDOl, MSINDO, was developed and reparameterized for the elements H, C F, Na-Cl, Sc-Zn, and Ga-Br [63-65],... 

We can derive the extended Huckel models and refinements from the Fock equations by assuming the Mulliken approximation for integrals,... 

Because the Mulliken approximation is less accurate for the kinetic energy, i.e. 

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. 

Applying the Mulliken approximation to the second term of (9b), we have... 

This still leaves the core-attraction terms, to which Barker and Eyring have applied the Mulliken approximation, and found inaccuracies of about... 

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. 

Using the Mulliken approximation in the three-center integrals, and the point-charge approximation for Vm it is shown in [222,224] that the MR operator matrix elements for a periodic system are given by... 

Two-center one-electron integrals Hfj,u are calculated using a modification of the Mulliken approximation. In... 

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