Population analysis basis functions

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

Table 9 shows the PP MO results for this interesting series of highly strained three-membered cyclic molecules. Here a detailed comparison is possible with the best results of an all-electron study, including d functions (also reported in Table 9). An analysis of this table reveals how all trends in population analysis, both in charges and overlap populations, are the same in the AE -I- d and in the simple PP calculations, with very few and very minor exceptions. PP predicts a charge donation to the aliphatic groups, while AE predicts a withdrawal, mainly due to the availability of d orbitals on sulphur, which can allocate extra electronic charge. As outlined in the general notes on population analysis (Section III.D) comparisons should be carried out on a relative basis and.

The information obtainable upon solution of the eigenvalue problem includes the orbital energies eK and the corresponding wave function as a linear combination of the atomic basis set xi- The wave functions can then be subjected to a Mulliken population analysis<88) to provide the overlap populations Ptj ... 

The IAM model further assumes the atoms in a crystal to be neutral. This assumption is contradicted by the fact that molecules have dipole and higher electrostatic moments, which can indeed be derived from the X-ray diffraction intensities, as further discussed in chapter 7. The molecular dipole moment results, in part, from the nonspherical distribution of the atomic densities, but a large component is due to charge transfer between atoms of different electronegativity. A population analysis of an extended basis-set SCF wave function of HF, for example, gives a net charge q of +0.4 electron units (e) on the H atom in HF for CH4 the value is +0.12 e (Szabo and Ostlund 1989). 

Mulliken Population Analysis. A partitioning scheme in which electrons are shared equally between different Basis Functions. 

To alleviate a number of these problems, Lowdin proposed that population analysis not be carried out until the AO basis functions tp were transformed into an orthonormal set of basis functions / using a symmetric orthogonalization scheme (Lowdin 1970 Cusachs and Politzer 1968)... 

This localization scheme permits tire assignment of hybridization both to the atomic lone pairs and to each atom s contributions to its bond orbitals. Hybridization is a widely employed and generally useful chemical concept even though it has no formal basis in the absence of high-syrnmetry constraints. Witli NBO analysis, tire percent s and p character (and d, f, etc.) is immediately evident from tire coefficients of tire AO basis functions from which the NAO or NBO is formed. In addition, population analysis can be carried out using the NBOs to derive partial atomic charges (NPA, see Section 9.1.3.2). 

The four secular equations H- — WG [ = 0 are solved as follows For a given cycle, an input electron configuration and charge are assumed for the metal, and the terms are computed. terms for ligand basis functions remain constant throughout the calculation. For each of the MO s calculated in the cycle, a Mulliken population analysis is performed, in which each overlap population is divided equally between the two basis functions involved.<16)... 

It is also possible to perform a basis set transformation from primitive basis functions to symmetry combinations of the KS MOs of the atoms or larger fragments that constitute a system. In that case the population matrix elements P v become more meaningful, because they reflect the involvement of the fragment MOs in the orbitals of the total system. A Mulliken population analysis in... 

Assigning atom charges and bond orders involves calculating the number of electrons belonging to an atom or shared between two atoms, i.e. the population of electrons on or between atoms hence such calculations are said to involve population analysis. Earlier schemes for population analysis bypassed the problem of defining the space occupied by atoms in molecules, and the space occupied by bonding electrons, by partitioning electron density in a somewhat arbitrary way. The earliest such schemes were utilized in the simple Hiickel or similar methods [256], and related these quantities to the basis functions (which in these methods are essentially valence, or even just p, atomic orbitals see Section 4.3.4). The simplest scheme used in ab initio calculations is Mulliken population analysis [257]. 

Mulliken population analysis is in the general spirit of the scheme used in the simple Hiickel method, but allows for several basis functions on an atom and does not require the overlap matrix to be a unit matrix. In ab initio theory each molecular orbital has a wavefunction ijj (Section 5.2.3.6.1) ... 

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