Orbital-Based Population Methods

Of the orbital-based methods, the earliest remains the most widely used method that developed by Mulliken and called the Mulliken population. The total number of electrons in a molecule N must equal the integral of p(r) over all spaces. For simplicity, we will examine the case of HF wavefunction then this integral can be 

The net atomic population neglects the electrons associated with the overlap between two atoms. Mulliken arbitrarily divided the overlap population equally between the two atoms, producing the gross atomic population 

---

Many semiempirical methods have been created for modeling organic compounds. These methods correctly predict many aspects of electronic structure, such as aromaticity. Furthermore, these orbital-based methods give additional information about the compounds, such as population analysis. There are also good techniques for including solvation elfects in some semiempirical calculations. Semiempirical methods are discussed further in Chapter 4. 

The NBO method can be used for ab initio calculations at the HE and any correlated levels as well as for DFT methods. The results do not strongly depend on the size of the basis set, which is one reason that the NBO method has replaced the outdated Mulliken population analysis, for example, for the calculation of atomic partial charges. But there is a price that has to be paid for the advantages. Like any orbital-based method for partitioning the electronic charge into atomic and bonding domains, the choice of the selection procedure has some arbitrary character that needs to be known in order to judge the quality of the results. 

The hypothesis that geometrical derivatives of the electric dipole moment can be used as effective charges of the atoms in a neutral diatomic molecule was initially proposed by Van Vleck. After decades of development the APT became customarily applied. In their spunky paper Lazzeretti et aV described a version of the APT approach that they claimed is probably superior to other physically based methods, and certainly more rehable on theoretical grounds than some loosely defined orbital-based approach. In their critical review of population methods they presented AIM favourably. 

The formation of C-C chemical bonds in a variety of solids, including some refractory dicarbides, has been considered by Li and Hoffman (1989) and Wijeyesekera and Hoffman (1984) based on EHT (extended Huckel theory) calculations. To our knowledge, these works are the only ones where the band analogues of bond populations, the so-called crystal orbital overlap populations (COOPs) have been calculated for refractory compounds. The most noticeable result is that, in spite of the evident crudeness of the nonself-consistent semiempirical EHT method, the calculations allow us to understand the nature of the phase transition from cubic to hexagonal structure which occurs in the ZrC, NbC, MoC,... series as the VEC increases. The increase of metal-to-metal bonding when going from cubic (NaCl-type) to hexagonal (WC-type) becomes evident. 

A common method of calculating the approximate photo-ionisation cross-sections is to use the Gelius intensity model [79]. Here the cross-section for a particular orbital, , is expressed in terms of atomic contributions based on population analysis. 

A simple and robust quantitative MO-type approach (as opposed to density approaches) is the ubiquitous Mulliken population analysis [40]. The key concept of this easily programmed and fast method is the distribution of electrons based on occupations of atomic orbitals. The atomic populations do not, however, include electrons from the overlap populations, which are divided exactly in the middle of the bonds, regardless of the bonding type and the electronegativity. As a consequence, differences of atom types are not properly accommodated and the populations per orbital can be larger than 2, which is a violation of the Pauli principle a simple remedy for this error is a Lowdin population analysis that... 

Since there is no operator that produces the atomic population, it is not an observable and so the procedure for computing N(k) is arbitrary. There are two classes of methods for computing the atomic population those based on the orbital population and those based on a spatial distribution. ... 

The first and most influential molecular-orbital calculation on metal-alkynyl complexes is that of Kostin and Fenske, who applied the Fenske-Hall method to the complexes FeCp(C=CH)(PH3)2 and FeCp-(C=CH)(C0)2 (11). They concluded that the M-CCH bonds in these complexes are nearly pure a in character. The large energy gap (ca. 15 eV) between the occupied metal orbitals and ir (C=CH) levels severely limits the ir-accepting quality of the latter, with the total electron population for the pair of tt orbitals being 0.22 e for FeCp(C=CH)(PH3)2 and 0.14 e" for FeCp(C=CH)(CO)2. The filled ir(C=CH) orbitals, in contrast, mix extensively with the higher-lying occupied metal orbitals these filled-filled interactions result in the destabilization of the metal-based orbitals. The HOMOs of both complexes possess substantial coefficients at the alkynyl jS-carbon this was noted to be consistent with the alkynyl-localized reactivity of these complexes. 

---