Mulliken’s analysis

Many qualitative and semiempirical studies followed Mulliken s analysis [6], but it was not until 1960 that the first ab initio calculation on CH2 was published [7] by J. M. Foster and S. F. Boys in the famous April 1960 issue of the Reviews of Modem Physics which contained the papers from a conference on Molecular Quantum Mechanics that was held at the University of Colorado in June of 1959. This is the same Boys who in 1950 introduced the idea of using oscillator functions or as we know them now, gaussian fionctions as basis functions for calculating the electronic structure of polyatomic molecules. 

The consistent total energy makes it possible to compute singlet-triplet gaps using RHF for the singlet and the half-electron calculation for the triplet. Koopman s theorem is not followed for half-electron calculations. Also, no spin densities can be obtained. The Mulliken population analysis is usually fairly reasonable. 

Let s compare the Mulliken population analysis for ethylene and fluoroethylene ... 

For the purposes of a purely theoretical analysis of molecular electronic structure, we need more detailed information. The term population analysis was introduced in a series of papers by Mulliken in 1955, but the basic ideas had already been anticipated by Mulliken himself, and by other authors. The technique has been widely applied since Mulliken s 1955 papers, because it is very simple and has the apparent virtue of being quantitative . The word quantitative seems to mean two different things to different authors ... 

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. 

In this chapter, we use the definitions of bond order and valence indices provided by Mayer [4-6] (for a historical account, see Ref. [6a] and for other types of bond indices, see Ref. [6b]). In terms of electronic structure theory, they represent an extension to Mulliken s population analysis. The bond order is defined as... 

In a Mulliken population analysis, the electron density of the overlap terms is equally divided between the two atoms. Since the overlap integrals S(2s F)(lsH) and S(2phydrogen atom is positively charged. 

As it is, it should be remembered that Fig. 5.2 has become a sort of Rosetta stone whose deciphering paves the way toward a better understanding of what falsely seems to be an inherent intricacy of Mulliken s population analysis. 

Knowledge of the molecular wavefunction enables us to determine the electron density at any given point in space. Here we inquire about the amount of electronic charge that can be associated in a meaningful way with each individual atom of a A -electron system. Our analysis covers Mulliken s celebrated population analysis [31], as well as a similar, closely related method. 

Mulliken s population analysis is rooted in the LCAO (linear combination of atomic orbitals) formulation it is not directly applicable to other types of wavefunctions. With Cr i representing the coefficient of the rth type of atomic orbital (li, 2s, etc.) of atom k in the ith molecular orbital, we describe the latter by... 

Mulliken s formula for Nk implies the half-and-half (50/50) partitioning of all overlap populations among the centers k,l,... involved. On one hand, this distribution is perhaps arbitrary, which invites alternative modes of handling overlap populations. On the other hand, Mayer s analysis [172,173] vindicates Mulliken s procedure. So we may suggest a nuance in the interpretation [44] departures from the usual halving of overlap terms could be regarded as ad hoc corrections for an imbalance of the basis sets used for different atoms. But one way or another, the outcome is the same. It is clear that the partitioning problem should not be discussed without explicit reference to the bases that are used in the LCAO expansions. 

We have learned about the unique ordering of the carbon net charges relative to one another. All methods using Mulliken s population analysis, both ab initio and semiempirical, no matter what basis sets are used to construct the wavefunctions, reproduce the following sequence of inductive effects ... 

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)... 

Whereas many scientists shared Mulliken s initial skepticism regarding the practical role of theory in solving problems in chemistry and physics, the work of London (6) on dispersion forces in 1930 and Hbckel s 7t-electron theory in 1931 (7) continued to attract the interest of many, including a young scientist named Frank Westheimer who, drawing on the physics of internal motions as detailed by Pitzer (8), first applied the basic concepts of what is now called molecular mechanics to compute the rates of the racemization of ortho-dibromobiphenyls. The 1946 publication (9) of these results would lay the foundation for Westheimer s own systematic conformational analysis studies (10) as well as for many others, eg, Hendrickson s (11) and Allinger s (12). These scientists would utilize basic Newtonian mechanics coupled with concepts from spectroscopy (13,14) to develop nonquantum mechanical models of structures, energies, and reactivity. 

Mulliken s population analysis has been thoroughly conducted to examine the net charge as well as the magnitude of covalent bondings. The author found that Mulliken s charge of Li in Li2.1V0.9O2 and Li1.1V0.9O2 and the BOP value for Li-O and V-O are different in their structure. It is contrary to the widely accepted picture of Li-intercalated compounds, and should be a very important consideration for the determination of battery properties such as OCV. The information should be helpful to investigate possibility of new electrode active materials. 

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