Population Mulliken

EH total energy change to the total (nuclear repulsion plus electronic) observed energy change for internal rotation in ethane. This is consistent with our earlier interpretation of EH total energy. 

We found that the electron densities and bond orders calculated in the simple Hiickel method were extremely useful for relating theory to observable molecular properties such as electron spin-resonance splittings or bond lengths. Hence, it is desirable that we find analogous quantities to describe the distribution of electrons in an aU-valence-electron method like the EH method. A number of suggestions have been made. The one we use is due to Mulliken [8]. It is the most widely used, and, as we shall see, it has an especially direct and useful connection with the EH method. 

Consider a real, normalized MO, l i, made up from two normaUzed AOs, Xj and xk - 

We square this MO to obtain information about electronic distribution  

If we integrate Eq. (10-11) over tbe electronic coordinates, we obtain (since j, and 

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One can note some interesting features from these trajectories. For example, the Mulliken population on the participating atoms in Figure 1 show that the departing deuterium canies a full electron. Also, the deuterium transferred to the NHj undergoes an initial substantial bond stretch with the up spin and down spin populations separating so that the system temporarily looks like a biradical before it settles into a normal closed-shell behavior. 

A simple measure of the election density distribution over the participating atoms is the Mulliken population [60]. For linear Li—H—Li the alpha spin is... 

Figure 7. Alpha Mulliken population on Li(2) as functions of time for different initial conditions.

Mulliken population analysis is a trivial calculation to perform once a self-consistent field has been established and the elements of the density matrix have been determined. 

In spite of its deficiencies, the Mulliken population scheme is very popular. One reason is that it is very easy to implement so it is available in many software packages. Probably the most important reason for its popularity is the fact... 

There is some ambiguity about Mulliken population analysis in the literature. This is because various software packages print different portions of the analysis and may name them slightly differently. The description here follows some of the more common conventions. 

Recent progress in this field has been made in predicting individual atoms contribution to optical activity. This is done using a wave-functioning, partitioning technique roughly analogous to Mulliken population analysis. 

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. 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

Despite these reservations, Mulliken population-derived atomic charges are easy to compute. Empirical investigation shows that they have various uses they provide approximate representation of the 3D charge distribution within a molecule. 

By default, Gaussian jobs perform a Mulliken population analysis, which partitions the total charge among the atoms in the molecule. Here is the key part of output for formaldehyde ... 

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

This exercise will examine other ways of computing charges other than Mulliken population analysis. Since atomic charge is not a quantum mechanical observable, all methods for computing it are necessarily arbitrary. We ll explore the relative merits of various schemes for partitioning the electron density among the atoms in a molecular system. 

The Mulliken scheme places the negative charge more or less evenly on the three carbons, and splits the positive charge among the hydrogens. Mulliken population analysis computes charges by dividing orbital overlap evenly between the two atoms involved. 

There is very little point in trying to obtain information from the 73 x 47 z= 3431 numbers that constitute the HF-LCAO coefficients for the occupied orbitals. Mulliken population indices are given next, together with Mulliken atomic charges (Figure 10.14). 

The KS-LCAO orbitals may be visualized by all the popular methods, or one may just focus on the Mulliken population analysis indices (Figure 13.5). 

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. 

Table 1. The 72-atom model examined by different theoretical methods. The energy differences (AE in kcal/mol) are calculated with respect to the lowest SCF energy. q(Fe) stands for Mulliken population charges on the Fe atoms q(S) and SS(b.i.) are the Mulliken population charges and the bond index for the bridging S atoms, respectively AEq is the calculated Mossbauer quadrupole splitting constant [mm/sec]. The PUHF spin states are those projected from the UHF wavefunction with 5 = 5,. 

Table II. Differences in Mulliken population analyses for an Si50i5Hi2 molecule and four combinations of point-ion cluster, using a 3-21G basis set...

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