Atomic orbitals gross population

Table 3.7. Surfac atomic orbital gross populations twofold adsorption of CO on Rh(lll).

A chemist also wants to know the gross populations on each atom. The gross population on A or B certainly includes the respective net population, but for each atom one must also include a part of the overlap population. The simplest procedure is to proportion the two parts equally to A and B, even when a and b are unequal i.e., Nab(< >A < >B) to eadi atom. Adding this to the net population of either atom gives a gross population N(A) or N(B), such that N(A) + N(B) = N. Therefore, a specific molecular orbital can be described as nonbonding either because N(A) N i.e., it is localised almost exclusively on one centre, or N(A) and N(B) are both significant but the overlap population 2 Nab ( >A ( >B) is approximately equal to zero. 

Here, Q- is the partial overlap population, namely, the electron population of the overlap region between the atomic orbitals y,- and Xj >n the MO atomic orbitals y,- and y, in the MO , and S,(, the overlap integral between atomic orbitals y,- and y . QH is the gross atomic population or the gross atomic charge on atom H and is given by the sum of the atomic orbital populations Qi. The net charge AQh is obtained from the difference between QH and the atomic number ZH, namely, the number of electrons in the neutral atom. 

As tne data in Table 6 show, the sum of the terms neglected in the Townes and Dailey method is by no means small. Cotton and Harris 3,) have attempted to re-vamp the method and after a series of approximations which at first sight appear more reasonable than the original ones, arrive at a series of equations in which the valence orbital populations of the Townes and Dailey method are replaced by the corresponding Mulliken gross atomic orbital populations. 

In order to obtain information concerning the bonding nature, we used the Mulliken population analysis (8). The number of electrons are partitioned into gross populations for the rth atomic orbital, n,-... 

A recent comparison (33.) of quantities calculated from molecular electronic wave functions for pyrrole and pyrazole indicated that ARCANA values compared more favorably with large-scale ab initio values than those calculated by other non-rigorous methods in the case of orbital energies of occupied molecular orbitals, gross atomic populations of heteroatoms, and total overlap populations including negative overlap populations between nonbonded atoms. 

One final definition of note is the gross atomic orbital population NfrJ, Eq. [13], which is the number of electrons in a particular atomic orbital. 

The (PS)., elements are the gross atomic orbital population of x.- The gross atomic population is then... 

Mulliken also noted another disturbing tendency in this population anal-ysis. - The gross atomic orbital population could have a value less than zero or greater than 2. A population greater than 2 appears to violate at least the spirit of the Pauli exclusion principle. A negative orbital population is even more difficult to interpret. Mulliken s examples all had very small negative values and were thus discounted as unimportant. However, the atomic population can be very negative—a population of —0.7 e for a sulfur d orbital in SFg... 

Note that this equation has no cross terms. The gross atomic orbital population is simply a sum over all MOs of the orbital coefficient squared, which means that only positive values will be obtained. Since the wavefunction is normalized, the atomic orbital population can be no larger than N(i), which is usually 2. Gross atomic populations are obtained by summing the atomic orbital population of all orbitals centered on a given atom. 

Davidson suggested that the wavefunction be projected onto a set of orbitals that have intuitive significance. These orbitals are a minimum set of atomic orbitals that provide the best least-squares fit of the first-order reduced-density matrix. Roby expanded on this idea by projecting onto the wavefunction of the isolated atom. One then uses the general Mulliken idea of counting the number of electrons in each of these projected orbitals that reside on a given atom to obtain the gross atomic population. 

Tnble 3.17. Gross population of Hartree-SNter-Fock Linear Combination of Atomic Orbital calculationi ) ). 

Here the indices A and B refer to atoms, k and I label atomic orbitals centered on atoms A and B, respectively, and i labels the molecular orbitals in the SCF approximation - which may be either doubly or singly occupied depending on the RHF or UHF case, respectively c,vt represents the coefficient of atomic orbital k in the molecular orbital i and Su is the overlap integral between atomic orbitals k and 1. The charge of the atom appears by comparison between the gross population Pa and the atomic number Z ... 

In Section 2.09.2.2, it was reported that, although inclusion of the sulfur 3sulfur atom, the gross charges on all atoms showed large variations. Table 5 lists the variations in charges on sulfur and... 

An often-used method of calculating atomic charges in molecules is the population analysis (47). The gross population of an atomic orbital is given by the Equations 21 and 22,... 

Equation (12c) also shows that the weight of aj is given by a Mulliken gross population of atomic orbital molecular orbital k. For equal a, it becomes 1 by normalization. 

The population method due to Pipek and Mezey [37] maximizes a certain sum of gross atomic Mulliken populations. The latter are not realistic when the LCAO basis includes diffuse atomic orbitals as is necessary for crystalhne sohds. 

In the column to the right the "gross population for each atomic orbital is reported. 

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