Mulliken gross atomic population

Expressions (4.3) and (4.4) conform to the symmetry of equation (4.1). In the following sections we give the Mulliken gross atomic populations (GAOP) [34] of the respective molecular spinors in terms of the spherical harmonics. 

The values are called the net atomic populations and the overlap population. Chemists speak of the charges on atoms in molecules, and Mulliken s second contribution was to propose a method of partitioning the overlap population between contributing atoms. He proposed that the overlap populations be divided equally between participating atoms, so giving the gross atomic populations of... 

Equation 6.55 represents the Mulliken population analysis, where the first sum in the parentheses containing only quadratic terms is called net atomic population, the second sum is the overlap population, and nA is the total sum of all electrons associated with atom A, the gross atomic population (Mulliken, 1955, 1962). Note the difference between Equation 6.54 and Equation 6.55 in the former the summation includes all AOs of the molecule, while in the latter the sum is confined to those AOs that belong to atom A. Obviously ... 

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. 

The net atomic population, n(k), does not include any of the electron density associated with the overlap population, N(k,l). It is clear that some of these electrons in the overlap population belong to atom k and the remaining electrons belong to atom /. Furthermore, these should be added to n k) and (/) to get the total number of electrons on each atom. Arbitrarily, Mulliken divided this overlap population evenly between the two atoms resulting in the gross atomic population N(k) for atom k, Eq. [10]. 

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... 

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. 

The earliest definition of atomic charges is related to Mulliken s population analysis [24] and assumes equipartition of overlap populations between the relevant atoms, yielding the gross atomic populations ... 

The charge attributed to an atom A within a molecule, defined as = Za — where Za is the atomic number of atom A and A is the electron density (see Atoms in Molecules) assigned to A. The calculation of q depends on the partitioning of the electron density. In the framework of Mulliken population analysis (see Electronic Wavefunctions Analysis), cja is related to the gross atomic population H q, where q is the gross... 

The quantities nr and nr s are used to calculate atom charges and bond orders. The Mulliken gross population in the basis function (fir is defined as the Mulliken net population nr (Eq. (5.211)) plus one half of all those Mulliken overlap populations nrh (Eq. (5.212)) which involve separated atoms Srs is very small) ... 

In this context, within a CNDO-like approach, one can suppose that the distributions tR of equation (27) are defined for each atomic center as the product of a Mulliken total gross shell population [9] and a nS distribution ... 

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... 

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. 

Table 22.7 Mulliken gross populations for bonding atomic spinors in TlAt and...

Notice that the shared electron density, P is divided equally between the two atoms in question. The gross atomic charge on each atom is simply the sum of all the q, belonging to that atom minus the nuclear charge of the atom on which orbital XfA. is located. This is called the Mulliken population analysis. The computed charge on an atom of a molecule is influenced by a number of factors such as the basis set chosen, the exact details of H , and whether electron correlation is taken into consideration or not. The Mulliken scheme is arbitrary in that it partitions the shared electron density equally between the two atoms. There are many other methods for population analysis, some perhaps preferable in that they do not appear to be as method and basis set dependent as the Mulliken scheme. 

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,-... 

The above arguments form the core of Mulliken s population analysis [83], and it is customary to call the atom-centered electrons the (atomic) net populations (NP) which, together with the overlap populations (OP), add up to the total electron number in the above example, this is 2 = NPi + NP2 + OP = 0.61 + 0.61 + 0.78. We might also cut the overlap population s)nnmetrically and add each half to the net populations, thereby defining (atomic) gross populations (GP) in the H2 example, this makes 2 = NPi + j x OP + NP2 + x OP = GPi + GP2 = 1.00 + 1.00 = 2, which is very simple. Within this scheme, the atomic charge Cj is the difference between the atomic number Z and the gross population cj = Z — GP = 1.00 - 1.00 = 0) such that the H atom is neutral in the H2 molecule, as expected. 

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