Atomic orbital populations

Atomic orbital populations and resulting QS in a crystal field model. 

In the disilane molecule, for example, the silicon atoms exhibit significant electron deficiency. The atomic orbital population and its equivalent in ethane is recorded in Table 3. 

Table 3. Atomic orbital population of silicon and carbon in disilane and ethane, respectively.

TABLE 10. MP2/TZDP Mulliken atomic orbital populations for X in XH3Ya ... 

In equation (A.77) is defined the atomic orbital population P,-. Summing all of the P, that belong to the same atom, I, yields the atomic population / [equation (A.78)]. The net charge qj on atom I is just the difference between the nuclear charge Zi and the atomic population,... 

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. 

Perhaps the most extensively studied metallo-bis(dithiolenes) that possess more elaborated dithiolene donors are those that utilize the dmit ligand, and DFT calculations have recently been used to investigate the nature of the ground-state one-electron levels of [Ni(dmit)2)], [Pt(dmit)2)], and [Pt(dmit)2)J (380). Despite the increased complexity of the dithiolene ligands in these complexes (Fig. 36), their electronic structure is remarkably similar to that of the simple or parent metallo-bis(dithiolenes) that have been discussed previously. This finding is most evident in the electron density contour plots of the HOMOs and LUMOs for [Ni(S2C2Me2)2], [Ni(dmit)2], and [Ni(H2 timdt)2)], and these are presented in Fig. 37. An atomic orbital population analysis reveals that the HOMO possesses b u symmetry, and is almost entirely comprised of out-of-plane ligand pn orbitals with only a few percent metal... 

Fig. 5.26. X-ray photoelectron, uv photoelectron, and angle-resolved carbon K and oxygen K x-ray emission spectra for carbonates, shown along with molecular-orbital assignments and atomic-orbital populations derived from the calculations of Connor et al. (1972) and orbital assignments derived from the calculations of Tossell (1976b) (after Tegeler et al., 1980 reproduced with the publisher s permission).

There has been considerable effort directed towards the assessment of the covalency or ionicity of various solids. The output of standard ab initio (SCF) Hartree-Fock-Roothaan calculations contains Mulliken (1955) charge distribution analysis parameters such as the atomic-orbital populations, net atomic charges, and bond overlap populations deseribed earlier (Chapter 3), which are often used to discuss the relative covalency or ionicity of materials. Considerable caution is required in using such parameters, however, since net atomic charges and other such quantities are not quantum-mechanical observables that is, they cannot even in principle be measured, and are highly basis-set dependent, as noted by Hehre et al. (1986 pp. 336-41). This is illustrated for molecules more relevant to mineralogy in Table 7.1, in which a number of properties of CO2 and SiOj are shown calculated at various basis-set levels. It is clear... 

One rather satisfactory way is to introduce all valence electrons in the molecular orbital approach without a hypothesis on the hybridization of atomic orbitals. Population analysis using the CNDO/2-SCF-MO showed the electronic configuration of the nitrogen atom of aromatic nitrogen cation systems (1, 40, 41, 42, 43, 44) in the ground state to be as follows (69G1078) ... 

Fig. 5 Measured x-ray emission spectrum of Ceo (heavy solid line) compared with carbon 2p DOS calculated by the DV-Xa method (thin solid Hne). Vertical line net 2p atomic orbital population calculated by the DV-Aa method. The thin solid line is the result of the convolution of 0.5-eV FWHM Lorentzian function and the net atomic orbital population. The peak at 270 eV is a substrate signal. (Taken from Kawai and Motoyama [20].)...

GAUSSIAN 90. (b) 2/) DOS by STO-3G, and (c) 3p DOS using the DV-Aa method, ymax indicates the maximum hight of the 3p or 2p net atomic orbital population. (Taken and modified from Kawai and Hashimoto [27].)... 

Strictly speaking, the 3p and 2p net atomic orbital populations calculated by the STO-3G basis using the GAUSSIAN 90 are plotted in Figs. 13a and b, respectively. The maximum values of I Qi 2 for 3p and 2p net atomic populations are also indicated in Figs. 13a and... 

Fig. 15 Effect of cluster choice on the calculation of oxygen 2p DOS (oxygen Ka x-ray emission spectrum) of CU2O. The 2p net atomic orbital population is broadened by 2.0-eV-FWHM Lorentzian function. (Taken from Sakai [29].)...

Table VI - Mulliken atomic orbital populations and net charges for [VV XgXg]2- complexes from DV calculations. NR is nonrelativistic model R is relativistic. Angular momentum subshells j= 1/2, 3/2, 5/2 and an extended ligand basis set of valence ns, np, nd and (n+l)s optimized functions are indicated.

It is instructive to study the vacant atomic orbital population in dimers and trimers. As mentioned in the Introduction, in the 80 s Bauschlicher et al. [10,11] came to the conclusion that the promotion of ns-electrons to np-orbitals leading to sp-hybridization is the main mechanism responsible for binding in alkaline-earth clusters. This conclusion was based on a study of the SCF Mulliken population analysis for tetramers, which are stable at the SCF level. At present, we can perform more precise analysis using the Natural Bond Orbital Analysis and calculate it at the electron correlation level. 

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. 

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