Population gross

T. Gross, Population Dynamics General Results from Local Analysis, Der Andere Verlag Tonning, Germany (2004). 

Before turning to the problem of calculating the spin Hamiltonian, we should consider how the results can be interpreted with respect to the bonding. From the spin Hamiltonian parameters, we shall obtain the values for Cji of the antibonding orbitals, and we need to know what relation these have to the Cj, of the bonding states. One approach to this problem, which seems fruitful, is the gross-population analysis developed by Mulliken (26). Consider first our three-electron example. In this case we split up the term... 

The columns a, bj denote the strong field Griffith parameters for the t2,-shell (values in Hartree). %Cr3d refers to the normalized total-gross population of the t2g orbitals for the average d3 configuration. Data taken from [23]. 

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 Figure 3 the gross population density of states (GPDOS) around the Fermi level for this Pt4 cluster is shown. Some features are apparent in this GPDOS. Around -10 eV, a band constructed of ca. 50% 5d and 50% 6s and 6p orbitals is visible. This band is referred to as the interstitial bonding orbital (IBO) by Kua and Goddard22. Above this IBO, the 5d band is visible with several maxima in the GPDOS up to the Fermi-level at -4.3 eV. In the same... 

Figure 3 The gross population density of states (GPDOS) calculated for a clean, tetrahedral Pt4 cluster.The interstitial bonding orbital (IBO) constituted mainly of 6s and 6p orbitals, mixed with 5d orbitals22,2) the riband includes a small amount of mixing with 6s and 6p orbitals,3) anti bonding combination of 6s, 6p and a small amount of Sd orbitals.

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 meaning of the quantities evaluated in this way is quite clear. With each occupied one-electron wave-function is associated an energy s which for closed shell systems represents the ionisation potential from that level s) and this ionisation potential is, using the technique of photoelectron spectroscopy (93) measurable in principle and often, already in practice. The total energy can also be calculated, and represents the energy of formation of the system from infinitely separated nuclei (or nuclei with cores) and electrons. Net orbital populations, bond populations and gross populations are readily defined. 

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

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 ... 

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. 

Instead of apportioning the electrons into net populations in basis functions and overlap populations for pairs of basis functions, it is convenient for some purposes to apportion the electrons among the basis functions only, with no overlap populations. Mulliken proposed that this be done by splitting each overlap population equally between the basis functions Xr and Xs- For each basis function Xr> this gives a gross population Nr in Xr that equals the net population n, plus one-half the sum of the overlap populations between Xr and all other basis functions ... 

One finds these other contributions to the gross population of 02 0.00 from 0.25 from 3ai, zero from Ibi and Ifej. Addition of these contributions gives a gross population (or occupation number) of 1.83 for OZs. Carrying out the calculation for the other basis functions, one finds (Problem 15.22c) the following gross populations (where the contributions are listed in the order Xa, 2ai, 1 2,3fli, lZ i) Nqu= 2-00 + 0.00... 

Addition of the gross populations for all basis functions centered on atom B gives the gross atomic population for atom B ... 

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,... 

R labels the atoms, k any other parameter of the basis funetions), the same proeedure as used in the definition of MuUiken gross populations,... 

---