Fractional occupation number

We may generalize this by introducing an occupation number (number of electrons), n, for each MO. For a single determinant wave function this will either be 0, 1 or 2, while it may be a fractional number for a correlated wave function (Section 9.5). 

Equation 13 has an important implication a clathrate behaves as an ideally dilute solution insofar as the chemical potential of the solvent is independent of the nature of the solutes and is uniquely determined by the total solute concentrations 2K yK1.. . 2x yKn in the different types of cavities. For a clathrate with one type of cavity the reverse is also true for a given value of fjiq (e.g. given concentration of Q in a liquid solution from which the clathrate is being crystallized) the fraction of cavities occupied 2kVk s uniquely determined by Eq. 13. When there are several types of cavities, however, this is no longer so since the individual occupation numbers 2k2/ki . ..,2k yKn, and hence the total solute concentration... 

The occupation numbers yMi, VM2> VP2 follow from Eq. 24" with the appropriate values of the constants CKi- The propane fraction of the gas bound in the hydrate according to Eq. 26" is then equal to... 

Here i/j. ng l 2mK is the fractional occupation number of the 2ma- sp/n-orbitals of Shell K and is a suitably averaged electron interaction matrix (cf. the usual... 

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

So the highest occupied Kohn-Sham orbital has a fractional occupation number Hohenberg-Kohn theorem applied to the non-interacting system. The proof of... 

Obviously, vanishes, unless Up + Uq + rir + tig = 2. Sufficient though not necessary for Eq. (176) is that the Up are equal to 0,, or 1 that is, now open-shell states with fractional NSO occupation numbers are also possible. This implies via Eq. (174) that the only nonvanishing elements of >-2 are those... 

The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a D-based theory only needs to incorporate electron correlation. It does not rely on the concept of a fictitious noninteracting system. Consequently, the scheme is not expected to suffer from the above mentioned limitations of KS methods. In fact, the correlation energy in 1-RDM theory scales homogeneously in contrast to the scaling properties of the correlation term in DPT [14]. Moreover, the 1-RDM completely determines the natural orbitals (NOs) and their occupation numbers (ONs). Accordingly, the functional incorporates fractional ONs in a natural way, which should provide a correct description of both dynamical and nondynamical correlation. 

The ensemble search in Eq. (82) is the Kohn-Sham procedure, generalized to allow fractional orbital occupation numbers [55, 57-59]. Equation (82) can... 

If X is the (fractional) occupation number of the 5 f orbital in the core, we may re-state the condition of applicability of the band treatment in the following way (Koelling ) ... 

In fact, there is another approach to DFT that allows fractional electron numbers, namely the extended Kohn-Sham (EKS) scheme [23,26,27]. ft allows the use of fractional occupation numbers fi 0 < fi < 1, hence... 

EKS approach, based on the more formal Janak s type of argumentation, doesn t put these restrictions on the ground-state occupation numbers below HOMO. In what follows in this chapter we will use the fractional occupation of HOMO only, in which case both approaches agree. Using Eqn (4) for HOMO energy with respect to (A — 1 + /) electrons, and for HOMO energy Ar+i with respect to (N + /) electrons, one can express I and A in the following form ... 

In Refs [10, If] we have shown that Eqn (30) is an expression for the first-order shell correction term in the EKS-DFT frame. As we pointed it out, the extended version [26,27] of the Kohn-Sham scheme [46] is appropriate because it allows fractional occupation numbers, thus permitting the... 

Differences in Afor different AB5Hn compounds compared with A for CeCosHs are listed in Table III. The values of these numbers (see Table III), calculated using the fractional site occupations for the 0 phase, can be compared with the experimentally determined entropy differences listed in Table I. The calculated configurational entropy differences (see Table III) agree satisfactorily with the experimental data (see Table I) currently available for seven ABsHn compounds. Structures of some ABsHn compounds deduced from neutron diffraction data (4) are listed in Table I. For compounds whose structures have not been determined, the occupation numbers listed in Table III are in best agreement with the thermodynamic data. 

The site fractions of the Si-containing species are one site occupied by SiH4 out of a total of 32 sites, and two sites out of 32 occupied by Si2H4. The site fraction of open sites is 29/32 = 0.906. As is seen in Eq. 11.8, it is necessary to divide the site fraction of each species by the site occupancy number ok to convert to a molar concentration. The concentration of SiH4 (number per unit area) is equal to that of Si2H4. 

CFP (9.11) also have a simple algebraic form. In the previous paragraph we discussed the behaviour of coefficients of fractional parentage in quasispin space and their symmetry under transposition of spin and quasispin quantum numbers. The use of these properties allows one, from a single CFP, to find pertinent quantities in the interval of occupation numbers for a given shell for which a given state exists [92]. 

In its most general physical use, occupation number is an integer denoting the number of particles that can occupy a well-defined physical state. For fermions it is 0 or 1, and for bosons it is any integer. This is because only zero or one fermion(s), such as an electron, can be in the state defined by a specified set of quantum numbers, while a boson, such as a photon, is not so constrained (the Pauli exclusion principle applies to fermions, but not to bosons). In chemistry the occupation number of an orbital is, in general, the number of electrons in it. In MO theory this can be fractional. 

Nesbet, R.K. (1997). Fractional occupation numbers in density-functional theory, Phys. Rev. A 56. 2665-2669. 

In order to compute local variables for a particular site in a molecule an approach is used which is based on the fractional occupation number concept. The original idea to exploit fractional occupation numbers in the framework of DFT is... 

The first step in the dynamics, the S Sj deactivation, is completed in only 52 1 fs, presenting the expected monoexponential decay profile, as can be seen in Figure 8-1 la. This figure shows the fraction of trajectories in each state between 0 and 200 fs for the 35 trajectories computed. Between 20 and 30 fs and again at 37 fs it is possible to observe a revival of the S2 state occupation. The fraction of trajectories in Sj is not shown in the figure for sake of clarity. It is just complementary to the fraction of trajectories in S2. Thus, a revival in S2 is companied by a decrease in Sj occupation. The revivals in the S2 occupation occur when the total number of... 

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