Generalized occupation numbers

Imposing the particle conservation condition, which introduces a new Fermi level A, we can define the so-called generalized occupation numbers... 

In that way the averaging procedure is closed which allows us to determine the generalized occupation numbers Ui and 6rii connected with standard occupation numbers rii through the relation ... 

For the extension to two dimensions we consider a square lattice with nearest-neighbor interactions on a strip with sites in one direction and M sites in the second so that, with cyclic boundary conditions in the second dimension as well, we get a toroidal lattice with of microstates. The occupation numbers at site i in the 1-D case now become a set = ( ,i, /25 5 /m) of occupation numbers of M sites along the second dimension, and the transfer matrix elements are generalized to... 

In the more general case of several LCAOs, where P has been calculated according to the occupation numbers, we have... 

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). 

In general whether we are discussing symmetrical or antisymmetrical states, the numbers nx are usually termed the occupation numbers, and if we are given a complete spectrum of one-particle states, indicated by A, then the set of occupation numbers assigned to the values of A, specifies the state of the system of JY identical particles just as well as the assignment of a A-value to each particle. Thus we may use the notation just as well as A to specify the state, and so write... 

Creation and Annihilation Operators.—In the last section there was a hint that the theory could handle problems in which populations do not remain constant. Thus < , < f>s 2 is the probability density in 3A -coordinate space that the occupation numbers are , and the general symmetrical state, Eq. (8-101), is one in which there is a distribution of probabilities over different sets of occupation numbers the sum over sets could easily be extended to include sets corresponding to different total populations N. 

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

It has been suggested that quasi-particle wave functions do not deviate much from LDA wave functions [26], Furthermore, in the evaluation of momentum densities shown in Figure 9, the characteristics of the quasi-particle states dominantly reflect on the occupation number densities which should be evaluated by using the general quasi-particle Green s function. In GWA, however, the corresponding occupation number densities are... 

Electronegativity was considered to be dependent both on the hybridization state of an orbital, and also on its electron occupation an empty orbital must be more electronegative than a singly, and even more so than a doubly, occupied orbital. Equation 3 quantifies this dependence on occupation number, or more generally, on charge, q. 

To perform excited-state calculations, one has to approximate the exchange-correlation potential. Local self-interaction-free approximate exchange-correlation potentials have been proposed for this purpose [73]. We can try to construct these functionals as orbital-dependent functionals. There are different exchange-correlation functionals for the different excited states, and we suppose that the difference between the excited-state functionals can be adequately modeled through the occupation numbers (i.e., the electron configuration). Both the OPM and the KLI methods have been generalized for degenerate excited states [37,40]. 

Since site a can be either empty [with probability 1 - P(a)] or occupied [with probability P(a)], 9 is the average occupation number for the site a. Clearly, 0 S 6 1. When forming the sumn in Eq. (2.1.7) or (2.1.8), we sum over all average quantities 0,. and obtain the average occupation number for the entire molecule. Clearly 0[Pg.28]

We say that the sites are identical in a weak sense whenever the three PFs 0(1,0,0), 0(0,1,0), and 0(0,0, 1) have the same value. This is identical to the requirement that the single-site intrinsic constant is the same for any specific site. In this case we can replace these three PFs by three times one representative PF, as is done on the rhs of Eq. (2.2.22). We shall say that the sites are identical in a strict sense whenever the PF of any given occupation number is independent of the specific group of occupied sites. For instance, in an equilateral triangle all PFs with two sites occupied are equal. Hence we can replace the sum on the rhs of Eq. (2.2.23) by three times one representative PF. This cannot be done, in general, for a linear arrangement of the three sites, in which case 0(1,1,0) is different from 0(1,0,1), even when the sites are identical in the weak sense (see Chapter 5). Similarly, for... 

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

Because polynomials of orbital occupation number operators do not depend on the off-diagonal elements of the reduced density matrix, it is important to generalize the Q, R) constraints to off-diagonal elements. This can be done in two ways. First, because the Q, R) conditions must hold in any orbital basis, unitary transformations of the orbital basis set can be used to constrain the off-diagonal elements of the density matrix. (See Eq. (56) and the surrounding discussion.) Second, one can replace the number operators by creation and annihilation operators on different orbitals according to the rule... 

Figure 7.3 Possible assignment of different orbitals in a completely general MCSCF formalism. Frozen orbitals are not permitted to relax from their HF shapes, in addition to having their occupation numbers of zero (virtual) or two (occupied) enforced...

The action of S on I na np> gives thus a term proportional to I na np> plus a sum over occupation number vectors, where the spin functions of an alpha orbital and a beta orbital have been flipped. Only permutations of the singly occupied orbitals are included. Inanp> is thus in general not an... 

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