Occupation numbers

In the previous section we discussed light and matter at equilibrium in a two-level quantum system. For the remainder of this section we will be interested in light and matter which are not at equilibrium. In particular, laser light is completely different from the thennal radiation described at the end of the previous section. In the first place, only one, or a small number of states of the field are occupied, in contrast with the Planck distribution of occupation numbers in thennal radiation. Second, the field state can have a precise phase-, in thennal radiation this phase is assumed to be random. If multiple field states are occupied in a laser they can have a precise phase relationship, something which is achieved in lasers by a teclmique called mode-locking Multiple frequencies with a precise phase relation give rise to laser pulses in time. Nanosecond experiments... 

Flere the zero point energy is ignored, which is appropriate at reasonably large temperatures when the average occupation number is large. In such a case one can also replace the sum over by an integral. Each of the triplet n can take the values 0, 1, 2,. . ., co. Thus the sum over can be replaced by an... 

From equation (A2.2.145). the average occupation number of an ideal Bose gas is... 

This model for a fluid was introduced by Lee and Yang [97]. The system is divided into cells with occupation numbers... 

No more than one particle may occupy a cell, and only nearest-neighbour cells that are both occupied mteract with energy -c. Otherwise the energy of interactions between cells is zero. The total energy for a given set of occupation numbers ] = (n, of the cells is then... 

The relationship between tlie lattice gas and the Ising model follows from the observation that the cell occupation number... 

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

Here n,. = 0, 1 or 2 is the occupation number of tlie orbital ((),. in the state being studied. The kinetic energy... 

Kruger and Rosch implemented within DFT the Green s matrix approach of Pisani withm an approximate periodic slab enviromnent [180]. They were able to successfiilly extend Pisani s embeddmg approach to metal surfaces by smoothing out the step fiinction that detenuines the occupation numbers near the Fenui level. 

Flead and Silva used occupation numbers obtained from a periodic FIF density matrix for the substrate to define localized orbitals in the chemisorption region, which then defines a cluster subspace on which to carry out FIF calculations [181]. Contributions from the surroundings also only come from the bare slab, as in the Green s matrix approach. Increases in computational power and improvements in minimization teclmiques have made it easier to obtain the electronic properties of adsorbates by supercell slab teclmiques, leading to the Green s fiinction methods becommg less popular [182]. 

Finally, and probably most importantly, the relations show that changes (of a nonhivial type) in the phase imply necessarily a change in the occupation number of the state components and vice versa. This means that for time-reversal-invariant situations, there is (at least) one partner state with which the phase-varying state communicates. 

Properties can be computed by finding the expectation value of the property operator with the natural orbitals weighted by the occupation number of each orbital. This is a much faster way to compute properties than trying to use the expectation value of a multiple-determinant wave function. Natural orbitals are not equivalent to HF or Kohn-Sham orbitals, although the same symmetry properties are present. 

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... 

As an example, for M = 4, we list the six equivalence classes of the four sites in terms of their occupation numbers ... 

To further demonstrate the power of the kinetic lattice gas approach we review briefly the work on precursor-mediated adsorption and desorption [60,61]. We consider an adsorbate in which, in addition to the most strongly bound chemisorbed (or physisorbed) adsorbed state, the adparticles can also be found in intrinsic or extrinsic precursor states. One introduces three sets of occupation numbers, , = 0 or 1, = 0 or 1, and /, = 0 or 1, depending... 

In the standard lattice gas model of adsorption we assume that the surface of the solid remains inert, providing adsorption sites. This implies that the state of the surface before adsorption and after desorption is the same. This is not the case if the surface reconstructs or lifts the reconstruction upon adsorption. Such a situation we want to describe. We introduce occupation numbers for the surface = 0 or 1, depending on whether the surface... 

In this context it turned out to be useful to investigate data in terms of the difference between the external and internal temperature of the system [43,44]. The external temperature is the temperature given from outside and used in the Metropolis sampling for the acceptance of moves of the monomers. The internal temperature, in contrast to the external temperature, is given by the occupation number of the states of a free bond in equihbrium. 

The 2 s and the 1 are called occupation numbers. In standard molecular orbital theory, occupation numbers are 0, 1 or 2 and they tell us the occupancy of a given orbital. 

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

I have introduced the occupation numbers vi and U2 (where ui = 2 and U2 = 1 in this simple case) to emphasize the symmetry of the electronic energy expression. 

The orbitals and orbital energies produced by an atomic HF-Xa calculation differ in several ways from those produced by standard HF calculations. First of all, the Koopmans theorem is not valid and so the orbital energies do not give a direct estimate of the ionization energy. A key difference between standard HF and HF-Xa theories is the way we eoneeive the occupation number u. In standard HF theory, we deal with doubly oecupied, singly occupied and virtual orbitals for which v = 2, 1 and 0 respectively. In solid-state theory, it is eonventional to think about the oecupation number as a continuous variable that can take any value between 0 and 2. 

Table 4.4 Natural orbital occupation numbers for the distorted acetylene model in Figure 4.11. Only the occupation numbers for the six central orbitals are shown...

The orbital occupation numbers n, (eigenvalues of the density matrix) will be between 0 and 1, corresponding to the number of electrons in the orbital. Note that the representation of the exact density normally will require an infinite number of natural orbitals. The first N occupation numbers N being the total number of electrons in the system) will noraially be close to 1, and tire remaining close to 0. 

This corresponds to eq. (6.5) with occupation numbers of exactly 1 or 0. The missing kinetic energy from eq. (6.4) is thus due to die occupation numbers deviating from being exactly 1 or 0. 

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