Variable occupation numbers

In the case of variable occupation numbers, we are interested in the term... 

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. 

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... 

In the preceding sections, the occupation vectors were defined by the occupation of the basis orbitals jJ>. In many cases it is necessary to study occupation number vectors where the occupation numbers refer to a set of orbitals cfe, that can be obtained from by a unitary transformation. This is, for example, the case when optimizing the orbitals for a single or a multiconfiguration state. The unitary transformation of the orbitals is obtained by introducing operators that carry out orbital transformations when working on the occupation number vectors. We will use the theory of exponential mapping to develop operators that parameterizes the orbital rotations such that i) all sets of orthonormal orbitals can be reached, ii) only orthonormal sets can be reached and iii) the parameters are independent variables. 

Generally speaking, the representation in terms of occupation numbers is considered to be an independent quantum-mechanical representation, distinct from the coordinate (or momentum) one. In that case, the occupation numbers for one-particle states are dynamic variables, and operators are the quantities that act on functions of these variables. In this section, second-quantization representation is directly related to coordinate representation in order that in what follows we may have a one-to-one correspondence between quantities derived in each of these representations. 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

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... 

Equations (163) allow one to derive equations for the proton mean occupation numbers and thermodynamic functions in the mean-field approximation. Substituting variables 8hmi- where i = 1, 2, 3 into Eqs. (157)-(159), we get... 

For many-electron molecules, the Hartree-Fock wavefunction that is computed by conventional electronic structure packages, such as GAUSSIAN, can be expanded from singleparticle molecular orbitals, i /i(r), that are themselves constructed from atom-centered gaussians that are functions of coordinate-space variables. The phase information that is contained in the molecular orbitals is necessary to define the wavefunction in momentum-space. In other words, the density in coordinate-space cannot be Fourier transformed into the density in momentum-space. Rather, within the context of molecular orbital theory, the electron density in momentum space is obtained by a Fourier-Dirac transformation of all of the v /i (r) s, followed by reduction of the phase information, weighting by the orbital occupation numbers. 

The electro-chemical reactions are assumed to take place in the fluid phase. The explicit introduction of the occupation numbers (0,-, S,- 0,- = 1) as variables is a natural description of the possible lack of catalytic sites and methanol cross over effects. Due to the use of expensive catalysts, electrochemical saturation effects may be reached by fuel cells in some... 

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

The time scale for electron dynamics studied above is very short, typically shorter than 1 femtosecond, during which the nuclei are supposed to stay still. In the above process, the electron loss has been taken into account through the change of the population of natural orbitals and subsequently the change of the coefficients of the Slater determinants. Therefore the same set of self-consistent field molecular orbitals (SCF MO) may be used for the short time period. However, for a longer time scale, it would be better to redetermine the molecular orbitals under the gradual loss of electronic population. This demands an efficient way to determine the molecular orbitals with variable and fractional occupation numbers, although this is not a new problem for quantum chemistry. 

This result reveals that the occupation numbers have the familiar Fermi function form but with the energy Ct, rather than the single-particle energy Ck, as the relevant variable. Using this expression for k in the BCS gap equation at finite temperature, Eq. (8.26), we find... 

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