Occupation-number formalism

As a particularly simple example of this formalism, let us consider the spin states of a single electron with respect to the 2-direction. In occupation number representation these states may be written as... 

D-functional theory, explicitly given in terms of natural orbitals and their occupation numbers, has emerged as an alternative method to conventional approaches for considering the electronic correlation. This chapter has introduced important basic concepts for understanding the NOF formalism. We have also offered a brief characterization of almost all references concerning this theory published hitherto. 

The averaged occupation numbers h, are a formal ingredient in Strutinsky s averaging method. Combining Eqs (15) and (17) yields their explicit form ... 

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 ... 

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...

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

In second-quantization formalism, operators in coordinate space are replaced by operators defined in the space of occupation numbers... 

Quasispin formalism can be used in describing the properties of the occupation number space for an arbitrary pairing state. For one shell of equivalent electrons, these pairing states can be chosen to be two one-particle states with the opposite values of angular momentum projections. 

Thus, Hj q describes directly the transition from the quasidiscrete state to fragments. According to second quantization formalism, it acts in an occupation-number space. 

Although a Slater-determinant reference state 4> cannot describe such electronic correlation effects as the wave-function modification required by the interelec -tronic Coulomb singularity, a variationally based choice of an optimal reference state can greatly simplify the -electron formalism. 4> defines an orthonormal set of N occupied orbital functions occupation numbers = 1. While () = 1 by construction, for any full A-electron wave function T that is to be modelled by it is convenient to adjust (T T) > 1 to the unsymmetrical... 

To illustrate the modifications of UHF formalism, it is convenient to consider pure spin symmetry for a single Slater determinant with Nc doubly occupied spatial orbitals Xi and N0 singly occupied orbitals y". The corresponding UHF state has Na mj = occupied spin orbitals and Np rns = — J, occupied spin orbitals f. The number of open-shell and closed-shell orbitals are, respectively Na = Na — Np > 0 and Nc = Np. Occupation numbers for the spatial orbitals are nc = 2, n ° = 1. If all orbital functions are normalized, a canonical form of the RHF reference state is defined by orthogonalizing the closed- and open-shell sets separately. 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

V-clcctron state T, correlation energy can be defined for any stationary state by Ec = E — / o, where Eo = ( //1) and E = ( // 4 ). Conventional normalization ) = ( ) = 1 is assumed. A formally exact functional Fc[4>] exists for stationary states, for which a mapping — F is established by the Schrodinger equation [292], Because both and p are defined by the occupied orbital functions occupation numbers nt, /i 4>, E[p and E[ (p, ] are equivalent functionals. Since E0 is an explicit orbital functional, any approximation to Ec as an orbital functional defines a TOFT theory. Because a formally exact functional Ec exists for stationary states, linear response of such a state can also be described by a formally exact TOFT theory. In nonperturbative time-dependent theory, total energy is defined only as a mean value E(t), which lies outside the range of definition of the exact orbital functional Ec [ ] for stationary states. Although this may preclude a formally exact TOFT theory, the formalism remains valid for any model based on an approximate functional Ec. 

These considerations lead one to suggest a modified cluster model that takes advantage of the fact that local density models admit partial occupation numbers (RSGK). We formally broaden in energy each levd by a and apply Fermi statistics to this continuous system. We add infinitesimal fractions of an electron to the broadened levels in order, until all the dectrons are used up, yidding a precise Fermi energy, f, and the various occupation numbers... 

In the occupation number representation within the second quantization formalism, which will be used here, the 1 -RDM takes the form ... 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

Figure 3 Plot of correlation energy (Ecor, obtained from a CASSCF calculation see text) versus the M d contribution (in terms of percentage) in the bonding eg orbital in a series of octahedral MLe complexes, with L = HjO, NH3, F, CP, Br, and P, and M a transition metal with a formal (a) or c/ (b) occupation number. Solid lines connect metals with a formal charge (+3) dashed lines connect metals with a formal charge (-I-4). For simplicity, the formal charges on the metals have been omitted from the plots.

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