Number occupation operator representation

The eigenvalue equation of the representation of the effective Hamiltonian operators (28) in the base of the number occupation operator of the slow mode is characterized by the equation... 

Eq. (4.22) thus gives an operator representation of the mapping from at to at One of the advantages of using the transformation in Eq. (4.22) is that it allows operator manipulations of formulae. Consider for example an occupation number vector I n k> with occupation vector n referring to orbitals in the transformed basis. This vector can be written... 

It is to be remarked that these operators can act only on states of the system expressed in occupation number representation, as explicitly appearing in the definitions, Eqs. (8-105), (8-106), (8-112), and (8-114). We can multiply any one of these operators by a scalar factor, so that we can also define the following operators ... 

Both in Eq. (8-149) and Eq. (8-147), we have written the function in the center of the integrand simply for ease of visual memory in fact both /(q) and F(q,q ) commute with all the B-operators and their positions are immaterial. The B-operators operate on vectors > in occupation number space, so that we can evaluate the matrix elements of F in occupation number representation, viz., Eq. (8-145), either from Eq. (8-147) or from Eq. (8-149). 

The matrix of the projection operator in occupation number representation has a typical element... 

Here the projection operator P is multiplied by the distribution probability te , and the result summed over all states >. A typical element of the matrix of this operator in occupation number representation, called the density matrix, is... 

The corresponding operator expression for the equilibrium entropy in occupation number representation is then also seen to be... 

The expectation value of the density operator, and, indeed, all the components of the density matrix, are stationary in time for an ensemble set up in terms of energy eigenstates. IT we use occupation number representation to set up the density matrix, it is at once seen from Eq. (8-187) that it also is independent of time ... 

Now we shall consider the time dependence of the ensemble average of any operator B not explicitly a function of time, Tr FVR. Because the trace is independent of the representation, we choose the one most convenient, which turns out to be the occupation number representation whose eigenvectors are eigenvectors of H. Thus we write... 

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... 

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 energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

For practical calculations it is often convenient to use the occupation number representation. In terms of boson annihilation(a ) and creationfaja ) operators satisfying... 

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... 

The relation to the density matrix elements in the spin-orbital occupation numbers representation recovers from noticing that the rows of indices of spin-orbitals ki,k2,..., / y = K (defining a row of creation operators. ..a a , forming a basis Slater determinant) can be in the same manner considered as a set of electronic coordinates in the spin-orbital representation as is the list xi, X2,. .., xjv, ... 

The resolvent in eq. (1.208) is called the one-electron Green s function and the notation for it reads G (z). The integration contour may be set in such a way that it encloses all the poles of the resolvent corresponding to the occupied MOs giving by this the required total projection operator. In the spin-orbital occupation number and the second quantization representations related to each other, one can write the operator projecting to the occupied (spin)-MO as an operator of the number of particles in it. Indeed, the expression... 

Occupation Number Representation of the Harmonic Oscillator. The Hamiltonian H for the harmonic oscillator, Eq. (3.4.1), can be rewritten in terms of ladder operators a + and a, which resemble the angular momentum ladder operators [6]. Substituing Eq. (3.4.2) into Eq. (3.4.1), H can be rewritten in terms of the momentum operator p (in the x direction) and the position operator x ... 

As we have taken the groupings A,B etc., to refer to true linked clusters TA, Tg etc., the operators T"A, Tg must appear as physically connected entities in the occupation number representations. Eft, Eg etc., will also then appear as connected entities - as a consequence of the multi-commutator expansion generated by eqs. (3.8). Since the groupings are... 

The Fock space approach has the potential advantage of exploiting the fact that operators written in the occupation number representation are independent of electron number, so that all the manipulations involving H and can be performed at the operator level first, which is somewhat simpler and more transparent than working with the matrix-elements involving functions > and. ... 

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. 

A unique feature of the occupation number representation is that the number of electrons does not appear in the definition of the Hamiltonian operator in this form as it does in the wavefunction form. This is because all of the occupation information resides in the bras and kets. This is true for any operator in second quantized form. This feature is used to advantage in theories that allow the number of particles to change, and to a more limited extent in the calculation of electron affinities and ionization potentials. It is less important to the MCSCF method but it is useful to remember that the bras and kets contain all of the occupation information. Other details of the wavefunction, such as the AO and MO basis set information, are included in the integrals that are used as expansion coefficients in the second quantized representation of the operator. 

In the spin-representation the two, three and four electron functions of the basis are simple products of fermion operators. Therefore, the upper limit of their occupation number is one. This upper bound value has also been adopted for the elements in a spin-adapted basis of representation. We know also that the diagonal elements must be positive. Finally, we know the value not only of the trace but also of the partial traces of the spin-adapted matrices (18, 19, 20). 

All operators in the second-quantization representation act on functions depending on occupation numbers. The occupation number operator will be denoted by Nnf. It is diagonal in the occupation number states representation... 

It is obvious that 0o cannot serve as a vacuum in the strict sense of the traditional hole-particle formalism, since the valence orbitals in are partially occupied. A straightforward cluster expansion in the occupation number representation from tpo would thus entail two problems (a) there is no natural choice of vacuum to effect a cluster expansion, and (b) the occupation number representation of cluster operators would refer to orbital excitations with respect to the entire oi thus necessitating the considerations of virtual functions which are by themselves combination of functions. If we want to formulate a many-body theory using if>o as the reference function, we need constructs where these cause no problems. 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

It is convenient to use the Kubo time correlation function representation (Kubo et al., 1957) to evaluate the rate constant W . for radiationless transitions between two electronic states a) and a ). The variable in the time correlation function is the time derivative of the number operator which specifies the electron occupancy of the state a). In the canonically transformed representation,... 

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