Representation occupation number

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... 

Li occupation number representation the matrix of P n B has the typical element... 

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... 

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... 

With this definition, the classical entropy per system equals the ensemble average of the expectation value of 8 in occupation number representation. 

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... 

We have carried out tins discussion in occupation number representation or coordinate representation each with a definite number N of particles. Similar results follow for the Fock space representation and the properties of grand ensembles. Averages over grand ensembles are also independent of time when the probabilities > are independent of time, whether the observable commutes with H or not. 

The wave function of a many-electron system can be written as a Slater determinant (see App. B). However, a more convenient notation is provided by using the occupation number representation, whereby the A-electron de-terminantal function takes the form (March et al. 1967, Raimes 1972)... 

Here ri. . .nk) is the occupation number representation of the Slater determinant where n, is the occupation of orbital i. The total number of orbitals is k and N is the total number of electrons. 

In terms of p the average value of an observable O can be written (in occupation number representation)... 

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

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, ... 

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... 

Silverstone and Sinanoglu/487, already in 1965, indicated how a cluster—expansion satisfying size-extensivity with respect to all the electrons (feature (a2)) can be effected. There is also a related work by Roby/497. Transcribed into occupation number representation, this amounts to a cluster expansion of the closed—shell type with respect to each determinant... 

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 configurations are expressed in terms of the occupation-number representation, where they are characterised by the occupation numbers Up of the orbitals ). Wp is either 1 or 0. The base configuration is... 

We now apply the variational method to the M-electron problem. In order to ensure antisymmetry we express the IV-electron Hamiltonian in the occupation-number representation. 

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

B.T. Pickup, The occupation number representation and the many-body problem, in Handbook of Molecular Physics and Quantum Chemistry, vol. 1 Fundamentals, ed. S. Wilson, P.F. Bernath 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. 

It is interesting to draw a distinction between the two aspects of correlation which we have considered so far in terms of the second-quantization method for systems of N identical Fermi particles. Those methods (which are but a more effective and general way of formulating Cl) rest upon the occupation-number representation given the set of all possible single-particle states (spinorbitals), one builds a complete set of N-particle states. .. > by constructing Slater determinants or... 

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