Number-conserving operators

If the P/Q operators are number conserving operators, the propagator is called a Polarization Propagator (PP). It may be viewed as the response of property P to perturbation Q. For the case where P = Q = r (the position operator), the propagator describes the response of the dipole moment (4fo r 4fo) to a linear field F = Ft. 

A proof of the bounds for the occupation numbers will be given in Section II. F. Expectation values of (particle-number conserving) operators are easily expressed in terms of the density matrices. For example, for the energy... 

For fermions, A has to be a number conserving operator for Eq. (31b) to hold.] The superket in this space, D(f)>, associated with the density matrix, is here a vector corresponding to the density matrix in the usual //-space, and obeys the... 

Due to its definition by direct sums and tensor products, the space C, with the canonical inner product for sums and products of Hilbert spaces, is a Hilbert space itself. Since the Hamiltonian H is a particle number conserving operator (we only consider non-relativistic S3rstems), the application of H to the component state 0j) will not lead out of the corresponding component space Hj. It is easily seen that also the application of H to the extended state Ip) then does not lead out of the space C. Thus the above defined linear space Y is a subspace of C. A closed subspace of a Hilbert sp2ice, however, is a Hilbert space itself. This concludes the proof of the proposition. 

The tools needed for evaluating the above matrix elements of the super-operator resolvent are based upon the idea of operators (of the same type as A and B ) forming complete sets (Manne, 1977 Dalgaard, 1979). For example, if A and are number-conserving operators (e.g., r s), then the set of operators (oL>P>y>- p>q>r>—)... 

Since r is a number-conserving operator, the reference state 0> and the state m> must contain the same number N of electrons. The poles of this so-called polarization propagator (PP) thus occur at the excitation energies E = ( , — 0) of Ihe system described by 0>, while the corresponding residues give the squares of the electric dipole transition moments <0 r m). ... 

It is customary to take tj = -I-1 for operators that contain only creation-annihilation pairs (e.g. typical 1- and 2-body operators in the Hamiltonian) such number-conserving operators are said to be of Bose type. For operators that contain odd numbers of creation/annihilation factors, thus changing particle numbers, t) is instead given the value —1 and the operators are said to be of Fermi type. 

Another very important choice occurs when A = al(t ), B = and, since these are not number-conserving operators, ij = — 1. The time form of the corresponding electron propagator is then, with T = t — t as in (13.2.9),... 

Similar rules may be derived for creation operators (Problem 13.7), but are less simple and, since a creation operator dJ(N) working on a ket function is equivalent to its adjoint (destruction operator), transferred to the bra, it is sufficient to know (13.4.5). For number-conserving operators, which contain equal numbers of d and d factors, the result is always N-independent Slater s rules are thus valid whatever the number of particles, and no N-dependence appears in the second-quantization Hamiltonian (3.6.9). 

When A and B in (13.5.19) are (1-electron) number-conserving operators of the type... 

The most common application of EOM procedures is to the determination of excitation energies and related transition quantities. In this case the lower sign is chosen in (13.6.18), since 0 and O must be number-conserving operators, and the simplest operator basis to consider is that in which and t]1 are elementary hole-particle pairs, = ai,a, = and = a-a = E]i, where i labels a spin-orbital that appears in the reference function while m refers to a complementary virtual set. To preserve a parallel with earlier equations, it is then convenient to write (13.6.19) in the form... 

Apart from the particle-number operators (1.3.12), the simplest number-conserving operators are the elementary excitation operators O/Mq, for which we shall occasionally use the notation... 

First-quantization operators conserve the numbCT of electrons. Following the discussion in Section 1.3, such operators are in the Fock space represented by linear combinations of operators that contain an equal number of creation and annihilation rqierators. The explicit form of these number-conserving operators depends on whether the first-quantized operator is a one-electron operator or a two-electron operator. One-electron operators are discussed in Section 1.4.1 and two-electron operators in Section 1.4.2. Finally, in Section 1.4.3 we consider the second-quantization representation of the electronic Hamiltonian operator. 

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