Polarization propagator excitation operator

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

Both P and Q are sums of excitation operators (with weighting coefficients p and 9 )- Thus, P and Q applied to 0> create a polarization of 0> and we call P 6 a polarization propagator. In the special case where P and Q are both single particle-hole excitations, i.e. only one term in Eqs (5) and (6), we talk about the particle-hole propagator. It is important to note that only the residues of the polarization propagator and not of the particle-hole propagator determine transition moments (Oddershede, 1982). We must have the complete summations in Eqs (5) and (6) in order to represent the one-electron operator that induces the transition in question. 

In describing polarization propagator methods it is instructive to start out with the simplest consistent method of the kind, namely the random-phase approximation (RPA). Within the framework we use here, RPA is described as the approximation to the general equation of motion (Eq. (58)) in which we set h = hj and assume 0> = HF>, that is, use the simplest truncation in both Eqs (64) and (89). It is convenient to split hj up into p-h and h-p excitation operators... 

The resulting second-order polarization propagator approximation (SOPPA) was first described in its present form by Nielsen et al. (1980). Splitting h4 up into 2p-2h and 2h-2p excitation operators as was done for h2 in Eqs (95)-(97)... 

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). ... 

In Section 10.1 we will illustrate this for ground-state expectation values such as Eq. (4.25) and many others and in Section 10.2 for sum-over-states expressions such as Eq. (4.74) and many others. In the rest of the chapter we will discuss methods in which approximations are made to the exact matrix representation of the linear response function or polarization propagator given in Eq. (3.159). This equation is exact as long as a complete set of excitation and de-excitation operators hn is used and the reference state is an eigenfunction of the imperturbed Hamiltonian. 

In Section 9.2 it was mentioned that the simplest approximation for an excited state 4 ° ) is to represent it by one singly excited determinant Approximating at the same time the groimd-state wavefunction with the Hartree-Fock determinant 0 and the Hamiltonian by the Hartree-Fock Hamiltonian F, Eq. (9.15), the excitation energies En — E become equal to orbital energy differences ta — and the transition moments (4 q° O 4 ° ) become simple matrix elements of the corresponding one-electron operator d in the molecular orbital basis ( d ) [see Exercise 10.ll. The spectral representation of the polarization propagator, Eq. (3.110), thus becomes approximated as... 

Excitation energies and transition moments can in principle be obtained as poles and residua of polarization propagators as discussed in Section 7.4. However, only in the case that the set of operators hn in Eq. (7.77) is restricted to single excitation and de-excitation operators q i,qai is it computationally feasible to determine all excitation energies. This restricts this approach to single-excitation-based methods like the random phase approximation (RPA) discussed in Sections 10.3 and 11.1 or time-dependent density functional theory (TD-DFT). 

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