Electron propagator operators

If the P/Q operators correspond to removal or addition of an electron, the propagator is called an electron propagator. The poles of the propagator (where the denonainator is zero) correspond to ionization potentials and electron affinities. 

Thus the matrix elements of the electron propagator are related to field operator products arising from the superoperator resolvent, El — H), that are evaluated with respect to N). In this sense, electron binding energies and DOs are properties of the reference state. 

Applications of electron propagator methods with a single-determinant reference state seldom have been attempted for biradicals such as ozone, for operator space partitionings and perturbative corrections therein assume the dominance of a lone configuration in the reference state. Assignments of the three lowest cationic states were inferred from asymmetry parameters measured with Ne I, He I and He II radiation sources [43]. 

When the operators A and B in Eq. (2.7) are sin q)le creation and annihilation operators the resulting propagator is called electron (nopagator or one-particte Green s function, and = -t-1. Collecting all these creation and amiihi-lation operators in a row vector a, the electron propagator can be expressed as. 

R. Flores-Moreno et al., Assessment of transition operator reference states in electron propagator calculations. J. Chem. Phys. 127, 134106/1-8 (2007)... 

J.V. Ortiz et al., Electron propagator calculations with a transition operator reference state. Chem. Phys. Lett. 103, 29-34 (1983)... 

Let us assume that we have a system of electrons in a single determinant state in which, say, the state pk (k = mo) is occupied (other states may be either occupied or empty). This electron propagates interacting with some external potential (for example that induced by nuclei). Under the action of this potential the electron scatters into a state cpk> (k = mV). In the absence of the magnetic field the spin projection does not change so that o = o. This process is represented by the product of the Fermi operators ... 

The zeroth order expression for the electron propagator is obtained by using po from eqn. (21) as the density operator and using the inner projection... 

This demonstrates the equivalence of this development to that of the undilated electron propagator. The structural equivalence at any order of perturbation theory between the dilated propagator G(r/) and the unrotated case, demonstrated by the corresponding choice of projection manifolds and density operators, guarantees the ease with which the dilated propagator coalesces into the undilated one for 0 = 0 at any level of approximation. For example, choosing p = pQ, h = h1 h3, and h = h1 h3 with... 

The standard approach to the calculation of the propagators (A.4, A.5) is perturbation theory with respect to the electron-electron coupling constant a = e /(hc) on the basis of the interaction picture. Technically this results in an expansion of expectation values of interacting field operators in powers of expectation values of the free (or noninteracting) field operators i o and Ag. The structure of this expansion can be summarised in a set of formal rules, the Feynman rules. For instance for the electron propagator one obtains ... 

Consider first the case where P and Q are simple creation and annihilation operators, e.g. P = af and Q — a . The residue <0 P nj> is then nonvanishing only if m > contains one electron less than 0 > and the residue of the last term vanishes unless m > contains one electron more than 0>. The poles of the first term are thus the ionization potentials for state 0> while the poles of the second term are the electron affinities. The af a, propagator is called the electron propagator and will be discussed in more detail in Section VI. 

Eq. (58) represents the starting point for all approximate propagator methods. Even though in the derivation we only discussed the linear response functions or polarization propagators, a similar equation holds for the electron propagator. The equation for this propagator has the same form but there are differences in the choice of h and in the definition of the binary product (Eq. (52)), which for non-number-conserving, fermion-like operators should be... 

Obviously, those are the same considerations as we went through in order to obtain Eqs (90) and (91) and the electron propagator method and the ADC are thus equivalent methods. Using n = 2 in Eq. (93) we determine and n — 3 gives The U matrix in Eq. (93) corresponds to the transition matrix (cf. Eq. (75)). Both and only contain C and D terms (see Eqs (87), (88), (90) and (91), i.e. hj = hj alone. From Eq. (63) we see that we may classify the operators in hj as 2p-lh (two-particle, one-hole) and 2h-lp operators, and the n = 3 ADC approach, corresponding to the third-order electron propagator method, is therefore referred to as the extended 2p-lh Tamm-Dancoff approximation (TDA) (Walter and Schirmer, 1981). A fourth-order approximation to the ADC equations has also been described (Schirmer et ai, 1983) but not yet tested in actual applications. 

Not all electron propagator calculations are based upon an order-by-order evaluation of the propagator. Redmon et al. (1975) included the hs operator manifold, without computing all terms in in the next order of perturbation... 

The indices r and s refer to general, orthonormal spin orbitals, s (x), respectively, where x is a space-spin coordinate. Integration techniques required in a Fourier transform from the time-dependent representation require that the limit with respect to 77 be taken [1, 4], Matrix elements of the corresponding field operators, aj and as, are evaluated with respect to an N-electron reference state, N), and final states with N 1 electrons identified by the indices m and n. Elements of the electron propagator matrix are energy dependent. A pole occurs when E equals a negative VDE, Eq(N) — E (N — 1), or a negative A.E, E (N +1) - Eo(N). 

Details of the derivation of general expressions for energy shifts at a given order can be found in Mohr et al. (1998). Contractions between pairs of fermion or boson field operators AM lead to electron and photon propagator functions. The exact electron propagator in a static external field is homogeneous in time and appears as... 

Prior to the present work, the universal choice of the EOM P-space for ionization potentials " was just the simple electron removal operators and a , the 1-block. This is effectively also the customary choice in the propagator and diagrammatic Green s function methods. 

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