Electron Propagator in Higher Orders

Improved descriptions of single-electron processes require treatments beyond the geometric approximation given by the Haxtree-Fock electron propagator in Eq. (4.25). More elaborate approximations of the equation of motion (9.1) 

Such a treatment can, with advantage, be expressed in terms of the superoperators introduced in Eq. (4.19) and in terms of a basis of field operators. The basis of fermion-like operators Xj = a, aj[aja, ,a aja, a ap, - is chosen, such that the electron field operators correspond to the SCF spin orbitals. The field operator space supports a scalar product (XjlXj) = ([A , X,]+) = Tr /9[Xl,Xj]+, where p is the density operator defined in Eq. (4.33). The superoperator identity and the superoperator hamiltonian operate on this space of fermion-like field operators and, in particular, Xi HXj) = [x/, [H,Xj - J. ) = Tt p[xI[H,X ] U.  

Perturbation theory starts with a partitioning of the hamiltonian, and thus of the superoperator hamiltonian, into an unperturbed part and a perturbation  

This expression in terms of the SCF spin orbital basis is readily obtained from Eqs. (4.3), (4.4), and (4.24). 

The field operator space can be partitioned by the projection operator 

---

The excitation propagator is of importance for the understanding of electronic excitation spectra, polarizabilities, indirect nuclear spin-spin coupling tensors, and many other quantities. It has been treated in higher order approximations and is capable of yielding predictive results. An approach analogous to the one followed for the electron propagator is quite feasible. 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

The proposals found here can be seen as the result of a two-way strategy for the treatment of large molecules. First, we improve on the accuracy of the very efficient second order approximation. In addition, we introduce approximations that lower considerably the required computer resources for the use of higher-order approximations to the electron propagator within the quasiparticle approach. 

In this way, the density differences depict the critical role of even the seemingly small correlation and relaxation effects incorporated by the higher order decouplings of the dilated electron propagator. The basis set employed in this calculation is the (10s/6p) Be basis used in many calculations /22,25,26/. 

All in all, the bi-orthogonal dilated electron propagator offers a simple extension of the real electron propagator technique and with the incorporation of higher order decouplings like the E3, E ADC(3) etc. and suitably large and flexible basis sets should offer same power and effectiveness in the treatment of metastable anions and cations as done by its real counterpart for stable bound systems. 

---