Propagation matrix

Since the dimension of the principal propagator matrix may be large, it is impractical to calculate the inverse matrix in eq. (10.115) directly. In practice the propagator is therefore calculated in two steps, by first solving for an intermediate vector X (corresponding to U in eq. (10.50)). 

The propagator matrix is energy-dependent poles occur when E equals an ionization energy, Eo N)-En N-l),oi an electron affinity, Eo(N)-Em(N+1). Dyson orbitals (DOs) for ionization energies are defined by... 

If u is partitioned into the primary space, a, and an orthogonal space of product operators, f, the partitioned form of the propagator matrix reduces to... 

Partitioning the operator manifold can lead to efficient strategies for finding poles and residues that are based on solutions of one-electron equations with energy-dependent effective operators [16]. In equation 15, only the upper left block of the inverse matrix is relevant. After a few elementary matrix manipulations, a convenient form of the inverse-propagator matrix emerges, where... 

Elements of the zeroth-order, inverse-propagator matrix are... 

The poles correspond to Koopmans s theorem.) The inverse-propagator matrix and its zeroth-order counterpart therefore are related through... 

Thus, the electron propagator matrix elements can be written as,... 

The notation is meant to suggest that the frequency is variable and depends on the propagator matrix elements. The following criteria have proved valuable in choosing the variable coefficients of eq. IV.5 (1) at low temperature, the VQRS reference should weight the region around the potential minimum most heavily, and (2) at high temperature, our approximation should approach the classical limit ... 

Using superoperators in combination with (1.1), we get the electron propagator matrix... 

By knowing the trajectory of a spin set, its individual density matrix can be calculated at any time points. The key to the simulation is the determination of the propagating matrix (see Section 3.5). The FID and spectrum of a spin set upon the individual trajectory (one scan) can be determined from the actual values of the time-dependent density matrix. 

The retarded and advanced Green functions can be expressed in terms of the electron propagator matrix that is supposed to be known from numerical calculations [55, 56]. Usually, it can be found, for example, from GAUSSIAN program [22] whose output provides all necessary information for the... 

The basis set is chosen such that around the Zth Dyson pole, the electron propagator matrix is diagonal. Consequently, the eigenstate /) can be presented as a linear expansion over atomic orbitals with coefficients ck(l) that depend on the number of the Dyson pole, Z, and the number of the atomic orbital, k, so that... 

Formally, if one has the experimental values of the dielectric tensor e, the magnetic permeability tensor /jl, and the optical rotation tensors p and p for the substrate, one can construct first the optical matrix M, then the differential propagation matrix A, and C, which, to repeat, is the x component of the wavevector of the incident wave. Once A is known, the law of propagation (wave equation) for the generalized field vector ift (the components of E and H parallel to the x and y axes) is specified by Eq. (2.15.18). Experimentally, one travels this path backwards. 

Consider the relationship between A and the dielectric tensor e. In ellipsometry, there is reflection and transmission by the surface (z = 0) of a semi-infinite anisotropic substrate (biaxial crystal) into an isotropic ambient (air, for z<0). Suppose that this semi-infinite anisotropic medium (the crystal) is homogeneous and that its optical matrix M is independent of z (if A does depend on z—that is, on how far into the crystal one goes—then the problem becomes much more difficult). If the optical matrix M of the substrate is independent of z, then so is the differential propagation matrix A if A is independent of z and has a value (, to be found below, the solution of Eq. (2.15.25) is given by... 

So far, however, one still needs an expression for the reflection matrix that shows how to extract from it the tensor elements for the refractive index tensor of the biaxial medium. We seek the reflection matrix R for the semi-infinite anisotropic biaxial medium. Using Eq. (2.15.8) and Eq. (2.15.21), we can relate the 4x4 differential propagation matrix A to the dielectric tensor e from Eqs. (2.15.21) and (2.15.24). Then it can be shown that... 

The limit with respect to rj is taken because of integration techniques required in a Fourier transform from the time-dependent representation. Indices r and s refer to general, orthonormal spin-orbitals, r x) and os(x), respectively, where x is a space-spin coordinate. Matrix elements of the corresponding field operators, al and as/ depend on the N-electron reference state, N), and final states with N 1 electrons, labeled by the indices m and n. The propagator matrix is energy-dependent poles occur when E equals a negative VDE, Eq(N) — En(N — 1), or a negative VAE, Em(N +1) — Eq(N). 

A one-electron, zeroth-order Hamiltonian defines a set of reference eigenfunctions (spin-orbitals) and eigenvalues (e) such that the matrix elements of the corresponding inverse propagator matrix read... 

AH matrices are of dimension 6Z and the harmonic propagator matrix G(0)(q m) is diagonal. The problem of calculating the phonon propagators thus reduces to the calculation of the self-energy matrices S(q ioif) that contain all anharmonic information. It is not difficult to demonstrate that the self-energy matrix is a Hermitian function of oif from which it follows that its analytic continuation in the complex frequency plane, in the neighborhood of the real axis, has the form... 

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