Response equations interaction wave functions

To compute the interacting RPA density-response function of equation (32), we follow the method described in Ref. [66]. We first assume that n(z) vanishes at a distance Zq from either jellium edge [67], and expand the wave functions (<) in a Fourier sine series. We then introduce a double-cosine Fourier representation for the density-response function, and find explicit expressions for the stopping power of equation (36) in terms of the Fourier coefficients of the density-response function [57]. We take the wave functions <)),(7) to be the eigenfunctions of a one-dimensional local-density approximation (LDA) Hamiltonian with use of the Perdew-Zunger parametrization [68] of the Quantum Monte Carlo xc energy of a uniform FEG [69]. 

Next, we consider the even weaker second-order perturbation of the three sublevels of T that is due to Recall that the analogous second-order contributions from are neglected in this approach, as they are generally believed to be small, and the first-order contributions have already been included. It is seen from inspection of Equation 3.3 that the operator can mix each of these three triplet wave functions with singlet, other triplet, and quintet wave functions. This interaction has no first-order effect on the energies -D, -Dy, and -D of the three substates, T,, T, and T, respectively, but in second order. Equation 3.14 with V = IP, they will be affected somewhat and become -D , -Dy, and -Z) . If we continue to define the zero-field splitting parameters by D = 3DJ/2 and E = (DJ - Dy )/2, they can be compared with the values observed. In Section 3.3, we noted the difficulties involved in attempts to evaluate these values accurately in this fashion, due to the very large number of states over which the summation in Equation 3.14 is necessary, and we commented on alternative methods of evaluation such as response theory. 

---