Right-hand first- order wave function

In the non-degenerate case it was an inhomogeneous equation with the nonvanishing right hand part, which could be used to determine the first order energy /A1) and the expansion coefficients of the first order wave function It is not like this in the... 

The point is that the vectors k 4 satisfying the unperturbed Schrodinger equation, if used to expand 44 make the right hand side disappear and the equation becomes a uniform one. The only thing we can do is to use it to determine the proper expansion coefficients of the zeroth order wave function b p in terms of the degenerate subspace as well as the first order energy. (The first order wave function is usually not calculated/considered in the degenerate case.)... 

Problem 23-3. Let H be a perturbation, such that H (x) = —b for 0 x a/2 and H (x) — +6 for a/2 x a, which is applied to a particle in a one-dimensional box (Eqs. 14-6 and 14-7). Obtain the first-order wave function. Show qualitatively that this function is such that the probability of finding the particle in the right-hand half of the box has been increased and explain in terms of classical theory. (Hint Use the symmetry about the point x = a/2.)... 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

This initial guess may then be inserted on the right-hand sides of the equations and subsequently used to obtain new amplitudes. The process is continued until self-consistency is reached. For the special case in which canonical Hartree-Fock molecular orbitals are used, the Fock matrix is diagonal and the T2 amplitude approximation above is exactly the same as the first-order perturbed wave-function parameters derived from Moller-Plesset theory (cf. Eq. [212]). In that case, the Df and arrays contain the usual molecular orbital energies, and the initial guess for the T1 amplitudes vanishes. 

The second term on the right-hand side is zero because generates only determinants in the same space as T, leading to matrix elements satisfying Eq. (3). Thus, the evaluation of a second-order property in the EOM-CC formulation requires only the first-order perturbed wave function, which is obtained by projection of Eq. (8) onto the set of determinants generated by T ... 

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