RRGM

Moiseyev N, Friesner R A and Wyatt R E 1986 Natural expansion of vibrational wave functions RRGM with residue algebra J. Chem. Phys. 85 331... 

Such a method was first proposed by Wyatt and co-workers.43 7,56 In their so-called recursive residue generation method (RRGM), both eigenvalues and overlaps are obtained using the Lanczos algorithm, without explicit calculation and storage of eigenvectors. In particular, the residue in Eq. [41] can be expressed as a linear combination of two residues ... 

Both RRGM and SLP have been used to compute various transition amplitudes with high efficiency and accuracy. Their applications, which have been reviewed in the literature,56 57 59 include laser-molecule interaction,43 44 99 correlation functions,45 104 absorption and emission spectra,100 103 105-107 intramolecular energy transfer,108-115 vibrational assignment,103 116 117 and reaction dynamics. ... 

Expansion of Vibrational Wave Functions RRGM with Residue Algebra. 

Computation of transition probabilities. In addition to survival amplitudes, the RRGM can also be used to compute state-to-state time-dependent transition amplitudes. If we denote the initial state at t = 0 as i), then the state that evolves from this initial state is i(t)) = U(t) i), where U t) is again the evolution operator. At time t, the amplitude for finding state /) in this evolved state is given by A,f i) = (f U(t) i). If we know the eigenvectors, i / ), and eigenvalues, Ea, for this Hamiltonian, then the transition amplitude can be written... 

We will now recast this amplitude into a form that is more convenient for RRGM calculations. It may not be obvious, but this transition amplitude can be written in terms of the difference between two survival amplitudes. This result can be shown as follows. If we define two orthonormal transition vectors, w ) = [ i) + /)]/ /2, and v ) = [ i) - /)]/V/2, then the transition amplitude can be expresed as... 

RRGM summary. In this section, we have reviewed the recursive residue generation method for the computation of the lineshape function and the transition probabilities. The RRGM aims at computation of the set of residues and eigenvalues ra, Ea. The steps are summarized as follows ... 

In the previous section, we showed that the RRGM can be used to calculate individual state-to-state transition probabilities, Ptft). Another way of studying the IVR processes involves the explicit calculation of the time evolution of the initial state i) (75,76). We will begin by recalling that the solution to the Schrodinger time-dependent equation can be written in terms of the evolution operator (propagator)... 

Traditional RRGM calculations. In the previous subsection, we showed that the wave operator does not generally give an accurate expression for an eigenstate, although it does efficiently build an active space. We now need to extract from the active space (containing several thousand states) the spectral information associated with various CH overtones, nvf), with n = 1, 2,. . . , 6. All of the calculations presented in this section were obtained using a 207,000 state primitive basis set. 

Figure 6 Residues "(6vl a) and energies of the eigenstates a) obtained using the RRGM (3000 recursions) in an active space containing 4000 states.

EOMCC = equation-of-motion coupled-clu.ster FOD = fourth-order differencing MBGF = many-body Green s function RR = resonance Raman RRGM = recursive-residue-generation method SIL = short-time iterative Lanczos SOD = second-order differencing. 

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