Rayleigh quotient method

These include the Rayleigh quotient method" and variational transition state theory (VTST).46 9 xhg 0 called PGH turnover theory and its semiclassical analog/ which presents an explicit expression for the rate of reaction for almost arbitrary values of the friction function is reviewed in Section IV. Quantum rate theories are discussed in Section V and the review ends with a Discussion of some open questions and problems. 

As shown by TalkneP there is a direct connection between the Rayleigh quotient method and the reactive flux method. Two conditions must be met. The first is that phase space regions of products must be absorbing. In different terms, the trial function must decay to zero in the products region. The second condition is that the reduced barrier height pyl" 1. As already mentioned above, differences between the two methods will be of the order e P. ... 

Rayleigh quotient method has been used only in the spatial diffusion limited regime but not in the energy diffusion limited regime (see the next Seetion). 

Residual minimization method (RMM-DIIS). Wood and Zunger [27] proposed lo minimize the norm of the residual vector instead of the Rayleigh quotient. This is an unconstrained minimization condition. Each minimization step starts with the evaluation of the preconditioned residual vector K for the approximate eigenstate... 

Rate of change of observables, 477 Ray in Hilbert space, 427 Rayleigh quotient, 69 Reduction from functional to algebraic form, 97 Regula fold method, 80 Reifien, B., 212 Relative motion of particles, 4 Relative velocity coordinate system and gas coordinate system, 10 Relativistic invariance of quantum electrodynamics, 669 Relativistic particle relation between energy and momentum, 496 Relativistic quantum mechanics, 484 Relaxation interval, 385 method of, 62 oscillations, 383 asymptotic theory, 388 discontinuous theory, 385 Reliability, 284... 

In all of the various VB methods that have been suggested involving nonlocal orbitals it is obvious that the orbitals must be written as linear combinations of AOs at many centers. Thus one is always faced with some sort of nonlinear minimization of the Rayleigh quotient. 

It is impossible to do justice within the limited extent of one chapter to all theoretical developments. For example, I will omit methods relating to the Fokker-Planck equation representation of the dynamics. This includes the method of adiabatic elimination discussed extensively in Ref. 33 or the approach based on the Rayleigh quotient, developed by Talkner (34,35). There are a number of reviews, monographs, and special journal issues devoted to the theory of activated rate processes (5,13,14,36-40), the interested reader is urged to consult them. I will also omit any quantum theory of activated rate processes. The thread which connects the material presented in this chapter will be the use of the Hamiltonian equivalent form of the STGLE and more general forms to derive the classical theory of activated rate processes. 

As a second example, it is instructive to derive the Kramers stationary flux function which serves as a basis for practical application in the Rayleigh quotient variational method (34,35). In principle there are an infinity of stationary flux functions, as any function in phase space which is constant along a classical trajectory will be stationary. Kramers imposed in addition the boundary condition that the flux is associated with particles that were initiated in the infinite past in the reactant region. Following Pechukas (69), one defines (68) the characteristic function of phase points in phase space Xr, which is unity on all phase space points of a trajectory which was initiated in the infinite past at reactants and is zero otherwise. By definition x, is stationary. The distribution function associated with the characteristic function Xr projected onto the physical phase space is then... 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

The fact that the Rayleigh quotient for the ZORA wave fiinction yields the Dirac eigenvalue logically leads into another, even simpler, approach than lORA to the improvement of ZORA, and one that preceded lORA the scaled ZORA method (van Lenthe et al. 1994). It is the renormalization terms in the lORA Hamiltonian that present the main difficulty. But for any given function, renormalization may be achieved by a simple scaling. We therefore make the approximations... 

The method established by iterating (11.5.19) is known as the inverse-iteration method with the Rayleigh quotient or simply as the Rayleigh method [7,9]. As should be clear from our discussion, the Rayleigh method is just Newton s method applied to the eigenvalue problem (11.4.28). 

The variational quantum Monte Carlo method (VMC) is both simpler and more efficient than the DMC method, but also usually less accurate. In this method the Rayleigh-Ritz quotient for a trial function 0 is evaluated with Monte Carlo integration. The Metropolis-Hastings algorithm " is used to sample the distribution... 

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