Rayleigh quotient

Band-by-band conjugate-gradient (CG) techniques. The sequential CG technique minimizes the Rayleigh quotient... 

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... 

The energy in variational calculations is obtained by minimizing the Rayleigh quotient. In the case of basis set (49), this quotient has the following form ... 

We may now substitute i/ (1, 2) into the Rayleigh quotient and obtain an estimate of the total energy,... 

These considerations may now be used to simplify the Rayleigh quotient for 4, and we see that... 

These are displayed with an arbitrary overall normalization. This is unimportant in the Rayleigh quotient so long as the functions normalizations are correct relative to one another. The real normalization constant depends upon the overlaps, of course. 

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. 

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. 

The Rayleigh quotient with respect to a function h is defined as ... 

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. ... 

A useful trial variational function is the eigenfunction of the operator L for the parabolic barrier which has the form of an error function. The variational parameters are the location of the barrier top and the barrier frequency. The parabolic barrierpotential corresponds to an infinite barrier height. The derivation of finite barrier corrections for cubic and quartic potentials may be found in Refs. 44,45,100. Finite barrier corrections for two dimensional systems have been derived with the aid of the Rayleigh quotient in Ref 101. Thus far though, the... 

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). 

RDFT can be derived in close analogy to the constrained-search argument by Levy [18], valid for standard DFT, if the Rayleigh quotient is replaced by the unsymmetric formula (

universal functional is defined by... 

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