Normal mode Hamiltonian

Weinstein A 1973 Normal modes for nonlinear Hamiltonian systems Inv. Math. 20 47... 

An instability of the impulse MTS method for At slightly less than half the period of a normal mode is confirmed by an analytical study of a linear model problem [7]. For another analysis, see [2]. A special case of this model problem, which gives a more transparent description of the phenomenon, is as follows Consider a two-degree-of-freedom system with Hamiltonian p + 5P2 + + 4( 2 This models a system of two springs con-... 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

The fact that both the local- and the normal-mode limits are contained within the algebraic approach allows one to study in a straightforward way the transition from one to the other. It is convenient to use, for this study, the local basis [Eq. (4.17)] and diagonalize the Hamiltonian for two identical bonds... 

The transition from the local- to the normal-mode limit is described by the parameter Xx2/A. When this parameter is zero, the Hamiltonian (4.28) is in the local limit, when the parameter is large the spectrum approaches the normalmode limit. It is convenient to define the dimensionless locality parameter as... 

The normal-mode Hamiltonian for linear triatomic molecules... 

Within van Roosmalen s scheme, it is not possible to construct simple diagonal Hamiltonians with the degeneracies required by bent normal-mode molecules. These molecules must therefore be dealt with by numerically diagonalizing the Hamiltonian matrix as discussed in the following sections. 

The normal mode Hamiltonians for bent triatomic molecules, 101... 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

To account for photochemical processes, we adopt a simple model that was proposed by Seidner and Domcke for the description of cis-trans isomerization processes [164]. In addition to the normal-mode expansion above, they introduced a Hamiltonian exhibiting torsional motion. The diabatic matrix elements of the Hamiltonian are given as... 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

There is a one to one correspondence between the imperturbed fi equencies CO, C0j j = 1,. .., N,. .. appearing in the Hamiltonian equivalent of the GLE (Eq. 3) and the normal mode frequencies. The diagonalization of the potential has been carried out exphcitly in Refs. 88,90,91. One finds that the imstable mode frequency A is the positive solution of the Kramers-Grote Hynes (KGH) equation (7). This identifies the solution of the KGH equation as a physical barrier fi-equency. 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

The third result was the establishment of a connection between the TST and GLE viewpoints by Poliak." He solved for the normal modes of the Hamiltonian Eq. (7) and then used the result in a calculation of the reaction rate through the multi-dimensional TST. Surprisingly, he recovered the Kramers-Grote-Hynes... 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

Scalable Resonances in Global Treatment of IR, Raman, Overtone, and SEP Data for Standard Normal-Mode Hamiltonian... 

M. Quack In answer to the question by Prof. Kellman on exact analytical treatments of anharmonic resonance Hamiltonians, I might point out that to the best of my knowledge no fully satisfactory result beyond perturbation theory is known. Interesting efforts concern very recent perturbation theories by Sibert and co-workers and by Duncan and co-workers as well as by ourselves using as starting point internal coordinate Hamiltonians, normal coordinate Hamiltonians, and perhaps best, Fermi modes [1]. Of course, Michael Kellman himself has contributed substantial work on this question. Although all the available analytical results are still rather rough approximations, one can always... 

The solution yields [1], beside the final set of normal modes frequencies, the final nonadiabatic form of the fermionic Hamiltonian, that is ... 

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