Nonlinear Hamiltonians

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

O. Gonzales and J. C. Simo. On the stability of symplectic and energy-momentum conserving algorithms for nonlinear Hamiltonian systems with symmetry. Comp. Meth. App. Mech. Engin., 134 197, 1994. 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

Nonlinear Hamiltonian system, geometric transition state theory, 200-201 Nonlinear thermodynamics coefficients linear limit, 36 entropy production rate, 39 parity, 28-29... 

F> obtained by solving the equation is consistent with the F> used to calculate the reaction field. Having established an effective nonlinear Hamiltonian, one may solve the Schrodinger equation by any standard (or nonstandard) manner. The common element is that the electrostatic free energy term Gp is combined with the gas-phase Hamiltonian Hq to produce a nonlinear Schrodinger equation... 

The structure of the contribution is as follows. In Section 1.5.2 we discuss the structure of effective nonlinear Hamiltonians for the solute. In Section 1.5.3 we present a two-step formulation of the QM problem, with the corresponding Hartree-Fock (HF) equation. In Section 1.5.4 we introduce the fundamental energetic quantity for the QM solvation models while in Section 1.5.5 extensions beyond the HF approximation are presented and discussed. 

The last fundamental aspect characterizing PCM methods, i.e. their quantum mechanical formulation, is presented by Cammi for molecular systems in their ground electronic states and by Mennucci for electronically excited states. In both contributions, particular attention is devoted to the specific aspect characterizing PCM (and similar) approaches, namely the necessity to introduce an effective nonlinear Hamiltonian which describes the solute under the effect of the interactions with its environment and determines how these interactions affect the solute electronic wavefunction and properties. 

The coefficients W determine the probabilities of third-order nonlinear optical processes in an unbounded crystal. An analogous expression can be derived for the coefficients determining the probabilities of fourth-order nonlinear optical processes. As already mentioned the derivation for multilevel molecules is rather complicated and has not yet been obtained. However, the simplicity of the final result, that is the simplicity of the nonlinear Hamiltonian, determines the simplicity of the calculations of nonlinear processes. Note also that a similar polariton approach can be applied for consideration of nonlinear processes in low-dimensional nanostructures (chains, quantum wells). For such structures just resonances of the pumping radiation with polaritons of low-dimensional structure and not with excitons will determine the resonances in the absorption of light as well as resonances in nonlinear processes. 

The time-dependent Schrodinger equation may be obtained by rewriting Frenkel variation principle for the case of nonlinear Hamiltonian as follows ... 

The idea that solitons might play an important role in biopolymers comes from Davydov. In his article [37], he studied alpha-helical proteins and applied some achievements of nonlinear mathematics to biology. As for DNA, the nonlinear physics started in 1980 [10] when the first nonlinear Hamiltonian of DNA, as well as possibility of sohtonic solution, was suggested. A crucial experimental research was explained in Ref. [38], representing victory of nonlinear over linear DNA physics. 

This effective Hamiltonian is very similar to that of the self-consistent Madelung potential (SCMP) model, Eq. (5.14). This is a nonlinear Hamiltonian, in the sense that it depends on the wave function 5. ... 

A.2 Time-Dependent Schrddinger Equation for Nonlinear Hamiltonians... 

This allows one to calculate (G) as the ground state eigenfunction of an effective nonlinear Hamiltonian... 

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