Hamiltonian equation dynamics

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

Floquet theory principles, 35—36 single-surface nuclear dynamics, vibronic multiplet ordering, 24—25 Barrow, Dixon, and Duxbury (BDD) method, Renner-Teller effect tetraatomic molecules, Hamiltonian equations, 626-628 triatomic molecules, 618-621 Basis functions ... 

ABBA molecules, 631-633 HCCS radical, 633-640 perturbative handling, 641-646 theoretical principles, 625-633 Hamiltonian equation, 626-628 vibronic problem, 628-631 Thouless determinantal wave function, electron nuclear dynamics (END) ... 

The dynamics across this linearized relative TS is exactly comparable to the dynamics across the linearized usual TS. Like the usual TS, a relative TS defines two regions in phase space an outer region and an inner region with > 0 and < 0, for the Hamiltonian equation, Eq. (49)]. However, the full dynamics may be qualitatively different, precisely because of the relative nature of the equilibrium and the occurrence of Coriolis terms in the relative frame. 

A most important result in rate theory was the proof14 that the classical dynamics of 5 governed by the Hamiltonian equation ... 

To establish a relationship between the Hamiltonian equation (10) and the actual enzymatic system one performs a molecular dynamics simulation to obtain the force F(t) acted upon the reaction coordinate. Then the force autocorrelation function , which is proportional to the friction kernel y(t), is related to the parameters of the fictitious medium of Equation (10) through... 

The vectors and R +i/2 are vectors ofAfCO, 1) i.i.d. random numbers, with y > 0 the usual Langevin dynamics friction parameter. Setting y = 0 reduces the scheme to the usual RATTLE scheme for solving holonomically constrained Hamiltonian equations of motion, whereas if y is chosen large then this will be expected to cause instability in the scheme, as we are not solving the OU process exactly. The... 

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