Hamiltonian equations model

Hamiltonian equations, 627-628 II electronic states, 632-633 triatomic molecules, 587-598 minimal models, 615-618 Hartree-Fock calculations ... 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

ABBA models, 631-633 Hamiltonian equations, 626-628 HCCS radical, 633-640 perturbative handling, 641-646 theoretical background, 625-626 triatomic molecules ... 

Hamiltonian equations, 610—615 minimal models, 615-618 multi-state effects, 624 pragmatic models, 618—621 spectroscopic properties, 598-610 linear molecules ... 

Thermostated dynamical systems are deterministic systems with non-Hamiltonian forces modeling the dissipation of energy toward a thermostat [48]. The non-HamUtonian forces are chosen in such a way that the equations of... 

We first construct an adiabatic approximation for the classical Hamiltonian equations of motion. We start from an unconstrained model with a classical... 

A further issue arises in the Cl solvation models, because Cl wavefunction is not completely variational (the orbital variational parameter have a fixed value during the Cl coefficient optimization). In contrast with completely variational methods (HF/MFSCF), the Cl approach presents two nonequivalent ways of evaluating the value of a first-order observable, such as the electronic density of the nonlinear term of the effective Hamiltonian (Equation 1.107). The first approach (the so called unrelaxed density method) evaluates the electronic density as an expectation value using the Cl wavefunction coefficients. In contrast, the second approach, the so-called relaxed density method, evaluates the electronic density as a derivative of the free-energy functional [18], As a consequence, there should be two nonequivalent approaches to the calculation of the solvent reaction field induced by the molecular solute. The unrelaxed density approach is by far the simplest to implement and all the Cl solvation models described above have been based on this method. 

The atomic motions change the configurations of the atoms (g f , gf), while the external forces, the forces between the atoms of the same molecule and the forces which act in every collision, change the velocities (and therefore also the momenta p, , p ). The corresponding changes in the phase of the gas model are expressed by the Hamiltonian equations of motion 69... 

Now, the two-state Hamiltonian with approximations made can be simulated near Rx by the model Hamiltonian equation (26) with parameters cos 6 - h a - , Ae - Aeeft. The transition probability P can be obtained from equations (27) and (22) provided interference is neglected ... 

All the methods we have thus far considered use Monte Carlo or molecular dynamics simulations. An alternative approach to get thermodynamic quantities in an effective Hamiltonian+discrete model is based on the use of the integral equation RISM method. The RISM-SCF method proposed by Ten-no and coworkers (Ten-no et al., 1993, 1994 Kawata et al., 1995) combines a QM description of the solute with a RISM description of the whole system in a way which deserves attention. 

In the previous section, we have shown that switching the picture from the nearly integrable Hamiltonian to the Hamiltonian with internal structures may make it possible to solve several controversial issues listed in Section IV. In this section we shall examine the validity of an alternative scenario by reconsidering the analyses done in MD simulations of liquid water. As mentioned in Section III, since a water molecule is modeled by a rigid rotor, and has both translational and rotational degrees of freedom. So, the equation of motion involves the Euler equation for the rigid body, coupled with ordinary Hamiltonian equations describing the translational motions. The precise Hamiltonian is therefore different from that of the Hamiltonian in Eq. (1), but they are common in that the systems have internal structures, and the separation of the time scale between subsystems appears if system parameters are appropriately set. 

To obtain some insight into the behavior of the solutions of the Hamiltonian equation (10), we performed a numerical simulation of a model system 23 we assumed that V(s) is a symmetric double well, we coupled, v to 1000 harmonic oscillators cok with frequencies ranging from 10 to 1000 cm, and symmetrically to one oscillator Qpv. Even though the simulation is completely classical, we obtained instructive results that illustrate several of the points we have mentioned in this section. 

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