Modified Hamiltonian

Ehrenfest trajectory for three-dimensional D + H2 generated by the RWP method, that is, the modified Hamiltonian operator f H). Dotted curves in (b) correspond to the Ehrenfest trajectory determined by the usual Schrddinger equation. See text for further details. 

This means that the average of / (f) can be computed using a constrained simulation for a modified Hamiltonian... 

In order to be able to determine re we must consider rotations. A straightforward procedure is a generalization of that discussed in the previous section, and it is called the cranking method. The method has been used extensively in nuclear physics (Schaaser and Brink, 1984). It consists in evaluating the expectation value of the modified Hamiltonian... 

If one is interested in changes of the solute molecule, or if the structure of the surrounding solvent can be neglected, it may be sufficient to regard the solvent as a homogeneous dielectric medium, as was done in the older continuum theories, and to perform a quantum mechanical calculation on the molecule with a modified Hamiltonian which accounts for the influence of the solvent as has been done by Hylton et al. 18 5>. Similarly Yamabe et al. 186> substituted dipole-moment operators for the solvent in their perturbational treatment of solvent effects on the activation energy in the NH3 + HF reaction. 

The relativistic adiabatic connection formula is based on a modified Hamiltonian H g) in which not only the electron-photon coupling strength is multiplied by the dimensionless scaling parameter g but also a g-dependent, multiplicative, external potential is introduced. 

We add the operator —p, ER to the total molecular Hamiltonian. According to Eq. (3.1), the electronic Hamiltonian of the molecule in the field due to the solvent is then He — p ER. The electronic Schrodinger equation is then solved using this modified Hamiltonian. This leads to a self-consistent solution where the electronic wave function and the electronic energy are modified due to the solvent field. Thus, polarization of the molecular electronic density (as described approximately above) is automatically included in this approach. 

Here the Bloch-modified Hamiltonian HB is obtained by integration by parts of the kinetic energy integral,... 

Let us treat the proton dynamics starting from the modified Hamiltonian (153), that is [149,150],... 

Later methods made adjustments to external forces to account for periodic boundary conditions and introduced suitable modifications of the Hamiltonian or the Newtonian equations of motion [75-78]. Considerable progress has been made since those early efforts, both with the original [79-83] and modified Hamiltonian approaches [84]. However, many subtle issues remain to be resolved. These issues concern the non-Hamiltonian nature of the models used in NEMD and the need to introduce a thermostat to obtain a stationary state. Recently Tuckerman et al. [25] have considered some statistical mechanical aspects of non-Hamiltonian dynamics and this work may provide a way to approach these problems. Although the field of NEMD has been extensively explored for simple atomic systems, its primary applications lie mainly in treating nonequilibrium phenomena in complex systems, such as transport in polymeric systems, colloidal suspensions, etc. We expect that there will be considerable activity and progress in these areas in the coming years [85]. 

If the species is charged then an appropriate Bom term must also be added. The reaction field model can be incorporated into quantum mechanics, where it is commonly referred to as the self-consistent reaction field (SCRF) method, by considering the reaction field to be a perturbation of the Hamiltonian for an isolated molecule. The modified Hamiltonian of the system is then given by ... 

For high precision calculations, and especially for the isotope shift, it is necessary to include also the motion of the nucleus in the center-of-mass (CM) frame. A transformation to CM plus relative coordinates yields the additional — fx./M)V V2 mass polarization term in the modified Hamiltonian... 

To cope with this inconsistency, we introduce the modified Hamiltonian (operator) matrix for the case wq 0, cf. Eq. (1.14),... 

Theorem 3.1 Suppose the Hamiltonian H and modified Hamiltonian H are smooth functions globally defined on a convex, compact subset 3 of and suppose that the exact solution and numerical approximations (for h sufficiently small) are confined to B. Then, we have asymptotically for h- 0 that... 

Fig. 3.3 The isoenergy contours are shown for the double well model dark curves) together with the corresponding curves for the Verlet modified Hamiltonian light curves)...

In Collisional Verlet, the momenta are adjusted at the collision point to preserve the kinetic energy (and hence the total energy). It is possible to project to some other manifold using projection techniques like those mentioned previously, as discussed in [40]. In particular, one may use the backward error analysis to obtain a modified Hamiltonian Hh corresponding to the Verlet method with stepsize h, then to project during collisions not onto the energy surface, but onto the modified energy surface, so that... 

As we have seen in the Chap. 3, when a symplectic method is applied to a molecular dynamics problem it induces a perturbed Hamiltonian (energy) function. For the Verlet scheme the modified Hamiltonian is... 

Note that when a symplectic integration method is used, we have, from the discussion in Chap. 3, a perturbed energy function and, moreover, from Theorem 3.1, the error in energy H is OQf), nonetheless the perturbations are large for a large stepsize h. In the Shadow Hybrid Monte-Carlo (SHMC) method [2, 3, 188], the accept-reject test is based on the modified Hamiltonian Hh (see Chap. 3), derived from the Baker-Campbell-Hausdorff expansion. SHMC can improve efficiency by decreasing the rejection rate. 

A practically identical model has been proposed independently by Weinstein and his CO workers the interaction field modified Hamiltonian (IFMH) model [141]. These authors discussed in details the case of the HF. .. HCOF complex and compared the IFMH results with perturbational calculations. It is interesting to note that in the course of the IFMH calculation the first iteration and the fully converged results yield practically identical interaction energy at least for the actual case. On the other hand, individual components of the interaction energy differ considerably in the two calculations, which shows the importance of charge reorganizations. 

Similarly to the case of the mutually consistent field (MCF) or interaction field modified Hamiltonian (IFMH) approaches we start from the group function equations for the solute (S) and solvent (B) subsystems (c.f. Sect. 3). 

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