High-frequency modes

As discussed above the errors in the trajectory are correlated with the missing rapid motions. In contrast to the friction approach of estimating the variance, which may affect long time phenomena, we identify our errors as the missing ( filtered ) high frequency modes. We therefore attempt to account approximately for the fast motions by choosing the trajectory variance accordingly. 

We describe a simple computational example to demonstrate two key features of the new protocol Stability with respect to a large time step and filtering of high frequency modes. In the present manuscript we do not discuss examples of rate calculations. These calculations will be described in future publications. 

The obtained PES forms the basis for the subsequent dynamical calculation, which starts with determining the MEP. The next step is to use the vibrationally adiabatic approximation for those PES degrees of freedom whose typical frequencies a>j are greater than a>o and a>. Namely, for the high-frequency modes the vibrationally adiabatic potential [Miller 1983] is introduced,... 

The adiabatic approximation in the form (5.17) or (5.19) allows one to eliminate the high-frequency modes and to concentrate only on the low-frequency motion. The most frequent particular case of adiabatic approximation is the vibrationally adiabatic potential... 

Although the rotation barrier is chiefly created by the high-frequency modes, it is necessary to consider coupling to low-frequency vibrations in order to account for subtler effects such as temperature shift and broadening of tunneling lines. The interaction with the vibrations q (with masses and frequencies m , tu ) has the form... 

Now, recall that for weak hydrogen bonds the high-frequency mode is much faster than the slow mode because 0 m 20 00. As a consequence, the quantum adiabatic approximation may be assumed to be verified when the anharmonic coupling parameter aG is not too strong. Thus, neglecting the diabatic part of the Hamiltonian (22) and using Eqs. (18) to (20), one obtains... 

There are two kinds of damping that are considered within the strong anharmonic coupling theory the direct and the indirect. In the direct mechanism the excited state of the high-frequency mode relaxes directly toward the medium, whereas in the indirect mechanism it relaxes toward the slow mode to which it is anharmonically coupled, which relaxes in turn toward the medium. 

The Hamiltonian of this model with a high-frequency mode coh and a single low-frequency exchange mode coi is given by Eq. (4.2.2) in which components have the following form ... 

The high-frequency mode is coupled through the anharmonicity coefficient y with the low-frequency mode which is a resonant one due to harmonic interaction with the surface reservoir. For simplicity, the last term is written in the Gaitler-London approximation (compare with more general Eq. (4.1.8) where this restriction is absent). The required GF of the high-frequency mode can be obtained from the... 

Thus, the required expression for the GF of the high-frequency mode taking into account the anharmonic coupling of the latter with the exchange deformation mode (characterized by a well-defined value of the reorientation barrier AU) takes the form (4.2.13) with restricted summation over the quantum numbers = <7 = 0, 1,. .. N of the subbarrier states. It is then expedient to rewrite Eq. (4.2.14) in the following form ... 

Considering only biquadratic anharmonic coupling, the dephasing of local vibrations was treated in the special case that only high-frequency modes underwent collectivization174 and subsequently with the allowance also made for collectivized low-frequency modes.138 175 It should be emphasized that the possibility for... 

This expression refers to diagrams without closed high-frequency loops (Fig. A3.1 a). Thus, provided the inequality fihQK 1 is valid, the GF defined above can be written in the form involving no high-frequency mode operators ... 

To illustrate the anharmonic contribution to RPFR from a particular high frequency mode treated in the ZPE approximation, for example a CH/CD stretch, we recall the oscillator energy neglecting Go is expressed... 

For these reactions of hydrogen, it is the isotope effect on the high frequency vibrational modes in the diatomic reactant and tri-atomic transition states which dominate in the calculation of the isotope effects using the TS model. Excitation into upper vibrational levels for these high frequency modes is negligible and the zero point energy approximation is appropriate (see Section 4.6.5.2 and Fig. 4.1). 

Fig. 2. The different kinds of interactions of the intramolecular high frequency mode that may be seen in the infrared spectra. Interactions with the substrate (a) will boA affect the vibration frequency and the lifetime of the vibrational excitation.

There exists an extensive literature on theoretical calculations of the vibrational damping of an excited molecule on a metal surface. The two fundamental excitations that can be made in the metal are creation of phonons and electron-hole pairs. The damping of a high frequency mode via the creation of phonons is a process with small probability, because from pure energy conservation, it requires about 6-8 phonons to be created almost simultaneously. 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

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