Nuclear Quantum-Mechanical Effects

One can also ask about the relationship of the FMS method, as opposed to AIMS, with other wavepacket and semiclassical nonadiabatic dynamics methods. We first compare FMS to previous methods in cases where there is no spawning, and then proceed to compare with previous methods for nonadiabatic dynamics. We stress that we have always allowed for spawning in our applications of the method, and indeed the whole point of the FMS method is to address problems where localized nuclear quantum mechanical effects are important. Nevertheless, it is useful to place the method in context by asking how it relates to previous methods in the absence of its adaptive basis set character. There have been many attempts to use Gaussian basis functions in wavepacket dynamics, and we cannot mention all of these. Instead, we limit ourselves to those methods that we feel are most closely related to FMS, with apologies to those that are not included. A nice review that covers some of the... 

In the simple semi-classical generalization of TST, electronic and nuclear quantum mechanical effects are included via... 

We do not attempt a derivation of eq 25 here (see, e.g., ref (8)), but rather display just enough detail to make clear how the matrix elements actually calculated (see below) are related to the formalism. We note that eq 26 includes both electronic and nuclear quantum mechanical effects, as reflected in... 

Where k is the transmission factor, < x >xs is the average of the absolute value of the velocity along the reaction coordinate at the transition state (TS), and P = l/keT ( vhere ke is the Boltzmann constant and T the absolute temperature). The term AG designates the multidimensional activation free energy that expresses the probability that the system vill be in the TS region. The free energy reflects enthalpic and entropic contributions and also includes nonequilibrium solvation effects [4] and, as will be shown below, nuclear quantum mechanical effects. It is also useful to comment here on the common description of the rate constant as... 

Simulating the Fluctuations of the Environment and Nuclear Quantum Mechanical Effects 11177... 

Z.7 Dynamics, Tunneling and Related Nuclear Quantum Mechanical Effects 11195... 

Most of the AIMD simulations described in the literature have assumed that Newtonian dynamics was sufficient for the nuclei. While this is often justified, there are important cases where the quantum mechanical nature of the nuclei is crucial for even a qualitative understanding. For example, tunneling is intrinsically quantum mechanical and can be important in chemistry involving proton transfer. A second area where nuclei must be described quantum mechanically is when the BOA breaks down, as is always the case when multiple coupled electronic states participate in chemistry. In particular, photochemical processes are often dominated by conical intersections [14,15], where two electronic states are exactly degenerate and the BOA fails. In this chapter, we discuss our recent development of the ab initio multiple spawning (AIMS) method which solves the elecronic and nuclear Schrodinger equations simultaneously this makes AIMD approaches applicable for problems where quantum mechanical effects of both electrons and nuclei are important. We present an overview of what has been achieved, and make a special effort to point out areas where further improvements can be made. Theoretical aspects of the AIMS method are... 

How well do these quantum-semiclassical methods work in describing the dynamics of non-adiabatic systems There are two sources of errors, one due to the approximations in the methods themselves, and the other due to errors in their application, for example, lack of convergence. For example, an obvious source of error in surface hopping and Ehrenfest dynamics is that coherence effects due to the phases of the nuclear wavepackets on the different surfaces are not included. This information is important for the description of short-time (few femtoseconds) quantum mechanical effects. For longer timescales, however, this loss of information should be less of a problem as dephasing washes out this information. Note that surface hopping should be run in an adiabatic representation, whereas the other methods show no preference for diabatic or adiabatic. 

To summarize, in this article we have discussed some aspects of a semiclassical electron-transfer model (13) in which quantum-mechanical effects associated with the inner-sphere are allowed for through a nuclear tunneling factor, and electronic factors are incorporated through an electronic transmission coefficient or adiabaticity factor. We focussed on the various time scales that characterize the electron transfer process and we presented one example to indicate how considerations of the time scales can be used in understanding nonequilibrium phenomena. 

The fluctuations of the electrostatic energy gap can also provide an interesting insight into nuclear quantum mechanical (NQM) effects. That is, we can use an approach that is formally similar to our previous treatment of electron transfer (ET) reactions in polar solvents [1] where we considered ET between the solute vi-bronic channels (for example, Ref. [1]). Our starting point is the overall rate constant... 

The nature of the repulsive force that arises when two closed-shell systems approach each other can be described in many ways, one of which is as follows. At larger distances the electron clouds of two atoms attract each other by dispersion forces, while at short distances the Pauli exclusion principle drives electrons with the same quantum numbers away from the space they are trying to share, so that a local deshielding of the nuclei occurs, and repulsion arises. Note that the Pauli principle does not imply forces , but only the purely quantum mechanical effect of mutual electron avoidance. At equilibrium, which results from a balance between dispersion attraction and nuclear repulsion forces, there is usually a net gain in energy. The parametric potential must therefore consist of a repulsive (the exponential) and an attractive (mostly, m = 6) term. 

Prom this derivation, we obtain three validity conditions for the gradient approximation (a) smallness of nonadiabaticity (b) longer timescale of nuclear dynamics, together with (c) absence of purely quantum mechanical effects, such as tunneling, in the nuclear degrees of freedom, where the last one being one of the central assumptions in the MQC treatment. 

In molecular dynamics simulations, quantum mechanics is used to calculate the energy of the quantum system but the motion of the whole system is treated classically. Taking account of the quantum character of the motion does not invalidate eqn (16.1), but it gives rise to nuclear quantum mechanical (NQM) effects which change AG and the transmission coefficient k. Zero point energy effects contribute to AG while dynamical effects such as tunnelling and transitions to excited vibrational states contribute to k. Various approaches have been proposed to evaluate these contributions. 

The latter method, called the PI-FEP/UM approach, allows accurate primary and secondary kinetic isotope effects to be computed for enzymatic reactions. These methods are illustrated by applications to three enzyme systems, namely, the proton abstraction and reprotonation process catalyzed by alanine race-mase, the enhanced nuclear quantum effects in nitroalkane oxidase catalysis, and the temperature (in)dependence of the wild-type and the M42W/G121V double mutant of dihydrofolate dehydrogenase. These examples show that incorporation of quantum mechanical effects is essential for enzyme kinetics simulations and that the methods discussed in this chapter offer a great opportunity to more accurately model the mechanism and free energies of enzymatic reactions. 

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