Quantum nuclear effects

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

PATH INTEGRAL METHODS FOR THE TREATMENT OF NUCLEAR QUANTUM EFFECTS... 

We can expect to see future research directed at QM/MM and ab initio simulation methods to handle these electronic structure effects coupled with path integral or approximate quantum free energy methods to treat nuclear quantum effects. These topics are broadly reviewed in [32], Nuclear quantum effects for the proton in water have already received some attention [30, 76, 77]. Utilizing the various methods briefly described above (and other related approaches), free energy calculations have been performed for a wide range of problems involving proton motion [30, 67-69, 71, 72, 78-80]. 

In contrast to the quasi-classical approaches discussed in the previous chapters of this review, Eq. (114) represents a description of nonadiabatic dynamics which is semiclassically exact in the sense that it requires only the basic semiclassical Van Vleck-Gutzwiller approximation [3] to the quantum propagator. Therefore, it allows the description of electronic and nuclear quantum effects. 

Including nuclear quantum effects in ab initio molecular dy-... 

Including nuclear quantum effects in ab initio molecular dynamics the quantum wavepacket ab initio molecular dynamics (QWAIMD) method... 

As the strength of D/A coupling increases, governed by T y, a number of adjustments to the TST rate constant formulation may be required. If nuclear quantum effects are minor, the LZ model may be applied to cases of arbitrary 7jy magnitude, expressed either in terms of a diabatic or adiabatic basis [8J. The relative merits of the two bases (as well as limitations in the applicability of the LZ model) have been discussed recently in conjunction with the analysis of electron transfer from strongly-coupled D/A initial states prepared optically [39, 65]. 

Hwang et al.131 were the first to calculate the contribution of tunneling and other nuclear quantum effects to enzyme catalysis. Since then, and in particular in the past few years, there has been a significant increase in simulations of QM-nuclear effects in enzyme reactions. The approaches used range from the quantized classical path (QCP) (e.g., Refs. 4,57,136), the centroid path integral approach,137,138 and vibrational TS theory,139 to the molecular dynamics with quantum transition (MDQT) surface hopping method.140 Most studies did not yet examine the reference water reaction, and thus could only evaluate the QM contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (e.g., Refs. 4,57,136) concluded that the QM contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

Billeter, S.R., et al. (2001). Hybrid approach for including electronic and nuclear quantum effects in molecular dynamics simulations of hydrogen transfer reactions in enzymes. J. Chem. Phys. 114, 6925-6936... 

Nevertheless, these methods are mostly applied with fixed charges (even if these are chosen in a sophisticated way) and with pairwise additivity approximation as well as with the neglect of nuclear quantum effects. Suggestions for polarizable models appeared in literature mainly for water [23], The quality of potential parameterization varies from system to system and from quantity to quantity, raising the question of transferability. Spontaneous events like reactions cannot appear in simulations unless the event is included in the parameterization. Despite these problems, it is possible to reproduce important quantities as structural, thermodynamic and transport properties with traditional MD (MC) mainly due to the condition of the nanosecond time scale and the large system size in which the simulation takes place [24],... 

Agarwal, P. K., Billeter, S. R., Hammes-Schieeer, S. (2002) Nuclear quantum effects and enzyme dynamics in dihydrofolate reductasecatalysis, J. Phys. Chem. B 106, 3283-3293. 

Recently, CHj has been the subject of ab initio integral path calculations, which include nuclear quantum effects such as zero-point motion and tunneling in full dimensionality and go beyond the harmonic approximation. These studies confirm the above results, in that while protons undergo large-amplitude pseudorotational motion and become scrambled and statistically equivalent, this motion is concerted and the situations in which the vibrating nuclear skeleton having an H2 moiety attached to a CH3 tripod are the most important contributors to the overall appearance of the cation. 

Empirical potentials, which are usually functions that are fit to experimental data, tend to predict the net effect of a variety of phenomena over a range of conditions, and are consequently less accurate than ah initio PES for describing N-body interactions. The virial coefiicients that are calculated Irom an input interaction potential (empirical or ab initio PES) without modification are known as classical virial coefficients because they do not include nuclear quantum effects explicitly. Virial coefficients computed using an effective potential such as the Quadratic Feynman-Hibbs (QFH) [1] that includes a quantum correction are known as semi-classical virial coefficients. 

The main reason for introducing quantum TSTs was that simpler descriptions of the rate constant had treated the nuclei as classical particles. While this is often an appropriate point of view, it fails completely in some cases, and almost always when hydrogen is involved in bond breaking or formation. Indeed, strong nuclear quantum effects have been observed in many reactions and recently even in several enzymatic reactions [19-21]. In such situations, one should treat some or all nuclei quantum mechanically. The rate constant is a dynamic quantity and hence very difficult to compute quantum mechanically because solving the time-dependent Schrodinger equation exactly is possible only for a handful of atoms [22]. Fortunately, the longtime quantum dynamic effects on the thermal rate constant of most chemical reactions... 

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