On-the-fly method

Note that, in the two cases discussed above, chemical intuition was essential in properly defining superstates such that an appropriate separation of timescales was obtained. For more complex systems, intuition alone will not be sufficient and considerable effort might be required to identify an exploitable gap in the characteristic timescales of the system. There is thus a need to develop on-the-fly methods to appropriately define superstates based on MD data alone. Efforts toward this goal are presently under way [31]. 

While quantum mechanical simulation of nuclear motion will become more practical in the future, classical mechanical molecular dynamics will remain an important tool for simulating large molecular systems for many years to come. Ab initio determination of forces will play an increasingly large role. But a system of N atoms requires at least 10 points to completely map out y(R) (ten points along each degree of freedom). For N of order 100, it is clearly prohibitive to comprehensively tabulate < y(R) in advance (in the absence of simplifications such as pairwise additivity). By contrast, a 1-ns trajectory with 1-fs time steps requires 10 evaluations of < y(R) and its derivatives, a very formidable task but far more accessible than the alternative. Thus, it will be essential in the future to develop on-the-fly methods for ab initio calculation of forces [4]. 

The above semiclassical methods can be used to compute absorption spectra when the propagation involves a single PES. An important advantage comes if we are able to use only local information, since this opens the route to the so-called on-the-fly methods, where the heavy electronic calculations are performed only at the points of the PES touched by the trajectories at selected time steps. 

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

Full quantum wavepacket studies on large molecules are impossible. This is not only due to the scaling of the method (exponential with the number of degrees of freedom), but also due to the difficulties of obtaining accurate functions of the coupled PES, which are required as analytic functions. Direct dynamics studies of photochemical systems bypass this latter problem by calculating the PES on-the-fly as it is required, and only where it is required. This is an exciting new field, which requires a synthesis of two existing branches of theoretical chemistry—electronic structure theory (quantum chemistiy) and mixed nuclear dynamics methods (quantum-semiclassical). 

While simulations reach into larger time spans, the inaccuracies of force fields become more apparent on the one hand properties based on free energies, which were never used for parametrization, are computed more accurately and discrepancies show up on the other hand longer simulations, particularly of proteins, show more subtle discrepancies that only appear after nanoseconds. Thus force fields are under constant revision as far as their parameters are concerned, and this process will continue. Unfortunately the form of the potentials is hardly considered and the refinement leads to an increasing number of distinct atom types with a proliferating number of parameters and a severe detoriation of transferability. The increased use of quantum mechanics to derive potentials will not really improve this situation ab initio quantum mechanics is not reliable enough on the level of kT, and on-the-fly use of quantum methods to derive forces, as in the Car-Parrinello method, is not likely to be applicable to very large systems in the foreseeable future. 

With the characterized mechanism, the next key question is the origin of its catalytic power. A prerequisite for this investigation is to reliably compute free energy barriers for both enzyme and solution reactions. By employing on-the-fly Born-Oppenheimer molecular dynamics simulations with the ab initio QM/MM approach and the umbrella sampling method, we have determined free energy profiles for the methyl-transfer reaction catalyzed by the histone lysine methyltransferase SET7/9... 

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

Another major, future advance in the quantum chemical computation of potential energy surfaces for reaction dynamics will be the ability to routinely compute the energies of molecular systems on the fly . The tedious and time-consuming process of fitting computed quantum chemical values to functional forms could be avoided if it were possible to compute the PES as needed during a classical trajectory or quantum dynamics calculation. For many chemical reactions, it should be practical in the near future to prudently select a sufficiently rapid and accurate electronic structure method to facilitate dynamics computations on the fly. 

In recent years, dynamic calculations of both the electronic and the molecular structure of complex molecular systems have started to become feasible. " These methods are based on the general idea that the electronic structure of the system is to be calculated on the fly as the nuclei move, while the nuclei respond to the forces determined from the dynamically calculated electronic structure. This assumes that the system moves on the lowest electronic state, and transitions between states are either ignored (because they are well separated in energy) or treated semiclassi-cally. 

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