Direct dynamics

So-called ab initio molecular dynamics techniques in which the DFT (usually just in its LDA approximation) is combined with a classical mechanical treatment of the nuclear (ion) motion have been a very popular way of studying condensed matter, i.e., the dynamics of liquids and solids. These techniques may, for instance, be used to study dynamic processes such as binding and atom diffusion [240] at surfaces, and in principle also reactivity at surfaces, without resorting to the usual procedure, namely that of determining the potential energy surface first and then doing the nuclear dynamics afterwards. 

The DFT scheme can be put in a molecular dynamics framework by considering the wavefunction as a variable and forming the lagrangian 

Once this eigenstate of the electronic hamiltonian is reached, we can use the Hellmann-Feynman theorem for calculating the force on the nuclei (the ions), i.e., if an eigenstate of the electronic hamiltonian is reached, then the force on ion number k is given as 

In order to reach the eigenstate, it is necessary to iterate the above equations in a given time-step, before the nuclei are moved according to Newton s mechanics 

since the hamiltonian depends parametrically on the nuclear coordinates, we can let the orbital dynamics follow the nuclear motion provided that the 

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Baldridge K K, Gordon M S, Steckler R and Truhlar D G 1989 Ab initio reaction paths and direct dynamics calculations J. Phys. Chem. 93 5107... 

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be Just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrddinger equation are derived by employing the time-dependent variational principle (TDVP). 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

Direct dynamics attempts to break this bottleneck in the study of MD, retaining the accuracy of the full electronic PES without the need for an analytic fit of data. The first studies in this field used semiclassical methods with semiempirical [66,67] or simple Hartree-Fock [68] wave functions to heat the electrons. These first studies used what is called BO dynamics, evaluating the PES at each step from the elech onic wave function obtained by solution of the electronic structure problem. An alternative, the Ehrenfest dynamics method, is to propagate the electronic wave function at the same time as the nuclei. Although early direct dynamics studies using this method [69-71] restricted themselves to adiabatic problems, the method can incorporate non-adiabatic effects directly in the electionic wave function. 

The Gaussian wavepacket based spawning method, mentioned above, has also been used in direct dynamics where it is called ab initio multiple spawning... 

By its nature, the application of direct dynamics requires a detailed knowledge of both molecular dynamics and quantum chemistry. This chapter is aimed more at the quantum chemist who would like to use dynamical methods to expand the tools at theh disposal for the study of photochemistry, rather than at the dynamicist who would like to learn some quantum chemishy. It hies therefore to introduce the concepts and problems of dynamics simulations, shessing that one cannot strictly think of a molecule moving along a trajectory even though this is what is being calculated. 

To demonstrate the basic ideas of molecular dynamics calculations, we shall first examine its application to adiabatic systems. The theory of vibronic coupling and non-adiabatic effects will then be discussed to define the sorts of processes in which we are interested. The complications added to dynamics calculations by these effects will then be considered. Some details of the mathematical formalism are included in appendices. Finally, examples will be given of direct dynamics studies that show how well the systems of interest can at present be treated. 

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

The Helgaker-Chen algorithm results in very large steps being possible, and despite the extra cost of the required second derivatives, this is the method of choice for direct dynamics calculations. A number of systems have been treated, and a review of the method as applied to chemical reactions is given in [2]. 

A more comprehensive Gaussian wavepacket based method has been introduced by Martinez et al. [35,36,218]. Called the multiple spawning method, it has already been used in direct dynamics studies (see Section V.B), and shows much promise. It has also been applied to adiabatic problems in which tunneling plays a role [219], as well as the interaction of a... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Importantly for direct dynamics calculations, analytic gradients for MCSCF methods [124-126] are available in many standard quantum chemistiy packages. This is a big advantage as numerical gradients require many evaluations of the wave function. The evaluation of the non-Hellmann-Feynman forces is the major effort, and requires the solution of what are termed the coupled-perturbed MCSCF (CP-MCSCF) equations. The large memory requirements of these equations can be bypassed if a direct method is used [233]. Modem computer architectures and codes then make the evaluation of first and second derivatives relatively straightforward in this theoretical framework. 

The present high cost of full CASSCF direct dynamics means that it is not possible to use such calculations to run large numbers of trajectories. As a result it cannot be used to build up experience of the types of effects to be found in dynamical studies of organic photochemistry, and in their interpretation. This problem can be remedied by performing calculations using the MMVB force field [63,64]. 

The multiple spawning method described in Section IV.C has been applied to a number of photochemical systems using analytic potential energy surfaces. As well as small scattering systems [36,218], the large retinal molecule has been treated [243,244]. It has also been applied as a direct dynamics method. 

Jones et al. [144,214] used direct dynamics with semiempirical electronic wave functions to study electron transfer in cyclic polyene radical cations. Semiempirical methods have the advantage that they are cheap, and so a number of trajectories can be run for up to 50 atoms. Accuracy is of course sacrificed in comparison to CASSCF techniques, but for many organic molecules semiempirical methods are known to perform adequately. 

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). 

Being able to ntn direct dynamics calculations will add an extra, important, tool to help chemists understand photochemical systems. This chapter has outlined the present standpoint of the theory and practice of such calculations showing that, although much work remains to be done, they are already bringing new insight to mechanistic studies of photochemistry. 

Jones, A.G., Ejaz, T. and Graham, P., 1999. Direct dynamic observation of phase transformations during zeolite crystal synthesis. In Industrial Crystallization 99. (Rugby Institution of Chemical Engineers). Cambridge, 13-15 September 1999. Paper 95, p. 98. 

Traditionally, trajectory calculations were only performed on previously calculated (or empirically estimated) potential energy surfaces. With the increased computational speed of modern computers, it has also become possible to employ direct dynamics trajectory calculations [34, 35]. In this method, a global potential energy surface is not needed. Instead, from some... 

However, the direct dynamics calculations are computationally expensive, and cannot employ particularly high levels of electron correlation or large basis sets. If certain regions of the potential cannot be treated to within the required accuracy using a computationally affordable level of theory, the results may have unacceptably large errors. Nevertheless, direct dynamics calculations have played and will play a critical role in the discovery and analysis of competing pathways in chemical reactions. 

Direct dynamics trajectory calculations at the MP2/6-31-FG level of theory were then used to explore the reaction dynamics of this system [63]. Sixty-four trajectories were started from the central barrier shown at A in Fig. 11, with initial conditions sampled from a 300 K Boltzmann distribution. Of the 31 trajectories that moved in the direction of products, four trajectories followed the MEP and became trapped in the hydrogen-bonded [CH3OH ... 

The second channel, producing CO, was first observed by Seakins and Leone [64], who estimated 40% branching to this channel. Later measurements by Lockenberg et al. [65] and Preses et al. [66] concluded the branching to CO is 18%. Note that decomposition of formaldehyde formed in reaction (26a) is not a possible source of CO due to the large barrier for formaldehyde decomposition. Marcy et al. [67] recently combined time-resolved Lourier spectroscopy experiments with direct dynamics classical trajectory calculations to examine the mechanism of the CO product channel. They observed two pathways for CO formation, neither of which involve crossing a TS. 

For both statistical and dynamical pathway branching, trajectory calculations are an indispensable tool, providing qualitative insight into the mechanisms and quantitative predictions of the branching ratios. For systems beyond four or five atoms, direct dynamics calculations will continue to play the leading theoretical role. In any case, predictions of reaction mechanisms based on examinations of the potential energy surface and/or statistical calculations based on stationary point properties should be viewed with caution. 

The ability of SFE-FTIR to perform a variety of extraction methods is a definite advantage, especially for the study of complex mixtures containing analytes of varying solubility. For analytes which are readily solubilised in C02, direct dynamic and direct static-dynamic SFE-FTIR methods are quite successful. Elimination of the trapping process reduces both analysis time and potential analyte loss arising from... 

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