Born-Oppenheimer approximation simulations

The applicability of the Born-Oppenheimer approximation for complex molecular systems is basic to all classical simulation methods. It enables the formulation of an effective potential field for nuclei on the basis of the SchrdJdinger equation. In practice this is not simple, since the number of electrons is usually large and the extent of configuration space is too vast to allow accurate initio determination of the effective fields. One has to resort to simplifications and semi-empirical or empirical adjustments of potential fields, thus introducing interdependence of parameters that tend to obscure the pure significance of each term. This applies in... 

How important the breakdown of the Born-Oppenheimer approximation is in limiting our ability to carry out ab initio simulations of chemical reactivity at metal surfaces is the central topic of this review. Stated more provocatively, do we have the correct theoretical picture of heterogeneous catalysis. This review will restrict itself to a consideration of experiments that have begun to shed light on this important question. The reader is directed to other recent review articles, where aspects of this field of research not mentioned in this article are more fully addressed.10-16... 

We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory co t) for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. 

In all dynamical simulations presented so far, it has been assumed that the electrons stay in their ground state throughout the whole process, i.e. the simulations have been based on the Born-Oppenheimer approximation. Still, at metal surfaces with their continuous spectrum of electronic states at the Fermi energy electron-hole (e-h) pair excitations with arbitrarily small energies are possible. However, the incorporation of electronically nonadiabatic effects in the dynamical simulation of the interaction dynamics of molecules with surface is rather difficult [2, 109, 110]. Hence the role of electron-hole pairs in the adsorption dynamics as an additional dissipation channel is still unclear [4],... 

In classical MD simulations it is often assumed that the nuclei move on a potential energy surface. Implicit in this assumption is that the Born-Oppenheimer approximation applies that electronic and nuclear motion can... 

Car-Parrinello techniques have been used to describe classical variables whose behavior, like quantum electrons in the Born-Oppenheimer approximation, is nearly adiabatic with respect to other variables. In simulations of a colloidal system consisting of macroions of charge Ze, each associated with Z counterions of charge —e, Lowen et al. [192] eliminated explicit treatment of the many counterions using classical density functional theory. Assuming that the counterions relax instantaneously on the time-scale of macroion motion, simulations of the macroion were performed by optimizing the counterion density at each time step by simulated annealing. 

Section 2 of this chapter notes will be devoted to the framework for separation of the ionic and electronic dynamics through the Born-Oppenheimer approximation. Atomic motion, with forces on the ions at each timestep evaluated through an electronic structure calculation, can then be propagated by Molecular Dynamics simulations, as proposed by first-principle Molecular Dynamics. This allows for a description of the electronic reorganisation following the atomic motion, e.g. bond rearrangements in chemical reactions. 

The simulation of a system of classical nuclei and electrons under the Born-Oppenheimer approximation can then be performed within a Molecular Dynamics framework as follows ... 

Our primary goal was the simulation of entire atomic systems, thus made of electrons and nuclei. As mentioned earlier (see Sect. 2.3), in a large class of systems (e.g. not too high temperature) one can decouple the motion of nuclei and electrons within the Born-Oppenheimer approximation. The previous section was then devoted to the Density Functional Theory solution of the electronic structure problem at fixed ionic positions. By computing the Hellmann-Feynman forces (11) we can now propagate the dynamics of an ensemble of (classical) nuclei as described in Sect. 2.3, using e.g. the velocity verlet algorithm [117]. 

As most of the electronic structure simulation methods, we start with the Born-Oppenheimer approximation to decouple the ionic and electronic degrees of freedom. The ions are treated classically, while the electrons are described by quantum mechanics. The electronic wavefunctions are solved in the instantaneous potential created by the ions, and are assumed to evolve adiabatically during the ionic dynamics, so as to remain on the Born-Oppenheimer surface. Beyond this, the most basic approximations of the method concern the treatment of exchange and correlation (XC) and the use of pseudopotentials. XC is treated within Kohn-Sham DFT [3]. Both the local (spin) density approximation (LDA/LSDA) [16] and the generalized gradients approximation (GGA) [17] are implemented. The pseudopotentials are standard norm-conserving [18, 19], treated in the fully non-local form proposed by Kleinman and Bylander [20]. 

In this review, almost all of the simulations we have described use only classical mechanics to describe the nuclear motion of the reaction system. However, a more accurate analysis of many reactions, including some of the ones that have already been simulated via purely classical mechanics, will ultimately require some infusion of quantum mechanical methods. This infusion has already taken place in several different types of reaction dynamics electron transfer in solution, > i> 2 HI photodissociation in rare gas clusters and solids,i i 22 >2 ° I2 photodissociation in Ar fluid,and the dynamics of electron solvation.22-24 Since calculation of the quantum dynamics of a full solvent is at present too time-consuming, all of these calculations involve a quantum solute in a classical solvent. (For a system where the solvent is treated quantum mechanically, see the quantum Monte Carlo treatment of an electron transfer reaction in water by Bader et al. O) As more complex reaaions are investigated, the techniques used in these studies will need to be extended to take into account effects involving electron dynamics such as curve crossing, the interaction of multiple electronic surfaces and other breakdowns of the Born-Oppenheimer approximation, the effect of solvent and solute polarization, and ultimately the actual detailed dynamics of the time evolution of the electronic degrees of freedom. 

An essential feature of MM methods for the treatment of biomole-cular systems is their computational efficiency. The inclusion of polarizability into the model increases the computational demand due to the addition of dipoles or additional charges centers and, in the context of MD simulations, the requirement for shorter integration time steps. In addition, for every energy or force evaluation it is necessary to solve for all the polarizable degrees of freedom in a self-consistent manner. Traditionally, this is performed via a self-consistent field (SCF) calculation based on the Born-Oppenheimer approximation in which the induced polarization is solved iteratively until a satisfactory level of convergence is achieved. With the Drude model, this implies that the Drude... 

The use of classical simulations on the basis of the Born-Oppenheimer approximation is problematic when dealing with metals, even in the absence of electrochemical reactions involving the transfer of electrons, which can certainly not be approximated by classical mechanics. 

Born-Oppenheimer approximation. In particular, they analyzed the time evolution of the ionization potential and the dependence of the dynamics on the initial temperature of the Aga molecule. The simulations were performed by use of the Verlet algorithm in its velocity form. 

Combining a microscopic electronic theory with molecular dynamics simulations in the Born-Oppenheimer approximation, Bennemann, Garcia, and Jeschke presented the first theoretical results for the ultrafast structural changes in the silver trimer [135]. They determined the timescale for the relaxation from the linear to a triangular structure initiated by a photodetachment process and showed that the time-dependent change of the ionization potential (IP) reflects in detail the internal degrees of freedom. 

The QM/MM Hamiltonian can be used to cany out Molecular Dynamics simulations of a complex system. In the case of liquid interfaces, the simulation box contains the solute and solvent molecules and one must apply appropriate periodic boundary conditions. Typically, for air-water interface simulations, we use a cubic box with periodic boundary conditions in the X and Y directions, whereas for liquid/liquid interfaces, we use a rectangle cuboid interface with periodic boundary conditions in the three directions. An example of simulation box for a liquid-liquid interface is illustrated in Fig. 11.1. The solute s wave function is computed on the fly at each time step of the simulation using the terms in the whole Hamiltonian that explicitly depend on the solute s electronic coordinates (the Born-Oppenheimer approximation is assumed in this model). To accelerate the convergence of the wavefunction calculation, the initial guess in the SCF iterative procedure is taken from the previous step in the simulation, or better, using an extrapolated density matrix from the last three or four steps [39]. The forces acting on QM nuclei and on MM centers are evaluated analytically, and the classical equations of motion are solved to obtain a set of new atomic positions and velocities. 

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