Born-Oppenheimer surfaces

One may consider the above equation as a generalization of Born-Oppenheimer dynamics in which electrons always stay on the Born-Oppenheimer surface. For a given conformation of nuclei, the numerical value of the fictitious mass associated with electronic degrees of freedom determines how far the electron density is allowed to deviate from the Born-Oppenheimer one. Each consecutive step along the trajectory, which involves electronic and nuclear degrees of freedom, can be obtained without determining the exact Born-Oppenheimer electron density. 

The most serious problem with MM as a method to predict molecular structure is convergence to a false, rather than the global minimum in the Born-Oppenheimer surface. The mathematical problem is essentially still unsolved, but several conformational searching methods for approaching the global minimum, and based on either systematic or random searches have been developed. These searches work well for small to medium-sized molecules. The most popular of these techniques that simulates excitation to surmount potential barriers, has become known as Molecular Dynamics [112]. 

For finite values of /n the system moves within a limited width, given by the fictitious electronic kinetic energy, above the Born-Oppenheimer surface. Adiabacity is ensured if the highest frequency of the nuclear motion separated from the lowest frequency associated with the fictitious motion of the electronic degrees of freedom cofm. It can be shown [30] that eo in is proportional to the gap Eg ... 

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

The potential energy is illustrated in Fig. 6.3a. While one can in principal calculate the exact quantum mechanical Born-Oppenheimer surface, the figure presents a semi-empirical surface constructed to yield the exact spectral properties of reactants and products and the correct activation energy (taken as the difference in energy between the energy (potential) in the reactant valley (x = oo) and the maximum of the MEP (minimum energy pathway).)... 

For the finite-temperature simulations, the temperature of the Si ions were controlled with a chain of five, linked Nose-Hoover thermostats." Because the electrons are always quenched back onto the Born-Oppenheimer surface after every timestep, no additional thermostat is needed for the electrons. Details of the configurations were similar to those with the CP scheme, except that the in-plane cells consisted of 16 atoms per layer and the basic timestep of the simulation was 100 a.u. 

Separation of the movement of the nuclei and electrons. This is possible because the electrons move much more rapidly (smaller mass) than the nuclei. The position of the nuclei is fixed for the calculation of the electronic Schrodinger equation (in MO calculations the nuclear positions are then parameters, not quantum chemical variables). Born-Oppenheimer surfaces are energy vs. nuclear structure plots which are (n + 1)-dimensional, where n is 3N- 6 with N atoms (see potential energy surface). 

In contrast to the subsystem representation, the adiabatic basis depends on the environmental coordinates. As such, one obtains a physically intuitive description in terms of classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been studied using QCL dynamics in this basis. These include the reaction rate and the kinetic isotope effect of proton transfer in a polar condensed phase solvent and a cluster [29-33], vibrational energy relaxation of a hydrogen bonded complex in a polar liquid [34], photodissociation of F2 [35], dynamical analysis of vibrational frequency shifts in a Xe fluid [36], and the spin-boson model [37,38], which is of particular importance as exact quantum results are available for comparison. 

The Vienna Ab Initio Simulation Package (VASP) program33,34 was developed to carry out calculations to obtain the fluctuation trajectory of the selected models. The VASP program uses a rather traditional self-consistency scheme to evaluate the instantaneous electronic ground-state at each molecular dynamics (MD) step so that the wavefunction can be converged to the Born-Oppenheimer surface at each time-step. 

Consider a quantum center (i.e., a molecule or a subpart of a molecule) embedded in a classical molecular environment. Defining with rn the nuclear coordinates of the quantum center and with x the coordinates of the atoms providing the (classical) perturbing field we can expand [26] the perturbed Hamiltonian matrix H of the quantum center on the Born-Oppenheimer surface as... 

Bakken, V. Millam, J. M. Schlegel, H. B. Ab initio classical trajectories on the Born-Oppenheimer surface Updating methods for Hessian-based integrators, J. Chem. Phys. 1999, 111, 8773-8777. 

The techniques collectively termed molecular mechanics (MM) employ an empirically derived set of equations to describe the energy of a molecule as a function of atomic position (the Born—Oppenheimer surface). The mathematical form is based on classical mechanics. This set of potential energy functions (usually termed the force field) contains adjustable parameters that are optimized to fit calculated values of experimental properties for a known set of molecules. The major assumption is, of course, that these parameters are transferable from one molecule to another. Computational efficiency and facile inclusion of solvent molecules are two of the advantages of the MM methods. 

For finite values of p, the system moves within a given thickness of jE " above the Born-Oppenheimer surface. Adiabacity is ensured if the highest frequency of the nuclear motion... 

Most potential energy functions used in molecular simulation studies are based on an empirical representation of Born-Oppenheimer surfaces [87]. The ground state of a molecular system is described by a continuous potential energy, function of the coordinates of its atoms assumed as point charges, with the following general form ... 

Equation [73] has the same form as the equations of motion for molecules with constrained internal coordinates, and we already know that such equations can be solved effectively using the SHAKE algorithm4 ° Equations [72] and [73] play a key role in the Car-Parrinello method and enable one to run the dynamics for both ionic and electronic degrees of freedom in parallel. With carefully chosen effective mass p and a small time step, the electronic state adjusts itself instanteously to the nuclear configuration (Born-Oppenheimer principle), and, therefore, the atomic dynamics is computed along the system s Born-Oppenheimer surface. Note that there is no need to carry out the costly matrix-diagonalization procedure for performing electronic structure calculations. 

Introduction Conical Intersection Seams as Analogs of Born-Oppenheimer Surfaces... 

A practical representation of the full wave function, or the constituent orbitals, involves basis functions. The electronic wave function (or electron density) is parameterized by linear basis function coefficients or nonlinear parameters, such as the position or widths of Gaussian basis function. The array c will denote the collection of all wave function parameters, both linear and nonlinear unless otherwise specified. The ground electronic Born-Oppenheimer surface, is given by... 

In the simulated annealing method, minimization of c) is mapped onto a statistical mechanical problem by considering (c) to be the potential energy governing a fictitious thermal system. If this fictitious thermal system can be brought to thermal equilibrium at sufficiently low temperature, then the variables c will be brought to the minimum of (c), i.e. to the Born-Oppenheimer surface. 

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