Born-Oppenheimer approximation calculations

As ab initio MD for all valence electrons [27] is not feasible for very large systems, QM calculations of an embedded quantum subsystem axe required. Since reviews of the various approaches that rely on the Born-Oppenheimer approximation and that are now in use or in development, are available (see Field [87], Merz ]88], Aqvist and Warshel [89], and Bakowies and Thiel [90] and references therein), only some summarizing opinions will be given here. 

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

Vhen calculating the total energy of the system, we should not forget the Coulomb inter-ction between the nuclei this is constant within the Born-Oppenheimer approximation Dr a given spatial arrangement of nuclei. When it is desired to change the nuclear positions,... 

The total energy in an Molecular Orbital calculation is the net result of electronic kinetic energies and the interactions between all electrons and atomic cores in the system. This is the potential energy for nuclear motion in the Born-Oppenheimer approximation (see page 32). 

The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the... 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

Use of the Born-Oppenheimer approximation is implicit for any many-body problem involving electrons and nuclei as it allows us to separate electronic and nuclear coordinates in many-body wave function. Because of the large difference between electronic and ionic masses, the nuclei can be treated as an adiabatic background for instantaneous motion of electrons. So with this adiabatic approximation the many-body problem is reduced to the solution of the dynamics of the electrons in some frozen-in configuration of the nuclei. However, the total energy calculations are still impossible without making further simplifications and approximations. 

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

Force constants, crude Born-Oppenheimer approximation, hydrogen molecule, minimum basis set calculation, 545-550 Forward peak scattering, electron nuclear... 

To be specific we consider electron transfer from a reactant in a solution, such as [Fe(H20)6]2+, to an acceptor, which may be a metal or semiconductor electrode, or another molecule. To obtain wavefunc-tions for the reactant in its reduced and oxidized state, we rely on the Born-Oppenheimer approximation, which is commonly used for the calculation of molecular properties. This approximation is based on the fact that the masses of the nuclei in a molecule are much larger than the electronic mass. Hence the motion of the nuclei is slow, while the electrons are fast and follow the nuclei almost instantaneously. The mathematical consequences will be described in the following. 

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. 

The problem, as Woolley addressed it, is that quantum mechanical calculations employ the fixed, or "clamped," nucleus approximation (the Born-Oppenheimer approximation) in which nuclei are treated as classical particles confined to "equilibrium" positions. Woolley claims that a quantum mechanical calculation carried out completely from first principles, without such an approximation, yields no recognizable molecular structure and that the maintenance of "molecular structure" must therefore be a product not of an isolated molecule but of the action of the molecule functioning over time in its environment.47... 

In Chapter 4 we will learn to calculate the equilibrium constant for an exchange reaction like Equation2.15 using the Born-Oppenheimer approximation. If, in addition, the adiabatic correction is included, the equilibrium constant calculated in the Born-Oppenheimer approximation must be multiplied by a correction factor containing the energy difference AAC. 

The Born-Oppenheimer approximation applies here just as it does for a single molecule. Calculate the electronic energy as a function of nuclear configuration. The problem is shown schematically in Fig. 4.5. Corresponding to certain configurations... 

In principal one can calculate the electronic energy as a function of the Cartesian coordinates of the three atomic nuclei of the ground state of this system using the methods of quantum mechanics (see Chapter 2). (In subsequent discussion, the terms coordinates of nuclei and coordinates of atoms will be used interchangeably.) By analogy with the discussion in Chapter 2, this function, within the Born-Oppenheimer approximation, is not only the potential energy surface on which the reactant and product molecules rotate and vibrate, but is also the potential... 

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