Born-Oppenheimer electronic potential energy

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

At present, reaction path methods represent the best approach for utilizing ab initio electronic structure theory directly in chemical reaction dynamics. To study reaction dynamics we need to evaluate accurately the Born-Oppenheimer molecular potential energy surface. Our experience suggests that chemical reaction may take place within in a restricted range of molecular configurations (i.e., there is a defined mechanism for the reaction). Hence we may not need to know the PES everywhere. Reaction path methods provide a means of evaluating the PES for the most relevant molecular geometries and in a form that we can use directly in dynamical calculations. 

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

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). 

Another assumption made in the straightforward application of classical mechanics to atomic motions is the Born-Oppenheimer approximation. That is, for each atomic geometry there is a single electronic potential energy surface under whose influence the atoms move. In reality, there are geometries at which more than one surface can play a role. The simplest such case is where two potentials (in the so-called diabatic picture) cross each other. 

The Born-Oppenheimer regime is attained when the electronic eigenenergies ( (0) =i are well separated so that the nuclear motion takes place on a single electronic potential energy surface, let us say E (0). This may be described by the effective nuclear Hamiltonian... 

The calculation of partition functions and of K is based on the following assumptions the Bom-Oppenheimer approximation is valid, that is, the electronic potential energy is independent of isotopic substitution the partition function can be factored into contributions from translational, rotational, and vibrational motions the translational and the rotational motions of the molecule can be treated in the classical approximation the vibrations are harmonic. Corrections to the effect of anharmonicity and to the Born-Oppenheimer approximation on isotopic exchange equilibria have been discussed by Wolfsberg (1969a, b) and by Kleinman and Wolfsberg (1973,1974), respectively. 

Before discussing applications of the SE-SCM (and variants thereof) to the description of cluster fission, we note that for atomic and molecular clusters microscopic descriptions of energetics and dynamics of fission processes, based on modern electronic structure calculations in conjunction with molecular dynamics simulations (where the classical trajectories of the ions, moving on the concurrently calculated Born-Oppenheimer (BO) electronic potential-energy surface, are obtained via integration of the Newtonian... 

Applying the Born-Oppenheimer approximation, we treat the nuclei as stationary, so is a constant. We can evaluate the nucleus-electron potential energy... 

Having recovered the potential surface from the solutions of the Born-Oppenheimer electronic problem, we can now proceed to solve the equation for nuclear motion. The Dirac-type equation for the nuclei can easily be reduced to the corresponding nonrelativistic equation by following the same reduction as we did for taking the nonrelativistic limit of the Dirac equation in section 4.6. Doing this, we abandon all pretense of Lorentz invariance for this part of the system, but we know from experiment that the nuclear relative motion in molecules takes place at rather low energies where relativistic effects may safely be neglected. 

Here r and R denote collectively the electronic and nuclear coordinates, respectively, Te is the kinetic-energy operator of the electrons, and V(r, R) represents the Coulombic interaction potential of all particles. The Born-Oppenheimer adiabatic electronic states depend parametrically on the nuclear coordinates R. The Vn(R) define the adiabatic electronic potential-energy surfaces. 

As discussed above, in Born-Oppenheimer (BO) direct dynamics the potential energy V(q) and derivatives dVIdq for each step of the trajectory integration are obtained by optimizing the electronic wavefunction. For a Born-Oppenheimer electronic structure theory calculation the electronic energy Ee(q) is determined variationally from... 

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. 

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). 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

Evaluating the energy e for different values of R gives the effective potential for the nuclei in the presence of the electron. This function is called the Born-Oppenheimer potential surface or just the potential surface. In order to evaluate e(R) we have to determine HAA, HAB, and SAB. These quantities, which can be evaluated using elliptical coordinates, are given by... 

In the vibrational treatment we assumed, as usually done, that the Born-Oppenheimer separation is possible and that the electronic energy as a function of the internuclear variables can be taken as a potential in the equation of the internal motions of the nuclei. The vibrational anharmonic functions are obtained by means of a variational treatment in the basis of the harmonic solutions for the vibration considered (for more details about the theory see Pauzat et al [20]). 

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