Born-Oppenheimer energy

Fig. 4.5 Schematic projection of the energetics of a reaction. The diagram shows the Born-Oppenheimer energy surface mapped onto the reaction coordinate. The barrier height AE has its zero at the bottom of the reactant well. One of the 3n — 6 vibrational modes orthogonal to the reaction coordinate is shown in the transition state. H and D zero point vibrational levels are shown schematically in the reactant, product, and transition states. The reaction as diagrammed is slightly endothermic, AE > 0. The semiclassical reaction path follows the dash-dot arrows. Alternatively part of the reaction may proceed by tunneling through the barrier from reactants to products with a certain probability as shown with the gray arrow...

Comparison of Vibrational Frequencies - (in cm ) of H2 Calculated from Non-Born-Oppenheimer Energies [121] with the Experimental Values of Dabrowski [125] and with the results of Wolniewicz, Obtained Using the Conventional Approach Based on the Potential Energy Curve"... 

Table 1. The Born-Oppenheimer energy and the parallel and perpendicular components of the polarizability for the EF Ut state of H2 at selected distances R...

If Rx and R2 are the YX and XY distances in a molecule YXY and 8 is the YXY angle, then the total Born-Oppenheimer energy E=T+ Vhas a potential energy Vgiven by53... 

The U(R2, C ) is the Born-Oppenheimer energy, which now explicitly depends on the c coordinates and consists of the nucleus-nucleus, electron-nucleus, and electron-electron interaction energy terms. Note that the real electron kinetic energy is included in this last term. The equation of motion for the particles (both real and fictitious) are then obtained from the extended Lagrangian (Eq. [71]) and reads as follows ... 

A global property function is usually expressed as the expectation value of an operator or as the derivative of such an expectation value with respect to an internal or external parameter of the system. In the Born-Oppenheimer approximation, the electronic wave function depends parametrically upon the coordinates of the n nuclei, and therefore a set of the 3 -6 linearly independent nuclear coordinates constitutes the natural variables for such a choice of the potential function. However, the manifold M on which the gradient vector field is bound can be defined on a subset of 1R provided q < 3n-6, for example the intrinsic reaction coordinate (unstable manifold of a saddle point of index 1 of the Born-Oppenheimer energy hypersurface) or the reduced reaction coordinate. 

The Born-Oppenheimer energy depends on nuclear coordinates R because the electronic Hamiltonian H or density functional depends on R through Coulombic electron-nuclear... 

The success of any molecular simulation method relies on the potential energy function for the system of interest, also known as force fields [27]. In case of proteins, several (semi)empirical atomistic force fields have been developed over the years, of which ENCAD [28,29], AMBER [30], CHARMM [31], GRO-MOS [32], and OPLSAA [33] are the most well known. In principle, the force field should include the electronic structure, but for most except the smallest systems the calculation of the electronic structure is prohibitively expensive, even when using approximations such as density functional theory. Instead, most potential energy functions are (semi)empirical classical approximations of the Born-Oppenheimer energy surface. 

Let Un(R) be the total Born-Oppenheimer energy for a given diatomic system in its nth electronic state, and T (R) = (n T n) and V (R) = (n V n) are the corresponding electronic kinetic and potential functions. Evidently, these are functions of the intemuclear distance R (Slater, 1963). 

S. Lifson and P. S. Stern,/. Chem. Phys., 77, 4542 (1982). Born-Oppenheimer Energy Surfaces of Similar Molecules Interrelations between Bond Lengths, Bond Angles, and Frequencies of Normal Vibrations in Alkanes. 

In Quantum Mechanical calculations, the energy is computed from the exact hamiltonian. It is then possible to build a Born-Oppenheimer energy surface which can be used later to perform lattice dynamics or to study the reaction path of a displacive phase transition. These methods give access to the electron density, the spin density and the density of states which are useful to predict electric and optical properties as well as to analyze the bonding. Recently, methods combining a quantum mechanical calculation of the potential and the Molecular Dynamics scheme have been developed after the seminal work of R. Car and M. Parrinello. 

The critical question is how accurate (or inaccurate) the Born-Oppenheimer energy of a molecule defined as above is. Recently Takahashi and Takatsuka have addressed this question and reported a semiclassical analysis of this matter [413]. They explicitly showed that the fifth order term is also exactly zero, and therefore the lowest-order correction to the Born-Oppenheimer energy must be of the order of... 

In contrast to previous chapters we have until now in this section shown explicitly that the electronic wavefunctions depend not only on the position vectors of the electrons rj but parametrically also on the internuclear distance R. In the following, we will not indicate the obvious dependence on the electronic coordinates for the sake of more compact formulas, but continue with showing the dependence of the electronic wavefunctions and Born-Oppenheimer energies on the internuclear distance. 

Figure 20.1 Born-Oppenheimer Energy as a Function of internuciear Distance for a Diatomic Moiecuie (Schematic).

Figure 20.3 shows the Born-Oppenheimer energy of hJ as a function of tab for the two states of lowest energy. We denote the molecular orbital for the ground state by... 

Figure 20.3 The Born-Oppenheimer Energy of the Ground State and Rrst Excited State of the Hydrogen Molecule Ion as a Function of Internuclear Distance.

The zero-order Born-Oppenheimer energy is a sum of two hydrogen-molecule-ion electronic energies plus %n -... 

For the He2 molecule we let Z = 2. Electron-electron repulsion terms are omitted from this zero-order Hamiltonian and the constant nuclear repulsion term has also been omitted. At the end of the calculation we must add f rm to the electronic energy to obtain the zero-order Born-Oppenheimer energy. 

The Born-Oppenheimer energy acts as a potential energy for vibration. 

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