Potential energy internuclear

Finally we have to remember to add on the nuclear repulsion and, if we repeat the calculation for a range of values of the internuclear separation, we arrive at the potential energy curves shown in Figure 4.3 for the ground-state (singlet)... 

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... 

The aim of this work is to obtain the four lowest E curves and wavefunctions of BH at the same level of accuracy and to bring out the interplay of ionic, Rydberg and valence states at energies and internuclear distances which were not previously investigated. We have therefore made use of a method, already put forward by us [16,17] to determine at once quasi-diabatic and adiabatic states, potential energy cnrves and approximate nonadiabatic couplings. We have analogously determined the first three E+ states, of which only the lowest had been theoretically studied... 

The relationships between bond length, stretching force constant, and bond dissociation energy are made clear by the potential energy curve for a diatomic molecule, the plot of the change in the internal energy AU of the molecule A2 as the internuclear separation is increased until the molecule dissociates into two A atoms ... 

In general three position variables will be needed to specify the potential energy of the reaction system. These may be the X-Y, Y-Z, and X-Z internuclear distances or two internuclear distances and the included angle. Even in this relatively simple case, four dimensions would be required for generation of the potential energy surface. However, if we restrict our attention to linear configurations of these atoms, it is possible... 

Schematic representation of the potential energy of a system comprised of three atoms in a linear configuration as a function of the internuclear separation distances. The dashed line represents the reaction X + YZ — XY+ Z.

Figure 1.7 The potential energy of the hydrogen molecule as a function of internuclear distance.

During initialization and final analysis of the QCT calculations, the numerical values of the Morse potential parameters that we have used are given as De = 4.580 eV, re = 0.7416 A, and (3 = 1.974 A-1. Moreover, the potential energy as a function of internuclear distances obtained from the analytical expression (with the above parameters) and the LSTH [75,76] surface asymptotically agreed very well. 

Although the energy of a chemical bond as a function of internuclear distance can be represented by the potential energy curve shown in Figure 3.2, neither the Wolfsberg-Helmholtz nor the Ballhausen-Gray approximation is a function that possesses a minimum. However, the approximation of Cusachs is a mathematical expression that does pass through a minimum. 

When potential energy depends on more than one geometric parameter, i.e. internuclear distances, the word surface is used in place of curve, and the plot is known as potential energy surface (PES). 

Bringing the M and M molecules together, however, results in a potential energy minimum at an internuclear distance, shown as req. This minimum corresponds to the formation of the bound excimer. 

We next consider the expression for k in the classical formalism. According to the Franck-Condon principle, internuclear distances and nuclear velocities do not change during the actual electron transfer. This requirement is incorporated into the classical electron-transfer theories by postulating that the electron transfer occurs at the intersection of two potential energy surfaces, one for the reactants... 

Consider a stable diatomic molecule with nuclei denoted as A and B. The Born-Oppenheimer potential V for such a molecule will depend on the internuclear distance rAB and will typically have the form shown in Fig. 3.1. The potential energy has a minimum at r0, which is often referred to as the equilibrium internuclear distance. As the distance rAB increases, the potential V increases and finally reaches a limiting value where the molecule is now better described as two separated atoms (or depending on the electronic state of the system, two separated atomic species one or both of which may be ions). The difference in energy between the two separated atoms and the minimum of the potential is the dissociation energy De of the molecule. As the internuclear distance of the diatomic molecule is decreased... 

Consider now the vibration of the AB diatomic molecule. The potential energy of vibration depends on the displacement of the diatomic molecule internuclear distance from its equilibrium value, S = r-re. The kinetic energy involves the velocity (time derivative), S = d(r-re)/dt, and is given by... 

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

Theoretical calculation of any atomic or molecular property through application of computational methods based on quantum mechanics or other sophisticated approach is typically practicable through approximate methods. The internuclear potential energy V(i ) independent of mass is conventionally derived from the results of computations of molecular electronic structure according to a scheme of wave mechanics,... 

When Dunham [4,5] presented formula 8 for vibration-rotational terms, he derived a functional V + V2 explicitly because in his JBKW formulation the addend V2 results from exact solution of an integral. In contrast, Dunham assumed a functional K(K+l), equivalent to /(/- -1) in contemporary notation, to contain a quantum number K, now J, for rotational angular momentum. To generate an effective potential energy comprising both internuclear potential... 

---