Potential energy functions, diatomic

The force constant that is associated with the stretching vibration of a bond is often taken as a measure of the strength of the bond, although it is more correctly a measure of the curvature of the potential energy function around the minimum (Figure 2.1) that is, the rigidity of the bond. For a diatomic molecule, the frequency of vibration v is determined by the force constant k and the reduced mass /x = + m2), where m and m2 are the masses of... 

The potential energy function U(R) that appears in the nuclear Schrodinger equation is the sum of the electronic energy and the nuclear repulsion. The simplest case is that of a diatomic molecule, which has one internal nuclear coordinate, the separation R of the two nuclei. A typical shape for U(R) is shown in Fig. 19.1. For small separations the nuclear repulsion, which goes like 1 /R, dominates, and liniR >o U(R) = oo. For large separations the molecule dissociates, and U(R) tends towards the sum of the energies of the two separated atoms. For a stable molecule in its electronic ground state U(R) has a minimum at a position Re, the equilibrium separation. 

Even for a diatomic molecule the nuclear Schrodinger equation is generally so complicated that it can only be solved numerically. However, often one is not interested in all the solutions but only in the ground state and a few of the lower excited states. In this case the harmonic approximation can be employed. For this purpose the potential energy function is expanded into a Taylor series about the equilibrium separation, and terms up to second order are kept. For a diatomic molecule this results in ... 

Klein-Rydberg method phys chem A method for determining the potential energy function of the distance between the nuclei of a diatomic molecule from the molecule s vibrational and rotational levels. klTn rid,berg. meth ad )... 

Varshni, Y. P. (1957). Comparative study of potential energy functions for diatomic molecules. Rev. Mod. Phys. 29, 664-682. 

To obtain the allowed energy levels, Ev, for a real diatomic molecule, known as an anharmonic oscillator, one substitutes the potential energy function describing the curve in Fig. 3.2c into the Schrodinger equation the allowed energy levels are... 

Bonding in a diatomic molecule may be described by the curve given in Fig. 2.5, which represents the potential energy (K(r)) as a function of the structure (r). The bonding force constant, k, is given by the second derivative of the potential with respect to the structural parameter r, which corresponds to the curvature of the potential energy function. The anharmonicity can be described by higher-order derivatives. 

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

The coefficients m,f x,f Be, a , De,. .. are the quantities determined from a spectroscopic analysis of a diatomic molecule, and our problem is to relate these to the force constants in the potential energy function,... 

Potential, chemical, 202 Potential energy, 211 diatomic molecule, 191 Potential energy functions, 391 Potential energy surfaces, 191 Precession, 155... 

Correlate the shape of the effective potential energy function between the nuclei with trends in the bond length and bond energies of diatomic molecules (Section 3.5, Problems 17-20). 

More than 20 and 10 years have passed since the publication of two well-known books, Spectra of Diatomic Molecules by G. Herzberg and Diatomic Interaction Potential Theory by J. Goodisman, respectively, which are devoted to the problem of diatomic potential energy functions, or, in other words, the theory of diatomic interactions. Time after time excellent and superior reviews have appeared, and we can refer to the literature of Varshni, Stwalley, Le Roy, Carney, Kolos, Winn, and others. However, their number is not so many. The theory of diatomic interactions occupies the simplest position within the general theory of inter-molecular interactions, for which there remain many theoretical problems. Unsolved problems also concern the theory of diatomic interactions, and the Everest of the complete understanding of the nature of diatomic interactions remains unsubjugated ... 

The function U(R) is denoted as the diatomic potential energy function. Usually it is represented by a curve (see Fig. 1) and characterized by... 

It is evident that p is a parameter which is chosen from the criterion of the best reproduction of a real diatomic potential energy function U(R). There are some possibilities to choose from to satisfy this criterion. The first possibility is to choose p such that (Goble and Winn, 1979) p = -(k3R3)/Qk2) - 1 -a - 1. From Eqs. (9) and (10) it follows that... 

In Section II we discussed the Taylor series of diatomic potentials near the equilibrium separations. The Pade approach is another method to represent analytically a real diatomic potential energy function, but in recent years this powerful and elegant approach surrendered its position, probably because its force has been insufficient when applied in practice. 

Figure 3. Potential energy function of a diatomic molecule in its electronic ground state. Vq is the dissociation energy, D, in units of he, and r, is the equilibrium position.

---