Potential energy function, curvature

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... 

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. 

The first derivatives of a potential energy function define the gradient of the potential and the second derivatives describe the curvature of the energy surface (Fig. 3.4). In most molecular mechanics programs the potential functions used are relatively simple and the derivatives are usually determined analytically. The second derivatives of harmonic oscillators correspond to the force constants. Thus, methods using the whole set of second derivatives result in some direct information on vibrational frequencies. 

Such a potential energy function gives rise to the famihar parabolic curve (Figure 22) where the curvature of the function is related to the force constant. The success of this simple harmonic model in treating surface atom vibrations lies in the relatively small displacement of surface atoms during a period of vibration. For some crystal properties, such as thermal expansion at elevated temperature, anharmoitic contributions to the potential must be included for an accurate description. 

The perturbation caused by the applied electric field results in a change of the curvature of the potential energy function, thus causing a change in the transition frequency. 

Figure 7. Left column. The potential energy functions V — /c, (solid lines) and their curvatures (dotted lines) for different values of c c = 2 (linear oscillator), and c — 4,6,8 (strongly non-linear oscillators). Middle column. Typical sample paths of Brownian oscillators, a = 2, with the potential energy functions shown on the left. Right column Typical sample paths of Levy oscillators, a = 1. On increasing m the potential walls become steeper, and the flights become shorter in this sense, they are confined.

Let us now consider a system in which the potential-energy function remains finite at x — + °° or at x — — °° or at both limits, as shown in Figure 9-3. For a value of W smaller than both F(+°°) and F(—oo) the argument presented above is valid. Consequently the energy levels will form a discrete set for this region. If W is greater than F(+ ), however, a similar argument shows that the curvature will be such as always to return the wave function to the x axis, about which it will... 

Figure 12. Relationship between dissociation energy and curvature of the potential energy function at the equilibrium represented by force constant ke or vibrational frequency cOe.

It is also important to recognize that differentiation of Eq. 4.4 twice yields d Vjdx = k. Thus the force constant of the spring is equal to the curvature of the potential-energy function, which is shown in Figure 4.2. 

The above treatment assumes harmonic vibrational potentials. A harmonic potential is parabolic and its second derivative, or the curvature of the potential energy function, or the force constant, is the same for compression or elongation of the bond distance or for small and large displacements from equilibrium. These assumptions may be realistic as long as the vibrational amplitude is small, but certainly are not valid on... 

A drop of water that is placed on a hillside will roll down the slope, following the surface curvature, until it ends up in the valley at the bottom of the hill. This is a natural minimization process by which the drop minimizes its potential energy until it reaches a local minimum. Minimization algorithms are the analogous computational procedures that find minima for a given function. Because these procedures are downhill methods that are unable to cross energy barriers, they end up in local minima close to the point from which the minimization process started (Fig. 3a). It is very rare that a direct minimization method... 

These discrepancies result (a) from the harmonic approximation used in all calculations [to,- (theory) > v, (exp)], (b) the known deficiencies of minimal and DZ basis sets to describe three-membered rings [polarization functions are needed to describe small CCC bond angles a>,(DZ + P) > w,(DZ) > to,(minimal basis)] and (c) the need of electron correlated wave functions to correctly describe the curvature of the potential energy surface at a minimum energy point [ [Pg.102]

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