Quadratic potential energy functions

In a molecular mechanics force field, the starting point for describing bond stretching and compression is the harmonic approximation. The simplest approach is the use of a quadratic potential energy function as shown in Eq. [3] ... 

As we discussed in Chap. 2, the solutions to the Schrodinger equation for a quadratic potential energy function of coordinate x are the harmonic-oscillator wavefunctions,... 

Consider the physical significance of the additional terms in (4.67) as compared to (4.39). The fourth term on the right side of (4.67) represents a shift in the vibrational levels. The constant involves the third and fourth derivatives of V evaluated at Re, and is therefore a consequence of the deviation of the potential energy function from the (quadratic) harmonic-oscillator potential ... 

A quadratic function defines a symmetric parabola and therefore cannot exactly reproduce the true relationship between the distortion of a bond length or valence angle and the energy needed to effect that distortion. However, a central assumption in the application of simple molecular mechanics models is that distortions from ideal values are small and in such cases it is only necessary that the potential energy function be realistic in the region of the ideal value. This is shown in Fig. 17.8.1, where a quadratic curve is compared to a Morse potential that is believed to more accurately reflect the relationship between bond length distortion and energy cost. 

Figure 28-7 In force field calculations, different levels of approximations are used to reproduce the stretching and compression of chemical bonds The plot shows a Morse potential energy function siipenmposed with various power series approximations (quadratic, cubic, and quartic functions) Note that the bottoms of the curves, representing the bond length for most chemical bonds of interest to medicinal chemists, almost overlap exactly. This nearly perfect fit in the bonding region is the reason simple harmonic functions can be used to calculate tx>nd lengths for unstrained molecular structures in the force lield method.

As in the BK procedure, the electrical properties and the potential energy surface may be expanded as a Taylor series in the normal coordinates. Orders of perturbation theory are defined in the same way as for the non-resonant case. Electrical property terms that are quadratic, cubic,. .. in the normal coordinates are taken to be first-order, second-order,. .. terms in the potential energy function that are... 

The relationship between the potential function K(R) and the observable spectroscopic parameters is summarized in Figure 2. The harmonic vibration frequencies are obtained as the eigenvalues of a secular determinant involving the quadratic force constants and the atomic masses and molecular geometry (the F and G matrices of Wilson s well-known formalism) by a calculational procedure discussed in detail by Wilson, Decius, and Cross.1 The eigenvectors determine the normal coordinates Q in terms of which the kinetic and quadratic potential energy terms are both diagonal (R = LQ). The various anharmonidty constants and vibration/rotation interaction constants are obtained in terms of the... 

The simplest functional form of the angle-bending potential energy function includes only a quadratic term (the harmonic approximation) as in Eq. [6] ... 

In the context of molecular dynamics involving chemical bonds, the linear stability condition hQ < 2 is often quoted as the stepsize limiting condition where Q is assumed to the frequency of the fastest (highest energy) bond stretch, i2p, present in the molecular dynamics model. This is realistic because the bond stretches are the highest frequency components in general in these models, and the potential energy function used for the bond stretch is typically quadratic in the separation distance ... 

Fig. 7.7. A ball oscillating in a potential energy well (scheme), (a) and (b) show the normal vibrations (normal modes) about a point /fo = being a minimum of the potential energy function V(/ o + ) of two variables = (xj, X2). This function is first approximated by a quadratic function i.e., a paraboloid V X, X2)- Computing the normal modes is equivalent to such a rotation of the Cartesian coordinate system (a), that the new axes (b) xj and x become the principal axes of any section of V by a plane V = const (i.e., ellipses). Then, we have V(xi,X2) = V Rq = 0) + j/ti (xj) + k2 The problem then becomes equivalent to the two-dimensional harmonic oscillator (cf.,...

The MM method will be seen to be the most useful one for obtaining force constants that can be used for different conformations of the molecule. In this approach the force constants are obtained from the second derivatives of an assumed potential energy function consisting of quadratic bonded tenns and non-quadratic non-bonded terms. Although present MM functions are too crude to be spectroscopically reliable, a new method for deriving such functions holds the promise of providing a reliable vibrational force field for the polypeptide chain. 

The energy level expression of Eq. (22.2-29) is only a first approximation. The power series expression for the vibrational potential energy function V was truncated at the quadratic term and the internuclear distance was replaced by its equilibrium value. One additional term of the power series can be kept or an alternate representation of the... 

An assumed potential energy function must both force a collection of atoms to assume a realistic polymer backbone structure, and create reasonable conformational dynamics. We have performed simulations wherein the bonds are kept at a distance of about 1.53A by a potential quadratic in the separation between successive bonds. Similarly, a quadratic potential keeps the bond angles near the tetrahedral value. Each bond experiences a rotational potential as shown in Figure 6. Several other features of the true potential are omitted as not contributing qualitatively to the mechanics. Substituent groups are considered as collapsed onto the carbon centers of the backbone. Excluded volume forces between remote carbons are omitted, as are hydrodynamic interactions. 

Like the equilibrium structure, the average structure has a well-defined physical meaning. The vibrational effects contained in the moments of inertia may be divided into harmonic e (har) and anharmonic e (anhar) contributions, which depend, respectively, on the quadratic and cubic part of the potential energy function. To evaluate the average stmctures, the moments of inertia for the average configuration, denoted by / (o = a, b, c), are required. These may be obtained from the effective moments of inertia by correcting for the e (har) effects ... 

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