Symmetrical Potential Functions

Historically, the most common technique used has been the linear variation method. In this procedure, the wave functions are expressed as linear combinations of harmonic-oscillator basis functions 

In addition, the number of basis functions required to obtain a satisfactory representation depends on the choice of the harmonic frequency for the basis. Stated in an- 

One set of reduced coordinates used leads to a Schrodinger equation 

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The potential energy is often described in terms of an oscillating function like the one shown in Figure 10.9(a) where the minima correspond to the relative orientations in which the interactions are most favorable, and the maxima correspond to unfavorable orientations. In ethane, the minima would occur at the staggered conformation and the maxima at the eclipsed conformation. In symmetrical molecules like ethane, the potential function reflects the symmetry and has a number of equivalent maxima and minima. In less symmetric molecules, the function may be more complex and show a number of minima of various depths and maxima of various heights. For our purposes, we will consider only molecules with symmetric potential functions and designate the number of minima in a complete rotation as r. For molecules like ethane and H3C-CCI3, r = 3. 

It is convenient to use spherical polar coordinates (r, 0, ) for any spherically symmetric potential function v(r). The surface spherical harmonics V,1" satisfy Sturm-Liouville equations in the angular coordinates and are eigenfunctions of the orbital angular momentum operator such that... 

Since the fictitious particle moves in a central force field described by a spherically symmetric potential function U(r), its angular momentum is conserved. Therefore, the motion of the fictitious particle will be in a plane defined by the velocity and the radius vectors. The Lagrangian may then be conveniently expressed in polar coordinates as... 

The fit to the data for trimethylene imine (rms deviation 2.5 cm-1) is not as good as has generally been obtained for molecules with symmetric potential functions. In this molecule there is a second pathway by which its two forms (Fig. 4.19) may be interconverted, namely, via the N-H inversion vibration. This vibration has the same symmetry properties as the ring-puckering. Consequently, harmonic, cubic and quartic cross terms are allowed in the potential. Neglect of these terms is doubtless one reason for the deviations observed when the data are fitted one-dimension-ally. 

A slight isotopic dependence of the effective one-dimensional potential function is found for trimethylene imine-N-d. However, this effect is of the same order as that observed for molecules with symmetric potential functions, and in this case is within the uncertainty of the potential functions. The barrier measured from the lowest well was 443 cm-1 and the energy difference between wells 90 cm-1, compared to 441 cm-1 and 95 cm-1 for the parent compound. 

Table 4.18A. Four-membered ring molecules with symmetric potential functions...

Furthermore, we can extend the theorem to a collection of point particles interacting with each other in any desired way but influenced by external forces only through a spherically symmetric potential function. If we describe such a system by using the polar coordinates of each particle, the Lagrangian function is... 

Since site-site potentials (also called atom-atom potentials) have been widely used in computer simulation and theoretical studies of molecular liquids, we describe the application of MTS methods to a model potential of this type. MTS methods are completely general, and can be applied equally well to other types of potential functions. In site-site models the potential energy between a pair of molecules is the sum of the potential energies between pairs of sites on different molecules. These sites are usually atoms or groups of atoms, which interact with sites on other molecules through spherically symmetric potential functions. Thus the potential energy between molecules 1 and 2 takes the form... 

The Shifted Force Potential. When two sites interact through a spherically symmetric potential function, they exert upon each other equal and opposite forces that act along the vector f between them. (A wavy underline indicates a vector quantity.) The magnitude of the force is given by F=-(du/dr). 

The contribution of the electron to the diamagnetic susceptibility of the system can be calculated by the methods of quantum-mechanical perturbation theory, a second-order perturbation treatment being needed for the term in 3C and a first-order treatment for that in 3C". In case that the potential function in 3C° is cylindrical symmetrical about the s axis, the effect of 3C vanishes, and the contribution of the electron to the susceptibility (per mole) is given... 

Values of w0+1 are given in Table I. There are also included data for un-symmetric molecules such as HC1. For these a reasonable potential function... 

In cubic close-packing each molecule is surrounded by twelve others, whose interaction with the central molecule can be represented by a potential function of cubic point-group symmetry in case that the twelve molecules are spherically symmetrical or oriented at random. The energy change produced by this potential function,/say, is... 

Fig. 81.—Potential energy associated with bond rotation as a function of angle, (a) Symmetrical potential according to Eq. (23) (b) and (c) potential energy functions with lowest minimum at 0=0 corresponding to the planar zigzag form of a polymethylene chain. These curves were calculated by Taylor. 0...

T dtc) and the d Zn(rffc)2, indicate a relatively great stabiUty for electronic states with symmetrical orbital functions. It parallels the maxima in ionisation potentials of the elements with half and completely filled subshells. 

A much improved description of PE melts is obtained with a more elaborate potential energy function, based on a spherically symmetric potential energy... 

In the above treatment of the problem of the particle in a box, no consideration was given to its natural symmetry. As the potential function is symmetric with respect to the center of the box, it is intuitively obvious that this position should be chosen as the origin of the abscissa. In Fig. 4b, x =s 0 at the center of the box and the walls are symmetrically placed at x = 1/2. Clearly, the analysis must in this case lead to the same result as above, because the particle does not know what coordinate system has been chosen It is sufficient to replace x by x +1/2 in the solution given by Eq. (68). This operation is a simple translation of the abscissa, as explained in Section 1.2. The result is shown in Fig. 4b, where the wave function is now given by... 

Thus the hydrogen-bonding is apparently dependent, amongst other factors, upon the nature of the alkyl substituents. Perhaps the more highly branched secondary alkyl substituents increase the hydrophobic environment about the hydrogen bond, thereby favoring a symmetrical interaction. In any event, it is apparent that this interaction is best described by a very broad, shallow potential function. 

For bending vibrations, the representation of the potential function, V(.v), in terms of Morse potentials [Eq. (6.5)] is not appropriate. The coordinate 5 is now s = aQ (Figure 6.7). The potential must be symmetric under s - s. However, one can make use of another correspondence between potentials and algebras. Consider the Poschl-Teller (1933) potential of Figure 6.8,... 

Y, and Z are connected by bonds of fixed length joined at fixed valence angles, that atoms W, X, and Y are confined to fixed positions in the plane of the paper, and that torsional rotation 0 occurs about the X-Y bond which allows Z to move on the circular path depicted. If the rotation 0 is "free such that the potential energy is constant for all values of 0, then all points on the circular locus are equally probable, and the mean position of Z, i.e., the terminus of , lies at point z. The mean vector would terminate at z for any potential function symmetric in 0 for any potential function at all, except one that allows absolutely no rotational motion, the vector will terminate at a point that is not on the circle. Thus, the mean position of Z as seen from W is not any one of the positions that Z can actually adopt, and, while the magnitude ll may correspond to some separation that W and Z can in fact achieve, it is incorrect to attribute the separation to any real conformation of the entity W-X-Y-Z. Mean conformations tiiat would place Z at a position z relative to the fixed positions of W, X, and Y have been called "virtual" conformations.i9,20it is clear that such conformations can never be identified with any conformation that the molecule can actually adopt... 

Here a = X and y(a, z) = Jo dt e is the incomplete gamma function. It can be noted that for the axially symmetric potential with a longitudinal field, the only dependence on X is the trivial one in Xp, while in the nonaxially symmetric potential obtained with a transversal field the relaxation rate will strongly depend on X through F(oc), which is plotted in Figure 3.6. 

Assuming an axially symmetric potential, the anisotropy energy of n) will be an even function of the longitudinal component of the magnetic moment s n. The averages we need to calculate are aU products of the form = (n =i (cn ))a> where the c are arbitrary constant vectors. Introducing the polar and azimuthal angles of the spin d, tp), we can write as... 

When is expressed in spherical coordinates as t i(r, 0, cp), then reflection through the origin is accomplished by replacing 0 and cp by (it — 0) and (it + cp), respectively. (r cannot change sign as it is just a distance.) In other words, the parity of the wave function is determined only by its angular part. For spherically symmetric potentials, the value of l uniquely determines the parity as... 

One source of information on intermolecular potentials is gas phase virial coefficient and viscosity data. The usual procedure is to postulate some two-body potential involving 2 or 3 parameters and then to determine these parameters by fitting the experimental data. Unfortunately, this data for carbon monoxide and nitrogen can be adequately represented by spherically symmetric potentials such as the Lennard-Jones (6-12) potential.48 That is, this data is not very sensitive to the orientational-dependent forces between two carbon monoxide or nitrogen molecules. These forces actually exist, however, and are responsible for the behavior of the correlation functions and - In the gas phase, where orientational forces are relatively unimportant, these functions resemble those in Figure 6. On the other hand, in the liquid these functions behave quite differently and resemble those in Figures 7 and 8. 

Figure 5.2 Typical spherically symmetrical cavity potential function between guest and cell. (Reproduced from McKoy, V., Sinanoglu, O., J. Chem. Phys38, 2946 (1963). With permission from the American Institute of Physics.)...

A similar analysis of data obtained from molecules with asymmetric end groups is more complicated. Apart from the problems connected with the separability of the torsional motion from the framework vibration, experience shows that several more terms have to be included in the Fourier series to describe the torsional potentials properly. On the other hand, the electron-diffraction data from asymmetric molecules usually contain more information about the potential function than data from the higher symmetric cases. In conformity with the results obtained for symmetric ethanes the asymmetric substituted ethanes, as a rule, exist as mixtures of two or more conformers in the gas phase. Some physical data for asymmetric molecules are given in Table 4. The electron-diffraction conformational analysis gives rather accurate information about the positions of the minima in the potential curve. Moreover, the relative abundance of the coexisting conformers may also be derived. If the ratio between the concentrations of two conformers is equal to K, one may write... 

B. Spectral Function of a Dipole Reorienting in a Local Axi symmetric Potential... 

In this chapter, dielectric response of only isotropic medium is considered. However, in a local-order scale, such a medium is actually anisotropic. The anisotropy is characterized by a local axially symmetric potential. Spatial motion of a dipole in such a potential can be represented as a superposition of oscillations (librations) in a symmetry-axis plane and of a dipole s precession about this axis. In our theory this anisotropy is revealed as follows. The spectral function presents a linear combination of the transverse (K ) and the longitudinal (K ) spectral functions, which are found, respectively, for the parallel and the transverse orientations of the potential symmetry axis with... 

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