Potential function torsion

In molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. 

Torsional barriers are referred to as n-fold barriers, where the torsional potential function repeats every 2n/n radians. As in the case of inversion vibrations (Section 6.2.5.4a) quantum mechanical tunnelling through an n-fold torsional barrier may occur, splitting a vibrational level into n components. The splitting into two components near the top of a twofold barrier is shown in Figure 6.45. When the barrier is surmounted free internal rotation takes place, the energy levels then resembling those for rotation rather than vibration. 

The most useful general form of the torsional potential function V(4>) is... 

Figure 6.45 Torsional potential function, V(4>), showing a two-fold barrier...

In the chapter on vibrational spectroscopy (Chapter 6) 1 have expanded the discussions of inversion, ring-puckering and torsional vibrations, including some model potential functions. These types of vibration are very important in the determination of molecular structure. 

Chapter 12. The forces which act on the atoms to maintain them in certain equilibrium positions are associated with changes in bond lengths and angles, and, furthermore in the present application, torsional angles around specific chemical bonds. Once a potential function has been established the so-called steric energy of the molecule can, in principle, be evaluated. 

This fact allows the effective relaxation of steric repulsion. The potential barrier for the motion around the C—C single bonds is smaller than that corresponding to the motion around the central C=C bond. Using the potential functions computed for these motions, and assuming a Boltzmann distribution, average torsional angles of 7.7 and 7.1, at 300 K, are obtained for rotations around Cl—C3 and C1=C2, respectively. This torsional motion seems to be due to the nonplanar structure observed experimentally. 

The first three terms, stretch, bend and torsion, are common to most force fields although their explicit form may vary. The nonbonded terms may be further divided into contributions from Van der Waals (VdW), electrostatic and hydrogen-bond interactions. Most force fields include potential functions for the first two interaction types (Lennard-Jones type or Buckingham type functions for VdW interactions and charge-charge or dipole-dipole terms for the electrostatic interactions). Explicit hydrogen-bond functions are less common and such interactions are often modeled by the VdW expression with special parameters for the atoms which participate in the hydrogen bond (see below). 

Rigid-geometry ab initio MO calculations of 86 torsional isomers of the dimethylphosphate anion (CH30)2P02 led to the determination of parameters for the Lennard-Jones type of nonbonded interaction, two- and three-fold torsional, and electrostatic interaction potential functions (215). Extension of this approach to full relaxation ab initio and MM schemes will be extremely useful, not only for phosphorus but also for other heteroatoms. 

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... 

The potential function for internal rotation of a group of identical atoms is usually assumed to have the form U(a)= U0( 1 — coswx), where n is the number of minima and U0 is the barrier height. For a torsional vibration, the vibration frequency is given by an expression analogous to... 

It is common practice to describe torsional rotations around single bonds and those around multiple bonds with the same type of potential function but with very different force constants. The function must be able to describe multiple minima. Generally, a Fourier expansion of the torsional angle with only cosine terms is used (Eq. 2.23),... 

The choice of series is not only dependent on the type of molecular motion. For example the power series may be convenient for an accurate description of the potential function close to the minima, while a Fourier series is convenient for describing potential barriers to torsional motion. 

In open chain molecules the torsional angles are the obvious independent variables for the potential function. But even in the simplest case, that is ethane, the origin of the torsional barrier is not fully understood, though ab initio calculations represent the experimental barrier fairly well26 For larger molecules theoretical predictions for the potential functions are often based upon semiempirical molecular mechanics calculations27,2S ... 

Open chain molecules have been widely studied using the electron-diffraction method and with considerable success. But quantitive barrier calculations meet with substantial difficulties. In cases with torsional barriers higher than, say 4 kJ/mol, the electron-diffraction method provides information mainly on the regions of the potential function near the minima. For lower barriers the method is usually not sufficiently sensitive to changes in the assumptions on V(0). If the barrier is, say 2 kJ/mol or less, the electron-diffraction results may in many cases be indistinguish-ably like free rotation. Accordingly in order to use the electron-diffraction method successfully for the study of torsional motion, support as to the choice of potential functions may favorably be obtained from other methods, as for example from microwave spectroscopy. 

The factors influencing the conformational stability in open chain molecules have previously been treated extensively in review articles (see for example Ref.6 and 107 ). The aim of the present section is to study torsional potential functions of a series of molecules of principal importance, in particular related to the results of electron-diffraction investigations. The bulk contents of information obtainable from an electron-diffraction intensity curve of a molecule carrying out torsional motion, are not concerned with the torsional motion at all. The part of the intensity curve giving information about the torsion, is distributed over the same range of the intensity curve as where the torsional independent information may be obtained. In the RD-curve the contribution from the torsional dependent part is more clearly separated. To illustrate this and the general influence of torsional motion, three simple molecules with three-fold torsional barriers have been selected (Figs. 3-5)... 

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... 

Torsional potential function. The torsional parameters were adjusted to give the best representation of the methyl and nitro group rotation in DMNA. 

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