The Torsional Energy

Etors describes part of the energy change associated with rotation around a B—C bond in a four-atom sequence A—B—C—D, where A—B, B—C and C—D are bonded. 

Looking down the B—C bond, the torsional angle is defined as the angle formed by 

The torsional energy is fundamentally different from Lstr and F bend in three aspects  

To encompass the periodicity, Eton is written as a Fourier series. 

For the ethylene molecule, the rotation around the C=C bond must be periodic by 180°, and thus only n = 2,4, etc., terms can enter. The energy cost for rotation around a double bond is of course much higher than that for rotation around a single bond in 

Systems with bulky substituents on sp -hybridized atoms are often found to have four minima, the anti conformation being split into two minima with torsional angles of approximately 170°. Other systems, notably polyfluoroalkanes, also split the gauche -minima info two, often ealled gnuchp. tangle of approximately +50°) and ortho (angle 

It is customary to shift the zero point of the potential by adding a factor of 1 to each term. Most rotational profiles resemble either the ethane or ethylene examples above, and a popular expression for the torsional energy is 

The and — signs are chosen so that the onefold rotational term has a minimum for an angle of 180°, the twofold rotational term minima for angles of 0° and 180°, and the threefold rotational term minima for angles of 60°, 180° and 300° (— 60°) as illustrated 

As mentioned in Section 2.2.3, the out-of-plane energy may also be described by an 

Molecules that are composed of atoms having a maximum valency of 4 (as essentially all organic molecules) are with a few exceptions found to have rotational profiles showing at most three minima. The first three terms in the Fourier series eq. (2.9) are sufficient for qualitatively reproducing such profiles. Force fields which are aimed at large systems often limit the Fourier series to only one term, depending on the bond type (e.g. single bonds only have cos (3u ) and double bonds only cos (2u))). 

For each combination of four atom types, A, B, C and D, there are generally three torsional parameters to be determined, yABCo 

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The value of the torsional energy increment has been variously estimated, but TORS = 0.42 kcal mol was settled on for the bond contribution method in MM3, In the full statistical method (see below), low-frequency torsional motion should be calculated along with all the others so the empirical TORS inererneut should be zero. In fact, TORS is not zero (Allinger, 1996). It appears that the TORS inererneut is a repository for an energy eiror or errors in the method that are as yet unknown. 

The multiple minima nature of the bending energy and the low barriers for interconversion resemble the torsional energy for organic molecules. An expansion of bend in tenns of cosine or sine functions of the angle is therefore more natural than a... 

Table 2.3 Comparison of functional forms used in common force fields. The torsional energy, [ors is in all cases given as a Fourier series in the torsional angle...

Table 2.3 gives a description of the functional form used in some of the common force fields. The torsional energy is written as a Fourier series, typically of order three, in all cases. Many of the force fields undergo developments, and the number of atom types increases as more and more systems become parameterized thus Table 2.3 may be considered as a snapshot of the situation when the data were collected. The universal type force fields, described in Section 2.3.3, are in principle capable of covering molecules composed of elements from the whole periodic table, these have been labelled as all elements . 

Assignment of these spectra involves solution of a highly redundant, interlocking puzzle. We seek assignments consistent with all available information. We can calculate the torsional energies and wavefunctions for various one-dimensional... 

The Effect of Pseudopotential on the Torsional Energy Levels of Hydrogen Peroxide and Deuterium Peroxide... 

The effect of pseudopotential on the torsional energy levels of hydrogen 401... 

Figure 2.3 Fourier decomposition of the torsional energy for rotation about the C-O bond of fluoromethanol (bold black curve, energetics approximate). The Fourier sum (A) is composed of the onefold (o), twofold (o), and threefold ( ) periodic terms, respectively. In the Newman projection of the molecule, the oxygen atom lies behind the carbon atom al center...

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