Thermodynamic Properties for Applications in Chemical Industry via Classical Force Fields - Multiscale Molecular Methods in Applied Chemistry - page 212

X

u RB
torsion

c n cos n ð'Þ¼ ð'Þ; (24)

n

where c n are the dihedral force constants of order n . An equivalent torsional
potential is based on the Fourier cosine series expansion:

X

1
2 V n ð

u torsion ð'Þ¼

1 þ cos ð n ' d n ÞÞ; (25)

n

where ' is the dihedral angle as shown in Fig. 2 . V n are the torsional rotation force
constants, d n the phase factors, and n the multiplicity or number of function minima

upon a rotation of 2 . The specified number of terms in the series expansion varies

for different force fields. Common choices are the first three terms of the expansion
and terms with selected multiplicity from one to six [ 53 ].

p

2.2.4 Improper Torsion

A special type of torsional potential is employed to enforce geometrical constraints
like planarity, e.g., in aromatic rings, or to prevent transitions between chiral
structures. This potential is usually referred to as improper torsion or out-of-plane
bending. Improper torsion acts between four atoms in a branched structure. There
are several approaches to describe this potential. E.g., to maintain the improper
dihedral at 0 or p , the torsional potential of the form

u 2 p
improper

ð'Þ¼ V n ð 1 cos ð 2 'ÞÞ (26)

can be used. Another route to incorporate the out-of-plane bending motion is to
define an angle

between a bond from the central atom and the plane defined by the
central atom and the other two atoms; cf. Fig. 2 . With this definition, a harmonic
potential can be constructed:

c

1
2 k c ðc c 0 Þ

2

u har

improper ðcÞ¼

; (27)

where c 0 its equilibrium value. k c is a constant that

determines the stiffness of the potential.

c is the improper angle and

2.2.5 Valence Coordinate Cross Terms

Some force fields include cross terms to account for the coupling between different
intramolecular interactions. E.g., it has been found that, upon decrease of a bond