Biomedical Engineering Reference
In-Depth Information
where
k
is the force constant, and the subscript 'e' refers to the equilibrium value where
the molecule is at rest. A variation on the theme is given by
2 sin
2
θ
ABC,e
cos θ
ABC
−
cos θ
ABC,e
2
k
ABC
U
ABC
=
(4.13)
4.4.3 Dihedral Motions
Next we must consider the dihedral angle ABCD between the four bonded atoms A, B, C
and D (Figure 4.3).
A
ABCD
C
B
D
Figure 4.3
Dihedral angle
Some authors divide these into
proper
dihedrals, where we might expect full rotation
about the connecting bond B-C, and
improper
dihedrals where the rotation is limited. For
example, if C-D were a C-H fragment of a methyl group we would be expect full rotation
about B-C and a three-fold symmetry in the potential energy term. A -CH-CH- linkage in
a benzene ring would only show a moderate flexing from its planar value (angle zero).
If we use χ to denote the ABCD angle, then a popular dihedral potential is given by
U
0
2
U
=
(1
−
cos (
n
(χ
−
χ
e
)))
(4.14)
Here
n
is the periodicity parameter, which would be 3 for a methyl group. Angle χ
e
is the
equilibrium torsional angle. Amore complicated example is given by
V
1
N
d
V
2
N
d
V
3
N
d
U
=
(1
+
cos (
n
1
χ
−
g
1
))
+
(1
+
cos (
n
2
χ
−
g
2
))
+
(1
+
cos (
n
3
χ
−
g
3
))
The
V
's are energy terms, the
n
's are periodicity parameters, the
g
's are phase parameters
and
N
d
is a constant that depends on the number of bonds.
Some authors treat improper dihedrals in the same way as bond bending, and take a
contribution to the molecular potential energy as
2
k
ABCD
χ
ABCD
−
χ
e,ABCD
2
1
U
ABCD
=
(4.15)
where χ is the dihedral angle, as above.
