Table 5.1 Some of the expressions commonly used to calculate the energy terms given in equation
(5.1)
Term Expression Model for
EBend EBend¼^1
2 ku(uu^0 )
(^2) Bond bending
whereu 0 is the ideal bond angle, that
is, the minimum energy positions of
the three atoms.
Atomic
movement
Atomic
movement
θ
ETorsion ETorsion¼^12 kf(1þcos (m(fþfoffset)) Rotation about a single bond
wherekfis the energy barrier to
rotation about the torsion anglef,m
is the periodicity of the rotation and
foffsetis the ideal torsion angle
relative (minimum energy positions
for the atoms) to a staggered
arrangement of the two atoms.
Torsion
φ
EvdW EvdW¼e[(rminr)
12
2(rminr )^6 ] Van der Waals non-bonded interactions
whererminis the distance between two
atomsiandjwhen the energy is at a
minimumeandris the actual distance
between the atoms. This equation is
known as the Lennard-Jones 6–12
potential. The ( )^6 term in this
equation represents attractive forces,
whilst the ( )^12 term represents short
range repulsive forces between the
atoms.
0
Potential
energy
r
r
min
ε
The separation distance
between atoms i and j
Repulsion
betweeni and j
Attraction between
atoms i and j
ECoulombic ECoulombic¼
qiqj
Drij Electrostatic coulombic interactions
whereqiandqjare the point charges
on atomsiandj,rijis the distance
between the charges andDis the
dielectric constant of the medium
surrounding the charges.
rij
Attractive
or repulsive
interactions
i j
model on which the energy term is based (Table 5.1). These expressions may be
the equations given in Table 5.1 but, depending on the nature of the system beng
modelled, other equations may be a more appropriate way of mathematically
describing the mechanical or electrical model.
The values of the parametersr,r 0 ,k,... etc used in the expressions for the
energy terms in Equation (5.1) are either obtained/calculated from experimental
observations or calculated using quantum mechanics using best fit methods.
Experimental calculations are based on a wide variety of spectroscopic tech-
niques, thermodynamic data measurements and crystal structure measurements
MOLECULAR MECHANICS 101