Introduction to Force Fields

Chemistry 380.37
May 2013
Dr. Jean M. Standard
May 14, 2013
Molecular Mechanics – Introduction to Force Fields
Molecular mechanics is a method in which an atom is treated as a single unit consisting of the electrons and the
nucleus; that is, the electrons are not treated explicitly. A bond between two atoms is described as two masses
joined by a spring. In molecular mechanics, the energy of a molecule is determined from an empirical function
called a force field that depends upon the coordinates of the atoms that comprise the molecule.
The Molecular Mechanics Force Field
A typical molecular mechanics force field consists of terms that describe bond stretching, angle bending, torsional
motion, nonbonded interactions, and electrostatic interactions,
U =
∑ U s + U b
bonds angles
∑ + ∑ U t + ∑ U nb + ∑ U el . (1)
torsions
nonbond
interactions
charge pairs
Evaluation of the force field for a particular molecular geometry provides the steric or strain energy, U. The force
field energy is sometimes denoted by V to represent that it acts as the potential energy in classical mechanics
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simulations.
The Stretching Energy
The stretching energy U s describes the energy of a bond between two connected atoms. A typical functional form
for the stretching energy involves a harmonic function,
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U s = 1 2 k s,AB ( r AB − r AB,eq ) 2 , (2)
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where r AB is the instantaneous bond length between atoms A and B, r AB,eq is the equilibrium (or strain-free) bond
length, and k s,AB is the stretching force
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constant for the A-B bond. The magnitude of the force constant depends
upon the type of atoms involved in the bond. A plot of the stretching energy U s for a typical bond is shown in
Figure 1.
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€
600
€
500
Us (kJ/mol)
400
300
200
100
0
-0.5 -0.3 -0.1 0.1 0.3 0.5
r–r eq (Å)
Figure 1. Typical behavior of the stretching energy for a bond.

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In some force fields (such as MMFF94), cubic and/or quartic corrections may be included to account for the
anharmonicity of the stretching potential. In such cases, the stretching energy takes a more complicated form,
U s = 1 2 k s,AB r AB − r AB,eq
Here, the cubic stretching force constant is
( ) 2 + 1 2 k (3)
s,AB ( r AB − r AB,eq ) 3 + 1 2 k (4 )
s,AB ( r AB − r AB,eq ) 4 . (3)
(3)
k s,AB
and the quartic stretching force constant is
(4)
k s,AB .
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The Bending Energy
The bending energy U b describes the energy corresponding to variations of the angle between two bonds. A typical
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function for the bending energy is
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U b = k b,ABC ( θ ABC −θ ABC,eq ) 2 , (4)
where θ is the instantaneous bond angle, θ ABC,eq is the equilibrium bond angle, and k b,ABC is the bending force
constant for angle A-B-C. A plot of
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the bending energy for a typical bond angle is shown in Figure 2.
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2.5
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2.0
Ub (kJ/mol)
1.5
1.0
0.5
0.0
-10.0 -5.0 0.0 5.0 10.0
θ−θ eq (deg)
Figure 2. Typical behavior of the bending energy for a bond angle.
In some force fields (including MMFF94), cubic and/or quartic corrections are added to the bending potential to
improve the accuracy and transferability of parameters.
The Torsional Energy
The torsional energy U t is often described by the function
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V t = V 1
2
( 1+ cosω ) + V 2
2
( 1+ cos 2ω ) + V 3
2
( 1+ cos 3ω ) + … , (5)
where ω is the torsional angle, and V 1 , V 2 , and V 3 are empirical constants. A typical plot for the torsional energy
of ethane is shown in Figure 3.
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3
1.4
1.2
1.0
Ut (kJ/mol)
0.8
0.6
0.4
0.2
0.0
0 60 120 180 240 300 360
ω (degrees)
Figure 3. Typical behavior of the torsional energy for a rotation about the C-C bond in ethane.
In this case, minima occur when the hydrogens on the adjacent carbons of ethane are staggered, and maxima occur
when the hydrogens are eclipsed.
The Non-bonded Interaction Energy
Another important term in a general molecular mechanics force field is the nonbonded interaction energy, or van der
Waals interaction energy. The non-bonded interaction energy U nb,AB between two nonbonded atoms A and B is
usually described by a Lennard-Jones 6-12 potential,
U nb,AB € = − a AB
+ b AB
6 12 , (6)
where a AB and b AB are empirical parameters for atoms A and B and r AB is the distance between atoms A and B. A
typical plot of the non-bonded interaction
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energy between two atoms is shown in Figure 4.
r AB
r AB
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€
25
€
20
15
Unb (kJ/mol)
10
5
0
-5
-10
1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
r AB (Å)
Figure 4. Typical behavior of the non-bonded interaction energy.

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The Electrostatic Energy
The Coulomb interaction potential between two charged particles is
U el =
q A q B
4πε 0 r AB
, (7)
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where q A and q B are the charges on particles A and B and r AB is the distance between atoms A and B. The
attractive interaction between opposite charges in shown in Figure 5(a) and the repulsive interaction between like
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charges is shown in Figure 5(b).
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Electrostatic Energy
Electrostatic Energy
0 2 4 6 8 10
r AB (Å)
0 2 4 6 8 10
r AB (Å)
(a) (b)
Figure 5. Typical behavior of the electrostatic energy for (a) opposite charges and (b) like charges.
Determination of Force Field Parameters
The parameters that make up the force field generally are derived from experimental data. This can be done in many
different ways; thus, there are many different force fields available in the literature. Some have been constructed to
deal with small organic molecules, others have been generated to focus especially on biomolecules, and others are of
general utility.
Sample Force Field Parameters
Molecular mechanics (MM) calculations rely on force fields constructed from empirical data. The force field
parameters are obtained from experimental or quantum mechanical results. Shown in Tables 1 and 2 are some
example bond stretching and angle bending force field parameters. These parameters are from the MM3 force field.
Table 1. Example bond stretching parameters, MM3 force field
Bond type
r AB,eq
k s,AB
(Å)
(kJ mol –1 Å –2 )
Csp 3 –Csp 3 €
1.523
€
1330
Csp 3 –Csp 2 1.497 1330
Csp 2 =Csp 2 1.337 2890
Csp 2 =O 1.208 3250
Csp 3 –Nsp 3 1.438 1540
Note that the stretching force constants for double bonds are in general larger than the stretching force constants for
single bonds.

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Table 2. Example angle bending parameters, MM3 force field
Angle type
θ ABC,eq
k b,ABC
(deg.)
(kJ mol –1 deg –2 )
Csp 3 –Csp 3 –Csp 3 €
109.47
€
0.041
Csp 3 –Csp 3 –H 109.47 0.033
H–Csp 3 –H 109.47 0.029
Csp 3 –Csp 2 –Csp 3 117.2 0.041
Csp 3 –Csp 2 =Csp 2 121.4 0.051
Csp 3 –Csp 2 =O 122.5 0.042
Here, one of the most important things to note is that the angle bending force constants are much smaller than the
bond stretching force constants. Also, note that even for a tetrahedral center (such as at an sp 3 -hybridized carbon),
the angle bending force constants depend upon the end atoms. Because of the smaller H atoms one the ends, an H-
C-H force constant will be smaller than a C-C-C force constant.