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Chapter 2
Molecular Mechanics
Basic Concepts
 Molecule: Collection of atoms held together by
forces
 Molecular Mechanics:
– Potential Energy is described as a function of a
coordinate X
– Matching a function to a set of data points by
varying its parameters
Series Expansion
 The Taylor series is a representation of a
function as an infinite sum of terms calculated
from the values of its derivatives at a single
point.

f ( n ) a 
n
x  a 
f x   
n!
n 0
 The Fourier series is a mathematical tool used
for analyzing periodic functions by decomposing
such a function into a weighted sum of much
simpler sinusoidal component functions

1
f x   a0   an cosn x   bn sin n x 
2
n 1
 Electrostatic Potential
V (r )  
qi q j
40 rij
i, j are charged particles (electrons & protons)
i, j
Simple Molecular Mechanics Force Field
ki
V (r )   (li  li ,eq ) 2
bonds 2
ki
2
  ( i   i ,eq )
angles 2
Bond-stretching
Angle-bending
Vn
 
(1  cos( n   )) Torsions
torsions 2

  12    6  q q 

ij
ij
i j 






    4 ij







r
r
4

r



i 1 j i 1
ij 
ij 
0 ij






N
N
Bonding Stretching
 Taylor Expansion
dV
1 d 2V
r  req  2
V (r )  V (req ) 
dr r req
2! dr
1 d 3V
3



r

r

eq
3
3! dr r r
eq
 Hook’s Law (Harmonic Potential)
1
2
V (rAB )  k AB rAB  rAB,eq 
2
 Morse Function
 AB rAB  rAB ,eq  2
V (rAB )  DAB [1  e
]
r  r 
2
eq
r  req
Taylor
Harmonic
Morse
req
req
r-req
A
B
r
B
Valence Angle Bending
 Typical force field function for angle strain energy
V ( ABC ) 


1
( 3)
( 4)
 ABC   ABC,eq   k ABC
 ABC   ABC,eq 2  
k ABC  k ABC
2
2
  ABC   ABC,eq 
C

C
A
B
Torsions
 Typical force field as an expansion of a Fourier
series
1
j 1
V ( ABCD )   V j , ABCD [1   1 cos j ABCD  j , ABCD 
2  jABCD
D
B
A
C
H CH3 
Vtorsions
CH3
H
H
H
j =1
j =3
j =2
j =4
Van der Waals Interactions
 In the absence of a permanent charge,
  12   6 
V (r )  4 AB  AB    AB  
 rAB   rAB  