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V
(
r
N
)
=
∑
bonds
k
b
(
l
−
l
0
)
2
+
∑
angles
k
a
(
θ
−
θ
0
)
2
{\displaystyle V(r^{N})=\sum _{\mbox{bonds}}k_{b}(l-l_{0})^{2}+\sum _{\mbox{angles}}k_{a}(\theta -\theta _{0})^{2}}
+
∑
dihedrals
k
ϕ
[
1
+
cos
(
n
ϕ
−
δ
)
]
{\displaystyle +\sum _{\mbox{dihedrals}}k_{\phi }[1+\cos(n\phi -\delta )]}
+
∑
j
=
1
N
−
1
∑
i
=
j
+
1
N
{
ϵ
i
,
j
[
(
r
0
i
j
r
i
j
)
12
−
2
(
r
0
i
j
r
i
j
)
6
]
+
q
i
q
j
4
π
ϵ
0
r
i
j
}
{\displaystyle +\sum _{j=1}^{N-1}\sum _{i=j+1}^{N}\left\{\epsilon _{i,j}\left[\left({\frac {r_{0ij}}{r_{ij}}}\right)^{12}-2\left({\frac {r_{0ij}}{r_{ij}}}\right)^{6}\right]+{\frac {q_{i}q_{j}}{4\pi \epsilon _{0}r_{ij}}}\right\}}
The functional form of the AMBER force field is[ 1]
V
(
r
N
)
=
∑
bonds
k
b
(
l
−
l
0
)
2
+
∑
angles
k
a
(
θ
−
θ
0
)
2
{\displaystyle V(r^{N})=\sum _{\mbox{bonds}}k_{b}(l-l_{0})^{2}+\sum _{\mbox{angles}}k_{a}(\theta -\theta _{0})^{2}}
+
∑
torsions
1
2
V
n
[
1
+
cos
(
n
ω
−
γ
)
]
{\displaystyle +\sum _{\mbox{torsions}}{\frac {1}{2}}V_{n}[1+\cos(n\omega -\gamma )]}
+
∑
j
=
1
N
−
1
∑
i
=
j
+
1
N
{
ϵ
i
,
j
[
(
r
0
i
j
r
i
j
)
12
−
2
(
r
0
i
j
r
i
j
)
6
]
+
q
i
q
j
4
π
ϵ
0
r
i
j
}
{\displaystyle +\sum _{j=1}^{N-1}\sum _{i=j+1}^{N}\left\{\epsilon _{i,j}\left[\left({\frac {r_{0ij}}{r_{ij}}}\right)^{12}-2\left({\frac {r_{0ij}}{r_{ij}}}\right)^{6}\right]+{\frac {q_{i}q_{j}}{4\pi \epsilon _{0}r_{ij}}}\right\}}
Note that despite the term force field, this equation defines the potential energy of the system; the force is the derivative of this potential with respect to position.
The OPLS (Optimized Potentials for Liquid Simulations) force field was developed by Prof. William L. Jorgensen at Purdue University and later at Yale University .
The functional form of the OPLS force field is very similar to that of AMBER :
E
(
r
N
)
=
E
b
o
n
d
s
+
E
a
n
g
l
e
s
+
E
d
i
h
e
d
r
a
l
s
+
E
n
o
n
b
o
n
d
e
d
{\displaystyle E\left(r^{N}\right)=E_{\mathrm {bonds} }+E_{\mathrm {angles} }+E_{\mathrm {dihedrals} }+E_{\mathrm {nonbonded} }}
E
b
o
n
d
s
=
∑
b
o
n
d
s
K
r
(
r
−
r
0
)
2
{\displaystyle E_{\mathrm {bonds} }=\sum _{\mathrm {bonds} }K_{r}(r-r_{0})^{2}\,}
E
a
n
g
l
e
s
=
∑
a
n
g
l
e
s
k
θ
(
θ
−
θ
0
)
2
{\displaystyle E_{\mathrm {angles} }=\sum _{\mathrm {angles} }k_{\theta }(\theta -\theta _{0})^{2}\,}
E
d
i
h
e
d
r
a
l
s
=
V
1
2
[
1
+
cos
(
ϕ
−
ϕ
0
)
]
+
V
2
2
[
1
−
cos
2
(
ϕ
−
ϕ
0
)
]
+
V
3
2
[
1
+
cos
3
(
ϕ
−
ϕ
0
)
]
+
V
4
2
[
1
−
cos
4
(
ϕ
−
ϕ
0
)
]
{\displaystyle E_{\mathrm {dihedrals} }={\frac {V_{1}}{2}}\left[1+\cos(\phi -\phi _{0})\right]+{\frac {V_{2}}{2}}\left[1-\cos 2(\phi -\phi _{0})\right]+{\frac {V_{3}}{2}}\left[1+\cos 3(\phi -\phi _{0})\right]+{\frac {V_{4}}{2}}\left[1-\cos 4(\phi -\phi _{0})\right]}
E
n
o
n
b
o
n
d
e
d
=
∑
i
>
j
f
i
j
(
A
i
j
r
i
j
12
−
C
i
j
r
i
j
6
+
q
i
q
j
e
2
4
π
ϵ
0
r
i
j
)
{\displaystyle E_{\mathrm {nonbonded} }=\sum _{i>j}f_{ij}\left({\frac {A_{ij}}{r_{ij}^{12}}}-{\frac {C_{ij}}{r_{ij}^{6}}}+{\frac {q_{i}q_{j}e^{2}}{4\pi \epsilon _{0}r_{ij}}}\right)}
with the combining rules
A
i
j
=
A
i
i
A
j
j
{\displaystyle A_{ij}={\sqrt {A_{ii}A_{jj}}}}
C
i
j
=
C
i
i
C
j
j
{\displaystyle C_{ij}={\sqrt {C_{ii}C_{jj}}}}
^ Cornell WD, Cieplak P, Bayly CI, Gould IR, Merz KM Jr, Ferguson DM, Spellmeyer DC, Fox T, Caldwell JW, Kollman PA (1995). "A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules". J. Am. Chem. Soc . 117 : 5179–5197. doi :10.1021/ja00124a002 . {{cite journal }}: CS1 maint: multiple names: authors list (link )
![{\displaystyle +\sum _{\mbox{dihedrals}}k_{\phi }[1+\cos(n\phi -\delta )]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73b4889d2f15e4166d292fb20de651eeb81b5e0f)
![{\displaystyle +\sum _{j=1}^{N-1}\sum _{i=j+1}^{N}\left\{\epsilon _{i,j}\left[\left({\frac {r_{0ij}}{r_{ij}}}\right)^{12}-2\left({\frac {r_{0ij}}{r_{ij}}}\right)^{6}\right]+{\frac {q_{i}q_{j}}{4\pi \epsilon _{0}r_{ij}}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2da2c7e5fc96f4c095a9c78694ab9f77782fbf15)
![{\displaystyle +\sum _{\mbox{torsions}}{\frac {1}{2}}V_{n}[1+\cos(n\omega -\gamma )]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e165d6edb76b2b3a07a851128e42eefc257aaf23)
![{\displaystyle E_{\mathrm {dihedrals} }={\frac {V_{1}}{2}}\left[1+\cos(\phi -\phi _{0})\right]+{\frac {V_{2}}{2}}\left[1-\cos 2(\phi -\phi _{0})\right]+{\frac {V_{3}}{2}}\left[1+\cos 3(\phi -\phi _{0})\right]+{\frac {V_{4}}{2}}\left[1-\cos 4(\phi -\phi _{0})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a00afbfcddbab6d802bbff2f7db61d1af129abbf)
