Gonnet, Pedro

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Mathematical Institute Eprints Archive

A Short Note on the Fast Evaluation of Dihedral Angle
Potentials and their Derivatives
Pedro Gonnet
September 26, 2011
Dihedral angle potentials, which are used in many force fields for molecular
dynamics simulations, model the energy of twisting a bond as a function of the
angle φijkl between the surfaces spanned by the particles pi, pj and pk, and pj ,
pk and pl respectively.
The potential function may have several forms. In the amber [4], cff [6],
charmm [7], dreiding [8], gromos [9] and shapes [1] force fields, the potential
is defined as
v1(φ) = K [1 + cos(nφ− δ)] , (1)
where K is the energy of the potential, n its multiplicity and δ its phase shift.
The multiplicity n is an integer and δ is usually an integer multiple of pi.
In the Unified Force Field (uff) [11], the potential function is generalized as
v2(φ) = K
n∑
k=0
ck cos(kφ), (2)
or, in Desmond [2], as
v3(φ) = K
n∑
k=0
ck cos(kφ− φ0), (3)
where φ0 is the equilibrium dihedral angle.
The first (1) and third (3) forms can be converted to the second form (2)
when δ and φ0 are integer multiples of pi. For all other values of δ and φ0, all
three forms generalize to the form
v4(φ) = K
n∑
k=0
[ak cos(kφ) + bk sin(kφ)] , (4)
where the coefficients ak and bk can be obtained by applying the equivalence
cos(α± β) = cos(α) cos(β)∓ sin(α) sin(β). (5)
In practice, the angle φ is obtained by computing its sine and/or cosine
from the vectors pi − pj , pk − pj and pl − pk, and computing its inverse sine,
cosine or tangent. In namd [10] and gromacs [5], φ is computed using the
1
inverse tangent and then inserted into (1). In Desmond, the sine and cosine are
computed using vector calculus and (3) is evaluated by repeated application of
the equivalence (5), thus requiring no evaluations of any trigonometric functions
or their inverses.
The entire computation can, however, be drastically simplified. Assuming
that we have computed x = cos(φ) using vector calculus, we can then re-write
(1)-(4) in terms of x:
v1(x) = K
[
1 + Tn(x) cos(δ) + Un−1(x)(1− x2)1/2 sin(δ)
]
,
v2(x) = K
n∑
k=0
ckTk(x),
v3(x) = K
n∑
k=0
ck
[
Tk(x) cos(φ0) + Uk−1(x)(1− x2)1/2 sin(φ0)
]
,
v4(x) = K
n∑
k=0
[
akTk(x) + bkUk−1(x)(1− x2)1/2
]
,
where Tk(x) and Uk(x) are the Chebyshev polynomials of the first and second
kind which satisfy
Tk(x) = cos
(
k cos−1(x)
)
, Uk−1(x) =
sin
(
k cos−1(x)
)
(1− x2)1/2
.
Using the derivatives of the Chebyshev polynomials
d
dx
Tk(x) = kUk−1(x) and
d
dx
Uk−1(x) =
kTk(x)− xUk−1(x)
x2 − 1 ,
we can write the corresponding derivatives of the potentials with respect to x
as
d
dx
v1(x) = K
[
1 + nUn−1(x) cos(δ) + (2xUn−1(x)− nTn(x)) (1− x2)−1/2 sin(δ)
]
,
d
dx
v2(x) = K
n∑
k=0
ckkUk−1(x),
d
dx
v3(x) = K
n∑
k=0
ck
[
kUk−1(x) cos(φ0) + (2xUk−1(x)− kTk(x)) (1− x2)−1/2 sin(φ0)
]
,
d
dx
v4(x) = K
n∑
k=0
[
akkUk−1(x) + bk (2xUk−1(x)− kTk(x)) (1− x2)−1/2
]
.
Note that when δ and φ0 are integer multiples of pi, the terms in sin(δ) and
sin(φ) can be dropped.
2
The advantage of this formulation is that the Chebyshev polynomials can be
evaluated using the three term recurrence relations
T0(x) = 1 T1(x) = x, Tk(x) = 2xTk−1(x)− Tk−2(x),
U0(x) = 1 U1(x) = 2x, Uk(x) = 2xUk−1(x)− Uk−2(x)
i.e. in 2n − 2 additions and 2n − 1 multiplications to obtain all Tk(x), Uk(x)
for k = 0 . . . n. Alternatively, the polynomails Uk(x) for odd/even k can be
computed as twice the sum of Tj(x), for all odd/even 0 ≤ j ≤ k. Furthermore,
v2(x) and
d
dx
v2(x) (as well as v3(x) and
d
dx
v3(x) when φ0 is an integer multiple
of pi) can be evaluated concurrently using the Clenshaw algorithm [3].
Although the relation of such potentials to polynomials in cos(φ) is not
entirely new [12], the connection to Chebyshev polynomials provides an efficient
and general recipe for their evaluation.
1 Acknowledgments
The author would like to thank John Pearson, Mark Richardson and Asgeir
Birkisson for the many fruitful discussions on this and other topics. This work
was supported by the Swiss National Science Foundation Individual Support
Fellowships PBEZP2-127959 and PA00P2 134146.
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A Short Note on the Fast Evaluation of Dihedral Angle\ud Potentials and their Derivatives

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