Biomedical Engineering Reference
In-Depth Information
the
y
and
z
axes. Finally we rotate by ψ about the new
z
axis. The relationship between a
position vector before and after the three rotations is
⎛
⎝
⎞
⎠
R
cos φ cos ψ
−
sin φ cos θ sin ψ
sin φ cos ψ
+
cos φ cos θ sin ψ
sin θ sin ψ
R
=
−
cos φ sin ψ
−
sin φ cos θ cos ψ
−
sin φ sin ψ
+
cos φ cos θ cos ψ
sin θ cos ψ
sin φ sin θ
−
cos φ sin θ
cos θ
(9.7)
There are two technical problems with the use of Euler angles. First, sampling the angles
at random does not give a uniform distribution; it is necessary to sample ψ , cos θ and ψ .
Second, there are a total of six trigonometric function evaluations per rotation.
A preferred alternative makes use of
quaternions
, which are four-dimensional unit
vectors. A quaternion
q
is usually written in terms of the scalar quantities
q
0
,
q
1
,
q
2
,
q
3
as
=
q
0
q
3
T
q
q
1
q
2
and the components satisfy
q
0
+
q
1
+
q
2
+
q
4
=
1
They can be related to the Euler angles as
q
0
=
cos
2
θ cos
2
(φ
+
ψ)
q
1
=
sin
2
θ cos
2
(φ
−
ψ)
(9.8)
q
2
=
sin
2
θ sin
2
(φ
−
ψ)
q
3
=
cos
2
θ sin
2
(φ
+
ψ)
The rotation matrix can be written as
⎛
⎞
q
0
+
q
1
−
q
2
−
q
3
2 (
q
1
q
2
+
q
0
q
3
)
2 (
q
1
q
3
−
q
0
q
2
)
⎝
⎠
R
R
=
2 (
q
1
q
2
−
q
0
q
3
)
q
0
−
q
1
+
q
2
−
q
3
2 (
q
2
q
3
+
q
0
q
1
)
(9.9)
2 (
q
1
q
3
+
q
0
q
2
)
2 (
q
2
q
3
−
q
0
q
1
)
q
0
−
q
1
−
q
2
+
q
3
and all that is necessary is to generate four suitable random numbers.
9.7 Flexible Molecules
Monte Carlo simulations of flexible molecules are difficult to perform unless the system
is small or some of the internal degrees of freedom are kept fixed. The simplest way to
generate a new configuration is to perform random changes to the Cartesian coordinates of
individual atoms in the molecule but it is a common experience that very small changes
are needed in order to produce an acceptable Boltzmann factor (in the MMC sense).
References
Metropolis, N. and Ulam, S. (1949)
J. Am. Stat. Assoc.
,
44
, 335.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N.
et al.
(1953)
J. Chem. Phys.
,
21
, 1087.
