Molecular dynamics thesis

v)
2a
2
±
9(a
•
v)
2
4 a
( )
4
−
2
v
2
a
2
.
(29)
*
In the first configuration an arbitrary value is used. In the configurations after that, the ending value of
the previous configuration is used as initial trial time.
**
The square of 
∆
r  is used because calculating square roots is then avoided. Calculating square roots is
a time consuming operation for a CPU.
***
It is possible to determine the desired value of t exactly by solving eqn. (24) for 
∆
r  = R, but this
involves calculating a number of square roots.

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20
So ( r )
2
  is not a monotonous function if
9(a
•
v)
2
4 a
( )
4
−
2
v
2
a
2
>
0 ,
(30)
which finally leads to cos(v ,a ) < 
−
8 9
and cos(v ,a ) >
8 9
. The second solution 
corresponds to a negative timestep and does not need to be considered.
What does all this mean? Fig. 6 shows two examples of ( r )
2
as a function of 
∆
t.
Figure 6. ( r )
2
as a function of 
∆
t for a case (dashed curve) in which
cos(v ,a )
>
−
8 9
and a case (solid curve) in which cos(v ,a ) <
−
8 9
.
The dashed curve is an example in which cos(v ,a ) > 
−
8 9
. In this case, ( r )
2
increases 
monotonously with 
∆
t . If the procedure described above is followed, ( r )
2
will always 
correspond to the desired R 
2
after a number of iterations, no matter what the initial value 
of 
∆
t is. The full curve is an example where cos(v ,a ) < 
−
8 9
. The curve has a local 
maximum at 
∆
t
1 
and a local minimum at 
∆
t
2 
. If this is the case, three situations can be 
distinguished, depending on the value of R 
2
. In the first situation R 
2
is smaller than 
the minimum of ( r )
2
at 
∆
t
2 
, in the second situation R 
2
lies between the minimum at 
∆
t
2 
and the maximum at 
∆
t
1
, and in the third situation R 
2
is larger than the maximum at 
∆
t
1 
. 
These three situations and the situation where cos(v ,a ) > 
−
8 9
, mentioned above, are 
illustrated in fig. 7. Trajectory 1 in fig. 7 corresponds to the situation in which R 
2
is 
smaller than the minimum at 
∆
t
2 
. In this case the correct value for 
∆
t will always be 
found. For trajectory 2 this is not the case. If the initial trial time is large, the computer 
could select point B or C, points the atom passes on its way back after it has made the 
turn. It should select point A. This is done by limiting the timestep to 
∆
t
1 
and reducing it 
further as needed. Atoms with type 3 trajectories should also be treated differently. These 
atoms never move more than a radial distance R away from their starting point, but they 
can travel a total length of up to 3R , see fig. 8.

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21