Essentials of Computational Chemistry

74 3 SIMULATIONS OF MOLECULAR ENSEMBLES
where we have used the relationship between velocity and momentum
v = p
m
Similarly, the relationship between two momentum vectors is
t2
p(t2) = p(t1) + m
t1
(3.11)
a(t)dt (3.12)
where a is the acceleration. Equations (3.10) and (3.12) are Newton’s equations of motion.
Now, we have from Newton’s Second Law
a = F
m
(3.13)
where F is the force. Moreover, from Eq. (2.13), we have a relationship between force
and the position derivative of the potential energy. The simple form of the potential energy
expression for a harmonic oscillator [Eq. (2.2)] permits analytic solutions for Eqs. (3.10) and
(3.12). Applying the appropriate boundary conditions for the example in Figure 3.1 we have
k
q(t) = b cos
m t
(3.14)
and
p(t) =−b √ mk sin
k
m t
(3.15)
These equations map out the oval phase space trajectory depicted in the figure.
Certain aspects of this phase space trajectory merit attention. We noted above that a
phase space trajectory cannot cross itself. However, it can be periodic, which is to say it
can trace out the same path again and again; the harmonic oscillator example is periodic.
Note that the complete set of all harmonic oscillator trajectories, which would completely fill
the corresponding two-dimensional phase space, is composed of concentric ovals (concentric
circles if we were to choose the momentum metric to be (mk) −1/2 times the position metric).
Thus, as required, these (periodic) trajectories do not cross one another.
3.3.2 Non-analytical Systems
For systems more complicated than the harmonic oscillator, it is almost never possible to
write down analytical expressions for the position and momentum components of the phase
space trajectory as a function of time. However, if we approximate Eqs. (3.10) and (3.12) as
q(t + t) = q(t) + p(t)
t (3.16)
m