Handbook of Solvents - George Wypych - ChemTech - Ventech!

2.1 Solvent effects on chemical systems 19
If we are capable of integrating the equations of movement of all the particles which
constitute a system, we can find their paths and velocities and we can evaluate the properties
of the system in determined time intervals. Thus, we can find how the system being studied
evolved as time moves forward. In the first simulations by Molecular Dynamics of a condensed
phase, 34 use was made of potentials as simple as the hard sphere potential, under
which the constituent particles move in a straight line until colliding elastically. The collisions
happen when the separation between the centers of the spheres is equal to the diameter
of the sphere. After each collision, the new velocities are obtained by making use of the
principle of conservation of the linear moments. But a chemical system requires more elaborate
potentials under which the force, which at every instant acts between two atoms or
molecules, changes in relation to the variation of the distance between them. This obliges us
to integrate the equations of movement of the system in very small time intervals, in which it
is assumed that the force which acts on each atom or molecule is constant, generally lying
between 1 and 10 femtoseconds. For each of these intervals, the positions and velocities of
each of the atoms is calculated, after which they are placed in their new positions and, once
again, the forces are evaluated to obtain the parameters of a new interval, and so on, successively.
This evolution in time, which usually requires the evaluation of hundreds of thousands
of intervals of approximately 1 femtosecond each, allows us to know the properties of
the system submitted to study during the elapse of time. In fact, the task commences by fixing
the atoms which make up the system being studied in starting positions, and later move
them continuously whilst the molecules being analysed rotate, the bond angles bend, the
bonds vibrate, etc., and during which the dispositions of the atoms which make up the system
are tabulated at regular intervals of time, and the energies and other properties which
depend on each of the conformations, through which the molecular system makes its way
with the passage of time, are evaluated. Molecular Dynamics is Chemistry scrutinized each
femtosecond.
Monte Carlo methods
The first simulation by computer of a molecular system was carried out using this method.
It consists of generating configurations of a system introducing random changes in the position
of its constituents.
In order to obtain a good convergence in the sequence of configurations, Metropolis et
al. 35 suggested an interesting approach. This approach avoids the generation of a very long
random configurations as follows: instead of choosing random configurations and then
weighing them according to the Boltzmann factor, one generates configurations with a
probability equal to the Boltzmann factor and afterwards weigh them evenly. For this purpose
once a new configuration is generated the difference in the potential energy with respect
to the previous one is computed (ΔU) and a random number 0 ≤r ≤1 is selected. If the
Boltzmann factor exp(-ΔU/kT)>r then the new configuration is accepted, if exp(-ΔU/kT)