Handbook of Solvents - George Wypych - ChemTech - Ventech!

472 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
( rt)
μψ��
,
∂V
=− +∑Λ ψ ( rt , )
*
ik k
[8.97]
∂ψ
i
( rt , )
k
In eq. [8.97] the Λ are the Lagrange multipliers introduced to satisfy eq. [8.94]. It is worth
noticing that, while the nuclear dynamics [8.95] can have a physical meaning, that is not
true for the dynamics associated with the {αv} and {ψi}: this dynamics is fictitious, like the
associated “masses” μ.
Ifμ and the initial conditions {ψi} 0 and {dψi/dt} 0 are chosen such that the two classical
sets of degrees of freedom (nuclear and electronic) are only weakly coupled, the transfer of
energy between them is small enough to allow the electrons to adiabatically follow the nuclear
motion, then remaining close to their instantaneous BO surface. In such a metastable
situation, meaningful temporal averages can be computed. The mentioned dynamics is
meant to reproduce what actually occurs in real matter, that is, electrons adiabatically following
the nuclear motion.
QM potentials have been widely used in molecular dynamics simulation of liquid water
using the CP DFT algorithm. See, for example, refs. [72,73].
8.7.2.2 Semi-classical simulations
Computer simulations are methods addressed to perform “computer experimentation”. The
importance of computer simulations rests on the fact that they provide quasi-experimental
data on well-defined models. As there is no uncertainty about the form of the interaction potential,
theoretical results can be tested in a way that is usually impossible with results obtained
by experiments on real liquids. In addition, it is possible to get information on
quantities of no direct access to experimental measures.
There are basically two ways of simulating a many-body system: through a stochastic
process, such as the Monte Carlo (MC) simulation, 74 or through a deterministic process,
such as a Molecular Dynamics (MD) simulation. 75,76 Numerical simulations are also performed
in a hybridized form, like the Langevin dynamics 42 which is similar to MD except
for the presence of a random dissipative force, or the Brownian dynamics, 42 which is based
on the condition that the acceleration is balanced out by drifting and random dissipative
forces.
Both the MC and the MD methodologies are used to obtain information on the system
via a classical statistical analysis but, whereas MC is limited to the treatment of static properties,
MD is more general and can be used to take into account the time dependence of the
system states, allowing one to calculate time fluctuations and dynamic properties.
In the following, we shall briefly describe the main features of MD and MC methodologies,
focusing the attention to their use in the treatment of liquid systems.
Molecular dynamics
Molecular Dynamics is the term used to refer to a technique based on the solution of the
classical equation of motion for a classical many-body system described by a many-body
Hamiltonian H.
In a MD simulation, the system is placed within a cell of fixed volume, usually of cubic
shape. A set of velocities is assigned, usually drawn from a Maxwell-Boltzmann distribution
suitable for the temperature of interest and selected to make the linear momentum
equal to zero. Then the trajectories of the particles are calculated by integration of the classical
equation of motion. It is also assumed that the particles interact through some forces,