BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 125

That is Lagrange multipliers introduced in order to satisfy the constraints in Eq.
(99). Then the equation of kinetic energy:

1 1 1



∑ ∑∑

2

∫

3



2



2

K = m ψ + dr MR + ma

(102)

i II νν

2 2 2

i I ν

Ù

is obtained. Based on the technique mentioned, CPMD extends MD beyond the
usual pair-potential approximation. In addition, it also extends the application of
DFT to much larger systems [129, 141].

1.3.2.1.3 HYBRID METHODS (MD + MC)

For some complex systems Monte Carlo simulations have very low acceptance
rates except for very small trial moves and hence become quite inefficient. Mo-
lecular dynamics simulations may not allow the system to develop sufficiently in
time to be useful, however, molecular dynamics methods may actually improve a
Monte Carlo investigation of the system. A trial move is produced by allowing the
molecular dynamics equations of motion to progress the system through a rather
large time step. Although such a development may no longer be accurate as a mo-
lecular dynamics step, it will produce a Monte Carlo trial move, which will have
a much higher chance of success than a randomly chosen trial move. In the actual
implementation of this method some testing is generally advisable to determine
an effective value of the time step.

1.3.2.1.4 AB INITIO MOLECULAR DYNAMICS

No discussion of molecular dynamics would be complete without at least a brief
mention of the approach pioneered by many researchers, which combines elec-
tronic structure methods with classical molecular dynamics. In this hybrid scheme
a fictitious dynamical system is simulated in which the potential energy is a func-
tional of both electronic and ionic degrees of freedom. This energy functional is
minimized with respect to the electronic degrees of freedom to obtain the poten-
tial energy surface to be used in solving for the trajectories of the nuclei. This
approach has proven to be quite fruitful with the use of density functional theory
for the solution of the electronic structure part of the problem and appropriately
chosen pseudo potentials. These equations of motion in relation with molecular
dynamics simulation, can then be solved by the usual numerical methods, for
example, the Verlet algorithm, and constant temperature simulations can be per-
formed by introducing thermostats or velocity rescaling. This ab initio method is
efficient in exploring complicated energy landscapes in whichboth the ionic posi-
tions and electronic structures are determined simultaneously.