Handbook of Solvents - George Wypych - ChemTech - Ventech!

470 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
three-body effects can be partially included in a pair-wise approximation, leading it to give
a good description of liquid properties. This can be achieved by defining an “effective” pair
potential, able to represent all the many-body effects. To do this, eq. [8.86] can be re-written
as:
eff
∑ 1() i ∑∑
2 ( ij )
[8.88]
V ≈ v r + v r
i
i
j> i
The most widely used effective potential in computer simulations is the simple
Lennard-Jones 12-6 potential (see eq. [8.70]), but other possibilities are also available; see
Section 8.6 for discussion on these choices.
8.7.2.1 Car-Parrinello direct QM simulation
Until now we have focused the attention to the most usual way of determining a potential interaction
to be used in simulations. It is worth mentioning a different approach to the problem,
in which the distribution of electrons is not treated by means of an “effective”
interaction potential, but is treated ab initio by density functional theory (DFT). The most
popular method is the Car-Parrinello (CP) approach, 68,69 in which the electronic degrees of
freedom are explicitly included in the description and the electrons (to which a fictitious
mass is assigned) are allowed to relax during the course of the simulation by a process called
“simulated-annealing”. In that way, any division of V into pair-wise and higher terms is
avoided and the interatomic forces are generated in a consistent and accurate way as the
simulation proceeds. This point constitutes the main difference between a CP simulation
and a conventional MD simulation, which is preceded by the determination of the potential
and in which the process leading to the potential is completely separated from the actual
simulation. The forces are computed using electronic structure calculations based on DFT,
so that the interatomic potential is parameter-free and derived from first principles, with no
experimental input.
Let us consider a system for which the BO approximation holds and for which the motion
of the nuclei can be described by classical mechanics. The interaction potential is given
by:
()
VR = ψ | Hψ
� | [8.89]
0 0
where H is the Hamiltonian of the system at fixed R positions and ψ 0 is the corresponding
instantaneous ground state. Eq. [8.89] permits to define the interaction potential from first
principles.
In order to use eq. [8.89] in a MD simulation, calculations of ψ 0 for a number of configurations
of the order of 10 4 are needed. Obviously this is computationally very demanding,
so that the use of certain very accurate QM methods (for example, the configuration
interaction (CI)) is precluded. A practical alternative is the use of DFT. Following Kohn and
Sham, 70 the electron density ρ(r) can be written in terms of occupied single-particle
orthonormal orbitals:
() r =∑ i()
r
ρ ψ
i
2
[8.90]