Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 471
A point on the BO potential energy surface (PES) is then given by the minimum with respect
to the ψ i of the energy functional:
*
[ { ψi}{ , I}{ , αv} ] = ∑∫ ψi() ⎢−
∇ ⎥ ψi() r + U[ ρ(){ r , RI}{
, αv}
]
3
2
V R d r r
i
Ω
⎡
⎣
2
h
2m
⎤
⎦
[8.91]
where {R I} are the nuclear coordinates, {α v} are all the possible external constraints imposed
on the system (e.g., the volume Ω). The functional U contains the inter-nuclear Coulomb
repulsion and the effective electronic potential energy, including external nuclear,
Hartree, and exchange and correlation contributions.
In the conventional approach, the minimization of the energy functional (eq. [8.91])
with respect to the orbitals ψ i subject to the orthonormalization constraint leads to a set of
self-consistent equations (the Kohn-Sham equations), i.e.:
⎧ 2
h
⎫
2 ∂U
⎨−
∇ + ⎬ ψi() r = εiψ i()
r
[8.92]
⎩
2m
∂ρ()
r
⎭
whose solution involves repeated matrix diagonalizations (and rapidly growing computational
effort as the size of the system increases).
It is possible to use an alternative approach, regarding the minimization of the functional
as an optimization problem, which can be solved by means of the simulated annealing
procedure. 71 A simulated annealing technique based on MD can be efficiently applied to
minimize the KS functional: the resulting technique, called “dynamical simulated annealing”
allows the study of finite temperature properties.
In the “dynamical simulated annealing,” the {R i}, {α v} and {ψ i} parameters can be
considered as dependent on time; then the Lagrangian is introduced, i.e.:
with:
L = ∑ ∫d
r + ∑ M R + ∑ −V
R
1 3 2 1 1 2
μ ψ� �
μ α� ψ , , α
2
2 2
∫
Ω
i
Ω
( ) ( )
3 *
d rψ rt , ψ rt , = δ
i j ij
[ { }{ }{ } ]
i I I v v i I v
I
v
[8.93]
[8.94]
In eq. [8.94] the dot indicates time derivative, MI are the nuclear masses andμ are arbitrary
parameters having the dimension of mass.
Using eq. [8.94] it is possible to generate a dynamics for {Ri}, {αv} and {ψi} through
the following equations:
MR
��
I I =−∇ R V
[8.95]
I
∂
μ α��
V
v v =−
∂αv
⎛ ⎞
⎜
⎟
⎝ ⎠
[8.96]