Quantum Mechanical Insights into Biological Processes at the Electronic Level - Computational Modeling of Biological Systems - page 154

In quantum nuclear dynamics, .R;t/ can be represented as a Gaussian
wavepacket, which is the object that needs to be propagated in the dynamics:

.ıt/ 2
2

@ 2
@t 2 .R;r/ C :::

.R;t C ıt/ .R;t/ C ıt @

@t .R;t/ C

" #

X

2
2M a r

.R;t/ i ıt
„

„

2

R a C E.R/

.R;t/; (44)

a

However, this kind of dynamics for the entire biological molecule is prohibitively
expensive. Usually, nuclei are considered classical objects, as long as the dynamics
involves only one PES at a time. As a compromise, there is also mixed quantum-
classical nuclear dynamics, where only some of the nuclei in the system are treated
as quantum particles (i.e., delocalized wave-packets).
We will first consider the ab initio adiabatic dynamics in which nuclei are treated
as classical particles moving on a single adiabatic Born-Oppenheimer PES, one
PES at a time. This kind of dynamics also can be approximated by the classical
force field formalism, if the force field parameters are available for the system. In
ab initio adiabatic dynamics, the Schr odinger equation for the electrons is solved
with nuclear coordinates being constant parameters. The nuclei are then exposed
to the potential provided by the electrons, and move according to this ab initio
potential, obeying the classical equations of motion. The forces acting on nuclei
are calculated on-the-fly, at every step of the dynamics by solving the electronic
Schr odinger equation, and utilizing the Hellmann-Feynman theorem:

* ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ +

@ H
@X n

@E
@X n D

F X n D ‰ ‰ : (45)

The ab intio PES may be fitted to an analytical function, in order to solve for
the forces. This is the foundation of the Born-Oppenheimer molecular dynamics
(BOMD) method [ 33 , 34 ], implemented in Gaussian , and also in the stand-alone
classical trajectory program VENUS . The typical time-step in BOMD is on the order
of a few femtoseconds. Adiabatic molecular dynamics can be coupled with classical
treatment of the larger system surrounding the quantum region, in a QM/MM
fashion.
Car-Parinello molecular dynamics (CPDM) is another variant [ 35 ]. In contrast
to BOMD, CPMD explicitly introduces the electronic degrees of freedom (usually
provided by DFT) as fictitious dynamical variables. In other words, the Kohn-
Sham molecular orbitals are chosen as the dynamical variables to represent the
electronic degrees of freedom in the system. Electrons are also assigned a fictitious
mass. An extended Lagrangian for the system is then written, and it leads to a
system of coupled equations of motion for both nuclei and electrons. The method
works in conjunction with DFT and plane-wave basis set. CPMD is computationally
expensive, and most likely not usable for sizable systems that might interest bio-
chemists. There is also a cheaper version of CPMD, based on atom-centered density
matrix propagation, ADMP, implemented in Gaussian . ADMP uses Gaussian basis
functions, and works with semiempirical, HF, and pure and hybrid DFT methods.