The problem: Nuclear and electronic motions have widely differing time scales.
Classical approaches track atomic motions and uses a macroscopic force field to include electronic effects. Quantum approaches track electronic effects in a field of fixed nuclear positions. Adding an iterative method, one can include the motion of the nuclear positions.
Can these be combined?
Should they be combined?
One method to combine the electronic and nuclear motions is Quantum Molecular Dynamics first proposed by Car and Parinello.
The approach is to treat the electronic degrees of freedom in a manner similar to the particle coordinates, by assigning a fictitious mass to the functions describing the electronic degrees of freedom. This is usually a Molecular Orbital derived from traditional Hartree-Fock (Wave Function Functionals) using both the usual molecular Hamiltonian, a semi-empirical Hamiltonian, or a Kohn-Sham Orbital derived from Density Functional Theory (DFT) formalism. (See the DFT discussion Density Functional Theory )
This approach is also referred to as "Dynamic Simulated Annealing" This is because rather than using a Monte Carlo approach to describe the changing electronic state, Car and Parinello used a molecular dynamics approach to implement simulated annealing. The MD equations of motion are used to move around configuration space. In this case, the degrees of freedom of configuration space are the coordinates of the atomic centers and the electronic degrees of freedom.
The initial step is to construct a Lagrangian, from which the equations of motion are derived.
The Lagrangian for a collection of particles labelled by i, is defined as;
where T is the kinetic energy and V is the potential energy.
Using classical mechanics, the equations of motion of the system can be determined using a variational approach giving;
where
are generalized coordinates.
As with Car-Parinello we use the following Lagrangian,
where
For QMD, the generalized coordinates are
where
is the position vector of the atomic centers,
are the atomic velocities,
are the electron coordinates,
are any external constraints on the system, and
is the electronic wave function of the molecular
system.
One last problem is to include the constraint (holonomic) that the
wave functions be normalized;
Note; these constraints result in 'forces' on the coefficients.
The forces can be included using Lagrange multipliers,
. A holonomic constraint is an algebraic function
of the coordinates and constants. I.e. the constraint is finite
and integrable. Including the normalization of the wave function
using Lagrange multipliers, the equations of motion become,
where the first equation describes the evolution of the electronic
degrees of freedom, the second equation describes the evolution
of the nuclear centers and the third equation describes the action
of the any external force that is described by coordinates
.
To develop and solve the equations the following approximations are used;
Using Kohn-Sham orbitals, the electron density can be written as;
where
is the occupation number.
can be evaluated using Kohn-Sham orbital theory
by solving
For the DFT approach, that is, using DFT for the electronic degrees
of freedom, the potential energy can be written as;
In this expression, U is the Kohn-Sham functional
where
Including the normalization of the wave function using Lagrange
multipliers
, the equations of motion become,
Solving these equations gives the time evolution of the motion of the system, within the framework of the following approximations;
A simple Euler method (known as the Verlet algorithm
see the discussion
Classical Modeling: Deterministic Methods ) to
obtain the wave-function coefficients as a function of time is
used. Note that this method must always stay close to the
Born-Oppenheimer potential surface. To do this, a self consistent
calculation must be performed periodically (every n time steps).
Thus, this calculation has many of the same features as the
traditional Hartree-Fock scheme presented above. The new feature
is that no post Hartree-Fock scheme is necessary, but is replaced
by an
step, namely the molecular dynamics step. Here N
is the number of atoms, not electrons, in the system. This molecular
dynamics step requires the computation of all the forces on atom i
due to all the other atoms and electrons in the system.
The Kohn-Sham equations, with the local density approximation(LDA)
are written as;
These can be solved in the normal manner using a variety of schemes,
expansion in plane wave sets, numerical basis sets, etc.
If one does not use a DFT approach, the wave function can be determined
via the traditional Hartree-Fock formalism. If we write the wave function
describing the electronic degrees of freedom as
That is, expand the wave function in terms of single particle basis functions.
The coefficients (
) contain the temporal dependence of the wave function.
Using this formalism, the term,
becomes;
That is, the derivatives with respect to the wave functions are
replaced by derivatives with respect to the variational coefficients,
.
The procedure then is to obtain an initial state, a point in configuration space, for both the electronic and nuclear degrees of freedom as well as the gradient.
The point is moved in configuration space using Molecular Dynamics with the appropriate constraints. As the point in configuration space proceeds to a minimum on the BO surface, the point gains " kinetic energy". This is a fictious energy, hence the term fictious Lagrangian.
Iterative application of the motion of the point in configuration space reaches the minimum of the energy surface.
The recipe to go from
The trick here is that at each time step, the equations for both the coordinates of the atomic centers and the electronic degrees of freedom are stepped forward in time. Thus, the evolution of the system includes both the nuclear and electronic parts of the system. Periodic quenching of the wave function is necessary to keep the system on the BO surface. Periodic quenching means performing an SCF calculation to recompute the coefficients for the wave function.
Some Results
QMD was first used to look at ionic systems where classical approaches doesn't always converge. In fact, QMD looks promising to investigate large systems involving charge transfer. In particular, covalent systems like C60 clusters, Si clusters and other semi-conductor systems. More information can be found in the references. In another approach D. S. Wallace, studied conjugated polymers. In this work, the calculation of the LCAO electronic wave function using the CNDO/2 semi-empirical Hamiltonian. The procedure chosen by Wallace was to sequentially relax the positions of the atomic centers followed by a calculation of the electronic wave function.
Additional information on the derivation of the Kohn-Sham equations can be found here