Lecture: Quantum Molecular Dynamics Quantum Molecular Dynamics

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;

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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;

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where tex2html_wrap_inline200 are generalized coordinates.

As with Car-Parinello we use the following Lagrangian,

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where

For QMD, the generalized coordinates are
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where tex2html_wrap_inline204 is the position vector of the atomic centers, tex2html_wrap_inline206 are the atomic velocities, tex2html_wrap_inline208 are the electron coordinates, tex2html_wrap_inline210 are any external constraints on the system, and tex2html_wrap_inline212 is the electronic wave function of the molecular system.

One last problem is to include the constraint (holonomic) that the wave functions be normalized;
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Note; these constraints result in 'forces' on the coefficients. The forces can be included using Lagrange multipliers, tex2html_wrap_inline216 . 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,
eqnarray32
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 tex2html_wrap_inline218 .

To develop and solve the equations the following approximations are used;

Using Kohn-Sham orbitals, the electron density can be written as;
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where tex2html_wrap_inline222 is the occupation number. tex2html_wrap_inline224 can be evaluated using Kohn-Sham orbital theory by solving
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For the DFT approach, that is, using DFT for the electronic degrees of freedom, the potential energy can be written as;
eqnarray106
In this expression, U is the Kohn-Sham functional
eqnarray111
where

Including the normalization of the wave function using Lagrange multipliers tex2html_wrap_inline216, the equations of motion become,
eqnarray130

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 tex2html_wrap_inline282 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;
eqnarray142
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
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That is, expand the wave function in terms of single particle basis functions. The coefficients ( tex2html_wrap_inline230) contain the temporal dependence of the wave function.

Using this formalism, the term,
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becomes;
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That is, the derivatives with respect to the wave functions are replaced by derivatives with respect to the variational coefficients, tex2html_wrap_inline236.

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 tex2html_wrap_inline238

  1. Begin with a system in which all the constraints are satisfied to within the specified tolerance.
  2. Evaluate all the external forces.
  3. Move all the atomic centers without constraints according to the external forces.
  4. Evaluate the constraints.
  5. Iterate the procedure until all constraints are within specified tolerances.

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



Author: Ken Flurchick