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7.4 Density Functional Molecular Dynamics (DFMD)
Car and Parrinello: ``ab initio molecular dynamics''
for the dynamical simulation of quantum mechanical systems [CAR 85].
Born-Oppenheimer approximation:
- the heavy atomic cores (``ions'')
consisting of nucleus and inner electronic shells are treated
as classical particles
- the valence and conduction electrons are
represented by wave functions which assume the configuration
of least energy in the momentary field created by the ions
and all other valence and conduction electrons.
Let
denote the wave functions of the electrons;
the electron density at position
is then
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(7.60) |
The classical ions residing at positions
produce a
potential field
in which the electronic wave
functions take on their minimal energy form.
Total energy of the electrons:
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(7.61) |
with
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(7.62) |
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(7.63) |
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(7.64) |
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(7.65) |
Approximation for : usually local density
expression (see [CAR 85]).
Procedure:
Apply variational calculus:
Kohn-Sham equations [KOHN 65]; then
solve those by an iterative method.
Slow!
Better:
To find the variational minimum, use the Simulated Annealing method
of Kirkpatrick et al. [KIRKPATRICK 83]. (See the chapter on Stochastics).
Still better:
Dynamical Simulated Annealing by Car and Parrinello:
Let denote an abstract ``mass''
assigned to each electronic wave function .
The ``kinetic energy'' due to the change of in time is
then
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(7.68) |
Now introduce a Lagrangian function
describing the ``motion''of .
Here is the ionic mass, and the Lagrange multipliers
allow for the conditions 7.67. From the Lagrangian the
``equations of motion''are derived:
The second equation describes the classical dynamics of the ions.
The first equation represents the abstract ``motion'' in the
space of the electronic degrees of freedom.
By keeping the ``kinetic energy'' of this motion
small we can constrain the -subsystem always close to the
momentary minimum of the energy surface defined by the slowly varying
ionic configuration.
To control the temperature of the
- rescale all
from time
to time, or
- use one of the thermostats available from statistical-mechanical
simulation; see Chapter 6.
The electronic degrees of freedom should adjust quickly to the varying
energy landscape
mass small compared to the ionic masses;
e. g. atomic mass unit.
Applications:
- amorphous silicon [CAR 85]
- lithium [WENTZCOVICH 91]
microclusters of alkali metals [VITEK 89]
- molten carbon [GALLI 90B]
- ionic melts [GALLI 90A]
For current applications, see the web or
reviews such as [VITEK 89,MAKRI 99,MAZZONE 99,OHNO 99].
Next: 8. Hydrodynamics
Up: 7. Quantum Mechanical Simulation
Previous: 7.3 Wave Packet Dynamics
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001