The Nosé-Hoover chain thermostat is defined by specifying three
parameters: A target kinetic energy, a frequency and a chain length. For the
ions, given the target temperature
, the target kinetic energy is just
, where
is the number of degrees of freedom involved in a common
thermostat. For example, if there is one thermostat on the entire ionic system,
then
, where
is the number of constraints to
which the atoms are subject. The frequency for the ionic thermostat should be
chosen to be some characteristic frequency of the ionic system for which one
wishes to insure equilibration. In water, for example, one could choose the O-H
bond vibrational frequency. (Having a precise value for this frequency is not
important, as one only wishes to insure that the thermostat will couple to the
mode of interest.) The choice of chain length is not terribly important as it
only determines how many extra thermostats there will be to absorb energy from
the system. Usually a chain length of 4 is sufficient to insure effective
equilibration. Longer chains may be used in situations where heating or cooling
effects are more dramatic.
For the electrons, the target kinetic energy is not usually known a
priori as it is for the ions. However, by performing a short run without
thermostats, one can determine a value about which the electron kinetic energy
`naturally' fluctuates and take this as the target value. While the precise
value is not important, a little experience goes a long way, as a choice that
is either too small or too large can cause spurious damping of the ions or
departures from the Born-Oppenheimer surface, respectively. A good choice for
the frequency of the electron thermostat can be made based on
, the maximum frequency in the phonon spectrum. The frequency of the
electron thermostat should be at least 2-3 times this value to avoid coupling
between the ions and the electron thermostats. As an example, for silicon, the
highest frequency in the phonon spectrum is 0.003 a.u., so a good choice for
the electron thermostat frequency is 0.01 a.u. The chain length of the electron
thermostat can be chosen in the same way as for the ions. 4 is usually
sufficient, however longer chains may be used if serious heating is expected.
In addition, the electron thermostats have an extra parameter that scales the
number of dynamical degrees of freedom for the electrons. (
, where
is the desired electron kinetic energy and
is the
number of dynamical degrees of freedom for the electrons - see Eq. (3.4) in
Ref.[5]). The default value is the true number of dynamical
degrees of freedom
, where
for
orthonormality constraints and
for norm constraints. When this number is
very large, it may not be possible to integrate the electron chain thermostats
stably using a frequency above that top of the phonon spectrum. Should this be
the case in your problem, then the number of dynamical degrees of freedom
should be scaled to some smaller number such that the system can once again be
integrated stably. This parameter has no other effect that to change the
relative time scales between the first element of the electron thermostat chain
and the other elements of the chain.
In addition to the basic parameters defining the chains themselves, one needs
to specify two more parameters related to the integration of the thermostated
equations of motion. The first is the order
of the Suzuki integrator.
Experience shows that the choice
is sufficient for most
applications. Finally, one must specify the number of times the Suzuki
integrator will be applied in a given update. This is the parameter
which determines the basic Suzuki time step
=
,
where
is the time step being used in the MD run.
or 3
is usually large enough to give stable integration. If more stable integration
is required, try
or make
larger.