Standard molecular dynamics generates the microcanonical or
ensemble
where in addition the total momentum is conserved [13].
The temperature is not a control variable and cannot be preselected and fixed.
But it is evident that also within molecular dynamics
the possibility to control the
average temperature (as obtained from the
average kinetic energy of the nuclei and the energy equipartition theorem)
is welcome for physical reasons.
A deterministic algorithm of achieving temperature control
in the spirit of extended system dynamics [30]
by a sort of dynamical friction mechanism was devised by
Nosé and Hoover [26,27], see
e.g. Refs. [12,26,13] for reviews of this technique.
Thereby, the canonical or
ensemble is generated
in the case of ergodic dynamics.
It is well-known that the standard Nosé-Hoover thermostat method
suffers from non-ergodicity problems for certain classes
of Hamiltonians, such as the harmonic oscillator [27].
A closely related technique, the so-called
Nosé-Hoover-chain thermostat [32],
cures that problem and assures ergodic sampling of phase space even
for the pathological harmonic oscillator.
This is achieved by thermostatting the original thermostat by another
thermostat, which in turn is thermostatted and so on.
In addition to restoring ergodicity even with only a few
thermostats in the chain, this technique is found to
be much more efficient in imposing the desired temperature.
The underlying equations of motion read
The desired average physical temperature is given by
and
denotes the number
of dynamical degrees of freedom to which the nuclear thermostat chain is
coupled (i.e. constraints imposed on the nuclei have to be subtracted).
It is found that this choice requires a very accurate integration
of the resulting equations of motion
(for instance by using a high-order Suzuki-Yoshida integrator [33]).
The integration of these equations of motion
is discussed in detail in Ref. [33]
using the velocity Verlet algorithm.
One of the advantages of the velocity Verlet integrator is that it can be easily used
together with higher order schemes for the thermostats.
Multiple time step techniques can be used and time reversibility of the
overall algorithm is preserved.
Integrate Thermostats for dt/2
V(:) := V(:) + dt/(2*M(:))*F(:)
R(:) := R(:) + dt*V(:)
Calculate new forces F(:)
V(:) := V(:) + dt/(2*M(:))*F(:)
Integrate Thermostats for dt/2
The choice of the ``mass parameters'' assigned to the thermostat
degrees of freedom should be made such that
the overlap of their power spectra and the ones of
the thermostatted subsystems is maximal [33].
The relations