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Parrinello-Rahman dynamics

In the method of Parrinello and Rahman [79,80], the equations of motion for atomic and lattice degrees of freedom are derived from the following Lagrangian:

$\displaystyle {\cal{L}}(s,h,\dot{s},\dot{h}) = \frac{1}{2}\sum_i^N m_i \dot{s_i}^t G \dot{s_i} -V(s,h) + \frac{1}{2}W Tr(\dot{h}^t \dot{h}) - p_{ext}\Omega,$ (6.35)

where $ s_i$ is a position vector in fractional coordinates for atom $ i$, $ h$ is the matrix formed by lattice vectors, tensor $ G$ is defined as $ G=h^t h$, $ p_{ext}$ is the external pressure, $ \Omega$ is the cell volume ( $ \Omega = det h$), and $ W$ is a constant with dimensionality of mass. Integrating equations of motion based on Lagrangian defined in eq. 6.35 generates $ NpH$ ensemble with enthalpy $ H=E+p_{ext} \Omega$ being the constant of motion. Parrinello-Rahman method can be combined with numerical thermostats such as Langevin thermostat (see Sec. 6.62.5), or Nosé-Poincaré method [65,72] to generate $ NpT$ ensemble.

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