About OOPSE OOPSE News Features Download CVS Mailing Lists Documentation Authors Funding Results	Constant-pressure integration with a flexible box (NPTf) There is a relatively simple generalization of the Nosé-Hoover-Andersen method to include changes in the simulation box shape as well as in the volume of the box. This method utilizes the full $3 \times 3$ pressure tensor and introduces a tensor of extended variables ( $\overleftrightarrow{\eta}$ ) to control changes to the box shape. The equations of motion for this method differ from those of NPTi as follows: $$\dot{{\bf r}} = {\bf v} + \overleftrightarrow{\eta} \cdot \left( {\bf r} - {\bf R}_0 \right),$$ (4.39) $$\dot{{\bf v}} = \frac{{\bf f}}{m} - (\overleftrightarrow{\eta} + \chi \cdot \mathsf{1}) {\bf v},$$ (4.40) $$\dot{\overleftrightarrow{\eta}} = \frac{1}{\tau_{B}^2 f k_B T_{\mathrm{target}}} V \left( \overleftrightarrow{\mathsf{P}} - P_{\mathrm{target}}\mathsf{1} \right) ,$$ (4.41) $$\dot{\mathsf{H}} = \overleftrightarrow{\eta} \cdot \mathsf{H} .$$ (4.42) Here, $\mathsf{1}$ is the unit matrix and $\overleftrightarrow{\mathsf{P}}$ is the pressure tensor. Again, the volume, $\mathcal{V} = \det \mathsf{H}$ . The propagation of the equations of motion is nearly identical to the NPTi integration: moveA: $$\overleftrightarrow{\mathsf{P}}(t) \leftarrow \left\{{\bf r}(t)\right\}, \left\{{\bf v}(t)\right\} ,$$ $${\bf v}\left(t + h / 2\right) \leftarrow {\bf v}(t) + \frac{h}{2} \left( \frac{{\bf f}(t)}{m} -... ...i(t)\mathsf{1} + \overleftrightarrow{\eta}(t) \right) \cdot {\bf v}(t) \right),$$ $$\overleftrightarrow{\eta}(t + h / 2) \leftarrow \overleftrightarrow{\eta}(t) + \frac{h \mathcal{V}(t)}... ...ft( \overleftrightarrow{\mathsf{P}}(t) - P_{\mathrm{target}}\mathsf{1} \right),$$ $${\bf r}(t + h) \leftarrow {\bf r}(t) + h \left\{ {\bf v} \left(t + h / 2 \right)... ...rrow{\eta}(t + h / 2) \cdot \left[ {\bf r}(t + h) - {\bf R}_0 \right] \right\},$$ $$\mathsf{H}(t + h) \leftarrow \mathsf{H}(t) \cdot e^{-h \overleftrightarrow{\eta}(t + h / 2)} .$$ OOPSE uses a power series expansion truncated at second order for the exponential operation which scales the simulation box. The moveB portion of the algorithm is largely unchanged from the NPTi integrator: moveB: $$\overleftrightarrow{\mathsf{P}}(t + h) \leftarrow \left\{{\bf r} (t + h)\right\}, \left\{{\bf v}(t + h)\right\}, \left\{{\bf f}(t + h)\right\} ,$$ $$\overleftrightarrow{\eta}(t + h) \leftarrow \overleftrightarrow{\eta}(t + h / 2) + \frac{h \mathca... ... \left( \overleftrightarrow{P}(t + h) - P_{\mathrm{target}}\mathsf{1} \right) ,$$ $${\bf v}\left(t + h \right) \leftarrow {\bf v}\left(t + h / 2 \right) + \frac{h}{2} \left( \f... ... h)\mathsf{1} + \overleftrightarrow{\eta}(t + h)) \right) \cdot {\bf v}(t + h),$$ The iterative schemes for both moveA and moveB are identical to those described for the NPTi integrator. The NPTf integrator is known to conserve the following Hamiltonian: $$H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( \fra... ...m{target}}}{2} \mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2.$$ (4.43) This integrator must be used with care, particularly in liquid simulations. Liquids have very small restoring forces in the off-diagonal directions, and the simulation box can very quickly form elongated and sheared geometries which become smaller than the cutoff radius. The NPTf integrator finds most use in simulating crystals or liquid crystals which assume non-orthorhombic geometries.
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	Updated on January 16, 2006