# 7.2 NPH Molecular Dynamics

Andersen (1980) introduced an additional (synthetic'') energy that was coupled to changes of the system volume. Putting and (with a generalized mass which may be visualized as the mass of some piston) he wrote the Hamiltonian as
 (7.10)

Introducing scaled position vectors he derived the equations of motion
 (7.11) (7.12)

These equations of motion conserve the enthalpy, as appropriate in an isobaric ensemble.

Parrinello and Rahman (1980) extended the NPH method of Andersen to allow for non-isotropic stretching and shrinking of box sides. Important applications are structural phase transitions in solids.

Scaling of the position vectors now follows the equation instead of .

The matrix describes the anisotropic transformation of the basic cell. The cell volume is
 (7.13)

The additional terms in the Hamiltonian are
 (7.14)

and the equations of motion are
 (7.15) (7.16)

where is a metric tensor, is a virtual mass (of dimension mass), and the pressure/stress tensor is defined as
 (7.17)

Morriss and Evans (1983) devised another type of NPH dynamics. They suggested to constrain the pressure not by an inert piston but by a generalized constraint force in the spirit of Gaussian dynamics.

The same idea may be carried over to constrain in addition to the pressure. In this way one arrives at a molecular dynamics procedure. (See Allen-Tildesley, Ch. 7.)
F. J. Vesely / University of Vienna