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