Van der Waals' density gradient theory

This can be considered as the first density-functional theory.

The grand potential of an interface is expressed as

$\Omega =\int dr\omega (\rho (r))+c\int dr(\nabla \rho )^{2}$,

where a local approximation is employed in the first term ($\omega$ being the grand potential density of the bulk system), and the variation in the density profile enters in the second term. This second term is the integral of the square of the density gradient, with a proportionality constant that is termed the influence parameter.

References

1. J. D. van der Waals and P. Kohnstamm "Lehrbuch der Thermostatik", Verlag Von Johann Ambrosius Barth, Leipzig (1927)
2. J.S. Rowlinson and B. Widom "Molecular Theory of Capillarity". Dover 2002 (originally: Oxford University Press 1982)