Van der Waals' density gradient theory

Van der Waals' density gradient theory can be considered to be the first density-functional theory.

The grand potential of an interface is expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \int dr \omega(\rho(r)) + c \int dr (\nabla\rho)^2 } ,

where a local approximation is employed in the first term (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)