Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 61 / d / i / Solution

$x_{\sigma }=|\nabla {\rho }_{\sigma }|/{\rho }_{\sigma }^{4/3}$ is a dimensionless reduced density gradient. It measures inhomogeneity relative to the local density scale, vanishes for a uniform gas, and allows a generalized gradient approximation correction to distinguish rapidly varying density from slowly varying density with the same local value.

The parameter $\beta$ controls the correction strength in the Becke 1988 exchange functional; it was fitted to atomic exchange energies. The large-gradient denominator simultaneously produces the intended asymptotic behavior, so its role is more constrained than an arbitrary multiplicative adjustment.^[ref]