= 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.<Density-functional exchange-energy approximation with correct asymptotic behavior by Axel D. Becke>[Reduced-gradient form, fitted parameter and asymptotic construction]{ref}
Back to article page
