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

$x_{\sigma }=|\nabla {\rho }_{\sigma }|/{\rho }_{\sigma }^{4/3}$ 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]

For typical atoms and molecules, the Becke 1988 exchange functional improves exchange energies and commonly improves molecular energetics relative to exchange-only local density approximation, because it accounts for density gradients and corrects important nonuniformity errors. This is a general trend, not a theorem for every observable or system.

A comparison of total molecular predictions also depends on the correlation functional, basis, spin treatment and geometry. B88 is an exchange approximation, not by itself a complete specification of an exchange-correlation calculation.

For a tail ${\rho }_{\sigma }\sim A_{\sigma }{e}^{-ar}$,At large $r$, the B88 correction integrand becomesSumming the spin densities and dividing by total density givesin atomic units. The exponentially decaying LDA term is negligible in this limit. The parameter $\beta$ cancels from the leading asymptotic term.

In the usual exchange-hole representation, the hole around an electron integrates to charge deficit $-1$. Far outside a finite system, the localized hole therefore looks like a unit opposite charge at distance $r$. The Coulomb interaction has leading scale $-1/r$, and the exchange pair energy carries the usual factor $1/2$ to avoid double counting, giving exchange energy per electron $-1/(2r)$.

This energy-per-electron tail should not be confused with the exchange or exchange-correlation potential, whose leading coefficient is generally $-1/r$ in the corresponding exact finite-system setting. Local energy-density representations also require a specified gauge; the question uses the conventional exchange-hole interpretation.