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

For fixed electron-electron interaction and electron number, a nondegenerate ground-state electron density determines the external potential up to an additive constant, and hence determines the Hamiltonian and ground-state observables.

Suppose two potentials $v,v^{\prime }$ differing by more than a constant have different ground states $\Psi ,{\Psi }^{\prime }$ but the same density $\rho$. Strict Rayleigh-Ritz variational principle inequalities giveAdding produces the contradiction $E+E^{\prime }<E+E^{\prime }$. This proves the first Hohenberg-Kohn theorem under the stated nondegeneracy assumptions. A constant potential shift changes energy by $N$ times that constant without changing the density.^[ref]

Far from a neutral bound system, the slowest-decaying electron-removal amplitude is controlled by the first ionization energy $I=E_{N-1}-E_N>0$. Its asymptotic wavefunction satisfies a free-particle-like equation at negative energy $-I$, giving a decay exponent $\kappa =\sqrt{2m_{e}I}/\hslash$. Density is quadratic in the amplitude, soor $a=2\sqrt{2I}$ in atomic units. Power-law radial factors and angular dependence do not alter this leading exponential coefficient. It is the ionization energy, not the electron affinity, that controls this tail.

The Wigner-Seitz radius $r_s$ is the radius of a sphere containing one electron at the local uniform density:In atomic units, the unpolarized homogeneous electron gas has exchange energy per electron $-0.4582/r_s$. The local density approximation applies that uniform-gas result at each position using the local density, then integrates the result weighted by $\rho$. It assumes local uniform-gas behavior rather than resolving nonlocal exchange explicitly.

Substitution givesFor $\rho \propto e^{-ar}$, this decays as $-C{e}^{-ar/3}$, with possible subleading radial prefactors. It therefore vanishes exponentially rather than with the required algebraic $-c/r$ exchange-energy tail.

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