Lieb–Oxford inequality

Template:MOS In quantum chemistry and physics, the Lieb–Oxford inequality provides a lower bound for the indirect part of the Coulomb energy of a quantum mechanical system. It is named after Elliott H. Lieb and Stephen Oxford.

The inequality is of importance for density functional theory and plays a role in the proof of stability of matter.

Introduction

In classical physics, one can calculate the Coulomb energy of a configuration of charged particles in the following way. First, calculate the charge density Template:Math, where Template:Math is a function of the coordinates Template:Math. Second, calculate the Coulomb energy by integrating:

$\frac{1}{2}\underset{{ℝ}^3}{\int }\underset{{ℝ}^3}{\int }\frac{\rho (x)\rho (y)}{|x-y|}{\mathrm{d}}^{3}x{\mathrm{d}}^{3}y.$

In other words, for each pair of points Template:Math and Template:Math, this expression calculates the energy related to the fact that the charge at Template:Math is attracted to or repelled from the charge at Template:Math. The factor of Template:Frac corrects for double-counting the pairs of points.

In quantum mechanics, it is also possible to calculate a charge density Template:Math, which is a function of Template:Math. More specifically, Template:Math is defined as the expectation value of charge density at each point. But in this case, the above formula for Coulomb energy is not correct, due to exchange and correlation effects. The above, classical formula for Coulomb energy is then called the "direct" part of Coulomb energy. To get the actual Coulomb energy, it is necessary to add a correction term, called the "indirect" part of Coulomb energy. The Lieb–Oxford inequality concerns this indirect part. It is relevant in density functional theory, where the expectation value ρ plays a central role.

Statement of the inequality

For a quantum mechanical system of Template:Math particles, each with charge Template:Math, the Template:Math-particle density is denoted by

$P(x_1,\dots ,x_N).$

The function Template:Math is only assumed to be non-negative and normalized. Thus the following applies to particles with any "statistics". For example, if the system is described by a normalised square integrable Template:Math-particle wave function

$\psi \in L^2({ℝ}^{3N}),$

then

$P(x_1,\dots ,x_N)=|\psi (x_1,\dots ,x_N){|}^2.$

More generally, in the case of particles with spin having Template:Math spin states per particle and with corresponding wave function

$\psi (x_1,{\sigma }_1,\dots ,x_N,{\sigma }_N)$

the Template:Math-particle density is given by

$P(x_1,\dots ,x_N)=\sum _{{\sigma }_1=1}^q\cdots \sum _{{\sigma }_N=1}^q|\psi (x_1,{\sigma }_1,\dots ,x_N,{\sigma }_N){|}^2.$

Alternatively, if the system is described by a density matrix Template:Math, then Template:Math is the diagonal

$\gamma (x_1,...,x_N;x_1,...,x_N).$

The electrostatic energy of the system is defined as

$I_P=e^2\sum _{1\le i<j\le N}\underset{{ℝ}^{3N}}{\int }\frac{P(x_1,\dots ,x_i,\dots ,x_j,\dots ,x_N)}{|x_i-x_j|}{\mathrm{d}}^3{x}_1\cdots {\mathrm{d}}^3{x}_N.$

For Template:Math, the single particle charge density is given by

$\rho (x)=|e|\sum _{i=1}^N\underset{{ℝ}^{3(N-1)}}{\int }P(x_1,\dots ,x_{i-1},x,x_{i+1},\dots ,x_N){\mathrm{d}}^3{x}_1\cdots {\mathrm{d}}^3{x}_{i-1}{\mathrm{d}}^3{x}_{i+1}\cdots {\mathrm{d}}^3{x}_N$

and the direct part of the Coulomb energy of the system of Template:Math particles is defined as the electrostatic energy associated with the charge density Template:Math, i.e.

$D(\rho )=\frac{1}{2}\underset{{ℝ}^3}{\int }\underset{{ℝ}^3}{\int }\frac{\rho (x)\rho (y)}{|x-y|}{\mathrm{d}}^{3}x{\mathrm{d}}^{3}y.$

The Lieb–Oxford inequality states that the difference between the true energy Template:Math and its semiclassical approximation Template:Math is bounded from below as

Template:NumBlk where Template:Math is a constant independent of the particle number Template:Math. Template:Math is referred to as the indirect part of the Coulomb energy and in density functional theory more commonly as the exchange plus correlation energy. A similar bound exists if the particles have different charges Template:Math. No upper bound is possible for Template:Math.

The optimal constant

While the original proof yielded the constant Template:Math, Lieb and Oxford managed to refine this result to Template:Math. Later, the same method of proof was used to further improve the constant to Template:Math. It is only recently that the constant was decreased to Template:Math. With these constants the inequality holds for any particle number Template:Math.

The constant can be further improved if the particle number Template:Math is restricted. In the case of a single particle Template:Math the Coulomb energy vanishes, Template:Math, and the smallest possible constant can be computed explicitly as Template:Math. The corresponding variational equation for the optimal Template:Math is the Lane–Emden equation of order 3. For two particles (Template:Math) it is known that the smallest possible constant satisfies Template:Math. In general it can be proved that the optimal constants Template:Math increase with the number of particles, i.e. Template:Math, and converge in the limit of large Template:Math to the best constant Template:Math in the inequality (Template:EquationNote). Any lower bound on the optimal constant for fixed particle number Template:Math is also a lower bound on the optimal constant Template:Math. The best numerical lower bound was obtained for Template:Math where Template:Math. This bound has been obtained by considering an exponential density. For the same particle number a uniform density gives Template:Math.

The largest proved lower bound on the best constant is Template:Math, which was first proven by Cotar and Petrache. The same lower bound was later obtained in using a uniform electron gas, melted in the neighborhood of its surface, by Lewin, Lieb & Seiringer. Hence, to summarise, the best known bounds for Template:Math are Template:Math.

The Dirac constant

Historically, the first approximation of the indirect part Template:Math of the Coulomb energy in terms of the single particle charge density was given by Paul Dirac in 1930 for fermions. The wave function under consideration is

$\psi (x_1,{\sigma }_1,\dots ,x_N,{\sigma }_N)=\frac{\det ({\phi }_i(x_j,{\sigma }_j))}{\sqrt{N!}}.$

With the aim of evoking perturbation theory, one considers the eigenfunctions of the Laplacian in a large cubic box of volume Template:Math and sets

${\phi }_{\alpha ,k}(x,\sigma )=\frac{{\chi }_{\alpha }(\sigma ){\mathrm{e}}^{2\pi \mathrm{i}k\cdot x}}{\sqrt{|\mathrm{\Lambda }|}},$

where Template:Math forms an orthonormal basis of Template:Math. The allowed values of Template:Math are Template:Math with Template:Math. For large Template:Math, Template:Math, and fixed Template:Math, the indirect part of the Coulomb energy can be computed to be

$E_P(\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{c})=-C|e{|}^{2/3}{q}^{-1/3}{\rho }^{4/3}|\mathrm{\Lambda }|,$

with Template:Math.

This result can be compared to the lower bound (Template:EquationNote). In contrast to Dirac's approximation the Lieb–Oxford inequality does not include the number Template:Math of spin states on the right-hand side. The dependence on Template:Math in Dirac's formula is a consequence of his specific choice of wave functions and not a general feature.

Generalisations

The constant Template:Math in (Template:EquationNote) can be made smaller at the price of adding another term to the right-hand side. By including a term that involves the gradient of a power of the single particle charge density Template:Math, the constant Template:Math can be improved to Template:Math. Thus, for a uniform density system Template:Math.

Further reading

• Template:Cite book