Density Functional Theory for Two-Dimensional Homogeneous Materials

Abstract

We study Density Functional Theory models for systems which are translationally invariant in some directions, such as a homogeneous 2-d slab in the 3-d space. We show how the different terms of the energy are modified and we derive reduced equations in the remaining directions. In the Thomas–Fermi model, we prove that there is perfect screening, and provide decay estimates for the electronic density away from the slab. In Kohn–Sham models, we prove that the Pauli principle is replaced by a penalization term in the energy. In the reduced Hartree–Fock model in particular, we prove that the resulting model is well-posed, and give some properties of the minimizer.

Appendix A. A Lieb–Thirring Inequality

In this section, we explain how to use Theorem 2.6 to obtain a Lieb–Thirring type inequality [18, 19]. We state our result in the general dimension $d\in \mathbb {N}\setminus \left\{ 0\right\}$.

Proposition A.1

Let $d\in \mathbb {N}\setminus \left\{ 0\right\}$ and $G\in {{\mathcal {S}}}(L^2(\mathbb {R}^d))$ be a positive operator. Then, for any $s\in \mathbb {N}\setminus \left\{ 0\right\}$,

(35)

with the constant

$$\begin{aligned} K := \dfrac{ K_\mathrm{LT}(d+s)^{1 + \frac{s}{d}} }{ ( 2c_{\mathrm{TF}}(s))^{\frac{s}{d}} } \left( \dfrac{s}{d+s} \right) ^{s/d} \dfrac{d}{d+s}. \end{aligned}$$

Here, $K_\mathrm{LT}(d+s)$ is the usual Lieb–Thirring constant in dimension $d+s$, that is the best constant in the inequality

$$\begin{aligned} \forall \gamma \in {{\mathcal {S}}}\left( L^2(\mathbb {R}^{d+s})\right) , \quad 0 \le \gamma \le 1, \quad K_\mathrm{LT}(d+s) \int _{\mathbb {R}^{d+s}} \rho _\gamma ^{1 + \frac{2}{d+s}} \le \mathrm{Tr}_{d+s}( - \Delta \gamma ).\nonumber \\ \end{aligned}$$

(36)

Proof

Consider $G \in {{\mathcal {S}}}(L^2(\mathbb {R}^d))$ , $G \ge 0$ such that $\mathrm{Tr}(G^{1 + \frac{2}{s}}) < \infty$ and consider the optimal $\gamma \in {{\mathcal {S}}}(L^2(\mathbb {R}^{d+s}))$ as in (30). Then, the Lieb–Thirring inequality (36) applied to $\gamma$ gives (after the appropriate per unit surface normalization)

$$\begin{aligned} \frac{1}{2} \mathrm{Tr}_d(- \Delta _d G) + c_{\mathrm{TF}}(s) \mathrm{Tr}_d(G^{1 + \frac{2}{s}}) = \frac{1}{2} \underline{\mathrm{Tr}}_{d+s}(- \Delta _{d+s} \gamma ) \ge \frac{1}{2} K_\mathrm{LT}(d+s) \int _{\mathbb {R}^d} \rho _\gamma ^{1 + \frac{2}{d+s}}, \end{aligned}$$

Since $\rho _\gamma = \rho _G$ then, for all $G \in {{\mathfrak {S}}}_{1 + \frac{2}{s}}(\mathbb {R}^d)$ such that $\mathrm{Tr}(- \Delta G) < \infty$, we get $\rho _G \in L^{1 + \frac{2}{d+s}}(\mathbb {R}^d)$, and

$$\begin{aligned} \mathrm{Tr}_d(- \Delta _d G) + 2 c_{\mathrm{TF}}(s) \mathrm{Tr}_d(G^{1 + \frac{2}{s}}) \ge K_\mathrm{LT}(d+s) \int _{\mathbb {R}^d} \rho _G^{1 + \frac{2}{d+s}}. \end{aligned}$$

Performing the scaling $G_{\lambda } = \lambda G$ and optimizing over $\lambda$ gives the result. $\square$

Proposition A.1 corresponds to the Lieb-Thirring inequality for operators in ${{\mathcal {S}}}(L^2(\mathbb {R}^{s+d}))$ in a semi-classical limit, when the semi-classical limit dilation is only performed in the first s variables (see also [20] for similar arguments).

This type of inequalities was recently studied in [10], where it is shown that for all $d \ge 0$ and $1 \le p \le 1 + \frac{2}{d}$, there is an optimal constant $K_{p,d}$ so that

$$\begin{aligned} K_{p,d} \Vert \rho _G \Vert _p^{\frac{2p}{d(p-1)}} \le \Vert G \Vert _{{{\mathfrak {S}}}^{q}}^{\frac{p(2-d) + d}{d(p-1)}} \mathrm{Tr}_d (- \Delta _d G), \quad \text {with} \quad q := \dfrac{2p + d - dp}{2 + d - dp}. \end{aligned}$$

It is proved that this constant is the dual constant of the usual Lieb–Thirring constant $L_{\gamma , d}$with $\gamma =q/(q-1)$. The case in Proposition A.1 corresponds to the choice

$$\begin{aligned} p = 1 + \dfrac{2}{d+s} \quad \text {with} \quad s \in \mathbb {N}, \quad \text {so that} \quad q = 1 + \frac{2}{s}. \end{aligned}$$

This corresponds to the dual constant $L_{\gamma , d}$ with $\gamma = \frac{q}{q-1} = 1 + \frac{s}{2}$. In particular, since $s \ge 1$, we have $\gamma \ge \frac{3}{2}$. In this regime, it is known that the best constant is the semi-classical one: $L_{\gamma , d} = L_{\gamma , d}^\mathrm{sc}$, hence $K_{p,d} = K_{p, d}^\mathrm{sc}$. This proves that the optimal constant K in the inequality (35) is the semi-classical one. In particular, we have

$$\begin{aligned} \frac{1}{2} \mathrm{Tr}_d(- \Delta _d G) + c_{\mathrm{TF}}(s) \mathrm{Tr}_d(G^{1 + \frac{2}{s}}) \ge c_\mathrm{TF}(d+s) \int _{\mathbb {R}^d} \rho _G^{1 + \frac{2}{d+s}}. \end{aligned}$$

In other words, the energy in the rHF model is always greater than the energy in the TF model.