Density Functional Theory for the Electron Gas and for Jellium

TL;DR

This paper investigates the relationship between the exact density functionals for the electron gas and jellium, showing their exchange-correlation functionals are identical, which supports using jellium simulations to develop electron gas functionals.

Contribution

It demonstrates that the exchange-correlation functionals for inhomogeneous electron gas and jellium are the same, justifying the use of jellium in functional development.

Findings

Exchange-correlation functionals are identical for both systems.

Supports using jellium simulations to inform electron gas functionals.

Addresses issues related to the thermodynamic limit.

Abstract

Density Functional Theory relies on universal functionals characteristic of a given system. Those functionals in general are different for the electron gas and for jellium (electron gas with uniform background). However, jellium is frequently used to construct approximate functionals for the electron gas (e.g., local density approximation, gradient expansions). The precise relationship of the exact functionals for the two systems is addressed here. In particular, it is shown that the exchange - correlation functionals for the inhomogeneous electron gas and inhomogeneous jellium are the same. This justifies theoretical and quantum Monte Carlo simulation studies of jellium to guide the construction of functionals for the electron gas. Related issues of the thermodynamic limit are noted as well.

Equations

$${\hat{H}}_e=\sum _{\alpha =1}^N\frac{{\hat{p}}_{\alpha }^2}{2m}+\frac{1}{2}{e}^2\int d\mathbf{r}d{\mathbf{r}}^{\prime }\frac{\hat{n}(\mathbf{r})\hat{n}({\mathbf{r}}^{\prime })-\hat{n}(\mathbf{r})\delta \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}.$$

$$\hat{n}(\mathbf{r})=\sum _{\alpha =1}^N\delta \left(\mathbf{r}-{\hat{\mathbf{q}}}_{\alpha }\right).$$

$${\hat{H}}_j=\sum _{\alpha =1}^N\frac{{\hat{p}}_{\alpha }^2}{2m}+\frac{1}{2}{e}^2\int d\mathbf{r}d{\mathbf{r}}^{\prime }\frac{\left(\hat{n}(\mathbf{r})-n_b\right)\left(\hat{n}({\mathbf{r}}^{\prime })-n_b\right)-\hat{n}(\mathbf{r})\delta \left(\mathbf{r}-{\mathbf{r}}^{\prime }\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}.$$

$${\hat{H}}_j={\hat{H}}_e+\int d\mathbf{r}{v}_b\left(\mathbf{r}\right)\hat{n}(\mathbf{r})+E_b$$

$$v_b\left(\mathbf{r}\right)=-e^2\int d{\mathbf{r}}^{\prime }\frac{n_b}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}.$$

$$E_b=\frac{1}{2}\int d\mathbf{r}d{\mathbf{r}}^{\prime }\frac{{\left(n_{b}e\right)}^2}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}=-\frac{1}{2}\int d\mathbf{r}{n}_b{v}_b\left(\mathbf{r}\right).$$

$$\beta {\Omega }_e\left(\beta ,V\mid \mu \right)=-\ln \sum _{N=0}^{\infty }T{r}^{(N)}{e}^{-\beta \left({\hat{H}}_e-\int d\mathbf{r}\mu (\mathbf{r})\hat{n}(\mathbf{r})\right)},$$

$$\beta {\Omega }_j\left(\beta ,V\mid \mu \right)=-\ln \sum _{N=0}^{\infty }T{r}^{(N)}{e}^{-\beta \left({\hat{H}}_j-\int d\mathbf{r}\mu (\mathbf{r})\hat{n}(\mathbf{r})\right)},$$

$$\mu (\mathbf{r})=\mu -v_{ex}\left(\mathbf{r}\right).$$

$$\beta {\Omega }_j\left(\beta ,V\mid \mu \right)=\beta E_b+\beta {\Omega }_e\left(\beta ,V\mid \mu -v_b\right).$$

$$X_e\left(\beta ,V\mid \mu \right)\equiv \sum _{N=0}^{\infty }T{r}^{(N)}{e}^{\beta {\Omega }_e}{e}^{-\beta \left({\hat{H}}_e-\int d\mathbf{r}\mu (\mathbf{r})\hat{n}(\mathbf{r})\right)}\hat{X},$$

$$X_j\left(\beta ,V\mid \mu \right)\equiv \sum _{N=0}^{\infty }T{r}^{(N)}{e}^{\beta {\Omega }_j}{e}^{-\beta \left({\hat{H}}_j-\int d\mathbf{r}\mu (\mathbf{r})\hat{n}(\mathbf{r})\right)}\hat{X},$$

$$X_j\left(\beta ,V\mid \mu \right)=X_e\left(\beta ,V\mid \mu -v_b\right)$$

$$n_e\left(\mathbf{r},\beta ,V\mid \mu \right)=-\frac{\delta \beta {\Omega }_e\left(\beta ,V\mid \mu \right)}{\delta \beta \mu (\mathbf{r})},n_j\left(\mathbf{r},\beta ,V\mid \mu \right)=-\frac{\delta \beta {\Omega }_j\left(\beta ,V\mid \mu \right)}{\delta \beta \mu (\mathbf{r})}.$$

$$n_j\left(\mathbf{r},\beta ,V\mid \mu \right)=n_e\left(\mathbf{r},\beta ,V\mid \mu -v_b\right).$$

$$F_e\left(\beta ,V\mid n_e\right)={\Omega }_e\left(\beta ,V\mid \mu \right)+\int d\mathbf{r}\mu (\mathbf{r})n_e\left(\mathbf{r},\beta ,V\mid \mu \right),$$

$$F_j\left(\beta ,V\mid n_j\right)={\Omega }_j\left(\beta ,V\mid \mu \right)+\int d\mathbf{r}\mu (\mathbf{r})n_j\left(\mathbf{r},\beta ,V\mid \mu \right).$$

$$F_j\left(\beta ,V\mid n_j\right)$$

$$F_j\left(\beta ,V\mid n_j\right)=F_e\left(\beta ,V\mid n_j\right)+\int d\mathbf{r}{v}_b(\mathbf{r})n_j\left(\mathbf{r}\mid \mu \right)+\beta E_b.$$

$$F_j\left(\beta ,V\mid n\right)=F_e\left(\beta ,V\mid n\right)+\int d\mathbf{r}{v}_b(\mathbf{r})n\left(\mathbf{r}\right)+\beta E_b.$$

$$F=F_0+F_H+F_{xc}$$

$$F_{eH}\left(\beta ,V\mid n\right)=\frac{1}{2}{e}^2\int d\mathbf{r}d{\mathbf{r}}^{\prime }\frac{n(\mathbf{r})n({\mathbf{r}}^{\prime })}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|},$$

$$F_{jH}\left(\beta ,V\mid n\right)=\frac{1}{2}{e}^2\int d\mathbf{r}d{\mathbf{r}}^{\prime }\frac{\left(n(\mathbf{r})-n_b\right)\left(n({\mathbf{r}}^{\prime })-n_b\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}.$$

$$F_{eH}\left(\beta ,V\mid n\right)+\int d\mathbf{r}{v}_b(\mathbf{r})n\left(\mathbf{r}\right)+\beta E_b=F_{jH}\left(\beta ,V\mid n\right).$$

$$F_{jxc}\left(\beta ,V\mid n\right)=F_{exc}\left(\beta ,V\mid n\right).$$

$$\frac{\delta F_j\left(\beta ,V\mid n\right)}{\delta n(\mathbf{r})}=\mu (\mathbf{r})=\frac{\delta F_e\left(\beta ,V\mid n\right)}{\delta n(\mathbf{r})}+v_b(\mathbf{r}).$$

$$\frac{\delta F_j\left(\beta ,V\mid n\right)}{\delta n(\mathbf{r})}=\mu (\mathbf{r})=\mu -v_{ex}\left(\mathbf{r}\right).$$

$$\frac{\delta F_e\left(\beta ,V\mid n\right)}{\delta n(\mathbf{r})}=\mu (\mathbf{r})-v_b(\mathbf{r})=\mu -\left(v_{ex}(\mathbf{r})+v_b\left(\mathbf{r}\right)\right).$$

$$F_{exc}\left(\beta ,V\mid n\right)=\int d\mathbf{r}{f}_{exc}\left(\mathbf{r},\beta ,V\mid n\right).$$

$$f_{exc}\left(\mathbf{r},\beta ,V\mid n\right)=f_{exc}\left(\mathbf{r},\beta ,V\mid n\right){\mid }_{n\left(\mathbf{r}\right)}+\int d{\mathbf{r}}^{\prime }\frac{\delta f_{exc}\left(\mathbf{r},\beta ,V\mid n\right)}{\delta n\left({\mathbf{r}}^{\prime }\right)}{\mid }_{n\left(\mathbf{r}\right)}\left(n\left({\mathbf{r}}^{\prime }\right)-n\left(\mathbf{r}\right)\right)$$

Full text

I Introduction

Keith Gubbins has been an inspiration to students and colleagues for more than fifty years, through direct mentoring and collaborations, and indirectly through his extensive innovative publications. My own collaboration was very early in my career as a new faculty member in the Physics Department at the University of Florida. Keith had just finished his book with Tim Reed showing the relevance of statistical mechanics to applications in chemical and other engineering fields. He continued to demonstrate the importance of formulating practical approximations to complex systems through their firm foundation in basic statistical mechanics. He also demonstrated by example the need to ignore the artificial interfaces between applied and basic disciplines, taking recent developments in the mathematics and physics literature to formulate innovative and accurate descriptions of complex systems, via the reinforcement of theory, simulation, and experiment. An example is his early application of non-local classical density functional theory (DFT) for adsorption in carbon slit pores, verified by simulation and including a discussion of problems of extracting experimental parameters needed for the calculation Gubbins97 . The presentation here has in common the tool of density functional theory for application to a complex system (warm dense matter). But the system is quite different - confined electrons - and hence quantum effects can dominate except at the highest temperatures. The emphasis is on formal relationships to guide practical applications, so it is hoped that Keith will appreciate some of his spirit in the following.

The historical development of DFT has followed two quite different paths, one for quantum systems at zero temperature (e.g., electrons in atoms, molecules, and solids) Hohenberg64 , ParrWang89 , Dreizler90 , Jones15 , and one for temperature dependent classical non-uniform fluids (e.g., two phase liquids, porous media) Evans92 , Lutsko10 . The former has focused on determination of the ground state energy whereas the latter has focused on the classical free energy of thermodynamics. Their relationship was established by Mermin who extended the original ground state theorems of Hohenberg and Kohn to finite temperatures for quantum systems Mermin65 , Eschrig10 described by the Gibbs ensemble of statistical mechanics. This early history is briefly reviewed in references Jones15 ,Evans92 . Recently the two paths have rejoined in the effort to describe conditions of warm, dense, matter WDM . These are solid density conditions but temperatures for electrons ranging from zero temperature ground state to well above the Fermi temperature. They include domains with atomic and molecular coexistence, and with association and dissociation chemistry relevant. A central ingredient is the (intrinsic) free energy functional $F$, consisting of a non-interacting contribution $F_{0}$, a Hartree contribution, $F_{H}$, and a remainder called the exchange - correlation free energy functional, $F_{xc}$. The exact functional form for the Hartree contribution is known and the contribution from $F_{0}$ is treated exactly (numerically) within the Kohn - Sham formulation of DFT KS65 extended to finite temperatures. Hence the primary challenge for applications is construction of the exchange - correlation density functional.

An important constraint is its equivalence with the corresponding functional for jellium (electrons in a uniform neutralizing background Vignale05 ) when evaluated at a uniform density. Although still a difficult quantity to determine, the latter has been studied widely by approximate theoretical methods Vignale05 , Sjostrom13 and more recently by quantum simulation methods across the temperature - density plane Dornheim16 . An accurate fitting function for practical applications now exists KSDT15 . Its utility for DFT is within the ”local density approximation”, first proposed in reference KS65 for the zero temperature energy functional and extended to finite temperatures for the free energy functional. It assumes that the non-uniform system $F_{xc}$ can be represented at each point by the uniform jellium $F_{xc}$ evaluated at the density for that point.

The objective here is to clarify the precise relationship of the DFT functionals for the two different systems, electron gas and jellium, for general non-uniform densities. The primary new result is that the exchange - correlation functionals for the two systems are equivalent, and the total free energies (intrinsic plus external potential) are the same. In the analysis it noted that jellium is usually considered in the thermodynamic limit, $\overline{N}\rightarrow\infty,$ $V\rightarrow\infty,$ $\overline{N}/V$ where $\overline{N}$ and $V$ are the average particle number and volume, respectively. It has been proved that the jellium free energy is well-defined in this limit Lieb75 , whereas that for the electron gas is not (it does not scale as the volume for large system size due in part to lack of charge neutrality). Nevertheless, the exchange - correlation component of the free energy for the electrons does have a proper thermodynamic limit as a consequence of the equivalence demonstrated here. In the next section, Hamiltonians for the isolated electron gas and jellium are defined. Next, the statistical mechanical basis for DFT is described for the grand canonical ensemble. The grand potential (proportional to the pressure) is defined as a functional of a given external potential $v_{ex}\left(\mathbf{r}\right)$ (occuring through a local chemical potential $\mu(\mathbf{r})=\mu-v_{ex}\left(\mathbf{r}\right)$), and the corresponding functionals for the inhomogeneous electron gas and inhomogeneous jellium are related. Associated with $\mu(\mathbf{r})$ are the densities $n_{e}(\mathbf{r})$ and $n_{j}(\mathbf{r})$ for the two systems. Their relationship is established. Next, the strict concavity of the grand potential functionals assures a one - to - one relationship of the densities to $\mu(\mathbf{r})$ so that a change of variables is possible. This is accomplished by Legendre transformations which define the free energy density functionals. It is noted that the free energy density functional obtained in this way is precisely that of Mermin’s DFT. The free energies differ by the potential energy of the background charge. It is shown that this cancels the differences between the orresponding Hartree free energy contributions, resulting in the desired equivalence of the exchange - correlation functionals.

II Inhomogeneous electron and jellium functionals

The Hamiltonian for an isolated system of $N$ electrons in a volume $V$  is given by

[TABLE]

where the number density operator is

[TABLE]

The position and momentum operators for electron $\alpha$ are $\widehat{\mathbf{q}}_{\alpha}$ and $\widehat{\mathbf{p}}_{\alpha}$, respectively. A caret over a symbol denotes the operator corresponding to that variable. The related jellium Hamiltonian is

[TABLE]

The constant $n_{b}$ denotes the density of a uniform neutralizing background for the electrons. When the grand canonical ensemble is considered it is given by $n_{b}=\overline{N}/V$, where $\overline{N}$ is the average particle number. The two Hamiltonians are seen to be related by

[TABLE]

with

[TABLE]

The second term on the right side of (4) is the potential of interaction between the electrons and the background, and the third term is the background self energy

[TABLE]

Equation (3) is the usual definition of jellium as the electron system plus a uniform neutralizing background.

II.1 Grand potential functionals

Now consider the addition of an external single particle potential $v_{ex}\left(\mathbf{r}\right)$ to the electron and jellium Hamiltonians. The equilibrium properties for the corresponding inhomogeneous systems are defined by the grand canonical potentials

[TABLE]

[TABLE]

Here, the local chemical potential is defined by

[TABLE]

These grand potentials are functions of the inverse temperature $\beta$ and the volume $V$, and functionals of the local chemical potential $\mu(\mathbf{r})$. The functionals themselves, $\Omega_{e}\left[\beta,V\mid\cdot\right]$ and $\Omega_{j}\left[\beta,V\mid\cdot\right]$, are characterized by $\widehat{H}_{e}$ and $\widehat{H}_{j}$ respectively. Since the latter two are different, the functionals are different. However, from (4) they have the simple relationship

[TABLE]

In fact all average properties in the corresponding grand ensembles have a similar relationship. For example, a property represented by the operator $\widehat{X}$ has the averages

[TABLE]

[TABLE]

so with (4) and (10) they are related by

[TABLE]

II.2 Free energy density functionals

The local number densities are

[TABLE]

The derivative of $\Omega_{j}\left(\beta,V\mid\mu\right)$ is taken at constant $n_{b}$. Using (10) or (13) it is seen that the number densities are related by

[TABLE]

It can be shown that $\Omega_{e}\left(\beta,V\mid\cdot\right)$ and $\Omega_{j}\left(\beta,V\mid\cdot\right)$ (again with constant $n_{b}$) are strictly concave functionals so that (14) defines the one to one invertible relationships $n_{e}\Longleftrightarrow\mu$ and $n_{j}\Longleftrightarrow\mu$. Consequently, a change of variables $\beta,V,\mu\rightarrow\beta,V,n_{e}$ for the inhomogeneous electron system and $\beta,V,\mu\rightarrow\beta,V,n_{j}$ for the inhomogeneous jellium are possible. The corresponding Legendre transformations then define the free energy functionals of these densities

[TABLE]

[TABLE]

These are precisely the density functionals of DFT (e.g., defined by Mermin Mermin65 ).

Their relationship follows using (4) and (15)

[TABLE]

and using (15) again gives the desired result.

[TABLE]

Since $\mu\left(\mathbf{r}\right)$ is arbitrary so also is $n_{j}\left(\mathbf{r}\mid\mu\right)$ and (19) can be written more simply as

[TABLE]

III Exchange-correlation functional equivalence

Traditionally the free energy density functional is separated into a non-interaction contribution, $F_{0}$, a mean-field Hartree contribution, $F_{H}$, and the remaining exchange-correlation contribution, $F_{xc}$

[TABLE]

Clearly $F_{0}$ is the same for the electron and jellium systems, as follows from (20) since $v_{b}(\mathbf{r})=0$ in this case). However, the Hartree terms (defined as the average intrinsic internal energy with pair correlation function equal to unity) are different

[TABLE]

[TABLE]

In particular, their system size dependence is quite different. For example, in the uniform density limit (assuming a spherical volume) $F_{eH}\left(\beta,V\mid n\right)\rightarrow C\left(n_{e}e\right)^{2}V^{5/3}$ (with $C=\left(4\pi\right)^{2}\left(1/15\right)\left(3/\left(4\pi\right)\right)^{5/3}\simeq\allowbreak 0.967$), whereas $F_{jH}\left(\beta,V\mid n\right)$ vanishes in this limit. This difference is the reason (along with charge neutrality) why $F_{j}\left(\beta,V\mid n\right)$ has a proper thermodynamic limit Lieb75 while $F_{e}\left(\beta,V\mid n\right)$ does not (e.g., it does not scale linearly with the volume).

At first sight this seems at odds with (20) since the left side has a thermodynamic limit whereas each term on the right separately does not. However, the second and third terms cancel the singular volume dependence of $F_{eH}$ so that

[TABLE]

This observation, together with the equavalence of the non-interacting contributions, leads to the equivalence of the exchange - correlation contributions for the electron gas and jellium

[TABLE]

Note that this equivalence applies for general inhomogeneous densities, extending the familiar relationship for uniform systems.

These are two different inhomogeneous systems, yet their correlations are the same for every admissable density. The functionals are each ”universal” in the sense that their forms are independent of the underlying external potential. However, the potential associated with the chosen density of their argument is different in each case. To see this consider the functional derivative of (20) with respect to the density

[TABLE]

Hence the density $n$ is determined from the jellium functional by

[TABLE]

Alternatively the same density is determined from the electron functional by

[TABLE]

For jellium the potential is $v_{ex}\left(\mathbf{r}\right)$ while for the electron gas it is $v_{ex}\left(\mathbf{r}\right)+v_{b}\left(\mathbf{r}\right)$. The two solutions to (27) and (28) have the same relationship as expressed in (15).

IV Discussion

Practical approximations for the electron gas exchange - correlation functional are typically introduced at the level of its density, defined by

[TABLE]

A formal functional expansion about the density at point $\mathbf{r}$ can be performed

[TABLE]

[TABLE]

The coefficients are evaluated at the ”uniform density” $n\left(\mathbf{r}\right)$, i.e. all functional density dependence is evaluated at the same value. Consequently, the lead term is just the uniform electron gas exchange

• correlation free energy per unit volume

[TABLE]

This is known as the ”local density approximation. Similarly the subsequent terms in (30) are the response functions for the uniform electron gas. From the above analysis all of these can now be identifed with those for jellium, which has been studied extensively. As noted in the Introduction, an accurate analytic fit for $F_{jxc}\left(\beta,V,n_{e}\right)$ is now available across the entire $\beta,n_{e}$ plane KSDT15 , so the local density approximation is known explicitly. Similarly, the first few response functions for jellium are known as well Vignale05 .

More generally approximations for exchange - correlations away from the uniform limit (e.g., generalized gradient approximations) can be addressed for inhomogeneous jellium as well. While this is a difficult problem it is placed in a more controlled thermodynamic context due to charge neutrality and extensivity.

V Acknowledgements

This research was supported by US DOE Grant DE-SC0002139.