Derivation of the Weizs\"acker Density Functional from Probability Theory

TL;DR

This paper proves that the Weizsäcker potential is an exact component of the universal density functional in ground-state density functional theory, using probability theory and kinetic energy analysis without approximations.

Contribution

It provides a rigorous, approximation-free derivation of the Weizsäcker potential as an exact term in DFT's universal functional.

Findings

Weizsäcker potential is exact in the universal functional.

The proof relies solely on probability theory and kinetic energy form.

Other kinetic energy terms are also examined.

Abstract

We demonstrate that the Weizs\"acker potential is an exact term in the universal functional in density functional theory (DFT) for the ground state of a system with $N$ electrons. This proof uses no approximations or physical arguments, and follows from the form of kinetic energy of the ground state and probability theory. We also examine the form of the other terms in the kinetic energy.

Equations

$$F_N[n]=T_N[n]+V_{ee,N}[n]$$

$$\Psi (r_1,...,r_N)=\sqrt{\rho (r_1,...,r_N)}{e}^{i\varphi (r_1,...,r_N)}$$

$$\begin{array}{rl}T& =\int d^3{r}_1...\int d^3{r}_N{\Psi }^{\ast }(r_1,...,r_N)\left(-\frac{1}{2}\sum _i{\nabla }_i^2\right)\Psi (r_1,...,r_N)\\ & =-\frac{N}{2}\int d^3{r}_1...\int d^3{r}_N{\Psi }^{\ast }(r_1,...,r_N){\nabla }_1^2\Psi (r_1,...,r_N)\\ & =\frac{N}{2}\int d^3{r}_1...\int d^3{r}_N\rho (r_1,...,r_N)\left({\left|\frac{1}{2}{\nabla }_1\log (\rho (r_1,...,r_N))\right|}^2+{\left|{\nabla }_1\varphi (r_1,...,r_N)\right|}^2\right)\end{array}$$

$$T=\frac{N}{8}\int d^3{r}_1...\int d^3{r}_N\rho (r_1,...,r_N){\left|{\nabla }_1\log (\rho (r_1,...,r_N))\right|}^2$$

$$\rho (r_1,...,r_N)=\rho (r_2,...,r_N|r_1)\rho (r_1)$$

$$\begin{array}{rl}& \log (\rho (r_1,...,r_N))=\log (\rho (r_2,...,r_N|r_1))+\log (\rho (r_1))\\ ⟹& {\nabla }_1\log (\rho (r_1,...,r_N))={\nabla }_1\log (\rho (r_2,...,r_N|r_1))+{\nabla }_1\log (\rho (r_1))\end{array}$$

$$\begin{split}T&=\frac{N}{8}\int d^{3}r_{1}...\int d^{3}r_{N}\rho(r_{1},...,r_{N})\Bigl{(}{\left|\nabla_{1}\log(\rho(r_{1}))\right|}^{2}\\ &+2\nabla_{1}\log(\rho(r_{1}))\cdot\nabla_{1}\log(\rho(r_{2},...,r_{N}|r_{1}))+{\left|\nabla_{1}\log(\rho(r_{2},...,r_{N}|r_{1}))\right|}^{2}\Bigr{)}\\ \end{split}$$

$$\begin{array}{rl}{T}_1& =\frac{N}{8}\int d^3{r}_1...\int d^3{r}_N\rho (r_1,...,r_N){\left|{\nabla }_1\log (\rho (r_1))\right|}^2\\ & =\frac{N}{8}\int d^3{r}_1...\int d^3{r}_N\rho (r_2,...,r_N|r_1)\rho (r_1){\left|{\nabla }_1\log (\rho (r_1))\right|}^2\\ & =\frac{N}{8}\int d^3{r}_1\rho (r_1){\left|{\nabla }_1\log (\rho (r_1))\right|}^2\\ & =\frac{N}{8}\int d^{3}r\rho (r){\left|\nabla \log (\rho (r))\right|}^2\\ & =\frac{N}{8}\int d^{3}r\frac{{\left|\nabla \rho (r)\right|}^2}{\rho (r)}\end{array}$$

$$T_1=\frac{1}{8}\int d^{3}r\frac{{\left|\nabla n(r)\right|}^2}{n(r)}$$

$$\begin{array}{rl}{T}_2& =\frac{N}{4}\int d^3{r}_1...\int d^3{r}_N\rho (r_1,...,r_N){\nabla }_1\log (\rho (r_1))\cdot {\nabla }_1\log (\rho (r_2,...,r_N|r_1))\\ & =\frac{N}{4}\int d^3{r}_1...\int d^3{r}_N\rho (r_2,...,r_N|r_1){\nabla }_1\log (\rho (r_1))\cdot {\nabla }_1\log (\rho (r_2,...,r_N|r_1))\\ & =\frac{N}{4}\int d^3{r}_1...\int d^3{r}_N{\nabla }_1\rho (r_1)\cdot {\nabla }_1\rho (r_2,...,r_N|r_1)\\ & =\frac{N}{4}\int d^3{r}_1{\nabla }_1\rho (r_1)\cdot {\nabla }_1\left(\int d^3{r}_2...\int d^3{r}_N\rho (r_2,...,r_N|r_1)\right)\\ & =\frac{N}{4}\int d^3{r}_1{\nabla }_1\rho (r_1)\cdot {\nabla }_{1}1=0\end{array}$$

$$\begin{array}{rl}{T}_3& =\frac{N}{8}\int d^3{r}_1...\int d^3{r}_N\rho (r_1,...,r_N){\left|{\nabla }_1\log (\rho (r_2,...,r_N|r_1))\right|}^2\\ & =\frac{N}{8}\int d^3{r}_1...\int d^3{r}_N\rho (r_2,...,r_N|r_1)\rho (r_1){\left|{\nabla }_1\log (\rho (r_2,...,r_N|r_1))\right|}^2\\ & =\int d^3{r}_1\rho (r_1)T(r_1)=\int d^{3}r\rho (r)T(r)\end{array}$$

$$T(r)=\frac{N}{8}\int d^3{r}_2...\int d^3{r}_N\rho (r_2,...,r_N|r){\left|\nabla \log (\rho (r_2,...,r_N|r))\right|}^2$$

$$g_{ij}(r)=\int d^3{r}_2...\int d^3{r}_N\rho (r_2,...,r_N|r){\nabla }_i\log (\rho (r_2,...,r_N|r)){\nabla }_j\log (\rho (r_2,...,r_N|r))$$

$$T(r)=\frac{N}{8}\sum _i{g}_{ii}(r)$$

$$T_3=\frac{N}{8}\int d^{3}r\rho (r)\sum _i{g}_{ii}(r)=\frac{1}{8}\int d^{3}rn(r)\sum _i{g}_{ii}(r)$$

Full text

I Universal Functional

The Hohenberg-Kohn theorems state that the ground state of $N$ electrons in a potential is determined uniquely by the electron density [1] . This implies the existence of a universal functional that accounts for the kinetic energy and electron-electron potential energies.

[TABLE]

Where $T_{N}[n]$ is the kinetic energy and $V_{ee,N}[n]$ is the electron-electron interaction energy.

The existence of the universal functional implies that any term in $F_{N}[n]$ found using first principles must be an exact term in the the universal functional.

II Kinetic Energy

The wavefunction $\Psi$ for a system on $N$ electrons is given by:

[TABLE]

Where $\rho$ is the joint probability, $\phi$ is the phase, and $r_{n}$ is the position of the $n$-th electron.

The kinetic energy is given by [2]:

[TABLE]

Where we have used the fact that the wavefunction is antisymmetric to change variables for the Laplacian of the position of each electron to operate only on $r_{1}$ N times.

No other terms in the electron-electron interactions or the potential depend on the phase $\phi$. If we wish to find the ground state, we minimize the energy, and the minimum of the phase dependent term in our kinetic energy is zero and occurs when the phase is a constant.

The kinetic energy for the ground state is then:

[TABLE]

III The Weizsäcker Functional

The joint probability distribution $\rho(r_{1},...,r_{N})$ can be rewritten as a single electron probability distribution and a conditional probability:

[TABLE]

This gives us:

[TABLE]

The kinetic energy is then:

[TABLE]

Let us denote the three terms in the order they are shown as $T_{1}$, $T_{2}$ and $T_{3}$ so that the kinetic energy is $T=T_{1}+T_{2}+T_{3}$. The first term can be simplified:

[TABLE]

Where the conditional probability $\rho(r_{2},...,r_{N}|r_{1})$ factors out and integrates to $1$, and we relabel $r_{1}$ as $r$.

The density is equal to the number of electrons times the probability distribution of one electron, $n(r)=N\rho(r)$, so we obtain the kinetic energy term:

[TABLE]

Which is the Weizsäcker functional[3].

This means the Weizsäcker functional is an exact term in the universal density functional.

IV Other Kinetic Energy Terms

We now examine the other terms in (7). The second term $T_{2}$:

[TABLE]

$T_{2}$ vanishes. $T_{3}$ gives us:

[TABLE]

Where we relabeled $r_{1}$ to $r$ and defined a kinetic energy function given by:

[TABLE]

The information metric [4] for the conditional probability distribution of all but one electron positions on that of the position of one electron, is given by:

[TABLE]

We then see that the kinetic energy function $T(r)$ is proportional to the trace of the information metric:

[TABLE]

So that $T_{3}$ is given as:

[TABLE]

This result substantiates previous work that finds the kinetic energy is proportional to the information metric [5], and explorations of the relationship between Density Functional Theory and information physics more generally [6].