The Hohenberg-Kohn Theorems: Foundations of DFT

Density Functional Theory (DFT) revolutionized materials science and computational chemistry by providing a computationally tractable way to study the electronic structure of many-electron systems. At its heart lie the Hohenberg-Kohn (HK) theorems, which establish the theoretical groundwork for using the electron density as the fundamental variable.

The First Hohenberg-Kohn Theorem: Uniqueness of the Ground State

The ground-state electron density uniquely determines the external potential and thus the Hamiltonian of a system.

This theorem states that for any system of interacting electrons in an external potential, the ground-state electron density, $\rho_0(r)$ , is a unique functional of the external potential, $v(r)$ . This means that if you know the ground-state density, you can, in principle, determine everything about the system, including the Hamiltonian and all its properties.

The first HK theorem can be proven by contradiction. Assume there are two different external potentials, $v_1(r)$ and $v_2(r)$ , that yield the same ground-state electron density, $\rho_0(r)$ . Let the corresponding Hamiltonians be $H_1 = T + V_{ee} + \int v_1(r) \hat{\rho}(r) dr$ and $H_2 = T + V_{ee} + \int v_2(r) \hat{\rho}(r) dr$ , where $T$ is the kinetic energy operator and $V_{ee}$ is the electron-electron interaction operator. Let $\Psi_1$ and $\Psi_2$ be the ground-state wavefunctions for $H_1$ and $H_2$ , respectively. By the variational principle, the ground-state energy for $H_1$ is $E_1 = \langle \Psi_1 | H_1 | \Psi_1 \rangle$ . If we use $\Psi_1$ as a trial wavefunction for $H_2$ , we get $E_2' = \langle \Psi_1 | H_2 | \Psi_1 \rangle = \langle \Psi_1 | H_1 - \int v_1(r) \hat{\rho}(r) dr + \int v_2(r) \hat{\rho}(r) dr | \Psi_1 \rangle = E_1 + \int (v_2(r) - v_1(r)) \rho_1(r) dr$ , where $\rho_1(r)$ is the ground-state density corresponding to $H_1$ . Since $\Psi_1$ is the ground state for $H_1$ , $E_1 < E_2'$ . Thus, $E_2' = \langle \Psi_1 | H_2 | \Psi_1 \rangle > E_1$ . However, if $v_1(r) \neq v_2(r)$ , then $\rho_1(r) \neq \rho_0(r)$ (the ground-state density for $H_2$ ), which contradicts the assumption that both potentials yield the same ground-state density $\rho_0(r)$ . Therefore, the ground-state density uniquely determines the external potential, and consequently, the Hamiltonian and all its properties.

What fundamental quantity does the first Hohenberg-Kohn theorem state uniquely determines the Hamiltonian of a system?

The ground-state electron density.

The Second Hohenberg-Kohn Theorem: Variational Principle

The second HK theorem provides a variational principle for the ground-state energy. It states that for any trial density $\rho(r)$ that has the same number of electrons as the true ground-state density, the energy functional $E[\rho(r)] = T[\rho] + V_{ee}[\rho] + \int v(r) \rho(r) dr$ will have its minimum value when $\rho(r)$ is the true ground-state density, $\rho_0(r)$ . This minimum energy is the ground-state energy, $E_0$ . The functional $E[\rho(r)]$ can be written as $E[\rho] = F_{HK}[\rho] + \int v(r) \rho(r) dr$ , where $F_{HK}[\rho] = T[\rho] + V_{ee}[\rho]$ is the universal functional of the electron density, independent of the external potential. This theorem is crucial because it allows us to search for the ground-state energy by minimizing an energy functional with respect to the electron density, rather than the complex many-electron wavefunction.

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The second HK theorem is the cornerstone for practical DFT calculations. It allows us to reformulate the many-body problem into a minimization problem over the electron density. The challenge then becomes finding the exact form of the universal functional $F_{HK}[\rho]$ , which includes the kinetic energy and electron-electron interaction terms. The Kohn-Sham ansatz addresses this by introducing fictitious non-interacting electrons that yield the same ground-state density as the real interacting system. This leads to the Kohn-Sham equations, which are a set of single-particle Schrödinger-like equations.

The Hohenberg-Kohn theorems are exact, but their practical application relies on approximations for the exchange-correlation functional.

Implications and Limitations

The HK theorems provide a rigorous justification for DFT. They show that the ground-state properties of a system are determined by its electron density, which is a much simpler quantity than the many-electron wavefunction. This simplification makes DFT applicable to systems with hundreds or even thousands of atoms, which would be intractable with traditional wavefunction-based methods. However, the theorems themselves do not provide the explicit form of the energy functionals, particularly the exchange-correlation functional, which must be approximated. The accuracy of DFT calculations heavily depends on the quality of these approximations.

What is the main challenge in applying the Hohenberg-Kohn theorems in practice?

Finding the exact form of the universal functional, especially the exchange-correlation part.

Learning Resources

Density Functional Theory - Wikipedia(wikipedia)

Provides a comprehensive overview of DFT, including its historical development and the foundational Hohenberg-Kohn theorems.

Hohenberg-Kohn Theorems - Chemistry LibreTexts(documentation)

A clear explanation of the two Hohenberg-Kohn theorems with mathematical derivations suitable for students.

Introduction to Density Functional Theory - Materials Science(tutorial)

An interactive tutorial that introduces DFT concepts, including the HK theorems, within the context of materials science.

The Hohenberg-Kohn Theorems - Lecture Notes(documentation)

Lecture notes from a renowned expert (Walter Kohn) explaining the theoretical underpinnings of DFT and the HK theorems.

A Brief History of Density Functional Theory(paper)

A historical perspective on the development of DFT, highlighting the significance of the Hohenberg-Kohn theorems.

Density Functional Theory: A Practical Introduction(video)

A video lecture that provides a practical introduction to DFT, touching upon the theoretical basis including the HK theorems.

The Hohenberg-Kohn Theorems - University of Cambridge(documentation)

Detailed lecture notes covering the Hohenberg-Kohn theorems and their implications for electronic structure calculations.

Introduction to Density Functional Theory (DFT)(video)

A video explaining the core concepts of DFT, including the fundamental theorems that make it a powerful tool.

Density Functional Theory (DFT) - Computational Chemistry(video)

An introductory video to DFT in computational chemistry, explaining its principles and the role of the HK theorems.

Hohenberg-Kohn Theorems - Quantum Mechanics(documentation)

Lecture notes focusing on the quantum mechanical underpinnings of DFT, with a specific section on the Hohenberg-Kohn theorems.