Hohenberg–Kohn Theorems¶

Density Functional Theory (DFT) is built upon two remarkable theorems proved by Pierre Hohenberg and Walter Kohn in 1964. These theorems established that the ground-state electron density contains all the information required to describe a many-electron system.

Before their work, it was widely believed that the many-electron wavefunction was the only quantity capable of determining molecular properties. Hohenberg and Kohn demonstrated that the much simpler electron density could serve the same purpose.

These two theorems form the theoretical foundation of modern Density Functional Theory.

Why Are These Theorems Important?¶

The electronic wavefunction for a molecule containing \(N\) electrons depends on the coordinates of every electron,

\[ \Psi(\mathbf r_1,\mathbf r_2,\ldots,\mathbf r_N). \]

For large molecules, this function becomes extraordinarily complex.

The electron density,

\[ \rho(\mathbf r), \]

depends only on the three spatial coordinates,

\[ (x,y,z), \]

regardless of the number of electrons.

The Hohenberg–Kohn theorems prove that this much simpler quantity is sufficient to determine the ground-state properties of the system.

First Hohenberg–Kohn Theorem¶

Statement¶

The ground-state electron density uniquely determines the external potential acting on the electrons (apart from an additive constant).

Mathematically,

\[ \boxed{ \rho(\mathbf r) \Longrightarrow V_{ext}(\mathbf r) } \]

Since the external potential is determined,

the positions and charges of the nuclei are also determined.

Consequently,

Electron Density
        │
        ▼
External Potential
        │
        ▼
Hamiltonian
        │
        ▼
Wavefunction
        │
        ▼
All Ground-State Properties

This means that there is a one-to-one correspondence between the ground-state electron density and the molecular system.

Physical Meaning¶

Suppose two different molecules somehow had exactly the same ground-state electron density.

The First Hohenberg–Kohn Theorem proves that this is impossible.

A unique electron density corresponds to one unique external potential, and therefore to one unique molecular Hamiltonian.

As a result,

the molecular geometry,
the electronic structure,
and all ground-state properties

are uniquely determined by the electron density.

Second Hohenberg–Kohn Theorem¶

The first theorem tells us that the density uniquely determines the system.

The second theorem tells us how to find the correct density.

Statement¶

The correct ground-state electron density minimizes the total energy functional.

Mathematically,

\[ \boxed{ E[\rho] \ge E_0 } \]

where

\(E[\rho]\) is the energy calculated from any trial density,
\(E_0\) is the exact ground-state energy.

This is analogous to the variational principle used in Hartree–Fock theory, except that the optimization is performed with respect to the electron density rather than the wavefunction.

The Energy Functional¶

Unlike Hartree–Fock, which minimizes the energy with respect to molecular orbitals,

DFT minimizes an energy functional,

\[ \boxed{ E[\rho] = T[\rho] + V_{ne}[\rho] + J[\rho] + E_{xc}[\rho] } \]

where

Term	Meaning
\(T[\rho]\)	Electronic kinetic energy
\(V_{ne}[\rho]\)	Electron–nucleus attraction
\(J[\rho]\)	Classical electron–electron repulsion
\(E_{xc}[\rho]\)	Exchange–correlation energy

Among these terms, the exchange–correlation energy is the only quantity whose exact form is unknown.

Variational Principle in DFT¶

The Second Hohenberg–Kohn Theorem allows DFT to use a variational procedure similar to Hartree–Fock.

Starting from an initial guess for the electron density,

Gaussian repeatedly

evaluates the total energy,
improves the electron density,
recalculates the energy,

until the minimum-energy density is obtained.

Thus,

Initial Density
        │
        ▼
Calculate Energy
        │
        ▼
Improve Density
        │
        ▼
Recalculate Energy
        │
        ▼
Repeat Until Minimum

The final density corresponds to the ground state of the system.

What the Hohenberg–Kohn Theorems Do Not Tell Us¶

Although the theorems prove that the electron density is sufficient,

they do not provide a practical method for obtaining it.

Specifically, they do not tell us

how to calculate the kinetic energy accurately,
how to evaluate the exchange–correlation energy,
or how to solve the resulting equations efficiently.

A practical computational framework is still required.

Enter the Kohn–Sham Method¶

In 1965, Walter Kohn and Lu Jeu Sham introduced a practical solution.

Instead of working directly with the interacting many-electron system,

they replaced it with a fictitious system of non-interacting electrons that has exactly the same ground-state electron density.

This approach leads to the Kohn–Sham equations, which retain the computational efficiency of Hartree–Fock while incorporating electron correlation through an exchange–correlation functional.

Nearly every modern DFT program, including Gaussian, is based on the Kohn–Sham formulation.

Relationship to Gaussian¶

During a DFT calculation, Gaussian uses the Hohenberg–Kohn theorems as the theoretical foundation.

The program then applies the Kohn–Sham formalism to compute the electron density and total electronic energy through an iterative SCF procedure.

The Hohenberg–Kohn theorems justify why this approach works, while the Kohn–Sham equations describe how the calculation is performed.

Summary¶

The Hohenberg–Kohn theorems establish the theoretical foundation of Density Functional Theory by proving that the ground-state electron density uniquely determines all ground-state properties of a molecular system. They also show that the exact ground-state density is the one that minimizes the total energy functional. Although these theorems do not provide a practical computational method, they justify the use of electron density as the central quantity in DFT and pave the way for the Kohn–Sham formulation used in modern quantum chemistry software.

Next Section¶

The next chapter introduces the Kohn–Sham Equations, which transform the theoretical principles of Density Functional Theory into a practical computational method that can be implemented efficiently in Gaussian.