Kohn–Sham Equations¶

The Hohenberg–Kohn theorems established that the ground-state electron density uniquely determines all properties of a molecular system. However, the theorems do not provide a practical method for calculating that density.

In 1965, Walter Kohn and Lu Jeu Sham introduced a practical formulation of Density Functional Theory known as the Kohn–Sham method.

Rather than solving the complicated interacting many-electron problem directly, the Kohn–Sham approach replaces it with an equivalent system of non-interacting electrons that reproduces exactly the same ground-state electron density.

This elegant idea allows Density Functional Theory to use computational techniques very similar to those employed in Hartree–Fock theory while incorporating electron correlation through an exchange–correlation functional.

Today, nearly every DFT calculation performed with Gaussian—including B3LYP, PBE, M06, and many other functionals—is based on the Kohn–Sham formalism.

The Central Idea¶

Instead of solving the true interacting electronic system,

Electron ↔ Electron ↔ Electron ↔ Electron

Kohn and Sham proposed solving an equivalent system of non-interacting electrons,

Electron      Electron      Electron

      ↓ Effective Potential ↓

Produces the Same Electron Density

Although the electrons in this fictitious system do not interact directly, they move within an effective potential that incorporates the effects of electron–electron interactions.

The key requirement is that the electron density of the non-interacting system must be identical to that of the real interacting system.

The Kohn–Sham Equation¶

The one-electron Kohn–Sham equations are

\[ \boxed{ \left[ -\frac12\nabla^2 + V_{\mathrm{eff}}(\mathbf r) \right] \phi_i(\mathbf r) = \varepsilon_i \phi_i(\mathbf r) } \]

where

\(\phi_i\) are the Kohn–Sham orbitals,
\(\varepsilon_i\) are the orbital energies,
\(V_{\mathrm{eff}}\) is the effective potential experienced by each electron.

Notice that this equation has almost the same mathematical form as the Hartree–Fock equation.

The Effective Potential¶

The effective potential is written as

\[ \boxed{ V_{\mathrm{eff}} = V_{ne} + V_H + V_{xc} } \]

It contains three contributions.

Term	Meaning
\(V_{ne}\)	Electron–nucleus attraction
\(V_H\)	Classical electron–electron (Hartree) repulsion
\(V_{xc}\)	Exchange–correlation potential

Together, these terms describe the environment in which each Kohn–Sham electron moves.

Electron–Nucleus Attraction¶

The first contribution,

\[ V_{ne}, \]

describes the attraction between negatively charged electrons and positively charged nuclei.

This is the same interaction that appears in Hartree–Fock theory and is responsible for binding electrons to atoms and molecules.

Hartree Potential¶

The second contribution,

\[ V_H, \]

represents the classical electrostatic repulsion between electrons.

It is obtained directly from the electron density and accounts for the average Coulomb interaction among electrons.

This term is analogous to the Coulomb contribution in Hartree–Fock theory.

Exchange–Correlation Potential¶

The third contribution,

\[ V_{xc}, \]

is the defining feature of Density Functional Theory.

It accounts for all quantum mechanical effects that are not included in the classical Hartree potential, including

exchange interactions,
electron correlation,
many-body quantum effects.

Unlike the other terms, the exact form of \(V_{xc}\) is unknown.

Instead, practical DFT calculations use approximate exchange–correlation functionals.

Developing accurate approximations for this term is one of the central goals of modern Density Functional Theory.

Kohn–Sham Orbitals¶

Although Density Functional Theory is fundamentally based on the electron density,

the Kohn–Sham method introduces molecular orbitals,

\[ \phi_1,\phi_2,\ldots,\phi_N, \]

which closely resemble Hartree–Fock orbitals.

These orbitals are mathematical constructs used to reproduce the correct electron density.

The electron density is then obtained from the occupied Kohn–Sham orbitals,

\[ \boxed{ \rho(\mathbf r) = \sum_i^{occ} |\phi_i(\mathbf r)|^2 } \]

for a spin-restricted system.

Thus,

the orbitals are not the final objective—they are an efficient means of obtaining the electron density.

Similarity to Hartree–Fock¶

The computational workflow of Kohn–Sham DFT is remarkably similar to that of Hartree–Fock.

Both methods

use molecular orbitals,
expand orbitals in Gaussian basis functions,
construct matrices,
solve eigenvalue equations,
iterate using the SCF procedure.

The primary difference lies in the treatment of electron–electron interactions.

Hartree–Fock	Kohn–Sham DFT
Uses the Fock operator	Uses the Kohn–Sham operator
Exact exchange	Approximate exchange–correlation
No dynamic correlation	Includes approximate correlation
Wavefunction-based	Density-based

Because of these similarities, the same numerical algorithms can be used for both methods.

SCF Procedure in Kohn–Sham DFT¶

A Kohn–Sham calculation follows the familiar SCF cycle.

Read Molecular Geometry
          │
          ▼
Choose Basis Set
          │
          ▼
Generate Initial Electron Density
          │
          ▼
Construct Kohn–Sham Matrix
          │
          ▼
Solve Kohn–Sham Equations
          │
          ▼
Obtain New Orbitals
          │
          ▼
Update Electron Density
          │
          ▼
Check Convergence
          │
     Yes ─┴─ No
          │
          ▼
   Final Electronic Energy

This workflow is essentially identical to the Hartree–Fock SCF algorithm, with the effective potential replacing the Hartree–Fock Fock operator.

Why Is the Exchange–Correlation Functional So Important?¶

The success of a DFT calculation depends largely on how accurately the exchange–correlation energy is approximated.

Different functionals make different approximations for this quantity.

As a result,

different functionals may predict different energies,
optimized geometries may vary slightly,
reaction energies and activation barriers can differ.

Choosing an appropriate exchange–correlation functional is therefore one of the most important decisions in practical DFT calculations.

Connection to Gaussian¶

When a Gaussian input contains

# B3LYP/6-31G

Gaussian

constructs the basis functions,
builds the Kohn–Sham matrix,
evaluates the exchange–correlation functional,
performs SCF iterations,
reports the converged electronic energy.

Thus, every B3LYP calculation in Gaussian is fundamentally a Kohn–Sham DFT calculation.

Summary¶

The Kohn–Sham method transforms Density Functional Theory into a practical computational framework by replacing the interacting many-electron system with an equivalent system of non-interacting electrons moving in an effective potential. The resulting Kohn–Sham equations closely resemble the Hartree–Fock equations and are solved using the same basis sets, matrix methods, and SCF procedure. The major distinction lies in the exchange–correlation potential, which incorporates quantum mechanical exchange and electron correlation through approximate functionals. This formulation underlies nearly all modern DFT calculations performed with Gaussian.

Next Section¶

The next chapter examines Exchange–Correlation Functionals, the only approximate component of the Kohn–Sham formalism. We will explore why this term cannot be determined exactly, how different approximations are constructed, and why the B3LYP functional has become one of the most widely used choices in computational chemistry.