What density functional theory changes

A density can organize the ground-state problem

A two-electron wavefunction depends on both electron positions and spins. The density depends on one position and tells us the expected number of electrons per unit volume. It discards much visible detail, so it is surprising that ground-state energy can be formulated in terms of the density.

The Hohenberg–Kohn formulation establishes an exact density variational principle for electrons in an external potential. An exact universal functional exists under the stated ground-state conditions. That result changes how the problem can be organized, but it does not hand us a tractable exact formula for every molecule.

Practical density functional approximations supply specific formulas. Their accuracy must be evaluated as approximations. A failure of PBE on a stretched molecule reveals a limitation of that particular approximation and its chosen solution.

Reading: Sandbox Physics · P. Hohenberg and W. Kohn

Kohn–Sham orbitals separate the difficult energy

Kohn–Sham theory introduces an auxiliary noninteracting system with the desired density. Its orbital kinetic energy is $T_s$ . Add attraction to the nuclei and the classical Coulomb self-energy of the total density. Everything still needed to recover the interacting energy belongs to $E_{\mathrm{xc}}$ , the exchange-correlation functional. This includes the difference between interacting and auxiliary kinetic energies.

Minimizing the energy yields self-consistent one-electron equations. They resemble HF equations, but the effective exchange-correlation potential comes from the chosen functional. The orbitals are computational tools for constructing the density. Their energies are not a list of independent interacting-electron energies to be summed into a total.

E[\rho]=T_s[\rho]+\int v_{\mathrm{ext}}(\mathbf r)\rho(\mathbf r)\,d^3r+E_{\mathrm H}[\rho]+E_{\mathrm{xc}}[\rho]+E_{\mathrm{nn}}

The external potential is nuclear attraction; nuclear repulsion is added explicitly. The exact exchange-correlation functional would make this decomposition exact.

E_{\mathrm H}[\rho]=\frac12\iint\frac{\rho(\mathbf r)\rho(\mathbf rprime)}{|\mathbf r-\mathbf rprime|}\,d^3r\,d^3rprime

The factor of one half counts each pair once. Exact exchange-correlation must also remove the unphysical self-interaction contained in this classical density expression.

Reading: Sandbox Physics · W. Kohn and L. J. Sham

Select a functional and preserve its numerical settings

PBE is a generalized-gradient approximation. It uses spin densities and their spatial gradients to approximate exchange-correlation energy. Its published form is fixed; choosing PBE does not mean fitting its parameters to the small hydrogen data shown here.

The saved comparison uses spin-restricted Kohn–Sham PBE in STO-3G, generated with PySCF 2.14.0 and Libxc 7.1.2. The unpruned atomic integration grid has 99 radial points and 590 angular points per shell. The energy-convergence tolerance is one trillionth of a hartree. The repository preserves a denser-grid check and a reproducible input script.

Spin-restricted and spin-unrestricted Kohn–Sham calculations are different searches. Allowing different alpha and beta orbitals can change the dissociation curve, just as extra spin freedom matters in HF. The plotted PBE reference always uses the restricted search, so the spin constraint remains fixed throughout the functional comparison.

Reading: Sandbox Physics · J. P. Perdew · W. Kohn and L. J. Sham

A lower PBE number has a different meaning

At $R=1.4$ bohr, restricted PBE/STO-3G gives approximately $-1.152064373$ hartree. Same-basis FCI is $-1.137275944$ . The PBE number is lower because the approximate functional energy is not the expectation of the same Hamiltonian in a normalized trial wavefunction. The wavefunction upper-bound argument therefore does not apply.

This example does not rank PBE by proximity to an experimental molecular energy: both the small basis and the approximate functional affect that different question. It compares method behavior at a fixed geometry and basis. To understand why restricted approximate functionals struggle at dissociation, the final chapter examines a sharper condition on a single hydrogen atom with fractional spin.

Reading: Sandbox Physics · J. P. Perdew · A. J. Cohen

Try the prediction

Read PBE as a named numerical calculation

Select the PBE curve and inspect its energy difference from same-basis FCI near equilibrium.
Follow the fixed geometry points toward separation. Compare the change in error with the wavefunction-method curves.
Open the calculation details and identify the basis, spin restriction and grid. Explain which change would require a new reference dataset.

Exercises and explanations

If an exact density functional exists, why do different practical functionals predict different energies?

Read the explanation

Existence does not supply the exact formula. PBE and other approximations replace the unknown exchange-correlation functional with different tractable expressions. Numerical basis and grid choices add further controlled approximations.

Can a PBE energy below finite-basis FCI violate the variational theorem?

Read the explanation

The wavefunction theorem bounds the expectation of one specified Hamiltonian over normalized states. Approximate PBE evaluates a different approximate energy functional, so that theorem does not compare its value with same-basis FCI.

Why should a DFT comparison record its integration grid?

Read the explanation

Exchange-correlation contributions are numerically integrated over space. A coarse grid can alter energies and derivatives even when the SCF equations are tightly converged. Grid convergence measures the remaining integration error against the claimed method difference.

Sources and calculation scope

Sandbox Physics, FR13 reference inputs, raw numerical records and reproduction guide (2026)

Original generated data: H2/STO-3G method energies, forces and occupations; independent Psi4 checks; basis and DFT-grid checks; fixed-orbital fractional-spin data. The guide links scripts, raw outputs, runtime versions and data licenses.

P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas (1964)

Ground-state density and the exact density variational principle. This existence result does not provide an approximate functional.

W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects (1965)

Auxiliary noninteracting orbitals and the separation of kinetic, electrostatic and exchange-correlation energies.

J. P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple (1996)

Published PBE generalized-gradient functional, with the 1997 erratum. The method-error experiment selects PBE; its saved numerical record fixes the library, grid and spin restriction.

A. J. Cohen, P. Mori-Sánchez and W. Yang, Fractional spins and static correlation error in density functional theory (2008)

Locked preprint v1, 12 May 2008; published as J. Chem. Phys. 129, 121104, DOI 10.1063/1.2987202. The final chapter studies the exact constancy condition and a newly computed finite-basis illustration of the hydrogen-atom figure.