Questions: Introduction to Density Functional Theory: From Wavefunctions to Electron Density

Question 1

A colleague claims that DFT is fundamentally less rigorous than wavefunction methods because it avoids the exact many-body Schrödinger equation. Which response best refutes this claim?

• A. DFT is equally rigorous because it uses the same Hartree-Fock equations, just reformulated in terms of density
• B. The Hohenberg-Kohn theorems are exact — the ground-state energy is provably a unique functional of the electron density, so DFT's theoretical foundation is as rigorous as any wavefunction method
• C. DFT is less rigorous, but this is acceptable because computational efficiency outweighs theoretical exactness
• D. DFT avoids the Schrödinger equation through empirical fitting of density functionals to experimental data

Answer: (B)

The Hohenberg-Kohn theorems prove rigorously that the exact ground-state energy is a unique functional of the electron density — this is an exact theorem, not an approximation. The only approximation in practical DFT is the exchange-correlation functional E_xc[ρ], because its exact form is unknown. The theoretical basis of DFT is no less rigorous than wavefunction methods; both ultimately derive from quantum mechanics. Option A is wrong because DFT replaces, rather than reformulates, the HF equations.

Question 2

In Kohn-Sham DFT, the real system of interacting electrons is replaced by a fictitious system of non-interacting electrons with the same density. What is the purpose of this substitution?

• A. To avoid the Pauli exclusion principle, which makes multi-electron wavefunctions antisymmetric and hard to compute
• B. To reduce the system to a single electron, which can be solved exactly with the hydrogen atom solution
• C. To allow the kinetic energy and classical Coulomb energy to be computed straightforwardly, leaving only the unknown exchange-correlation energy to be approximated
• D. To convert the wavefunction into an orbital-free representation where no basis set is needed

Answer: (C)

For non-interacting electrons, the kinetic energy decomposes into a sum of one-electron terms and is easily computed from single-particle orbitals. The classical electron-electron repulsion (Hartree energy) is also tractable. What remains — the difference between the true interacting kinetic energy and the non-interacting kinetic energy, plus all non-classical many-body effects — is swept into E_xc[ρ]. This is the only term requiring approximation. The Kohn-Sham approach makes DFT computationally similar in cost to HF while, in principle, capturing all correlation effects through E_xc.

Question 3

The Hohenberg-Kohn theorems prove that the exact ground-state energy is a unique functional of the electron density; the only source of error in practical DFT calculations is the approximate exchange-correlation functional.

• T. True
• F. False

Answer: True

This is the key distinction that separates DFT's theoretical basis from its practical implementation. The theorems themselves are mathematically exact — any two systems with the same ground-state electron density have the same ground-state energy, and the true density minimizes the energy functional. The gap between theory and practice lies entirely in E_xc[ρ]: since its exact form is unknown, approximations like LDA, GGA, and hybrid functionals must be used. Calling DFT 'inherently approximate' conflates the exact theorem with the practical functional approximation.

Question 4

A more expensive functional (e.g., a hybrid functional) generally gives more accurate results than a cheaper one (e.g., GGA) for any molecular property.

• T. True
• F. False

Answer: False

While Jacob's ladder describes a general improvement in accuracy going from LDA → GGA → hybrid → double-hybrid, this trend is not universal across all properties and systems. For example, LDA can outperform GGA for certain solid-state properties, and hybrid functionals like B3LYP sometimes underperform PBE for metal surfaces. Dispersion-dominated systems require specialized corrections (DFT-D3, ωB97X-D) regardless of functional rung. Functional selection is problem-specific: B3LYP is reliable for organic molecules, PBE for solids, range-separated hybrids for charge-transfer systems. More expensive does not automatically mean more accurate.

Question 5

Why does replacing the 3N-variable many-electron wavefunction with the three-variable electron density provide a computational advantage in DFT? What is the key theoretical result that makes this substitution valid?

Model answer: A many-electron wavefunction depends on 3N spatial coordinates (three per electron), making it exponentially harder to represent as the system grows. The electron density ρ(r) always depends on just three spatial variables regardless of how many electrons the system has. The Hohenberg-Kohn theorems validate this substitution by proving that the ground-state electron density uniquely determines the external potential and therefore all ground-state properties — so no information relevant to the ground state is lost by switching from the wavefunction to the density as the fundamental variable.

This computational advantage is dramatic in practice: for a system of 100 electrons, the wavefunction lives in a 300-dimensional space while the density is always three-dimensional. The Hohenberg-Kohn first theorem guarantees this isn't a lossy compression — the density contains all the same physical information as the full wavefunction for ground-state properties. This is why DFT scales as roughly N³–N⁴ with system size (similar to HF) rather than the exponential scaling of exact correlated methods.