Questions: Density Functional Theory for Molecular Structure

Question 1

A molecule has 50 electrons. Compared to specifying the full many-electron wavefunction, how many spatial coordinates does the electron density ρ(r) depend on?

• A. 50 — one per electron
• B. 150 — three per electron
• C. 3 — regardless of the number of electrons
• D. It depends on the basis set used in the calculation

Answer: (C)

The electron density ρ(r) is a function of three spatial coordinates (x, y, z) regardless of how many electrons are present. This is DFT's core advantage: the wavefunction of a 50-electron system depends on 150 spatial coordinates, making optimization exponentially harder, while the density always lives in 3D space. DFT replaces the 3N-dimensional wavefunction optimization with a 3D density optimization.

Question 2

A researcher uses DFT with the PBE (GGA) functional and obtains excellent geometry predictions. A critic says 'DFT is inherently approximate because it only approximates the wavefunction.' What is the most accurate response to this critique?

• A. The critic is right — DFT approximates the wavefunction and so errors are unavoidable regardless of functional choice
• B. The Hohenberg-Kohn theorems guarantee the exact energy is a functional of density; the approximation lies in the exchange-correlation functional, not in DFT's conceptual foundation
• C. DFT is not approximate — the Kohn-Sham equations solve the exact Schrödinger equation for the real interacting system
• D. DFT accuracy is limited only by basis set completeness, not the choice of exchange-correlation functional

Answer: (B)

DFT does not approximate a wavefunction — it does not produce one at all. The Hohenberg-Kohn theorems establish that the exact ground-state energy is in principle determined by the electron density alone. The practical approximation is the exchange-correlation functional: the exact form is unknown, so LDA, GGA, and hybrid functionals are used as approximations. Option C is wrong because the Kohn-Sham equations describe a fictitious non-interacting system, not the real interacting electrons.

Question 3

The Hohenberg-Kohn theorem guarantees that practical DFT calculations yield the exact ground-state energy for any molecular system.

• T. True
• F. False

Answer: False

The Hohenberg-Kohn theorem establishes that the exact ground-state energy is uniquely determined by the electron density — in principle. In practice, the exact exchange-correlation functional is unknown. Every practical DFT calculation uses an approximation (LDA, GGA, hybrid, etc.), which introduces errors. The theorem guarantees the existence of an exact density-based route to the answer; it does not guarantee that our current functional approximations achieve it.

Question 4

In the Kohn-Sham DFT scheme, a fictitious system of non-interacting electrons is constructed to reproduce the same electron density as the real interacting system.

• T. True
• F. False

Answer: True

This is the central insight of the Kohn-Sham approach. The real many-electron problem is mapped onto an auxiliary set of non-interacting electrons whose density exactly matches the real system. These non-interacting electrons satisfy one-electron Schrödinger-like equations (the Kohn-Sham equations), which are far more tractable. All the complexity of electron-electron interaction is folded into the exchange-correlation functional.

Question 5

Why does DFT scale more favorably with molecular size than exact wavefunction-based methods like full configuration interaction?

Model answer: DFT works with the electron density — a function of only 3 spatial coordinates — rather than the N-electron wavefunction, which depends on 3N coordinates. The Kohn-Sham equations reduce the problem to N coupled one-electron equations, giving approximately O(N³) scaling. Full configuration interaction must account for all electron correlation explicitly, scaling exponentially with N. GGA and hybrid DFT functionals typically scale as O(N³), making calculations on hundreds of atoms feasible.

The 3D density formulation is the key: no matter how many electrons a molecule has, the density is always a function of (x, y, z). This dimensional reduction is what makes DFT practical for large systems like drug molecules or extended solids, where wavefunction methods become computationally prohibitive.