Can Traditional DFT Calculate Excited-State Properties?

Ground-state density functional theory (DFT) is designed to find the electronic ground state. That calculation can provide useful ground-state properties, but it does not automatically produce excited-state energies, spectra, or relaxed excited-state structures. To study those properties, you need an excited-state method built on or alongside DFT.

The most common choice for molecular electronic excitations is time-dependent DFT (TD-DFT). For certain states, a separate self-consistent-field calculation with a changed orbital occupation—often called ΔSCF—can also be useful. Neither method makes every excited state equally reliable, so the character of the excitation matters.

What ground-state DFT does and does not tell you

In Kohn–Sham DFT, the calculation solves an auxiliary one-electron problem whose density is used to represent the interacting system’s ground-state density. The Kohn–Sham orbital energies and orbitals are computational tools; they are not generally a direct list of measurable electron-removal, electron-addition, or excitation energies.

In particular, subtracting the energy of an occupied Kohn–Sham orbital from that of a virtual orbital is not generally a quantitative excitation energy. The difference can sometimes help identify a likely orbital transition, but the energy of an actual excited state includes the response of the electrons and depends on the exchange-correlation approximation.

Ground-state DFT can still provide a starting geometry, electron density, and orbitals for an excited-state calculation. These are useful inputs, not a substitute for an excited-state treatment.

How TD-DFT calculates excitations

TD-DFT applies time-dependent perturbation theory to the DFT description. In its commonly used linear-response form, it calculates excitation energies and transition properties from the response of the electron density to a perturbation, such as light. It uses the chosen exchange-correlation functional, usually with a set of ground-state orbitals as its starting point.

A typical TD-DFT calculation can report vertical excitation energies, transition dipole moments, oscillator strengths, and excited-state transition densities. A vertical excitation is evaluated at the ground-state geometry, without allowing the nuclei to move during the electronic transition. The oscillator strength is a dimensionless measure related to how strongly a transition couples to light; a transition with zero or very small oscillator strength can be weak or absent in an ordinary absorption spectrum.

For example, suppose a calculation reports a vertical excitation at $3.00\,\text{eV}$ with oscillator strength $f=0.10$. The approximate wavelength corresponding to that photon energy is

$$\lambda \approx \frac{1239.84\,\text{eV}\,\text{nm}}{3.00\,\text{eV}} = 413.3\,\text{nm}.$$

This places the transition in the visible region. The value $f=0.10$ indicates a transition with nonzero but not especially strong intensity compared with a strongly allowed transition. It is not a probability that the molecule will be excited in every experiment; observed intensities also depend on conditions such as concentration and spectral broadening.

When ΔSCF is useful

In ΔSCF, you perform separate self-consistent calculations for the ground state and an excited electronic configuration, then take their energy difference:

$$\Delta E = E_{\text{excited}} - E_{\text{ground}}.$$

This can be effective when the target state can be represented by a clear occupation change and the calculation converges to the intended state. It may also let you optimize an excited-state geometry and examine its electron density. However, the method can converge to the wrong state, and an occupation pattern does not always uniquely identify the desired physical state. Open-shell states may also require attention to spin contamination and spin-state treatment.

TD-DFT and ΔSCF are not interchangeable recipes. TD-DFT is usually the more direct choice for absorption energies and oscillator strengths. ΔSCF can be attractive for selected excited-state energies, densities, or geometries, but its reliability depends on state tracking and the chosen functional.

Which properties can you calculate?

Property	Common approach	Main qualification
Vertical excitation energy	TD-DFT	Accuracy depends on state character and functional.
Absorption intensity	TD-DFT	Use oscillator strengths; compare with a spectrum using suitable broadening.
Excited-state geometry	Excited-state TD-DFT optimization or ΔSCF	Track the same electronic state throughout optimization.
Emission energy	Relaxed excited-state and ground-state calculations	Include geometry relaxation; a vertical absorption energy is not an emission energy.
Excited-state density	State-specific or response-density analysis	A transition density is not the same thing as the ordinary excited-state density.
Bonding or charge redistribution	Excited-state density and orbital analysis	Interpret orbitals as qualitative aids, not uniquely observable quantities.

A vertical absorption energy is evaluated at the ground-state geometry; an emission energy is usually evaluated at a relaxed excited-state geometry. Comparing them without accounting for nuclear relaxation can therefore be misleading.

Important limitations

Conventional local and semilocal functionals often underestimate the energy of long-range charge-transfer excitations. Global hybrid functionals can improve some cases, while range-separated hybrids are often considered for long-distance charge transfer. Results still need validation for the system at hand.

Adiabatic TD-DFT, the widely used approximation based on an instantaneous exchange-correlation response, also has difficulty with many states dominated by double excitations. Near conical intersections or other regions where electronic states are strongly coupled, a basic linear-response calculation may not describe the physics well. A different or more specialized method may be needed, such as a multireference approach.

Common mistakes

• Treating a Kohn–Sham orbital gap as a spectrum. It is not generally the excitation energy; calculate the excitation with an appropriate excited-state method.
• Calling a vertical excitation a relaxed-state energy. Vertical energies hold the nuclei fixed. Geometry optimization changes the energy and may change state ordering.
• Assuming every TD-DFT state is trustworthy. Inspect transition character, charge transfer, oscillator strength, and functional dependence.
• Confusing transition and excited-state densities. A transition density describes coupling between states; an excited-state density describes charge in a particular excited state.
• Assuming a changed occupation guarantees the right ΔSCF state. Verify the converged state and monitor it during geometry changes.

For a routine molecular absorption spectrum, TD-DFT is often a practical first method. For a specific excited-state density or optimized structure, consider state-specific approaches such as ΔSCF and check that they represent the state you intend to study. If the state is strongly charge-transfer, doubly excited, or near a state crossing, benchmark against a more suitable method or experimental data where possible.