How to Calculate the Singlet–Triplet Gap and Open-Shell Character

The singlet–triplet gap compares the energies of two electronic states with different total spin. Calculating it is straightforward once the states and geometry are defined; interpreting it requires more care, especially for molecules with diradical character.

The key distinction is between the energy gap and the open-shell character. A small gap can be consistent with a diradical, but it does not by itself prove that either state has a particular electronic structure.

Define the gap and its sign

A common convention for specified singlet and triplet states, $S$ and $T$, is

$$\Delta E_{ST}=E(S)-E(T),$$

where the states being compared must be identified explicitly. In standard spectroscopic notation, $S_0$ is the singlet ground state, $S_1$ is the first excited singlet, and $T_1$ is the lowest triplet. Thus, a gap between the singlet ground state and the lowest triplet is $E(S_0)-E(T_1)$, while a comparison of the first excited singlet and lowest triplet is $E(S_1)-E(T_1)$. With the convention above, a positive gap means the triplet is lower in energy; a negative gap means the singlet is lower. Some authors instead define $E(T)-E(S)$, so always give the formula, not just the number.

Also specify what “gap” means. A vertical gap compares states at the same nuclear geometry. An adiabatic gap compares their energies at separately optimized geometries. If you include zero-point vibrational or thermal corrections, state that too; those corrections produce a different quantity from the electronic-energy gap.

Calculate the two state energies

For a vertical gap, choose one geometry, calculate the singlet and triplet energies there using a consistent electronic-structure method and basis set, and subtract. In linear-response TD-DFT from a singlet reference, if the states are identified correctly, the vertical difference between the first excited singlet and lowest triplet can be calculated from their excitation energies:

$$\Delta E_{ST}=\omega_{S_1}-\omega_{T_1}.$$

The common ground-state energy cancels. This is valid only when both excitation energies refer to the same reference geometry and ground state. Check that the states are the ones you intend to compare: state ordering can change, and an excited state can have a different character from one geometry to another.

For an adiabatic gap, optimize the singlet and triplet separately, then subtract their total energies. Use the same method, basis set, and energy conventions for both. When the singlet has substantial diradical character, a conventional closed-shell reference may describe it poorly; the numerical result can then depend strongly on the electronic-structure method. Compare methods appropriate to the system rather than assuming one functional will be reliable for every diradical.

Assess open-shell singlet character

An open-shell singlet has two electrons occupying different orbitals, with their spins coupled to give total spin zero. It differs from a closed-shell singlet, where the relevant electrons are paired in one orbital. Both have singlet multiplicity, so multiplicity alone does not distinguish them.

A useful diagnostic is the occupation of natural orbitals. For spin-summed spatial natural orbitals, occupations are between 0 and 2. In an idealized closed-shell pair, the occupied and unoccupied orbitals have occupations near 2 and 0. For a strongly diradical pair, the two important orbitals have occupations closer to 1 and 1. Intermediate values indicate partial diradical character. These are qualitative guides, not universal cutoffs; report the occupations and how they were obtained.

For this purpose, natural orbitals from a suitable correlated, multiconfigurational calculation are generally more informative than ordinary Kohn–Sham orbital occupations, which are normally assigned as 0 or 2 for a closed-shell determinant. Natural transition orbitals can help interpret a TD-DFT excitation by showing its dominant hole and particle patterns, but a complicated or strongly mixed excitation may not have a single simple “open-shell percentage.”

Broken-symmetry DFT: useful but limited

An unrestricted calculation can produce a broken-symmetry $M_S=0$ solution in which the two spin densities localize differently. This often provides a practical model for an open-shell singlet. However, the determinant is generally not an eigenfunction of total spin: its expectation value $\langle S^2\rangle$ can differ from zero. Its energy is therefore not automatically the energy of a pure singlet state.

A common workflow is to compare the broken-symmetry solution with an unrestricted triplet and inspect the spin densities and $\langle S^2\rangle$. Approximate spin-projection or exchange-coupling schemes can then be used if suitable for the problem. These are model-dependent approximations; state their use and assumptions. For a quantitative gap in a strongly multireference system, consider methods designed to describe several important configurations, such as a suitable active-space or multireference approach, and check whether the chosen active space captures the relevant orbitals.

Worked example

Suppose a TD-DFT calculation at one geometry gives an $S_1$ excitation energy of 2.10 eV and a $T_1$ excitation energy of 1.84 eV from the same singlet ground-state reference. Under the convention $\Delta E_{ST}=E(S_1)-E(T_1)$,

$$\Delta E_{ST}=2.10-1.84=0.26\ \text{eV}.$$

The positive value means $T_1$ lies 0.26 eV below $S_1$ at that geometry. It is a vertical gap, not an adiabatic gap. If a separate wavefunction analysis gives two relevant natural-orbital occupations of 1.15 and 0.85, those occupations are consistent with appreciable open-shell character: they are much closer to 1 and 1 than to 2 and 0. They do not, by themselves, establish that the gap is accurate or quantify a universal diradical percentage.

Common mistakes

• Leaving the sign convention unstated. Report the equation and the sign interpretation.
• Mixing vertical and adiabatic energies. Do not subtract excitation energies at one geometry and describe the result as a relaxed-state gap.
• Comparing states with different references or methods. Keep the calculation protocol consistent, and verify state identities rather than relying only on state numbers.
• Calling a broken-symmetry determinant a pure singlet. Check spin contamination and describe any projection or correction used.
• Inferring diradical character from a small gap alone. Use orbital occupations, spin densities, and other electronic-structure diagnostics together.
• Reporting excessive precision. Functional, basis-set, geometry, and state-character errors can exceed the last few reported digits; give units and sensible precision.