Excited states with TDDFT¶

Ground-state SCF gives you one state: the lowest. To predict a UV/Vis spectrum, a photochemical pathway, or the colour of a dye, you need the excited states and how brightly each one absorbs. Time-dependent density-functional theory (TDDFT) is the workhorse for this: it takes a converged ground-state Kohn-Sham reference and returns vertical excitation energies and oscillator strengths at a cost only a small multiple of the ground-state SCF. This page runs TDDFT on formaldehyde two ways, the full Casida linear-response equation and its Tamm-Dancoff approximation (TDA), compares them on the same molecule, and the theory section below derives where the eigenproblem comes from and what the approximation drops.

vibe-qc builds the excitation problem from the ground-state molecular orbitals you already have, so the workflow is always two steps: converge an SCF, then hand its MO energies, MO coefficients, and density to the TDDFT solver.

import math
import numpy as np
from vibeqc import (
    Atom, Molecule, BasisSet, RKSOptions, run_rks,
    run_tddft_tda, run_tddft_casida,
)

# Formaldehyde H2CO: C at origin, O up the z-axis, two H in the xz-plane.
ang2bohr = 1.8897259886
rCO = 1.205 * ang2bohr          # C=O bond
rCH = 1.111 * ang2bohr          # C-H bond
half = math.radians(116.5) / 2  # half the H-C-H angle
Hx, Hz = rCH * math.sin(half), -rCH * math.cos(half)

mol = Molecule([
    Atom(8, [0.0,  0.0, rCO]),   # O
    Atom(6, [0.0,  0.0, 0.0]),   # C
    Atom(1, [ Hx,  0.0, Hz]),    # H
    Atom(1, [-Hx,  0.0, Hz]),    # H
])
n_occ = 8                        # 16 electrons / 2
basis = BasisSet(mol, "6-31g*")

# Step 1: converged ground-state reference (B3LYP).
opts = RKSOptions()
opts.functional = "B3LYP"
gs = run_rks(mol, basis, opts)
assert gs.converged, "ground state must converge before TDDFT"

mo_e = np.asarray(gs.mo_energies)
mo_c = np.asarray(gs.mo_coeffs)
dens = np.asarray(gs.density)

# Step 2a: Tamm-Dancoff approximation.
tda = run_tddft_tda(
    mol, basis, mo_e, mo_c, n_occ,
    n_states=5, functional="B3LYP", density_ao=dens,
)

# Step 2b: full Casida linear response on the same reference.
cas = run_tddft_casida(
    mol, basis, mo_e, mo_c, n_occ,
    n_states=5, functional="B3LYP", density_ao=dens,
)

for res in (tda, cas):
    print(f"\n=== {res.method} (B3LYP kernel) ===")
    print(f"{'state':>5} {'E/eV':>8} {'lambda/nm':>10} {'f_osc':>9}")
    for s in res.states:
        print(f"{s.index:>5} {s.excitation_energy_ev:8.4f} "
              f"{s.wavelength_nm:10.1f} {s.oscillator_strength:9.5f}")

Running it prints the lowest five singlet excitations from each method. These are the real numbers from the run, B3LYP/6-31G* on the geometry above, ground state converged in 34 iterations to E = -114.43887772 Ha (HOMO-LUMO gap 6.16 eV):

=== TDA (B3LYP kernel) ===
state     E/eV  lambda/nm     f_osc
 4.1116      301.5   0.00000
 9.1021      136.2   0.36210
 9.2420      134.2   0.00432
10.2013      121.5   0.03364
10.3771      119.5   0.00000

=== Casida (B3LYP kernel) ===
state     E/eV  lambda/nm     f_osc
 4.0906      303.1   0.00000
 9.0529      137.0   0.42018
 9.1606      135.3   0.00914
 9.8107      126.4   0.31225
10.3709      119.6   0.00000

Reading the spectrum¶

A TDDFT table is two columns of physics: where a state sits (the excitation energy, equivalently a wavelength) and how visible it is (the oscillator strength f, which is what an absorption spectrum actually plots). A state with f = 0 is dark; it exists but does not absorb a photon to first order.

The lowest excitation of formaldehyde, 4.09 eV in Casida (303 nm, near-UV) with f = 0, is the textbook n -> pi* transition: an electron promoted from the oxygen lone pair (the HOMO) into the empty C=O pi antibonding orbital (the LUMO). It is symmetry-forbidden, hence dark, and it is the state that makes formaldehyde photochemistry interesting. Experiment places it near 3.94 eV and TD-B3LYP benchmarks land at 3.9 to 4.0 eV, so 4.09 eV is the expected GGA-hybrid result for this basis.

The second state, 9.05 eV with f = 0.42, is the bright pi -> pi* transition (HOMO into the next virtual). Its large oscillator strength is the dominant feature any computed formaldehyde UV spectrum would show. The dominant_amplitudes field on each state names the leading orbital pair behind the excitation, so you can label a peak as “HOMO -> LUMO” rather than guessing:

state = cas.states[0]
occ, virt, amp = state.dominant_amplitudes[0]
print(f"leading: occ {occ} -> virt {virt}, weight {amp:.3f}")
# leading: occ 8 -> virt 1, weight -0.999   (HOMO -> LUMO)

TDA versus full Casida¶

The two solvers describe the same physics at two levels of theory. Casida solves the complete linear-response problem; TDA throws away the coupling between excitations and de-excitations (the B matrix in the theory below). The practical question is how much that costs you, and the answer on formaldehyde is “very little for the states that matter”:

State	Character	TDA / eV	Casida / eV	TDA - Casida
1	n -> pi* (dark)	4.1116	4.0906	+0.021
2	pi -> pi* (bright)	9.1021	9.0529	+0.049
3	sigma -> pi*	9.2420	9.1606	+0.081
4	mixed	10.2013	9.8107	+0.391
5	dark	10.3771	10.3709	+0.006

Two patterns are worth internalizing. First, TDA sits systematically above Casida: dropping the de-excitation coupling removes a downward shift, so TDA energies are upper bounds relative to the full response. Second, the gap is tiny for the low, well-separated valence states (21 meV on the n -> pi*) and grows only where states couple strongly or mix (state 4, where the 0.39 eV difference flags that the de-excitation block is doing real work).

This is why TDA is the default in much of the literature for valence excitations: it is cheaper, it is numerically more robust because the eigenproblem is a plain symmetric diagonalization rather than the squared Casida form, and it is immune to the triplet-instability collapse that can drive a full-response excitation energy imaginary near a near-degenerate reference. Reach for full Casida when you want oscillator strengths right to a few percent or when comparing against experimental intensities; reach for TDA for fast, stable state orderings and excited-state geometries.

The XC kernel and why the reference matters¶

TDDFT excitation energies inherit every strength and weakness of the ground-state functional, plus one extra ingredient: the exchange-correlation kernel f_xc, the response of the XC potential to a density perturbation. In vibe-qc you switch the kernel on by passing both functional= and density_ao=; the adiabatic LDA/GGA kernel (ALDA/AGGA) is then built on the same DFT grid the ground state used. Omit density_ao and you get the bare orbital-energy-difference response with only the exchange admixture, which is not a full TDDFT-functional result, and vibe-qc warns you so it cannot pass silently.

The choice of B3LYP here is deliberate. Pure GGAs such as PBE have a narrow HOMO-LUMO gap on formaldehyde and the ground-state SCF oscillates rather than converging cleanly (a real SCF difficulty of the small-gap GGA on this system, not a vibe-qc bug). The 20 percent exact exchange in B3LYP opens the gap, the SCF converges in 34 iterations, and the hybrid TDDFT kernel (exact exchange plus the ALDA part of the functional, the Bauernschmitt-Ahlrichs composition) is the canonical, well-benchmarked choice for organic chromophores. If you do want a pure-functional spectrum, converge the ground state first with a level shift or a tighter accelerator, then feed the converged MOs to the solver exactly as above.

One sharp edge: range-separated hybrids are refused, not run wrong. Their position-dependent exact exchange cannot be folded into a single global coefficient over ordinary two-electron integrals, so passing, for example, functional="wb97x" raises NotImplementedError rather than quietly returning meaningless energies. Use a global hybrid or a pure functional.

Open-shell and periodic variants¶

The closed-shell pair above has two siblings for the cases where a single RKS reference is not enough. Both build the same excitation problem from a different ground state:

run_tddft_tda_uhf(...) takes separate alpha and beta MO energies and coefficients from a UHF or UKS ground state and solves the spin-unrestricted TDA (UCIS for a UHF reference). It is the route to excitations of radicals, triplets, and other open-shell species. The polarized f_xc kernel is not yet plumbed in this driver, so a hybrid reference gets its exact-exchange admixture but no f_xc term, and the function warns when that approximation is in effect.
run_tddft_tda_periodic(result, basis, n_occ=..., ...) takes a converged Gamma-point periodic SCF result and runs the same TDA solver on its Bloch orbitals. It is intended for vacuum-padded molecular cells and large supercells, where periodic-image interactions in the excitation kernel are negligible.

Both return the same TDDFTResult container, so the reading code above is unchanged.

Citations¶

Because TDDFT routes several distinct papers, running with an output= path writes the usual .bibtex and .references siblings carrying the exact methods to cite:

tda = run_tddft_tda(
    mol, basis, mo_e, mo_c, n_occ,
    n_states=3, functional="B3LYP", density_ao=dens,
    output="h2co_tddft",
)
# writes h2co_tddft.bibtex + h2co_tddft.references

The TDDFT-specific keys that land are Runge-Gross 1984 (the TDDFT existence theorem), Casida 1995 (the linear-response working equations), Hirata-Head-Gordon 1999 (the Tamm-Dancoff approximation), and Bauernschmitt-Ahlrichs 1996 (the hybrid kernel), alongside the B3LYP functional and 6-31G* basis citations the ground state already carried. See the citations guide for the full provenance surface.

Theory¶

The sections below build the excitation problem from the ground up: why a ground-state functional can predict excited states at all, the linear-response derivation that produces the Casida eigenproblem, the Tamm-Dancoff approximation that simplifies it, and where the oscillator strength comes from.

From ground-state DFT to excitations¶

Static Kohn-Sham DFT (see Molecular DFT) is a ground-state theory: the Hohenberg-Kohn theorems are about the lowest state of a given external potential. The bridge to excited states is the Runge-Gross theorem (1984), the time-dependent analogue: for a system evolving from a fixed initial state, the time-dependent external potential is a unique functional of the time-dependent density. That legitimizes a time-dependent Kohn-Sham system whose density matches the true one, and the poles of its density response to a weak perturbation are the exact vertical excitation energies. TDDFT is the practical realization: perturb the ground-state density at frequency omega, ask where the response diverges, and read off the spectrum.

The Casida eigenproblem¶

Linearizing the time-dependent Kohn-Sham equations about the ground state, and expanding the response in the space of occupied-to-virtual orbital transitions (labeled by an occupied index i and a virtual index a), turns “find the poles of the response” into a matrix eigenvalue problem. The excitation energies omega are the eigenvalues of the coupled non-Hermitian system

\[\begin{split} \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ -\mathbf{B}^{*} & -\mathbf{A}^{*} \end{pmatrix} \begin{pmatrix} \mathbf{X} \\ \mathbf{Y} \end{pmatrix} = \omega \begin{pmatrix} \mathbf{X} \\ \mathbf{Y} \end{pmatrix}, \end{split}\]

where X collects the excitation amplitudes (occupied -> virtual) and Y the de-excitation amplitudes (virtual -> occupied). The two orbital-rotation Hessian blocks in the canonical MO basis are

\[ A_{ia,jb} = \delta_{ij}\delta_{ab}(\varepsilon_a - \varepsilon_i) + 2\,(ia|jb) - c_x\,(ij|ab) + (ia|f_\text{xc}|jb), \]

\[ B_{ia,jb} = 2\,(ia|jb) - c_x\,(ib|ja) + (ia|f_\text{xc}|jb). \]

The diagonal of A is just the orbital-energy difference of the bare transition; the integral terms are the response corrections. The Hartree term 2(ia|jb) couples transitions through the Coulomb field, c_x is the fraction of exact exchange (1 for Hartree-Fock, 0 for a pure functional, the admixture alpha for a global hybrid such as 0.20 for B3LYP), and (ia|f_xc|jb) is the adiabatic XC kernel, the piece that distinguishes TDDFT from time-dependent Hartree-Fock. Because the adiabatic kernel is local and frequency-independent, the same matrix element enters both A and B.

For real orbitals the non-Hermitian problem is recast into the symmetric form vibe-qc actually solves,

\[ (\mathbf{A}-\mathbf{B})^{1/2}\,(\mathbf{A}+\mathbf{B})\,(\mathbf{A}-\mathbf{B})^{1/2}\,\mathbf{Z} = \omega^{2}\,\mathbf{Z}, \]

whose eigenvalues are the squared excitation energies. The square root of A - B requires that matrix to be positive definite, which holds for a stable ground state; a negative eigenvalue there signals a triplet or singlet instability of the reference, and the squared form makes that failure visible as an imaginary omega.

The Tamm-Dancoff approximation¶

The Tamm-Dancoff approximation (TDA) sets B = 0: it discards the coupling between excitations and de-excitations entirely. The coupled non-Hermitian problem then collapses to a single Hermitian eigenvalue problem

\[ \mathbf{A}\,\mathbf{X} = \omega\,\mathbf{X}, \]

a plain symmetric diagonalization of the A block alone. When the kernel has no f_xc (pure exact exchange, c_x = 1) this is exactly configuration interaction singles (CIS); with the DFT kernel it is TDA-DFT. Dropping B is rarely a large error for low valence states (the formaldehyde table above quantifies it at tens of meV), and it buys three things: lower cost, a numerically more stable eigenproblem, and immunity to the triplet-instability collapse, because the missing A - B square root can no longer go imaginary. The trade is slightly worse oscillator strengths and energies that are upper bounds relative to full response.

Oscillator strengths¶

An excitation energy tells you where a peak is; the oscillator strength tells you how tall it is. It is built from the transition dipole moment between the ground state and excited state I, which in the AO basis comes from the transition density assembled from the excitation amplitudes,

\[ P^{0I}_{\mu\nu} = \sum_{ia} X^{I}_{ia}\,(C_{\mu i} C_{\nu a} + C_{\mu a} C_{\nu i}), \qquad \boldsymbol{\mu}_{0I} = \operatorname{tr}\!\big(\mathbf{P}^{0I}\,\hat{\boldsymbol{\mu}}\big). \]

The dimensionless oscillator strength in the length gauge is then

\[ f_I = \tfrac{2}{3}\,\omega_I\,|\boldsymbol{\mu}_{0I}|^{2}, \]

with omega and the dipole in atomic units. A symmetry-forbidden transition has a vanishing transition dipole and so f = 0 (the formaldehyde n -> pi* state), while a strongly dipole-allowed transition such as pi -> pi* carries most of the spectral weight. The length-gauge form above is accurate to roughly 5 percent for valence excitations, which is why intensities are usually quoted to two significant figures rather than more.

Resources¶

The formaldehyde example runs in well under a second on one core after the ground-state SCF, on the same order as the RKS reference itself. The cost driver is the four-index AO-to-MO integral transformation and the diagonalization of the transition-space matrix, both of which scale with the number of occupied-virtual pairs; the ALDA/AGGA kernel adds one grid pass per occupied-virtual pair. For the small 6-31G* basis here that is negligible, but it grows quickly with basis size, so request only as many n_states as you need.

References¶

TDDFT existence theorem. E. Runge and E. K. U. Gross, “Density-Functional Theory for Time-Dependent Systems,” Phys. Rev. Lett. 52, 997 (1984). The time-dependent analogue of the Hohenberg-Kohn theorem that makes TDDFT well-defined.
Casida linear-response equations. M. E. Casida, “Time-Dependent Density Functional Response Theory for Molecules,” in Recent Advances in Density Functional Methods, Part I, ed. D. P. Chong, 155-192 (World Scientific, 1995). The working matrix formulation solved here.
Tamm-Dancoff approximation. S. Hirata and M. Head-Gordon, “Time-dependent density functional theory within the Tamm-Dancoff approximation,” Chem. Phys. Lett. 314, 291 (1999). Introduces and benchmarks the B = 0 approximation; the source of the TDA-B3LYP reference values.
Adiabatic hybrid kernel. R. Bauernschmitt and R. Ahlrichs, “Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory,” Chem. Phys. Lett. 256, 454 (1996). The composition of the hybrid TDDFT kernel as exact exchange plus f_xc.
Excited-states review. A. Dreuw and M. Head-Gordon, “Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules,” Chem. Rev. 105, 4009 (2005). A practical survey of TDDFT, its failures (charge-transfer, Rydberg, doubly excited states), and where TDA helps.

Next¶

Natural orbitals cover the ground-state density analysis that complements the natural transition orbitals you can extract from a TDDFT state with compute_nto, the most compact orbital picture of an excitation.
Molecular DFT is the ground-state reference every TDDFT run starts from; the functional and grid choices there set the accuracy ceiling for the excitations.
Open-shell SCF is the UHF/UKS reference that feeds run_tddft_tda_uhf for radicals and triplets.