TDDFT, TDA and CIS excited states

vibe-qc computes vertical excitation energies, oscillator strengths, transition dipoles and natural transition orbitals (NTOs) from a converged ground-state reference by linear-response time-dependent density-functional theory (TDDFT), in its full Casida form or in the Tamm-Dancoff approximation (TDA). On a Hartree-Fock reference the TDA is configuration interaction singles (CIS). The energies are production for restricted and unrestricted references with LDA, GGA and global-hybrid kernels; the finite-difference excited-state gradient is a preview that covers only the closed-shell RHF/CIS surface. Two defects in the current build affect printed numbers and are described under Known defects. The label vocabulary is defined in Methods: theory and status. The tutorial Excited states with TDDFT works one molecule through the physics; this page is the reference.

Quick start

Water at PBE0/6-31G*, the TDA and the full Casida equation, through run_job. The excitation problem is solved after the SCF converges and written into the .out.

from vibeqc import Atom, Molecule, run_job

ang = 1.8897259886
mol = Molecule([
    Atom(8, [0.0, 0.0, 0.1173 * ang]),
    Atom(1, [0.0, 0.7572 * ang, -0.4692 * ang]),
    Atom(1, [0.0, -0.7572 * ang, -0.4692 * ang]),
], charge=0, multiplicity=1)

result = run_job(
    mol, basis="6-31g*", method="rks", functional="pbe0",
    tddft=True, tddft_type="tda", tddft_n_states=4, tddft_spectrum=True,
    output="h2o_tda",
)
run_job(
    mol, basis="6-31g*", method="rks", functional="pbe0",
    tddft=True, tddft_type="casida", tddft_n_states=4,
    output="h2o_casida",
)

On a vibe-qc 0.17.10 development build (Apple M5 Max) each job takes on the order of ten seconds on an otherwise idle machine (several times that on a busy one), most of it the XC-kernel evaluation (see Cost). The SCF energy is -76.3238384517 Ha and the .out carries this block (Casida below it):

  ## TD-DFT excited states (TDA (pbe0))
  State   E (eV)  λ (nm)   f_osc  Dominant transition
      1   8.4355   147.0  0.0330  5→1 (0.999), 5→6 (0.016), 5→4 (0.015)
      2  10.4117   119.1  0.0000  5→2 (0.999), 5→7 (0.033), 5→3 (0.026)
      3  10.9988   112.7  0.2089  4→1 (0.990), 3→2 (0.125), 5→5 (0.042)
      4  13.1764    94.1  0.1700  4→2 (0.982), 3→1 (0.176), 3→4 (0.036)

  ## TD-DFT excited states (Casida (pbe0))
      1   8.4037   147.5  0.0279  5→1 (0.999), 5→4 (0.027), 5→8 (0.016)
      2  10.4037   119.2  0.0000  5→2 (0.999), 5→7 (0.037), 5→3 (0.030)
      3  10.9262   113.5  0.2429  4→1 (0.996), 3→2 (0.078), 5→5 (0.018)
      4  13.1031    94.6  0.2248  4→2 (0.987), 3→1 (0.151), 4→7 (0.032)

In the dominant-transition column the occupied index counts orbitals from the lowest (5 is the HOMO) and the virtual index from the LUMO (1 is the LUMO), both 1-based. State 1 is therefore HOMO to LUMO. State 2 has \(f = 0\) because the transition is dipole-forbidden by symmetry. The full Casida energies lie 0.03 to 0.08 eV below the TDA here, the usual size of the \(\mathbf B\) matrix effect for valence states. The oscillator-strength column is printed as the code computes it; see the defect note before quoting it.

The run also writes the usual h2o_tda.bibtex and h2o_tda.references siblings. tddft_spectrum=True adds a Gaussian-broadened text spectrum to the .out (see Output).

Note

run_job cannot attach the TDDFT result to a pybind11 SCF result object (the RHF/RKS/UHF/UKS result types have no __dict__), so result.tddft is usually not available. Read the .out and the QVF archive, or call the solvers directly as shown in The library route.

Theory

A weak time-dependent field perturbs the ground-state density, and the poles of the linear density response are the excitation energies. Written in the basis of occupied-virtual orbital pairs \((ia)\) for a closed-shell reference, the poles solve the non-Hermitian Casida equation (Casida, Mark E. (1995). *Time-Dependent Density Functional Response Theory for Molecules.* Recent Advances in Density Functional Methods, Part I (World Scientific), 155–192. doi:[10.1142/9789812830586_0005](https://doi.org/10.1142/9789812830586_0005); the underlying existence theorem is Runge, Erich and Gross, E. K. U. (1984). *Density-Functional Theory for Time-Dependent Systems.* Physical Review Letters, **52**(12), 997–1000. doi:[10.1103/PhysRevLett.52.997](https://doi.org/10.1103/PhysRevLett.52.997))

\[\begin{split} \begin{pmatrix} \mathbf A & \mathbf B \\ \mathbf B & \mathbf A \end{pmatrix} \begin{pmatrix} \mathbf X \\ \mathbf Y \end{pmatrix} = \omega \begin{pmatrix} \mathbf 1 & \mathbf 0 \\ \mathbf 0 & -\mathbf 1 \end{pmatrix} \begin{pmatrix} \mathbf X \\ \mathbf Y \end{pmatrix}, \end{split}\]

with the spin-adapted singlet blocks that vibeqc.tddft builds,

\[\begin{split} \begin{aligned} A_{ia,jb} &= \delta_{ij}\delta_{ab}(\varepsilon_a-\varepsilon_i) + 2(ia|jb) - c_x\,(ij|ab) + (ia|f_{xc}|jb),\\ B_{ia,jb} &= 2(ia|jb) - c_x\,(ib|ja) + (ia|f_{xc}|jb). \end{aligned} \end{split}\]

Here \(\varepsilon\) are the ground-state orbital energies, \((pq|rs)\) are two-electron integrals in chemist’s notation over the ground-state orbitals, \(c_x\) is the exact-exchange fraction and \(f_{xc}\) is the adiabatic exchange-correlation kernel. The approximations that enter are the adiabatic approximation (the kernel is local in time, so it has no frequency dependence and cannot describe states of double-excitation character) and the choice of functional in \(f_{xc}\). The exchange fraction is \(c_x = 1\) for a Hartree-Fock reference, \(0\) for a pure functional, and the functional’s admixture for a global hybrid, following Bauernschmitt, Rüdiger and Ahlrichs, Reinhart (1996). *Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory.* Chemical Physics Letters, **256**(4–5), 454–464. doi:[10.1016/0009-2614(96)00440-X](https://doi.org/10.1016/0009-2614(96)00440-X) (B3LYP 0.20, PBE0 0.25). The kernel is the LDA or GGA second functional derivative from libxc, contracted on the reference’s DFT grid.

The code solves the equation in the reduced form \((\mathbf A-\mathbf B)^{1/2}(\mathbf A+\mathbf B)(\mathbf A-\mathbf B)^{1/2}\mathbf Z=\omega^2\mathbf Z\), which needs \(\mathbf A-\mathbf B\) to be positive definite and clamps its negative eigenvalues to zero. That holds for a stable ground state; an unstable reference (triplet or orbital instability) makes the result meaningless, so run the SCF stability analysis first.

Setting \(\mathbf B=\mathbf 0\) gives the Tamm-Dancoff approximation (Hirata, So and Head-Gordon, Martin (1999). *Time-dependent density functional theory within the Tamm-Dancoff approximation.* Chemical Physics Letters, **314**(3–4), 291–299. doi:[10.1016/S0009-2614(99)01149-5](https://doi.org/10.1016/S0009-2614(99)01149-5)): a Hermitian eigenproblem for \(\mathbf A\) alone, which is faster, immune to the triplet-instability collapse of the full equation, and usually within 0.1 to 0.3 eV of it for valence states. On a Hartree-Fock reference the TDA is CIS (Foresman, James B., Head-Gordon, Martin, et al. (1992). *Toward a systematic molecular orbital theory for excited states.* Journal of Physical Chemistry, **96**(1), 135–149. doi:[10.1021/j100180a030](https://doi.org/10.1021/j100180a030)), the variational configuration interaction of singly excited determinants. The singlet and triplet CIS matrices, implemented reference-agnostically in vibeqc.excited.cis_excitations, are

\[ A^{\text{singlet}}_{ia,jb} = \delta_{ij}\delta_{ab}(\varepsilon_a-\varepsilon_i) + 2(ia|jb) - (ij|ab), \qquad A^{\text{triplet}}_{ia,jb} = \delta_{ij}\delta_{ab}(\varepsilon_a-\varepsilon_i) - (ij|ab). \]

The oscillator strength in the length gauge is \(f_I=\tfrac23\,\omega_I\,|\boldsymbol\mu_{0I}|^2\) in atomic units, where the transition dipole is contracted from the transition density built from the eigenvector (for the full equation, the vector \(\mathbf X+\mathbf Y\), normalised to unit length). Natural transition orbitals (Martin, Richard L. (2003). *Natural transition orbitals.* Journal of Chemical Physics, **118**(11), 4775–4777. doi:[10.1063/1.1558471](https://doi.org/10.1063/1.1558471)) compress an excitation into its most compact orbital description: the singular value decomposition \(\mathbf X=\mathbf U\,\boldsymbol\sigma\,\mathbf V^{\mathsf T}\) of the \(n_\text{occ}\times n_\text{virt}\) amplitude matrix gives hole orbitals \(\mathbf C_\text{occ}\mathbf U\), particle orbitals \(\mathbf C_\text{virt}\mathbf V\) and weights \(\sigma_k\) (with \(\sum_k\sigma_k^2=1\) for a normalised TDA vector). A single weight near 1 means the excitation is one hole-particle pair.

TDDFT is the standard tool for low-lying valence excitations of closed-shell molecules. In this implementation the cost is dominated by the XC kernel, not by the integrals (see Cost). It fails for charge-transfer and long-range states with a pure or low-exchange functional (the kernel lacks the \(-1/R\) asymptote), for Rydberg states in bases without diffuse functions, for states with double-excitation character, and near conical intersections. Range-separated hybrids, the usual remedy for charge-transfer states, are not available in the response kernel here.

Options

run_job keywords (molecular):

Keyword	Default	Effect
tddft	False	Run the excitation calculation after the SCF converges. Needs an SCF that produces MO coefficients (RHF, RKS, UHF, UKS, ROHF TDA). Skipped without a message if the SCF did not converge.
tddft_n_states	5	Number of roots.
tddft_type	"tda"	"tda" or "casida".
tddft_spectrum	False	Append a Gaussian-broadened text spectrum to the .out (sigma 0.15 eV, 200 grid points, normalised to a maximum of 1).
nto	False	Compute NTOs for every state and embed them in the QVF archive. Runs inside the QVF writer, so it needs output_qvf=True.
tddft_molden	False	With nto=True, also write one Molden file per state and role, <stem>.nto_S<n>_hole.molden and <stem>.nto_S<n>_electron.molden.
tddft_gradient	False	Finite-difference S1 gradient on the RHF/CIS surface only. See Excited-state gradients.

run_periodic_job accepts tddft and tddft_n_states only; the solver is always the TDA at the Gamma point. The library entry points, all importable from vibeqc:

Function	Reference	Notes
run_tddft_tda(molecule, basis, mo_energies, mo_coeff, n_occ, *, n_states=5, functional=None, density_ao=None, output=None, grid_options=None)	RHF, RKS	Singlets. functional=None is CIS.
run_tddft_casida(...)	RHF, RKS	Same signature; full linear response.
run_tddft_tda_uhf, run_tddft_casida_uhf	UHF, UKS	Take separate alpha and beta energies, coefficients, occupations and density_alpha_ao / density_beta_ao.
run_tddft_tda_rohf(molecule, basis, mo_energies, mo_coeff, n_docc, n_socc, *, n_states=5, output=None)	ROHF	Doubly-occupied to virtual space only.
run_tddft_tda_periodic(result, basis, *, n_occ, n_states=5, functional=None, output=None)	periodic Gamma	See Limits and gates.
compute_nto(state, mo_coeff, n_occ), compute_nto_uhf(...)		Return NTOResult; compute_nto_uhf returns a dict with "alpha" and "beta" entries.
vibeqc.excited.cis_excitations(eps, n_occ, ovov, oovv, *, spin="singlet", n_states=5)	any	Singlet or triplet CIS from orbital energies and the \((ia\vert jb)\), \((ij\vert ab)\) blocks.
vibeqc.excited_gradient.cis_state_gradients_fd, make_hf_cis_energy_fn	RHF	State-tracked finite-difference gradients.

Pass density_ao (and, for the KS functional, the reference’s grid_options) so the \(f_{xc}\) kernel is included. Without a density and with a hybrid functional the solver includes only the \(c_x\) exchange term and emits a UserWarning that the energies are not full TDA/TDDFT values; with a pure functional and no density the kernel is silently absent. The TDA switches to an iterative Davidson-type solver when the pair space exceeds 2000 and at most 10 percent of the roots are requested (RuntimeWarning if a root residual exceeds \(10^{-4}\)); otherwise it diagonalises densely. The full Casida equation is always dense.

The command line offers vibeqc tddft <xyz> [--basis def2-svp] [--functional ""] [--n-states 5] [--casida] [--charge 0] [--multiplicity 1]. With no functional it runs HF/CIS; with --multiplicity greater than 1 it uses UHF or UKS and the unrestricted solvers. Quote a basis name that contains * in the shell.

Cost

For a DFT reference the time is spent in the exchange-correlation kernel, not in the response matrices. The kernel is evaluated one occupied-virtual pair at a time (one grid contraction per pair, then a loop over pairs of pairs), so it grows with roughly the square of \(n_\text{occ}n_\text{virt}\) and with the grid size. Measured for the water PBE0 case above, one TDA solve for four states, on a machine that was also busy with another job (so read the ratios, not the absolute seconds):

Basis	Functions	Occupied-virtual pairs	SCF	TDA with \(f_{xc}\)	TDA without \(f_{xc}\)
STO-3G	7	10	10 s	4 s	0.01 s
6-31G*	18	65	27 s	38 s	0.01 s

Without the kernel (the Hartree-Fock/CIS case, or density_ao omitted) the same solve took a hundredth of a second, so the whole difference is the kernel. A separate profile of a water TDA solve attributes more than 99 percent of the time to the kernel routine and shows wall time growing from seconds in STO-3G to minutes in def2-SVP. Expect a DFT excitation calculation to cost several times the SCF and to grow much faster than it as the basis grows; a HF/CIS calculation is close to free once the reference exists. Time a small basis first, keep tddft_n_states modest (the kernel cost does not depend on it, but the memory for the dense response matrices does), and do not extrapolate a STO-3G timing to a production basis.

The library route

Use the solvers directly when you want the result objects, NTOs or triplets. Same water molecule, PBE0/6-31G*, plus HF/CIS singlets and triplets:

import numpy as np
import vibeqc as vq
from vibeqc.runner import apply_ks_grid_default
from vibeqc.tddft import make_eri_provider
from vibeqc.excited import cis_excitations

ANG = 1.8897259886
mol = vq.Molecule([
    vq.Atom(8, [0.0, 0.0, 0.1173 * ANG]),
    vq.Atom(1, [0.0, 0.7572 * ANG, -0.4692 * ANG]),
    vq.Atom(1, [0.0, -0.7572 * ANG, -0.4692 * ANG]),
], 0, 1)
basis = vq.BasisSet(mol, "6-31g*")
n_occ = mol.n_electrons() // 2

opts = vq.RKSOptions()
opts.functional = "pbe0"
apply_ks_grid_default(opts)          # the grid run_job would use
gs = vq.run_rks(mol, basis, opts)

td = vq.run_tddft_tda(
    mol, basis, np.asarray(gs.mo_energies), np.asarray(gs.mo_coeffs), n_occ,
    n_states=4, functional="pbe0", density_ao=np.asarray(gs.density),
    grid_options=opts.grid,
)
for s in td.states:
    print(s.index, f"{s.excitation_energy_ev:.4f}", s.dominant_amplitudes[0][:2])

nto = vq.compute_nto(td.states[0], np.asarray(gs.mo_coeffs), n_occ)
print("NTO weights", np.round(nto.hole_weights[:3], 4))

# CIS on a Hartree-Fock reference, singlet and triplet
hf = vq.run_rhf(mol, basis)
prov = make_eri_provider(np.asarray(vq.compute_eri(basis)),
                         np.asarray(hf.mo_coeffs), n_occ)
args = (np.asarray(hf.mo_energies), n_occ, prov.ovov(), prov.oovv())
print("singlet", np.round(cis_excitations(*args, spin="singlet", n_states=3).excitation_energies_ev, 4))
print("triplet", np.round(cis_excitations(*args, spin="triplet", n_states=3).excitation_energies_ev, 4))

1 8.4355 (5, 1)
2 10.4117 (5, 2)
3 10.9988 (4, 1)
4 13.1764 (4, 2)
NTO weights [0.9998 0.0111 0.0104]
singlet [ 9.6338 11.457  12.4118]
triplet [ 8.6546 10.7573 10.8161]

The library numbers reproduce the run_job block above. HOMO-LUMO is a single NTO pair (weight 0.9998). The HF/CIS singlet energies agree with an independent Hartree-Fock TDA reference calculation to better than \(10^{-3}\) eV. Each TDDFTState carries excitation_energy, excitation_energy_ev, wavelength_nm, oscillator_strength, transition_dipole (atomic units), dominant_amplitudes and the full excitation_vector.

Output

• .out: a ## TD-DFT excited states (<TDA|Casida|TDA-ROHF> (<functional>)) table with state, energy in eV, wavelength in nm, oscillator strength and the three leading transitions. With tddft_spectrum=True, a ## UV/Vis spectrum block. If the solver raises, the block reads FAILED: <ExceptionType>: <message> and the job continues; the SCF result is unaffected.

• QVF (with output_qvf=True): a spectra.uvvis section with energies, intensities, wavelengths and the method label. With nto=True, paired wavefunction.gto sections wf_nto_S<n>_hole and wf_nto_S<n>_electron with orbital_kind="natural" and the NTO weights as occupations, which vibe-view renders as orbitals. The periodic route does not write TDDFT data into the QVF.

• Citations (.bibtex, .references): Runge-Gross 1984 and Casida 1995 for any TDDFT run, Hirata-Head-Gordon 1999 for the TDA, Bauernschmitt-Ahlrichs 1996 when a hybrid exchange fraction enters the kernel, and Martin 2003 when nto is on, next to the functional and basis entries.

• Files for NTOs: the Molden files above when tddft_molden=True.

Excited-state gradients

The gradient of an excited state is the derivative of \(E_\text{SCF}+\omega_I\). vibeqc.excited_gradient.cis_state_gradients_fd differentiates that sum by central differences (default step \(10^{-3}\) Å) and follows the root across the displaced geometries by maximum overlap of the CIS amplitude vector, so a root flip does not corrupt the slope. It costs about \(6N\) additional SCF plus CIS evaluations. The only energy function shipped is make_hf_cis_energy_fn, the closed-shell RHF reference with singlet or triplet CIS and no ECP. Its coordinates are in Ångström. Verified on H\(_2\) at 0.74 Å, RHF/6-31G*:

import numpy as np
from vibeqc.excited_gradient import make_hf_cis_energy_fn, cis_state_gradients_fd

fn = make_hf_cis_energy_fn([1, 1], "6-31g*", n_states=2)
coords_angstrom = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 0.74]])
g = cis_state_gradients_fd(fn, coords_angstrom, states=[0, 1])   # 0 = ground, 1 = S1
print(np.round(g[0][:, 2], 6), np.round(g[1][:, 2], 6))

[-0.007561  0.007561] [ 0.198837 -0.198837]

The values are Ha/bohr. states uses 0 for the ground state and \(k\ge1\) for the \(k\)-th CIS root. For your own surface (a DFT kernel, say), build a callable that maps Ångström coordinates to a CISStateSet and pass it as energy_fn.

run_job(tddft=True, tddft_gradient=True) wraps this for the S1 state and prints a block headed Excited-state gradient (S1, RHF/CIS(TDA) surface, FD, step=1e-3 Å); see the defect note below before using it. vibeqc tddft-opt <xyz> minimises the S1 energy with L-BFGS-B on the same HF/CIS energy function. Its --functional option is printed but not used: the surface is always HF/CIS, whatever the option says. Treat it as an exploration convenience.

Known defects in this build

Warning

Two defects in vibe-qc 0.17.10 development builds change printed numbers. Both were found by comparing against an independent program and by re-running the library calls; the excitation energies are not affected.

1. Oscillator strengths are too large. The transition dipole is contracted as \(2\sum_{ia}X_{ia}\langle i|\mathbf r|a\rangle\), whereas the spin-adapted closed-shell singlet is \(\sqrt2\sum_{ia}X_{ia}\langle i|\mathbf r|a\rangle\). Restricted (RHF/RKS) oscillator strengths are therefore a factor of 2 too large, and unrestricted (UHF/UKS) ones a factor of 4, relative to Hartree-Fock TDA in PySCF on the same geometry (water RHF/6-31G*: 0.0375 versus 0.0187 for the first state; OH radical UHF/6-31G*: 0.0122 versus 0.0031). Excitation energies agree to \(10^{-4}\) eV. Ratios of intensities within one run, and which states are bright or dark, are unaffected. Do not quote absolute f values or molar absorptivities until this is fixed.

2. run_job(tddft_gradient=True) differentiates the wrong geometry. The runner passes bohr coordinates to a function that expects Ångström, so the gradient is evaluated at a geometry stretched by a factor of 1.8897. For H\(_2\) at 0.74 Å it prints 0.0131 Ha/bohr where the library call above, with the correct Ångström input, gives 0.1988 Ha/bohr. Use cis_state_gradients_fd directly with Ångström coordinates until the runner is fixed.

Limits and gates

• Range-separated hybrids (for example wB97X, CAM-B3LYP): not implemented. The response would need error-function-attenuated \((ij|ab)\) integrals, not a single global scale on the regular ones. The library raises NotImplementedError; under run_job the SCF completes and the .out block reads FAILED: NotImplementedError: TDDFT/TDA with the range-separated hybrid 'wb97x' is not implemented ... while the job exits normally. Use a global hybrid (PBE0, B3LYP) or a pure functional.

• Double hybrids: the perturbative doubles correction is not applied; the result is the bare response of the hybrid reference and a UserWarning says so.

• Kernels: LDA, GGA and global-hybrid exchange fractions only. The kernel builder documents that meta-GGA kernels are unavailable.

• ROHF: TDA only, over doubly-occupied to virtual excitations, labelled TDA-ROHF in the .out with a note that excitations out of or into the singly-occupied orbitals are not included. tddft_type="casida" on ROHF and any tddft=True on ROKS raise NotImplementedError before the SCF starts (issue #361). Verified for ROKS.

• Spin: run_tddft_tda and run_tddft_casida return singlets. Triplets are available through cis_excitations(spin="triplet") on an HF reference.

• Full-grid external XC functionals (for example SKALA, which is experimental): NotImplementedError, because the provider exposes no pointwise second functional derivative. OpenTrustRegion with tddft=True: ValueError.

• Skipped runs: the block is skipped after an unconverged SCF and for any result that carries no MO-energy attributes.

• Excited-state gradient: tddft_gradient=True requires an RHF reference, tddft_type="tda" and multiplicity 1; an RKS, Casida or open-shell request raises NotImplementedError before the calculation (issue #570), as does a molecular ECP basis. A CIS energy function for an odd electron count raises NotImplementedError.

• Periodic: run_periodic_job(tddft=True) calls run_tddft_tda_periodic on the Gamma-point orbitals and writes the table under ## TD-DFT excited states (TDA, Gamma-point). Three limits. First, the wrapper builds the response from open-space integrals of the unit-cell basis and passes no density, so there is no periodic image sum and no \(f_{xc}\) kernel; it is meaningful for vacuum-padded molecular cells and large supercells only, and a pure functional gives a bare response without exchange or correlation kernel. Second, it is refused with NotImplementedError on BIPOLE, on slab GDF, on periodic ROHF/ROKS and for full-grid external XC, and the basis-free semiempirical periodic routes reject the keyword. Third, and verified: on the GDF Gamma route the result object (PeriodicKRHFGDFResult) stores mo_coeffs with a leading k-point axis (shape (1, 2, 2) for H\(_2\) in a 12 bohr cell at STO-3G), and the wrapper reads that first axis as the basis size. The run_periodic_job call then writes FAILED: ValueError: No virtual orbitals available for TDDFT (n_basis == n_occ) into the .out, and so does a direct run_tddft_tda_periodic(result, basis, n_occ=1). Passing an object that carries the \(k=0\) block (mo_energies[0], mo_coeffs[0]) works around it. Treat periodic TDDFT as unverified for GDF results until this is fixed.

• Cost and memory: see Cost. The solvers hold the dense AO two-electron tensor and its MO blocks, \(O(n_\text{bf}^4)\) and \(O(n_\text{occ}^2n_\text{virt}^2)\). The pre-flight memory estimator includes the TDDFT terms.

See also

• Excited states with TDDFT: worked formaldehyde example, theory derivation and the open-shell and periodic variants.

• Ionisation potentials from the electron propagator and Green’s-function ionisation energies: the ionisation counterpart.

• Functionals: which exchange fraction each hybrid carries.

• Open-shell references and SCF convergence: the reference the response is built on.

• Citations, Output files and QVF and vibe-view.

• Methods: theory and status and the feature matrix.