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UMRSF-TDDFT: spin-flip excited states from a UHF triplet reference

Mixed-Reference Spin-Flip TDDFT (MRSF-TDDFT) gives balanced, low-spin-contaminated singlet and triplet excited states — including the near-degenerate, diradical-character situations (conical intersections, bond breaking, polyenes) where ordinary TDDFT fails. Standard MRSF spin-flips out of a high-spin ROHF triplet reference. UMRSF ("U" for unrestricted) does the same physics on an unrestricted (UHF) triplet reference instead. You reach for it when you want the MRSF target states but prefer — or need — a UHF reference determinant rather than ROHF orbitals. This tutorial computes single-point UMRSF-TDDFT excitation energies for butadiene (C₄H₆) at BHHLYP / 6-31G*.

A little theory

MRSF-TDDFT starts from a triplet (M_s = +1) reference and applies a spin-flip response operator: flipping one β-spin excitation down mixes the two open-shell reference configurations, so the resulting response states are proper singlets and triplets built from a common set of orbitals. Mixing the two reference determinants (the "mixed reference") is what removes the spin contamination that plagues plain spin-flip TDDFT and delivers a balanced description of states that are about to cross.

The only thing UMRSF changes is the reference wavefunction: instead of a restricted-open-shell (ROHF) triplet, it uses an unrestricted (UHF) triplet, and the response is solved with the unrestricted spin-flip kernel. In OpenQP the switch is two coordinated keywords — [scf] type=uhf (the "U") and [tdhf] type=umrsf — with the reference forced to high spin ([scf] multiplicity=3). Everything downstream (the BHHLYP functional, the basis, the requested number of roots) is identical to ordinary MRSF. For the derivation and the full method family, see the OpenQP manual.

Input-file style

The runnable deck is inputs/c4h6_umrsf_energy.inp — butadiene in 6-31G*, a UHF triplet reference, an unrestricted-MRSF response, energy only. Annotated:

[input]
system=
 6   -1.901080641    0.114577198    0.000000000   # C  (Angstrom)
 6   -0.574900598   -0.402237631    0.000000000   # C
 6    1.901080641   -0.114577198    0.000000000   # C
 6    0.574900598    0.402237631    0.000000000   # C
 1   -2.755248674   -0.545115910    0.000000000   # H
 1   -2.071562576    1.183086840    0.000000000   # H
 1    2.755248674    0.545115910    0.000000000   # H
 1   -0.441348756   -1.477941036    0.000000000   # H
 1    2.071562576   -1.183086840    0.000000000   # H
 1    0.441348756    1.477941036    0.000000000   # H
charge=0
runtype=energy          # single-point excitation energies only
method=tdhf             # linear-response (TD) engine; MRSF lives here
functional=bhhlyp       # half-and-half functional, the usual MRSF choice
basis=6-31g*

[scf]
type=uhf                # UHF reference -- this is the "U" in UMRSF
multiplicity=3          # high-spin (triplet) reference to spin-flip from

[tdhf]
type=umrsf              # unrestricted mixed-reference spin-flip response
nstate=5                # solve 5 target roots
multiplicity=1          # target the SINGLET manifold

What each key does:

  • [input] method=tdhf selects the time-dependent (linear-response) engine; the flavor of response (plain TDDFT, MRSF, or UMRSF) is chosen in [tdhf].
  • [input] runtype=energy — this deck is energy-only. functional=bhhlyp and basis=6-31g* are the standard MRSF functional/basis pairing.
  • [scf] type=uhf is the defining choice: the reference determinant is unrestricted. Swapping this to type=rohf (with [tdhf] type=mrsf) recovers ordinary MRSF-TDDFT — the geometry, functional, basis, and root count stay the same.
  • [scf] multiplicity=3 puts the reference in the high-spin (triplet) state. Spin-flip methods must start from high spin; this is the determinant the response flips out of, not a state you interpret.
  • [tdhf] type=umrsf turns the response into unrestricted mixed-reference spin-flip, the partner of the UHF reference.
  • [tdhf] nstate=5 requests five roots.
  • [tdhf] multiplicity=1 tells the solver to return the singlet target states (set multiplicity=3 to collect triplet targets instead).

The four keywords that make this "U"MRSF rather than plain MRSF, at a glance:

section keyword value
[scf] type uhf ← the "U"
[scf] multiplicity 3 (triplet reference)
[tdhf] type umrsf
[tdhf] multiplicity 1 (singlet targets)

Python style

The equivalent calculation with the OpenQP Python API is inputs/c4h6_umrsf_energy.py. The compact job.theory.mrsf(...) helper always sets an ROHF reference and type=mrsf, so it cannot express the UHF/umrsf variant. Instead the script writes the same [input] / [scf] / [tdhf] keywords the deck uses directly, via job.update(...) — the documented raw-section style for calculations the model-specific helpers do not cover.

from oqp.openqp import OpenQP

# Butadiene geometry (Angstrom), same atoms/order as the .inp deck.
butadiene = """
6   -1.901080641    0.114577198    0.000000000
6   -0.574900598   -0.402237631    0.000000000
6    1.901080641   -0.114577198    0.000000000
6    0.574900598    0.402237631    0.000000000
1   -2.755248674   -0.545115910    0.000000000
1   -2.071562576    1.183086840    0.000000000
1    2.755248674    0.545115910    0.000000000
1   -0.441348756   -1.477941036    0.000000000
1    2.071562576   -1.183086840    0.000000000
1    0.441348756    1.477941036    0.000000000
"""

job = OpenQP("c4h6_umrsf", silent=1)
job.molecule(butadiene, charge=0)

# UMRSF-TDDFT = method=tdhf + UHF triplet reference + [tdhf] type=umrsf.
job.update({
    "input": {"method": "tdhf", "functional": "bhhlyp",
              "basis": "6-31g*", "runtype": "energy"},
    "scf":   {"type": "uhf", "multiplicity": 3},                 # UHF high-spin reference
    "tdhf":  {"type": "umrsf", "nstate": 5, "multiplicity": 1},  # 5 singlet targets
})

# Serial run for an ordinary (non-MPI) Python invocation.
job.control(usempi=False)

mol = job.run()
results = mol.get_results()

print("Reference (SCF) energy:", results["energy"])
print("UMRSF state energies (Hartree, relative to reference):")
for i, e in enumerate(results["td_energies"]):
    print(f"  state {i}: {e:+.6f}")

Each dictionary passed to job.update(...) maps one-to-one onto a [section] of the input deck, so the three entries here are exactly the [input], [scf], and [tdhf] blocks above. job.control(usempi=False) runs serially, which is what you want for a plain python … invocation. To recover ordinary MRSF you would change two values — "scf": {"type": "rohf", …} and "tdhf": {"type": "mrsf", …}.

Run it

Input-file style (from the inputs/ folder):

cd umrsf-tddft/inputs
openqp c4h6_umrsf_energy.inp

Python style:

cd umrsf-tddft/inputs
python c4h6_umrsf_energy.py

Both need OpenQP installed (pip install openqp) and produce the same numbers.

Reading the output

A UMRSF-TDDFT single point produces one reference (SCF) energy plus a list of excited-state energies, one per requested root:

  • In the log file (<project>.log) look for the converged SCF energy of the UHF triplet reference, then the table of nstate MRSF roots with their excitation energies (and oscillator strengths).
  • From Python, mol.get_results() returns a dict: results["energy"] is the reference energy and results["td_energies"] is the list of state energies the script iterates over. (mol.get_scf_energy() returns the same SCF reference energy.)
  • With [tdhf] multiplicity=1 the five roots are singlet target states; the gaps between successive td_energies are the singlet excitation energies of butadiene. Re-run with multiplicity=3 to obtain the triplet manifold from the same reference.

Manual