QMRSF-DK

QMRSF-DK builds a spin-adapted response space from a quintet reference and dresses its closed-shell block with exact exchange. From one quintet ROHF/ROKS calculation it returns the complete active space of four electrons in four orbitals, CAS(4,4): 20 singlets, 15 triplets, and one quintet.

Background

The four singly occupied orbitals of a quintet reference define the active space. Single spin flips from the eight complementary Ms = ±1 reference determinants generate every two- and four-open-shell configuration; the two Ms = ±2 determinants supply the six closed-shell configurations that single spin flips cannot reach, through a dressed kernel. The resulting blocks are small enough to diagonalize directly, so no Davidson iteration is needed and the response costs little beyond the reference SCF.

The method targets systems whose frontier manifold is strongly correlated: open-shell singlet ground states, antiaromatic transition states, and the triplet-pair states of singlet fission, where the singlet, triplet, and quintet descriptions of the same pair must be available on an equal footing.

Exchange partition and the S3R default

A hybrid functional assigns two-electron integrals to the exact-exchange class by their orbital indices. That assignment is not preserved by rotations inside a degenerate active-orbital pair, so a reference with an exactly or nearly degenerate frontier pair gives excitation energies that depend on the orientation the SCF happened to converge to.

QMRSF-DK evaluates three exchange partitions in the same run:

Convention	Description
value-based	the original index-based assignment; orientation dependent
S3R	the covariant convention used for the reported result
Haar	the average over the full pair-rotation group; also covariant

S3R is the production result. The value-based and Haar spectra are printed alongside it as diagnostics, together with a covariance gate: the largest change of any excitation energy when the detected degenerate pairs are rotated. The covariant conventions return machine zero for that gate; the value-based one exposes the orientation freedom, which reaches several electronvolts in degenerate systems.

Pairs are detected from the SOMO energies. Every adjacent pair degenerate to within 1e-2 hartree is averaged, which covers symmetry-enforced degeneracies and the distance-degenerate frontier pairs of a separated dimer. A reference with no such degeneracy has no orientation freedom to remove, and the same construction is then applied to the frontier pair alone, so the definition stays continuous as a degeneracy is approached.

Requirements

Setting	Required value
[input] runtype	energy
[input] method	tdhf
[scf] type	rohf
[scf] multiplicity	5
[tdhf] type	qmrsf_dk
[tdhf] multiplicity	1

The pre-flight checker reports any of these that is missing before the calculation starts.

Minimal example

Square H4, whose degenerate frontier pair is exactly what the covariant partition is for:

[input]
system=
   1   0.700000000   0.700000000   0.000000000
   1  -0.700000000   0.700000000   0.000000000
   1  -0.700000000  -0.700000000   0.000000000
   1   0.700000000  -0.700000000   0.000000000
charge=0
runtype=energy
basis=sto-3g
functional=bhhlyp
method=tdhf

[guess]
type=huckel

[scf]
type=rohf
multiplicity=5

[tdhf]
type=qmrsf_dk
hfscale=0.5

examples/QMRSF-DK/H4_QMRSF-DK_S3R_ENERGY.inp in the code repository is this input.

Range-separated references

A range-separated functional treats alpha + beta*erf(mu*r12) of the electron interaction by exact exchange, so its exchange operator is

K = alpha * K[1/r12] + beta * K[erf(mu*r12)/r12]

rather than a single scaled exchange. QMRSF-DK uses that operator in both places where the exchange fraction enters, the reference core exchange and the dressed kernel, and applies the covariant partition to the whole operator rather than to its short-range part alone. Setting beta = 0 reproduces the global-hybrid result exactly.

[input]
functional=camb3lyp
...

[tdhf]
type=qmrsf_dk

The reference alpha, beta, and mu come from the functional. [tdhf] cam_alpha, cam_beta, and cam_mu override them for the kernel; a negative value inherits the corresponding reference value independently. Only a resolved set that remains incomplete or invalid (for example, a nonpositive range parameter when range separation is active) is refused.

Orientation dependence is larger with a range-separated reference than with a global hybrid, because more of the exchange operator carries the noncovariant assignment, so the covariant partition matters more there.

Output

The log reports the three spectra, the leading configurations of each state, the covariance gate, and the detected degenerate pairs with their doublet anisotropy. Machine-readable results are written next to the log as <log>.qmrsf_dk.json, which carries the S3R singlet, triplet, and quintet states with their total energies and excitation energies. Every manifold is reported against the lowest singlet, so the singlet-triplet and singlet-quintet gaps appear directly. The singlet excitation energies are also published through the standard OQP::td_energies result.

qmrsf_dk_active.dat holds the active-space integrals for an independent check of the same construction. It carries the plain integrals only, so it does not reproduce a range-separated spectrum on its own.

Limitations

• QMRSF-DK is currently invoked through sectioned .inp input. The concise .oqp grammar and the high-level OpenQP Python helpers do not yet expose [tdhf] type=qmrsf_dk.
• Energies only; gradients and other runtypes are not implemented.
• The response covers the active O -> O space. Excitations that need correlation from outside the four frontier orbitals, such as the ionic pi pi* states of benzene, are overestimated.
• The active space is fixed at CAS(4,4) by the quintet reference.