How the acceptor closes the gap: acceptor-strength effects in para-methoxy push-pull dyes

1. What this note explains

In a push-pull chromophore the HOMO is concentrated on the donor end and the LUMO is concentrated on the acceptor end. Changing the donor mainly moves the HOMO; changing the acceptor mainly moves the LUMO. This note holds the donor fixed as para-methoxy and varies the acceptor across three strengths: cyano (CN), dicyanovinyl (DCV), and the tricyanodihydrofuran (TCF) acceptor from the DCDHF series.

The note reports how the HOMO and the LUMO move separately across that series, and how the HOMO-LUMO gap closes as the acceptor becomes stronger.

2. The scaffold

The series is para-methoxy push-pull dyes. The three molecules are:

MeO-Ph-CN: p-methoxybenzonitrile.
MeO-Ph-DCV: p-methoxybenzylidenemalononitrile.
MeO-Ph-TCF: OMe-substituted DCDHF, made by replacing the NMe₂ donor of DCDHF-Me2 with OMe.

The acceptors are ordered CN < DCV < TCF by increasing acceptor strength. The donor is OMe throughout. The π-path is not held fixed: CN is a nitrile on the ring, DCV inserts a vinylidene spacer, and TCF is a dihydrofuran acceptor, so acceptor strength and conjugation length move together. Changes in the frontier-orbital energies are therefore not an acceptor-only effect.

This series is not the complementary experiment to the BMN donor-strength note, which held Ar–CH=C(CN)₂ fixed and varied X. OMe is a stronger donor than that series’ H parent.

3. What was computed

Every molecule was optimized at B3LYP/def2-SVP, then the 12 lowest singlets were computed with full-response TD-DFT at def2-TZVP with both CAM-B3LYP and B3LYP, in Psi4 1.9.1.^1–4 The protocol is identical to the recent BMN donor-strength note; here it is applied to an acceptor-strength ladder instead.

Two quantities are reported for each molecule and functional: the self-consistent HOMO and LUMO energies (from which the Kohn-Sham gap is ε(LUMO) − ε(HOMO)), and the lowest TD-DFT vertical excitation energy S₁. The HOMO-LUMO gap is a ground-state orbital spacing; S₁ is an excited-state energy. They trend together here, but they are not the same quantity.

4. The frontier-orbital picture

CAM-B3LYP / def2-TZVP

B3LYP / def2-TZVP

HOMO LUMO HOMO−1 · LUMO+1 connector value: ε(LUMO) − ε(HOMO), eV

Figure 1. Frontier-orbital energies of MeO-Ph-CN, MeO-Ph-DCV, and MeO-Ph-TCF under CAM-B3LYP and B3LYP (def2-TZVP). The panels share a vertical scale. Solid bars: HOMO and LUMO; dashed bars: HOMO−1 and LUMO+1. The connector value is the Kohn-Sham HOMO-LUMO gap. Exact values are in Table 1.

The absolute gap is strongly functional-dependent. For MeO-Ph-CN the CAM-B3LYP gap is 7.94 eV while the B3LYP gap is 5.34 eV — a spread of 2.60 eV for the same molecule and geometry. Range-separated hybrids such as CAM-B3LYP lower the donor-localized HOMO and raise the acceptor-localized LUMO relative to B3LYP, widening the gap; this is the characteristic behavior for a charge-transfer system. The trend with acceptor strength is nearly identical under the two functionals: both show the gap closing as the acceptor becomes stronger, and both order the acceptors the same way.

5. The numbers

Table 1 gives the frontier energies, the Kohn-Sham gap, and the lowest TD-DFT excitation for every molecule and functional. The HOMO and LUMO columns contain the same values that appear in Figure 1. The S₁ column is the lowest vertical excitation energy. For DCV and TCF it is the HOMO→LUMO excitation and is smaller than the Kohn-Sham gap because the excited-state calculation includes electron-hole interaction and relaxation. For CN it is HOMO→LUMO+1, so it is not the HOMO-LUMO excitation energy.

acceptor	functional	ε(HOMO) (eV)	ε(LUMO) (eV)	gap (eV)	S₁ (eV)	S₁ character
CN	CAM-B3LYP	-8.15	-0.21	7.94	5.03	HOMO→LUMO+1 (67.8%)
DCV	CAM-B3LYP	-8.06	-1.86	6.20	3.93	HOMO→LUMO (96.8%)
TCF	CAM-B3LYP	-7.95	-2.27	5.68	3.55	HOMO→LUMO (96.8%)
CN	B3LYP	-6.78	-1.44	5.34	4.84	HOMO→LUMO+1 (69.9%)
DCV	B3LYP	-6.78	-2.97	3.81	3.70	HOMO→LUMO (97.3%)
TCF	B3LYP	-6.66	-3.32	3.34	3.13	HOMO→LUMO (96.3%)

Table 1. Kohn-Sham frontier-orbital energies, Kohn-Sham gaps, and lowest TD-DFT vertical excitations for the acceptor series. The S₁ character column is the dominant amplitude. DCV and TCF have a HOMO→LUMO lowest excitation. For CN the lowest excitation is HOMO→LUMO+1, so the S₁ energy is not the HOMO-LUMO excitation energy even though the HOMO-LUMO gap is reported.

From CN to TCF the LUMO drops by -2.06 eV under CAM-B3LYP and -1.89 eV under B3LYP, while the HOMO changes by only 0.20 eV and 0.12 eV respectively. The gap closes by 2.26 eV under CAM-B3LYP and 2.01 eV under B3LYP. Most of the gap closure comes from lowering the LUMO, not from moving the HOMO.

6. The LUMO moves more than the HOMO

The LUMO is localized on the acceptor fragment, so it is directly perturbed by changing the acceptor. The HOMO is localized on the para-methoxy donor, which is identical across the series, so it feels the acceptor only through the conjugated bridge. The result is that the LUMO shifts roughly an order of magnitude more than the HOMO from CN to TCF.

The lowest excitation also red-shifts as the acceptor strengthens: by 1.48 eV under CAM-B3LYP and 1.70 eV under B3LYP from CN to TCF. The direction matches the gap closure. The magnitude is smaller than the gap closure because S₁ is an excited-state response and not the raw orbital spacing.

7. Reproducibility

All calculations use Psi4 1.9.1⁴ in the recorded conda environment (environment-psi4_19.yml), on Linux, gas phase, no solvent model. The protocol follows the recent BMN donor-strength note: geometry optimization at B3LYP/def2-SVP (C1), then full-response TD-DFT for the 12 lowest singlets at def2-TZVP with CAM-B3LYP and B3LYP.^1–3 MeO-Ph-CN and MeO-Ph-DCV converged with gau_tight. MeO-Ph-TCF did not satisfy the ultra-tight displacement criterion, so its final geometry was taken from the preceding gau step; the forces were below 1×10⁻⁵ au and the structure is planar.

Starting structures were built analytically because no force-field toolkit is available in the environment. The DCV acceptor was twisted 30° about the aryl-acceptor single bond. The TCF acceptor was started planar because twisting the rigid dihydrofuran made gau_tight stall on displacement criteria. The CN acceptor in MeO-Ph-CN is linear and defines no acceptor plane, so its starting structure is planar; the diagnostic reported is the angle between the aryl plane and the C(aryl)-C(nitrile) bond vector. The canonical run command is research/meo-acceptor-ladder/run_all.sh. It is resumable: each stage skips files that already exist, so a fresh clone can rebuild every committed artifact from the starting geometries and the environment.

The frontier-orbital energies were extracted from the Psi4 output logs by orbital_gaps.py (occupied) and extract_frontier.py (virtual), with the latter cross-checking occupied energies against the former to 1 meV. The logs themselves are not committed, but their sha256 hashes are recorded in results/frontier_orbitals.json. The inline figure was rendered by make-figure.mjs, the hero PNG by render-figure-png.mjs, and every cited number resolves from metrics.json via generate-metrics.mjs; both generators support --check to verify that the committed artifacts match the generator’s output.

8. Where the model stops

These are gas-phase vertical excitations of isolated molecules with no solvent model, and charge-transfer states are strongly solvatochromic. Nothing here has vibrational structure or computed band widths. The series does not isolate acceptor strength from conjugation length, so the CN→DCV→TCF gap closure is not a pure acceptor-strength effect. TD-DFT carries a typical 0.2–0.3 eV error for charge-transfer states,⁵ so the computed gap and excitation energies are indicative rather than exact. Measured solution spectra of these compounds would test the computed trend in a way this calculation cannot.

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