One donor, one acceptor, one new band: push–pull chromophores and charge transfer

Aniline is a benzene ring carrying an amino group; its first absorption band sits at 265 nm, far into the ultraviolet, and it is feeble. Nitrobenzene is the same ring carrying a nitro group; its lowest bright band is weaker still. Neither molecule absorbs anything of consequence past about 330 nm. Now put both substituents on one ring, para to each other — para-nitroaniline — and a new band appears at 311 nm that neither parent molecule owns, carrying an oscillator strength of 0.32: roughly ten times brighter than aniline’s band and twenty times brighter than nitrobenzene’s. Two weak ultraviolet absorbers combine into one strong near-visible one, and the whole is emphatically not the sum of its parts.

The pigments post built the rule that color is an electronic energy gap in the 1.8–3.1 eV window, and its Figure 2 made a promise: conjugation length is only one of the dials that set an organic chromophore’s gap, and the donor/acceptor dial would be “taken up in a later post.” This is that post. The mechanism is the donor–π–acceptor (D–π–A, or push–pull) architecture, and it is how dye chemists move an absorption band without growing the molecule — the working principle behind azo dyes, many sunscreen actives, and essentially every organic nonlinear-optical chromophore. Everything below is computed, not sketched: four molecules through the same TD-DFT pipeline, with the full data record in the repo (calcs/uvvis-pushpull/).

1. The push–pull architecture

The design has three parts. A donor group (here –NH₂) holds a high-lying filled orbital — nitrogen’s lone pair, conjugated into the ring — which pushes the HOMO up. An acceptor group (here –NO₂) holds a low-lying empty π* orbital, which pulls the LUMO down. A conjugated π bridge (here the benzene ring) lets the two ends talk to each other. Raising the floor and lowering the ceiling squeezes the HOMO–LUMO gap from both sides at once, which red-shifts the transition further than either substituent manages alone (Figure 1).

But the energy gap is only half of the design. Because the donor localizes the HOMO on one end of the molecule and the acceptor localizes the LUMO on the other, the HOMO→LUMO excitation physically moves an electron across the molecule — from the amino end to the nitro end. That is an intramolecular charge-transfer (CT) transition, the molecular sibling of the ligand-to-metal and intervalence transfers that §5 of the pigments post found in Prussian blue and the iron oxides. The electron traverses a real distance, the transition dipole is correspondingly large, and — by the machinery of the molar absorptivity post — a large transition dipole means a large oscillator strength and a fast radiative clock. Push–pull buys a redder and brighter band with one design.

2. The calculation

Four molecules isolate the two dials: benzene (bare bridge), aniline (donor only), nitrobenzene (acceptor only), and para-nitroaniline (pNA, donor and acceptor together). Each was built analytically, optimized at B3LYP/def2-SVP in C1 symmetry with tight convergence, and then run through full TD-DFT (RPA, not the Tamm–Dancoff approximation) for the twelve lowest singlet states in the larger def2-TZVP basis,¹ with two functionals per molecule: B3LYP, a global hybrid,² and CAM-B3LYP, its range-separated correction³ — the reason for running both is §5. Everything ran in psi4 1.11;⁴ the core of the excited-state call is Code 1.

escf, wfn = psi4.energy(func, molecule=m2, return_wfn=True)   # func: b3lyp / cam-b3lyp
res = tdscf_excitations(wfn, states=12, tda=False,            # full RPA, 12 singlets
                        r_convergence=1e-5, maxiter=120)
for st in res:
    e_ev = float(st["EXCITATION ENERGY"]) * HA2EV
    f    = float(st["OSCILLATOR STRENGTH (LEN)"])             # length gauge

Code 1. The excited-state step, per molecule and functional: converge the ground-state Kohn–Sham determinant, then solve the full TD-DFT (RPA) equations for the twelve lowest singlets and read off each state’s excitation energy and length-gauge oscillator strength.

Two derived quantities do the interpretive work. First, each state’s dominant occupied→virtual excitation is located in space by the hole–particle centroid separation: how far, in ångströms, the center of the departing electron density sits from the center of the arriving density. A local π→π* state moves the electron nowhere (separation near zero); a CT state moves it across the molecule. States with separation ≥ 2 Å and appreciable brightness are labeled CT by a deliberately conservative auto-classifier; subtler assignments (dark n→π* versus symmetry-forbidden π→π*) are made by hand. Second, each molecule’s lowest bright state gets a radiative lifetime from the two-level relation the absorptivity post derived, \(A_{21} = 1/\tau_{\text{rad}} = 0.667\,\tilde\nu^{2} f\) with \(\tilde\nu\) in cm⁻¹.

One stated shortcut: the basis carries no diffuse functions (def2-TZVP, not def2-TZVPD). Diffuse sets are the recommended practice for CT and Rydberg states, so the absolute energies below — especially the deep-UV states — would shift somewhat with them. The B3LYP-versus-CAM-B3LYP comparison, which is the methodological point, is robust to this.

3. Donor alone, acceptor alone

Table 1 collects the lowest bright band of each molecule under each functional, and Figure 1 shows the broadened CAM-B3LYP spectra.

Table 1. The lowest bright excited state (f ≥ 0.01) of each molecule under each functional: wavelength, energy, oscillator strength, two-level radiative lifetime, and band assignment. The push–pull molecule’s lowest bright state is the charge-transfer band — the reddest and by far the brightest entry in the table, with the shortest radiative lifetime of the three substituted rings.

The bare bridge sets the baseline. Benzene’s two lowest singlets (computed at 231 and 205 nm with B3LYP) are the textbook symmetry-forbidden ¹B₂ᵤ and ¹B₁ᵤ states — oscillator strengths numerically zero — and its first bright absorption is the degenerate ¹E₁ᵤ pair at 176 nm, off the left edge of Figure 1’s window. Experimentally that band sits near 179 nm in the gas phase;⁵ the computed values land within 0.1–0.2 eV of it. An unsubstituted ring is transparent until the deep ultraviolet.

A donor alone red-shifts, weakly. Conjugating the amino lone pair into the ring pushes the HOMO up, and aniline’s first π→π* lands at 265 nm (B3LYP) / 255 nm (CAM-B3LYP) — a large shift from 176 nm, but with oscillator strength only ≈ 0.04, and a correspondingly lazy radiative lifetime of 24–29 ns. The vapor-phase experiment puts this band near 282 nm.⁶ The donor moved the gap; it did not produce a strong band.

An acceptor alone red-shifts differently. The nitro group drags in its own low-lying π* and lone-pair states: nitrobenzene’s lowest state is a dark n→π* near 320 nm (f ≈ 0), its lowest nominally bright band (283/255 nm) is weak, f ≈ 0.014–0.018, and its genuinely strong π→π* sits one state higher at 259 nm (B3LYP) / 240 nm (CAM-B3LYP) with f ≈ 0.2 — the band the classic vapor-and-solution study of nitrobenzene, which observed it near 252 nm, assigned as intramolecular charge transfer toward the nitro group.⁷ The weak lowest bright state’s radiative lifetime stretches to 54–83 ns — exactly the weak-absorber-is-slow-emitter lockstep the absorptivity post predicts.

Figure 1. Broadened CAM-B3LYP absorption spectra of the four molecules (Gaussian broadening, FWHM 0.35 eV). Benzene is flat across the whole window — its first allowed band lies at 174 nm, off the left edge. Aniline (donor only) and nitrobenzene (acceptor only) each raise structure in the 220–260 nm region but nothing beyond ~310 nm. Para-nitroaniline’s charge-transfer band at 282 nm is a new feature that neither parent shows: the reddest and strongest band in the figure, produced by the donor and acceptor acting together.

4. The band neither parent owns

Para-nitroaniline’s lowest bright state is the payoff. It is a nearly pure HOMO→LUMO excitation (96.6% single-configuration weight with B3LYP), it lands at 311 nm (B3LYP) / 282 nm (CAM-B3LYP) — to the red of every bright band either parent molecule has — and it carries f ≈ 0.32–0.38, the largest oscillator strength of any computed state below 5 eV in the entire four-molecule study. The hole–particle centroid separation for this state is 2.6 Å: the excitation physically relocates an electron by more than a bond length, from the amino end toward the nitro end. The local π→π* states of the same molecule stay at or below 1.3 Å on the same diagnostic (B3LYP values). This is charge transfer measured, not asserted.

The two-level radiative lifetime makes the brightness concrete: 3–5 ns, against 24–29 ns for aniline and 54–83 ns for nitrobenzene. On the absorptivity post’s inverse lockstep between absorption strength and emission time, the push–pull band is the fast, strong end of the family — and it is the same transition dipole, computed from the same wavefunctions, driving both numbers.

The ground-state geometry quietly corroborates the story (Table 2). In isolated aniline the amino nitrogen is genuinely pyramidal — the lone pair only partly commits to the ring, and the N–H bonds pucker 14.4° out of plane. In pNA the same group flattens to 4.5°, and the nitro group holds exactly coplanar with the ring. The push–pull conjugation is strong enough to iron the donor nearly flat: the molecule pre-organizes its own π system to make the donor-to-acceptor communication — and hence the CT transition — as strong as possible.

Table 2. Optimized-geometry diagnostics (B3LYP/def2-SVP, all fully relaxed in C1). Isolated aniline’s donor is genuinely pyramidal; in the push–pull molecule the same –NH₂ is ironed nearly flat and the acceptor –NO₂ sits coplanar, maximizing conjugation across the bridge.

5. Two functionals, one diagnostic

Why run every molecule twice? Because charge-transfer states are the textbook failure mode of standard TD-DFT. A global hybrid like B3LYP, with a fixed fraction of exact exchange, lacks the correct long-range −1/R attraction between the separated hole and electron, and therefore places CT states too low — sometimes catastrophically so.⁸ Range-separated functionals like CAM-B3LYP switch in full exact exchange at long interelectronic distance precisely to fix this.³ A push–pull molecule computed with both functionals is therefore a built-in stress test: local states should barely move between them, and CT states should move a lot.

That is exactly what the data do. Benzene’s E₁ᵤ shifts by +0.09 eV from B3LYP to CAM-B3LYP; aniline’s local π→π* by +0.19 eV. Para-nitroaniline’s CT band shifts by +0.42 eV (3.98 → 4.40 eV, 311 → 282 nm) — several times the local states’ movement (Figure 2). Nitrobenzene is the instructive middle case: its low-lying π→π* states already shove density toward the nitro group (hole–particle separations of 1.9–2.0 Å, just under the CT threshold), and they shift by 0.39–0.47 eV — functional sensitivity tracks CT character even where the classifier’s label doesn’t change. Plotting the B3LYP→CAM-B3LYP shift of each matched bright state against its hole–particle separation (Figure 3) makes the pattern plain: states that move the electron under ~1 Å shift by ≲ 0.2 eV, and states that move it ~2 Å or more shift by 0.3–0.5 eV. The disagreement between two functionals is itself a CT diagnostic — you can see the missing long-range exchange turning on.

Figure 2. Para-nitroaniline’s charge-transfer band computed with B3LYP (blue, 311 nm) and CAM-B3LYP (green, 282 nm). The 0.42 eV disagreement between the two functionals is several times what the same pair produces for the local π→π* bands of benzene or aniline — the signature of B3LYP’s missing long-range exchange for charge-separated states. The dashed line marks the vapor-phase experimental band at 292 nm: B3LYP overshoots to the red (−0.26 eV), CAM-B3LYP lands slightly blue (+0.16 eV).

Figure 3. The B3LYP→CAM-B3LYP energy shift of each bright state, plotted against that state’s hole–particle centroid separation (states matched across functionals by their dominant orbital transition; separations from the B3LYP calculation). Spatially local excitations (left) barely feel the functional change; excitations that move the electron 2 Å or more (right) shift by 0.3–0.5 eV. Nitrobenzene’s nominally “local” low states sit in between — their density already leans toward the nitro group — which is exactly where a CT-sensitivity diagnostic should place them.

Validation against experiment keeps the comparison honest (Table 3). No number below was tuned; the experimental values are literature gas-phase or vapor band maxima^5–7,9.

Table 3. Computed vertical excitation energies against experimental band maxima (computed − experiment in the last two columns). Absolute errors are typical TD-DFT, a few tenths of an eV either way. The pattern is the point: for the CT band, B3LYP errs low — the known global-hybrid underestimation — while CAM-B3LYP lands slightly high, and the two functionals bracket the measurement.

One caution on that table: pNA is strongly solvatochromic. Its CT band slides from 292 nm in vapor to roughly 380 nm in water, because the enormous excited-state dipole — the electron has, after all, just crossed the molecule — is stabilized by polar solvent^9,10. The gas-phase computed values here are compared to gas-phase experiment; neither should be matched against an aqueous spectrum.

6. The optimizer on a saddle

A methodological embarrassment is worth recording, because it is the kind that produces quietly wrong papers. The first geometry optimization returned aniline and pNA with perfectly planar amino groups — pyramidalization exactly 0.0°, nitrogen angle-sum exactly 360°. That is not the minimum; it is the planar inversion transition state, the top of the umbrella-flip barrier.

The root cause was the starting guess. The molecules were built analytically, and the initial amino group placed its two N–H hydrogens symmetrically on opposite sides of the ring plane. That arrangement preserves a mirror symmetry whose consequence is that the gradient along the pyramidalization coordinate is zero by symmetry at the planar geometry — so the optimizer, which only ever follows the gradient downhill, converged to the saddle and reported success. Re-seeding both hydrogens on the same side (an “umbrella” start, symmetry deliberately broken) let the optimization roll off the saddle into the true pyramidal minimum: 14.4° for aniline, 4.5° for pNA. Same functional, same basis, same optimizer, same convergence criteria — the only change was the symmetry of the starting point.

The excited-state story survived the correction almost untouched: aniline’s bright band moved 269 → 265 nm and the pNA CT band 312 → 311 nm (B3LYP). But the geometry claims — Table 2’s entire donor-flattening narrative — would have been silently wrong, with every diagnostic reading a clean 0.0°. An optimizer converging is not the same thing as an optimizer finding a minimum; symmetric starting guesses find symmetric stationary points, whatever their character.

7. What this does and does not show

The limits, stated plainly. These are vertical excitation energies compared against experimental band maxima — a standard but imperfect pairing, since vibrational structure shifts a band’s maximum away from its vertical energy. The basis omits diffuse functions, which matter most for exactly the CT and high-lying states of interest; the absolute numbers would move with a diffuse-augmented basis, though the two-functional comparison would not. The calculations are gas-phase, and §5’s solvatochromism caution applies to any comparison with solution data. And the two-level radiative lifetimes inherit every caveat of the absorptivity post’s idealization — real photophysics adds non-radiative channels that only shorten the observed lifetime. Within those limits, the qualitative structure — the emergent CT band, its brightness, its 2.6 Å electron displacement, its functional sensitivity — is exactly the physics the D–π–A design predicts.

The dial behind the dye catalog

The pigments post established that an organic colorant is a conjugated π system whose HOMO–LUMO gap sits in the visible window, and that lengthening the conjugation shrinks the gap. This post computed the second dial: keep the bridge fixed and install a donor and an acceptor at its ends, and the gap closes from both sides while the transition acquires charge-transfer character — redder and brighter, the two things a dye chemist wants and the particle-in-a-box argument alone cannot deliver. Para-nitroaniline’s 2.6 Å of electron displacement is also why its excited state carries a huge dipole moment, which is why pNA is the fruit-fly molecule of organic nonlinear optics — the field the Pockels-effect post introduced through the same transition-dipole matrix element. Push the same design further — stronger donors, stronger acceptors, longer bridges — and the CT band walks out of the ultraviolet and into the visible window, where the azo yellows and reds of the modern palette live. The full data record, geometries, and scripts for every number in this post are in the repo under calcs/uvvis-pushpull/.

References