Charge-Transfer States in TDDFT

Adapted from my Physics H190 presentation at Berkeley (2021), on A. Dreuw, J. L. Weisman, and M. Head-Gordon, J. Chem. Phys. 119, 2943 (2003).

A charge-transfer excitation moves an electron from one molecule, the donor, to another molecule, the acceptor. These excitations sit at the center of photosynthesis, organic solar cells, and light-emitting diodes, and TDDFT with a local functional can place a charge-transfer state several electron volts below where it belongs. The failure has a clean algebraic origin that this note works out.

The Exact Limit

Consider a donor and an acceptor separated by a large distance $R$, and the excitation that moves one electron from the highest occupied orbital of the donor to the lowest empty orbital of the acceptor. At infinite separation this costs the ionization potential of the donor minus the electron affinity of the acceptor. At finite $R$ the transferred electron and the hole it left behind attract each other as two point charges, which lowers the energy by $1/R$ in atomic units. The lowest charge-transfer excitation therefore obeys

$$\omega_{CT}(R) \geq IP_D - EA_A - \frac{1}{R}$$

Keep the $1/R$ in mind, because the failure below comes down to whether a method can produce it.

Linear Response in TDHF

The CIS note stated the TDHF eigenvalue problem without deriving it. Let’s derive it here, since the derivation shows where every term in the matrices comes from. The time-dependent Hartree-Fock equation in matrix form is written as

$$i\frac{\partial}{\partial t}\pmb{C} = \pmb{F}\pmb{C}$$

where $\pmb C$ collects the molecular orbital coefficients and $\pmb F$ is the Fock matrix. Define the density matrix $\pmb P = \pmb C \pmb C^\dagger$. Differentiating $\pmb P$ and substituting the equation above together with its adjoint gives an equation of motion for the density alone,

$$i\frac{\partial}{\partial t}\pmb{P} = \pmb{F}\pmb{P} - \pmb{P}\pmb{F}$$

Now shine weak light on the molecule. The perturbation is a one-electron operator oscillating at frequency $\omega$, written in Fourier components as

$$g_{pq} = \frac{1}{2}\left[f_{pq}\exp(-i\omega t) + f^*_{qp}\exp(i\omega t)\right]$$

Both matrices then split into a static part and a first-order response,

$$\pmb P = \pmb P^{(0)} + \pmb P^{(1)}, \qquad \pmb F = \pmb F^{(0)} + \pmb F^{(1)}$$

Notice that $\pmb F$ responds twice. It contains the external field $g$ directly, and it also depends on the density, so a change in $\pmb P$ feeds back through

$$\Delta F_{pq} = \sum_{st}\frac{\partial F_{pq}}{\partial P_{st}}\,P^{(1)}_{st}$$

Writing the response of the density with amplitudes that oscillate at the same frequency, substituting everything into the equation of motion, and collecting the coefficients of $\exp(-i\omega t)$ and $\exp(+i\omega t)$ separately, we arrive at

$$\begin{pmatrix} \pmb{A} & \pmb{B} \\ \pmb{B}^* & \pmb{A}^* \end{pmatrix} \begin{pmatrix} \pmb{X} \\ \pmb{Y} \end{pmatrix} = \omega \begin{pmatrix} \pmb{1} & 0 \\ 0 & -\pmb{1} \end{pmatrix} \begin{pmatrix} \pmb{X} \\ \pmb{Y} \end{pmatrix}$$

with the same $\pmb A$ and $\pmb B$ blocks as in the CIS note. The physical picture is that the excitation energies are the frequencies at which the density can oscillate on its own, with no field driving it. TDDFT follows the identical route. The only change is that the derivative of the Fock matrix brings in the exchange-correlation kernel in place of Hartree-Fock exchange.

The Ethylene Dimer Test

Dreuw, Weisman, and Head-Gordon made the failure concrete with a dimer of ethylene and tetrafluoroethylene pulled apart along the intermolecular axis, a pair that supports low-lying states transferring an electron from one molecule to the other. At a separation of 4 Å, TDDFT with the local SVWN functional puts the lowest charge-transfer state at 1.71 eV. The same functional’s own values for the ionization potential and electron affinity, inserted into the bound above, demand at least 9.54 eV at that distance. The method disagrees with itself by almost 8 eV.

The distance dependence fails as well. Computed along $R$, the curves from the local functionals SVWN and LB94 are nearly flat, while CIS rises with the correct $1/R$ shape. Hybrid functionals land in between, recovering part of the rise in proportion to the exact exchange they carry.

The Missing Term

Let’s see where the $1/R$ lives in the equations. For a global hybrid functional that mixes a fraction $c_{HF}$ of Hartree-Fock exchange, the diagonal block of the response matrix reads, in chemists’ notation for the two-electron integrals,

$$A_{ia,jb} = \delta_{ij}\delta_{ab}(\epsilon_a - \epsilon_i) + (ia|jb) - c_{HF}(ij|ab) + (1-c_{HF})(ia|f_{xc}|jb)$$

where $(pq|rs)$ places the orbital product $p(\pmb r)\,q(\pmb r)$ on one electron and $r(\pmb r')\,s(\pmb r')$ on the other. Now let orbital $i$ live on the donor and orbital $a$ on the acceptor, with the two molecules far enough apart that their orbitals do not overlap. Every integral containing the product $i(\pmb r)\,a(\pmb r)$ vanishes, which removes the Coulomb term $(ia|jb)$, the kernel term, and the whole $\pmb B$ matrix. One two-electron term survives. In $-c_{HF}(ij|ab)$ the products $ij$ and $ab$ each sit on a single molecule, and the diagonal integral $(ii|aa)$ becomes the electrostatic attraction between the hole density on the donor and the electron density on the acceptor, which tends to $1/R$ at large separation. The surviving matrix element is

$$A_{ia,ia} \longrightarrow (\epsilon_a - \epsilon_i) - \frac{c_{HF}}{R}$$

The $1/R$ attraction between the separated electron and hole enters the theory only through nonlocal exchange, scaled by however much of it the functional carries. A pure local functional has $c_{HF} = 0$, so its charge-transfer energies are bare orbital energy differences, completely flat in $R$. A hybrid recovers a fraction $c_{HF}$ of the correct slope, which matches the ordering of the curves in the dimer test.

The flat curve explains the shape of the failure. The size comes from the orbital energies themselves. Approximate functionals let each electron partially repel itself, an artifact called self-interaction error, and Hartree-Fock exchange is the piece that would cancel it. The occupied eigenvalues consequently sit too high, and $\epsilon_a - \epsilon_i$ lands far below the true $IP_D - EA_A$. With the exact exchange-correlation potential the Kohn-Sham eigenvalues would supply the correct limit, but the functionals in everyday use do not come close for this quantity.

Partial Fixes

If a fraction of exact exchange helps, full exact exchange looks tempting. The catch is that exchange and correlation are only physical as a pair. The exact exchange hole is delocalized, model correlation holes are local, and combining the exact hole with a local model spoils the cancellation between the two, so functionals with one hundred percent exact exchange perform poorly elsewhere. The resolution the field settled on is range separation. A range-separated functional such as CAM-B3LYP or $\omega$B97X keeps a modest fraction of exact exchange at short range and switches smoothly to full exact exchange at long range, which restores the complete $1/R$ tail while leaving short-range thermochemistry intact. For a quick diagnosis, a CIS calculation remains the classic sanity check, since CIS carries full exact exchange and produces the $1/R$ rise by construction.

Feel free to contact me if there is an error or confusion!