Time-Dependent DFT

Ground-state DFT rests on Hohenberg-Kohn: the density determines the potential, so one function of three variables carries everything the many-body wavefunction knows about the ground state. Runge and Gross (1984) proved the moving-picture version: for a given initial state, the time-dependent density $\rho (\mathbf{r},t)$ determines the time-dependent potential — the whole movie of the density fixes the movie of the system. Same bargain as before: trade the exponentially large wavefunction for the density, and pay for it with a functional you have to approximate.

The Kohn-Sham move extends the same way. Invent a fictitious system of non-interacting electrons in orbitals ${\phi }_j(\mathbf{r},t)$, choose its potential so it reproduces the interacting density at every instant, and you get the time-dependent Kohn-Sham equations:

Structurally it is the SCF equation with a time derivative where the orbital energy used to be. The catch hides in $v_{xc}[\rho ](\mathbf{r},t)$: in principle it depends on the density's entire history, not just its present value. That memory is where the hard many-body physics went to live.

Everything genuinely new in TDDFT is concentrated in one object. Ask how the exchange-correlation potential at point $\mathbf{r}$ responds when the density wobbles at point ${\mathbf{r}}^{\prime }$ — a functional derivative:

This is the xc kernel. It rides alongside the Hartree response $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$ (which is exact and free), and together they say how the effective potential pushes back when the density moves. Every TDDFT excitation energy differs from a bare orbital-energy gap only through these two terms — the tables below measure that difference directly.

In practice almost everyone makes the adiabatic approximation: evaluate the ground-state functional's derivative at the instantaneous density and pretend there is no memory,

Cheap, frequency-independent, and surprisingly good for valence excitations. What it throws away is real: a frequency-independent kernel can shift and mix the single-excitation poles it inherits from the Kohn-Sham system, but it cannot create new poles — so states with substantial double-excitation character simply do not exist in adiabatic TDDFT's spectrum. Memory effects and double excitations are the standing IOUs of the whole enterprise.

How do you get a spectrum out of the TDKS equations? Two roads, one answer.

Road 1: real time. Kick the molecule with a spatially uniform electric-field impulse at $t=0$, then just run the TDKS equations forward and record the dipole moment as it rings. Fourier transform the ring and every peak sits at an excitation energy, with the peak weight telling you how brightly that state absorbs. This is what real-time TDDFT codes do, and it is linear response performed experimentally on the simulation itself: hit it, listen.

Road 2: frequency domain. Don't propagate anything. Linearize the TDKS equations around the ground state, project onto occupied-virtual orbital pairs, and the poles of the response become the eigenvalues of a matrix — Casida's equation. One diagonalization, every discrete excitation at once, machine precision.

Same physics, different linear algebra: road 1 wants many cheap matrix-vector applications of the dynamics; road 2 wants one expensive eigendecomposition. The demonstration below runs both on the same water molecule and watches the FFT peaks land on the Casida roots to within the FFT's own resolution.

The response-function view. Define $\chi (\omega )$ as the density-density response of the interacting system — how much $\rho$ moves per unit of applied potential at frequency $\omega$ — and ${\chi }_0(\omega )$ as the same object for the non-interacting Kohn-Sham system. The KS response is known explicitly: it has a pole at every occupied-to-virtual gap,

But the KS electrons feel the density move through the Hartree potential and $v_{xc}$, and that feedback closes a loop: the response responds to itself. Summing the loop gives a Dyson-type equation,

with $v_H=1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$. Read it as a renormalization machine: ${\chi }_0$ supplies raw poles at the bare gaps ${\epsilon }_a-{\epsilon }_i$, and the kernel shifts each pole and mixes poles that talk to each other, producing the true excitation energies. Discretize this equation in the particle-hole basis and you get exactly the Casida matrix; the kernel matrix elements are its off-diagonals. The table below shows the renormalization pole by pole — bare gap, then shifted root — on real numbers.

The smallest instance of the whole theory. Take H₂ in a minimal basis: one occupied orbital, one virtual, so the particle-hole space is one-dimensional and every matrix in sight is a $1\times 1$ — a number. Two numbers, in fact: the bare gap $\Delta ={\epsilon }_u-{\epsilon }_g$, and the one kernel matrix element $K$ (the transition density interacting with itself through Hartree-plus-$f_{xc}$).

For a local kernel the same $K$ lands in both Casida blocks: $A=\Delta +K$ and $B=K$. The Casida eigenvalue is ${\omega }^2=(A-B)(A+B)$, and the kernel cancels out of the first factor:

That's adiabatic TDDFT in one line: start from the bare gap squared, add a correction proportional to gap times kernel. Small $K$ expands to $\omega \approx \Delta +K$ — the kernel as first-order perturbation theory on the gap. The Tamm-Dancoff approximation keeps $A$ alone: ${\omega }^{\mathrm{TDA}}=\Delta +K$ exactly, which is why TDA sits slightly above the full answer. With PySCF's actual numbers for H₂/STO-3G below, the hand formula reproduces the library solver to six decimal places.