The Runge–Gross theorem is a foundational result in time‑dependent density functional theory (TDDFT). Formulated by Erich Runge and E. K. U. Gross in 1984, the theorem establishes a one‑to‑one correspondence between the time‑dependent external scalar potential acting on a quantum many‑body system and the resulting time‑dependent particle density, provided the initial many‑body wavefunction is fixed. Consequently, the external potential (up to an additive purely time‑dependent function) is a unique functional of the density, which underpins the existence of a density‑based formulation of the time‑dependent Schrödinger equation.
Statement of the theorem
Consider an interacting system of $N$ electrons described by the time‑dependent Schrödinger equation
$$ i\hbar\frac{\partial}{\partial t}\Psi(t)=\hat H(t),\Psi(t), $$
with Hamiltonian
$$ \hat H(t)=\hat T+\hat W+\int d\mathbf r, v(\mathbf r,t),\hat n(\mathbf r), $$
where $\hat T$ is the kinetic‑energy operator, $\hat W$ the electron‑electron interaction, $v(\mathbf r,t)$ the external scalar potential, and $\hat n(\mathbf r)$ the density operator. Let $\Psi_0=\Psi(t_0)$ be the many‑body state at an initial time $t_0$.
If two external potentials $v(\mathbf r,t)$ and $v'(\mathbf r,t)$ are Taylor‑expandable about $t_0$ and generate the same time‑dependent density $n(\mathbf r,t)$ for all $t$ in a finite interval $[t_0,t_0+T]$ while starting from the same $\Psi_0$, then the potentials differ at most by a purely time‑dependent function $c(t)$:
$$ v'(\mathbf r,t)=v(\mathbf r,t)+c(t). $$
Thus, the mapping
$$ {v(\mathbf r,t),\Psi_0} \longleftrightarrow n(\mathbf r,t) $$
is invertible (up to $c(t)$).
Historical background
The theorem was published in Physical Review Letters 52, 997 (1984) as “Time‑Dependent Density Functional Theory”. It extended the earlier Hohenberg–Kohn theorem (1964), which guarantees a similar one‑to‑one correspondence for ground‑state densities in static density functional theory (DFT). Runge and Gross’s work provided the rigorous justification for using the time‑dependent density as the basic variable in a formally exact theory of electronic dynamics.
Significance in TDDFT
- Existence of a universal functional: The theorem implies that, in principle, all observables of an interacting many‑electron system can be expressed as functionals of the time‑dependent density alone, provided the initial state is known.
- Kohn‑Sham construction: It justifies the introduction of a non‑interacting reference system (the time‑dependent Kohn‑Sham system) that reproduces the exact interacting density, enabling practical computational schemes.
- Linear‑response theory: The Runge–Gross framework underlies the widely used linear‑response TDDFT formalism for calculating excitation energies and optical spectra.
Conditions and limitations
- Taylor‑expandability: The external potentials must be analytic in time around the initial point; non‑analytic driving fields (e.g., sudden switches) fall outside the original proof.
- Fixed initial state: The one‑to‑one mapping holds only for a given many‑body initial wavefunction; changing the initial state requires a separate mapping.
- Additive gauge freedom: The theorem does not determine the purely time‑dependent function $c(t)$, reflecting the gauge freedom of the scalar potential.
Extensions and related results
Subsequent work has relaxed some of the original assumptions. Notable developments include:
- Vignale’s theorem (1995): Demonstrated the mapping for vector potentials in current‑density functional theory.
- Maitra, Burke, and co‑workers (2002–2005): Explored the existence of the functional for non‑analytic potentials and for finite‑temperature ensembles.
- Rigorous mathematical treatments (2007–2020): Provided proofs for broader classes of potentials using functional‑analytic techniques and addressed issues of uniqueness in the presence of degeneracies.
Applications
The Runge–Gross theorem forms the theoretical basis for numerous computational methods in quantum chemistry and condensed‑matter physics, including:
- Real‑time propagation TDDFT for ultrafast electron dynamics.
- Linear‑response TDDFT for optical absorption and excitation spectra of molecules, solids, and nanostructures.
- Time‑dependent current‑density functional theory (TDCDFT) for magnetic and spin‑dependent phenomena.
References
- E. Runge and E. K. U. Gross, “Density‑functional theory for time‑dependent systems,” Phys. Rev. Lett. 52, 997 (1984).
- M. Ehrenreich and H. Cohen, “Fundamentals of Time‑Dependent Density Functional Theory,” Adv. Quantum Chem. 55, 1‑45 (2008).
- G. Vignale, “Current‑density functional theory of many‑electron systems,” Phys. Rev. Lett. 74, 3237 (1995).
The Runge–Gross theorem remains a central pillar of TDDFT, providing the theoretical justification for representing the full many‑body dynamics of electrons through the evolution of the one‑particle density.