Constrained DFT

Charge- and spin-constrained DFT, and the electron-transfer couplings that follow from it.

The response connection

A cDFT constraint couples a Lagrange multiplier \( \lambda \) to a fragment-weighted density operator. The derivative

\[ \frac{\partial N}{\partial \lambda} \]

— how much charge moves per unit constraint potential — is a susceptibility. So cDFT probes the same object as RPA and GW and attenuated MP2, through a different coupling.

Implementation

• Fragment charge and spin constraints via a grid-Becke weight operator
• A nested Lagrange-multiplier solve (Wu–Van Voorhis): an inner SCF at fixed \( \lambda \), an outer Newton iteration on \( \lambda \) itself

The nesting is what makes cDFT more expensive than a plain SCF — each outer step is a full converged inner solve.

Electron-transfer coupling

Once you have two charge-localized diabatic states, the coupling \( H_{ab} \) between them follows from a non-orthogonal determinant overlap, computed via Löwdin biorthogonalization.

That gives the matrix element governing electron-transfer rates in Marcus theory, from states that are constructed rather than guessed.

A caveat

The cdft_lambda_tol convergence tolerance interacts with the coupling calculation in a way worth checking: a loosely converged \( \lambda \) produces diabatic states that are not quite the ones you asked for, and \( H_{ab} \) inherits that error. Tighten it before trusting a coupling.