Multistate metadynamics for automatic exploration of conical intersections

We introduce multistate metadynamics for automatic exploration of conical intersection seams between adiabatic Born-Oppenheimer potential energy surfaces in molecular systems. By choosing the energy gap between the electronic states as a collective variable the metadynamics drives the system from an arbitrary ground-state configuration toward the intersection seam. Upon reaching the seam, the multistate electronic Hamiltonian is extended by introducing biasing potentials into the off-diagonal elements, and the molecular dynamics is continued on a modified potential energy surface obtained by diagonalization of the latter. The off-diagonal bias serves to locally open the energy gap and push the system to the next intersection point. In this way, the conical intersection energy landscape can be explored, identifying minimum energy crossing points and the barriers separating them. We illustrate the method on the example of furan, a prototype organic molecule exhibiting rich photophysics. The multistate metadynamics reveals plateaus on the conical intersection energy landscape from which the minimum energy crossing points with characteristic geometries can be extracted. The method can be combined with the broad spectrum of electronic structure methods and represents a generally applicable tool for the exploration of photophysics and photochemistry in complex molecules and materials.

Figure 1

Scheme of the algorithm. (a) Starting in the ground-state minimum, forces resulting from the bias potential $V_G$ using energy-gap CV drive the system toward the intersection seam (yellow trajectory). (b) Upon reaching the seam, an off-diagonal coupling $V_{\mathrm{ge}}$ (red) is added to the electronic Hamiltonian, opening a gap in the vicinity of the intersection point. The metadynamics subsequently explores configurations with nearly degenerate ground- and excited-state energies until the next intersection point is reached (c).

Figure 2

Multistate metadynamics trajectory for furan: (a) The energy gap $\mathrm{\Delta }{E}_{\mathrm{meta}}$ along the trajectory and (b) the off-diagonal bias potential $V_{\mathrm{ge}}$. Periods in which $V_{\mathrm{ge}}$ is updated ($\mathrm{\Delta }{E}_{\mathrm{meta}}<\epsilon$) are highlighted in light blue.

Figure 3

Conical intersection energy landscapes of three different trajectories: Energies of the ground (black) and first excited (red) states along the metadynamics trajectory for furan. A moving average with a window of 25 fs has been applied for better visibility. Upon reaching the CI after 1.0–1.5 ps, the system remains confined to the CI seam and explores basins (dotted plateaus) corresponding to different MECPs. For each plateau, the optimized structures of the MECPs together with their branching space vectors, energies relative to the ground-state minimum (in eV), and values of the 3D-Wiener number are given. (a) Multiple ring puckering, (b) ring opening and fragmentation, and (c) combinations of both types are observed.