Analytical gradients and derivative couplings for the TDDFT-1D method.

We derive and implement analytic gradients and derivative couplings for time-dependent density functional theory plus one double (TDDFT-1D) which is a semiempirical configuration interaction method whereby the Hamiltonian is diagonalized in a basis of all singly excited configurations and one doubly excited configuration as constructed from a set of reference Kohn-Sham orbitals. We validate the implementation by comparing against finite difference values. Furthermore, we show that our implementation can locate both optimized geometries and minimum-energy crossing points along conical seams of S1/S0 surfaces for a set of test cases.

Electron transfer and spin-orbit coupling: Can nuclear motion lead to spin selective rates?

We investigate a spin-boson inspired model of electron transfer, where the diabatic coupling is given by a position-dependent phase, eiWx. We consider both equilibrium and nonequilibrium initial conditions. We show that, for this model, all equilibrium results are completely invariant to the sign of W (to infinite order). However, the nonequilibrium results do depend on the sign of W, suggesting that photo-induced electron transfer dynamics with spin-orbit coupling can exhibit electronic spin polarization (at least for some time).

On the inclusion of one double within CIS and TDDFT.

We present an improved approach for generating a set of optimized frontier orbitals (HOMO and LUMO) that minimizes the energy of one double configuration. We further benchmark the effect of including such a double within a rigorous configuration interaction singles or a parameterized semi-empirical time-dependent density functional theory Hamiltonian for a set of test cases. Although we cannot quite achieve quantitative accuracy, the algorithm is quite robust and routinely delivers an enormous qualitative improvement to standard single-reference electronic structure calculations.

Modeling nonadiabatic dynamics with degenerate electronic states, intersystem crossing, and spin separation: A key goal for chemical physics.

We examine the many open questions that arise for nonadiabatic dynamics in the presence of degenerate electronic states, e.g., for singlet-to-triplet intersystem crossing where a minimal Hamiltonian must include four states (two of which are always degenerate). In such circumstances, the standard surface hopping approach is not sufficient as the algorithm does not include Berry force. Yet, we hypothesize that such a Berry force may be crucial as far as creating chiral induced spin separation, which is now a burgeoning field of study. Thus, this Perspective highlights the fact that if one can generate a robust and accurate semiclassical approach for the case of degenerate states, one will take a big step forward toward merging chemical physics with spintronics.

Analytic gradients and derivative couplings for configuration interaction with all single excitations and one double excitation-En route to nonadiabatic dynamics.

We present analytic gradients and derivative couplings for the simplest possible multireference configuration interaction method, CIS-1D, an electronic structure Ansatz that includes all single excitations and one lone double excitation on top of a Hartree-Fock reference state. We show that the resulting equations are numerically stable and require the evaluation of a similar number of integrals as compared to standard CIS theory; one can easily differentiate the required frontier orbitals (h and l) with minimal cost. The resulting algorithm has been implemented within the Q-Chem electronic structure package and should be immediately useful for understanding photochemistry with S0-S1 crossings.

Frozen-mode small polaron quantum master equation with variational bound for excitation energy transfer in molecular aggregates.

The small polaron quantum master equation (SPQME) is a powerful method for describing quantum dynamics in molecular systems. However, in the slow-bath regime where low-frequency vibrational modes dominate the dynamics, the fully dressed small polaron coordinates lead to errors in the SPQME theory. Furthermore, low-frequency modes also cause infrared divergence in the SPQME method, making the theory applicable only to systems described by spectral densities of the super-Ohmic form. In this study, we propose to treat these low-frequency vibrations as dynamically arrested "frozen" modes in a semiclassical representation and apply the small polaron representation only to the high-frequency vibrations. Furthermore, we show that a variational polaron approach can be utilized to determine the frequency upper bound of the frozen modes, allowing dynamical simulations free of manually tuned parameters. This frozen-mode SPQME is applied to models describing excitation energy transfer (EET) in molecular aggregates and comprehensively compared with the quasiadiabatic path integral method a well as the Redfield theory to demonstrate the applicability of this new method. We show that errors due to slow baths in the original SPQME theory are significantly reduced by the frozen-mode approximation. More significantly, we show that the new approach successfully extends the SPQME theory to be applicable to systems with the Drude-Lorentz spectral density, resulting in a great expansion of the applicability of the SPQME theory for EET problems. In summary, we demonstrate a "frozen-mode" SPQME that provides efficient and accurate simulations of EET dynamics of molecular systems in a broad parameter regime.

A demonstration of consistency between the quantum classical Liouville equation and Berry's phase and curvature for the case of complex Hamiltonians.

Although the quantum classical Liouville equation (QCLE) arises by cutting off the exact equation of motion for a coupled nuclear-electronic system at order 1 (1 = ℏ0), we show that the QCLE does include Berry's phase effects and Berry's forces (which are proportional to a higher order, ℏ = ℏ1). Thus, the fundamental equation underlying mixed quantum-classical dynamics does not need a correction for Berry's phase effects and is valid for the case of complex (i.e., not just real) Hamiltonians, where exotic features can arise in the course of electronic relaxation. Furthermore, we also show that, even though Tully's surface hopping model ignores Berry's phase, Berry's phase effects are included automatically within Ehrenfest dynamics. These findings should be of great importance if we seek to model coupled nuclear-electronic dynamics for systems with odd numbers of electrons and spin-orbit coupling, where the complex nature of the Hamiltonian is paramount.

On the accuracy of the LSC-IVR approach for excitation energy transfer in molecular aggregates.

We investigate the applicability of the linearized semiclassical initial value representation (LSC-IVR) method to excitation energy transfer (EET) problems in molecular aggregates by simulating the EET dynamics of a dimer model in a wide range of parameter regime and comparing the results to those obtained from a numerically exact method. It is found that the LSC-IVR approach yields accurate population relaxation rates and decoherence rates in a broad parameter regime. However, the classical approximation imposed by the LSC-IVR method does not satisfy the detailed balance condition, generally leading to incorrect equilibrium populations. Based on this observation, we propose a post-processing algorithm to solve the long time equilibrium problem and demonstrate that this long-time correction method successfully removed the deviations from exact results for the LSC-IVR method in all of the regimes studied in this work. Finally, we apply the LSC-IVR method to simulate EET dynamics in the photosynthetic Fenna-Matthews-Olson complex system, demonstrating that the LSC-IVR method with long-time correction provides excellent description of coherent EET dynamics in this typical photosynthetic pigment-protein complex.

Incorporating Berry Force Effects into the Fewest Switches Surface-Hopping Algorithm: Intersystem Crossing and the Case of Electronic Degeneracy.

We present a preliminary surface-hopping approach for modeling intersystem crossing (ISC) dynamics between four electronic states: one singlet and one (triply degenerate) triplet. In order to incorporate all Berry force effects, the algorithm requires that, when moving along an adiabatic surface associated with the triplet manifold, one must also keep track of a quasi-diabatic index (akin to a "ms" quantum number) for each trajectory. For a simple model problem, we find that a great deal of new physics can be captured by our algorithm, setting the stage for larger, more realistic (or perhaps even ab initio) simulations in the future.

The Simplest Possible Approach for Simulating S0- S1 Conical Intersections with DFT/TDDFT: Adding One Doubly Excited Configuration.

A simple combination of density functional theory/time-dependent density functional theory (DFT/TDDFT) and configuration interaction is presented to fix the incorrect topology of the S0- S1 conical intersection (CI) and allow a description of bond making and bond breaking in photoinduced dynamics. The proposed TDDFT-1D method includes one lone optimized doubly excited configuration in addition to the DFT/TDDFT singly excited states within the context of a large configuration interaction Hamiltonian. Results for ethylene and stilbene are provided to demonstrate that this ansatz can yield physically meaningful potential energy surfaces near S0- S1 avoided crossings without changing the vertical excitation energies far from the relevant crossings. We also investigate the famous linear water example to show that the algorithm calculates the correct topology of the S0- S1 CI and yields the correct geometric phase.