Calculation of quasi-diabatic states within the DFT/MRCI(2) framework: The QD-DFT/MRCI(2) method.

We describe a procedure for the calculation of quasi-diabatic states within the recently introduced DFT/MRCI(2) framework [S. P. Neville and M. S. Schuurman, J. Chem. Phys. 157, 164103 (2022)]. Based on an effective Hamiltonian formalism, the proposed procedure, which we term QD-DFT/MRCI(2), has the advantageous characteristics of being simultaneously highly efficient and effectively black box in nature while directly yielding both quasi-diabatic potentials and wave functions of high quality. The accuracy and efficiency of the QD-DFT/MRCI(2) formalism are demonstrated via the simulation of the vibronic absorption spectra of furan and chlorophyll a.

A DFT/MRCI Hamiltonian parameterized using only ab initio data: I. valence excited states.

A new combined density functional theory and multi-reference configuration interaction (DFT/MRCI) Hamiltonian parameterized solely using the benchmark ab initio vertical excitation energies obtained from the QUEST databases is presented. This new formulation differs from all previous versions of the method in that the choice of the underlying exchange-correlation (XC) functional employed to construct the one-particle (orbital) basis is considered, and a new XC functional, QTP17, is chosen for its ability to generate a balanced description of core and valence vertical excitation energies. The ability of the new DFT/MRCI Hamiltonian, termed QE8, to furnish accurate excitation energies is confirmed using benchmark quantum chemistry computations, and a mean absolute error of 0.16 eV is determined for the wide range of electronic excitations included in the validation dataset. In particular, the QE8 Hamiltonian dramatically improves the performance of DFT/MRCI for doubly excited states. The performance of fast approximate DFT/MRCI methods, p-DFT/MRCI and DFT/MRCI(2), is also evaluated using the QE8 Hamiltonian, and they are found to yield excitation energies in quantitative agreement with the parent DFT/MRCI method, with the two methods exhibiting a mean difference of 0.01 eV with respect to DFT/MRCI over the entire benchmark set.

Few-femtosecond electronic and structural rearrangements of CH4+ driven by the Jahn-Teller effect.

The Jahn-Teller effect (JTE) is central to the understanding of the physical and chemical properties of a broad variety of molecules and materials. Whereas the manifestations of the JTE in stationary properties of matter are relatively well studied, the study of JTE-induced dynamics is still in its infancy, largely owing to its ultrafast and non-adiabatic nature. For example, the time scales reported for the distortion of CH4+ from the initial Td geometry to a nominal C2v relaxed structure range from 1.85 fs over 10 ± 2 fs to 20 ± 7 fs. Here, by combining element-specific attosecond transient-absorption spectroscopy and quantum-dynamics simulations, we show that the initial electronic relaxation occurs within 5 fs and that the subsequent nuclear dynamics are dominated by the Q2 scissoring and Q1 symmetric stretching modes, which dephase in 41 ± 10 fs and 13 ± 3 fs, respectively. Significant structural relaxation is found to take place only along the e-symmetry Q2 mode. These results demonstrate that CH4+ created by ionization of CH4 is best thought of as a highly fluxional species that possesses a long-time-averaged vibrational distribution centered around a D2d structure. The methods demonstrated in our work provide guidelines for the understanding of Jahn-Teller driven non-adiabatic dynamics in other more complex systems.

Ultrafast Molecular Frame Quantum Tomography.

We develop and experimentally demonstrate a methodology for a full molecular frame quantum tomography (MFQT) of dynamical polyatomic systems. We exemplify this approach through the complete characterization of an electronically nonadiabatic wave packet in ammonia (NH_{3}). The method exploits both energy and time-domain spectroscopic data, and yields the lab frame density matrix (LFDM) for the system, the elements of which are populations and coherences. The LFDM fully characterizes electronic and nuclear dynamics in the molecular frame, yielding the time- and orientation-angle dependent expectation values of any relevant operator. For example, the time-dependent molecular frame electronic probability density may be constructed, yielding information on electronic dynamics in the molecular frame. In NH_{3}, we observe that electronic coherences are induced by nuclear dynamics which nonadiabatically drive electronic motions (charge migration) in the molecular frame. Here, the nuclear dynamics are rotational and it is nonadiabatic Coriolis coupling which drives the coherences. Interestingly, the nuclear-driven electronic coherence is preserved over longer timescales. In general, MFQT can help quantify entanglement between electronic and nuclear degrees of freedom, and provide new routes to the study of ultrafast molecular dynamics, charge migration, quantum information processing, and optimal control schemes.

Machine Learning Seams of Conical Intersection: A Characteristic Polynomial Approach.

The machine learning of potential energy surfaces (PESs) has undergone rapid progress in recent years. The vast majority of this work, however, has been focused on the learning of ground state PESs. To reliably extend machine learning protocols to excited state PESs, the occurrence of seams of conical intersections between adiabatic electronic states must be correctly accounted for. This introduces a serious problem, for at such points, the adiabatic potentials are not differentiable to any order, complicating the application of standard machine learning methods. We show that this issue may be overcome by instead learning the coordinate-dependent coefficients of the characteristic polynomial of a simple decomposition of the potential matrix. We demonstrate that, through this approach, quantitatively accurate machine learning models of seams of conical intersection may be constructed.

Time-Resolved X-ray Photoelectron Spectroscopy: Ultrafast Dynamics in CS2 Probed at the S 2p Edge.

Recent developments in X-ray free-electron lasers have enabled a novel site-selective probe of coupled nuclear and electronic dynamics in photoexcited molecules, time-resolved X-ray photoelectron spectroscopy (TRXPS). We present results from a joint experimental and theoretical TRXPS study of the well-characterized ultraviolet photodissociation of CS2, a prototypical system for understanding non-adiabatic dynamics. These results demonstrate that the sulfur 2p binding energy is sensitive to changes in the nuclear structure following photoexcitation, which ultimately leads to dissociation into CS and S photoproducts. We are able to assign the main X-ray spectroscopic features to the CS and S products via comparison to a first-principles determination of the TRXPS based on ab initio multiple-spawning simulations. Our results demonstrate the use of TRXPS as a local probe of complex ultrafast photodissociation dynamics involving multimodal vibrational coupling, nonradiative transitions between electronic states, and multiple final product channels.

A laboratory frame density matrix for ultrafast quantum molecular dynamics.

In most cases, the ultrafast dynamics of resonantly excited molecules are considered and almost always computed in the molecular frame, while experiments are carried out in the laboratory frame. Here, we provide a formalism in terms of a lab frame density matrix, which connects quantum dynamics in the molecular frame to those in the laboratory frame, providing a transparent link between computation and measurement. The formalism reveals that in any such experiment, the molecular frame dynamics vary for molecules in different orientations and that certain coherences, which are potentially experimentally accessible, are rejected by the orientation-averaged reduced vibronic density matrix. Instead, molecular angular distribution moments are introduced as a more accurate representation of experimentally accessible information. Furthermore, the formalism provides a clear definition of a molecular frame quantum tomography and specifies the requirements to perform such a measurement enabling the experimental imaging of molecular frame vibronic dynamics. Successful completion of such a measurement fully characterizes the molecular frame quantum dynamics for a molecule at any orientation in the laboratory frame.

A perturbative approximation to DFT/MRCI: DFT/MRCI(2).

We introduce a perturbative approximation to the combined density functional theory and multireference configuration interaction (DFT/MRCI) approach. The method, termed DFT/MRCI(2), results from the application of quasi-degenerate perturbation theory (QDPT) and the Epstein-Nesbet partitioning to the DFT/MRCI Hamiltonian matrix. The application of QDPT obviates the need to diagonalize the large DFT/MRCI Hamiltonian; electronic energies are instead obtained as the eigenvalues of a small effective Hamiltonian, affording an orders of magnitude savings in the computational cost. Most importantly, the DFT/MRCI(2) approximation is found to be of excellent accuracy, furnishing excitation energies with a root mean squared deviation from the canonical DFT/MRCI values of less than 0.03 eV for an extensive test set of organic molecules.

Electronic relaxation and dissociation dynamics in formaldehyde: pump wavelength dependence.

The effect of the incident UV pump wavelength on the subsequent excited state dynamics, electronic relaxation, and ultimate dissociation of formaldehyde is studied using first principles simulation and Coulomb explosion imaging (CEI) experiments. Transitions in a vibronic progression in the à ← X̃ absorption band are systematically prepared using a tunable UV source which generates pulses centered at 304, 314, 329, and 337 nm. We find, both via ab initio simulation and experimental results, that the rate of excited state decay and subsequent dissociation displays a prominent dependence on which vibronic transition in the absorption band is prepared by the pump. Our simulations predict that nonadiabatic transition rates and dissociation yields will increase by a factor of >100 as the pump wavelength is decreased from 337 to 304 nm. The experimental results and theoretical simulations are in broad agreement and both indicate that the dissociation rate plateaus rapidly after ≈2 ps following an ultrafast sub-ps rise.

Resolving competing conical intersection pathways: time-resolved X-ray absorption spectroscopy of trans-1,3-butadiene.

Time-resolved X-ray absorption spectroscopy is emerging as a uniquely powerful tool to probe coupled electronic-nuclear dynamics in photo-excited molecules. Theoretical studies to date have established that time-resolved X-ray absorption spectroscopy is an atom-specific probe of excited-state wave packet passage through a seam of conical intersections (CIs). However, in many molecular systems, there are competing dynamical pathways involving CIs of different electronic and nuclear character. Discerning these pathways remains an important challenge. Here, we demonstrate that time-resolved X-ray absorption spectroscopy (TRXAS) has the potential to resolve competing channels in excited-state non-adiabatic dynamics. Using the example of 1,3-butadiene, we show how TRXAS discerns the different electronic structures associated with passage through multiple conical intersections. trans-1,3-Butadiene exhibits a branching between polarized and radicaloid pathways associated with ethylenic "twisted-pyramidalized" and excited-state cis-trans isomerization dynamics, respectively. The differing electronic structures along these pathways give rise to different XAS signals, indicating the possibility of resolving them. Furthermore, this indicates that XAS, and other core-level spectroscopic techniques, offer the appealing prospect of directly probing the effects of selective chemical substitution and its ability to affect chemical control over excited-state molecular dynamics.

Removing the Deadwood from DFT/MRCI Wave Functions: The p-DFT/MRCI Method.

The combined density functional theory and multireference configuration interaction (DFT/MRCI) method is a powerful tool for the calculation of excited electronic states of large molecules. There exists, however, a large amount of superfluous configurations in a typical DFT/MRCI wave function. We show that this deadwood may be effectively removed using a simple configuration pruning algorithm based on second-order Epstein-Nesbet perturbation theory. The resulting method, which we denote p-DFT/MRCI, is shown to result in orders of magnitude saving in computational timings, while retaining the accuracy of the original DFT/MRCI method.

Vacuum Ultraviolet Excited State Dynamics of the Smallest Ketone: Acetone.

We combined tunable vacuum-ultraviolet time-resolved photoelectron spectroscopy (VUV-TRPES) with high-level quantum dynamics simulations to disentangle multistate Rydberg-valence dynamics in acetone. A femtosecond 8.09 eV pump pulse was tuned to the sharp origin of the A1(n3dyz) band. The ensuing dynamics were tracked with a femtosecond 6.18 eV probe pulse, permitting TRPES of multiple excited Rydberg and valence states. Quantum dynamics simulations reveal coherent multistate Rydberg-valence dynamics, precluding simple kinetic modeling of the TRPES spectrum. Unambiguous assignment of all involved Rydberg states was enabled via the simulation of their photoelectron spectra. The A1(ππ*) state, although strongly participating, is likely undetectable with probe photon energies ≤8 eV and a key intermediate, the A2(nπ*) state, is detected here for the first time. Our dynamics modeling rationalizes the temporal behavior of all photoelectron transients, allowing us to propose a mechanism for VUV-excited dynamics in acetone which confers a key role to the A2(nπ*) state.

Sub-7-femtosecond conical-intersection dynamics probed at the carbon K-edge.

Conical intersections allow electronically excited molecules to return to their electronic ground state. Here, we observe the fastest electronic relaxation dynamics measured to date by extending attosecond transient-absorption spectroscopy (ATAS) to the carbon K-edge. We selectively launch wave packets in the two lowest electronic states (D0 and D1) of C2H4 + The electronic D1 → D0 relaxation takes place with a short time constant of 6.8 ± 0.2 femtoseconds. The electronic-state switching is directly visualized in ATAS owing to a spectral separation of the D1 and D0 bands caused by electron correlation. Multidimensional structural dynamics of the molecule are simultaneously observed. Our results demonstrate the capability to resolve the fastest electronic and structural dynamics in the broad class of organic molecules. They show that electronic relaxation in the prototypical organic chromophore can take place within less than a single vibrational period.

Capturing roaming molecular fragments in real time.

Since the discovery of roaming as an alternative molecular dissociation pathway in formaldehyde (H2CO), it has been indirectly observed in numerous molecules. The phenomenon describes a frustrated dissociation with fragments roaming at relatively large interatomic distances rather than following conventional transition-state dissociation; incipient radicals from the parent molecule self-react to form molecular products. Roaming has been identified spectroscopically through static product channel-resolved measurements, but not in real-time observations of the roaming fragment itself. Using time-resolved Coulomb explosion imaging (CEI), we directly imaged individual "roamers" on ultrafast time scales in the prototypical formaldehyde dissociation reaction. Using high-level first-principles simulations of all critical experimental steps, distinctive roaming signatures were identified. These were rendered observable by extracting rare stochastic events out of an overwhelming background using the highly sensitive CEI method.

Propagative block diagonalization diabatization of DFT/MRCI electronic states.

We present a framework for the calculation of diabatic states using the combined density functional theory and multireference configuration interaction (DFT/MRCI) method. Due to restrictions present in the current formulation of the DFT/MRCI method (a lack of analytical derivative couplings and the inability to use non-canonical Kohn-Sham orbitals), most common diabatization strategies are not applicable. We demonstrate, however, that diabatic wavefunctions and potentials can be reliably calculated at the DFT/MRCI level of theory using a propagative variant of the block diagonalization diabatization method (P-BDD). The proposed procedure is validated via the calculation of diabatic potentials for LiH and the simulation of the vibronic spectrum of pyrazine. In both cases, the combination of the DFT/MRCI and P-BDD methods is found to correctly recover the non-adiabatic coupling effects of the problem.

Electron transfer in photoexcited pyrrole dimers.

Following on from previous experimental and theoretical work [Neville et al., Nat. Commun. 7, 11357 (2016)], we report the results of a combined electronic structure theory and quantum dynamics study of the excited state dynamics of the pyrrole dimer following excitation to its first two excited states. Employing an exciton-based analysis of the Ã(π3s/σ*) and B̃(π3s/3p/σ*) states, we identify an excited-state electron transfer pathway involving the coupling of the Ã(π3s/σ*) and B̃(π3s/3p/σ*) states and driven by N-H dissociation in the B̃(π3s/3p/σ*) state. This electron transfer mechanism is found to be mediated by vibronic coupling of the B̃ state, which has a mixed π3s/3p Rydberg character at the Franck-Condon point, to a high-lying charge transfer state of the πσ* character by the N-H stretch coordinate. Motivated by these results, quantum dynamics simulations of the excited-state dynamics of the pyrrole dimer are performed using the multiconfigurational time-dependent Hartree method and a newly developed model Hamiltonian. It is predicted that the newly identified electron transfer pathway will be open following excitation to both the Ã(π3s/σ*) and B̃(π3s/3p/σ*) states and may be the dominant relaxation pathway in the latter case.

The simulation of X-ray absorption spectra from ground and excited electronic states using core-valence separated DFT/MRCI.

We present an extension of the combined density functional theory (DFT) and multireference configuration interaction (MRCI) method (DFT/MRCI) [S. Grimme and M. Waletzke, J. Chem. Phys. 111, 5645 (1999)] for the calculation of core-excited states based on the core-valence separation (CVS) approximation. The resulting method, CVS-DFT/MRCI, is validated via the simulation of the K-edge X-ray absorption spectra of 40 organic chromophores, amino acids, and nucleobases, ranging in size from CO2 to tryptophan. Overall, the CVS-DFT/MRCI method is found to yield accurate X-ray absorption spectra (XAS), with consistent errors in peak positions of ∼2.5-3.5 eV. Additionally, we show that the CVS-DFT/MRCI method may be employed to simulate XAS from valence excited states and compare the simulated spectra to those computed using the established wave function-based approaches [ADC(2) and ADC(2)x]. In general, each of the methods yields excited state XAS spectra in qualitative and often quantitative agreement. In the instances where the methods differ, the CVS-DFT/MRCI simulations predict intensity for transitions for which the underlying electronic states are characterized by doubly excited configurations relative to the ground state configuration. Here, we aim to demonstrate that the CVS-DFT/MRCI approach occupies a specific niche among numerous other electronic structure methods in this area, offering the ability to treat initial states of arbitrary electronic character while maintaining a low computational cost and comparatively black box usage.

Efficient calculation of X-ray absorption spectra using Chebyshev-Slepian filter diagonalisation.

The efficient, yet accurate, simulation of X-ray absorption spectra represents a significant challenge for ab initio electronic structure methods. Conventional approaches involve the explicit calculation of all core-excited states spanning the energy range of interest, even though only a small number of these states will contribute appreciably to the spectrum. We here report a different approach, based on a time-independent Chebyshev filter diagonalization scheme, which allows for the X-ray absorption spectrum to be computed without the explicit calculation of the core-excited eigenstates. Furthermore, in a subsequent postprocessing calculation, selected peaks may be analyzed via the calculation of natural transition orbitals, if desired. The scheme presented here is based on a refinement of the time-independent Chebyshev filter diagonalization approach. Previous formulations of this method have been characterized by a requirement for significant "user input" via the (sometimes unintuitive) tuning of various numerical parameters. To circumvent this, we introduce a new class of filters based on discrete prolate spheroidal sequences. We demonstrate that the resulting method, which we term Chebyshev-Slepian filter diagonalization, makes filter diagonalization essentially a black-box procedure. The Chebyshev-Slepian filter diagonalization method is implemented at the second-order algebraic diagrammatic construction level of theory and validated through the calculation of the X-ray absorption spectra of trifluoroacetonitrile and 1,4-benzoquinone.

A general approach for the calculation and characterization of x-ray absorption spectra.

We present a general approach for the calculation and assignment of X-ray absorption spectra based on electronic wavepacket propagations performed using explicitly time-dependent electronic structure calculations. Such calculations have the appeal of yielding the entire absorption spectrum for the cost of a single set of electronic wavepacket propagations, obviating the need to explicitly calculate large numbers of core-excited states. The spectrum can either be calculated from the Fourier transform of the time-dependent dipole moment or from the Fourier transform of the wavepacket autocorrelation function. We propose that calculating the absorption spectrum using the latter approach will generally be the preferred option. This method has two important advantages. First, the autocorrelation functions can be obtained for twice the propagation time, resulting in a halving of the computational effort required to calculate the spectrum relative to the time-dependent dipole moment approach. Second, using the tools of filter diagonalisation, the autocorrelation functions may be used to determine the time-independent final core-excited states underlying the peaks of interest in the spectrum. The proposed scheme is validated by calculating and characterizing the X-ray absorption spectra of benzene and trifluoroacetonitrile at the time-dependent second-order algebraic diagrammatic construction level of theory.

Vacuum ultraviolet excited state dynamics of the smallest ring, cyclopropane. II. Time-resolved photoelectron spectroscopy and ab initio dynamics.

The vacuum-ultraviolet photoinduced dynamics of cyclopropane (C3H6) were studied using time-resolved photoelectron spectroscopy (TRPES) in conjunction with ab initio quantum dynamics simulations. Following excitation at 160.8 nm, and subsequent probing via photoionization at 266.45 nm, the initially prepared wave packet is found to exhibit a fast decay (<100 fs) that is attributed to the rapid dissociation of C3H6 to ethylene (C2H4) and methylene (CH2). The photodissociation process proceeds via concerted ring opening and C-C bond cleavage in the excited state. Ab initio multiple spawning simulations indicate that ring-opening occurs prior to dissociation. The dynamics simulations were subsequently employed to simulate a TRPES spectrum, which was found to be in excellent agreement with the experimental result. On the basis of this agreement, the fitted time constants of 35 ± 20 and 57 ± 35 fs were assigned to prompt (i) dissociation on the lowest-lying excited state, prepared directly by the pump pulse, and (ii) non-adiabatic relaxation from higher-lying excited states that lead to delayed dissociation, respectively.