The anion photoelectron spectrum and diabatization of tetrazolyl.

The potential energy surface of tetrazolyl [cyclic (N4CH)] has a conical intersection seam between the two lowest-energy electronic states near the ground state minimum geometry. This work treats that molecule. The potential energy surfaces used in this study are based on a least-squares fitting procedure that includes ab initio energies, energy gradients, and derivative couplings described using polynomials up to fourth-order and ab initio data obtained from multireference configuration interaction wave functions. A five-electronic-state description was generated with a root mean square absolute energy error of 9.6 cm-1, compared to 326.8 cm-1 when only second-order terms were used. The time-independent multimode vibronic coupling in the KDC approximation was used to simulate and analyze the anion ultraviolet photoelectron spectrum of tetrazolide.

Quantum Dynamics of Photodissociation: Recent Advances and Challenges.

Recent advances in constructing accurate potential energy surfaces and nonadiabatic couplings from high-level ab initio data have revealed detailed potential landscapes in not only the ground electronic state but also excited ones. They enabled quantitatively accurate characterization of photoexcited reactive systems using quantum mechanical methods. In this Perspective, we survey the recent progress in quantum mechanical studies of adiabatic and nonadiabatic photodissociation dynamics, focusing on initial state control and product energy disposal. These new insights helped to understand quantum effects in small prototypical systems, and the results serve as benchmarks for developing more approximate theoretical methods.

Toward a Unified Analytical Description of Internal Conversion and Intersystem Crossing in the Photodissociation of Thioformaldehyde. I. Diabatic Singlet States.

The photodissociation of thioformaldehyde is an archetypal system for the study of competition between internal conversion and intersystem crossing, which involves its two singlet states (S0 and S1) and two triplet states (T1 and T2). In order to perform accurate dynamic simulations, either quantum or quasi-classical, it is essential to construct an analytical representation for all necessary electronic structure data. In this work, a diabatic potential energy matrix (DPEM), Hd, for the two singlet states (S0 and S1) is reported. The analytical form of DPEM is symmetrized and constructed to reproduce adiabatic energies, energy gradients, and derivative couplings obtained from high-level multireference configuration interaction wave functions. The Hd is fully saturated in the molecular configuration space with a trajectory-guided point sampling approach. This Hd can provide the accurate description of the photodissociation of thioformaldehyde on its singlet states and is also a necessary part for incorporating the spin-orbit couplings into a unified diabatic framework. Preliminary quasi-classical trajectory simulations show that a roaming mechanism also exists in the molecular dissociation channel of thioformaldehyde.

Nonadiabatic Reactive Quenching of OH(A2Σ+) by H2: Origin of High Vibrational Excitation in the H2O Product.

The nonadiabatic dynamics of the reactive quenching channel of the OH(A2Σ+) + H2/D2 collisions is investigated with a semiclassical surface hopping method, using a recently developed four-state diabatic potential energy matrix (DPEM). In agreement with experimental observations, the H2O/HOD products are found to have significant vibrational excitation. Using a Gaussian binning method, the H2O vibrational state distribution is determined. The preferential energy disposal into the product vibrational modes is rationalized by an extended Sudden Vector Projection model, in which the h and g vectors associated with the conical intersection are found to have large projections with the product normal modes. However, our calculations did not find significant insertion trajectories, suggesting the need for further improvement of the DPEM.

Permutation invariant polynomial neural network based diabatic ansatz for the (E + A) × (e + a) Jahn-Teller and Pseudo-Jahn-Teller systems.

In this work, the permutation invariant polynomial neural network (PIP-NN) approach is employed to construct a quasi-diabatic Hamiltonian for system with non-Abelian symmetries. It provides a flexible and compact NN-based diabatic ansatz from the related approach of Williams, Eisfeld, and co-workers. The example of H3 + is studied, which is an (E + A) × (e + a) Jahn-Teller and Pseudo-Jahn-Teller system. The PIP-NN diabatic ansatz is based on the symmetric polynomial expansion of Viel and Eisfeld, the coefficients of which are expressed with neural network functions that take permutation-invariant polynomials as input. This PIP-NN-based diabatic ansatz not only preserves the correct symmetry but also provides functional flexibility to accurately reproduce ab initio electronic structure data, thus resulting in excellent fits. The adiabatic energies, energy gradients, and derivative couplings are well reproduced. A good description of the local topology of the conical intersection seam is also achieved. Therefore, this diabatic ansatz completes the PIP-NN based representation of DPEM with correct symmetries and will enable us to diabatize even more complicated systems with complex symmetries.

Representation of Diabatic Potential Energy Matrices for Multiconfiguration Time-Dependent Hartree Treatments of High-Dimensional Nonadiabatic Photodissociation Dynamics.

Conventional quantum mechanical characterization of photodissociation dynamics is restricted by steep scaling laws with respect to the dimensionality of the system. In this work, we examine the applicability of the multi-configurational time-dependent Hartree (MCTDH) method in treating nonadiabatic photodissociation dynamics in two prototypical systems, taking advantage of its favorable scaling laws. To conform to the sum-of-product form, elements of the ab initio diabatic potential energy matrix (DPEM) are re-expressed using the recently proposed Monte Carlo canonical polyadic decomposition method, with enforcement of proper symmetry. The MCTDH absorption spectra and product branching ratios are shown to compare well with those calculated using conventional grid-based methods, demonstrating its promise for treating high-dimensional nonadiabatic photodissociation problems.

Internal conversion and intersystem crossing dynamics based on coupled potential energy surfaces with full geometry-dependent spin-orbit and derivative couplings. Nonadiabatic photodissociation dynamics of NH3(A) leading to the NH(X3Σ-, a1Δ) + H2 channel.

We simulate the photodissociation of NH3 originating from its first excited singlet state S1 into the NH2 + H (radical) and NH + H2 (molecular) channels. The states considered are the ground singlet state S0, the first excited singlet state S1 and the lowest-lying triplet state T1, which permit for the first time a uniform treatment of the internal conversion and intersystem crossing. The simulations are based on a diabatic potential energy matrix (DPEM) of S0, S1 coupled by a conical intersection seam, as well as a potential energy surface (PES) for T1 coupled by spin-orbit coupling (SOC) to the two singlet states. The DPEM and PES are fitted to ab initio electronic structure data (ESD) including energies, energy gradients, and derivative couplings. The DPEM also defines an adiabatic to diabatic state (AtD) transformation, which is used to transform the singular adiabatic SOC into a smooth function of the nuclear coordinates in the diabatic representation, allowing the diabatic SOC to be fit to an analytical functional form. ESD and SOC data obtained from these surfaces can serve as input for either quantum or semi-classical characterization of the nonadiabatic dynamics. Using the SHARC suite of programs, nonadiabatic simulations based on over 40 000 semi-classical trajectories assess the convergence of our results. The production of NH + H2 is not direct, but is only achieved through a quasi-statistical dissociation mechanism after internal conversion to the ground electronic state. This leads to a much lower yield comparing with the main NH2 + H channel. The NH(X3Σ_) radical produced through the intersystem crossing from S0 to T1 is rare (∼0.2%) compared to NH(a1Δ) due to the process being spin forbidden.

Unified Description of the Jahn-Teller Effect in Molecules with Only Cs Symmetry: Cyclohexoxy in Its Full 48-Dimensional Internal Coordinates.

The two lowest potential energy surfaces of cyclohexoxy which are coupled by conical intersections and the spin-orbit interaction are determined in the full 48-dimensional internal coordinate space using a feedforward neural network to fit a diabatic potential energy matrix. The electronic structure data are obtained at the multireference configuration interaction with single- and double-excitation level. Underlying parallels between these coupled surfaces and those of the alkoxy radicals methoxy and isopropoxy are established. Earlier work by Dillon and Yarkony is extended. While the parallels would have been challenging to appreciate using the concept of the Jahn-Teller active modes, they are readily seen in terms of two internal modes centered at the conical intersection: g the energy difference gradient vector and h the interstate coupling gradient vector. In other words, g and h vectors provide a unified description of the Jahn-Teller effect in molecules exhibiting C3v and quasi-C3v symmetries. A spectral simulation in the full 48-vibrational-internal coordinate space is reported. This spectrum is obtained using recently developed algorithms designed to increase the size of the systems that can be treated with a time-independent vibronic coupling approach.

Conical intersection seams in spin-orbit coupled systems with an even number of electrons: A numerical study based on neural network fit surfaces.

In this work, we consider the existence and topography of seams of conical intersections (CIs) for two key singlet-triplet systems, including a uniformly scaled spin-orbit interaction. The basic one triplet and one singlet state system denoted as (S0,T1) and the two singlets and one triplet system denoted as (S0,S1,T1) are treated. Essential to this analysis are realistic electronic structure data taken from a recently reported neural network fit for the 1,21A and 13A states of NH3, including Hsf (spin-free) and Hso (spin-orbit) surfaces derived from high quality ab initio wavefunctions. Three types of seams for the (S0,S1,T1) system are reported, which depend on the choice of the electronic Hamiltonian, He. The nonrelativistic CI seam [He = Hsf, (S0,S1)], the energy minimized nonrelativistic singlet-triplet intersection seam [He = Hsf, (S0,T1)], and the fully relativistic seam in the spin-diabatic representation (He = Htot = Hsf + Hso) are reported as functions of R(N-H). The derivative couplings are computed using He = Htot and Hsf from the fit data. The line integral of the derivative coupling is employed to juxtapose the geometric phase in the relativistic, He = Htot, and nonrelativistic, He = Hsf, cases. It is found for the (S0,T1) system that there is no CI in the spin-adiabatic representation, while for the (S0,S1,T1) system, CI can only be formed for two pairs of spin-adiabatic electronic states. The geometric phase effect thus needs to be handled with care when it comes to spin-nonconserving dynamics simulations.

High-fidelity first principles nonadiabaticity: diabatization, analytic representation of global diabatic potential energy matrices, and quantum dynamics.

Nonadiabatic dynamics, which goes beyond the Born-Oppenheimer approximation, has increasingly been shown to play an important role in chemical processes, particularly those involving electronically excited states. Understanding multistate dynamics requires rigorous quantum characterization of both electronic and nuclear motion. However, such first principles treatments of multi-dimensional systems have so far been rather limited due to the lack of accurate coupled potential energy surfaces and difficulties associated with quantum dynamics. In this Perspective, we review recent advances in developing high-fidelity analytical diabatic potential energy matrices for quantum dynamical investigations of polyatomic uni- and bi-molecular nonadiabatic processes, by machine learning of high-level ab initio data. Special attention is paid to methods of diabatization, high fidelity construction of multi-state coupled potential energy surfaces and property surfaces, as well as quantum mechanical characterization of nonadiabatic nuclear dynamics. To illustrate the tremendous progress made by these new developments, several examples are discussed, in which direct comparison with quantum state resolved measurements led to either confirmation of the observation or sometimes reinterpretation of the experimental data. The insights gained in these prototypical systems greatly advance our understanding of nonadiabatic dynamics in chemical systems.

Full-dimensional quantum stereodynamics of the non-adiabatic quenching of OH(A2Σ+) by H2.

The Born-Oppenheimer approximation, assuming separable nuclear and electronic motion, is widely adopted for characterizing chemical reactions in a single electronic state. However, the breakdown of the Born-Oppenheimer approximation is omnipresent in chemistry, and a detailed understanding of the non-adiabatic dynamics is still incomplete. Here we investigate the non-adiabatic quenching of electronically excited OH(A2Σ+) molecules by H2 molecules using full-dimensional quantum dynamics calculations for zero total nuclear angular momentum using a high-quality diabatic-potential-energy matrix. Good agreement with experimental observations is found for the OH(X2Π) ro-vibrational distribution, and the non-adiabatic dynamics are shown to be controlled by stereodynamics, namely the relative orientation of the two reactants. The uncovering of a major (in)elastic channel, neglected in a previous analysis but confirmed by a recent experiment, resolves a long-standing experiment-theory disagreement concerning the branching ratio of the two electronic quenching channels.

Enabling a Unified Description of Both Internal Conversion and Intersystem Crossing in Formaldehyde: A Global Coupled Quasi-Diabatic Hamiltonian for Its S0, S1, and T1 States.

In our recent work, a diabatic Hamiltonian that couples the S0 and S1 states of formaldehyde was constructed using a robust fitting-and-diabatizing procedure with artificial neural networks, which is capable of representing adiabatic energies, energy gradients, and derivative couplings over a wide range of geometries including seams of conical intersection. In this work, based on the diabatization of S0 and S1, the spin-orbit couplings between singlet states (S0, S1) and triplet state T1 are also determined in the same diabatic representation. The diabatized spin-orbit couplings are then fit with a symmetrized neural-network functional form. The ab initio spin-orbit couplings are well reproduced in large configuration space. Together with the neural-network-based potential energy surface for T1, the full quasi-diabatic Hamiltonian for the S0, S1, and T1 states is completed, enabling a unified description of both internal conversion and intersystem crossing in formaldehyde. The vibrational levels on the three adiabatic states are found to be in good agreement with known experimental band origins.

Enabling complete multichannel nonadiabatic dynamics: A global representation of the two-channel coupled, 1,21A and 13A states of NH3 using neural networks.

Global coupled three-state two-channel potential energy and property/interaction (dipole and spin-orbit coupling) surfaces for the dissociation of NH3(Ã) into NH + H2 and NH2 + H are reported. The permutational invariant polynomial-neural network approach is used to simultaneously fit and diabatize the electronic Hamiltonian by fitting the energies, energy gradients, and derivative couplings of the two coupled lowest-lying singlet states as well as fitting the energy and energy gradients of the lowest-lying triplet state. The key issue in fitting property matrix elements in the diabatic basis is that the diabatic surfaces must be smooth, that is, the diabatization must remove spikes in the original adiabatic property surfaces attributable to the switch of electronic wavefunctions at the conical intersection seam. Here, we employ the fit potential energy matrix to transform properties in the adiabatic representation to a quasi-diabatic representation and remove the discontinuity near the conical intersection seam. The property matrix elements can then be fit with smooth neural network functions. The coupled potential energy surfaces along with the dipole and spin-orbit coupling surfaces will enable more accurate and complete treatment of optical transitions, as well as nonadiabatic internal conversion and intersystem crossing.

Neural Network Based Quasi-diabatic Representation for S0 and S1 States of Formaldehyde.

A neural network based quasi-diabatic potential energy matrix Hd that describes the photodissociation of formaldehyde involving the two lowest singlet states S0 and S1 is constructed. It has strict complete nuclear permutation inversion symmetry encoded and can reproduce high level ab initio electronic structure data, including energies, energy gradients, and derivative couplings, with excellent accuracy. It has been fully saturated in the configuration space to cover all possible reaction pathways with a trajectory-guided point sampling approach. This Hd will not only enable the accurate full-dimensional dynamic simulations of the photodissociation of formaldehyde involving S0 and S1 but also provide a crucial ingredient for incorporating spin-orbit couplings into a diabatic framework, thus ultimately enabling the study of both internal conversion and intersystem crossing in formaldehyde on the same footing.

Impact of Diabolical Singular Points on Nonadiabatic Dynamics and a Remedy: Photodissociation of Ammonia in the First Band.

Several recent publications have pointed out a potentially severe drawback in some widely used diabatization methods based on the electronic properties of molecules. In a diabatic representation defined by a property-based method, artificial singularities may arise due to the defining equation of the adiabatic-to-diabatic (AtD) transformation. Such diabolical singular points (DSPs) may seriously affect nuclear dynamics if they lie in the relevant configuration space. Their impact is demonstrated here using the A-band photodissociation of ammonia as an example. To this end, quantum dynamics calculations are performed based on a diabatic potential energy matrix (DPEM) constructed using the generalized Mulliken-Hush method, which is based on dipoles. These property-based results are compared with the results obtained with a DPEM determined using derivative coupling explicitly. A DSP seam is found near the Franck-Condon region, which results in a complete failure to reproduce the absorption spectrum. A modification of the generalized Mulliken-Hush method is proposed to remove the DSPs while preserving the conical intersection, which leads to an accurate reproduction of the absorption spectrum and the NH2(Ã)/NH2(X̃) product branching ratio.

Compact Bases for Vibronic Coupling in Spectral Simulations: The Photoelectron Spectrum of Cyclopentoxide in the Full 39 Internal Modes.

We report an algorithm to automatically generate compact multimode vibrational bases for the Köppel-Domcke-Cederbaum (KDC) vibronic coupling wave function used in spectral simulations of moderate-sized molecules. As a full quantum method, the size of the vibronic expansion grows exponentially with respect to the number of vibrational modes, necessitating compact bases for moderate-sized systems. The problem of generating such a basis consists of two parts: one is the choice of vibrational normal modes, and the other is the number of phonons allowed in each mode. A previously developed final-state-biased technique addresses the former part, and this work focuses on the latter part: proposing an algorithm for generating an optimal phonon distribution. By virtue of this phonon distribution, compact and affordable bases can be automatically generated for systems with on the order of 15 atoms. Our algorithm is applied to determine the nonadiabatic photoelectron spectrum of cyclopentoxide in the full 39 internal modes.

On the nonadiabatic collisional quenching of OH(A) by H2: a four coupled quasi-diabatic state description.

A four-state diabatic potential energy matrix (DPEM), Hd, for the description of the nonadiabatic quenching of OH(A2Σ+) by collisions with H2 is reported. The DPEM is constructed as a fit to adiabatic energies, energy gradients, and derivative couplings obtained exclusively from multireference configuration interaction wave functions. A four-adiabatic-electronic-state representation is used in order to describe all energetically accessible regions of the nuclear coordinate space. Partial permutation-inversion symmetry is incorporated into the representation. The fit is based on electronic structure data at 42 882 points, described by over 1.6 million least squares equations with a root mean square (mean unsigned) error of 178(83) cm-1. Comparison of ab initio and Hd determined minima, saddle points, and energy minimized points on C2v, Cs, C∞v, and C1 (noncoplanar) portions of two conical intersection seams are used to establish the accuracy of the Hd.

Construction of Quasi-diabatic Hamiltonians That Accurately Represent ab Initio Determined Adiabatic Electronic States Coupled by Conical Intersections for Systems on the Order of 15 Atoms. Application to Cyclopentoxide Photoelectron Detachment in the Full 39 Degrees of Freedom.

We present, for systems of moderate dimension, a fitting framework to construct quasi-diabatic Hamiltonians that accurately represent ab initio adiabatic electronic structure data including the effects of conical intersections. The framework introduced here minimizes the difference between the fit prediction and the ab initio data obtained in the adiabatic representation, which is singular at a conical intersection seam. We define a general and flexible merit function to allow arbitrary representations and propose a representation to measure the fit-ab initio difference at geometries near electronic degeneracies. A fit Hamiltonian may behave poorly in insufficiently sampled regions, in which case a machine learning theory analysis of the fit representation suggests a regularization to address the deficiency. Our fitting framework including the regularization is used to construct the full 39-dimensional coupled diabatic potential energy surfaces for cyclopentoxy relevant to cyclopentoxide photoelectron detachment.

The generality of the GUGA MRCI approach in COLUMBUS for treating complex quantum chemistry.

The core part of the program system COLUMBUS allows highly efficient calculations using variational multireference (MR) methods in the framework of configuration interaction with single and double excitations (MR-CISD) and averaged quadratic coupled-cluster calculations (MR-AQCC), based on uncontracted sets of configurations and the graphical unitary group approach (GUGA). The availability of analytic MR-CISD and MR-AQCC energy gradients and analytic nonadiabatic couplings for MR-CISD enables exciting applications including, e.g., investigations of π-conjugated biradicaloid compounds, calculations of multitudes of excited states, development of diabatization procedures, and furnishing the electronic structure information for on-the-fly surface nonadiabatic dynamics. With fully variational uncontracted spin-orbit MRCI, COLUMBUS provides a unique possibility of performing high-level calculations on compounds containing heavy atoms up to lanthanides and actinides. Crucial for carrying out all of these calculations effectively is the availability of an efficient parallel code for the CI step. Configuration spaces of several billion in size now can be treated quite routinely on standard parallel computer clusters. Emerging developments in COLUMBUS, including the all configuration mean energy multiconfiguration self-consistent field method and the graphically contracted function method, promise to allow practically unlimited configuration space dimensions. Spin density based on the GUGA approach, analytic spin-orbit energy gradients, possibilities for local electron correlation MR calculations, development of general interfaces for nonadiabatic dynamics, and MRCI linear vibronic coupling models conclude this overview.

Accurate Neural Network Representation of the Ab Initio Determined Spin-Orbit Interaction in the Diabatic Representation Including the Effects of Conical Intersections.

A method for fitting ab initio determined spin-orbit coupling interactions, in the Breit-Pauli approximation, based on quasidiabatic representations using neural network fits is reported. The algorithm generalizes our recently reported neural network approach for representing the dipole interaction. The S0, S1, and T1 states of formaldehyde are used as an example. First, the two singlet states S0 and S1 are diabatized with a modified Boys Localization diabatization method. Second, the spin-orbit coupling between singlet and triplet states is transformed to the diabatic representation. This removes the discontinuities in the adiabatic representation. The diabatized spin-orbit couplings are then fit with smooth neural network functions. The analytic representation of spin-orbit coupling interactions in a diabatic basis by neural networks will make accurate full-dimensional quantum dynamical treatment of both internal conversion and intersystem crossing possible, which will help us to gain better understanding of both processes.