Compensation States Approach in the Hybrid Diabatization Scheme: Extension to Multidimensional Data and Properties.

The diabatization of reactive systems for more than just a couple of states is a very demanding problem and generally requires advanced diabatization techniques. Especially for dissociative processes, the drastic changes in the adiabatic wave functions often would require large diabatic state bases, which quickly become impractical. Recently, we addressed this problem by the compensation states approach developed in the context of our hybrid diabatization scheme. This scheme utilizes wave function as well as energy data in combination with a diabatic potential model. In regions where the initial diabatic state basis becomes insufficient for an appropriate representation of the adiabatic states, new model states are generated. The new model states compensate for the state space not spanned by the initial diabatic basis. Such a compensation state is obtained by projecting the initial diabatic state space out of the adiabatic wave function. This yields a very efficient basis representation of the electronic Hamiltonian. The present work presents two new aspects. First, it is shown how other operators like the spin-orbit operator in the framework of the Effective Relativistic Coupling by Asymptotic Representation (ERCAR) can be evaluated in this compact model state space without losing the correct wave function information and accuracy. Second, the extension of the approach to multidimensional potential energy surface models is presented for methyl iodide including the C-I dissociation coordinate and the angular H3C-I bending coordinates.

The effective relativistic coupling by asymptotic representation approach for molecules with multiple relativistic atoms.

The Effective Relativistic Coupling by Asymptotic Representation (ERCAR) approach is a method to generate fully coupled diabatic potential energy surfaces (PESs) including relativistic effects, especially spin-orbit coupling. The spin-orbit coupling of a full molecule is determined only by the atomic states of selected relativistically treated atoms. The full molecular coupling effect is obtained by a diabatization with respect to asymptotic states, resulting in the correct geometry dependence of the spin-orbit effect. The ERCAR approach has been developed over the last decade and initially only for molecules with a single relativistic atom. This work presents its extension to molecules with more than a single relativistic atom using the iodine molecule as a proof-of-principle example. The theory for the general multiple atomic ERCAR approach is given. In this case, the diabatic basis is defined at the asymptote where all relativistic atoms are separated from the remaining molecular fragment. The effective spin-orbit operator is then a sum of spin-orbit operators acting on isolated relativistic atoms. PESs for the iodine molecule are developed within the new approach and it is shown that the resulting fine structure states are in good agreement with spin-orbit ab initio calculations.

Hydrogen-iodine scattering. II. Rovibronic analysis and collisional dynamics.

Our recently published [Weike et al., J. Chem. Phys. 159, 244119 (2023)] spin-orbit coupled diabatic potential energy model for HI is used in a thorough analysis of bound and quasi-bound states as well as elastic and inelastic processes in H + I collisions. The potential energy model, designed explicitly for studying scattering, accurately describes the various couplings in the system, which lead to complex dynamics. Ro-vibronic bound and quasi-bound states related to the adiabatic electronic ground state and an excited electronic state are analyzed. Calculations using the full 104 × 104 diabatic matrix model or a single adiabatic state are compared in order to investigate approximations in the latter. Elastic and inelastic scattering cross sections as well as thermal rates between the ground and first excited fine structure levels of iodine are computed for collision energies up to 12 500 cm-1. Resonances related to the quasi-bound states are analyzed in terms of their energy, width, lifetime, and decay probabilities. The effect of different resonances on the thermal rates is discussed. Resonances between 30 000 and 40 000 cm-1 are also studied for selected values of the total angular momentum, in particular their decay probabilities into different final states of iodine and hence their potential effect on branching ratios in photodissociation of HI.

Hydrogen-iodine scattering. I. Development of an accurate spin-orbit coupled diabatic potential energy model.

The scattering of H by I is a prototypical model system for light-heavy scattering in which relativistic coupling effects must be taken into account. Scattering calculations depend strongly on the accuracy of the potential energy surface (PES) model. The methodology to obtain such an accurate PES model suitable for scattering calculations is presented, which includes spin-orbit (SO) coupling within the Effective Relativistic Coupling by Asymptotic Representation (ERCAR) approach. In this approach, the SO coupling is determined only for the atomic states of the heavy atom, and the geometry dependence of the SO effect is accounted for by a diabatization with respect to asymptotic states. The accuracy of the full model, composed of a Coulomb part and the SO model, is achieved in the following ways. For the SO model, the extended ERCAR approach is applied, which accounts for both intra-state and inter-state SO coupling, and an extended number of diabatic states are included. The corresponding coupling constants for the SO operator are obtained from experiments, which are more accurate than computed values. In the Coulomb Hamiltonian model, special attention is paid to the long range behavior and accurate c6 dispersion coefficients. The flexibility and accuracy of this Coulomb model are achieved by combining partial models for three different regions. These are merged via artificial neural networks, which also refine the model further. In this way, an extremely accurate PES model for hydrogen iodide is obtained, suitable for accurate scattering calculations.

Simulation of the photodetachment spectra of the nitrate anion (NO3-) in the B̃ 2E' energy range and non-adiabatic electronic population dynamics of NO3.

The photodetachment spectrum of the nitrate anion (NO3-) in the energy range of the NO3 second excited state is simulated from first principles using quantum wave packet dynamics. The prediction at 10 K and 435 K relies on the use of an accurate full-dimensional fully coupled five state diabatic potential model utilizing an artificial neural network. The ability of this model to reproduce experimental spectra was demonstrated recently for the lower energy range [A. Viel, D. M. G. Williams and W. Eisfeld, J. Chem. Phys. 2021, 154, 084302]. Analysis of the spectra indicates a weaker Jahn-Teller coupling compared to the first excited state. The detailed non-adiabatic dynamics is studied by computing the population dynamics. An ultra-fast non-statistical radiationless decay is found only among the Jahn-Teller components, which is followed by a slow statistical non-radiative decay among the different state manifolds. The latter is reproduced perfectly by a simple first order kinetics model. The dynamics in the second excited state is not affected by the presence of a conical intersection with the first excited state manifold.

Development of a fully coupled diabatic spin-orbit model for the photodissociation of phenyl iodide.

The theoretical treatment of the quantum dynamics of the phenyl iodide photodissociation requires an accurate analytical potential energy surface (PES) model. This model must also account for spin-orbit (SO) coupling. This study is the first step to construct accurate SO coupled PESs, namely, for the C-I dissociation coordinate. The model is based on the Effective Relativistic Coupling by Asymptotic Representation (ERCAR) method developed over the past ten years. The SO-free Hamiltonian is represented in an asymptotic diabatic basis and then combined with an atomic effective relativistic coupling operator determined analytically. In contrast to the previously studied cases (HI, CH3I), the diabatic basis states are due to excitations in the phenyl fragment rather than the iodine atom. An accurate analytical model of the ab initio reference data is determined in two steps. The first step is a simple reference model describing the data qualitatively. This reference model is corrected through a trained artificial neural-network to achieve high accuracy. The SO-free and the fine structure states resulting from this ERCAR model are discussed extensively in the context of the photodissociation.

A general method for the development of diabatic spin-orbit models for multi-electron systems.

Spin-orbit (SO) coupling can have significant effects on the quantum dynamics of molecular systems, but it is still difficult to account for accurately. One promising way to do this is to devise a diabatic SO model combined with the molecular potential energy. Few such models have been developed utilizing spatial and time-reversal symmetry. These models are tedious to derive and are specific for the molecular symmetry and included spin states. Here, we present a relatively simple approach to construct such models for various spin states with S≠12 from a basic one-electron SO case with S=12. The multi-electron fine structure states are expressed in terms of Slater determinants of single-electron spin functions (spinors). The properties of all single-electron matrix elements over the SO operator are derived and expressed as Taylor expansions in terms of symmetry-adapted nuclear coordinates. The SO matrix elements for the multi-electron case are then obtained from these single-electron matrix elements using the Slater-Condon rules. This yields the full SO matrix and symmetry properties of the multi-electron matrix elements in a straightforward way. The matrix elements are expressed as symmetry-adapted polynomials up to arbitrary order. This approach is demonstrated first for an abstract model of two electrons in a set of p orbitals in a C3v symmetric environment and then applied to set up a diabatic model for the photodissociation of methyl iodide (CH3I). The high accuracy of this new approach is demonstrated in comparison to an available analytic SO model for CH3I.

Accurate quantum dynamics simulation of the photodetachment spectrum of the nitrate anion (NO3 -) based on an artificial neural network diabatic potential model.

The photodetachment spectrum of the nitrate anion (NO3 -) is simulated from first principles using wavepacket quantum dynamics propagation and a newly developed accurate full-dimensional fully coupled five state diabatic potential model. This model utilizes the recently proposed complete nuclear permutation inversion invariant artificial neural network diabatization technique [D. M. G. Williams and W. Eisfeld, J. Phys. Chem. A 124, 7608 (2020)]. The quantum dynamics simulations are designed such that temperature effects and the impact of near threshold detachment are taken into account. Thus, the two available experiments at high temperature and at cryogenic temperature using the slow electron velocity-map imaging technique can be reproduced in very good agreement. These results clearly show the relevance of hot bands and vibronic coupling between the X̃ 2A2 ' ground state and the B̃ 2E' excited state of the neutral radical. This together with the recent experiment at low temperature gives further support for the proper assignment of the ν3 fundamental, which has been debated for many years. An assignment of a not yet discussed hot band line is also proposed.

Complete Nuclear Permutation Inversion Invariant Artificial Neural Network (CNPI-ANN) Diabatization for the Accurate Treatment of Vibronic Coupling Problems.

A recently developed scheme to produce accurate high-dimensional coupled diabatic potential energy surfaces (PESs) based on artificial neural networks (ANNs) [ J. Chem. Phys. 2018, 149, 204106 and J. Chem. Phys. 2019, 151, 164118] is modified to account for the proper complete nuclear permutation inversion (CNPI) invariance. This new approach cures the problem intrinsic to the highly flexible ANN representation of diabatic PESs to account for the proper molecular symmetry accurately. It turns out that the use of CNPI invariants as coordinates for the input layer of the ANN leads to a much more compact and thus more efficient representation of the diabatic PES model without any loss of accuracy. In connection with a properly symmetrized vibronic coupling reference model, which is modified by the output neurons of the CNPI-ANN, the resulting adiabatic PESs show perfect symmetry and high accuracy. In the present paper, the new approach will be described and thoroughly tested. The test case is the representation and corresponding vibrational/vibronic nuclear dynamics of the low-lying electronic states of planar NO3 for which a large number of ab initio data is available. Thus, the present results can be compared directly with the previous studies.

Vibronically and spin-orbit coupled diabatic potentials for X(2P) + CH4 → HX + CH3 reactions: Neural network potentials for X = Cl.

Vibronically and spin-orbit (SO) coupled diabatic potentials for the Cl(2P) + CH4 → HCl + CH3 reaction are constructed based on a recently developed approach [T. Lenzen and U. Manthe, J. Chem. Phys. 150, 064102 (2019)]. Diabatic potentials and couplings describing the entrance channel of the reaction are obtained based on ab initio data using a diabatization by an ansatz scheme. A detailed investigation of the electronic structure in the entrance channel using multireference configuration interaction (MRCI), coupled cluster [CCSD/CCSD(T)], and SO-MRCI calculations is presented. Neural networks using permutationally invariant polynomials as inputs are employed to represent the elements of the diabatic potential energy matrix. The same set of diabatic states is also used in the transition state region and all four exit channels. Here, the lowest adiabatic potential energy surface (PES) derived from the diabatic model is chosen to reproduce an adiabatic PES recently developed by Li and Guo. The accuracy of the resulting PES is evaluated, and the properties of the newly developed coupled diabatic potentials are analyzed in detail.

Diabatic neural network potentials for accurate vibronic quantum dynamics-The test case of planar NO3.

A recently developed scheme to produce high-dimensional coupled diabatic potential energy surfaces (PESs) based on artificial neural networks (ANNs) [D. M. G. Williams and W. Eisfeld, J. Chem. Phys. 149, 204106 (2019)] is tested for its viability for quantum dynamics applications. The method, capable of reproducing high-quality ab initio data with excellent accuracy, utilizes simple coupling matrices to produce a basic low-order diabatic potential matrix as an underlying backbone for the model. This crude model is then refined by making its expansion coefficients geometry-dependent by the output neurons of the ANN. This structure, strongly guided by a straightforward physical picture behind nonadiabatic coupling, combines structural simplicity with high accuracy, reproducing ab initio data without introducing unphysical artifacts to the surface, even for systems with complicated electronic structure. The properties of diabatic potentials obtained by this method are tested thoroughly in the present study. Vibrational/vibronic eigenstates are computed on the X̃ and à states of NO3, a notoriously difficult Jahn-Teller system featuring strong nonadiabatic couplings and complex spectra. The method is investigated in terms of how consistently it produces dynamics results for PESs of similar (fitting) quality and how the results depend on the ANN size and ANN topography. A central aspect of this work is to understand the convergence properties of the new method in order to evaluate its predictive power. A previously developed, high-quality model utilizing a purely (high-order) polynomial ansatz is used as a reference to showcase improvements of the overall quality which can be obtained by the new method.

Quantum dynamics and geometric phase in E⊗e Jahn-Teller systems with general Cnv symmetry.

E ⊗ e Jahn-Teller (JT) systems are considered the prototype of symmetry-induced conical intersections and of the corresponding geometric phase effect (GPE). For decades, this has been analyzed for the most common case originating from C3v symmetry and these results usually were generalized. In the present work, a thorough analysis of the JT effect, vibronic coupling Hamiltonians, GPE, and the effect on spectroscopic properties is carried out for general Cnv symmetric systems (and explicitly for n = 3-8). It turns out that the C3v case is much less general than often assumed. The GPE due to the vibronic Hamiltonian depends on the leading coupling term of a diabatic representation of the problem, which is a result of the explicit n, α, and ß values of a CnvEα ⊗ eß system. Furthermore, the general existence of n/m (m∈N depending on n, α, and ß) equivalent minima on the lower adiabatic sheet of the potential energy surface (PES) leads to tunneling multiplets of n/m states (state components). These sets can be understood as local vibrations of the atoms around their equilibrium positions within each of the local PES wells symmetrized over all equivalent wells. The local vibrations can be classified as tangential or radial vibrations, and the quanta in the tangential mode together with the GPE determine the level ordering within each of the vibronic multiplets. Our theoretical predictions derived analytically are tested and supported by numerical model simulations for all possible Eα ⊗ eß cases for Cnv symmetric systems with n = 3-8. The present interpretation allows for a full understanding of the complex JT spectra of real systems, at least for low excitation energies. This also opens a spectroscopic way to show the existence or absence of GPEs.

Neural network diabatization: A new ansatz for accurate high-dimensional coupled potential energy surfaces.

A new diabatization method based on artificial neural networks (ANNs) is presented, which is capable of reproducing high-quality ab initio data with excellent accuracy for use in quantum dynamics studies. The diabatic potential matrix is expanded in terms of a set of basic coupling matrices and the expansion coefficients are made geometry-dependent by the output neurons of the ANN. The ANN is trained with respect to ab initio data using a modified Marquardt-Levenberg back-propagation algorithm. Due to its setup, this approach combines the stability and straightforwardness of a standard low-order vibronic coupling model with the accuracy by the ANN, making it particularly advantageous for problems with a complicated electronic structure. This approach combines the stability and straightforwardness of a standard low-order vibronic coupling model with the accuracy by the ANN, making it particularly advantageous for problems with a complicated electronic structure. This novel ANN diabatization approach has been applied to the low-lying electronic states of NO3 as a prototypical and notoriously difficult Jahn-Teller system in which the accurate description of the very strong non-adiabatic coupling is of paramount importance. Thorough tests show that an ANN with a single hidden layer is sufficient to achieve excellent results and the use of a "deeper" layering shows no clear benefit. The newly developed diabatic ANN potential energy surface (PES) model accurately reproduces a set of more than 90 000 Multi-configuration Reference Singles and Doubles Configuration Interaction (MR-SDCI) energies for the five lowest PES sheets.

Vibronic eigenstates and the geometric phase effect in the 2E″ state of NO3.

The 2E″ state of NO3, a prototype for the Jahn-Teller effect, has been an enigma and a challenge for a long time for both experiment and theory. We present a detailed theoretical study of the vibronic quantum dynamics in this electronic state, uncovering the effects of tunnelling, geometric phase, and symmetry. To this end, 45 vibronic levels of NO3 in the 2E″ state are determined accurately and analyzed thoroughly. The computation is based on a high quality diabatic potential representation of the two-sheeted surface of the 2E″ state developed by us [W. Eisfeld et al., J. Chem. Phys. 140, 224109 (2014)] and on the multi-configuration time dependent Hartree approach. The vibrational eigenstates of the NO3- anion are determined and analyzed as well to gain a deeper understanding of the symmetry properties of such D3h symmetric systems. To this end, 61 eigenstates of the NO3- anion ground state are computed using the single sheeted potential surface of the 1A1 state published in the same reference quoted above. The assignments of both the vibrational and vibronic levels are discussed. A simple model is proposed to rationalize the computed NO3 spectrum strongly influenced by the Jahn-Teller couplings, the associated geometric phase effect, and the tunnelling. Comparison with the available spectroscopic data is also presented.

An improved spin-orbit coupling model for use within the effective relativistic coupling by asymptotic representation (ERCAR) method.

An improved atomic spin-orbit model is presented, which is designed to be used within the framework of the effective relativistic coupling by asymptotic representation method. This method is used for the generation of highly accurate coupled potential energy surfaces (PESs) to represent the fine structure energies of appropriate systems. The approach is demonstrated using CH3I and its photodissociation as a typical example. The method is based on a specific diabatization of electronic spin-space ("spin-free") states with respect to the asymptote at which a single relativistic atom is separated from a molecular non-relativistic fragment. Thus, the relativistic coupling effects can be treated entirely within the atomic framework. So far, an effective spin-orbit coupling operator which only accounts for intra-state coupling within each atomic spin-space state was used. In the present work, this approach is extended to account for inter-state couplings among different atomic spin-space states as well. It is shown that this extended approach improves the accuracy of the PESs significantly for higher excited states and also enhances the accuracy of low energy states. In particular, it improves the representation of the spin-orbit induced conical intersection among the 3Q0 and 1Q1 states of CH3I, which is of high relevance for the nonadiabatic quantum dynamics of the photodissociation.

Block-diagonalization as a tool for the robust diabatization of high-dimensional potential energy surfaces.

Robust diabatization techniques are key for the development of high-dimensional coupled potential energy surfaces (PESs) to be used in multi-state quantum dynamics simulations. In the present study we demonstrate that, besides the actual diabatization technique, common problems with the underlying electronic structure calculations can be the reason why a diabatization fails. After giving a short review of the theoretical background of diabatization, we propose a method based on the block-diagonalization to analyse the electronic structure data. This analysis tool can be used in three different ways: First, it allows to detect issues with the ab initio reference data and is used to optimize the setup of the electronic structure calculations. Second, the data from the block-diagonalization are utilized for the development of optimal parametrized diabatic model matrices by identifying the most significant couplings. Third, the block-diagonalization data are used to fit the parameters of the diabatic model, which yields an optimal initial guess for the non-linear fitting required by standard or more advanced energy based diabatization methods. The new approach is demonstrated by the diabatization of 9 electronic states of the propargyl radical, yielding fully coupled full-dimensional (12D) PESs in closed form.

Development of multi-mode diabatic spin-orbit models at arbitrary order.

The derivation of diabatic spin-orbit (SO) Hamiltonians is presented, which are expanded in terms of nuclear coordinates to arbitrary order including the treatment of multi-mode systems, having more than one mode of the same symmetry. The derivation is based on the microscopic Breit-Pauli SO operator and the consequent utilization of time reversal and spatial symmetry transformation properties of basis functions and coordinates. The method is demonstrated for a set of (2)E and (2)A1 states in C(3v)* (double group) symmetry, once for a 3D case of one a1 and one e mode and once for a 9D case of three a1 and three e coordinates. It is shown that the general structure of the diabatic SO Hamiltonian only depends on the basis states and is strictly imposed by time reversal symmetry. The resulting matrix can be expressed easily by a power series using six parametrized structure matrices as expansion coefficients multiplied by the associated monomials in terms of symmetrized coordinates. The explicit example presented here provides a full-dimensional diabatic SO model for methyl halide cations, which will be studied in the future.

A new approach for the development of diabatic potential energy surfaces: Hybrid block-diagonalization and diabatization by ansatz.

A new diabatization method is presented, which is suitable for the development of accurate high-dimensional coupled potential energy surfaces for use in quantum dynamics studies. The method is based on the simultaneous use of adiabatic wave function and energy data, respectively, and combines block-diagonalization and diabatization by ansatz approaches. It thus is called hybrid diabatization. The adiabatic wave functions of suitable ab initio calculations are projected onto a diabatic state space and the resulting vectors are orthonormalized like in standard block-diagonalization. A parametrized diabatic model Hamiltonian is set up as an ansatz for which the block-diagonalization data can be utilized to find the optimal model. Finally, the parameters are optimized with respect to the ab initio reference data such that the deviations between adiabatic energies and eigenvalues of the model as well as projected state vectors and eigenvectors of the model are minimized. This approach is particularly advantageous for problems with a complicated electronic structure where the diabatic state space must be of higher dimension than the number of calculated adiabatic states. This is an efficient way to handle problems with intruder states, which are very common for reactive systems. The use of wave function information also increases the information content for each data point without additional cost, which is beneficial in handling the undersampling problem for high-dimensional systems. The new method and its performance are demonstrated by application to three prototypical systems, ozone (O3), methyl iodide (CH3I), and propargyl (H2CCCH).

Full-dimensional diabatic potential energy surfaces including dissociation: the ²E″ state of NO3.

A scheme to produce accurate full-dimensional coupled diabatic potential energy surfaces including dissociative regions and suitable for dynamical calculations is proposed. The scheme is successfully applied to model the two-sheeted surface of the (2)E″ state of the NO3 radical. An accurate potential energy surface for the NO3⁻ anion ground state is developed as well. Both surfaces are based on high-level ab initio calculations. The model consists of a diabatic potential matrix, which is expanded to higher order in terms of symmetry polynomials of symmetry coordinates. The choice of coordinates is key for the accuracy of the obtained potential energy surfaces and is discussed in detail. A second central aspect is the generation of reference data to fit the expansion coefficients of the model for which a stochastic approach is proposed. A third ingredient is a new and simple scheme to handle problematic regions of the potential energy surfaces, resulting from the massive undersampling by the reference data unavoidable for high-dimensional problems. The final analytical diabatic surfaces are used to compute the lowest vibrational levels of NO3⁻ and the photo-electron detachment spectrum of NO3⁻ leading to the neutral radical in the (2)E″ state by full dimensional multi-surface wave-packet propagation for NO3 performed using the Multi-Configuration Time Dependent Hartree method. The achieved agreement of the simulations with available experimental data demonstrates the power of the proposed scheme and the high quality of the obtained potential energy surfaces.

Toward spin-orbit coupled diabatic potential energy surfaces for methyl iodide using effective relativistic coupling by asymptotic representation.

The theoretical treatment of state-state interactions and the development of coupled multidimensional potential energy surfaces (PESs) is of fundamental importance for the theoretical investigation of nonadiabatic processes. Usually, only derivative or vibronic coupling is considered, but the presence of heavy atoms in a system can render spin-orbit (SO) coupling important as well. In the present study, we apply a new method recently developed by us (J. Chem. Phys. 2012, 136, 034103, and J. Chem. Phys. 2012, 137, 064101) to generate SO coupled diabatic PESs along the C-I dissociation coordinate for methyl iodide (CH3I). This is the first and mandatory step toward the development of fully coupled full-dimensional PESs to describe the multistate photodynamics of this benchmark system. The method we use here is based on the diabatic asymptotic representation of the molecular fine structure states and an effective relativistic coupling operator. It therefore is called effective relativistic coupling by asymptotic representation (ERCAR). This approach allows the efficient and accurate generation of fully coupled PESs including derivative and SO coupling based on high-level ab initio calculations. In this study we develop a specific ERCAR model for CH3I that so far accounts only for the C-I bond cleavage. Details of the diabatization and the accuracy of the results are investigated in comparison to reference ab initio calculations and experiments. The energies of the adiabatic fine structure states are reproduced in excellent agreement with ab initio SO-CI data. The model is also compared to available literature data, and its performance is evaluated critically. This shows that the new method is very promising for the construction of fully coupled full-dimensional PESs for CH3I to be used in future quantum dynamics studies.