Accurate diabatization based on combined-hyperbolic-inverse-power-representation: 1,2 2A' states of BeH2.

A diabatic potential energy matrix (DPEM) for the two lowest states of BeH2+ has been constructed using the combined-hyperbolic-inverse-power-representation (CHIPR) method. By imposing symmetry constraints on the coefficients of polynomials, the complete nuclear permutation inversion symmetry is correctly preserved in the CHIPR functional form. The symmetrized CHIPR functional form is then used in the diabatization by ansatz procedure. The ab initio energies are reproduced with satisfactory accuracy. In addition, the CHIPR-based DPEM also reproduces the local topology of a conical intersection. Future work will focus on a complete four-state diabatic representation with emphasis on the long-range interactions and spin-orbit couplings, which will enable accurate quantum scattering calculations for the Be+(2P) + H2 → BeH+(X1Σ+) + H(2S) reaction.

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.

The OpenMolcas Web: A Community-Driven Approach to Advancing Computational Chemistry.

The developments of the open-source OpenMolcas chemistry software environment since spring 2020 are described, with a focus on novel functionalities accessible in the stable branch of the package or via interfaces with other packages. These developments span a wide range of topics in computational chemistry and are presented in thematic sections: electronic structure theory, electronic spectroscopy simulations, analytic gradients and molecular structure optimizations, ab initio molecular dynamics, and other new features. This report offers an overview of the chemical phenomena and processes OpenMolcas can address, while showing that OpenMolcas is an attractive platform for state-of-the-art atomistic computer simulations.

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.

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.

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.

A fundamental invariant-neural network representation of quasi-diabatic Hamiltonians for the two lowest states of H3.

The fundamental invariant neural network (FI-NN) approach is developed to represent coupled potential energy surfaces in quasidiabatic representations with two-dimensional irreducible representations of the complete nuclear permutation and inversion (CNPI) group. The particular symmetry properties of the diabatic potential energy matrix of H3 for the 1A' and 2A' electronic states were resolved arising from the E symmetry in the D3h point group. This FI-NN framework with symmetry adaption is used to construct a new quasidiabatic representation of H3, which reproduces accurately the ab initio energies and derivative information with perfect symmetry behaviors and extremely small fitting errors. The quantum dynamics results on the new FI-NN diabatic PESs give rise to accurate oscillation patterns in the product state-resolved differential cross sections. These results strongly support the accuracy and efficiency of the FI-NN approach to construct reliable diabatic representations with complicated symmetry problems.

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.

Exclusive Neural Network Representation of the Quasi-Diabatic Hamiltonians Including Conical Intersections.

We propose a numerically simple and straightforward, yet accurate and efficient neural networks-based fitting strategy to construct coupled potential energy surfaces (PESs) in a quasi-diabatic representation. The fundamental invariants are incorporated to account for the complete nuclear permutation inversion symmetry. Instead of derivative couplings or interstate couplings, a so-called modified derivative coupling term is fitted by neural networks, resulting in accurate description of near degeneracy points, such as the conical intersections. The adiabatic energies, energy gradients, and derivative couplings are well reproduced, and the vanishing of derivative couplings as well as the isotropic topography of adiabatic and diabatic energies in asymptotic regions are automatically satisfied. All of these features of the coupled global PESs are requisite for accurate dynamics simulations. Our approach is expected to be very useful in developing highly accurate coupled PESs in a quasi-diabatic representation in an efficient machine learning-based way.

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.

Extending the Representation of Multistate Coupled Potential Energy Surfaces To Include Properties Operators Using Neural Networks: Application to the 1,21A States of Ammonia.

Fitting coupled adiabatic potential energy surfaces using coupled diabatic states enables, for accessible systems, nonadiabatic dynamics to be performed with unprecedented accuracy, when compared with on-the-fly dynamics. On-the-fly dynamics has advantages, not the least of which is the ability to compute molecular properties including electric dipole moments, transition dipole moments, and spin-orbit couplings. The availability of these terms extends the range of processes that can be treated with on-the-fly methods. In this work we use the example of fitting electric dipole and transition dipole moments of the 1,21A states of ammonia to show how to bring these advantages to the fit-coupled-surface method using a diabatic representation.

On the Impact of Singularities in the Two-State Adiabatic to Diabatic State Transformation: A Global Treatment.

Diabatizations based on molecular properties can remove the singularity in the derivative coupling at a conical intersection in the diabatic representation. Yet, they can also create new singularities in the derivative couplings because of the defining equations of the adiabatic to diabatic state transformation. In the iconic two-state case, these singularities occur at points termed diabolical singular points and form a seam of dimension Nint-2, where Nint is the number of internal degrees of freedom. This seam is of the same dimension as the conical intersection seam, but is distinct. Here, the global topography of the diabolical singularity seam of 1,21A states of ammonia is reported using a Boys localization (BL) dipole-based diabatization and juxtaposed with a previously reported global representation of the coupled electronic state potential energy surfaces. The principal finding is that the seam of BL diabatization-induced singularities passes very close to the key saddle point on the 21A potential energy surface which connects the 21A equilibrium structure with the NH2(X̃,Ã) + H channel. The construction and detailed study of the reported diabolical singularity seam is made possible by a recently constructed analytic representation of the dipole and transition dipole moment surfaces.

Two-state diabatic potential energy surfaces of ClH2 based on nonadiabatic couplings with neural networks.

A general neural network (NN)-fitting procedure based on nonadiabatic couplings is proposed to generate coupled two-state diabatic potential energy surfaces (PESs) with conical intersections. The elements of the diabatic potential energy matrix (DPEM) can be obtained directly from a combination of the NN outputs in principle. Instead, to achieve higher accuracy, the adiabatic-to-diabatic transformation (ADT) angle (mixing angle) for each geometry is first solved from the NN outputs, followed by individual NN fittings of the three terms of the DPEM, which are calculated from the ab initio adiabatic energies and solved mixing angles. The procedure is applied to construct a new set of two-state diabatic potential energy surfaces of ClH2. The ab initio data including adiabatic energies and derivative couplings are well reproduced. Furthermore, the current diabatization procedure can describe well the vicinity of conical intersections in high potential energy regions, which are located in the T-shaped (C2v) structure of Cl-H2. The diabatic quantum dynamical results on diabatic PESs show large differences as compared with the adiabatic results in high collision energy regions, suggesting the significance of nonadiabatic processes in conical intersection regions at high energies.

Neural network based quasi-diabatic Hamiltonians with symmetry adaptation and a correct description of conical intersections.

In a previous paper, we have demonstrated that artificial neural networks (NNs) can be used to generate quasidiabatic Hamiltonians (Hd) that are capable of representing adiabatic energies, energy gradients, and derivative couplings. In this work, two additional issues are addressed. First, symmetry-adapted functions such as permutation invariant polynomials are introduced to account for complete nuclear permutation inversion symmetry. Second, a partially diagonalized representation is introduced to facilitate a better description of near degeneracy points. The diabatization of 1, 21A states of NH3 is used as an example. The NN fitting results are compared to that of a previous fitting with symmetry adapted polynomials.

Representation of coupled adiabatic potential energy surfaces using neural network based quasi-diabatic Hamiltonians: 1,2 2A' states of LiFH.

An analytic quasi-diabatic representation of ab initio electronic structure data is key to the accurate quantum mechanical description of non-adiabatic chemical processes. In this work, a general neural network (NN) fitting procedure is proposed to generate coupled quasi-diabatic Hamiltonians (Hd) that are capable of representing adiabatic energies, energy gradients, and derivative couplings over a wide range of geometries. The quasi-diabatic representation for LiFH is used as a testing example. The fitting data including adiabatic energies, energy gradients and interstate couplings are obtained from a previously fitted analytical quasi-diabatic potential energy matrix, and are well reproduced by the NN fitting. Most importantly, the NN fitting also yields quantum dynamic results that reproduce those on the original LiFH diabatic Hamiltonian, demonstrating the ability of NN to generate highly accurate quasi-diabatic Hamiltonians.

Observation of the geometric phase effect in the H + HD → H2 + D reaction.

Theory has established the importance of geometric phase (GP) effects in the adiabatic dynamics of molecular systems with a conical intersection connecting the ground- and excited-state potential energy surfaces, but direct observation of their manifestation in chemical reactions remains a major challenge. Here, we report a high-resolution crossed molecular beams study of the H + HD → H2 + D reaction at a collision energy slightly above the conical intersection. Velocity map ion imaging revealed fast angular oscillations in product quantum state-resolved differential cross sections in the forward scattering direction for H2 products at specific rovibrational levels. The experimental results agree with adiabatic quantum dynamical calculations only when the GP effect is included.

Application of Clustering Algorithms to Partitioning Configuration Space in Fitting Reactive Potential Energy Surfaces.

A large number of energy points add great difficulty to construct reactive potential energy surfaces (PES). To alleviate this, exemplar-based clustering is applied to partition the configuration space into several smaller parts. The PES of each part can be constructed easily and the global PES is obtained by connecting all of the PESs of small parts. This divide and conquer strategy is first demonstrated in the fitting of PES for OH3 with Gaussian process regression (GPR) and further applied to construct PESs for CH5 and O+CH4 with artificial neural networks (NN). The accuracy of PESs is tested by fitting errors and direct comparisons with previous PESs in dynamically important regions. As for OH3 and CH5, quantum scattering calculations further validate the global accuracy of newly fitted PESs. The results suggest that partitioning the configuration space by clustering provides a simple and useful method for the construction of PESs for systems that require a large number of energy points.

Construction of diabatic energy surfaces for LiFH with artificial neural networks.

A new set of diabatic potential energy surfaces (PESs) for LiFH is constructed with artificial neural networks (NNs). The adiabatic PESs of the ground state and the first excited state are directly fitted with NNs. Meanwhile, the adiabatic-to-diabatic transformation (ADT) angles (mixing angles) are obtained by simultaneously fitting energy difference and interstate coupling gradients. No prior assumptions of the functional form of ADT angles are used before fitting, and the ab initio data including energy difference and interstate coupling gradients are well reproduced. Converged dynamical results show remarkable differences between adiabatic and diabatic PESs, which suggests the significance of non-adiabatic processes.