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

Accurate incorporation of hyperfine coupling in diabatic potential models using the effective relativistic coupling by asymptotic representation approach

The accurate treatment of relativistic couplings like spin-orbit (SO) coupling into diabatic potential models is highly desirable. We have been developing the effective relativistic coupling by asymptotic representation (ERCAR) approach to this end. The central idea of ERCAR is the representation of the system using an asymptotic diabatic direct product basis of atom and fragment states. This allows to treat relativistic coupling operators like SO coupling analytically. This idea is extended here to the incorporation of hyperfine (HF) coupling into the diabatic potential model. Hyperfine coupling is due to the magnetic dipole-dipole and the Fermi contact interaction as well as the electric quadrupole interaction. The corresponding operators can be expressed in terms of the angular momentum operators for nuclear spin Î and for total angular momentum Ĵ of the atomic fine structure states. The diabatic basis of an existing ERCAR model is complemented by nuclear spinors and the HF coupling operators are easily evaluated in that basis. Diagonalization of the resulting full diabatic ERCAR model yields the HF energies and states for any molecular geometry of interest. The new method is demonstrated using an existing accurate diabatic potential model for hydrogen iodide (HI) [N. Weike, A. Viel and W. Eisfeld, Hydrogen-iodine scattering: I. Development of an accurate spin-orbit coupled diabatic potential energy model, J. Chem. Phys., 2023, 159, 244119] to see the effects of hyperfine coupling. The HF coupling effect of the 2P3/2 ground state and spin-orbit excited 2P1/2 state of iodine combined with the 2S1/2 ground state of hydrogen are added to the ERCAR Hamiltonian. It is shown that each fine structure state is split by the hyperfine interactions into sets of seven hyperfine states. The fine structure ground state at the global minimum is split into three degenerate groups of hyperfine states with splittings of 152 and 76 MHz.

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.

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.

Representation and conservation of angular momentum in the Born-Oppenheimer theory of polyatomic molecules

This paper concerns the representation of angular momentum operators in the Born-Oppenheimer theory of polyatomic molecules and the various forms of the associated conservation laws. Topics addressed include the question of whether these conservation laws are exactly equivalent or only to some order of the Born-Oppenheimer parameter κ = (m/M)^1/4 and what the correlation is between angular momentum quantum numbers in the various representations. These questions are addressed in both problems involving a single potential energy surface and those with multiple, strongly coupled surfaces and in both the electrostatic model and those for which fine structure and electron spin are important. The analysis leads to an examination of the transformation laws under rotations of the electronic Hamiltonian; of the basis states, both adiabatic and diabatic, along with their phase conventions; of the potential energy matrix; and of the derivative couplings. These transformation laws are placed in the geometrical context of the structures in the nuclear configuration space that are induced by rotations, which include the rotational orbits or fibers, the surfaces upon which the orientation of the molecule changes but not its shape, and the section, an initial value surface that cuts transversally through the fibers. Finally, it is suggested that the usual Born-Oppenheimer approximation can be replaced by a dressing transformation, that is, a sequence of unitary transformations that block-diagonalize the Hamiltonian. When the dressing transformation is carried out, we find that the angular momentum operator does not change. This is a part of a system of exact equivalences among various representations of angular momentum operators in Born-Oppenheimer theory. Our analysis accommodates large-amplitude motions and is not dependent on small-amplitude expansions about an equilibrium position. Our analysis applies to noncollinear configurations of a polyatomic molecule; this covers all but a subset of measure zero (the collinear configurations) in the nuclear configuration space.

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.

Novel methodology for systematically constructing global effective models from ab initio-based surfaces: A new insight into high-resolution molecular spectra analysis

In this paper, a novel methodology is presented for the construction of ab initio effective rotation-vibration spectroscopic models from potential energy and dipole moment surfaces. Non-empirical effective Hamiltonians are obtained via the block-diagonalization of selected variationally computed eigenvector matrices. For the first time, the derivation of an effective dipole moment is carried out in a systematic way. This general approach can be implemented quite easily in most of the variational computer codes and turns out to be a clear alternative to the rather involved Van Vleck perturbation method. Symmetry is exploited at all stages to translate first-principles calculations into a set of spectroscopic parameters to be further refined on experiment. We demonstrate on H₂CO, PH₃, CH₄, C₂H₄, and SF₆ that the proposed effective model can provide crucial information to spectroscopists within a very short time compared to empirical spectroscopic models. This approach brings a new insight into high-resolution spectrum analysis of polyatomic molecules and will be also of great help in the modeling of hot atmospheres where completeness is important.

Diabatic States of Molecules

Quantitative simulations of electronically nonadiabatic molecular processes require both accurate dynamics algorithms and accurate electronic structure information. Direct semiclassical nonadiabatic dynamics is expensive due to the high cost of electronic structure calculations, and hence it is limited to small systems, limited ensemble averaging, ultrafast processes, and/or electronic structure methods that are only semiquantitatively accurate. The cost of dynamics calculations can be made manageable if analytic fits are made to the electronic structure data, and such fits are most conveniently carried out in a diabatic representation because the surfaces are smooth and the couplings between states are smooth scalar functions. Diabatic representations, unlike the adiabatic ones produced by most electronic structure methods, are not unique, and finding suitable diabatic representations often involves time-consuming nonsystematic diabatization steps. The biggest drawback of using diabatic bases is that it can require large amounts of effort to perform a globally consistent diabatization, and one of our goals has been to develop methods to do this efficiently and automatically. In this Feature Article, we introduce the mathematical framework of diabatic representations, and we discuss diabatization methods, including adiabatic-to-diabatic transformations and recent progress toward the goal of automatization.

Analyzing Grid-Based Direct Quantum Molecular Dynamics Using Non-Linear Dimensionality Reduction

Grid-based schemes for simulating quantum dynamics, such as the multi-configuration time-dependent Hartree (MCTDH) method, provide highly accurate predictions of the coupled nuclear and electronic dynamics in molecular systems. Such approaches provide a multi-dimensional, time-dependent view of the system wavefunction represented on a coordinate grid; in the case of non-adiabatic simulations, additional information about the state populations adds a further layer of complexity. As such, wavepacket motion on potential energy surfaces which couple many nuclear and electronic degrees-of-freedom can be extremely challenging to analyse in order to extract physical insight beyond the usual expectation-value picture. Here, we show that non-linear dimensionality reduction (NLDR) methods, notably diffusion maps, can be adapted to extract information from grid-based wavefunction dynamics simulations, providing insight into key nuclear motions which explain the observed dynamics. This approach is demonstrated for 2-D and 9-D models of proton transfer in salicylaldimine, as well as 8-D and full 12-D simulations of cis-trans isomerization in ethene; these simulations demonstrate how NLDR can provide alternative views of wavefunction dynamics, and also highlight future developments.

Machine Learning Approach to Calculate Electronic Couplings between Quasi-diabatic Molecular Orbitals: The Case of DNA

Diabatization of one-electron states in flexible molecular aggregates is a great challenge due to the presence of surface crossings between molecular orbital (MO) levels and the complex interaction between MOs of neighboring molecules. In this work, we present an efficient machine learning approach to calculate electronic couplings between quasi-diabatic MOs without the need of nonadiabatic coupling calculations. Using MOs of rigid molecules as references, the MOs that can be directly regarded to be quasi-diabatic in molecular dynamics are selected out, state tracked, and phase corrected. On the basis of this information, artificial neural networks are trained to characterize the structure-dependent onsite energies of quasi-diabatic MOs and the intermolecular electronic couplings. A representative sequence of DNA is systematically studied as an illustration. Smooth time evolution of electronic couplings in all base pairs is obtained with quasi-diabatic MOs. In particular, our method can calculate electronic couplings between different quasi-diabatic MOs independently, and thus, this possesses unique advantages in many applications.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Machine Learning for Electronically Excited States of Molecules

Electronically excited states of molecules are at the heart of photochemistry, photophysics, as well as photobiology and also play a role in material science. Their theoretical description requires highly accurate quantum chemical calculations, which are computationally expensive. In this review, we focus on not only how machine learning is employed to speed up such excited-state simulations but also how this branch of artificial intelligence can be used to advance this exciting research field in all its aspects. Discussed applications of machine learning for excited states include excited-state dynamics simulations, static calculations of absorption spectra, as well as many others. In order to put these studies into context, we discuss the promises and pitfalls of the involved machine learning techniques. Since the latter are mostly based on quantum chemistry calculations, we also provide a short introduction into excited-state electronic structure methods and approaches for nonadiabatic dynamics simulations and describe tricks and problems when using them in machine learning for excited states of molecules.

Simulation of UV absorption spectra and relaxation dynamics of uracil and uracil-water clusters

Despite many studies, the mechanisms of nonradiative relaxation of uracil in the gas phase and in aqueous solution are still not fully resolved. Here we combine theoretical UV absorption spectroscopy with nonadiabatic dynamics simulations to identify the photophysical mechanisms that can give rise to experimentally observed decay time constants. We first compute and theoretically assign the electronic spectra of uracil using the second-order algebraic-diagrammatic-construction (ADC(2)) method. The obtained electronic states, their energy differences and state-specific solvation effects are the prerequisites for understanding the photodynamics. We then use nonadiabatic trajectory-surface-hopping dynamics simulations to investigate the photoinduced dynamics of uracil and uracil-water clusters. In contrast to previous studies, we found that a single mechanism - the ethylenic twist around the C[double bond, length as m-dash]C bond - is responsible for the ultrafast component of the nonradiative decay, both in the gas phase and in solution. Very good agreement with the experimentally determined ultrashort decay time constants is obtained.

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.

UV absorption spectra of DNA bases in the 350-190 nm range: assignment and state specific analysis of solvation effects

The theoretical assignment of electronic spectra of polyatomic molecules is a challenging problem that requires the specification of the character of a large number of electronic states. We propose a procedure for automatically determining the character of electronic transitions and apply it to the study of UV spectra of DNA bases in the gas phase and in the aqueous environment. The procedure is based on the computation of electronic wave function overlaps and accounts for an extensive sampling of nuclear geometries. Novelties of this work are the theoretical assignment of the electronic spectra of DNA bases up to 190 nm and a state specific analysis of solvation effects. By accounting for different effects contributing to the total solvent shift we obtained a good agreement between the computed and experimental spectra. Effects of vibrational averaging, temperature and solvent-induced structural changes shift excitation energies to lower values. Solvent-solute electrostatic interactions are state specific and strongly destabilize nRyd states, and to lesser extent nπ* and πRyd states. Altogether, this results in the stabilization of ππ* states and destabilization of nπ*, πRyd and nRyd states in solution.

Diabatic States at Construction (DAC) through Generalized Singular Value Decomposition

A procedure, called generalized diabatic-at-construction (GDAC), is presented to transform adiabatic potential energy surfaces into a diabatic representation by generalized singular value decomposition. First, we use a set of localized, valence bond-like configuration state functions, called DAC, as the basis states. Then, the adiabatic ground and relevant excited states are determined using multistate density functional theory (MSDFT). GDAC differs in the opposite direction from traditional approaches based on adiabatic-to-diabatic transformation with certain property restraints. The method is illustrated with applications to a model first-order bond dissociation reaction of CH₃OCH₂Cl polarized by a solvent molecule, the ground- and first-excited-state potential energy surfaces near the minimum conical intersection for the ammonia dimer photodissociation, and the multiple avoided curve crossings in the dissociation of lithium hydride. The GDAC diabatization method may be useful for defining charge-localized states in studies of electron transfer and proton-coupled electron transfer reactions in proteins.

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.

Diabatic-At-Construction Method for Diabatic and Adiabatic Ground and Excited States Based on Multistate Density Functional Theory

We describe a diabatic-at-construction (DAC) strategy for defining diabatic states to determine the adiabatic ground and excited electronic states and their potential energy surfaces using the multistate density functional theory (MSDFT). The DAC approach differs in two fundamental ways from the adiabatic-to-diabatic (ATD) procedures that transform a set of preselected adiabatic electronic states to a new representation. (1) The DAC states are defined in the first computation step to form an active space, whose configuration interaction produces the adiabatic ground and excited states in the second step of MSDFT. Thus, they do not result from a similarity transformation of the adiabatic states as in the ATD procedure; they are the basis for producing the adiabatic states. The appropriateness and completeness of the DAC active space can be validated by comparison with experimental observables of the ground and excited states. (2) The DAC diabatic states are defined using the valence bond characters of the asymptotic dissociation limits of the adiabatic states of interest, and they are strictly maintained at all molecular geometries. Consequently, DAC diabatic states have specific and well-defined physical and chemical meanings that can be used for understanding the nature of the adiabatic states and their energetic components. Here we present results for the four lowest singlet states of LiH and compare them to a well-tested ATD diabatization method, namely the 3-fold way; the comparison reveals both similarities and differences between the ATD diabatic states and the orthogonalized DAC diabatic states. Furthermore, MSDFT can provide a quantitative description of the ground and excited states for LiH with multiple strongly and weakly avoided curve crossings spanning over 10 Å of interatomic separation.