Direct diabatization of electronic states by the fourfold-way: Including dynamical correlation by multi-configuration quasidegenerate perturbation theory with complete active space self-consistent-field diabatic molecular orbitals

Fast and accurate excited states predictions: machine learning and diabatization

The efficiency of machine learning algorithms for electronically excited states is far behind ground-state applications. One of the underlying problems is the insufficient smoothness of the fitted potential energy surfaces and other properties in the vicinity of state crossings and conical intersections, which is a prerequisite for an efficient regression. Smooth surfaces can be obtained by switching to the diabatic basis. However, diabatization itself is still an outstanding problem. We overcome these limitations by solving both problems at once. We use a machine learning approach combining clustering and regression techniques to correct for the deficiencies of property-based diabatization which, in return, provides us with smooth surfaces that can be easily fitted. Our approach extends the applicability of property-based diabatization to multidimensional systems. We utilize the proposed diabatization scheme to achieve higher prediction accuracy for adiabatic states and we show its performance by reconstructing global potential energy surfaces of excited states of nitrosyl fluoride and formaldehyde. While the proposed methodology is independent of the specific property-based diabatization and regression algorithm, we show its performance for kernel ridge regression and a very simple diabatization based on transition multipoles. Compared to most other algorithms based on machine learning, our approach needs only a small amount of training data.

Nonadiabatic Coupling in Trajectory Surface Hopping: How Approximations Impact Excited-State Reaction Dynamics

Photochemical reactions are widely modeled using the popular trajectory surface hopping (TSH) method, an affordable mixed quantum-classical approximation to the full quantum dynamics of the system. TSH is able to account for nonadiabatic effects using an ensemble of trajectories, which are propagated on a single potential energy surface at a time and which can hop from one electronic state to another. The occurrences and locations of these hops are typically determined using the nonadiabatic coupling between electronic states, which can be assessed in a number of ways. In this work, we benchmark the impact of some approximations to the coupling term on the TSH dynamics for several typical isomerization and ring-opening reactions. We have identified that two of the schemes tested, the popular local diabatization scheme and a scheme based on biorthonormal wave function overlap implemented in the OpenMOLCAS code as part of this work, reproduce at a much reduced cost the dynamics obtained using the explicitly calculated nonadiabatic coupling vectors. The other two schemes tested can give different results, and in some cases, even entirely incorrect dynamics. Of these two, the scheme based on configuration interaction vectors gives unpredictable failures, while the other scheme based on the Baeck-An approximation systematically overestimates hopping to the ground state as compared to the reference approaches.

Diabatic Valence-Hole States in the C2 Molecule: "Putting Humpty Dumpty Together Again"

Despite the long history of spectroscopic studies of the C₂ molecule, fundamental questions about its chemical bonding are still being hotly debated. The complex electronic structure of C₂ is a consequence of its dense manifold of near-degenerate, low-lying electronic states. A global multi-state diabatic model is proposed here to disentangle the numerous configuration interactions that occur within four symmetry manifolds of excited states of C₂ (¹Π_g, ³Π_g, ¹Σ_u⁺ , and ³Σ_u⁺ ). The key concept of our model is the existence of two "valence-hole" configurations, 2σg22σu11πu33σg2 for ^1,3Π_g states and 2σg22σu11πu43σg1 for ^1,3Σ_u⁺ states, that are derived from 3σ_g ← 2σ_u electron promotion. The lowest-energy state from each of the four C₂ symmetry species is dominated by this type of valence-hole configuration at its equilibrium internuclear separation. As a result of their large binding energy (nominal bond order of 3) and correlation with the 2s²2p² + 2s2p³ separated-atom configurations, the presence of these valence-hole configurations has a profound impact on the global electronic structure and unimolecular dynamics of C₂.

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.

Non-adiabatic quantum dynamics without potential energy surfaces based on second-quantized electrons: Application within the framework of the MCTDH method

A first principles quantum formalism to describe the non-adiabatic dynamics of electrons and nuclei based on a second quantization representation (SQR) of the electronic motion combined with the usual representation of the nuclear coordinates is introduced. This procedure circumvents the introduction of potential energy surfaces and non-adiabatic couplings, providing an alternative to the Born-Oppenheimer approximation. An important feature of the molecular Hamiltonian in the mixed first quantized representation for the nuclei and the SQR representation for the electrons is that all degrees of freedom, nuclear positions and electronic occupations, are distinguishable. This makes the approach compatible with various tensor decomposition Ansätze for the propagation of the nuclear-electronic wavefunction. Here, we describe the application of this formalism within the multi-configuration time-dependent Hartree framework and its multilayer generalization, corresponding to Tucker and hierarchical Tucker tensor decompositions of the wavefunction, respectively. The approach is applied to the calculation of the photodissociation cross section of the HeH⁺ molecule under extreme ultraviolet irradiation, which features non-adiabatic effects and quantum interferences between the two possible fragmentation channels, He + H⁺ and He⁺ + H. These calculations are compared with the usual description based on ab initio potential energy surfaces and non-adiabatic coupling matrix elements, which fully agree. The proof-of-principle calculations serve to illustrate the advantages and drawbacks of this formalism, which are discussed in detail, as well as possible ways to overcome them. We close with an outlook of possible application domains where the formalism might outperform the usual approach, for example, in situations that combine a strong static correlation of the electrons with non-adiabatic electronic-nuclear effects.

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.

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.

Synergetic Effects of Triplet-Triplet Annihilation and Directional Triplet Exciton Migration in Organic Crystals for Photon Upconversion

In condensed solids, triplet exciton migration and succeeding triplet-triplet annihilation (TTA) are major bottleneck processes for efficient photon upconversion (UC) using sunlight excitation. We theoretically investigated the reaction times of TTA and the triplet-triplet energy transfer (TTET) as the elementary processes of triplet exciton migration in organic crystals of two molecular species: 9,10-diphenylanthracene (DPA) and its double-strapped alkyl  derivative (C7-sDPA) as the models of a recently reported crystalline system of TTA-UC by Kamada et al. The reaction times calculated based on Marcus theory clarified that the dimensionality of TTET and synergetic effects of TTA and TTET are responsible for the high UC quantum yield as well as their triplet lifetimes.

Determining whether diabolical singularities limit the accuracy of molecular property based diabatic representations: The 1,21A states of methylamine

An efficient, easily implemented method for locating singularities attributable to the failure of the defining equations in a molecular property based diabatization, termed diabolical singular points, is reported. For two state diabatizations, the singular points form a seam of dimension N ^int - 2, where N ^int is the number of internal degrees of freedom. The dynamical outcomes of nuclear trajectories that reach the region of this seam are flawed. The algorithm easily identifies these otherwise hard to anticipate regions of fallaciously large derivative coupling. The fact that the algorithm is easily incorporated into a two state diabatization code based on molecular properties makes it a practical tool for determining whether the existence of diabolical singularities is relevant to the problem being considered. The algorithm is illustrated using a multireference single and double excitation configuration interaction description of the 1,2¹A states of CH₃NH_2.

Electronic spectrum and characterization of diabatic potential energy surfaces for thiophenol

We present an accurate simulation of the UV spectrum and a diabatization of three singlet potential surfaces along four coordinates.

Diabatization for Time-Dependent Density Functional Theory: Exciton Transfers and Related Conical Intersections

Intermolecular exciton transfers and related conical intersections are analyzed by diabatization for time-dependent density functional theory. The diabatic states are expressed as a linear combination of the adiabatic states so as to emulate the well-defined reference states. The singlet exciton coupling calculated by the diabatization scheme includes contributions from the Coulomb (Förster) and electron exchange (Dexter) couplings. For triplet exciton transfers, the Dexter coupling, charge transfer integral, and diabatic potentials of stacked molecules are calculated for analyzing direct and superexchange pathways. We discuss some topologies of molecular aggregates that induce conical intersections on the vanishing points of the exciton coupling, namely boundary of H- and J-aggregates and T-shape aggregates, as well as canceled exciton coupling to the bright state of H-aggregate, i.e., selective exciton transfer to the dark state. The diabatization scheme automatically accounts for the Berry phase by fixing the signs of reference states while scanning the coordinates.

Constructing diabatic representations using adiabatic and approximate diabatic data--Coping with diabolical singularities

We have recently introduced a diabatization scheme, which simultaneously fits and diabatizes adiabatic ab initio electronic wave functions, Zhu and Yarkony J. Chem. Phys. 140, 024112 (2014). The algorithm uses derivative couplings in the defining equations for the diabatic Hamiltonian, H(d), and fits all its matrix elements simultaneously to adiabatic state data. This procedure ultimately provides an accurate, quantifiably diabatic, representation of the adiabatic electronic structure data. However, optimizing the large number of nonlinear parameters in the basis functions and adjusting the number and kind of basis functions from which the fit is built, which provide the essential flexibility, has proved challenging. In this work, we introduce a procedure that combines adiabatic state and diabatic state data to efficiently optimize the nonlinear parameters and basis function expansion. Further, we consider using direct properties based diabatizations to initialize the fitting procedure. To address this issue, we introduce a systematic method for eliminating the debilitating (diabolical) singularities in the defining equations of properties based diabatizations. We exploit the observation that if approximate diabatic data are available, the commonly used approach of fitting each matrix element of H(d) individually provides a starting point (seed) from which convergence of the full H(d) construction algorithm is rapid. The optimization of nonlinear parameters and basis functions and the elimination of debilitating singularities are, respectively, illustrated using the 1,2,3,4(1)A states of phenol and the 1,2(1)A states of NH3, states which are coupled by conical intersections.

Nonintuitive Diabatic Potential Energy Surfaces for Thioanisole

Diabatization of potential energy surfaces is a technique that enables convenient molecular dynamics simulations of electronically nonadiabatic processes, but diabatization itself is nonunique and can be inconvenient; the best methods to achieve diabatization are still under study. Here, we present the diabatization of two electronic states of thioanisole in the S-CH3 bond stretching and C-C-S-C torsion two-dimensional nuclear coordinate space containing a conical intersection. We use two systematic methods: the (orbital-dependent) 4-fold way and the (orbital-free) Boys localization diabatization method. These very different methods yield strikingly similar diabatic potential energy surfaces that cross at geometries where the adiabatic surfaces are well separated and do not exhibit avoided crossings, and the contours of the diabatic gap and diabatic coupling are similar for the two methods. The validity of the diabatization is supported by comparing the nonadiabatic couplings calculated from the diabatic matrix elements to those calculated by direct differentiation of the adiabatic states.

Modeling cusps in adiabatic potential energy surfaces

A method for modeling cusps on adiabatic potential energy surfaces without the need for any adiabatic-to-diabatic transformation is presented and shown to be successfully applied to the (2)A″ state of NO2. The more complicated case of a system with permutationally equivalent crossing seams is also examined and illustrated by considering the two first (2)A' states of the nitrogen trimer.

Structure and properties of the precursor/successor complex and transition state of the CrCl²⁺/Cr²⁺ electron self-exchange reaction via the inner-sphere pathway

The electron self-exchange reaction CrCl(OH2)5(2+) + Cr(OH2)6(2+) → Cr(OH2)6(2+) + CrCl(OH2)5(2+), proceeding via the inner-sphere pathway, was investigated with quantum-chemical methods. Geometry and vibrational frequencies of the precursor/successor (P/S) complex, (H2O)5Cr(III)ClCr(II)(OH2)5(4+)/(H2O)5Cr(II)ClCr(III)(OH2)5(4+), and the transition state (TS), (H2O)5CrClCr(OH2)5(4+‡), were computed with density functional theory (DFT) and conductor polarizable continuum model hydration. Consistent data were obtained solely with long-range-corrected functionals, whereby in this study, LC-BOP was used. Bent and linear structures were computed for the TS and P/S. The electronic coupling matrix element (H(ab)) and the reorganizational energy (λ) were calculated with multistate extended general multiconfiguration quasi-degenerate second-order perturbation theory. The nuclear tunneling factor (Γ(n)), the nuclear frequency factor (ν(n)), the electronic frequency factor (ν(el)), the electron transmission coefficient (κ(el)), and the first-order rate constant (k(et)) for the electron-transfer step (the conversion of the precursor complex into the successor complex) were calculated based on the imaginary frequency (ν(‡)) of the TS, the Gibbs activation energy (ΔG(‡)), H(ab), and λ. The formation of the precursor complex via water substitution at Cr(OH2)6(2+) was also investigated with DFT and found to be very fast. Thus, the electron-transfer step is rate-determining. For the substitution reaction, only a bent TS structure could be obtained. The overall rate constant (k) was estimated as the product K(A)k(et), whereby K(A) is the equilibrium constant for the formation of the ion aggregate of the reactants Cr(OH2)6(2+) and CrCl(OH2)5(2+), Cr(H2O)6·CrCl(OH2)5(4+) (IAR). k calculated for the bent and linear isomers agrees with the experimental value.

Full-dimensional potentials and state couplings and multidimensional tunneling calculations for the photodissociation of phenol

Full-dimensional potentials and state couplings were developed for the photodissociation of phenol. We also present multidimensional tunneling calculations at the transition state on the shoulder of the first conical intersection.