Nonadiabatic Coupling Matrix Elements Research Articles

Non-adiabatic coupling matrix elements in a magnetic field: Geometric gauge dependence and Berry phase.

Non-adiabatic coupling matrix elements (NACMEs) are important in quantum chemistry, particularly for molecular dynamics methods such as surface hopping. However, NACMEs are gauge dependent. This presents a difficulty for their calculation in general, where there are no restrictions on the gauge function except that it be differentiable. These cases are relevant for complex-valued electronic wave functions, such as those that arise in the presence of a magnetic field or spin-orbit coupling. In addition, the Berry curvature and Berry force play an important role in molecular dynamics in a magnetic field and are also relevant in the context of spin-orbit coupling. For methods such as surface hopping, excited-state Berry curvatures will also be of interest. With this in mind, we have developed a scheme for the calculation of continuous, differentiable NACMEs as a function of the molecular geometry for complex-valued wave functions. We demonstrate the efficacy of the method using the H2 molecule at the full configuration-interaction (FCI) level of theory. In addition, ground- and excited-state Berry curvatures are computed for the first time using FCI theory. Finally, Berry phases are computed directly in terms of diagonal NACMEs.

Electronic structure of ground and low-lying excited states of BaLi+ molecular ion: spin-orbit effect, radiative lifetimes and Franck-Condon factor

The study of BaLi+ and its reactivity plays a crucial role in advancing our understanding of chemical bonding or reaction mechanisms. The aim of this work is to represent a complete and extended theoretical study of BaLi+ molecular ion including ground and highly excited electronic states of 1,3Σ, 1,3Π and 1,3Δ symmetries, dissociated to the first seven dissociation limits. The corresponding potential energy curves (PECs), permanent and transition dipole moments have been investigated. These calculations were performed using the multireference configuration interaction (MRCI) method in combination with optimized basis sets and non-empirical pseudopotentials (ECP) for both Ba and Li atoms. Afterwards, the spin–orbit (SO) operator is incorporated in valence MRCI calculation using optimized relativistic spin–orbit pseudopotentials and 16 Ω states are generated and splitted into Λ-S states. The SO effect gives rise to a more complicated structure of electronic states presented in PEC and permanent and transition dipole moments. Nonadiabatic coupling matrix elements between the five lowest 1Σ+ states are also presented for the nonrelativistic results. Based on the vibrational radiative lifetime and Franck–Condon calculation, the possibilities of laser cooling of this system have been also discussed.

Doping-mediated excited state dynamics of diphosphine-protected M@Au12 (M = Au, Ir) superatom nanoclusters.

Doping heterometal atoms into ligand-protected gold superatom nanoclusters (Aun NCs) is proposed to further diversify their geometrical and electronic structures and enhance their photoluminescence properties, which is attributed to the mixing and effects between atoms. However, the fundamental principles that govern the optoelectronic properties of the doped Aun NCs remain elusive. Herein, we systematically explored two prototypical 8-electron Aun (n = 11 and 13) NCs with and without Ir dopant atoms using comprehensive ab initio calculations and real-time nonadiabatic molecular dynamics simulations. These doped Aun NCs maintain their parent geometrical structures and 8-electron superatomic configuration (1S21P6). Strong core-shell (Ir-Aun) electronic coupling significantly expands the energy gap, resulting in a weak nonadiabatic coupling matrix element, which in turn increases the carrier lifetime. This increase is mainly governed by the low-frequency vibration mode. We uncovered the relationship between electronic structures, electron-vibration, and carrier dynamics for these doped Aun NCs. These calculated results provide crucial insights for the atomically precise design of metal NCs with superior optoelectronic properties.

Machine Learning Representations of the Three Lowest Adiabatic Electronic Potential Energy Surfaces for the ArH2+ Reactive System.

In this work, we present Gaussian process regression machine learning representations of the three lowest coupled 2A' adiabatic electronic potential energy surfaces of the ArH2+ reactive system in full dimensionality. Additionally, the nonadiabatic coupling matrix elements were calculated. These adiabatic potentials and their nonadiabatic couplings are necessary ingredients in the theoretical investigation of the nonadiabatic reaction dynamics of the Ar + H2+ → ArH+ + H and Ar+ + H2 → ArH+ + H reactions, as well as the competing charge transfer process, Ar + H2+↔ Ar+ + H2. Accurate ab initio electronic structure calculations (ic-MRCI+Q/aug-cc-pVQZ), whereby the effect of spin-orbit coupling in Ar+ has been accounted for through the state interaction method, serve as input for the machine learning training process. The potential energy surfaces are fitted with high accuracies, with root-mean-square errors on the order of 10-7 eV for the three surfaces, which meet the requirements for chemical dynamics at low temperature. It was found that quite a large number of training points (of the order of 5000 ab initio points) are needed in order to achieve these accuracies due to the complex topography of these electronic surfaces.

He+ +N2 Charge Transfer Reaction: An Ab initio Analysis.

An ab initio analysis on the involved potential energy surfaces is presented for the investigation of the charge transfer mechanism for the He+ +N2 system. At high collision energy, as many as seven low-lying electronic states are observed to be involved in the charge transfer mechanism. Potential energy surfaces for these low-lying electronic states have been computed in the Jacobi scattering coordinates, applying multireference configuration interaction level of theory and aug-cc-pVQZ basis sets. Asymptotes for the ground and various excited states are assigned to mark the entrance (He+ +N2 ) and charge transfer channels (He+N2 + ). Nonadiabatic coupling matrix elements and quasi-diabatic potential energy surfaces have been computed for all seven states to rationalize the available experimental data on the charge transfer processes and to facilitate dynamics studies.

Non-adiabatic interactions in H[formula omitted] + C[formula omitted] system: An ab initio study

Diabatic surfaces generated for the ground state 11Σ+(11A′) as well as for the first excited electronic state 21Σ+(21A′) have been quantified for the C3 collision with H+ system employing the MRCI/aug-cc-pVQZ method. These collisions are significant in understanding the mechanism of energy transfer in astrophysics and molecular physics. For studying the dynamics of the interaction between the charge transfer and inelastic processes, properties such as non-adiabatic coupling matrix elements, and mixing angle have been determined. The computed surface and their properties will be useful in studying charge partitioning between the inelastic and charge transfer channels by wave packet quantum dynamics.

Intersystem crossing and intramolecular triplet excitation energy transfer in spiro[9,10-dihydro-9-oxoanthracene-10,2′-5′,6′-benzindan] investigated by DFT/MRCI methods

Recent experimental studies of a spiro-linked anthracenone (A)–naphthalene (N) compound (AN) in butyronitrile (BuCN) solution (Dobkowski et al., J. Phys. Chem. A 2019, 123, 6978) proposed an excited-state energy dissipation pathway [1ππ* (N) + 1ππ*(A)]↝1 nπ* (A)↝3 nπ* (A)↝3ππ*(N). However, a detailed theoretical study employing combined density functional theory and multireference configuration interaction methods, performed in the present work, suggests that the photoexcitation decay follows a different pathway. In BuCN solution, the intersystem crossing (ISC) follows the well-established El-Sayed rule and involves the 3ππ*(A) state which is found to be the lowest excited triplet state localized on the anthracenone moiety. Because the Dexter triplet excitation energy transfer (TEET) to the first excited triplet state of the naphthalene subunit is forbidden in C2 v symmetry, it is mandatory to go beyond the Condon approximation in modeling this process. Nonadiabatic coupling matrix elements were computed to obtain a TEET rate different from zero. Our calculations yield time constants of 5 ps for the 1 nπ* (A)↝3ππ* (A) ISC and of 3 ps for the subsequent 3ππ* (A)↝3ππ* (N) TEET in BuCN whereas the energy dissipation involving the 3 nπ*(A) state as an intermediate occurs on a much longer time scale.

πσ*-Mediated Nonadiabatic Tunneling Dynamics of Thiophenols in S1: The Semiclassical Approaches.

The S-H bond tunneling predissociation dynamics of thiophenol and its ortho-substituted derivatives (2-fluorothiophenol, 2-methoxythiophenol, and 2-chlorothiphenol) in S1 (ππ*) where the H atom tunneling is mediated by the nearby S2 (πσ*) state (which is repulsive along the S-H bond extension coordinate) have been investigated in a state-specific way using the picosecond time-resolved pump-probe spectroscopy for the jet-cooled molecules. The effects of the specific vibrational mode excitations and the SH/SD substitutions on the S-H(D) bond rupture tunneling dynamics have been interrogated, giving deep insights into the multidimensional aspects of the S1/S2 conical intersection, which also shapes the underlying adiabatic tunneling potential energy surfaces (PESs). The semiclassical tunneling rate calculations based on the Wentzel-Kramers-Brillouin (WKB) approximation or Zhu-Nakamura (ZN) theory have been carried out based on the ab initio PESs calculated in the (one, two, or three) reduced dimensions to be compared with the experiment. Though the quantitative experimental results could not be reproduced satisfactorily by the present calculations, the qualitative trends among different molecules in terms of the behavior of the tunneling rate versus the (adiabatic) barrier height or the number of PES dimensions could be rationalized. Most interestingly, the H/D kinetic isotope effect observed in the tunneling rate could be much better explained by the ZN theory compared to the WKB approximation, indicating that the nonadiabatic coupling matrix elements should be invoked for understanding the tunneling dynamics taking place in the proximity of the conical intersection.

NAC-TDDFT: Time-Dependent Density Functional Theory for Nonadiabatic Couplings

First-order nonadiabatic coupling (NAC) matrix elements (fo-NACMEs) are the basic quantities in theoretical descriptions of electronically nonadiabatic processes that are ubiquitous in molecular physics and chemistry. Given the large size of systems of chemical interests, time-dependent density functional theory (TDDFT) is usually the first choice of methods. However, the lack of many-electron wave functions in TDDFT renders the formulation of NAC-TDDFT for fo-NACMEs conceptually difficult. Because of this, various variants of NAC-TDDFT have been proposed in the literature from different standing points, including the Hellmann-Feynman-like expression and auxiliary/pseudo-wave function (AWF)-, equation-of-motion (EOM)-, and time-dependent perturbation theory (TDPT)-based formulations. Based on critical analyses, the following conclusions are made here: (1) The Hellmann-Feynman-like expression, which is rooted in exact wave function theory, is hardly useful due to huge demand on basis sets. (2) Although most popular, the AWF variants of NAC-TDDFT are not theoretically founded and become ambiguous particularly for the fo-NACMEs between two excited states, although they do agree with the EOM and TDPT variants under the Tamm-Dancoff approximation. (3) The TDPT variant of NAC-TDDFT is theoretically most rigorous but suffers from numerical instabilities on the one hand and does not differ to a significant extent from the EOM variant on the other hand. (4) As such, the EOM variant of NAC-TDDFT for the fo-NACMEs between the ground and excited states and between two excited states is solely the right choice in practice. These formal analyses are fully supported by numerical experimentations, taking azulene as a showcase. The proper implementation of the EOM variant of NAC-TDDFT is also highlighted, showing that the fo-NACMEs between the ground and excited states and between two excited states are computationally very much the same as the analytic energy gradients of DFT and TDDFT, respectively. Possible future developments of the EOM variant of NAC-TDDFT are also highlighted. Its extensions to spin-adapted open-shell TDDFT and proper treatment of spin-orbit couplings (which are another source of force for electronically nonadiabatic processes) are particularly warranted in the near future.

Internal conversion of singlet and triplet states employing numerical DFT/MRCI derivative couplings: Implementation, tests, and application to xanthone.

We present an efficient implementation of nonadiabatic coupling matrix elements (NACMEs) for density functional theory/multireference configuration interaction (DFT/MRCI) wave functions of singlet and triplet multiplicity and an extension of the Vibes program that allows us to determine rate constants for internal conversion (IC) in addition to intersystem crossing (ISC) nonradiative transitions. Following the suggestion of Plasser et al. [J. Chem. Theory Comput. 12, 1207 (2016)], the derivative couplings are computed as finite differences of wave function overlaps. Several measures have been taken to speed up the calculation of the NACMEs. Schur's determinant complement is employed to build up the determinant of the full matrix of spin-blocked orbital overlaps from precomputed spin factors with fixed orbital occupation. Test calculations on formaldehyde, pyrazine, and xanthone show that the mutual excitation level of the configurations at the reference and displaced geometries can be restricted to 1. In combination with a cutoff parameter of tnorm = 10-8 for the DFT/MRCI wave function expansion, this approximation leads to substantial savings of cpu time without essential loss of precision. With regard to applications, the photoexcitation decay kinetics of xanthone in apolar media and in aqueous solution is in the focus of the present work. The results of our computational study substantiate the conjecture that S1 T2 reverse ISC outcompetes the T2 ↝ T1 IC in aqueous solution, thus explaining the occurrence of delayed fluorescence in addition to prompt fluorescence.

Fast and Accurate Computation of Nonadiabatic Coupling Matrix Elements Using the Truncated Leibniz Formula and Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory.

We present a fast and accurate numerical algorithm for computing the first-order nonadiabatic coupling matrix element (NACME). The algorithm employs the truncated Leibniz formula (TLF) approximation within the finite-difference method, which makes it easily applicable in connection with any wave function-based methodology. In this work, we used the algorithm in connection with the recently developed mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT, MRSF for brevity). The accuracy is assessed for NACME between the singlet electronic states of a dissociating hydrogen molecule. It is demonstrated that an intermediate approximation, TLF(1), affords a negligible numeric error on the order of ∼10-10 a.u. while enabling a fast computation of NACME. As the MRSF method yields the correct description of the dissociation curves of H2 for all the electronic states involved, the numeric TLF(1)/MRSF NACME values are in excellent agreement with the reference analytical values obtained by the full configuration interaction. For polyatomic molecules, the MRSF NAC vectors agree very closely with the MRCISD NAC vectors. Hence, the proposed protocol is a promising tool for the evaluation of NACMEs.

Deciphering Benzene–Heterocycle Stacking Interaction Impact on the Electronic Structures and Photophysical Properties of Tetraphenylethene-Cored Foldamers

Deciphering Benzene–Heterocycle Stacking Interaction Impact on the Electronic Structures and Photophysical Properties of Tetraphenylethene-Cored Foldamers

Analytical derivatives of the individual state energies in ensemble density functional theory. II. Implementation on graphical processing units (GPUs).

Conical intersections control excited state reactivity, and thus, elucidating and predicting their geometric and energetic characteristics are crucial for understanding photochemistry. Locating these intersections requires accurate and efficient electronic structure methods. Unfortunately, the most accurate methods (e.g., multireference perturbation theories such as XMS-CASPT2) are computationally challenging for large molecules. The state-interaction state-averaged restricted ensemble referenced Kohn-Sham (SI-SA-REKS) method is a computationally efficient alternative. The application of SI-SA-REKS to photochemistry was previously hampered by a lack of analytical nuclear gradients and nonadiabatic coupling matrix elements. We have recently derived analytical energy derivatives for the SI-SA-REKS method and implemented the method effectively on graphical processing units. We demonstrate that our implementation gives the correct conical intersection topography and energetics for several examples. Furthermore, our implementation of SI-SA-REKS is computationally efficient, with observed sub-quadratic scaling as a function of molecular size. This demonstrates the promise of SI-SA-REKS for excited state dynamics of large molecular systems.

Three states global fittings with improved long range: singlet and triplet states of H.

Full dimensional analytical fits of the coupled potential energy surfaces for the three lower singlet and triplet adiabatic states of H are developed, providing analytic derivatives and non-adiabatic coupling matrix elements. The fits are highly accurate and include an improved description of the long range interactions, including new terms for the description of the long range in the diatomic fits and the atom-diatom dissociation channels. The fits are based on the DIM formalism including three body terms in Hamiltonian matrix elements, each of them obeying S2 permutational symmetry, where the positive charge is placed in either of the three hydrogen atoms, but the full system obeys S3 permutational symmetry, invariant under all permutations of the nuclei. The ab initio points used in the fitting are obtained from a complete basis set extrapolation, made for all electronic states. Total root mean square errors of the fits are 27 and 12 cm-1, for the singlet and triplet states, respectively. The errors in the channels are lower than 2 cm-1 and 6 cm-1 for the H + H and H+ + H2 channels respectively. The new fits have been used to calculate the rovibrational bound states of the lowest singlet and lowest triplet states showing very good agreement with previous calculations in the literature.

Fast estimation of the internal conversion rate constant in photophysical applications.

An efficient method for estimating non-adiabatic coupling matrix elements (NACME) and rate constants for internal conversion (kIC) is presented. The method, based on Plotnikov's theory, requires only calculations of the electronic wave functions and the corresponding electronic excitation energies. Computationally expensive calculations of the derivatives of the electronic wave function with respect to the nuclear coordinates are avoided. When the main accepting modes of the electronic excitation energy are X-H vibrations, the present method can be used for estimating the efficiency of the energy transfer between donor and acceptor molecules. It can also be used in studies of the influence of hydrogen bonding or solvent effect on fluorescence quenching, in studies of vibronic effects of TADF (thermally activated delayed fluorescence) emitters, and for calculating kIC. Here, kIC and NACME are calculated for free-base porhyrin, magnesium porphyrin, azulene, naphthalene, pyrene and fluorenone interacting with a solvent molecule. Reverse kIC and NACME are further calculated for the T1→ T2 transition of dibenzothiophene-S,S-dioxide (PTZ-DBTO2), which is used in TADF applications. Finally, we estimate the efficiency of the energy transfer between two large porphyrinoid dimers.

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.

Some ab initio thoughts on the bonding in O3H

We study the ground state of O3H (= OaObOcH) with single (RCCSD(T)) and multi (MRCI) reference correlation methods in order to shed some light on its bonding mechanism in connection with its low dissociation energy and rather long bond distance (OaOb−OcH). For such a task all three dissociation/formation paths were considered (O2 + OH, O + O2H, and O3 + H) and the associated nonadiabatic coupling matrix elements were examined. It appears that the excited states of the above asymptotic fragments participate in the equilibrium wavefunction of O3H in a way that results in a symmetry broken structure.

Ab initio Calculations of the Lowest $$^{1}\Sigma _{g}^{ + }$$ States of the Na2 Dimer

The ground and 10 lowest excited $$^{1}\Sigma _{g}^{ + }$$ adiabatic electronic states of the Na2 dimer are calculated using the pseudopotential method. The use of the basis [7s6p5d4f] of atomic orbitals makes it possible to extend the range of available internuclear distances up to 1.7–50 A. It is found that the theoretical values of the Te and De constants are in a good agreement with the experimental ones. Herein you will find the sample calculations of the radial non-adiabatic coupling matrix elements enable to transform the basis of the adiabatic states to quasidiabatic one. It is found also that the Le Roy modified radius scales down the left boundary of an asymptotic range for the electronic state with the (3s + 5p) dissociation limit and for the higher states.

Analytic non-adiabatic couplings for the spin-flip ORMAS method

Analytic non-adiabatic coupling matrix elements (NACME) are derived and implemented for the spin-flip occupation restricted multiple active space configuration interaction (SF-ORMAS-CI) method. SF-ORMAS is a general spin correct implementation of the SF-CI method and has been shown to correctly describe various stationary geometries, including regions of conical intersections. The availability of non-adiabatic coupling allows a fuller examination of non-adiabatic phenomena with the SF-ORMAS method. In this study, the implementation of the NACME is tested using two model systems, MgFH and ethylene. In both cases, the SF-ORMAS method exhibits good qualitative agreement with established multi-reference methods, suggesting that SF-ORMAS is a suitable method for the study of non-adiabatic chemical phenomena.

A Robust and Unified Solution for Choosing the Phases of Adiabatic States as a Function of Geometry: Extending Parallel Transport Concepts to the Cases of Trivial and Near-Trivial Crossings.

We investigate a simple and robust scheme for choosing the phases of adiabatic electronic states smoothly (as a function of geometry) so as to maximize the performance of ab initio non-adiabatic dynamics methods. Our approach is based upon consideration of the overlap matrix (U) between basis functions at successive points in time and selecting the phases so as to minimize the matrix norm of log(U). In so doing, one can extend the concept of parallel transport to cases with sharp curve crossings. We demonstrate that this algorithm performs well under extreme situations where dozens of states cross each other either through trivial crossings (where there is zero effective diabatic coupling), or through non-trivial crossings (when there is a non-zero diabatic coupling), or through a combination of both. In all cases, we compute the time-derivative coupling matrix elements (or equivalently non-adiabatic derivative coupling matrix elements) that are as smooth as possible. Our results should be of interest to all who are interested in either non-adiabatic dynamics, or more generally, parallel transport in large systems.