Exact numerical computation of a kinetic energy operator in curvilinear coordinates

Comparison of curvilinear coordinates within vibrational structure calculations based on automatically generated potential energy surfaces

For floppy molecules showing internal rotations and/or large amplitude motions, curvilinear internal coordinates are known to be superior to rectilinear normal coordinates within vibrational structure calculations. Due to the myriad definitions of internal coordinates, automated and efficient potential energy surface generators necessitate a high degree of flexibility, supporting the properties arising from these coordinates. Within this work, an approach to deal with these challenges is presented, including key elements, such as the selection of appropriate fit functions, the exploitation of symmetry, the positioning of grid points, or elongation limits for different coordinates. These elements are tested for five definitions of curvilinear coordinates, with three of them being generated in an automated manner. Calculations for semi-rigid molecules, namely H2O, H2CO, CH2F2, and H2CNH, demonstrate the general functionality of the implemented algorithms. Additional calculations for the HOPO molecule highlight the benefits of these curvilinear coordinates for systems with large amplitude motions. This new implementation allowed us to compare the performance of these different coordinate systems with respect to the convergence of the underlying expansion of the potential energy surface and subsequent vibrational configuration interaction calculations.

Accurate Vibrational and Ro-Vibrational Contributions to the Properties of Large Molecules by a New Engine Employing Curvilinear Internal Coordinates and Vibrational Perturbation Theory to Second Order

The unbiased comparison between theory and experiment requires approaches more sophisticated than the basic harmonic-oscillator rigid-rotor model, for taking into account vibrational averaging effects and ro-vibrational couplings in molecules of increasing size. Second-order vibrational perturbation theory based on curvilinear internal coordinates (ICs) offers a remarkable compromise between accuracy and computational cost, thanks to the reduction of mode-mode couplings with respect to their counterparts based on Cartesian coordinates. Therefore, we have developed, implemented, and validated a general engine employing ICs, which allows the accurate evaluation of vibrational averages and ro-vibrational couplings for molecules containing up to about 50 atoms beyond the harmonic approximation. After validation of the new tool for relatively small molecules, the effectiveness of ICs has been demonstrated for some flexible and/or quite large molecular bricks of life.

Variational Vibrational States of Methanol (12D)

Full-dimensional (12D) vibrational states of the methanol molecule (CH3OH) have been computed using the GENIUSH-Smolyak approach and the potential energy surface from Qu and Bowman (2013). All vibrational energies are converged better than 0.5 cm-1 with respect to the basis and grid size up to the first overtone of the CO stretch, ca. 2000 cm-1 beyond the zero-point vibrational energy. About 70 torsion-vibration states are reported and assigned. The computed vibrational energies agree with the available experimental data within less than a few cm-1 in most cases, which confirms the good accuracy of the potential energy surface. The computations are carried out using curvilinear normal coordinates with the option of path-following coefficients, which minimize the coupling of the small- and large-amplitude motions. It is important to ensure tight numerical fulfillment of the C3v(M) molecular symmetry for every geometry and coefficient set used to define the curvilinear normal coordinates along the torsional coordinate to obtain a faithful description of degeneracy in this floppy system. The reported values may provide a computational reference for fundamental spectroscopy, astrochemistry, and for the search of the proton-to-electron mass ratio variation using the methanol molecule.

VSCF/VCI theory based on the Podolsky Hamiltonian

While the vibrational spectra of semi-rigid molecules can be computed on approaches relying on the Watson Hamiltonian, floppy molecules or molecular clusters are better described by Hamiltonians, which are capable of dealing with any curvilinear coordinates. It is the kinetic energy operator (KEO) of these Hamiltonians, which render the correlated calculations relying on them rather costly. Novel implementation of vibrational self-consistent field theory and vibrational configuration interaction theory on the basis of the Podolsky Hamiltonian are reported, in which the inverse of the metric tensor, i.e., the G matrix, is represented by an n-mode expansion expressed in terms of polynomials. An analysis of the importance of the individual terms of the KEO with respect to the truncation orders of the n-mode expansion is provided. Benchmark calculations have been performed for the cis-HOPO and methanimine, H2CNH, molecules and are compared to experimental data and to calculations based on the Watson Hamiltonian and the internal coordinate path Hamiltonian.

Efficient vibrationally correlated calculations using n-mode expansion-based kinetic energy operators

Due to its efficiency and flexibility, the n-mode expansion is a frequently used tool for representing molecular potential energy surfaces in quantum chemical simulations. In this work, we investigate the performance of n-mode expansion-based models of kinetic energy operators in general polyspherical coordinate systems. In particular, we assess the operators with respect to accuracy in vibrationally correlated calculations and their effect on potential energy surface construction with the adaptive density guided approach. Our results show that the n-mode expansion-based operator variants are reliable and systematically improvable approximations of the full kinetic energy operator. Moreover, we introduce a workflow to generate the n-mode expanded kinetic energy operators on-the-fly within the adaptive density guided approach. This scheme can be applied in studies of species and coordinate systems, for which an analytical form of the kinetic energy operator is not available.

Determining internal coordinate sets for optimal representation of molecular vibration

Arising from the harmonic approximation in solving the vibrational Schrödinger equation, normal modes dissect molecular vibrations into distinct degrees of freedom. Normal modes are widely used as they give rise to descriptive vibrational notations and are convenient for expanding anharmonic potential energy surfaces as an alternative to higher-order Taylor series representations. Usually, normal modes are expressed in Cartesian coordinates, which bears drawbacks that can be overcome by switching to internal coordinates. Considering vibrational notations, normal modes with delocalized characters are difficult to denote, but internal coordinates offer a route to clearer notations. Based on the Hessian, normal mode decomposition schemes for a given set of internal coordinates can describe a normal mode by its contributions from internal coordinates. However, choosing a set of internal coordinates is not straightforward. While the Hessian provides unique sets of normal modes, various internal coordinate sets are possible for a given system. In the present work, we employ a normal mode decomposition scheme to choose an optimal set. Therefore, we screen reasonable sets based on topology and symmetry considerations and rely on a metric that minimizes coupling between internal coordinates. Ultimately, the Nomodeco toolkit presented here generates internal coordinate sets to find an optimal set for representing molecular vibrations. The resulting contribution tables can be used to clarify vibrational notations. We test our scheme on small to mid-sized molecules, showing how the space of definable internal coordinate sets can significantly be reduced.

Vibrationally correlated calculations in polyspherical coordinates: Taylor expansion-based kinetic energy operators

The efficiency of quantum chemical simulations of nuclear motion can in many cases greatly benefit from the application of curvilinear coordinate systems. This is rooted in the fact that a set of smartly selected curvilinear coordinates may represent the motion naturally well, thus decreasing the couplings between motions in these coordinates. In this study, we assess the validity of different Taylor expansion-based approximations of kinetic energy operators in a (curvilinear) polyspherical parametrization. To this end, we investigate the accuracy as well as the numerical performance of the approximations in time-independent vibrational coupled cluster and full vibrational interaction calculations for several test cases ranging from tri- to penta-atomic molecules. We find that several of the proposed schemes reproduce the vibrational ground state and excitation energies to a decent accuracy, justifying their application in future investigations. Furthermore, due to the restricted mode coupling and their inherent sum-of-products form, the new approximations open up the possibility of treating large molecular systems with efficient vibrational coupled cluster schemes in general coordinates.

Smolyak Scheme for solving the Schrödinger equation: Application to Malonaldehyde in Full Dimensionality

In 1963 Smolyak introduced an approach to overcome the exponential scaling with respect to the number of variables of the direct product size [S. A. Smolyak Soviet Mathematics Doklady, 4, 240 (1963)]. The main idea is to replace a single large direct product by a sum of selected small direct products. It was first used in quantum dynamics in 2009 by Avila and Carrington [G. Avila and T. Carrington, J. Chem. Phys., 131, 174103 (2009)]. Since then, several calculations have been published by Avila and Carrington and by other groups. In the present study, and to push the limit to larger and more complex systems, this scheme is combined with the use of an on-the-fly calculation of the kinetic energy operator and a Block-Davidson procedure to obtain eigenstates in our home-made Fortran codes, ElVibRot and Tnum-Tana. This was applied to compute the tunneling splitting of malonaldehyde in full dimensionality (21D) using the potential of Mizukami et al. [W. Mizukami, S. Habershon, and D.P. Tew, J. Chem. Phys. 141, 1443-10 (2014)]. Our tunneling splitting calculations, 21.7±0.3 cm-1 and 2.9±0.1 cm-1 , show excellent agreement with the experimental values, 21.6 cm-1 and 2.9 cm-1 for the normal isotopologue and the mono-deuterated one, respectively.

Using Collocation to Solve the Schrödinger Equation

We review the collocation approach to the solution of the Schrödinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule-surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.

Exact quantum dynamics developments for floppy molecular systems and complexes

Molecular rotation, vibration, internal rotation, isomerization, tunneling, intermolecular dynamics of weakly and strongly interacting systems, intra-to-inter-molecular energy transfer, hindered rotation and hindered translation over surfaces are important types of molecular motions. Their fundamentally correct and detailed description can be obtained by solving the nuclear Schrödinger equation on a potential energy surface. Many of the chemically interesting processes involve quantum nuclear motions which are 'delocalized' over multiple potential energy wells. These 'large-amplitude' motions in addition to the high dimensionality of the vibrational problem represent challenges to the current (ro)vibrational methodology. A review of the quantum nuclear motion methodology is provided, current bottlenecks of solving the nuclear Schrödinger equation are identified, and solution strategies are reviewed. Technical details, computational results, and analysis of these results in terms of limiting models and spectroscopically relevant concepts are highlighted for selected numerical examples.

Perturb-Then-Diagonalize Vibrational Engine Exploiting Curvilinear Internal Coordinates

The present paper is devoted to the implementation and validation of a second-order perturbative approach to anharmonic vibrations, followed by variational treatment of strong couplings (GVPT2) based on curvilinear internal coordinates. The main difference with respect to the customary Cartesian-based formulation is that the kinetic energy operator is no longer diagonal, and has to be expanded as well, leading to additional terms which have to be taken into proper account. It is, however, possible to recast all the equations as well-defined generalizations of the corresponding Cartesian-based counterparts, thus achieving a remarkable simplification of the new implementation. Particular attention is paid to the treatment of Fermi resonances with significant number of test cases analyzed fully, validating the new implementation. The results obtained in this work confirm that curvilinear coordinates strongly reduce the strength of inter-mode couplings compared to their Cartesian counterparts. This increases the reliability of low-order perturbative treatments for semi-rigid molecules and paves the way toward the reliable representation of more flexible molecules where small- and large-amplitude motions can be safely decoupled and treated at different levels of theory.

Intramolecular Vibrational Redistribution in Formic Acid and its Deuterated Forms

The intramolecular vibrational relaxation dynamics of formic acid and its deuterated isotopologues is simulated on the full-dimensional potential energy surface of Richter and Carbonnière [F. Richter and P. Carbonnière, J. Chem. Phys. 148, 064303 (2018)] using the Heidelberg MCTDH package. We focus on couplings with the torsion vibrational modes close to the trans- cis isomerisation coordinate from the dynamics of artificially excited vibrational mode overtones. The C-O stretch bright vibrational mode is coupled to the out-of-the plane torsion mode in HCOOH, where this coupling could be exploited for laser-induced trans-to- cis isomerisation. Strong isotopic effects are observed: deuteration of the hydroxyl group, i.e., in HCOOD and DCOOD, destroys the C-O stretch to torsion mode coupling whereas in DCOOH, little to no effect is observed.

Quantum dynamics with curvilinear coordinates: models and kinetic energy operator

In order to simplify the numerical solution of the time-dependent or time-independent Schrödinger equations associated with atomic and molecular motions, the use of well-adapted coordinates is essential. Usually, this set of curvilinear coordinates leads to a Hamiltonian operator that is as separable as possible. Although their corresponding kinetic energy operator (KEO) expressions can be derived analytically for small systems or special kinds of coordinates, a numerical and exact approach allows one to compute them in terms of sophisticated curvilinear coordinates. Furthermore, the numerical approach enables one to easily define reduced-dimensionality or constrained models. We present here a recent implementation of this numerical approach that allows nested coordinate transformations, therefore leading to great flexibility in the definition of the curvilinear coordinates. Furthermore, this implementation has no limitations in terms of numbers of atoms or coordinate transformations. The quantum dynamics of the cis-trans photoisomerization of part of the retinal chromophore illustrates the construction of the coordinates and KEO part of a three-dimensional model. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Performance of a black-box-type rovibrational method in comparison with a tailor-made approach: Case study for the methane-water dimer

The present work intends to join and respond to the excellent and thoroughly documented rovibrational study of X. G. Wang and T. Carrington, Jr. [J. Chem. Phys. 154, 124112 (2021)] that used an approach tailored for floppy dimers with an analytic dimer Hamiltonian and a non-product basis set including Wigner D functions. It is shown in the present work that the GENIUSH black-box-type rovibrational method can approach the performance of the tailor-made computation for the example of the floppy methane-water dimer. Rovibrational transition energies and intensities are obtained in the black-box-type computation with a twice as large basis set and in excellent numerical agreement in comparison with the more efficient tailor-made approach.

Interplay of Rotational and Pseudorotational Motions in Flexible Cyclic Molecules

Solutions to the time-independent nuclear Schrödinger equation associated with the pseudorotational motion of three flexible cyclic molecules are presented and discussed. Structural relaxations related to the pseudorotational motion are described as functions of a pseudorotation angle ϕ which is formulated according to the definition of ring-puckering coordinates originally proposed by Cremer and Pople ( J. Am. Chem. Soc. 1975, 97 (6), 1354-1358). In order to take into account the interplay between pseudorotational and rotational motions, the rovibrational Hamiltonian matrices are formulated for the rotational quantum numbers J = 0 and J = 1. The rovibrational Hamiltonian matrices are constructed and diagonalized using a Python program developed by the authors. Suitable algorithms for (i) the construction of one-dimensional cuts of potential energy surfaces along the pseudorotation angle ϕ and (ii) the assignment of the vibrorotational wave functions (which are needed for the automatic calculation of rotational transition energies J = 0 → J = 1) are described and discussed.

On neglecting Coriolis and related couplings in first-principles rovibrational spectroscopy: Considerations of symmetry, accuracy, and simplicity. II. Case studies for H2O isotopologues, H3+, O3, and NH3

For centuries, it has been known that vibrational and rotational degrees of freedom are in general not separable. Nevertheless, surprisingly little is known about the best strategies for approximately separating these degrees of freedom in practice-even in the case of semirigid molecules, where the separation is most meaningful. There is also some confusion in the literature about the proper way to quantify the magnitude of the Coriolis (i.e., rotation-vibration) coupling in rovibrational Hamiltonians or its effect on the rovibrational eigenenergies. In this study, a vibrational-coordinate-independent metric is proposed to quantify the magnitude of the Coriolis contribution to the rovibrational Hamiltonian. The impact of Coriolis coupling on the rovibrational eigenenergies is computed numerically exactly, using both full and various truncated Hamiltonians. The role played by the choice of the vibrational coordinate system-and especially by the choice of "embedding" or body-fixed frame-is examined extensively, both numerically and analytically. This investigation targets several molecular prototypes, all of which serve as important benchmarks for the high-resolution spectroscopic community. Most of these are triatomic molecules, including water (H₂¹⁶O), its deuterated isotopologues (D₂¹⁶O and HD¹⁶O), H₃⁺, and ozone (¹⁶O₃), but the tetratomic ammonia molecule (¹⁴NH₃) is also investigated. These studies provide important insight into the nature of Coriolis coupling under various circumstances. The findings of this study also have significant practical ramifications, vis-à-vis the use of simplifying numerical approximation techniques in nuclear-motion computations.

Fingerprint region of the formic acid dimer: variational vibrational computations in curvilinear coordinates

Curvilinear kinetic energy models are developed for variational nuclear motion computations including the inter- and the low-frequency intra-molecular degrees of freedom of the formic acid dimer. The coupling of the inter- and intra-molecular modes is studied by solving the vibrational Schrödinger equation for a series of vibrational models, from two up to ten active vibrational degrees of freedom by selecting various combinations of active modes and constrained coordinate values. Vibrational states, nodal assignment, and infrared vibrational intensity information is computed using the full-dimensional potential energy surface (PES) and electric dipole moment surface developed by Qu and Bowman [Phys. Chem. Chem. Phys., 2016, 18, 24835; J. Chem. Phys., 2018, 148, 241713]. Good results are obtained for several fundamental and combination bands in comparison with jet-cooled vibrational spectroscopy experiments, but the description of the ν₈ and ν₉ fundamental vibrations, which are close in energy and have the same symmetry, appears to be problematic. For further progress in comparison with experiment, the potential energy surface, and in particular, its multi-dimensional couplings representation, requires further improvement.

Using collocation to study the vibrational dynamics of molecules

In this paper, I review collocation methods for solving the time-independent and the time-dependent Schroedinger equation. Unlike traditional variational methods, collocation methods do not require integrals and quadrature. Either collocation or quadrature is necessary if the potential does not have a special form. If the basis is a direct product of univariate bases and the quadrature grid is also a direct product, there exist variational methods that do not require quadrature approximations for potential energy matrix elements. These methods, however, do require storing, in computer memory, vectors with as many components as there are quadrature points. For this reason direct-product variational methods are poor for problems with more than five atoms. There are well established ideas for reducing the size of the basis in a variational calculation. Three such ideas are: 1) prune the direct product basis; 2) use basis functions that are products of multivariate functions; 3) optimise the basis functions (e.g. Multiconfiguration time-dependent Hartree). Reducing the basis size, however, is not enough to the make variational methods tractable because, for all three of these ideas, quadrature rears its ugly head. Collocation is an attractive alternative to variational methods.

Low-rank sum-of-products finite-basis-representation (SOP-FBR) of potential energy surfaces

The sum-of-products finite-basis-representation (SOP-FBR) approach for the automated multidimensional fit of potential energy surfaces (PESs) is presented. In its current implementation, the method yields a PES in the so-called Tucker sum-of-products form, but it is not restricted to this specific ansatz. The novelty of our algorithm lies in the fact that the fit is performed in terms of a direct product of a Schmidt basis, also known as natural potentials. These encode in a non-trivial way all the physics of the problem and, hence, circumvent the usual extra ad hoc and a posteriori adjustments (e.g., damping functions) of the fitted PES. Moreover, we avoid the intermediate refitting stage common to other tensor-decomposition methods, typically used in the context of nuclear quantum dynamics. The resulting SOP-FBR PES is analytical and differentiable ad infinitum. Our ansatz is fully general and can be used in combination with most (molecular) dynamics codes. In particular, it has been interfaced and extensively tested with the Heidelberg implementation of the multiconfiguration time-dependent Hartree quantum dynamical software package.

Quantum dynamical simulations of intra-chain exciton diffusion in an oligo (para-phenylene vinylene) chain at finite temperature

We report on quantum dynamical simulations of exciton diffusion in an oligo(para-phenylene vinylene) chain segment with 20 repeat units (OPV-20) at finite temperature, complementary to our recent study of the same system at T = 0 K [R. Binder and I. Burghardt, J. Chem. Phys. 152, 204120 (2020)]. Accurate quantum dynamical simulations are performed using the multi-layer multi-configuration time-dependent Hartree method as applied to a site-based Hamiltonian comprising 20 electronic states of Frenkel type and 460 vibrational modes, including site-local quinoid-distortion modes along with site-correlated bond-length alternation (BLA) modes, ring torsional modes, and an explicit harmonic-oscillator bath. A first-principles parameterized Frenkel-Holstein type Hamiltonian is employed, which accounts for correlations between the ring torsional modes and the anharmonically coupled BLA coordinates located at the same junction. Thermally induced fluctuations of the torsional modes are described by a stochastic mean-field approach, and their impact on the excitonic motion is characterized in terms of the exciton mean-squared displacement. A normal diffusion regime is observed under periodic boundary conditions, apart from transient localization features. Even though the polaronic exciton species are comparatively weakly bound, exciton diffusion is found to be a coherent-rather than hopping type-process, driven by the fluctuations of the soft torsional modes. Similar to the previous observations for oligothiophenes, the evolution for the most part exhibits a near-adiabatic dynamics of local exciton ground states (LEGSs) that adjust to the local conformational dynamics. However, a second mechanism, involving resonant transitions between neighboring LEGSs, gains importance at higher temperatures.

Quantum and Quantum-Classical Studies of the Photoisomerization of a Retinal Chromophore Model

We report an in-depth analysis of the photo-induced isomerization of the 2-cis-penta-2,4-dieniminium cation: a minimal model of the 11-cis retinal protonated Schiff base chromophore of the dim-light photoreceptor rhodopsin. Based on recently developed three-dimensional potentials parametrized on ab initio multi-state multi-configurational second-order perturbation theory data, we perform quantum-dynamical studies. In addition, simulations based on various quantum-classical methods, among which Tully surface hopping and the coupled-trajectory approach derived from the exact factorization, allow us to validate their performance against vibronic wavepacket propagation and, therefore, a purely quantum treatment. Quantum-dynamics results uncover qualitative differences with respect to the two-dimensional Hahn-Stock potentials, widely used as model potentials for the isomerization of the same chromophore, due to the increased dimensionality and three-mode correlation. Quantum-classical simulations show, instead, that three-dimensional model potentials are capable of capturing a number of features revealed by atomistic simulations and experimental observations. In particular, a recently reported vibrational phase relationship between double-bond torsion and hydrogen-out-of-plane modes critical for rhodopsin isomerization efficiency is correctly reproduced.

First-Principles Quantum and Quantum-Classical Simulations of Exciton Diffusion in Semiconducting Polymer Chains at Finite Temperature

We report on first-principles quantum-dynamical and quantum-classical simulations of photoinduced exciton dynamics in oligothiophene chain segments, representative of intrachain exciton migration in the poly(3-hexylthiophene) (P3HT) polymer. Following up on our recent study (Binder R.; Burghardt, I. Faraday Discuss. 2020, 221, 406), multilayer multiconfiguration time-dependent Hartree calculations for a short oligothiophene segment comprising 20 monomer units (OT-20) are carried out to obtain full quantum-dynamical simulations at finite temperature. These are employed to benchmark mean-field Ehrenfest calculations, which are shown to give qualitatively correct results for the present system. Periodic boundary conditions turn out to significantly improve earlier estimates of diffusion coefficients. Using the Ehrenfest approach, a series of calculations are subsequently carried out for larger lattices (OT-40 to OT-80), leading to estimates for temperature-dependent mean-squared displacements, which are found to exhibit a near-linear dependence as a function of time. The resulting diffusion coefficient estimates are an increasing function of temperature, whose detailed functional form depends on the degree of static disorder. With a realistic static disorder parameter (σ_s ≃ 0.06 eV), the diffusion coefficients decrease from D ∼ 1 × 10^-2 cm² s^-1 to D ∼ 1 × 10^-3 cm² s^-1, in qualitative agreement with experimental data for P3HT. The dynamical scenario obtained from our simulations shows that exciton migration in P3HT-type chains is a largely adiabatic process throughout the temperature regime we investigated (i.e., T = 50-300 K). The resulting picture of exciton migration is a coherent, but not bandlike, motion of an exciton-polaron driven by fluctuations induced by low-frequency modes. This process acquires partial hopping character if static disorder becomes prominent and Anderson localization sets in.

First-principles description of intra-chain exciton migration in an oligo(para-phenylene vinylene) chain. I. Generalized Frenkel-Holstein Hamiltonian

A generalized Frenkel-Holstein Hamiltonian is constructed to describe exciton migration in oligo(para-phenylene vinylene) chains, based on excited state electronic structure data for an oligomer comprising 20 monomer units (OPV-20). Time-dependent density functional theory calculations using the ωB97XD hybrid functional are employed in conjunction with a transition density analysis to study the low-lying singlet excitations and demonstrate that these can be characterized to a good approximation as a Frenkel exciton manifold. Based on these findings, we employ the analytic mapping procedure of Binder et al. [J. Chem. Phys. 141, 014101 (2014)] to translate one-dimensional (1D) and two-dimensional (2D) potential energy surface (PES) scans to a fully anharmonic, generalized Frenkel-Holstein (FH) Hamiltonian. A 1D PES scan is carried out for intra-ring quinoid distortion modes, while 2D PES scans are performed for the anharmonically coupled inter-monomer torsional and vinylene bridge bond length alternation modes. The kinetic energy is constructed in curvilinear coordinates by an exact numerical procedure, using the TNUM Fortran code. As a result, a fully molecular-based, generalized FH Hamiltonian is obtained, which is subsequently employed for quantum exciton dynamics simulations, as shown in Paper II [R. Binder and I. Burghardt, J. Chem. Phys. 152, 204120 (2020)].

On neglecting Coriolis and related couplings in first-principles rovibrational spectroscopy: considerations of symmetry, accuracy, and simplicity

The rotation-vibration (Coriolis) coupling contribution to variationally computed rovibrational energy levels is investigated, employing triatomic AB[Formula: see text] molecules as models. In particular, calculations are performed for H[Formula: see text][Formula: see text]O, across a range of vibrational and rotational excitations, both with and without the Coriolis contribution. A variety of different embedding choices are considered, together with a hierarchy of increasingly severe approximations culminating in a generalized version of the so-called "centrifugal sudden" method. Several surprising and remarkable conclusions are found, including that the Eckart embedding is not the best embedding choice.

First-principles quantum simulations of exciton diffusion on a minimal oligothiophene chain at finite temperature

High-dimensional multiconfigurational quantum dynamics simulations are carried out at finite temperature to simulate exciton diffusion on an oligothiophene chain, representative of a segment of the poly(3-hexylthiophene) (P3HT) polymer. The ab initio parametrized site-based Hamiltonian of Binder et al. [Phys. Rev. Lett., 2018, 120, 227401] is employed to model a 20-site system, including intra-ring and inter-ring high-frequency modes as well as torsional modes which undergo thermal fluctuations induced by an explicit harmonic oscillator bath. The system-bath dynamics is treated within the setting of a stochastic mean-field Schrödinger equation. For the 20-site excitonic system, a total of 20 Frenkel states and 248 modes are propagated using the multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) method. The resulting dynamics can be interpreted in terms of the coherent motion of an exciton-polaron quasi-particle stochastically driven by torsional fluctuations. This dynamics yields a near-linear mean squared displacement (MSD) as a function of time, from which a diffusion coefficient can be deduced which increases with temperature, up to 5.7 × 10-3 cm2 s-1 at T = 300 K.

Stochastic modeling of macromolecules in solution. I. Relaxation processes

A framework for the stochastic description of relaxation processes in flexible macromolecules, including dissipative effects, is introduced from an atomistic point of view. Projection-operator techniques are employed to obtain multidimensional Fokker-Planck operators governing the relaxation of internal coordinates and global degrees of freedom and depending upon parameters fully recoverable from classic force fields (energetics) and continuum models (friction tensors). A hierarchy of approaches of different complexity is proposed in this unified context, aimed primarily at the interpretation of magnetic resonance relaxation experiments. In particular, a model based on a harmonic internal Hamiltonian is discussed as a test case.

Toward breaking the curse of dimensionality in (ro)vibrational computations of molecular systems with multiple large-amplitude motions

Methodological progress is reported in the challenging direction of a black-box-type variational solution of the (ro)vibrational Schrödinger equation applicable to floppy, polyatomic systems with multiple large-amplitude motions. This progress is achieved through the combination of (i) the numerical kinetic-energy operator (KEO) approach of Mátyus et al. [J. Chem. Phys. 130, 134112 (2009)] and (ii) the Smolyak nonproduct grid method of Avila and Carrington, Jr. [J. Chem. Phys. 131, 174103 (2009)]. The numerical representation of the KEO makes it possible to choose internal coordinates and a body-fixed frame best suited for the molecular system. The Smolyak scheme reduces the size of the direct-product grid representation by orders of magnitude, while retaining some of the useful features of it. As a result, multidimensional (ro)vibrational states are computed with system-adapted coordinates, a compact basis- and grid-representation, and an iterative eigensolver. Details of the methodological developments and the first numerical applications are presented for the CH₄·Ar complex treated in full (12D) vibrational dimensionality.

Non-adiabatic mass correction to the rovibrational states of molecules: Numerical application for the H 2 + molecular ion

General transformation expressions of the second-order non-adiabatic Hamiltonian of the atomic nuclei, including the kinetic-energy correction terms, are derived upon the change from laboratory-fixed Cartesian coordinates to general curvilinear coordinate systems commonly used in rovibrational computations. The kinetic-energy or so-called "mass-correction" tensor elements are computed with the stochastic variational method and floating explicitly correlated Gaussian functions for the H 2 + molecular ion in its ground electronic state. {Further numerical applications for the ⁴ He 2 + molecular ion are presented in the forthcoming paper, Paper II [E. Mátyus, J. Chem. Phys. 149, 194112 (2018)]}. The general, curvilinear non-adiabatic kinetic energy operator expressions are used in the examples, and non-adiabatic rovibrational energies and corrections are determined by solving the rovibrational Schrödinger equation including the diagonal Born-Oppenheimer as well as the mass-tensor corrections.

Direct Computation of the Quantum Partition Function by Path-Integral Nested Sampling

In the present work we introduce a computational approach to the absolute rovibrational quantum partition function using the path-integral formalism of quantum mechanics in combination with the nested sampling technique. The numerical applicability of path-integral nested sampling is demonstrated for small molecules of spectroscopic interest. The computational cost of the method is determined by the evaluation time of a point on the potential energy surface (PES). For efficient PES implementations, the path-integral nested sampling method can be a viable alternative to the direct-Boltzmann-summation technique of variationally computed rovibrational energies, especially for medium-sized molecules and at elevated temperatures.

The kinetic energy operator for distance-dependent effective nuclear masses: Derivation for a triatomic molecule

The kinetic energy operator for triatomic molecules with coordinate or distance-dependent nuclear masses has been derived. By combination of the chain rule method and the analysis of infinitesimal variations of molecular coordinates, a simple and general technique for the construction of the kinetic energy operator has been proposed. The asymptotic properties of the Hamiltonian have been investigated with respect to the ratio of the electron and proton mass. We have demonstrated that an ad hoc introduction of distance (and direction) dependent nuclear masses in Cartesian coordinates preserves the total rotational invariance of the problem. With the help of Wigner rotation functions, an effective Hamiltonian for nuclear motion can be derived. In the derivation, we have focused on the effective trinuclear Hamiltonian. All necessary matrix elements are given in closed analytical form. Preliminary results for the influence of non-adiabaticity on vibrational band origins are presented for H₃⁺.

Conformational Dynamics Guides Coherent Exciton Migration in Conjugated Polymer Materials: First-Principles Quantum Dynamical Study

We report on high-dimensional quantum dynamical simulations of photoinduced exciton migration in a single-chain oligothiophene segment, in view of elucidating the controversial nature of the elementary exciton transport steps in semiconducting polymers. A novel first-principles parametrized Frenkel J aggregate Hamiltonian is employed that goes significantly beyond the standard Frenkel-Holstein Hamiltonian. Departing from a nonequilibrium state created by photoexcitation, these simulations provide evidence of an ultrafast two-timescale process at low temperatures, involving exciton-polaron formation within tens of femtoseconds (fs), followed by torsional relaxation on an ∼400  fs timescale. The second step is the driving force for exciton migration, as initial conjugation breaks are removed by dynamical planarization. The quantum coherent nature of the elementary exciton migration step is consistent with experimental observations highlighting the correlated and vibrationally coherent nature of the dynamics on ultrafast timescales.

Fast and slow excited-state intramolecular proton transfer in 3-hydroxychromone: a two-state story?

The photodynamics of 3-hydroxychromone in its first-excited singlet electronic state (bright state of ππ* character) is investigated with special emphasis given to two types of reaction pathways: the excited-state intramolecular-proton-transfer coordinate and the hydrogen-torsion coordinate linking the excited cis and trans isomers. A newly-found conical intersection with the second-excited singlet electronic state (dark state of nπ* character) is suspected to be, to some extent, the reason for the slower rate constant. This hypothesis based on quantum-chemistry calculations is supported by quantum-dynamics simulations in full dimensionality. They show significant transfer of electronic population and provide consistently a vibronic interpretation for the forbidden band in the UV absorption spectrum.

Communication: General variational approach to nuclear-quadrupole coupling in rovibrational spectra of polyatomic molecules

A general algorithm for computing the quadrupole-hyperfine effects in the rovibrational spectra of polyatomic molecules is presented for the case of ammonia (NH₃). The method extends the general variational approach TROVE [J. Mol. Spectrosc. 245, 126-140 (2007)] by adding the extra term in the Hamiltonian that describes the nuclear quadrupole coupling, with no inherent limitation on the number of quadrupolar nuclei in a molecule. We applied the new approach to compute the nitrogen-nuclear-quadrupole hyperfine structure in the rovibrational spectrum of NH314. These results agree very well with recent experimental spectroscopic data for the pure rotational transitions in the ground vibrational and ν₂ states and the rovibrational transitions in the ν₁, ν₃, 2ν₄, and ν₁ + ν₃ bands. The computed hyperfine-resolved rovibrational spectrum of ammonia will be beneficial for the assignment of experimental rovibrational spectra, further detection of ammonia in interstellar space, and studies of the proton-to-electron mass variation.