Adiabatic-to-diabatic Transformation Research Articles

Non-adiabatic coupling in the potential energy surfaces of SO2 molecule.

To investigate the potential energy surfaces and the coupling between the adiabatic states of SO2 molecules, it is necessary to consider the non-adiabatic coupling terms (NACTs), where the Born-Oppenheimer approximation breaks down. In this work, we analyze the conical intersections between 1 1A1 and 1 1B2 states (the A' states in Cs symmetry) and 1 1A2 and 1 1B1 states (the A'' states in Cs symmetry) using NACTs and adiabatic-to-diabatic transformation (ADT) angles. Our results confirm reasonable interaction between 1 1A1 and 1 1B2 states and strong interaction between 1 1A2 and 1 1B1 states.

A beyond Born-Oppenheimer treatment of C6H6 + radical cation for diabatic surfaces: Photoelectron spectra of its neutral analog using time-dependent discrete variable representation.

We employ theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation (C6H6 +) for the first time and explore the workability of the time-dependent discrete variable representation (TDDVR) method for carrying out dynamical calculations to evaluate the photoelectron (PE) spectra of its neutral analog. Ab initio adiabatic PESs and nonadiabatic coupling terms are computed over a series of pairwise normal modes, which exhibit rich nonadiabatic interactions starting from Jahn-Teller interactions and accidental conical intersections/seams to pseudo Jahn-Teller couplings. Once the electronic structure calculation is completed on the low-lying five doublet electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of the cationic species, diabatization is carried out employing the adiabatic-to-diabatic transformation (ADT) equations for the five-state sub-Hilbert space to compute highly accurate ADT angles, and thereby, single-valued, smooth, symmetric, and continuous diabatic PESs and couplings are constructed. Subsequently, such surface matrices are used to perform multi-state multi-mode nuclear dynamics for simulating PE spectra of benzene. Our theoretical findings clearly depict that the spectra for X̃2E1g and B̃2E2g-C̃2A2u states obtained from BBO treatment and TDDVR dynamics exhibit reasonably good agreement with the experimental results as well as with the findings of other theoretical approaches.

Study of non‐adiabatic interactions among low‐lying electronic states of HeH2+ with its implication in its dissociation into various species

AbstractPossible interplay of various atomic and molecular systems like He+, He, H+, H, H2+, H2, HeH+, HeH2+ in the interstellar spaces in various phases of the creation and development of the universe as well as in the present interstellar spaces have been the subject of experimental and theoretical investigations for quite a long period. A chemical network involving these species has been considered. The molecular system HeH2+, being a weakly bound species, may readily give the dissociation products (H, HeH+), (He, H2+), (He+, H2) and (He, H, H+) through different dissociation channels, as has been argued from the contour diagram of the ground as well as non‐adiabatically interacting excited electronic state. A seam of Conical Intersections (CI) between two lowest excited states of linear HeH2+, as a result of non‐adiabatic interactions, has been demonstrated for the first time, through calculation and plot of non‐adiabatic coupling term (NACT) between these two states. Calculations of NACT's, followed by construction of the adiabatic‐to‐diabatic transformation (ADT) angles, ADT matrix and calculation of topological (Berry) phases have been presented to confirm the presence of Jahn‐Teller(JT) CI in two representative configuration spaces(CS) in this seam: (i) one away from the equilibrium geometry hosting JT‐CI between two excited states 22A′ and 32A′ and (ii) another CS near the equilibrium geometry hosting JT‐CI between excited states 22A′ and 32A′ as well as a closely spaced JT‐CI between excited states 32A′ and 42A′. Suitable diabatizations have been performed for a range of CS covering these two CI, which help in understanding several dissociation channels of HeH2+ involving the above mentioned excited states, leading to an increase in the yield of dissociation products (He, H, H+). Distribution of Mulliken partial charges, as obtained from Mulliken population analysis have been analyzed for justifying the existence of HeH2+, the possibility of different dissociation products of HeH2+ in different dissociation channels as well as the justification of the CI‐induced charge transfer in excited states.

Beyond Born-Oppenheimer constructed diabatic potential energy surfaces for F + H2 reaction.

First principles based beyond Born-Oppenheimer theory has been implemented on the F + H2 system for constructing multistate global diabatic Potential Energy Surfaces (PESs) through the incorporation of Nonadiabatic Coupling Terms (NACTs) explicitly. The spin-orbit (SO) coupling effect on the collision process of the F + H2 reaction has been included as a perturbation to the non-relativistic electronic Hamiltonian. Adiabatic PESs and NACTs for the lowest three electronic states (12A', 22A', and 12A″) are determined in hyperspherical coordinates as functions of hyperangles for a grid of fixed values of the hyperradius. Jahn-Teller (JT) type conical intersections between the two A' states translate along C2v and linear geometries in F + H2. In addition, A' and A″ states undergo Renner-Teller (RT) interaction at collinear configurations of this system. Both JT and RT couplings are validated by integrating NACTs along properly chosen contours. Subsequently, we have solved adiabatic-to-diabatic transformation (ADT) equations to evaluate the ADT angles for constructing the diabatic potential matrix of F + H2, including the SO coupling terms. The newly calculated diabatic PESs are found to be smooth, single-valued, continuous, and symmetric and can be invoked for performing accurate scattering calculations on the F + H2 system.

ADT: A Generalized Algorithm and Program for Beyond Born-Oppenheimer Equations of "N" Dimensional Sub-Hilbert Space.

The major bottleneck of first principle based beyond Born-Oppenheimer (BBO) treatment originates from large number and complicated expressions of adiabatic to diabatic transformation (ADT) equations for higher dimensional sub-Hilbert spaces. In order to overcome such shortcoming, we develop a generalized algorithm, "ADT" to generate the nonadiabatic equations through symbolic manipulation and to construct highly accurate diabatic surfaces for molecular processes involving excited electronic states. It is noteworthy to mention that the nonadiabatic coupling terms (NACTs) often become singular (removable) at degenerate point(s) or along a seam in the nuclear configuration space (CS) and thereby, a unitary transformation is required to convert the kinetically coupled (adiabatic) Hamiltonian to a potentially (diabatic) one to avoid such singularity(ies). The "ADT" program can be efficiently used to (a) formulate analytic functional forms of differential equations for ADT angles and diabatic potential energy matrix and (b) solve the set of coupled differential equations numerically to evaluate ADT angles, residue due to singularity(ies), ADT matrices, and finally, diabatic potential energy surfaces (PESs). For the numerical case, user can directly provide ab initio data (adiabatic PESs and NACTs) as input files to this software or can generate those input files through in-built python codes interfacing MOLPRO followed by ADT calculation. In order to establish the workability of our program package, we selectively choose six realistic molecular species, namely, NO2 radical, H3+, F + H2, NO3 radical, C6H6+ radical cation, and 1,3,5-C6H3F3+ radical cation, where two, three, five and six electronic states exhibit profound nonadiabatic interactions and are employed to compute diabatic PESs by using ab initio calculated adiabatic PESs and NACTs. The "ADT" package released under the GNU General Public License v3.0 (GPLv3) is available at https://github.com/AdhikariLAB/ADT-Program and also as the Supporting Information of this article.

Conical intersections and nonadiabatic coupling terms in 1,3,5-C6H3F3 +: A six state beyond Born-Oppenheimer treatment.

In order to circumvent numerical inaccuracy originating from the singularity of nonadiabatic coupling terms (NACTs), we need to perform kinetically coupled adiabatic to potentially coupled diabatic transformation of the nuclear Schrödinger Equation. Such a transformation is difficult to achieve for higher dimensional sub-Hilbert spaces due to inherent complicacy of adiabatic to diabatic transformation (ADT) equations. Nevertheless, detailed expressions of ADT equations are formulated for six coupled electronic states for the first time and their validity is extensively examined for a well-known radical cation, namely, 1,3,5-C6H3F3 + (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states (X̃2E'', Ã2A2 '', B̃2E', and C̃2A2 '), where several types of nonadiabatic interactions, like Jahn-Teller conical intersections (CI), accidental CIs, accidental seams (series of degenerate points), and pseudo Jahn-Teller interactions can be observed over the Franck-Condon region of nuclear configuration space. Those interactions are depicted by exploring degenerate components of C-C asymmetric stretching, C-C symmetric stretching, and C-C-C scissoring motion (Q9x, Q9y, Q10x, Q10y, Q12x, and Q12y) to compute complete active space self-consistent field level adiabatic PESs and NACTs as implemented in the MOLPRO quantum chemistry package. Subsequently, we perform the ADT using our newly devised fifteen (15) ADT equations to locate the position of CIs, verify the quantization of NACTs, and to construct highly accurate diabatic PESs.

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

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

A beyond Born-Oppenheimer treatment of five state molecular system [formula omitted] and the photodetachment spectra of its anion

A five-state multimode non-adiabatic dynamics has been performed on NO3 by constructing accurate diabatic potential energy surfaces (PESs) in vibronically coupled manifold using beyond Born-Oppenheimer (BBO) theory. In this article, we have considered the photodetachment of NO3- to the doublet states (X∼2A2′, A∼2E″ and B∼2E′) of NO3 and determined the vibronic couplings and conical intersections of the latter. Using ab initio calculated adiabatic PESs and Non-adiabatic Coupling Terms, we have carried out adiabatic-to-diabatic transformation (ADT) in five-state sub-Hilbert space to obtain ADT angles and thereby, constructed single-valued, smooth, symmetric and continuous diabatic PESs. The explicit analytic expressions for diabatic PESs and coupling elements are provided. Subsequently, we have performed nuclear dynamics starting from the electronic ground state of NO3- to simulate photodetachment spectra. The overall spectral profiles for 2A2′ and 2E″ states show reasonably good agreement with experimental results and that of 2E′ state with other theoretical calculations.

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

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

Five Electronic State Beyond Born-Oppenheimer Equations and Their Applications to Nitrate and Benzene Radical Cation.

We present explicit form of Adiabatic to Diabatic Transformation (ADT) equations and expressions of non-adiabatic coupling terms (NACTs) for a coupled five-state electronic manifold in terms of ADT angles between electronic wave functions. ADT matrices eliminate the numerical instability arising from singularity of NACTs and transform the adiabatic Schrödinger equation to its diabatic form. Two real molecular systems NO3 and C6H6+ (Bz+) are selectively chosen for the demonstration of workability of those equations. We examine the NACTs among the lowest five electronic states of the NO3 radical [X̃2A2' (12B2), Ã2E″ (12A2 and 12B1) and B̃2E' (12A1 and 22B2)], in which all types of non-adiabatic interactions, that is, Jahn-Teller (JT) interactions, Pseudo Jahn-Teller (PJT) interactions, and accidental conical intersections (CIs) are present. On the other hand, lowest five electronic states of Bz+ [X̃2E1g (12B3g and 12B2g), B̃2E2g (12Ag and 12B1g), and C̃2A2u (12B1u)] depict similar kind of complex feature of non-adiabatic effects. For NO3 radical, the two components of degenerate in-plane asymmetric stretching mode are taken as a plane of nuclear configuration space (CS), whereas in case of Bz+, two pairs are chosen: One is the pair of components of degenerate in-plane asymmetric stretching mode, and the other one is constituted with one of the components each from out-of-plane degenerate bend and in-plane degenerate asymmetric stretching modes. We calculate ab initio adiabatic potential energy surfaces (PESs) and NACTs among the lowest five electronic states at the CASSCF level using MOLPRO quantum chemistry package. Subsequently, the ADT is performed using those newly developed equations to validate the positions of the CIs, evaluate the ADT angles and construct smooth, symmetric, and continuous diabatic PESs for both the molecular systems.

Beyond Born-Oppenheimer treatment for the construction of triple-sheeted accurate diabatic Hamiltonian matrix of F+H2 system

We intend to study the non-adiabatic interactions among the three lowest adiabatic states (12A′, 12A″, and 22A′) of F+H2 triatomic reactive system in hyperspherical coordinates for a fixed hyperradius at ρ = 7.5 bohr as functions of hyperangles, θ (0° ≤ θ ≤ 90°) and ϕ (0° ≤ ϕ ≤ 360°). The adiabatic potential energy surfaces are calculated using MRCI level of methodology whereas the non-adiabatic coupling terms between those states are calculated from the analytic gradient methods implemented in MOLPRO quantum chemistry package. The ground (12A′) and the first excited (12A″) states exhibit conical intersection (CI) and seam of CI along C2v geometries, whereas the first (12A″) and the second (22A′) excited states undergo Renner-Teller coupling at linear geometries. We carry out adiabatic-to-diabatic transformation (ADT) by solving ADT equations to obtain ADT angles for constructing single-valued, continuous and symmetric 3 × 3 diabatic potential energy matrix so that subsequent accurate scattering calculations can be performed.

Ab - initio non-adiabatic couplings among three lowest singlet states of H3+: Construction of multisheeted diabatic potential energy surfaces

We calculate the adiabatic potential energy surfaces and non-adiabatic interactions among the three lowest singlet states (11A', 21A' and 31A') of H3+ in hyperspherical coordinates for a fixed hyperradius, ρ = 9 bohr as functions of hyperangles, θ (0 < θ < 90°) and ϕ (0 < ϕ < 360°). All ab initio calculations are performed using MRCI level of methodology implemented in quantum chemistry package, MOLPRO. The ground (11 A') and the first excited (21 A') states exhibit several conical intersections as functions of ϕ for θ > 70°. Subsequently, we carry out adiabatic to diabatic transformation (ADT) to obtain ADT angles for constructing single-valued, continuous, smooth and symmetric 3 × 3 diabatic potential energy matrix to perform accurate scattering calculations.

Ab initio constructed diabatic surfaces of NO2 and the photodetachment spectra of its anion.

A thorough investigation has been performed for electronic structure, topological effect, and nuclear dynamics of NO2 molecule, where the adiabatic potential energy surfaces (PESs), conical intersections between the ground (X(2)A1) and the first excited state (A(2)B2), and the corresponding non-adiabatic coupling terms between those states are recalculated [Chem. Phys. 416, 11 (2013)] to achieve enough accuracy in dynamics. We employ beyond Born-Oppenheimer theory for these two state sub-Hilbert space to carry out adiabatic to diabatic transformation (ADT) to obtain the ADT angles and thereby, to construct single-valued, smooth, and continuous diabatic PESs. The analytic expressions for the adiabatic PESs and ADT angles are provided to represent a two-state three-mode diabatic Hamiltonian of NO2 for performing nuclear dynamics to calculate the photo-electron spectra of its anion. It appears that not only Jahn-Teller type coupling but also Renner-Teller interaction contributes significantly on the overall spectrum. The coupling between the electronic states (X(2)A1 and A(2)B2) of NO2 is essentially through the asymmetric stretching mode, where the functional form of such interaction is distinctly symmetric and non-linear.

Study of Topological Effects Concerning the Lowest A″ and the Three A' States for the CO2(+) Ion.

A study of the topological effects, viz., the Jahn-Teller (JT) and Renner-Teller (RT) effects, in CO2(+) has been carried out by calculating nonadiabatic coupling terms (NACTs) at the state-averaged CASSCF level using the cc-pVTZ basis set for the lowest three A' states and one A″ state along a circular contour. Using the NACTs, the privileged adiabatic-to-diabatic transformation (ADT) angles (γ12) for 1A' and 2A' states of CO2(+) have been calculated along various circular contours. Employing one of the oxygen atoms as the test particle exposed two conical intersections (ci) located on each side of the CO diatom. The main purpose of this study is to explore the possibility of forming reliable diabatic potential energy surfaces for this system. Success in achieving this goal is guaranteed by the ability to calculate quantized privileged ADT angles along closed contours covering large regions in configuration space (see, e.g., J. Phys. Chem. A 2014 , 118 , 6361 ). The calculations were carried out for two and three JT states. In most cases very nice quantization has been achieved although the calculations were frequently done, as required, for large regions in configuration space (sometimes ≥18 Å(2)). In one case, for which the quantization was not gratifying, the inclusion of the RT effect modified it considerably.

Jahn–Teller and coupled Jahn–Teller/Renner–Teller effects in the calculation of adiabatic-to-diabatic transformation angle for the lowest three 2A′ states of NH2 (NHH)

We report here on the first study of topological effects for the NHH system, as carried out by treating simultaneously the two dominant effects of this system, namely, the Jahn–Teller (JT) effect and the Renner–Teller (RT) effect. Both the effects were treated rigorously as demanded by the Born–Oppenheimer approach. No approximations were made and in those cases where convergence was required, it was achieved by including the required number of states. The study concentrates on calculating the privileged adiabatic-to-diabatic transformation (ADT) angle γ12, along closed contours, which is the only needed angle to carry out the ADT in the case of relatively low energies. For this purpose, three coupled A′-states are usually considered and only in the last two extreme cases, where the area in configuration space becomes relatively large, namely 15–35 Å2, we had also to include an A″ state (the second Δ-state), a situation that enforces the more elaborate (JT/RT) effect. In this paper, we also report on potential energies as calculated along the above-mentioned contours. Among them are considered the energies associated with the two adiabatic Δ-states, 1A′ and 1A″, of different symmetry and therefore are responsible for the RT effect. These states are expected to be degenerate along the collinear axis. It was revealed that these states, in contrast to the Renner theory, are not degenerate along a finite interval of the collinear axis at the vicinity of the JT conical intersection (JT-CI). In other words, the JT-CI annihilates the RT degeneracy along this interval.

The excited states of K3 cluster: The molecular symmetry adapted non-adiabatic coupling terms and diabatic Hamiltonian matrix

We calculate the adiabatic potential energy surfaces (PESs) and the non-adiabatic coupling terms (NACTs) for the excited electronic states of K3 cluster by MRCI approach using MOLPRO. The NACTs are adapted with molecular symmetry to assign appropriate IREPs so that the elements of the Hamiltonian matrix are totally symmetric. We incorporate those NACTs into three-state adiabatic-to-diabatic transformation (ADT) equations to obtain ADT angles for constructing continuous, single-valued, smooth and symmetric diabatic Hamiltonian matrix, where its elements are fitted with analytic functions. Finally, we demonstrate that the dressed diabatic and adiabatic-via-dressed diabatic PECs show prominent topological effect over dressed adiabatic curves.

Construction of Diabatic Hamiltonian Matrix from ab Initio Calculated Molecular Symmetry Adapted Nonadiabatic Coupling Terms and Nuclear Dynamics for the Excited States of Na3 Cluster

We present the molecular symmetry (MS) adapted treatment of nonadiabatic coupling terms (NACTs) for the excited electronic states (2(2)E' and 1(2)A1') of Na3 cluster, where the adiabatic potential energy surfaces (PESs) and the NACTs are calculated at the MRCI level by using an ab initio quantum chemistry package (MOLPRO). The signs of the NACTs at each point of the configuration space (CS) are determined by employing appropriate irreducible representations (IREPs) arising due to MS group, and such terms are incorporated into the adiabatic to diabatic transformation (ADT) equations to obtain the ADT angles. Since those sign corrected NACTs and the corresponding ADT angles demonstrate the validity of curl condition for the existence of three-state (2(2)E' and 1(2)A1') sub-Hilbert space, it becomes possible to construct the continuous, single-valued, symmetric, and smooth 3 × 3 diabatic Hamiltonian matrix. Finally, nuclear dynamics has been carried out on such diabatic surfaces to explore whether our MS-based treatment of diabatization can reproduce the pattern of the experimental spectrum for system B of Na3 cluster.

The molecular symmetry adapted non - adiabatic coupling terms and diabatic Hamiltonian matrix

We calculate the adiabatic Potential Energy Surfaces (PESs) and the Non - Adiabatic Coupling Terms (NACTs) for the excited electronic states (22 E' and 12 A'1) of Na3 cluster at the MRCI level by using ab initio quantum chemistry package (MOLPRO), where the NACTs are adapted with Molecular Symmetry (MS) by employing appropriate Irreducible Representations (IREPs). Such terms are incorporated into the Adiabatic to Diabatic Transformation (ADT) equations to obtain the ADT angles to construct the continuous, single - valued, symmetric and smooth 3 × 3 diabatic Hamiltonian matrix.

Conical intersections between [formula omitted] and [formula omitted] electronic states of [formula omitted

We explore both the general symmetry-allowed accidental (SAA) conical intersections (CIs) and Renner–Teller (RT) interactions prevailed between the ground (X2A1) and first excited (A2B2) electronic states of NO2 in the configuration space of normal mode coordinates. Global ab initio potential-energy surfaces and Non Adiabatic Coupling Term (NACT) between those states are reported. For each of the three pairs of normal mode (Q1,Q2,Q3), calculated NACT show singularity at different positions, which are used to calculate Adiabatic-to-Diabatic Transformation (ADT) angles and thereby, the diagonal elements of ADT matrix display (i) sign change for odd number of SAA CI (s) and (ii) no sign change for even number of SAA CIs as well as Renner–Teller interactions. Similar to Jahn–Teller CI, the existence of SAA CIs are, further, confirmed by Longuet-Higgins’ phase change.

Dressed Adiabatic and Diabatic Potentials for the Renner–Teller/Jahn–Teller F + H2System

We follow a suggestion by Lipoff and Herschbach (Mol. Phys. 2010, 108, 1133) and compare dressed potentials to get insight regarding the low-energy dynamics (e.g., cold reaction) taking place in molecular systems. In this particular case we are interested in studying the effect of topological effects on the interacting atoms. For this purpose we consider dressed adiabatic and adiabatic-via-dressed diabatic potentials in the entrance channel of reactive systems. In a recent study of this kind for the F + H2 system (J. Chem. Phys. 2012, 136, 054104), we revealed that a single Jahn-Teller conical intersection is expected to have only a mild effect on the dynamics. This fact implies that the Born-Oppenheimer approximation is expected to be valid for this system at least for low enough energies. In the present article this study is extended to include also the Renner-Teller effect as produced by the two lower degenerate Π states. As a result we consider three electronic states which enforce the use of the adiabatic-to-diabatic transformation (ADT) matrix A. The results indicate that the topological effects as produced by the extended Renner/Teller-Jahn/Teller system are strong to the level that, most likely, abolishes the Born-Oppenheimer approximation for this system, all this in contrast to our previous findings (see above publication).