Nonadiabatic Coupling Terms Research Articles

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.

Analytic non-adiabatic derivative coupling terms for spin-orbit MRCI wavefunctions. II. Derivative coupling terms and coupling angle for KHeA2Π1/2⇔KHeB2Σ1/2.

A method for calculating the analytic nonadiabatic derivative coupling terms (DCTs) for spin-orbit multi-reference configuration interaction wavefunctions is reviewed. The results of a sample calculation using a Stuttgart basis for KHe are presented. Additionally, the DCTs are compared with a simple calculation based on the Nikitin's 3 × 3 description of the coupling between the Σ and Π surfaces, as well as a method based on Werner's analysis of configuration interaction coefficients. The nonadiabatic coupling angle calculated by integrating the radial analytic DCTs using these different techniques matches extremely well. The resultant nonadiabatic energy surfaces for KHe are presented.

Analytic non-adiabatic derivative coupling terms for spin-orbit MRCI wavefunctions. I. Formalism.

Analytic gradients of electronic eigenvalues require one calculation per nuclear geometry, compared to at least 3n + 1 calculations for finite difference methods, where n is the number of nuclei. Analytic nonadiabatic derivative coupling terms (DCTs), which are calculated in a similar fashion, are used to remove nondiagonal contributions to the kinetic energy operator, leading to more accurate nuclear dynamics calculations than those that employ the Born-Oppenheimer approximation, i.e., that assume off-diagonal contributions are zero. The current methods and underpinnings for calculating both of these quantities, gradients and DCTs, for the State-Averaged MultiReference Configuration Interaction with Singles and Doubles (MRCI-SD) wavefunctions in COLUMBUS are reviewed. Before this work, these methods were not available for wavefunctions of a relativistic MRCI-SD Hamiltonian. Calculation of these terms is critical in successfully modeling the dynamics of systems that depend on transitions between potential energy surfaces split by the spin-orbit operator, such as diode-pumped alkali lasers. A formalism for calculating the transition density matrices and analytic derivative coupling terms for such systems is presented.

Non-adiabatic coupling and conical intersection(s) between potential energy surfaces for HeH2 +

By computing the non-adiabatic coupling terms and the adiabatic-to-diabatic transformation angle along closed contours in nuclear configuration space using the CASSCF method and the aug-cc-pVTZ basis set for the lowest three electronic states of HeH2 +, we explore the conical intersection between states in near collinear and noncollinear geometries, particularly in the C 2v geometry.

Topological studies related to molecular systems formed during the Big Bang: H3+ as an example

AbstractIn the present article are analyzed the non‐adiabatic coupling terms (NACT) for two molecular systems, namely H3+ and H3. In contrast to previous occasions in which the NACTs are studied along (closed) circular contours usually surrounding conical intersections (ci), in the present article are studied distribution of the NACTs in (planar) configuration spaces (CS). The motivation for this study has to do with a novel idea being mentioned earlier (Molec. Phys., 116, 2435 [2018]; ArXiv:1801.00103) that NACTs are like a Glue (eventually) associated with the ability of creating molecules and/or protecting them from breaking up. It was found that the distributions of the NACTs due to the two molecules are similar as long as the attention is given to regions close to their equilateral cis, but then they behave significantly different in other regions. In case of H3+, the NACTs are distributed rather uniformly whereas, in case of H3 they become spiky the closer they approach the diatom axis. The main conclusion of this study is that the glue which has its origin in the NACTs is most likely to be effective in case of H3+ that explains the creation and later survival of this molecule.

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.

Fast Overlap Evaluations for Nonadiabatic Molecular Dynamics Simulations: Applications to SF-TDDFT and TDDFT.

Fast overlap integral algorithms for the spin-flip time-dependent density functional theory (SF-TDDFT) and the linear response (LR)-TDDFT were proposed on the basis of determinant factorization (DF) and the truncated Leibnitz formula (TLF). These in turn allow efficient computation of nonadiabatic coupling terms (NACTs) in nonadiabatic molecular dynamics simulations. The TLF(0), TLF(1), and TLF(2) were proposed according to the truncation order. The DF and TLF(1) or TLF(2) provide a four order combined performance improvement to the conventional method without introducing additional errors in the finite difference approximation. On the other hand, the DF and TLF(0) provide a five orders performance improvement making it the most efficient algorithm for NACT calculations so far with errors slightly larger than those of the finite difference approximation. The same techniques can be also applicable to other determinantal wave functions.

Beyond Born-Oppenheimer treatment on spectroscopic and scattering processes

We review the formulation of beyond Born-Oppenheimer (BBO) theory based on first principle for the construction of diabatic potential energy surfaces (PESs) both for few important spectroscopic systems, viz., Na3 cluster, NO2 radical as well as scattering process like D+ + H2. The essential theoretical development leading to the BBO equations are thoroughly discussed. It has been found that the above molecular systems posses numerous nonadiabatic interactions that range from Jahn-Teller, Renner-Teller types of conical intersections along with strong pseudo Jahn-Teller couplings between various electronic states. We have calculated the adiabatic PESs and nonadiabatic coupling terms for those systems and subsequently performed adiabatic-to-diabatic transformation to construct smooth, symmetric and continuous diabatic potential energy matrix. Nuclear dynamics has been performed on the diabatic PESs of Na3 and NO2 to simulate the photoelectron spectra that match quite well with the experimentally measured ones. Moreover, we have carried out reactive scattering dynamics on the adiabatic and diabatic surfaces of system to reproduce experimental cross sections for reactive charge and non-charge transfer processes.

Ab initio surface hopping excited-state molecular dynamics approach on the basis of spin-orbit coupled states: An application to the A-band photodissociation of CH3 I.

Ab initio molecular dynamics approach has been extended to multi-state dynamics on the basis of the spin-orbit coupled electronic states that are obtained through diagonalization of the spin-orbit coupling matrix with the multi-state second-order multireference perturbation theory energies in diagonal elements and the spin-orbit coupling terms at the state-averaged complete active space self-consistent field level in off-diagonal elements. Nonadiabatic transitions over the spin-orbit coupled states were taken into account explicitly by a surface hopping scheme with utilizing the nonadiabatic coupling terms calculated by numerical differentiation of the spin-orbit coupled wavefunctions and analytical nonadiabatic coupling terms. The present method was applied to the A-band photodissociation of methyl iodide, CH3 I + hv → CH3 + I (2 P3/2 )/I* (2 P1/2 ), for which a pioneering theoretical work was reported by Amatatsu, Yabushita, and Morokuma. The present results reproduced well the experimental branching ratio and energy distributions in the dissociative products. © 2018 Wiley Periodicals, Inc.

Topological Effects in Vibronically Coupled Degenerate Electronic States: A Case Study on Nitrate and Benzene Radical Cation.

We carry out detailed investigation for topological effects of two molecular systems, NO3 radical and C6H6+ (Bz+) radical cation, where the dressed adiabatic, dressed diabatic, and adiabatic-via-dressed diabatic potential energy curves (PECs) are generated employing ab initio calculated adiabatic and diabatic potential energy surfaces (PESs). We have implemented beyond Born–Oppenheimer (BBO) theory for constructing smooth, single-valued, and continuous diabatic PESs for five coupled electronic states [J. Phys. Chem. A2017,121, 6314–6326]. In the case of NO3 radical, the nonadiabatic coupling terms (NACTs) among the low-lying five electronic states, namely, X̃2A2′ (12B2), A~2E″ (12A2 and 12B1), and B~2E′ (12A1 and 22B2), bear the signature of Jahn–Teller (JT) interactions, pseudo JT (PJT) interactions, and accidental conical intersections (CIs). Similarly, Bz+ radical cation also exhibits JT, PJT, and accidental CIs in the interested domain of nuclear configuration space. In order to generate dressed PECs, two components of degenerate in-plane asymmetric stretching modes are selectively chosen for both the molecular species (Q3x–Q3y pair for NO3 radical and Q16x–Q16y pair for Bz+ radical cation). The JT coupling between the electronic states is essentially originated through the asymmetric stretching normal mode pair, where the coupling elements exhibit symmetric and nonlinear functional behavior along Q3x and Q16x normal modes.

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.

Non-adiabatic coupling in the ozone molecule

ABSTRACTThe question whether a Berry phase should be included in the calculation of rovibronic states of the ozone molecule in its electronic ground state has been addressed in the present work. Since several conical intersections connect the three lowest singlet states, a phase of π generated by a symmetry-demanded conical intersection between states 2 and 3 might be present, or it might be compensated by further conical intersections. Hyperspherical coordinates were employed here as they provide a natural choice for the cyclic coordinate that is needed to investigate the presence or absence of such a topological phase. On a hyperspherical grid, we have computed the electronic energies of the three lowest singlet states using a multi-reference configuration interaction procedure. Non-adiabatic coupling terms (NACTs) along the cyclic variable were evaluated numerically. Investigation of the behaviour of the adiabatic-to-diabatic transformation angles, using the quantisation criterion of the NACTs [M. Baer and A. Alijah, Chem. Phys. Lett. 319, 489 (2000)], shows that a geometrical phase is retained. This phase has no practical effect on the well-established energy values of the rovibrational states located deeply in the three equivalent potential wells, but will come into action as highly excited states reach the dissociation region.

Propagation of nonstationary electronic and nuclear states: attosecond dynamics in LiF

ABSTRACTRapid optical excitation of a molecule produces a nonstationary state localised in the Franck–Condon region. To move out of that region, one needs to propagate both the electronic and the nuclear state. We formulate the motion on a grid of nuclear coordinate. The coupling to the electric field is fully included in the Hamiltonian used for propagation. We use perturbation theory to analyse the results of dynamics from one grid point to another. The nonadiabatic coupling terms arise from propagating the electronic states. We apply the formalism to the simple case of a diatomic molecule in an approximate but accurate scheme that allows performing computation on a limited number of grid points. As the coherent dynamics unfolds, we expand the grid in the direction of the wave packet motion with the quantum chemical calculations of the electronic structure performed ‘on the fly’. The LiF molecule excited by a one-cycle IR pulse is used as a computational example. The 30-fs propagation through the crossing of the ionic and covalent states is overall adiabatic. The role of electron–nuclear coherences is emphasised.

Topological study of the H3++ molecular system: H3++ as a cornerstone for building molecules during the Big Bang

ABSTRACTThe present study is devoted to the possibility that tri-atomic molecules were formed during or shortly after the Big Bang. For this purpose, we consider the ordinary H3+ and H3 molecular systems and the primitive tri-atomic molecular system, H3++, which, as is shown, behaves differently. The study is carried out by comparing the topological features of these systems as they are reflected through their non-adiabatic coupling terms. Although H3++ is not known to exist as a molecule, we found that it behaves as such at intermediate distances. However, this illusion breaks down as its asymptotic region is reached. Our study indicates that whereas H3+ and H3 dissociate smoothly, the H3++ does not seem to do so. Nevertheless, the fact that H3++ is capable of living as a molecule on borrowed time enables it to catch an electron and form a molecule via the reaction H3++ + e → H3+ that may dissociate properly:Thus, the two unique features acquired by H3++, namely, that it is the most primitive system formed by three protons and one electron and topologically, still remain for an instant a molecule, may make it the sole candidate for becoming the cornerstone for creating the molecules.

Surface Hopping Dynamics beyond Nonadiabatic Couplings for Quantum Coherence.

Description of correct electron-nuclear couplings is crucial in modeling of nonadiabatic dynamics. Within traditional semiclassical or mixed quantum-classical dynamics, the coupling between quantum electronic states and classical nuclear trajectories is governed by nonadiabatic coupling vectors coupled to classical nuclear momenta. This enables us to develop a very powerful nonadiabatic dynamics algorithm, namely, surface hopping dynamics, which can describe the splitting of nuclear wave packets and detailed balance. Despite its efficiency and practicality, it suffers from the lack of quantum decoherence due to incorrect accounts for the electron-nuclear coupling. Here we present a new surface hopping algorithm based on the exact electron-nuclear correlation from the exact factorization of molecular wave functions. This algorithm demands comparable computational costs to existing surface hopping methods. Numerical simulations with two-state models and a multidimensional multistate realistic molecule show that the electron-nuclear coupling beyond the nonadiabatic coupling terms can describe the quantum coherence properly.

Practical and reliable approximation of nonadiabatic coupling terms between triplet electronic states using only adiabatic potential energies

The nonadiabatic coupling terms (NACTs, Fij’s) between triplet electronic states can be well approximated with a very simple equation suggested recently [J. Chem. Phys. 146, 064107 (2017)], using only adiabatic potential energies (Via and Vja). Furthermore, the second-order differentiation can be evaluated accurately from the energy gaps, (Via-Vja), at just three consecutive geometries, even at a point exhibiting severe singularity in the behaviour of NACTs. The performance of this approximation method was investigated for the couplings between the 33Π and 43Π states of NaK, between the 13B3u and 23B3u states of N2, and between the 23A″ and 33A″ states of methylene (CH2).

Introducing time-dependent molecular fields: a new derivation of the wave equations

This article is part of a series of articles trying to establish the concept Molecular Field. The theory that induced us to introduce this novel concept is based on the Born-Huang expansion as applied to the Schroedinger equation that describes the interaction of a molecular system with an external electric field. Assuming the molecular system is made up of two coupled adiabatic states the theory leads from a single spatial Curl Equation, two space-time Curl equations and one single space-time Divergent equation to a pair of decoupled Wave Equations usually encountered within the theory of fields. In the present study the Wave Equations are derived for an external field having two features: (a) its intensity is high enough; (b) its duration is short enough. For this situation the study reveals that the just described interaction creates two fields that coexist within a molecule: one is a novel vectorial field formed via the interaction of the external field with the Born-Huang non-adiabatic coupling terms (NACT) and the other is an ordinary, scalar, electric field essentially identical to the original external field. Part of the present study is devoted to the visualization of the outcomes via two intersecting Jahn-Teller cones which contain NACTs that become singular at their intersection point, also known as the point of Conical Intersection. This description is reminiscent of a metaphoric cosmic Black-Hole which swallows one type of an electric field and creates a new type. Finally, the fact that eventually we are facing a kind of a cosmic situation may bring us to speculate that singular NACTs are a result of cosmic phenomena. Thus, if indeed this singularity is somehow connected to reality then, like other singularities in physics, it is formed at (or immediately after) the Big Bang (and because of that, eventually the ability to form molecules).

Beyond Born-Oppenheimer theory for ab initio constructed diabatic potential energy surfaces of singlet H3+ to study reaction dynamics using coupled 3D time-dependent wave-packet approach.

The workability of beyond Born-Oppenheimer theory to construct diabatic potential energy surfaces (PESs) of a charge transfer atom-diatom collision process has been explored by performing scattering calculations to extract accurate integral cross sections (ICSs) and rate constants for comparison with most recent experimental quantities. We calculate non-adiabatic coupling terms among the lowest three singlet states of H3+ system (11A', 21A', and 31A') using MRCI level of calculation and solve the adiabatic-diabatic transformation equation to formulate the diabatic Hamiltonian matrix of the same process [S. Mukherjee et al., J. Chem. Phys. 141, 204306 (2014)] for the entire region of nuclear configuration space. The nonadiabatic effects in the D+ + H2 reaction has been studied by implementing the coupled 3D time-dependent wave packet formalism in hyperspherical coordinates [S. Adhikari and A. J. C. Varandas, Comput. Phys. Commun. 184, 270 (2013)] with zero and non-zero total angular momentum (J) on such newly constructed accurate (ab initio) diabatic PESs of H3+. We have depicted the convergence profiles of reaction probabilities for the reactive non-charge transfer, non-reactive charge transfer, and reactive charge transfer processes for different collisional energies with respect to the helicity (K) and total angular momentum (J) quantum numbers. Finally, total and state-to-state ICSs are calculated as a function of collision energy for the initial rovibrational state (v = 0, j = 0) of the H2 molecule, and consequently, those quantities are compared with previous theoretical and experimental results.

On the adiabatic representation of Meyer-Miller electronic-nuclear dynamics.

The Meyer-Miller (MM) classical vibronic (electronic + nuclear) Hamiltonian for electronically non-adiabatic dynamics-as used, for example, with the recently developed symmetrical quasiclassical (SQC) windowing model-can be written in either a diabatic or an adiabatic representation of the electronic degrees of freedom, the two being a canonical transformation of each other, thus giving the same dynamics. Although most recent applications of this SQC/MM approach have been carried out in the diabatic representation-because most of the benchmark model problems that have exact quantum results available for comparison are typically defined in a diabatic representation-it will typically be much more convenient to work in the adiabatic representation, e.g., when using Born-Oppenheimer potential energy surfaces (PESs) and derivative couplings that come from electronic structure calculations. The canonical equations of motion (EOMs) (i.e., Hamilton's equations) that come from the adiabatic MM Hamiltonian, however, in addition to the common first-derivative couplings, also involve second-derivative non-adiabatic coupling terms (as does the quantum Schrödinger equation), and the latter are considerably more difficult to calculate. This paper thus revisits the adiabatic version of the MM Hamiltonian and describes a modification of the classical adiabatic EOMs that are entirely equivalent to Hamilton's equations but that do not involve the second-derivative couplings. The second-derivative coupling terms have not been neglected; they simply do not appear in these modified adiabatic EOMs. This means that SQC/MM calculations can be carried out in the adiabatic representation, without approximation, needing only the PESs and the first-derivative coupling elements. The results of example SQC/MM calculations are presented, which illustrate this point, and also the fact that simply neglecting the second-derivative couplings in Hamilton's equations (and presumably also in the Schrödinger equation) can cause very significant errors.

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.