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We employ theoretically "exact" and numerically "accurate" Beyond Born–Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation ( C 6 H 6 + ) 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 ( X ̃ 2 E 1 g , B ̃ 2 E 2 g , and C ̃ 2 A 2 u ) 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 X ̃ 2 E 1 g and B ̃ 2 E 2 g − C ̃ 2 A 2 u 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. [ABSTRACT FROM AUTHOR]

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- C 6 H 3 F 3 + (TFBZ+). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states ( X ̃ 2 E ′ ′ , Ã 2 A 2 ′ ′ , B ̃ 2 E ′ , and C ̃ 2 A 2 ′ ), 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. [ABSTRACT FROM AUTHOR]

In this article, Beyond Born–Oppenheimer (BBO) treatment is implemented to construct diabatic potential energy surfaces (PESs) of 1,3,5-C6H3F3+ over a series [eighteen (18)] of two-dimensional (2D) nuclear planes constituted with eleven normal modes (Q2, Q9x, Q9y, Q13x, Q13y, Q18x, Q18y, Q10x, Q10y, Q12x and Q12y) to include all possible nonadiabatic interactions among six coupled electronic states (X~;2E′′, , B~;2E′ and). We had formulated explicit expressions of adiabatic to diabatic transformation (ADT) equations [S. Mukherjee, J. Dutta, B. Mukherjee, S. Sardar and S. Adhikari, J. Chem. Phys., 2019, 150, 064308] for the same system forming six state sub-Hilbert space and at present, these ADT equations are solved by incorporating MRCI level ab initio adiabatic PESs and CP-MCSCF calculated nonadiabatic coupling terms (NACTs) to derive diabatic PESs and couplings. Such single-valued, smooth, symmetric and continuous diabatic surface matrices are utilized to carry out multi-state multi-mode nuclear dynamics with the help of time-dependent discrete variable representation (TDDVR) methodology to compute the photoelectron (PE) spectra of 1,3,5-C6H3F3. Our theoretically calculated spectra for X~;2E′′, and states using BBO treatment and TDDVR dynamics show peak by peak correspondence with the experimental results as well as better than the findings of the multi-configuration time-dependent Hartree (MCTDH) method. [ABSTRACT FROM AUTHOR]

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 (X²A1) and the first excited state (A²B2), 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 photoelectron 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 (X²A1 and A²B2) of NO2 is essentially through the asymmetric stretching mode, where the functional form of such interaction is distinctly symmetric and non-linear. [ABSTRACT FROM AUTHOR]

When a set of three states is coupled with each other but shows negligibly weak interaction with other states of the Hilbert space, these states form a sub-Hilbert space. In case of such subspace [J. Chem. Phys. 124, 074101 (2006)], (a) the adiabatic-diabatic transformation (ADT) condition, ∇vector A+τvector A=0 [Chem. Phys. Lett. 35, 112 (1975)], provides the explicit forms of the nonadiabatic coupling (NAC) elements in terms of electronic basis function angles, namely, the ADT angles, and (b) those NAC terms satisfy the so-called curl conditions [Chem. Phys. Lett. 35, 112 (1975)], which ensure the removal of the NAC elements [could be singular also at specific point(s) or along a seam in the configuration space] during the ADT to bring the diabatic representation of the nuclear Schrödinger equation with a smooth functional form of coupling elements among the electronic states. Since the diabatic to adiabatic representation of the Hamiltonian is related through the same unitary transformation (∇A+τA=0), it could be quite interesting to explore the nature of the nonadiabatic coupling terms starting from a diabatic Hamiltonian and, thereafter, to formulate the extended Born–Oppenheimer (EBO) equation for those adiabatic states transformed from diabatic ones. We consider a three-state diabatic potential matrix constructed for the excited states of Na3 cluster [J. Chem. Phys. 88, 6068 (1988)] at the pseudo-Jahn–Teller model situation, which can reproduce experimentally measured vibrationally resolved absorption lines [Surf. Sci. 156, 770 (1985)] with appropriate choice of coupling parameters, analytically calculate the nonadiabatic coupling elements along with their curls, and numerically evaluate the ADT angles to explore the nature of its nonadiabaticity. While formulating the single surface beyond the BO equation, our theoretical derivation demonstrates that the existence of zero curls of the NAC terms is a necessity. Indeed, when the energy gap between the third state (12 A1′/22 A1′) and the doubly degenerate states (22 E′/32 E′) of the model Hamiltonian for Na3 cluster is considered to be either identically or approximately zero, the curl for each NAC element naturally approaches zero, leading to a theoretically valid EBO equation. We demonstrate the numerical validity of the EBO equation by calculating the nonadiabatic effects on the photoabsorption spectrum starting with the initial wave function located on the ground electronic state and compare with the corresponding diabatic spectrum when the three states are either degenerate at a point or approaching to form three-state degeneracy at the same point. Finally, we calculate the vibrational eigenspectrum of the ground adiabatic state by using (so to say) theoretically and numerically valid EBO equation to compare with those experimentally measured and BO/geometric phase calculated spectra (Tables I-III). [ABSTRACT FROM AUTHOR]

We demonstrate the workability of a parallelized algorithm of the time-dependent discrete variable representation (TDDVR) method to explore the detailed dynamical aspects of vibronic interaction in two three-state model Hamiltonians (X 2E1g, B 2E2g, C 2A2u and B 2E2g, D 2E1u, E 2B2u) of benzene radical cation along with a preliminary investigation on its five electronic states (X 2E1g, B 2E2g, C 2A2u, D 2E1u, and E2B2u). Since those electronic states are interconnected through a series of conical intersections, we have used six and nine vibronically important modes for the three- and five-state Hamiltonians, respectively, in order to perform the quantum dynamics on such system. The population profiles calculated by using our TDDVR approach show reasonably good agreement with the results obtained by exact quantum mechanical (multiconfiguration time-dependent Hartree) method, whereas the corresponding (calculated) photoabsorption spectra originating from various electronic states agree well with the experimental ones. It is important to note that the parallelized algorithm of our TDDVR approach reduces the computation cost by more than an order of magnitude compared to its serial analog. The TDDVR approach appears to be a good compromise between accuracy and speed for such large molecular system, where quantum mechanical description is needed in a restricted region. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Validity of curl τ ̃ ≃ 0 ̃ at the close vicinity of conical intersection points is explored. • The magnitudes of curl arising from Jahn-Teller interactions are either zero or negligible. • The other curls due to pseudo Jahn-Teller couplings attain reasonably higher magnitude. • The eigenvalues of nonadiabatic coupling matrix are found to be gauge invariant. Abstract Formulation of single-surface extended Born-Oppenheimer (EBO) equation is a long-standing problem for three or higher dimensional sub-Hilbert space mainly due to the difficulty to include the contribution of off-diagonal elements of anti-symmetric nonadiabatic coupling matrix ( τ → ) as the diagonal ones. Diagonalization of a vector matrix is possible only when its components commute with each other (i.e. curl τ → = 0 → ) and thereby, that vector matrix ( τ → ) can be expressed as a product of a vector function and an antisymmetric scalar matrix. For two electronic state sub-Hilbert space (Abelian case), since curl τ → = 0 → naturally, such factorization is readily obtained. In non-Abelian cases, for systems with three or more than three sub-Hilbert space, such factorization of nonadiabatic coupling matrix (NACM) may be achievable approximately (Sarkar and Adhikari, 2006; Sarkar and Adhikari, 2008), which will be explored further both theoretically and numerically. In this article, we take an attempt to generalize the single-surface EBO equations for higher dimensional sub-Hilbert space by selecting two realistic molecular systems containing Jahn-Teller (JT) conical intersections (CIs). The curl equations and gauge invariance (GI) of the eigenvalues of the NACM for NO 3 radical and 1,3,5-C 6 H 3 F 3 + (TFBZ+) radical cation are computed numerically, which clearly indicates the possibility to formulate single-surface EBO equations. [ABSTRACT FROM AUTHOR]

In order to explore spectroscopic properties of a floppy molecule, it is necessary to compute accurate diabatic potential energy surfaces (PESs) around the Franck–Condon region of nuclear configuration space. In such cases, it is desirable to include the non-adiabatic coupling terms (NACTs) through Jahn–Teller (JT) and pseudo Jahn–Teller (PJT) effects as well as other accidental conical intersections (CIs) prevailing among the electronic states of the molecule. In this work, we investigate NACTs among the five lowest doublet electronic states of NO3radical [and 22B2)], in which all those effects (JT, PJT and CIs) have prominent contributions. The study is carried out in terms of normal modes (Qis) of NO3and the CIs (JT and accidental) are located in a plane of nuclear configuration space of two such coordinates (QiandQj). We have calculatedab initioadiabatic PESs and NACTs of those five states of NO3at the MRCI level as functions of pairwiseQis. Further, to explore the nature of a given CI in terms of normal coordinates contributing to NACTs within a particular planeQi-Qj, we have performed CP-MCSCF study computing analytic NACTs and characterised their contributions along variousQks. [ABSTRACT FROM PUBLISHER]

The major portion of the algorithmof the time-dependent discretevariable representation (TDDVR) method is recently parallelized usingthe shared-memory parallelization scheme with the aim of performingdynamics on relatively large molecular systems. Because of the astronomicalimportance of naphthalene and anthracene, we have investigated theirradical cations as models for theoretical simulation of complex photoelectronspectra and nonradiative decay process using the newly implementedparallel TDDVR code. The strong vibronic coupling among the six lowestdoublet electronic states makes these polynuclear hydrocarbons dynamicallyimportant. The aim of the present investigation is to show the efficiencyof our current TDDVR algorithm to perform dynamics on large dimensionalquantum systems in vibronically coupled electronic manifold. Boththe sequential and the parallelized TDDVR algorithms are almost linearscalable for an increase in number of processors. Because a significantspeed-up is achieved by cycling in the correct way over arrays, allof the simulations are performed within a reasonable wall clock time.Our theoretical spectra well reproduce the features of the correspondingexperimental analog. The dynamical outcomes, for example, population,photoelectron spectra, and diffused interstellar bands, etc., of ourquantum-classical approach show good agreement with the findings ofthe well-established quantum dynamical method, that is, multi configurationtime-dependent Hartree (MCTDH) approach. [ABSTRACT FROM AUTHOR]

We explore the workability of a parallelized algorithm of time‐dependent discrete variable representation (TDDVR) methodology formulated by involving "classical" trajectories on each DOF of a multi‐mode multi‐state Hamiltonian to reproduce the population dynamics, photoabsorption spectra and nuclear dynamics of the benzene radical cation. To perform such dynamics, we have used a realistic model Hamiltonian consists of five lowest electronic states (X2E1g, B2E2g, C2A2u, D2E1u, and E2B2u) which are interconnected through several conical intersections with nine vibrational modes. The calculated nuclear dynamics and photoabsorption spectra with the advent of our parallelized TDDVR approach show excellent agreement with the results obtained by multiconfiguration time‐dependent Hartree method and experimental findings, respectively. The major focus of this article is to demonstrate how the "classical" trajectories for the different modes and the "classical" energy functional for those modes on each surface can enlight the time‐dependent feature of nuclear density and its' nodal structure. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011 [ABSTRACT FROM AUTHOR]

We have performed molecular dynamics on the three isomers of the difluorobenzene radical cation (C6H4F+2) after excitation from the ground state to a specific higher electronically excited state by using our recently implemented parallelized Time-Dependent Discrete Variable Representation (TDDVR) methodology. A five-state eleven-mode realistic model Hamiltonian for o-C6H4F+2and two separate five-state ten-mode Hamiltonians for m- and p-isomer of the same radical cation are considered, where those five electronic states are interconnected through several conical intersections in the vicinity of the Franck–Condon (FC) region and thus the dynamics for each case become complex. The photoelectron, mass analyzed threshold ionization spectra and population profiles obtained by using our TDDVR approach show reasonably good agreement with the results obtained by multiconfiguration time dependent Hartree (MCTDH) method. It is worthwhile to mention that the parallelized TDDVR algorithm reduces the computation time by more than an order of magnitude compared to its serial analog and, therefore, such approach appears to be a good compromise between accuracy and speed for a large molecular system. [ABSTRACT FROM AUTHOR]

We are investigating the molecular dynamics of the allene system using a parallelised Time Dependent Discrete Variable Representation (TDDVR) methodology by employing a corresponding cation ( [image omitted]) where its two electronic surfaces (A2E and B2B2) are vibronically coupled with each other. In fact the allene radical cation exhibits a three-surface system due to the presence of degeneracy in the A2E state. Our initial investigation is carried out on a linear vibronically coupled model Hamiltonian consisting of 11 vibrationally active modes with two potential energy surfaces. We included both the bilinear and quadratic coupled terms in the effective three-surface Hamiltonian of the [image omitted] radical cation. The spectral profiles obtained from the higher order Hamiltonian show better agreement with the experimental spectrum than the result corresponding to the linearly coupled one. Along with the spectral calculation, we also analyse the nuclear cum population dynamics of the same system. The TDDVR calculated spectral profile as well as the population dynamics show reasonably good agreement with the results calculated by an exact quantum mechanical (multiconfiguration time-dependent Hartree, MCTDH) approach as well as experimental measurement. It appears that the TDDVR approach for those large systems where the quantum mechanical description is needed in a restricted region, is a good compromise between accuracy and speed. [ABSTRACT FROM AUTHOR]

We recently implemented our parallelized quantum-classical dynamical approach, known as the Time-Dependent Discrete Variable Representation (TDDVR) method, which is applied to the spectroscopically important hexafluorobenzene (HFBz) radical cation, where several conical intersections exist in their seven lowest excited electronic states (S 1 1 B 2u , S 2 1 E 1g , S 3 1 B 1u , S 4 1 E 1u , and S 5 1 A 2u ) considering degeneracy among potential energy surfaces (PESs), to demonstrate their various dynamical aspects. This new parallel version shows almost linear scalability with increasing number of computing processors. To get photoelectron (PE) spectra, Mass-Analyzed Threshold Ionization (MATI) spectra, population dynamics, and many other dynamical observables, the first-principles dynamics is applied at the state-of-the-art level to the corresponding Hamiltonian, where the Jahn-Teller (JT) and pseudo-Jahn-Teller (PJT) type interactions are involved in those coupled seven electronic states. The quantum-classical method is used to thoroughly analyze the effects of these couplings on the nuclear dynamics of the involved electronic states, and the findings are compared with those observables obtained from experiments. Intrinsic dynamical properties are explained using the reduced densities of the wave packet (WP) in a coupled electronic manifold. The PE and MATI spectra of HFBz computed using TDDVR are found to be in good agreement with earlier experimental data and other theoretically simulated spectra.

In this article, Beyond Born-Oppenheimer (BBO) treatment is implemented to construct diabatic potential energy surfaces (PESs) of 1,3,5-C 6 H 3 F 3 + over a series [eighteen (18)] of two-dimensional (2D) nuclear planes constituted with eleven normal modes ( Q 2 , Q 9 x , Q 9 y , Q 13 x , Q 13 y , Q 18 x E' and ). We had formulated explicit expressions of adiabatic to diabatic transformation (ADT) equations [S. Mukherjee, J. Dutta, B. Mukherjee, S. Sardar and S. Adhikari, Q 18 y , Q 10 x , Q 10 y , Q 12 x and Q 12 y ) to include all possible nonadiabatic interactions among six coupled electronic states (X̃ 2 E'', , B̃ 2 E' and ). We had formulated explicit expressions of adiabatic to diabatic transformation (ADT) equations [S. Mukherjee, J. Dutta, B. Mukherjee, S. Sardar and S. Adhikari, J. Chem. Phys. , 2019, 150 , 064308] for the same system forming six state sub-Hilbert space and at present, these ADT equations are solved by incorporating MRCI level ab initio adiabatic PESs and CP-MCSCF calculated nonadiabatic coupling terms (NACTs) to derive diabatic PESs and couplings. Such single-valued, smooth, symmetric and continuous diabatic surface matrices are utilized to carry out multi-state multi-mode nuclear dynamics with the help of time-dependent discrete variable representation (TDDVR) methodology to compute the photoelectron (PE) spectra of 1,3,5-C 6 H 3 F 3 . Our theoretically calculated spectra for X̃ 2 E'', and states using BBO treatment and TDDVR dynamics show peak by peak correspondence with the experimental results as well as better than the findings of the multi-configuration time-dependent Hartree (MCTDH) method.

We employ theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) treatment to construct diabatic potential energy surfaces (PESs) of the benzene radical cation (C 6 H 6 + ) 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 (X̃ 2 E 1g , B̃ 2 E 2g , and C̃ 2 A 2u ) 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 X̃ 2 E 1g and B̃ 2 E 2g -C̃ 2 A 2u 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.

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-C 6 H 3 F 3 + (TFBZ + ). While implementing this formulation, we compute ab initio adiabatic potential energy surfaces (PESs) and NACTs within the low-lying six electronic states (X̃ 2 E '' , Ã 2 A 2 '' , B̃ 2 E ' , and C̃ 2 A 2 ' ), 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 (Q 9x , Q 9y , Q 10x , Q 10y , Q 12x , and Q 12y ) 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.