Single surface beyond Born–Oppenheimer equation for a three-state model Hamiltonian of Na3 cluster

Photoelectron spectra of benzene: Can path dependent diabatic surfaces provide unique observables?

While carrying out Beyond Born-Oppenheimer theory based diabatization, the solutions of adiabatic-to-diabatic transformation equations depend on the paths of integration over two-dimensional cross-sections of multi-dimensional space of nuclear degrees of freedom. It is shown that such path-dependent solutions leading to diabatic potential energy surface matrices computed along any two different paths are related through an orthogonal matrix, and thereby, those surface matrices should provide unique observables. While exploring the numerical validity of the theoretical framework, we construct diabatic Hamiltonians for the five low-lying electronic states (X̃2E1g, B̃2E2g, and C̃2A2u) of benzene radical cation (C6H6+) along three different approaches of contour integration over two dimensional nuclear planes constituted by seven non-adiabatically active normal modes. Three different diabatic surface matrices are further employed to generate the photoelectron spectra of the benzene molecule (C6H6). It is interesting to note that the spectral peak positions and intensity patterns for all three cases are almost close to each other and also exhibit very good agreement with the experimental results.

Beyond Born-Oppenheimer Constructed Diabatic Potential Energy Surfaces for HeH2. J Phys Chem A 2023;127:3832-3847. [PMID: 37098130 DOI: 10.1021/acs.jpca.3c01047]

First-principles based beyond Born-Oppenheimer theory has been employed to construct multistate global Potential-Energy Surfaces (PESs) for the HeH₂⁺ system by explicitly incorporating the Nonadiabatic Coupling Terms (NACTs). Adiabatic PESs and NACTs for the lowest four electronic states (1²A', 2²A', 3²A' and 4²A') are evaluated as functions of hyperangles for a grid of fixed values of the hyperradius in hyperspherical coordinates. Conical intersection between different states are validated by integrating the NACTs along appropriately chosen contours. Subsequently, adiabatic-to-diabatic (ADT) transformation angles are determined by solving the ADT equations to construct the diabatic potential matrix for the HeH₂⁺ system which are smooth, single-valued, continuous, and symmetric and are suitable for performing accurate scattering calculations for the titled system.

Beyond Born-Oppenheimer based diabatic surfaces of 1,3,5-C6H3F3+ to generate the photoelectron spectra using time-dependent discrete variable representation approach

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, Q_9x, Q_9y, Q_13x, Q_13y, Q_18x, Q_18y, Q_10x, Q_10y, Q_12x and Q_12y) to include all possible nonadiabatic interactions among six coupled electronic states (X̃²E'', , B̃²E' 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 X̃²E'', 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.

The role of electron-nuclear coupling on multi-state photoelectron spectra, scattering processes and phase transitions

We present first principle based beyond Born-Oppenheimer (BBO) theory and its applications on various models as well as realistic spectroscopic and scattering processes, where the Jahn-Teller (JT) theory is brought in conjunction with the BBO approach on the phase transition of lanthanide complexes. Over one and half decades, our development of BBO theory is demonstrated with ab initio calculations on representative molecules of spectroscopic interest (NO2 radical, Na3 and K3 clusters, NO3 radical, C6H6+ and 1,3,5-C6H3F3+ radical cations) as well as triatomic reactive scattering processes (H+ + H2 and F + H2). Such an approach exhibits the effect of JT, Renner-Teller (RT) and pseudo Jahn-Teller (PJT) type of interactions. While implementing the BBO theory, we generate highly accurate diabatic potential energy surfaces (PESs) to carry out quantum dynamics calculation and find excellent agreement with experimental photoelectron spectra of spectroscopic systems and cross-sections/rate constants of scattering processes. On the other hand, such electron-nuclear couplings incorporated through JT theory play a crucial role in dictating higher energy satellite transitions in the dielectric function spectra of the LaMnO3 complex. Overall, this article thoroughly sketches the current perspective of the BBO approach and its connection with JT theory with various applications on physical and chemical processes.

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 C_2v 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.

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. A 2017, 121, 6314-6326]. In the case of NO₃ radical, the nonadiabatic coupling terms (NACTs) among the low-lying five electronic states, namely, X̃ ²A₂ ^' (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 (Q _3x -Q _3y pair for NO₃ radical and Q _16x -Q _16y 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 Q _3x and Q _16x normal modes.

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), òE″ (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⁺ [X̃²E_1g (1²B_3g and 1²B_2g), B̃²E_2g (1²A_g and 1²B_1g), and C̃²A_2u (1²B_1u)] 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.

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.

Jahn-Teller intersections induced by introduction of bending in linear polyatomics: study with HCNH, a selected molecular system

Demonstration of onset of Jahn-Teller (JT) intersections, characterized by a topological phase of π, on introduction of bending in the collinear tetra-atomic C(2)H(2)(+) cation, originally a Renner-Teller (RT) system, has raised interest in the study of the generality of this phenomenon. This interest has initiated similar study by shifting one external light atom from the collinear molecular axes of systems such as HCNH and HC(2)O. Recent studies have revealed that slightly bent HCNH poses a specialty in this regard and thus demands focused attention. In the present work we performed a combined study using both the potential intersections and the angular nonadiabatic coupling terms (NACTs) and report two new results: (i) The first result is the appearance of the JT conical intersections (ci's) between the two lowest states of a slightly bent HCNH molecule (1(2)A' and 1(2)A", originated from collinear X(2)Π state) only with certain nonplanar configurations, in contrast to its appearance in C(S) configuration space (molecular plane) for slightly bent C(2)H(2)(+), HC(2)O, or some other tetra-atomics. This is also associated with the first time demonstration of Berry phase for such a single isolated ci for HCNH molecule. (ii) For energetically higher potentials of slightly bent HCNH, the present study reveals the existence of the intersection of the state 1(2)A" with the state 2(2)A' (originated from the collinear 1(2)Σ(+) state); in contrast to the (1,2) ci this appears in the molecular plane. The search for ci's has been performed by varying the distances of the two H-atoms (designated as H(C) and H(N)) from the C-N axis as well as the dihedral angle φ between the two planes (H(C),C,N) and (C,N,H(N)). Existence of these JT ci's results in deterioration of a two-state Hilbert subspace (HSS) model in diabatization, while the proper choice of an n (>2)-state HSS circumvents this problem.

Space-time contours to treat intense field-dressed molecular states

In this article we consider a molecular system exposed to an intense short-pulsed external field. It is a continuation of a previous publication [A. K. Paul, S. Adhikari, D. Mukhopadhyay et al., J. Phys. Chem. A 113, 7331 (2009)] in which a theory is presented that treats quantum effects due to nonclassical photon states (known also as Fock states). Since these states became recently a subject of intense experimental efforts we thought that they can be treated properly within the existing quantum formulation of dynamical processes. This was achieved by incorporating them in the Born-Oppenheimer (BO) treatment with time-dependent coefficients. The extension of the BO treatment to include the Fock states results in a formidable enhancement in numerical efforts expressed, in particular, in a significant increase in CPU time. In the present article we discuss an approach that yields an efficient and reliable approximation with only negligible losses in accuracy. The approximation is tested in detail for the dissociation process of H(2) (+) as caused by a laser field.