Construction of Beyond Born‐Oppenheimer Based Diabatic Surfaces and Generation of Photoabsorption Spectra: The Touchstone Pyrazine (C₄N₂H₄)

Abstract

We construct theoretically "exact" and numerically "accurate" Beyond Born-Oppenheimer (BBO) based diabatic potential energy surfaces (PESs) of pyrazine (C 4 N 2 H 4 ) molecule involving lowest four excited adiabatic PESs (S 1 to S 4 ) and nonadiabatic coupling terms (NACTs) among those surfaces as functions of nonadiabatically active normal modes (Q 1 , Q 6a , Q 9a and Q 10a ) to compute its photoabsorption (PA) spectra. Those adiabatic PESs are calculated using CASSCF as well as MRCI based methodologies, where NACTs are obtained from CP-MCSCF approach. Employing ab initio quantities (adiabatic PESs and NACTs), it is possible to depict the conical intersections (CIs) and develop matrices of diabatic PESs over six normal mode planes. Once single-valued, smooth, symmetric and continuous 2 × 2 and 4 × 4 diabatic surface matrices are in hand for the first time, such matrices are used to perform multi-state multi-mode nuclear dynamics with the aid of Time-Dependent Discrete Variable Representation (TDDVR) methodology initializing the product type wavefunction on 1 1 B 1u (S 1 ) and 1 1 B 2u (S 2 ) states to obtain the corresponding PA spectra. TDDVR calculated spectra for those states (S 1 and S 2 ) obtained from BBO based 2 × 2 and 4 × 4 diabatic surface matrices show good and better agreement with the experimental results, respectively. Both of these calculated results depict better peak progression over the existing profiles of Multi-Configuration Time-Dependent Hartree (MCTDH) dynamics over 2 × 2 Vibronic Coupling Model (VCM) Hamiltonian.

Beyond Born–Oppenheimer Constructed Diabatic Potential Energy Surfaces for HeH2+

“…Therefore, with Λ elementary rotational matrices one can have Λ! possible ways of taking products to form the A matrix: − A = P n { A 12 false( normalΘ 12 false) · A 13 false( normalΘ 13 false) ... A N − 1 , N false( normalΘ N − 1 , N false) } , goodbreak0em1em⁣ n = 1 , ... , normalΛ ! where P n is the n th permutation between two rotational matrices and normalΛ = C 2 N = N ( N − 1 ) 2 . {Θ ij }s are the ADT angles dependent on the nuclear coordinates.…”

“…Therefore, with Λ elementary rotational matrices one can have Λ! possible ways of taking products to form the A matrix: − where P n is the n th permutation between two rotational matrices and . {Θ ij }­s are the ADT angles dependent on the nuclear coordinates.…”

Curl condition: Existence of sub‐Hilbert space for molecular species or chemical processes

“…The left panel depicts Mathematical Curls, (A) Z 12 ρϕ and (C) Z 34 ρϕ , whereas the right one presents the associated ADT Curls, (B) C 12 ρϕ , and (D) C 34 ρϕ along the ϕ coordinate at three different ρ values for Q 9a À Q 1 normal mode pair. For the Curls associated with (A,B) "1-2" and (C,D) "3-4" couplings, the sharp peaks close to ϕ ¼ 2π=3 and ϕ ¼ π=2, respectively, are magnified in the inset diagrams.…”

“…In addition, their workability have been demonstrated on several molecules of spectroscopic interest [29,30,[32][33][34][35], various triatomic scattering processes [35][36][37] and phase transitions in perovskite solids [38]. A program package, 'ADT' [31] has been developed for formulation of ADT equations through symbolic manipulations as well as numerical solutions of those equations for N coupled electronic states and K nuclear degrees of freedom.…”

“…Since ADT Curl is "trivially" zero (0) for a two state SHS (Abelian case) over the entire CS, we choose a realistic system possessing two state SHS to affirm the validity of Mathematical Curl visá-vis Curl Condition numerically. In any case, for the first time, we explore the workability of Curl Condition for realistic molecular species/ chemical processes/solid state systems constituted with three to six coupled SHS (non-Abelian cases), where the ADT angles from ab initio NACTs [27][28][29][30]32,34,36,44] are used to re-calculate NACTs for evaluating ADT Curls and Mathematical Curls. This article is presented in the following way: Section 2 presents a brief illustration of BBO theory for a complete matrix/Hilbert space.…”