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A rigorous approach to the formulation of extended Born‐Oppenheimer equation for a three‐state system
Author(s) -
Sarkar Biplab,
Adhikari Satrajit
Publication year - 2008
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.21870
Subject(s) - born–oppenheimer approximation , diabatic , adiabatic process , conical intersection , curl (programming language) , hilbert space , coordinate system , schrödinger equation , transformation (genetics) , physics , manifold (fluid mechanics) , quantum , mathematical physics , mathematics , quantum mechanics , ground state , chemistry , geometry , computer science , mechanical engineering , biochemistry , molecule , gene , programming language , engineering
If a coupled three‐state electronic manifold forms a sub‐Hilbert space, it is possible to express the non‐adiabatic coupling (NAC) elements in terms of adiabatic–diabatic transformation (ADT) angles. Consequently, we demonstrate: (a) Those explicit forms of the NAC terms satisfy the Curl conditions with non‐zero Divergences; (b) The formulation of extended Born‐Oppenheimer (EBO) equation for any three‐state BO system is possible only when there exists coordinate independent ratio of the gradients for each pair of ADT angles leading to zero Curls at and around the conical intersection(s). With these analytic advancements, we formulate a rigorous EBO equation and explore its validity as well as necessity with respect to the approximate one (Sarkar and Adhikari, J Chem Phys 2006, 124, 074101) by performing numerical calculations on two different models constructed with different chosen forms of the NAC elements. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009