University of Illinois Urbana-Champaign

TRANSFORMATION OF QUANTUM MECHANICAL OPERATOR MATRIX FROM CARTESIAN TO CYLINDRICAL NORMAL COORDINATES IN QUASI-DIABATIC BASIS

Vasilyev, Oleg A.

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6886.pdf
FB03_1.pdf

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https://hdl.handle.net/2142/122345

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Title
TRANSFORMATION OF QUANTUM MECHANICAL OPERATOR MATRIX FROM CARTESIAN TO CYLINDRICAL NORMAL COORDINATES IN QUASI-DIABATIC BASIS
Author(s)
Vasilyev, Oleg A.
Contributor(s)
  • Stanton, John F.
  • Miller, Terry A.
  • Sharma, Ketan
Issue Date
2023-06-23
Keyword(s)
Theory and Computation
Abstract
As described in the previous talk, if one is analyzing a spectrum involving rotational and fine structure, the cylindrical representation of the potential energy matrix in the quasi-diabatic basis, Vcyl, is more convenient due to the simple form of rovibronic Hamiltonian. It also facilitates the identification of the symmetry of the vibronic basis functions.ᵃ On the other hand, ab initio parameterization of the vibronic Hamiltonian is typically performed in the Cartesian representation, Vcₐrt. The two matrix representations for a molecule with a simple, linear Jahn–Teller E × e effect can be written as Vcyl=[1/2wQ+Q- kQ+ kQ- 1/2wQ+Q-] Vcart=[1/2w(Qa²+Qb²)+kQa -kQb -kQb 1/2w(Qa²+Qb²)-kQa] where Q± = Qₐ ± iQb are the normal modes. Here the corresponding electronic basis sets are related as Φ± = 1/√2(Φₐ ± iΦb). Transforming Vcart into Vcyl through coordinate substitution together with the basis set transformation becomes tedious for higher-order expansions, especially considering multimode and multistate problems. It is therefore desirable to develop a general procedure that can be used for the transformation of operators from the Cartesian form to the cylindrical one. In this talk, we will show that the potential energy operator represented in a tensor form can be naturally transformed from Cartesian to cylindrical representation through a series of tensor-matrix multiplications, provided that the two matrices transforming the vibrational normal coordinates and the electronic basis set are known. In general, this method can be used to transform any quantum mechanical operator matrix. We demonstrate the effectiveness of this method with calculations for NO₃ and CH₃O.
Publisher
International Symposium on Molecular Spectroscopy
Type of Resource
Text
Genre of Resource
Conference Paper / Presentation
Language
eng
Handle URL
https://hdl.handle.net/2142/122345
DOI
https://doi.org/10.15278/isms.2023.6886

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Abstracts & Presentations - 2023 International Symposium on Molecular Spectroscopy PRIMARY