Abstract

Spectra of high-symmetry molecules contain fine and superfine level cluster structure related to J-tunneling between hills and valleys on rovibronic energy surfaces (RES). Such graphic visualizations help disentangle multi-level dynamics, selection rules, and state mixing effects including widespread violation of nuclear spin symmetry species. A review of RES analysis compares it to that of potential energy surfaces (PES) used in Born–Oppenheimer approximations. Both take advantage of adiabatic coupling in order to visualize Hamiltonian eigensolutions. RES of symmetric and D2 asymmetric top rank-2-tensor Hamiltonians are compared with Oh spherical top rank-4-tensor fine-structure clusters of 6-fold and 8-fold tunneling multiplets. Then extreme 12-fold and 24-fold multiplets are analyzed by RES plots of higher rank tensor Hamiltonians. Such extreme clustering is rare in fundamental bands but prevalent in hot bands, and analysis of its superfine structure requires more efficient labeling and a more powerful group theory. This is introduced using elementary examples involving two groups of order-6 (C6 and D3~C3v), then applied to families of Oh clusters in SF6 spectra and to extreme clusters.

Highlights

  • A key mathematical technique for atomic or molecular physics and quantum chemistry is matrix diagonalization for quantum eigensolution

  • Before describing tensor eigensolution techniques and rovibronic energy surfaces (RES), a brief review is given of potential energy surface (PES) to put the tensor RES in a historical and methodological context

  • The latter have only recently been seen in highly excited rovibrational spectra [2] and present challenging problems of symmetry analysis to sort out a plethora of tunneling resonances and parameters for so many resonant states

Read more

Summary

Overview of Eigensolution Techniques for Symmetric Molecules

A key mathematical technique for atomic or molecular physics and quantum chemistry is matrix diagonalization for quantum eigensolution. We are motivated to seek ways to visualize more of the physics of molecular eigensolutions and their spectra This leads one to explore digital graphical visualization techniques that provide insight as well as increased computational power and thereby complement numerically intensive approaches [1]. Following this is a discussion of mixed-rank tensors that exhibit 12-fold and 24-fold monster-clusters. It uses underlying duality between internal and external symmetry states and their operations. A main idea is that symmetry operators “know” the eigensolutions of their algebra and of all Hamiltonian and evolution operators made of gk’s

Computer Graphical Techniques
Unitary Multipole Functions and Operators
Tensor and Elementary Matrix Operators
Fano–Racah Tensor Algebra
Tensor Eigensolution and Legendre Function RE Surfaces
Angular Momentum Cones and RES Paths
Reduced Matrix and RES Scaling
Asymmetric Top and Rank-2 RES
H AsymTop
Symmetry Labeling of Asymmetric Top Eigenstates
Tunneling between RES-Path States
Tensor Eigensolutions for Octahedral Molecules
Tensor Symmetry Considerations
Numerical Assignment of Symmetry Clusters
Criteriaf orC1 Level Clustering
Introducing Dual Symmetry Algebra for Tunneling and Superfine Structure
Abelian Symmetry Analysis
Operator Expansion of Cn Symmetric Hamiltonian
Spectral Resolution of Cn Symmetry Operators
Spectral Resolution of Cn Symmetric Hamiltonian
Non-Abelian Symmetry Analysis
Operator Expansion of D3 Symmetric Hamiltonian
Spectral Resolution of D3 Symmetry Operators
Sorting Commuting Subalgebras
Resolving All-commuting Class Subalgebra
Spectral Resolution of Dual Groups D3 and D3
Spectral Resolution of D3 Hamiltonian
Global-Lab-Relative G versus Local-Body-Relative G Base State Definition
Global versus Local Eigenstate Symmetry
Symmetry Correlation and Frobenius Reciprocity
Coset Structure and Factored Eigensolutions
Octahedral Symmetry Analysis
Octahedral Characters and Subgroup Correlations
Resolving Hamiltonians with C4 Local Symmetry
Orthogonality-Completeness of Local Symmetry Parameters
Resolving Hamiltonians with C3 Local Symmetry
Spectral Resolution of full Oh Symmetry
Resolving Hamiltonians with C2v Local Symmetry
Local Sub-Group Tunneling Matrices and Their Inverse
Rotor-With-Gyro Model of Internal Rotation
10. Summary and Conclusions
Classical D3 Modes
Comparing K-Matrix and H-Matrix Formulation
K-Matrix Eigensolutions for Broken Local Symmetry

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.