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Ab initio calculation of spin-orbit splitting for vibronic wave functions
Lan Cheng and John Stanton
Department of Chemistry, University of Texas at Austin
Spin-orbit and Jahn-Teller coupling
• Spin-orbit interaction
Energy level splitting
Spin-orbit and Jahn-Teller coupling
• Spin-orbit interaction
• Jahn-Teller effect
Energy level splitting
Geometrical distortionVibronic coupling
Spin-orbit and Jahn-Teller coupling
• Spin-orbit interaction
• Jahn-Teller effect
• “Vibronic quenching” of spin-orbit splitting
Energy level splitting
Geometrical distortionVibronic coupling
See, for example, Child, Longuet-Higgins, Phil. Trans. Roy. Soc. A 254, 259 (1961).Barckholtz, Miller, Int. Rev. Phys. Chem. 17, 435 (1998).
Spin-orbit splitting for vibronic wave functions: a perturbational view
Spin-orbit splitting for vibronic wave functions: a perturbational view
Vibronic wave function
Spin-orbit splitting for vibronic wave functions: a perturbational view
Example: the origin band of CH3O:
Vibronic wave function
Spin-orbit splitting for vibronic wave functions: a perturbational view
Example: the origin band of CH3O:
Vibronic wave function
Vibrational
Electronic
Spin-orbit splitting for vibronic wave functions: a perturbational view
Example: the origin band of CH3O:
Vibronic wave function
Vibrational
Electronic
Spin-orbit splitting for vibronic wave functions: a perturbational view
Example: the origin band of CH3O:
127 cm-1c1≈0.8363 cm-1 c2≈0.35
“Vibronic quenching” of spin-orbit splitting
Vibronic wave function
Vibrational
Electronic
Spin-orbit splitting for vibronic wave functions: the variational approach
• Köppel-Domcke-Cederbaum (KDC) quasidiabatic model Hamiltonian
Schmidt-Kluegmann, Köppel, Schmatz, Botschwina, Chem. Phys. Lett. 369, 21 (2003).
Spin-orbit splitting for vibronic wave functions: the variational approach
• Köppel-Domcke-Cederbaum (KDC) quasidiabatic model Hamiltonian
Electronic spin-orbit coupling parameterInter-state coupling
Schmidt-Kluegmann, Köppel, Schmatz, Botschwina, Chem. Phys. Lett. 369, 21 (2003).
Electronic spin-orbit splitting: quantum-chemical methods
• EOMIP-CCSD approach
Klein, Gauss, J. Chem. Phys. 129, 194106 (2008).
Balanced treatment of electron correlation for the two states
Electronic spin-orbit splitting: quantum-chemical methods
• EOMIP-CCSD approach
• Exact two-component (X2C) spin-orbit matrix
elements Coupling between scalar relativity and spin-orbit coupling
Klein, Gauss, J. Chem. Phys. 129, 194106 (2008).
Li, Xiao, Liu, J. Chem. Phys. 137, 154114 (2012).Filatov, Zou, Cremer, J. Chem. Phys. 139, 014106 (2013).Cheng, Gauss, J. Chem. Phys. submitted (2014).
Balanced treatment of electron correlation for the two states
Electronic spin-orbit splitting: Benchmark results
• Level splittings (in cm-1) for 2π radicalsOH SH SeH TeH
Calculated 134 374 1730 3769
Experiment 139 377 1764 3840
Error -4% -1% -2% -2%
FO ClO BrO IO
Calculated 196 321 991 2162
Experiment 197 322 975 2091
Error -1% 0% 2% 3%
Spin-orbit splitting: B2E’ and A2E’’ states of NO3
• Electronic spin-orbit splitting (in cm-1)
B2E’ A2E’’
EOMIP-CCSD 122 0
Spin-orbit splitting: B2E’ and A2E’’ states of NO3
• Electronic spin-orbit splitting (in cm-1)
• Electronic spin-orbit splitting for B2E’ state is of normal magnitude
• A2E’’ state has “no” spin-orbit splitting ?!??!
B2E’ A2E’’
EOMIP-CCSD 122 0
Vibronic levels: state of CH3O
Quadratic Hamiltonian Quartic Experiment
00(E) 66 63 63 62
61(A1) 736 710 685 682
61(A2) 988 967 944
31(E) 1031 1044 1047 1043
31(E) 1097 1109 1110 1107
EOMIP-CCSDT/ANO1 linear and quadratic force constantsEOMIP-CCSD/ANO0 cubic and quartic force constantsSpin-orbit parameter of 127 cm-1 calculated at X2C-EOMIP-CCSD level
Mode 3: C-O stretching; Mode 6: H-C-H bending
Experimental results from: Foster, Misra, Lin, Damo, Carter, Miller, J. Phys. Chem. 92, 5914 (1988).Temps, Molecular Dynamics and spectroscopy by stimulated emission pumping (1995).
Vibronic levels: state of CH3OQuadratic Hamiltonian Quartic Experiment
61(E) 1281 1250 1230 1224
61(E) 1289 1277 1232
51(A2) 1426 1332 1344
51(A1) 1526 1414 1433
21(E) 1436 1368 1360 1365
21(E) 1482 1407 1428 1413
51(E) 1548 1496 1519 1517
51(E) 1567 1504 1523
Mode 2: Umbrella ; Mode 3: C-O stretching;Mode 5: H-C-O rocking; Mode 6: H-C-H bending
Outlook
• Spin-orbit effect for B2E’ state of NO3
• Dispersed fluorescence spectra for CH3O
• Jürgen Gauss• Werner Schwalbach
• Takatoshi Ichino
• The work has been supported the NSF grant (CHE1012743).
Acknowledgements