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 distortion Vibronic coupling

Spin-orbit and Jahn-Teller coupling Spin-orbit interaction Jahn-Teller effect “Vibronic quenching” of spin-orbit splitting Energy level splitting Geometrical distortion Vibronic 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 You should use phi here to be consistent with the last slide, or Psi on the last slide

Spin-orbit splitting for vibronic wave functions: a perturbational view

Spin-orbit splitting for vibronic wave functions: a perturbational view Example: the origin band of CH3O: Origin not original Will have to explain subscripts to audience

Spin-orbit splitting for vibronic wave functions: a perturbational view Electronic Vibrational Example: the origin band of CH3O: Subscript on CH3O (also on other slides)

Spin-orbit splitting for vibronic wave functions: a perturbational view Electronic Vibrational Example: the origin band of CH3O:

Spin-orbit splitting for vibronic wave functions: a perturbational view Electronic Vibrational Example: the origin band of CH3O: “Vibronic quenching” of spin-orbit splitting 63 cm-1 c1≈0.83 c2≈0.35 127 cm-1

Spin-orbit splitting for vibronic wave functions: the variational approach Köppel-Domcke-Cederbaum (KDC) quasidiabatic model Hamiltonian Might point out that your wf coefficients on the last slide came from a variational calculation Point out use of CARTESIAN basis here instead of angular momentum basis where SOC is on the diagonal and real 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 Should be INTER Electronic spin-orbit coupling parameter Inter-state coupling Schmidt-Kluegmann, Köppel, Schmatz, Botschwina, Chem. Phys. Lett. 369, 21 (2003).

Electronic spin-orbit splitting: quantum-chemical methods EOMIP-CCSD approach Balanced treatment of electron correlation for the two states Klein, Gauss, J. Chem. Phys. 129, 194106 (2008).

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 Balanced treatment of electron correlation for the two states 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).

Electronic spin-orbit splitting: Benchmark results Level splittings (in cm-1) for 2π radicals OH SH SeH TeH Calculated 134 374 1730 3769 Experiment 139 377 1764 3840 Error -4% -1% -2% Might comment on systematic degradation as Z grows 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

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

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 1097 1109 1110 1107 EOMIP-CCSDT/ANO1 linear and quadratic force constants EOMIP-CCSD/ANO0 cubic and quartic force constants Spin-orbit parameter of 127 cm-1 calculated at X2C-EOMIP-CCSD level Mode 3: C-O stretching; Mode 6: H-C-H bending Is this numbering system the usual one? Also, you might put 0,63 and 0,62 in the first row 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 CH3O Quadratic Hamiltonian Quartic Experiment 61(E) 1281 1250 1230 1224 1289 1277 1232 51(A2) 1426 1332 1344 51(A1) 1526 1414 1433 21(E) 1436 1368 1360 1365 1482 1407 1428 1413 51(E) 1548 1496 1519 1517 1567 1504 1523 Numbers are awesome. Missing experimental numbers? 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

Acknowledgements Jürgen Gauss Werner Schwalbach Takatoshi Ichino The work has been supported the NSF grant (CHE1012743).