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