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THEORETICAL STUDIES OF HIGHLY EXCITED VIBRATIONAL STATES USING VAN VLECK PERTURBATION THEORY University of Wisconsin Ned Sibert, Anne McCoy, Darin Burleigh, Sai Ramesh, Xiaogang Wang, Marc Joyeaux, Christoph Iung, and Christof Jung On σ-Type Doubling and Electron Spin in the Spectra of Diatomic Molecules J. H. Van Vleck Department of Physics, University of Wisconsin Received 31 January 1929 1958 BAND INTENSITIES IN TETRAHEDRAL COMPLEXES BALLHAUSEN, C.J.; LIEHR, ANDREW D. 1964 A PERTURBATION METHOD SUITABLE FOR HIGHER ORDER CALCULATIONS Louck, James D. 1993 LARGE AMPLITUDE MOTIONS IN THE WATER MOLECULE .Sage, M. L. 1991 Calculation of IR Intensities of Highly Excited Vibrational States in HCN Using Van Vleck Perturbation Theory McCOY, ANNE B.; SIBERT, EDWIN L. III 1964 CONTRIBUTION OF ROTATION-VIBRATION INTERACTION TO THE SPIN-DOUBLET SEPARATION IN $2\Pi$ DIATOMIC MOLECULES James, Thomas C. 2003 A PRIMER ON DUNHAM'S APPROACH TO ANALYSIS OF SPECTRA OF FREE DIATOMIC MOLECULES OGILVIE, J.F. 1961 THEORETICALLY-CALCULATED VIBRATIONAL-ROTATIONAL SPECTRUM OF $H_{2}$, HD, AND $D_{2}$ COOLEY, JAMES W. 1978 VAN VLECK TRANSFORMATION TO TENTH ORDER AS AN ALTERNATIVE TO THE CONTACT TRANSFORMATION THORVALD, PEDERSEN 1978 VAN VLECK TRANSFORMATION APPLIED TO THE BORN-OPPENHEIMER SEPARATION JORGENSEN, F.; THORVALD, PEDERSEN 1994 DEFINITIVE EVALUATION OF NONADIABATIC VIBRATIONAL AND ADIABATIC EFFECTS FROM VIBRATION-ROTATIONAL SPECTRA OF DIATOMIC MOLECULES OGILVIE, J.F. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 next Overview of Theoretical Spectroscopy Potential Energy Surfaces Hˆ ( N ) jN E Nj jN Dynamics and Spectroscopy Diagonalize H ] ] T 1 HT K H11 H12 H13 H14 K11 0 0 0 H21 H22 H23 H24 0 K22 0 0 H31 H32 H33 H34 0 0 K33 0 H41 H42 H43 H44 0 0 0 K44 Overview of Theoretical Spectroscopy Potential Energy Surfaces Hˆ ( N ) jN E Nj jN Effective Hamiltonians Dynamics and Spectroscopy FE08 Effective Vibrational Rotational Hamiltonian in the Presence of a Stark Field B. Ram Prasad and Mangala Sunder Krishnan The standard theory for analysing high resolution Vibrational-RotationalTorsional spectra of semi-rigid and non-rigid molecules is based on perturbation theory, which leads to the concept of effective Hamiltonians. Though there are several ways of obtaining molecular Hamiltonians, the method proposed by Van Vleck has proven to be quite useful in the area of molecular spectroscopy. Perturbation Theory 1 T HT K H11 H12 H13 H14 K11 K12 K13 K14 H21 H22 H23 H24 K21 K22 K23 K24 H31 H32 H33 H34 K31 K32 K33 K34 H41 H42 H43 H44 K41 K42 K43 K44 This only works for when there are no degenerate states. Wave functions for zero order Hamiltonian Wave functions for full Hamiltonian Solution to Vibrational Example if w1 2w2 Solution to Vibrational Example if w1=2w2 Simple Solution to Vibrational Example if w1=2w2 |4,0> |3,2> |2,4> |1,6> |0,8> Polyads N=2n1+n2=constant=8 Simple Solution to Vibrational Example if w1=2w2 H= Each block corresponds to a polyad of states Van Vleck Perturbation theory is a simple way to transform the matrix to this form. Visualizing Wave Functions for SCCl2 Probability distributions of a select eigenstate plotted as a function of the Q5 and Q6 coordinates for increasing values of Q1 going from (a)-(f). Van Vleck Perturbation theory is a simple way to transform H to a desired form. If we write T as 1 T e T HT e i S iS , then T is unitary if S is Hermitian. He i S e i [ S , ] H K One solves for S by expanding H and K in powers of . e in [ S ( n ) , ] e i2 [ S ( 2 ) , ] i [ S (1) , ] e H K Result of 2nd order Van Vleck PT CHBr3 normal mode CH stretch-bend states Nt Polyad Labels 1 2 3 Ei Ej CHBr3 post Van Vleck CH stretch-bend states Nt Labels 1 2 3 Ei Ej Comparison of 2nd, 4th, 6th, and 8th order Van Vleck PT band origins (cm-1) for CH4 Xiao-Gang Wang and E. L. Sibert, JCP 124, 4510 (1999). Comparison of 2nd, 4th, 6th, and 8th order Van Vleck PT band origins (cm-1) for CH4 We are pushing the limits of CVPT so that it can be used to accurately calculate highly excited states. Rectilinear vs Curvilinear Approaches Rectilinear CVPT results for H2CO vibrations Curvilinear CVPT results for H2CO vibrations Two Current Research Directions 1) Vibrations coupled to additional DOF 2) Creating good basis sets Vibrations Coupled to Other Degrees of Freedom Many Hamiltonians can be written in the form H H vib H x H cpl CVPT allows one to simplify the analysis via the following transform. K e e i S i S ( H vib H x H cpl )e H vib e i S Hx e K vib H x K cpl i S i S H cpl e i S Vibrations coupled to other DOF Hamiltonians that can be written in the form H H vib H x H cpl include: Rotation-vibration of H2O, H2CO, SO2, CH4, laser-molecule for HCN, H2O, H2CO, CHBr3, CHF3 Isomerization HCN vibration-torsion CH3OH and deuterated analogues condensed phase vibrational relaxation Minimum energy positions vs torsion angle Methanol quartic expansion in internal coordinates where expansion coefficients are functions of the torsion angle calculated at the CCSD(T) level using the cc-pVTZ basis Basic Approach Comparison of experimental and theoretical fundamentals and E-A splittings D (in cm-1 ). Comparison of experimental and theoretical fundamentals and E-A splittings D (in cm-1 ). Creating Good Basis Sets Molecular Physics, Vol. 103, (2005), 149–162. SAI G. RAMESH and EDWIN L. SIBERT Harmonic Basis 0 0 2000 1000 4000 Size of Vij Ei 3000 4000 Ej Post Van Vleck 0 0 2000 1000 4000 Size of Vij Ei 3000 4000 Ej CHF3 Results We are using CVPT to build good basis sets for variational calculations. C. Iung and F. Ribeiro, and E. Sibert With these fundamental frequencies we were unable to identify polyad quantum numbers. HFCO results of convergence tests. Conclusions Van Vleck perturbation theory remains an important tool for treating molecular vibrations as well as vibrations coupled to many other DOF. A study of the vibrations of fluoroform with a sixth order nine-dimensional potential: a combined perturbative-variational approach SAI G. RAMESH and E. L. SIBERT, Mol. Phys, 103, 2005, 149–162 Quantum, semiclassical and classical dynamics of the bending modes of acetylene E. L. Sibert and Anne B. McCoy JCP (1996) The Sibert Group