Theory of Molecular Frequencies

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					National Aeronautics and Space Administration




                           Theory of Molecular Frequencies




                                  H. M. Pickett
           Jet Propulsion Laboratory, California Institute of Technology
National Aeronautics and Space Administration



                                    SPFIT and SPCAT

•   Fitting and prediction programs for JPL Molecular Spectroscopy Catalog
      – Core software developed to replace special purpose spectroscopy programs
•   Introduced well thought-out organization of the program with modular
    function calls and data structures. Uses spherical tensor formulation of
    rotation and spin operators, so diagonal and off-diagonal matrix operators
    are calculated consistently.
•   Originally written in FORTRAN-77, but recoded to ‘c’ for dynamic memory
    allocation and portability. Currently tested with Microsoft Visual C++ and
    gnu gcc.
•   Many features have been added, but I have tried to keep input formats
    backward compatible.
•   Current capabilities: 280 vibronic states, 9 spins, Euler series, internal
    rotation, spin interactions with high-order symmetry (e.g. CH3O).



Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                    2
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                   Nature of Spectroscopic Fitting and Prediction
•   Human Spectroscopic Interface:
      – Line Frequency, Upper-state Quantum Numbers, Lower-state Quantum Numbers
      – Parameters and matrix operators:
               H = Σk Parameterk Operatork
          • Where H is the Hamiltonian matrix and Operator is a matrix operator of type
            k, and Parameterk is a numerical value
          • Usually H is factorable into finite dimension Hermitian sub-blocks
          • Each block of H must be diagonalized to give energy
•   Computer View:
      – Line Frequency = EQ’,q’ – EQ”,q” where Q labels a block of the Hamiltonian and q
        labels the index for a particular eigenvector within block Q
      – Line Frequency can be an energy or can represent lines that do not have any
        absorption strength.
      – SPFIT is restricted to cases where H is a purely real matrix



Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                            3
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                   Automated Spectroscopic Fitting and Prediction
•   There is no automated algorithm for assigning quantum numbers to an
    observed line frequency
      – SPFIT/SPCAT defines a mapping of quantum numbers to Q and q, although the
        determination of q is much more arbitrary.
      – The quantum number assignment is the most important task where the skill of a
        well-trained spectroscopist is needed
•   Several Graphical User Interfaces have been developed to aid assignment
    that use SPFIT/SPCAT as computational core
•   Options for Gaussian quantum program have been developed to ouput
    calculations in SPCAT format
•   Fitting line frequencies involves least square fitting of line frequencies to the
    Parameters and can be very non-linear
      – Need a good starting point




Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                         4
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                                   Predicting Intensities

•   Absorption cross-section (units = m2) × density (m-3) = absorption coefficient
    (m-1)
•   Integrated absorption cross-section (units = Hz m2) is directly related to
    spontaneous emission rate through the Einstein detailed balancing
      – Units of JPL catalog are MHz (nm)2
•   Cross-section is proportional to |μ|2, where μ is the transition dipole
      – μ = Σk μ-parameterk Operatork
      – Relative signs of the dipole parameters can be important




Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                  5
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                               Predicting Intensities (con.)

•   The dipole parameters are scalar constants obtained usually from Stark
    effect (shifts of line frequencies in an electric field) or from Zeeman effect
    (shifts of line frequencies in a magnetic field)
      – Dipole parameters can include first and second order Hermann-Wallis
        corrections (centrifugal distortion effects on the dipole)
      – SPCAT can output contributions to the transition dipole as well as intensities,
        allowing fitting of Stark effect in the presence of centrifugal distortion
      – SPCAT computations of Dipole operators use the same routines as computation
        of Energy operators (an anti-bugging advantage).
      – User can specify both electric and magnetic dipoles




Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                           6
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                      Recent Extensions to SPFIT/SPCAT (1 / 3)

•   New option for blends
      – Old option: successive lines in .lin file with the same frequency are considered
        weighted blend if uncertainty is the same
      – New: if last line in the .lin file with the same frequency has uncertainty xerr that is
        at lest 2 times larger, then the rms width of the blend is included in the fit with
        uncertainty of xerr. The quantum numbers for this last line are ignored
                        9 9 0 8 8 0         234123.44 0.4          1.0
                        9 9 1 8 8 1         234123.44 0.4          1.0
                        9 9 1 8 8 1         234123.44 2.0          1.0

•   New spin coupling scheme for 3-fold through 6-fold symmetric tops with
    hyperfine, e.g. CH3O. Allows full symmetry to be applied with no states that are
    not allowed by spin statistics.
      – Augments earlier capability to couple 2 equivalent spins together so that proper ortho-
        para symmetry is retained, e.g. ClOCl


Oct. 19-20, 2006                   Caltech Spectroscopy Workshop                                  7
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                     Recent Extensions to SPFIT/SPCAT (2 / 3)

•   New option for increased phase flexibility
      – Standard phase convention for Wang block is (i)s, where s = 0,1,2,3 for ee, oe,
        oo,eo, respectively
          • This makes all even-order operators real and all odd-order operators
             imaginary
      – Earlier versions were very permissive, allowing user to specify imaginary
        operators that are actually anti-symmetric because i was ignored
      – We then required that all operators diagonal in v be real and several applications
        to fail that had ‘worked before.’
      – We were able to identify 8 different Wang block phase conventions
      – The best phase convention is selected behind the scenes, based on the
        parameter set selected
          • For example, with the standard phase PaPb, Pa and Pb only PaPb is real,
             but with a different convention all 3 operators can be real.
          • The phase convention can also be forced using a new option-line parameter


Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                          8
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                      Recent Extensions to SPFIT/SPCAT (3 / 3)

•   Inclusion of Euler series
      – Provides better convergence than standard power series, particularly for light
        molecules like water
•   Improved capabilities for l-doubling basis
      –   Explicit way to specify lz operator
      –   Extension of K-origin ‘operator’ to l-doubling basis
      –   Fix bugs in treatment of K=0 for l-doubling basis
      –   Provide capability for distinct Δl ≠ 0 operators with ΔK Δl < 0 and ΔK Δl > 0
•   Provide capability for calculation of (εab – εba) (SaNb + NbSa – SbNa – NaSb)
      – Since the diagonal tensor components of spin-orbit parameters, e.g εaa, can be
        quite large, εab and εba can have a significant effect on the spectrum




Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                           9
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                                                   Methanol Energies, A State
         3000
                                                                      •    Energy is periodic in rho K with period
                                                                           of 3
         2500                                                         •    Energies of E states have phase shift
                                                                           of ±1 in rho K relative to A state
         2000                                                  V_3
                                                                      •    v = 3 has an avoided crossing with v=4
                                                               v=0         near K = 0
                                                               v=1
Energy




                                                               v=2
         1500
                                                               v=3
                                                               v=4
                                                               v=5
         1000                                                  v=6




         500




           0
            -1.5   -1   -0.5      0      0.5   1    1.5
                               rho * K


Oct. 19-20, 2006                                      Caltech Spectroscopy Workshop                             10
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                          Future Extensions to SPFIT/SPCAT

•   Current program is mature, and don’t I expect to make sweeping changes
      – Current capabilities of SPFIT appear adequate for astrophysics needs
      – Limiting resource is spectroscopic expertise
•   We need to explore if there are better ways of handling torsional states
    above the barrier along with states below the barrier
      – Perturbations not enfolded into the pure torsional problem are mixing across
        vibrations that are not well characterized
•   We would like to cross-compare different programs that calculate internal
    rotation (program A generates noise-free energies and/or lines, program B
    fits these lines)
      – JPL has started these tests and it looks feasible and useful




Oct. 19-20, 2006                  Caltech Spectroscopy Workshop                        11

				
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