Computing ro-vibrational levles of methane from the potential energy force field (surface)

Xiao-Gang Wang1, Tucker Carrington Jr1, and Edwin L. Sibert III2,

1Chemistry Department, Queen's University 2Department of Chemistry and Theoretical Chemistry Institute, University of Wisconsin, Madison,

International Workshop

"Spectroscopy of methane and planetary applications"

Hotel Kyriad Dijon Est - Mirande

Dijon, France, November 8-10, 2010

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One of my first project as a graduate student is to analyze v = 4 room temperature methane spectra. This is P=8. Pentacontakipentad

High-resolution spectrum is a cryptic book from heaven. – Qingshi Zhu

It takes a few days to record a high-resolution spectrum, but years to analyse. – Qingshi Zhu

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Theoretical computation of molecular spectra

H

Potential energy surface

Molecular spectra

Effective Hamiltonian

Contact transformation

Fit with effective Hamiltonian solve

diagonalize diagonalize H

link

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Theoretical computation of molecular spectra

H

Potential energy surface

Molecular spectra

Effective Hamiltonian

Contact transformation

Fit with effective Hamiltonian solve

diagonalize diagonalize H

link

Too many parameters ! diverge and resonance perfect but too costly

  Standard normal coordinates Q is rectilinear : linear combinations of Cartesians

rectilinear and curvilinear normal coordinates

  Curvilinear normal coordinates Q : linear combinations of internal coordinates (R)

  L is obtained by diagonalizaing GF matrix in symmetry coordinates (S)

  l is obtained by diagonalizaing Hessian matrix in cartesian coordinates

KEO in rectilinear normal coordinates

  KEO in rectilinear Q (Watson 1968)

  The key is to expand

  This KEO has been used extensively by the spectroscopists.   Analytical contact transformation by Papousek and Aliev (1982) to derive expressions of

vibration and ro-vibration spectroscopic constants in terms of force field and structural

parameters like .

  Sibert et al found that curvilinear normal coordinates lead to faster convergence sometimes.

KEO in curvilinear normal coordinates

  KEO in curvilinear Q (Pickett 1972)

  G is Wilson's G-matrix if internal coordinate R is used.   G(Q), A(Q), µ(Q) are computed from the B matrix.

  For methane, B matrix is a 15x15 matrix. Use Eckart condition to fix the 6 rows of

rotations and translations.

[1] Wang and Sibert, JCP (1999), [2] Wang and Sibert, Spectrochim. Acta. A, (2002)

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Expand the Hamiltonian and group terms

  This KEO plus a quartic force field are the approximate Hamiltonian used in all our CVPT study

are terms not included

centrifugal terms

Coriolis terms

  We exapand this KEO up to 4th order (p2q4) with finite difference method.

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Canonical Van Vleck perturbation theory (CVPT) – contact transformation

  Expand the Hamiltonian

  Transform the Hamiltonian so that becomes diagonal within the polyad

  Numerical transformations (Sibert, JCP 1988, Comp. Phys. Comm. 1988)

  Rotation becomes 2D degenerate harmonic oscillator

CVPT is an approximate method with small number of parameters that is applicable to all polyads.

CVPT studies of methane : paper I

  4th order CVPT most levels converge better than 1 cm-1 except for high n4 modes.   In Wang and Sibert 1999 JCP, the 4th order order CVPT was used to fit 130 levels of

CH4, CD4, 13CH4, CH3D, 13CH3D, CHD3, 13CHD3, CH2D2

  with 0.7 cm-1 rmsd and fit only 12 force field parameters and

  predictions good for 8000 cm-1.

  Fitting method of Mardis and Sibert, JMS, 187, 167 (1998). approximate Hessians by

products of gradients.

  5 cubic   13 cubic

  33 quartic

'eff' fits 47 parameters to 89 experimental J=0 levels

CVPT studies of methane : paper II

  In Wang and Sibert, Spectrochim. Acta. A, (2002), rotational levels are computed at 6th order CVPT   Tetrahedral centrifugal splitting is caused by J4, a H(4) term.

  Improvement in the current work,   Fit the levels with 6th order CVPT

  Fit the low-J rotational and vibrational levels together

  Include all levels up to octad of CH4 and J=0 octad levels of 13CH4 observed with P(1)

lines[1,2].

  Include some observed tetradecad J=0 and 1 levels of CH4 and 13CH4, in particular [3,4].

CVPT studies of methane : current work

[1] Albert et al. Chem. Phys. 2009, 356, 131 [2] Niederer, Quack, private communication (Octad fit of 13CH4)

[3] Nikitin et al., J. Q. S. R. T., 2009, 110, 964 (isotopic shift)

[4] Nikitin et al., J. Q. S. R. T., 2010, 110, 2211 (GOSAT 2009 assignments)

  Fit 92 levels with 16 parameters   Fit J=0 and 1 levels of CH4 and 13CH4

  0.6 cm-1 rmsd

Georges98 assign it as Q(1) Nikitin09 assign it as P(1)

Georges et al., J. M. S., 1998, 187, 13 Nikitin et al., J. Q. S. R. T., 2009, 110, 964

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Assignment, assignment and assignment!

  Directly observed levels with definite assignments are good inputs to refine the potential.

  Recent two-temperature works of Campargue et al. and Votava et al. deduce E'' and J''

  But upper state J' and are still unknown in many cases. Is it P, Q, R ? What sym ?

  Exception is J'' = 0. We know J'=1 and = A2

  Votava et al. (2009) assigned 9 R(0) lines levels of icosad.

  Hippler and Quack (2001) assigned nu2 + 2nu3 lines of isocad.

Voltava et al., PCCP, 2010, 25K and 81K two-temperature method. These are rare and precious observed levels for refining the potential energy surface.

Important differences between theory and experiment.

Good agreement, confirming the assignment.

Another theoretical level nearby

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Overview of our contracted-iterative variational method

1) Polyspherical coordinates and KEO

2) Contracted stretch functions and bend functions

3) Iterative (Lanczos) diagonalization to solve the 5D bend

problem

  The method is applied to, with many improvements along the way,

CH4 vibration, Wang and Carrington, JCP(2003) CH4 rotation, Wang and Carrington, JCP (2004) CH3D, CHD3 rotation, Wang and Carrington, JCP (2005) CH5+ vibration, Wang and Carrington, JCP (2008)

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Variational method

Expand the solution in basis functions

Construct the Hamiltonian matrix

Diagonalize the Hamiltonian matrix (Lanczos algorithm)

A direct product basis scales exponentially with dimension

103 basis for 3-atom, 106 basis for 4-atom,

109 basis for 5-atom, 1012 basis for 6-atom

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Iterative (Lanczos) eigensolver

  Only need to compute matrix-vector product   Memory cost scales as N ( store a few vectors)

  Time cost scales as N 2

  Radau vectors are similar to bond vectors, but stretch-bend momentum coupling is zero.

  4 distances and 5 angles :

  Almost all variational calculations of methane uses Radau vectors.

Polyspherical coordinates for Radau vectors

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  are angular momentum operators in a space-fixed frame

  no momenta coupling

The KEO in space-fixed frame with Radau vectors : easy

  Start from the space-fixed frame, simple KEO,

coupling!

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The KEO in polyspherical coordinates

  Derive the body-fixed KEO from space-fixed KEO by chain rule, in terms of 12 coordinates:

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Simple kinetic matrix elements

  Spherical harmonic type basis

to remove the KEO singularity

  simple matrix elements

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Complicated bend potential matrix elements

  Every index is strongly coupled by the potential

  Potential matrix is computed by Gauss quadrature: 2D example

  We do not store this matrix, but compute the matrix-vector products

  DVR basis for the stretch : potential integral is diagonal

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A 9D product basis (vibration)

  Product basis functions

  It is impossible to diagonalize with a 9D product basis

  But there is still a solution: contracted basis

  would need 33x109 basis functions,

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Contracting the product basis

  Solve the bend problem

  Use products of eigenfunctions of reduced-dimension Hamiltonians. Rewrite,

  Solve the stretch problem

  The lowest solutions are the contracted basis: huge reduction in size

  The coupling is dealt with in the contracted basis

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and matrix

is factorizable and easy to compute

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matrix : F-matrix method

We do not store the V matrix, but store the F-matrix.

Distribute over hundreds of processors. No communication.

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New calcultions increase bend basis from 280 to 436 to converge 4nu4 down by further 7 cm-1. See [1].

[1]Bowman, Carrington, Meyer, Mol. Phys, 106, 2145 (2008)

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c

cal.-exp.

Theory undershoots expt.

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Future work : a good fitting strategy

  A J = 0 calculation took 3 weeks in 2003 a) 20 days computing bend functions with 1 cpu (95% of time) b) 13 hours to compute DV on 11 cpu. c) 1.5 day for diagonalizing final basis on 1 cpu

Nowadays, it is a few hours with all the improvements and parallelization.

  But fortunately, bend functions are not changed in the fitting since the ab initio potential is very good. 10 minutes a cycle

  Compute these gradients in parallel. N-times more cpu's. -->1000 cpus

  A collaboration with Quack group : a good functional form with small number of parameters is important for fitting.

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Acknowledgment

  National Science and Engineering Research Council and NSF for funding.

  Campargue, Niederer, Quack and Boudon for discussions.