Vibration-rotation spectra from first principles Lecture 1: Variational nuclear motion calculations

1. Vibration-rotation spectra from first principles Lecture 1: Variational nuclear motion calculations Jonathan Tennyson Department of Physics and Astronomy University College London OSU, February 2002

2. “(Variational calculations) will never displace the more traditional perturbation theory approach to calculating ….. vibration-rotation spectra” Carter, Mills and Handy, J. Chem. Phys.,99, 4379 (1993)

4. Rotation-vibration energy levels • The conventional view: • Separate electronic and nuclear motion, • The Born-Oppenheimer approximation • Vibrations have small amplitude • Harmonic oscillations about equilibrium • Rotate as a rigid body • Rigid rotor model Improved using perturbation theory

5. But Small amplitude vibrations often poor approximation What about dissociation? Equilibrium not always a useful concept What about multiple minima? Perturbation theory may not converge Diverges for J > 7 for water For high accuracy need electron-nuclear coupling Important at the 1 cm-1 level for H-containing molecules

6. Variational approaches: Ein > Ein+1 • Internal coordinates: Eckart or Geometrically defined • Exact nuclear kinetic energy operator • within the Born-Oppenheimer approximation • Vibrational motion represented either by • Finite Basis Representation (FBR) or • Grid based Discrete Variable Representation (DVR) • Solve problem using Variational Principle • Potentials either ab initio or from fitting to spectra

7. Variational approaches • Treats vibrations and rotations at the same time • Interpret result in terms of potentials • Only assume rigorous quantum numbers: • n, J, p, symmetry (eg ortho/para) • Give spectra if dipole surface available • Include all perturbations of energy levels and spectra • Yield models that can be transferred between isotopomers Provide a complete theoretical treatment with no assumptions

8. Internal coordinates: Orthogonal coordinates for triatomics Orthogonal coordinates have diagonal kinetic energy operators. Important for DVR approached

9. Hamiltonians for nuclear motion Laboratory fixed: 3N coordinates Translation, vibration, rotation not separately identified Space fixed: remove translation of centre-of-mass 3N-3 coordinates Vibration and rotation not separately identified Body fixed: fix (“embed”) axis system in molecule 3 rotational coordinates (2 also possible) 3N-6 vibrational coordinates (or 3N-5)

10. Hamiltonians for nuclear motion Laboratory fixed: Useless for variational calculations due to continuous translational “spectrum”. Used for Monte Carlo methods. Space fixed: Requires choice of internal coordinates. Vibration and rotation not separately identified. Widely used for Van der Molecules. Body fixed: Requires choice of internal axis system. Vibrational and rotational motion separately identified. Singularities! New Hamiltonian for each coordinate/axis system Same for J=0

11. Diatomic molecules: 1 vibrational mode stretch Hamiltonian: Numerical solution: trivial on a pc Eg LEVEL by R J Le Roy, University of Waterloo Chemical Physics Research Report CP-642R (2001) http://scienide.uwaterloo.ca/~leroy/level/

12. Triatomics: 3/4 vibrational mode 3 degrees of freedom (4 for linear molecules) New mode: bend Hamiltonian: many available, some general Numerical solution: general programs available Eg BOUND, DVR3D, TRIATOM See CCP6 program library http://www.dl.ac.uk/CCP/CCP6/library.html

13. Tetratomics 6 vibrational degrees of freedom New mode: torsion New mode: umbrella Hamiltonian: available for special cases Numerical solution: results for low energies No published general programs

14. Pentatomics 12 degrees of freedom New modes: book, ring puckering, wag, deformation, etc Hamiltonian: for very few special cases eg XY4 systems, polyspherical coordinates (polyspherical coordinates are orthogonal coordinates formed by any combination of Radau and Jacobi coordinates) Numerical solution: almost none (CH4)

15. Vibrating molecules with N atoms 3N-6 degrees of freedom Modes: all different types Hamiltonian: not generally available but see J. Pesonen, Vibration-rotation kinetic energy operators: A geometric algebra approach, J. Chem. Phys., 114, 10598 (2001). Numerical solution: awaited for full problem But MULTIMODE by S Carter & JM Bowman gives solutions for semi-rigid systems using SCF & CI methods plus approximations http://www.emory.edu/CHEMISTRY/faculty/bowman/multimode/

16. Triatomics: general form of the Born-Oppenheimer Hamiltonian KV vibrational kinetic energy operator KVR vibration-rotation kinetic energy operator (null if J=0) V the electronic potential energy surface Steps in a calculation: choose… • …a potential (determines accuracy) • …coordinates (defines H) • …basis functions for vibrational motion

17. Effective Hamiltonian after intergration over angular and rotational coordinates. Case where z is along r1 Vibrational KE Vibrational KE Non-orthogonal coordinates only Rotational & Coriolis terms Rotational & Coriolis terms Non-orthogonal coordinates only Reduced masses (g1,g2) define coordinates

18. General coordinates r2 q r1 Choice of g1 and g2 defines coordinates

19. Body-fixed axes: Embeddings implemented in DVR3D r2 embedding r1 embedding bisector embedding (d)NEW! z-perpendicular embedding

20. Basis functions. General functions: Floating spherical Gaussians Non-orthogonal Stretch functions: Morse oscillator (like) Harmonic oscillators Spherical oscillators, etc Must be complete set Problems as R 0 Bending functions: Associate Legendre functions Jacobi polynomials Coupling to rotational function ensures correct behaviour at linearity Rotational functions: Spherical top functions, DJMK Complete set of (2J+1) functions

21. Performing a Variational Calculation: • Construct individual matrix elements • Construct full Hamiltonian matrix • Diagonalize Hamiltonian: get Ei and

22. Matrix elements Hnm = < n | T + V | m > Can often obtain matrix elements over Kinetic Energy operator analytically in closed form For general potential function, V, need to obtain matrix elements using numerical quadrature For Polynomial basis functions, Pn, use M-point Gaussian quadrature to give Points, xi, Weights, wi < n | V | m > = Si wi Pn(xi) Pm(xi) V(xi) Scales badly (~MN) with number of modes, N

23. Grid based methods Discrete Variable Representation (DVR) uses points and weights of Gaussian quadrature. Wavefunction obtained at grid of points, not as a continuous function. DVR is isomorphic to an FBR

24. DVR versus FBR • DVR advantages • Diagonal in the potential (quadrature approximation) • < a| V | b > = dab V(xa) • Sparse Hamiltonian matrix • Optimal truncation and diagonalization • based on adiabatic separation • Can select points to avoid singularities • DVR disadvantages • Not strictly variational (difficult to do small calculation) • Problems with coupled basis sets • Inefficient for non-orthogonal coordinate systems Transformation between DVR and FBR quick & simple

25. Matrix diagonalization • Matrices usually real symmetric • Diagonalization step rate limiting for triatomics, a N3. • Intermediate diagonalization and truncation • major aid to efficiency. Iterative versus full matrix diagonalizer • Is matrix sparse? • How many eigenvalues required? • Are eigenvectors needed? • Is matrix too large to store?

26. Rotational excitation • 2J+1 spherical top functions, DJkM, form a complete set. • Rotational parity, p, divides problem in two: • Two step variational procedure essential for treating high J: • First step: diagonalize J+1 “vibrational” problems assuming • k, projection of J along z axis, is good quantum number. • Second step: diagonalize full Coriolis coupled problem • using truncated basis set. • Can also compute rotational constants directly as expectation values.

27. Transition intensities • Compute linestrength as • Sij = |St < i | mt | j >|2 • where m is dipole surface (not derivatives) • | i > and | j > are variational wavefunctions • Rotational and vibrational spectra at same time • Only rigorous selection rules: • DJ = +/- 1, p = p’ • DJ = 0, p = 1- p’ • (ortho  ortho, para  para). • All weak transitions automatically included. • Best done in DVR • Expensive (time & disk) for large calculation • More accurate than experiment?

28. The DVR3D program suite: triatomic vibration-rotation spectra Potential energy Surface, V(r1,r2,q) J Tennyson, NG Fulton & JR Henderson, Computer Phys. Comm., 86, 175 (1995). Dipole function m(r1,r2,q)

30. Why calculate VR spectra? • Test potential energy surfaces • construct potentials • Predict assign spectra • lab, astronomy, etc • Calculate transition intensities • physical data from observed spectra eg n, T,….. • atmospheric studies, astrophysics, combustion …. • Generate bulk data • partition functions  specific heats, opacities • JANAF, astrophysics, etc • Link with reaction dynamics • eg HCN  HNC • H3+ + hn  H2 +H+ • Quantum ``chaology'' • Classical dynamics of highly excited molecules is chaotic

31. Potentials:Ab initioorSpectroscopically determined

34. M-Dwarf Stars Oxygen rich, cool stars: T = 2000 – 4000 K Spectra dominated by molecular absorptions H2O, TiO, CO most important Water opacity Viti & Tennyson computed VT2 linelist: All vibration-rotation levels up to 30,000 cm-1 Giving ~ 7 x 108 transitions

35. Absorption by steam at T = 3000 K Ludwig 3.0 Hitran linelist 2.0 Absorption (cm-1 atm-1 at STP) 1.0 0.0 0 500 1000 Frequency (cm-1) JH Schryber, S Miller & J Tennyson, JQSRT, 53, 373 (1995)

36. The Sun: T = 5760 K

37. Sunspots T=3200K H2, H2O, CO, SiO T=5760K Diatomics H2, CO, CH, OH, CN, etc Molecules on the Sun Sunspots Image from SOHO : 29 March 2001

38. Sunspot, T ~ 3200 K Penumbra, T ~ 4000 K

39. Sunspot: N-band spectrum Sunspot lab L Wallace, P Bernath et al, Science, 268, 1155 (1995)

40. Assigning a spectrum with 50 lines per cm-1 • Make ‘trivial’ assignments • (ones for which both upper and lower level known experimentally) • 2.Unzip spectrum by intensity • 6 – 8 % absorption strong lines • 4 – 6 % absorption medium • 2 – 4 % absorption weak • < 2 % absorption grass (but not noise) • 3.Variational calculations using ab initio potential • Partridge & Schwenke, J. Chem. Phys., 106, 4618 (1997) • + adiabatic & non-adiabatic corrections for Born-Oppenheimer approximation • 4.Follow branches using ab initio predictions • branches are similar transitions defined by • J – Ka = na or J – Kc = nc, n constant Only strong/medium lines assigned so far OL Polyansky, NF Zobov, S Viti, J Tennyson, PF Bernath & L Wallace, Science, 277, 346 (1997).

41. Sunspot: N-band spectrum Sunspot Assignments lab L-band & K-band spectra also assigned

42. Variational calculations: Assignments using branches Spectroscopically Determinedpotential Accurate but extrapolate poorly Error / cm-1 Ab initio potential Less accurate but extrapolate well J

43. The Future: • PDVR3D: • DVR3D program for parallel computers, • Eg Cray-T3E or IBM SP2 • H2O • All J = 0 states to dissociation (> 1000 states) • 20 minutes wallclock on 64 Cray T3E processors • All J > 0 up to dissociation. Scales as (J+1). • Needs reliable potentials! HY Mussa and J Tennyson, J. Chem. Phys., 109, 10885 (1998).