The ‘Multimode’ Approach to Challenging Problems in Vibrational Spectroscopy Joel M. Bowman, Stuart Carter, Xinchuan Huang and Nicholas Handy Important collaborators: Bastiaan Braams and Anne McCoy Support from the Office of Naval Research and the National Science Foundation

The challenges of ab initio spectroscopy 1. General and practical methods to solve the Schroedinger equation for nuclear motion 2. General and practical methods to obtain high quality ab initio-based potentials and dipole moment surfaces to use in step 1

Acetylene Hamiltonian in valence coordinates      H H C C r CH 1 r CH 2 r CC M. Bramley and N. C. Handy, J. Chem. Phys. 98, 1378 (1993)

Kinetic Energy Operator Coded in “RVIB4” Colwell, Carter and Handy

Jacobi Coordinates

Hamiltonian See CCP6 Library for a 4-atom code in Jacobi- Law,Tennyson, Hutson

Normal Coordinates Redux In the 80s and 90s great progress was made in using curvilinear coordinates. Triatomics are a “solved problem”. Extensions to tetraatomics made in the 90s, but choice of coordinates became problem-specific, e.g., bond lengths and angles, umbrella, Radau, Jacobi, polyspherical, etc. Kinetic energy operators are coordinate-specific and can be quite complex. Not clear how this approach would extend beyond 4 atoms or special case pentatomics, e.g., CH 4. “RVIB3” (Carter-Handy), “RVIB4” (C-H-Colwell), “DVR3D” (Tennyson), see CCP6, Carrington, Guo, Bowman, Leforstier, Light, etc. Bring back Normal Coordinates and the Hamiltonians based on them

Start with the Watson Hamiltonian Issues: Huge matrices and large dimensional integration

The Hamiltonian Matrix Huge matrices and large dimensional integration Numerical quadrature done in N dimensions? e.g., 6 dimensions for H 3 O +, 9 for H 3 O 2 -, 15 for H 5 O 2 + Matrix dimension is ca 10 N e.g., 10 6 for H 3 O +, 10 9 for H 3 O 2 -, for H 5 O 2 +

MULTIMODE w/Stuart Carter at Emory 1996 The one-mode representation of the potential has V (1) terms. i.e., “cuts” through the hyperspace of normal coordinates with one coordinate varying. The two-mode representation contains those terms plus the V (2) terms, etc. n-mode representation of the potential

How this helps So only have to do 3 dimensional quadratures and the matrix is quite sparse, i.e., lots of zeros. Dimensionality of quadrature space is nMR in general and matrix fills “slowly” as n increases

n-mode representation in MULTIMODE For an n-mode representation of V in a problem with N modes there are N!/[n!(N-n)!] grids of dimension n. Numerical integration is over these grids. Thus, the dimensionality of quadratures is n < N; currently n max = 6 Many matrix elements are zero. The potential may be directly calculated on these quadrature grids or on sparser grids and interpolated. Highly parallel procedure

H-Matrix Dimension Direct diagonalization feasible for dim ≈ 20,000 Iterative methods used for dim up to 100,000 Use symmetry to block diagonalize H Pick reference geometry to exploit max symmetry

MM should be exact for triatomics a Partridge and Schwenke (1997 ) b MM - Carter and Bowman (1998). H 2 O J = 3

Vibrational energies of CH 4 (J=0) a State H.O. 2MR3MR4MR Radau b ZPE 4 (F 2 ) 2 (E) 1 (A 1 ) (F 2 ) a Carter and Bowman, J. Chem. Phys. (1999) using Taylor-Lee-Martin ab initio force field. b Yu, J. Chem. Phys. (2003) Recent benchmark calculations by Carrington using Radau

What about V? H 2 CO  H 2 +CO, H+HCO JPC (2004) (OH - )H 2 O JACS (2004) H 5 O 2 + JCP (2004) (H 2 O) 2 (done) O( 3 P)+C 3 H 3 JTCC (2005) CH 5 + JCP (2003), in progress C( 3 P, 1 D)+C 2 H 2 in progress H 5 + JCP (2005) CH 3 OH (done) MP2,CCSD(T), MRCI ab initio calculations done (MOLPRO) on grids or using “direct dynamics”. Fits in inter- nuclear distances enforcing permutational symmetry. ca ab initio energies feasible on our 100 cpu cluster - DURIP “Multinode”

Floppy hydrated systems H 3 O + (Huang, Carter, … ) H 3 O 2 - (Braams, Huang, Carter, McCoy*,…) H 5 O 2 + (Braams, Huang, Carter, McCoy*,…) *DMC calculations, “easy” for the ground state, but require the real McCoy (expertise) for excited states.

H3O+H3O+ Ammonia like inversion motion - a challenge for MM Where to put the reference geometry/what normal modes to use? We picked the saddle point - D 3h and had to use a large grid in the imaginary frequency mode to span the two minima.

H3O+H3O+ Ab initio-based (CCSD(T)) potential energy surface in two modes - contour values in cm -1. Using a form for the potential developed by Leonard, Carter and Handy for NH 3

Comparison of calculated and experimental vibrational splittings (cm -1 ) in H 3 O + and D 3 O + using MM and RVIB4 H3O+H3O+ D3O+D3O+ Halonen and co-workers have developed an even more accurate PES and get better agreement with experiment. Exp - Oka, Sears, Saykally Theory - X. Huang, S. Carter and J. M. Bowman, J. Chem. Phys. 118, 5431 (2003). [J >> 0 Chakrobarty, Truhlar, Bowman, Carter (2004)]

David Nesbitt, private communication, JCP (2005)

H 3 O 2 - aka (OH - )H 2 O Experiments by Johnson and co-workers Full dim PES (Huang, Braams, Bowman) MM-Reaction Path version* (Huang, Carter,JMB) DMC calculations (McCoy) *Carter and Handy (2002), Miller, Handy, Adams (1988), Hougen, Bunker, Johns (1970)

OH - [H 2 O] Exp and Previous Theory H 3 O 2 -  H 2 O + OH - Structure and HO:Xantheas (1995) Estimated binding energy: 8,300 – 9,900 cm -1 H-atom transfer barrier: 50 – 100 cm -1 Klopper (2002) H delocalization : 300 K, Tuckerman, Parrinello (1997) Ar-Predissociation IR spectrum: Johnson et al. (2002) Potential minimum C 1 symmetry

H-transfer barrier and  HO in H 3 O 2 - Barrier height = 74 cm -1 in good agreement w/Sampson & Klopper Mod e SP621i MIN Results at MIN are close to Xantheas’ 1995 MP2 calculations and in rough “agreement” with exp. But HO results at SP disagree with experiment. IR intensities also disagree.

H 3 O 2 - PES CCSD(T)/aug-cc-pVTZ energies Using MOLPRO Use MM-grids to generate configs Fit (Braams) using a basis that is symmetric wrt interchange of like atoms - use all internuclear distances as basic variables; Morse, SPF, etc coefficients (incomplete 7 th order terms) Standard Deviation : 16 cm -1 Average fitting error : 9 cm -1 Range (cm -1 )No. of PointsRMS Error (cm -1 ) 0 – – – – –

Torsional potential (cm -1 ) along C 2 -path (Like H 2 O 2 with a bridging H-atom-w/smaller torsional barriers) ZPE is much greater than the barrier separating the two equivalent minima.

Harmonic Frequencies (cm -1 ) along C 2 -path H3O2-H3O2- D3O2-D3O2- OO-stretch/wag “avoided crossing” OO-stretch/wag “avoided crossing” absent for D 3 O 2 -.

Bridging H-atom stretch The double HO approx gives very small intensity for this mode at the min. But a much larger intensity at the saddle point. (But an imaginary frequency.)

H 5 O The Current Limit

Potential and dipole moment in full dimensionality Vibrational calculations -MM and DMC-McCoy Experiments (Johnson group) ? JCP (in press)

Erratum: JMB Talk on Tuesday (Bowman instantly had second thoughts but it was too late…) Bowman stated that no experimental spectra had been reported for H 5 +