Outline

1. Introduction

2. User community and accuracy needs

3. Which large-amplitude motions

4. Which tools = which sym. operations

5. Example (in progress) C2H3+

High-Barrier Multi-DimensionalTunneling Hamiltonians

Acknowledgements

Main collaborators on theoretical development of tunneling Hamiltonians 1980-2009

Nobukimi Ohashi (Japan) since 1984Laurent H. Coudert (France) since 1986

Yung-Ching Chou (Taiwan) since 2003Melanie Schnell (Germany) since 2006

High-Barrier Multi-DimensionalTunneling Formalism

Goal

To parameterize high-resolution spectra of

floppy molecules to experimental accuracy

without knowledge of the potential surface

Diatomic Example: Parameterize rotational levels as BJ(J+1) – DJ2(J+1)2 instead of starting from an ab initio potential curve V(r)

High-Barrier Multi-Dimensional Tunneling Formalism

The Anti-Quantum-Chemist

The Anti-Quantum-Chemist is:

A great antagonist expected to fill the world with wickedness but to be conquered forever by the Quantum Chemist at his second coming.

(from a reliable dictionary)

User Community Hierarchy

High-barrier tunneling Hamiltonians

Spectrum Analyzers v J Ka Kc

Radio Astronomers , E, Int, molecule

Physical Effects People splittings =E = E(vJ Ka Kc ) - E(vJ Ka Kc ) torsion, inversion, H-transfer, BO failure

Primary Users

Secondary Users

Parameterize to measurement accuracy

Modern microwave measurements: 20 000.002 MHz or 200 000.02 MHz1 part in 107 = 8 significant figures

Requires energy calculations: with 8 + 1 + 1 = 10 significant figures = E' E" and no computational error

Molecule List + How to Classify LAMs

Use Vocabulary from Reciprocating Engine:

(converts) to-and-fro of piston (to) circular motion of crankshaft

Piston = N atom inversion, H atom transfer, H-bond exchange Crankshaft = CH3 top, OH2 top, H migration

If Point Group, Permutation-Inversion Group, and types of motion are the same, thenthe Tunneling Hamiltonian is the same.

Molecule Year Groups LAMs

H2N-NH2 1981 C2 G16(2) 1 torsion + 2 inversion

HO-OH 1984 C2 G4(2) 1 torsion

HF-HF 1985 Cs G4(2) 1 H-bond exchange

H2O-H2O 1985 Cs G16 2 tors + 1 H-b exch CH3-NH2 1987 Cs G12

(m) 1 torsion + 1 inversionC2H3

+ 1987 C2v G12 1 H-migration (tor)(HCCH)2 1988 C2v G16 1 torsion CH3NHD 1990 C1 G6 1 torsion + 1 inversion(CH3OH)2 1993 C1 G36 2 torsion + 1 H-bond + 1 lone pair exchange (H2O)2H+ 1994 C2 G16

(2) 1 tor + 2 inversionsNa3 1997 C2v G12 1 pseudorotation (tor)

(CH3O)2P(=O)CH3 2002 C1 G18 2 torsion + 1 “inversion”

CH3CHO S1 2004 C1 G6

1 torsion + 1 inversion CH=OCH3-C H 2006 Cs G12

(m) CH O 1 torsion + 1 H transfer

(CH3)3SnCl 2008 C3v G162 3 torsions

cis/trans HCCH 2010 C2h G4(8)

1 torsion + 2 LAM bends

Tools

1. Point group at equilibrium the number of frameworks

2. Tunneling Hamiltonian matrix itself

3. Permutation-Inversion group symmetry as transformation of variables

4. Time reversal complex conjugation

5. Extended Permutation-Inversion groups

Framework (E.B.Wilson 1935):

Ball-and-stick model with atoms labeled

Different frameworks cannot be oriented to make all atom labels match.

HaOHb has 1 frameworkNHaHbHc has 2 frameworks (inversion)HaCbCcHd has 2 frameworks (break CH bonds)

CH3NH2 has “6” frameworks (no bond breaking)

6 Benzene out-of-plane orbitals

E W 0 0 0 WW E W 0 0 0 0 W E W 0 0 0 0 W E W 0 0 0 0 W E WW 0 0 0 W E

H =

Tunneling matrix has three kinds of elements:non-tunneling E, tunneling splitting W, and 0

|1;pz

|2;pz

|3;pz

|4;pz

|5;pz

|6;pz

Some LAM-Rotation Tunneling Concepts:

Use for LAM angular (periodic) variablesand for LAM to-and-fro () motions

Framework basis functions = v,K set of fcns|v(s,s)n|KJMn for each framework n

Htun = Tunneling Hamiltonian = i ViRi = f0(s,s) + f1x(s,s)Jx + f1y(s,s)Jy + f1z(s,s)Jz + f2x(s,s)Jx

2 + f2xy(s,s)(JxJy+JyJx) + …

Tunneling matrix elements = Tn =|v(s,s)1|KJM1 |Htun||v(s,s)n|KJMn

1 2 3 4 …… n

Hrot T2 T3 T4 …… Tn

T2† Hrot Ti Tj …… Tk

T3† Ti

† Hrot Tp …… Tq

………………………………………………………………………………………

Tn† Tk

† Tq† …….… Hrot

Framework #

Hrot dimension(2J+1)(2J+1)

Tn = 1 ntunneling matrix(2J+1)(2J+1)

We want some Tn = 0

Multi-Dimensional Tunneling Hamiltonian Matrix of dimension: (2J+1)n(2J+1)n

P puts 1JK'nJK" element also in anotherposition in H (with possible phase change)

1;J,K'|Htun|n;J,K" = P1;J,K'|Htun|n;J,K" = = P1;J,K'|PHtun|Pn;J,K" == (-1)K'+K"p;J,-K'|Htun|q;J,-K"

Permutation-Inversion symmetryoperations as coordinate transformationsExample for PI operation P:

P (1,2,1,2,,,) == (1+2/3,2-2/3,-2,+1,-+,,)

Molecule Year Lines/Parameters/rms

H2N-NH2 1981HO-OH 1984HF-HF 1985H2O-H2O 1990 173 / 22 / 90 kHz (47)CH3-NH2 2004 1523IR+MW / 53 /18 kHz(346MW)C2H3

+ 2009 322 / 11 / 0.05 cm-1 (HCCH)2 1988CH3NHD 1990(CH3OH)2 1999 (H2O)2H+ 1994 Na3 1997

Year Lines/parameters/rms

(CH3O)2P(=O)CH3 2003 609 / 54 / 8 kHz

CH3CHO S1 2004 136 / 11 / 0.002 cm-1

CH=O CH3-C H 2008 2578 / 37 / 15 kHz CH O

(CH3)3SnCl 2008

cis/trans HCCH 2010

H2 H1CaCbH3

H1

H3CaCbH2

H3

H2CaCbH1

H1CaCbH2

H3

H3CaCbH1

H2

H2CaCbH3

H1

H migration in C2H3+

Parameterization of H-migration tunneling splittings is similar to the parameterization of methyl-top internal rotation splittings

Int. rot. splittings Fa1 cos(2/3)(K-)

H-migration split. h2 2

amplitude periodicity

These two parameters and their higher-order J and K corrections occur in the off-diagonal matrix elements of the tunneling Hamiltonian

Six frameworksNearest neighbor tunneling only 1 2 3 4 5 6

1 Hrot T2 0 0 0 T2 2 T2

† Hrot T2 0 0 0

3 0 T2† Hrot T2 0 0

4 0 0 T2†

Hrot T2 0 5 0 0 0 T2

† Hrot T2 6 T2

† 0 0 0 T2† Hrot

C2H3+ Tunneling Matrix

(water dimer formalism)

size ofHrot T2

(2J+1)x(2J+1)

C2H3+ Data

IR: Crofton, Jagod, Rehfuss, Oka, J. Chem. Phys. 91 (1989) 5139

MW: Bogey, Cordonnier, Demuynck, Destombes, Ap.J. 399 (1992) L103

C2H3+ Tunneling Theory

K = 0: Hougen, JMS 123 (1987) 197

K = 0,1: Cordonnier, Coudert JMS 178 (1996) 59

Laurent Coudert’s tunneling treatmentuses h2, 2 parameters to fit 3 splittings:

Cordonnier, Coudert JMS 178 (1996) 59

Transition Obs. Splitting Calc. Splitting

909 – 818 -0.350(30) -0.341 MHz110 – 101 -0.480(20) -0.492 MHz312 – 303 -0.350(20) -0.351 MHz5 lines < 0.3 < 0.26 MHz

Obs. MW splittings from Lille Bogey, Cordonnier, Demuynck, Destombes Ap.J. 399 (1992) L103-L105

Can we use Coudert’s existing program to fit Oka et al.’s 500 assigned lines in the 3 C-H stretching region ???

Gabrys, Uy, Jagod, Oka, Amano on C2H3+

in J. Phys. Chem. 99 (1995) 15611, say:

“We hope that the observed splittings shown in Table 4 and Figure 3 will be quantitatively explained some day.”

Unknown factor:

What is the magnitude of randomvibrational perturbations at 3 ?

Tunneling splittings 0.1 to 0.2 cm-1

If perturbations < 0.05 cm-1, try to fit.

If perturbations > 0.2 cm-1, it’s hopeless.

Next slide shows attempt to fit K = 0 splittings.K = 1 and 2 fits are similar, but with differentamplitudes, damping, and periods.

-0.2000

0.0000

0.2000

0.4000

0 4 8 12 16 20 24

J value

cm-1

C2H3+ K = 0 A-E Splittings

Calculated Observed

Conclusion (as of 20 June 2009):

We have in hand a theoretical high-barrier tunneling-formalism fitting program for Oka’s C2H3

+ H-migration rotational energy levels.

Before this Columbus meeting, there were some problems with the fit. At this Columbus meeting Laurent Coudert, Takeshi Oka, and I (a) were able to …(b) were not able to …