A tour of the vibrational motion of some unusual molecules

• Acteylene/vinylidiene

• H3O+

Acknowledgements: Shengli Zou, Stuart Carter, Xinchuan Huang, Jaime Rheineker, Alex Brown,Department of Energy and Office of Naval Research

Exp

eri

men

ts K. M. Ervin, J . Ho, and W. C. Lineberger, J . Chem.Phys. 91, 5974 (1989).

M. P. Jacobson, J . P. O'Brian, R. J . Silbey, and R.W. Field, J . Chem. Phys. 109, 121 (1998); (b) M.P. Jacobson and R.W. Field, J . Phys. Chem A,104, 3073 (2000).

J . Levin, H. Feldman, A. Baer, D. Ben-Hamu, O.Heber, D. Zajfman, and Z. Vager, Phys. Rev. Lett.81, 3347 (1998).

In search of the vinylidene needle in the acetylene

haystack

In search of the vinylidene needle

in the acetylene haystack

Ervin, Ho and Lineberger (1989)

H2C2- + hv “vinylidene”+ e-

Active Normal modes of Vinylidene

4

C3

1

C2C2

4

C3

1

6- rocking 6 - scissors

J. Levin, H. Feldman, et al.

Phys. Rev. Lett. 81, 3347 (1998).

H2C2- + hv “vinylidene”+ e-

Study of Unimolecular Reactions by Coulomb Explosion Imaging: The Nondecaying Vinylidene

“The data analysis given here shows unambiguouslythat a large part (,50%) of the molecules measured3.5 ms after their production as vinylidene isomers retainsthe vinylidene geometry. This is inconsistent with thegenerally accepted concept of the vinylidene being ashort-lived isomer which decays into the linear isomerwithin a few picoseconds.”

Th

eory

an

d

Calc

ula

tion

s

T. Carrington Jr., L. M. Hubbard, H. F. Schaefer, III, and W. H. Miller (1984).

Rxn path Hamiltonian, semi-classical tunneling estimate of lifetime 1 ps.

N-y Chang, M-y. Shen, and C-h. Yu, (1997)

CCSD(T) and CBS limit calculation of geometries, barrier height and energetics.

J . F. Stanton and J . Gauss (1999)

Ab initio and PT calculations of anharmonic effects in vinylidene and vinylidene

anion.

S. Carter, I. M. Mills, and J. N. Murrell (1980); .L. Halonen, M.S. Child, and S.

Carter (1982); J . F. Stanton, (2000)

Six degree of freedom potential energy surface

T. Germann and W. H. Miller (1998).

Isomerization "resonances" in v-a using negative imaginary potential

R. Schork and H. Köppel (2001)

Wavepacket calculations in 5 dof but using a negative imaginary potential. CCD(T)

ab initio calculations.

R. L. Hayes, E. Fattal, N. Govind E. A. Carter (2001).

Ab initio molecular dynamics calculations for fully deuterated system.

Tunneling Picture -unimolecular decay

16058 cm -1

900 cm -1

“Isomerization coordinate”

“Barrier recrossing in the vinylidene–acetylene isomerization reaction:A five-dimensional ab initio quantum dynamical investigation”

Rainer Schork and Horst Koeppel (2001)

5 dof wavepacket calculations of vinylidene with an absorbing potential just beyond TS - new ab initio calculations

Summary of calculations 2001

• Direct dynamics (classical) C2D2 - much re-crossing

of the isomerization barrier• RD Time-dependent wavepacket, with absorbing

potential - long-lived states. (“Good” agreement with exp photodetachment spectrum with artificial broadening.)

• Better characterization of energetics and saddle point

What would an exact quantum calculation tell us?Is such a calcualation feasible?

Acetylene Exact Hamiltonian

H

H

C C

rCH1 rCH2

rCC

M. Bramley and N. C. Handy, J. Chem. Phys. 98, 1378 (1993)

Acetylene Exact Hamiltonian

Acetylene/ Vinylidene Coordinates

r1

r2

φ

R

H

H C

C

Acetylene/ Vinylidene Isomerization

C’ C

H

H

rHH

H

H

C’

C2R

rcc

H

H

C’

C

rcc

rHH

rHH

rcc

1. Isomerization “easy” to describe 2. Permutational symmetry easy to incorporate3. Hamiltonian is relatively simple

Acetylene/ Vinylidene Energetics

16058

900

“Isomerization coordinate”

Energetics of the Potential Surface

Acetylene Vinylidene Saddle Point

Energy/cm-1 0 16058 16919

R/bohr 0 2.25 1.77

/deg - 0 (8 0) 3(58)

r(rHH)/bohr 6.8 3.54 4.5

r(rCC)/bohr .30 .46 .4

a Potentia l du e to Stanto n base d o n th e potent ial Carte r et al.

Exact Hamiltonian (J=0)

r1

r2

φ

R

H

H C

C

H = TR

+ Tr

+ Tr

+

j

μR

R

+

j

μ

r

+

j

μ

r

+ V (

,

, φ , R , r

, r

)

r

j

=

r

j

+

r

j

Diagonalization of H

Let Hop be the Hamiltonian operator and

Let {} be a complete orthonormal basis

€

Hi , j = φi Hop φ j

Always use a finite size basis, say N. Then

the H-matrix is N x N. For a 2-variable problem,

the direct-product space is of order N1xN2 and

the order of H is N1xN2. Thus if we used

10 functions per mode for a six degree-of

freedom problem the order would be 106.

Challenges

Guo and co-workers (2002)

Used force-field (no vinylidene), eigenvalues (only)

Direct-product grid (DVR),H-matrix of order 44 x 106

Reduced to 11 x 106 using symmetry. Lanczos method

Used to get eigenvalues only up up to 13000 cm-1.

CPU time: 90 hours on a DEC alpha EV6 workstation

Large-amplitude dofs: three angular, R, rHH

Density of states at 20000 cm-1: ~10 per cm-1

Our Diagonalization Strategy

• Don’t aim for spectroscopic accuracy

• Succesive diagonalization method

• Matrix diagonalizations of order 104

• Check robustness of results

Investigate the nature of molecular eigenstates above the threshold for isomerization to vinylidene

H3D =

r j 12

2

2μRRcut2 +

r j 1

2

2μ1r1e2 +

r j 2

2

2μ2r2e2

€

+V3D (θ1,θ2,φ,Rcut )

yj1,j2j12,K=0

θ1,θ2,φ( ) = C(j1j2j12;m1−m1K =0m1

∑ )Yj1m1(θ1,0)Yj2−m1(θ2,φ)

The 3 dof Hamiltonian J = 0

The angular basis

Make linear combinations that are eigenfunctions of parity and then use the symmetry in CC-HH

V3D is from the full potential with R fixed at Rcut and minimized with respect to rHH and rCC

Four dof Hamiltonian

€

H 4 D = H 3D +r j 12

2 1

2μ R R2−

1

2μ R Rcut2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟−

1

2μ R

∂ 2

∂R2

€

+V4 D (θ1 ,θ 2 ,φ ,R)

€

−V4 D (θ1 ,θ 2 ,φ ,Rcut )

Combine 3D angular eigs with sine

basis in R and diagonalize H4D

€

H 2 D = −1

2μ r1

∂ 2

∂r12

−1

2μ r2

∂ 2

∂r22

+V2 D (r1 ,r2 )

Two dof Hamiltonian

Use a 1d cut for CC and generally no potential for HH, use sine basis instead.

Final Step

Combine 4D eigs of H4D with 2D eigs

of H2D

€

H 6D = H 4 D + H 2 D

€

+ r

j 12 1

2μ r1r1

2−

1

2μ r1r1e

2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+

r j 2

2 1

2μ r2r2

2−

1

2μ r2r2 e

2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

+V6D (θ1 ,θ 2 ,φ ,R,r1 ,r2 )

€

−V4 D (θ1 ,θ 2 ,φ ,R)

€

−V2 D (r1 ,r2 )

Need ca 100 2D x 300 4D = 30 000.

Diagonalize in the middle of the 4D basis

Test of the new codeLow-lying states of acetylene

Assignment Present Guo Present Diff

00000 5702.2 5702.9 0.0 0.7

00020 6909.4 1207.2 1206.5 0.8

00002 7042.0 1339.8 1339.4 0.4

01000 7676.0 1973.8 1973.8 -0.0

00040 8158.3 2456.1 2453.5 2.7

00022 8191.5 2489.4 2486.6 2.7

00022 8288.2 2586.1 2579.4 6.7

00004 8354.8 2652.7 2650.9 1.8

01020 8872.4 3170.2 3167.9 2.3

01002 8989.6 3287.4 3286.1 1.3

10000 9051.8 3349.6 3349.3 0.3

11111 9366.4 3664.2 3644.7 19.5

11111 9397.8 3695.7 3682.3 13.3

Results I.

rHH 6.28 acetylene 3.54 vinlyidene

R 0 acetylene2.25 vinlyidene

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

6000 7000 8000 9000 10000 11000 12000 13000

Length (bohr)

E (cm-1

)

<rHH

>

<R>

(a)

Results II.

rHH 6.28 acetylene 3.54 vinlyidene

R 0 acetylene2.25 vinlyidene1.0

2.0

3.0

4.0

5.0

6.0

7.0

21000 21500 22000 22500 23000

Length (bohr)

E (cm-1

)

<rHH

>

<R>

(b)

Wavefunction Plots (R,rHH)

2.50 3.75 5.00 6.25 7.50

3.5

3.0

2.5

2.0

1.5

1.0

0.5

rHH (bohr)

E = 21680 cm-1

2.50 3.75 5.00 6.25 7.50

3.5

3.0

2.5

2.0

1.5

1.0

0.5

rHH (bohr)

E = 21709 cm-1

Wavefunction Plots (R,2)

-1.0 -0.5 0.0 0.5 1.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

cos (2)R (bohr)

E = 21680 cm-1

-1.0 -0.5 0.0 0.5 1.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

cos (2)R (bohr)

E = 21709 cm-1

Simulated Photodetachment SpectraStanton-Carter potential has incorrect vinylidene CC-

stretchso we did a new potential surface, just submitted to CPL

Shengli Zou and Joel M. Bowmana

A new ab initio potential energy surface describing

acetylene/vinylidene isomerization

A new potential energy surface for C2H2 that describes

acetylene/vinylidene isomerization is reported. The surface is an

accurate, least-squares fit to nearly 10000 symmetry-equivalent, ab initio

electronic calculations obtained at the CCSD(T) level of theory, with an

aug-cc-pVTZ basis. The ab initio geometries and normal-mode

frequencies of the acetylene and vinylidene minima, and saddle point are

reproduced very well by the fitted potential energy surface. Full-

dimensional calculations of low-lying acetylene vibrational energies are

also reported using a new code and compared to experiment.

All calculations on S-C potentialhave been re-done on new

potential

0

1

2

3

4

5

6

7

20000 20400 20800 21200 21600 22000

RrHH

Energy (cm-1)

Leng

th (

boh

r)

Some conclusions• Acetylene/vinylidene isomerization is a symmetric

double well

• Molecular eigenstates with vinylidene character

exist

• Doublet structure exists (ground state splitting is a few

cm-1)

• QM study of highly excited states of tetratomics is

possible in

full dimensionalitySome open questions

• How extensive is the vinylidene “spectrum”?• What are signatures of vinylidene states?• Is the double well and all the symmetry responsible

for the ‘divided’ spectrum

Proton transfer in waterthe ‘Zundl’ ion H5O2

+

H3O+ + H2O -> H2O + H3O+

O

2.259Å

1.827Å

O

1.831Å

The hydronium ion H3O+

Inversion doublets” in the spectrum observed and calculated

By our group for the first time in full dimensionality (2002)

MULTIMODEBased on “Watson Hamiltonian” - normal coordinates

and the following crucial representation of the potential

€

V(Q1 ,Q2 ,...,QN ) = Vi(1)(Qi )

i

∑ + Vij(2 )

ij

∑ (Qi ,Q j )+ Vijk(3)(Qi ,Q j ,Qk )

ijk

∑ + Vijkl(4 )

ijkl

∑ (Qi ,Q j ,Qk ,Ql )+ ...

• Vibration self-consistent field• “Virtual” state CI• Check convergence wrt above representation• “No limits”• Needs a reference geometry Usually a minimum, but saddle points ok

New Potential

€

V (r1,r2 ,r3,θ2,θ3 ,β ) = Cijklmn f (r1)i f (r2 ) j f (r3 )k g(θ2 )l g(θ3 )m h(n,β )i , j ,k ,l ,m,n

∑

Comparison of e xperimental and calculated vibrational energies (cm-1) of H3O+

and D3O+ obtained from RVIB4 and MULTIMODE (MM) using PES-2. (+) and (-)

indicate the p arity of each doublet. Predicted values listed under Exp are

indicated in parenthesis, as explained in the text.H3O

+

MM RVIB4 EXP(+) (-) (+) (-) (+) (-)

Split Exp.Split

GroundState

0 41 0 41 0.0 55.35 41 55.35a

2 578 918 580 917 581.17 954.40 337 373.23 b

2ν2 1421 1971 1421 1971 1475.84 550

1623 1675 1623 1673ν4

1623 1675 1623 16731625.95 1693.87 50 67. 92c

ν1 3387 3420 3386 3418 3445.00 3491.17 33 46. 16d

3527 3559 3522 3550ν3

3523 3552 3522 35503535.56 3574.29 28 38. 73e

D3O+

MM RVIB4 EXP(+) (-) (+) (-) (+) (-)

Split SplitExp.

GroundState

0 10 0 10 0.0 15. 35 10 15. 35f

ν2 469 631 469 631 453.74 645.13 162 191.39 g

2ν2 974 1329 974 1328 354

11 93 1207 11 93 11 93ν4

11 93 1207 11 93 120613

ν1 2449 2458 2448 2457 9

2621 2629 2618 2624ν3

2618 2626 2618 26242629.65 2639.59 6 9.9 4h

An ab initio potential energy surface and vibrational energies of H3O+ and its

isotopomers

Huang, Carter, Bowman