Theoretical Studies of the Fundamental and Overtone

Spectrum of the Water Dimer

D. A. MatthewsJ. F. Stanton

J. Vázquez

The University of Texas at Austin

The Water Dimer

Simple system to study hydrogen bonding. The first step to understanding bulk liquid

water. Important in atmospheric processes such as

formation of H2SO4 (i.e. acid rain). Plays some role in absorption of solar

radiation in atmosphere.

VPT2 Force field may be expressed as a Taylor expansion about the

equilibrium geometry:

Both Rayleigh-Schrödinger and van Vleck approaches give the same “dressed” Hamiltonian in second order:

The diagonal terms of this operator give the VPT2 energies.

€

H = 12

ωk (pk2 + qk

2)k

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥+ 1

6φlmnqlqmqn

l,m,n∑

⎡

⎣ ⎢

⎤

⎦ ⎥+ 1

24φklmnqkqlqmqn

k,l ,m,n∑ + Bα pα

2

α∑

⎡

⎣ ⎢

⎤

⎦ ⎥

€

˜ H = 12

ωk (pk2 + qk

2)k

∑ + 124

φklmnqkqlqmqnk,l,m,n∑ + Bα pα

2

α∑

− 18

φknrφklmωkωlωm (qnqr pl pm + pl pmqnqr + 2

3 δmrδnl ) + ωk (ωl2 + ωm

2 −ωk2)qlqmqnqr

(ωl + ωm + ωn )(ωl + ωm −ωn )(ωl −ωm + ωn )(−ωl + ωm + ωn )k,l ,m,n ,r∑

VPT2

An example is F2: a local approximation to the force field using CCSD(T) give reasonable levels for both methods…

Gives better results than “exact” variational methods for truncated polynomial force fields.

E

RF-F

VPT2

Also, VPT2 is exact for a Morse oscillator, so it is ideal for stretching modes.

But when using the global CCSD(T) force field, the “exact” variational method falls apart.

E

RF-F

!

Resonance Artifact of perturbation theory. Fermi Resonances:

Affect states coupled by cubic force constants (e.g. 21 and 1+3+8 in the water dimer).

Darling-Dennison Resonances:

Affect states coupled by quartic force constants and coriolis coupling constants (e.g. 21 and 23 in water).

€

ωa ≈ ωb + ωc

€

ωa + ωb ≈ ωc + ωd

Resonance

a

2b

c

d+ e

ωa≈ 2ωb

ωc≈ ωd+ ωe

First, Fermi resonances are found, and resonant states are added to the effective Hamiltonian. The off-diagonal elements (first order interactions) are given by multiples of the cubic force constants.

abb

Kaaac,Kaccc

ade

bbc Kbbde

cde

Resonance2ωe≈ 2ωc

ωc+ ωb≈ 2ωd

Second, Darling-Dennison resonances are found and states added to the effective Hamiltonian. The diagonal and off-diagonal elements between these states include second order quartic, bicubic, and coriolis contributions as in the dressed Hamiltonian, except that terms involving cubic constants between Fermi states are removed.

2e

2c

c+ b

2d

Keecc Keecb Keedd

Kcccb,Kcbbb

Kccdd

Kcbdd

Resonance

FermiFirst and second order interactions between Fermi and Darling-Dennison resonant states may be non-zero, leading to mixing of these states. When the effective Hamiltonian is fully formed, it is diagonalized to give the final levels.

aee, acb,Kbbcc, Kdecc,etc.

Darling-Dennison

Results: Fundamentalsaug-cc-pVTZa ANO1a Experimentb

Mode (cm-1) I (km/mol) (cm-1) I (km/mol) (cm-1) I (km/mol)v1 3711 62 3737 46 3735 49v2 3634 5 3654 5 3660 3v3 3591 148 3620 131 3601 100v4 1614 39 1628 47 1619 13v5 1603 68 1611 63 1599 27v6 304 7 308 21 311v7 144 64 147 38 143v8 121 163 108 189 103v9 3725 58 3748 49 3745 45v10 495 77 501 90 523v11 122 113 110 111 108v12 85 54 86 57 88a) Using CCSD(T) frozen core, and VPT2.b) Experimental frequencies from J. Phys. Chem. A, 109 (17), 4005 (2005), Intensities from Ne matrix: Y. Bouteiller and J. P. Perchard, Chem. Phys. 305, 1 (2004).

Results: Two-quantum OH Levelsaug-cc-pVTZa ANO1a Experimentb

Dominant State (cm-1) I (km/mol) (cm-1) I (km/mol) (cm-1) Assignment

v1+v2 7412 0.01 7462 0.022v9 7378 0.03 7424 0.01v1+v3 7311 0.06 7358 0.07v2+v3 7236 0.85 7286 0.85 7282 no assignmentv1+v3/2v2 7162 2.03 7207 2.08 7193 2v2 or v1+v32v2 7132 0.03 7174 0.002v3 7026 0.09 7086 0.07v1+v9 7443 0.17 7491 0.19v3+v9 7311 0.01 7363 0.10

7240 resonance?

v2+v9 7193 3.09 7234 3.04 7250given as 2v1, but probably v2+v9

a) Using CCSD(T) frozen core, VPT2, and diagonalizing Darling-Dennison resonances.b) Gas phase frequencies from Nesbitt et al., J. Chem. Phys. 122, 194316 (2005).

Acknowledgements

John StantonJuana Vázquez

The Robert A. Welch FoundationThe Camille and Henry Dreyfus Foundation