Vibration-rotation spectra from first principles

Lecture 2: Calculations of spectroscopic accuracy

Jonathan Tennyson

Department of Physics and Astronomy

University College London

OSU, February 2002

“Experiments (are) measured to tenths of wave numbers… this level of accuracy in a calculation is meaningless”

Freisner, Bentley, Menou and Leforestier, J. Chem. Phys. 99, 324 (1993)

For triatomics accuracy determined by:• The potential energy surface• The validity of a potential (ie the Born-Oppenheimer approximation)

Potentials:• from electronic structure calculations• spectroscopically determined

Potentials: Ab initio or Spectroscopically determined

Using spectra to improve a potential?

1. Guess form eg V(r1,r2,) = ci fi (r1,r2,)

2. Compute obscalc and standard deviation3. Compute derivatives.

Hellman-Feynman theorem d < n | H | n > /dc = < n | dH/dc | n > gives d < n | V | n > /dci = < n | fi (r1,r2,) | n >

4. Repeat calculation with improved V(r1,r2,) Guesses improved using specialist techniques eg I-NoLLS: a program for interactive nonlinear least-squares fitting of the

parameters of physical models, M.M. Law & J. M. Hutson, Comp. Phys. Commun., 102, 252 (1997).

Fitting to spectroscopic data

• Best start: high quality ab initio calculation (starting point usually determines quality of fit).• Final fit usual in 2 – 3 iterations (But many tests first!)• Usually fit energy levels rather than spectra• Fit vibrational and rotational data simultaneously (Essential for light molecules)• Born-Oppenheimer approximation !?• Fit 20 – 30 parameters (only).

Spectroscopically determined water potentials

Reference Year vib/cm-1 Nvib Emax /cm-1

Hoy, Mills & Strey 1972 214 25 13000

Carter & Handy 1987 2.42 25 13000

Halonen & Carrington 1988 5.35 54 18000

Jensen 1989 3.22 55 18000

Polyansky et al (PJT1) 1994 0.6 40 18000

Polyansky et al (PJT2) 1996 0.94 63 25000

Partridge & Schwenke 1997 0.33 42 18000

Shirin et al 2002 0.10 99 25000

mportant to treat vibrations and rotations

Spectroscopically determined potentialPolyansky, Jenson & Tennyson (PJT1), J. Chem. Phys., 101, 7561 (1994)

Fit:1600 term valueswith J up to 14

a New experimental value: 10899.64 cm-1

Ab initio accuracy better than 1cm1 • Adiabatic or Born-Oppenheimer Diagonal Correction (BODC)

• Non-adiabatic corrections for vibration and rotation

• Electronic (kinetic) relativistic effect

• Relativistic Coulomb potential (Breit effect)

• Radiative correction (Lamb shift or qed)

Can BO electronic structure calculations be done this accurately?

Variational rotation-vibration calculations with exact kinetic energy operator accurate to better than 0.001 cm1

H3+

H2OH2S

HCN/HNC

Molecule considered at high accuracy

Ab initio Born-Oppenheimer potentials for H3+

Year Authors Emin / Eh E / cm

1975 Carney & Porter 1.33519 1900

1980 Schinke et al 1.34023 790

1985 Burton et al 1.34188 430

1986 Meyer et al 1.34309 160

1990/2 Frye et al 1.343828 9

1994 Rohse et al 1.3438336 1

1998 Cencek et al a 1.3438355 0.04

For spectroscopy shape is more important than magnitude

a Also electronic relativistic correction, ~ 3 cm1

Adiabatic effects in H3+

The Born-Handy approximation

mode Eobs / cm-1 BO +Vad

011 2521.409 0.11 0.24 100 3178.290 1.30 0.40 020 4778.350 0.00 0.50 022 4998.045 0.30 0.64 111 5554.155 1.40 0.50

1 2992.505 1.46 0.36 2 2205.869 0.47 0.25 3 2335.449 +0.47 0.14

1 2736.981 1.04 0.28 2 1968.169 +0.58 0.11 3 2078.430 0.74 0.18

Ab initio vibrational band origins

H2D+

H3+

D2H+

Non-adiabatic effects in diatomics

P.R. Bunker and R.E. Moss, Mol. Phys., 33, 417 (1977)

Vibrational KE

Vibrational KENon-orthogonal coordinates only

Rotational & Coriolis terms

Rotational & Coriolis terms Non-orthogonal coordinates only

Effective Hamiltonian after intergrationover angular and rotational coordinates.Case where z is along r1

Reduced masses (g1,g2) define coordinates

Non-adiabatic effects in the ST Hamiltonian

mode Eobs / cm1 BO +Vad v nuc

011 2521.409 0.11 0.24 +0.056 100 3178.290 1.30 0.40 +0.025 020 4778.350 0.00 0.50 +0.020 022 4998.045 0.30 0.64 +0.010 111 5554.155 1.40 0.50 0.000

1 2992.505 1.46 0.36 0.020 2 2205.869 0.47 0.25 0.050 3 2335.449 +0.47 0.14 +0.090

1 2736.981 1.04 0.28 +0.001 2 1968.169 +0.58 0.11 +0.023 3 2078.430 0.74 0.18 0.004

Ab initio vibrational band origins

H2D+

H3+

D2H+

O.L. Polyansky and J. Tennyson, J. Chem. Phys., 110, 5056 (1999).

J Ka Kc J Ka Kc Eobs / cm-1 BO +Vad v nuc + KNBO

3 2 1 3 2 2 2225.501 0.385 0.245 0.062 0.044

3 2 1 2 0 2 2448.627 0.521 0.259 0.011 0.076

2 2 0 2 2 1 2208.417 0.435 0.242 0.050 0.068

2 2 1 2 0 2 2283.810 0.521 0.239 +0.030 0.059

2 2 0 1 0 1 2381.367 0.573 0.250 +0.008 0.060

3 3 1 2 1 2 2512.598 0.647 0.250 +0.075 0.099

2 0 2 3 1 3 2223.706 0.418 0.163 +0.050 +0.068

2 2 1 3 1 2 2242.303 0.753 0.151 +0.140 +0.095

2 1 2 2 2 1 2272.395 0.420 0.168 +0.035 +0.099

2 2 0 2 1 1 2393.633 0.320 0.162 +0.140 +0.087

3 3 1 3 2 2 2466.041 0.224 0.164 +0.190 +0.080

3 3 1 2 2 0 2596.960 0.185 0.177 +0.167 +0.077

3 3 0 2 2 1 2602.146 0.203 0.172 +0.167 +0.080

2

3

H2D+ : ab initio spectra

Rotational non-adiabatic effectsUse nuc given by nuclear massExplicit inclusion of effect via rotational g-factors

PR Bunker & RE Moss, J. Mol. Spectrosc., 80, 217 (1980)

Preliminary results for H3+

Calculations for all observed levels, J up to 15Reproduces observations to better than 0.001 x J2 cm1

for vibrational ground state

OL Polyansky, MA Kostin, J Tennyson, BT Sutcliffe, I Paidarova & SPA Sauer, to be published

Ab initio predictions of water vibrational fundamentals

Reference Method Year v1 v2 v3

Bucknell and Handy SCF 1974 1728 4045 4139

Bartlett et al MBPT 1979 1610 3702 3789

Knowles et al CASSCF 1982 1645 3691 3794

Martin et al QCISD(T) 1992 1595 3657 3756

Kedziora and Shavitt MRCISD 1997 1604.6 3650.5 3746.9

Partridge and Schwenke CCSD(T) 1997 1597.4 3660.5 3758.2

Experimental Values 1594.746 3657.053 3755.929

Reference Year Barrier height/cm1 Comment

Carter and Handy 1987 11493 Spectroscopic Empirical Jensen 1989 11246 Spectroscopic EmpiricalPolyansky et al (PJT2) 1994 10966 Spectroscopic Empirical Lanquetin et al 1999 11154 Effective Hamiltonian Partridge & Schwenke (PS) 1997 11155 Ab initio Partridge & Schwenke 1997 11128 Spectroscopic Empirical PS + adiabatic + relativistic 1998 11192 Ab initio Csaszar et al 1998 11046 70 Extrapolated ab initio Tarczay et al 1999 11127 35 High accuracy ab initio Kain et al 2000 11105 5 Corrected ab initio Valeev et al 2001 11119 12 Ab initio (MP2 – R12)

Water: Barrier to linearity

Achieving a “perfect” ab initio potential

Need to consider: (for water)• SCF at full basis set limit (done)• Valence CI to full basis set limit (by extrapolating from large basis calculation)• Extension of CI to full CI limit (only possible with v. small, eg DZP, basis set) • Core – valence correlation

New high accuracy extrapolated ab initio calculations in progressPolyansky, Csaszar, Tennyson, Barletta, Shirin, Zobov & Schwenke

The future: explicit inclusion of r12 into the wavefunction

Ab initio: vibrational errors

Ab initio + Adiabatic: vib. errors

Ab initio + adiabatic + relativistic

MVD1 Csaszar, Kain, Polyansky, Zobov and Tennyson, Chem. Phys. Lett., 293, 317 (1998).

Ab initio +Gaunt1 +Breit2 Obs / cm1

(010) 1598.19 +0.10 +0.04 1594.75 (020) 3158.49 +0.18 +0.09 3151.63 (030) 4677.22 +0.21 +0.10 4666.79 (040) 6148.29 +0.20 +0.05 6134.01 (050) 7561.09 +0.10 0.10 7542.44 (060) 8894.52 0.16 0.35 8869.95 (101) 7249.52 +1.60 +1.32 7249.82 (201) 10612.70 +2.34 +1.94 10613.35 (301) 13829.31 +3.07 +2.54 13830.94 (401) 16896.50 +3.87 +3.20 16898.84 (501) 19776.00 +4.44 +4.04 19781.10

Relativistic electronic potential effects in water

1 Gaunt correction: 1 electron approximation2 Breit correction: full calculation

H.M. Quiney, P. Barletta, G. Tarczay, A.G. Csaszar, O.L. Polyansky and J. Tennyson, Chem. Phys. Lett., 344, 413 (2001).

(also D2)

2s, 2p

2p3/2

2p1/2

2s1/2

2s1/2

The hydrogen atom: n = 2 levels

Fine structure Non-relativistic

0.365 cm-1

2p1/2

0.035 cm-1

Lamb shift

2p3/2

Schrodinger Dirac QED

Ab initio + Lamb Obs / cm1

(010) 1598.19 0.09 1594.75 (020) 3158.49 0.18 3151.63 (030) 4677.22 0.29 4666.79 (040) 6148.29 0.43 6134.01 (050) 7561.09 0.60 7542.44 (060) 8894.52 0.86 8869.95 (101) 7249.52 +0.37 7249.82 (201) 10612.70 +0.54 10613.35 (301) 13829.31 +0.71 13830.94 (401) 16896.50 +0.83 16898.84 (501) 19776.00 +1.01 19781.10 (601) 22519.69 +1.19 22529.44 (701) 25105.51 +1.29 25120.28

One-electron Lamb shift effects in water

P. Pyykko, K.G. Dyall, A.G. Csaszar, G. Tarczay, O.L. Polyansky and J. Tennyson, Phys. Rev. A, 63, 024502 (2001)

BO / cm1 +BODC1 + Non-adiabatic

v nuc2 diag3 full4

(010) 1597.60 0.46 0.19 0.06 0.07 (020) 3157.14 0.94 0.38 0.12 0.15(100) 3661.00 0.55 0.46 0.72 0.70 (030) 4674.88 1.43 0.55 0.18 0.23(110) 5241.83 0.16 0.65 0.77 0.76 (040) 6144.64 2.00 0.71 0.23 0.30(120) 6784.56 0.23 0.83 0.83 0.84(200) 7208.80 1.25 0.88 1.39 1.37 (002) 7450.86 1.47 0.90 1.47 1.57(050) 7555.62 2.71 0.84 0.28 0.32

Born-Oppenheimer corrections for water

1 Born-Oppenheimer diagonal correction using CASSCF wavefunction2 Non-adiabatic correction by scaling vibrational mass, V

3 Two parameter diagonal correction4 Full treatment by Schwenke (J. Phys. Chem. A, 105, 2352 (2001).)

J. Tennyson, P. Barletta, M.A. Kostin, N.F.Zobov, and O.L. Polyansky, Spectrachimica Acta A (in press).

Assignments using branches

Ab initio potential

J

Err

or /

cm-1

Determined potentialSpectroscopically

Variational calculations:

Polyad structure in water absorption spectrum

Long pathlength Fourier Transform spectrum recorded by R Schmeraul

R. Schermaul, R.C.M. Learner, J.W. Brault, A.A.D. Canas, O.L. Polyansky, D. Belmiloud, N.F. Zobov and J. TennysonJ. Molec. Spectrosc. (in press)

Weak lines

New experimental measurements

REIMS data

• Carleer et al.

• Bruker F.T.S

• Range :13200 25020 cm-1

• T : 291 K

• p(H2O) : 18.5 hPa

• pathlength ~ 602.32 m

• Number of new lines : 2286

IMPERIAL data (R.A.L)

• Schermaul et al.

• Bruker F.T.S

• Range :13350 14750 cm-1

• T : 294.4 K

• p(H2O) : 23.02 hPa

• pathlength ~ 800.75 m

• Number of lines : 3179

• Number of new lines : 963

Weak water lines Very difficult to record

Only a few weak lines in HITRAN

Also Kitt Peak archive data Also spectra 8000 – 13500 cm

Water vapour spectrum: new assignments in the blue

Long pathlength FTSM Carleer et al,

J. Chem. Phys., 111, 2444 (1999)

Vibrational mode Previous worka This workb band originLocal Normal lines levels lines levels cm 1

(4,2)1 (115) 10 5 22513.(7,0)+0 (700) 5 2 90 39 22529.296(7,0)0 (601) 42 20 57 15 22529.445(6,0)2 (521) 16 10 22630.(7,0)1 (611) 16 10 23947.(8,0)+0 (800) 24 20 25120.(8,0)0 (701) 12 6 23 18 25120.278

Water: Rotation-Vibration spectra in the near ultra violet

a C. Camy-Peyret et al, J. Mol. Spectrosc., 113, 208 (1985).b N.F. Zobov et al, J. Chem. Phys., 113, 1546 (2000).

Intensity data compared to HITRAN-96by polyad for spectral region 8500 – 15800 cm-1

HITRAN underestimates intensity of strong lines!D Belmiloud et al, Geophys. Res. Lett., 27, 3703 (2000).

Numbers are ratio of total intensity to HITRAN

Polyad Integrated absorbance

Spectral linefits

Ab Initio calculation

Correction Giver et al.

1.26 1.31 0.92

1.19 1.21 1.04 1.14

1.26 1.25 1.25 1.09

1.06 1.04 0.96

Frequency / cm-1

Water absorption by the atmosphere:

Standard Model

W Zhong, JD Haigh, D Belmiloud, R Schermaul & J Tennyson, Quart. J. Roy. Metr. Soc., 127, 1615 (2001)

Frequency / cm-1

Water absorption by the atmosphere:correction of Giver et al (2000)

Frequency / cm-1

Water absorption by the atmosphere:Effect of weak water lines

Frequency / cm-1

Water absorption by the atmosphere:Effect of ESA-WVR linelist

Missing absorption due to water:First estimates

Theory

Experiment

Radiative Transfer Model

Atmosphericabsorption

In the red and visible :

Unobserved weak lines have a significant effect : ~ 3 Wm-2

Estimated additional 2.5-3 % absorption in the near I.R/Red.

Estimated additional 8-11 % absorption in the ‘Blue’ ? Underestimate of strong lines even more important : ~ 8 Wm-2

Estimated additional 8 % absorption in the near I.R/Red.

Missing absorption due to water:Outstanding issues

In the near infrared and red:

Contributions due to H218O, H2

17O and HDO.

Possible role of water dimer (H2O)2.

In the blue and ultraviolet:

Are H216O line intensities also underestimated?

Contribution due to weak lines

Sensitivity of vibrational band origins

Effect Contribution / cm-1

H2O H2S H3+

BO convergence + 30 + 30 +/ 0.03

Relativistic correction (1e) 19 +/ 0.03

Darwin term (2e) 0.8 a

Gaunt correction + 5

a

Breit correction + 6 + 0.03 a

QED +1.3 + 1.5 a

Adiabatic correction (BODC) + 5 + 2 +/ 1.5

Non-adiabatic correction 4 ~ 3 0.5 a Unknown, assumed negligible

Water assignments using variational calculations

• Long pathlength absoption (T = 296K) 11000 - 25000 cm-1

Fourier Transform and Cavity Ring Down

• Laboratory emisson spectra (T =1300 1800K) 400 – 6000 cm-1

• Absorption in sunspots (T = 3200 K) N band, L band, K band 10-12m 3 m 2 m

25000 new lines assigned

Dataset of 12000 measured H216O energy levels

J. Tennyson, N.F. Zobov, R. Williamson, O.L. Polyansky & P.F. Bernath,J. Phys. Chem. Ref. Data, 30, 735 (2001).

Oleg Polyansky

Nikolai Zobov

Maxim Kostin

Paolo BarlettaMizuho Tanaka

Roman Tolchenov