Calculation of isotopic dipole moments with spectroscopic accuracy

José Rachid Mohallem(with Antonio Arapiraca)

Departamento de FísicaUniversidade Federal de Minas Gerais, Brasil

rachid@fisica.ufmg.br

BELO HORIZONTE – MINAS GERAIS

LABORATÓRIO DE ÁTOMOS E MOLÉCULAS ESPECIAIS

● Theory of the interaction of positrons and positronium with atoms and molecules: bound and scattering states.

● Theory of nonadiabatic effects in small molecules; corrections to vibrational spectra.

► New line: Isotopic dipole moments and other molecular properties of interest in astrochemistry and radioastronomy.

Motivation from basic physics

Apolar molecules do not have permanent dipole moments. Right?

Wrong! They can have isotopic dipole moments!

↓

In consequence, they can exhibit pure rotational spectra.

Isotopic dipole moments: electronic and vibrational effects

~ 10-3 debye

• For HD the smaller Bohr radius of D causes polarization of the electronic cloud. There is no vibrational symmetry breaking.

~ 10-2 debye

• For CH3CD3 vibrations break the molecular symmetry, in average, generating a net polarization.

For comparison

● The dipole moment of the water molecule is

1.86 debye !!!

Motivation from radioastronomy

● Detection of molecules in space from pure rotational spectra

(the rotation-vibration spectrum is difficult to detect on earth because of the atmosphere; it is normally observed in space telescopes)

● Abundance of deuterium in space (????)

Theoretical approach

● Fixed-nuclei Born-Oppenheimer electronic theory can only approach the vibrational effect.

● But as we will see, the electronic isotopic effect is necessary to reach the experimental accuracy and to identify isotopic trends.

● We treat the two effects in a joint approach.

Electronic approach: the FNMC Hamiltonian[Mohallem et al, Theochem 709, 11 (2004); Chem. Phys. Lett. 501, 575 (2011)]

● Usually the generator of electronic states and PES is the BO Hamiltonian (a.u.)

● We introduce the effective FNMC Hamiltonian

which makes the electronic wf and the PES dependent of the nuclear masses and is thus responsible for the electronic isotopic effect.

∑ ∑ +∇−=m

A

n

iBOA

A

iAel HP

mPH )

2(

2

nucelall

n

i

iBOel VRPESRHVHH +=Φ=Φ+∇−== ∑ )(,)(,

2

2

εε

rrmH

mHatomHydrogen BO

1

2

1

222

2222

−∇−=−∇−∇−=+∇−=→µ

Vibrational approach: the ZPV correction (ZPVC)[Astrand et al, Theo. Chem. Acc. 103, 365 (2000); J. Chem. Phys. 112, 2655 (2000)]

● The nuclear wave function and the molecular properties are expanded perturbatively around an effective geometry.

● The effective geometry is obtained variationally as the minimum of the PES plus the ZP vibrational energy → first-order corrections to properties vanish.

● The property calculated on the effective geometry, µeff, already has the ZPV correction.

● Second-order corrections to properties are evaluated as

effkk keffave Q

∂∂+= ∑ 2

2

4

1 µω

µµ

FNMC + ZPVC as an upgrade of Dalton 2.0[Arapiraca et al, J. Chem. Phys. 135, 244313 (2011)][Dalton: www.daltonprogram.org]

● ZPVC is a facility of the Dalton code.

● FNMC was implemented into Dalton by Dan Jonsson during a visit to UFMG.

● The upgrade was optimized by A. Arapiraca to yield reliable values of isotopic dipole moments in SCF-HF and DFT electronic approaches. Post-HF available methods as RAS and CAS-MCSCF generate unstable second derivative of properties.

● Numerical derivative step length (SL) is critical.

Results: Isotopic dipole moment of HD (debye) [Arapiraca et al, J. Chem. Phys. 135, 244313 (2011)]

FNMC-SCF eq eff av 10 4aug-cc-pVDZ 7.9 8.3 8.1

aug-cc-pVTZ

Spherical 7. 3 7.2 7.3

Cartesian 4. 7 4.8 4.9

aug-cc-pVQZ

Spherical 5. 7 5.9 6.1

Cartesian 3. 7 3.9 4.0

(3s, 2p, 1d)a

Spherical 11.6 12.6 9.0

Cartesian 9. 5 9.4 9.4

Wolniewiczc 8.36

Cafierod 8.31

BO 0.0 0.0 0.0

Experimentalb 8.8328

HT (spherical), this work 17 12.5

a. Assafrão and Mohallem, J. Phys. B: At. Mol. Opt. Phys. 40, F85 (2007)b. Drakopoulos and Tabisz, Phys. Rev. A 36, 5556 (1987)c. Wolniewicz, Can. J. Phys. 54, 672 (1976)d. Cafiero and Adamowicz, Phys. Rev. Lett. 89, 073001 (2002)

Results: Isotopic dipole moment of CH3CD3

[Arapiraca et al, J. Chem. Phys. 135, 244313 (2011)]

BO FNMC

Step length eff vib ave eff vib ave

RAS 0.048 -0.0028 0.0150 0.0122 -0.0043 0.0150 0.0107

0.049 0.0147 0.0119 0.0147 0.0104

0.05 0.0142 0.0114a 0.0150 0.0107

0.051 0.0145 0.0117 0.0140 0.0097

0.052 0.0140 0.0112 0.0142 0.0099

SCF any -0.0038 0.0163 0.0125 -0.0053 0.0163 0.0110

Experiment b 0.0108617(5)

c 0.01078(9)

a. C. Puzzarini and P. Taylor, J. Chem. Phys. 122, 054315 (2005)b. I. Ozier and W. L. Meertz, Can. J. Phys. 62, 1844 (1984)c. E. Hirota and C. Matsumura, J. Chem. Phys. 55, 981 (1971)

Results: CH3CD3 (continued), CH2CD2 and isotopomers.

[Arapiraca et al, J. Chem. Phys. 135, 244313 (2011)]

BO FNMC Experiment

System eff vib ave eff vib ave

12CH3

13CD3 -0.0038 0.0165 0.0127 -0.0053 0.0165 0.0112 0.01094(11) a

13CH3

12CD3 -0.0038 0.0163 0.0125 -0.0056 0.0163 0.0107 0.01067(10)a

CH2CD2 (asym.) -0.0003 0.0114 0.0111 -0.0020 0.0114 0.0094 0.0091(4) b

CHDCHD (cis) -0.0001 0.0125 0.0124 -0.0013 0.0125 0.0112 —

a. E. Hirota et al, J. Chem. Phys. 66, 2660 (1977)b. E. Hirota et al, J. Mol. Spectrosc. 89, 223 (1981)

Results: Isotopic dipole moment of isotopomers of the propane moleculeArapiraca and Mohallem, submited to TCA

System BO FNMC Experiment

eff vib ave eq eff vib ave exp

C3H8 0.0747 0.0114 0.0861 0.0875 0.0742 0.0117 0.0859 0.0848(20)a

CH3CD2CH3 0.0760 0.0242 0.1002 0.0866 0.0745 0.0244 0.0989 0.095(2)a

CD3CH2CD3 0.0773 -0.0046 0.0727 0.0884 0.0782 -0.0046 0.0736 0.076(2)a

CHD2CH2CHD2 0.0747 -0.0112 0.0635 0.0891 0.0763 -0.0112 0.0651 —

a. J. S. Muenter and V. W. Laurie, J. Chem. Phys. 45, 855 (1966)

Results: Isotopic dipole moment of isotopomers of the water molecule

• For water, electronic correlation is important so that DFT-B3LYP-aug-cc-pVDZ is used (possible benchmarks)

System BO FNMC Experiment

eff vib ave eq eff vib ave

H2O -1.8598 0.0064 -1.8534 -1.8556 -1.8613 0.0064 -1.8549 1.85498(9)a

D2O -1.8588 0.0046 -1.8542 -1.8548 -1.8593 0.0064 -1.8547 1.8545(4)b

HDO -1.8590 0.0059 -1.8531 -1.8552 -1.8601 0.0059 -1.8542 1.8517(5)a

a. Shostak et al, J. Chem. Phys. 94, 5875 (1991)b. Dyke and Muenter, J. Chem. Phys. 59, 3125 (1973)

Results: Isotopic dipole moment of isotopomers of the Benzene molecule

• Why benzene? → Polycyclic aromatic hydrocarbon ISM (H and D)

• Best results (apparently) for SCF-HF with 6-31+G(d,p)

System BO FNMC Experiment

eff vib ave eq eff vib ave

C6DH5 0.0005 0.0097 0.0102 -0.0014 -0.0010 0.0097 0.0087 0.0081(28)a

C6D2H4 0.0007 0.0163 0.0170 -0.0020 -0.0015 0.0165 0.0150

C6D3H3 0.0010 0.0191 0.0201 -0.0029 -0.0018 0.0191 0.0173

a. Fliege and Dreizler, Z. Naturforsch 42A, 72 (1987)

Exploratory application: Detecting benzene in space (from pure rotational spectra)?

● To simulate the radio emission spectrum of the benzene molecule we need the rotational constants (frequency) and the dipole moments (intensity).

● Rotational constants are well evaluated either with BO or FNMC● Detection depends on line intensity and time of observation

(signal/noise relation improves)

In the following figures:► Experimental frequencies from experiment: [Oldani et al, CPL 108, 7 (1984) for C6H5D and JMS 190, 31 (1988) for C6H4D2]

► Experimental intensities (EC): from PGOPHER with experimental input ►Theoretical data: from PGOPHER with theoretical input

Simulated emission spectrum for C6H5D

Simulated absorption spectrum for C6H5D

Next

● “Calibration” of functionals and basis functions for DFT calculations

● Applications to molecules of interest, particularly to polycyclic aromatic hydrocarbons.

Staff and Collaborators Positron● Profa. Denise Assafrão (DF-UFES) ● Luis Poveda, Post-Doc ● Adriano Dutra, PhD student● Paulo Henrique Amaral, Ms student

High resolution molecular spectroscopy (nonadiabatic effects)● Profa. Cristina P. Gonçalves (UESB) ● Prof. Alex Alijah (GSMA-CNRS, Reims, France) ● Leonardo Diniz, PhD student

Isotopic effects, astrochemistry and radioastronomy● Prof. Sérgio Pilling (UNIVAP)● Dr. Antonio F. C. Arapiraca● Paulo Oliveira, Undergraduate student

Acknowledgements

● To the organizers of HighRus-2012

● To CNPq and Fapemig (Brazilian agencies) for support.

Thank you all!!!!

Electronic approach: the adiabatic approximation

llkkkkkAB

R TET χχεµ ,,

2

2=

−++∇−

02 ,

2

=

−++∇− kkkk

AB

R ET χεµ

∑Φ=Ψl

ll RRrRr )(),(),( χ

)(),(),( RRrRr χΦ=Ψ