Trabalho apresentado no XVII Symposium on High Resolution Molecular Spectroscopy (HighRus-2012), 2012, Zelenogorsk-Russia. Anals of XVII Symposium on High Resolution Molecular Spectroscopy (HighRus-2012), 2012.

1. Calculation of isotopic dipole moments
with spectroscopic accuracy
José Rachid Mohallem
(with Antonio Arapiraca)
Departamento de Física
Universidade Federal de Minas Gerais, Brasil
rachid@fisica.ufmg.br

2. BELO HORIZONTE – MINAS GERAIS

3. 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.

4. 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.

5. 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 ~ 10-2 debye
breaking.
• For CH3CD3 vibrations break the
molecular symmetry, in average,
generating a net polarization.

6. For comparison
● The dipole moment of the water molecule is
1.86 debye !!!

7. 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 (????)

8. 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.

9. 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.)
n
∇ i2
H el = H BO = −∑ + Vall , H el Φ = ε ( R )Φ, PES = ε ( R) + Vnuc
i 2
● We introduce the effective FNMC Hamiltonian
m n
∇ i2
H el = −∑ (∑ PA PA ) + H BO
A i 2m A
which makes the electronic wf and the PES dependent of the nuclear
masses and is thus responsible for the electronic isotopic effect.
∇2 ∇2 ∇2 1 ∇2 1
Hydrogen atom → H = − + H BO = − − − =− −
2m 2m 2 r 2µ r

10. 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
1  ∂2µ 
µ ave = µeff +∑  2
 
k 4ω k  ∂Qk  eff

11. 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.

12. Results: Isotopic dipole moment of HD (debye)
[Arapiraca et al, J. Chem. Phys. 135, 244313 (2011)]
FNMC-SCF  eq  eff  av  10  4 
aug-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
Wolniewicz c 8.36
Cafiero d 8.31
BO 0. 0 0. 0 0. 0
Experimental b 8. 8328
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)

13. 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 0.0114 a 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)

14. 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
12 13 a
CH 3 CD 3 -0.0038 0.0165 0.0127 -0.0053 0.0165 0.0112 0.01094(11)
13 12
CH 3 CD 3 -0.0038 0.0163 0.0125 -0.0056 0.0163 0.0107 0.01067(10) a
b
CH 2 CD 2 (asym.) -0.0003 0.0114 0.0111 -0.0020 0.0114 0.0094 0.0091(4)
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)

15. Results: Isotopic dipole moment of isotopomers of
the propane molecule
Arapiraca and Mohallem, submited to TCA
System BO FNMC Experiment
 eff  vib  ave  eq  eff  vib  ave  exp
C 3 H8 0.0747 0.0114 0.0861 0.0875 0.0742 0.0117 0.0859 0.0848(20) a
CH 3 CD 2 CH 3 0.0760 0.0242 0.1002 0.0866 0.0745 0.0244 0.0989 0.095(2) a
CD 3 CH 2 CD 3 0.0773 -0.0046 0.0727 0.0884 0.0782 -0.0046 0.0736 0.076(2) a
CHD 2 CH 2 CHD 2 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)

16. 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)

17. 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
C 6 DH5 0.0005 0.0097 0.0102 -0.0014 -0.0010 0.0097 0.0087 0.0081(28) a
C6 D2 H 4 0.0007 0.0163 0.0170 -0.0020 -0.0015 0.0165 0.0150
C6 D3 H 3 0.0010 0.0191 0.0201 -0.0029 -0.0018 0.0191 0.0173
a. Fliege and Dreizler, Z. Naturforsch 42A, 72 (1987)

18. 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

19. Simulated emission spectrum for C6H5D

20. Simulated absorption spectrum for C6H5D

21. Next
● “Calibration” of functionals and basis functions for DFT calculations
● Applications to molecules of interest, particularly to polycyclic
aromatic hydrocarbons.

22. 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

23. Acknowledgements
● To the organizers of HighRus-2012
● To CNPq and Fapemig (Brazilian agencies) for support.
Thank you all!!!!

24. Electronic approach: the adiabatic approximation
Ψ (r , R) = ∑ Φ l (r , R) χ l ( R)
l
 ∇2 
 − R + Tk ,k + ε k − E  χ k = Tk ,l χ l
 2µ AB 
Ψ (r , R) = Φ (r , R) χ ( R)
 ∇2 
−
R
+ Tk ,k + ε k − E  χ k = 0
 2 µ AB 