Vibrationally averaged post Born-Oppenheimer isotopic dipolemoment calculations approaching spectroscopic accuracyA. F. C. Arapiraca, Dan Jonsson, and J. R. Mohallem Citation: J. Chem. Phys. 135, 244313 (2011); doi: 10.1063/1.3671940 View online: http://dx.doi.org/10.1063/1.3671940 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i24 Published by the American Institute of Physics. Related ArticlesAn activated scheme for resonance energy transfer in conjugated materials J. Chem. Phys. 135, 244512 (2011) Efficient electron dynamics with the planewave-based real-time time-dependent density functional theory:Absorption spectra, vibronic electronic spectra, and coupled electron-nucleus dynamics J. Chem. Phys. 135, 244112 (2011) Quantum entanglement between electronic and vibrational degrees of freedom in molecules J. Chem. Phys. 135, 244110 (2011) Zero kinetic energy photoelectron spectroscopy of jet cooled benzo[a]pyrene from resonantly enhancedmultiphoton ionization J. Chem. Phys. 135, 244306 (2011) The infrared spectrum of NNCO+ trapped in solid neon J. Chem. Phys. 135, 224307 (2011) Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 135, 244313 (2011)

Vibrationally averaged post Born-Oppenheimer isotopic dipolemoment calculations approaching spectroscopic accuracy

A. F. C. Arapiraca,1,2 Dan Jonsson,3 and J. R. Mohallem1,a)

1Laboratório de Átomos e Moléculas Especiais, Departamento de Física, ICEx, Universidade Federal deMinas Gerais, P.O. Box 702, 30123-970, Belo Horizonte, MG, Brazil2Centro Federal de Educação Tecnológica de Minas Gerais, CEFET-MG, Campus X, 35.790-000,Curvelo, MG, Brazil3Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø,N-9037 Tromsø, Norway

(Received 10 September 2011; accepted 2 December 2011; published online 30 December 2011)

We report an upgrade of the Dalton code to include post Born-Oppenheimer nuclear mass correc-

tions in the calculations of (ro-)vibrational averages of molecular properties. These corrections are

necessary to achieve an accuracy of 10−4 debye in the calculations of isotopic dipole moments. Cal-

culations on the self-consistent field level present this accuracy, while numerical instabilities com-

promise correlated calculations. Applications to HD, ethane, and ethylene isotopologues are imple-

mented, all of them approaching the experimental values. © 2011 American Institute of Physics.

[doi:10.1063/1.3671940]

I. INTRODUCTION

Isotopic dipole moments, specially those arising from

substitution of H atoms with D (deuterium) atoms, have been

investigated in the last decades in both the experimental and

theoretical fields, as they permit some asymmetric isotopo-

logues (isotopomers) to display pure rotational spectra, ab-

sent in the corresponding symmetric species. This feature im-

plies important changes in the spectroscopic properties of the

isotopologues. The isotopic dipole moments come from two

different effects, namely, electronic distribution asymmetries

caused by the smaller Bohr radius of D as compared with H,

and average asymmetries during vibrations.

The prototype system is HD. The most accurate mea-

surement of its dipole moment, relative to the R(0) rotational

transition line in the far-infrared rotational absorption spec-

troscopy, amounts to 8.83(0.28) × 10−4 debye at 295 K.1

Calculations of the HD dipole moment need to go beyond

the Born-Oppenheimer (BO) approximation, since electronic

asymmetry must be taken into account. The small size of HD

allows theoreticians to approach the problem on quasi-exact

levels. Most calculations refer, however, to starting points

in which the concept of molecular structure is firmly estab-

lished, say an adiabatic type approach. Thus, Blinder consid-

ered a symmetry breaking perturbative term to the BO Hamil-

tonian obtaining a HD dipole moment of 8.89 × 10−4 debye;2

Wolniewicz performed standard BO-based calculations with

nonadiabatic corrections that lead to 8.36 × 10−4 debye;3

Ford and Browne resorted also to a perturbative approach

but derived the instantaneous dipole moment from rotation-

vibration dipole transition probabilities, yielding 8.29 × 10−4

debye;4 Thorson et al. predicted a value of 8.51 × 10−4 debye

(Refs. 5 and 6) from a symmetry adapted BO Hamiltonian.

a)Author to whom correspondence should be addressed. Electronic mail:rachid@fisica.ufmg.br.

Similar approaches exploring features other than the appear-

ance of the permanent dipole moment from Pack,7 Gonçalves

and Mohallem,8 and Strasburger9 have been proposed. On the

other hand, Cafiero and Adamowicz obtained a value of 8.31

× 10−4 debye from a fully nonadiabatic approximation10 in

which the dipole moment is not derived from molecular struc-

ture parameters (after all a full nonadiabatic molecule in vac-

uum would display spherical symmetry), instead deriving it

from an applied electric field. This issue is controversial from

both conceptual11 and technical12 points of view.

Subsequently, measurements for larger systems followed.

Hirota and Matsumura measured the dipole moment of the

isotopologue CH3CD3 of ethane by microwave spectroscopy

as 0.01078(9) debye.13 A more recent measurement by

Ozier and Meertz with molecular beam electric spectroscopy

yielded 0.0108617(5) debye.14 The two values differ by 10−4

debye, the same order of magnitude as the HD dipole mo-

ment. Later, Hirota et al. (1977) focused on the two iso-

topologues of ethane obtained by replacing one 12C atom by13C (see Table III for values).15 Hirota et al. reported the

value of 0.0091(4) debye for the isotopologue CH2CD2 of

ethylene.16 Hollenstein et al. focused on the isotopologues

CH3D and CD3H of methane in Fourier transform interfer-

ometry experiments, obtaining respectively, 0.00557(10) and

0.00569(14).17 They also performed diffusion Monte Carlo

calculations leading to, respectively, 0.0068(4) and 0.0066(5)

debye. Again, the difference between the isotopologues is

about 10−4 debye so that it seems reasonable to consider the

spectroscopic accuracy to be of the same order of magnitude.

Also important to cite is the measurement for monodeuter-

ated benzene, C6H5D, by Fliege and Dreizler19 also with mi-

crowave spectroscopy. Finally, a measurement for the iso-

topologue HC2D of acetylene was made by Matsumura et al.leading to 0.01001(15) debye,20 which should be compared

to the BO calculation of 0.0105 debye from Cazzoli et al.21

0021-9606/2011/135(24)/244313/5/$30.00 © 2011 American Institute of Physics135, 244313-1

244313-2 Arapiraca, Jonsson, and Mohallem J. Chem. Phys. 135, 244313 (2011)

This list may be incomplete in view of the large number of

isotopologues studied in experiments.

Isotopic substitutions, as well as simply the evaluation of

post-BO effects, are also relevant to the permanent dipole mo-

ments of polar molecules, such as H2O, HDO, D2O, and LiD.

Hobson et al. recently discussed the feasibility of approaching

this problem within the adiabatic approximation, through the

inclusion, in the molecular Hamiltonian, of a term coupling

the dipole moment to an external electric field.22 The connec-

tion of these studies to the goal of obtaining accurate dipole

moment surfaces23 is also relevant.

Except for diatoms, vibrational dipole moments can, in

principle, be obtained within the BO approximation, through

the use of different reduced nuclear masses in the nuclear

equation for different isotopologues. This procedure results in

different nuclear wavefunctions and, in consequence, differ-

ent average properties. Usually, the post-BO electronic con-

tribution is neglected in comparison with the BO vibrational

contribution, as in applications to CH3CD3 and SiH3SiD3.18

However, as we consider the dipole moments of different iso-

topologues of the same molecule, for example, it becomes

clear that full agreement with experiments can be achieved

only with the consideration of the electronic contribution,

which is on the order of 10−3 debye.

In this context, we report in Secs. II–IV a general

methodology starting from the standard adiabatic approxima-

tion and involving electronic as well as vibrational symme-

try breaking effects in the calculation of molecular proper-

ties. The method is applied to calculations of isotopic dipole

moments of several isotopologues, with the goal of attaining

spectroscopic accuracy.

II. COMPUTATIONAL METHODS

A possibility of performing (ro-)vibrational averages of

properties in molecular calculations was introduced by the

variation-perturbation approach of Astrand et al.24–26 based

on an expansion of a molecular property around an effec-

tive geometry, which is the minimum of the surface de-

fined by adding the zero-point energy to the potential energy

function.24 In this approach, the (ro-)vibrational average of a

property such the dipole moment μ, for instance, in the case

of a diatomic molecule, is calculated as

μave = μeff +

�∂2μ

∂R2

�eff

4mωeff

, (1)

where m is the reduced mass, ω is the harmonic frequency,

and R is the corresponding bond distance. The subscript effmeans that the corresponding quantity is evaluated using the

effective geometry. This procedure accounts for the anhar-

monicity of the potential to second order and can be accom-

plished from knowledge of the cubic force field (involved in

the determination of the effective geometry).24

This approach has been adapted to the DALTON 2.0

(Ref. 28) code within the BO approximation.25, 26 BO cal-

culations of the isotopic dipole moments of SiH3SiD3 and

CH3CD3 with this method were reported by Puzzarini and

Taylor.18 For this last isotopologue of C2H6, the calculated

dipole moment is (at 0 K) 0.0114 debye (as compared with

the experimental value of 0.0109 (Ref. 14) cited above). The

difference between calculation and experiment is again of the

order of 10−4 debye. Thus, the introduction of post-BO effects

is interesting not only in obtaining isotopic dipole moments

in cases the BO-based approach is unable to treat, but also to

the attempt of achieving spectroscopic accuracy. Here, we re-

port an upgrade of the DALTON 2.0 code28 to introduce finite

nuclear mass corrections (FNMC) in (ro-)vibrational calcula-

tions of molecular properties and apply it to isotopic dipole

moments.

FNMC (Refs. 29–31) can be introduced into any com-

putational routine for electronic molecular calculations by

changing the BO Hamiltonian HBO to

H = −�

A

��

i

PA

∇2i

2MA

PA

�+ HBO. (2)

In this equation, MA is the mass of a generic nucleus A,

while i represents electrons; PA projects the full electronic

wavefunction (generally in the LCAO, linear combination of

atomic orbitals, form) over the space of atomic functions of

A. It can be shown32 that the first term, which accounts for

FNMC, introduces the appropriate reduced electronic mass

in the calculations, being responsible for electronic symme-

try breaking, which generates the electronic isotopic dipoles.

This change has been made in the present work as an upgrade

of DALTON 2.0, in order to perform (ro-)vibrational calcula-

tions beyond the crude fixed-nuclei BO approximation. The

theory of (ro-)vibrational effects is a straightforward repeti-

tion of that in Refs. 24–26, just replacing the BO Hamiltonian

by Eq. (2) above. A quantity can be evaluated in a particu-

lar configuration, as the minimum geometry or the effective

geometry, and thus averaged over vibrational states. Dipole

moments are evaluated as usual with

−→μ = �A

ZA−→R A −

�ρ−→r d3r, (3)

in standard notation, where now the electronic density ρ is

affected by FNMC. This approach is alternative to that pre-

sented in Ref. 22 and a comparison between the two ap-

proaches would be interesting. We postpone this issue to fu-

ture work, however, since our aim here is to focus on systems

that do not have large permanent dipole moments in their ref-

erence structure.

III. CALCULATIONS

In a typical molecular-orbital DALTON 2.0 application,

the effective geometry is obtained on both SCF-Hartree-

Fock and MC-SCF (multiconfiguration-self-consistent-field)

-RAS (restricted-active-space) or -CAS (complete-active-

space) levels. For this, we input the equilibrium geome-

try parameters−→R eq obtained from full-CI (configuration-

interaction) calculations for HD and single-double-CI for the

other systems, with the aug-cc-pVDZ basis set (except for

HD, see below). A crucial detail in (ro-)vibrational calcula-

tions with Dalton is the sensible choice of the step length

(SL) for the numerical evaluation of the second derivative in

Eq. (1) in correlated calculations. Previous applications18, 26

244313-3 Post-BO isotopic dipole moments J. Chem. Phys. 135, 244313 (2011)

suggest using the value SL = 0.05 as a reference. A final

fundamental procedure to preserve symmetry during geom-

etry optimizations is to tighten the convergence thresholds.

While SCF calculations were found to be free of prob-

lems as regards evaluation of the second derivative, we noted

that the same is not true on the correlated levels. In fact,

the CAS/RAS results do not converge, with variation of SL,

on the fourth decimal place (in debye), needed to match the

experimental accuracy. We choose applications to CH3CD3

below in order to illustrate this issue, as comparison with pre-

vious BO calculations with this molecule18 is enlightening.

Fortunately, repeating a reported observation of Ruud et al.in the calculations of nuclear magnetic shielding constants,27

SCF calculations already give the desired accuracy for the iso-

topic dipole moments. In view of this, we limit the present

results to those obtained using SCF, leaving a study of the

influence of correlation effects to future work.

In what follows, we use the terminology μeq, μeff, μvib,

and μave, with μave = μeff + μvib, to mean, respectively, the

equilibrium geometry, effective geometry, vibrational contri-

bution, and vibrationally averaged isotopic dipole moments.

A. Dipole moment of HD

The HD dipole moment, pointing from D to H, is about

one order of magnitude lower than those of the other systems

considered here, so we choose this system for a first assess-

ment of our approach in comparison with experiments. For

HD, we have shown previously that, to obtain a reliable dipole

moment curve as function of the internuclear distance R, a

specific and quite small (3s, 2p, 1d) basis set has to be used.33

To consider vibrational averages with the present approach, a

basis set study was made, as shown in Table I, where only the

FNMC-SCF results are displayed.

Default basis sets for DALTON 2.0 are spherical Gaus-

sian functions but, as we move from double zeta (DZ) to

triple zeta (TZ) or larger, it becomes important to differ-

entiate from Cartesian basis set calculations (because they

have different numbers of d and f basis functions). De-

spite the importance of using the largest basis sets aug-

cc-pVQZ for minimum and effective geometry optimiza-

tion (Req and Reff), the estimate for the dipole moment

can get worse as we augment the basis set by the varia-

tional criterion. Among the aug-cc-pV series, the best per-

formance seems to be for aug-cc-pVTZ, but it still loses to

the basis set of Ref. 33 which furnishes 9.0 × 10−4 debye,

our assumed converged value, as we compare with the ex-

perimental value 8.83(28) × 10−4 debye at 295 K.1 In

Table I, we also display the result 12.5 × 10−4 debye for the

dipole moment of HT (T is for tritium) that is to be com-

pared with a previously fully nonadiabatic calculation, 11.11

× 10−4 debye.10 Note that here μave is largely dominated by

μeff, since vibrations do not break the molecular symmetry.

B. Dipole moments of CH3CD3 and carbon isotopicvariants

We turn now to systems that present both the elec-

tronic and vibrational contributions to the isotopic dipole mo-

TABLE I. Equilibrium geometry, effective geometry, and vibrationally av-

eraged SCF isotopic dipole moment of HD against various basis sets, in units

of 10−4 debye.

FNMC-SCF μeq μeff μAv

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) (Ref. 33)

Spherical 11.6 12.6 9.0a

Cartesian 9.5 9.4 9.4

Other calculations

8.89 (Ref. 2)

8.36 (Ref. 3)

8.29 (Ref. 4)

8.51 (Ref. 6)

8.31 (Ref. 10)

BO 0.0 0.0 0.0

Experimental (Ref. 1) 8.83(28)

HT (spherical), this work 17 12.5

aPresent converged value.

ment, which have the order of magnitude of 10−2 debye.

In this section, we consider tri-deuterated ethane, CH3CD3

(Table II), and its carbon isotopic variants 12CH313CD3 and

13CH312CD3 (Table III). Their dipole moments point from the

center of charge of the D atoms to the center of charge of the

H atoms, lying along the C–C bond. Bonding of three D atoms

to the heavier carbon isotope in 12CH313CD3 increases further

the dipole moment, while for 13CH312CD3 the two effects are

competitive, giving a slightly lower dipole moment, as com-

pared to CH3CD3. For all of them, the problems with basis

sets mentioned above are minimized and become less impor-

tant than the inclusion of post-BO effects. We then choose the

aug-cc-pVDZ basis set, which has the advantage of allowing

comparisons with BO calculations in Ref. 18 in an applica-

tion to CH3CD3. This case furnishes a good assessment to the

issues involved. In Ref. 18, the vibrationally averaged BO-

RAS dipole moment of 0.0114 debye was considered as an

improvement over the SCF one, 0.0125 debye, as compared

to 0.01086 debye from experiment, see Table II. We repeated

the RAS calculations at both BO and FNMC levels, with the

1s orbitals of the carbon atoms frozen and including all single

and double excitations from the seven highest occupied or-

bitals to the seven lowest unoccupied orbitals of the dominant

configuration (see Table II), as in Ref. 18. As we vary the SL

in the vicinity of the recommended SL = 0.05, we note the

lack of convergence in the fourth decimal place at both BO

and FNMC levels. On the contrary, the SCF calculations are

free of oscillations and the SCF-FNMC result, 0.0110 debye

agrees with experiment to within 10−4 debye, so that we take

this value as our best evaluation in this case. In view of this,

we use the SCF with aug-cc-pVDZ approach in the following

applications.

244313-4 Arapiraca, Jonsson, and Mohallem J. Chem. Phys. 135, 244313 (2011)

TABLE II. RAS and SCF step length study of the isotopic dipole moment of CH3 CD3 in units of debye.

BO FNMC

Step length μeff μvib μave μeff μvib μave

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 0.0110b

Experiment (Ref. 14) 0.0108617(5)

(Ref. 13) 0.01078(9)

aValue reported in Ref. 18.bPresent converged value.

As we consider the carbon isotopic variants, it is notable

that the BO calculations can hardly explain the experimental

differences among them. The slight increase for 12CH313CD3

as well as the decrease for 13CH312CD3 in the experimental

and calculated dipole moments, related to CH3CD3, shown in

Table III, are understood only in terms of the electronic con-

tribution, as in the HD case. In fact, the relative motion of

the two carbon atoms is not able to break the symmetry of

the molecule. On the other hand, contrary to HD, the dipole

moments are not dominated by the effective geometry con-

tributions μeff, that is, the vibrational contributions, μvib, are

larger than the effective geometry ones, μeff.

Again, in all cases stable SCF calculations perform bet-

ter than the corresponding numerically unstable RAS calcula-

tions, as compared to experiment.

C. Dipole moments of CH2CD2 and isotopomers

Among the three isotopomers of ethylene, shown in

Figure 1, the trans-isotopomer has no permanent isotopic

dipole moment due to its symmetry. The cis-isotopomer

CHDCHD possesses a dipole moment perpendicular to the

carbon double bond and the asymmetric one, CH2CD2, has

its dipole moment parallel to the double bond.

Calculations are made in the same way as for the sys-

tems discussed previously; the present theoretical result for

CH2CD2 agrees very well with experiment, see Table III.

The BO dipole moments are larger and almost the same for

CH2CD2 and CHDCHD. We have found no experimental or

other theoretical evaluations of the dipole moment for the cis-

isotopomer in the literature, so that the present calculation,

giving 0.0112 debye, stands as the first reference value for

this quantity.

IV. DISCUSSION AND FINAL REMARKS

It is clear from the detailed search around SL = 0.05 in

Table II that neither the BO-RAS nor the FNMC-RAS results

converge, even to the second significant figure. It is also clear

that the problem lies in the unstable RAS vibrational correc-

tions. It therefore becomes impossible to choose particular

RAS values to compare with experiments at the level of accu-

racy we seek here. In particular, we noted that BO-RAS does

not even capture the correct trend, i.e., of a decreasing dipole

moment on moving from 12CH313CD3 to 13CH3

12CD3 for

different values of SL. Yet, FNMC-RAS results do follow the

right trend, but are unstable under variations in SL. The SCF

results, on the other hand, are fully stable (independent of the

SL, at least in large ranges around SL = 0.05). The BO-SCF

calculations are not accurate enough but the FNMC-SCF cal-

culations agree with experiment to within 10−4 debye. For all

systems treated here we noted that the SCF optimal geome-

tries are closer to the experimental ones than those calculated

by correlated methods. This could be part of an explanation

for the good performance of SCF. Of course, these conclu-

sions were drawn from calculations of isotopic dipole mo-

ments and do not necessarily apply to other quantities.

It is particularly notable that the favorable FNMC results

come from effective geometry dipole moments. In Tables II–

III, one can see that BO-RAS values of μeff are all the same

for the different isotopologues, differing just in the vibrational

average corrections. For ethylene, we noted that the poor

TABLE III. SCF isotopic dipole moment of 12CH313CD3, 13CH3

12CD3, and CH2CD2 in both conformations asym-

metric and cis, in units of debye.

BO FNMC

System μeff μvib μave μeff μvib μave Experiment

12CH313CD3 −0.0038 0.0165 0.0127 −0.0053 0.0165 0.0112a 0.01094(11) (Ref. 15)

13CH312CD3 −0.0038 0.0163 0.0125 −0.0056 0.0163 0.0107a 0.01067(10) (Ref. 15)

CH2CD2 (asym.) −0.0003 0.0114 0.0111 −0.0020 0.0114 0.0094a 0.0091(4) (Ref. 16)

CHDCHD (cis) −0.0001 0.0125 0.0124 −0.0013 0.0125 0.0112a ...

aPresent converged values.

244313-5 Post-BO isotopic dipole moments J. Chem. Phys. 135, 244313 (2011)

FIG. 1. The cis, trans, and asymmetric conformations of ethylene iso-

topomers with their dipole moments indicated by arrows.

evaluation of μeff is responsible for the BO-RAS calculation

departing from experiment even more than the BO-SCF cal-

culation, differently from the previous cases. Nevertheless,

FNMC values for μeff differ from each other and from the

BO values. Note that, once the effective geometry results are

calculated on the FNMC level, it is irrelevant in the SCF cal-

culations whether the vibrational corrections are evaluated on

the BO or FNMC level. Furthermore, on the SCF level, for

which we find experimental agreement, it becomes clear that

FNMC plays a fundamental role in reaching this goal. Despite

of this, the convenient separation of strictly non-BO electronic

effects and BO or FNMC vibrational effects is not fully cor-

rect, as shown by the calculations on ethane isotopologues.

There is clearly an interference of the two effects.

Calculations of regular dipole moments are known to be

less dependent on the correlation level than on the basis set,35

suggesting that the atomic cores play a dominant role in the

separation of the centers of charge. FNMC are one-center

type corrections, that is, they have an atomic character.29

These features might explain why isotopic dipole moments

are shown here to be even less dependent on correlation.

Regarding the CAS/RAS numerical problems, we spec-

ulate that their cause lies in the use of the same step length

for all normal coordinates, when doing the property average,

which may be problematic if the modes have very different

frequencies. In particular, the rotation of the methyl group for

the ethane isotopomers might be problematic. Ideally, the step

length should be scaled with the frequency but a detailed in-

vestigation of this issue is beyond the scope of the present

paper, being a topic of future study.

Spectroscopic identification of hydrogen isotopes is

very important in many fields and especially important for

Astrophysics.34 In this paper, we presented a theoretical

method to identify and calculate isotopic dipole moments that

is accurate enough to evaluate post-BO effects. The method

exploits, on the electronic level, standard quantum chemical

approximations, which makes it appropriate to further appli-

cations to larger systems. It is also shown that electronic sym-

metry breaking beyond the BO approximation is crucial to

obtain spectroscopic accuracy and that the SCF level of cal-

culations seems to be appropriate for this goal.

ACKNOWLEDGMENTS

A.F.C.A. acknowledges Professors Amary Cesar, Peter

Taylor, and Pedro Vazquez and Dr. Luciano Vidal for use-

ful discussions and advice about running Dalton. We thank

Professor Ronald Dickman for a review of the paper. CNPq,

Fapemig and Capes (Brazil) are thanked for financial support.

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