Post Born–Oppenheimer isotopic effects and zero-point vibrational averages were previously inbodied in calculations of the dipole moments of isotopic species of some apolar molecules within the HF-SCF approximation (Arapiraca, 2011) [27]. Many other molecules, however, demand the inclusion of electronic correlation for this goal. Here, DFT calculations are reported for the isotopic effects on dipole moments of molecules with increasing permanent dipole moments, namely propane ( 0.1 debye), propyne (0.7 debye) and water (1.9 debye). The results account well for the experimental values and isotopic trends of the dipole moments of these molecules. 2014 Elsevier

1. DFT vibrationally averaged isotopic dipole moments of propane, propyne
and water isotopologues
A.F.C. Arapiraca a,b
, J.R. Mohallem a,⇑
a
Laboratório de Átomos e Moléculas Especiais, Departamento de Física, ICEx, Universidade Federal de Minas Gerais, PO Box 702, 30123-970 Belo Horizonte, MG, Brazil
b
Centro Federal de Educação Tecnológica de Minas Gerais, CEFET-MG, Campus I, 30421-169 Belo Horizonte, MG, Brazil
a r t i c l e i n f o
Article history:
Received 18 April 2014
In final form 19 June 2014
Available online 27 June 2014
a b s t r a c t
Post Born–Oppenheimer isotopic effects and zero-point vibrational averages were previously inbodied in
calculations of the dipole moments of isotopic species of some apolar molecules within the HF-SCF
approximation (Arapiraca, 2011) [27]. Many other molecules, however, demand the inclusion of elec-
tronic correlation for this goal. Here, DFT calculations are reported for the isotopic effects on dipole
moments of molecules with increasing permanent dipole moments, namely propane (l ’ 0:1 debye),
propyne (l ’ 0:7 debye) and water (l ’ 1:9 debye). The results account well for the experimental values
and isotopic trends of the dipole moments of these molecules.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
Isotopic substitution, with particular attention here to deutera-
tion (H !D), is able to change the electronic structure of molecules
and even to break the molecular point group symmetry [1–3].
These effects can either generate small isotopic dipole moments
in otherwise apolar molecules [4–16] or change the regular dipole
moments of polar ones [17–22]. In the first case new spectroscopic
possibilities are yielded, since the isotopologues start presenting
pure rotational spectra. These effects have prospective importance
for high-resolution molecular spectroscopy as well as for astro-
chemistry [23].
In order to approach measurements, [4–22], theoretical meth-
ods must include at least vibrational averaging [24–26,16,27] with
the aim of reaching spectroscopic accuracy. More sophisticated
methods apply to the simplest isotopologue, HD [28–33]; for larger
systems they are limited by their computational burden and are
not necessarily more accurate [34–36]. Once vibrations are consid-
ered, it becomes possible to distinguish among isotopologues
because of the average vibrational asymmetries, a procedure that
can be accomplished on the Born–Oppenheimer (BO) level of
approximations.
In a recent paper [27], hereafter referred to as I,we have shown
that including vibrational averaging is not enough, however, to
attain the desired 10À3
debye accuracy, which is approached only
with the introduction of post-BO finite-nuclear-mass-corrections
(FNMC) [37–39]. Taking advantage of the vibrational averaging
facility of the Dalton 2.0 code [40], FNMC was updated to it [39]
and applications was done in I to some apolar molecules on the
Hartree–Fock-self-consistent-field (HF-SCF) level.
At the stage of electronic calculations, Dalton 2.0 admits
HF-SCF, complete (or restricted)-active space-multi-configura-
tional-SCF (MC-SCF-CAS or -RAS) and density-functional-theory
(DFT) approaches. Pure isotopic dipole moments that appear under
isotopic substitution in apolar molecules should depend very little
on the electronic correlation, since the polarization of the elec-
tronic cloud is due to an atomiclike reduced mass effect [38]. This
explains why HF-SCF gives such good results for the isotopic dipole
moments of the molecules present in I, namely HD and isotopo-
logues of C2H4 and C2H6, which are likely to be discribed by a single
determinantal wavefunction. The numerical instabilities on the
Dalton MC-SCF calculations reported in I are not still understood
but might have their source on the smallness of the corrections
beyond HF-SCF, see I.
Calculations on the HF-SCF level are not appropriate to general
molecules, however; electronic correlation must be relevant in the
context of isotopic dipole moment calculations whenever it is rel-
evant yet in the context of BO electronic calculations. It can be
advanced that hydrocarbon molecules presenting triple bonds or
showing small permanent dipole moments, as well as molecules
having lone electron pairs, lie in this case.
So it becomes important to consider further this point in DFT
calculations of a series of molecules as compared to the HF-SCF
ones. Our choice was to study propane (C3H8), propyne (C3H4)
and water isotopologues. Propane presents a quite small dipole
moment of about 0:1 debye in its normal (non isotopic) structure.
Propyne has an average dipole moment of 0:8 debye and a triple
http://dx.doi.org/10.1016/j.cplett.2014.06.037
0009-2614/Ó 2014 Elsevier B.V. All rights reserved.
⇑ Corresponding author. Fax: +55 31 34095600.
E-mail address: rachid@fisica.ufmg.br (J.R. Mohallem).
Chemical Physics Letters 609 (2014) 123–128
Contents lists available at ScienceDirect
Chemical Physics Letters
journal homepage: www.elsevier.com/locate/cplett

2. bond. As the total vibrational plus post-BO correction has an order
of magnitude of 10À2
debye, it is expected that the isotopic effect
will affect substantially the dipole moment of propane and propy-
ne, which is confirmed experimentally [18]. Finally, besides appli-
cation to the water molecule has intrinsic relevance in view of its
importance to life, its polar character and the presence of the lone
electron pair makes it another interesting subject to the present
study. Previous applications aiming at introducing vibrational
effects as well as relativistic and adiabatic corrections on the dipole
moments of water isotopologues were found [34,35], so that com-
parison with other theoretical approaches is possible in this case.
Some isotopologues of these molecules have their dipole moments
measured, so we can again perform direct comparison of our
results with experiments.
Hence, in this letter we intend primarily to calculate the isoto-
pic changes on the dipole moments of isotopologues of the mole-
cules cited above with spectroscopic accuracy, which means
performing vibrational averages beyond the BO approximation.
Secondly, we are interested in checking the performance of our
methodology as applied to these systems which require different
levels of electronic correlation. In view of the previously reported
numerical problems with MC-SCF applications, we resort to DFT
to introduce electronic correlation. The appropriateness of this
choice is discussed in the last two sections.
2. Theory and computational methods
The approach for performing vibrational averages of molecular
properties in Dalton 2.0 is the variation-perturbation method of
Astrand et al. [41,25,42]. The first step consists in obtaining effec-
tive molecular geometry parameters defined as
Qeff
k ¼ Qeq
k þ
1
4x2
k
X
l
Feq
kll
xl
; ð1Þ
which corresponds to the minimum of the surface defined by add-
ing the zero-point energy to the potential energy surface (PES) [41].
This step is referred to as ZPVC, zero point vibrational correction. In
this equation, Qk is the normal coordinate k having a harmonic fre-
quency xk; Fklm is the cubic force field and the superscript eq means
equilibrium geometry. A molecular property is then expanded
around this effective geometry so that the first order correction
vanishes. To second order, the vibrational average of a property as
the dipole moment l, for instance, is calculated as
lave ¼ leff þ
1
4
X
k
1
xk
@2
l
@Q2
k
!
eff
; ð2Þ
where leff is the dipole moment in the effective geometry, or the
ZPVC dipole moment.
This approach is a facility of the Dalton 2.0 [40] code within the
BO approximation [25,42]. In what concerns FNMC [37–39], it can
be introduced into any computational routine for electronic molec-
ular calculations by changing the BO Hamiltonian HBO to (in atomic
units, a.u.)
H ¼ À
X
A
X
i
PA
r2
i
2MA
PA
!
þ HBO: ð3Þ
In this equation, MA is the mass of a generic nucleus A while i stands
for 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. The first term, which accounts for
FNMC, introduces the appropriate reduced electronic mass in the
calculations [38], being responsible for electronic symmetry break-
ing, which generates the electronic isotopic dipoles. This Hamilto-
nian has been upgraded to Dalton 2.0, as reported in I. The
consequent theory of vibrational effects is a straightforward repeti-
tion of that in Refs. [41,25,42], just replacing the BO Hamiltonian by
Eq. (3) above. In the first applications in I, the method showed to be
able to approach the necessary accuracy of 10À3
debye in calcula-
tions of the isotopic dipole moments of HD, ethane, ethylene and
their isotopologues. These calculations where performed on the
HF-SCF (Hartree–Fock-self-consistent-field) level with aug-cc-pVDZ
basis set (except for HD, for which larger basis sets have been
tested), because the correlated multi-reference calculations
MCSCF-RAS or -CAS, allowed in Dalton 2.0, showed to be numeri-
cally unstable for these applications. DFT calculations are verified
to be rather stable so that we resort to DFT to develop the present
study.
Dipole moments can be evaluated in a particular configuration,
as in the minimum or in the effective geometry. As usual,
l!
¼
X
A
ZA R
!
A À
Z
q r
!
d
3
r; ð4Þ
in standard notation, where now the electronic density q carries the
isotopic signature through FNMC. In the next step, l
!
is then aver-
aged over vibrational states by the method cited above.
3. Calculations
3.1. Choice of DFT functionals and basis sets
In the tables, the averaged dipole moment is represented as
lave ¼ leff þ l00
vib; ð5Þ
where l00
vib stands for the second term on the rhs of Eq. 2. It should
not be identified with the total vibrational contribution to the
dipole moment, however, since leff also accounts for vibrational
effects (for this reason we have changed the notation lvib of I to
l00
vib here).
The basis sets appropriate to dipole moment calculations are
well known. Based on our previous study on apolar molecules in
I (see also [26]) and on another study for water [43] we choose
the 6–31G⁄⁄
and aug-cc-pVDZ (aug-cc-pVXZ, X = D,T,Q for water)
as possible basis sets. The choice of the appropriate DFT functional
for the present applications, on the other hand, was more careful.
For each system we checked some potentially good functionals,
namely B3LYP, B3P86 and B3PW91, in calculations with the stan-
dard isotopologue and elect B3LYP as the one approaching better
the experimental result. This study is displayed in Table 1.
A complicating factor in the evaluation of lave is that the differ-
ent steps of the calculations can depend in different ways on the
methodology. The equilibrium geometries, which are inputs for
effective geometry calculations, can be optimized on the same
level of electronic calculations, on different levels, or simply taken
from the literature. Ideally, in order to assess the performance of
each method, the first choice is preferred, but it is not necessarily
the best choice for attaining experimental agreement. These fea-
tures are better illustrated in the following applications.
3.2. Propane isotopologues
Propane is an interesting system to investigate here. Concerning
the present study, the single-bonded propane molecule should
keep some resemblance with the apolar single- or double-bonded
hydrocarbon molecules studied in I. Different from those, however,
in its standard isotopic composition, C3H8, propane has already a
small permanent dipole moment in its lower energy configuration
(C2v) of about 0:1 debye, so that vibrations (contributing an order
of 10À2
debye) as well as deuteration (contributing an order of
10À3
debye) are able to change substantially this value. This
124 A.F.C. Arapiraca, J.R. Mohallem / Chemical Physics Letters 609 (2014) 123–128

3. consideration is supported by a survey of experimental dipole
moment data for its isotopologues. The first measurement of the
dipole moment of C3H8 was made in the sixties by Lide, with
microwave Stark spectroscopy [44]. Latter, Muenter and Laurie
repeated the same experiment but considering various deuterated
isotopologues [18]. As usual, they took the average values among
low-lying rotational lines in the ground vibrational state as the
permanent dipole moment of the ground state.
To the best of our knowledge, there is no theoretical evaluation
of the propane dipole moment, either in its normal or deuterated
species, other than calculations on equilibrium geometries for nor-
mal C3H8. In this case there are many data in the NIST database,
from HF-SCF, MP2 (2nd-order Möller–Plesset perturbation theory)
and DFT calculations with a broad range of basis sets [45]. The
results vary from 0:014 debye for DFT-LSDA, with a STO-3G basis
set, to 0:101 debye for DFT-PBEPBE with a Sadlej_pVTZ basis set.
It can be confirmed there that the quality of the dipole moment
calculations is more dependent on the basis set than on the corre-
lation method.
Here we intend not only to introduce post-BO vibrational aver-
aging on normal C3H8 but also to account to experimental values of
dipole moment for different isotopologues. The isotopic species
considered here are C3 H8, CH3CD2CH3, CD3CH2CD3 and CHD2CH2-
CHD2, according to Figure 1, in the ground state staggered confor-
mations. In the first three cases our aim is to compare with the only
available experiments. The situation in which the equilibrium
geometry BO contribution, pointing as shown in Figure 1, fully
add up to the FNMC equilibrium contribution is represented by
the last specie; that is why it has been added to the measured spe-
cies among some other possibilities. The contrary situation, where
the two contributions compete in the absence of vibrations, is
already represented by CH3CD2CH3 (recall that the smaller Bohr
radius, or equivalently, the larger mass of D makes its ‘‘valence’’
higher than that of H).
Dipole moments are calculated on the HF-SCF and DFT levels
with the combination B3LYP/aug-cc-pVDZ (see Table 1) and the
results shown in Table 2. At the first sight, the DFT performance
is deceptive in comparison to HF-SCF, as the B3LYP values of lave
are all overestimated by about 0:01 debye, which apparently
would turn the inclusion of FNMC irrelevant (note that in this case
BO calculations seem to predict the correct isotopic trend, though
this could be seen as a coincidence). However, this failure is clearly
seen in Table 2 to be connected with bad evaluations of leff , a con-
sequence of the fact that DFT gives worse starting equilibrium and
effective geometries than the corresponding HF-SCF ones, see [45].
As we use in Eq. (5) values of leff coming from HF-SCF and l00
vib
from DFT, resulting in lMix
ave in the table, we obtain the best overall
agreement with experiments, pointing to a non-negligible contri-
bution of the electronic correlation already for propane. The differ-
ences are now in the third decimal place so that FNMC becomes
important to agree with the isotopic trend.
As expected, the experimental results show that deuteration
changes substantially (more than 10% from normal propane) the
dipole moment. On the other hand, deuteration does not break
the C2v symmetry of any propane isotopologue considered here,
so that all contributions to the dipole moment have the same
direction, pointing downwards as shown if Figure 1. leff differs
to less from lBO
eq ¼ 0:0879 debye and is affected by both vibrational
and electronic (FNMC) effects, so that the final results become a
complicated combination of the two effects. Yet the relative values
of leff and l00
vib can be rationalized from two extreme situations.
CH3CD2CH3 has the smallest leff among the deutered isotopo-
logues and the largest l00
vib. This means a larger and less rigid C—
C—C bond angle. The opposite CD3CH2CD3 has the largest leff and
the smallest l00
vib, meaning a smaller and more rigid C—C—C bond
angle. These features seem to be consistent with the smaller Bohr
radius of D. We conclude that the two effects, electronic and vibra-
tional, are competitive in the present application, with predomi-
nance of the vibrational effect. These remarks clearly applies to
CHD2CH2CHD2 for which a reasoning based just on equilibrium
geometry features would predict wrongly the largest dipole
moment as discussed just bellow.
Overall, the l00
vib contribution is the responsible for capturing
the experimental trend on both BO and FNMC approaches. On
the other hand, the inclusion of FNMC is fundamental to approach
better the experimental values, and its differences from BO come
mainly from leff as observed already. This connection of FNMC
and effective geometries also appeared in previous applications
in I, particularly for hydrocarbons.
For the isotopologue CHD2CH2CHD2 there is no experimental
result to compare, hence the theoretical lMix
ave ¼ 0:0672 debye value
presented here stands as the first predictions for this quantity.
3.3. Propyne isotopologues
Propyne has a triple C–C bond so that electronic correlation is
expected to be important. Hence it becomes a good subject to com-
pare HF-SCF and DFT calculations. The dipole moment results are
Table 1
Vibrationally averaged FNMC and BO dipole moments, lave ðlave;BOÞ, for different
basis sets and DFT functionals for the standard (non-deuterated) isotopologues.
B3LYP B3P86 B3PW91 Experiment
C3H8 [18]
6–31G⁄⁄
0.0502 0.0504 0.0507 0.0848(10)
(0.0498) (0.0509) (0.0504)
aug-cc pVDZ 0.0942 0.0944 0.0947
(0.0994) (0.0947) (0.0952)
C3H4 [18]
6–31G⁄⁄
0.7752 0.7939 0.7918 0.7806(20)
(0.7729) (0.7915) (0.7894)
aug-cc pVDZ 0.8812 0.8957 0.8881
(0.8766) (0.8922) (0.8846)
H2O [22]
6–31G⁄⁄
1.9483 2.0380
(2.0077) –a
(2.0355) 1.85498(9)
aug-cc-pVDZ 1.8549 1.8593 1.8552
(1.8534) (1.8578) (1.8537)
aug-cc-pVTZ 1.8568 1.8580 1.8519
(1.8553) (1.8565) (1.8507)
aug-cc-pVQZ 1.8560 1.8550 1.8491
(1.8534) (1.8540) (1.8481)
a
Not converged.
Figure 1. The propane isotopologues: C3H8 (no deuteration), CH3CD2CH3 (deute-
rium in positions 1,2), CD3CH2CD3 (deuterium in positions 3 to 8), CHD2CH2CHD2
(deuterium in positions 5 to 8).
A.F.C. Arapiraca, J.R. Mohallem / Chemical Physics Letters 609 (2014) 123–128 125

4. important per se since propyne has been observed in radioastro-
nomic spectra [46].
The more recent measurement of the dipole moment of propy-
ne in its normal composition seems to be from Ware et al., namely
0:7840 debye [47]. Previously, Muenter and Laurie had measured
the isotopic dipole moment for four species, CH3CCH, CH3CCD,
CD3CCH and CD3CCD, displayed in the last column of Table 3
[18]. For consistency, we keep the values of Muenter and Laurie,
even for CH3CCH, as our experimental references. In [45] we can
see that minimum geometry DFT calculations tends to overesti-
mate the above value to about 0:8 debye while HF and MP2 give,
in average, smaller values than the experimental ones. We did
not find calculations for the different isotopic species of C3H4.
HF-SCF and DFT with B3LYP/6–31G⁄⁄
(see Table 1) calculations
were performed here for the same isotopic species measured in
[18]. The improvement towards experimental agreement, in com-
parison to the application to propane, is remarkable. The improve-
ment is also clearly due to the introduction of electronic
correlation, since for propyne HF-SCF does not show the same per-
formance as for propane and the molecules treated in I. On the
other hand, both HF-SCF and DFT yield the correct isotopic trend.
This feature cannot, however, be clearly connected to the inclusion
of FNMC in this application, since the BO calculations also seem to
generate the correct trend.
Deuteration of the three ‘‘umbrella’’ H atoms is shown to be
innocuous in changing the dipole moment. The effective changes
occur when the H isotope in the tail of the molecule is replaced.
Both leff and l00
vib increase for the lighter H isotope in this position,
pointing for the stretching of the C—H bond along the molecular
axis as being determinant for the dipole moment variations.
3.4. Water isotopologues
The available experimental data of the dipole moments of
water isotopologues come from high accurate MBER (molecular
beam eletric resonance) spectra. Dyke and Muenter measured
D2O obtaining 1:8545ð4Þ debye [21], while Shostak et al. reported
results for H2O, 1:85498ð9Þ debye, and HDO, 1:8517ð5Þ debye
[22]. Concerning calculations, it is well known that MP2 and
DFT (with B3LYP, B3PW91 and VdW-DF functionals) [43,48]
approach the experimental value for H2O, but in equilibrium
geometry configurations. This fact advances that in the case of
water isotopologues ZPVC should affect the dipole moments (in
fact, any other geometry-dependent molecular property) in less
extent than the previous systems. Mengel and Jensen calculated
state-dependent dipole moment values of H2O from a refined
dipole moment surface [49]. A study of the dipole moment of
HDO against the bond angle was made in our group previously,
without vibrational averaging [50]. In view of its importance,
however, considerable effort has been expended to reproduce
theoretically the dipole moment of H2O in its ground vibrational
state. Thus, Russel and Spackman performed a MP2-ZPVC calcula-
tion leading to 1:8573 debye [24], while Astrand et al. performed
a MC-SCF-ZPVC calculation obtaining 1:8936 debye [25]. More
recently, Lodi et al. approached the problem from a ‘‘more ab ini-
tio’’ point of view, building an accurate BO dipole moment surface
for H2O with which they could average the dipole moment on the
ground vibrational state, obtaining 1:8583 debye without the
inclusion of relativistic corrections and 1:8540 debye including
them [34], quite close to the experimental value. However, Hob-
son et al. later added to the Lodi et al. calculation the diagonal-
BO-correction (DBOC), which lead to the value of 1:8560 debye,
departing from the experimental value relatively to the previous
one [35]. Of course this situation is not satisfactory. It seems that
a proper assessment to this problem must take into account the
isotopic effect through FNMC which per se involves adiabatic cor-
rections like DBOC. For this we performed (non-relativistic) calcu-
lations for the three measured isotopologues on various levels of
correlation, namely HF-SCF, MC-SCF-CAS and DFT, for different
basis sets and DFT functionals.
Table 2
HF-SCF and DFT-B3LYP effective (leff ) and vibrationally averaged (lave) dipole moments of propane isotopologues. All calculations done with the aug-cc-pVDZ basis set.
System (Experiment)b
HF-SCF B3LYP Mixa
leff ðleff ;BOÞ l00
vib ðl00
vib;BOÞ lave ðlave;BOÞ leff ðleff;BOÞ l00
vib ðl00
vib;BOÞ lave ðlave;BOÞ lMix
ave
C3H8 0.0741 0.0120 0.0861 0.0810 0.0132 0.0942
0.0848(20) (0.0746) (0.0115) (0.0861) (0.0820) (0.0124) (0.0994) 0.0873
CH3CD2CH3 0.0743 0.0249 0.0992 0.0811 0.0256 0.1067
0.095(20) (0.0760) (0.0242) (0.1002) (0.0832) (0.0247) (0.1079) 0.0999
CD3CH2CD3 0.0781 À0.043 0.0738 0.0857 À0.0011 0.0846
0.076(20) (0.0772) (-0.045) (0.0727) (0.0850) (-0.0014) (0.0836) 0.077
CHD2CH2CHD2 0.0763 À0.0110 0.0635 0.0838 À0.0091 0.0747
(0.0746) (-0.0112) (0.0634) (0.0825) (-0.0094) (0.0731) 0.0672
a
Mix means adding leff from HF-SCF to l00
vib from DFT to yield lMix
ave. The experimental error of 0:002 debye is considered conservative by the authors. The BO equilibrium
geometry dipole moment is lBO
eq ¼ 0:0879 debye for all isotopologues.
b
[18].
Table 3
HF-SCF and DFT-B3LYP effective (leff ) and vibrationally averaged (lave) dipole moments of propyne isotopologues. All calculations done with the 6–31G⁄⁄
basis set.The
experimental error of 0:002 debye is considered conservative by the authors. The BO equilibrium geometry dipole moment is lBO
eq ¼ 0:716 debye for all isotopologues.
System HF-SCF B3LYP Experiment [18]
leff ðleff;BOÞ l00
vib ðl00
vib;BOÞ lave ðlave;BOÞ leff ðleff;BOÞ l00
vib ðl00
vib;BOÞ lave ðlave;BOÞ
CH3CCH 0.6702 0.0309 0.7011 0.7194 0.0558 0.7752
(0.6679) (0.0309) (0.6988) (0.7168) (0.0561) (0.7729) 0.7806(20)
CH3CCD 0.6691 0.0197 0.6890 0.7173 0.0374 0.7547
(0.6655) (0.0197) (0.6852) (0.7131) (0.0378) (0.7509) 0.7678(20)
CD3CCH 0.6704 0.0350 0.7054 0.7193 0.0606 0.7799
(0.6707) (0.0350) (0.7056) (0.7194) (0.0609) (0.7803) 0.7841(20)
CD3CCD 0.6693 0.0237 0.6930 0.7172 0.0422 0.7594
(0.6683) (0.0238) (0.6921) (0.7158) (0.0426) (0.7584) 0.7722(20)
126 A.F.C. Arapiraca, J.R. Mohallem / Chemical Physics Letters 609 (2014) 123–128

5. Again, contrary to the cases of purely isotopic dipole moments
in I, HF-SCF does not work well for water isotopologues. Instead,
it overestimates the H2O dipole moment up to 7% in average, for
many choices of basis sets. Confirming a previous BO application
[25], MC-SCF-CAS becomes stable, but still overestimates de H2O
dipole moment up to 2%, even with the inclusion of FNMC. We fur-
ther observe that BO calculations on any level and for most basis
sets use to predict a wrong isotopic trend. Particularly it uses to
reverse the relative values of the dipole moments of H2O and D2O.
On the other hand, DFT reproduces very well the experimental
isotopic dipole moments of water as long as we start from the very
accurate equilibrium geometry of Lodi et al. [34]. Considering first
different functionals and basis sets, we observed an outstanding
performance of the B3P86/aug-cc-pVQZ combination(see Table 1),
which might be seen as a result of an apparent convergence of the
series aug-cc-pVXZ with increasing size of the basis set X-zeta fea-
ture. However, as we change to the B3PW91 functional, this con-
vergence fails and the best results turn to be for the aug-cc-pVDZ
basis set, a feature of the DZ basis set already observed in previous
calculations in I (see also [43,26]). Overall, all tested combinations
of functionals and basis sets agree within 10À2
debye with experi-
ments but the results closer to experiments appear for the B3LYP/
aug-cc-pVDZ combination, shown in Table 4.
Turning to these results, we note that the agreement for H2O
and D2O is remarkable; the result for HDO is a little worse, though
confirming the isotopic trend. Since the calculations in the equilib-
rium geometry wrongly predict a larger dipole moment for HDO,
this seems to be the source for this worsening. In fact, the same ill-
ness appears to affect the effective geometry dipole moment of
HDO.
Overall, as discussed above, the contribution of l00
vib is of lower
importance for water, being of the same order of magnitude of the
FNMC contribution but with an opposite sign. In consequence, BO
vibrational averaging alone predicts a wrong isotopic trend.
Instead, FNMC is fundamental here and yields the proper trend
on both the MC-SCF-CAS (not shown in the table) and DFT levels
for any basis set.
4. Discussion and conclusions
It is clear from the tables that going beyond the BO approxima-
tion mainly affects the leff component of the dipole moment for all
systems. The l00
vib component is calculated with the same accuracy
on the BO level.
The application to propane isotopologues represents a limit
case in which HF-SCF and DFT give essentially equivalent results.
The accordance with experiments is within the experimental
errors, taken as conservative by the authors [18]. Since FNMC
effects are within 10À3
debye and noting that the BO calculations
already give the proper isotopic trend, the role of FNMC here is
unclear (recall however that FNMC does not introduce appreciable
computational extra burden to the calculations). Finally, it comes
out that the presence of many vibrational modes embodies differ-
ent situation in which BO-vibrational and post-BO isotopic effects
can compete or combine to the magnitude of the dipole moment.
Unfortunately this feature cannot be discussed quantitatively in
terms of leff and l00
vib, as already pointed out. On the other hand,
it becomes clear that FNMC effects are mainly connected to leff ,
a remark already noted in the applications in I to apolar hydrocar-
bons, while electronic correlation is mostly connected to l00
vib. The
possibility of mixing different methods to account better for the
different steps of the calculations turns out as the most important
methodological aspect of this application. Of course this point
must be subject to further tests.
Turning to the case of propyne isotopologues, we note that the
inclusion of correlation via DFT becomes vital in order to agree
with experiments. We connect this feature to the presence of a tri-
ple bond as well as an already substantial non-isotopic dipole
moment of about 0:8 debye. That is, HF-SCF calculations become
no longer satisfactory in this case. Once again BO results give the
proper isotopic trend, though now on the DFT level. It seems that
the role of FNMC here is to move the results towards the experi-
mental ones, except for CD3CCH for which FNMC and BO practi-
cally coincide. It is also notable that for this simpler molecule, as
compared to propane, leff and l00
vib have always the same sign, con-
tributing to enlarge the dipole moment in all cases.
For water, HF-SCF gives fully wrong results; electronic correla-
tion is needed even for agreement within 10À1
debye, that is, even
for equilibrium geometry calculations as iswell known [43]. This
need is clearly connected to the existence of the lone electron pair.
BO-DFT calculations, on the other hand, fails in predicting the rel-
ative dipole moments for the C2v isotopologues H2O and D2O. The
reason, as proposed in the previous section, is that the vibrational
effect for water is much lower than in the other cases here and in I
(except for HD). This fact can be understood as we note that the
main contribution to lvib in H2O and isotopologues comes from
the bending mode, the dependence of l with the bond angle is
almost linear [50], so that in average the vibrational effect is small.
In consequence, the full lave is of the same order, 10À3
D, of the
FNMC contribution and almost the same for the three isotopo-
logues. Fortunately, DFT-FNMC-ZPVC calculations retrieve the cor-
rect isotopic trend and yields a remarkable agreement for H2O and
D2O as well. The result for the non-C2v HDO does not have the
Table 4
DFT-B3LYP BO and FNMC equilibrium leq, effective (leff ) and vibrationally averaged (lave) dipole moments of water isotopologues. All calculations done with the aug-cc-pVDZ
basis set.
System leq DFT (BO) leq DFT (FNMC) Other Experiment
leff l00
vib lave leff l00
vib lave
H2O 1.8541 1.8598 À0.0064 1.8534 1.8556 1.8613 À0.0064 1.8549 1:8573a
1:85498ð9Þe
1:8936b
1:8583c
1:8540c
1:8560d
D2O – 1:8588 À0:0046 1:8542 1:8548 1:8593 À0:0064 1:8547 1:8545ð4Þf
HDO – 1:8590 À0:0059 1:8531 1:8601 1:8601 À0:0059 1:8542 1:8517ð5Þe
a
From [24].
b
From [25].
c
From [34].
d
From [35].
e
From [22].
f
From [21].
A.F.C. Arapiraca, J.R. Mohallem / Chemical Physics Letters 609 (2014) 123–128 127

6. same quality, despite being well acceptable. We guess that to reach
the same accuracy of H2O and D2O we should re-optimize its
geometry accounting for the electronic symmetry breaking,
that is, in a FNMC adiabatic PES. Considering that DBOC (as a per-
turbative correction to the PES) is unable to break the electronic
symmetry and that the FNMC facility is not available for a high
quality BO-PES that must be used, we have to postpone this check.
This last point raises the question that, being FNMC needed to
account for the isotopic trend up to 10À3
debye accuracy, the
results from Lodi et al. [34] should be subject to re-evaluation, spe-
cially in what concern the amounts of vibrational and relativistic
corrections. In fact, as Hobson et al. add DBOC to their calculations
with H2O their result flees from the experimental value [35]. It is
remarkable that the vibrational correction to the dipole moment
of the water molecule reported by Lodi et al. is one order of mag-
nitude smaller than ours, as well as than the ZPVC results from
[24,25]. So, this point deserves to be resumed in the future.
Finally a few words about the comparison with experimental
results. It is clear that the theoretical–experimental agreement
may not be a good guide on methodological tests, in view of possi-
ble error cancelations in calculations. The distribution of experi-
ments over decades also creates problems for such comparisons.
In fact, microwave dipole moment measurements are almost
always made relative to the standard OCS dipole moment. Before
1968 this standard was in error so that earlier measurements
should be corrected for this fact. An example is the propyne dipole
moment measured by Muenter and Laurie in 1966, l ¼ 0:7806ð20Þ
D [18], and 20 years after by Ware et al. l ¼ 0:7840 D [47]. The two
values may only be compared when the proper OCS calibration is
taken into account. Since we are primarily interested on evaluating
the FNMC effect for obtaining the isotopic trend we kept the earlier
values for the propyne isotopologues. However, considering both
the prior as the present application, we believe that the volume
of results, all pointing in the direction of the experimental isotopic
trends, supports our procedure.
In conclusion, we have shown that the inclusion of isotopic
FNMC and vibrational average effects, combined with electron cor-
relation via DFT, is very important to the calculation of accurate
values and isotopic trends of dipole moments of general isotopo-
logues. Furthermore, the stable and best performance of DFT as
compared to the other methods available in Dalton recommends
DFT for future studies on the same subject.
Acknowledgements
We thank Conselho Nacional de Pesquisas (CNPq) for financial
support and people from CENAPAD-MG for their help in the
computations.
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