The polar tensors, effective charges, and infrared intensities of X2CY molecules
A. B. M. S. Bassi and R. E. Bruns
Citation: The Journal of Chemical Physics 62, 3235 (1975); doi: 10.1063/1.430874
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The polar tensors, effective charges, and infrared intensities of
X 2CY molecules
A. B. M. S. Bassi and A. E. Bruns
Instituto de Quimica, Universidade Estadual de Campinas, Campinas, SP, Brasil
(Received 3 September 1974)
The experimental polar tensors and effective charges for the X 2 CY molecules where X=F,CI and
Y =O,S have been calculated. The polar tensors, with the exception of 3Px/iJxy, and the effective
charges of anyone of these molecules can be calculated from these values for the other three
molecules. The effective charges of carbon, oxygen, and sulfur are not transferable although those for
fluorine and chlorine are transferable within the calculated experimental errors. The effective charge
values appear to show trends which reflect the electronegativities of the atoms in the molecules. The
sums of the gas phase infrared fundamental intensities of these molecules are shown to be
interrelated.
ap,Jay ..
INTRODUCTION
The infrared gas phase intensities of the fundamental
vibrations of XeCY molecules (X = F, CI; Y = 0, S) are of
special interest because the integrated intensities of any
one of these four molecules can be calculated using the
intensities of the other three molecules and the normal
coordinate transformations of all four molecules. 1,2,S
Although the relationship between the intensities is difficult to express in analytical form, a very simple relationship
(ap/arl)CI2CO -
(ap/ ar lh 2 co
= (ap/arl)CIaCS -
(ap/arl)FaCS rj
=1,2, ••• ,6
(1)
exists involving the dipole moment derivatives of these
molecules with respect to the internal coordinates. A
similar relationship is expected to exist between the
polar tensor elements as defined by Biarge, Herranz,
and Morcillo 4 and recently reviewed by Person and Newton.i Using the nomenclature of the latter authors this
relationship involving the derivatives of the molecular
dipole moment with respect to the atomic Cartesian coordinates is investigated. The effective charge parameter for atom a, ~ .. , which is related to the atomic polar tensor elements by the equation
has been shown by King, Mast, and Blanchette6 to be
apprOXimately transferable for various hydrocarbons.
It is of interest here to determine if the effective charges
are transferable within this XaCY group of molecules
and if a relationship for these parameters of a form
similar to Eq. (1) is applicable. The effective charges
are of special importance as they can be determined
without knowledge of the normal coordinate transformation. In principle, one only needs to know the sum of
the infrared fundamental intensities of the molecule plus
a rotational term dependent only on the permanent dipole moment and the principal moments of inertia.
CALCULATIONS
The polar tensor for atom a can be represented by
The Journal of Chemical Physics, Vol. 62, No.8, 15 April 1975
apy/ay ..
apg/ay ..
(3)
The total polar tensor P x is then a juxtaposition of all
the atomic polar tensors of the molecule. For the XaCY
molecules we express P x as
(4)
The elements of the polar tensors were determined using
values of the dipole moment derivatives with respect to
the normal coordinates calculated from the measured intensities of Hopper et al. 7,8 For F aCO we have interchanged the assignment of lis and lis of Ref. 7 following
the suggestion of McKean 9 who has re-examined these
data. Thus our values for the intensities, r s and r 5 ,
correspond to the assignment of Overend and Scherer, 10
where liS = 584 cm- 1 and 115 = 626 cm- I • The values of
these derivatives form the 3 x (3N - 6) matrix defined by
Person and Newtons as
Po= (P'ItPoa··· P QSN-6) ,
where
(5)
(6)
The (11 = ± 1 depending on the sign of the dipole moment
derivative. As the proper choices for the (11 have resisted experimental solution for the XaCY molecules we
have calculated Px elements using the signs of the (11 indicated by the CNDO results of previous calculations. I, 2,11
The polar tensors are then calculated directly from P Q
through the matrix equation
(7)
The L-1 and U matrices are determined by the transformations from normal coordinates to symmetry coordinates (S=LQ) and from internal coordinates to symmetry coordinates (S=UR). The product DMl/2=B defines the relationship between the internal coordinates
and the atomic Cartesian coordinates, R= Bx. Also
I Ml/2 = J3 and Pp of the rotational term of Eq. (7) have
Copyright © 1975 American I nstitute of Physics
3235
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A. B. M. S. Bassi and R. E. Bruns: Infrared intensities of X2 CY molecules
3236
z
RESULTS
The polar tensors of FzCO, ClaCO, FzCS, and ClaCS
are presented in Table II. These values can be used to
emphasize several interesting general properties of
polar tensors which have been pointed out previously. 4,5
It can be seen that the sum of the atomic polar tensors
yields the null matrix for each of these molecules verifying the equation
y
N
'"
p(a)-o
i...Jx-,
(9 )
<>=1
FIG. 1. The Cartesian coordinate systems for the X2CY molecules.
been given in Ref. 5.
The geometries, dipole moment values and symmetry
coordinates employed in these calculations are identical
to those used previously. I,a, 11 For ClaCO, FaCS, and
ClaCS the L- I matrices of Refs. 7 and 8 were used. The
L- 1 matrix of Overend and Scherer lo corresponding to
the assignment of lI3 = 584 and lI5 = 626 cm- I was employed
for the F zCO polar tensor. The Cartesian coordinate
systems and their relationship to molecular geometry
are presented in Fig. 1. Transformation of the dipole
moment derivatives between the two coordinate systems
illustrated there were performed by matrices which
rotate one system into the other.
To test for the transferability of polar tensor elements, error limits were determined from the errors
in the intensities, r j , given in Refs. t7) and t8). Converting these errors to errors in the ap/aQj allows an
estimate of a set of propagated errors in P x through
Eq. (7). As estimates of errors in the normal coordinate transformations have not been published, our errors do not include contributions from these sources.
The actual errors in P x are expected to be somewhat
larger than those estimated here.
Although we have not recalculated the inverse of the
normal coordinate transformations to insure consistency
between our definitions and those used in Refs. t7) and
t8), these transformations can be partially verified using
polar tensors. As pointed out by King et al. s the sum
over all the fundamental intensities is related to the effective charges, the masses of the atoms, rna, and a
rotational term n through the equation
~Ak=K{~~~/ma-n}.
(8)
As ~! is the trace of P~ (P~)' the ~a are in part determined by L -I. A comparison of each side of Eq. (8) is
given in Table I. The excellent agreement shown there
indicates that no major inconsistencies apparently exist
between our definitions of the various coordinates and
those used in the experimental work. As these definitions can be ambiguous, the usefulness of Eq. (8) is to
be emphasized.
which was first pointed out by Morcillo and co-workers. 4
The polar tensors for chlorine and fluorine in the righthanded Cartesian coordinate system (X', yl, Z') of
Fig. 1 are presented in Table III. As the Z' and X'
axes are parallel and perpendicular to the CX 1 bond
this coordinate system is convenient for analyzing the
polar tensor elements corresponding to the CF and CCl
stretching and bending motions. Such movements are
either parallel or perpendicular to these axes. As a
result, the off-diagonal elements for the halogen atoms
in Table III are smaller relative to the diagonal elements
than in the coordinate system (X, Y, Z). The effective
charges ~a evaluated from the matrices in Tables II and
III are listed in Table IV. The traces of these matrices
and the effective charges ar~, of course, invariant to
the coordinate transformation (X, Y, Z) - (X', y', Z ').
The effective charges for carbon, oxygen, and sulfur
are certainly not transferable. The differences between
such values for different molecules exceeds the estimated experimental errors by large amounts. Such
large differences are not easily explained by inaccurate
normal coordinate transformations which would tend to
augment the estimated errors in Table IV. However,
the effective charges for chlorine and fluorine appear
to be invariant to oxygen or sulfur substitution. The
effective charges of either halogen differ by about O.Ole
similar to the range observed by King et al. for the effective charges of hydrogen in methyl chloride, bromide,
and iodide.
While the polar tensor elements and effective charges
are not generally transferable within this group of molecules, relationships similar to those observed for the
dipole moment derivatives with respect to the internal
TABLE 1. Comparison of intensity sums calculated directly
from experimental data and those from the effective charges
(km mole-l)~
X 2CY
L;Ak
K{ L:~=d~/ma - o}
F 2CO
Cl 2CO
F 2CS
Cl 2CS
854
641
609
390
852
641
609
392
aK = Nrr/3c 2 = O. 702 X 10 13 mole-I sec 2 km -2. The values of KO
for F 2CO and Cl 2CO are 1. 2 and 1. 3 km mole-I. KrI. values
for F 2 CS and Cl 2CS are negligible. These values are calculated from the equation rI.= (P;+p;J/In+ (P;+p~)IIyy+ (p~+p~)1
I..,. where p" is the x component of the permanent dipole moment and In is a principal moment of inertia.
J. Chem. Phys., Vol. 62, No.8, 15 April 1975
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3237
A. B. M. S. Bassi and R. E. Bruns: Infrared intensities of X 2 CY molecules
TABLE II. The polar tensors of FzCO, F 2CS, Cl 2CO and Cl2CS (e). a,b
pJeC)
X2CY
°
(1.90:0.04
F 2CO
0.53±0.01
Cl 2CO
°
°
0.20 ±O. 05
(1.99:0.04
0
C·36:0.04
F 2CS
("28:0.07
Cl 2CS
°
(-0.49:0.01
(-.0.71:0.01
2.10:0.0J
1. 54:0.
co.59:0.01
oJ
-0.87:0.0)
°
-0.30±0.01
0
(-0.70:0.02
-0.20±0.02
-0.98:0.0J
0
0
-0.23±0.03
0
c·0.ll:0.02
(-0.62:0.02
)c
D.l2:0' 02
0
-0.64:0.0J
-0.40±0.01
°
-,,,:,J
co.58:0.03
0.04 ±O. 01
0
1. 55 :0. 05
-0.33:0.01)
°
°
0.00 ±O. 01
-O.62±0.OI
0
-0.28±0.02
0
-0.34:0.02 )
-0.33 ±0.01)
0
-0.04±0.01
-0.04±0.02
° 2.01:0.0)
° , ,
- 0.15 ± O. 03
°
-0.11±0.00
-0.31±0.00
0.12±0.01
°
piXt )"
ppl
0
-0.68±0.01
°
-0.35:0.02)
0
-0.39±0.01
0.06±0.01
-0.39±0.02
"The values of the polar tensor elements are multiples of the electronic change e. 1 DA -I = O. 2082e = 3. 335 X 10-20 coulomb.
See Ref. 5 for a discussion of this unit.
t>rhese polar tensors were calculated using symmetry coordinates, geometries and the dipole moment values given in Ref. 1,
2, and 11. The L- I matrices and Po matrices are formed from the values of Refs. 7, 8, and 10. For F2CO the values of rs
and I's have been interchanged as described in the text. The following signs were used for the values of the aplaQ( FzCO,
CI 2CO, and F 2CS, CJ'j =+ 1 for i=2 and CJ'j=-l for i=l, 3, 4, 5, and 6; Cl 2CS, CJ'j =+1 for i=2 and 3 and CJ'j =-1 for i =1, 4, 5,
and 6. It should be noted that these signs are arbitrary and depend upon the definitions of the coordinate axes, symmetry
coordinates, positive sense of the dipole moments, and the phases of the normal coordinates.
"The diagonal polar tensor elements for X2 are equal to those of XI' The off-diagonal elements of X2 are equal but have opposite signs to those of XI'
coordinates appear to be valid, at least within our calculated experimental errors. For example, the ap~ laz Cl
elements for the oxygen and sulfur matrices are identical to the dipole moment derivatives ap lareo and ap I
ares, where reo and res are the internal displacement
coordinates of the CO and CS bonds, As these latter
derivatives are related to each other through Eq. (1) an
analogous relationship exists for the ap.laz Cl •
~Cl (CI2CO) - ~Cl (F 2CO) = ~Cl (ClaCS) - ~Cl (FaCS). (11)
1
Cl1 )
pi
Cl4
)
(F 2CS) ,
4
The agreement between the polar tensor elements is
very good. All values are consistent within the error
limits of Table V except those for apx lax s. As the difference between the experimental and calculated values
of this element (0.0ge) is only slightly larger than the
sum of the estimated propagated (0.04e) and experimental
(ClaCO) - pi"'a) (F 2CO)
= P.iClS ) (CI 2CS) -
3
There are three distinct cases in which these equations
are valid: (1) 0'1 = 0!3 =CI and 0!2 = 0!4 = F, (2) 0'1 = O!a =0
and 0!3=0'4=S, and (3) 0!1=0'2=0'3=0!4=C, Thepolar
tensor elements and the effective charges of Cl2CS calculated from the values for F 2CO, F 2CS, and ClaCO using Eqs. (10) and (11) are compared with the experimental values in Tables V and VI.
In general, such relationships can be expressed as
P.i
2
(10)
and
TABLE III. The halogen atomic polar tensors pi~) for F 2CO, Cl 2CO, F 2CS, and Cl 2CS for coordinate system (X', y" z,). a,b
C
35 O 03
:
.
°
-0.11±0.00
o
0.07±0.03
piCPI ) (CI2CO)
-0.15:0.04
o
0.00±0.01
(
-0.05±0.04
o
0.04:0.03)
-0.33 ±0.03
(
-0.98±0.03
-0.14:0.04 )
0
0
0.20:0.03)
-0.04±0.01
0.14±0.03
0
-0.98±0.03
pig ll ) (CI2CS)
o
0.07:0.03 )
0.06±0.01
- O. 83 ±O. 04
o
-0.86±0.04
"The units are e. See Footnote a, Table II.
~he diagonal elements for Fz(Cl z) are equal to those of FI(CI I). The off-diagonal elements of
F 2 (Cl2) have identical magnitudes and opposite signs to those of F 1(Clt ).
J. Chem. Phys., Vol. 62, No.8, 15 April 1975
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3238
A. B. M. S. Bassi and R. E. Bruns: Infrared intensities of X 2 CY molecules
TABLE IV. The experimental effective charges of
F 2CO, F 2CS, Cl2CO, and Cl 2CS. a
TABLE VI. The experimental and calculated effective charges
of Cl 2CS. a
X2CY
X 2CY
!;c
!;s
!;Cl
Exptl
Calculated from
Eq. (11)
2.02 ±O. 09
0.78±0.04
O. 87±0. 04
2.08±0.21
0.77±0.10
0.87±0.09
1;6
d
!;Cl
4.08±0.37
0.61±0.06
0.76 ±O. 07
4.37 ± O. 87
0.60±0.15
0.76 ±0.15
3.98 ±1.12
0.69 ± 0.18
0.77±0.18
F 2CO
Cl 2CO
F 2CS
Cl 2CS
F 2CO
Cl 2CO
F 2CS
Cl 2CS
I;c
2.88 ±O. 06
2.53±0.09
2.43 ±0.06
2.02±0.09
l;y
1. 04 ± 0.03
1.16 ±O. 04
0.65 ±O. 03
0.78 ±O. 04
I;x
1.05±0.02
0.86±0.03
1. 06 ± 0.03
0.87 ± O. 04
I;~
I;~
!;5c
8.33±0.37
6.40 ±O. 45
5.91±0.30
4.08±0.37
1. 09 ± 0.05
1. 35 ±O. 09
0.43 ±O. 03
0.61 ± O. 05
1.10±0.05
0.74±0.06
1.13 ±O. 07
0.76 ±O. 07
Exptl
Calculated from
Eq. (11)
Calculated from
Eq. (12)
2
aUnits of e, see Footnote a, Table II.
aUnits of e, see Footnote a, Table II.
(0.02e) errors, small errors in the normal coordinate
transformations of the XzCY molecules could account for
the discrepancy. This polar tensor element is, in part,
determined by the ap/aa value of the Bl symmetry species. The estimate of ap/aa for ClzCS using Eq. (1)
(r f = a =CF in-plane bending coordinate) also differs
from the experimental value by an amount slightly greater than the propagated error.z An estimate of the errors in the normal coordinates, at least in the Bl symmetry species, appears warranted.
molecules are related by the equation
The agreement between the calculated and experimental effective charges of ClzCS is excellent, the largest
difference of 0.06e occurs for the carbon effective
charge. Not only are the differences smaller than the
errors propagated for the calculated values (Table VI)
they are substantially smaller than the experimental
errors. Included in Tables IV and VI are values of ~~ •
The values in Table IV are the squares of the experimental ~CI' Table VI includes these values of ~~ for
ClzCS and estimates of these quantities using Eq. (11)
and
Inspection of the values in Table VI leads to the conclusion that Eq. (12) is also valid.
DISCUSSION
Recent measurement 12 of the dipole moment of F zCS
(0.08 D) using the Stark effect confirms the bond moment estimate of 0.05 ± 0.2 D made by Hopper et al. 8
This evaluation implies that the dipole moments of these
TABLE V.
(,28:0.07
~hese
calculated using an analogous equation as the derivatives of the dipole moment with respect to the internal
coordinates can be calculated using Eq. (1).
That the bond moment hypothesis for the equilibrium
dipole moments is valid can be concluded from Eq. (13).
However acceptance of Eq. (10) does not imply that this
model can be extended to experimental data dependent
on vibrational distortions of the equilibrium geometry.
Indeed the differing values of ap,jax", and apjay", where
a =0, S (and of apx./ax~ and apy./ay~ where a = F, CI) are
measures of deviations from results predicted by a
bond moment modeL The latter implies that such diagonal polar tensor elements are expected to be identical.
That apx/ax", and apy/ay", (apx./ax~ and apy./ay~ for
chlorine and fluorine) differ by large amounts indicates
that changes in the electronic charge distributions are
quite different for bending motions in and out of the XZ
(or X'Z') plane. Nonzero values of the off-diagonal
halogen polar tensor elements in the X' y' Z' system
(Table III) also point out the inaccuracy of the bond moment hypothesis.
The polar tensor values indicate the complexity of dipole moment changes for these molecules. Table II
illustrates that the inequality
apy/ay> apx/ax> aPe/az ,
holds for all the oxygen and sulfur values but that no
(,45:0.12
°
°
-0.21±0.07
piClt )
piS)
CO.12:0.02
-0.15±0.03
°
Calc'
It is not surprising that the atomic polar tensors can be
Comparison of the calculated and experimental polar tensors of Cl 2CS (e).
piC)
Exptl
(13)
°
( - 0.58;0.03
0.04±0.01
1. 55 :0. oJ
°
(-0.21:0.04
°
-0. 35 : 0. 02 )
°
-0.39±0.01
°
-0.34±0.04)
0.06±0.01
- 0.77:0.oJ
°
- 0.39±0.02
(- 0.61;0.05
0.07±0.04
0.07 ±o. 02
1.45:0.1J
- 0.75:0.oJ
-0.33±0.05
°
°
°
values were calculated using Eq. 10 and the polar tensor values for F CO, Cl CO and F CS listed in Table II.
2
2
-0.34 ±o. 04
2
J. Chern. Phys., Vol. 62, No.8, 15 April 1975
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A. B. M. S. Bassi and R. E. Bruns: Infrared intensities of X 2 CY molecules
similar inequality generally describes the carbon polar
tensor values. Rotation to the X I Z' system simplifies
analysis of the halogen polar tensors in that their values
follow the inequality
3239
or sulfur effective charges although they are not transferable helps to obscure such a preference.
The importance of effective charges should not be understated. As Table VI illustrates the effective charges
for ClaCS can be calculated directly from those of FaCO,
ClzCO, and FliCS. The agreement between the experimental and calculated ~ is not only better than our propagated error but well within the experimental error determined from the Cl2CS measurements alone. Hence,
by Eq. (8) the sums of the intensities for ClzCS are related to those of FaCO, ClaCO, and FaCS and the rotational terms n for the four molecules. As far as practical applications are concerned it is not necessary that
the effective charges (or the polar tensor elements) be
transferable. If empirical equations such as (10) and
(11) can be found for other groups of molecules the individual infrared intensities can be calculated, as has
been done for ClaCS, 3 without their direct measurement.
Furthermore the effective charges appear to relate with
chemically intuitive quantities such as the electronegativities. The effective charge for fluorine is about 0.1ge
larger than that of chlorine. Substituting fluorine for
chlorine lowers the effective charges of oxygen and sulfur by 0.13e. Also, the carbon effective charge in Table
IV increases as more electronegative atoms are bonded
to carbon.
(II
In the XZ system the diagonal X element is more negative than the Z element for F aCO whereas the reverse is
true for F zCS.
The effective charge values derived from the polar
tensor elements are potentially very useful for the interpretation of infrared intensities. The charge for
hydrogen was found to be relatively invariant for the
series of hydrocarbons (CH 4 , CaHe, CeHe, and CaH4)
studied by King et al. 6 Their values for ~H range from
0.16 to 0.18e. Consideration of the experimental errors
of these quantities indicates that ~H is transferable for
these molecules. The values of ~H (0.13e) extracted by
these workers for the methyl halides (CH 3 X; X=CI, Br,
and I) are somewhat lower than those for the hydrocarbons. Thus the hydrogen effective charge appears to be
invariant for the series of methyl halides. However ~H
does not appear generally transferable. King and coworkers 6 report ~H values of 0.33 and 0.37e for CaHa
and HCN and Person and Newton5 calculate 0.21e for ~H
of HaCO. Although the experimental uncertainties may
be large in some cases it seems unlikely, in the light
of our results, that the effective hydrogen charge is invariant for all these molecules.
The eff ective charges for the XaCY molecules appear
to contain values which are transferable and those which
are not. Differences in ~F or ~CI values in Table IV are
similar to the small deviations observed for ~H in the
methyl halide series or in the group of hydrocarbons
exclusive of acetylene. Simultaneous acceptance of
Eqs. (11) and (12) with Q!1 = Q!3 = Cl and l1'a = 11'4 = F implies that the effective charges of both fluorine and
chlorine are transferable. That both equations are
valid when 11'1 = l1'a= and 11'3 = 11'4 = S or when 11'1 = l1'a = I1's
= 0!4 = C is not clear. These equations are all valid within the propagated errors of Table VI although the effective charge values of all the atoms except the halogens
differ by amounts greater than the experimental error
tolerances for transferability. It is interesting to note
that the ~~ value calculated by Eq. (12) differs from
the experimental value by an amount greater than the
error of the experimental value (O.06e) whereas this
value from Eq. (11) falls well within this error limit.
The values of ~~, on the other hand, calculated from
either Eqs. (11) or (12) yield values within the 0.37e experimental error. A preference for either Eq. (11) or
(12) involving the effective charges of oxygen, sulfur,
or carbon awaits more exact treatments of the experimental errors. Certainly the proximity of the oxygen
°
ACKNOWLEDGMENT
The authors would like to thank Professor W. B. Person for his encouragement of this investigation. Reference 12 was pointed out to us by Dr. P. M. Kuznesof.
Financial support of the Funda~~o de Amparo l\. Pesquisa
do Estado de S~o Paulo (Grant No. 71/1266) and the
Financiadora de Estudose Projetos (FINEP) is acknowledged.
IR . E. Bruns and R. K. Nair, J. Chern. Phys. 58, 1849
(1973).
2R • E. Bruns, J. Chern. Phys. 58, 1855 (1973).
3R. E. Bruns, J. Mol. Struct. 26, 124 (1975).
4J • F. Biarge, J. Herranz and J. Morcillo, Anales Real Soc.
Espan. Fis. Quim. (Madrid), A57, 81 (1961).
5W. B. Person and J. H. Newton, J. Chern. Phys. 61, 1040
(1974).
6W. T. King, G. B. Mast, andP. P. Blanchette, J. Chern.
Phys. 56, 4440 (1972).
7M. J. Hopper, J. W. Russell, and J. Overend, J. Chem.
Phys. 48, 3765 (1968).
8M. J. Hopper, J. W. Russell, and J. Overend, Spectrochim.
Acta A, 28, 1215 (1972).
9n. C. McKean, private co.mmunication as described in R. E.
Bruns and W. B. Person, J. Chem. Phys. 58, 2585 (1973).
10J . Overend and J. R. Scherer, J. Chern. Phys. 32, 1296
(1960).
11n. C. McKean, R. E. Bruns, W. B. Person, and G. A. Segal, J. Chern. Phys. 55, 2890 (1971).
12A. J. Careless, H. W. Kroto, and B. M. Lansborg, Chem.
Phys. 1, 371 (1973).
J. Chern. Phys., Vol. 62, No.8, 15April1975
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