CCFDF Interpretation - American Chemical Society

Article pubs.acs.org/JPCA

How Accessible Is Atomic Charge Information from Infrared Intensities? A QTAIM/CCFDF Interpretation Arnaldo F. Silva, Wagner E. Richter, Helen G. C. Meneses, Sergio H. D. M. Faria, and Roy E. Bruns* Instituto de Química, Universidade Estadual de Campinas, 13083-970, Campinas, SP, Brazil ABSTRACT: Infrared fundamental intensities calculated by the quantum theory of atoms in molecules/charge−charge flux−dipole flux (QTAIM/CCFDF) method have been partitioned into charge, charge flux, and dipole flux contributions as well as their charge−charge flux, charge−dipole flux, and charge flux−dipole flux interaction contributions. The interaction contributions can be positive or negative and do not depend on molecular orientations in coordinate systems or normal coordinate phase definitions, as do CCFDF dipole moment derivative contributions. If interactions are positive, their corresponding dipole moment derivative contributions have the same polarity reinforcing the total intensity estimates whereas negative contributions indicate opposite polarities and lower CCFDF intensities. Intensity partitioning is carried out for the normal coordinates of acetylene, ethylene, ethane, all the chlorofluoromethanes, the X2CY (X = F, Cl; Y = O, S) molecules, the difluoro- and dichloroethylenes and BF3. QTAIM/CCFDF calculated intensities with optimized quantum levels agree within 11.3 km mol−1 of the experimental values. The CH stretching and in-plane bending vibrations are characterized by significant charge flux, dipole flux, and charge flux−dipole flux interaction contributions with the negative interaction tending to cancel the individual contributions resulting in vary small intensity values. CF stretching and bending vibrations have large charge, charge−charge flux, and charge−dipole flux contributions for which the two interaction contributions tend to cancel one another. The experimental CF stretching intensities can be estimated to within 31.7 km mol−1 or 16.3% by a sum of these three contributions. However, the charge contribution alone is not successful at quantitatively estimating these CF intensities. Although the CCl stretching vibrations have significant charge−charge flux and charge−dipole flux contributions, like those of the CF stretches, both of these interaction contributions have opposite signs for these two types of vibrations.

■

INTRODUCTION Fundamental infrared intensities have long been known to be sensitive probes of molecular electronic structures. Because the simplest model for describing these structures consists of assigning net charges to all the atoms in a molecule, infrared intensities have often been compared with and correlated with atomic charges. First attempts in this direction assumed a bond moment model1 for which each bond in the molecule was described by a dipole moment to explain bending or deformation intensities and effective charges to treat bond stretching intensities. Later these concepts were extended into electroptical and equilibrium charge−charge flux (ECCF) procedures2−5 and applied to hydrocarbons among other molecules. Using the ECCF models and also taking advantage of the fact that symmetry restricts out-of-plane vibrations to have zero charge fluxes, an infrared charge determined completely from experimental structure and spectral data, was proposed for linear and planar molecules6,7 and is currently being extended to some nonplanar molecules.8 The introduction of the polar tensor formulizm9,10 for analyzing experimental intensities focused on an atoms in molecule approach to interpret intensities. Electronic structure changes during vibrations were described using a charge−charge flux-overlap model6,7,11,12 for which dipole moment changes during molecular vibrations are caused by the displacements of equilibrium charges on the atoms, intramolecular charge transfer among atoms and a quantum mechanical interference term that © 2012 American Chemical Society

had no apparent classical analogue. Furthermore, the molecular intensity sum has long been known to be a sum of atomic contributions13 that are mass weighted invariants of the atomic polar tensors, called effective atomic charges.14 This charge and one of its fellow invariants, the mean dipole moment derivative, are often correlated with atomic charges calculated by quantum chemical procedures.15 In fact, the mean dipole moment derivative is popularly known as a GAPT (generalized atomic polar tensor) charge16 owing to its mathematical properties that are similar to those of an atomic charge. Bader has proposed a quantum theory of atoms in molecules (QTAIM) for which first principle physical criteria are used to define atomic boundaries.17 Among its many applications dipole moment derivatives have been described in terms of atomic contributions.18 Our group has used these ideas to partition the dipole moment derivatives of several families of molecules in terms of charge, charge flux, and dipole flux (CCFDF) contributions to the dipole moment derivatives.19,20 This QTAIM/CCFDF model has two contributions that are equal to the charge and charge flux contributions of the CCFO model. However, the overlap contribution of the CCFO model is replaced by an atomic dipole flux contribution that has a simple classical physics interpretation. This makes it possible to Received: May 8, 2012 Revised: June 21, 2012 Published: June 22, 2012 8238

dx.doi.org/10.1021/jp304474e | J. Phys. Chem. A 2012, 116, 8238−8249

The Journal of Physical Chemistry A

Article

determine how accessible atomic charge information within the QTAIM framework is from experimental infrared intensities. If the dynamic flux contributions to the intensities are small relative to the charge contribution, one can expect very high correlations between the infrared intensities and atomic charges. It is the aim of this study to investigate the importance of each contribution to the fundamental infrared intensities of molecules. Besides our choice of obtaining atomic charges and dipoles using QTAIM, this assessment will depend on the quantum levels used to calculate electronic densities and the molecules included in our investigation. As such, only molecules for which all the fundamental intensities have been measured experimentally were chosen so that quantum levels providing the most accurate intensity results could be determined. The molecules included the acetylene, ethylene, and ethane hydrocarbons, all the chlorofluoromethanes, the X2CY (X = F, Cl; Y = O, S) molecules, the difluoro- and dichloroethylenes, and BF3, an electron deficient molecule.

■

After substitution in eq 1 their contributions to the infrared intensity are 2 ⎡ ⎛ ∂p ⎞ 2 ⎛ ∂p ⎞ 2 ⎛ Nπ ⎞⎢⎛ ∂pσ ⎞ ⎜ ⎟ ⎜ ⎟ + ⎜⎜ σ ⎟⎟ + ⎜⎜ σ ⎟⎟ Aj = ⎝ 3c 2 ⎠⎢⎜⎝ ∂Q j ⎟⎠ ∂Q j ⎠ ⎝ ⎝ ∂Q j ⎠DF ⎣ C CF

⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎛ ∂p ⎞ + 2⎜⎜ σ ⎟⎟ ⎜⎜ σ ⎟⎟ + 2⎜⎜ σ ⎟⎟ ⎜⎜ σ ⎟⎟ ⎝ ∂Q j ⎠C⎝ ∂Q j ⎠DF ⎝ ∂Q j ⎠C⎝ ∂Q j ⎠CF ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎤ ⎥ + 2⎜⎜ σ ⎟⎟ ⎜⎜ σ ⎟⎟ ⎥ ∂ ∂ Q Q ⎝ j ⎠CF⎝ j ⎠DF⎦

The first three squared terms represent the charge, charge flux, and dipole flux contributions to the jth fundamental vibrational intensity and of course are always positive. The last three terms correspond to interactions between charge, charge flux, and dipole flux contributions and can be positive when both derivatives are of the same sign reinforcing the intensity or negative when they have opposite signs decreasing the intensity. Our previous QTAIM studies analyzed intensities in terms of charge, charge flux, and dipole flux contributions to the dipole moment derivatives. On the other hand, the partition of the intensity in eq 5 can be expressed as

QTAIM CONTRIBUTIONS TO INTENSITIES

The infrared intensity of the jth fundamental band is proportional to the square of the dipole moment derivative with respect to the normal coordinate, Qj

⎞2 ⎛ Nπ ⎜ ∂p ⃗ ⎟ Aj = 2 ⎜ 3c ⎝ ∂Q j ⎟⎠

Aj = A(C) + A(CF) + A(DF) + A(CxCF) + A(CxDF) j j j j j

(1)

+ A(CFxDF) j

For most molecules with high symmetry the vibrations are accompanied by dipole moment changes that are restricted to only one of the Cartesian coordinate axis, σ = x, y, or z. The molecular dipole moment can be determined from the QTAIM atomic charges, qi, and atomic dipoles, mi, by pσ =

∑ qiσi + ∑ mi,σ i

with the sum being taken over all the atoms in the molecule. Its dipole moment derivative with respect to the jth normal coordinate is ∂pσ ∂Q j

=

∑ qi i

∂σi + ∂Q j

∑ i

∂qi ∂Q j

σi +

∑ i

■

∂mi , σ ∂Q j

CALCULATIONS Electronic structure calculations were carried out on an AMD 64 Opteron workstation at the optimized theoretical geometries for each molecule using the GAUSSIAN03 program21 that provided the infrared intensities and the Hessian matrix. The Hessian matrix contains information on the atoms' critical points and was appropriately formatted by homemade software for input to the MORPHY98 program,22 which calculated the atomic charges and dipoles. For the difluoro- and dichloroethylenes nonautomatic integration was performed with the flowing integration parameters: tn1 = 90, tn2 = 30, tn3 = 50, tn4 = 70, tn5 = 50, tn6 = 80, tn7 = 10−8, tn8 = 3, u1 = 0.2, u2 = 0.5, u3 = 15, u4 = 0.4, w1 = 0.1, and w2 = 20. All other integration parameters were default. Automatic integration was applied for the remaining molecules, but the charges and dipoles of the central atoms were corrected as described previously.23 CCFDF contributions to the dipole moment derivatives in terms of atomic Cartesian coordinates and elements of the

(3)

The first term is a weighted sum of equilibrium atomic charges for which the weights correspond to the relative atomic displacements for the jth normal coordinate. The second is a weighted sum of charge fluxes for which the loadings are the equilibrium Cartesian coordinates of the atoms. The third term is a simple sum of the derivatives of the σth Cartesian component of the atomic dipoles with respect to the jth normal coordinate. As such, the QTAIM charge−charge flux−dipole flux (CCFDF) model partitions this Cartesian component of the total dipole moment derivative into charge (C), charge flux (CF) and dipole flux (DF) contributions, ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎛ ∂p ⎞ = ⎜⎜ σ ⎟⎟ + ⎜⎜ σ ⎟⎟ + ⎜⎜ σ ⎟⎟ ∂Q j ⎝ ∂Q j ⎠C ⎝ ∂Q j ⎠CF ⎝ ∂Q j ⎠DF

(6)

and is very convenient for two reasons. First, the sum of these contributions yields the fundamental intensity so one can directly examine the impact of each of the above contributions on this experimental observable. Second, the signs of the intensity contributions do not depend on Cartesian coordinate definitions, normal coordinate phases, or molecular orientations. Their signs only depend on the relative polarities of their dipole moment changes. If these polarities are the same, the interaction is positive and reinforces the sum of the charge, charge flux and dipole flux contributions. If the polarities are opposite, the contribution is negative and decreases the CCFDF intensity estimate.

(2)

i

(5)

∂pσ

(4) 8239



Article

best described by a combination of the QCISD method and the cc-pVTZ basis set. Taking all the molecules into account 75% had most accurate results with the QCISD electron correlation treatment but with a variety of basis sets. Also the MORPHY98 program resulted in smaller integration errors for electron densities from the QCISD wave functions than those obtained at the MP2 level. For these reasons only the QCISD method was applied to all the molecular calculations in this paper. In only two cases was the basis set that provided the most accurate result not used. For trans-C2H2F2 the best agreement with experiment was found with the 6-31G(2d,2p) basis, but a critical point miss location prohibited the evaluation of the QTAIM parameters. For this molecule the second most accurate basis set, 6-31G(3d,3p), was used. For the C2H2 molecule, a similar problem occurred (actually a non-nuclear attractor was found29) with the most accurate basis set, ccpVTZ, so it was substituted by the 6-311++G(3d,3p) basis.

molecular polar tensor, are related in the equations ∂pσ

∑ σi

= qA +

∂σA

i

∂qi ∂σA

+

∑ i

∂mi , σ ∂σA

(7)

and ∂pν

=

∂σA

∑ νi i

∂qi ∂σA

+

∑ i

∂mi , ν ∂σA

(8)

where σ = x, y, z, and the atomic charges and dipole components are given by qi and mi,σ, respectively. The charge flux contribution has the derivative of the charge of the ith atom of the molecule relative to the displacement of a given atom A with respect to its equilibrium position along each Cartesian axis σ, as can be seen in eqs 7 and 8. This derivative can be calculated numerically by the equation: ∂qi

=

∂σA

■

qi+ + qi− 2ΔσA

RESULTS Table 1 contains the experimental fundamental intensities, the theoretical intensities calculated directly from the molecular wave functions by the Gaussian program, and the intensities calculated using the QTAIM/CCFDF model. The agreement between the experimental intensities and those calculated from the molecular wave function is excellent with a root-meansquare (rms) error of 11.4 km mol−1. Furthermore, the intensities calculated using the CCFDF contributions in eqs 5 and 6 have a rms error of only 5.0 km mol−1 with those calculated directly from the wave functions. Table 2 contains the charge, charge flux, and dipole flux contributions as well as their interactions grouped according to the type of vibration. An exploratory analysis of this data using both univariate and multivariate techniques was carried out. Particularly informative were the results of the correlation and principal component analyses. A large percentage of the data variance can be described by a very strong negative correlation of the charge flux−dipole flux interaction with both the charge and dipole flux contributions. This correlation is shown in Figure 1 where the charge flux−dipole flux interaction is plotted against the sum of the charge and dipole fluxes. The regression line shown in this figure has a slope of −1.01, indicating that the sum of the positive charge flux and dipole flux contributions is almost completely canceled by the negative charge flux−dipole flux interaction. If the cancellation were perfect, the QTAIM/CCFDF estimate of the fundamental intensities would be given by the sum of the charge, charge− charge flux, and charge−dipole flux contributions. A graph of the sum of these quantities against the CCFDF intensities is given in Figure 2. A large portion of the charge, charge−charge flux, and charge−dipole flux sums fall close to or a little below the line indicating exact agreement with the QTAIM/CCFDF intensity estimates. Some of the sums are even negative. Six vibrations have charge, charge−charge flux, and charge−dipole flux sums that are much smaller than the estimated intensities. In fact, these points are the same as the six points that lie a little above the regression line for the charge flux and dipole flux sum vs the charge flux−dipole flux interaction contribution. These vibrations all belong to the X2CY molecules and BF3. The two lowest outliers in Figure 2 represent the out-of-plane bends of the isoelectronic F2CO and BF3 molecules. The other outlier points belong to the CS stretching vibrations in F2CS and Cl2CS and the CCl stretching vibrations in Cl2CO and Cl2CS. Note that most of the CF stretching points appear to fall closer

(9)

where q+i and q−i are, respectively, the charge of atom i after its displacement in the positive and negative directions of σ (x, y, z). This displacement of atom α in both directions of each axis in relation to the Cartesian equilibrium position is performed to simulate the vibrational motion of the atom in the molecule. Similarly, the derivative that represents the dipole flux contribution was numerically calculated by the equation:

∂mi , σ ∂σA

=

mi+, σ + mi−, σ 2ΔσA

(10)

m−i,σ

where and are, respectively, the σth dipole moment component of the ith atom after its displacement in the positive and negative directions of σ (x, y, z). Atomic charges and dipoles were obtained at the optimized equilibrium geometry and at distorted geometries with atoms displaced by 0.01 Å along the negative and positive directions of all three Cartesian coordinates axis. The PLACZEK24 program was used to prepare all necessary input to run the GAUSSIAN03 single point calculations, which are used to obtain the atomic charges and dipoles with MORPHY98. The program also computes the dipole moment derivatives, as well as their charge, charge flux, and dipole flux contributions. The molecular polar tensor and its CCFDF contributions in Cartesian coordinates are converted to normal coordinates by PLACZEK using the following equations: + mi,σ

(CF) PQ = PX L′ = (P(C) + P(DF) X + PX X )L ′

(11)

(CF) PQ = P(C) + P(DF) Q + PQ Q

(12)

(CF) (DF) where PQ, P(C) are the molecular polar tensor and Q , PQ , and PQ its charge, charge flux, and dipole flux contributions in normal coordinates, and L′ is the conversion matrix obtained from Gaussian03. The electron correlation treatment levels and basis sets were chosen on the basis of the most accurate agreement with the experimental intensities. For the difluoro- and dichloroethylenes, the X2CY and BF3 molecules, a systematic study was carried out with over 20 basis sets, including the k-nlmG Pople25 and Dunning26 basis sets, at both the MP2 and QCISD electron correlation treatment levels. Previous work by our group on the chlorofluoromethanes27 and C2H2, C2H4, and C2H628 showed that their fundamental infrared intensities are

8240


8241

B2

B1

cis-C2H2Cl2 A1

B2

B1

1,1-C2H2F2 A1

Bu

B2 trans-C2H2F2 Au

B1

cis-C2H2F2 A1

v CH v CC δ CH v CCl δ CCl v CH δ CH v CCl δ CCl

Q1 Q2 Q3 Q4 Q5 Q8 Q9 Q10 Q11 Q12

Q1 Q2 Q3 Q4 Q5 Q7 Q8 Q9 Q10 Q11 Q12

Q6 Q7 Q9 Q10 Q11 Q12

δ CH δ CF v CH δ CH v CF δ CF

v CH v CC δ CH v CF δ CF v CH v CF δ CH δ CF δ CH δ CF

Q1 Q2 Q3 Q4 Q5 Q8 Q9 Q10 Q11 Q12

v CH v CC δ CH v CF δ CF v CH δ CH v CF δ CH δ CF

Qj

3077 1587 1179 714 173 3087 1303 857 571 697

3070 1728 1360 926 550 3154 1302 955 438 803 611

873 325 3116 1274 1160 338

3135 1714 1263 1009 239 3122 1375 1130 768 755

νj (cm )

−1

26.50 0.10 18.40 0.10 10.70 19.70 57.20 5.50 37.60

42.20 216.00 0.00 64.80 5.10 8.60 190.10 23.50 0.60 60.30 0.30

56.70 12.70 9.50 14.70 217.70 1.50

3.70 42.00 29.40 50.70 1.80 3.70 20.00 84.90 25.40 34.30

exp

2.46 24.74 0.18 17.84 0.35 10.17 26.93 58.64 5.56 41.12

3.70 223.30 4.00 64.90 4.40 0.00 189.00 17.00 1.30 60.30 2.00

53.30 13.80 10.80 32.90 257.84 5.70

10.26 38.34 26.39 58.02 2.41 0.87 24.15 94.05 26.89 27.70

WF

Aj (km/mol)

3.05 30.78 0.43 17.97 0.06 7.32 7.13 64.54 5.84 38.75

3.48 214.79 2.52 62.83 3.97 0.02 177.50 15.48 1.50 59.79 4.31

50.99 23.72 3.22 10.41 214.31 5.03

10.80 37.55 25.57 58.51 2.35 0.91 18.30 99.83 30.70 28.93

CCFDF

6-31G(3d,3p)

6-31G(2d,2p)

6-31G(3d,3p)

6-31G(2d,2p)

basis set

E

CHCl3 A1

B2

B1

CH2Cl2 A1

E

CH3Cl A1

CF4 T2

E

CHF3 A1

B2

B1

CH2F2 A1

v CH v CCl δ ClCCl δ HCCl v CCl

δ HCH v CCl

v CH δ HCH v CCl δ ClCCl v CH

v CH δ HCH v CCl v CH δ HCH

v CF δ FCF

v CH v CF δ FCF δ HCF v CF δ FCF

v CH δ HCH v CF δ FCF v CH δ HCH δ HCH v CF

Q1 Q2 Q3 Q4a Q5a

Q1 Q2 Q3 Q4 Q6 Q7 Q8 Q9

Q1 Q2 Q3 Q4a Q5a Q6a

Q3b Q4b


Q1 Q2 Q3 Q4 Q6 Q7 Q8 Q9

Qj

3034 678 366 1219 773

3137 1430 714 282 3195 896 1268 757

2967 1355 732 3042 1455 1015

1298 632

3035 1141 700 1378 1157 508

2948 1508 1111 528 3014 1178 1435 1090

νj (cm )

−1

0.30 4.40 0.50 15.40 111.20

6.90 0.60 8.00 0.60 0.00 1.20 26.20 95.00

19.40 7.60 23.44 5.12 5.93 1.97

414.63 4.33

27.40 129.00 14.00 45.00 262.50 2.50

26.70 0.00 60.70 4.90 41.60 4.50 10.50 269.10

exp

0.00 5.30 0.40 23.50 132.05

8.20 0.00 12.40 0.50 0.10 0.90 42.60 117.60

23.30 13.70 24.30 6.00 4.90 2.05

406.73 5.77

35.13 103.80 14.67 65.34 264.86 3.00

43.78 1.44 101.17 5.30 40.90 20.95 23.32 227.87

WF

0.04 5.27 0.36 23.46 131.94

8.21 0.00 12.43 0.53 0.08 0.92 42.48 117.48

23.31 13.70 24.22 5.94 4.90 2.05

406.51 5.75

35.00 103.73 14.66 65.31 264.71 2.99

43.73 1.45 101.10 5.28 40.74 20.89 23.30 227.74

CCFDF

Aj (km/mol)

cc-pVTZ

cc-pVTZ

cc-pVTZ

cc-pVTZ

cc-pVTZ

cc-pVTZ

basis set

Table 1. Experimental and Theoretical Fundamental Infrared Intensity Values (km mol−1) Calculated Directly from the QCISD Molecular Wavefunctions and from Their QTAIM/CCFDF Parameters

The Journal of Physical Chemistry A Article


8242

CH4 T2

B2 BF3 A2″ E′

B1

B2 F2CS A1

B1

B2 Cl2CO A1

B1

B2 Cl2CS A1

B1

F2CO A1

Bu

trans-C2H2Cl2 Au

1368 787 526 1189 417 622

Q1 Q2 Q3 Q4 Q5 Q6

Q2 Q 3a Q 4a

v CS v CF δ FCF v CF δ SCF δ SCF

δ BF ν BF δ BF

Q 3b Q4b

1827 567 285 850 440 580

Q1 Q2 Q3 Q4 Q5 Q6

v CO v CCl δ ClCCl v CCl δ OCCl δ OCCl

v CF δ HCH

1137 505 220 816 284 473

Q1 Q2 Q3 Q4 Q5 Q6

v CS v CCl δ ClCCl v CCl δ SCCl δ SCCl

3019 1311

720 1505 482

1928 965 584 1249 626 774

895 226 3090 1200 817 245

νj (cm−1)

Q1 Q2 Q3 Q4 Q5 Q6

Q6 Q7 Q9 Q10 Q11 Q12

Qj

v CO v CF δ FCF v CF δ OCF δ OCF

δ CH δ CCl v CH δ CH v CCl δ CCl

Table 1. continued

22.50 11.17

74.40 368.50 10.65

390.40 8.90 6.70 201.50 0.30 1.30

245.30 14.50 0.10 376.50 0.20 4.90

210.80 13.80 0.00 162.90 0.30 2.40

381.70 56.40 5.20 370.80 7.00 30.60

48.50 0.50 11.30 16.90 94.70 0.10

exp

23.20 10.17

94.30 374.80 11.35

393.70 3.70 8.00 207.00 2.60 8.10

261.25 18.20 0.16 406.12 1.17 12.18

199.90 16.40 0.50 191.70 0.10 1.20

391.20 48.50 4.90 362.00 8.00 35.60

42.88 0.45 12.17 21.60 101.50 3.80

WF

Aj (km/mol)

23.08 10.03

95.40 376.60 11.60

393.35 3.72 8.01 206.81 2.62 8.14

261.25 18.20 0.16 406.12 1.17 12.18

199.80 16.30 0.60 192.20 0.10 1.20

391.20 48.60 4.80 362.10 8.10 35.80

32.05 0.23 12.19 18.51 95.29 7.94

CCFDF

cc-pVTZ

6-31G(2d,2p)

4-31G

4-31G

D95

6-31G(2d,2p)

6-31G(3d,3p)

basis set

Eu

b3u C2H6 A2u

b2u

C2H2 ∑u Πu C2H4 b1u

E

CCl3F A1

B2

B1

CCl2F2 A1

E

CClF3 A1

CCl4 T2

CH2 CH2 CH2 CH2 CH2

str def str rock def

Q5 Q6 Q7a Q8a Q9a

2915 1379 2996 822 1472

2990 1444 3106 810 949

Q7 Q9 Q10 Q11 Q12 CH2 CH2 CH2 CH2 CH2

str scis str rock wag

3282 730

1085 539 347 846 384 243

1095 475 665 261 1152 432 442 915

1109 785 476 1212 562 348

793 310

Q3 Q5a


Q1 Q2 Q3 Q4 Q6 Q7 Q8 Q9


Q3b Q4b

262

Q6a

CH str CH bend

v CF v CCl δ ClCCl v CCl δ FCCl δ ClCCl

v CF v CCl δ ClCCl δ ClCCl v CF δ FCF δ FCF v CCl

v CF v CCl δ FCF v CF δ FCF δ ClCF

v CCl δ ClCCl

νj (cm−1)

Qj

47.80 4.00 61.60 3.05 6.70

13.90 10.10 25.50 0.30 82.10

70.20 88.50

158.00 1.10 0.30 197.30 0.05 0.00

284.30 0.10 12.30 2.60 183.30 0.10 0.20 324.80

493.90 31.50 0.00 296.75 1.65 0.00

107.33 0.07

0.05

exp

53.34 0.74 62.42 2.65 7.37

13.40 8.58 20.57 0.04 91.12

84.66 90.78

179.30 1.50 0.00 234.35 0.02 0.00

293.30 0.00 11.50 0.00 211.60 0.50 0.10 375.20

473.40 34.30 0.00 293.80 2.50 0.05

137.50 0.00

0.20

WF

55.95 1.45 61.51 5.30 3.78

11.97 8.76 21.60 0.20 89.22

89.66 78.81

179.25 1.53 0.03 234.16 0.07 0.01

293.06 0.02 11.48 0.03 211.49 0.47 0.11 374.95

470.90 34.28 0.01 293.64 2.53 0.02

137.38 0.01

0.12

CCFDF

Aj (km/mol)

cc-pVTZ

cc-pVTZ

6-311++G(3d,3p)

cc-pVTZ

cc-pVTZ

cc-pVTZ

cc-pVTZ

basis set




Article

CCFDF WF cc-pVTZ 29.32 3.11 99.23 33.07 2.69 1.54

Doubly degenerate. bTriply degenerate. a

E

WF

29.30 3.10 99.30 35.30 2.70 1.60 24.70 0.90 95.00 30.50 4.35 1.30 3031 1490 1059 3132 1498 1206 Q1 Q2 Q3 Q4a Q 5a Q 6a v CH δ HCH v CF v CH δ HCH CH3F A1

Table 1. continued

Qj

νj (cm−1)

exp

Aj (km/mol)

CCFDF

basis set

Qj

νj (cm−1)

exp

Aj (km/mol)

basis set

to the line representing exact agreement than do those of the other types of vibrations. A high positive correlation (0.82) is found between the charge flux and dipole flux contributions to the infrared intensities. This is consistent with the large negative correlations found between the charge flux and dipole flux contributions to the dipole moment derivatives for the fluorochloromethanes,27,30 X2CY,31 and difluoro- and dichloroethylenes.32 Also a large negative correlation coefficient of −0.74 is found between the charge−charge flux and charge− dipole flux interaction intensity contributions consistent with the negative charge flux−dipole flux interaction intensity contributions. The principal component analysis indicated that the different types of vibrations have different kinds of electronic structure rearrangements during molecular vibrations. These differences can be seen in Figure 3 that presents a bar graph with the six intensity contributions and their sums for the CH, CF, and CCl stretching and bending motions. The vertical bars in the graph correspond to the 10% and 90% quantiles, i.e., 80% of the results fall between the lower and upper limits of the bars. CH Stretching Vibrations. The CH stretches are characterized by large positive charge flux and dipole flux contributions and a very negative charge flux−dipole flux interaction. As can be seen in the CH stretching group on the left in Figure 3, these significant contributions tend to cancel each other and their bar columns are much larger than the black one for the calculated QTAIM/CCFDF total intensity. The averages of the charge flux and dipole flux contributions are 313.7 and 400.5 km mol−1, respectively, compared to the charge flux−dipole flux interaction average of −697.9 km mol−1. The sum of these averages is 16.3 km mol−1. This is comparable to the average of the experimental CH stretching intensities in Table 1, 21.4 km mol−1. So the degree of cancellation of these three contributions will have a considerable impact on the final calculated CCFDF intensity values. Of all the contributions to the CH stretching intensities the charge contribution is by far the smallest. With the exception of acetylene, all the charge contributions to the QTAIM/CCFDF CH stretching intensities fall between 0.0 and 7.7 km mol−1. The large hydrogen effective charge calculated by the G sum rule from the fundamental intensities of acetylene relative to those determined for the other hydrocarbons has often been documented in the literature12,33 and interpreted as being due to the large static charge on hydrogen. Although the largest charge contribution to a CH stretching intensity in Table 2 occurs for acetylene, 25.2 km mol−1, it is only about a third of the experimental intensity of 70.2 km mol−1. The charge flux, dipole flux, and their interaction furnish the largest contributions to this CH stretching intensity (out of scale in Figure 3). The sum of the charge flux and dipole flux contributions, 3214 km mol−1 is almost completely canceled by their interaction, −3194 km mol−1. Furthermore, the charge−charge flux and charge−dipole flux interactions are of opposite sign and similar magnitudes, 424 and −379 km mol−1. So the sum of all these contributions to the CH stretching intensity of acetylene, 65 km mol−1, is small but still much larger than the charge contribution. Assuming a positively charged hydrogen atom in acetylene as calculated by QTAIM from the QCISD/6-311++G(3d,3p) wave function, the positive charge−charge flux interaction and the negative charge−dipole flux interaction for the CH stretch in Table 2 correspond to the dipole moment directional 8243


C2

0.65 0.31 0.13 0.39 0.15 1.04 0.39 2.39 0.02 0.30 1.69 2.16

21.42 6.49 11.85 0.36 10.94 11.41 12.87 2.46 1.25 0.79 0.47 2.49

2.44 8.05 0.04

62.90 9.58 106.37 33.97 123.54 21.40 515.30 2.94 44.28 58.85

13.32

65.69

Qj

Q1 Q8 Q9 Q1 Q7 Q1 Q8 Q9 Q3 Q1 Q4 Q1

Q3 Q9 Q11 Q6 Q10 Q3 Q9 Q11 Q3 Q9 Q12 Q6

Q 10 Q 12 Q4

Q4 Q 10 Q 11 Q4 Q8 Q2 Q4 Q2 Q4 Q3

Q5

Q 12

molecule

cis-C2H2F2 cis-C2H2F2 trans-C2H2F2 1,1-C2H2F2 1,1-C2H2F2 cis-C2H2Cl2 cis-C2H2Cl2 trans-C2H2Cl2 CH4 CH3F CH3F CH2F2

cis-C2H2F2 cis-C2H2F2 cis-C2H2F2 trans-C2H2F2 trans-C2H2F2 1,1-C2H2F2 1,1-C2H2F2 1,1-C2H2F2 cis-C2H2Cl2 cis-C2H2Cl2 cis-C2H2Cl2 trans-C2H2Cl2

trans-C2H2Cl2 trans-C2H2Cl2 CH4

8244

cis-C2H2F2 cis-C2H2F2 trans-C2H2F2 1,1-C2H2F2 1,1-C2H2F2 F2CO F2CO F2CS F2CS CH3F

cis-C2H2F2

cis-C2H2F2 4.71

0.02

39.26 22.03 79.62 27.00 292.92 11.20 202.30 164.97 73.28 34.63

243.00 1.00 27.15

32.64 23.15 0.00 0.00 97.11 133.80 9.36 0.00 15.70 294.86 0.00 0.00

378.52 36.55 631.20 138.59 424.74 387.24 166.91 530.82 300.77 137.83 187.13 164.89

CF2

23.97

5.17

35.82 134.32 175.53 75.74 474.12 32.50 111.40 170.45 265.26 66.82

165.07 1.04 73.38

37.73 9.49 80.70 59.96 98.71 199.55 11.73 37.63 4.79 185.30 30.68 16.66

481.08 20.61 705.11 89.28 416.27 503.84 117.40 624.23 484.09 275.85 328.74 323.45

DF2

−400.56 −2.04 −89.27 −75.00 −108.79 −236.44 −90.43 −745.33 −38.20 −300.20 −335.37 −278.84 −96.20 −0.71 −21.25

56.85 −15.70 −61.86 −9.33 −65.72 −95.44 24.57 19.24 4.88 24.18 7.60 12.89 −40.13 −5.79 −3.25 94.93 71.75 273.29 101.45 484.04 52.60 479.10 44.74 216.76 125.42 −16.59 −79.37

−52.88 24.51 0.00 0.00 65.18 78.15 −21.95 0.00 −8.84 −30.50 0.00 0.00 48.69 5.68 1.98 −99.39 −29.06 −184.06 −60.57 −380.47 −30.90 −645.80 −44.01 −113.93 −90.29

35.18

1.14

−70.18 −29.64 0.00 0.00 −195.81 −326.80 −20.95 0.00 −17.34 −467.50 0.00 0.00

35.43 5.03 19.37 −11.75 −15.89 −45.80 −13.48 77.25 6.36 18.14 47.09 52.84

−31.43 −6.69 −18.33 14.64 16.05 40.15 16.07 −71.24 −5.01 −12.82 −35.53 −37.73

−853.45 −54.89 −1334.26 −222.46 −840.96 −883.42 −279.96 −1151.27 −763.15 −389.98 −496.05 −461.88

2CxDF

2CxCF

2CFxCF

28.93

CHF3

Q3

Q4

CF Bend 2.35 CH2F2

Q6 Q8

Q3 Q9 Q2 Q5 Q3 Q1 Q4 Q1 Q6 Q1

C2H6 C2H6

CF Stretch 58.51 CH2F2 99.83 CH2F2 214.31 CHF3 62.83 CHF3 177.50 CF4 48.60 CClF3 362.10 CClF3 3.72 CCl2F2 206.81 CCl2F2 99.23 CCl3F

18.51 7.94 10.03

Q2 Q5 Q2 Q8 Q2 Q5 Q2 Q8 Q5 Q9 Q11 Q9

CH Bend 25.57 CH3F 18.30 CH3F 30.70 CH2F2 50.99 CH2F2 10.41 CH3Cl 2.52 CH3Cl 15.48 CH2Cl2 59.79 CH2Cl2 0.43 C2H2 7.13 C2H4 38.75 C2H4 32.05 C2H6

Qj Q6 Q1 Q1 Q4 Q1 Q6 Q1 Q3 Q10 Q7 Q7 Q5

CH str

molecule CH2F2 CHF3 CH3Cl CH3Cl CH2Cl2 CH2Cl2 CHCl3 C2H2 C2H4 C2H4 C2H6 C2H6

10.80 0.91 3.22 3.48 0.02 3.05 7.32 12.19 23.08 29.32 33.07 43.73

Aj

97.67

38.70

150.04 162.66 311.33 308.80 660.81 390.07 422.03 226.27 242.93 111.79

1.94 0.58

0.95 0.37 3.91 7.80 2.95 0.84 5.80 8.48 25.19 1.43 0.50 0.62

5.39 6.59 0.24 1.14 2.22 5.03 7.71 25.19 1.50 0.57 1.05 0.31

C2

5.40

0.78

5.21 78.04 3.26 73.41 88.53 116.02 106.11 6.02 124.76 28.62

225.37 3.95

1.82 21.67 12.91 1.63 118.84 19.01 66.29 147.32 0.00 573.04 14.09 21.30

117.08 132.78 33.52 247.43 68.51 191.36 98.14 1784.99 774.04 8.52 518.39 69.37

CF2

70.17

23.08

0.01 124.79 85.82 52.78 14.93 77.82 47.52 0.14 102.54 66.67

310.15 25.51

0.31 47.60 19.09 10.96 30.08 32.01 32.48 7.32 14.89 660.54 6.77 53.99

221.47 221.16 123.46 369.90 159.51 267.01 155.44 1428.77 1135.40 50.89 875.36 232.65

DF2

45.94

11.01

−55.90 −225.33 63.68 −301.12 −483.73 425.47 −423.23 73.83 −348.18 −113.13

41.87 3.03

2.63 5.64 14.20 −7.13 −37.42 −8.01 −39.23 −70.71 0.00 −57.34 −5.31 7.28

−50.25 −59.17 5.69 33.53 24.65 62.02 55.02 424.11 68.26 4.41 −46.65 −9.26

2CxCF

−165.58

−59.78

2.14 284.95 −326.92 255.32 198.69 −348.45 283.22 −11.35 315.65 172.66

−49.12 −7.71

−1.09 −8.36 −17.27 18.49 18.83 10.39 27.46 15.76 38.73 61.56 3.68 −11.58

69.11 76.36 −10.93 −41.00 −37.61 −73.26 −69.24 −379.44 −82.67 −10.78 60.63 16.96

2CxDF

−38.94

−8.51

−0.40 −197.37 −33.44 −124.48 −72.72 −190.03 −142.01 −1.85 −226.21 −87.36

−528.77 −20.07

−1.51 −64.23 −31.39 −8.45 −119.58 −49.34 −92.80 −65.69 0.00 −1230.47 −19.54 −67.82

−322.06 −342.72 −128.67 −605.06 −209.07 −452.08 −247.03 −3193.96 −1874.94 −41.64 −1347.27 −254.07

2CFxCF

Aj

14.66

5.28

101.10 227.74 103.73 264.71 406.51 470.90 293.64 293.06 211.49 179.25

1.45 5.30

3.11 2.69 1.45 23.30 13.70 4.90 0.00 42.48 78.81 8.76 0.20 3.78

40.74 35.00 23.31 5.94 8.21 0.08 0.04 89.66 21.60 11.97 61.51 55.95

Table 2. Charge, Charge Flux, Dipole Flux, and Their Interaction Contributions to the Fundamental Infrared Intensities Calculated by the QTAIM/CCFDF Parameters



8245

0.02

562.40

104.67

8.00

32.42

Q2

Q2

Q1

Q1

Q1

Q1

Q5

Q6

v CC(cis‑C2H2Cl2)

v CO(F2CO)

v CO(Cl2CO)

v CS(Cl2CS)

v CS(F2CS)

δ OC−F(F2CO)

δ OC−F(F2CO)

608.70

95.50

3.25

66.68

Q2

(Cis‑C2H2F2)

0.63 5.01 7.61 0.30 0.05 2.36

V Cc(1.1‑C2H2F2)

V Cc

Q5 Q 11 Q7 Q3 Q3 Q4

Q4 Q 10 Q2 Q4 Q2 Q4 Q3 Q3 Q9

cis-C2H2Cl2 cis-C2H2Cl2 Cl2CS Cl2CS Cl2CO Cl2CO CH3Cl CH2Cl2 CH2Cl2

cis-C2H2Cl2 cis-C2H2Cl2 trans-C2H2Cl2 Cl2CS Cl2CO CH2Cl2

103.95 110.36 35.62 2.47 123.04 29.20 20.12

Q7 Q 12 Q5 Q 10 Q 12 Q3 Q3

trans-C2H2F2 trans-C2H2F2 1,1-C2H2F2 1,1-C2H2F2 1,1-C2H2F2 F2CO F2CS

5.45 0.81 0.80 4.30 0.22 85.34 5.97 11.96 12.24

C2

Qj

molecule

Table 2. continued

0.00

10.90

619.02

1610.00

215.72

674.20

20.66

333.65

78.90

0.22 7.22 0.00 0.00 0.11 0.02

9.24 30.06 0.00 229.80 12.43 362.46 11.13 5.68 68.65

0.58 0.00 1.76 0.59 0.00 3.60 11.97

CF2

349.40

104.80

115.37

536.10

425.18

485.20

1.31

138.68

174.46

1.05 6.30 5.22 0.10 0.02 0.91

1.29 2.72 9.10 0.60 0.08 66.00 0.73 5.37 0.89

37.09 68.26 18.56 5.90 84.51 26.00 3.25

DF2

−6.90 18.10 0.70 23.50 1.94 −309.33 −5.72 −11.04 −15.66

−5.30 2.97 5.50 −3.30 0.26 −150.10 −4.19 −16.03 −6.61

−534.47

0.00

64.40 −922.30

−200.00

−1858.00

131.30 −122.31

−227.60 283.32

421.91

−300.53

10.42

0.00

−67.50

−605.71

−1143.80

−0.33 1044.70

−1.30

−430.22

−1231.50

−234.65

47.60 −192.33

−0.95 −13.50 0.00 0.00 −0.10 −0.28

−9.31 0.00 11.43 −3.73 0.00 −19.50 −12.48

−124.19 −173.59 −51.42 7.64 −203.94 −55.10 16.18

−1.62 −11.24 −12.61 0.30 −0.07 −2.93

2CFxCF

2CxDF

298.32

−32.01

0.74 12.03 0.00 −0.10 0.15 0.45

14.19 9.88 0.20 −62.70 3.28 351.75 16.30 16.49 57.97

15.59 0.00 −15.84 −2.41 0.00 20.60 −31.03

2CxCF CH str

molecule

35.80

8.10

393.35

199.80

261.25

391.20

30.78

214.79

δ B−F(BF3)

v B−F(BF3)

δ B−F(BF3)

δ SC−F(F2CS)

δ SC−F(F2CS)

δ OC−Cl(Cl2CO)

δ OC−Cl(Cl2CO)

δ SC−Cl(Cl2CS)

CF Bend 23.72 CHF3 5.03 CF4 3.97 CClF3 1.50 CClF3 4.31 CCl2F2 4.80 CCl2F2 8.01 CCl Stretch 17.97 CHCl3 64.54 CHCl3 16.30 CCl4 192.20 CClF3 18.20 CCl2F2 406.12 CCl2F2 24.22 CCl3F 12.43 CCl3F 117.48 trans-C2H2Cl2 CCl Bend 0.06 CHCl3 5.84 CCl4 0.23 CCl2F2 0.60 CCl2F2 0.16 CCl3F 0.53 CCl3F Other 37.55 δ SC−Cl(Cl2CS)

Aj

Q4

Q3

Q2

Q6

Q5

Q6

Q5

Q6

Q5

Q3 Q4 Q3 Q4 Q3 Q6

Q2 Q5 Q3 Q2 Q2 Q7 Q2 Q4 Q 11

Q6 Q4 Q3 Q5 Q8 Q9

Qj

65.20

595.60

660.80

39.45

0.06

94.32

16.42

6.00

3.30

3.60 1.09 31.27 0.00 1.50 0.22

16.58 17.20 18.07 66.79 9.02 242.82 1.41 98.02 7.34

35.44 68.71 14.14 43.28 20.92 18.38

C2

0.10

5.30

0.00

0.00

2.53

0.00

6.47

0.00

0.10

0.15 0.05 8.56 0.17 0.61 0.09

4.27 96.26 141.62 21.49 0.11 252.62 2.02 188.19 67.08

1.12 6.13 0.69 3.61 0.78 1.95

CF2

25.30

7.30

254.00

11.75

0.05

38.71

6.71

1.80

1.30

2.83 1.68 26.31 0.31 4.76 0.00

14.75 6.11 19.63 48.35 7.94 146.73 1.87 69.16 1.29

27.86 69.97 8.74 47.48 11.31 24.94

DF2

5.70

−112.50

0.00

0.00

0.80

0.00

−20.62

0.00

−1.20

1.47 0.46 32.73 −0.02 1.91 −0.28

16.82 81.37 101.17 75.77 −2.03 495.34 3.38 271.63 44.38

12.59 41.04 −6.25 25.00 −8.10 11.98

2CxCF

−81.20

−131.50

−819.40

−43.06

−0.11

−120.85

−20.99

−6.60

−4.10

−6.39 −2.71 −57.37 0.03 −5.34 −0.05

−31.28 −20.50 −37.66 −113.65 −16.92 −377.51 −3.26 −164.67 −6.16

−62.85 −138.68 −22.23 −90.66 −30.76 −42.82

2CxDF

−3.50

12.40

0.00

0.00

−0.71

0.00

13.18

0.00

0.70

−1.30 −0.58 −30.02 −0.46 −3.41 0.03

−15.87 −48.50 −105.45 −64.47 1.90 −385.05 −3.89 −228.17 −18.63

−11.17 −41.42 4.92 −26.18 5.96 −13.96

2CFxCF

11.60

376.60

95.40

8.14

2.62

12.18

1.17

1.20

0.10

0.36 −0.01 11.48 0.03 0.03 0.01

5.27 131.94 137.38 34.28 0.02 374.95 1.53 234.16 95.29

2.99 5.75 0.01 2.53 0.11 0.47

Aj




Article

charge flux and dipole flux contributions are 75.2 and 80.3 km mol−1 compared with an average of −140.0 km mol−1 for the charge flux−dipole flux interaction. The sum of these three contributions, 15.5 km mol−1, is more than twice the average of the experimental CH bending vibrations in Table 2, 6.0 km mol−1. The quantile bar for the charge contributions is the smallest one as occurs for the CH stretches. The CH bending charge contributions range from 0.04 to 25.2 km mol−1, the largest value also occurring for acetylene. As for the CH stretching vibrations, the charge−charge flux and charge−dipole flux contributions to the CH bends tend to cancel one another having a correlation coefficient of −0.86 although they have absolute values considerably smaller than those for the charge flux, dipole flux and the charge flux−dipole flux interaction. The charge−charge flux and charge−dipole flux interaction contributions have small average values, −1.4 and −2.1 km mol−1. The charge contribution for the CH bending intensity in acetylene, 25 km mol−1, is very similar to the one for the CH stretch. However, owing to symmetry the charge flux contribution is zero. Consequently, the charge−charge flux and charge flux−dipole flux contributions are also zero. The dipole flux and charge−dipole flux interaction contributions, 15 and 39 km mol−1, respectively, sum to 54 km mol−1, a little more than twice the charge contribution to the bending intensity. Figure 4b shows parallel dipole moment directional changes for the charge and dipole flux contributions, consistent with this positive charge−dipole flux interaction value. CF Stretching Vibrations. In contrast to the CH stretching and bending vibrations, the charge, charge−charge flux, and charge−dipole flux interaction contributions are dominant in determining the QTAIM/CCFDF intensities. The positive charge flux and dipole flux contributions tend to cancel the negative charge flux−dipole flux interactions as observed for the CH vibrations. The sum of the charge flux and dipole flux terms has a −0.99 correlation coefficient with the charge flux− dipole flux interaction contribution. The sum of the averages of the charge flux and dipole flux contributions, 184.6 km mol−1, is about the same size but of opposite sign to the average of the charge flux−dipole flux interaction, −169.0 km mol−1. Their sum is only 15.6 km mol−1, much smaller than the average of the charge contributions of 198.3 km mol−1.As a consequence, the QTAIM/CCFDF estimates of the CF stretching intensities can be approximated by the sum of the charge, charge−charge flux and charge−dipole flux contributions. Figure 5 contains a bar plot of these sums and the experimental CF stretching intensities. As can be seen, there the agreement is very good for all the CF stretches. The rms error is 31.7 km mol−1 compared with an average CF stretching intensity of 194.7 km mol−1, giving a relative error of 16.3%. This small percentage error occurs because the sum of the charge, charge−charge flux, and charge−dipole flux is much larger than the sum of the charge flux, dipole flux, and charge flux−dipole flux interaction contributions. Included in Figure 5 are the charge contributions to the CF stretching intensities. As can be seen, there the charge contribution alone is incapable of correctly accounting for the experimental CF stretching intensities even though it furnishes one of its most dominant contributions. CF Bending Vibrations. Figure 3 shows that all six contributions to the QTAIM/CCFDF intensities for the CF bending vibrations are much smaller than those for the CF stretches. In contrast to the CH stretches and bends and the CF stretches no strong negative correlation is found between the sum of the charge flux and dipole flux terms and the charge

Figure 1. Graph of the sum of the charge flux and dipole flux contributions (CF+DF) vs the charge flux−dipole flux interaction (2 CFxDF).

Figure 2. Graph of the sum of the charge, charge−charge flux interaction, and charge−dipole flux interaction contributions (C+2CxCF+2CxDF) vs the QTAIM/CCFDF intensity values.

changes shown in the schematic diagram in Figure 4a. The opposite directions for the charge flux and dipole flux contributions to the dipole moment changes are consistent with the negative charge flux−dipole flux interaction contribution to the intensity. It should be noted here that the charge−charge flux and charge−dipole flux interaction contributions in Table 2 tend to cancel one another, having a correlation coefficient of −0.99 for all the CH stretches. The charge−charge flux contributions have an average value of 15.8 km mol−1 compared to the charge−dipole flux average of −12.8 km mol−1. Of course, if this cancellation were perfect as well as the one occurring for the charge flux−dipole flux sum and their interaction, the intensity value would be completely determined by the equilibrium atomic charges. CH Bending Vibrations. The charge flux, dipole flux, and charge flux−dipole flux interactions also provide the most important contributions to the CH bending intensities although they are not as large as the ones found for the CH stretches as can be observed in Figure 3. The sum of the charge flux and dipole flux terms has a −1.00 correlation coefficient with the charge flux−dipole flux interaction indicating almost perfect cancellation of these contributions. The average values of the 8246



Article

Figure 3. Charge, charge flux, dipole flux, and their cross interactions as contributions for the fundamental IR band intensities for each kind of normal mode. Their sum (total) is also represented. The vertical bars correspond to the 10% and 90% quantiles; i.e., 80% of the results fall between the lower and upper limits of the bars.

Figure 5. Bar graph of the sum of the charge, charge−charge flux interaction, and charge−dipole flux interaction contributions and the experimental intensities of the CF stretching vibrations. The numbers on the abscissa indicate the order of the CF dtretching vibrations in Table 2.

The charge and dipole flux provide the largest positive contributions that partially compensate the negative charge−dipole flux contribution. The sums of the charge, dipole flux, and charge− dipole flux interaction range from 0.0 to 39.6 km mol−1. These values are smaller than the charge contributions that range from 2.5 to 123.0 km mol−1. Figure 4c shows the directional changes in the dipole moment contributions for the asymmetric bending of F2CO assuming a negative atomic charge for fluorine as calculated by

Figure 4. Schematic diagrams of charge, charge flux, and dipole flux polarities for the (a) CH stretch in C2H2, (b) CH bend in C2H2, (c) CF asymmetric bend in F2CO, (d) CF stretch in CH3F, and (e) CCl stretch in CH3Cl.

flux−dipole flux interaction for the CF bends. The most important contribution is the mostly negative charge−dipole flux interaction with values ranging from −203.9 to +16.2 km mol−1. 8247



Article

QTAIM/CCFDF intensity values for the bond stretches are much larger for the polar double bonds than for the nonpolar ones in agreement with the experimental intensity values. The Q6 normal coordinates in the X2CY molecules and the Q12 one in 1,1-C2H2F2 are out-of-plane vibrations so the charge flux contribution and its interactions are zero by symmetry. As a result, only the charge, dipole flux, and the charge−dipole flux contributions are nonzero. For all these vibrations the charge− dipole flux contributions are negative and efficiently cancel the sums of the charge and dipole flux contributions. In F2CO these contributions are very large. The sum of the charge and dipole flux contributions, 958.1 km mol−1, is almost completely canceled by the charge−dipole flux interaction of −922.3 km mol−1, resulting in a small calculated CCFDF intensity of 35.8 km mol−1. Cancellation of the charge−dipole flux interaction and the sum of the charge and dipole flux contributions also occur for Cl2CO, F2CS, Cl2CS, and 1,1 C2H2F2 although the absolute values of the individual contributions are smaller than those in F2CO. This accounts for the very small experimental intensity values of these out-of plane bends even though they occur in quite polar molecules. Other Vibrations. The BF3 stretching vibration, Q3, has a very large charge contribution (595.6 km mol−1) as do the CO stretching (Q2, 562.4 km mol−1) and CF asymmetric stretching vibrations (Q5, 515.3 km mol−1) in the isoelectronic F2CO molecule. Furthermore, the charge flux and dipole flux contributions for these F2CO vibrations are very large, ranging from 111.4 to 674.2 km mol−1. However, the very small charge flux and dipole flux contributions calculated for the BF3 stretching motion, 5.3 and 7.3 km mol−1, may be indicative of highly ionic bonds for which these contributions might be expected to be small. Indeed, very small charge flux values, 1.1 and 0.6 km mol−1, and dipole flux values, 5.6 and 13.6 km mol−1, have been calculated by our group for NaCl and LiF, respectively. The BF3 out-of-plane bend (Q2) has a zero charge flux and large positive values for the charge and dipole flux contributions and a larger negative contribution from the charge−dipole flux interaction. As also occurs for the F2CO out-of-plane bend, these values partially cancel and result in a relatively small QTAIM/CCFDF intensity for this vibration, in agreement with the experimental observation, even though the bonds in BF3 are very polar.

QTAIM at the 6-31G(2d,2p) level. Because the charge−charge flux intensity interaction is positive (Table 2), contributions to the dipole moment changes for charge and charge flux are in the same direction. The charge−dipole flux and the charge flux−dipole flux intensity interactions are negative, indicating an opposite polarity change for the dipole flux. CCl Stretching Vibrations. The charge contributions to the CCl stretching intensities are intermediate between the ones for the CH and CF stretches, as would be expected from electronegativity considerations. The charge flux, dipole flux, and their interaction, like those for the CF stretches, although appreciable have smaller contributions than those for the CH stretches, as can be seen by the 10%−90% quantile bars in Figure 3. On the other hand, the charge−charge flux and the charge−dipole flux interactions for the CCl stretches, like those of the CF stretches, are of larger magnitudes than those for the CH stretches. Furthermore, both these significant charge−charge flux and charge−dipole flux contributions have opposite signs for the CF and CCl stretching vibrations. The charge−charge flux interactions have quantile bars in Figure 3 that represent mostly positive values for the CCl stretches but negative ones for the CF stretches. And for both stretches the quantile bars for the charge−dipole flux interactions have predominantly opposite signs to those of the charge−charge flux interactions. Whereas the charge−charge flux interactions for the CCl stretches vary from −62.7 to 495.3 km mol−1 with only two negative contributions, they vary from −377.5 to +5.5 km mol−1 with just three positive contributions for the charge−dipole flux contributions. For the CF stretches charge−charge flux values range from +425.5 to −645.8 with only three positive values whereas the charge−dipole flux values are between −348.5 and +479.1 km mol−1 with three negative values. This indicates that the charge and charge flux contributions to the dipole moment derivatives have opposite polarities for the CF stretches but parallel ones for the CCl stretches. This situation is diagrammed in Figure 5 for methyl fluoride and chloride assuming that both the fluorine and chlorine atoms are negatively charged as calculated by QTAIM. Stretching the highly polar CF bond results in intramolecular charge rearrangement so that fluorine becomes less negative; i.e., the charge and charge fluxes have opposite polarities. On the other hand, stretching the less polar CCl bond results in electron transfer to the chorine atom with the charge and charge flux dipole moment change contributions having the same polarity. For both stretches the charge flux−dipole flux interaction contributions to the intensities are negative, indicating that electronic density is polarized in an opposite direction to the intramolecular electron transfer. CCl Bending Vibrations. The CCl bends have by far the smallest CCFDF contributions to the infrared intensities when compared to the others discussed in this report. The charge, charge flux, dipole flux, and charge−charge flux terms are all small and positive ranging from 0.0 to 32.7 km mol−1. The charge−dipole flux and the charge flux−dipole flux interactions are mostly all negative ranging from +0.3 to −57.4 km mol−1. Double Bond Vibrations. The CC nonpolar bonds in cis-C2H2F2 and cis-C2H2Cl2 have much smaller absolute values for their charge, charge flux, charge−charge flux, charge−dipole flux, and charge flux−dipole flux intensity contributions than the CC polar bond in 1,1-C2H2F2, as could be expected owing to the large charge asymmetry of the carbon double bond. This is also true for the CO and CS stretching vibrations in F2CO, Cl2CO, F2CS, and Cl2CS. As a result, the

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CONCLUSIONS The QTAIM/CCFDF model results indicate how difficult it might be to extract atomic charge information from infrared fundamental intensities. For the most promising vibrations, the CF stretches and bends, equilibrium atomic charge displacements furnish one of the dominant contributions to the total intensities but the charge interactions with charge flux and dipole flux are also just as important. However, the sum of the charge contributions with these interactions does allow quite accurate estimates of the CF stretching intensities. For the CH and CCl stretching and bending vibrations the charge contributions are not even dominant. For the CH stretches and bends the charge contributions are dwarfed by charge flux and dipole flux contributions as well as their interactions. This is even true for acetylene, considered to have among the most acidic hydrogen atoms of all the hydrocarbons. In contrast, the ECCF2−5 model proposes that out-of-plane polar tensor elements (or perpendicular ones for linear molecules) can be interpreted as charges because they do not contain charge 8248



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flux contributions owing to symmetry and the ECCF model does not contemplate atomic dipoles. This approach is attractive because these charges can be determined from experimental infrared intensities. Dinur and Hagler34 have shown that the outof-plane polar tensor elements correspond to charges in the asymptotic regime for the pairwise interactions of nonbonded atoms. They have called these polar tensor elements “apparent charges” and have shown they are equivalent to a sum of a charge contribution and a dipole flux one as found by the CCFDF model.

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(25) Jensen, F. Introduction to Computationl Chemistry, 2nd ed.; John Wiley and Sons: New York, 2007; Chapter 4. (26) Dunning, T. H., Jr. J. Chem. Phys. 1989, 90, 1007. (27) Silva, A. F.; Silva, J. V., Jr; Haiduke, R. L. A.; Bruns, R. E. J. Phys. Chem. A 2011, 115, 12572. (28) Meneses, H. G. C. Master's Dissertation, Universidade Estadual de Campinas, 2010. (29) Martin, A.; Blanco, M. A.; Costales, P.; Mori, P. Phys. Rev. Lett. 1999. (30) Silva, J. V., Jr.; Haiduke, R. L. A.; Bruns, R. E. J. Phys. Chem. A 2006, 110, 4839. (31) Faria, S. H. D. M.; Silva, J. V., Jr; Haiduke, R. L. A.; Vidal, L. N.; Vazquez, P. A. M.; Bruns, R. E. J. Phys. Chem. A 2007, 111, 7870. (32) Silva, J. V., Jr.; Faria, S. H. D. M.; Haiduke, R. L. A.; Bruns, R. E. J. Phys. Chem. A 2007, 111, 515. (33) Kim, K.; King, W. T. J. Mol. Struct. 1979, 57, 201. (34) Dinur, U.; Hagler, A. T. J. Chem. Phys. 1989, 91, 2949.

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.F.S. and W.E.R. thank CNPq (Conselho Nacional de Desenvolvimento Cientı ́fico e Tecnológico) for graduate student fellowships and R.E.B. thanks CNPq for a research fellowship. We are grateful to FAPESP (Fundaçaõ de Amparo à Pesquisa do Estado de São Paulo) for partial financial support of this work.

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REFERENCES

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CCFDF Interpretation - American Chemical Society

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