Molecular Quadrupole Moments for the Series of Fluoro- and

Hence it appears that having the best possible geometry is not a crucial factor to obtain reliable quadrupole moments for these aromatic systems. Furt...
0 downloads 0 Views 311KB Size
Download PDF
6524

J. Phys. Chem. 1996, 100, 6524-6530

Molecular Quadrupole Moments for the Series of Fluoro- and Chlorobenzenes Jesu´ s Herna´ ndez-Trujillo Departamento de Fı´sica y Quı´mica Teo´ rica, Facultad de Quı´mica, UniVersidad Nacional Auto´ noma de Me´ xico, Me´ xico D.F. 04510, Me´ xico

Alberto Vela* Departamento de Quı´mica, DiVisio´ n de Ciencias Ba´ sicas e Ingenierı´a, UniVersidad Auto´ noma Metropolitana-Iztapalapa, A.P. 55-534, Me´ xico D.F. 09340, Me´ xico ReceiVed: December 5, 1995X

Ab-initio molecular quadrupole moments for benzene and the complete series of fluoro- and chlorobenzenes are reported. In addition, quadrupole moments for two methylbenzenes and for naphthalene are also presented. Molecular geometries are fully optimized using a local spin density approximation in a density functional theory code (DGAUSS) that solves the Kohn-Sham equations within the linear combination of Gaussian type orbitals approach. These geometries are used to calculate the RHF/6-31G** permanent electric moments. It is found that the employment of a reasonable geometry is sufficient to obtain reliable quadrupole moments. Calculated values for the quadrupole moment tensor component perpendicular to the aromatic ring (Qzz) are in excellent agreement with available experimental data. Nuclear and electronic contributions to Qzz are also calculated and compared against experiment. The main trends of the results, i.e., the variations of Qzz with the nature and number of substituents in the aromatic ring, are discussed in terms of charge redistributions in the molecules. Standard basis set effects within the RHF scheme are analyzed for the series of fluorobenzenes. It is concluded that for the type of molecules studied here it is very important to use polarization functions, while diffuse functions do not improve the calculated quadrupole moment values.

I. Introduction The interactions between aromatic molecules play an important role in determining the properties of many systems in biology, chemistry, and material science.1 It has been found that aromatic-aromatic interactions in the interior of proteins are important in stabilizing their tertiary structure.2 In chemistry, many experimental and theoretical reports are devoted to the study of complexes involving aromatic molecules.3,4 In particular, X-ray,5,6 NMR,7,8 and neutron and synchrotron diffraction experiments9 for aromatic nonpolar fluorobenzenes mixed with aromatic nonpolar hydrocarbons have shown the existence of these complexes in the solid and liquid states. Furthermore, it has been established that in these nonpolar mixtures quadrupole-quadrupole interactions play a major role in the formation and stabilization of molecular complexes.10,11 There are several ways12-15 of approaching the problem of calculating intermolecular interaction energies in aromatic-aromatic complexes, some of which require the knowledge of central multipole moments. One possible source of quadrupole moments is experiment, but unfortunately, it has been a difficult task to get reliable values; as a result, available data are very scarce. Hence, there is a clear need for calculating them with good accuracy and reasonable computing time. In this work, we present a theoretical methodology to calculate molecular quadrupole moments of aromatic molecules that involves both density functional theory (DFT) and restricted Hartree-Fock (RHF) calculations, avoiding the use of very large basis sets. The molecules examined here are the whole series of fluorobenzenes and chlorobenzenes. With relatively little additional computational effort, the procedure proposed allows the study of larger aromatic systems. To illustrate this latter statement, quadrupole moments of two methylbenzenes and naphthalene X

Abstract published in AdVance ACS Abstracts, March 1, 1996.

0022-3654/96/20100-6524$12.00/0

are also reported. In section II, the effects of basis set and geometry on the calculated quadrupole moments, at the RHF level, are discussed and the method of calculation is presented. In section III, results are presented and discussed; this includes a comparison (i) against experimental quadrupole moments, (ii) against other calculations reported in the literature, and (iii) between fluoro and chloro-substituted benzenes. Also, nuclear and electronic contributions to the diagonal elements of the quadrupole moment tensor are discussed. II. Calculation of Quadrupole Moments A. Basis Set Effects and Geometry Optimization. At the SCF level, several theoretical works on small- and mediumsized molecules have shown the importance of including polarization functions in the calculation of quadrupole moments. For example, quadrupole moments obtained for small molecules at the RHF level16 show that polarized basis sets provide much better results than when these functions are not included. On the other hand, the electrical properties of small molecules (three to six atoms) are described better when diffuse functions are included.17 Several works analyze basis set effects on the quality of the calculated quadrupole moments for some aromatic molecules.18-20 To assess the situation for the series of fluorobenzenes, we have performed RHF calculations with a standard geometry (SG) to obtain the quadrupole moment tensor component perpendicular to the aromatic ring (throughout this work will be denoted as Qzz). Several standard split-valence basis sets, ranging from STO-3G to 6-311G, and including either polarization functions, diffuse functions or both, are used. It is worth noting that, in the present methodological approach, attention is focused on the quality of permanent electric moments provided by standard split-valence basis sets. Therefore, no effort is spent on testing full double-ζ basis sets or changing orbital exponents of the polarization and diffuse functions. © 1996 American Chemical Society

Molecular Quadrupole Moments for Some Halobenzenes

J. Phys. Chem., Vol. 100, No. 16, 1996 6525 TABLE 1: Effect of Geometry on Qzz for Benzene and the Series of Fluorobenzenes:a Comparison between Standard (RHF//SG) and Optimized Geometry (RHF//RHF) Using 6-31G** Basis Setsb molecule

Qzz RHF//SG

Qzz RHF//RHF

benzene fluorobenzene 1,2-difluorobenzene 1,3-difluorobenzene 1,4-difluorobenzene 1,2,3-trifluorobenzene 1,2,4-trifluorobenzene 1,3,5-trifluorobenzene 1,2,3,4-tetrafluorobenzene 1,2,3,5-tetrafluorobenzene 1,2,4,5-tetrafluorobenzene pentafluorobenzene hexafluorobenzene

-8.550 -6.392 -4.650 -3.258 -2.470 -1.947 -0.069 0.448 1.961 3.008 3.607 6.138 9.659

-8.288 -6.082 -4.356 -3.015 -2.155 -1.741 0.156 0.571 2.076 3.059 3.733 6.092 9.433

a All molecules are centered on their centers of mass, and the z axis is perpendicular to the aromatic ring. b All values are in buckinghams.

Figure 1. Basis set effect on the diagonal quadrupole moment tensor component perpendicular to the aromatic ring for (a) benzene, (b) 1,3,5trifluorobenzene, and (c) hexafluorobenzene. All calculations are RHF/ /SG. Standard notation is used to denote different basis sets. In all cases, the total energy (not shown) decreases monotonically when the basis sets are ordered as depicted. The horizontal solid lines correspond to the experimental quadrupole moments; horizontal dashed lines are their associated error bars (from references in Table 4).

Certainly, this does not mean that the standard split-valence basis set that produces the best quadrupole moments will in turn provide results of similar quality for other molecular properties such as dipole-dipole polarizabilities. The standard geometry (SG) is defined as follows: all molecules are planar; the C-C, C-H and C-F bond distances are 1.379, 1.085, and 1.33 Å, respectively, and all C-C-X bond angles (X ) H, F) are set equal to 120°. This C-C bond length corresponds to the average of the experimental C-C in benzene21 and hexafluorobenzene;22 the C-H bond length is that reported experimentally for benzene,21 and the C-F bond distance is the experimental value for hexafluorobenzene.22 A representative set of results is shown in Figure 1, where the experimental Qzz values are also depicted. For the three molecules in Figure 1, the calculated quadrupole moments approach the experimental

values as the standard basis set size is enlarged and polarization functions are included, while the inclusion of diffuse functions produces an overestimation of |Qzz|. For 1,3,5-trifluorobenzene two basis sets with diffuse functions are slightly better than those with only polarization functions. Considering the small value of Qzz for this molecule, this effect can be considered negligible. Results similar to those displayed in Figure 1 have been found for the whole series of fluorobenzenes. Thus, for the large molecules studied here it is very important to use polarization functions, while the employment of only diffuse functions is not sufficient to obtain reliable quadrupole moment values. This contradicts a recent report23 of extensive calculations for benzene where it is concluded that diffuse functions with small orbital exponents are essential for obtaining a good Qzz. However, some doubt should be cast over these conclusions since the calculated Qzz values were compared against an experimental value24 that was later changed by the same group of experimentalists,25 to what is considered the best experimental value. The results in Figure 1 using a 6-31G** basis set are very close to those obtained with larger basis sets, and they compare very well with available experimental results. Hence, this basis set is considered as the more adequate for the calculation of quadrupole moments and will be used in the method reported here (see below). Using this basis set, the effect of geometry on Qzz has been evaluated for the whole series of fluorobenzenes, and the results are shown in Table 1. It can be seen that geometry-optimized Qzz values (6-31G**//6-31G**) are very close to those using standard geometry (6-31G**//SG); the largest difference is 0.31 B (buckingham) for fluorobenzene and 1,4-difluorobenzene. Hence it appears that having the best possible geometry is not a crucial factor to obtain reliable quadrupole moments for these aromatic systems. Further improvement on the calculation of quadrupole moments might be accomplished by enhancing the level of theory. The effort should be directed toward the inclusion of electron correlation instead of improving the optimized geometry. The computational effort to perform 6-31G**//6-31G** calculations for the molecules shown in Table 1 is very high. Furthermore, if one desires to calculate larger aromatic systems, these computational needs are even more demanding. On the other hand, geometry optimizations using a local spin density approximation to the exchange-correlation energy functional in DFT have been shown to be very reliable for small molecules and benzene,26-31 and the computational demands are smaller than those needed at the RHF level. It is then desirable to design a calculation methodology that blends an affordable computa-

6526 J. Phys. Chem., Vol. 100, No. 16, 1996 tional effort for geometry optimization with what has been shown here to be the necessary element to obtain reliable quadrupole moments at the SCF level, namely, the use of a standard split-valence double-ζ polarized basis set. B. Method of Calculation. From the results described in the previous section, the method of calculation of quadrupole moments proposed in this paper is the following. 1. Geometry Optimization. This is carried out using DFT with double-ζ basis sets with polarization functions (DZVP);32 A1 auxiliary basis sets are used to represent the electron density and the exchange-correlation potential and energy. The local density approximation (LDA) to the exchange-correlation energy functional used is that due to Vosko-Wilk-Nusair (VWN).33 The optimized orbital DZVP basis sets used in the present work are the following: (41) for hydrogen, (621/41/1) for carbon and fluorine, and (6321/521/411) for chlorine. The grid for the numerical integrations consists of 25 radial mesh points for H, 30 for C and F, and 35 for Cl. For each atom, there are 110 angular points per radial shell. In this step, the calculations are done using DGAUSS.28 2. Calculation of the Quadrupole Moment Tensor. With the geometry optimized from the previous step, the quadrupole moment tensor within RHF theory is obtained by a single-point calculation with 6-31G** basis sets. This is done using CADPAC.34 Following Pople’s notation,35 the present calculation methodology will be denoted as RHF/6-31G**//S-VWN/DZVP. All calculations (DFT and RHF) have been performed on a Cray YMP/4-432 supercomputer and HP-730 workstations, and the Quantum Chemistry programs where used through the Unichem environment.36 III. Results and Discussion A. Optimized Molecular Geometries. Although this work is focused on the calculation of quadrupole moments, the geometries obtained in the first step of the method described above will be briefly discussed here. Some molecules are taken as examples; comparison with experimental results and with other theoretical approaches is presented. Optimized geometries for the 28 aromatic molecules studied here can be found in the Supporting Information. In Tables 2 and 3, calculated bond distances and angles for benzene, four fluorobenzenes, and one chlorobenzene are reported; these values are compared against experimental distances, obtained by either X-ray or electron diffraction, and against other theoretical calculations. Symmetry equivalent bond distances and angles reported in Tables 2 and 3 are average values. This is because DFT geometry optimizations are done in Cartesian coordinates and to grid limitations.37,38 For all molecules considered here, bond distances have standard deviations less than 0.001 Å, with the exceptions of 1,2,3,5tetrafluorobenzene and pentafluorobenzene, whose deviations are 0.007 and 0.004 Å, respectively. The agreement with experimental results is good since the difference between measured and theoretical bond distances lies in the third figure, and there are only few exceptions where the error lies in the second figure. Comparison against other previous theoretical results shows (see Tables 2 and 3) that, overall, bond distances reported here are in good agreement with those in the literature. For bond angles, the standard deviation is less than 0.3°. The standard deviations for dihedral angles (not in Table 3) never exceed 0.7°. Considering the nature of DFT geometry optimizations mentioned above, we conclude that these deviations from planarity are negligible. It can be concluded that the geometries obtained here are in good agreement with experimental data

Herna´ndez-Trujillo and Vela TABLE 2: Calculated Bond Distances and Comparison with Experimental and Other Theoretical Results for Some of the Molecules Studied in This Worka molecule

C-C distance

C-X distance

(1.399)b

benzene

1.3950 [1.383],c [1.385]d C1-C2: 1.3891 (1.3852)e [1.374]f C2-C3: 1.3935 (1.3972)e [1.3833]f C3-C4: 1.3951 (1.3992)e [1.3848]f

fluorobenzene

C1-C2: 1.3883 (1.383)g [1.374]f C3-C4: 1.3899 (1.385)g [1.375]f C4-C5: 1.3936 (1.405)g [1.384]f

1,3-difluorobenzene

1,3,5-trifluorobenzene

1.3883 (1.389)h [1.375]f

hexafluorobenzene

1.3931 (1.394)i [1.375]b 1.4016 (1.393)j [1.383]b

hexachlorobenzene

1.1006 (1.101)b [1.070],c [1.076]d C2-H2: 1.0980, (1.093)e [1.0693]f C3-H3: 1.1002, (1.093)e [1.0714]f C4-H4: 1.0983 (1.093)e [1.0711]f C-F: 1.3419 (1.355)e [1.369]f C1-F1: 1.3390, (1.324)g [1.365]f C2-H2: 1.0960, (1.107)g [1.067]f C4-H4: 1.0975, (1.107)g [1.069]f C5-H5: 1.0996, (1.107)g [1.071]f C-H: 1.0965 (1.069)h [1.066]f C-F: 1.3358 (1.339)h [1.360]f 1.3249 (1.327)i [1.349]b 1.7116 (1.715)j [1.734]b

a The calculated distances are average values over equivalent bonds; values in parentheses and brackets refer to experimental and other theoretical results, respectively; distances are expressed in angstroms; labeling of atoms is that shown in Figure 2. b Reference 46. c Reference 47. d Reference 48. e Reference 49. f Reference 50. g Reference 51. h Reference 52. i Reference 53. j Reference 54.

TABLE 3: Calculated Bond Angles and Comparison with Experimental and Other Theoretical Results for Some of the Molecules Studied in This Worka molecule

bond angle

benzene fluorobenzene

C-C-C: 120 C6-C1-C2: 122.144 (123.41),b [122.4],c [122.3]d C1-C2-C3: 118.603 (118.01),b [118.4],c [118.5]d C2-C3-C4: 120.344 (120.18),b [120.4],c [120.3]d C3-C4-C5: 119.958 (120.21),b [119.7],c [119.8]d H2-C2-C3: 121.729 [121.9],c [121.8]d H3-C3-C4: 120.292 [120.0]d 1,3 difluorobenzene C6-C1-C2: 122.459 (120.9),e [122.5]d C1-C2-C3: 117.337 (120.1),e [117.4]d C3-C4-C5: 118.341 (118.6),e [118.4]d C4-C5-C6: 121.058 (120.9),e [120.8]d F3-C3-C4: 119.253 (119.5),e [119.1]d H4-C4-C5: 121.746 [121.9]d 1,3,5-trifluorobenzene C6-C1-C2: 122.972 (123.7),f [122.7]d C1-C2-C3: 117.026, [117.3]d hexafluorobenzene C-C-C: 120 hexachlorobenzene C-C-C: 120 a The calculated angles are average values over equivalent angles; values in parentheses and brackets are experimental and theoretical results respectively; angles are expressed in degrees. Labeling of atoms is that shown in Figure 2. b Reference 49. c Reference 48. d Reference 50. e Reference 51. f Reference 52.

and that it is safe to use them for the calculation of the quadrupole moments, i.e., the second step of the proposed method of calculation. B. Quadrupole Moments. The quadrupole moment tensor is given by

Q ) 1/2∫dr ρ(r)(3rr - r21)

(1)

where F(r) is the total charge density of the molecule. The reference frame is centered on the center of mass of the molecule and oriented along the principal quadrupole moment tensor axes,

Molecular Quadrupole Moments for Some Halobenzenes

J. Phys. Chem., Vol. 100, No. 16, 1996 6527

Figure 2. Molecular orientations where the quadrupole moment tensor is diagonal. For hexasubstituted molecules, X may be H, F, or Cl; otherwise, X may be F or Cl. For 1,2,4-substituted benzenes, the angle R gives the orientation of the quadrupole moment principal axes; R is 42.2° and 37.8° for X ) F and Cl, respectively.

TABLE 4: Comparison of the Calculated Qzz Component of the Quadrupole Moment Tensor with Experimental Resultsa molecule

this workb

experiment

benzene fluorobenzene 1,3,5-trifluorobenzene hexafluorobenzene 1,3,5-trichlorobenzene 1,3,5-trimethylbenzene hexamethylbenzene naphthalene

-8.56 -6.33 +0.57 +9.55 -1.90 -7.53 -6.30 -13.57

-8.48 ( 0.36c -6.30 ( 0.30d +0.94 ( 0.12e +9.5 ( 0.5e -3.237 ( 0.33f -9.6 ( 0.8e -7.2 ( 0.6e -13.49 ( 1.50g

a All values are in buckinghams. b All molecules are centered on their centers of mass, and the z axis is perpendicular to the aromatic ring. c Reference 25. d Reference 55. e Reference 24. f Reference 41. g Reference 56.

Figure 3. Comparison between the quadrupole moment tensor component obtained at the RHF/6-31G**//RHF/6-31G** level (Qzz RHF//RHF) and by the method proposed in this work (Qzz RHF//KS) for benzene and the whole series of fluorobenzenes. Calculated values are given in Tables 1 and 5. The slope, intercept, and correlation coefficient of the linear fit are 0.9838, 0.1047, and 0.999 98, respectively.

i.e., where the quadrupole moment tensor is in diagonal form. The resulting z axis for this choice of reference frame is oriented perpendicularly to the aromatic ring. Figure 2 shows the specific selection of axes used in this work. In Figure 3, RHF calculations for the whole series of fluorobenzenes under the scheme 6-31G**//6-31G** (see Table 1) are compared against results obtained with the proposed method of calculation. It can be seen that a very satisfactory linear fit (slope of 0.9838 and correlation coefficient of 0.999 98)

is found between the two sets of calculations. In Table 4, a comparison between available experimental results and those calculated in this work is presented. The agreement with experiment is excellent. For benzene, fluorobenzene, and hexafluorobenzene the calculated quadrupole moments lie within the reported experimental errors. Also, it may be noted that the correct sign for the very small quadrupole moment of 1,3,5trifluorobenzene is reproduced. Since the number of experimental quadrupole moments available for halobenzenes is small, to establish the reliability of the method proposed here, two methylbenzenes and naphthalene were also calculated. The results are also shown in Table 4. It is encouraging that for naphthalene, a large aromatic molecule, the calculated quadrupole moment is within experimental error. It should be noted that the results for 1,3,5-trisubstituted benzenes are not as good as those for the other molecules. Comparison against other calculations, mainly for benzene, is also possible. Augspurger

6528 J. Phys. Chem., Vol. 100, No. 16, 1996

Herna´ndez-Trujillo and Vela

TABLE 5: Diagonal Quadrupole Moment Tensor Components for Benzene, Fluorobenzenes and Chlorobenzenesa,b molecule

Qxx

Qyy

Qzz

benzene fluorobenzene 1,2-difluorobenzene 1,3-difluorobenzene 1,4-difluorobenzene 1,2,3-trifluorobenzene 1,2,4-trifluorobenzene 1,3,5-trifluorobenzene 1,2,3,4-tetrafluorobenzene 1,2,3,5-tetrafluorobenzene 1,2,4,5-tetrafluorobenzene pentafluorobenzene hexafluorobenzene chlorobenzene 1,2-dichlorobenzene 1,3-dichlorobenzene 1,4-dichlorobenzene 1,2,3-trichlorobenzene 1,2,4-trichlorobenzene 1,3,5-trichlorobenzene 1,2,3,4-tetrachlorobenzene 1,2,3,5-tetrachlorobenzene 1,2,4,5-tetrachlorobenzene pentachlorobenzene hexachlorobenzene

4.28 7.80 -0.35 8.50 -10.57 4.61 -10.21 -0.26 7.31 -6.66 -13.44 3.49 -4.72 0.86 5.31 10.25 -9.61 10.04 10.26 0.91 10.15 -2.15 -7.25 5.06 0.19

4.28 -1.47 4.95 -5.33 12.88 -2.69 10.18 -0.31 -9.27 3.61 9.74 -9.60 -4.83 7.01 2.78 -5.10 13.19 -3.44 -7.07 0.98 -6.42 3.95 8.38 -3.87 0.22

-8.56 -6.33 -4.60 -3.17 -2.31 -1.92 0.03 0.57 1.96 3.05 3.70 6.11 9.55 -7.87 -8.09 -5.15 -3.58 -6.60 -3.19 -1.90 -3.73 -1.80 -1.13 -1.19 -0.41

a All values are in buckinghams. b All molecules are centered on their centers of mass, and the z axis is perpendicular to the aromatic ring.

and Dikstra20 have reported derivative RHF calculations (DHF/ TZ2P) and obtained a Qzz value for benzene of -9.442 B (1.3450 B ) 1 au), which appears to be too negative. Results including electron correlation19 through CISD lead to a benzene Qzz value of -8.776 B, which is close to experiment. Laidig obtained -8.288 and +9.434 B for the Qzz of benzene and hexafluorobenzene, respectively.39 It appears that the quadrupole moments reported here are in better agreement with experiment than those previously reported. Thus, it is concluded that the method of calculation described above is an efficient and reliable calculation strategy to obtain quadrupole moments for large planar aromatic molecules. Table 5 reports the three components of the diagonal quadrupole moment tensor for the molecules studied. Figure 4 shows the behavior of Qzz against the particular molecules (in lexicographic order) and against the number of substituents in the aromatic ring. The main trends of these results can be qualitatively explained in terms of the inductive effects caused by the substituent and the difference in electronegativity between fluorine and chlorine atoms. It has been shown40 that Qzz is related to the electron density along the z axis. This fact can be seen in Figure 4. When the number of fluorine or chlorine atoms in the molecule increases, the electron density along the z axis decreases and, consequently, Qzz increases. On going from a fluorinated to the corresponding chlorinated benzene, the smaller electronegativity of the latter implies a smaller reduction of the electronic density or inductive effect and hence a more negative Qzz. Clearly, this effect becomes more pronounced as the number of substituents increases, culminating in a large Qzz difference between the hexa-substituted benzenes. Note that within each of the three groups of isomers, the observed changes for Qzz are quite important; the effect of changing the position of the substituents is more pronounced for chlorobenzenes. These changes between isomers cannot be explained solely by induction effects; probably a resonant effect, where electron density is drawn back toward the aromatic ring, is also present.

Figure 4. (a) Qzz for benzene and the whole series of fluoro- and chlorobenzenes, in lexicographic order; (b) Qzz for the complete series of fluorobenzenes and chlorobenzenes against the number of substituents in the aromatic ring. Solid lines correspond to the least squares fits; the correlation coefficients for the linear fits are 0.9856 and 0.8499 for fluoro- and chlorobenzenes, respectively.

Figure 4b shows that there is a linear relation (correlation coefficient ) 0.9856) between Qzz and the number of fluoro substituents; this linear relation is improved if (i) Qzz values for each set of isomers are averaged (correlation coefficient ) 0.9983) or (ii) the plot contains only Qzz for the five Dnh molecules, n ) 2, 3, 6 (correlation coefficient ) 0.9999). This behavior suggests that for fluorobenzenes there is a certain background or basic additivity for Qzz. Using the same criteria, it has been found that methylbenzenes also display the same level of additivity.24 For chloro-substituted benzenes, the linearity is not as good as in the fluoro case (correlation coefficient ) 0.8494). If Qzz is plotted against the number of substituents only for the Dnh molecules or using the averaged Qzz values for the isomers, the linear fitting is slightly improved (correlation coefficients are 0.9215 and 0.9898, respectively). That chloro-substituted benzenes do not follow an additivity behavior as closely as fluorobenzenes do can also be appreciated in Figure 4b. In chlorobenzenes, Qzz does not change smoothly on going from one molecule to another, as in fluorobenzenes. It seems that 1,2-, 1,2,3-, and 1,2,3,4-chlorobenzenes are out of the general trend indicated by the other chloro-substituted molecules. This “irregularity” for the chloro case, as well as the Qzz changes within the three groups of isomers for both,

Molecular Quadrupole Moments for Some Halobenzenes

J. Phys. Chem., Vol. 100, No. 16, 1996 6529

TABLE 6: Nuclear and Electronic Contributions to Qzz for Benzene and Fluoro- and Chloro-Substituted Benzenes (Values in Parentheses Correspond to Experimental Data)a,b molecule

nuclear

electronic

benzene fluorobenzene 1,2-difluorobenzene 1,3-difluorobenzene 1,4-difluorobenzene 1,2,3-trifluorobenzene 1,2,4-trifluorobenzene 1,3,5-trifluorobenzene 1,2,3,4-tetrafluorobenzene 1,2,3,5-tetrafluorobenzene 1,2,4,5-tetrafluorobenzene pentafluorobenzene hexafluorobenzene chlorobenzene 1,2-dichlorobenzene 1,3-dichlorobenzene 1,4-dichlorobenzene 1,2,3-trichlorobenzene 1,2,4-trichlorobenzene 1,3,5-trichlorobenzene 1,2,3,4-tetrachlorobenzene 1,2,3,5-tetrachlorobenzene 1,2,4,5-tetrachlorobenzene pentachlorobenzene hexachlorobenzene 1,3,5-trimethylbenzene hexamethylbenzene

-258.00 (-257.43)c -378.97 -487.51 -522.54 -544.77 -603.01 -672.15 -653.88 (-692.73)c -788.19 -779.96 -834.92 -964.67 -1125.77 (-1128.04)c -533.47 -777.38 -871.35 -1019.61 -1136.03 -1335.30 -1394.39(-1395.36)d -1615.74 -1663.11 -1785.21 -2116.01 -2543.94 -802.84(-801.84)d -1349.13(-1344.49)c

249.44 (247.45)c 372.64 482.91 519.37 542.46 601.09 672.19 654.45 (693.66)c 790.15 783.01 838.62 970.79 1135.32 (1137.53)c 525.56 769.29 866.20 1016.03 1129.43 1332.10 1392.5(1392.06)d 1612.01 1710.66 1784.08 2114.82 2543.54 795.29(792.25)d 1342.83(1337.27)c

a All values are in buckinghams. b All molecules are centered on their centers of mass, and the z axis is perpendicular to the aromatic ring. c Reference 24. d Reference 41.

might be explained through a detailed molecular orbital or a distributed moment analysis.12,13 These analyses are a probable line for future work and, hence, beyond the scope of the present work. The molecular quadrupole moment has two contributions: nuclear and electronic. Since the quadrupole moment tensor is traceless, for an axially symmetric molecule, the knowledge of only one tensor component is sufficient to unambiguously specify the quadrupole moment tensor. Thus, taking the z axis along the principal axis of symmetry, eq 1 reduces to N E QZZ ) QZZ + QZZ ) ∑ZR(zR2 - xR2) - [〈z2〉 - 〈x2〉]

(2)

R

where the first term corresponds to the nuclear contribution and the second term is the electronic contribution to Qzz. Ritchie41 and Vrbancich and Ritchie24 have calculated nuclear and electronic contributions to Qzz for some aromatic molecules; they used experimental geometries to calculate the nuclear contribution, and by subtracting from the experimental total quadrupole moment, the electronic contribution was obtained. In Table 6, the nuclear and electronic contributions to Qzz calculated in this work for the series of fluoro- and chlorobenzenes and for two methylbenzenes are presented. For comparison, the corresponding values obtained in refs 24 and 41 for six of these molecules are also shown. The agreement between the two sets of values is excellent. For the nuclear contribution, this agreement is a consequence of the good quality geometries resulting from the previous step. For benzene, N E , corresponds to a displacement of electronic | > QZZ |QZZ charge from the C-H bonds toward the aromatic ring, and consequently, Qzz is negative. The opposite is observed for hexafluorobenzene, where the charge displacement is from the aromatic ring to the C-F bonds, and accordingly, Qzz is positive. This leads to the conclusion that for 1,3,5-trifluorobenzene,

TABLE 7: Calculated Dipole Moments for Some Fluoroand Chloro-Substituted Benzenesa

a

molecule

this work

experimental

fluorobenzene 1,2-difluorobenzene 1,3-difluorobenzene pentafluorobenzene chlorobenzene 1,2-dichlorobenzene 1,3-dichlorobenzene

1.73 2.92 1.73 1.64 2.09 3.07 1.96

1.60b 2.46b 1.51b 1.44c 1.69b 2.5b 1.72b

All values are in debyes. b Reference 57. c Reference 58.

where Qzz is very small, the displacements of electronic charge from and toward the aromatic ring are nearly balanced. The same analysis holds for the D3h and D6h chloro- and methylsubstituted benzenes, where the nuclear and electronic contributions to Qzz in Table 6 indicate that the electronic charge is more displaced toward the aromatic ring in 1,3,5-substituted benzene than in hexa-substituted benzene, where both contributions are balanced. A similar interpretation may be done for Qxx ) Qyy for the seven axial aromatic molecules just discussed; by increasing the number of substituents on the aromatic ring, the Qxx ) Qyy value becomes more negative (for fluorine) or less positive (for methyl), indicating an electronic charge withdrawal from the aromatic ring. This rough picture of electronic distribution agrees with that obtained by Laidig,39 in his study of the quadrupole moment tensor for benzene and hexafluorobenzene using the theory of atoms in molecules, and with Stone,12 who used a distributed multipole analysis. Clearly, the method of calculation used here also produces molecular dipole moments. For fluoro- and chlorobenzenes, where reliable experimental gas phase data were found in the literature, calculated dipole moments are presented in Table 7. Comparison between experimental and calculated results shows that the latter are always larger than the former and they deviate considerably (a maximum of 23.4% and a minimum of 7.8%). These results should not be that surprising since HF dipole moments deviate strongly and are too large compared with experiment.35 Results in Table 7 contrast with the excellent agreement between the calculated and experimental quadrupole moments in Table 4. Probably, a reasonable description of the electron density at large distances from the nuclei is responsible for the very acceptable quadrupole moments provided by the present approach. Previous reports on the calculation of the quadrupole moment for benzene19 indicate that electron correlation (through a CISD calculation) is important. This contrasts with the results presented here, where good quadrupole moments were obtained without including electron correlation. We have performed DFT quadrupole moment tensor calculations (with DZVP and DZVP2 basis sets) for fluorobenzenes, using several levels of theory ranging from LDA to SCF-generalized gradient approximation (GGA) calculations. It has been found that as the level of theory is improved, the quadrupole moments approach the experimental value from below, but never getting as close as the RHF calculation. For benzene, the best Qzz value is -9.37 B, which is 10% away from experiment and corresponds to that obtained with Becke’s exchange energy functional42 and Perdew’s correlation functional43 (BP). The poor performance of GGA in calculating quadrupole moments was one of the motivations to propose the RHF single-point calculation. In this context, it should be mentioned that dipole moments obtained by DFT are much better than those in Table 7. To further study and understand the effect of electron correlation on the calculation of quadrupole moments, it would be useful to perform MP244

6530 J. Phys. Chem., Vol. 100, No. 16, 1996 and hybrid DFT calculations,45 which have shown to yield good results for small- and medium-sized molecules. IV. Summary Molecular quadrupole moments for benzene, the complete series of fluoro- and chlorobenzenes, two methylbenzenes, and naphthalene have been calculated at the RHF level using DFToptimized geometries. The quadrupole moments reported here are in excellent agreement with experiment. It is also shown that the quadrupole moments provided by the present approach for planar aromatic molecules belonging to highly symmetric point groups behave almost linearly with the number of substituents on the aromatic ring. This reflects an almost additive behavior of this permanent electric moment for these systems. The nature of the substituent is reflected by the different slopes of the straight line. It is concluded that the proposed method of calculation (RHF/6-31G**//S-VWN/DZVP) is an efficient and reliable strategy to obtain quadrupole moments for large planar aromatic molecules. Acknowledgment. We gratefully acknowledge the Direccion General de Servicios de Computo Academico of UNAM for the use of the Cray YMP/4-432 supercomputer and UAM-I for the use of HP-730 workstations. We thank Cray-UNAM Program of Sponsorship for Research and Development (Grant SC-101294). J.H.-T. is grateful for financial support from the Consejo Nacional de Ciencia y Tecnologı´a and to Miguel Costas for his comments. We extend our sincere appreciation to Gabriel Cuevas for his interest, valuable comments, and suggestions for this work. Supporting Information Available: Optimized local spindensity geometries for benzene, the complete series of fluoroand chlorobenzenes, naphthalene, 1,3,5-trimethylbenzene, and hexamethylbenzene (10 pages). Ordering information is given on any current masthead page. References and Notes (1) Hunter, C. A. Angew. Chem., Int. Ed. Engl. 1993, 32, 1584. (2) Burkley, S. K.; Petsko, G. A. AdV. Protein Chem. 1988, 39, 125. (3) Duncan, W. A.; Swinton, F. L. Trans. Faraday Soc. 1966, 62, 1082. (4) Price, S. L.; Stone, A. J. J. Chem. Phys. 1987, 86, 2859. (5) Dahl, T. Acta Chem. Scand. 1971, 25, 1031. (6) Dahl, T. Acta Chem. Scand. 1973, 27, 995. (7) Suhm, M. A.; Weingartner, H. Chem. Phys. Lett. 1989, 159, 193. (8) Duer, M. J. J. Chem. Soc., Faraday Trans. 1993, 89, 823. (9) Williams, J. H. Acc. Chem. Res. 1993, 26, 593. (10) Brown, N. M. D.; Swinton, F. L. J. Chem. Soc., Chem. Commun. 1974, 770. (11) Herna´ndez-Trujillo, J.; Costas, M.; Vela, A. J. Chem. Soc., Faraday Trans. 1993, 89, 2441. (12) Stone, A. J.; Alderton, M. Mol. Phys. 1985, 56, 1047. (13) Fowler, P. W.; Buckingham, A. D. Chem. Phys. Lett. 1991, 176, 11. (14) Kitaura, K.; Morokuma, K. Int. J. Quantum Chem. 1976, 10, 325.

Herna´ndez-Trujillo and Vela (15) Hobza, P.; Selzle, H. L.; Schlag, E. W. Chem. ReV. 1994, 94, 1767. (16) Schaefer, H. F., III. The Electronic Structure of Atoms and Molecules; Addison-Wesley Pub. Co.: New York, 1972. (17) Dikstra, C. E. Ab Initio Calculations of the Structures and Properties of Molecules; Elsevier Science Publishers B. U.: New York, 1988. (18) Chablo, A.; Cruickshank, D. W. J.; Hinchliffe, A.; Munn, R. W. Chem. Phys. Lett. 1981, 78, 424. (19) Ha, T. K. Chem. Phys. Lett. 1981, 79, 313. (20) Augspurger, J. D.; Dikstra, C. E. Mol. Phys. 1992, 76, 229. (21) Langseth, A.; Stoicheff, B. P. Can. J. Phys. 1956, 34, 350. (22) Boden, N.; Davies, P. P. Mol. Phys. 1973, 25, 81. (23) Hobza, P.; Selzle, H. L.; Schlag, E. W. J. Am. Chem. Soc. 1994, 116, 3500. (24) Vrbancich, J.; Ritchie, G. L. D. J. Chem. Soc., Faraday Trans. 2 1980, 76, 648. (25) Dennis, G. R.; Ritchie, G. L. D. J. Phys. Chem. 1991, 95, 656. (26) Fitzgerald, G.; Andzelm, J. J. Phys. Chem. 1991, 95, 10531. (27) St-Amant, A.; Salahub, D. R. Chem. Phys. Lett. 1990, 169, 387. (28) Andzelm, J.; Wimmer, E. J. Chem. Phys. 1992, 96, 1280. (29) Versluis, L.; Ziegler, T. J. Chem. Phys. 1988, 88, 322. (30) Labanowski, J. K.; Andzelm, J. W. Density Functional Methods in Chemistry; Springer-Verlag: Berlin, 1991. (31) Lee, C.; Fitzgerald, G.; Yang, W. J. Chem. Phys. 1993, 98, 2971. (32) Godbout, N.; Salahub, D. R.; Andzelm, J.; Wimmer, E. Can. J. Phys. 1992, 70, 560. (33) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (34) Amos, R. D.; Rice, J. E. Cadpac: The Cambridge Analytical DeriVatiVes Package, Issue 5.0; Cambridge, 1991. (35) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J. Chem. Phys. 1993, 98, 5612. (36) Unichem Chemistry Codes, C. R. I., 1991. (37) Goursot, A.; Pa´pai, C. A.; Daul, C. A. Int. J. Quantum Chem. 1994, 52, 799. (38) Baker, J.; Andzelm, J.; Scheiner, A.; Delley, B. J. Chem. Phys. 1994, 101, 8894. (39) Laidig, K. E. Chem. Phys. Lett. 1991, 185, 483. (40) Bader, R. F. W.; Chang, C. J. Phys. Chem. 1989, 93, 2946. (41) Ritchie, G. L. D. Chem. Phys. Lett. 1982, 93, 410. (42) Becke, A. D. J. Chem. Phys. 1986, 84, 4524. (43) Perdew, J. P. Phys. ReV. B 1986, 33, 8822. (44) Wolinski, K.; Sadlej, A. J.; Karlstro¨m, G. Mol. Phys. 1991, 72, 425. (45) Becke, A. D. J. Chem. Phys. 1993, 98, 1372. (46) Tamagaua, K.; Iijima, T.; Kimura, M. J. Mol. Struct. 1976, 30, 243. (47) Almlo¨f, J.; Feagri, K., Jr. J. Am. Chem. Soc. 1983, 105, 2965. (48) Bock, C. W.; Trachtman, M.; George, P. J. Comput. Chem. 1985, 61, 592. (49) Portalone, G.; Schultz, G.; Domenicano, A.; Hargittai, I. J. Mol. Struct. 1984, 118, 53. (50) Boggs, J. E.; Pang, F.; Pulay, P. J. Comput. Chem. 1982, 3, 344. (51) Van Schaick, E. J. H.; Geise, H. J.; Mijlhoff, F. C.; Renes, G. J. Mol. Struct. 1973, 16, 389. (52) Almenningen, A.; Hargittai, I.; Brunvoll, J.; Domenicano, A.; Samdal, S. J. Mol. Struct. 1984, 116, 199. (53) Almenningen, A.; Bastiansen, O.; Seip, R.; Seip, H. M. Acta Chem. Scand. 1964, 18, 2115. (54) Brown, G. M.; Strydom, O. A. Acta Crystallogr., Sect. B 1974, B30, 801. (55) Dennis, G. R.; Gentle, I. R.; Ritchie, G. L. D. J. Chem. Soc., Faraday Trans. 2 1983, 79, 529. (56) Calvert, R. L.; Ritchie, G. L. D. J. Chem. Soc., Faraday Trans. 2 1980, 76, 1249. (57) CRC Handbook of Physics and Chemistry; 74th ed.; CRC: Boca Raton, 1993-1994. (58) Doraiswamy, S.; Charma, D. D. Pramana 1974, 2, 219.

JP953576X