2090
Jon Applequist and James R. Carl
S(i) = alklfl(3
+
a,k,f,(u) alkl + a&, Now consider two samples, letting a second subscript on CY denote the sample, while the first subscript will continue to denote the component
(Al) From the mean value theorem for integrals there must exist at least one point EO in the domain such that
Substitution of the expressions A1 and A2 and collection of terms yields
The condition for (A2) to be satisfied is independent of the concentration times path length term C Y . Consequently, all area-normalized spectra for any solution containing only these two absorbing components at V O intersect at the same point. This point is defined by condition A3. Similarly, if three absorbing components at EO are present it may be shown that condition A2 is satisfied provided fl(Y0)
=
f*GO)
= f3GO)
i e . , provided the normalized molar absorptivities of all three components are equal a t EO.
References and Notes J. T. Bulmer and H. F. Shurell, J. Phys. Chem., 77, 256 (1973). J. Lascombe, J. Devaure, and M. L. Josien, J. Chim. Phys.. 61, 1271 (1964). R. C. Lord, B. Nolin, and H. D. Stidham, J. Amer. Chem. Soc.. 77, 1365 (1955). E. D. Becker. Spectrochim. Acta. 14, 743 (1959). 8. B. Howard, C. F. Jumper, and M. T. Emmerson, J. Mol. Spectrosc.. 10, 117 (1963). U. Liddei and E. B. Becker, J. Chem. Phys.. 25, 173 (1956). C. F. Jumper, M. T. Emmerson, and B. 8. Howard, J. Chem. Phys.. 35, 1911 (1961). T. Gramstad and 0.Mundheim, Spectrochim. Acta. Part A , 28, 1405 (1972). A. I. Biggs, Trans. Faraday Soc.. 50, 800 (1954). E. F. H. Brittain, W. 0. George, and C. J. H. Wells, "Introduction to Molecular Spectroscopy," Academic Press, London, 1970, p 110. M. D. Cohen and E. Fisher, J. Chem. Soc.. 3044 (1962). J. R. Morrey, J. Phys. Chem.. 66, 2149 (1962). T. Nowicka-Jankowska, J. lnorg. Nucl. Chem.. 33, 2043 (1971). G. C. Pimentel and A. L. McClellan. "The Hydrogen Bond," W. H. Freeman, San Francisco, Calif., 1960, pp 95 and 101. S.N. Vinogradov and R. H. Linneli, "Hydrogen Bonding," Van Nostrand-Reinhold, New York, N. y . , 1971, p 64. R. N. Jones, et a/.. National Research Council of Canada Bulletin No. 11, 12 (1968): No. 13 (1969). P. J. Burchill and J. A. McRae, Aust. J. Chem.. 24, 187 (1971). J. J. Kankare, Ana/. Chem.. 42, 1322 (1970). B. Higman, "Applied Group-Theoretic and Matrix Methods," Dover Publications, New York, N. Y., 1964, p 149, M. J. D. Powell, "Minimum of a Function of Several Variables," Program 60, Quantum Chemistry, Program Exchange, Indiana University, Bloomington, Ind. M. J. D. Powell. Comput. J.. 7, 155 (1964). A. I. Vogel, "A Textbook of Practical Organic Chemistry," 3rd ed, Longmans, London, 1956, pp 163 and 176. E. W. Washburn, "International Critical Tables of Numerical Data," Ill, McGraw-Hili, New York, N, Y., 1928, p 28. J. T. Bulmer and H. F. Shurvell, Can. Spectrosc.. 16,94 (1971). K. Nagano and D. E. Metzier, J. Amer. Chem. Soc.. 89, 2891 (1967). R. E. Kagarise, Spectrochim. Acta. 19, 629 (1963).
The Polarizability of the CN Group from a Dipole Interaction Treatment of Experimental Polarizabilities of Nitriles Jon Applequist* and James R. Carl Department of Biochemistry and Biophysics, lowa State University, Ames, lowa 50010
(Received January 2, 1973)
Publication costs assisted by the National lnstitute of General Medical Sciences
The atom dipole interaction model described previously for treating molecular polarizabilities is modified by regarding the CN group in nitriles as a single isotropic or anisotropic unit. A range of "optimum" polarizability parameters for the CN group is obtained by adjusting parameters to fit the experimental polarizabilities of seven nitriles. It is concluded that no significant advantage is gained over the earlier treatment in which the C and N atoms were regarded as separate isotropic units, and that the latter model is slightly superior in fitting available molecular anisotropy data.
Introduction The experimentally observed polarizabilities of molecules provide information on the polarizabilities of the atoms or groups of atoms in the molecules. However, the information obtained depends strongly on the nature of the assumptions made in interpreting the experimental The Journal of Physicai Chemistry, Voi. 77, No. 17, 1973
data, Thus we have recently found1 that atom polarizabilities obtained on the basis of a noninteracting, or additive, model are significantly larger than those obtained on the basis of an "atom dipole interaction model," in which the atoms are treated as isotropically polarizable particles located a t their nuclei. It was concluded that the
2091
Polarizability of the CN Group from a Dipole Interaction Treatment atom polarizabilities from the latter model were the more physically realistic. Even so, the physical significance of polarizabilities of parts of a molecule obtained from such a model is limited because (i) the exchange of electrons between coval~ently bonded parts is not fully consistent with the view that these parts can be assigned separate polarizabilitierg; (ii) the treatment of the induced dipole in an atom or group as a point dipole neglects the significant spatial extent of the actual charge distribution; and (iii) it is usually necessary to make arbitrary assumptions about the isotropy and location of the polarizability in order to keep the number of adjustable parameters within reasonable bounds. The CN group in nitriles is taken here as an example. Since there are six valence electrons shared between the C and N atoms, one might ask, what is the physical significance of polarizabilities assigned to these atoms? On the other hand, if one regards the CN group as a single polarizable point in order to avoid this problem of electron exchange, at what point should the polarizability be located, and what assumptions should be made concerning the isotropy (or anisotropy) of the group? The purpose of this paper is to explore these questions for the CN group by means of numerical calculations of molecular polarizabilities based on alternative assumptions regarding the assignment of polarizabilities in the CN group. Such calculations for the case where C and N are treated as separate, isotropic units were reported previously.1 It was found possible to obtain rather good agreement with observed polarizabilities for seven nitriles with this model, though a broad range of polarizabilities of C and N were found to be equally “optimal” in fitting the experimental data. These results will be compared here with similar calculations in which the CN group is treated as a single polarizable unit.
Models The theory described previously1 allows one to calculate the polarizability tensor of a molecule when the polarizability tensors and coordinates of the units making up the molecule are given. The primary physical assumption is that contained in the Silbersteinz theory, namely, that the units interact by way of their dipole fields. As in the previous study, the approach here is to adjust certain parameters for the polarizabilities of the units to obtain an optimum fit to the experimental polarizabilities of a set of molecules of known geometry. The set considered here consists of seven nitriles which contain, in addition to the CN group, only the atoms C, H, and C1. In these calculations it is assumed that the latter three atoms have the isotropic polarizabilities found previously from a study of alkanes and Inalomethanes. For the CN group three different models are compared. Model I. The C and N atoms are treated as separate isotropic units located at their nuclei. The calculations for this model were discussed earlier,l but some of them are presented again here for comparison with the other models. The adjustable parameters in this case are the two scalar polarizabilities ac and a~ of the C and N atoms. Model 1%The CN group is treated as a single isotropic unit whose scalar polarizability ~ C is N an adjustable parameter. A second adjustable parameter is the distance of this unit from the nucleus of the carbon atom to which the CN is attached, referred to as the C-CN distance. Model III. The CN group is treated as a single anisotropic unit located at the midpoint of the CN bond. The
2.01 1.9’
3 a
1.8
‘
1.7.
1.5.
1.4.
-
1.5
1.6
I
I
I
1
I
I
I
1.7
1.8
1.9
2.0
2.1
2.2
2.3
C-CN D i s t a n c e
Contours of the error parameter S for model II. Figures indicate S value. Circles indicate points at which calculations were made, giving the contours by interpolation.
Figure 1.
polarizability tensor a C N of the CN group is assumed to have cylindrical symmetry about the bond axis (specified by the unit vector u), and thus takes the form3
+
(YCN Z;ICN[(l - 6,,/3)I ~ C ~ U U ] (1) where I is the unit tensor and ~ C and N B C N are the mean polarizability and anisotropy, respectively, defined by
CYCN
+
(cu,, 2aL)/3
6CN = (cull
-
@l)/ECN
(2)
(3)
where a,,and a 1 are the components parallel and perpendicular, respectively, to the CN bond. The adjustable parameters chosen for this model are ~ C and N B C N . The location of the polarizable point could have been regarded as a third adjustable parameter, but this was considered undesirable, since it seemed doubtful that more significant polarizability parameters could be obtained in this way and since this would have entailed a great increase in computer time to find the optimum.
Calculations The nitriles included in this study are CHsCN, CHsCHzCN, (CH3)&HCN, (CH3)3CCN, CHz(CN)z, CHZClCN, and CC13CN. The same set was used in our earlier study,l and the references to experimental data on polarizabilities and molecular structures are given there. The wavelength a t which all polarizability data apply is 5893 A. For each of the above models, the optimum values of the adjustable parameters for the CN group were taken as those which minimize the sum S of the squares of relative (fractional) deviations of the calculated mean molecular polarizabilities from the experimental values for the seven nitriles. The minimum in S was located by means of the error contours shown in Figures 1 and 2 and in Figure 3 of ref 1. The calculations were performed as described previously.l The use of relative, rather than absolute, deviaThe Journal of Physical Chemistry, Vol. 77, No. 1 7 , 1973
2092
Jon Applequist and James R. Carl TABLE I: Parameter Sets for CN Group Parameter set
ac, A 3
O ~ NA3 ,
la
0.22
Ib
0.36
IC
0.75
0.85 0.52 0.1 1
~ C N A3 I
Ila
2.00
2.00
1.82 (1.58)
1.82 (1.69)
A3
6C N tions means that it is the relative deviations that tend to be uniform a t the optimum. This is not necessarily the most objective optimum that might be established with a set of molecules in which the proportion of the polarizability coming from CN groups varies, but is adopted here for simplicity and for uniformity with our previous procedure. The error contours for all three models show that there is no single well defined optimum parameter set, but rather a long, narrow locus of parameters that are almost equally optimal. For models I1 and III, S is about 0.008 in the optimum region, corresponding to an rms deviation of 3.4%. For model I, S was found1 to be about 0.01 along the optimal locus, corresponding to an rms deviation of 3.8%. Table I gives values of the parameters located near the optimum for each model and selected to illustrate the trends in calculated polarizabilities. The values in parentheses have S values slightly above the minimum, but less than 0.01. Table I1 shows the calculated mean polarizabilities ti and the principal components 011, 012, 013 for the seven nitriles included in the optimization using the parameters in Table I. When 012 = 013 by symmetry, 013 is omitted, and 011 is the component parallel to the symmetry axis. It is seen that in practically all cases h is insensitive to the parameters (reflecting the fact that the parameter sets are all nearly optimal) and that, for ti, the three models tend to agree among themselves as well as or better than they do with the experimental values. The calculated values of the components show considerable variation among the parameter sets given in the tables. A comparison with the experimental components might provide a further criterion for an optimum fit. This proves to be difficult to achieve because none of the models predicts the anisotropies as well as it does the mean polarizabilities, and because the components are known experimentally only for the cylindrically symmetric molecules CHsCN, (CHs)&CN, and CC13CN. However, a rough idea of the validity of the various parameters can be The Journal of Physical Chemistry, Vol. 77, No. 17, 1973
3.56 1.16 1.81
A
IIb IIC
~ C N I
Figure 2. Contours of the error parameter S for model I I i .
C-CN,
S‘
23.0 8.72 3.32
6CN
llla
2.45
-0.60
IIIb lllc
(2.05) (1.75)
(-0.05) (i-0.50)
92.9 30.3 6.24
gained by comparing experimental and calculated values of 011 - a2 , the quantity determined by the Kerr constant measurements for these three molecules.4 Table I gives the sum S’ of the squares of relative deviations of this quantity for each parameter set. The best value, S’ = 1.16, is found for model I and corresponds to an rms deviation of 62%. This is typical of the ability of the isotropic atom model to predict anisotropies of a variety of molecu1es.l Models I1 and I11 show considerably poorer agreement for the anisotropy using the polarizability parameters in the optimum region for h, and the S’ values in Table I suggest the optimum parameters for 011 - 012 would not coincide with the optimum in h for these models. As a further test of the physical significance of the C N polarizability parameters, the polarizability of cyanogen, NCCN, has been calculated using the optimum parameters found for the nitriles. The results are given in Table 11. The agreement with experiment is significantly poorer for all parameter sets than was found for the nitriles. This may well result from a difference in the electronic structure of this compound, as is reflected also in the C-C bond lengths: 1.38 8, for cyanogen us. 1.46 8, for nitriles.5
Discussion A striking fact is that all of the models adopted here are about equally successful in accounting for molecular polarizabilities of nitriles. No particular advantage has been gained in treating the CN group as a single unit (models I1 and III), in spite of the extensive electron sharing between the C and N atoms. In fact, by the criterion of S’, the isotropic atom model (model I) is somewhat superior to the others in accounting for anisotropies. A virtue of model I is that it ascribes polarizability to two points in the CN group instead of one, a fact which should be significant in treating the interaction of this group with other atoms close by. It is possible that this virtue offsets errors due to neglect of electron exchange between atoms. These calculations shed some light on the inherent anisotropy of the CN group. For CH&N, (CH3)3CCN, CC13CN, and NCCN the calculated molecular anisotropy is the wrong sign when ~ C Nis -0.6, as seen in the results for parameter set IIIa. The correct sign is obtained when 8cN becomes more positive. The results for models I and I1 are in agreement with this, since model I effectively as-
2093
Polarizability of the CN Group from a Dipole Interaction Treatment TABLE 11:
Calculated Polarizabilities (A31 of Nitriles at 5893 A Calcd
Compound
la
4.10 6.00 3.15 6.22 8.45 5.69 4.51 8.20 9.32 9.29 5.98 9.91 10.29 9.72 5.88 3.83 8.36 5.45 6.18 8.59 5.79 4.1 5 10.29 10.06 10.40 3.41 6.73 1.75
Ib
4.15 6.67 2.90 6.25 8.99 5.48 4.27 8.22 9.83 9.07 5.75 9.88 10.62 9.51 6.07 3.32 9.52 5.36 6.22 9.81 5.68 3.90 10.28 10.49 10.17 3.36 7.48 1.31
IC
Ila
lib
IIC
Illa
lllb
lllc
4.07 6.84 2.68 6.10 9.05 5.16 4.10 8.04 9.78 8.81 5.54 9.60 10.29 9.26 6.08 2.91 10.46 4.88 6.09 9.17 5.40 3.71 10.03 10.18 9.95 3.44 7.83 1.25
4.13 4.96 3.72 6.23 7.87 5.76 5.07 8.16 9.49 8.56 6.41 9.8'4 9.62 9.96 5.96 4.94 7.29 5.66 6.20 8.27 5.63 4.70 10.30 9.31 10.80 4.11 5.14 3.60
4.15 5.58 3.43 6.26 8.35 5.62 4.81 8.14 9.21 9.05 6.17 9.79 9.97 9.70 6.01 4.40 8.14 5.48 6.22 8.56 5.68 4.44 10.29 9.79 10.54 3.87 5.32 3.14
4.13 6.07 3.17 6.26 8.77 5.44 4.56 8.1 1 9.39 9.00 5.94 9.71 10.16 9.49 6.00 3.89 8.88 5.23 6.20 8.83 5.60 4.18 10.23 10.05 10.31 3.50 5.22 2.64
4.17 3.73 4.39 6.20 7.54 5.73 5.32 8.08 9.94 7.57 6.74 9.74 8.41 10.41 6.03 6.24 5.95 5.90 6.18 8.26 5.36 4.92 10.21 7.94 113 4 4.52 3.59 4.98
4.13 4.77 3.81 6.21 7.72 5.76 5.16 8.15 9.57 8.39 6.49 9.84 9.46 10.03 5.95 5.11 7.04 5.69 6.19 8.21 5.57 4.79 10.29 9.11 10.88 4.21 5.23 3.70
4.10 5.53 3.39 6.24 8.21 5.75 4.76 8.21 9.31 9.16 6.16 9.93 10.25 9.77 5.93 4.30 7.97 5.52 6.21 8.47 5.77 4.38 10.36 9.98 10.55 3.96 6.53 2.68
*
Exptl
4.48 5.74 3.85 6.24
8.05
9.59 10.71 9.03 5.79
6.10
10.42 10.70 10.29 4.84f 7.59g 3.478
a Perpendicular to CCC plane. Perpendicular to HCCN plane. Perpendicular to HCH plane. Perpendicular to CCCl plane. e Linear structure with bond lengths C-C 1.380 A, C-N 1.157 A;5 interunit distance in model I I taken as 2(C-CN) 1.380 A. f H. E. Watson and K. L. Ramaswamy, Proc. Roy. SOC., Ser. A, 156, 144 (1936). gK. L. Wolf, H. Briegieb, and H. A. Stuart, Z. Phys. Chem. 8, 17, 429 (1929); values adjusted to give &from footnote t, keeping 011 a2 fixed.
-
-
signs a positive anisotropy to the CN group, according to an elementary result of the Silberstein theory for diatomic molecules,l and model I1 assigns ~ C N= 0. Since all the models agree on this point, there is reasonable assurance that BCN is nearly zero or positive, and not strongly negative. These calculations have attempted to arrive at polarizability parameters for the CN group using information on molecular structures and mean polarizabilities, with some help from Kerr effect anisotropy data. It seems remarkable that a broad range of parameters, along the optimal locus, show an almost equally good fit to the data. It will be of interest to see whether other properties dependent on atom and group polarizabilities are similarly insensitive to these parameters. Calculations of optical rotations
of certain nitriles indicate that this is true to a certain extent for this property also.6 Acknowledgments. This investigation was supported by a research grant from the National Institute of General Medical Sciences (GM-13684) and an undergraduate research participation grant from the National Science Foundation (GY-5990) supporting J. R. C. References and Notes (1) J. Applequist, J. R., Carl, and K.-K. Fung, J. Amer. Chem. SOC.,94, 2952 (1972). (2) L. Silberstein, Phil. Mag., 33, 92, 215, 521 (1917). (3) J. G. Kirkwood, J. Chem. Phys., 5,479 (1937). (4) R. J. W. LeFevre, B. J. Orr, and G. L. D. Ritchie, J. Chem. SOC., 2499 (1965). (5) "Tables of Interatomic Distances and Configuration in Molecules and Ions," Chem. SOC.Spec. Pub/., No. 11 (1958). (6) J. Appiequist,J. Amer. Chem. SOC., submitted for publication.
The Journal ot Physical Chemistry, Vol. 77,
No. 17, 1973
