N. K. Sanyal, P. Ahmad, and L. Dixit
2552
A Quantum Mechanical Treatment of Bond and Molecular Polarizabilities of Some Substituted Hydrocarbons with Ring and Chain Structures Nitish K. Sanyal,* Parvez Ahmad, and L. Dixit Department of Physics, University of Gorakhpur Gorakhpur, 273001, India (Received September 5 7972, Revised Manuscript Received May 3, 1973)
The one-dimensional semiempirical &function potential model of chemical binding, proposed by Lippincott and Stutman, has been applied to calculate the average molecular polarizabilities of some substituted hydrocarbons with ring and chain structures. The residual atomic polarizability degrees of freedom of these molecules vary from 7 to 58. Results have been discussed in the view of available experimental values reported by LeFevre and others.
Introduction Lippincott and Stutmanl proposed the use of a onedimensional &function potential model to calculate the bond and molecular polarizabilities of simple systems. The model has been successfully applied to simple polyatomic molecules by Lippincott, et a1.,2 Nagarajan,3 and Sanyal, et aL4 These studies include such molecular geometries for which the residual atomic polarizability degrees of freedom (ndf, defined in ref 1) is relatively small (5-13). Beran and Kevan5 have used this model to calculate the molecular polarizabilities of fluorocarbons, substituted fluorocarbons, ethers, esters, ketones, and aldehydes. Recently, Puranchandra and Ramamurthy6 have extended the use of the &function model for the calculation of molecular polarizabilities of some substituted benzenes. In the present communication, we have examined the applicability of the &function potential model of chemical binding to relatively large organic molecules with ring and chain structures. The results have been compared with known experimental values of average molecular polarizabilities. Polarizability Calculations The general expression for molecular polarizability expressed in Cartesian coordinates is given by
e is the electronic charge, $0 is the ground-state &function wave function, and E denotes the energy. Equation 2 is difficult to evaluate in general, except in the case of diatomic systems where the accurate wave functions give rise to calculated polarizabilities that show reasonable agreement with the experimental ones.?,8 To calculate the molecular polarizabilities for the general class of polyatomic molecules, Lippincott and Stutman’ used the semiempirical &function model of chemical binding.gJ0 The d-function model was obtained by replacing the coulomb potentials in the Schrodinger equation of a molecular system with &function potentials. The molecular wave functions are obtained from linear combinations of atomic b-function wave functions. The Journal ofPhysfcal Chemistry, Vol. 77, No. 27, 7973
On the basis of the variational treatmentll first introduced by Hylleraas12 and Hasse,13 the x x component of the polarizability is expressed in the form cy,,
4nA a0
= --[(xl
-
XI* - ( n - I)(x,
- a)(xl (3)
where x is the coordinate of any one of n equivalence class of electrons which falls in the first equivalence class, f is the average coordinate of any one of these electrons, A is the &function strength determinedg from the reduced electronegativity of the atom, and a0 is the radius of the first Bohr orbit. Since the 6-function wave function does not allow any interaction between the coordinates, (xi f ) ( x z - 3) = 0. The model with the mean &function strength predicts f = 0, so eq 3 becomes
(4) or equivalently
Molecular polarizability consists of parallel and perpendiculai components of the constituent bond polarizabilities. The bond parallel component is obtained from the contributions by the bonding and nonbonding electrons of the valence shell. The contribution of bonding electrons is calculated by using a linear combination of atomic 6-function wave functions representing the nuclei involved in the bond; i.e., the expectation value of electronic position squared (2) along the bond axis is calculated, and this is used to evaluate the parallel component of bond polarizability a ,,b from the equation Qllh
=
4nA,27T2 -[x-]
(6)
a0
where
n is the bond order, R is the internuclear distance a t the equilibrium configuration, and
where ni and N i ( i = 1, 2 ) represent the principal quantum number and the number of electrons making the con-
Bond and Molecular Polarizabilities
2553
taken from the work of Sutton.18 A bond order of 2.5 was used in nitrogen-oxygen (NO2) bonds owing to the coordinate characteristic of the bonding. In order to discuss the computed results, it is worthwhile to consider the classes of molecules tackled pertaining to single, double, and triple rings and chain structures individually. Single-Ring Molecules. In this class, the molecules examined are benzamide, aniline, p-toluidine, toluene, p xylene, mesitylene, durene, hexamethylbenzene, pyridine, The charge density in the bond region then should be reand halogen, nitro, and cyno derivatives of some of the lated to the per cent covalent character, u above molecules. The ndf value for these molecules has c = e-1/4(~,(10) been taken as 18 except in the case of pyridine, where it is 15. Here a contrived but justified consideration has been where X1 and Xz are the electronegativities of the atoms 1 made that only nine bonds forming the ring are taken into and 2, respectively, on the Pauling14 scale. The corrected account in the evaluation of ndf of benzene ( n d f = 3N value of parallel component of bond polarizability is given 2nb = 36 - 18 = 18) and it is assumed that whatever subby stitutions are made on the benzene ring, the n d f value re%p = a l l b g (11) mains constant. I t is evident from the results that in most of the moleThe contribution of nonbonded electrons is calculated cules our calculated values are in reasonable agreement by evaluating the contribution of electrons in the valence with experimental value^.^^-^^ The calculated value of shell of each atom not involved in the bonding. Such calfor toluene (123.091, see Appendix) is in culations are made on the basis of the L e ~ i s - L a n g m u i r ~ ~ polarizability ,~~ agreement with the experimental values reported by Lanoctet rule modified by Linnettl7 in terms of a double dolt-Bornsteinlg (122.6) and LeFevre20 (118.33). quartet of electrons. This can be expressed as Double-Ring Molecules. Molecular polarizabilities of naphthalene and its derivatives have been discussed here. The ndf value of this class has been determined on the where f, is the fraction of the valence electrons in the j t h same considerations as in the case of benzene. In naphatom not involved in the bonding and aj is the atomic thalene, there are 18 atoms and 16 bonds (forming the polarizability ofjth atom. ring) and hence n d f = 54 - 32 = 22. The calculated value The perpendicular component of the bond polarizability (163.94) of naphthalene is in good agreement with the exwas obtained by an empirical approach made by Lippinperimental value (164.6) of L e F e ~ r e . ~Ing the case of decott and Stutman,l which is expressed as rivatives, experimental values are not available for the 3N - 2nb comparison but the calculated values seem to be reasonC 2 N 1 / = -_ c 1 . (13) N I ably accurate. Triple-Ring Molecules. Anthracene and its derivatives where N is the number of atoms and nb is the number of belong to this class where three benzene rings are fused to bonds in the molecule. Taking into account the polarity each other. The n d f value of anthracene has been evalucorrection, the sum of the perpendicular components of ated from the following considerations. ( a ) One benzene the bond polarizability is given as ring has 18 atomic polarizability degrees of freedom. (b) The second ring fused in the benzene ring contributes only four to the ndf value of the resulting molecule, as in or the case of naphthalene. The structure of anthracene shows two side rings attached by a third middle ring. The contribution to the n d f value by the two side rings will be 36 (by “a”) and by the where ndf = (3N - 2nb), the residual atomic polarizabilmiddle ring only 2 (by “b”). Hence the total value of ndf ity degrees of freedom obtainable from the consideration for anthracene will be 38. It is evident from Table I that of symmetry and geometry of the molecular type. the calculated values of molecular polarizabilities of anNow the average molecular polarizability with no bond thracene and its derivatives are in reasonable agreement polarity corrections can be written as with experimental values.30331 Molecular polarizabilities of phenazine and thianthrene have also been calculated. The structures of these molecules are and with bond polarity correction
tribution to the binding, respectively. For a heteronuclear bond, a polarity correction must be made in the parallel component of the bonding electrons to account for the charge density introduced by the electronegativity difference of the atoms. The degree of polarity p defined by Pauling14 is given as = 1 - e-114 (XI- XA2 (9)
x,)’
Equation 17 was used in the present calculation of average molecular polarizabilities. Results and Discussion Calculated values of molecular polarizabilities (in 10-25 cm3) of formamide, acetamide, and the molecules of ring structure are given in Table I. The bond lengths were
These structures may be treated as consisting of two benzene rings attached by two nitrogen and two sulfur atoms, respectively. Hence the n d f value of these molecules will be 36, the contribution of two benzene rings only. In the case of phenazine, the calculated value (234.87) is in excellent agreement with the experimental value (234.3).28 Molecules of Chain Structure. Table I1 represents the molecular polarizabilities (in cm3) of some alkyl The Journal of Physical Chemistry, Vol. 77, No. 2 1 , 1973
2554 TABLE I:
N. K. Sanyal, P. Ahmad, and L. Dixit
Calculated Values of Molecular Polarizabilities of Some Substituted Hydrocarbons
Formamide Acetamide Benzarnide Aniline p-Fluoroaniline p-Chloroaniline p- Bro moan iline p- I odoaniline p-Nitroaniline Difluoroaniline Dichloroaniline Dibromoaniline Diiodoaniline p-Toluidine Dimethylaniline Toluene
47.599 97.807 231.758 208.667 208.622 231.359 241.484 266.665 237.351 208.577 254.052 274.301 324.663 245.416 284.314 224.249
6.918 6.918 6.918 2.772 7.172 14.869 19.602 28.442 13.327 8.400 23.794 33.260 50.940 2.972 2.972 0.000
54.767 63.254 140.818 143.397 135.296 157.068 165.394 176.344 135.629 129.062 169.288 185.599 207.795 142.089 141.1 48 145.023
36.428 55.993 126.495 118.346 1 1 7.030 134.432 142.160 157.150 128.769 115.346 149.044 164.387 194.466 130.159 142.812 123.091
p-Fluorotoluene p-Chlorotoluene p-Brornotoluene p-lodotoluene p-Cyanotoluene p-Nitrotoluene p-Xylene Mesitylene Nitromesitylene Durene Hexamethylbenzene Pyridine Naphthalene 1,5-DichIoronaphthalene 1,5-Dinitronaphthalene @-Naphthol Anthracene 9-Chloroanthracene 9,10-Dichloroanthracene 9-Bromoanthracene 9,lO-Dibrornoanthracene 9-Nitroanthracene 9 , l O-Dinitroanthracene
224.270 246.942 257.067 276.262 261.317 260.507 258.269 300.838 324.334 338.478
4.200 11.897 16.630 25.470 2.970 10.355 0.000 0.000 10.355 0.000
136.862 158.333 166.509 177.256 147.937 136.937 143.478 142.359 136.596 141.512
409.803 169.444 307.307
0.000 2.972 0.000
333.211
cm3)
38.8 53.886 127.466 115.3 115.13 134.96 145.466
151.83
21 21 21 22 23 23 23 ... 23 ... 24
...
... . .
121.755 139.057 146.735 159.663 137,402 135.926 133.916 147.732 157.095 159.999
... 134.666 152.333 122.6 118.33 1 1 7.0 137.0 148.0 171 .O 139.0 141.0 137.0 153.76 ... 174.0
140.314 121.115 184.520
184.700 97.843 163.942
208.1 91.8 164.6
27 28 29
23.794
209.332
188.779
...
...
363.506 337.001 435.958
20.710 3.946 0.000
175.162 178.476 322.486
186.459 173.141 252.800
...
...
... 253.6
... 30
470.212
1 1 .897
339.188
273.766
273.466
30
496.775
23.794
354.802
291.790
...
...
495.030
16.630
349.772
287.144
536.179
33.260
375.706
315.048
...
...
485.040
10.355
308.045
267.813
...
...
509.772
20.710
298.030
276.171
...
...
474.245 442.458 487.267 396.690 465.768
2.972 7.893 3.946 5.944 24.266
325.126 312.103 316.574 301.992 343.011
267.447 254.152 269.262 234.870 277.682
...
139.0
...
283.2
23 24 19 20 25 25 25 25 25 25 26 27 ...
27
30
9-Cyano-
anthracene Anthraquinone Anthraldehyde Phenazine Thianthrene
bromides, calculated with the aid of the &function model, and those reported by LeFevre, et al.32 The ndf values of this series have been determined by the same relation (ndf = 3N - 2nb), which varies from 7 to 58. Calculated values of substituted n-alkanes show a continuous increasing deviation from the LeFevre values. This deviation increases regularly by approximately 2 units for the addition of each CH2 group. The causes of this discrepancy are not clear. It may be either due to the weakness of this model to take The Journal of Physical Chemistry, Vol. 77, No. 21, 1973
283.2 ... ...
31 ...
...
234.3
28
...
...
into account the increased interaction with lengthening of the hydrocarbon chain or inaccuracies in experimental values. Further experimental work is needed to settle this point. Table III shows the bond parallel component of polarizabilities with bond polarity corrections. Results show that this component may be transferred from one molecular system to another having similar chemical bonds. In nitroanthracene, the value of bond parallel component for
2555
Bond and Molecular Polarizabilities
Acknowledgment. The authors are thankful to Professor
TABLE II: Molecular Polarizabilities of Bromine-Substituted n-Alkanes cm3)
70.128 1 08.349 11 45.427
1' 83.076
220.720 258.375 2 96.025 333.674 371.324 408.973 446.623 fl84.272 634.870 7 10.1 69
16.630 16.630 16.630 16.630 16.630 16.630 16.630 16.630 16.630 16.630 16.630 16.630 16.630 16.630
74.719 95.532 117.262 139.338 161.582 183.919 206.314 228.747 251.206 273.684 296.1 77 318.681 408.766 453.834
53.826 73.503 93.106 113.015 132.978 152.974 172.989 193.014 213.053 263.096 253.143 273.161 353.422 393.544
D. Sharma for his continued interest in the project. Financial assistance received by two of us (P. A. and L. D.) 55.3 72.8 90.7 108.6 126.5 144.4 162.3 180.2 198.1 216.0
... 251 .a 323.4 359.2
from the University Grants Commission and Council of Scientific and Industrial Research, New Delhi, in the form of research fellowships is gratefully acknowledged.
Appendix Sample calculations of bond and molecular polarizabilities for toluene are given here. O
= 0.337257 X 10-16 cm2
Reference 32
the C-N bond is 19.55 and in cyanoanthracene the value for the C=N bond is 21.89, which is in accordance with the observation that x and u electrons do not contribute equally to the bond parallel component of polarizability and that the contribution of electrons in the x orbital is less than those of the electrons in the orbital. Individual contribution of the x and u orbitals to the polarizability can possibly be estimated from the calculations based on molecular orbital theory. Further investigations can only provide this information. The excellent agreement obtained between the calculated and exper>imentalvalues indicates the wide applicability of the &function model of chemical binding to complicated molecular systems such as ring structures and complex metal carbonyl, 33,34 etc. TABLE Ill: Bond Parallel Component of Polarizabilities Molecules
Formamide Acetamide Benzamide Aniline p-Nitroaniline p-Toluidine Toluene p-Cyanotoluene p-Nitrotoluene p-Xylene Mesitylene Nitromesitylene Pyridine Naphthalene 1,5-Dinitronaphthalene Anthracene 9-Nitroanthracene 9 , l O-Dinitroanthracene 9-Cyanoanthracene Anthraquinone Anthraldehyde Phenazine a
3
C-H=1.11A,(C-C),=1.39A,C-C=1.52A,AH=1.00au, A c = 0.846 au, (YH = 5.92 x 10-25 cm3, cyc = 9.78 x 1 0 - 2 5 cm3, XH = 2.1, X C = 2.5, and ndf = 18. For the C-H bond R2
a
H
(C-C),"
c-c
... ...
15.031 16.134 15.272 15.948 ici.688 15.688 15.688 16.134 16.134 15.275 15.866 15.882 16.589 16.092 16.603 16.006 16.304 16.561 16.808 16.128
38.218 20.048
...
... 20.905 22.254 22.254 22.254 20.964 23.423 21.500
... ... ...
... ... ... 17.526
... 19.009 ...
7.9106 X
cm3
= (7.9106)(0.960789)(10-25) = 7.600417 x For 8 C-H bonds a = 60.8032 X lO-25cm3 Ly
=
p
bu
cm3
For the (C-C), bond f 2 = 0.495218 X cm2 (4)(1)(0.846)(0.245241) 10-24 cm3 = = 0.529 15.6879 X Ly
,p
- Ly
= (15.6879)(1) = 15.6879 x
bL7
cm3 Cm3
cm3) C-H
C-N
7.208 7.113 7.601 7.601 7.601 7.601 7.601 7.601 7.601 7.113 7.113 7.113 6.902 6.649
13.362 14.036 12.123 15.724 15.307 18.045
6.649 6.649 6.649 6.649 6.649 6.649 6.649 5.226
... ...
C-N
... ... ... t
.
.
c=o
N-H
16.876 14.625 16.716 ...
5.076 4.794 4.794 4.863 4.863 4.863
...
...
,..
... ... ...
... 22.414
...
20.081
... ...
N-0
... ... ... ...
11.662
... ...
9.51 1 ...
...
...
19.556 13.245
...
...
, . .
13.636
...
...
...
...
...
9.823
...
... 6.182
...
...
19.556
9.511
18.045
...
...
...
... ...
21.889
...
...
...
... ...
12.454 15.190
...
... 13.460
...
... ...
...
...
...
10.471 ...
Represents the carbon-carbon bond of ring structure.
The Journalof Physical Chemistry, Vol. 77, No. 21, 7973
2556
P. Gallezot, Y. Ben Taarit, and B. lrnelik
For 9 (C-C), bonds a l = 141.1911 X
(8) H. J. Kolker and M. Karplus, J. Chem. Phys., 39, 201 1 (1963). (9) E. R. Lippincott and M. 0. Dayhoff, Spectrochim. Acta, 18, 807 (1960). (10) A. A. Frost, J. Chem. Phys., 22, 1613 (1954); 23, 985 (1955); 25, 1150, 1154 (1956). (11) J. W. Hirschfeider, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gasses and Liquids," Wiley, New York, N. Y . , 1967, pp 942-946. (12) E. A. Hylleraas, Z. Phys., 65, 209 (1930). (13) H. R. Hasse, Proc. Cambridge Phii SOC., 26, 542 (1930); 27, 66 (1931), (14) L. Pauling, "The Nature of the Chemical Bond," Cornell University Press, Ithaca, N. Y., 1960. (15) G. N. Lewis, J. Amer. Chem. SOC.,38, 762 (1916). (16) I. Langmuir, J. Amer. Chem. SOC.,38, 2221 (1916). (17) J. W. Linnett, J. Amer. Chem. SOC.,83, 2643 (1961). (18) L. E. Sutton, Chem. SOC. Spec. Pub/., No. 11 (1958); No. 18 (1965). (19) H. H. Landolt and R. Bornstein, "Zahlenwerte and Functionen," 6 Auflage, 3 Teil, Springer-Verlag, Berlin, 1950, pp 509-517. (20) R. J. W. LeFevre and L. Radom, J. Chem. SOC.6,1295 (1967). (21) M. J. Aroney, R. J. W. LeFevre, and A. N. Singh, J. Chem. SOC., 3179 (19651. (22) M . j.Aroney, R. J. W. LeFevre, L. Radom, and G. L. D. Ritchie, J. Chem. SOC.6,507 (1968). 123) M. J. Aronev. K. E. Calderbank. R. J. W. LeFevre. and R. K . Pi~, erens, J. Chem. SOC.6,~561(1968). (24) R. J. W. LeFevre, L. Radom, and G. L. D. Ritchie, d. Chem. SOC. 6, 913 (1969). (25) M. J. Aroney, K. E. Calderbank, R. J. W. LeFevre, and R. K. Pierens, J. Chem. SOC.6,159 (1969). (26) K. E. Calderbank, R. J. W. LeFevre, and R. K. Pierens, J. Chem. SOC.8, 968 (1969). (27) M. J. Aroney, H. H. Huang, R J. W. LeFevre, and G. L D. Ritchie, J. Chem. Soc. 6 ,416 (1966). (28) J. Hurley and R. J. W. LeFevre, J. Chem. Soc. 6, 824 (1967). (29) C. G. LeFevreand R. J. W. LeFevre, J. Chem. SOC., 1641 (1955). (30) R. J. W. LeFevre, L. Radom, and G. L. D. Ritchie, J. Chem. SOC. 6.775 (1968). (31) P', H. Gore, J. A. Hoskins, R. J. W. LeFevre, L. Radorn, and G . L. D. Ritchie, J. Chem. SOC.6, 227 (1969). (32) C. G. LeFevre, R. J. W. LeFevre, and A. J. Williams, J. Chem. SOC.,4188 (1965). (33) G. Nagarajan and M. W. Lindauer, Phys. Rev. A, 5 , 557 (1972). (34) N. K. Sanyal, P. Ahmad, and L.Dixit, paper presented at the First International Conference on Quantum Chemistry held at Menton, France, July 1973.
cm3
For the C-C bond
9 = 0.589818 X
CY
=
bC
cm2
= (22.2541)(1) = 22.2541 x
io-25 cm3
Total bond parallel polarizability of toluene is Za = 224.2485 X 10-25cm3 Za.=O (There are no nonbonding electrons in C-H and C-C bonds.)
\
References a n d Notes (1) (2) (3) (4) (5) (6) (7)
E. R. Lippincott and J. M. Stutman, J. Phys. Chem., 88, 2926 (1964). E. R. Lippincott, G. Nagarajan, and J. M. Stutman, J. Phys. Chem.. 70, 78 (1966). G. Nagarajan, Z. Naturforsch. A, 21, 864 (1966). N. K. Sanyal, L. Dixit, and A. N. Pandey, lndian J. Pure Appl. Phys.. 10, 329 (1972). J. A. Beran and L. Kevan, J. Phys. Chem., 73, 3860 (1969). B. Purnachandra Rao and V. Ramamurthy, Curr. Sci., 41, 15 ( 1972), R. M. Stevens, R. M. Pitzer, and W. N. Lipscomb, J. Chem. Phys., 38, 550 (1963); R. M. Stevens and W. N. Lipscomb, ibid., 40, 2238 (1964); 41, 184, 3710 (1964).
- - - I
Location of Nickel Ions in Y Zeolites. II. Influence of Various Reagents Adsorption on Nickel Positions P. Gallezot, Y. Ben Taarit, and B. Imelik" institut de Recherches sur ia Catalyse CNRS, 69,700 Villeurbanne, France (Received March 22 1973) Publication costs assisted by the Centre National de la Recherche Scientihque
The crystal structures of NiY zeolite containing various sorbed phases ("3, NO, C5HbN, CloH8, C4H8, C2H4, CO) were determined in order to localize the Ni2+ ions. Neither C10H8, C4H8, and C2H4 nor CO changes the distribution of Ni2+ ions previously determined for the dehydrated zeolite. NH3 can readily extract the cations from the SI sites. On the other hand, NO and C5H5N generate only a slow migration process favored by the presence of residual water molecules. The structure of Ni-NO complex is discussed and a mechanism for the cation motion is proposed. -
Introduction In order to understand the behavior and catalytic activity of zeolites better, the positions of the cations in the framework have to be known. However, it is not sufficient to determine these positions after a given pretreatment since the cations may migrate under the influence of adThe Journal of Physical Chemistry, Voi. 77,
No. 27, 7973
sorbates or reagents, as was shown in a previous investigation for copper-exchanged zeolites.1 After dehydration, Cu2+ ions are mainly found in SI, sites but they migrate very readily toward the supercages if large molecules such as pyridine, naphthalene, or butene are admitted into the solid. In part I of this work,2 nickel ion positions were determined in Y zeolites as a function of both exchange level
