J. Chem. Inf. Comput. Sci. 1998, 38, 1171-1176
`
`1171
`
`Correlation and Prediction of the Refractive Indices of Polymers by QSPR
`
`Alan R. Katritzky,*,§ Sulev Sild,§,‡ and Mati Karelson*,‡
`
`Center for Heterocyclic Compounds, University of Florida, Gainesville, Florida 32611-7200, and Institute of
`Chemical Physics, University of Tartu, 2 Jakobi Street, EE 2400 Tartu, Estonia
`
`Received May 25, 1998
`
`A general QSPR model (R2 ) 0.940, s ) 0.018) was developed for the prediction of the refractive index for
`a diverse set of amorphous homopolymers with the CODESSA program. The five descriptors, involved in
`the model, are calculated from the structure of the repeating unit of the polymer. The average prediction
`error by this model is 0.9%.
`
`INTRODUCTION
`
`The refractive index n is a basic optical property of
`polymers that is directly related to other optical, electrical,
`and magnetic properties. The refractive index is also widely
`used in material science. The specific refractive index
`increment (dn/dc) is an important parameter in light scattering
`measurements of dilute polymer solutions, which can be used
`for the determination of molecular weight, size, and shape.1
`Importantly, the refractive index can indicate the potential
`of a polymer for a specific purpose. A satisfactory quantita-
`tive structure-property relationship (QSPR) that would allow
`quantitative prediction of the refractive index of as yet
`unsynthesized polymers would clearly be of significant
`utility.
`In principle, combining the QSPR method with
`pattern recognition techniques should make possible the
`theoretical prediction of structures with desired property
`values.
`Theoretical methods for calculating the refractive indices
`of polymers generally utilize equations formulated by (i)
`Lorentz and Lorentz (eq 1) and (ii) Gladstone and Dale (eq
`2). Both approaches require the availability (or theoretical
`estimation) of molar refraction and molecular volume data.
`A good summary of early attempts to estimate the molar
`refraction of polymers using group contributions was pro-
`vided by Krevelen.1 In a recent review, Askadskii2 proposed
`several semiempirical equations for the calculation of various
`physical properties of polymers and copolymers (with
`accuracy usually within 3-5%). The calculation of refrac-
`tive index is based on eq 1, where the molecular refraction
`(RLL) is calculated as a sum of corresponding atom and bond
`contributions, and the volume (V) is estimated as a van der
`Waals volume of the compound divided by the average
`coefficient of molecular packing.
`) n2 - 1
`(cid:226)V
`n2 + 2
`) (n - 1)(cid:226)V
`
`RLL
`
`RGD
`
`(1)
`
`(2)
`
`can be easily performed provided all the necessary incre-
`ments are known from the experimental data for every
`structural element. However, interactions between functional
`groups can introduce significant errors in predicted refractive
`index values. Agrawal and Jenekhe3 demonstrated that the
`refractive index of (cid:240)-conjugated polymers predicted by
`existing group contribution methods can have deviations from
`experimental values as high as 22%. The source of these
`discrepancies is believed to be large optical dispersion and
`(cid:240)-electron delocalization effects in conjugated polymers. To
`overcome this problem, Yang and Jenekhe4 developed new
`Lorentz and Lorentz molar refraction group contributions for
`24 functional groups commonly found in conjugated poly-
`mers. They successfully used these new RLL data to calculate
`the refractive indices of 33 conjugated polymers (with an
`average error of 0.9%).4
`Some of the shortcomings and limitations of group
`contribution methods can be avoided by using the theoretical
`QSPR approach. The quantum-chemical descriptors used
`in this approach encode information about the electronic
`structure of the molecule and thus implicitly account for the
`cooperative effects between functional groups, charge re-
`distribution, and possible hydrogen bonding in the polymer.
`The only previously published QSPR relationship for the
`prediction of refractive index was developed by Bicerano
`(R2 ) 0.955) for a set of 183 polymers, with 10 descriptors
`involved.5 These descriptors included three different topo-
`logical indices, the total number of rotational degrees of
`freedom (both of the polymer backbone and the side groups),
`and several constitutional descriptors such as the number of
`fluorine atoms, the number of chlorine atoms bonded to an
`aromatic ring, the number of sulfur atoms, and the number
`of hydrogen bonding moieties, etc. Alternative topological
`descriptors for polymers have been developed in the frame-
`work of the topological extrapolation method (TEM) by
`Mekenyan et al.6 and used to calculate the refractive index
`within a homologous series of polymers.
`The QSPR method has already been applied in the
`framework of the CODESSA program7 to predict success-
`fully various physical properties of low molecular weight
`compounds; early examples were summarized in our review,8
`for later examples see refs 9-11. This approach was
`extended to calculate appropriate descriptors for the repeating
`
`The main advantage of using the group contributions
`method is its simplicity. Prediction with reasonable accuracy
`
`§ Center for Heterocyclic Compounds, University of Florida.
`‡ University of Tartu.
`
`10.1021/ci980087w CCC: $15.00 © 1998 American Chemical Society
`Published on Web 10/22/1998
`
`BOREALIS EXHIBIT 1090
`
`Page 1 of 6
`
`

`
`1172 J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998
`
`KATRITZKY ET AL.
`
`Figure 1. The plot of the best five parameter correlation for refractive index.
`
`units of polymers which were subsequently used to develop
`correlations for the glass transition temperatures of poly-
`mers.12,13 For a set of 22 relatively low molecular weight
`fluorinated polymers and copolymers, the glass transition
`temperatures were correlated with four descriptors (R2 )
`0.928).12 Glass transition temperatures for a structurally
`diverse data set of 88 high molecular weight homopolymers
`were described by a five descriptors model (R2 ) 0.946).13
`The refractive indices of a diverse set of 125 common
`low molecular weight organic compounds were successfully
`correlated by the CODESSA approach in a general QSPR
`relationship (R2 ) 0.945).14 Five descriptors were involved
`in this model: HOMO-LUMO energy gap, quantum-
`chemically (AM1 method) calculated lowest (absolute value)
`electron-nucleus attraction energy for a carbon atom, total
`charge-weighted partial positively charged surface area,
`surface area of hydrogen donor atoms, and gravitation index
`(calculated over all bonds).
`In the present study, a new
`QSPR relationship is developed for the refractive indices of
`a diverse set of polymers. The descriptors selected for this
`polymer data set are then compared with the descriptors
`selected in our previous study14 for correlation of the
`refractive indices of low molecular weight compounds.
`
`METHODOLOGY
`
`The refractive index data for 95 essentially amorphous
`polymers, measured at room temperature (298 K), were taken
`from a published compilation (Table 1).5 The polymers
`
`chosen for the data set cover a wide range of refractive index
`values and represent a diverse set of chemical structures.
`The majority of the polymers fall
`into the classes of
`homochain polymers (only carbon atoms in the main chain)
`and polyoxides, but several polyamides and polycarbonates
`were also included. The data set contained large subsets of
`polyethylenes, polyacrylates, polymethacrylates, polysty-
`renes, polyethers, and polyoxides. The entire set was
`characterized by a high degree of structural variety; the
`functionalities represented in the side chains include halides,
`cyanides, carboxylates, acetates, amides, ethers, alcohols,
`hydrocarbon chains, aromatic, and nonaromatic rings.
`For high molecular weight polymers it is at best extremely
`difficult to calculate descriptors directly. Instead, we used
`the repeating unit end-capped by hydrogen atoms as the small
`representative model structure. All polymer chains were
`assumed to be terminated by a hydrogen atom.
`The three-dimensional structure of the repeating unit for
`each polymer was drawn and preoptimized using the PC-
`MODEL program.15 The preoptimized structures were then
`fully optimized with the semiempirical AM1 method16 using
`the MOPAC 6.0 program17 to obtain the necessary quantum-
`chemical descriptors for the further calculations. More than
`800 constitutional, topological,18 geometrical, and quantum
`chemical19 descriptors were calculated for the repeating unit
`from the results of the semiempirical calculations using the
`CODESSA (COmprehensive DEscriptors for Structural and
`Statistical Analysis)7 program.
`
`Page 2 of 6
`
`

`
`REFRACTIVE INDICES OF POLYMERS BY QSPR
`
`J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998 1173
`
`Table 1. Experimental and Calculated Refractive Index Values
`compound
`
`representative structure
`
`poly(ethylene)
`poly(acrylic acid)
`poly(methyl acrylate)
`poly(ethyl acrylate)
`poly(vinyl alcohol)
`poly(vinyl chloride)
`poly(acrylonitrile)
`poly(vinyl acetate)
`poly(styrene)
`poly(2-chlorostyrene)
`poly(2-methylstyrene)
`poly(propylene)
`poly(ethoxyethylene)
`poly(n-butyl acrylate)
`poly(vinyl hexyl ether)
`poly(1,1-dimethylethylene)
`poly(methyl methacrylate)
`poly(ethyl methacrylate)
`poly(isopropyl methacrylate)
`poly(2-chloroethyl methacrylate)
`poly(phenyl methacrylate)
`poly(tetrafluoroethylene)
`poly(chlorotrifluoroethylene)
`poly(oxymethylene)
`poly(oxyethylene)
`poly((cid:15)-caprolactam)
`poly(ethylene terephthalate)
`poly(vinyl n-octyl ether)
`poly(vinyl n-decyl ether)
`poly(vinyl n-pentyl ether)
`poly(vinyl 2-ethylhexyl ether)
`poly(vinyl n-butyl ether)
`poly(vinyl isobutyl ether)
`poly(vinyl sec-butyl ether)
`poly(isobutyl methacrylate)
`poly(n-hexyl methacrylate)
`poly(n-butyl methacrylate)
`poly(4-methyl-1-pentene)
`poly(vinyl chloroacetate)
`poly(n-propyl methacrylate)
`poly[oxy(2,6-dimethyl-1,4-phenylene)]
`poly(p-xylylene)
`poly(vinyl butyral)
`poly(vinyl benzoate)
`poly(N-vinylpyrrolidone)
`poly[oxy(methylphenylsilylene)]
`poly(vinylidene fluoride)
`poly(trifluoroethyl acrylate)
`poly(2,2,2-trifluoro-1-methylethyl methacrylate)
`poly(trifluoroethyl methacrylate)
`poly(N-methyl methacrylamide)
`poly(N-vinylcarbazole)
`poly(R-vinylnaphthalene)
`poly(styrene sulfide)
`poly(pentabromophenyl methacrylate)
`poly(phenyl R-bromoacrylate)
`poly(2,6-dichlorostyrene)
`poly(chloro-p-xylylene)
`poly((cid:226)-naphthyl methacrylate)
`poly(sec-butyl R-bromoacrylate)
`poly(2-bromoethyl ethacrylate)
`poly(methyl R-bromoacrylate)
`poly(ethylmercaptyl methacrylate)
`poly(benzyl methacrylate)
`poly[oxy(methyl-n-hexylsilylene)]
`poly(propylene oxide)
`poly(3-butoxypropylene oxide)
`poly(3-hexoxypropylene oxide)
`poly(4-fluoro-2-trifluoromethylstyrene)
`poly(propylene sulfide)
`poly(p-bromophenyl methacrylate)
`poly(vinylidene chloride)
`poly(pentachlorophenyl methacrylate)
`
`HCH2CH2H
`HCH2CH(COOH)H
`HCH2CH(COOMe)H
`HCH2CH(COOEt)H
`HCH2CH(OH)H
`HCH2CH(Cl)H
`HCH2CH(CN)H
`HCH2CH(OCOMe)H
`HCH2CH(C6H5)H
`HCH2CH(C6H4Cl)H
`HCH2CH(C6H4Me)H
`HCH2CH(Me)H
`HCH2CH(OEt)H
`HCH2CH(COOC4H9)H
`HCH2CH(OC6H13)H
`HCH2C(Me)2H
`HCH2C(Me)(COOMe)H
`HCH2C(Me)(COOEt)H
`HCH2C(Me)(COOCH(Me)2)H
`HCH2C(Me)(COOC2H4Cl)H
`HCH2C(Me)(COOC6H5)H
`HCF2CF2H
`HCFClCF2H
`HOCH2H
`HOCH2CH2H
`H(CH2)5C(O)NHH
`H(CH2)2OC(O)C6H4COOH
`HCH2CH(OC8H17)H
`HCH2CH(OC10H21)H
`HCH2CH(OC5H11)H
`HCH2CH(OCH2CH (Et)(C4H9))H
`HCH2CH(OC4H9)H
`HCH2CH(OCH2CH(Me)2)H
`HCH2CH(OCH(Me)(Et))H
`HCH2C(Me)(COOCH2CH(Me)2)H
`HCH2C(Me)(COOC6H13)H
`HCH2C(Me)(COOC4H9)H
`HCH2C(CH2CH(Me)2)H
`HCH2CH(OC(O)CH2Cl)H
`HCH2C(Me)(COOC3H7)H
`HOC6H2(Me)2H
`HCHdCHC6H4H
`HCH2CH(OC(O)C3H7)H
`HCH2CH(OC(O)C6H5)H
`HCH2CH(NC4OH6)H
`HOSi(Me)(C6H5)H
`HCH2CF2H
`HCH2CH(COOCH2CF3)H
`HCH2CH(Me)(COOCH(Me)CF3)H
`HCH2C(Me)(COOCH2CF3)H
`HCH2C(Me)(CONMe)H
`HCH2CH(NC12H8)H
`HCH2CH(C10H9)H
`HSCH2CH(C6H5)H
`HCH2C(Me)(C6Br5)H
`HCH2C(Br)(COOC6H5)H
`HCH2CH(C6H3Cl2)H
`HCHdCHC6H4ClH
`HCH2C(Me)(COOC10H9)H
`HCH2C(Br)(COOCH(Me)(Et))H
`HCH2C(Et)(COOC2H4Br)H
`HCH2C(Br)(COOMe)H
`HCH2C(Me)(COSEt)H
`HCH2C(Me)(COOCH2C6H5)H
`HOSi(Me)(C6H13)H
`HOCH(Me)CH2H
`HOCH(CH2OC4H9)CH2H
`HOCH(CH2OC6H13)CH2H
`HCH2CH(C6H3F(CF3))H
`HSCH(Me)CH2H
`HCH2C(Me)(COOC6H4Br)H
`HCH2CCl2H
`HCH2C(Me)(COOC6Cl5)H
`
`exp. n
`
`1.4760
`1.5270
`1.4790
`1.4685
`1.5000
`1.539
`1.5200
`1.4670
`1.5920
`1.6098
`1.5874
`1.4735
`1.4540
`1.4660
`1.4591
`1.5050
`1.4893
`1.4850
`1.4728
`1.5170
`1.5706
`1.3500
`1.3900
`1.4800
`1.4563
`1.5300
`1.5750
`1.4613
`1.4628
`1.4590
`1.4626
`1.4563
`1.4507
`1.4740
`1.4770
`1.4813
`1.4830
`1.4650
`1.5130
`1.4840
`1.5750
`1.6690
`1.4850
`1.5775
`1.5300
`1.5330
`1.4200
`1.4070
`1.4185
`1.4370
`1.5398
`1.6830
`1.6818
`1.6568
`1.7100
`1.6120
`1.6248
`1.6290
`1.6298
`1.5420
`1.5426
`1.5672
`1.5470
`1.5679
`1.4450
`1.4570
`1.4580
`1.4590
`1.4600
`1.5960
`1.5964
`1.6000
`1.6080
`
`calcd n
`
`1.4714
`1.4927
`1.4852
`1.4830
`1.4731
`1.5353
`1.5405
`1.5001
`1.5988
`1.6184
`1.5946
`1.4791
`1.4818
`1.4760
`1.4670
`1.4738
`1.4852
`1.4811
`1.4804
`1.5093
`1.5779
`1.3379
`1.4249
`1.4730
`1.4752
`1.5112
`1.5621
`1.4598
`1.4516
`1.4689
`1.4630
`1.4746
`1.4736
`1.4783
`1.4927
`1.4664
`1.4740
`1.4724
`1.5251
`1.4757
`1.5961
`1.6486
`1.5104
`1.5786
`1.5361
`1.5827
`1.4023
`1.4061
`1.4069
`1.4104
`1.5211
`1.6816
`1.6500
`1.6337
`1.7009
`1.6271
`1.6206
`1.6208
`1.6274
`1.5311
`1.5300
`1.5400
`1.5525
`1.5702
`1.4779
`1.4736
`1.4548
`1.4562
`1.4825
`1.5674
`1.6024
`1.5688
`1.6555
`
`¢n
`0.0046
`0.0343
`0.0062
`-0.0145
`0.0269
`0.0037
`-0.0205
`-0.0331
`-0.0068
`-0.0086
`-0.0072
`-0.0056
`-0.0278
`-0.0100
`-0.0079
`0.0312
`0.0041
`0.0039
`-0.0076
`0.0077
`-0.0073
`0.0121
`-0.0349
`0.0070
`-0.0189
`0.0188
`0.0129
`0.0015
`0.0112
`-0.0099
`-0.0004
`-0.0183
`-0.0229
`-0.0043
`-0.0157
`0.0149
`0.0090
`-0.0074
`-0.0121
`0.0083
`-0.0211
`0.0204
`-0.0254
`-0.0011
`-0.0061
`-0.0497
`0.0177
`0.0009
`0.0116
`0.0266
`0.0187
`0.0014
`0.0318
`0.0231
`0.0091
`-0.0151
`0.0042
`0.0082
`0.0024
`0.0109
`0.0126
`0.0272
`-0.0055
`-0.0023
`0.0249
`-0.0166
`0.0032
`0.0028
`-0.0225
`0.0286
`-0.0060
`0.0312
`-0.0475
`
`Page 3 of 6
`
`

`
`1174 J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998
`
`KATRITZKY ET AL.
`
`Table 1 (Continued)
`
`compound
`poly(N-benzyl methacrylamide)
`poly(trifluorovinyl acetate)
`poly(tert-butyl-methacrylate)
`poly(vinyl methyl ether)
`poly(3,3,5-trimethylcyclohexyl methacrylate)
`poly(3-methylcyclohexyl methacrylate)
`poly(4-methylcyclohexyl methacrylate)
`poly(ethyl R-chloroacrylate)
`poly(N-methylmethacrylamide)
`poly(methyl R-chloroacrylate)
`poly(1,3-dichloropropyl methacrylate)
`poly(cyclohexyl R-bromoacrylate)
`poly(1-phenylethyl methacrylate)
`poly(2,3-dibromopropyl methacrylate)
`poly(o-chlorobenzyl methacrylate)
`poly(o-methoxystyrene)
`poly(p-methoxystyrene)
`poly(ethylene succinate)
`poly(vinyl formate)
`poly(2-fluoroethyl methacrylate)
`poly(cyclohexyl R-chloroacrylate)
`poly(2-bromoethyl methacrylate)
`
`representative structure
`HCH2C(Me)(CONHCH2C6H5)
`HCF2CF(OC(O)Me)H
`HCH2C(Me)(COOC(Me)3)H
`HCH2CH(OMe)H
`HCH2C(Me)(COOC9H17)H
`HCH2C(Me)(COOC7H13)H
`HCH2C(Me)(COOC7H13)H
`HCH2C(Cl)(COOEt)H
`HCH2C(Me)(CONMe)H
`HCH2C(Cl)(COOMe)H
`HCH2C(Me)(COOC3H5Cl2)H
`HCH2C(Br)(COOC6H11)H
`HCH2C(Me)(COOCH(C6H4)Me)H
`HCH2C(Me)(COOC3H5Br2)H
`HCH2C(Me)(COOCH2C6H4Cl)H
`HCH2CH(C6H4OMe)H
`HCH2CH(C6H4OMe)H
`HCH2CH(OC(O)C2H4COOH)H
`HCH2CH(OC(O)H)H
`HCH2C(Me)(COOC2H4F)H
`HCH2C(Cl)(COOC6H11)H
`HCH2C(Me)(COOC2H4Br)H
`
`exp. n
`1.5965
`1.3750
`1.4638
`1.4670
`1.4850
`1.4947
`1.4975
`1.5020
`1.5135
`1.5170
`1.5270
`1.5420
`1.5487
`1.5739
`1.5823
`1.5932
`1.5967
`1.4744
`1.4757
`1.4768
`1.5320
`1.5426
`
`calcd n
`1.5918
`1.3891
`1.4773
`1.4816
`1.4810
`1.4804
`1.4815
`1.5184
`1.5119
`1.5222
`1.5343
`1.5331
`1.5596
`1.5618
`1.5810
`1.5821
`1.5881
`1.4670
`1.4962
`1.4582
`1.5085
`1.5280
`
`¢n
`0.0047
`-0.0141
`-0.0135
`-0.0146
`0.0040
`0.0143
`0.0160
`-0.0164
`0.0016
`-0.0052
`-0.0073
`0.0089
`-0.0109
`0.0121
`0.0013
`0.0111
`0.0086
`0.0074
`-0.0205
`0.0186
`0.0235
`0.0146
`
`Table 2. Best Five Parameter Correlation for Refractive Indexa
`¢X
`X
`1.000
`0.118E-01
`0.574E-03
`0.167E-01
`0.477E-03
`-0.260
`
`0.154E-02
`0.362E-04
`0.513E-03
`0.480E-04
`0.298E-01
`
`t-test
`
`-7.682
`15.881
`32.556
`9.939
`-8.740
`
`a (R2 ) 0.940, F ) 282.13, and s2 ) 0.000 313).
`
`descriptor
`
`intercept
`HOMO-LUMO energy gap
`AM1 heat of formation
`max nuclear repulsion for a C-H bond
`partial negative surface area [Zefirov’s PC]
`relative number of F atoms
`
`The QSPR models were developed using both the heuristic
`and the best multilinear regression analysis methods avail-
`able in the framework of the CODESSA program.9 In both
`cases, a preselection of descriptors was carried out, by
`removing descriptors having an essentially constant value
`for all structures. The final QSPR model was selected on
`the basis of the highest correlation coefficient (R2), the lowest
`standard error, and the relevance of involved descriptors to
`refractive index as a physical phenomena. Another important
`criteria for the model selection was the intercept value, since
`the refractive index in a vacuum is unity. Assuming that
`all the descriptors involved in the QSPR model have zero
`values in a vacuum, the intercept of the respective (multi)-
`linear relationship should be determined by the refractive
`index of a vacuum. Therefore, we also used a modified best
`multilinear regression analysis program that fixed the
`intercept value to one during regression analysis. The
`stability of every potential model was tested against the cross-
`2 describesvalidated correlation coefficient (RCV2 ). The RCV
`
`
`the stability of a regression model obtained by focusing on
`the sensitivity of the model to the elimination of any single
`data point.
`
`RESULTS AND DISCUSSION
`
`The final QSPR model with a correlation coefficient of
`0.940 was developed from a preselected pool of more than
`655 CODESSA descriptors. The model consisted of four
`quantum-chemical descriptors and one constitutional descrip-
`tor as follows: (i) HOMO-LUMO energy gap, (ii) AM1
`calculated heat of formation, (iii) maximum nuclear repulsion
`
`for a C-H bond, (iv) partial negative surface area (PNSA)
`calculated from Zefirov’s partial charges, and (v) the relative
`number of F atoms (for details, see Table 2).
`The HOMO-LUMO energy gap is defined as the energy
`difference between highest occupied molecular orbital
`(HOMO) and the lowest unoccupied molecular orbital
`(LUMO). The refractive index and the HOMO-LUMO
`energy gap are both related to the polarizability of the
`molecule. A small difference between HOMO and LUMO
`energies usually means that the molecule is easily polarized.
`This particular descriptor was also involved in our previous
`QSPR treatment of low molecular organic compounds.
`Herve et al. showed that an empirical relationship exists
`between the refractive index and the energy gap in semi-
`conductors.20
`formation reflects the
`The AM1 calculated heat of
`thermodynamic stability of the polymer.
`Its emergence in
`the correlation equation for the refractive index is possibly
`connected with the “looseness” of the electrons in the
`molecule that is interacting with the electromagnetic radia-
`tion. The positive value of the corresponding regression
`coefficient (cf. Table 2) indicates that the compounds that
`are less stable (higher heats of formation) possess higher
`refractive indices. Apparently, the electronic distribution in
`these molecules is, on average, more flexible to interact with
`light.
`The maximum nuclear repulsion for a C-H bond is the
`maximal nuclear repulsion energy (Enn) between a pair of
`bonded carbon and hydrogen nuclei. This nuclear repulsion
`energy is calculated by eq 3, where Z is the nuclear (core)
`
`Page 4 of 6
`
`

`
`REFRACTIVE INDICES OF POLYMERS BY QSPR
`
`J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998 1175
`
`Table 3. Descriptor Coefficients Calculated for Three Subsets
`sets I, II
`sets II, III
`1.000
`1.000
`-0.120E-01
`-0.118E-01
`0.586E-03
`0.570E-03
`0.168E-01
`0.165E-01
`0.458E-03
`0.484E-03
`-0.2320
`-0.255
`
`sets I, III
`1.000
`-0.127E-01
`0.548E-03
`0.170E-01
`0.431E-03
`-0.2471
`
`descriptor
`
`intercept
`HOMO-LUMO energy gap
`AM1 heat of formation
`max nuclear repulsion for a C-H bond
`partial negative surface area [Zefirov’s PC]
`relative number of F atoms
`
`charge and the R is the distance between the carbon and
`hydrogen atoms. This descriptor depends on the reciprocal
`of the C-H bond length and thus possibly encodes the
`information about the hybridization of the carbon atoms,
`since the carbon-hydrogen bond length varies depending
`whether the carbon atom is in the sp3, sp2, or sp hybridization
`state.
`
`Enm(CH) ) ZCZH
`RCH
`
`(3)
`
`The partial negative surface area (PNSA) is an electro-
`statical descriptor calculated from the Zefirov’s partial
`charges and is defined as a sum over the surface areas of
`the negatively charged atoms. This descriptor encodes
`information about the charge distribution in the repeating
`unit. The PNSA is dependent on the size of the repeating
`unit; the squared correlation coefficient of 0.735 shows a
`moderate intercorrelation between PNSA and the molecular
`weight of repeating unit. Thus PNSA also describes mo-
`lecular size related bulk properties of repeating units linked
`into the polymer chain.
`The relative number of F atoms is defined as a ratio
`between the number of fluorine atoms and the total number
`of atoms in the repeating unit. This descriptor is required
`due to the extraordinary chemical nature of the fluorine.
`Fluorine containing polymers have usually very low refrac-
`tive index values, and the negative slope for the relative
`number of fluorine atoms is in good agreement with this
`trend. The use of quantum-chemical descriptors alone
`appears to overestimate the refractive index for this set of
`polymers.
`The model as described above shows a standard error of
`0.018. The average prediction error is 0.9%, and the highest
`prediction error is 3.2%. The cross-validated correlation
`2 ) 0.934) shows the stability of the model.
`coefficient (RCV
`An alternative method for cross-validation was also used to
`test the stability of the model. The data set of experimental
`refractive indices was divided into three subsets according
`to their magnitude. When two of the subsets were combined
`and the QSPR model recalculated using the same descriptors
`but newly optimized regression coefficients, the predicted
`refractive indices for the third subset gave a squared
`correlation coefficient of 0.906. We applied similar proce-
`dures to calculate the squared correlation coefficients (0.959
`and 0.951) for the other two subsets. The average correlation
`coefficient for the three subsets was 0.938, and the descriptor
`coefficients were essentially constant (see Table 3).
`A comparison between the QSPR model developed in the
`present paper for polymers with the model previously found14
`for low molecular weight organic compounds shows that the
`HUMO-LUMO energy gap is a common descriptor for both
`data sets. Several of the other descriptors describe similar
`
`types of physicochemical interactions. Thus, both QSPR
`models include electrostatic descriptors which describe the
`charge distribution in the molecule or repeating unit of the
`polymer (partial negative surface area for low molecular
`weight organic compounds, partial positive surface area, and
`hydrogen donor dependent surface area for polymers). Also,
`the lowest E-N attraction for a C atom (for low molecular
`weight organic compounds) and the strongest nuclear repul-
`sion for a C-H bond (for polymers) are both descriptors
`that can be related to the hybridization of the carbon atoms.
`The differences in the descriptors selected for the low and
`high molecular weight models may in part be done to the
`variation of physical interactions in different media, e.g., solid
`phase versus liquid phase.
`Bicerano’s QSPR model consisted of 10 topological and
`constitutional descriptors;5 our QSPR model is quite distinct
`as it comprises four general quantum-chemical descriptors,
`augmented by one constitutional descriptor. Bicerano’s
`model implies that the refractive index for a vacuum should
`be 1.885. The comparison of statistical parameters shows
`better statistical quality in Bicerano’s model (R2 ) 0.955
`versus R2 ) 0.940, s ) 0.0157 versus s ) 0.0177), but this
`is not surprising in view of the fact that the number of
`descriptors involved in this correlation equation is twice as
`large (10 instead of five) as in our equation. We have also
`attempted to correlate topological and constitutional descrip-
`tors with the refractive index and verified that results
`comparable with Bicerano’s QSPR model5 can be reproduced
`if a sufficient number of topological and constitutional
`descriptors is used. On the other hand, improvement of
`results by increasing the number of descriptors in the
`correlation equation should be considered with care, since
`overfitting and chance correlations may in part be due to
`such an approach.
`
`CONCLUSION
`
`It is evident that the QSPR approach can be applied to
`develop successful QSPR models for polymers. The five-
`parameter QSPR model, proposed in present study, can
`predict the refractive index values of structurally diverse
`polymers with a significant degree of confidence (the average
`prediction error is 0.9%). The model employs only theoreti-
`cal descriptors calculated from structure of repeating units
`and is thus applicable to not yet synthesized polymers.
`Therefore, this QSPR model should be useful in development
`of new polymeric materials.
`
`ACKNOWLEDGMENT
`
`This work was partially supported by the U.S. Army
`Research Office (Grant No. DAAH 04-95-1-0497) and NSF
`(Grant No. CHE-9629854). We thank Dr. Yilin Wang for
`help in the preparation of this manuscript.
`
`Page 5 of 6
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`

`
`1176 J. Chem. Inf. Comput. Sci., Vol. 38, No. 6, 1998
`
`KATRITZKY ET AL.
`
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