J . Phys. Chem. 1994,98, 9078-9082
9078
Conformational Characteristics of Poly(oxymethy1ene) Based upon ab Initio Electronic Structure Calculations on Model Molecules Grant D. Smith Thermosciences Institute, RTC 230-3, NASA AMES Research Center, Moffett Field, California 94035
Richard L. Jaffe NASA AMES Research Center, Moffett Field, California 94035
Do Y. Yoon* IBM AImaden Research Center, 650 Harry Rd., San Jose, California 951 20-6099 Received: February 8, 1994; I n Final Form: May 25, 1994’
A revised rotational isomeric state (RIS) model for poly(oxymethy1ene) (POM) chains has been developed on the basis of the conformational energetics of the model molecules, dimethoxymethane (DMM) and 1,3-dimethoxydimethyl ether (DEE), as determined from a b initio electronic structure calculations. Due to the strong preference for gauche (g) conformations in P O M chains, the energy minima for the g states were found to be deep and narrow, necessitating the inclusion of preexponential factors for the statistical weights in the R I S model that deviate from unity. Therefore, both the energy minima and preexponential factors for all of the important conformations of the model molecules were estimated by performing a b initio electronic structure calculations using a D95+(2df,p) basis set. Electron correlation effects were included a t the MP2 level. The revised R I S model, with all parameters estimated from a b initio calculations, predicts the unperturbed chain dimensions and the dipole moment of P O M chains as a function of temperature in good agreement with experiments. Therefore, it is demonstrated that a b initio calculations a t the level of theory employed (D95+(2df,p) basis set a t the MP2 level) are sufficient to provide accurate estimates of the conformational energetics of P O M chains. Moreover, it is also shown that proper description of the conformation-dependent properties of P O M chains requires consideration of not only the energy minima but also the energy well profile for each of the important conformations.
Introduction Crystallographic studies (e.g., refs 1-4) indicate that poly(oxymethylene) (POM) chains, illustrated in Figure 1, form helices comprised of gauche (g) conformations of the same sign of the C.-O-C.-O bonds. As discussed in the preceding papers in this issue, experiments and ab initio electronic structure calculations on dimethoxymethane (DMM), shown in Figure 1, also indicate a predominance of the gauche conformation of the C--O-C-Obond in the gas phase. Our ab initio calculations on DMM,S performed with a D95+(2df,p) basis set with MP2 treatment of electron correlation, indicated that the lowest energy conformation, gg (denoting two gauche bonds of the same sign), is about 2.5 kcal/mol lower in energy than the gt conformation. The corresponding gauche energy of -2.5 kcal/mol (the energy difference between the gg and gt conformers) is significantly greater in magnitude than values determined from analysis of the dipole moment of DMM by means of a rotational isomeric state (RIS) model, where values ranging from -1.4 to-1.74 kcal/ moF-8 have been obtained. However, ab initio studies of the conformational energy wells of DMMS show the gg well to be deep and narrow while the gt well is broad and shallow. Analyses of the energy wells indicate that a preexponential factor of about 2.5 is required for the gt conformer, relative to a preexponential factor of unity for the ggconformer, in order to correctly describe the conformer populations. The conformer populations calculated by including preexponential factors yielded the dipole moment and NMR vicinal coupling constant for DMM as a function of temperature in good agreement with experiment.5 Therefore, it can be argued that appropriate ab initio calculations allow one to obtain a more detailed understanding of the conformational *Abstract published in Aduance ACS Abstracts, July
15, 1994.
0022-3654/94/2098-9078%04.50/0
poly(oxymethy1ene)(POW
dimethoxymethane(DMM)
1,3-dimethoxydimethylether (DDE)
Figure 1. Chemicalstructure of poly(oxymethy1ene) and model molecules for poly(oxymethy1ene).
properties of DMM and related molecules than was previously possible. In this paper we have applied the results of our ab initio calculations of the conformational energies of model molecules to construct a revised rotational isomeric state (RIS) model for POM. Predictions of the mean-square chain dimensions and dipole moment for POM are compared with experimental values, and the sensitivity of the results to various parameters is discussed. Ab Initio Electronic Structure Calculations
In the preceding papers we demonstrated that DMM conformer populations determined from ab initio electronic structure calculations reproduce experimental dipole moments and NMR 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 36, I994 9079
Conformational Characteristics of Poly(oxymethy1ene)
TABLE 1: Conformational Enereies of DDE ~~
energy (kcal/mol)b 4-31G
D95+(2df,p) SCF MP2
D95**
conformep
type SCF MP2 SCF MP2 minimum 0.00 0.00 0.00 0.00 0.00 0.00 2.36 3.05 2.11 2.81 minimum 4.05 4.62 2.36 2.07 2.23 1.80 2.01 1.65 minimum minimum 7.31 8.68 4.27 5.77 3.86 5.52 2.35 2.33 3.45 3.30 2.75 2.64 gt yg; &?-+&?-_g+g+-&?-ET saddle point 0 The labels g+ and g refer to gauche conformations with a + or - sense, respectively. The g labels refers to a gauche conformation of either sign, but a pair refers to the same sign. The subscripted labels + and - indicate conformations distorted significantly (2Oo-3O0) from a normal (e.g., DDE gggg) gauche conformation. b Energies are relative to the lowest energy conformer (gggg). gggg gm g+g+g+&?-
TABLE 2
DDE Conformer Geometries torsional angles (deg)" 4-31G
conformer Egg gtgg g+g+&T+g t yIT+E-+E-x+IT+-E-EIT
41 (COCO) 112.9 118.4 115.6 -6.8 1 1 2.8
D95**
42 (OCOC)
43 (COCO)
44 (OCOC)
107.9 -3.3 126.8 120.6 96.2
107.9 112.7 -14.9 120.6 -96.0
112.9 111.4 -1 13.9 -7.2 -1 12.3
61 (COCO) 112.2 112.1 113.3 -3.0 110.0
42 (OCOC)
43 (COCO)
107.1 8.8 117.8 114.0 98.7
107.1 111.2 -79.2 114.0 -95.8
44
(OCOC) 112.2 111.5 -1 10.0 -3.1 -109.6
valence angles (deg)b gggg
gtgg g+g+&?-+&?tggt
115.1 115.3 115.6 113.5
112.8 109.7 114.3 109.6
115.3 115.1 116.8 116.4
112.8 113.5 112.8 109.6
115.1 115.0 114.8 113.5
a Values are from SCF optimized geometries for the basis set indicated. Torsional angles are relative to t = Oo. Values are from D95** SCF optimized geometries. The torsional and valence angles are listed in order of consecutive bonds along the backbone. 61 is the torsional angle between and 02, 4 2 is betweeen 02 and 03, etc.
vicinal coupling constants. This agreement demonstrates the accuracy of the ab initio conformational energies provided an adequate basis set (e.g., D95+(2df,p)) is used and electron correlation effects (MP2 level or better) are considered. For complete calibration of a RIS model for POM, it was necessary to also determine conformational energies for selected conformationsof l ,3-dimethoxydimethyiether(DDE), illustrated in Figure 1. These calculations were performed a t the same level of ab initio electronic structure theory as for DMM (see ref 5 for details). These energies are tabulated in Table 1, and the torsional and valence geometries are given in Table 2. As was found for DMM,S the gauche conformation of the C-.O-C--O bond is strongly favored over the trans conformation, as can be seen by comparing the energies of the gggg and gtgg conformers. As the size of the basis set increases, the preference for the gauche conformation decreases. This trend was also seen for DMM. Electron correlation effects are important, as indicated by comparing the MP2 and S C F energies for a given basis set. For DMMS, little change in relative conformer energies with improved treatment of electron correlation was seen beyond the MP2 level. We therefore assume that MP2 treatment is adequate for DDE.
RIS Model Representation. Following F l ~ r ywe , ~considered the following first-order and second-order interactions in POM: the first-order gauche interaction with statistical weight u and energy E,; secondorder g'g' interactions centered on CH2 and 0 with statistical weights and $b and energies E+, and E b , respectively; and second-order g'f interactions centered on CH2 and 0 with statistical weightsw, and wb and energies E,, and E,, respectively. The representation of the conformers of DMM and DDE in terms of these interactions and the resulting RIS energies for the conformers are shown in Table 3. Energy Parameters. From the energies of the DMM gt and tt conformer (relative to gg), we obtain E , = -3.4 kcal/mol and
TABLE 3: RIS Representation of DMM and DDE Conformers conformer RIS representation' ab initio energy RIS energy'sb DMM gg gt g+&?tt
0 -Es-E+, E,, - E*. -2E, - E+,
gggg gtgg g+g+&?-+r tggt
0 -E, - E+, - EA E, - EA -2E, - 2E+,
0.00 2.81 4.2 6.26
0.0 2.8 4.2 6.2
0.00 2.81 1.65 5.52
0.0 2.8 1.6 5.6
DDE
For DMM conformers energies are relative to theggconformer, and for DDE conformers energiesare relativeto theggggconformers. Energies are in kcal/mol. E, = -3.4, E+, = 0.6, E A = 0.0, E,, = 4.8, and E , = 1.6, all in kcal/mol. The conformation labeled g + g is the saddle point between the g+t and tg conformers. E+, = 0.6 kcal/mol, the latter value indicating the presence of a second-order CH2 centered gg interaction. A second-order $a interaction, although of much higher energy (E+, = 1.54 kcal/ mol), was also seen by Miyasaka et al.9 based upon molecular mechanics studies of DMM and DDE. The gauche energy, defined for DMM as the difference between the gg and gt conformer energies, is given as E , E+, = -2.8 kcal/mol. As discussed previou~ly,~ we believe a value of -2.5 kcal/mol, which lies within the estimated uncertainty of k0.3 kcal/mol of the ab initio value, to be the best estimate of this energy. Accordingly, we utilize a value of E,, = -3.1 kcal/mol in the RIS calculations described below. Previously,s we showed that for DMM a preexponential factor for thegt conformation off@ = 2.5 (relative to a value of unity for theggconformer) was required to accurately represent configuration space. For the polymer RIS model, we therefore assign a first-order preexponential factor offi = 2.5 to all trans conformations. From the energy of the DMM g+gconformation (saddle point) we obtain a value E,, = 4.8 kcal/
+
9080 The Journal of Physical Chemistry, Vol. 98, No. 36, 1994
Smith et al.
TABLE 4 parameter E, E+, Eo, Eh E,
Figure 2. Statistical weight matrices for POM.
mol. Again, as discussed previously,s our best estimate for the energy of the g + g conformation is 0.3 kcal/mol less than the a b initiovalue, yielding a value E,, = 4.5 kcal/mol. A second-order preexponential factor of f,, = 3 was assigned to all C-centered g + g conformations, as indicated by the value of fg+*-determined for DMM.5 The energy of the DDE gtgg conformer (relative to gggg) indicates no significant second-order interaction associated with the 0-centered gg conformation, yielding Eh = 0.0 kcal/mol. The second lowest energy conformation of DDE is the g + g + g + g conformation, whose energy yields E, = 1.6 kcal/mol. As was seen for the energy of the gt conformer of DMM5 (relative to g g ) , the energy of the DDE g + g + g + g conformer (relative to gggg) shows a dependence on the size of the basis set, with the energy difference decreasing with increasing basis set size. The effect is weaker, however, than was seen for the gt conformer of DMM. We therefore use a value of E, = 1.5 kcal/mol, slightly lower than the value of 1.6kcal/mol obtained from the ab initio energies but well within the estimated uncertainty of f0.3 kcal/m01.~The wb interaction is much more favorable than the w, interaction becauseof the relatively small size of the oxygen atoms (compared to CH2/CHj groups) which are pentane eclipsed in this conformation. The subscript indicates a distortion of the torsional angle by 20'-30' from values in the gggg conformation (see Table 2). A conformer of equal energy, g+g+-g-g, is separated from the g + g + g + g conformer by a barrier of about 0.7 kcal/mol, as shown in Table 1. Analysis of the shape of the g+g+g+g/g+g+-g-g well, similar to that conducted for conformers of DMM,5 indicates that when these conformations are described by a singleg+g+ggconformation, a second-order preexponential factor off, = 2 is required. Statistical Weight Matrices. The two 3 X 3 statistical weight matrices (one for each unique bond pair) corresponding to the RIS model enumerated above are shown in Figure 2. The statistical weights are given by
"+"
u
= exp(-E,/kT)
(3) (4)
The final RIS parameters for POM are summarized in Table 4.
Torsional Fluctuations The RIS representation of POM given in Table 3 assumes that each conformation is well represented by a single, minimumenergy geometry. It has been demonstrated that inclusion of thermal fluctuations of the torsional geometry about the minimumenergy geometry for helical molecules can result in a significant decrease in the calculated characteristic ratio, especially for highly extended chains.10 We have included thermal fluctuations about the gauche minima by treating each gauche state in the statistical weight matrix as three states, one with the minimum-energy geometry &with a relative weight of 0.5 and the other two with geometries 4gf 1 5 O and relative weights of 0.25. The procedure
ft fw. f,
RIS Parameters for POM value 0.6 kcal/mol 4.5 kcal/mol 0.0 kcal/mol 1.5 kcal/mol
parameter C-O bond length C-O bond dipole moment C-0-C valence angle 0-C-O valence angle gauche torsional angle
2.5
P
3
A&
-3.1 kcal/mol
value 1.39 A 1.33 D 1120 113O
&107O 0.25 150
2
TABLE 5 DMM and DDE Conformer Dipole Moments conformer ba hb p (h = 1.24) p (h= 1.33) DMM' gg ggd gt gtd g'r
0.27 (0.34) 1.87
1.19
(2.05)
(1.24)
2.43 2.55
1.17
e
it
1.28
0.06 (0.01) 1.96 (2.05) 2.35 2.71
DDV
1.40 1.38 1.38 1.94 1.59 1.62 Dipole moments are in debye. C-O bond dipole moment which reproduces the magnitude of the conformer dipole moment. From MP2 electrondensitiesusinga D95+(2df,p) basisset and D95**SCFoptimized geometries. For comparison, the dipole moments for the SCF optimized geometries of the gg and f gconformers using the SCF electron densities are 0.23 and 2.00, respectively. d From MP2 electron densities using a D95+(2df,p) basis set and D95** MP2 optimised geometries. e The conformation labeled g + g is the saddle point between the g+t and tg conformers.f From SCF electron densities using a D95+(2df,p) basis et and D95** SCF optimized geometries. gggg gtgg g+g+g+g rggt
1.40 1.61 1.25
1.33 1.55 1.21
used in determining the weights and geometries of the gauche states and the resulting statistical weight matrices is given in the Appendix.
RIS Predictions With the RIS energy and geometric parameters given in Table 4, standard methods' were employed in calculating the characteristic ratio and mean-square dipole moment ratio in POM. The geometric parameters given in Table 4 correspond to the a b initio values for the DDE gggg (central bonds) found in Table 2, with one exception. The C-0-C valence angle was reduced by 3 O to give good agreement with the MP2 geometry of the DMM gg conformer.
Dipole Moment Model Molecules. In order to predict the mean-square dipole moment of a polymer from the RIS model, it is necessary to associate a dipole moment with each backbone bond. Commonly, the bond dipole is considered to lie along the bond. Given the dipole moment and optimized geometries of the conformers of DMM and DDE (from a b initio calculations), it is possible to determine the value of the bond dipole moment for the C-0 bond which best reproduces the individual conformer dipole moments. The dipole moments of various conformers of DMM and DDE, as determined from ab initio electronic structure calculations, are shown in Table 5. Using a bond dipole moment of 1.24 D, which was determined to give the best representation of the MP2 density dipole moment of the gt conformer of DMM, with the MP2 optimized geometry, the resulting dipole moments (using ab initio geometries) for the remaining DMM conformers were calculated and are shown in Table 5. The bond dipole moment reproduces the dipole moments of the tt conformer and gg conformer to within about 0.3 D. This discrepancy is quite large for the gg conformer (about 100% of the ab initio value) and is
The Journal of Physical Chemistry, Vol. 98, No. 36, 1994 9081
Conformational Characteristics of Poly(oxymethy1ene)
0 .-E!
-
0.35. ' :----A 0.30
'
'
1
'
'
I
'
"
1
"
25
'
'
'
--A
;------- B
_____
0.25
10
-
-.-.IB ----.-c
-C
-.-.-.
0
experimental
....--D -.-.--E
.-*.
lo
$
0.05
: tL
0.00 440
"
"
"
'
460
"
480
"
500
"
"
520
i 540
temperature (K) Figure 3. Mean-square dipole moment ratio for POM as a function of temperature. Experimental melt values are shown. Case A: RIS predictions with torsional fluctuations, parameters given by Table 4. Case B: Same as A, without torsional fluctuations. Case C: Same as A,fr = 1.0.
important because of the high gg population. For DDE, using bond a dipole of po = 1.33 D, which best reproduces the dipole moment of the gggg conformer, the dipole moments of the remaining conformers were determined and are shown in Table 5. The representation is only fair, indicating the limitation of a bond dipole representation. It is also worth noting that the somewhat different bond dipole moments are required to reproduce the important DMM and DDE conformer dipole moments. Using a bond dipole of po = 1.33 D from the DDE ggggconformer, the POM RIS model (with the parameters given in Table 4) gives a root-mean-square dipole moment for DDE of 1.46 D a t 298 K, in good agreement with experimental (solution, nonpolar solvent) value of 1.52 D." Another group6 reports an experimental DDE dipole moment of 1.41 D (solution, nonpolar solvent) at 298 K. Over the range of experimental data (298353 K)," the predicted temperature dependence of the dipole moment of DDE from the RIS model, d In (p2)/dT X 1000, is 0.7, in good agreement with theexperimentalvalueof 0.8.l1 Using a bond dipole of po = 1.33 D, agreement with experiment for higher molecular weight oligomersll was found to be only fair, with the RIS predictions indicating a somewhat stronger temperature dependence of the dipole moment than was seen experimentally. A value of po = 1.33 D was used in the polymer dipole moment calculations. POM. The predicted mean-square dipole moment ratio, ( p 2 )/ np02, for POM is shown as a function of temperature in Figure 3. Here, ( p 2 ) is the mean-square dipole moment of the polymer (which can be experimentally measured), n is the number of C-0 bonds, and po is the dipole moment of the bond. Experimental values for polymer melts12 at these temperatures are also shown. We have renormalized the experimental mean-square dipole moments using a C-0 bond dipole of 1.33 D, slightly larger than the PO = 1.31 D used in ref 12. The predicted values (case A) are in reasonable agreement with experimental values, with the predicted values being 15-20% lower than experiment. The predicted temperature dependence, d In (w2)/dTX 1000, is 4.0 over the temperature range of the experimental data, which compares well with the experimental value of 4.4. Also shown are the predicted mean-square dipole moment ratios without torsional fluctuations (case B). These values average about 5% lower than the corresponding values with torsional fluctuations, indicating only a minor influence of fluctuations on the predicted polymer mean-squaredipole moment. Figure 3 alsodemonstrates the dependence of the mean-square dipole moment of POM on the parametersf, and f,. For case C,f,was set to unity, while for case D (not shown), fw, was set to unity. It can be seen that the mean-square dipole moment depends strongly on&, with the predicted values being significantly reduced when fi = 1.O. In
.
--.-.-
e-._
.-
--..........
5
280
300
320
340
360
380
temperature (K) Figure 4. Characteristic ratio for POM as a function of temperature. Experimental values are shown (with error bars where reported). Case A: RIS predictions with torsional fluctuations, parameters given by Table 4. Case B: Same as A, without torsional fluctuations. Case C: Same as A , J = 1.0. Case D: Same as A, f, = 1.0. Case E Same as A,f, and f, = 1.0. contrast, the mean-square dipole moment was found to show little dependence on f,, with the predictions for case D being essentially coincident with those for case A. These results indicate that increasing the population of trans bonds increases the meansquare dipole moment of the polymer. This accounts for the strong positive temperature dependence of the mean-square dipole moment. However, increasing the number of 0-centered g+gsequences has little influence on the mean-square dipole moment. The fraction of C-centered g'g- sequences, which is sensitive to fw,, is small because of their high energy and hence has little influence on the conformational properties of the polymer.
Characteristic Ratio of POM The predicted characteristic ratio, C, = (R2)/n12, of POM (case A) is shown as a function of temperature in Figure 4. Here, ( Rz) is the mean-square end-to-end distance for the chains, n is the number of C-0 bonds, and 1 is the bond length. Experimental values extrapolated from good (polar) solvents13J4are also shown. Agreement between predicted and experimental values is quite good. The temperature dependenceof the predicted characteristic ratio,dln (RZ)/dTX 1000,is-4.8at363K, whichisinqualitative agreement with experiment in that it is largeand negative. Figure 4 also shows the predicted characteristic ratio for the model without torsional fluctuations (case B). The predicted characteristic ratio values range from 15% higher than the predicted values with fluctuations a t the lower temperatures to 6% higher a t the higher temperatures. This result is consistent with the observation that fluctuations have a greater influence on the characteristic ratio of more extended, helical chains.I0 Also shown in Figure 4 is influence of the preexponential factors ft and f,, in cases C, D, and E. Case C corresponds toft = 1.O, case D to f, = 1.0, and case E to both factors equal to unity. The characteristic ratio can be seen to strongly depend on f, or the population of 0-centered g+g- sequences. This is in contrast to the mean-square dipole moment ratio, which was found to be nearly independent of thef,. The characteristic ratio shows a fairly weakdependence on the population of trans bonds, indicated by the fact that the characteristic ratio increase only about 1015% when fi is reduced from 2.5 to 1. In contrast, the meansquare dipole moment depends strongly on the population of trans bonds. When populations of both the 0-centered g+g-sequences and trans bonds is reduced (case E), the characteristic ratio increases significantly. For this case, neglect of torsional fluctuations increases the characteristic ratio by another 2030% (not shown).
9082 The Journal of Physical Chemistry, Vol. 98, No. 36, 1994 Influence of Intermolecular Interactions The RIS predictions of the mean-square dipole moment and characteristic ratio of POM do not take into account possible effects of intermolecular interactions. Comparison of NMR vicinal coupling in DMM from gas phase, nonpolar solvent and polar solvent measurements appears to indicate that the gauche energy decreases somewhat in a polar environment.l5 This is consistent with the more polar trans-containing conformers (see Table 5) being stabilized in the polar media. Such an effect may be present in POM melts and is not inconsistent with the fact that the measured mean-square dipole moments of POM melts are somewhat greater than our RIS predictions. The characteristic ratio of POM was measured in polar solvents, so again some polar effects might be expected. However, RIS predictions reveal that thecharacteristicratioofPOMshowsonly a weakdependence on the population of trans bonds. We therefore expect that the influence of moderate intermolecular polar interactions on the characteristic ratio of POM would be relatively minor.
Conclusions Representations of the conformational energetics of DMM and DDE from ab initio electronic structure calculations have been applied in the development of a revised RIS model for POM that accurately predicts experimentally measured conformationdependent properties of the polymer. In developing the RIS description, differences in the shapes of the energy wells, in addition to the energy minimum, need to be considered for the important conformations. The mean-square dipole moment of POM was found to depend strongly on the fraction of trans bonds and was found to be nearly independent of the fraction of 0-centered g+g- sequences. The strong positive temperature dependence of the mean-square dipole moment is a result of the increase in the fraction of trans bonds with increasing temperature. In contrast, the characteristic ratio was found to depend much more strongly on the fraction of 0-centered g+g-sequences. The strong negative temperature dependence of the characteristic ratio is primarily a consequence of the increase in 0-centered g+gsequences with increasing temperature. It is possible to incorporate the preexponential factors of the POM RIS statistical weights into “adjusted” energy parameters which subsequently reproduce the statistical weights a t some arbitrary temperature using unity preexponential factors. For example, at 400 K, these adjusted energies are E,, = -2.4 kcal/ mol (with a corresponding gauche energy of -1.8 kcal/mol) and E , = 1.0 kcal/mol. Use of these adjusted energies, however, leads to significantly different temperature dependence of the chain dimensions and dipole moments from those predicted by the model including preexponential factors. However, the predictions of the adjusted model may still lie within the limits of experimental uncertainties, and hence such a RIS model may be adequate for practical purposes. In such a case, the energies for the conformations yielded by the “adjusted” energy parameters should not be taken as the differences in the conformer energies, but instead should be considered as differences in the averaged energies. This distinction is of particular importance when using the conformational energy differences in parametrizing a molecular mechanics force field.
Acknowledgment. The authors thank Professor R. H. Boyd for making dipole moment measurements on POM oligomers available. G.D.S. is grateful for support provided by NASA through Eloret Contract NAS2- 14031,
Appendix. Torsional Fluctuations From conformational energy given by eq 1 of ref 5 (see also Figure 1 of ref 5 ) , we determined the standard deviation of the torsional angle for the gauche state of DMM about the minimum. We then represented each gauche state in the statistical weight matrices as three states, one with the energy minimum geometry, $, with weight 1 - 2p, and the others with geometry & f A4, with weight p. The quantities A 4 and p were determined so as to yield the same standard deviation in 4 as obtained from eq 1. The augmented statistical weight matrices are shown in Figure 5. Whenp = O,t&ematricesreducedtothoseofFigure2.Selecting a value for p, and determining the appropriate value for A#, we found that the predicted characteristic ratio and mean-square dipole moment ratio for the polymer were not sensitive to the particular value of 0.1 Ip I0.5. We therefore chose p = 0.25, yielding A 4 = 15’ at 300 K. These values were used in all RIS calculations which included fluctuations.
References and Notes (1) Tadakoro, H.;Yasumoto, T.; Murahashi, S.; Nitta, I. J . Polym. Sci. 1960, 44, 266. (2) Carazollo, G . A. J . Polym. Sci., Part A 1963, I, 1573. (3) Takahashi, Y.; Tadokoro, H. J. Polym. Sci.,Polym. Plrys. Ed. 1979, 17, 123. (4) Carzollo, G.; Mammi, A. J . Polym. Sci., Part A 1963, 1, 965. (5) Smith, G. D.; Jaffe, R. L.; Yoon, D. Y. J . Phys. Chem., preceding paper in this issue. (6) Uchida, T.; Kurita, Y.; Kubo, M. J . Polym. Sci. 1956, 19, 365. (7) Flory, P. J. Sratistical Mechanics of Chain Molecules: Interscience: New York, 1969. (8) Abe, A.; Mark, J. E. J . Am. Chem. SOC.1976,98, 6468. (9) Miyasaka, T.; Kinai, Y.; Imamura, Y. Makromol. Chem. 1981,182, 3533.
(10) Mansfield, M. L. Macromolecules 1983, 16, 1863. (11) Boyd, R. H. Unpublished data. (12) Porter, C. H.; Lawler, J. H. L.; Boyd, R. H. Macromolecules 1970,
3, 308.
(13) Stockmayer, W. H.; Chan, L. L. J. Polym. Sci., Part A-2 1966, 4 , 437. (14) Kokle, V.; Billmeyer, F. W., Jr. J. Polym. Sci., Part B 1965, 3, 47. (15) Abe, A.; Inomate, K.; Tanisawa, E.; Ando, I. J . Mol. Struct. 1990, 238, 315.
