228
J. H. Newton and W. 8.Person
The Journal of Physical Chemistry, Vol. 82,No. 2, 1978
(23) K. Fueki, D. F. Feng, L. Kevan, and R. Christoffersen, J. Phys. Chem., 75,2297 (1971). (24) K. Fueki, D.F. Feng, and L. Kevan, J. Am. Chem. Soc., 95, 1398 (1973). (25) K. Fueki, D. F. Feng, and L. Kevan, J. Phys. Chem., 78, 393 (1974). (26) A. D. Buckingham, Discuss. Faraday Soc., 24, 151 (1957). (27) S. Goldman and R. Bates, J . Am. Chem. SOC.,94, 1476 (1972).
F is ~ the vector sum of the permanent dipole moment of a molecule in the bulk liquid plus the average electric moment induced by hindering the rotation of its neighbors relative to itself. (29) M. F. Fox and E. Hayon, Chem. Phys. Lett., 25, 511 (1974). (30) C. J. F. Bottcher, 0. C. van Belle, P. Bordewijk, and A. Rip, “Theory of Electric Polarization“, Vol. 1, Elsevier, Amsterdam, 1973, pp
(28) Q
113-127.
Dipole Moment Derivatives and Infrared Intensities. 5. Atomic Polar Tensors from CH4 James H. Newton and Willis B. Person* Department of Chemlstty, Universlty of Florida, Gainesvllls, Florida 326 11 (Received July 21, 1977)
Atomic polar tensors for carbon and hydrogen atoms are derived from the experimental integrated molar absorption coefficients in methane. These tensors compare very favorably with those calculated by CND0/2 and ab initio quantum mechanical methods. The apparent sign discrepancy between earlier reports comparing the latter two methods has been identified, and both theoretical calculations are now in very good quantitative agreement with experimental values. The resulting consistency allows the choice of the “best” atomic polar tensor values for C and H atoms, with the expectation that they can be “transferred” to these atoms in other hydrocarbons and used to predict the integrated molar absorption coefficients for their fundamental bands.
Introduction Recently, a number of reasonably successful prediction9 have been made for the infrared intensities of the fundamental absorption bands for molecules containing fluorine atoms, using an F-atom polar tensor transferred from the experimental polar tensor derived from the CH3F m~lecule.l-~Because of the relative neglect of the intensities of hydrocarbons, there may be some impression that the polar tensor model is less successful in predicting the intensities of fundamental vibrations of hydrocarbons than it is for vibrations of fluorocarbons. We are withholding our own opinion on this subject until further tests are made, but we think this subject is one that certainly deserves careful study. One problem with attempts to predict intensities of hydrocarbons has been the lack of a good initial estimate of the H-atom polar tensor to be transferred from one hydrocarbon to another. We might expect to obtain our H-atom polar tensor from methane, but there has been some difficulty in deciding about the correct signs of the dipole derivatives for that molecule.8 We have, therefore, reinvestigated the interpretation of the intensities of the CH4 and CD4molecules in order to determine definitively the H-atom (and C-atom) polar tensors for the methane molecule. Calculation of the Atomic Polar Tensors The procedure for calculating the atomic polar tensors from the experimental values of the integrated molar absorption coefficients,A,, has been described several time previou~ly.l-~The components of the dipole moment derivatives with respect to the normal coordinates, ap,/aQi (ap/aQ, in e = 3.20 X 10-2Ai (km mol-l)/gi as given previously3), for each particular assumption about signs (for example, both ap,/aQ3 and ap,/aQ4 positive) define a P polar tensor in normal coordinate space. T%e polar tensor (composed of juxtaposed atomic polar tensors’s) in space-fixed Cartesian coordinate space for this nonpolar molecule (hence, no “rotation correction”) is given by
P,
= P,L-’UB
(11
(see ref 1 and 2 for definitions.) The transformation matrices, L-l, U,and B are obtained in the course of a normal coordinate calculation made using the force field of Duncan and Mills,g with symmetry coordinates as defined by Shimanouchi.lo The symmetry coordinates, force field, and other values from the normal coordinate calculation are summarized in the Appendix; the Cartesian coordinate axes, numbering of atoms, and orientation of the CH4 molecule is shown in Figure 1. These recalculations are necessary in order to ensure that the sign conventions are consistent throughout the treatment. The atomic polar tensors calculated from the experimental data for two [(-,-) and (-,+)I of the four possible assumptions for the signs of ap,/aQ3and ap,/aQ4 are given in Table I. The other two sign assumptions [(+,+) and (+,-)I may be eliminated from further consideration on the basis of their disagreement between the experimental values derived on the basis of those signs with the values calculated quantum mechanically.
Discussion In order to reach some decision about which of the several possible polar tensors are correct for methane, we may compare the different possible sets of ap,/aSja values from the experimental data with values calculated from approximate and ab initio quantum mechanical calculatiorma Such calculations from the 1iteraturellJ’ did not agree on which sign choice was correct (see the discussion in ref 8). We have, therefore, repeated the CNDO/2 calculations ourselves, using coordinate definitions consistent with those we used in the normal coordinate calculation. The results are compared with experiment, with the approximate CNDO/2 calculations by Segal and Klein,ll and with the ab initio calculation by Meyer and Pulay12 in Table 11. We see in Table I1 that the agreement between our CNDO/2 calculation and the Meyer and Pulay ab initio calculation is very good indeed. The apparent discrepancy
0022-3654/78/2082-0226$01.00/00 1978 American Chemical Society
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978 227
Atomic Polar Tensors from CH4
i
T2
Flgure 1. Coordinate axes and molecular orientation for CH4 used for this normal coordinate calculation.
(0)
(b)
Figure 2. The definition of the S4 symmetry coordinate of CH4 used by (a) Segal and Klein and (b) Meyer and Pulay. The z axes for both definitions are in the same direction.
between Segal and Klein and Pulay and Meyer discussed previously8 is nothing more than a difference in the definition of the symmetry coordinate S4 used in the two treatments. The two different definitions are shown in Figure 2. Thus, all three theoretical treatments agree very well with each other, and all agree surprisingly well with the experimentalvalues obtained from the (-,-) sign choice. Incidentally, the experimental values ap,/aQ3, and apx/aQrrare obtained from four independent measurements: from CH4 and CD4 by Ruf13 and from CHI and CD4 by Saeki, Mizuno, and Kondo.14 The experimental values are in good agreement. The standard deviations of these four different measurements yield the propagated uncertainies shown in the Table I1 for ap,/aS,, and in Table I for the elements of the polar tensors. Although the agreement between calculated ap,.aSj, values and experimental values shown in Table I1 clearly indicates that ap,/aQ3, and ap,/aQ, are both negative, the small magnitudes of the values in that table, and the notorious difficulty in quantitative prediction of dipole moment derivatives from approximate quantum mechanical treatments,8suggests that there may still be some doubts remaining about the certainty of the (-,-I sign choice. These doubts may be dispelled by careful examination of Table I, comparing the atomic polar tensors calculated by the CND0/2 method (as described previously') with those obtained from the experimental data with the two different sets of signs. In particular the values deduced from the experimental data for the carbon-atom polar tensor are significantly different for the two different sign choices under consideration. It is quite clear that the values calculated using the CNDOIB method agree with the (-,-) sign choice and do not agree with the values derived from the (-,+I sign choice. Similarly the values we derive from the Meyer-Pulay calculated values using of ap,/aS,, (arranged as a polar tensor Ps112)
P,
PSUB (2) agree with the same C-atom polar tensor. Thus we believe the examination of intensity data in the form of atomic polar tensors may be useful in trying to decide such delicate matters as the sign choice to be preferred. It should be emphasized that the sign choice we have made here (-,-I is the same as the one preferred by Kondo and Saeki15on the basis of the least-squares fit of the two =
n 0
m?
u53
8 8 8.1 I
d
c.1 rl mg,
0 1 0I 8 0
+I
m
o
3 r i
v38v38
c o d
0.+I 0 +I 0 0 0 0
I
W
n * 0
$8 8 0 0 0
00 0
m s
m
2 00 8 +'
h
N
Y
5:
h
",
x
PIx
0
+I
228
J. H. Newton and W. E. Person
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978
TABLE 11: Comparison of Experimental and Calculated Values of 8 d a S ' s for CHAd Calcd Exptla
CNDO/2 CNDO/2 ab initio (-9-1 (-,+ 1 (SK)b (this work) (MPP -0.131 - 0.203 -0.157 k 0.008 -0.124 rt 0.034 -0.13 aP, /aS,,(CH) 0.068 2 0.002 - 0.050 -0.068 -0.073 rt 0.004 0.050 ~P,/~S,,(G HCH) a The experimental values are average values derived from the intensities for CH, and CD, reported in ref 1 3 and 14. Error limits are twice the standard deviation. Taken from ref 11. Note that the calculated sign for ap,.aS,, from this reference is "+ ", due to a different definition of S, than used for the other two calculations. See text. Taken from ref 12. d Units are e.
*
TABLE I11 : Comparison of the Calculated and Experimental apla Q's and Integrated Band Intensities for Infrared-Active Fundamentals of CH, and CD, CH, apx/aQ,, (e u - ' l Z ) apXIaQ4, (e u - ' I 2 ) Exptla CND0/2b
PMC
+0.155 k 0.008 -0.146 -0.203
CD,
apxlaQax(e u-'")
+ 0.110* 0.002
aPxlaQ4, ( e u-'")
k0.105 * 0.010 -0.116 - 0.140
-0.086 - 0.098
k0.082 + 0.001
- 0.048 -0.078
A , (kmmol-l) A , (km mol-') A , (km mol-') A , (km mol-') Exptla 65.7 ?: 4.2 35.4 * 0.8 32.1 ? 3.0 19.7 i 0.2 CND0/2b 62.1 21.7 39.6 11.3 PMC 120.9 28.2 57.6 17.9 a Values obtained by averaging data from ref 1 3 and 14. Calculated using CNDO/2 tensors in Table I. Calculated using ab initio tensors in Table I. ~
ap/aS values to the experimental intensity data for CH4 and all its deuterated isotopic derivatives. (They were also aided in their choice by a CND0/2 calculation.) I t is of some interest to calculate the dipole derivatives ap,/aQi,, and hence A , values, from the polar tensors (or apx/aS,,values) calculated by CND0/2 and by Meyer and Pulay, in order to compare calculation and experiment by examining the fit to the actual measured experimental data. This comparison is given in Table III. We see that the much-maligned CNDO/2 calculation predicts intensities that are closer to the experimental values than does the ab initio calculation by Meyer and Pulay even though the latter is expected to be a much better calculation. It should be noted here that the values quoted from Meyer and Pulay were those calculated with the basis set that gave the minimum total energy for CH4 (calculation 15).12 These calculated dipole derivatives do not give the best fit to the experimental infrared intensities, providing yet another example where the minimum energy criterion for choice of best basis functions does not provide wave functions that also yield the best dipole moment derivatives and infrared intensities. Prasad and Singh have recently suggested that one criterion for choosing the preferred signs for the ap/aQ,'s be that the signs should be preferred that resulted in a maximum value for the difference between the square of the effective charge [c2 on the central atom (here carbon) and the sum of EH2 values for the terminal atoms (here hydrogen).16 The values of the invariants for the atomic polar tensors from CH4 are listed in Table IV. We see there that our preferred sign choice leads to the smallest absolute value of l[c2 - 4 6 ~ ~and 1 , is the minimum not the maximum value. Since we do not believe there is any physical reason to prefer a maximum or minimum value for this function, we suggest that this criterion (maximum I[c2 - 4tH21)be abandoned. Incidentally, it is of some interest to note that Prasad and Singh apparently did obtain the same sign choice (-,-) as we did, using their criterion. However, one must realize that the signs of ap/aQivalues are meaningless without the detailed knowledge of all the definitions used to obtain
TABLE IV: Atomic Polar Tensor Invariants Derived from CHAa
-
~~
Hydro- t c 22Exptl signs Carbon gen 4 & ~ (-,-) (preferred; see text) p z 0.000 0.041 j F 0.014 -0.004 5 0.024 0.166 F Z 0.0006 0.027 -0.109 (-,+
1
0' 0,000
p
0.301 5 0.521 E 2 0.271
0.028 0.003 0.136 0.018 t 0 . 1 9 7
P 2 0.000
CNDOI2
0.024 0.036 -0.009 & 0.062 0.128 E 2 0.0038 0.016
p
MP (ref 12)
0' 0,000 &
E'
0.077 0.133 0,018
0.055 -0.019 0.195 0.038
See ref 2 for definitions of &,the effective charge,jF, the mean dipole derivative, and p 2 , the anisotropy. Here jT and are in units of e and p 2 and t 2 are in units of e 2 . a
TABLE AI : Definition of Internal Coordinatesa R , = Sr,, R, = 6 r l , R, = 8r1, €2, = 6 r , 5
R, = R, = 6 a 2 , , R, = Sa,,,
R, = 6ff3,s R,,= sa,,, a The subscripts refer to the atoms (forming the internal coordinate) shown in Figure 1;rij is a CiHj bond and ffjjk is a HiCjHk angle. R5
=
'@',I3
them. Comparing Prasad and Singh's values of Ea2 with ours shows that their (-,-) convention for the signs of ap/aQ/s corresponds to our (-,+) convention. We believe there is an inconsistency between the definitions in the normal coordinate calculation used by Prasad and Singh16 and those used by Meyer and Pulay,12 with which they compared in reaching their decision about sign choices.
The Journal of Physical Chemistry, Vol. 82, No. 2, 1978 229
Atomic Polar Tensors from CH,
TABLE AII: D e f i n i t i o n s of t h e S y m m e t r y Coordinates in Terms of t h e Unnormalized U M a t r i x a
R, a,
SI
e
G:
fy
fz a
1.0 0 0 3.0
i;SRb
Taken from r e f 10.
R4
1.0
1.0
0 0 -1.0 0 -1.0 0
-1.0
0
0 0 0 0 0 0
;:
R,
R2 1.0 0 0 2.0
0 0 0 0
0 0
- 1.0 0 - 1.0 0
1.0
- 1.0
0 0
0 0
CH,a ( m d y n / A )
F a
5.842
0.0
0.0 0.0 0.0 0.0
0.486
0.0 0.0 0.0
0.486
0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0
5.383 0.206
0.206 0.458
0.0 0.0
T h e angle b e n d i n g force constants have been weighted
by ( r 0 C H ) ' a n d the stretch-bend i n t e r a c t i o n f o r c e constants by r ° C H . T h e force f i e l d was t a k e n from r e f 9.
TABLE AIV: N o r m a l Coordinate T r a n s f o r m a t i o n Matrices for CH, a n d CD, Element
R6
R,
R8
0 0
0
0 -1.0
0 0
1.0
2.0
0 1.0 0 2.0
0 0 1.0
-1.0 0 1.0 0 -1.0 0 1.0 1.0
-1.0
0
2.0
0
0 -1.0 0
-1.0
-2.0
1.0
0 -1.0 1.0
0 0 1.0
R9
0 1.0
R,o 0 -1.0
-1.0
-1.0
0
0 -1.0 0
-1.0
0 1.0
1.0
0
0 1.0
- 1.0 1.0
1.0
Redundant coordinate.
TABLE AIII: S y m m e t r y F o r c e Constants for A, E
R5
CH,
0.704627
1.724948 0.035642 - 0.035642 1.724941
1.219865 0.037817 - 0.037817 1.219865
1.049622
0.779423 -0.010493 - 0.269383 1.168279
-0.151891 1.551079 a T h e L elements - - r the f y a n d U n i t s are u - ' with t h e f x b l o c k .
Acknowledgment. We are grateful to the National Science Foundation (Grant No. CHE-74-21471)for partial financial support of this work.
CD4
0.9961 10
- 0.04084
near-Hartree-Fock calculation. Both calculations are also in surprisingly good agreement with experiment. The best values for the H- and C-atom polar tensors in CHI are believed to be given in the (-,-) column of Table I. The polar tensor for H(2) in the z' bond-axis coordinate system may be transferred to other hydrocarbons, according to the scheme used for F-atom polar tensors in fl~orocarbons,l-~ to predict the infrared intensities of hydrocarbons. We have tested its transferability in some CH,F, ( x + y = 4) molecules with moderate success.6
b l o c k are identic1
Once again, we must emphasize the importance of a consistent treatment all the way from the experimental data to the final interpretation. Table IV serves one other important purpose; namely, it summarizes the important invariant properties of the atomic polar tensors from methane. This table contains essentially the same information as is contained in less compact form in Table I. For example, a comparison of the calculated values oft;^ with the experimental values derived from the two sign choices again leads to the conclusion that the (-,-) choice is preferred.
Conclusion Once again we see that it is very easy to have apparent inconsistencies in signs due to inconsistencies in the arbitrary sign conventions occurring in the calculation of dipole moment derivatives, unless one person carries through the entire calculation from start to finish. When the sign conventions are consistent, the values of ap,/aSj, (or of polar tensor elements) calculated by the approximate CNDOIB method are in surprisingly good agreement with the results obtained by Meyer and Pulay12 in their
Appendix Normal Coordinates of CH4 and CD4. The internal coordinate definitions used in the normal coordinate calculations of CHI and CD4are given in Table AI. The symmetry coordinates defined by ShimanouchilOare given in Table AIL The force field defined by Duncan and Mills@is given in Table AIII. The normal coordinate transformation matrices are given in Table AIV. The CH bond length was taken from ref 9. References and Notes (1) The procedure and concepts were introduced by (a) J. F. Biarge, J. Herranz, and J. Morcillo, An. R . SOC.ESP.Fls. Qulm., Ser. A , 57, 81 (1961); see also (b) W. B. Person and J. H. Newton, J. Chem. Phys., 61, 1040 (1974) (Part Iof this series); (c) J. H. Newton, Ph.D Thesis, University of Florida, 1974. (2) J. H. Newton and W. B. Person, J. Chem. Phys., 64, 3036 (1976). (Part 11 of this series.) (3) W. B. Person and J. Overend, J. Chem. Phys., 66, 1444 (1977). (4) B. J. Krohn, W. B. Person, and J. Overend, J. Chem. Phys., 65, 969 (1976). (5) E. J. Krohn, W. B. Person, and J. Overend, J. Chem. Phys., In press. (6) J. H. Newton, R. A. Levine, and W. B. Person, J. Chem. Phys., 67, 3282 (1977). (Part I11 of this series.) (7) J. H. Newton and W. 8. Person, J. Chem. Phys., in press. (Part IV of this series.) (8) See the discussion of CH, in the review by W. B. Person and D. Steele in "Molecular Spectroscopy", Vol. 2, The Chemical Society, London, 1974, p 357 ff. (9) J. L. Duncan and I.M. Mills, Specfrochlm. Acta, 20, 523 (1964). (10) T. Shimanouchi, "Physical Chemistry; An Advanced Treatise", Vol. IV, H. Eyring, D. Henderson, and W. Jost, Ed., Academic Press, New York, N.Y., 1970. (11) G. A. Segal and M. L. Klein, J. Chem. Phys., 47, 4236 (1967). (12) W. Meyer and P. Pulay, J. Chem. Pbys., 56, 2109 (1972). (13) E. T. Ruf, M.S. Thesis, University of Mlnnesota, 1959. See also J. Heicklen, Spectrochlm. Acta, 17, 201 (1961). (14) S. Saeki, M. Mizuno, and S.Kondo, Spectrochlm. Acta, Par! A , 32, 403 (1976). (15) S. Kondo and S. Saeki, Spectrochlm. Acta, Part A, 29,735 (1973). (16) P. L. Prasad and S. Singh, J. Chem. Phys., 66, 1621 (1977).
