J. Phys. Chem. 1984, 88, 2394-2397
2394
Dipole Moment Derivatives, Polar Tensors, and Effective Charges of Silicon Tetrafluoride Kwan Kim Department of Chemistry, College of Natural Sciences, Seoul National University, Seoul 151, Korea (Received: June 23, 1983)
Preferred signs for the dipole moment derivatives of SiF4are selected on the basis of comparison of CNDO calculated values with the experimental alternatives. The apparent sign discrepancy from earlier reports has been identified. The effective atomic charge for fluorine was found to be xFIc= 0.574, in good agrement with that in other fluorine compounds.
Introduction The analysis of integrated infrared intensities in terms of atomic polar tensors1B2 and effective atomic charges3 has revealed a hitherto unnoticed order in the structural parameters that control vibrational transition moments. The atomic polar tensor for atom CY in a molecule is defined as the conjugate (transposed) gradient of the molecular dipole moment vector, P,written in various notations as’ aP,lax,
p,~=v,*p
[
z apy/ax, 3P,lax,
aP,la Y , aP,laZ, aPy/aY, apy/aza] aP,la Y , aP,lazoi
(1)
and the square of the effective atomic charge is defined as one-third of the sum of squares of the polar tensor components; that is Xa2
= (1 /3)V,*PVUP
(2)
The analysis of the integrated intensities of a variety of hydrocarbons, fluorocarbons, and other materials with diverse structures revealed that the effective charges for the hydrogen and fluorine atoms in these systems fell within the fairly narrow range of values3t5xHje= 0.088 f 0.015 and x F j e = 0.57 f 0.05, respectively. Furthermore, the components of their polar tensors were sufficiently insensitive to structural differences that they could be used to predict meaningfully accurate vibrational transition moments in hydrocarbons and various fluorine Reasonably successful predictions for the infrared intensities of the fundamental absorption bands for molecules containing fluorine atoms encourage further testing of the atomic polar tensors and effective charges for various fluorine compounds. In this paper we report the interpretation of the intensities of the SiF, molecule in order to determine if the effective fluorine charge in SiF4 is really close to those of other fluorine compounds. Previously Prasad et aLl2 reported the effective fluorine charge of SiF4. However, their criterion for choosing the preferred signs for the (1) W. B. Person and J. H. Newton, J . Chem. Phys., 61, 1040 (1974). (2) J. F. Biarge, J. Herranz, and J. Morcillo, An. R. SOC.ESP.Fis. Quim. Ser. A . 57., 81 - (1961). (3) W. T. Kiig, G: B. Mast, and P. P. Blanchette, J . Chem. Phys., 56,4440 (1972); 58, 1272 (1973). (4) This definition differs from the original one, ref 3, by the factor of (5) Either the natural (atomic) or the International System (SI) of units
.-
is used throughout this paper. For a summary of these units and conversion factors to other intensity units, see W. B. Person, S. K. Rudys, and J. H. Newton, J . Phys. Chem., 79, 2525 (1975). (6) B. J. Krohn, W. B. Person, and J. Overend, J . Chem. Phys., 67, 5091 (1977); 65, 969 (1976). (7) W. B. Person and J. Overend, J . Chem. Phys., 66, 1442 (1977).
(8) J. H. Newton, R. A. Levine, and W. B. Person, J . Chem. Phys., 67, 3282 (1977). (9) J . H. Newton and W. B. Person, J . Phys. Chem., 82, 226 (1978). (IO) W. B. Person and J. H. Newton, J . Mol. Struct., 46, 105 (1978). (11) J. D. Rogers and J. J. Hillman, J . Chem. Phys., 75, 1085 (1981). (12) P. L. Prasad and S. Singh, J . Chem. Phys., 66, 1621 (1977).
0022-3654/84/2088-2394$01.50/0
dipole derivatives with respect to normal coordinates, that the correct signs correspond to values of the atomic effective charges which show a maximum difference for the central and terminal atoms, was criticized as not valid in general by many investigat o r ~ . ~ We ~ , ’have, ~ therefore, reinvestigated the polar tensor and effective charge of fluorine atoms in SiF4. This study includes, in addition, the results of a C N D 0 / 2 calculation of the atomic polar tensors in SiF4. It was found that the effective fluorine charge is xF = 0.574, a value that is more representative of that in various dlorine compounds.
Method of Calculation Starting with the experimental measurement, we note that the integrated molar infrared absorption coefficient, Ai, for the ith fundamental band is expressed as follows in the rigid rotorharmonic oscillator approximation:
’*
Ai = (Ndi/3cZ)IdP/dQi12 (3) Here, N is Avogadro’s number, di is the degeneracy of the ith fundamental vibrational mode, c is the velocity of light, and aP/aQi is the vector dipole moment derivative with respect to the ith normal coordinate. Thus, the absolute values of aP/aQi are obtained from the measured integrated intensities. For molecules with any reasonable symmetry, the direction of dP/eQi is parallel to only one coordinate axis (X, Y,or Z ) , with either plus or minus sign, designated by the symbol 6,. Thus, we can write
aP/aQi = 6,(dP,/aQi)t (4) Here t is X,Y,or Z in the molecule-fixed Cartesianlcoordinate system parallel to the principal axes of inertia, and t is the unit vector for that coordinate, with 6, = f l . In terms of these coordinates, we can write the 3 X (3N - 6) matrix of dipole derivatives with respect to the normal coordinates obtained from the experimental intensities. We call this matrix the PQmatrix. The normal coordinates are related to the 3 N - 6 internal symmetry coordinates Siby15
S = LQ or Q = L-lS (5) Here S and Q are the column vectors of Siand Qi coordinates, respectively, and L is the (3N - 6) X (3N - 6 ) normal-coordinate transformation matrix. Thus, we can write the 3 X (3N - 6) matrix of dipole derivatives with respect to symmetry coordinates (P,) as16 P, = P&-’ or PQ = P,L (6) We may transform from internal symmetry coordinates Sj to the (13) J. H. Newton and W. B. Person, J. Phys. Chem., 82, 226 (1978). (14) A. B. M. S.Bassi and R. E. Bruns, J . Phys. Chem., 80,2768 (1976). (15) E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, “Molecular Vibrations”, McGraw-Hill, New York, 1955. (16) Equation 6 is just the matrix expression of the chain rule for partial
differentiation:
a p / a s , = :(aP/aQ,)
0 1984 American Chemical Society
(aQ,/as,)
The Journal of Physical Chemistry, Vol. 88, No. 11, 1984 2395
Dipole Moment Derivatives of Silicon Tetrafluoride
3N
- 6 ordinary internal coordinates Rk by
S = UR or R = U’S (7) Here U’ is the transpose of U, since this transformation matrix is orthogonal.ls Thus PR = P,U or P, = PRU’ (8) All of the coordinate transformations above have been between various molecule-fixed coordinate systems. Now let us consider the transformation to a space-fixed Cartesian coordinate system. In considering this transformation, one must now write explicitly the six “Eckart conditions” (see ref 15) defining the translations and rotations of the whole molecule. The transformation from the internal coordinates to the 3N ordinary space fixed Cartesian coordinates is given by15 R/p = (B/B)(X)
(9)
Here R is the (3N - 6) X 1 column matrix of the internal coordinates; p is the 6 X 1 column matrix of the Eckart conditions so that R/p is the “augmented” 3N X 1 column vector; B is a (3N - 6) X 3N transformation matrix whose elements are defined by Bij = aRi/dXj;and B is the corresponding 6 X 3N matrix relating the p matrix to X. Thus, the 3 X 3N polar tensor P, is P, = PRB + Po@ (10)
z I
Figure 1. Coordinate axes and molecular orientation for SiFI used in normal-coordinate calculation. TABLE I: Structural Data and Definition of Internal and Symmetry Coordinates for SiF,
masses ( u ) :a rnsi = 27.976928, m F = 18.9984033 structure: R s i ~ =0.1552 nm internal coordinatesC R , = fir,, R6 = ‘01214 R7 = f i f f 2 l S R , = fir13 R, = 6rI4 R” = f i % I S R , = 6r,, R 9 = 6cu314 R , = Sa213 Rin = fia,is
Here we have defined a new tensor P, with elements aP,/api. Since a translation of the whole molecule does not change the dipole moment of the molecule, aP,/api is zero for p l , p2, and p3, the translations along X , Y, or 2 for the three rotations (p4, ps, and p6) about the principal axes of inertia ( X , Y, and 2)we may use the relationship given by Biarge et aLz In this way we may obtain the polar tensor P, from the experimental data by transforming from PQto P, using eq 6, 8, and 10 to obtain’ P, = PQL-IUB+ Pp@ (1 1) where the explicit implication of P, is given above in eq 1. We note that we may also transform from P, to PQby using the inverse relations to obtain’ PQ = P,AU’L (12) where the 3N X 3N - 6 matrix A is the inverse transformation from internal coordinates to Cartesian coordinates, defined so that BA = 13N4, a 3N - 6 square identity matrix. Equation 12 has particular interest in that it is especially easy to calculate the polar tensor P, from quantum-mechanical procedures. One simply calculates the dipole moment at equilibrium and again after a small displacement in the well-defined space-fixed X direction of the LY atom. The difference in the X component divided by the displacement is taken to be the approximately the derivative
ap,/ax, APJAX, (13) The change in the total dipole moment vector when the LY atom is displaced in the X direction also gives aP,/aX, and aP,/aX,. Similarly we can evaluate the other terms in the polar tensor P,“ and combine them to obtain P,. Using eq 12, we can then convert the calculated polar tensor to PQ,which can be compared with the experimental value. The polar tensor values for SiF4 were calculated by means of eq 1 1 using PQvalues of Schatz et a l l 7 The rotational contribution to P,, P,@,is zero because of the null equilibrium dipole moments. The Cartesian coordinate axes, numbering of atoms, and orientation of the SiF4 molecule are shown in Figure 1 . The equilibrium structural data, and the definition of the internal and symmetry coordinates, are listed in Table I. The L matrix for SiF4 was calculated by using the harmonic force field reported by McDowell et a1.l’ These values are given in Table 11. The B matrix was evaluated by using Wilson’s rnethod.l5 ( 1 7 ) P. N. Schatz and D.F. Hornig, J . Chem. Phys., 21, 1516 (1953). (18) R. S. McDowell, M. J. Reisfeld, C. W. Patterson, B. J. Krohn, M. C . Vasquez, and G. A. Laguna, J . Chem. Phys., 77, 4337 (1982). (19) A. H. Wapstra and N. B. Gove, Nucl. Data Tables, A9, 265 (1971). (20) K. Hagen and K. Hedberg, J . Chem. Phys., 59, 1549 (1973).
Symmetry Coordinates
S,= (1/2)(R1t R , + R , + R,) S, = (1/2)(R, - R , - R , + R,,)
A,
E
S, = ( l / 1 2 1 ‘ z ) ( 2 R s - R 6 - R 7t 2 R , - R , - R I n ) S, = ( 1 / 6 ’ / ’ ) ( 2 R ,- R , - R 4 ) Ss=(1/12”2)(2R,-R,-R7-2R,+R,+R,,) S, = (1/21”)(R3- R,) S, = ( l / 2 ) ( R 6- R , + R , -Ria) S, = (1/12”*)(3R, - R , - R , - R 4 ) S, = ( 1 / 6 “ ’ ) ( R , R 6 + R , - R , - R , - R , , ) S,,d= (1/6”’)(RS t R , t R7 + R , t R , R,”)
Fx
F, F,
+
+
a Reference 19. Reference 20. The subscripts refer to the atoms shown in Figure I ;rij is a SiiFj bond and aijh is a FiSijFk angle.
TABLE 11: Harmonic Force Field, Normal Coordinates, and Integrated Intensities in SiF, A
VI
E
V2
Harmonic Frequencies (cm-’)a 807.1 F ”3 26 7 v4
1044.2 389.8
Harmonic Force Constants (N m-’)b F K,, 636.66 728.34 K4s 31.64 64.00 Kss 107.00 0 64.00
K,, K,, Kn K,,
A
E
Normal Coordinates ( U - I ’ ~ Y Q2
Qi
S,
A
E rxd
s, s,
Q,
Q3
0.229425
A,(v,)
0.256 042
0.017991 0.280471
0.316 181 -0.210201
Intensities (Km.mol-’)e 591 A,(IJ.,)
114
Reference 18. Reference 18. The angle bending force constants have been weighted by 1 A 2 and the stretch-bend interaction force constant by 1 A . The indices labeling the normal coordinates correspond t o the labels identifying the normal frequencies above. The K and I. elements for the I:, and F, block are identical with the V X block. e Reference 17. a
Results and Discussion Prasad et a1.I2 reported the preferred sign of dipole moment derivatives with respect to normal coordinates, dP/dQ3 and dP/dQ4, for SiF4. However, as mentioned above recalculations
2396
The Journal of Physical Chemistry, Vol. 88, No. 11, 1984
Kim
TABLE 111: Atomic Polar Tensors for SiF, Given in Multiples of e exptla (-- b
++I
PxSi
0
t1.219
0
pxF(z)e
pxF(3)
calcd CNDOi2C
(Tfj
22.190 0 ~2.190 t2.190
30.425 0 ~0.425
0”
F1.358 0
0 70.752 0
20.596 0
~ 0 . 4 2 5 ,0”16] t0.116 0 30.466
20.117
T0.793
2.590 0 0 2.590 0
] ]
[
-0.445 [0 0 -0.985 [0 0.191
0 0
-0.445
0 0 -0.445 0
0.191
0 -0.512
1
The experimental tensors are derived from the intensities of SiF, in ref 1 7 . The signs are the signs of the 3PjaQi’s. For example ( 3 7 ) would mean that the signs of aP/aQ, ahd aP/aQ, are both either negative or positive. These tensors were calculated by using the INCNDO program 141 from QCPE. The value with upper sign (+1.219) is, for example, the tensor element obtained from the signs of aP/aQi’s corresponding to the upper sign combination (--) in two sets of sign choices ( T T ) . See note b above. e The atomic polar tensor of fluorine atom number 2 in Figure 1. a
TABLE IV: Comparison of Experimental and Calculated Values of aP/aSj’s for SiF, in Units of e exptl (Tsja
aP,/aS,,(SiF) aP,/aS,,(6 FSiF) a
~1.811 t0.587
(Ti)
calcd CNDO/2
70.915 t0.76 1
-1.215 +0.798
TABLE V: Effective Fluorine Charges for Various Compounds in Natural Units (e) SiF, 0.574a CHF, 0.59gb BF, NF, F,CO F,CS
See the notes in Table 111. e
are necessary in order to ensure that the sign conventions are consistent throughout the treatment. The atomic polar tensors in SiF4calculated from the experimental intensity data taken from the work of Schatz et al.17 are given in Table 111 as a function of sign choice for the aP/aQ,. In addition, Table I11 contains the calculated values from the C N D 0 / 2 quantum-mechanical calculation. In order to reach some decision about which of the several possible polar tensors are correct for SiF,, we may compare the different possible sets of BP/aS, values from the experimental data with values calculated from the C N D 0 / 2 method. Those values are presented in Table IV. Comparison of the calculated results with the possible experimental results in Table IV leads us to prefer one sign combination. The values of dP/dS, calculated by the CNDO/2 method agree exceptionally well in magnitude and sign with the experimental dipole moment derivatives from the (-+) sign combination (aP/aQ, negative and aP/dQ4 positive). Although the agreement between calculated aP/aS, values and experimental values shown in Table IV clearly indicate that dP/dQ, and dP/dQ4 are negative and positive, respectively, the small magnitudes of the values in that table, and the notorious difficulty in quantitative prediction on dipole moment derivatives from approximate quantum-mechanical treatments, suggest that there may still be some doubts remaining about the certainty of the (-+) sign choice. Nevertheless, it appears clear that the two-sign assumption [(++) and (+-)I may be eliminated from further consideration on the basis of the disagreement in signs of the BP/dS, values. The doubt on the possibility of signs of dP/aQ3 and dP/dQ4 being both negative may be, however, dispelled by careful examination of Table 111, comparing the atomic polar tensors calculated by the CNDO/2 method with those obtained from the experimental data with the two different sets of signs (--) and (-+)I. In particular, the values deduced from the experimental data for the silicon atom polar tensor are significantly different for the two different sign choices under consideration. It is quite clear that the values calculated by using the C N D 0 / 2 method agree with the (-+) sign choice and do not agree with the values derived from the (--) sign choice. 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. As described above, Prasad et a1.12suggested that one criterion for choosing the preferred signs for the aP/dQ,’s be that the signs should be preferred which result in a maximum value for the
0.556c 0.525e 0.606g 0.6 1 2’
This work. Reference 31. Reference 12. Reference 8. Reference 34. Reference 35.
CH,F CWZ CF, SF, Reference 32.
Reference 33.
0.58d 0.6 1j f 0.60h 0.640j
Reference 8 Reference 8.
difference between the square of the effective charge xsF on the central atom (here silicon) and the sum of xF2values for the terminal atoms (here fluorine).21 The preferred sign choice for SF,(-+) gives xe2 - 4xFz= 3.478 ez, which corresponds to the maximum difference found by Prasad et al. However, 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 be abandoned. Incidently, it is of some interest to note that Prasad et a1.12 obtained the different sign choice (--) using their criterion although their values of effective charges were almost the same as we got. However, one must realize that the signs of aP/aQi values are meaningless without detailed knowledge of all the definitions used to obtain them. Comparing Prasad et al.’s values of xa2with ours shows that their (--) convention for the signs of aP/aQ,’s corresponds to our (-+) convention. We believe there was an inconsistency in the definitions of their normal-coordinate calculation. The force constants of SiF4, established by using Coriolis constants and isotopic frequency shifts as the necessary additional constraints to fix the three constants in the 2 X 2 F, symmetry block, have been reported in at least 10 different investigations.18~22-30 Unfortunately, these force fields have not been in good agreement. Moreover, all of the papers except McDowell et al.’sI8used observed rather than harmonic frequencies for the fundamentals. With modern high-resolution techniques the vibration-rotation bands of heavy spherical-top molecules can be resolved and accurate values of the Coriolis constants obtained, thus removing the uncertainty that is inevitable when these must be estimgted from unresolved band contours. We believe that a force field of McDowell et aL1*is reasonably accurate since they (21) This criterion differs from the original one, ref 12, by the factor of (see ref 4). (22) J. Heicklen and V. Knight, Spectrochim. Acra, 20, 295 (1964). (23) J. L. Duncan and I. M. Mills, Spectrochim. Acta, 20, 1089 (1964). (24) D. C. Mckean, Spectrochim. Acta, 22, 269 (1966). (25) I. W. Levin and S. Abramowitz, J . Chem. Phys., 44, 2562 (1966). (26) A. Ruoff, Spectrochim. Acta, Part A , 23, 2421 (1967). (27) I. W. Levin and S. Abramowitz, J . Res. Natl. Bur. Stand., Sect. A , 72,‘24i (1968). (28) R. J. H. Clark and D. M. Rippon, J. Mol. Spectrosc., 44,479 (1972). (29) F. Koniger and A. Muller, J . Mol. Spccrrosc., 65, 339 (1977). (30) F. Koniger, A. Muller, and W. J. Orville-Thomas, J. Mol. Srruct., 37, 199 (1977).
Dipole Moment Derivatives of Silicon Tetrafluoride TABLE VI: Comparison of Fluorine Atom Polar Tensors in SiF,, CF,, CH,F, and SF,
PzzFIe SiF, CF,a CH,F~ SF,C a
Reference 8 .
-0.793 -0.92 -0.93 -0.986
Reference 8.
( p x x F= PyyF)/e -0.425 -0.33 -0.26
-0.358
Reference 35.
have employed high-resolution techniques to obtain accurate values of the Coriolis constants and, in addition, examined the spectra of the overtone and combination bands in order to estimate the anharmonicity constants and the harmonic fundamental frequencies. Accordingly, McDowell et ale's force field was used in this work. Prasad et al.12 used the normal coordinates from the force field described by MckeamZ4 Apparently it is noticed that there is a sign disagreement in the values of normal coordinates between Prasad et al.'s and this work. Once again, we must emphasize the importance of a consistent treatment all the way from the experimental data to the final interpretation. The results reported here provide additional information on the nature of fluorine atom effective charges. Table V contains the effective charges of fluorine atoms for various compounds. As we have reported p r e v i o ~ s l ythe , ~ ~effective fluorine charges given in Table V reveal once again rather surprising features in view of conventional chemical wisdom. The effective charges for the fluorine atom in such diverse compounds fall within a narrow range of values. These results suggest that the effective charge may represent a particularly localized combination of chemical bond properties, whose magnitude is less sensitive to the nature of central atoms in their geometrical arrangement. This work provides additional information of the transferability of fluorine atom polar tensors. For spherical-top molecules, like (31) K. Kim and W. T. King, J. Chem. Phys., 73, 1967 (1980). (32) R. E. Bruns and A. B. M. S. Bassi, J . Chem. Phys., 64,3053 (1976). (33) J. H. Newton and W. B. Person, J. Chem. Phys., 64, 3036 (1976). (34) A. B. M. S. Bassi and R. E. Bruns, J . Chem. Phys., 62,3235 (1975). (35) K. Kim, R. S. McDowell, and W. T. King, J. Chem. Phys., 73, 36 (1980). (36) K. Kim, Ph.D. Thesis, Brown University, Providence, RI, 1980.
The Journal of Physical Chemistry, Vol. 88, No. 11, 1984 2397 SiF,, the polar tensor for a fluorine atom along the positive Z axis is a diagonal tensor with components as PZF = dPz/dZFand P x / = PyVF= aPx/cIXF. In Table VI the fluorine atom polar tensors in SiF4 are compared with the polar tensors in CF4, CH3F, and SF6. The results summarized in Table VI help quantify the transferability of fluorine atom polar tensors. The distinct tensor components, Pr,F and PxxF,each differ by less than 20% from the mean values in the four compounds listed. A closer inspection, however, indicates that some of these differences might nonetheless be meaningful, suggesting that the longitudinal component of the fluorine atom polar tensor Pz/ is more strongly influenced by nearest-neighbor Si-F, S-F, or C-F interactions, whereas the transverse component PxxFis more sensitive to next-nearest H-F or F-F interactions. To determine whether this reationalization provides a useful basis for interpreting the difference between atom polar tensors, however, would require more information than that provided here. In conclusion, we see that it is very easy to have apparent inconsistencies in signs due to inconsistencies in the arbitrary sign conventions occuring 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/aS, or of polar tensor elements calculated by the approximate CNDO/2 method are in surprisingly good agreement with experiment. The best values for the F and Si atom polar tensors in SiF, are believed to be given in the (-+) column of Table 111. The effective atomic charge for fluorine was found to be XF/e = 0.574, in good agreement with that in various fluorine compounds. This may, perhaps, be indicative of the fact that the problem of interpreting infrared intensities can be treated in a more simple and precise way by using the "atomic effective charge". Finally, we need more information to interpret the difference between atom polar tensors in various compounds. However, the polar tensor for F(2) in the Z bond-axis coordinate system may be transferred to other fluorinated silanes, according to the scheme used for F atom polar tensors in fluorocarbons,6-10 to predict the infrared intensities of fluorinated silanes.
Acknowledgment. Partial financial support from the Korea Science and Engineering Foundation is gratefully acknowledged. Registry No. SIF,, 7783-61-1; F, 14762-94-8.
