J. Phys. Chem. 1981, 85, 1998-2007
1998
IBC model was modified to include density dependence of ( P ) . The calculated relaxation rate of CS2*by CS2 in the liquid phase is too slow to explain the experimental data if only the collision frequency is scaled for density. Madigosky and Litovitz2argued that the relaxation of CS2* by collisions with CS2 in the gas phase was aided by the acceleration of the colliding partners over the intermolecular potential well. Many-body interactions make this well less deep in the liquid state so that ( P ) ( * is the z component in e A (Figure 1) of the dipole moment for the equilibrium configuration, and Iy = I, is the moment of inertia in u A2 about the y or x axis; this second term in eq 10 is the contribution from the p# matrix and is the “rotation correction”. Hence, two experimental values ( A , and A3) determine the values of the parameters a and b for symmetrical AB2 linear molecules, while four experimental values (Al, Az, A,, and p > ) determine the four polar tensor parameters a, a’, b, and b’for unsymmetrical ABC linear molecules. The numerical values of these polar tensor parameters depend upon the signs chosen for the ap,/aQj values. Table I1 liits all possible values for the parameters (derived from eq 10 and 11) for all possible signs for the dipole derivatives ap,/aQ,. One of the major contributions the analysis of intensity results from the ab initio quantum mechanical calculations is that the calculated values allow us to determine the probable signs that are correct for the ap,/aQi values, as described in more detail below. The elements (Ll1-l, Lgl-’, L1yl, and LS-l (eq 11))of the normal coordinate transformation matrix for ABC molecules depend on the values assumed for the force constants. The L-l matrix elements for AB2molecules and the values = Lzrz,? for ABC molecules depend only on the of masses and geometrical arrangements of the atom (see ref 14 for details). The force constants we have used to obtain the experimental results in Table I1 are defined in Table 111. Since we expect that the same values for the polar tensor parameters should be derived from experimental data from HCN and from DCN, we see that the experimental results definitely indicate that ap,/aQ, and ap,/aQ3 have opposite signs (as emphasized recently by Hornig15). An analysis of the combination and difference bands of CO, and NZ0l0 indicates quite definitely that the signs of ap,/aQz and dpL/aQ1are the same in both molecules and that ap,/ilQ, in N,O has a sign opposite to dp,/aQ,. Hence we can choose the italicized sets of polar tensor parameters in Table I1 as being those determined by experiment for HCN and NzO. For the other molecules the experimental data do not dictate the choice of any one set of results as being preferred over any other.
Quantum Mechanical Calculations We have used a 4-31G basis set with the Gaussian 70 to calculate the atomic polar tensors and force and for the central atom constants for these linear molecules. The force constant calculations are described in part 2. Although some elegant 0 a’) new techniques have recently been developed16to reduce P,B = 0 -(a + a’) 0 (8) the computational effort required to calculate dipole de-(b + b‘) rivatives, our calculations were made by the “brute force” As shown in more detail elsewhere,l* the atomic polar method described below. tensor parameters (a, a’, b, and b? are related (eq 4)to the Energies, wave functions, and dipole moments were ilpa/dQi dipole derivatives from Table I by: (1)for AB2 computed for each molecule at a number of configurations symmetric molecules around the equilibrium ge0metry.l’ Displacements of the nuclei from equilibrium ranged from -0.04 to +0.04 A in a = (a~~/aQz&mX-’ appropriate directions. Most of the dipole derivatives (9) (APTs) reported in this first paper were obtained by b =(l/fi)(a~,/a~~w,~-~ calculating the dipole moment for the experimental (2) For ABC unsymmetrical molecules equilibrium geometry and repeating the calculation for the a = (aPx/aQ2x)L2x2r-1 + P~O~AZA(~)/I, molecule with one outer atom (i) moved 0.02 A along the P,A=
(0(n +
Q’
= (apy/aQzy)L,y,;l
P,c=
:
(7)
)
+ p,omCzC(e)/zx Iy = 1,
(10)
b = -(ap,/aQl)Ll,-l - ( ~ P J ~ Q , ) ~ , , - ~ b’ =
+ ( ~ P J ~ Q ~ ) L , , - ~(11)
-(a~z/a~1)~13-1
Here mi is the mass in atomic mass units (u) of the ith
(15)D. F. Hornig, J. Chern. Phys., 68,5668 (1978). (16)For example, see (a) A. Komornicki and R. L. Jaffe, J. Chern. Phys., 71,2150(1979);(b) P. Pulay, G. Fogarasi, F. Pang, and J. E. Boggs, J. Am. Chem. Soc., 101,2550 (1979). (17)L. E. Sutton, Ed., “Tables of Interatomic Distances and Configuration in Molecules and Ions”, ibid.,Supplement; Special Publications 11 and 18,The Chemical Society, London, 1958,1965.
The Journal of Physical Chemistry, Vol. 85,No. 14, 198 1 2001
Intensities of Triatomic Molecules
TABLE 11: Summary of All Possible Values of Atomic Polar Tensor Elements for Some Linear Triatomic Molecules Derived from t h e Data from Table I (Units Are e ) APTs molecule
atom
co
0 C
T
0.236 i0.472
i1.137 t 2.274
1.185 2.322
S
7 0.084 k0.168
7 1.200
1.206 2.406
I
CS,
a (t)" Linear Symmetric AB, ( D - h )
C a (or a ' ) (+,t)C
(t,i)C
(t,l)d
to.009 t0.117 70.126
t0.103 70.002
N,NO
3N
r0.108 t0.139 ~0.247
$0.138 t0.107 ~0.245
HCN~
H N C D N C
t0.05 t 0.59 70.64
t0.237 70.321 t 0.084
t0.13
t 0.04
t0.237 70.321 t 0.084
cl
~0.154 t0.245 T 0.093
t 0.274
~0.193
N C BrCN
iff
t 2.400
(t,T)d
i
11
iii
iv
Linear Unsymmetric ABC ( C s u ) 0 S C
ClCN
(t)b
b (or b ' )
ocs
DCN~
b
Br N C
2.004 70.159 +2.163 7 1.744 k0.248 t 1.496
71.536 T 0.809 t 2.345 T 0.626 31.237 t1.863
2.00 0.23 2.17
2.01 0.16 2.17
1.54 0.83 2.35
1.54 0.81 2.35
1.75 0.32 1.54
1.75 0.29 1.54
0.64 1.25 1.89
0.66 1.25 1.89
i0.22 ~0.35
70.218 i0.074 t 0.292
0.23 0.85 0.98
0.39 0.46 0.38
0.24 0.84 0.96
0.40 0.45 0.32
i0.13 i0.35 70.48
T0.218 70.074 i 0.292
0.14 0.79 0.82
0.36 0.55 0.49
0.24 0.72 0.73
0.41 0.44 0.34
r0.077 30.194
30.323 t0.542 70.219
70.420 30.097 50.517
0.39 0.65 0.29
0.54 0.55 0.40
0.48 0.38 0.54
0.63 0.15 0.61
t0.273 T0.102 i0.173
70.114 t 0.293 70.179
70.176 70.124 t0.300
0.35 0.45 0.11
0.51 0.32 0.31
0.29 0.36 0.34
0.47 0.17 0.61
T
10.101
t0.50 T 0.46
t 0.226
~0.031
refers t o the sign of apx / a Q,, ; the upper sign in the values given for a is obtained by using the + sign for apx / a Q,, etc. The first t refers t o the sign of a p x / a Q,, ; the second t refers t o the sign of p z o ;both t h e upper signs Signs of apr/aQ3. o r both lower signs go together. The + 0.009 value for a from 0 of OCS comes when both signs are taken positive (+,+). The first t refers to the sign of a p r / a Q , ;the second t to the sign for a p t / a Q , ; the value of - 2.004 for b from 0 of OCS comes when both signs are taken positive (+,+). e Sign choice i is ( t , t, i, t ) for a p x / a Q , , , p r o , a p t / a Q , , a p z / a Q 3 ,respecPreferred values (italicized) are taken from Kim and King, tively; ii is ( t , i, 2, t ) ;iii is (t, t , t , T ) ; and iv is ( 2 , 7 , t , T). ref 24. a t
TABLE 111: Force Constants (in N m - l ; 100 N m - ' = 1 mdyn A - I ) Used to Calculated t h e Normal Coordinates for Some Linear Triatomic Molecules (AB, and ABC) molecule CO, CS,
frAB
frBC
frr
AB,, Symmetric 1550 + 130 750
+60
fff
source
58.1 23.1
a a
48.4 78.0 23.7 52 29.4 17.6 17.1
b a
ABC, Unsymmetrical OCS NNO HCN FCN ClCN BrCN ICN a
1614 1788 625 933 522 437 339
749 1139 1870 1696 1820 1766 1719
+lo4 +136 -20 -24 +42 -8.0 -19
c
d e
e d
E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, "Mole-
cular Vibrations", McGraw-Hill, New York, 1955. Y. Morino and T. Nakagawa, J. Mol. Spectrosc., 26, 496 (1968). G. Strey and I. M. Mills, Mol. Phys., 26, 129 V. K. Wang and J. Overend, Spectrochim. (1973). Acta, Part A, 29, 1623 (1973). (See also footnote e.) e D. H. Whiffen, Spectrochim. Acta, Part A , 34, 1173, 1183 (1978).
positive z axis (for example). The dipole derivative was calculated by apa/azi = [ p " ( ~ -) ~ Y R , ) ] / O . O ~ Here p"(R) is the a t h component of the dipole moment calculated a t configuration R (zi =: z , , ~+ 0.02 A, in this
example); pa(R,) is the corresponding dipole moment for the equilibrium configuration. This calculation was then repeated a t zi = ze,i- 0.02 8, and the calculated approximate derivatives were averaged. This calculation gives the three elements in the third column of the APT for the ith atom. The process is repeated to obtain the remaining APT elements for atom i and then for the other outer atom. The APT for the central atom is the negative sum of those for the outer atoms (eq 6 and 8). For the calculation of electrical anharmonicities reported at the end of this paper, calculations of the dipole moment were made at many more configurations. The computed dipole moments were fitted to the dipole function expanded in terms of valence coordinates: pr = pzo + ( d p / a R A B ) m A B + ( a p / a R B C ) m B C + (a2p/aRAB2)mABP
+ (a2p/aRBc2)mBC2 + (a2p/ a R m a R B c )
~
A
B
~
pX = pXo+ (ap/aa)Aa (12) Here M A B is the change in bond length from the computed equilibrium bond length, and a is the ABC bond angle (Figure 1). The first dipole derivatives with respect to the internal valence coordinates are related to the derivatives with respect to Cartesian coordinates (APT elements) by P, PRB + P,@ (13) It should be emphasized again that the values for com-
B
C
2002
The Journal of Physical Chemistry, Vol. 85, No. 14, 1981
puted atomic polar tensor elements that are presented in the tables were calculated for displacements about the experimental equilibrium configuration. They differ from the values fit to eq 12, as is discussed below. If there were zero “noise” (or computational error), we could derive the values of n parameters from n dipole terms by fitting eq 12. Examination of the errors that result from attempts to fit m parameters to n points, m > n, suggests that this idealistic situation is not possible. The data were fit by a least-squares procedure and the uncertainties in the parameters were evaluated. These uncertainties arise both from computational noise and from inadequacies of the model. The procedure that was followed was to estimate a set of parameters, p, use these to compute an error vector for the observables, AX, and then to calculate the Jacobian matrix, J, whose elements are the partial derivatives d X i / d p j . Assuming a linear relationship between AXi and the Apj values, the necessary correction to the parameters to produce a least-squares fit is givenla by Ap = (JtWJ)-’JtWAX (14) The matrix W may be used to weight individual points Xi, though in the present context there did not appear to be any ground for such differential weighting so we used the identity matrix. Here Jt is the transpose of the Jacobian matrix J; the other symbols are defined above. By eliminating the kth row of the Jacobian the variation in the kth parameter is suppressed. The standard deviation of the derived parameter based on fitting of the function X = f(p) is given by = ~(x)[(J~WJ)ii-~]l/’ (15) [&)I2 = VtWV/(n - m) where V is the residual vector, n is the number of observed values, and m is the number of parameters. The correlation between the ith and j t h parameters is (JtWJ)iY1# ( (JtWJ)ii-1(JtWJ)j~’)1/2 (16) Even though we are dealing with computed quantities and have endeavoured carefully to select the nuclear coordinates so as to optimize the determination of the parameters, some of the potential and dipole terms are very ill-defined. A t this point it is worth mentioning some difficulties in obtaining convergence that were encountered during the Gaussian wave function calculations. In some of these calculations using the standard three-point convergence routine3 the successive solutions oscillated in energy and did not reach convergence even in 40 cycles. We found many times that the situation could be improved by changing the scale factor used for the initial extended Huckel calculation from 1.3125, the value in the standard program, to 0.875. This change may often result in relatively easy convergence with the three-point convergence routine. Sometimes (for example in calculations for ClCN) the convergence was to an obviously erroneous solution with orbitals constructed from 3p, and 3p atomic orbitals that were not degenerate. We were ohen able to get convergence to a solution of proper symmetry using LINK 401 (instead of the recommended LINK 4023) even with the extended 4-31G basis. The polar tensor parameters calculated by the Gaussian 70 progrm for the 4-31G basis set by the approximate “brute force” method described above, evaluated for displacements away from the experimental equilibrium ge(18) G. Strang, “Linear Algebra and Ita Applications”, Academic Press, New York, 1976.
Person et al.
TABLE IV: Summary of Atomic Polar Tensor Elements for Some Triatomic Molecules Calculated by Using t h e Gaussian 7 0 Program with a 4-31G Basis Set atom a/e (or a ’ ) b/e (or b ’ ) OCO
scs
Linear Symmetric AB, ( D - h ) 0 -0.39 -1.398 C +0.78 2.9 24 0.005 -1.521 S C -0.010 +3.042
Linear Unsymmetric ABC ( C - v ) 0 -0.304 -1.97 S -0.133 -1.54 C +0.437 +3.51 N,NO -0.082 -0.785
ocs
2 N H N C F N C
HCN(DCN) FCN ClCN
cl N C
-0.249 +0.331 +0.321 -0.289 -0,032 +0.002 - 0.333 0.331 +0.147 -0.256 +0.109
-1.453 +2.238 +0.301 -0.169 -0.132 -0,875 - 0.293 + 1.168 -0.152 -0.397 -0.549
+
TABLE V: Comparison between Different Levels of Approximate Quantum Mechanical Calculation with Experimental Values for Atomic Polar Tensor Elements for CO,, N,O, and HCN (AllValues Given in e ) molecule CO,
N,O
a’
a AB, Type
b
b’
CNDO -0.51 Gaus 70 -0.39 exptl t0.236
- 2.09 - 1.40 t1.14
ABC Type CNDO“ -0.53 -0.50 Gaus 70 -0.08 -0.25 NHF” exptl -0.108 or -0.139 o r
-1.03 -1.96 -0.78 -1.45 -0.33 -1.11 t0.626 51.24
+0.138
HCN CNDO“ +0.28 G a u s 7 0 +0.32 NHF“ SCFd CId VCIC +0.237 exptlb
+0.107 -0.26 -0.29
~0.321
+0.06 +0.30 +0.28 +0.31 +0.26 +0.21 t0.218
-0.25 -0.17 -0.11 -0.11 +0.03 +0.08 k0.074
” Taken from Bruns and Person, ref 20, converted to be consistent with our conventions. Here NHF refers t o the “near Hartree-Fock” results taken from the calculation by A. D. McLean and M. Yoshimini, IBM J. Res. Develop. SuppZ., 12, 206 (1968). From Kim and King, ref 24. From Liu, Sando, North, Friedrich, and Chipman, ref 22, adjusted to be consistent with our conventions. Here VCI refers t o results from a configuration interaction calculation. From Gready, Bacskay, and Hush, ref 23, adjusted t o be consistent with our conventions. Here SCF refers t o a self-consistent field (Hartree-Fock) calculation; CI is a result from a configuration interaction calculation. ometry, are summarized in Table IV.
Comparison between Calculated and Experimental Polar Tensor Parameters There have been other quantum-mechanical calculations of dipole moment derivatives reported for COP,NzO, and HCN.1s23 A comparison between the various calculations (19) See J. A. Pople and D. L. Beveridge, “Approximate Molecular Orbital Theory”,McGraw-Hill,New York, 1970; G. A. Segal and M. C. Klein, J. Chem. Phys., 47, 4236 (1967); program QCPE 141, Quantum Chemistry Program Exchange, Indiana University.
Intenstties of Triatomic Molecules
and the experimental values for APTs for these three molecules is given in Table V. For C 0 2 and N20, the calculated values from the Gaussian 70 calculation agree quite well with the experimental values. As expected, the Gaussian 70 calculation gives results that are in somewhat better agreement with experiment than does the CND0/2 c a l c ~ l a t i o n Both . ~ ~ ~calculations ~ overestimate the magnitude for a for the oxygen atom of C02, but the relative error from the Gaussian 70 4-31G calculation is about half that from the CNDO/2 calculation. The magnitude of b for the 0 atom in COz is calculated by Gaussian 70 to be too large by 0.26e or about 23%, considerably better than the CNDO result. Note that the calculated parameter values for the oxygen atom in N20 (a’and b’in Table V) are (perhaps fortuitously) similar to those for that atom in C02, in agreement with the experimental results. (The value of b’for 0 in N 2 0 is predicted in the Gaussian 70 calculation to be slightly smaller than that predicted for 0 in COPand a’ is predicted to be slightly larger than a for 0 in COP,both in agreement with the experimental results.) In the Gaussian 70 calculation for NzO the calculated magnitude of a’is O.lle too large (78% error), and that for b’is 0.21e too large (17% error). The agreement with the experimental parameter values calculated for the N1 atom in N20 by the Gaussian 70 4-31G calculation is somewhat better than were the calculated values for the 0 atom, but again the difference between calculated and experimental values is on the order of 0.1-0.2e. The Gaussian 70 calculation for N20 is clearly a distinct improvement over the CNDO calculation. The results for HCN are of special interest. Here neither the CNDO nor the Gaussian 70 calculations agree with the relative signs for b and b’ from the experimental study. The discrepancy has been noted before13” and has recently received considerable attention in the l i t e r a t ~ r e . ’ ~ JI ~t - ~ has now become quite clear15921-23 that the problem is that the parameter for the CN bond (b’in our notation) is quite sensitive to the nature of the wave function, and calculations even at nearly the SCF level do not give the correct relative sign for b ’unless configuration interaction is included. When that difficult but necessary refinement is made in the calculation^,^^^^^ the calculated results do in fact agree quite well with the experimental values, as can be seen in Table V. The intensities in HCN have been the subject of a recent careful experimental study by Kim and King.24 These authors have remeasured experimental intensities for HCN and DCN with slightly different results from the earlier work (summarizedby Pugh and Rao8). We have used their values in our table here. They have also discussed the agreement between the experimental results and the calculated values in a comparison similar to our Table V. They report also calculated values for APTs based in a 6-31G basis set with the Gaussian 70 program. We have not included their results as a separate entree in Table V because they are identical with the calculated values we obtained with the Gaussian 70 program and the 4-31G basis set. We believe that the comparison in Table V establishes approximately the range of validity that can be expected for Gaussian 70 4-31G calculations. Surprises such as those found in HCN may be expected occasionally, but there is (20) R. E. Bruns and W. B. Person, J. Chem. Phys., 53, 1413 (1970). (21) A. B. M. S. Bassi, J . Chem. Phys., 68,5667 (1978). (22) B. Liu, K. M. Sando, C. S. North, H. B. Friedrich, and D. M. Chipman, J. Chem. Phys., 69,1425 (1978). (23) J. E. Gready, G. B. Bacskay, and N. S. Hush, Chem. Phys., 31, 467 (1978). (24)K. Kim and W. T. King, J. Chem. Phys., 71,1971 (1979).
The Journal of Physical Chemistty, Vol. 85, No. 14, 198 1 2003
often some warning of trouble given by trends in the calculated values (for example in b’for HCN in Table V) as the quality of the calculation improves. Keeping this in mind, we have compared the calculated values of APTs in Table IV with the possible experimental values in Table I1 to choose which of the latter values agrees best with the experiment. These experimental values for APTs are compared with the calculated ones in the summary Table VI for all the molecules of this study. This summary in Table VI for COO,CS2,and OCS is particularly interesting. The differences between the calculated and experimental values of the APTs for the S atom in CS2 are similar to those discussed above for the 0 atom in COP For OCS the magnitude of the b value is calculated to be 0.5 larger for the 0 atom in OCS than for the 0 atom in C02, in agreement with experiment, but the b‘value for the S atom in OCS is predicted to be the same as that for the S atom in CS2, rather than 0.6e more positive, as found by experiment. However, the agreement between calculated and experimental values of the APTs appears to be quite good, in general. For example, the calculation predicts correctly that a for the 0 atom should become about O.le more positive from COzto OCS and that a for the S atom should become about O.le more negative from CS2to OCS. Similarly then the agreement between calculated and experimental values for the equilibrium dipole moment 032) for OCS is very good. The results for the cyanogen halide molecules, XCN (X = H, F, C1, Br, and I) form an especially interesting example of the usefulness of the calculations in analyzing the experimental data. Here we had at least two problems with the calculations. The first was the difficulty already discussed in calculating b’for the N atom in HCN. The second is the difficulty with the convergence mentioned above for ClCN. In considering these molecules, we found it instructive to consider the trends in APTs from XCN molecules as a function of the electronegativity of the X atom (regarded as a parameter against which we display trends, rather than a fundamental cause of the trends). Such trends have been found reviously, for example, in the data from CH& molecules.12 Thus, we calculate A P T s (and the associated invariants of the polar tensor) for HCN (electronegativity of H, XH = 2.2)25 and for FCN ( x p = 4.0).25 These calculated parameters are as a plotted function of xx and connected by straight lines between the value calculated for these two atoms. The experimental APTs for HCN, ClCN, and BrCN are then plotted as a function of xx, choosing those parameter values from Table I1 that are similar for these similar molecules. (These are the values summarized in Table VI.) The resulting plots are shown in Figures 2-4. We see there that calculated and experimental trends appear to be satisfyingly linear and paralle, increasing our confidence that the sign choices leading to the experimental results for XCN molecules summarized in Figure 2-4 and in Table VI are indeed correct. The trends in atomic polar tensor parameters for atoms in XCN as the X atom is changed from hydrogen to bromine, as seen in Table VI or in Figures 2-4, satisfy our “chemical intuition” and provide a basis for extrapolation to predict properties of other XCN compounds. For example, we may easily predict the vibration intensities of ICN and FCN fundamental bands, and we believe we have a reasonable first basis for estimating APTs for the CN group in CH3CN and in other organic nitriles. We leave these further details for later development, however, when
F
(25)M.G. Day, Jr., and J. Selbin, “Theoretical Inorganic Chemistry”, 2nd ed, Reinhold, New York, 1969.
2004
Person et al.
7he Journal of Physical Chemistty, Vol. 85, No. 14, 1981
‘1 I .o
0 0-
- 0.2-
bx
-04. -06-
-0 8-
IO
2.0
3.0
40
Flgure 2. Plot of calculated and experimental values of parameters a and b (in e) for the X atom in XCN molecules as a function of the electronegativity of the X atom. Here X is the value calculated by the Gaussian 70 program; 0 is the selected experimental value (both from Table VI); is the line drawn through calculated values for HCN and FCN; and - is the line drawn through the experimental data. The vertical lines are drawn at the value of the electronegativityof the X atom; since the electronegativity of the I atom is the same as for the H atom, only one vertical line is drawn.
---
1.0
2.0
3.0
4.0
x* Flgure 4. Plot of calculated and selected experimental values of the effective charge 6, (in e) on the a atom for XCN molecules (5, Is also shown for CH,X molecules) as a function of the electronegativity of the X atom. See caption to Figure 2 for definition of symbols (B are is the line drawn through those points for CH3X molecules and experimental values). ---e-
* O 02 l
- I 04 IO
I
;
,
I ,
20
30
One further interesting result is shown in Table VII, which compared atomic polar tensor parameters for hydrogen atoms in different chemical environments. We see there that the parameters for the H atom in HCN are strikingly similar to those for the H atom in HCI and that they are considerably different from those for CHI. This result is also consistent with chemical intuition.
I
40
xx Figure 3. Plot of calculated and selected experimental values of the atomic polar tensor elements a’and b’(in e) for the N atom in XCN molecules as a function of the electronegativityof the X atom. See caption to Figure 2 for definition of symbols.
more experimental data become available to check against these predictions.
Interpretation. A Beginning Recently, King and Mast,%Deciwn Mast and Decius,= King,29 and Zilles30 have suggested that the change in dipole moment of the molecule as a result of moving an atom ( d p , / d x i , for example) can be interpreted as a sum of two (or perhaps three) terms: (1)motion of a fixed charge that moves with the molecules and (2) a charge flux, representing the rearrangement of electrons as a result of the movement of the ith nucleus. Here, we shall follow the treatments by Kinga and by Zillesm and consider two contributions to the charge flux: (1)a true “charge flux”, defined as the change in the Mulliken gross atomic charge31 (26)W. T.King and G. B. Mast, J. Phys. Chem., 80,2521 (1976). (27)J. C. Deciua, J. Mol. Spectrosc., 57, 348 (1975). (28)G.B.Mast and J. C. Decius, J. Mol. Spectrosc., 79,158 (1980). (29)W. T. King, “Effective Atomic Charges”, to be published in “Vibrational Intensities in Infrared and Raman Spectroscopy”, W. B. Person and G. Zerbi, Ed., Elsevier, Amsterdam, 1981. (30)B. Zilles, Ph.D. Thesis, University of Florida, 1980. (31)R. S.Mulliken, J.Chem. Phys., 23,1833,1841,2338,2343(1955); 36, 326 (1962).
The Journal of Physical Chemistry, Vol. 85, No. 14, 1981 2005
Intensities of Triatomic Molecules
TABLE VI: Comparison between Preferred Values of Experimental Polar Tensor Elements, Selected from Table I1 and the Calculated Values (Gaussian 7 0 ) Given in Table IV APTs
0 0 C C S S C C
exptl calcd exptl calcd exptl calcd exptl calcd
Linear Symmetric AB, (D&) - 0.236 -1.137 1.185 - 0.388 -1.398 1.562 + 2.274 2.322 +0.472 + 2.80 3.125 +0.78 1.206 +0.084 - 1.200 1.521 +0.005 -1.521 2.406 - 0.168 + 2.400 3.042 - 0.010 +3.042
-0.536 -0.747 +LO73 1.495 -0.344 -0.504 + 0.688 + 1.007
0.812 1.149 3.247 4.597 1.648 2.3 29 6.595 9.315
ocs
0
N,
Linear Unsymmetric ABC ( C m u ) - 1.536 1.543 -0.103 2.016 -0.304 - 1.97 - 0.809 0.809 +0.002 1.551 -0.133 - 1.54 + 2.345 2.349 + 0.101 +3.51 3.564 +0.437 -0.626 0.644 -0.108 -0.785 0.793 -0.082 1.253 -0.139 - 1.237 1.495 -0.249 - 1.453 1.895 +0.247 + 1.863 +0.331 +2.238 2.286 +0.218 0.400 + 0.237 +0.301 0.545 +0.321 0.460 -0.321 +0.074 - 0.169 0.442 -0.289 -0.292 0.315 +0.084 -0.132 0.139 -0.032
-0.581 -0.859 - 0.268 - 0.602 +0.849 +1.461 -0.281 -0.316 -0.505 -0.650 +0.786 + 0.967 + 0.231 +0.314 -0.189 -0.249 - 0.041 -0.065
2.053 2.775 0.658 1.980 5.03 5 9.443 0.268 0.494 1.206 1.450 2.611 3.637 0.0003 0.0004 0.156 0.014 0.141 0.010
+0.149 +0.145
N,NO
exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl calcd exptl exptl exptl
+0.154 +0.147 -0.245 - 0.256 + 0.093 co.109 +0.193 - 0.226 +0.033
- 0.4 20 -0.152 - 0.097 -0.397 +0.517 + 0.549 -0.176 -0.124 + 0.300
-0.043 +0.047 -0.196 -0.303 + 0.234 10.255 0.070 -0.192 0.122
0.329 0.089 0.022 0.020 0.180 0.194 0.136 0.010 0.071
- 0.582 -0.538
CO,
CS,
HCN
ClCN
BrCNb
0 S S C C
0 N N H H N N C C CI
cl N N C C Br N C
0.473 0.258 0.360 0.537 0.533 0.570 0.325 0.343 0.304
0 0
0 0
-0.035 - 0.202
-0.614 - 0.675
No calculations were made for BrCN. These a Invariants of the ith atomic polar tensor;see ref 1 2 b for definitions. values are the experimental values from Table I1 that we prefer based on Figures 2-4 (see text).
TABLE VII: Experimental Values for Atomic Polar Tensor Elements (in e ) for the Hydrogen Atom from Several Different (Symmetric ) Molecular Environments molecule
a
b
EH
Li H CH,F CH, HCI HCN HCrCH HF
-0.765 +0.042 + 0.064 +0.177 +0.237 + 0.283 + 0.413
-0.48 -0.143 -0.139 +0.193 +0.218 +0.187 +0.35
1.18 0.163 0.167 0.316 0.402 0.442 0.681
ref
a J. H. Newton and W. B. Person, J. Phys. Chem., 82, 226 (1978). K. Kim and W. T. King, ref 24. J. H. Newton and W. B. Person, ref 12b. W. B. Person and D. Steele, ref 13.
for a unit displacement of the atom, and (2) an "overlap" term due, among other things, to the rearrangement of electrons in lone pair orbitals (rehybridization). These terms are operationally defined as follows: Here Pf(C) is the calculated contribution to the atomic polar tensor of the ath atom due to movement of the fixed charge (defined here as the Mulliken gross atomic charge calculated at the experimental equilibrium configuration),
Px"(CF) is the calculated contribution from the charge flux (defined in ref 29 and 30 above), and Px"(0)is the contribution from the terms that are not give explicitly by the charge density term. The values of these three contributions are summarized in Table VI11 for these molecules. We cannot claim that the results presented here in Table VI11 are particularly enlightening at present, although we are convinced that the analysis of results in these terms will prove to be valuable as more data are assembled. The results for Cop,OCS, and CSpappear to suggest some very interesting trends, for example, in the charge flux and overlap contributions to b. We see that the Mulliken charge on the 0 atom is nearly the same for COzand OCS, but the contribution to a from the overlap term is calculated to be O.le more positive in OCS. Similarly the change in a for the S atom from CS2 to OCS is attributed to a change in the overlap contribution. For the b term in the 0 atom, the change from COz to OCS is entirely due to the change in the change flux term. A similar change is calculated (perhaps erroneously) for the charge flux contribution to b for the S atom from CSz to OCS, but this change is balanced by a compensating change in the overlap contribution. The values calculated for charge and charge flux contributions in N z O do not appear to be consistent with the results for COz, and raise some questions concerning the
2006
Person et al.
The Journal of Physical Chemistry, Vol. 85,No. 14, 1981
TABLE VIII: Summary of Charge, Charge Flux, and Overlap Contributions (in e ) t o the Calculated Atomic Polar Tensors Given in Table VI charge molecule
atom
OCO
scs 0cs N,NO
overlap
b
a
b
0 S 0 S
-0.480
-0.480
0
- 1.375 -0.971
-0.276
-1.951 -1.537
0.211 -0.392
3
0.002 0.420 -0.560 +0.428
HCN
H
FCN
N F N
ClCN
charge flux
a
+0.281
0.281
0
-0.515 0.259
+
-0.515 +0.259
+0.124 -0.514
+0.124 -0.514
+ 0.331
+0.331 -0.343
0 0 0 0 0 0 0 0 0
- 0.343 - 0.343 -0.216 +0.305 -0.388
cl N
-0.343 -0.216 +0.305 -0.388
0
-0.668 -0,625 -0.554 +0.218
total b
a
0.098
0.503
b
a
-0.388
-1.398
-0.831
0.005
-1.521
0.496 -0.262
-0.304 -0.133
-1.970 -1.540
-0.206 0.265
-0.911 -1.359
-0.082 -0.206
-0.785 -0.911
-0.010 +0.054 +0.345 -0.117 -0.158 -0.132
+0.530 -0.254
0.321 -0.289
0.301 -0.169
+0.136 +0.548
0.002 -0.333
-0.875 -0.293
+0.097 +0.227
0.147 -0.256
-0.152 -0.397
TABLE IX: Comparison of Calculated (by Eq 12) and Experimental Dipole Derivatives t o Second Order molecule ABC
ocs
a1P/arAB2,
no. of pts 15
exptlb
0.0709 (t0.0003) 0.0708 (t0.0004) 0.0713 (i 0.002) 0.150
calcdC
0.145
15 22
NNO
12
exptld calcdC FCN
P I 0 , eA
14
-0.286 (~0.00002) -0.035 - 0.202 0.354 (i 0.0002)
14
0.354 (t 0.0005)
calcdC HCN
15
exptla calcdC (Gaussian 70) calcd (VCI, Table V)
aPlarAB, e
eIA - 2.06 (t1.16) - 2.09 (+1.16) -1.174 (i 0.367) -3.00 (t2.31)
1.99 (50.012) 2.01 (t 0.036) 1.762 (i0.006) 1.45 (t0.05) 1.97 0.71 (r0.0006)
-0.35 (t 0.024)
0.63 0.78 2.45 (i 0.18)
(OY
- 0.29 (t 0.003)
- 0.614 -0.61 -0.67
-0.23 -0.30 -0.21
-0.94 (i 0.019)
-0.925 (t0.003) -0.99 (t 0.025) -1.54 -1.15 (t0.0008)
a% la a%,
g-
e/A
0.84 (t0.64) 1.09 (t0.77) 1.219 (t 0.163) 0.25 (t0.31)
-0.53 (t0.87) - 0.53 (t0.13) -0.56 (t1.06)
5.34 (t0.023)
(t 0.012)
(0)
-1.82
-0.253 (i0.002) -0.255
(iO.018)
0.006 (f 0.23)
(0lU
- 0.88 (t0.087)
(OY
-0.29
0.87 -0.669
- 0.94 (t 0.017)
alp I a ? B e 1 , elA
-1.45 -1.45
0.758 (~0.003) 0.825 (i0.017)
(t 0.0001)
apla ~ B C e,
0.16 (r0.16)
-0.16 (r0.004)
0.054 (t0.26)
0.15 (t 0.16)
+0.11 -0.17 + 0.08
Calculated A p / A r values at the experimental equilibrium A. Foord and D. H. Whiffen, ref 32. a Constrained value. From Table VI. geometry from Tables VI or IV; with ap/arAB = - b and aplarBc = b'.
quality of the calculations for N20 in spite of their apparent agreement with experiment as discussed above (see also part 2, following article in this issue). For the XCN molecules, it would appear that the poor fit of b'values to experiment may be related to a poor quality of the prediction of the charge flux term. Nevertheless, we present out results in Table VI11 to illustrate the form of these contributions, and for later comparison with a similar analysis of data from more closely related molecules.
Calculations of Electrical Anharmonicity Terms Finally, we should like to report on attempts, described above, to fit eq 12 to calculated dipole moments at several different configurations (12 to 22 different sets of values for rABand r B c ) for ABC linear triatomic molecules OCS, FCN, NzO, and HCN. The agreement shown between experimental APTs and calculated ones derived on the assumption that only linear terms are important in eq 12 is sufficiently good that it may not seem necessary to in-
clude the terms in eq 12 that are second order in the displacements. On the other hand, it will be shown in part 2 of this study (following paper) that surprisingly good anharmonic force fields are calculated with the 4-31G basis set, and there is a rapidly growing interest14in the intensities of combination bands which are dependent both on the anharmonic force field and on the higher derivatives of the dipole moment. Equation 12 was fitted to the computed data for a number of configurations (12-22) which is larger than the number of parameters (6) to be fitted. This fit yielded the calculated dipole parameters to second order and their dispersions shown in Table IX. Carbonyl sulfide has been thoroughly studied experimentally,32and experimental values for the parameters can be compared with the calculated results. Unfortunately, even in this case the quoted uncertainities in the experimentally derived parameted2are disappointingly large (32) A. Foord and D.
H.Whiffen, Mol. Phys., 26, 959 (1973).
J. Phys. Chem. 1981, 85, 2007-2012
because of the considerable difficulties still encountered in the analysis of intensity data to obtain these parameters. We emphasize that, despite the use of 15 calculated dipole moments from 15 different nuclear configurations, the values calculated for the 6 parameters are still not adequately determined by our model calculation. For example, a2p/arCs2= 1.1 f 0.8 D/A2 and a2p/(arcoar,,) = 0.5 f 0.9 D/A2 from the calculations. This suggests that an even lower limit of convergence will be required for the calculation of higher terms in the dipole power series expansion (eq 12) to be well determined than is now required for calculation of terms in the potential energy power series. However, we see that the convergence in the calcualted values improves considerably when 22 points are used in the fit to eq 12. Still, the quoted uncertainities in the experimental values for these parameters are also just as large as the dispersions in the calculated results. In fact, the calculated values for the second derivative terms (a2p/ariarj)agree with the experimental values for the OCS molecules for the most part within their mutual uncertainties, and the agreement between experiment and calculation may in fact be claimed to be better than that found for the first derivatives. Encouraged by these results for OCS from fitting eq 12 to include the electrical anharmonicity, we have also in-
2007
cluded in Table IX results from similar calculations for N20, FCN, and HCN. We are somewhat disturbed by the surprisingly large change (for OCS, for example) in the values calculated for the equilibrium dipole moment (p): and for the first derivatives (apz/arABand dpL/arBC)when we fit eq 12 rather than just calcualte A p / b at the experimental equilibrium as described above and summarized in Tables IV and VI. The main reason for these differences (see Table IX), we believe, is in fact that these parameters are evaluated to fit eq 12 at the calculated equilibrium geometry, instead of at the experimental equilibrium geometry. It is also possible that the dipole moment function is not really linear in the displacement coordinates. At any rate, the comparison in Table IX between the two sets of calculated values for these parameters (pe,apz/arAB,and apz/arBC) suggests the quality of agreement that can be expected. Acknowledgment. Partial financial support (for publication costs) from the National Science Foundation (Research Grant No. CHE-7818940) is gratefully acknowledged. W.B.P. is grateful to the Science Research Council of the United Kingdom for support as a Senior Visiting Fellow, and to the Chemistry Department of Royal Holloway College for the hospitality and support enjoyed during his stay here in 1978.
Ab Initio Calculations of Vibrational Properties of Some Linear Triatomic Molecules. 2. Anharmonic Force Fields D. Steele, Wlllls B. Person;
and K. G. Brownt
Department of chemistry, Royal Holloway College, University of London, Egham, Surrey, TW20 OEX, England (Received: Ju& 0, 1080)
The 4-31 Gaussian basis set has been used to calculate the anharmonic potential function including all cubic and the more important quartic terms for HCN, FCN, OCS, NzO, and COP Agreement with experiment is very satisfactory except possibly for N20. The relative values of the diagonal stretching terms are in close accord with Morse predictions. The interpretation of the frequencies of fundamental molecular vibrations and of the infrared and Raman intensities arising from excitation of these has always been severely hampered by lack of relevant data which are sensitive to the parameters chosen to describe the relevant property and by the difficulties in selecting the appropriate solution for the secular equation of the nth degree. The infrared intensity problem has been discussed in part 1.l To interpret molecular frequencies, approximations to molecular force fields are regularly made a t a variety of levels. These may be grouped into the following categories: (a) model fields within the quadratic approximation; (b) general quadratic force field; (c) simplified anharmonic fields; (d) general anharmonic fields to some order, usually 3 or 4, in the displacements. Intensities of fundamental bands can be very sensitive to the degree of approximation in category a, but, a t the present level of accuracy, anharmonicity leads to minor correction^.^,^ By contrast, *Visiting Professor and Senior Fellow 1978. Address correspondence to this author at the Department of Chemistry, University of Florida, Gainesville, FL 32611. Department of Chemistry, University of Detroit, Detroit, MI 48221. 0022-3654/81/2085-2007$01.25/0
the intensity of combination bands is zero in the limit of zero mechanical anharmonicity and for dipoles which are linear in the displacements. The relative contributions of mechanical anharmonicity and dipole nonlinearity varies from vibration to vibration and indeed due to uncertainties in anharmonic fields few systems have been adequately studied to investigate this problem. Ab initio calculations can play a very important role in the evaluation of force fields, just as they may in interpreting intensities. In an elegant series of studies by Blom, Altona, and colleagues one useful approach has been de~ e l o p e d . Using ~ the Gaussian 70 wave function package6 with the 4-31 G basis set, they calculated the molecular force fields of a variety of molecular systems. Typically (1) W. B. Person, K. G. Brown, D. Steele, and D. Peters, J . Phys. Chem., preceding article in this issue. (2) B. L. Crawford, Jr., and H. L. Dinsmore, J. Chem. Phys., 18,983 (1950). (3) S. J. Yao and J. Overend, Spectrochim. Acta, Part A, 32, 1059 (1976). (4) (a) Part 111, C. E. Blom, L. P. Otto, and C. Altona, Mol. Phys., 32, 1137 (1976); (b) Part IV, C. E. Blom and C. Altona, ibid.,33,875(1977). (5) W.J. Hehre, W. A. Latham, R. Ditchfield, M. D. Newton, and J. A. Pople, “Gaussian 70”, program 236, Quantum Chemistry Program Exchange, Indiana University, 1971.
0 1981 American Chemical Society
