J. Phys. Chem. 1994,98, 4498-4501
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Prediction of Infrared Intensities by Using Density Functional Theory. Applications to H20, HOO,CH4, and C2Hdt Kenvin D. Dobbs and David A. Dixon' DuPont Central Research and Development, Experimental Station E328. Wilmington, Delaware 19880-0328 Received: January 14, 1994; In Final Form: March 9, 1994"
Density functional theory at the local and gradient-corrected (BeckePerdew and Becke-Lee, Yang, Parr) levels has been used to predict the geometries, frequencies, and infrared intensities for H20, HOO, CH4, and C2H4. Large basis sets have been emplyed in this study. An extensive set of calculations was performed for H2O and HOO, whereas for CH4 and C2H4 calculations were only done at the highest level. The calculated infrared intensities at the BP/TZVPD level are ZI = 1.8,12 = 68.9, and = 49.9 km/mol for H2O; Zl = 11.8, Z2 = 33.8, and 13 = 16.8 km/mol for HOO; Z(t1,-str) = 60.3 and Z(tl,-bend) = 37.0 km/mol for CH4; and Z(v7) = 96.7, Z(v9) = 19.3, Z(v10) = 0.2, I(vl1) = 17.6, and Z(v12) = 9.3 km/mol for C2H4.
Introduction Infrared transitions are extremely useful for identifying trace components in many regimes where it is difficult to make other analytical measurementssuch as the upper and lower atmosphere or in combustion environments. Although the position of the band aids in identifying the species, the infrared intensities are needed to determine how much of a species is present. For a number of highly reactive species, such as radicals, it is very difficult to determine absolute infrared intensities experimentally. Before embarking on a set of difficult experiments, it would be useful to have a tool that can predict the intensities to within about a factor of 2. For example, the value of the infrared intensity determines the sensitivity with which the species can be detected. The lower the intensities, the less sensitive the technique. Knowledge of the sensitivity is very important when searching for minor species in the atmosphere, and this is extremely important in the design of the actual detection system. One example of such a species is the hydroperoxyl radical, HOO, which plays an important role in the photochemistry of the atmosphere. The vibrational-rotational constants and the location of the fundamental transitions are known acc~rately,'-~ but the infrared intensities were not available until recently.6~~ In fact, calculations predicted this radical to have particularly strong transitions which would make it easier to detect.8 However, the earlier predictions varied widely and were up to an order of magnitude too large.9 We have been interested in predicting the intensities of the infrared transitions of a molecule10 as this quantity is critical for predicting the global warming potential" of a compound. Thus, we are evaluating different theoretical procedures for such predictions. With the difficulty noted above for predicting the intensities for HOO with traditional molecular orbital methods, we decided to explore the possibility of using density functional methodsI2for predicting the infrared intensities. We have found that density functional methods are a computationally efficient means for predicting the geometries13 and molecular properties such as dipole moments and various p~larizabilitiesl~ for a wide range of compounds. With the soon-to-be wide availability of analytic second-derivative methods at the density functional level,'5 it is useful to see if such methods can be used to predict infrared intensities. Below we describe an extensive study for the simple molecules H20 and HOO with a variety of basis sets and Contribution No. 6753.
* To whom correspondence should be addressed. 0
Abstract published in Aduance ACS Abstracts, April 15, 1994.
DFT functionals and a more restricted set of results for CH4 and C2H4 at the highest level.
Computational Methods The density functional theory calculations were done with the program DGauss,l6 which employs Gaussian basis sets on a Cray YMP computer. The initial basis sets for C and 0 are triple-zeta in the valence space augmented with a set of polarization functions with theform(7111/41l/l).l7 Theauxiliaryfittingbasisset for the electron density and the exchange-correlation potential has the form [8/4/4]. For H, a polarized triple-zeta valence basis set was used with the form (31 1/1) together with an auxiliary fitting basis set of the form [4/ 11. This basis set is labeled TZVP. Next we generated a larger basis set by adding diffuse functions to the basis set and by changing the form of the d functions. For C and 0, we added the diffuse s recommended by Chong and co-workers for polarizability calculations18and then added diffuse p functions with exponents of 0.0361 for C and 0.0545 for 0 obtained by geometricextrapolation. The d functions were taken as two-term fits of STO's with STO exponents of 2.0 for C and 2.1 for 0 for the inner d and an exponent of 0.7 for the diffuse STO d.19 For H we used the p function recommended by Chong and co-workers and added a diffuses with an exponent of 0.0457 obtained by geometric extrapolation. This gives basis sets of the form (71 1 11/41 11/22) for C and 0 and (31 11/11) for H. This basis is labeled as TZVP+. The calculations were done at the local density functional level with the local potential of Vosko, Wilk, and Nusair20 and at the self-consistent gradient-corrected (nonlocal) level either with the nonlocal exchange potential of Becke2' together with the nonlocal correlation functional of Perdew22 (BP) or with the nonlocal correlation potential of Lee, Yang, and Parr23 with the exchange potential of Becke (BLYP). Geometries were optimized by using analytical gradients.24 Second derivatives were calculated by numerical differentiation of the analytic first derivatives. A two-point method with a finite difference of 0.01 au was used. Results and Discussion
The results for H2O and HOO are given in Tables 1 and 2, respectively. For H20, the DFT geometries are in reasonable agreement with the 0-H bond distance being too long by -0.01 5 A.25 The fact that the DFT methods give too long a bond distance for bonds to hydrogen has been noted by others.26 The agreement of the bond angle is better than 1'. The calculated vibrational frequencies are in good agreement with the experimentalvalues,Z7 including anharmonicity which means that they are too low as
0022-365419412098-4498SO4.5010 0 1994 American Chemical Societv
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The Journal of Physical Chemistry, Vol. 98, No. 17, 1994 4499
TABLE 1: H20 Geometry and Frequencies' 0.973 105.4 3690 1550 3813 8.3 85.7 57.4
0.972 104.4 3672 1598 3789 3.3 73.8 33.8
0.974 104.2 3634 1605 3746 2.4 73.7 29.5
0.975 104.3 3679 1573 3787 5.1 76.9 72.8
0.973 104.2 3657 1608 3762 1.8 68.9 49.9
0.974 104.1 3627 1611 3728 1.8 68.7 47.9
0.962 104.0 3850 1661 3978 5.3 71.9 71.4
0.9626 104.5 3896 1677 4026 3.5 77.2 28.8
* Geometry parameters in A and deg, frequencies in cm-l, and infrared intensities in km/mol. Geometry from ref Values in parentheses are harmonic frequencies from ref 27b. Infrared intensities from ref 28.
25.
0.958 104.5 3657 (3832) 1595 (1648) 3756 (3942) 2.24 53.6 44.6
Values of v from ref 27a.
TABLE 2 HOO Geometry and Frequencies. property TZVP/LDF TZVP/BP TZVP/BLYP TZVP+/LDF TZVP+/BP TZVP+/BLYP HF TZ2PF+/MP-2 CCSD(T)/TZ2PF36 r(0H) r(O0) 0 Yl YZ
V3
II I1
13
0.995 1.323 105.6 3380 1366 1190 13.7 27.1 26.1
0.991 1.352 104.7 3405 1366 1121 10.5 32.8 21.5
0.992 1.367 104.2 3397 1353 1071 11.9 35.8 20.1
0.995 1.317 105.5 3359 1382 1188 14.5 30.3 19.9
0.991 1.347 104.8 3391 1377 1118 11.8 33.8 16.8
0.991 1.362 104.4 3381 1365 1068 14.0 36.1 15.6
0.948 1.315 105.7 4103 1596 1274 67.4 53.2 46.3
0.974 1.319 104.3 3695 1447 1229 47.2 28.8 97.2
0.9704 1.3329 104.2 3696 1439 1138 36.9 39.8 31.7
Geometry parameters in A and deg, frequencies in cm-l, and infrared intensities in km/mol. Geometry from ref Infrared intensities from refs 6 and 7. compared to the harmonic values to which they are directly equivalent. As noted by Handy and co-workers,26 this is due to the fact that the bond length is too long. Inclusion of gradient corrections lowers the stretching frequencies slightly and raises the bend. The BLYP correction lowers the stretch more than does the BP correction. The infrared intensities at the TZVP/ LDFT level are too large as compared to the experimental values.28 Inclusion of the diffuse functions improves the agreement somewhat. Inclusion of nonlocal corrections at the TZVP level improves the agreement for I1 and I 2 but makes the value for I3 too small. Addition of diffuse functions a t the nonlocal level gives the best agreement with experiment. The value for I1 is a little too low (20%), and the value for I 2 is too high by about 28%. The agreement is even better for Z3 where the calculated values are about 10% too high. These results can be contrasted to the Hartree-Fockvalues.29-30 Witha very extensive basis set [8s6p4d/ 6 ~ 3 ~the 1 ,values for the frequencies are I1 = 15.1,12 = 97.7, and I3= 91.8 km/mol. These are clearly in much worse agreement with the experimental values than any of the DFT calculations. Even at the MP-2 level31 with an [8s6p4d/6s3p] basis set the values are Il = 5.2,12 = 12.1, and I3= 12.8 km/mol.30 Schaefer and co-workersg2have recently reported coupled cluster with a perturbative correction for triples (CCSD(T)) results with a DZP basis set for H2O. They obtain improved values for both I1 and I3at this level as compared to the MP-2 results, but the results are not in as good agreement with experiment as the DFT results are. With a smaller DZVP basis set, Sim et al.33 find I1 = 9.5, 1 2 = 65.1, and I3= 55.2 km/mol a t the DFT(BP) level. These DFT results are not as good as what we find even with the TZVP basis set, showing the need for diffuse functions at the DFT level that we have already noted. For HOO, the results are given in Table 2. Again the calculated value for r(O-H) is too long.5 The error in the calculated value for r ( 0 - 0 ) depends strongly on the level of calculation. At the LDFT level, the value is shorter than the experimental value by 0.012 8, with the TZVP basis set. Addition of diffuse functions leads to an additional decrease, and the bond distance is now 0.018 8, too short. Inclusion of nonlocal corrections leads to a bond distance that is too long. Improvement of the basis set leads to better agreement with experiment. There is clearly a difference in the behavior of the functionals with the BP giving significantly better agreement (0.012 8, too long) as compared to the BLYP functional (0.027 8,too long). The bond angles are
5.
Ia
Ib
1192 60 37
1378 87 47
* *
Frequencies from ref 6 .
TABLE 3: Geometry and Frequencies of CK. property TZVP+/BP CCSD(T)/DZP32 r(CW 1.097 1.0918 v(ad 2988 3076 1448 1579 v(e) 3107 3220 v(tda v(t2)b
expt 0.977 1.335 104.1 3436 1392 1098 4.5 1.3 13.0 3.6 7.9 & 2.0
exptb 1.094 2917 (3025) 1534 (1583) 3019 (3157) 1306 (1 368) 65.7 f 4.2 35.4 f 0.8
a Geometry parameters in A and deg, frequenciesincm-I, and infrared intensities in km/mol. r(C-H) from ref 25. Values for Y from ref 27. Values for harmonic frequencies from ref 37. Infrared intensities from ref 38.
in good agreement with the experimental value,' especially if nonlocal corrections are included. For HOO, the H-0 stretching frequency (VI)is too low as compared to the experimental value.6 The value for v2 is also somewhat lower than the experimental value whereas the value for v3 is somewhat higher. As expected, the value for v3 which has a significant 0-0stretching component is the most sensitive to thelevelofcalculation. TheLDFTand BPvaluesaresomewhat larger than the experimental value, and the BLYP value is lower consistent with the longer 0-0 bond a t the BLYP level. The calculated values of the intensities show some dependence on the level of calculation. The BP correction gives somewhat lower values than does the BLYP correction for I1 and 12, but the trend is reversed for I,. There is not as large an effect on the intensities for I1 and I 2 when diffuse functions are added as found for H20. In fact, diffuse functions raise the intensities for ZI and I2. The agreement with the experimental value^^.^ is not as good for HOO as for H2O. The best set of calculations (TZVP+/BP) gives values for I1 and 1 2 which are too high by about a factor of 2.6. The value of 4 is about a factor of 2.1 too high. Although this is not as good agreement as one would hope for, the results are significantly improved over H F results which give values of ZI = 67.4, Z2 = 53.2, and I3= 46.3 km/mol. To see if correlation corrections based on the U H F wave function would lead to improved intensities, we calculated these values at the MP-2 level with a large basis set. The calculations were done with the program Gaussian-92.34 The basis set for the geometry optimization is of the form TZ2PF.35 The frequencies and intensities were obtained with a larger basis set, TZ2PF+ where the "+" denotes the addition of diffuse functions. For
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The Journal of Physical Chemistry, Vol. 98, No. 17, I994
TABLE 4 CzH4 Geometries and Frequencies’ property BP/TZVP+ r(C-H) 1.093 1.334 r(C=C) freq vl(a,) v2 v3
v4(au) udblg) v6
~(blu) Udb2g) ~(b2u) VI0
~~~(bsu) VI2
e
exptb 1.08 1 1.334
Letters property L(HCH)
u(BP/TZVP+)
I(BP/TZVP+)
u(expt)c
w(expt)d
I(expt)e
3056 1637 1331 1014 3117 1195 912 918 3144 797 3045 1415
0.0 0.0 0.0 0.0 0.0 0.0 96.7
3026 1623 1342 1023 3103 1236 949 943 3106 826 2989 1444
3153 1655 1397 1044 3232 1245 969 959 3235 842 3147 1473
0.0 0.0 0.0 0.0 0.0 0.0 84.4 0.0 26.0 0.03 14.3 10.4
0.0 19.3 0.2 17.6 9.3
BP/TZVP+
exptb
116.4
117.3
I(HF)I
121 19 0.05 18 12
u(MP2)g 3192 1663 1379 1058 3263 1225 967 942 3292 826 3172 1473
I(MP2)g
91.6 17.2 0.01 10.8 8.3
Geometry parameters in 8, and deg, frequencies in cm-I, and infrared intensities in km/mol. Reference 25. c Reference 27a. d Reference 39. Reference 40. f Reference 29. g Reference 41.
comparison, we give the values for H20 in Table 1. For HOO, although the value for I2 is comparable to the BP value at the TZZPF+/UMPZ level, the value for ZIis still higher than the BP value by a factor of 4, and the value for 4 has become much worse, almost 6 times larger than the BP value. Thus, correlation at the MP-2 level does not really lead to improved values. Bartlett and c o - ~ o r k e r have s ~ ~ used the CCSD(T) method with a large TZ2PF basis set to calculate the intensities for HOO. Although they obtain the best values from molecular orbital theory, except for 12, their values are at least a factor of 2 higher as compared to our DFT values. For CH4 and C2H4, we decided to only calculate the intensities at the TZVP+/BP level. These results are summarized in Tables 3 and 4. The values of the frequencies for CH4 as compared to e~periment2~a.3~ follow the pattern observed by Handy and coworkers26 as do the geometry values.25 The infrared intensities for the tz bands are in excellent agreement with the experimental values,3*with thevalue for the stretch lower than the experimental value by 10% and the bend greater by a comparable amount. In contrast, the best HFvaluesZ9are much larger with II = 114 and 1, = 67 km/mol. At the CCSD(T)/DZP leve1,32 the calculated intensities are not in as good agreement with experiment as are the D M values. For C2H4, the geometry parameters are in good agreement with the experimental values25 with the caveat that the C-H bond is too long by 0.012 A. The frequencies aregenerally slightly lower than the harmonic values.39 The largest deviations are found for the C-H stretches as would be expected as these exhibit the most deviation from experiment for the bond length. Again, the infrared intensities are in good agreement with e ~ p e r i m e n t . ~ ~ The largest percent deviation is found for the vlo(b1,) mode although we note that this mode has a very low intensity. For C2H4, the H F values are in quite good agreement with the experimental values with the largest error found for the v7(blu) mode. Jordan and co-workers have published an extensive set of calculations at the MP-2 level.41 At the highest level, their values are similar to ours for most of the modes. The MP-2 value for the ulO(bZu)modeissmaller than theexperimentalvalueincontrast to our result; a similar result is found for the vll(b3,) mode. At the LDFT level with a TZVP basis set augmented with an additional set of polarization functions, Papai et al.42calculate 1 ~ = 9 5 . 7 , 1 ~ = 1 0 . 8 , Z l ~ = 0 . 3 ,11.5,andZ12=12.6km/mol, 111= which differ from our NLDFT results somewhat. These differences could easily be due to the differences in the functional.
Conclusion The above results show that the DFT method can be used to provide a way to predict infrared intensities for fundamental
transitions with good accuracy. Because of its computational scaling, it will be significantly cheaper than a b initio molecular methods yet will provide results in better agreement with experiment. It will be necessary to augment the basis sets with diffuse functions just as found at the HFlevel and in the prediction of N L O properties at the DFT level. It is also apparent that gradient (nonlocal) corrections are likely to be necessary to obtain the best agreement with experiment. The largest differences between the DFT results and experiment are found for HOO. Although the DFT results are not within the desired factor of 2, they are clearly better than the a b initio molecular orbital results based on a single-configuration wave function. One caveat that must be placed on the comparison between DFT and experiment for HOO is the accuracy of the experiment. Infrared intensities are notoriously difficult to measure in an absolute sense, and there is always the possibility of errors in the experimental measurements. Although errors in the intensity measurements for HOO could be present, we find no reason to suspect any problems with the experimental results.
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