J. Phys. Chem. 1995,99, 5927-5933
5927
Ab Initio Predictions of Vibrational Frequencies for Cationic Species Peter E. Peterson, Mahdi Abu-Omar, Thomas W. Johnson, Ricky Parham, Dennis Goldin, Charles Henry, 111, Andrew Cook, and Kevin M. Dunn* Department of Chemistry, Hampden-Sydney College, Hampden-Sydney, Virginia 23943 Received: October 24, 1994; In Final Form: January 31, 1995@
Predicted vibrational frequencies are reported for 15 small cationic species from Hartree-Fock calculations using the 4-21G and 6-31G* basis sets. Force constants from these calculations were corrected using empirical scale factors taken from a collection of similar neutral molecules. These scale factors reproduce 184 experimental frequencies for the neutral molecules with percent errors of 3.1% and 2.5% for the 4-21G and 6-3 l G * basis sets, respectively. When these scale factors are applied to the cations, 22 known experimental frequencies are predicted with percent errors of 4.5% and 2.0%, respectively.
1. Introduction Molecular ions are important as reactive intermediates, in the chemistry of interstellar space; in flames, and in electrical discharges. The past 15 years has witnessed extensive growth in the study of these ions. On the experimental side, velocity modulation techniques have made it possible to measure highresolution infrared spectra of molecular ions.’,2 On the theoretical side, advances in ab initio quantum chemistry have made possible the accurate prediction of infrared frequencie~.~As might be expected, theoretical predictions have played an active role in guiding the experimental worke4 Early theoretical calculations on the infrared spectra of molecular ions were done at the Hartree-Fock level of theory and configuration intera~tion.~ As in ab initio predictions of infrared frequencies of neutral molecules, the results are often scaled to correct for systematic errors in the theoretical method. Sometimes the scale factors are taken from similar molecules,6 and sometimes a uniform scale factor derived from experimental data on many molecules is e m p l ~ y e d . ~ As computational hardware and algorithms have improved, more sophisticated ab initio methods have been e m p l ~ y e d . ~ However, even given the quality of current theoretical methods and the availability of low-cost, high-speed computers, the highest quality calculations are available only for relatively small molecules. In theoretical predictions of infrared frequencies for neutral molecules, a great deal of success has been achieved using force constants calculated using modest levels of theory and scaling the force constants rather than the frequencies. If the force constants are expressed in valence internal coordinates, the scale factors are quite transferable from one molecule to another. For example, scale factors determined for benzene8 have been used to correct force constants for pyridinegand borazine (B3N3H,5),I0 which does not even contain carbon. The advantage of this approach over the application of a single uniform scale factor is that it can correct systematic errors which differ for chemically different types of force constants, for example, scaling X-H stretches differently from X-X stretches. The advantage over the scaling of frequencies based on similar modes in other molecules is that the force constant similarities are more easily justified than similarities between vibrational motions in similar molecules. Recently Florian and Johnson have used empirical scaling to correct the Hartree-Fock, MP2, and density functional force @
Abstract published in Advance ACS Abstracts, April 1 , 1995.
constants of formamide.” In this work, they scale the HF and DFT force constants to reproduce the MP2 force constants as closely as possible. Their comparison favors DFT over HF in the ability to reproduce MP2 interaction constants. However, the errors in the interaction constants results in only minor errors in the vibrational frequencies. They conclude that empirical scaling can be profitably used to correct DFT force constants, achieving accuracy comparable to MP2 results with less computational effort. As their study was limited to formamide alone, the transferability of DFT scale factors among similar molecules has not yet been established. It is also possible to scale frequencies mode by mode using experimental frequencies from a similar molecule. Lee and Schaefer have done this for HCNH+ deriving frequency scale factors from HCN and HNC.6 With this approach their HF predictions are in error by an average of 2.3% for four of the five predicted frequencies. However, they were unable to scale the prediction for the lowest bending frequency because of the absence of experimental data on the corresponding frequency of HNC. The main disadvantage of the empirical scaling is the need for a molecule “similar enough” to the one for which predicted frequencies are being sought. It is the purpose of the present work to determine a single, small set of empirical scale factors, derived from a collection of representative neutral molecules, which may be used to correct force constants for a wide variety of target molecules. The transferability of this set of scale factors will be tested by using them to predict the infrared frequencies of a collection of molecular ions.
2. Computational Methods The ab initio calculations were performed with the TEXAS programI2 which was augmented with a set of interface programs written at Hampden-Sydney. Two basis sets were used: the split-valence 4-21G basis set’3 and the split-valence plus polarization 6-31G*basis set.I4 The valence internal coordinates promoted by h l a y were used for the calculation of force constants by finite differences of analytical gradients.13 At each level of theory, the reference geometry was the optimized geometry at that level of theory. Force constants calculated at this level of theory are known to contain residual errors due to the finite size of the basis set, the neglect of electron correlation, and the neglect of anharmonicity. These errors have a systematic component, however, and may be corrected empirically by scaling the predictions.
0 1995 American Chemical Society QQ22-3654/95/2Q99-5927$09.QQ~Q
5928 J. Phys. Chem., Vol. 99, No. 16, 1995
The scaling scheme employed in the present work is to be distinguished from that used by DeFrees and M ~ L e a n .They ~ corrected the ab initio predictions by scaling the frequencies by a uniform scale factor of 0.89 derived by averaging the ratio of the experimental and theoretical frequencies for a collection of small neutral molecules; 165 frequencies were used in this average. Their method has the advantage that it is simple to apply this scale factor to subsequent work. It suffers in accuracy, however, because it does not take advantage of modespecific differences in the accuracy of the ab initio frequencies. The present work scales force constants rather than frequencies. The force constants are divided into several groups depending on their chemical identity (C-H stretch, bend, torsion, etc.). Within a group, the force constants are scaled by a common scale factor. Off-diagonal force constants are scaled by the geometric mean of the scale factors for the diagonal elements: F,,scaled V
-
112
- Fij(cicj)
The scale factors are refined by a least-squares procedure in which the factors are chosen to minimize the deviations between the predicted and experimental frequencies for a collection of neutral molecules. The least-squares procedure is weighted by the reciprocal of the experimental frequency, in effect minimizing the relative error rather than the absolute error. The current work included 184 frequencies for 27 isotopic substitutions of 15 molecules in the least-squares fit. The scale factors determined for the neutral molecules were then used to empircally correct the ab initio force constants for 14 isotopic substitutions of 8 molecular cations. The expectation was that the systematic errors in the cation calculations would be similar to those for the neutral molecules and that, consequently, the neutral scale factors would correct for these errors. The scaled cation frequencies, unlike those for the neutral molecules, are true predictions in that they do not depend on experimental data for the cations. 3. Results
Of immediate concern to the goals of this research was the selection of appropriate sets of force constants to be grouped together for scaling. The general procedure for this selection was to begin with a large number of independent scale factors, one for each type of force constant. For example, C-H, F-H, N-H, and 0 - H stretching force constants were scaled by separate scale factors. The scale factors were then optimized in a least-squares fit of the scaled theoretical frequencies to the experimental ones. Scale factors which had similar optimized values for both basis sets and could be described as chemically similar were subsequently grouped together in later fits. For example, the separate C-H and N-H stretching scale factors had close to the same values in the early fits and were subsequently combined into a single scale factor. In the final analysis, however, there was no objective criterion for whether two scale factors ought to have been combined. There was a tradeoff in many cases between the desire to limit the total number of scale factors, on the one hand, and to accurately reproduce the experimental frequencies, on the other. This was particularly true in the case of bending scale factors. A good case could be made for the use of separate linear and nonlinear bending scale factors on the grounds that the linear bending motions often involve multiple bonds, which are chemically distinct from the single bonds most often associated with nonlinear bending motions. However, linear bending scale factors involving C=C bonds turned out to be no closer to those
Peterson et al. TABLE 1: Final Combined Scale Factors force constant type
4-21G
6-3 1G*
tors bend out C-C stre C-F stre C-N stre C - 0 stre X-H streU F-H stre 0 - H stre C=C, C=C stre C=N, C e N stre C=O, C=O stre N=N, N I N stre
0.7752 0.7292 1.2486 0.9351 0.8686 0.9126 0.9439 0.8413 0.9829 0.9268 0.7625 0.7529 0.8806 0.8077
0.7309 0.7620 0.6462 0.8740 0.7810 0.8284 0.7934 0.8317 0.8340 0.81 15 0.1597 0.7242 0.1722 0.7141
“X=CorN.
involving C=O bonds than to the nonlinear bending scale factors. Consequently, it was not deemed helpful to employ a single linear bending scale factor and a single nonlinear bending scale factor. Furthermore, some of these types of force constants were represented in only one molecule of the set, and there was some concern that the molecule included might not be representative of the whole class. Therefore, it was decided to group linear and nonlinear bends into a single group for scaling purposes. Because of the variation in scaling requirements among the linear molecules and since the nonlinear molecules in the set outnumber the linear ones, they dominate the bending scale factor and consequently bending frequencies are not as accurately reproduced for the linear molecules as they are for the nonlinear molecules. The final set of scale factor types is listed in Table 1. The types of force constants refer to force constants expressed in the valence coordinates advocated by P u l a ~ . ’Three ~ types of bending constants are recognized: torsional constants, linear and nonlinear bending constants, and out-of-plane bending constants. Four types of X-X force constants, three types of X-H force constants, and four types of X=X force constants are recognized, making a total of 14 scale factors. The optimized values of these scale factors are also given in Table 1. It might be supposed that the number of scale factors could have been further decreased by combining, for example, the separate scale factors for C-H,N-H, and F-H stretching. But in the final scaling only C-H and N-H were combined because the F-H scale factor was quite different from that of C-H and N-H for the 4-21G basis set. While they might have been combined for the 6-31G* basis set, we wanted to have a single set of groupings which would be likely to be appropriate for basis sets other than the two used here. Therefore, scale factors were only combined if they had similar values in each of the two basis sets. That the scale factors are different from each other points to differences in the accuracy with which different types of vibrational motion are described at this level of theory. On average, force constants computed with the 4-21G basis set require less scaling (scale factors closer to 1.0) than the larger 6-31G* basis set owing to fortuitous cancellation of errors in the smaller basis set calculations. This observation can be misleading, however. The root-mean-square error is larger for the 4-21G basis set (43 cm-’) than for the 6-31G* basis set (33 cm-I). Furthermore, the 6-31G* scale factors are more nearly uniform than the 4-21G factors, pointing to less mode-specificity in the errors when the larger basis set is used. Thus the use of the larger basis set makes it easier to choose an appropriate scale factor for a bond (e.g., N=O) not explicitly included in Table 1.
J. Phys. Chem., Vol. 99, No. 16, 1995 5929
Vibrational Frequencies for Cationic Species
TABLE 3: Scaled Vibrational Frequencies for Small Nonlinear Neutral Molecules
TABLE 2: Scaled Vibrational Frequencies for Linear Neutral Molecules molecule N2
co HF DF HCN DCN HCCH
DCCD
HCCD
HCCN
HCCCCH
symmetry species
4-21G"
6-31G*a
exptlb
233 1 2143 3980 2885 796(+) 2066 3360 634(+) 1894 2641 761(+) 769 1959 3283 3388 564(+) 635(+) 1736 2411 2704 602(+) 764(+) 1830 2569 3342 225 581(+) 802(+) 877 2073 2348 3325 207(-) 531(+) 767(+) 781(+) 878 2009 2250 3331 3334
233 1 2143 3980 2885 776 2084 3341 620 1911 2625 693(+) 770 1966 3289 3379 566(+) 579 1747 2415 2690 574(+) 73 1 1839 2565 3339 239 538 741(+) 860 2077 2311 3331 246 594(+) 719(+) 771(+) 867 2020 2226 3335 3335
2331 2143 3958 2907 712 2097 3311 569 1925 2630 612 730 1974 3289 3374 505 537 1762 2439 270 1
518 678 1854 2584 3336 223 499 663 864 2079 2274 3327 23 1 482 627 630 874 2020 2184 3293 3329
(+) indicates a positive deviation from the experimental value of more than 10%. (-) indicates a negative deviation from the experimental value of more than 10%. Experimental values from ref 16.
Except for one, all of the scale factors are less than 1.0, reflecting the generalization that Hartree-Fock force constants are larger than experimental ones. The exception to this is in the out-of-plane scale factor for the 4-21G basis set, which is larger than 1.0. This scale factor corrects the umbrella force constant of ammonia and methylamine. A similarly large scale factor is seen in the case of aniline.I5 The out-of-plane vibration involves the motion of the lone pair electrons from one side of the plane to the other. The unpolarized 4-21G basis set produces lone pairs which are more like p orbitals than sp3 hybrids, resulting in planar or near-planar amine groups. Consequently, unpolarized basis sets systematically produce amine out-of-plane angles and force constants which are too small. The addition of polarization functions in the 6-31G* basis set overcorrects this deficiency, resulting in scale factors that are less than those for other force constants. Tables 2-5 compare the predicted and experimental vibrational frequencies for the neutral molecules. All experimental frequencies were taken from Shimanouchi.I6 Most of the assignments are straightforward, the exception being the umbrella vibration of ammonia, which is split by tunneling. Following Shimanouchi, we take the inversion fundamental of ammonia to be the average of the 1- O+ and 1+ 0- bands.
-
-
molecule
symmetry species
4-21G
6-31G*
exptP
1539 3657 3789 1129 2629 2780 1350 2702 3726 940 1591 3346 3493 719 1161 2376 2576 1304 1488 2914 301 1 986 1053 206 1 2228 1148 1299 1434 2183 2942 301 1 994 1015 1262 2100 2228 2989 1014 1082 1227 1289 1404 2141 2228 2967 3010
1594 3667 3774 1167 2644 2766 1397 2703 3723 972 1614 3365 3486 738 1172 2405 2567 1299 1486 2916 3012 98 1 1051 2063 223 1 1144 1293 1431 2186 2943 301 1 990 1012 1259 2102 223 1 2390 1010 1077 1221 1287 1401 2143 223 1 2968 301 1
1595 3657 3756 1178 2671 2788 1402 2727 3707
950b 1627 3337 3444 748b 1191 2420 2564 1306 1534 2917 3019 996 1092 2109 2259 1155 1300 1471 2200 2945 3017 1003 1036 1291 2142 2263 2993 1033 1090 1234 1333 1436 2202 2234 2974 3013
Experimental values from ref 16. Average of two values; see text.
In Tables 2-5, large percentage errors in the scaled theoretical frequencies are denoted with a (+) or (-) for deviations greater than &lo%, respectively. With one exception these are all lowfrequency bending modes of linear molecules. Only one of these frequencies shows a negative deviation. The only large deviation for a nonlinear molecule is the lowest totally symmetric bending vibration for perdeuterated propane, for which the experimental frequency is uncertain. This pattern of errors is consistent with the supposition that linear bending force constants probably ought to be assigned a scale factor independent of the bending scale factor for nonlinear molecules. This point will be addressed in the Discussion. Tables 6 and 7 show the scaled theoretical frequencies for linear and nonlinear cations. The experimental frequencies are not as widely available as for the neutral molecules, a fact which provided motivation for the present work in the first place. Experimental frequencies are available for HN2+,I7 HC0+,I8 HCNH+,I9 HCCCNHf,20 HzF+,ZI H30+,22 D30+,Z3 NH4+,24 NH3D+,25and ND4+.26 Two of the cations require special comment. First, the inversion fundamentals of H3O+ and D3O+ are split by tunneling
Peterson et al.
5930 J. Phys. Chem., Vol. 99, No. 16, 1995 TABLE 4: Scaled Vibrational Frequencies for Medium Nonlinear Neutral Molecules symmetry species
molecule CH3F C3b
a1 e a1 e al
e CH30H C,
a" a' a' a" a' a' a" a' a' a" a' a' a" a' a" a"
CH3NH2 C,
a' a" a' a' a" a' a' a' a"
a' a" a1 e' ai e"
CH~CHT D3d
a? ai
e" e'
a? ai e" e' a
4-21G
6-31G*
exptl"
1048 1085 1406 1436 2943 3017 307 966 1064 1067 1264 1388 1437 1445 2906 2956 302 1 3703 278 829 894 1038 1115 1248 1379 1435 1452 1584 287 1 2959 2993 3369 3459 278 782 988 1158 1350 1351 1435 1439 292 1 2925 2968 2993
1049 1 I45 1441 1443 2949 3022 299 1017 1060 1126 1316 1429 1442 1452 2905 2947 3015 3709 29 1 768 918 1044 1115 1291 1403 1439 1453 1609 2880 2960 2993 3403 3478 279 777 989 1168 1351 1384 1435 1440 2918 2924 2964 2986
1049 1182 1464 1467 2930 3006 295 1033 1060 1165 1345 1455 1477 1477 2844 2960 3000 3681 268 780
TABLE 5: Scaled Vibrational Frequencies for Neutral Propane molecule
symmetry species
1044 1130 1430 1473 1485 1623 2820 296 1 2985 3361 3427 289 822 995 1190 1379 1388 1468 1469 2896 2954 2969 2985
Experimental values from ref 16.
-
as in the case of NH3. As with ammonia, we have taken the average of 1O+ and 1+ 0- for comparison with the predicted frequencies. While this practice was defensible for ammonia, in which the splitting is a small fraction of the fundamental frequency, in oxonium the splitting is quite larger. For H30+, the infrared band centers are 954 cm-' and 526 cm-I, while for D30f they are at 645 cm-' and 438 cm-I. Nevertheless, the average of the two infrared bands should be a reasonable approximation to the harmonic frequency, and on these terms the predicted frequencies are quite close to the experimental averages. The second molecule requiring explicit comment is the nonclassical ion, CH5'. The minimum of this ion has C, symmetry at the Hartree-Fock level of theory.*' The minimum on the potential energy surface is essentialiy H2 bound to a methyl cation, HZ being coplanar with one of the methyl hydrogens. An alternative view is that the structure is similar to H3+ with a methyl group substituted for one of the hydrogens. With the 6-31G* basis set, the methyl bond lengths are 1.08 8, while the other two C-H bonds are longer, 1.23 8,. The H-H +
a1 b2
4-21G"
6-31G*O
exptlb
203 25 1 340 704 859 865 886 1049 1121 1150 1241 1298 1340 1356 1424 1425 1432 1440 1443 2914 2917 2925 2946 297 1 2976 2980 2987 145 185 283(- -) 512 63 1 66 1 723 834 915 92 1 953 1023 1032 1033 1035 1037 1052 1106 1176 2093 2095 2129 2185 2201 2204 2207 2212
200 25 1 343 706 859 860 887 1043 1131 1162 1253 1316 1371 1374 1427 1425 1434 1442 1448 2912 2913 2919 2934 2964 2972 2975 2977 143 184 285(-) 513 627 66 1 720 827 924 925 964 1025 1032 1033 1035 1038 1055 1134 1223 2094 2097 2126 2178 2197 2202 2205 2205
216 268 369 748 869 922 940 1054 1158 1I92 1278 1338 1378 1392 1451 1462 1464 1472 1476 2887 2887 2962 2967 2968 2968 2973 2977 143 172 332 544 659 688 712 862 945 949 959 1064 1064 1064 1064 1068 1086 1086 1203 208 1 208 1 2122 2149 222 1 2224 2225 2224
a (+) indicates a positive deviation from the experimental value of more than 10% (-) indicates a negative deviation from the experimental value of more than 10%. Experimental values from ref 16.
distance is 0.85 A, similar to the H-H distance in H3+ at the same level of theory (0.84 A). Evidently there is a C-Hz three-center bond in CH5+ which makes it problematic an to which scale factor ought to be used for the force constants involving this bond. Two reasonable choices would be to treat the H-H unit as a bond for scaling purposes. The other would be to consider it unbound and use bending coordinates (and scale factors) to describe the motion of the H atoms relative to the methyl group. We have used both alternatives in this work, resulting in two sets of predicted frequencies for CH5+ which differ only in the choice of scale factor used for the H-H subunit. Comfortingly, the predicted
Vibrational Frequencies for Cationic Species
J. Phys. Chem., Vol. 99, No. 16, 1995 5931
TABLE 6: Scaled Vibrational Frequencies for Linear Cations
TABLE 7: Scaled Vibrational Frequencies for Small Nonlinear Cations ~~
molecule
symmetry species
4-3 1G"
6-3 1G*"
exptlb
molecule
symmetry species
4-21G
6-31G*
exDtln ~~
HN:
n U
U
HCO+
n
a a HCNH+
n n U U U
HCCCN+
n
n #7
n U U
U U
771(+) 2272 3184 898 2222 3107 762(+) 901(+) 2088 3176 3445 218 613 750 868 902 2063 2338 3247 3484
704 2278 3293 865 2207 3142 685 849 2100 3213 3534 216 553 599 822 882 2066 2295 3298 3576
al bi ai a1 a1 bi a' a' a' al e
687 2258 3234 830 2184 3089 646 801 2156 3188 3283
ai
e a] e al e f2
e al
35 14
f2
(+) indicates a positive deviation from the experimental value of more than 10%. (-) indicates a negative deviation from the experia
f2
e ai
mental value of more than 10%. Experimental values from refs 1720.
f2
e
frequencies are almost identical except for the H-H "stretch" frequency which is 105 cm-' higher when the X-H stretching scale factor is used with the 4-21G basis and 44 cm-' higher when the X-H stretching scale factor is used with the 6-31G* basis. The better agreement in the 6-31G* frequencies comes from the fact that the stretching and bending scale factors are more alike with the 6-31G* basis than with the 4-21G basis. As with the neutral molecules, the greatest percent errors in the predicted frequencies are found in the low-frequency bending modes of the linear molecules. Also, agreement is considerably better with the 6-31G* basis set than with the 4-21G basis set, as was the case for the neutral molecules. The cause of this pattem will be discussed in the next section.
al e al al e a1 e e a1
e al al bi b2 a2 ai a] bz al bi a" a' a' a" a' a" a' a' a' a' a' a" a" a' a'
4. Discussion
Table 8 reports the analysis of the errors made in the scaled ab initio predicted frequencies. For the neutral molecules, the residual errors in the predicted frequencies are 3.0% and 2.4% for the 4-21G and 6-31G* basis sets, respectively. There is considerable differences in the accuracy of the predictions for linear molecules. Both basis sets perform better for nonlinear molecules than linear ones, in part because of the choice to scale linear and nonlinear bending force constants together. Since nonlinear molecules outnumber linear ones in the set used to determine the scale factors, the nonlinear molecules dominate the bending scale factor to the detriment of the linear ones. In fact, when the n frequencies (linear bends) are excluded, the errors for all the neutral molecules are very similar to the errors for the nonlinear molecules alone. Handy has shown that a proper treatment of linear bending motions requires a minimal treatment of electron correlation (e.g., MP2) and inclusion of f functions in the basis set.28 Including f functions in HF calculations generally drives the linear bending frequencies higher, resulting in poorer agreement with experiment than HF calculations without f functions. Both electron correlation and f functions must be included for substantial improvement in the accuracy of the bending frequencies. Empirical scaling of the HF force constants corrects systematic errors due to the neglect of electron correlation and the use
a"
a' a" a' a' a' a' a' a"
1121 3445 3449 821 2467 2527 982 2498 3447 672 1518 35 10 3637 511 1108 2483 2685 1462 1613 3189 3277 1098 1141 2256 2415 1275 1457 1564 2372 3214 3277 1106 1119 1387 2294 2415 3257 1118 1207 1375 1397 1540 2332 2415 3237 3277 84 890 1136 1201 1358 1406 1518 1937 2665 2918 3009 3082 84 908 1137 1201 1358 1406 1533 1937 2770 294 1 3010 3082
Experimental values from refs 21-26. see text. (I
1329 3299 3340 969 2399 2413 1163 2406 3319 745 1593 3401 3500 564 1157 2419 2577 1433 1636 3257 3387 1075 1158 2304 25 00 1258 1426 1577 2446 3294 3387 1083 1111 1387 2349 2500 3359 1109 1183 1345 1417 1544 2396 2500 3328 3387 135 95 1 1196 1223 1390 1411 1498 2099 2819 2972 3043 3111 135 959 1200 1223 1386 1412 1512 2099 2863 3003 3044 3111
3335 3349
740b 1632 3528 541b
1447
3343
2495
Average of two values:
5932 J. Phys. Chem., Vol. 99, No. 16, 1995
Peterson et al.
TABLE 8: Errors in the Calculated Frequencies rms error
mean error percent error no. of 4-21G 6-31G* 4-21G 6-31G* 4-21G 6-31G* data all no n linear nonlinear
40.9 32.9 61.2 32.9
31.6 25.3 45.3 26.3
all
72.1 66.3 62.4 82.2
37.7 38.0 43.5 29.2
no n linear nonlinear
Neutral Molecules 30.5 24.1 25.8 20.7 43.2 31.5 26.8 22.0 Cations 63.3 33.5 56.9 33.2 53.0 40.7 75.6 24.9
3.0 2.0 5.8 2.2
2.4 1.7 4.5 1.9
184 169 41 143
4.5 2.7 5.1 3.9
2.0 1.4 2.6 1.3
22 18 12 10
TABLE 9: Errors in the Calculated Frequencies: Scaled Force Constants vs. Scaled Frequencies present work
DeFrees and McLean
rms error
mean error
percent error
rms error
mean error
percent error
no. of data
37.0
33.3
2.0
60.9
49.0
3.3
18
of a finite basis set. Apparently the errors are less systematic in the case of linear bending force constants than for nonlinear bending force constants. Simply giving the linear molecules their own bending scale factor does not completely solve the problem, however. As discussed above, there does not seem to be a “common” linear bending scale factor as there is for the nonlinear molecules, especially for the 4-21G basis set. Better agreement for the linear molecules will come at the expense of employing perhaps several different linear bending scale factors for different types of molecules. For now, we have chosen to limit the number of different scale factors and to accept the limitations this places on the accuracy of predicted bending frequencies in linear molecules. The residual errors in the predicted cation frequencies are quite similar to those for the neutral molecules. This is all the more impressive when it is considered that the scale factors were optimized to reproduce the experimental data for the neutral molecules but not for the cations. As with the neutral molecules, the largest percent errors for the cations occur in the linear bending (JC)frequencies, and, when these are excluded, the percent errors for the cations are comparable to those for the neutral molecules. The small percent error for the linear cations is misleadingly good. In practice, the currently available experimental frequencies are overwhelmingly stretching fundamentals. The absence of experimental bending frequencies results in apparently good agreement for the linear cations, but this is likely to change as more experimental data becomes available. We now consider the relative advantages of scaling force constants rather than frequencies. Table 9 compares the relative errors for the set of cations common to the present work and the previous work of DeFrees and M ~ L e a n . Both ~ works employed the HF/6-31G* model and differed primarily in the method used for empirical scaling, with DeFrees and McLean using a uniform scale factor of 0.89 for all 6-31G* frequencies. The mean error for their cation frequencies is identical with the mean error they report for neutral molecules, 49 cm-1.3 As can be seen from the table, the scaling of force constants rather than frequencies results in a reduction in the root mean square and mean errors of 23 cm-’ and 17 cm-’, respectively. The percent error is lower by 1.3% when the force constants are scaled. The improvement in the predicted frequencies comes at the expense of a more elaborate correction scheme, however. It is quite easy to scale the frequencies regardless of whether the
force constants are in Cartesian or intemal coordinates. In order to scale the force constants, they must be expressed in intemal coordinates. While this is not expensive computationally, the use of intemal coordinates to express the Hessian is not usually available in the latest generation of ab initio codes. We are currently seeking to automate the scaling process so that the more popular codes can be used to scale the force constants. If we can make this easy to use, it has the potential to improve the accuracy with which modest ab initio methods can predict vibrational frequencies for both neutral molecules and cations.
5. Conclusions We have shown that the residual errors in Hartree-Fock vibrational frequencies are similar for cations and neutral molecules. We have determined empirical scale factors for the force constants of a set of neutral molecules and used these factors to scale the force constants of a set of cations. The agreement with experiment is better for nonlinear molecules than for linear molecules, with the largest deviations in the lowfrequency bending modes of the linear molecules and ions. This observation points to the possibility that a separate scale factor may be needed for linear and nonlinear bending force constants. We will be exploring this possibility in the future. By scaling force constants rather than frequencies, we have achieved better accuracy in the predicted frequencies for both ions and neutral molecules. However, the scaling of force constants is less amenable to automatic calculation because the force constants must be expressed in intemal, rather than Cartesian coordinates. The popularity of this method will depend in large measure on the development of an interface to the many ab initio programs which compute force constants in Cartesian coordinates. Acknowledgments. This research was supported by a grant from Research Corporation. M. Abu-Omar, P. Peterson, and T. W. Johnson received fellowship grants from HampdenSydney College. References and Notes (1) Gudeman, C. S.; Saykally, R. J. Annu. Rev. Phys. Chem. 1984,35, 387. (2) Saykally, R. J. Science 1988, 239, 157. (3) DeFrees, D. J.; McLean, A. D. J . Chem. Phys. 1985, 82, 333. (4) Maier, J. P., Ed. Ion and Cluster Ion Spectroscopy and Structure; Elsevier: Amsterdam, 1989. ( 5 ) Yamaguchi, Y.; Schaefer, H. F., IIIJ. Chem. Phys. 1980, 73, 2310. (6) Lee, T. J.; Schaefer, H. F., I11 J . Chem. Phys. 1984, 80, 2977. (7) Botschwina, P.; Sebald, P. Chem. Phys. 1990, 141, 311. Botschwina, P.; Horn, M.; Flugge, J.; Seeger, S . J . Chem. Soc., Faraday Trans. 1993, 89,2219. Botschwina, P.; Heyl, A.; Horn, M.; Flugge, J. 1.Mol. Specfrosc. 1994, 163, 127. (8) hlay, P.; Fogarasi, G.; Boggs, J. E. J . Chem. Phys. 1981, 74, 3999. (9) Pongor, G.; F’ulay, P.; Fogarasi, G.; Boggs, J. E. J . Am. Chem. SOC. 1984, 106, 2765. Sellers, H.; Pulay, P.; Boggs, J. E. J . Am. Chem. SOC. 1985, 107, 6487. (10) Lemert, R. F.; Dunn, K. M. Thirteenth Austin Symposium on Molecular Structure, Austin, TX, 1990. (11) Florian, J.; Johnson, B. G. J . Phys. Chem. 1994, 98, 3681. (12) Pulav, P. Theor. Chim.Acta 1979. 50. 299. (13) Pul&, P.; Fogarasi, G.; Pang, F.; Boggs, J. E. J . Am. Chem. SOC.
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