J . Phys. Chem. 1991,95,6871-6879
6871
Evaluation of Vibrational Force Fields Derived by Using Semiempirical and ab Inltio Methods D. M. Seeger, C. Korzeniewski,* and W. Kowalchyk Department of Chemistry, The University of Michigan, Ann Arbor, Michigan 48109 (Received: November I, 1990)
Modern semiempirical computational methods were surveyed for their ability to predict vibrational force fields. The MNDO, AMI, and PM3 methods were used to evaluate the harmonic force constants and Vibrational frequencies of a series of structurally related molecules. Full symmetry and symmetrized internal valence force constants are reported. Results are compared to the experimental harmonic force constants and to values obtained from minimal and extended basis ab initio calculations. The PM3 Hamiltonian is evaluated for the first time and predicts frequencies with greater accuracy than MNDO or AMI. The semiempirical results agree well with ab initio values.
Introduction Vibrational spectroscopy is employed routinely in all subdisciplines of chemistry and finds broad application in many other areas of science. Experimental spectra are useful in assigning molecular structure and in understanding the bonding properties of molecules. Of great value in spectral interpretation are methods for computing the energies of vibrational transitions and the intensities of spectral bands. Computational methods facilitate the assignment of experimental bands and permit the estimation of f a constants in caseawhere there is insufficient experimental data. In recent years, quantum mechanical methods have proven to be quite useful in evaluating vibrational force fields.'" It is most often observed that vibrational frequencies computed using quantum mechanical methods are higher in energy than experimental values, a result of neglecting the correlated motion of the electrons and from approximating the potential as a harmonic function. Errors arising from the first effect are minimized by using large basis sets and including electron correlation; however, the cost is quite high even for molecules containing only a few atoms. Anharmonic terms can be included in the analysis and thereby reduce errors resulting from the second effect, but again, the cost of calculating these terms is high. Studies Over the past twenty years have established that, when calculations are performed at the Hartrce-Fock (HF) level using medium-size basis sets (4-31G, for example), the discrepancy between the computed and experimental force constants is sufficiently systematic to permit the application of generalized scaling procedures which bring the computed spectrum into agreement with experiment.' This approach works well except in cases where the vibrations are sensitive to the off-diagonal elements or when correlation effects are Iarge.6" For such systems, refinement of structural models is achieved by evaluating force fields at increasingly higher levels of For molecules that are too large to be handled by ab initio methods, or when rapid and inexpensive solutions are required, semiempirical methods offer a compromise between cost and computational accuracy. Early attempts to apply quantum mechanical pracuiures in the simulation of vibrational spectra reported encouraging results using semiempirical methods,I2-l6 and it has been shown that force constants determined by using semiempirical approaches are comparable in accuracy to those obtained by using minimal basis ab initio calculation^.'^^^^'^^^^ Reant works employing semiempirical calculations to construct a force field for bacteriorhodopsin confirm the importance of these methods for understanding macromolecular structure.'O As there are many molecules of experimental interest that are too large for practical analysis by ab initio methods, an evaluation of the reliability of recently developed semiempirical methods for predicting vibrational force fields would be of value. Author to whom cormpondence should be a d d r d .
This paper reports the harmonic vibrational force fields of a series of simple polyatomic molecules computed by using the semiempirical MNDO, AM 1, and PM3 Results are compared to the experimental harmonic force amstants and to values obtained from minimal and extended basis ab initio calculations. The structurally related glyoxal, acrolein, butadiene, ethylene, and formaldehyde are among the molecules treated, as they have been the focus of early ~emicmpirical~~ and medium basis ab initio studies,' and permit the evaluation of generalized scale factors for the related force constants. The treatment is further extended to medium-size systems such as aniline and polyethylene. The performance of the PM3 Hamiltonian in predicting vibrational force fields is evaluated for the first time. Results of this work are intended to provide guidelines for the use of these semiempirical approaches when examining the vibrations of systems for which ab initio methods are unavailable or impractical.
Methods Except where noted, all semiempirical computations were performed with version 5.0 of the MOPAC system of programs" (1) Fogarasi, G.; Pulay, P. In VibrariomalSpectra andsrrucfun;Durig, J. R., Ed.; Elsevier: New York, 1985; Vol. 14, p 125. ( 2 ) Boatz, J. A.; Gordon, M. S.J. Phys. Chem. 1989, 93, 1819. (3) Haur, B. A. Jr.; Schaad, L.J.; Carsky, P.; Zahradnik, R. Chem. Rco. 1986.86.709.
(4) hlay, P.; Fogaresi. G.; Pongor, G.; Boggs, J. E.; Vargha, A. J . Am. Chrm. Soc. 1983,105,7037. (5) Simandiras, E. D.; Rice, J. E.; Lee,T. J.; Amos, R. D.; Handy, N. C. J . Chem. Phys. 1988,88, 3187. (6) Coffn. J. M.: Pulav, P. J. Phvs. Chem. 1991. 95. 118. (7j Hamilton, T..P.; hiay, P. 1.>hys. Chem. 1989,93,2341. (8) Guo, H.; Karplus, M. J. Chem. Phys. 1991, 94, 3679. (9) Ahrcn, A. M.; Gamll, R. L.;Jordan, K. D. 1.Phys. Chem. 1988,92, 6228.
(IO) Groesjean, M. F.; Taven, P.; Schulten, K. J. Phys. Chem. 1990,91, 8059. (1 1) Niahimura, Y.; Tauboi, M.; Kato, S.;Morokumura, K. In Prrcedhgs of the 8th International con/rence on Raman Spectrmmpy, Bordeaux 198a Luscombe, J., Juong, P. V., Eds.;Wiley-Heyden: Chicheater, England, 1982; p 703. (12) Pulpy, P.; Tbt%k,F. Mol. Phys. 1973, 25, 1153. (13) Pancheoko, Y. N.; Pulay, P.; TMk, F. J. Mol. Srrucr. 1976,34,283. (14) TWk, F.; Hegcdus, A.; Koa, K.; Pulay, P. J. Mol. Struet. 1976, 32, 93. (15) Fogarasi, G.; Pulay, P. J . Mol. Struct. 1977, 39, 275. (16) Blom, C. E.; Altona, C. Mol. Phys. 1976, 31, 1377. (17) Schlegel, B. H.; Wolfe, S.;Bemardi, F. J. Chem. Phys. 1975, 63, 3632. (18) (a) Dewar, M. J. S.;Thiel, W. J. Am. Chem. Soc. 1977, 99, 4899, 4907. (b) Dewar, M. J. S.;McKee, M. L.J. Am. Chem. Soe. 1977.99,5231. (19) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J . Am. Chem. Soc. 1985,107,3902. (20) Stewart, J. J. P. 1.Comput. Chem. 1989, IO, 209.
0022-3654 19 1-,12095-687 1 S02.50lO 63 1991 American Chemical Society I
6872 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991
Seeger et al.
TABLE I: F o m Comt~tsrad Frequencies of FormrMebyde ~~
~~
semiempirical type FII co FI2CO/CH sym F13CO/CH2 scis FZ2CH sym FZ3CH sym/CH2 scis F3) CHI scis Fu CH asym Fd5CH asym/CH2 rock Fss CHI rock Fw CHI wag y2 yj
AI AI AI (SC)
y4
BI (OP)
"I
vs B2 v6
BZ (ro)
exptb 12.91 0.665 0.414 4.962 -0.123 0.571 4.873 0.213 0.835 0.403
AM 1 18.115 1.051 0.555 5.587 -0.012 0.484 5.144 0.425 0.738 0.404
2782 1746 1500 1167 2843 1249
3121 2053 1443 1165 3085 1176
nonscaled MNDO 19.088 1.082 0.506 6.234 -0.062 0.513 5.736 0.416 0.763 0.463 3302 2115 1490 1215 3256 1210
scaled' PM3 16.579 0.428 0.275 5.103 -0.091 0.375 4.876 0.125 0.573 0.339
AM 1 13.065 0.809 0.495 4.589 -0.01 1 0.535 4.225 0.405 0.815 0.406
MNDO 12.874 0.762 0.424 4.589 -0.054 0.535 4.222 0.364 0.796 0.428
PM3 12.307 0.344 0.279 4.452 -0.100 0.520 4.254 0.138 0.794 0.383
2999 1987 1288 1098 3026 1069
2829 1745 1515 1167 2793 1237
2833 1745 1515 1167 2790 1237
2797 1753 1484 1167 2826 1260
ab initio nonscaled 4-2 1 14.866 0.699 0.430 5.062 -0.117 0.675 5.000 0.180 0.982 0.508
2974 1921 1702 1331 3060 1408
'Scaled according to methods of ref 4. Also scc scale factors of Table VI. bReference 4.
on a VAX Workstation 3200. The AMl,19 MNDO,'* and PM3m methods were employed in thesc studies. The keyword DFORCE was used to obtain harmonic frequencies and full Cartesian coordinate force constant matrices. Force constants and vibrational frequencies were evaluated at the theoretical equilibrium molecular geometry determined by using the indicated semiempirical method. While early vibrational studies advocate evaluating force constants at either the experimental molecular geometry2 or the empirically corrected theoretical geometry,'Ja the theoretical geometry was chosen here to indicate more clearly the magnitude of the errors expected when applying these methods to molecules for which accurate experimental geometries are unknown, or empirical scaling factors are unavailable. While using the MOPAC program it was noted that eigenvalues assigned to rotational degrees of freedom differed significantly from zero. To correct this problem the software was modified to ensure that the step size used in the evaluation of second derivatives was no larger than 0.05 A. This resulted in corrections to normal mode frequencies of less than 0.1 cm-'. Version 6.0 of MOPAC, in particular the eigenvector following routine,35was used to optimize the geometry and calculate the frequencies of polyethylene using the (C6H12)x repeat unit.'2 Ab initio calculations were performed using GAUSSIANB~ on an IBM on the San Diego CRAY Y-MP super3090 and GAUSSIANSS~~ computer. Harmonic force constants are re rted in both symmetrized internal valance (local) coordinatesfpand full symmetry coordinates. All force constants are reported in units of mdyn/A for bond stretching and bond-bond interaction force constants; mdyn A for angle bending, wagging, and torsions and all their interaction force constants; and mdyn for bond stretching-angle bending interaction force constants. The transformation of force constants from atom-based Cartesian coordinates to symmetrized internal valence coordinateswas effected according to the methods of Pulay and co-~orkers.'**~ Force constants were scaled using the scaled quantum mechanical force field method' and a least-squares fitting procedure.23
Results and Discussion Structurally Related Heavy Atom Hydrides. In analyzing the vibrations of the polyatomic molecules reported within, we found it most instructive to evaluate how force constants compare within a class of structurally related compounds. The re rted work draws upon early studies of Blom and Altona,'6v261pand Pulay (21) Hcimer, N. E.; Swanson, J. T.; Stewart, J. J. P.Q C Catalogue, ~ 1990, Program QCMW19. (22) May, P.; Fogarasi, G.; Pang, F.; Boggs, J. E.J . Am. Chem. Soc. 1979, 101, 2550. (23) Califano, S. Vfbrollonol Srores; Wiley: New York, 1976.
and co-worker~,'J~-'~*~ which focused upon deriving generalized scaling procedures to refine computed vibrational force fields. In early work, Blom and A l t ~ n a developed ~ ~ ~ ' a procedure which applied separate scale factors to the diagonal stretching, angle bending, torsional, and wagging force constants, and applied a single scale factor to all interaction terms. At about the same time, Pulay and co-workers reported scaled force fields using the semiempiricalCNDO/2 (complete neglect of differential overlap) method.l*15 An analysis of the out-of-plane vibrations of benzene and fluorobenzenesreproduced the sign of the experimental force constants and predictedthe values to within an order of magnitude. The errors in this analysis were sufficiently systematic that only three adjustable parameters were required to reproduce the experimental vibrational spectrum with a mean deviation of 16 cm-l.14 These investigators went on to use the CNDO/2 method to derive vibrational force fields for the cis and trans forms of glyoxal, acrolein, and 1,3-butadiene, and had similar success.13 These molecules have structural similarities which permit determination of scaling factors which are suitable for transfer among the set. Three scale factors were selected and applied to diagonal bond stretching terms, in-plane deformations, and out-of-plane deformations of each molecule. Force fields for these molecules were later reexamined at the H F level using Pulay's 4-21 basis set.' Ethylene and formaldehyde were included and scale factors appropriate for transfer among the five structurally related molecules were derived. This work formally introduced the scaled quantum mechanical (SQM) force field method which presents the most systematic way to scale both diagonal and off-diagonal force constants. In the present study, force fields of the structurally related molecules formaldehyde,ethylene, and the trans forms of glyoxal, acrolein, and 1,3-butadiene were examined by using the AMI, MNDO, and PM3 Hamiltonians. Vibrational frequencies and force constants are compared to results of Pulay's HF/4-21 study and are summarized in Tables I-V. HF/4-21 frequencies were computed after transforming the reported' matrix of scaled force constants to the unsculed representation using the indicated' scale factors. For the five molecules studied, computed vibrational frequencies which are most cloisely associated with the major stretching vibrational force constants were plotted against the experimental values. Results of frequencies associated with 0-0, C - C , and C-C stretching modes are shown in Figure 1. The ab initio method predicts the C 4 stretching frequency with the greatest accuracy among the methods tested, and all the methods (24) Blom, C. E.; Altona, C. Mol Phys. 1976,32, 1137. (25) Blom, C. E.; Altona, C. Mol Phys. 1977, 33,875. (26) Blom, C. E.; Altona, C. Mol Phys. 1977,31, 177. (27) Blom, C. E.; Altona, C.; Oakam, A. Mol Phys. 1977, 31,551.
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6873
Evaluation of Vibrational Force Fields
TABLE II: Force Constants md Frequencies of Etbylew semiempirical type Al, v , CHI sym str A,, v2 C = C str; CH2 scis A,. v, CH, scis: C 4 str CH; tw A;: BI, us CH2 asym str B,, ~6 CH2 rock B I U V7 CHI wag B, ug CH2 wag BZuv9 CHI asym str BZuuI0 CHI rock B3, v I I CH2 sym str B3uv12CHI scis
4
tYDc FI C 4 FI2C--C/CH sym F13C=C/CH2 scis FZ2CH sym F2, CH sym/CH2 scis F,, CH2 scis F u C--C tors Fs5 CH asym Fs CH asym/CH2 ro FMCH2 rock F77 CH2 OP sym FBICHI op asym FW CH asym F9,lo CH asym/CH2 ro asym Flo,loCH2 rock asym FII.11 CH sym F11,12 CH sym/CH2 sc asym F12,12 CH2 scis asym
exptb 3026 1630 1342 1023 3086 1220 949 940 3105 826 3021 1444
exW 9.395 0.363 -0,314 5.637 0.056 0.488 0.139 5.657 0.396 0.659 0.296 0.218 5.493 0.174 0.485 5.603 0.092 0.452
AM1 3210 1826 1388 874 3153 1167 1056 1068 3186 834 3218 1412
nonscaled MNDO 3430 1782 1425 870 3381 1182 1113 1099 3410 855 3434 1466 nonscaled MNDO 1 1.960 0.792 -0,445 6.720 0.100 0.489 0.374 6.202 0.350 0.582 0.327 0.437 6.205 0.297 0.505 6.687 0.127 0.455
AM1 12.797 0.8 18 -0.484 5.898 0.084 0.465 0.388 5.400 0.336 0.574 0.291 0.409 5.410 0.286 0.491 5.877 0.140 0.428
scaled' MNDO 3077 1600 1378 1023 3028 1173 95 1 939 3056 848 3073 1454
PM3 AM1 3158 3075 1829 1619 1307 1383 88 1 1023 3128 3021 1093 1181 938 1054 984 948 3146 3053 821 844 3138 3084 1328 1428 semiempirical PM3 12.508 0.364 -0.330 5.618 0.121 0.410 0.382 5.242 0.148 0.487 0.289 0.347 5.272 0.144 0.455 5.589 0.140 0.372
AM1 9.759 0.685 -0.427 5.417 0.08 1 0.475 0.133 4.960 0.326 0.587 0.230 0.322 4.969 0.277 0.502 5.398 0.101 0.438
scaled' MNDO 9.157 0.621 -0.386 5.398 0.089 0.480 0.129 4.982 0.31 1 0.572 0.238 0.319 4.985 0.264 0.496 5.372 0.1 13 0.447
PM3 3069 1628 1344 1023 3047 1171 98 1 916 3064 880 3056 1423
PM3 9.088 0.302 -0.301 5.330 0.126 0.471 0.129 4.974 0.154 0.559 0.251 0.300 5.002 0.150 0.522 5.303 0.146 0.428
ab initio nonscaled 4-21 3265 1783 1475 1158 3320 1364 1124 1102 3350 917 3244 1626 ab initio nonscaled 4-21 10.341 0.174 -0.359 6.008 0.089 0.588 0.166 5.912 0.242 0.760 0.403 0.310 5.958 0.120 0.558 5.986 0.120 0.546
O S e C footnote a of Table 1. Also scc scale factors of Table VI. *Reference 4. CReferences29 and 30. (Note that the force constants are experimental harmonic.)
2000
1800
-
U
9 1600 L
-a u
* 1400:
1000
1200
1400
1600
1800
2000
2200
Exprrlmrntrl Figure 1. Comparison of computed and experimental vibrational frequencies for modes associated with (2-0, C I C , and C C stretching motion appropriate for the molecules glyoxal, acrolein, 1,3-butadiene, ethylene, and formaldehyde. Vibrational frequencies were computed at the Hartree-Fock level using the 4-21 basis set ( O ) , and using the semiempirical PM3 (+), AM1 (0),and MNDO (*) methods. predict values which deviate from the experimental frequency in a systematic way. The scale factors associated with the C - 0 stretching force constant in acrolein, glyoxal, and formaldehyde (Table VI) have percent standard deviations of 1.5%. 1.1'76, and 0.7% for the MNDO,AMI,and PM3 Hamiltonians, respectively. Similar trends are observed for the C=C stretching frequencies,
2710
1810
2910
5010
5110
3210
JSSO 3450
Experimental Figure 2. Comparison of computed and experimental vibrational frequencies for symmetric CH2 stretching modes of glyoxal, acrolein, butadiene, ethylene, and formaldehyde computed at the HF/4-21 level (a), and using the PM3 (+), AMI (0),and MNDO (*) methods. where the a b initio result agrees most closely with experiment. The C-C stretching frequencies for acrolein and butadiene are best predicted by the PM3 method. Larger deviations exist among the scale factors associated with the C< stretching force constants (Table VI), as might be expected for these lower energy vibrations. Computed and experimental frequencies associated with inplane C-H modes are compared in Figures 2 and 3. The PM3 method predicts values which agrce most closely with experimental
Seeger et al.
6874 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 TABLE III: C l v o d Freaueocks
semiempirical type A, CH str B, CH str A, CO str B, CO str A, CH ro B, CH ro A, CC str, CCO dcf A, CCO def, CC str B, CCO def B, CH wag A, CH wag A, CC tors
exptb 2844 2835 1745 1732 1338 1312 1065 55 I 339 1048 80 1 127
nonscalcd MNDO 3265 3278 2129 2123 1408 1388 1 I46 594 416 1 IO5 848 -72
AM 1 3096 3115 2066 2070 1328 1298 1 I67 61 1 382 1067 794 22
PM3 2945 2957 1994 1989 1206 1 I75 1068 567 393 1027 790 9
AM 1 2831 2848 1743 1733 1271 1251 1047 579 367 1044 805 127
scaled0 MNDO 2833 2845 1758 1733 1283 1269 1068 542 380 1037 812 127
PM3 2835 2845 1761 1732 1232 1213 1047 574 407 988 874 120
ab initio nonscaled 4-2 1 3055 3056 1934 1933 1471 1445 1128 602 356 1182 913 199
"See footnote a of Table I. Also see scale factors of Table VI. *Reference 4.
TABLE I V Acrolein Freawacies
semiempircal type CHI asym CHv str CH sym CH f str co str C = C str, CHI scis CH sc,CH'rock C H ro, ~ C H scis ~ CH* ro, CH2 scis C-C str, CHVrock, CHI rock CHI ro, C-C str CCO def, C-C str, CCC def CCC dcf, CCO dcf CH tw, CH' wag CHf wag CHI wag CHVwag, CHI tw c-c tor
exptb 3102 3000 3000 2777 1723 1625 1422 1361 1276 1 159 913 564 324 993 980 959 589 158
AMI 3155 3177 3206 3084 205 1 1853 1422 1359 1298 1250 97 1 594 348 1076 93 1 1031 557
87
nonscaled MNDO 3397 3373 3427 3249 2112 1803 1477 1421 1344 1248 984 583 361 95 1 1128 1065 557 26
Scaled'
PM3 3122 305 1 3143 2925 1978 1848 1340 1203 1245 1 I68 925 570 359 910 983 1057 567 75
AMI 2968 2990 3017 2903 1724 1629 1389 1328 1257 1 I46 919 574 343 967 990 984 583 158
MNDO 3001 2978 3027 2869 1720 1624 1362 1394 1254 1162 917 557 340 1009 987 954 578 158
PM3 2962 3050 3030 2839 1745 1561 1406 1277 1244 1229 970 593 381 973 lo00
938 604 158
ab initio nonscaled 42lb 3477 3438 3377 3027 1900 1767 1588 1517 1399 1251 990 608 349 1 I63 1137 1118 669 183
'Scc footnote a of Table I. Also scc scale factors of Table VI. bRefercnce 4. TABLE V Butadiene Frequencies
semiempirical tYpc
B, CHz asym A, CHI asym B, CH' str A, CH' str B, CHI sym A, CHI sym A, C 4 , C-C Str B, C-C, CHI scis A, CHI scis B, CH2 scis, C==C str B, CH* rock A, CH' rock, C-C str A, C-C str, CHI rock B, CHI rock A, C C str, CH2 rock A, CCC dcf B, CCC def A, CHI tw, CH wag B, CH wag, CHI tw A, CHI wag B, CH2 wag B, CHI tw, CH wag A, CH wag, CHI tw A, C C tor
exptb 3102 3101 3056 3014 2985 3014 1643 1599 1442 1385 1296 1291 1205 99 1 890 513 30 1 1013 967 908 91 1 753 524 I63
AMI 3180 3184 3159 3142 3215 3217 1875 1832 1444 1403 I303 1291 1325 1003 963 538 324 952 99 1 1052 1054 703 490 87
nonscaled MNDO 3398 3401 3371 3357 3432 3432 1840 1766 1490 1459 1356 1346 1298 1034 96 1 527 327 97 3 1023 1 IO4 1 IO9 688 486 41
Scaled"
PM3 3137 3137 3041 3034 3149 3148 1873 1824 1377 1321 1208 1 I94 1267 962 914 530 334 924 942 1017 1041 706 502 76
* S a footnote a of Table 1. Also scc scaled factors of Table VI. bRcfercncc 4.
AMI 3046 3042 302 1 3006 3075 3077 1640 1606 1399 1373 1279 1268 1 I98 987 892 526 319 1019 975 914 916 748 514 163
MNDO 3047 3044 3020 3007 3075 3075 1664 1601 1391 1420 1298 1282 1 I99 99 1 895 502 313 1036 984 928 927 736 495 163
PM3 3075 3076 2982 2975 3086 3086 1690 1595 1387 I325 1223 1251 1203 977 916 533 339 1001 963 908 916 744 538 162
ab initio nonscaled 4-2Ib 3307 3305 3266 3260 3224 3223 1809 1742 1622 1543 1447 1422 1331 1114 948 555 320 1158 1121 1070 1075 863 585 160
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6875
Evaluation of Vibrational Force Fields TABLE YI: Valence Coordiunte Sule Factor@ type
formaldehyde
ethylene
C-H str C C str C = C str C 4 str CCC, CCH def CCO, CHf def CHv, CHI wag CHf wag C-C tor C=C tor, CH, tw mean sq dev
0.821
0.919
C-H str C C str C = C str C - 0 str CCC, CCH def CCO, CHf def CHv, CHI wag CHf wag C-C tor C - C tor, CH, tw mean sq dev
glyoxal AM 1 0.836 0.781
acrolein
0.763 0.702
0.721 1.023
0.927
1.104 0.788
0.958 35.338
1.004 29.1 0.736
1.368 38.1 0.803
29.5 MNDO 0.754 0.88 1 0.666
0.983 1.043
0.835 0.730
0.923 31.0
0.88 1 -3.270 1.382 36.6
24.6
averages
0.915 0.700 0.759
0.886 0.722 0.762 0.701 0.972 0.959 0.821 0.962 3.628 1.280 46.5
0.875 0.734 0.761 0.708 0.987 0.997 0.788 0.975 14.333 1.367
0.967 0.755 4.032 1.453 31.4
0.780 0.858 0.821 0.644 0.875 0.945 0.742 0.907 40.931 1.380 37.9
0.766 0.674
butadiene
0.775 0.842 0.800 0.661 0.925 0.941 0.725 0.904 20.662 1.421
0.803 0.786 0.812 0.917 0.703 17.772 1 SO0 33.4
PM3
C-H str C-C str C = C str C - 0 str CCC, CCH def CCO, CHf def CHV,CHI wag CHf wag C C tor C - C tor, CH, tw mean sq dev a References
0.872
0.949
0.927 0.930
0.727 0.742
0.756 1.148
1.387
1.067 0.866
1.129 12.5
0.926 222.8 1.348 33.7
54.7
0.962 0.993 0.740
0.943 1.064 0.627 0.745 1.126 1.128 0.849 1.013 5.024 1.268 54.5
0.931 0.996 0.698 0.748 1.102 1.194 0.840 1.023 77.7 1.336
1.031 0.804 5.370 1.393 42.2
4 and 22.
vibrational frequencies. Computed frequencies identified with symmetric CH2 stretching motion are plotted against the experimental values in Figure 2. Modes of formaldehyde appear lowest in energy, while modes of acrolein, butadiene, and ethylene are clustered within about a 75-cm-' range along the experimental frequency axis. The HF/4-21 frequency for acrolein falls well above the experimental frequency value and significantly outside the range of the other test molecules. Computed ab initio frequencies are higher in energy than those obtained by using the AM1 or PM3 method, but the slope of a least-squares fit line connecting the values obtained for a given computational method (excluding acrolein) is closest to unity for the ab initio method, suggesting that scale factors applied to the associated force constants will transform more readily among the related molecules for this method. Figure 3 shows the correlation between computed and experimental frequencies for vinyl and formyl C-H stretching modes.' Again, the PM3 method predicts values which agree mast closely with experimental vibrational frequencies. The experimental values for modes associated with vinyl groups are clustered between about 3000 and 3050 cm-I. Similar to the trend observed for the symmetric CHI stretching mode, the HF/C2I value for the C-H vinyl stretching mode of acrolein shows the largest deviation from experiment. Scale factors computed for each molecule are listed in Table VI. These values are applied to the diagonal force constants using the coordinate definitions employed by Pulay and co-workers. Interaction force constants are scaled using the square root of the product of the appropriate scale factors.4J2 Consistent with data in Figures 2 and 3, the semiempirical methods predict vibrational frequencies associated with C-H stretching motion which agree closely with the experimental values; hence, the scaled factors are somewhat closer to 1.0 than are the HF/4-21 values. Errors
0 e
a
a
0"
2750
2860
1960
5050
5160
5260
$350
5460
Exprrlmrntrl
Figure 3. Comparison of computed and experimental vibrational frequencies for formyl C-H stretching modes of glyoxal and acrolein, and the vinyl C-H stretching modes of butadiene and acrolein. Vibrational frcquencics were computed at the HartrctFock level using the 4-21 basii (D), and using the semiempirical PM3 (+), AM1 (0),and MNDO (*) methods.
associated with C-C and C - C stretching force constants vary among the three semiempirical methods with the PM3 method predicting values which agree best with the experimental values. Hydrides Containing One Heavy Atom. Results from analysis H#,NH,,and CH, are presented in Tables of the molecules H20, VII-X. Experimental harmonic frequencies are compared with calculated values. In all cases, the computed symmetry force
Seeger et al.
6876 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 TABLE VII: hirnnollic Force Coastrats d Frequencies of H20 semiempirical nonscaled exptd AMI MNDO PM3 CNDob harm. 8.645 17.55 f, 8.453 7.037 9.258 0.429 0.232 0.265 0.253 f, -0.100 La 0.225 0.234 0.315 0.377 0.319 0.920 0.960 0.786 0.866 fa 0.696
FI F12 F22 F33
type V I A, vZA~ v j BI
8.353 0.318 0.696 8.553 cxptd harm. 3832 1648 3943
7.302 0.331 0.920 6.772
AMI 3584 1885 3505
9.51 1 0.445 0.960 9.005
9.074 0.533 0.786 8.126
AMI 8.456 0.319 0.224 0.703
scaled' MNDO 8.456 0.231 0.253 0.679
PM3 8.452 0.419 0.353 0.702
17.78 0.818 0.438 17.32 semiempirical
8.666 0.3 17 0.703 8.137
8.687 0.358 0.679 8.225
8.871 0.499 0.702 8.033
CNDO 5592 1841 5609
AMI 3903 1648 3870
MNDO 3990 1743 3870
nonscaled MNDO PM3 3929 4085 1960 1648 3843 4049
scaled PM3 3944 1647 3826
ab initio STO-3G' 4-31G' 10.185 8.707 -0.475 -0.128 0.311 0.325 1.307 0.793 9.170 0.440 1.307 10.660
8.579 0.460 0.793 8.835
ab initio STO-3GC 4-31GC 4141 3873 2179 1750 4391 4018
'Force constants arc scaled by 1.202, 0.464 AMI; 0.913, 0.707 MNDO; 0.978, 0.893 PM3; for stretching, bending, respectively. bReference 12. 'Reference 17. dReference 31. (Using de)= 0.9572.) TABLE VIII: H8rmoaic Force Coastrats 8 d Frawncies of H 8 semiempirical nonscaled scaled' ab initio exptb harm. MNDO PM3 MNDO PM3 STO-3G' 4-3 1G' f, 4.284 5.271 1.889 4.286 4.285 5.467 3.903 f, -0.012 0.075 0.009 0.061 0.020 -0.089 -0.017 f,. 0.134 0.095 0.039 0.084 0.060 0.021 0.134 0.741 0.682 0.717 0.707 1.186 fa 0.754 0.853
4.347 4.305 0.199 0.085 0.682 0.7 17 0.707 1.880 4.225 4.265 semiempirical CXptb nonscaled scald' type harm MNDO PM3 MNDO PM3 3039 1813 2741 2731 vl,Al 2722 1235 1193 1214 1214 v ~ A I 1215 3010 1809 2715 2724 v ~ B I 2733
FlI 4.272
F12 0.190 F22 0.754 F,r 4.296
5.346 0.134 0.741 5.196
1.898
0.055
5.378 0.030 1.186 5.556
3.886 0.190 0.853 3.920
ab initio STO-3GC 4-31GC 3052 2593 1500 1268 3111 2614
'Force constants are scaled by 0.813, 0.967 MNDO, 2.269, 1.037 PM3; for stretching, bending, respectively. bRefercnce 31. (Using fie) = 1.355.) CRcference 17.
constants are of the same sign as the experimental values and agree to within an order of magnitude. For water, the AMI, MNDO, and PM3 methods predict harmonic frequencies which deviate from the experimental values by an average of 10.6%,2.5%, and 9.4% respectively. Frequencies computed by using the CNDO method show an average deviation of 33.3% while 17.2% and 3.1% deviations are observed for frequencies computed at the HF level by using the STO-3G and 4-31G basis sets, respectively. All semiempirical methods, including the early CNDO study, predict incorrectly the sign of the bond stretching interaction force constant. This trend is also observed for H$ (Table VIII), where the semiempirical methods predict positive values for the bond stretching interaction force constant. Large errors are associated with the S-H stretching force constant, and its value for the PM3 method indicates that it is wholly inappropriate for this molecule. In all cases, STO-3G values contain the largest overall error. The ammonia force field is presented in Table IX together with the harmonic frequencies. The semiempirical methods predict harmonic frequencies which are in better agrement with experiment than those observed at the HF level. Semiempirical methods tend to underestimate the value of the N-H stretching force constant, while ab initio values are larger than expected. However, vibrational frequencies associated with N-H stretching
0
400
800
1100
1600
2000
Experiment
Figure 4. Comparison of computed and experimental vibrational frequencies for the NHz group vibrations of aniline (except for the N-H stretching frequencies). Vibrational frequencies were computed at the HartrctFock level using 4-31G basis (O), 6-31GS basis (A), and 431G/6-31GS basii sets (e)and using the semiempirical PM3 (+),AMI ( O ) , and MNDO (*) methods. Note that the NH2 wag frequency at 4-31G/6-31G* basis sets is not shown its value is -592 cm-l.
motion are close to the experimental harmonic values owing to the large bond interaction force constants predicted by the semiempirical methods. Ab initio methods predict the sign of the bond interaction force constant incorrectly, but it is within the expected order of magnitude. All methods reproduce the u2 mode poorly, resulting from the large error associated with the angle bending force constant. Excluding the u2 mode, average percent deviations from experiment of 2.7%,4.3%, 3.8%,22.8%, and 6.3% are observed in the AMI, MNDO, PM3, STO-3G, and 4-31G results, respectively. Computational results for methane are summarized in Table X. Similar to ammonia, semiempirical methods underestimate the bond stretching force constants and overestimate the bond interaction force constant. The AM1 and PM3 methods predict frequencies of comparable accuracy to the HF/4-21 Again, the STO-3G basis set predicts vibrational frequencies which contain the largest errors. Aniline. In order to extend this survey to more complex molecules the vibrations of aniline were evaluated by using semiempirical and ab initio methods. Frequencies were evaluated by using the 4-31G and 6-31G* basis sets at the respective equilibrium geometrie~'~and at the 6-31GS level by using the 4-31G equilibrium geometry (4-31G/6-31GS). This latter apI
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6817
Evaluation of Vibrational Force Fields
TABLE I X Harmonic Force Constants and Frequencies of NH3 semiempirical nonscaled exptb harm AM 1 MNDO PM3 AMI 7.193 6.948 7.002 fr 7.052 6.779 0.242 0.212 0.407 fm 0.01 5 0.234 0.856 0.81 1 0.627 fu 0.632 0.701 0.082 0.060 0.004 fra 0.125 0.085 fru' 0.302 0.332 0.284 0.344 0.319 fa. -0.050 0.003 0.029 0.067 0.003
F11 F12
7.082 0.729 0.532 7.037 -0.177 0.682
F22 F33
FM
Fu
7.617 0.629 0.9 14 6.980 -0.224 0.827
7.248 0.749 0.707 6.545 -0.247 0.698 ~~~
tYPe VI AI Y2 AI v3
u4
exptb 3506 1022 3577 1691
E E
7.763 7.486 0.720 0.692 0.946 0.633 6.540 6.760 -0.340 -0.237 0.744 0.624 semiemDirica1
~
~~~~
PM3 7.015 0.41 1 0.668 0.004 0.314
7.459 0.537 0.681 6.835 -0.192 0.615
7.837 0.632 0.778 6.604 -0.310 0.613
scaled" MNDO 3595 1272 3536 1594
PM3 3678 1269 3478 1595
0.055
ab initio STO-3Ge 4-31Ge 10.791 8.115 -0.264 -0,044 0.808 0.467 -0.067 0.066 0.208 0.288 0.357 -0.07 1 10.263 0.647 1.522 1 1.055 -0.275 0.45 1
8.027 0.642 0.325 8.159 -0.222 0.538
~~~~~
nonsialed MNDO 3635 1473 3573 1848
AM 1 3535 1140 3465 1765
scaled" MNDO 7.043 0.208 0.637 0.051 0.243 0.022
PM3 3662 1398 3461 1758
AM 1 3591 1079 3522 1668
ab initio STO-3G' 4-31Ge 3699 4184 890 625 4522 3879 2073 1607
OForce constants are sclaed by 1.033, 0.894 AMI; 0.979, 0.744 MNDO 1.010, 0.823 PM3; for stretching, bending, respectively. bReference 32. e Reference
17.
TABLE X Harmonic Force Constants and Frequencies of CH,
semiempirical
FlI F22 F33 F34
Fu
fr
L
exptC 5.842 0.581 5.382 -0.225 0.547
AM 1 5.621 0.493 4.784 -0,330 0.602
nonscaled MNDO 5.524 0.477 4.804 -0.302 0.601
5.535 0.153
4.994 0.209
4.984 0.180
PM3 5.591 0.492 4.789 -0,306 0.553 4.988 0.201 ~
~
AMI 6.140 0.487 5.225 -0.342 0.595
scaleda MNDO 6.8 15 0.502 5.927 -0.345 0.633
PM3 6.514 0.493 5.580 -0.330 0.554
5.453 0.229
6.149 0.222
5.812 0.234
~~~~~
type A C H sym ~2 E HCH v3TCHasym U, T HCH Y,
exptC 3137 1567 3157 1356
AMI 3216 1412 3104 1380
PM3 3312 1452 3208 1363
Force
AMI 3077 1420 2969 1388
a constants are scaled by 1.012, 1.075 AM 1; 0.897, 1.009 MNDO e Reference 22. (Transformed into full symmetry coordinates here.)
pmach has been suggested as a cost-effective means for computing the vibrations of medium-size molecules by ab initio methods.' Aniline provides a test of the limits of this approach as the two basis sets predict radically different geometriesOm Vibrational frequencies appear in Tables XI and XII. Assignments were made following the work of Evans3 and WhiffenS3' The frequencies were separated into three groups: 24 substituent-insensitive ring vibrational modes, 6-substituent-sensitivering modes, and 6 modes associated with the NH2 group, presented in Tables VI1 and VIII, respectively. Table VI11 and Figure 4 compare NH2 group frequencies of aniline. As expected, large errors result from using the 4-31G geometry in the computation of force constants at the 6-31G* level, particularly in the NH2 wagging mode, as a negative vibrational frequency results. Figure 4 indicates that the PM3 method outperforms both the MNDO and AMI Hamiltonians and gives results which approach the accuracy of the ab initio values when evaluated at the 6-31G* level. An exception is the NH2 wagging mode for which the various methods predict values which differ widely. This mode is described well by 6-31G* basis set demonstrating the importance of polarization and is predicted well by the AMI method which
4-21' 5.722 0.699 5.546 -0.260 0.674
7.264 -0.075
5.488 0.043
5.590 0.044
~~
semiempirical nonscaled MNDO 3388 1443 3307 1437
STO-3Gb 7.038 0.836 7.339 -0.091 0.814
ab initio nonscaled 4-31Gb 5.617 0.674 5.445 -0.131 0.667
scaled" MNDO 3050 1406 2977 1400
PM3 3069 1451 2971 1362
ab initio nonscaled STO-3Gb 4-31Gb 3443 3076 1880 1688 3706 3185 1660 1503
4-2Ic 3104 1724 3274 1478
0.949, 1.062 PM3; for stretching, bending, respectively. Reference 17.
has been parametrized to account for nitrogen hybridi~ati0n.l~ Of the aniline vibrations associated with ring modes, the 6-31G' basis set predicts large errors in modes associated with C-H motion when evaluated at the 4-31G equilibrium geometry (Table XI). Polyethylene. A convenient feature of MOPAC software is the ability to evaluate the electronic properties of polymers. In early work, Dewar et al. showed that the vibrations of a polyethylene unit cell could be evaluated following energy minimization applying the Born von Kirmin cyclic boundary condition^.'^ Presented (28) Meyer, W.; Pulay, P. Theor. Chim Acra 1916, 32, 253. (29) Duncan, J. L.; McKcan, D. C.; Mallineon, P. D. J. Mol. Srnccr. 1973, 34, 221. (30) Fogarasi, G.; Pulay, P. Acra Chim. Acad. Sci. Hung. 1980,108,55. (31) Nibler, J. w.; Pimentel, G. C . J . Mol. Specrrcwc. 1968, 26, 294. (32) Hoy, H. R.;Mills, I. M.; Strey, G. Mol. Phys. 1972, 24, 1265. (33) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.;Pople, J. A. In Ab Idrio Molecular Orbital Theory; Wiley-Interscience: New York, 1986; p 232. (34) Allan, D. S.;Sceger, D.M.; Korzcniewski, C. Appl. Specrrasc. 1990, 44, 1579. (35) (a) Baker, J. J. Compur. Chem. 1986,7,385. (b) In MOPAC Software version 6.0, FORTRAN code, the comments in subroutine EF.FOR.
6078 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 TABLE XI: R i n g - S u k t i h r m t - I t i v e Frequeociea of Aniline tvDC CXd' AMI MNDO 3192 341 2 3089 CH asym str 3421 3205 CH sym str 3053 3193 3410 CH sym str 3048 3183 3403 304 1 CH asym str 3183 3401 CH asym str 3025 1784 1733 CC str 1603 1719 1590 1751 CC str 1666 1632 1503 CC str 1470 1588 1560 cc str 1398 1292 1324 CC str 1334 1393 1308 CH ip bend 1253 1276 1 I73 CH ip bend 1199 CH ip bend 1203 1152 1148 1137 CH ip bend 1084 1109 CH ip bend 1067 1028 1 I72 ring br 1201 990 624 ring def ip 658 619 CH op bend 999 968 1063 980 1032 CH op bend 957 930 874 96 1 CH op bend CH op bend 807 825 82 1 878 749 CH op bend 908 ring def op 638 690 631 ring def op 490 415 48 1
Seeger et al.
PM3 3068 308 1 3063 3052 3052 1791 1776 1580 1553 1327 1239 1169 1158 1116 1049 1I25 630 1012 970 909 766 839 634 467
4-31G 3358 3379 3352 3335 3333 1806 1781 1684 1653 1381 1529 1329 1232 1288 1142 905 710 1169 1138 1039 883 960 799 587
6-31G* 3370 3388 3362 3347 3345 1806 1791 1673 1640 1360 1496 1299 1196 1153 1130 892 68 1 1IO9 1087 992 848 927 772 555
4-31G/6-31G* 3402 3420 3395 3379 3377 1818 1790 1672 1638 1347 1487 1295 1200 1235 1131 892 680 1094 1073 967 829 904 763 558
PM3 1369 866 537 412 358 196
4-31G 1424 1111 597 418 475 25 1
6-31G* 1395 1084 574 41 1 457 246
4-3 1G/6-3 IG* I41 1 1081 575 40 1 45 1 24 1
3984 3855 1842 1132 375 348
3886 3788 1843 1258 722 25 1
4029 3904 1800 1102 -592 287
'Adapted from ref 38. TABLE XII: Ring-Subatihrmt-SensitiveFrequencies of Anlliw tYPe expt' AM 1 MNDO CN str 1278 1486 1448 CCC ip bend 812 943 943 CCC ip bend 526 565 537 390 448 CN ip bend 422 500 372 366 ring op def CN op wag 233 20 1 199
NH2 asym str NHz sym str NH2 scis NH2 rock NHz wag CN tors
3500 3418 1619 1054 664 216
3463 3492 1734 1227 672 266
NH2 Group Frequencies of Aniline - 3553 341 1 3584 3532 1830 1679 1269 1086 1005 989 167 229
'Adapted from ref 38. TABLE XIII: Frequeocies of Poiyethyhe type expt' MNDO CHzasym 2919 3252 CH2asym 2883 3206 CHI sym 2851 3291 CHI sym 2848 3272 CH2 scis 1468 1451 CHI scis 1440 1466 CHI wag 1370 I493 CHI twist 1295 1226 CHI wag 1176 1331 CH2 rock 1 I68 1I83 CC str sym 1 131 1187 CC str 1061 I200 CHI twist 1050 1117 CH2 rock 725 739 a Reference
AMI 3050 3002 3106 3076 1405 1412 1496 1183 1271 1182 1173 1208 1057 737
PM3 2971 2937 3031 3026 1385 1425 1463 1119 1179 1117 1159 1070 1008 750
4I.
4500
3500 0
-m 0) L
a
2500
0 1500
500
in Table XI11 are the 14 fundamental vibrational frequencies of polyethylene computed by using the MNDO, AMI, and PM3 (36) (a) Friwh, M. J.; Binkley, J. S.;Schlegel, H.B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.;Stewart, J. J. P.;Bobrowitz, F. W.; Rohlfing, C. M.; Kahn, L.R.; &Freer, D. J.; R.;Whitaide, R. A.; Fox, D. J.; Ruder, E.M.; Pople, J. 0 ~ ~ ~ 8 1 Carnegie-Mellon ~ ~ 8 6 : Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1986. (b) Friwh, M. J.; Head-Gordon, M.; Schlegcl, H. B.; Raghavachari, K.; Binkley, J. S.;Gonzales,C.; DeFreer, D. J.; Fox, D. J.; Whiteside, R.A.; Seeger, R.;Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Ruder, E. M.; Topiol, F.; Pople. J. OAUS~IANI)~: Carnegie-Mcllon Quantum Chemistry Publirhing Unit: Pittsburgh, PA, 1988.
wet,
I
1
500
I
I
7
1500
I
I
7
2600
I
I
-
I
3500
8
I
9
4500
Exprrlmontrl Harmonlc
Figure 5. Comparison of computed and experimental harmonic frequencies for a scrim of diatomic molecules. Vibrational frcqucnoisS were computed at the HartreeFock Iml using the 6-31G* (a),and using the semiempirical PM3 (+), AMI (0),and MNDO (*) methods. The symbol for the G-31G* value for H F obscures the PM3 symbol.
Hamiltonians. The PM3 method predicts stretching frequencies with considerably greater accuracy than the MNDO and AM1 methods with an average deviation of 3.3% compared to 11.7%
Evaluation of Vibrational Force Fields TABLE X I V H
d c Force Colrstmb and Frequencies of Some Diatomics semiempirical expt 5.751 22.980 4.768 9.660 3.041 7.956 19.033 7.510
H2
N2 F2 HF BH BF
co
Be0
H2 N2 F2
HF BH BF CO BCO a
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6879
exDt 4401 2360 923 4139 2366 1400 2170 1487
AMI 4343 2744 1365 446 1 3086 1921 2267
AM 1 5.599 3 1.070 10.425 1 1.220 5.173 14.971 20.778
W error -1.3 16.3 47.9 7.8 30.4 37.2 4.5
MNDO 5.475 30.973 16.068 11.907 4.372 12.253 22.943 7.923
semiempirical MNDO 96 error 4294 2740 1694 4595 2837 1737 2383 1527
-2.4 16.1 83.5 11.0 19.9 24.1 9.8 2.7
ab initid PM3 5.958 28.773 9.797 10.675
3-21G 6.447 28.185 9.443 9.298 3.262 8.844 21.061 10.153
21.491
PM3 4480 2641 1323 4351
96 error
2306
6.3
1.8
11.9 43.3 5.1
3-21G 4660 2614 1299 4062 245 1 1476 2312 1729
6-31W 6.41 1 3 1.490 8.674 10.702 3.437 8.808
24.020 10.330
ab initid % error 6-31G* 5.9 10.8
40.7 -1.9 3.6 5.4 6.5 16.3
4647 2763 1245 4358 2516 1473 2438 1744
%error 5.6 17.1 34.9 5.3 6.3 5.2 12.4 17.3
Reference 33.
TABLE X V Boron T M d e Frequencies
nonscaled type
exptb
AM1
A~YZ
279 371 819
E Y ~
151
319 403 952 152
AI YI A2 ~2
471 455 954 243
470 479
888 691 1454 480
1219 618 2108 429
AI
YI
E Y3 E ~4 A~ YI
E Y4
810
272
MNDO BBr3 313 399 966 140 BC13 516 456 1069 236 BF3 1113 674 1804 501
scaleda AMI
MNDO
277 39 1 828 147
27 1 414 a44 144
514 437 248
464 467 966 240
851 701 1503 476
887 666 1452 490
885
'Force constants arc scaled by for BBr3: 0.749, 0.941 AMI; 0.748, 1.079 MNDO for BC13: 1.244,0.832 AMI; 0.807, 1.048 MNDO for BF3: 0.484, 1.285 AMI; 0.635,0.977 MNDO, for stretching, bending, respectively. bReference43.
and 7.1% for the MNDO and AM1 methods, respectively. The accuracy of the PM3 method indicates that it may be useful for making initial vibrational assignments of polymers, particularly when used in combination with software that permits the motions associated with a given mode to be vi~ualized.~' Diatomic Molecules. For completeness, this report includes a listing of harmonic frequencies and force constants computed for a series of diatomic molecules. Semiempirical frequencies are compared to experimental values and to those obtained from ab initio H F calculations. The results are summarized in Table XIV and Figure 5 . In general, the computational methods predict frequencies which are higher in energy than the experimental values. Exceptions include hydrogen (MNDO and AMI) and hydrogen fluoride (3-21G). The accuracy with which the computational methods predict harmonic frequencies varies widely depending upon the nature of the atomic centers; however, for
a given molecule all methods show similar deviations. The frequencies of the boron-containing diatomic molecules are not reproduced well by the semiempirical methods, an effect contrasted by results obtained for boron trihalide compounds (Table XV). Here, the semiempirical methods reproduce the sign of both symmetry and symmetrized valence force constants, and the magnitude of the computed values agrees well with the experimental values.
Conclusion While long-term goals of quantum chemistry concern developing computational procedures capable of predicting experimental harmonic frequencies within 196, results approaching this measure of accuracy require high levels of theory and very large basis sets (e.g., inclusion off functions5);hence, these methods are extremely costly when applied to large molecules of low symmetry. Results of this study indicate that frequencies computed by using modem semiempirical methods compare well to values obtained at the H F level by using medium-size basis sets, and demonstrate the utility of these methods for deriving initial vibrational models of experimental systems. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research and to the University of Michigan Block Grant at the San Diego Supercomputing Site. We thank Dr. D. Allan, Dr. Kurt Hillig, and Dr. J. J. P. Stewart for technical assistance and C.K. gratefully acknowledges support from the Camille & Henry Dreyfus Foundation Distinguished New Faculty Program and a Dow Corning Assistant Professorship. (37) Whiffcn, D. H. J . Chem. Soc. 1956, 1350. (38) Evans, J. C. Spectrochim. Acta 1960, 16,428. (39) Bock, C. W.; Trachtman, M.;George, P. J . Mol. Struct. ( T H E 0 CHEW 1985, 122, 155. (40)Scc~er,D. M.;Koncnicwski, C. To be published. (41) Dewar, M.J. S.;Yamaguchi, Y.;Suck, S . H. Chem. Phys. 197!&13, 145. (42) MOPAC Software version 6.0 Rcferenct Manual.
(43) Lindcman, L. P.;Wilson, M.K.J. Chcm. Phys. 1956, 21, 242.
