J . A m . Chem. SOC.1989, 11 I , 1256-1 258
1256
Heat of Formation of Singlet Methylene G. A. Petemson* and Mohammad A. Al-Laham Contribution f r o m the Hall-Atwater Laboratories of Chemistry, Wesleyan University, Middletown, Connecticut 06457. Received May 2, 1988
Abstract: The heat of formation, AHf",, of C H 2 (IAJ has been determined by a b initio calculations including extrapolation to the complete basis set (CBSj limit. The calculated value for the heat of formation is h H f D o= 102.0 f 0.7 kcal/mol, which agrees with both the experimental upper limit of 102.6 kcal/mol and the average of several direct experimental measurements, 101.8 f 0.5 kcal/mol. The contribution of the Is, valence correlation energy is f 2 . 0 4 kcal/mol.
The heat of formation of singlet methylene is of considerable interest for studies of chemical reactions such as the rearrangement of propene to cyclopropane.' W e have determined this heat of formation using a b initio calculations including extrapolation to the complete basis set (CBS) limit.2 The calculated value, 102.0 f 0.7 kcal/mol, lies within the uncertainty in the experimental value, 101.8 f 0.5 kcal/mol.
Method and Results To find the heat of formation, A H f D Oof, C H I (]A1),we have studied to gas-phase reaction: H2 + C H , (IAl) CH,
-
The standard heats of formation at 0 K of H 2 and C H 4 are known by definition and experiment3d to be 0.0 and -15.97 f 0.1 kcal/mol, respectively. W e used the experimental geometry for the separate species in calculating the total energy. The internuclear distance for the hydrogen molecule3bis 0.7414 8,. The internuclear distance and H-C-H angle for C H 2 ( ] A l ) a r e 1.110 8, and 102.4', respectively,jb and for methane3bthe internuclear distance is 1.085 8, and the H-C-H angles a r e 109.47'. The calculated total energy change for the reaction must be corrected for the zero-point energy change including anharmonicity differences to find AHfoo. The harmonic zero-point energy can be calculated with the experimental fundamental frequencies. The vibrational frequency of hydrogen3bis 4160.2 cm-l. The symmetric and antisymmetric vibrational frequencies for singlet methylene are 2805.9 and 2864.5 cm-I, r e ~ p e c t i v e l yand , ~ ~ the bending freq ~ e n c is y ~1352.6 ~ cm-l. Methane3a has a singlet a t 2917 cm-I, a doublet at 1534 cm-I, and two triplets a t 1306 and 3019 cm-l. The anharmonicity corrections to the zero-point energies for the hydrogen molecule,3brneth~lene,~' and methane3a are 1 17.91, 397, and 903.7 cm-', respectively. Because of the lack of enough information about the anharmonicity correction for singlet methylene, we used the value for triplet methylene as a n approximation. The anharmonicity correction for the bending mode in triplet methylene (1 17.91 cm-I or 0.33 kcal/molj is undoubtedly too large for singlet methylene, but the error from using this value to obtain AHf', is clearly less than 0.3 kcal/mol. W e believe the error in the zero-point energy correction is closer to icO.1 kcal/mol. The total energy change for a chemical reaction can be expanded in a Mdler-Plesset perturbation e x p a n ~ i o n . ~A reliable ( I ) Doering, W. v. E.; Ehlhardt, W. J. A m . Chem. S o t . 1987, 109, 2697. (2) (a) Nyden, M. R.; Petersson, G. A. J . Chem. Phys. 1981, 75, 1843. (b) Petersson, G. A.; Nyden, M. R. J . Chem. Phys. 1981, 75, 3423. (c) Petersson, G. A.; Licht, S. L. J . Chem. Phys. 1981, 75, 4556. (d) Petersson, G. A.; Yee, A. K.; Bennett, A. J . Chem. Phys. 1985, 83, 5105. (e) Petersson, G. A.; Braunstein, M. J . Chem. Phys. 1985, 83, 5129. (0 Peterson, G. A,; Bennett, A.; Tensfeldt, T.; AI-Laham, M.; Shirley, W.; Mantzaris, J. J . Chem. Phys. 1988, 89, 2193. (3) (a) Dennison, D. Rea. M o d . Phys. 1940, 12, 175. (b) Herzberg, G. Spectra of Diatomic Molecules; Van Nostrand, Princeton, NJ, 1950. (c) Herzberg, G.; Johns, J. Proc. R. SOC.London A 1966, 107, 295. (d) Stull, D. R.; Prophet, H . J A N A F Thermodynamical Tables; NSRDS-NBS 31; National Bureau of Standards: Washington, DC, 1971. (e) Petek, H.; Nesbitt, D.; Ogilby, P. R.; Moore, C. B. J . Phys. Chem. 1974, 87, 5367. (f) Bunker, P.; Jensen, P. J . Chem. Phys. 1983, 79, 1224.
0002-7863/89/1511-1256$01.50/0
calculation of the energy change requires that each order of perturbation theory be converged with respect to expansion of the basis set and that enough terms have been included in the perturbation series to achieve a converged result. In Table I we present the self-consistent field (SCF) contribution to the heat of formation calculated with basis sets ranging from the simple STO-3G to the large quadruple-[ plus triple polarization plus 2f (QZ+TP+Zf) atomic pair natural orbital ( A P N O ) basis set.2d,2fAs the basis set is improved, the S C F contribution to the heat of formation converges monotonically from above. Extrapolation of these calculations on the basis of the convergence of higher angular momentum (Le. g, h, etc.) contributions gives the complete basis set (CBS) S C F limit.2d.2fThe small changes from the Q Z + T P result to the QZ+TP+2f result and from the QZ+TP+2f result to the CBS extrapolation suggest that the CBS S C F contribution to the reaction energy is converged to within ic0.1 kcal/mol. Table I1 examines the convergence of the correlation energy contribution to the heat of formation. First we examine the frozen core (FC) approximation in which we include only the correlation energy of the valence electrons. The first set of calculations in Table I1 shows that, as the basis set is improved, the direct Mdler-Plesset second-order (MP2) contribution converges to the complete basis limit from below. Since larger basis sets recover more of the correlation energy, they favor CH,, which has more electron pair interactions. Extrapolation2d-2fto the complete basis set (CBS2) limit using the N-I asymptotic convergence2a-2cof N-configuration pair natural orbital expansions5convergers from above. Evidently, the low-lying natural orbitals of C H , are not as well described by A P N O s and require more flexibility in the basis set. Correction for the basis set superposition error (BSSE) brings the two approaches into good agreement (AHfoofrom MP2(FC) = +17.46 f 0.24 kcal/mol). The third-order contribution, MP3(FC), is much smaller and converges rapidly with expansion of the basis set to -4.60 & 0.05 kcal/mol. The fourth-order contribution, MP4( FC), is still smaller but converges less rapidly with the expansion of the basis set to -0.17 f 0.3 kcal/mol. The above error estimates presume that adding g, h, i, etc., basis functions will have a smaller effect than the addition o f f functions. T h e N-' convergence of the expansion in N natural orbitals requires that this is true for each individual molecule but not necessarily for the small differences between molecular energies. However, we believe that this is a reasonable assumption. At first glance, the expansion in orders of M P perturbation theory appears to be sufficiently converged a t fourth order that higher orders will contribute less to A H f o othan the basis set truncation error in the second- and fourth-order contributions. Unfortunately, a more careful consideration of the convergence (4) (a) M d e r , C.; Plesset, M. S. Phys. Reo. 1934, 46, 618. (b) Pople, J. A.; Binkely, J. S.; Seeger, R. Int. J . Quantum Chem. Symp. 1976, IO, 1. (c) Bartlett, R. J.; Shavitt, I. Chem. Phys. Lett. 1977, 50, 190. (d) Pople, J . A,; Raghavachari, K.; Frisch, M. J.; Binkley, J. S.; Schleyer, P. v. R J . A m . Chem. SOC.1983, 105, 6389. (5) Shull, H.; Lowdin, P. 0. J . Chem. Phys. 1959, 30, 617.
0 1989 American Chemical Society
J . Am. Chem. SOC.,Vol. 1 1 1 , No. 4, 1989 1257
Heat of Formation of Singlet Methylene Table I. Convergence of the SCF Energy (hartree) for the Reaction H,
basis set STO-3G 4-31G 6-31G 6-31G**
H,
CH,
-1.11668 -1.126 73 -1.126 73 -1.131 28 -1 . I 3 0 28 -1.131 67 -1.13338 -1.133 38 -0.000 08 -1 . I 33 46
-38.371 98 -38.81005 -38.853 05 -38.876 16 -38.88623 -38.893 71 -38.894 93 -38.895 18 -0.000 15 -38.895 33
+ CH, ('A,)
-
CH4 and the Resulting AHrooCH, ('A,) (kcal/mol) CH4 reaction ~HrOoCH2 (IA,) -0.238 19 -0.20295 -0.200 75 -0.19426 -0.190 53 -0.187 89 -0.187 42 -0.18753 -0.000 07 -0.1 87 60
-39.726 85 -40.13972 -40.18053 -40.201 70 -40.207 0 4 -40.213 27 -40.21573 -40.21609 -0.000 30 -40.21639
DZ+P; APNO" TZ+DP; APNOb QZ+TP; APNO QZ+TP+Zt APNO 1-6 extrapolation CBS SCF "This is the same as the ILL] basis set in Table 11. bThis is the same as the {KK, KL. LL, LL'J basis set in Table 11. Table 11. Convergence of the Correlation Energy (hartree) for the Reaction H2 (kcal /mol)
method and basis set MP2( FC) 6-31G**
DZ+P; APNO" TZ+DP QZ+TP QZ+TP+2f CBS2(FC) DZ+P; APNO" TZ+DP QZ+TP QZ+TP+BSSE~ M P3 ( FC) 6-31G**
DZ+P; APNO" TZ+DP QZ+TP QZ+TP+2f M P4 ( FC) 6-31G**
DZ+P; APNO" TZ+DP QZ+TP QZ+TP+2f MP4( FC)/total 6-31G**
H2
+ CHI (]Al)
CH,
CH4
-
+ I 21.23 +99.11 +97.73 f93.66 +91.33 +89.66 f89.37 +89.44 +0.04 +89.48
CH., and the Resulting AHf', CH, ('A,) reaction -0.025 -0.026 -0.025 -0.027 -0.027
60 05 13 06 44
AHfOO CH2
+ 16.06
-0.026 -0.028 -0.029 -0.030 -0.030
34 01 32 40 40
-0.1 10 98 -0.1 15 67 -0.124 02 -0.13024 -0.140 5 1
-0.162 93 -0.16973 -0.178 47 -0.187 70 -0.198 35
-0.034 -0.034 -0.034 -0.034
22 54 69 76
-0.149 12 -0.153 87 -0.15601 -0.157 27
-0.216 -0.219 -0.220 -0.220
05 59 18 18
-0.032 7 1 -0.03 1 18 -0.029 48 -0.028 16
+20.52 +19.57 +18.50 +17.67
-0.005 52 -0.006 48 -0.006 31 -0.005 93 -0.005 93
-0.019 12 -0.020 32 -0.020 78 -0.019 78 -0.01956
-0,01822 -0.019 09 -0.0 19 67 -0.0 18 44 -0.018 16
+0.006 42 +0.007 72 +0.007 42 +0.007 27 +0.007 33
-4.03 -4.84 -4.66 -4.56 -4.60
-0.001 42 -0.001 59 -0.001 56 -0.001 54 -0.001 54
-0.005 -0.005 -0.006 -0.006 -0.006
-0.005 -0.006 -0.007 -0.007 -0.008
+0.001 3 1 +0.001 11 +O.OOO 96 +O.OOO 64 +O.OOO 27
-0.82 -0.70 -0.60 -0.40 -0.17
-0.033 28
-0.135 78
-0.01788
+11.22
68 96 52 74 86
79 44 12 64 13
-0.186 94
('AI)
+16.35 +15.77 +16.98 +17.22
-0,01722 +10.81 DZ+P; APNO" -0.03608 -0.141 96 -0.195 25 TZ+DP -0.037 19 -0.151 32 -0.205 26 -0.01675 +10.51 QZ+TP -0.037 87 -0.15676 -0.21378 -0.019 15 +12.02 QZ+TP+2f -0.037 87 -0.166 93 -0.224 64 -0.019 84 +12.45 "This frozen-core DZ+P size basis set is obtained from the {KK,KL, LL) full DZ+P basis set in ref 2f by deleting the 2s' and 2p' functions, which correlate the ( I s , 1s) pair, and the 4p" and 4d" functions. which correlate the (Is, 2p) pairs. bThis calculation included the correction for the basis set superposition error (BSSE) on both H2 and CH2.
suggests that this is not true. T h e (3a,, l b , ) near degeneracy in C H , makes the perturbation expansion relatively slowly convergent. As a result, the M P 3 contribution is larger for CH2 than for C H 4 (Table 11). However, the larger number of three-body interactions in C H 4 makes the M P 4 ( F C ) triple-excitation contribution (-0.005 69 hartree) much larger than that for C H , (-0.002 99 hartree). The remaining MP4(FC) contributions from single, double, and quadruple excitations a r e larger in C H , (0.003 87) than in C H 4 (-0.00244), indicating that the near-degeneracy effect in C H , will dominate again beyond fourth order so that the full CI AHf',, should be less than the M P 4 limit. W e have shown previously2ds2fthat including pair energies6 to infinite order and pair coupling terms' to third order gives a CBS(".3) correlation energy that is very close to a full C I . It is appropriate to test this approximation in the present application to determine the accuracy when dealing with the (3a,, l b , ) near degeneracy in CH,. The full C I correlation energy of C H , ('A,) with a Dunning D Z f P basis sets has been determined by (6) Sinanoglu, 0.Ada. Chem. Phys. 1964,6, 315. (b) Nesbet, R. K. Phys. Reu. 1968, 175, 2. (7) Kutzelnigg, W.; Shavitt, 1.; Roos, B. 0.;Siegbahn, P. E. M. Methods ofE'leclronic Structure Theory; Schaefer, H. F.; 111, Ed.; Plenum: New York, I977 ., . , . (8) Dunning, T. H. J . Chem. Phys. 1970, 53, 2823
Bauschlicher and T a y l ~ r . W ~ e have determined the MP2, MP3, MP4, and pair C I energies with the same basis set a t the same geometry. The difference between the full C I energy and the MP2 energy is -0.031 06 hartree. The sum of the M P 3 and M P 4 contributions is only -0.026 07 hartree, confirming a relatively large contribution of 4 . 0 0 4 99 hartree from fifth and higher orders of Mdler-Plesset perturbation theory. The sum of the pair C I energies and the third-order pair coupling terms, MP("s3), gives -0.034 99 hartree for the contribution beyond second order, indicating a contribution of +0.003 93 hartree from fourth and higher order pair coupling terms. The MP("x3) approximation is comparable in accuracy to the M P 4 approximation for CH, ('Al). However, extrapolation to the complete basis set limit reduces the discrepancy between the MP("s3)energy and the full C I energy. T h e difference between the CBS(",3) and the CBS2 energies (-0.029 07 hartree) is considerably closer to the full C I contribution beyond second order (-0.03 1 06 hartree). In a recent paperZfwe introduced a CBS(".3)/(KK, KL, LL, LL'] A P N O model chemistry employing D Z + P sets of APNOs for the (Is, Is), (Is, valence), and (valence, valence) intraorbital and (valence, valence) interorbital pairs. Using this model, we obtain a correlation energy contribution to the AHfoofor C H 2 ( ' A l ) of (9) Bauschlicher, C. W., Jr.; Taylor, P. R. J . Chem. Phys. 1986, 85, 6510.
1258 J . Am. Chem. SOC.,Vol. 111, No. 4, 1989
Petersson and A / - Laham
Table 111. Convergence of the C B S Correlation Energy (hartree) for the Reaction H, (kcal/ mol) method and basis set CBS2(FC) D Z + P = (LL};A P N O "
+ CHI ('A,)
-
C H 4 and the Resulting rlHfooC H 2 ( ] A , )
H2
CH2
CH4
-0.034 22 -0.034 22 -0.034 54 -0.034 54 0.034 76
-0.149 12 -0.15264 -0.15387 -0.155 47 -0.157 27
-0.2 16 05 -0.2 17 96 -0.21959 -0.2 19 93 -0.220 18
-0.032 -0.031 -0.031 -0.029 -0.028
71 10 18 92 16
+20.52 +19.52 +19.57 +18.78 +17.67
(KK, KL, LL) T Z + D P = (LL, LL') (KK, KL, LL, LL')
-0.040 -0.040 -0.040 -0.040
02 02 53 53
-0,17074 -0.174 98 -0. I77 06 -0.178 24
-0.231 -0.232 -0.235 -0.235
82 78 24 45
-0.021 06 -0.017 79 -0.01765 -0.01668
t13.22 +11.16 +11.08 + 10.47
(KK, KL, LL}; A P N O (KK, KL, LL, LL'J
-0.040 02 -0.040 53
-0.229 07 -0.233 17
-0.290 64 -0.293 63
-0.021 55 -0.01993
+13.52 +12.51
(KK, KL, LL) T Z + D P = (LL, LL'J (KK, KL, LL, LL'J QZ+TPtBSSE* CBS("t3)(FC) D Z + P = ILL}; A P N O "
CBS(".3)
reaction
AHfDo CH2 ( ' A I )
QThisfrozen-core D Z + P size basis set is obtained from the (KK, KL, LL}full D Z + P basis set in ref 2f by deleting the 2s' and 2p' functions, which correlate the ( I s , Is) pair, and the 4p" and 4d" functions, which correlate the ( I s , 2p) pairs. *This calculation included the correction for the basis set superposition error (BSSE) on both H2 and C H 2 . Table IV. Summary of the More Cost Effective Calculations of AHf', (kcal/mol) for CH, ( ' A , ) method basis set S C F MP2(FC) MP3(FC) CBS(",3)(FC) CBS(",3) STO-3G 121.23 6-31G** 93.66 109.72 105.69 TZ+DP 89.66 105.43 100.77 100.74 101.99' QZ+TP+2f 89.44 106.66 102.06 est limit 89.48 106.94 102.34 100.24 101.6 k 0.5 OThis calculation included the KK and KL basis functions for the inner shell.
+12.51 kcal/mol, in good agreement with the MP4(FC)/QZ+TP+2f results +12.45 kcal/mol and an estimated MP4(FC) limit of +12.69 kcal/mol. Unfortunately, the very close agreement is fortuitous. Detailed results obtained with our model chemistry are presented in Table I11 along with results obtained with the same method with several other basis sets for comparison. If we choose the D Z + P (LL}basis set as a starting point, then the addition of KL basis functions (optimized for atomic Is, valence pairs) or LL' basis functions (optimized for atomic valence-valence interorbital pairs) or correction for the BSSE will reduce the CBS2(FC) value for AHf', by about 1 kcal/mol. These basis set effects are roughly additive. Correction for BSSE cannot be part of a model chemistry, since the energy of a molecule becomes dependent on the reaction under study. However, the CBS2(FC)/QZ+TP+BSSE result is useful to demonstrate the lack of basis set saturation for C H 2 in our CBS/(KK, KL, LL, LL'J model chemistry. Evidence that this lack of saturation persists in higher order calculations comes from comparison of our CBS("t3)( F C ) / ( K K , KL, LL, LL'] correlation energy for C H 2 (-0.178 24 hartree) with the CASSCF-SOCI+Q/ [5~4p3d2flg,4~3p2d] result (-0.179 49 hartree) of Bauschlicher, Langhoff, and Taylor.lo Our C B S model chemistry clearly underestimates the valence-shell correlation energy of C H 2 . Nevertheless, it is a significant improvement over the direct M P 4 ( F C ) result with the same [4s4p3d,3s2p] basis set (-0.152 50 hartree). T h e effect of the lack of basis set saturation on the CBS("j3)/{KK, KL, LL, LL'I energy for C H , is probably to overestimate AHfoo by about 0.5 kcal/mol. Calculations a t the CBS("s31 level with and without the F C approximation indicate that the contribution of the inner shellvalence interactions to the heat of formation of C H , is +2.04 kcal/mol (Table 111). Direct M P 2 calculations give a n inner shell-valence contribution of +0.74 kcal/mol with the (KK, KL, LL} basis set and + I .09 kcal/mol with the {KK, KL, LL, LL'J basis set. (10) Bauschlicher, C. W., Jr.; Langhoff, S. R.; Taylor, P. R. J . Chem. Phys. 1987, 87, 387.
The CBS program uses G A U S S I A N ~ Z ~to' evaluate all necessary integrals and determine the S C F energy and then performs the C B S extrapolation of the correlation energy. All calculations employing [spd,sp] basis sets were performed on a D E C Micro Vax-I1 computer. The MP4(FC)/QZ+TP+2f calculations were performed on the Cray X - M P supercomputer a t the Pittsburgh Supercomputer Center, which was available through the N S F Office of Advanced Scientific Computing. Conclusions The total calculated complete basis set value for the heat of formation of singlet methylene is obtained by adding the C B S S C F contribution (Table I) to the CBS("$3)/(KK,KL, LL, LL'} A P N O correlation contribution (Table 111). Our C B S result for the heat of formation of methylene is 101.99 kcal/mol. The MP4(FC)/QZ+TP+2f calculation gives 101.89 kcal/mol, and the C B S S C F energy plus the estimated MP4(FC) limit gives 102.17 kcal/mol. Unfortunately, the agreement with the C B S result comes from the cancellation of a contribution from terms beyond fourth order (-2 kcal/mol), by a contribution from Is, valence interactions (+2 kcal/mol). Using our error estimates above, we conclude that AHfoo= +102.0 h 0.7 kcal/mol, which is in agreement with both the average of several direct experimental valuesI2 (101.8 f 0.5 kcal/mol) and the experimental upper limit'3 (102.6 kcal/mol). The final results for AHFoobtained with the most cost effective calculations are summarized in Table IV. The S C F and M P 2 contributions to l H f o oare large and must be converged with expansion of the basis set. The MP3 contribution is also significant but converges rapidly with basis set expansion. The core contribution is small and cancels contributions from higher orders of perturbation theory. The minimum requirements for a reliable calculation are reasonable estimation of the CBS limits for AE(SCF) and AE(,)(FC) and inclusion of AE(3)(FC). Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant No. C H E 8419381. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. W e are also grateful to Wesleyan University, Merck S h a r p and Dohme Research Laboratories, and Digital Equipment Corp. for providing funds toward the purchase of computer equipment used in this research. Registry No. H,, 1333-74-0; C H 4 , 74-82-8; C H I , 2465-56-7
(1 1) Binkley, J. S.; Frisch, M.; Raghavachari, K.; DeFrees, D.; Schlegel, H. B.; Whiteside, R.; Fluder, E.; Seeger, R.; Fox, D. J.; Head-Gordon, M.; Topiol, S. GAUSSIAN 82, revision H, Carnegie Mellon University. ( 12) (a) Simons, J. W.; Curry, R. Chem. Phys. Lett. 1976, 38, 171. (b) Lengel, R. K.; Zare, R. N. J. A m . Chem. SOC.1978,100,7495. (c) Feldmann, D.; Meier, K.; Zacharias, H.; Welge, K. H. Chem. Phys. Lett. 1978, 59, 171. (13) Hayden, C. C.; Neumark, D. M.; Shobatake, K.; Sparks, R. K.; Lee, Y. T. J . Chem. Phys. 1982, 76, 3607.
