J. Phys. Chem. 1083, 87, 4833-4839
4833
Ab- Initio Calculation of the Zero-Field Splitting Parameters of Vlnylmethylene David Feller, Weston Thatcher Borden, and Ernest R. Davldson" Department of Chemistry B
O IO, Unlvershy of Washington. Seattle, Washington 98 195 (Received: Mey 12, 1983)
Multiconfiguration self-consistent-field (MCSCF) and multireference configuration interaction (MR-CI)ab-initio calculations have been performed on the 3A" state of vinylmethylene. These calculations indicate a relatively flat potential energy surface along the asymmetric C-C bond distortion coordinate. In the vicinity of the predicted minimum the unpaired spin distribution in the a space is strongly dependent on the difference in the two C-C bond lengths. As a consequence, the computed values of 0.56 cm-' for D and 0.05 cm-' for E that were obtained at the optimized geometry of the anti isomer change by 0.1 and 0.01 cm-', respectively, at geometries which differ by less than 0.001 hartree in energy.
They employed a point spin model and again assumed a delocalized A system, as in the allyl radical. The assumption of no u-A interaction gave values of D for vinylmethylene and for a variety of other unsaturated carbenes that were in almost exact agreement with experiment. When Hutton and Roth replaced the delocalized a spin density with the localized one of Davis et al., they obtained the same value of D for both the syn and the anti isomer. This value was about 0.3 cm-' larger than the average of the experimental numbers for the two isomers. Although the wave function of Davis et al. gives too large a value of D, calculations that obtain agreement with the experimental value by assuming that the unpaired u spin H H H does not affect the A spin distribution seem physically 1 2 unreasonable. This is the vinylmethylene paradoxcalculations that are based on a physically incorrect model The calculation of Hutton et al. assumed that the unappear to give values of D much closer to experiment than paired u electron was localized a t C1 and that the a spin do calculations that include the effect of the unpaired u density was delocalized exactly as in the allyl radical. The electron on the A electrons. latter assumption is correct only if the T spin distribution In an effort to resolve this paradox, we have carried out is unaffected by the unpaired u spin. This assumption was large-scale configuration interaction (CI) calculations at called into question by the generalized valence bond (GVB) optimized geometries for both of the triplet isomers. Using calculations of Davis et aL2 Their calculations showed that the triplet CI wave functions, we have evaluated the the unpaired u spin produces considerable localization in spin-spin contribution to the zfs parameters. On the basis the a spin density. of the small contribution, 0.02 cm-', of the spin-orbit Davis et al. obtained a value of D in the range 0.44-0.51 contribution to the D value of methylene5 the spin-spin cm-', based on a semiempirical calculation using the exterm is also expected to be the dominant term in D for pression vinylmethylene. D = Cl2(*llHDl*l) -k c22(*21HD1*2) + 2clc2(*llHDl*2) Previous calculations suggest that ab initio methods should be capable of predicting reliable values of D and where c1 and c2 are the expansion coefficients of the two E for molecules the size of vinylmethylene. Ab-initio most important nonorthogonal valence bond resonance calculations of D and E for CH2 are in good agreement with structures in vinylmethylene. The structure corresponding the latest laser magnetic resonance values. Langhoff and to \kl localizes both unpaired electrons on C1.Because the Davidsod obtained D = 0.781 cm-' and E = 0.050 cm-l exchange integral between these two electrons in the triplet compared to the experimental values3 of D = 0.778 cm-' favors \kl, c1/c2 was computed to be 1.99 for equal C-C and E = 0.040 cm-'. bond lengths and 3.44 at the geometry favored by these In addition to methylene, previous examples of ab-initio was assigned authors. The one-center integral ( \klJHD(\kl) zfs calculations have included work on trimethylenethe value of D in methylene, for which the authors used methane7 (C(CH2)J and the 3A" state of formaldehyde.* 0.627 cm-'. However, the actual value of D for methylene In the former, a CI value of 0.020 cm-' was obtained for is 0.78 cm-l. Use of the latter value would have given a D, compared to the experimental value of 0.025 cm-l. The computed value of D in vinylmethylene that was too high. comparatively large size of the system precluded the use Another theoretical estimate of the zfs parameters in vinylmethylene comes from the work of Hutton and R ~ t h . ~ of a polarized basis set at the CI level. However, an SCF evaluation of D with a polarized basis setg suggests that
Introduction The smallest triplet carbene that has been found to exhibit geometrical isomerism is vinylmethylene. On the basis of approximate calculations of the zero-field splitting (zfs) parameters Hutton et al.' assigned the following experimental values (in cm-') of D and E to the anti (1) and syn (2) forms of the molecule: Dmti= 0.409, Emti= 0.022, D , = 0.458, and E,, = 0.019.
(1) Hutton, R. S.; Manion, M. L.; Roth, H. D.; Wasserman, E. J. Am. Chem. SOC. 1974,96, 4680. (2) Davis, J. H.; Goddard, W. A,; Bergman, R. G.J. Am. Chem. SOC. 1977, 99, 2427. (3! (a) Sears, T. J.; Bunker, P. R.; McKellar, A. R.; Evenson, K. M.; Jennings, D. A.; Brown, J. M. J. Chem. Phys. 1982, 77, 5348. (b) Wasserman, E.; Hutton, R. S.; Kuck, V. J.; Yager, W. A. J. Chem. Phys. 1971, 55, 2593. 0022-365418312087-4833$0 1.5010
(4) Hutton, R. S.; Roth, H. D. J. Am. Chem. SOC. 1982, 104, 7395. (5) Langhoff, S. R. J. Chem. Phys. 1974, 61, 3881. (6) Langhoff, S. R.; Davidson, E. R. Int. J. Quant. Chem. 1973, 7,759. (7) Feller, D.; Borden, W. T.; Davidson, E. R. J. Chem. Phys. 1981, 74, 2256. (8) Davidson, E. R.: Ellenbogen, J. C.; Langhoff, S. R. J . Chem. Phys. 1980, 73, 865.
0 1983 American Chemical Society
4034
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983
the previous CI result would be increased by 0.004 cm-', which would bring i t into very close agreement with experiment. In the case of formaldehyde the computed zfs parameters are also in relatively good agreement with experiment. In this molecule the oxygen atom produces a large spinorbit component which nearly cancels the spin dipoledipole term. Initial Geometry Optimizations The 2Az state of the allyl radical suffers from the socalled "doublet instability" phenomenon, which manifests itself in nonphysical, broken symmetry solutions of the restricted Hartree-Fock (RHF) equations even at symmetric geometries.l0 Consequently, geometry optimization at the RHF level leads to nonequivalent C-C bond lengths, a finding a t odds with the EPR spectrum of the radical. A multiconfiguration self-consistent-field (MCSF) wave function, consisting of all possible excitations of three electrons in the three-orbital conceptual minimal basis set A space, has been demonstrated" to be sufficiently flexible to restore the lost symmetry. Unrestricted Hartree-Fock (UHF) wave functions have also been shown to predict a symmetric allyl radical. In vinylmethylene the doublet instability phenomenon also results in spurious distortion of the two C-C bond lengths at the RHF level of theory. Thus, it was decided to optimize the geometries for each state of vinylmethylene with a A space MCSCF wave function. Our experience with previous calculations on the closely related allyl and trimethylenemethane systems has shown that for such hydrocarbons the STO-3G minimal basis12is an economical and reliable alternative to the slightly larger 3-21G basis.13 Thus, the STO-3G basis was chosen for the first round of optimizations. For the 3A" and 'A" states the MCSCF wave functions involved three electrons distributed among three A orbitals. The calculations on the closed-shell singlet (lA') involved four electrons and four orbitals in order to include the double excitation from u to A that is known to be important for singlet carbenes. SCF and MCSCF geometry optimizations were done with the GAMESS program from NRCC.14 Some MCSCF calculations were also performed with ALIS'~from Iowa State University. All CI and zfs parameter evaluations were done with the MELD program from this laboratory. For comparison purposes the optimal RHF geometries are shown in Figure 1alongside the corresponding optimal MCSCF geometries. The largest RHF to MCSCF bond length differences are seen to occur in the triplet, where the optimal RHF structure displays typical CC double and single bond lengths (1.32 and 1.47 A) while the MCSCF structure has more nearly equal bond lengths (1.38 and 1.42 A). The similarity of the A systems of 3A" vinylmethylene to that of 2A" allyl is evident in the RHF optimized CC bond lengths of the two molecules, which are within 0.01 8, of each other. A space MCSCF calculations on allyl give equal bond lengths which are the average of the two unequal bonds in vinylmethylene. In contrast to (9) Feller, D.; Borden, W. T.; Davidson, E. R., unpublished result. (IO) Pauldus, J.; Veillard, A. Mol. Phys. 1969,35,445. (11) Takada, T.;Dupuis, M. J . Am. Chem. SOC.1983,105, 1713. (12)Hehre, W. J.; Stewart, R. F.; Pople, J. A. J. Chem. Phys. 1969,5I, 2657. (13) Binkley, J. S.; Pople, J. A.; Hehre, W. J. J. Am. Chem. SOC.1980, 102, 939. (14) Dupuis, M.;Spangler, D.; Wendoloski, J. J. NRCC Software Catalog Vol. 1, Program GGOl (GAMES), 1980. (15) Elbert, S. T.; Cheung, L. M.; Ruedenberg, K. NRCC Software Catalog Vol. 1, Program QMOl (LIS), 1980.
Feller et al.
RHF
MCSCF
H
I
r
H
!
11.082 H
1,
Figure 1. RHF and three-orbitakhree-electron MCSCF optimal geometries with the STO-3G basis.
the triplet, inclusion of electron correlation effects tended to lengthen both CC bonds in the two vinylmethylene singlets. In the 'A" state the unpaired a electron is fairly well localized to the noncarbene terminal carbon, in qualitative accord with the exchange term, KO,,arguments of Davis et al. Of the three vinylmethylene states, the 'A' state shows the best agreement with the geometry assumed by these authors. The triplet geometry assumed by Davis et al. was based on standard bond lengths and bond angles. It differs by +0.14 and -0.04 8, in the long and short CC bonds from our optimized MCSCF results. Their HCC carbene bond angle is smaller by 6 O from the one we obtained for the triplet and smaller by 1 1 O than the one we obtained for the open-shell singlet. Contrary to the claim of these authors the RHF spin population in triplet vinylmethylene clearly shows a localized A distribution. As discussed above, such localization would be anticipated from the behavior of allyl, where numerous workers have reported the symmetry breaking of the SCF wave function. An STO-3G MCSCF normal mode vibrational analysis at the optimal triplet geometry produced a set of harmonic frequencies that closely matched the 3-21G MCSCF frequencies in allyl computed by Takada and Dupuis.l' The lowest energy mode (473 cm-') in vinylmethylene is a CCC bending motion just as it is in allyl (476 cm-'). The carbon-carbon stretching frequencies, 1105 and 1237 cm-' in vinylmethylene and 1093 and 1204 cm-' in allyl, are likewise close. However, the force constants and normal modes are quite different. At this level of theory vinylmethylene has unequal bond stretching force constants and localized CC stretching modes. The optimal STO-3G UHF triplet geometry CC bond lengths are even more nearly equal than the MCSCF bond lengths. The expectation value of S2was 2.38, indicating a moderate amount of spin contamination. Semiempirical MNDO16 UHF calculations actually predicted the bond (16) Dewar, M. S.;Thiel, W. J. Am. Chem. SOC.1977,99,4899.
Ab-Inltlo Calculations of Vinylmethylene
The Journal of Physical Chemistty, Vol. 87, No. 24, 7983 4035
TABLE I: Computed Carbon-Carbon Bond Lengths in 3A" Vinylmethylenea R(C,Cz)
R(CZC3)
AR
min STO
basis
MNDO UHF, S' = 2.3
1.354
1.402
0.048
STO-3G
RHF UHF, S z = 2.4 three-orbital/3-electron n MCSCF
1.472 1.406 1.423
1.317 1.404 1.378
-0.155 -0.002 -0.045
3-21G
UHF, S' = 2.3 three-orbitalithree-electronn MCSCF six-orbital/three-electronn MCSCF eight-orbital/eight-electron u / n MCSCF ten-orbital/ten-electron u / n MCSCF
1.385 1.395 1.393 1.411 1.408
1.385 1.375 1.376 1.396 1.384
0.000 -0.020 -0.017 -0.015 -0.0246
3-orbital/three-electronTI MCSCF 8-orbital/eight-electron U / R MCSCF
1.402 1.413
1.383 1.400
-0.019 -0.013
SV a
+d
wave function
Bond lengths are in angstroms.
Single and double excitations only.
to the carbene center to be 0.05 A shorter than the bond to the methylene group. MNDO also predicted an HCC bond angle at the carbene center of almost 170' and a long 1.10-A CH bond to the central carbon. This suggests the importance in the MNDO wave function of a hyperconjugated valence bond structure containing a triple bond to the carbene center. The ab-initio geometry optimizations were repeated with the 3-21G and [3s,2p/2s]" split valence (SV) basis sets in order to judge the effect of basis set enlargement. The latter basis was supplemented with one set of d polarization functions (exponent = 0.75) and employed a scale factor of 1.2 on hydrogen. The optimal bond lengths for these and the previously discussed calculations are displayed in Table 1. It is evident that while the differences between MCSCF and UHF decrease upon going to the larger bases they do not entirely dissappear. The 3Af' CCC and HCC (carbene) bond angles change by less than 2' in all these calculations. This certainly suggests that the 170' HCC bond angle predicted by MNDO calculations is unreasonably large. It should be noted that the 153' bond angle predicted by MNDO for 3B2methylene is also much too large. A second MCSCF wave function was tested that extended the number of orbitals among which excitations were permitted (the so-called "active space") to all six available in the a space with the 3-21G basis. The bonds lengths predicted by this wave function did not differ appreciably from the three-orbital/ three-electron results.
CI Energy Differences In order to determine the relative energies of 3A", lA", and 'A', CI calculations were first performed at the optimal STO-3G T space MCSCF geometries. The list of configurations treated for each state consisted of all single and selected double excitations from up to nine reference configurations. The space-orbital products, from which these spin-adapted reference configurations were formed, are listed in Table 11. For the two open shell states the complete list of reference configurations corresponds to the full three-orbital/three-electronset used in the smaller MCSCF wave functions. For the initial CI's the 3A" and 'A" reference spaces consisted of only five reference configurations formed from the first three space orbital products in Table 11. These were selected on the basis of the size of their coefficients in a small SD-CI. One indication of the quality of the reference space is the extent to which the sum of the squares of the reference configurations in the fiial CI approaches unity. The value ranged (17) Dunning, T. H., Jr.; Hay, P. J. In "Modern Theoretical Chemistry-; Schaefer 111,H. F., Ed.; Plenum Press: New York, 1977; Vol. 2.
TABLE 11: Reference Configurations Used in Generating the Sinele and Double Excitation SDaces state 3A"a
'A'
a
symmetry a' 1 , ..1011
symmetry a" 1 2 3
2...21 2...21 2...21 2...21 2...21 2...21 2...21 2...22 2...2 2...2
2 1 RHF 1 2 111 1 2 21 2 1 1 2 2 22
RHF 2
The same list was used for t h e ' A " state.
TABLE 111: Estimated Multireference Full CI Energies (hartree) with t h e SV + d Basis a t t h e STO-3G MCSCF Geometries
a
state
E( full CI)
A E , kcal/mol
)A" (anti) (syn) IA"( anti) 'A'( anti) ""(anti)
-116.197 - 116.198 -116.174 - 116.1 7 9 - 116.184
0.6 0.0 15.5 12.4 8. ga
Computed at t h e geometry of Davis e t al.
from 0.93 to 0.91 for the triplet, depending on the basis set and size of the CI. With the largest basis set used in this study the number of spin-adapted configurations exceeded 1.1million for the triplet state. Since this is larger than the number which our programs can treat variationally, we selected a subset of the double excitations based on a second-order perturbation theory estimate of their energy contribution. In order to improve the convergence of this truncated CI the virtual space RHF orbitals were transformed to K orbita l ~ . ' These ~ orbitals have been shown to mimic the frozen natural orbital set. An extrapolation to the full CI energy limit was made by use of the formula1g AE(ful1 CI) = AE(C1)* [ l + E(disc)/E(kept)]*co/(2*co - 1) where AE(C1) is the energy lowering from the variational calculation, and E(disc) and E(kept) are the second-order perturbation theory estimate of the energy corresponding to the configurations which were discarded and kept, respectively. The value of co is taken as the sum of the squares of the reference configurations in the final CI wave ~~
(18) Feller, D. F.; Davidson, E. R. J. Chem. Phys. 1981, 84, 3977. (19) Davidson, E. R.; Silver, D. W. Chem. Phys. Lett. 1977, 52, 403.
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The Journal of Physical Chemlstty, Vol. 87, No. 24, 1983
Feller et al.
TABLE IV: SCF and Multireferencea CI Zfs Parameters for t h e Anti Form of Triplet Vinylmethylene a t t h e STO-3G MCSCF Geometry basis
sv
config no. 1 4 339 8 168 15 366 19 241
1 8 1 695
SV
+ carbon d
1 18 199
530 0 9 8
DZP
1 1 7 952
11 4 9 434
energy -115.7793 - 115.9600 -115.9897 -116.0051 - 116.0091
%AE(SD)
D, c m - '
E, c m - '
0.0 69.7 85.3 93.5 95.5
0.663 0.694 0.675 0.669 0.664
0.069 0.067 0.072 0.074 0.074
0.0 79.5
0.657 0.674
0.065 0.064
0.0 72.0
0.653 0.680
0.063 0.061
-116.019 (est) -115.8210 - 116.0979
-116.163 (est) -115.8317 - 116.1071
-116.205 (est)
T h e reference space consisted of the five spin-adapted configurations which could be formed from the first three space orbital products listed in Table 11. a
function. Tests with the SV basis indicate that the estimated full CI energy has converged to about 0.001 hartree by the time roughly 85% of the multireference SD energy is variationally recovered. The majority of CI calculations reported here were run with the SV + d basis near the 80% level. This corresponds to variationally treating 20 00025 000 spin-adapted configurations. The extrapolated full CI energies with the SV + d basis set are given in Table 111. Although our final 3Af'(anti)-'A'(anti) energy gap (11.8 kcal/mol) is in exact agreement with the GVB-CI value quoted by Davis et al., this agreement is fortuitous. As mentioned above, their singlet geometry agrees quite well with our optimal structure, but their triplet geometry is substantially different. The estimated full CI energy a t their triplet geometry is over 8 kcal/mol higher than a t the STO-3G optimal geometry. Based on the nearly equal energies of lA' and l A , an out-of-plane hydrogen motion, which allows the two states to mix, might prove favorable on the lowest singlet surface. Although we have not pursued this point ourselves, calculations by Yoshimine20have confirmed that this is, in fact, the case. Zero-Field Splitting Parameters Past attempts to compute molecular zfs parameters by ab initio methods have shown these properties can be quite sensitive to the electronic distribution in the molecules. Thus, convergence of the theoretical estimate of D and E with regard to quality of CI and basis set size is sometimes discouragingly slow. Studies of both effects were conducted for vinylmethylene. Convergence with regard to quality of CI was investigated by means of a series of calculations with the Dunning-Hay SV basis a t the STO-3G MCSCF geometry. Because double excitations were selected on the basis of their energy contribution, there was no assurance that the most important configurations for other properties would be retained. Nevertheless, as seen in Table IV, for vinylmethylene the variation in D is only 0.03 cm-' for a variation in the number of configurations from 4000 to (20) Yoshimine; M., private communication.
19000. Although the largest CI still includes only 10% of the entire SD space, it recovers more than 95% of the estimated total SD-CI correlation energy. To the extent that the other 160000 configurations will enter the wave function with quite small coefficients, we hope that further changes in D would be small. Convergence with regard to basis set size was investigated by means of additional RHF and CI calculations using the SV + d and a [4s,2p,ld/3s,lp] basis set. The latter basis consisted of the Dunning21 contraction on carbon and a contraction22on hydrogen developed in this laboratory. The RHF values of D from Table IV show little change upon enlargement of the basis. D decreases by only 0.006 cm-l when d functions are added to carbon and less than 0.001 when hydrogen p functions are added. The same increments in basis set yielded CI changes in D of around 0.010 and 0.006 cm-', respectively. An uncertainty arises in trying to pinpoint the effect of basis set enlargement a t the CI level because of the difficulty in defining comparable truncated CI's with different basis sets. From the combined RHF and CI results it appears that the biggest effect of basis set enlargement beyond the Dunning SV level comes from adding d functions to carbon. Overall, the effect of CI on D is small as it is for methylene at its optimal geometry. Langhoff et al.% have shown that the SCF and CI values of D for CH2 cross near 130' when plotted against bond angle. Somewhat surprisingly, all of the computed values of D for vinylmethylene in Table IV are about 50% higher than the experimental value of 0.409 cm-'. Because of the sensitivity of D to the A spin distribution, which in vinylmethylene depends strongly on geometry, a study of the dependence of D on the difference in the C-C bond lengths, AR,seemed warranted. Five geometries along the asymmetric C-C stretch coordinate were used, with AR = -0.053,0.0,0.042,0.053, and 0.159 A. The bond angles and CH bond lengths were taken from the optimal STO-3G (21) Dunning, T. H., Jr. J. Chem. Phys. 1970, 53, 2823. (22) Stenkamp, L. Ph.D. Dissertation, University of Washington, 1975. (23) Langhoff, S. R.; Elbert, S. T.; Davidson, E. R. Znt. J. Quant. Chem. 1973, 7, 999.
The Journal of Physical Chemistty, Voi. 87, No. 24, 1983 4837
Ab-Initio Calculations of Vinylmethyiene
TABLE V : Spin Populations and Zfs Parameters Along t h e Asymmetric Stretch Coordinate Obtained with SCFIK Orbitals CI's Using Five Reference Configurationsa
c, AR6
-0.10 0.00 0.08 0.10 0.30
s
P(X) 0.24 0.81 0.24 0.76 0.22 0.55 0.22 0.48 0.20 0.29
c2
c3
P(Y) P(Z)
P(X)
P(X)
DC
E
0.76 0.76 0.76 0.77 0.77
-0.10 -0.11 -0.14 -0.14 -0.12
0.27 0.34 0.57 0.64 0.81
0.67 0.65 0.47 0.40 0.20
0.06 0.06 0.05 0.05 0.04
0.04 0.04 0.03 0.03 0.03
a The molecule lies in t h e yz plane with t h e hydrogen attached t o C, ( t h e central carbon) extending along the positive z axis. The molecule is in the anti conformation, The difference in CC bond lengths in bohrs. 1 bohr = D and E are given in cm-'. 0.5292 angstroms.
MCSCF geometry. The 4.053-A geometry is close to that of the STO-3G MCSCF triplet, while the 0.159-A geometry is close to that of IA". All calculations were done with the SV + d basis. Initially only five of the nine A configurations listed in Table I1 were used to define the CI zeroth-order reference space. The resulting spin populations and zfs parameters are given in Table V. The a spin density and, consequently, the value of D varies rapidly along the asymmetric stretch coordinate. A value of AR in the vicinity of +0.05 8, is necessary if agreement with experiment is to be achieved. No definitive indication of a preferred geometry can be obtained from the estimated full CI energies because the inherent uncertainties in the extrapolation procedure are as large as the 1-2-mhartree energy differences. At the AR = +0.05 A geometry two RHF solutions, differing in energy by about 1 mhartree, were obtained starting from different initial guesses. Just as for allyl, these had the unpaired a electron localized on different ends of the molecule. It might be hoped that differences in the K orbital sets obtained from these two RHF solutions would be overcome by the large multireference CI procedure we used; since, in the limit of a full CI, the energy and properties are invariant with respect to a unitary transformation among the orbitals. Unfortunately, starting from the RHF solution with a high A spin density on the methylene carbon (C& the CI spin populations a t the two terminal carbons were essentially reversed from the values displayed in Table V. Thus, it was decided to use an iterative natural orbital (INO)" procedure, coupled with an expansion of the reference space to all nine of the configurations in Table 11, in hope of arriving at converged spin distributions which were independent of the set of starting orbitals. When configuration selection is based on some fixed threshold of the configuration's energy contribution the number of configurations retained varies with each I N 0 cycle. Therefore, the variational CI energy was not a particularly good criterion for convergence of the procedure. We chose to use the value of the A spin population at the two terminal carbons instead. When the populations varied by less than 0.02 electrons the I N 0 cycles were stopped. As seen in Figure 2 and listed in Table VI, the I N 0 procedure significantly reduces the differences in the terminal carbon A spin populations compared with the original calculations, and also decreases the rate of change of spin population as a function of AR. The increased tendency of the a system to delocalize means that an even larger distortion in the positive AR direction would be (24) Bender, C. F.; Davidson, E. R. J. Phys. Chen. 1966, 70, 2675.
*I
E'*
,
1
pi M C S C F
0.0
n Spin Pop.
0.7
0.6
0.5
0.4
0.3
L
I
I
-0.05
0.0
0.05
0.15
0.1 ARCC
Flgure 2. A spin populations obtained from a five reference configuration C I using SCF/K orbitals and from a nine reference configuration C I using INO's plotted against the asymmetric stretch coordinate (in angstroms). SV d basis set three-orbital/three-electron and eight-orbitakight-electron MCSCF energies (in millihartrees) corresponding to the same geometries are plotted in the upper portion of the figure. 1 mhartree = 0.6 kcai/mol.
+
TABLE VI: Spin Populations Obtained with I N 0 CI's Using Nine Reference Configurations
-0.10 0.00 0.10
0.23 0.23 0.22
0.72 0.65 0.55
0.04 0.04 0.03
0.77 0.77 0.77
-0.13 -0.15 -0.15
0.39 0.48 0.58
a The molecule lies in the yz plane with the hydrogen attached t o C, ( t h e central carbon) extending along the positive z axis. The molecule is in the anti conformation. 6 The difference in CC bond lengths in bohrs. 1 bohr = 0.5292 angstroms.
needed in order to obtain a value of D which agreed with experiment since it appears that the methylene carbon (C,) must have an excess of ?r spin density in order to obtain this value of D. The estimated full CI energies resulting from the I N 0 procedure are again insufficiently precise to allow a determination of the optimal geometry. In the upper portion of Figure 2 the SV + d ?r MCSCF energies are plotted along the asymmetric C-C stretch coordinate. The total variation in energy is on the order of 1mhartree over the range of AR plotted. Given the difficulty in reproducing the experimental value of D for vinylmethylene, it might be asked if the agreement found for methylene and trimethylenemethane is fortuitous. Evidence that this is not the case comes in part from comparison of calculations with the experimental work of Wasserman et al.25on the series CHz, HC(CF3), and C(CF3)2. Wasserman et al. found only a few hundredths of a wavenumber variation in D. Since accurate (25) Wasserman, E.; Barash, L.; Yager, W. A. J. Am. Chem. SOC.1965, 87, 4974.
The Journal of Physical Chemlstty, Vol. 87, No. 24, 1983
4838
Feller et al.
TABLE VII: INO-CI Results a t the Optimal SV + d Eight-Orbital/Eight-Electron'A" MCSCF Geometries syn conformation
E(RHF) E(est full CI)
- 115.8170 -116.202
-115.8167 -116.204
D, cm-I
0.490
0.436
(first NO config) 0.587 (CI) E, c m - ' 8.'
.....
)i 1.984
Oa'
1.980
1
spin populationf Cl S
P(X) P(Y) P(Z) Cl P(X)
c3
P(X)
. 10.'
1.001
.11.
a02 1
anti conformation
0.036
(first NO config) 0.061 (CI)
0.049 (CI)
0.22 0.66 0.57 0.30
0.23 0.65 0.04 0.77
-0.14
-0.14
0.47
0.47
dipole moment, D Y 0.272 2
(first N O config) 0.563 (CI) 0.035
-0.145
(first NO config)
0.027 0.304
isotropic hyperfine coupling constantsb 20,24 45.91 HI -17.14 - 16.44 H* H3 -11.14 - 18.10 H.4 8.6 2 3.57
t 12.
M,eI
Flgure 3. Density contour plots for the five active cr natural orbitals in the eight-orbRal/elght-electron MCSCF calculations. The contours enclose 90, 70, 50, 30, and 10% of the probability density. The three 7r
orbitals are not shown.
zfs calculations on molecules containing nine first-row atoms present enormous computational difficulties, the fluorine atoms in the above series were replaced by hydrogen, to give CHz, HC(CH3),and C(CH3)2 With the SV + d basis the CI values for D are 0.79,0.76,and 0.79 cm-', respectively, suggesting the ab initio theory is capable of successfully reproducing the experimental values of D for a series of substituted methylene compounds. Attempted Refinements of the 3A" Geometry Another attempt was made to see if some relatively simple wave function might be found that predicted an optimal geometry with a positive AR. The most important feature missing from the three-orbital/three-electrona space MCSCF calculations is cr bond correlation. Therefore, two pairs of orbitals were added to the MCSCF active space, along with the singly occupied Q orbital. This gives an eight-orbital/eight-electronMCSCF with 1110 spinadapted configurations when up to quadruple excitations are allowed. Contour plots of the five active u orbitals are shown in Figure 3. It was necessary to start the MCSCF orbital optimization with localized orbitals in order to assure convergence to the desired wave function. The optimal bond lengths with the 3-21G and SV + d basis, given in Table I, are only slightly different from the pure a space results. Figure 2 shows the variation in the SV + d a / a MCSCF energies along the asymmetric C-C stretch coordinate. The variation is again less than 1 mhartree over
a The molecule lies in t h e yz plane with the hydrogen attached t o C , ( t h e central carbon) extending along the positive z axis. H, is attached t o t h e central carbon, H, and H, are attached to the methyl carbon. H, is attached to t h e carbene carbon. Coupling constants are in MHz.
the range of AR in the figure. MNDO (UHF) is the only method to predict a positive AR. Because MNDO showed an unusually long CH bond of the central carbon we also added the CH bond and a correlating orbital to the MCSCF active space. Allowing up to quadruple excitations among the ten active space orbitals would have generated almost 10 000 spin-adapted configurations and made geometry optimization prohibitively expensive. Thus, the number of excitations was limited to singles and doubles, so that only 1404 configurations were generated. As seen in Table I, the ten-orbital/ten-electron wave function also had little effect on the relative bond lengths. While the correlated CH bond did indeed lengthen by 0.02 A, this lengthening is no larger than is typically seen in such calculations. For example, when one of the CH bonds to the methylene carbon was correlated, its bond length also increased by 0.02 A. A final INO-CI determination of the syn and anti conformation zfs parameters and selected molecular properties for the 3Af' state was made at our best estimate of the geometry. The optimal eight-orbital/eight-electron MCSCF SV + d basis set bond lengths and angles were used except for the CH bond lengths which were set to the value 1.099 A determined in calculations which included CH bond correlation. The results of these calculations are given in Table VII. While D(syn) is found to be larger than D(anti), in accord with experiment, the difference between the two is only about half the value found by Hutton et al. An Estimate of the Effect of Vibrational Averaging If the triplet potential surface were asymmetric about the minimum and/or the zfs parameters were sufficiently
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983 4839
Ab-Initio Calculations of Vinylmethylene
vibrational averaging effect is expected. The 3-21G R space MCSCF potential surface was examined to see how its force constants compared to allyl. If the surface is fit to a bond energy model
E = (k1/2)6R12
D IC m-' I 0.6
1
0.5
0.4
-0.1 5
-0.1
-0.05
0.05
0.0
ai
ARC,
Flgure 4. a space MCSCF asymmetric stretch potential curve fw 3Aff vinylmethylene computed with the 3-21G basis. The corresponding estimated values of D are indicated by symbols. The horizontal bar shows the approximate position of the zerwoint vibrational energy of this mode. Differences in bond lengths are given in angstroms and energies are in millihartrees. 1 mhartree = 0.6 kcal/mol.
"+"
nonlinear along a vibrational mode, vibrational averaging could play a significant role in changing D from the value reported in Table VII. While a thorough treatment of this effect is beyond the scope of the present study, an estimate of the contribution from the C-C-C asymmetric stretch mode can be made. The zero-point energy associated with this mode is approximately 0.003 hartree. Using the 3-21G basis set and the three-orbitallthree-electron R space MCSCF wave function we computed a series of energies as a function of AR. These points are plotted in Figure 4. The associated values of D were estimated from polynomial expansions of the previously computed CI values in powers of the T spin population a t C1.Superimposing these estimates of D on the potential curve in Figure 4, one can see that D is approximately linear in AR and the potential curve is very symmetric. Thus, no appreciable
+ (k2/2)6R22 + klz6R16R2
where 6R1 and 6R2 are displacements of the ClC2and C2C3 bond lengths from their equilibrium values, then kl = 5.8, k2 = 7.1, and klz = 1.7 mdyn/A. The value of k12is unusually large and reflects the substantial change in R bonding as the relative bond lengths are changed. When the energy is reexpressed in terms of the symmetric stretch, Qs = (6R2 6R1)/2, and asymmetric stretch, QA = (6R2 6R1)/2 = (AR - AR,)/2, then
+
E = (Ks/2)*Qs2 + ( K A / ~ ) * &+AKSAQSQA ~
+ k2 + k12 = 14.6 mdyn/A KA = kl + k2 - k12 = 11.2 mdyn/A
Ks = kl
KSA
= kl - k2 = -1.3 mdyn/A
By comparison, a comparable calculation for allyl gives k1 = k2 = 7.8 and k12= 0.4 mdyn/A, so Ks = 16.0 and KA = 15.2 mdyn/A. Thus, vinylmethylene has a smaller asymmetric stretch force constant than allyl. With the larger SV carbon d basis, k1 and k2 are only slightly smaller, while k12increases to 2.7 mdyn/A. This leads to an even smaller KAof 10.0 mdyn/A and an even flatter potential surface as a function of AR. Estimates obtained from a multireference CI suggests that the full CI surface may be still flatter.
+
Conclusion The final computed zfs parameters in Table VI1 are still larger than the experimental values. However, the potential surface is relatively flat, and the spin density is very sensitive to the difference between the C-C bond lengths. Therefore, a relatively small error in the calculated equilibrium geometry could lead to a large error in the spin density and, hence, in D. Acknowledgment. This work was supported in part by grants from the National Science Foundation and the National Institute of Health. Registry No. Vinylmethylene, 463-49-0.
