J. Phys. Chem. 1986, 90, 2057-2060 of octahedra repeat by translation along the a axis; octahedra of the same height of adjacent chains are connected through O(2) and N(2) of the same CDCB*-dianion (see Figures 5 and 6). Octahedra centered on Na(2) are symmetric with respect to the centers of symmetry and occur in pairs, sharing the edge between the vertices at 0(22), and ‘O(22). Together with the DX molecules they form infinite chains extending along the a axis. These chains repeat by translation along the c axis; adjacent chains are connected through N(1) and N(4) of two CDCB2- dianions related by the c glide symmetry. On the whole the crystal may be considered a giant molecule consisting of stacks of CDCB2- dianions fully dipped into an extended network of hydrogen bonds (see Table 111).
2057
Acknowledgment. The authors are indebted to Quinto L. Mulazzani for the gas chromatographic analyses and to Rosa Simoni and Giancarlo Fini for the thermal measurements. Partial financial support from the Minister0 della Pubblica Istruzione, fondi 60%, and from the Istituto di Spettroscopia Molecolare del CNR, Bologna, is acknowledged. Registry No. CDCBNa2-4H20.0.SDX,100112-33-2.
Supplementary Material Available: A listing of observed and calculated structure factor amplitudes and thermal parameters of non-hydrogen atoms (13 pages). Order information is given on any current masthead page.
Ab Initio Molecular Orbital Calculations on “Isolated” Vibrational Frequencies in AMe, Radicals and Radical Ions (A = B-, C, N’, AI-, Si, P’) Ian Carmichael Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: October 29, 1985)
A linear correlation is found between ab initio estimates of the C-H bond lengths in AMe, radicals and radical ions (A = B-,C, N+, Al-, Si, P+) and the theoretically derived harmonic frequenciesfor deuterium-isolated C-H stretches. SCF molecular orbital calculations within the unrestricted Hartree-Fock approximation employing a split-valence basis set coupled with an analytic second-derivative procedure for force constants are used to obtain these frequencies. The calculated data are scaled to experiment by comparison with the well-characterized isolated vibrations in trimethylamine, trimethylphosphine, and trimethylborane. After scaling, relatively accurate predictions are possible for the so far unobserved isolated C-H vibrations in the AMe3 species listed above. For the substituted tert-butyl radical, absorptions in the vicinity of 2825 and 2905 cm-I are expected. The corresponding pair in the trimethylaluminum molecule should lie around 2890 and 2940 cm-I. For the systems considered, the C-D stretches for monodeuterated radicals and the C-Mu stretches for muonium-containingradicals are also isolated and can be estimated by another simple scaling argument. The preferred site of (formal) muonium replacement has been identified and estimates of the parameters in a simple expression for the dihedral-angle dependence of the proton (or muon) hyperfine splitting constant have been derived.
Introduction Recently the frequencies of a number of “deuterium-isolated” C-H stretching vibrations for simple alkanes have been recorded in the gas phase by Raman spectroscopy.’ For molecules in which all but one hydrogen have been replaced by deuterium the C-H stretching mode, v& is uncoupled from other vibrations and thus localized, hence the term “isolated”. Further a strong linear correlation was found between these frequencies and the corresponding C-H bond length, r,(CH), calculated by a b initio techniques2 Such a linear correlation between v& and experimentally derived bond lengths, r,(CH), has also been r e p ~ r t e d ~for- ~a wide range of organic molecules. In addition, the effect of other substituents such as methyl groups, halogens, or lone pairs on the theoretically determined bond lengths of adjacent C-H bonds has been d i ~ c u s s e d . Potential ~,~ applications of these correlations in support of microwave studies and structure-reactivity models are numerous. In the course of a recent theoretical investigation into the electronic structure of the tert-butyl radical and several analogous ( 1 ) Snyder, R. G.; Aljibury, A. L.; Strauss, H. L.; Casal, H. L.; Gough, K. M.; Murphy, W. F. J . Chem. Phys. 1984,81, 5352. (2) Snyder, R. G.; Aljibury, A. L.; Strauss, H. L., cited in ref 1 . (3) McKean, D. C.; Duncan, J. L.; Batt, L. Spectrochim. Acta, Part A
-. - - .
1973. 29. 1037. .. --- .
(4) McKean, D. C. Chem. SOC.Reu. 1978, 7 , 399. (5) McKean, D. C.; Boggs, J. E.; Schaffer, L. J . Mol. Struct. 1984, 116, 313.
0022-3654/86/2090-2057$01 S O / O
species a similar correlation has been uncovered between the ab initio optimized C-H bond lengths and the theoretically derived harmonic frequencies, -@, for the isolated stretching modes. This is a special case of the more general linear relation noted by Defrees et aL6 between computed values of r,(CH) and calculated C-H bond stretching force constants. In view of the above results, the existence of such a relation, reminiscent of Badger’s rule,’ is perhaps not surprising. However, the precision with which the linear relation holds for the AMe, species offers a simple method for the prediction of various isotopic “isolated” frequencies such as those pertaining in AMe2CH2Dand AMe2CH2Mu. While the modes in the latter group of molecules will not be observable by the conventional techniques of vibrational spectroscopy, an indication of their magnitude provides useful structural guidelines concerning, for example, specificity in substitutional sites. From electron spin resonance8 and muonium spin rotationg measurements the isotropic hyperfine coupling constants, aiso,and their temperature dependence, dlal/dT, have been recorded for CMe,, CMe2CHzD, and CMezCHzMu. The differences in dlal/dT among the isotopic species (regarding muonium as a light isotope of hydrogen of mass about mH/9) are obviously related to the characteristic vibrational amplitudes of each substituent. (6) Defrees, D. J.; Hassner, D. Z.; Hehre, W. J.; Peter, E. A,; Wolfsberg, M. J . Am. Chem. SOC.1978, 100, 641. (7) Badger, R. M. J . Chem. Phys. 1934, 2, 13; 1935, 3, 710. (8) Burkard, P.; Fischer, H. J . Mugn. Reson. 1980, 40, 335. (9) Roduner, E.;Strub, W.; Burkard, P.; Hochmann, J.; Percival, P. W.; Fischer, H.; Ramos, M.; Webster, B. C. Chem. Phys. 1982, 67, 275.
0 1986 American Chemical Society
2058
Carmichael
The Journal of Physical Chemistry, Voi,90, No. 10, 1986
TABLE I: UHF/3-21G Isolated Stretching Frequencies for AMe3 Species species r,(CH), 8, ucM, cm-I uCD, cm-I uCMu, cm-] BMe31.0945 3108 2276 8974 1.1018 3012 2204 8704
CMe3
1.0848 1.0913
3229 3143
2365 2299
9313 9076
NMe3+
1.0775 1.0824
3324 3266
2436 2388
9569 9409
AIMe,-
1.0916 1.0933
3151 3129
2305 2289
9094 9030
SiMe,
1.0852 1.0877
3236 3205
2368 2344
9332 9244
PMe,’
1.081 1 1.0843
329 1 3250
2408 2376
9479 9362
These amplitudes, in turn, may be gauged from the frequencies predicted by the above procedure.
Computational Details All calculations were performed at the U H F level ( R H F for closed-shell systems) with the GAUSSIAN 82 series of programs” modified to run under operating system VMS 4.0 on a VAX 11/780 superminicomputer. Note that under VMS 4.0, if the GAUSSIAN 82 routines must be recompiled, then the qualifier “NOOPT” should be applied. Unreliable results are otherwise produced. Standard internal basis sets were employed. A split-valence 3-2 1G description was adopted and geometry optimizations were obtained by analytic energy gradients and force constants by analytic second derivatives. To obtain accurate estimates of the vibration frequencies stringent criteria were set in the optimization routines. This practice is particularly important to accommodate the very low-frequency methyl torsions encountered in the AMe3 species. Results and Discussion With the introduction of analytic methods for the determination of first forces” and then force constantsI2 by repeated differentiation of the Hartree-Fock energy expression for a molecule, the opportunity has arisen to offer predictions of the equilibrium geometry and harmonic force field for moderately large systems. For the AMe3 open-shell species considered here BMe3-, CMe3, NMe3+,AlMe), SiMe3, and PMe3+,full geometry optimizations have been carried through at the UHF/3-21G level and the resulting parameter values have been reported.I3 From these equilibrium structures conventional normal coordinate analyses have been performed and estimates of the harmonic frequencies have been obtained. The validity of the Born-Oppenheimer approximation is assumed so that isotopic substitution may be readily performed on the theoretical model and the corresponding isolated frequencies may be easily obtained. This procedure is particularly convenient within the GAUSSIAN 82 routines since the Cartesian force constants may be archived for later retrieval. Table I collects the results of these calculations and assigns the computed isolated frequencies to the stretching of particular C-H bonds. In NMe,’, the heavy-atom framework is calculated to be planar and one C-H bond (the shorter) in each methyl group lies in this plane. In all other open-shell species studied, which are determined to be pyramidal, the unique C-H bond (the longer) lies on the opposite side of the plane formed by the three @-carbons to the central atom. The variation of the isolated C-H stretching frequencies with computed bond length is displayed in Figure 1. Similar linear plots may be readily constructed for the monosubstituted species, AMe2CH2Dand AMe,CH,Mu. For each plot (10) Binkley, J. S.; Frisch, M. J.; DeFrees, D. J.; Raghavachari, K.; Whiteside, R. A.; Schlegel, H. B.; Fluder, E. M.; Pople, J. A. GAUSSIAN 82, Carnegie-Mellon University: Pittsburgh, 1983. (11) Pulay, P. Mol. Phys. 1969, 17, 197. (12) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J. S. Int. J . Quanr. Chem. Symp. 1979, 13, 225. (13) Carmichael, I. J . Phys. Chem. 1985, 89, 4727.
Figure 1. Correlation between the calculated (UHF/3-21G) C-H bond lengths, re,,and the calculated deuterium-isolated C-H stretching frequencies, ?& for open-shell AMe3species: A, from shorter C-H bond, re(C-H2); 0 , from longer C-H bond; re(C-Hl).
TABLE I 1 Parameters of Linear Correlation between r . and Piso
isotope p,‘ amu H 0.929741 D Mu
1.724561 0.112564
s,*
cm-’/A d,b cm-’
-13081 -9618.8 -36211
17427 12803 48612
sp1i2
tlC
-1.26 X lo5 -0.998 -1.26 X IO’ -0.998 -1.21 X IO5 -0.998
+
“Reduced mass from m,m,/(m, m J . bParameters from functional form CiS0 = sr, + d. CLinear-correlationcoefficient. the best straight line, in a least-squares sense, through the calculated data was constructed from $so = sr, d (1)
+
In Table I1 the fitting parameters are collected together with estimates of the linear-correlation coefficient for the fits, -0.998 in each case. The table also shows that the slopes of the fitted lines scale by the square root of the C-X effective mass, p. Such a relation is expected to hold for truly isolated “diatomic” C-X stretching. Both Born-Oppenheimer effects (Le., the mass dependence of the zero-point energy) and adiabatic effects due to enhanced vibronic couping will act to shift the muonic frequencies from those reported here. On the basis of effective mass ratios increases between 1% and 2% can be expected. To calibrate the theoretical harmonic frequencies to experimentally observable vibrational peaks several procedures may be adopted. Pople et al.I4 have suggested the use of a single mode-independent scaling factor (0.89 for the present basis set) based on a statistical analysis of computations on a large number of neutral molecules. Alternatively, individual modes may be scaled with respect to the ratio of calculated to observed frequencies for known modes of “similar” molecule^.^^ For comparison with the present series of molecules, the deuterium isolated C-H stretching frequencies of trimethylamine are convenient and have been recorded.I6 These frequencies were obtained from the (14) Pople, J. A,; Schlegel, H. B.; Krishnan, R.; DeFrees, D. J.; Binkley, J. S., Frisch, M. J., Whiteside, R. A,; Hout R. F., Hehre, W. J. Int. J . Quant. Chem. Symp. 1981, IS, 269. (15) Colvin, M. E., Raine, G . P., Schaefer 111, H. F.; Dupuis, M. J . Chem. Phys. 1983, 79, 1551.
TABLE 111: RHF/3-21G Isolated C-H Stretching Frequencies for AMe3 Species species re(C-H), A t,”cm-’ v? cm-l BMe, NMe, AIMe,
1.0855 1.0916 1.0830 I .0929
3221 3169
298OC 2884
3243 3116
2952d 2799
1.0869 1.0899
3217 3184
PMe,
1.0829 3262 295P 1.0838 3247 2919 “Theoretical harmonic estimate. ’Experimental value. Reference
1
3400-
3300 m
‘E
A 8 1 TAU
3200 -
16. “Reference 15. eReference 17.
3100“111,
3000i
I
I b7
I 08
I
1.09
I
1.10
2060
The Journal of Physical Chemistry, Vo/. 90, No. 10, 1986
Carmichael
TABLE I V Fully Optimized RHF/3-21G Geometries for AMe, Species
A
species BMe, CMe3+ NMe,
rAc,
0, deg 90 90 105.16
d, deg
~ c H 8, ~ ,
~ c H 8, ~ ,
PI,
90 90 180
1.0855 1.0784 1.0929
1.0916 1.0912 1.0830
deg 114.37 113.68 112.81
P2,
1.5894 1.4736 1.4639
deg 110.1 1 108.70 109.43
x, deg 121.99 122.99 120.74
EWF, hartree -142.758834 -155,572743" -172.310270
AlMe, SiMe,' PMe,
1.9976 1.8746 1.9025
90 90 118.61
90 90 180
1.0869 1.0831 1.0838
1.0899 1.0890 1.0829
112.23 111.97 111.30
111.16 110.55 109.1 1
120.75 121.17 120.76
-358.810350 -405.376287 -457.195 697
'A normal coordinate analysis at this geometry produced an E pair with a small imaginary frequency suggesting either that numerical problems had been encountered or that a more stable conformation could be found. TABLE V: HF/3-21G Force Constant Differences in C-H Bonds of Nondanar AMe, SDecies species 0," deg 6k,b mdyn A-' pm
TABLE VIII: UHF Isotropic Hyperfine Coupling Constantsofor Methyl Protons in AMe3 Species
~
even electron NMe, PMe,
105.62 118.61
0.44 0.12
0.99 0.09
odd electron BMe,CMe, AIMe3SiMe, PMe,'
105.74 99.05 109.49 108.36 104.57
0.32 0.30 0.08 0.1 1 0.15
0.73 0.65 0.17 0.25 0.32
TABLE VI: HF/3-21G Force Constant Differences in C-H Bonds for Planar AMe, Soecies
0.20 0.47 0.12 0.23
0.61 1.28 0.30 0.59
odd electron NMe,'
0.21
0.49
TABLE VII: Predicted Isolated C-H Stretching Frequencies for AMe, Species species vpH,cm-I odd electron BMe3CMe3 NMe,'
2795, 2710 2905, 2825 2990, 2935
AIMe,SiMe, PMe3+
2835, 2815 2910, 2880 2950, 2925
even electron AIMe, CMe3+ %Me,+
2935, 2885 3040, 2880 2975, 2895
+ B(cosz 4 )
a ( l H I ) a('H2)
C N' Si Pc
4.3 0.4
1.2 3.5
3-21G 3-21G
1.59 2.63
0.06 0.25
A,,
B 0.2 4.1 0.4 4.1
-0.5 -0.5
2.1 3.1
(a('H))
exptl
2.2 2.5
2.27' 2.86d
0.6 1.0
0.62V 1.15'
For the trimethylsilyl radical, previously reported calc~lations'~ on the spin density distribution with the present small split-valence -0.5 mT and B 2.1 mT. Here the basis set suggest A,, parameters have been reduced to reflect proton hyperfine coupling in order to facilitate comparison with experiment. Previously, Fessendenzl has shown such isotopic scaling to be valid in the case of monodeuterated alkyl radicals. If free rotation is assumed, the expected average value for the proton hyperfine coupling constant is 0.6 mT, which may be compared with the experimental determination of 0.628 mT.22 The results of similar calculations for the trimethylphosphinium radical are listed in Table VIII. The use of such a limited split-valence basis to describe the peripheral spin density in these species is justified by the agreement obtained above and further supported by calculations currently in progress using basis sets of double-{ quality and beyond. For the other species, in particular the two radical anions, a more complete description is required. With the Dunning double-{ contractionz3of Huzinaga's (9s5p,4s) Gaussian expansions24for the tert-butyl radical we find A, = 0.2 mT and B = 4.1 mT. This leads to an averaged coupling constant of 2.2 mT, in close agreement with the experimental value of 2.27 mT.I9 Similar results for the trimethylaminium radical are also given in Table VIII. Further details of these spin density calculations will be published elsewhere, primarily because much more elaborate theoretical models and computational procedures have proved necessary to reliably estimate the spin density at the heavy nuclei.
-
-
Acknowledgment. The research described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-2775 from the Notre Dame Radiation Laboratory.
of the corresponding isolated vibrational frequencies in the substituted radicals. The large amplitude excursions in the muonic species lead to preferential placement in the "trans" bond of the pyramidal AMe3 species and enhance the intrinsic barriers to methyl rotation. In terms of the angle, 4, defined in Figure 2, the observed coupling constant may be expressedz0 as A = A,
basis [42/2Ib [42/2]
"Units are m T (1 T = lo4 G). *Double-( contraction due to Dunnix~g.~, 'Reference 19. dReference 25. 'Reference 22. fReference 26.
"Pyramidal angle (see Figure 2). b6k = k C H 2 - k C H , where C-HI is = re(C-Hl) - re(Cthe longer (and weaker) type of C-H bond. H2) in picometers.
even electron BMe, CMe3+ AIMe, SiMe,'
species
(2)
(20) Fischer, H. Free Radicals, Kochi, J. K., Ed.; Wiley: New York, 1973.
Registry No. BMe3-, 57442-98-5; CMe,, 1605-73-8; NMe,', 1716789-4; AIMe,-, 34076-42- 1; SiMe,, 16571-41-8; PMe,', 29997-99-7; BMe,, 593-90-8; CMe,', 14804-25-2; NMe,, 75-50-3; AIMe,, 75-24-1; SiMe,', 28927-31-3; PMe,, 594-09-2; D,,7782-39-0; muonium, 1258765-4; positive muon, 12587-62-1. (21) Fessenden, R. W. J . Chim. Phys. 1964, 61, 1570. (22) Krusic, P. J.; Meakin, P. J . Am. Chem. SOC.1976, 98, 228. (23) Dunning, T. H. J . Chem. Phys. 1970, 53, 2823. (24) Huzinaga, S.J . Chem. Phys. 1965, 42, 1293. (25) Fessenden, R. W.; Neta, P. J . Phys. Chem. 1972, 76, 2857. (26) Symons, M. C. R.; McConnachie, G. D. G. J . Chem. SOC.,Chem. Commun. 1982, 851.
