THE JOURNAL OF
PHYSICAL CHEMISTRY (Registered in U.
VOLUME64
S. Patent Office)
(0Copyright, 1961, by the American Chemical Society)
JANUARY 4, 1961
NUMBER 12
THE ANALYSIS OF MOLECULAR WAVE FUNCTIONS BY NUCLEAR MAGXETIC RESOKANCE SPECTROSCOPY'" BY MARTINK A R P L U S ' ~ ~ ~ Noyes Chemical Laboratory, University of Illinois, Urbana, Illinois Received January 67,1960
A valence-bond forniulation is presented for the relationship between molecular wave functions and the electron-coupled nuclear spin interactions observed by nuclear magnetic resonance spectroscopy. For non-bonded atoms, the magnitude of the coupling is shown to depend on the deviations from perfect pairing in the wave function. This result makes it possible to use the measured coupling constants in testing models for hyperconjugation. An application to the second-order hyperconjugation in ethane is given. For directly-bonded atoms, the coupling constant depends on parameters, such as orbital hybridization and bond polarization, in localized electron-pair functions. Use of this dependence is made to determine the bond polarization in several tetrahedral molecules.
A knowledge of the wave functions of atoms and molecules is of fundamental importance for the determination and interpretation of chemical properties. Since the techniques available at present are not sufficient for the calculation of exact wave functioiis for most systems of chemical interest, approximation methods have had t o be introduced. The inaccuracies in these approximate treatments are such that experimental comparisons are necevsary to justify or supplement the theory. I n this paper we wish to discuss two approaches to the use of the results from nuclear magnetic resonance spectroscopy (n.m.r.) in the study of molecular wave functions. One example is concerned with the determination of the validity of a theoretical model for hyperconjugation and the other with the evaluation of coefficients in a parametrized electron-pair function for molecules. In n.ni.r. one is concerned with measurement of the energy levels of a system of nuclear spin magnetic moments in an external magnetic field. If we briefly consider the simplest case of a bare proton in a magnetic field Ho of lo4 gauss, we obtain the energy levels shown in Fig. l b relative to an arbitrary zero in the absence of a field (Fig. la). Since the proton has spin I H = 1/2, there are two
energy levels ( M E = which are separated by an energy AE given by the equation 0 = gd"o
(1)
Here g H is the g factor for the proton (gH = 5.586), PN is the nuclear magneton (PN = 0.50504 X ergs gauss-l). For HO = lo4 gauss, the resulting AE value is 1.420 X cm.-'. By the application of a radiation field of the appropriate frequency (42.6 Mc.), transitions between the two levels are induced and the resonance absorption can be observed. If what has been described were the entire scope of n.m.r., the technique would be of considerable utility to the physicist for the measurement of nuclear g-factors, but of very little interest to the chemist. Since the protons commonly studied are in molecules surrounded by electrons and neighbored by other nuclei, the energy level scheme is modified significantly from that pictured in Fig. l b for the isolated proton. The electrons that are present shield the proton moment from the external field with the result that the energy level spacing is altered. For the field Ho, we now have
*
= gHBN( 1 -
UH)HO
-
(2)
as shown in Fig. IC. The quantity QH is the proton (1) (a) Work supported in part by a grant from the University of shielding parameter, which varies from 0.9 X Illinois Graduate Reseaich Board. (b) Alfred P. Sloan Foundation to -4.5 X in different molecules (Q Fellow in H, is 2.66 X If other nuclei with mag(2) Department of Chemiutry, Columbia University, New York netic moments are present, they also interact 27. N. Y. 1793
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1794
with the proton. In HD, the deuteron has a spin ID = 1 with three components (MD = - 1, 0, +1) with respect to an external field. The interaction energy of the proton and deuteron moments depends on their relative orientation with the result pictured in Fig. Id. The energy level diagram for the proton in the presence of the deuteron, as well as the two electrons, is given approximately by the equation E
= gnh(1
+
- c~)HoMH hAaDM&n MH = *I/* MD = 0, f l (3 1
where AHD is the coupling constant. As is evident from Fig. Id and eq. 3, there are now three proton transitions with AE equal to
a= gHpN(1-
Ho{
+tHD (4) -AHD
from which the absolute value of AHD can be determined ([AHDIin H D is 40.3 c.P.s.). Since the shielding parameter and coupling constant for protons and other nuclei vary from one molecule to another, their values should provide chemically useful information. The evident connection between the electrons and the shielding parameter suggests that its magnitude can be utilized for the study of electronic wave functions. Although considerable research has been done in this field, we do not concern ourselves with the results.a For the coupling constants, the connection with electronic properties is not self-evident. The direct dipole-dipole interaction between nuclear moments, which is important in solids, averages to zero in the rapidly tumbling molecules found in solution. The residual coupling that is observed in solution arises from the interaction between electrons and the nuclei. In the simple example of HD, the proton p and the deuteron d have their magnetic moments oriented in the local fields generated by the magnetic moments of the electrons in the atomic orbitals J/H and J/D, respectively. Since the electronic moments are coupled in an anti-parallel orientation by the electrostatic interaction which comprises the chemical bond, the proton spins have an indirect interaction of the form (p spin : $EIspin: J/D spin : d spin) or, diagrammatically, (p f : $H 4 : $D f : d J- ). This electronic interaction mechanism suggests that the coupling constant, like the shielding parameter, can serve to increase our understanding of molecular wave functions. For a molecule containing several nuclei with non-aero magnetic moments, the observed spectrum and its analysis can be considerably more complicated than in the H D example discussed above. The effective Hamiltonian for the nuclear spin wave functions has the form X = BNHO gn(l n
- un)ls + h
n
> n‘
ADD‘1n.L’
(5)
where gn is the g-factor for nucleus n, In, In! are the spin operators for nucleus n,n’ and 1,. is the z-component spin operator for nucleus n. The (3) For a relatively up-to-date review, Bee J. A. Pople. W. G . Schneider and H. J. Bernstein, “High Resolution Nuclear Magnetic Resonsnce,” McCraw-Hill Book Co., Inc., New York, 1959, especially Chapts. 7, 11 and 12.
VOl. 64
quantity Ann!, the coupling constant between nucleus n and n‘, is usually expressed in cycles per second (c.P.s.) and is defined as positive if the anti-parallel orientation of In and Inf is of lower energy than the parallel orientation. Techniques have been developed4 for obtaining unambiguous values of un and Ann#from the observed spectra. For some relatively complicated cases, it has been possible to determine not only the magnitude but also the relative signs of the coupling constant^.^ In the treatment given in the body of this paper, we make use of the fact that the magnetic resonance spectra have been analyzed and the coupling constants determined experimentally for the molecules considered.
Hyperconjugationin Ethane Although the concept of hyperconjugative interactions has long been utilized for rationalizing certain molecular properties,6 there is still considerable disagreement concerning its importance.’ One cause of this uncertainty is that many of the effects ascribed to hyperconjugation represent relatively small variations in molecular properties (heats of hydrogenation, bond lengths, dipole moments, reaction rates, etc.).6 Since there are several different explanations for the observed changes, their relation to hyperconjugation is not always clear. We here employ the coupling constants A”# between vicinal protons to elucidate the contribution made by hyperconjugation to the ground state energies and wave functions of molecules (especially ethane). Although all semi-quantitative attempts to study hyperconjugation have been in terms of the molecular orbital method (L.C.A.O. approximation),8 we base our treatment on a valence-bond approach and include u-type terms, as well as the usual r-orbital contributions. In terms of the valence bond model, hyperconjugation arises from presence in the ground state wave function of structures other than the one with perfect pairing. Correspondingly, the decrease in energy resulting from the inclusion of these additional structures is the hyperconjugation energy. Because the errors in the valence-bond model, as in most of the theory of many-electron systems, are difficult to assess by direct calculation, we employ an experimental criterion for the validity of our approach. Since the vicinal proton coupling constant AHHI is a sensitive function of the contribution of structures with deviations from perfect pairing, a comparison of calculated and experimental AH” values (4) Pople, et ol., ref. 3, Chapt. 6 and references therein. (5) H. 8. Gutowsky. C. H. Holm, A. Saika and G . A. Williams, J . A m . Chem. Soc., 79, 4596 (1957). (6) R. 5. Mulliken, J . Chem. Phys., 7 , 339 (1939). See V. A. Crawford, Qwrl.Rev., 3, 226 (1949) for a general mrvey; also J. W. Baker, “Hyperconjugation,” The Clarendon Press, Oxford, 1952. (7) A good sample of the vaned views maintained by different workers can be found in the papers of the Conference on Hyperconjugation held at Indiana University, June, 1958, and published in Tetrahedron. 6 , No. 2/3 (January, 1959). See also R. S. Mulliken. ibid., 6, 68 (1959). (8) R. S. Mulliken. C. A. Rieke and W. G . Brown. J . Am. Chem. Soc.. 6 3 , 41 (1941). Also see N. Muller and R. S. Mulliken, iMd., SO, 3489 (1958) for references t o recent calculations and slight modifications of the approach of Mulliken, Rieke and Brown.
Dec., 1960
ANALYSISOF MOLECULAR WAVEFUNCTIONS
is used to t,est the applicability of the valencebond model to the hyperconjugation problem. To express the coupling constants in terms of valence-bond theorylg we require the details of the electron-nuclear interactions that are involved. For prot,ons it has been shown'O that the dominant ternis in the Hamiltonian are given by the expression
ZERO FIELD Hn=O
F I E L D Ho (NO S H I E L D NG)
0), A". is zero for H and H' vicinal protons energy." Comparison of eq. 5 and 7 shows that [fw(p"#) = -1(2),13 Conversely, a non-zero Ann! can be written A") value implies that hyperconjugation is present in the ground state. To illustrate the formulation, we apply it to ethylenic compounds, for which the somewhat surprising result has been found14 that the trunsEquation 8 provides the required relationship proton coupling is always considerably larger than between the coupling constant and the ground that for the cis-protons (IAgH11-5-12 c.P.s., state wave function !PO. We now introduce the IAtI;PH"B1-12-18 c.P.s.). Approximating the molevalence-bond model for !Po,which can be expressed cule by a six-electron fragment with appropriately in the form hybridized orbitals,15 we can write five valencebond structures for the ground state. These are *d = C cj+j (9) diagrammed in Fig. 2. With coefficients cj dej where the $, are the non-ionic canonical valence- termined by the variational calculation and A bond structures and the cj are constant coefficients set equal to the value of 9 e.v. used in methanelg obtained by a variational procedure. Substi- one finds that AgHt = Jr6.1 C.P.S. and AP$ = tution of eq. 9 into eq. 8 specialized for protons +11.9 C.P.S. The relative values of the cis and trans coupling are in excellent agreement with the leads to the equations measured results. Also, in some cases, it has been possible to find indications that the two constants have the same relative sign, which corresponds to (10) the theory.I6 The somewhat smaller magnitude of where, for the superposition diagram of $j and the theoretical values, as compared with the $j, ij.1 is the number of islands and fj,l(p":) is experimental ones, may indicate that the true t,he exchange coefficient between the 1s orbitals ground state wave function has larger contributions from $j ( j > 0) than those calculated here. on H and H' From the form of eq. 10 and the meaning of the Also, the compounds in which AHHIwas measured exchange factor in balence-bond theory,I2 we (13) I n eq. 10 and this argument we have neglected direct electronic can see the relation between A"' and hypercon- interactions, which can yield a contribution to A"', on the order of jugation. If $o i s the perfect-pairing structure 0.1 C.P.S. S. Alexander, J . Chem. P h p . , 28,358 (1958); €1. S. Gutowsky, and there is no hyperconjugation (ie., cj = 0 for M.(14) Karplus and D. M.Grant, ibrd., 31, 1278 (1959), lists a number of (9) M. Karplusand 1).H. Anderson, J . Chem. Phys., SO, 6 (1959). (10) N. F. Ramsey, E'hys. Rev., 91, 303 (1953); M. J. Stephens, Proc. Roy. Soe. (London), A243, 274 (1957). (11) See Ramxey, ref. 10. There are a number of alternative6 to the somewhat complex average excitation energy approach. Some of these are discussed in M. Karplus, Rev. M o d . Phys., 38,455 (1960). (12) L. Pauling, J. C h m ~ Phys.. 1, 280 (1933).
compounds. (15) See M. &rplUB, ~ b z d . ,30, 11 (1959), where the details of the calculations are presented. (16) C. N. Banwell, A. D. Cohen, N. Sheppard and J. J. Turner, Proe. Chem. Soe., 266 (1959). Also A. D. Cohen and N. Sheppard. Proc. Roy. SOC.(London), 8 8 6 8 , 488 (1959); N. Sheppard and J. J. Turner, ibid., 8862, 506 (1959).
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1796
i
I
(4 60°)-3-6 and A"! (4 180°)-107 with 18 Such a correction would be in agreement the rotational average of vicinal coupling =
I
Vol. 64
A
=
C.P.S.
C.P.S.
constants, which in substituted ethanes is found t,o he in the range 6.0-7.4 c.p.s.18 TABLE I C n u r m m CONSTANTS AHHI (c.P.s.) FOR VICINALPROTONS "Estimated" angle
+
AHH' (ealcd.)
AHH' (expt.)
2-Bromo-4,4-dimethylcyclohexnnone5 (Axial-axial) -180' +9.2 12.2 (Axial-equat.) 60' +1.7 7.0 Z-or-Bromocholestan-3-one" ( Axial-axial) -180" +9.2 13.1 ( Axial-equat. ) 60" +1.7 6.3 Substituted ethanesb (tran.s) -180' +9.2 10-18 (gauche) 60' +1.7 1-3.5 C. H. Banwell, a E. J. Corey, private communication. A. D. Cohen, N. Sheppard and J . J. Turner, Proc. Chem. SOC., 266 (1959).
-
I
' 0
30'
I 60°
I
I
120' DIHEDRAL ANGLE 0 (DEG).
90'
I
150'
180'
Fig. 3.-The vicinal proton coupling constant A"' in ethane as a function of the dihedral angle qi. The open circles correspond to the values of d, for which A"' was calculated and the curve is drawn through the points on the assumption of a cos* + dependence.
contained various substituents, whose effect is neglected in the model. We now apply the corresponding model to ethane-like systems, which can be treated also as six-electron problems. Here we consider the vicinal proton coupling as a function of the HCC'/CC'H' dihedral angle @. The results obtained are plotted in Fig. 3, which shows that AH" is a sensitive function of the angle 4. To determine the accuracy of the theoretical curve, it would be necessary to have experimental coupling constants for a series of compounds with known dihedral angle. Unfortunately there does not appear to be a single ethane derivative in which the required angle has been measured. In spite of this lack of structural data, there are comparisons that can be made. For a number of compounds the known conformations are such that estimates of the dihedral angle can be made, with pairs of axial hydrogens corresponding to (p--18O0, and one equatorial and one axial hydrogen corresponding to 4-60' (see Table I). Recently, Banwell, et aLli6 have been able to determine the trans (+-180°) and gauche (4-60') coupling constants in substituted ethanes by using solvents of different dielectric constants to vary the proportions of rotational isomers (see Table I). These results, as well as the extensive measurements of Lemieux, et al., on acetylated sugarsll' confirm the theory in the finding that lAg$l > IAg$ej. The experimental data also suggest that, as in the ethylene case, the calculated couplings are somewhat low, i.e., that A", (17) Lemieux, Kullnig, Bernatein and Schneider, J . Am. Chem. Sac., We do not list the values because the lack of rigidity of the structures is such as to fix the angle + with even less accuracy than those given in Table I. 80, 6098 (1958).
Having seen that the contributions of deviations from perfect pairing obtained by a variational calculation lead to approximately correct values for the vicinal coupling constants, we can now attempt to employ the valence-bond model for an estimate of the hyperconjugat'ion energy of ethane. Table I1 gives the results for the HCCH fragment as a function of the angle (6. It is evident that $c, the perfect pairing structure (see Fig. 2) has a coefficient on the order of unity. The other structures, $i ( i = 1,2,3,4), which involve deviations from perfect pairing and the loss of effective bonds, make only a small contribution, with the sum of the absolute values of their coefficients between 0.025 and 0.063. Correspondingly, the st,abilization energy E h arising from these structures is also small (
