Nuclear Magnetic Shielding of Protons in Amides and the Magnetic

(nuclear magnetic resonance) shielding constants seems to offer some hope ... chemical shift between the two protons which are cis and trans with resp...
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1388

P. T. NARASIMHAN AND MAXT. ROGERS

Vol. 63

NUCLEAR MAGNETIC SHIELDING O F PROTONS I N AMIDES AND THE MAGNETIC ANISOTROPY OF T H E C=O BOND BY P. T. NARASIMHAN AND MAXT. ROGERS Kedzie Chemical Laboratory, Michigan State University, East Lansing, Michigan Received January 1 4 , 1960

The magnetic shielding of nuclei due to distant chemical groups may be calculated from a knowledge of the magnetic anisotropy of the bonds in these groups. The reverse procedure, namely, the calculation of the magnetic anisotropy of bonds from n.m.r. (nuclear magnetic resonance) shielding constants seems to offer some hope and has been employed here to estimate the magnetic anisotropy of the C=O bond from the available data on formamide and dimethylformamide. The internal chemical shift between the two rotons which are cis and trans with respect to the C=O bond in acetamide and similarly the shift between the two N-methyf groups in dimethylacetamide have also been calculated.

I. Introduction The magnetic shielding of a nucleus in a molecule due to the neighboring electron cloud has been the subject of several important theoretical investigations in recent years. 1-8 Using second-order perturbation theory Ramsey' has derived an equation for the shielding tensor in terms of the ground and excited state wave functions of the molecule in which the nucleus is present. The variational method has also been employed by several a u t h o r ~ ~and - ~ *many ~ of them have limited themselves to the case of the proton shielding in the hydrogen molecule. However, when one considers larger molecules the application of the complete perturbation theory or the variational procedure is a very difficult task indeed. Further, such an approach is not likely to be fruitful a t the present state of our knowledge concerning molecular wave functions. Thus it appears desirable to make the following approximations. We consider the electron cloud immediately surrounding the nucleus in question as contributing a shielding (11, where the subscript I denotes the local shielding. Other distant charge distributions in the molecule may then be thought of as contributing (Id to the total shielding of the nucleus. Thus we may write where (I is the net shielding of the nucleus in the molecule. The reason for this separation of the total shielding. into local and distant ones is that it is easier toestimate from a knowledge of the magnetic anisotropy of the distant groups. It is then necessary to carry out the perturbation or variational calculations for the region nearest to the nucleus only. I n this manner we can simplify an otherwise almost intractable problem. Stephen8 has recently made use of this procedure and has calculated the shielding constants of protons in methane, ethylene and acetylene. The variational procedure was used in his calculations for estimating (11, However, in the present work we shall not be (1) N. F. Ramsey, "Nuclear Moments," John Wiley and Sons, Inc., New York, N. Y., 1953. "Molecular Beams," Clarendon Press, Oxford, 1956. (2) H. M. McConnell, Ann. Rev. P h y s . Chem., 8,105 (1957). (3) J. F. Hornig and J. 0. Hirschfelder, J . Chem. Phys., 23, 474

(1955). (4) T.P. Drts and R. Bersohn, P h y s . Rev., 104, 849 (1956). (5) B. R. McGarvey, J . Chem. Phys., 27, 68 (1957). (6) J. A. Pople, Proc. Roy. Soc. (London),A239, 541, 550 (1957). 17) H.M. McConnell, J . Chem. Phys., 27, 226 (1957). (8) M . J. Stephen, Proc. Roy. Soc. (London),A243, 2G4 (1957).

concerned with the details for calculating (11, but since our interest is in the magnetic anisotropy (Ax) of bonds we shall consider in greater detail the quantity (Id, this being directly related to AX. The expression for (Id in terms of the magnetic anisotropy of distant groups can be derived from the general equation for the shielding tensor on the basis of both perturbation and variational methods as has been demonstrated recently by McConnell7 and Stephene8 In a classical sense, the externally applied uniform magnetic field may be thought of as inducing magnetic dipoles in these distant groups and the secondary field at the nucleus due to these dipoles as contributing to (Id. For purposes of calculation we may approximate these induced dipoles by point dipoles or ideal dipoles when the distances between the nucleus and these groups are large. I n high resolution n.m.r. spectroscopy, with which we are presently concerned, the samples are usually in the form of liquids or gases and the molecule containing the nucleus under consideration is assuming different orientations in the magnetic field due to molecular motions. The secondary field from the induced dipoles is thus averaged out and in order to observe a finite shielding (Id 0 the distant groups must be magnetically anisotropic and the molecular geometry and motions be such that these factors do not enable the condition (Id = 0 to be realized. Since the magnitude of the shielding contribution from distant groups depends on their magnetic anisotropy and molecular geometry (including changes in this) the question naturally arises whether one can obtain any information concerning the anisotropy from the experimental shielding data. In order to answer this question we have considered the n.m.r. shielding of protons which are cis and trans to the carbonyl oxygen atom in amides, and using the available anisotropy data,9t'0 we have been able to estimate the anisotropy of the C=O bond. From the data thus obtained calculations have been made of the difference in the shielding of the cis- and trans-protons in acetamide and dimethylacetamide. The results are encouraging and may serve to show the usefulness of this approach. However, before proceeding further, we shall consider the rather classical problem of the secondary field due to an induced dipole and thus

+

(9) J. Tillieu and J. Guy, J . Chem. Phys., 24, 1117 (1956). See alao J. Tillieu, Ann. Phvs., 2, 471, 631 (1957). (10) J. Baudet, J. Tillieu and J. Guy, Compt. rend., 244, 2920 (1957).

NUCLEAR MAGNETIC SHIELDING OF PROTONS IN AMIDES

Sept., 1959

derive a general expression for c d along purely classical lines in terms of the magnetic anisotropy of a bond when this bond is farther away from the nucleus whose shielding is being examined. 11. Magnetic

Shielding from an Anisotropic Bond Let us now consider the shielding of the nucleus N due to :In anisotropic bond A-B formed between two atoms A and B, We choose a point P somewhere along the bond as the location of the induced dipole due to the external uniform magnetic field

1389

where Ax1 = xle - xyy and 4x2 = xzz - xxx. If the radius vector does not lie in any of the three principal planes tJhen general expression (9) has t o he used. (1 - 3 co~*ez)xzs(9)

I n some cases the values of 6 and/or R may change with internal molecular motions and here one must use appropriate average values (see + or Ha which is taken to be applied along A-B (x-axis). Appendix) in the expressions (1 - 3 cos2 With P as the origin of Cartesian coardinates, and [ (1-3 cos2 O/R3)Iavas the case may be. denoting the radius vector between P and N by R, 111. Proton Shielding in Formamide and Diwe can write the secondary field at N due to the methylf’ormamide ideal dipole at P as * l The high resolution proton resonance spectrum of formamide with N14 has been obtained recently HN = - T p (2) by Piette, Ray and OggI2 using a double resonance where the tensor T written in dyadic form is technique and by Schneider13 with N16. The pro(boldface type indicates the tensor) ton resonance spectrum of dimethylformamide was obtained earlier by Phillips14 and Gutowsky and Holm.16 The evidence for restricted rotation (3) around the C-N bond in these amides a t low temperatures is conclusive and the two sites marked In equation 3, 1 is the unit dyad and R..R is the A and B (see Fig. 1) are magnetically non-equivadyad for the radius vector. Now, the components lent. Since the high resolution spectra a t low tem+ of the magnetic moment p are peratures show that the rate of rotation around the C-N bond is so slow as to give two distinct resonance lines for the A and B type protons we may be justified in considering the planar structure where xbxx,xbyyand xbssare the principal magnetic for these molecules with the groups “frozen” in susceptibilities of the bond A-B. The magnetic this position. Taking then, for example, the case of formamide (I) it is easy to show that the diffield H N a t the nucleus N is related to the externally ference in the shielding values of the two types of protons marked A and B is a result of the difapplied uniform magnetic field Ho by the shielding ference between gA and ?dB, since on account of local tensor UN as follows symmetry the ql values of these protons will be identical. It is assumed that the effect of distant HN = Ho - UNHO (5) groups, such as the carbonyl group, on (11 for protons and hence we may write p d as A and B is equal. Denoting the net shielding of these two types of protons by q Aand qB, we have f

f

+

f

f

f

f

-

-

*

f

UA

where the factor has been introduced in order to average the shielding due to molecular motions in liquids and gases. In the case of an axially symmetric bond, that is one with xlz xyu = xxxit is easily seen that

*

3 c0s2Os)

(7)

where 6. is the angle between the x-axis and the -t

radius vector R. In equation 7 Ax is the molar anisotropy, namely, xsl - xyy = xEz - xxx and La is the Avogadro number. Equation 7 has been used by McConnelP in discussing proton shifts in benzene and methyl halides. For a bond with xzz i. xyy =k xxx the following expression holds good provided the y-axis (or x-axis) can be chosen such that R lies in the x-y (or x-z) plane. Thus (for R lying in the x-y plane) we have (11) C. J. F. Bottcher, “Theory of Dielectric Polarization,” Elsevier Puhlishing Co., New Yotk, N . T.,1952.

- UB

=

UdA

- U-dB

(10)

Thus we have identified the observed shielding difference of these protons with the difference in the shielding contributions to these nuclei from the distant groups, namely C=O and C-H bonds (Fig. 1) H

H ,

’

(A)

PH3

H

(A)

‘c-N-~

0 H‘ (B) Formamide ( I ) HIC H , (A)

’ ‘c-K

Dirnethylformamide‘ (11) HIC CHB ( A )

\

C-N

/

0 H ‘ (B) Acetamide (111) Dimethylacetamide (IV) Fig. 1.-Structures of amides with magnetically non-equivalent sites marked A and B. (12) L. H. Piette, J. D. R a y and R.A. Ogg, J. Mol. Spectroscopy, 2 , 66 (1958). (13) W. G. Schneider (private communication). (14) W. D. Phillips, J . Chem. P h y s . , as, 1363 (1955). (15) H. S. Gutowsky a n d C. H. Holm, ibid., 26, 1228 (1956).

P. T. NARASIMHAN AND MAXT. ROGERS

1390

Vol. G3

and

+4*0A

a3 +2.0

x Z e C = O refers to the C=O bond and ~ , , C - ~ t the o C-H bond. Details of the method of evaluating the average values (equations similar to 13) are given in the Appendix. It will be seen from equation 11 that from a knowledge of the values of Axl,Axz and AXC-H one can calculate the internal chemical shift between the A and B protons in formamide. Similarly we can make use of equation 12 for dimethylformamide provided the principal susceptibilities of the I C=O and C-H bonds are known. Tillieu and Guyg 0.81 0.91 1.01 1.11 1.21 have calculated the principal susceptibilities of the C-H bond for various states of hybridization of Location of C=O dipole from carbon, A. A? negative. carbon using both Slater and Coulson-Duncanson wave functions while Baudet, et aZ.,lO have calFig. 1. culated similar values for the C=O cr-bond. HowFrom the known geometry of these molecules it ever, theoretical calculations are not available a t is possible to calculate this difference in shielding, present for the C=O bond and hence an a priori that is, the internal chemical shift between A and calculation or the internal chemical shift is unB protons, by using the expressions given earlier fortunately not possible. But this situation forces for the shielding due to anisotropic bonds. (We us to consider the question raised earlier in this have neglected here the contribution to from paper, namely, whether it is possible to obtain any steric, electrostatic and similar factors. T o a information regarding the magnetic anisotropy of certain extent these are included in the molecular bonds from the experimental shielding data. In geometry as well as the bond anisotropy.) Thus, fact, the magnetic anisotropy of the C=O bond in we have in the case of formamide these amides can be calculated from the observed data on internal chemical shift between the A and B type protons in formamide and dimethylformamide by making use of the available experimental data on magnetic susceptibilities of these compounds16 as well as the theoretically derived bond susceptibilities.9 The results thus obtained on the C=O bond have been employed in the calculation of the internal chemical shift of A and B The subscripts A, B, refer to the nuclei and 0 and type protons in acetamide (111) and dimethylH refer to the locations of the point dipoles of the acetamide (IV). Of course, one has to make the C=O and C-H bonds. The molecule is taken to be assumption that the electronic structure, and hence in the z-y plane (z-axis along the bonds) and Ax1 the principal susceptibilities of the C=O bond in and Axz refer to the C=O bond and are defined these molecules, are not appreciably different. To following equation 8. I n the case of the C-H u- a certain extent, this assumption may be justibond we have xEa xyu= xxx and hence we have fied on the basis of the additivity of bond susceptiused equation 7 while for the C=O bond, which in- bilities in these molecules. volves a ?r-bond, equation 8 has been used in IV. Results deriving expression 11. Tables I and I1 show the results of the calculaFor the evaluation of the internal shift between tions on the magnetic anisotropy of the C=O bond the protons of the two N-methyl groups in dimethyl- in formamide and dimethylformamide. In the calformamide equation 11 is not applicable since the rotation of these methyl groups around their re- culations pertaining to Table I AT was taken to be for both formamide and dimethylformspective N-C bonds causes the angles and distances negative while the sign of A F was taken to be positive between the C=O dipole, and similarly the C-H amide dipole of the CHO group, and these methyl protons in the calculations pertaining to Table 11. There is to vary. We must therefore make use of the an uncertainty in the sign of A ( I owing to the fact general expression (equation 9) and also use suit- that it has not been possible to decide experimena t which one of these sites the proton is more able averages. Thus for dimethylformamide (11) tally shielded than the other. The possibility of A? the following expression holds good being positive for one molecule, say formamide, and negative for the other can be ruled out on the basis of the extremely large magnitudes of the calculated susceptibility and anisotropy values (e.g., Ax1 = I

+

(16) P. W. Selwood, "MPgnetoohemiatry," Interscience Publishers, New York, N. Y.,1956.

NUCLEAR MAGNETIC SHIELDING OF PROTONS IN AMIDES

Sept., 1959

1391

-20.994 X lo-'; Ax2 = -92.278 X for the C=O bond. The data given in Tables I and I1 were obtained as follows. Equations similar to (11) and (12) were set up for the two compounds and solved in terms of one of the principal susceptibilities. The average susceptibility, xsv, of the C=O bond in these compounds can be obtained from the experimental molar susceptibility datals and the available bond susceptibility data. Since

bond lengths and angles used here have been taken from W heland. l7 Table I11 presents the results of calculations on the internal chemical shift between the A and B type protons in acetamide (using an equation similar to 11) and dimethylacetamide (using an equation similar to 12). The x values have been taken from Tables I and 11. I n these compounds the secondary field due to the C-C bond of the C-CHa group alone was taken into account while those from the three C-H bonds in this methyl group xsv = (xxx XYY Xes) (15) were neglected because of the larger distance. It may be pointed out here that for a given sign the three principal susceptibilities of the C=O bond of A b the calculated internal chemical shift values are easily evaluated. in these two molecules are not affected by the TABLE I choice in the location of the C=O dipole as long as PRINCIPAL SUSCEPTIB~LITIES OF THE C=O BONDDERIVED one takes consistent sets from Table I or Table 11.

; + +

FROM N . M . R .

INTERNAL CHEMICAL SHIFTDATAON FORMADIMETHYLFORMAMIDE ( A E NEGATIVE)

MIDE AND

Looation of the

c=o

dipole Magnetic aniaotropy from Value of of the C=O bond ( X 10') carbon AXCH

A.

( X 106)

0.81

f0.26 .49

0.91 1.01 1.11 1.21

+ + + + + + + + +

.26 .49

.26 .49 .26 .49 .26 .49

Ax1 +5.508 +5.771 +5.004 +5.305 +4.184 f4.370 +3.257 +3.389 +2,448 +2.532

Axz 19,724 f10.033 3.6.024 +6.200 +1.492 +1.360 -2.822 -3.194 -6.315 -6.879

TABLE I11 THEINTERNAL CHEMICAL SHIFT OF A AND B TYPE PROTONS IN ACETAMIDE AND DIMETHYLACETAMIDE (SEE FIG. 1)

Principal susceptibilities of the C=O bond

A s (calcd.)

ix

( X 109

xxx

XYY

-5.987 -6.105

-1.771 -1,843

i-3.737 +3.928

-3.688 -3.705 -0,940

-2.668 -2.810

+2.336 +2.495

-3.632 -3.800 -4.452 -4.664 -5.077 -5.321

4-0.552 -I-0.570

-0.790 +1.627 +1.919 +3.686 f4.090

AXCH

XES

-1.195 -1.275 + Z . 620 -2.789

Compound

(X

x values from Table I

10')

x values

from Table I1

AU- (obsd.)

( X 107)

'

4-3.10 +0.26 -1.63 .49 -1.78 +3.24 .26 Dimethyl-1.43 +2.31 f2.21 acetamide .49 -1.53 f2.41 a The AXCH values refer to those used in formamide and dimethylformamide calculations (see Tables I and 11). * Value not available.

Acetamide

+ + +

Discussion

TABLE I1 An examination of Tables I and I1 shows the PRINCIPAL SUSCEPTIBILITIES OF THE C= 0 BONDDERIVED dependence of the A x and x values on various FROM N . M . R . INTERNAL CHEMICAL SHIFTDATAON FORM- parameters such as the location of the C=O dipole AMIDE AND DIMETHYLFORMAMIDE ( A c POSITIVE) and the A x value of the C-H bond. While the

latter is due to an uncertainty in the theoretical calculations on the principal susceptibilities of the dipole Magnetic anisotropy Principal susceptibilities C-H bond using different wave functions (see from Value of of the C=O bond of the C=O bond AXCR ( X 10') oarbon ( X 109 ref. 9) the former arises from a very serious dif(A.) Ax1 Ax2 ( X 108) xxx XYY ficulty in the present method itself. We shall now -8.223 +0.2n - 5 . 1 1 8 +2.430 -0.669 -5.787 0.81 + .49 - 5 . 3 8 1 - 8 . 5 3 3 $ 2 . 5 5 5 - , 5 9 7 - 5 . 9 7 8 discuss this matter in greater detail. It was men0.91 + . 2 6 - 5 , 0 0 3 - 6 . 0 2 2 + 1 . 0 0 7 - , 0 1 2 - 5 . 0 1 5 tioned earlier that for mathematical simplicity we + .49 - 5 . 3 0 5 - 6 . 2 0 0 f 1 . 0 2 5 + , 1 3 0 - 5 . 1 7 5 idealize the induced dipoles by point dipoles and in 1.01 + . 2 F - 4 . 1 8 4 - 1 . 4 9 1 - 1 , 7 4 1 + ,952 - 3 . 2 3 2 fact such an idealization is valid when we consider + . 4 9 - 4 . 3 7 0 - 1 . 3 6 0 -1,890 $ 1 . 1 2 0 - 3 . 2 5 0 the field due to these dipoles a t very large distances. 1.11 + . 2 6 - 3 . 2 5 1 f 2 . 8 4 6 - 4 . 3 1 7 + 1 . 7 7 4 - 1 . 4 7 7 For actual calculations it is customary to locate + . 4 9 - 3 . 3 8 3 t . 3 . 2 1 2 - 4 . 6 0 9 + 1 . 9 8 6 - 1 . 3 9 7 this dipole somewhere along the bond by making 1.21 + . 2 6 - 2 . 4 4 8 + 6 . 3 1 4 - 6 . 3 6 5 + 2 . 3 9 7 - 0 . 0 5 1 use of certain assumptions. One such assumption + 49 - 2 . 5 3 2 + 6 . 8 7 9 - 6 . 7 7 0 + 2 . 6 4 1 + 0 . 1 0 9 is to locate the dipole a t the electrical center of The internal chemical shift data were taken from gravity of the electron distribution constituting the work of Piette, Ray and 0gg,l2 S ~ h n e i d e r , ' ~the bond. Unfortunately the task of locating the and Gutowsky and Holm.16 It is interesting to note dipole in this manner is not as trivial as it appears that before the results of Schneider on liquid forma- a t first sight. From theoretical studies on dipole mide (with N16) were available to us we extrapo- moments of bonds's it is well-known that the lolated the data of Piette, et al., on aqueous solutions cation of the center of gravity of a charge distribution in a bond is dependent among other things, of formamide by the equation on two important factors, namely, the hybridization AU = u A - uB = Aum(1 C ) + CK (16) and ionic character of the bond. At present these where C is the concentration of water and K is a con- two quantities are not as accurately known as one stant. The value of AT, = *2.2 X lo-' thus would desire and hence the location of the dipole obtained may be compared with the value of in this manner is subject to some error. rt2.23 X lo-' obtained by Schneider on the pure (17) G. W. Wheland, "Resonance in Organic Chemistry," John liquid. I n the present calculations the value of Wiley and Sons, Inc., New York, N. Y., 1955. The various A g = i 2.2 X lo-' has been used. (18) C. A. Coulson, "Valence," The Clsrendon Press, Oxford, 1052. Location of the

c=o

X5Z

I

-

-

P. T. NARASIMHAN AND MAXT. ROGERS

1392

"

y -2.0

t

0

X

- 4.0 -6.01 %ax

0.81 0.91 1.01 1.11 1.12 Location of C=O dipole from carbon, A. A? positive. Figs. 2(a) and 2(b).-Dependence of the calculated principal susceptibility values on the location of the carbonyl dipole,

I n the case of a purely covalent bond one can, perhaps without serious error, place the dipole along the bond a t a distance corresponding to the covalent radii of the two atoms forming the bond. In fact in the present calculations this assumption has been made for the C-H bond. But in the case of more polar bonds the electrical center of gravity is likely to shift toward the more electronegative atom while the exact location of this point is not easily determined. In the present case of the C=O bond, the oxygen atom being much more electronegative than carbon, the center of gravity is likely to be very near the oxygen atom and hence calculations have been made with the C=O dipole a t different locations near the oxygen as well as on the oxygen itself. Owing to this uncertainty in the location of the induced dipole the anisotropy and susceptibility values differ widely although it must be mentioned that the values themselves are of reasonable order of magnitude. We shall return to the question of locating the dipole and choosing the probable susceptibility values a little later. In Figs. 2a and 2b the calculated values of xxx, xyu and xeEfor the C=O bond are plotted against the distance between the dipole and the carbou atom. The two figures have been obtained using two different signs for AF. We shall now discuss the probable sign of A T in these molecules from the point of view of the present calculations and certain theoretical considerations. It has been shown recently by Poples that the C=O bond in aldehydes makes a large paramagnetic contribution in the C/

H

plane. An explanation of this is

1)ased oil the conceiitration of the ?r-bonding orbital Such a situation leads to stit electron deficiency on the 2 p r atomic orbital of carbon and large paramagnetic contributions in the H C b 0 plane may then arise from transitions which

on the oxygen atom.

involve the transfer of the C-0 g-electrons to the mitibonding r-orbitals. A similar argument can

Vol. 63

be extended to the case of the C=O bond in these and xyy amides. According to Table I the values (2-Y is the molecular plane) become diamagnetic as the dipole is moved towards oxygen while according to Table I1 xyu (and probably xal, also) becomes paramagnetic as the dipole is shifted toward oxygen. This trend in the latter case is in good agreement with the theoretical considerations mentioned above and hence we may conclude that A g is positive in these molecules. Further the calculated A? value in dimethylacetamide appears to be in better agreement with experiment when one uses x values from Table 11. The principal susceptibility of the C=O bond in urea in the direction perpendicular to the plane of the molecule (ie., the value of xxx) can be approximately calculated from the available molar anisotropy datalg and from the bond principal susceptibility data.gr10 The value thus obtained for xxxis -9.7 X lo+. It is therefore very likely that the C=O bond in these amides is also characterized by such a large diamagnetic susceptibility value for xxx. According to Table I such a value for xxx is possible only when the dipole is located farther away from the oxygen atom. This situation is certainly contradictory to simple electronegativity concepts. On the other hand, the results of Table I1 are in better agreement with the large diamagnetic value of xux particularly since this corresponds to the dipole being located nearer to the oxygen atom. Regarding the question of locating the dipole we may therefore conclude that no serious error will result by assuming the dipole to be located on the oxygen atom itself. In this case then the most probable values of anisotropy and susceptibility are: Ax1 --2.5 X lo+; Ax,+6.6 X lo-'; ~ X ~ m - 6 . 6X xyy"+2.5 X One must of course bear in mind xs.-O. the approximate nature of these values. It is indeed rather unfortunate that no direct experimental measurement of these quantities is possible a t present. Also, if the electronic center of gravity of the C=O bond in these molecules could be located more precisely by other methods it might then be possible to limit the choice in the x values more satisfactorily. However, these drawbacks may be overcome in the near future. From Table I11 the internal chemical shift for acetamide is found to be -+3.2 X lo-' (ACE taken to be positive). Experimental internal shift data for acetamide are not available a t present and it will be interesting to compare this value when data on this compound are forthcoming. In the present treatment we have considered the induced dipole effect to be the only factor in the shielding by distant groups and have neglected electronegativity effects. Also, the theoretical values of A T refer to the A and B type protons as being "frozen-in" while experimeiital values are obtained under conditions of exchange between A and B. Of course it is possible to determine the value of A c under conditions of extremely slow exchange, as for example, by working a t low temperatures and the value thus obtained should be compared with the theoretical results. Further, (19) K. Lonsdale, P i o r . Roy. Sac. ( L o n d o n ) , 8177, 272 (1941).

NUCLEAR MAGNETIC SHIELDING OF PROTONS IN AMIDES

Sept., 1959

1393

on the experimental side the measurements of internal chemical shifts might be affected by as1 sociation in the liquid state. Such association shifts have been observed in the case of formamide by Piette and eo-workers. It will therefore be desirable to obtain internal chemical shift data on these compounds by dissolving them in inert solvents and extrapolating the data to infinite dilution and also working a t low temperatures. It is probable that better agreement might then result by following the procedure outlined in the present paper for calculating the internal chemical shifts. Acknowledgments.-Our thanks are due to Dr. W. 0. Schneider for communicating the results of his measurements on formamide with N15 prior to publication. We also wish to thank the staff of the computer (MISTIC) laboratory of Michigan State University for their valuable cooperation. Fig. 3.-Evaluation of T , average values. Case 1: This research was supported in part by the Atomic Energy Commission through contract AT-( 11-1)- 5 < (90" - I)). The dipole is located a t 0 and the nucleus N is free to rotate around the axis A-€3. 151. Appendix When a given nucleus is involved in internal molecular motions the average shielding of that nucleus due to a point dipole must be used in equation 9 (above). The components T , = (1- 3cos20,/ R3)aV(where E = 2, y, x ) were evaluated numerically in the following manner. Case 1 : 2: < (90° - 7) (see Fig. 3).-Let us consider the case of a nucleus N which is free to rotate around the axis A-B, this axis being taken to be in the Z-Y plane. With the point dipole placed a t 0, the quantities of interest to us now are the radius vector R and Be. Since AN ( = r l ) and AB are chosen to be a t right angles to each other we can express a(the angle between NA and AO, L e . , rl and r2) in terms of 0 and 6(6 is the angle of rotation of rl in its plane). Thus it can be shown Fig. 4.-Evaluation of T , average values. Case 2: ( = that (90' - 7). The dipole is located a t 0 and the nucleus N

T

cos

ct =

sin j3 cos 4

(1-4)

and hence we can express R as R

=

(TIZ

+

7-2'

- 2nr2 sin j3 cos

and +)'I2

(2A)

cos ea = ( R 2

+ ra2 - r42)/2Rra

(3.4)

and hence we can write

Since 9 varies for 0 to 2 a we can find the average value by integrating the left-hand side of equation 4A after substituting equation 2A for R and dividing finally by 27r. Thus after rearranging we obtain

2n

1

Jg

3 - 4ra

to rotate around the axis A-B.

JC =

Now we have also

T, =

18 free

[(ra2- r42)2JC

+ 2(r?

-

+

E"

(a

+ b cos 4)-s/2 d+

(W

with a = r12 rp22 and b = -2razsinp. In order to evaluate T , in this case we may proceed as follows. Extend the line AB (Fig. 3) until it intersects the Y-axis a t a point, say P. As in the case of T , we may now consider OP as r3, N P as r4 and OA and NA as r2 and rll respectively. However, p is now the angle between PA and r2. Cos 0, can be expressed as in 3A and Tycan be obtained in a similar manner (4A). Case 2 : 5 = (90" -7) (see Fig. 4).-In this instance we can evaluate T, as in case 1 but T , is given by T, = (1 - y e Y ) = BY

1 [ J B - 374'1

COS' I)

Jc

(9A)

2n

where

Equation 9A is obtained by a method similar to that used in the derivation of (4A). J B and JC are as defined in equations 7A and 8A. Case 3: T , = [l-3 C O S ~ O ~ / (see R ~ ]Fig. ~ ~ 5).. We now consider the case (Fig. 5) in which the

P. T. NARASIMHAN AND MAXT. ROGERS

1394

I

Fig. 5.-Case 3: evaluation of T, average values. The dipole is located a t 0 and the nucleus N is free to rotate around the axis A-B which lies in the %-Y plane.

Vol. 63

the averages required for the compounds treated here. As an illustration let us consider dimethylformamide (Fig. 6). For the present we shall confine our attention to only one of the protons of the N-CH3 group (proton 1). The secondary field a t proton 1 due to the C k O dipole a t P is required. The Z-axis is chosen along the bond and the X-axis is perpendicular to the plane of the molecule (ie., Z and Y in the plane of the paper). An examiliation of Fig. 6 shows that T , can be obtained in a manner similar to Tyof equation 9A. Similarly T y can be obtained by the use of an expression similar to T, of equation 4A while T, can be obtained from equation 11A. The procedure can be similarly applied for proton (2) but in each case the Z-axis is chosen along the bond (here -0, for example) and X-axis perpendicular to the plane of the molecule. The evaluation of the integrals in equations 6A, ?A and 8A is necessary for the calculation of T,,Tyand T , in these cases. These integrals are of the elliptic type and cannot be evaluated in a closed form. I n the present work we have evaluated these integrals by numerical integration using the high speed electronic digital computer, MISTIC a t Michigan State University. The program used enables one to obtain J A , J B and Jc for the corresponding values of a and b which are punched on a data tape. I n the computer 179 d's are generated (0 to 2n are initially supplied) for the interval 0-27 and the corresponding funcare computed and tions, say (a b cos @)-'/z stored. The integration is then performed by taking seven functions successively and evaluating the integrals for each set (six intervals) by using a sixth degree polynomial approximation. The integral is then obtained by adding the 30 values thus obtained. As an example, we have given below the values of J A , J B and J c obtained for some typical values of a and b. The values of T i x ,T i yand T i z for this c%se(dimethylacetamide-dipole on oxygen, Le., 1.21 A. from carbon) have also been given.

+

f

Hz

Fig. 6.-Illustration of the details of calculating Te average values for proton HI in dimethylacetamide. The carbonyl dipole is located at P.

X-axis is chosen perpendicular to the axis AB of the rotating nucleus. Then the plane described by the rotating nucleus is perpendicular to the 2-Y plane. Now, using the relation cos2& =

(irl

sin + / R ) 2

(IOAI

and substituting equation 2A for R in (10A) and proceeding as before we find 1 - 6~1'~ JB -k T , = - 3rI2/b2X JA 4- b227r

'3

Again J A , J B and J c are as defined in equations 6A, 7A and 8A. The above three cases are sufficient to deal with

a

b

JA

J B

+16.0838 x 10-8

-2 4208 x 10-8

1.5734358

0.05241128 x 10'2

x

108

Jc

Ti*

T2V

TL.

6.36724015 X 1037

+1.4274

+1.1355 x 1022

-2.5629 x 1022

x

1022

Finally it may be pointed out here that the internal chemical shift values calculated by using equation 11 become almost identical with those obtained from (12) when a>>b.