Russell 5. Drago, Jeffrey I. Zink,' Robert M. Richman, and W. D. perry2'
University of Illinois Urbono, 61801
Theory of Isotropic Shifts in the NMR of Paramagnetic Materials Part / I
In Part I (published in this Journal last month), we discussed that aspect of electron relaxation processes that makes possible the detection of the nmr spectra of paramagnetic species. The average electron spin polarization was defined and the problems involved in determining its magnitude were discussed. The scalar (or contact) shift was then related to the electron density a t the nucleus. In this article our attention will be focused on the diuolar (or pseudocontact) shift. The principles developed in'the pievious article are essential for the understanding of this discussion.
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Termino!ogy lor Spin Delocalization
Before proceeding further. brief mention shmld be made of ,m, = -% by Thus a proton aligns both its magnetic moment and spin parallel to the field (Fig. 4a). For an electron in a magnetic field, we have a similar situation. The interaction between the electron magnetic moment, r e , and the applied field H i s represented by the Hamiltonian X = -&'H (44)
la>.
In order to keep the terminology for isotropic shifts consistent with that for the proton, we should refer to the electron in the lowest energy state to have &spin, (antiparallel to the field) and that the important thing, the magnetic moment, is parallel to the field. The Dipolar Interaction
The dipolar interaction between a nucleus and an electron is the second type of interaction which may cause a change in the magnetic field at the point in space where the nucleus being observed is located. The interaction gets the name "dipolar" because the mathematical characterization of the energy of interaction has the same form as the interaction energy arising from two macroscopic magnetic dipoles such as bar magnets. It is often called the "pseudocontact" interaction to distinguish it from the Fermi contact interaction where the electron is in "contact" with a nucleus via the radial part of the electronic wave functions. The dipolar interaction does not depend upon the electronic radial wave function in our treatment because it takes d a c e throuzh soace. The maenetic field of the electron megnetic dipole interacts directiy with the nuclear maenetic d i ~ o l ewithout the intervention of chemical bonds.-~hus,this interaction occurs whenever an unpaired electron and a magnetic nucleus both exist in a molecule. As will he shown, if the molecule is magnetically isotropic and is tumbling, the average field produced by the electron at the nucleus being observed is zero and no nmr pseudocontact shift occurs. However, effects of the dipolar coupling may still be observed in the shortened nuclear relaxation time. In the above discussion and in the derivation of the equations used to calculate the pseudocontact shift, a
where
r.
= -gAS
Since S = 'h for an electron, there are two allowed orientations in the external field and these are illustrated in Figure 4b. As in the case of the proton, the lowest energy state is that in which the magnetic moment of the electron is aligned with the field. It should be noted that this situation gives rise to m, = -% denoted by lp> for the lowest energy level. Thus, when an electron has its magnetic moment aligned with an external field, its spin is aligned antiparallel to the field. This difference arises simply because of the difference in the sign of rea n d r ~ .
T h i s is the second of a two-oart series on the Theorv , of Isotronie Shifrs in rhe S M K of Pnrarnagnetw Materials. T h e iirrr pnn ap. prarrd in the .June issue. .%I f~gurrs.rqLarions,literature ritarinns, and tootnorwart n t ~ r n ~ r r v d c ~ ~ n ~ c c u rhrouahour uvely 7~~
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(b) Figure 4. (a) Proton spin levels in a magnetic field. spin ot Electron spin levels in a magnetic field. spin of %.
'h. (b)
point charge model is employed. Of course it is not strictly correct to ignore the radial part of the electronic wave functions. If we assume that the electron has a high probability of being confined in a volume of space of very small radius compared to the distance separating it from the nucleus in question, we may approximate the electron as a point dipole located on one of the nuclei. For example, in transition metal complexes we assume the electron is located directly on the metal nucleus. We know that in fact it is in a molecular orbital consisting primarily of a metal d orbital. But the average radius of the d orbital is small compared to the distance from the metal atom to the ligand nuclei. The maximum error arising from the approximation for first row transition metal complexes is about ten percent and for most systems is much smaller (26). We will use the point dipole approximation in developing the theory of the electron-nuclear interaction. The classical energy of interaction between two magtetic dipolesp~and pz is
where r is the radius vector (i.e., it has direction and length) from nl to nz and r is the distance between the dipoles. The quantum mechanical Hamiltonian for the dipolar interaction, X d , may be obtained by replacing p, by the electron magnetic moment, - g&S. and pz by the nuclear magnetic moment, by g,PJ. For a system consisting of one electron and one proton we find
whose elements may either be calculated as shown in reference (6), pages 133-136, or determined experimentally. S.g is thus the total electron moment, including both spin and orbital contributions, which interacts with the external field H. Thus it is S.g which must interact with the nuclear moment in eqn. (47) xd = -g.P,P,S.g.T.I (49) In this most general case we are treating we have rapid electron spin relaxation and the spin S seen by the nucleus has an average polarization 8, developed previously and a direction q. The nuclear spin I has a magnitude Ih and a direction k parallel to the applied field, giving
% = - ~ , B , B & I.g.T.k)l,
In order to do the proper averaging to account for the tumbling of the molecule in solution, we must average the Hamiltonian over all angles through which the molecule tumbles. Our general expression for the average electron spin polarization was
+
where C = &HS(S 1)/3kT Note that g enters into eqn. (50) twice: ffg) in 3, determines the average electron spin polarization, and g determines the instantaneous orientation and magnitude of the total electron moment. Substituting eqn. (51) into eqn. (50) produces the dipolar Hamiltonian, to include the averaging overall orientations. For axial symmetry, we ohtain
(xJ,, =
-g~P.vB,C(f(g)q.g.T.k)~,~I, (52)
where 9 and Q, the angles over which we must average, measure the angle between the symmetry axis of the molecule and the magnetic field direction (k), and the rotation about the symmetry axis, respectively (see Fig. 7). For
which may also he written
where U is the unit dyadic (a three-hy-three matrix with ones on the diagonal and zeros elsewhere) and the term in brackets is abbreviated by T which is called the "dipolar coupling tensor." The notation follows that of McConnell (27) and will become clearer when we work out a specific example. I t can be shown that dipolar coupling averages to zero in a rapidly tumbling isotropic molecule (see Fig. 5). For a tensor quantity, this average is one-third the trace (sum of the diagonal elements), implying that T has a trace of zero. Consequently dipolar coupling does not contribute to the line positions of the liquid phase spectrum of magnetically isotropic molecules. Next we shall consider anisotropic molecules. The above discussion and the averaging carried out will he made more auantitative with the followine derivation of the pseudoconiact shift equation. Of the several ways in which magnetic anisotro~v . - mav. he treated we will choose the one most common use in the literature on the subject. We have written the zero-order electronic Zeeman interaction by the Hamiltonian
r
(50)
=
gp,~.S
(48)
This holds true for a free electron. In a molecule, however, i t is necessary to write the Hamiltonian in a form which shows the directional properties of the magnetic field within the molecule and magnetic effects arising from the electron's orhital angular momentum. In other words, our notation must reflect the fact that the electron may "feel" different magnetic fields when the molecule is aligned in different directions. We indicate this by writing g as a tensor. Our interaction thus may be written H = 6,S.g.H. For our present purpose, it is sufficient to recognize that g is no longer a scalar of magnitude 2.0023 hut a tensor
Figure 5 . The coupling of two magnetic dipoles may be thought of in terms of the field felt at one dipole due to the other. ( a ) Lines of flux emanating from a dipole-in this case, the average magnetic moment of a rapidly flipping unpaired spin. ( b ) The dot is a magnetic nucleus, and in this Orientation of an isotropic molecule. it feels a field parallel to the applied field. ( c ) In this orientation, the nucleus feels a field antiparailel to the applied field. When tumbling is rapid, we must average over these and all other orientations, in which case there is zero net field due to the electron. (dl Now consider an anisotropic molecule, e.g., M X I Y ? discussed in Figure 3 . W e have illustrated here the moments corresponding to the two different orientations of the molecule in 3 I c ) Let the nucleus be somewhere on the Y liaand. Averaaino over all orientations does not give zero. This is the case of electron relaxation slower than tumbling, so the net population of the two spin states of the electron is the same for ail orientations. even though the Z-component of the motion is changing. i e . . the molecule is tumbling so fast the electron population does not have time to readjust for the individual orientations. (el Here electron relaxation is faster than tumbling, so the electron has time to establish a Boltzmann equilibrium population of the spin states for each orientation. In case ( 1 ) at 5e we have a large moment associated with the highestfold Symmetry axis aligned with the field. in case ( 2 ) of 5e, the highestfold axis is perpendicular to the applied fieid so the net moment is smalier and there is a more equal population of the two spin states.
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symmetries other than axial, one must average over three angles instead of two. This averaging introduces some complexity into the form of the pseudocontact shift equation to he used, for, in some cases, 9, (when relaxation is fast compared to tumhling) must he included in the averaging (as written in eqn. (52)) while when relaxation is slow compared to tumhling, it must not be. Furthermore, the tumhling will determine which axis q we must use in eqn. (52). First, we shall illustrate how the axis q is determined for molecules rapidly tumbling in solution. The result we shall demonstrate is that the axis of the electron magnetic moment q will he simply k, the axis defined by the static magnetic field. In a solid or in a viscous liquid, the vector axis of quantization will depend on the orientation of the molecule as well as on the magnetic field direction and will thus in general he different from k. In other words, the electron spin magnetic moment will try to align itself with both the external field and the electron orbital moment (due to spin-orbit coupling) which are, in general, not parallel. Two possible axes of spin orientation, q' and q", are shown in Figure 6. These vectors may each he resolved into two components, one perpendicular to k(p) and one along k(a). In solution with rapid tumhling occurring many orientations will occur on the resonance time scale. In particular, both q' and q" could occur. In such a case, the perpendicular components p' and p" would cancel each other leaving the resultant axis of quantization to he along k. If completely random tumhling occurs, all of the perpendicular components will, on the average, cancel each other out, thus leaving k as the axis of quantization for any molecule rapidly tumbling in solution. Now we can return to the question of when S, and thus f(g) should he included in the averaging of eqn. (52). Again we shall examine molecules rapidly tumhling in solution. We may consider two limiting cases of electron relaxation relative to the tumbling: (1) the electrons relax much slower than the tumbling, and (2) the electrons relax much faster than the tumbling. Recall that molecular tumbling times are on the order of 10-l1 sec and relaxation times>ange from 10-9 to 10-l3 sec. Thus it is plausible that each of these limiting cases will be found in molecular systems. As shown above, the condition of fast tumhling means that the axis of quantization q will he k, the static magnetic field direction. However, each instantaneous orientation has a particular g value associated with it. Recall the case for single crystal esr measurements on a complex with axial symmetry where g2 = g cos2 0 g~~ sin2 8 with e being the angle between the
+
magnetic field and the symmetry axis. For case ( I ) , the electrons relax so slowly that they cannot follow the changes in the energy of the various Zeeman levels caused by changes in the g value. The Boltzman populations which will he established in this case will he estahlished for levels in which the energy is determined by the average of all of the g values, i.e., E = ga,B,HS~. In other words, the electrons will "feel" an average of all the g values corresponding to all of the orientations. Hence, the average spin polarization will he that given by eqn. (51) with f(g) = g,,. Hence, S, may be determined independently of the other components included in the averaging of eqn. (52) and is not included in the brackets of that equation. In case (2), however, the electrons do react fast enough to feel each different g value corresponding to each orientation. Hence, instantaneous Boltzmann populations are set up for each instantaneous orientation. In this case we cannot use one Boltzmann population corresponding to the average of all possible g's, hut instead must calculate all of the possible Boltzmann populations corresponding to all of the possible g's and then average these populations. T o do this we must include f(g) inside the averaging brackets. In this way we are taking into account the different Boltzmann populations and then averaging all of these. The explicit form of f(g) to he included in eqn. (52) will be discussed in detail later. It can he seen from eqn. (31) that we have obtained an expression for the dipolar Hamiltonian which is similar to the nuclear Zeeman Hamiltonian, X = g,,p,H.I, the difference being that the static magnetic field of the spectrometer H in the latter expression has been replaced by an effective magnetic field from the electron spin Heit which is given hy &C(f (g) q.g.T.k)s,n. This result is interpreted to mean that the nuclear spin experience~a new magnetic field arising from the electron, Heir, in addition to the static field from the spectrometer H. The divolar or "pseudocontact" shift is caused completely by tt;e presence of this additional magnetic field and is thus
in units of gauss. The Dipolar Shift for Tetragonal Nickel (11) Complexes We illustrate the calculation of the bracketed term in eqn. (53) with the example of a D4h Ni(II) complex of the type NiLnX2.Figure 7 illustrates the symmetry and identifies the angles and axis system to he used in the following discussion. In the molecular axis system, the g tensor is diagonal (6). (54) 0
Figure 6. Two possible vector axes 9' and 9" resolved into their perpendicular (p' and p") and parallel (a' and a") components relative to k, the axis defined by the spectrometer.
466
Figure 7. Axis system used in the calculation of the dipolar shifts. The axes labeled i, j, and k refer to the laboratory axis system where k is the direction of the static field of the spectrometer.
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0
gr
Because of the axial symmetry the two orthogonal axes A and v will have identical molecular properties. These are molecular axes perpendicular to K the highest fold symmetry axis. We d e f i n e v l k (i.e., w k = 0) and specify the direction of X by k.n = cos 8 , k.X = sin 0. In the molecular axis system the vector r connecting the paramagnetic nucleus and the nucleus we are lwking at in the nmr has components T cos x in the r direction, T sin x cos Q in the A direction, and r sin x sin Q in the v direction, as one may prove by simple geometry from Figure 7. The form of f(g) may he ohtained by comparing the general form of the Zeeman energy along an axis q, (for rapid tumhling q becomes k), Es, = BeSqf(g)H with the explicit form obtained using the g tensor Esq = Be S.gH = BeSqq.g.kH. We see that f(g) = 9.g.k. For very rapid tumhling, the axis of quantization q will he k. If the molecule were not tumhling, we could treat the electron as if it were responding to an effective field g k Holg, (instead of the
spectrometer field ~ H Q and ) the axis of quantization would be q = g.k/lg.kl. Using q = k for our rapid tumbling case, our expression for the shift becomes
Using Figure 7, we find that the components of vector k in the molecular axis system are
With the components of r calculated earlier, we can easily find the components of T (eqn. (47)). All that remains is to carry out all of the multiplications and then average over the indicated angles. Explicitly, our final expression for the shift (written in the molecular axis system), is
= %, (cosO) ) ~ Using the relations ( ~ 0 ~ =~ %,8 (sin28), = 0 and ( c o ~ ~=n %, ) ~we find (after much tedious algebra)
Note t h a t j f g = g L (i.e., magnetic isotropy), AH = 0. Note also that T has zero trace as stated previously. The first term in parentheses is called the "function of g values" and the last term the "geometric factor." If the metal complex does not have axial symmetly, another principal value of the g tensor and another angle over which averaging takes place must be used. The results of a calculation for this type of symmetry may be found in the literature (28). If the electron spin relaxation time is much slower than the tumbling time, the electron will "see" an average
In such a case (for example Fe(II) complexes with small ligands in nonviscous solvents) ffg) does not have to be averaged for each instantaneous orientation and the function ofg values in the final expression will simply be
The expressions for the dipolar shift under a variety of conditions of tumbling and electron spin relaxation are reported by Jesson (29). Several situations which cannot be treated by application of eqn. (52) are discussed by Kurland and McGawey (14). The applications of the pseudocontact contribution to the isotropic shift are numerous. It has been employed to study ion-pairing in solution (30-36), to provide thermodynamic data about interaction (36) to study the influence of solvent on the interaction, (33, 34) and to give estimates of the ion-pair distances ( 3 0 3 4 ) . Studies have also been aimed a t elucidating the nature of second coordination sphere interactions in solution (37, 38). The most extensive applications are those which involve the use of the dipolar shift from so-called shift reagents (lanthanide complexes) to provide information about conformations and structure in solution. This topic has been the subject of several recent reviews (39-41). Many of these applications are based on the assumption that the shift observed is solely pseudocontact in nature when lantbanide shift reagents are involved. In long chain molecules held together by o-bonds this assumption is quite good for atoms far removed from the donor center. At atom positions close to the donor atom and in systems with extensively delocalized r-systems this assumption is not correct. There often are small but significant contact contributions. Even in the tetrabutylammonium cation, ion-paired to several paramagnetic anions, contact contributions were observed (21). As should now be obvious, from our discussion, complications from the contact shift as well as uncertainty about molecular correlation times for tumbling, the stoichiometry, symmetry, and geometry of the species in solution all are potential sources of difficulty in this type of application. Literature Cited Zink. J. I., un~ubliahedcalculafim. McConnell, H.M.,andRobert.an, R E . . J C k m . Phys., 29.1361l1958). LaMsr.G.N..J. C k m . Phya., 43.1085 11965). Jesson, J . P.. J Chom. Phys., 47.579 11967). LaMar.G.N.,J. Chrm. Phva., 41.2992 (1W). LaMar,G.N.. J. Chem. Phva.. O.235(1968). Larsen, D . W., and Wahl. A. C.,Inarg. C h m , 4, 128111965). (33) Fanning. J.C.,and Drago,R,S., J A m o r Cham. Soc., 90,3987(19661. (34) Walker, I.M.,andDrago, RS.. J A m e r C h m . Soe, 90,6951 (19681. (35) W a l k e r . I . M . , h n f h a l . L . . and Querpshi. M.S..Inorg. C h m . . lO.2463(19711. (36)Lim. Y.Y., a n d D l a g o , R . S . . J A m e r C k m . Soc.. 9 4 8 4 11972). (37) Rettig, M . F.,and Drago. R, S., J. Amor. Cham. Soc., 88,2966 (1966). 138) Eaton. D.R.. Con. J. Chem., 47.2665(1969). 139) Mayo.9.C.. Cham Soe Re"., 2.49. (1973). (10) Von Ammon, R.. andfiseher. R. D..Angelo. Chrm. (Inti. Edit.). 11.675 11972). (41) Sievers, R. E., (Editor), ''Nuclear Mamefie Resonance Shift &agents," Academic PEW.I n e , NouYoxk, 1973.
(26) (27) I281 I291 I301 I311 (32)
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