Russell S. Drago, Jeffrey I. Zink,' Robert M. Richmon, and W. D. PerryZ University of Illinois Urbana. 61801
I
Theory of Isotropic Shifts in the NMR of Paramagnetic Materials Part I
In the early 1950's, it was felt that paramagnetic compounds would provide nuclear magnetic resonance spectra which were broadened too extensively to be of value. Then, in 1957, McConnell and Holm reported a well-resolved spectrum for paramagnetic nickelocene, [Ni(C5H&] (1). Since that time, a great deal of research has been reported in this area and the information from the study of the nmr of paramagnetic complexes has surpassed that obtained from the study of diamagnetic complexes. Our concern in this article is to give an under: standable presentation of the theory and the derivation of equations used to describe the nmr of paramagnetic compounds, and to give an elementary qualitative presentation of the fundamental principles of the nmr spectra of these compounds. This is particularly timely in view of the extensive application of paramagnetic complexes of lanthanides as shift reagents in organic chemistry (2). A familiarity with quantum mechanical operators on the level of Hanna (3) is recommended. The nmr snectra of naramametic transition metal com" plexes in solution are characterized by large, temperature dependent shifts in the resonance frequencies of the ligand nuclei either upfield or downfield from the frequencies in diamagnetic environments. The observed shift is referred to as the&otropic shift since i t is an average of what would be obtained for all possible orientations of the molecule with respect to the applied magnetic field: Anisotropy, in contrast to isotropy, is an orientation-dependent quality or quantity. As will be shown, the isotropic shift includes nuclear resonance shifts arising from two different interactions: the "Fermi hyperfine contact" or "scalar" interaction and the electron-nuclear "dipolar" or "pseudocontact" interaction. The total observed isotropic shift A",,, is the algebraic sum of the shifts caused by the contact mechanism (An,) and by the pseudocontact mechanism (A",). The contact shift, A , is directly proportional to the amount of unpaired electron spin density that is "in contact" with the nucleus. or in molecular orbital languaae, - -. to the value of the square of the electronic wave function a t the particular nucleus being investigated. The pseudocontact or dipolar shift, A",, is a through-space coupling This is the first of a two-part series on the Theory of Isotropic Shifts in the NMR of Paramagnetic Materials. The second part will appear in the July issue. All figures, eqqations, literature citations, and footnotesare numbered consecutively throughout. 'Present Address: Department of Chemistry, University of California, Los Angeles, California 90024. 2Present address: Department of Chemistry, Auburn University, Auburn, Alabama 36830. =Theelectron spin resonance (esr) experiment measures the energy required to flip the spin of an unpaired electron in a magnetic field. This electron can couple with nuclear magnetic moments by the same mechanisms described here to produce a hyperfine coupling A,, which is identical to the Ai defined above.
between ligand nuclei and unpaired electrons on the metal ion and depends upon the geometry of the complex, not on the type of bonding. Dipolar shifts occur only for complexes having magnetic anisotropy. In most complexes, both Av, and AD, contribute to the observed shift, but chemists have been able to design systems in which one of these two interactions dominates. For example, octahedral Ni(I1) and tetrahedral Co(I1) are magnetically isotropic, so Av, = 0. On the other hand, the unpaired electrons in lanthanide complexes are in deeply-buried 4f orbitals, so that delocalization onto ligands (and hence An,) is very small. Relaxation Processes The presence of an external magnetic field gives two energy levels for one unpaired electron. The lower energy level corresponds to the electron magnetic moment aligned with the field, and the upper energy level corresponds to the moment opposed to the field. As we shall show, on the average the level of lowest energy has an excess population of spins compared to the upper level. In this situation, one would expect to observe two extremely broad (vide infra) peaks in the nmr spectrum of a nucleus with nuclear spin 112 coupled to the electron, one of them arising from coupling with the up spin, the other from coupling with the down spin. The separation of the peaks would he given by Ai, the hyperfine coupling c o n ~ t a n tAi .~ is typically cm-' (kT at room temperature is 200 cm-I). The peak corresponding to the electron magnetic moment aligned with the field would he very slightly more intense, since this lower-energy state has a larger population. What is actually observed is a single peak, shifted much less than Ai (see Fig. 1). This is a result of electron relaxation or electron exchange, which is so rapid that the nucleus cannot respond to an "up" spin and a "down" spin, hut instead interacts with some sort of average of the two, giving rise to one nmr peak. The expected case of two shifted peaks has never been observed since for such slow electron relaxation the proton resonances are too broad to
Figure 1. (a) N M R of free ligand nucleus. ( b ) Hypothetic splitting and broadening of the resonance due to the two possible orientations of the unpaired electron. ( c ) Actual observed spectrum This scale is expanded SO that the ditference in the peak position from la) to ( c ) could be as much as several hundred parts per million.
Volume 5 1 . Number 6.June 1974 / 371
be discerned. This can be understood by a consideration of the lifetime of the excited state acrompanying an nmr transition (41. The factor influencing return to the ground state (relaxation) which is most important to the explanation of the spectra of paramagnetic complexes is the socalled spin-lattice relaxation process characterized by a time T I . This relaxation mechanism involves fluctuating magnetic fields which arise from molecular motion and which, when properly phased, cause the excited state to relax to the ground state in a non-radiative way. The extent of spin-lattice interaction depends on two factors: (1) the magnitude of local fields and ( 2 ) the rate of fluctuation of local fields. When the relaxation mechanism is very effective T I is very short; and, when the relaxation mechanism is not very effective, T I is very long. A short T I implies a very short lifetime of the excited state. The uncertainty principle states that AtAE
-
h
(2)
A short lifetime implies a small A t and consequently a large AE, that is, a large range of frequencies induce nmr transitions in this sample and a broad line results in the nmr spectrum. The converse of the above statement is also true. For most diamagnetic compounds, T I is long (on the order of several seconds) and thus sharp signals are observed. Paramagnetic substances are very efficient in causing relaxation of the excited nuclear spin states and, as a consequence, extremely broad lines are produced. The presence of the electron magnetic moment induces relaxation of the nuclear spin and nuclear magnetic moments and thus can broaden the resonances. Electrons relaxing with exactly the same frequency a s Ai are most efficient a t broadening the spectrum. At is typically about 108 Hz which is, stated another way, one cycle per sec. The rate of molecular tumbling in solution is characterized by a correlation time r, on the order of 10-I1 sec. The lifetime of an electron spin state is related to r,, the electron relaxation time, or T I , the electron exchange time (whichever is faster). re is a measure of the time taken for energy to he transferred to other degrees of freedom. T , is characteristic of the time s ~ e nhv t the liaand at a ~ a r t i c u At, the nucleus lar metal site. If 1/Tle > At o r l / r . will see only a population-weighted average of the two electron spin states. Furthermore, the nucleus will he less efficiently relaxed under these conditions. Thus, it is easy to see that the lifetime of the electron spin state necessary to yield nmr spectra ranges from 10-9 to 10-13 ~ e c . ~ An illustration of the effect of coordination to a paramagnetic metal ion upon the resonance spectmm of methyl amine is given in Figure 2 (5). In this octahedral Ni(II) complex, the magnetic anisotropy is zero and therefore the shift of the roto on resonance freauencies can he simply equated with the contact shifts. he fact that the two peaks shift in opposite directions is important in the interpretation of spin-delocalization mechanisms. For a detailed presentation of the fundamentals of nu-
>>
I -4716 Hz
I 0
I 56% Hz
Figure 2. On top is the nmr specturn of diamagnetic NH2CH3. Below is the contact shifted spectrum of Ni(NH2CHde2+ Note that some peaks shift far ~ p f i e i dwhile some shift far downfieid. The applied magnetic field here is 60 MHz.
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/ Journal of Chemical Education
clear magnetic resonance spectroscopy and elementary quantum mechanics, Carrington and McLachlan (6) is recommended. Average Electron Spin Polarization
As we have mentioned, if the electron spin relaxation is very slow, protons magnetically coupled to the electrons will he relaxed very efficiently giving rise to nmr lines too broad to be observed. If fast electron spin relaxation occurs, the coupled nuclei will not "feel" the electron spin states (aligned or anti-aligned), hut instead will react to an average electron spin which is called the "average electron spin polarization." In this section, we shall develop in detail the physics describing the average electron spin polarization for several cases important in chemistry and set the groundwork for later discussions of the types of coupling which may occur in molecules. When a molecule containing one or more unpaired electrons is placed in a strong static magnetic field, a net magnetization is induced in the molecule (71. The induced magnetization is a vector quantity either aligned with the static field or havine a resultant aliened with it. This induced magnetization, called paramagnetism, is ultimately. res~onsihlc for the larcc shifts observed in the . magnetic resonance spectra of che nuclei. The induced magnetization produces a new magnetic field in the vicinity of the molecule in addition to that pmdueed by the magnet of the spectrometer. The magnitude of the field at the point in space occupied by the nucleus we wish to study by nmr will thus he different from the field which would occur if the molecule were diamagnetic. It is the additional field produced by the induced magnetization which causes the isotropic shift. The magnitude of this additional field a t the nucleus will depend on the type of electron-nuclear coupling (scalar or dipolar) which in turn will depend upon the honding and geometry in the molecule. The scalar and dipolar mechanisms responsible for transmitting the magnetic effects of the electrons to nuclei in the molecule will he discussed in detail in later sections of this paper. Here we will devote our attention to the description of the origin and the calculation of the magnitude of the induced magnetization. For simplicity, we will devote our detailed discussion to a system containing one unpaired electron. We will indicate wherever necessary how the calculations may be extended to more than one electron. Furthermore, we assume a knowledge of the origin of magnetic resonance phenomena 18) and will only review enough detail to establish notation. For a more thorough discussion of magnetism, the reader is referred to reference (9). The magnetic moment of an electron r e is related to its spin S through eqn. (3). The quantity Be appearing in eqn. (3) is the Bohr magneton with magnitude eh/2mc and units of erg/G. m and e are, respectively, the mass and charge of the electron; h is Planck's constant divided by 2a; and c is the velocity of light. The quantity, g, called the electronic g factor, is unitless and for our purposes is defined by eqn. (3). The energy of interaction between the electron magnetic moment and a static magnetic field H is represented by the Hamiltonian Systems with slowly-relaxing electrons (117, 5 A,) have very broad nmr spectra. But, in the esr experiment, where one looks at electron spin transitions, this slow relaxation guarantees a longlived excited state, and thus (by eqn. (2)) a small line width. Thus, err peaks are sharp for these systems and broad for fasterrelaxing systems. It is, in fact, rare that one can perform an nmr and esr experiment on the same compound. The techniques are complementary.
The interaction produces two energy levels of energy gp,HS, (with S, = f 112). S, is often referred to as M,. It is the eigenvalue of the spin angular momentum S in the q direction. H is the magnitude of the magnetic field H. We need consider only the Zeeman levels; other perturbations such as hyperfine coupling arising from the nucleus are very small and may he neglected. When the system has attained thermal equilibrium, an excess of spins will exist in the lowest energy state. We may apply Boltzman statistics to find the ratio of the populations of alpha (high energy) to beta (low energy) spin states, given in eqn. (5). Nt' -- exp(-AElkT) = exp(-g@,.H/kT) (5) N,,
The usual Curie type approximation, valid in many systems of interest, is AE
