J. Phys. Chem. 1981, 85, 29-35
but the requisite I(7fgd)I would also be smaller than estimated above. Spectral evidence for and against the assignment of the origin of the 3B, state 1250 cm-l above the 3A, state can be noted. First, the TI So spectrum of both DMO-h6 and -de broadens somewhat to higher energy of the supposed 3B, origin, consistent with a second state in the region.27 However, since the S1 So origin is only -800 cm-l to higher energy, the T1 So system cannot be followed beyond the perturbation to any large extent. Further, the broadening may as well be associated with overlapping vibronic bands and Fermi resonances. Secondly, the isotope effect on the energy of the -0 + 1250
--
-
+
(27)R. M.Hochstrasser and C. Marzzaco, J. Chem. Phys., 49,971 (1968).
29
-
cm-l band is more readily explained in terms of a vibronic band of the 3A, 'A system. For the assignment to the 3B, origin, a shift w i d perdeuteration of 40 cm-l to lower energy would be required. We conclude that the available evidence does not warrant the identification of this feature as the origin of the 3B, state and that the 3B, state is low-lying but slightly above the 'A, state in DMO. Based on the model, this gives 2 X lo3 cm-' 5 E(3B,) - E(3A,) 5 5 x io3 cm-l. Acknowledgment. We acknowledge support for this work by the National Science Foundation under Grant CHE-77-11839 and by the Research Corporation. We also thank Professor H. Hope for providing us with a refined structure for DMO at 77 K.
Spin-Lattice Relaxation in the Tightly Coupled ABX Spin Systemt Andre Thevand, Guy Pourard, Universite de Provence, Centre de St. Jerome, Laboratoire de Chlmle Organique et Methodes Spectroscoplques, 13397 Marseille, Csdex 4, France
and Larry Werbelow" Chemistry Department, New Mexico School of Mines. Socorro, New Mexico 87801 (Received: August 12, 1980)
A magnetization mode formalism is introduced to quantify the perturbation-response characteristics of the longitudinal magnetization for the much studied ABX spin system. This formalism yields a highly illustrative and suggestive kinetic description for this three-spin one-half system. To render these expressions more practical in context of the FT NMR experiment, we discuss the influence of nonselective sampling pulses on the various magnetization modes. It is rationalized that, for certain flip angles, many of the multiplet asymmetry modes have zero intensity regardless of the initial perturbation employed. However, this does not imply that important spin correlations associated with these multiplet asymmetries can be neglected.
Introduction Simple kinetic arguments suggest that for a system comprising n-spin one-half nuclei, each diagonal element of the spin density operator (population) responds to a perturbation in the prescribed manner P-1
Pj(t) = ajo+ C aij exp(-Ajt) j=l
(1)
where Pi(t) denotes the population of state li) at time t. The preexponential coefficients, aij, are nonlinear admixtures of initial spin preparation and spectral density terms, whereas the characteristic time constants, Aj, are complicated combinations of the various spectal density terms alone. Even for a three-spin system, the apparent complexity of eq 1 obviates characterization and subsequent microscopic interpretations of the multidimensional spectrum of time constants. However, there exist certain symmetries between the various parameters appearing in eq 1which provide adequate simplification and commensurate compensation for the researcher who successfully weathers this superficial complexity. These simplifying symmetries, 'This work has been submitted by A.T. in partial fulfillment of the requirements for the degree of Doctorates Sciences, Universite de Provence, Marseille, France. 0022-3654/81/2085-0029$01 .OO/O
unveiled by Zeidler' and explained by Pypeq2 provide the rationale for magnetization mode descriptions3 of longitudinal magnetization. Recently, such concepts have been applied to the experimental investigation of weakly coupled first-order spin systems such as AX, AMX, AX2and AXQ.4 U;lfortunately, many homonuclear spin systems of chemical interest do not exhibit such spectral simplicity. In these instances, it might at first appear that the concepts of virtual spin5and magnetization mode analysis lack illustrative significance. Thus, in systems where strong but incomplete mixing of direct product eigenbasis occurs, the population kinetics are generally parameterized by initial rates of individual spectral components by either selective or nonselective inversion-recoveryexperimentsS6 (1) M. D. Zeidler, Ber. Bumenges. Phys. Chem., 72,481 (1968). (2)N. C.Pyper, Mol. Phys., 21, 1 (1971);22,433 (1972). (3)L. G. Werbelow and D. M. Grant, J. Chem. Phys., 63,544(1975); L. G. Werbelow and D. M. Grant, Adu. Magn. Reson., 9,189(1977),and references cited therein;L. G. Werbelow, D. M. Grant, E. P. Black, and J. M. Courtieu, J. Che. Phys., 69,2407 (1978). (4)For example: P. E. Fagerness, D. M. Grant, K. F. Kuhlmann, C. L. Mayne, and R. B. Parry, J. Chem. Phys., 63, 2525 (1975);J. M. Courtieu, D. M. Grant, and C. L. Mayne, ibid., 66, 2669 (1977);C. L. Mayne, D. W. Alderman, and D. M. Grant, ibid., 63,2514(1975);V. A. Daragan, Dokl. Akad. Nauk SSSR, 232,114 (1977). (5)L. G: Werbelow, G. Pouzard, and A. Thevand, J. Chim. Phys. Phys.-Chzm. Btol., 76,722 (1979).
0 1981 American Chemical Society
30
The Journal of Physical Chemistry, Vol. 85, No. 1, 1981 5.8
Thevand et ai. TABLE I: Relevant Zero Projections of Spherical Tensor Operators for the ABX and Analogous Analogous A , X and AMX SDin Svstems
2.5 111.3
3.7
ABX
AMX
Flgure 1. Idealized ABX spectrum characterized by the following parameters: (wA - 0,)/(27r) = JAB,Jm = 8 Hz, and JBx= -4 Hz. The various single quantum transitions between the spin states (see eq 2) are noted.
Such experiments result in linearized approximations to eq 1. Although theoretically sound, such an approach does not exploit any underlying kinetic symmetry. Furthermore, for a system of n-spin one-half nuclei, the number of well-resolved spectral components often exceeds the 2" - 1 independent characteristic time constants. The presence of these kinetic redundancies may unduly complicate the reduction of the perturbation-response data. Most importantly, analysis of initial recovery rates of individual multiplet components is quite difficult and often obscures recognition and hinders assessment of important multispin and related interference effects. It will be demonstrated that a magnetization mode analysis may, for certain spin systems such as for the ABX system considered in detail in this work, circumvent many of these conventional shortcomings. Of more general importance, it will be shown how such a description provides an optimal basis for discussion of nonequilibrium mixing effects induced by nonzero flip angles in the FT NMR inversion-recovery experiment.
Theory One may conveniently discuss the properties of the ABX spin system in terms of the diagonalized eigenbasis7 11)
I
I+++)
12)
I++-)
17)
I--+)
IS).: I---)
21/213)E (1 2'/'14)
1
+ C+)1/21+-+) + (1 - C+)1/21-++)
(2)
(1 + C+)'/21-++) - (1 - C+)'/21+-+)
+
21/2)5)E (1 C-)'/'I+--) 2'1216) I (1 C-)'I21-+-)
+
- (1 - C)'/'I-+-)
+ (1 - C-)'/21+--)
The direct product states are defined by the ordering,) .1 = J I ~ I ~ In I ~general, ). only states 11) and IS) and 12) and 17) are related by spin inversion symmetry. In the special limit where the scalar coupling constant, J M and J B X , are equal, all states are characterized by good spin inversion properties. The coefficients introduced in eq 2 are defined by the expressions
c'
=
[(WA - LOB)
T(JAX
- JBX)]/D*
(3)
where D* = [((LOA - LOB) f T ( J -~JBX))' ~ T ~ J ~ ' ] ' / ' and LO^ is the intrinsic Larmor frequency, yiBo(l - ui), of spin "i", An idealized ABX spectrum is presented in Figure 1. Values for C+ and C- have been arbitrarily chosen (0.85 and 0.37, respectively). The various single quantum transitions are labeled for future reference.
+
(6) R. Freeman, H. D. Hill, B. L. Tominson, and L. D. Hall, J.Chem. Phys., 61,4466 (1974);A. Briguet, J. C. Duplan, and J. Delman, J.Phys. (Paris),36,897 (1975);W.W.Bovee, Thesis, Delft University, 1975;D. Canet, Mol. Phys., 36,1731 (1978);H. Nery and D. Canet, ibid., 35,231
-,.
f\_". 1 R7R)
(7)J. Pople, W. Schneider, and H. Bernstein, "High Resolution NMR", McGraw-Hill, New York, 1959.
It is easily rationalized that if spins A and B are not equivalent, in principle it should prove possible to monitor seven independent kinetic measurables corresponding to the seven magnetization modes defined as
+ L86 + Lj7 + A88 + L85 + Lp3 + L18 + ~ 5 2 7 ) Y8 = 2'/'A.(L16 + Lpz + L85 + L86 + Ly8 + L85) v8 = 21/2(Lp2+ Lq8 - L85 - L$ - L86 - L95) = (Lp4 - L87)/ (1 s+) + (L86 - L18)/ (1 - 8-) + (L!7 - L&J/(1 + s+)+ (LE8 - L85)/(1 + s-) ut = (L94 - L17)/ (1 - 8 ') - (L86 L88) / (1 - s-)+ (L87 - LP3)/(1 + s+)- (LE8 - L86)/(1 + s-) = (Lp4 - L27)/ (1 - s+) + (L86 - L88)/ (1 - s-) 6% - L93)/ (1 + s+)- (Lt8 - L85)/ (1 + s-1 vy = Lp4 + L87 - Lg8 - L85 + Lp3 - LO8 + L97 (4) Up
= A(Lp4
V8
-
where S* = (1 - (C*)2)1/2, L t is the intrinsic line intensity associated with the multiplet component corresponding to the single quantum transition between states li) and b). The term, A, signifies deviation from thermal equilibrium. The various intensities noted in the above expressions are proportional to the weighted population difference between the states in question Lp4 = k(1 - S+)(Pi- P4) L&3 k ( l - s-)(P2- P6) L87 = k(1 + S+)(P3- P7) L88 = k(1 s-)(P,- P8) Lp3 = k(1 S+)(P,- P3) Q5 = k(1 S-)(P, - P5) L97 = k(1 - s+)(P4- P7) L& = k ( l - s-)(P6 - p8)
+
+ +
Lpz = k'(P1 - P2) L98 = k'(P7 - P8) Lo6 = YZk'(1 - c+c-- s+s-)(Pg - P6) LB5 = Y2k'(l + c+c-+ S+S-)(P3- P5) = %k'(1 + c+c-+ s+s-)(P4 - Pc) L!j5= Y2k'(l - c+c-- S+S-)(P4- PJ
(5)
The proportionality constants, k and k', are functions of various instrumental factors and gyromagnetic ratios. For homonuclear spin systems, k = k'. These magnetization modes are similar in structure to the "basic z magnetizations" defined by Bain.8 Operator equivalents of these linear combinations of state populations are provided in Table I. (8)A. D. Bain, J.Mugn. Reson., 37,209 (1980).
Spln-Lattice Relaxation in the ABX Spin System
The Journal of Physical Chemistry, Vol. 85, No. 1, 1981 31
The measurables, uy and vi, correspond to the summed intensity of the A% and X multiplets minus the respective thermal equilibrium values. The measurable, u;, is identified as the intensity of the two principal components minus the intensity of the four minor components of the X multiplet. This variable has a vanishing equilibrium value. The measurable, u:, corresponds to a weighted intensity difference between “A-type” and “B-type” transitions.’ The equilibrium value for this variable vanishes since Y A = Y B . If one decomposes the AB part of the ABX spectrum into a superposition of two AB quartets, the remaining identifications can be readily made: ug is the weighted intensity difference between A-type and B-type transitions for one quartet minus the relative difference for the other quartet; ut is the weighted intensity asymmetry of the A-type transitions minus the intensity asymmetry of the B-type transitions; and v i is the summed intensity of one of the quartets minus the summed intensity of the other quartet. Each of these latter variables is characterized by a vanishing equilibrium value. Unfortunately, only four of the modes can be conveniently defined in a manner that renders them independent of coupling strength (C*). In this system, it has been noted that there are seven independent time constants and, therefore, several degrees of kinetic redundancy. This is manifest in the realization that various modes are amenable to alternative correspondences. For example, v! is proportional to the asymmetry of the principal components of the X sextet, L!2 L!8. Utilizing these defining expressions,one can express the perturbation-response characteristics of the ABX spin system in the matrix form (-d/dt)uo(t) = I’uO(t) = QWQ-lQ(P(t) - P ( m ) ) (6) In this expression, P(t) is the vector of state populations, W is the characteristic transition probability matrix, and Q is the transformation matrix implicit in eq 4 and 5. Assuming that all multiplet components are well resolved (no partially overlapping lines) and extreme narrowing obtains, one is able to straightforwardly deduce the elements of the system’s evolution matrix, r, from the published l i t e r a t ~ r e . ~Even ~ ’ ~ with these simplifying assumptions, it is necessary to specify 10 unique spectral densities for the ABX spin system whenever considering dipolar and random-field-like interactions. These spectral densities include three random field terms, j A , j B , and j,, where ji
= (~;)~/2J~(b,(i,t)b,(i,O)) dt
The notation incorporated in these expressions is standard.3 If we further introduce the following simplifications
we can write the elements of the evolution matrix, A, in the form given by eq 8, 8a, and 8b. A more detailed description of the transient characteristics of the ABX longitudinal magnetizations, appropriate for frequency-dependent spectral densities, is available upon request from the authors. Equation 6 and 8 are correct for describing the timedependent intensities of various combinations of multiplet components in the conventional slow-passage experiment. Recent studies11J2have clearly demonstrated that caution must be exercised whenever the perturbation-response characteristics of coupled spin systems are analyzed with Fourier-transform methodologies. This is especially important if one is interested in the information available from multiplet asymmetry studies. As noted in eq 4 and 5, the various magnetization modes are defined in terms of intrinsic or adiabatic line intensities. Each of these intensities, L$, is simply related to a single population difference, Pi - Pj. However, in the Fourier-transform experiment, the intensities of the various lines, L p , generally cannot be described by a single population difference. For example, for a nonselective sampling pulse (flip angle a),the observable intensity of component (coherence)L F is given by the following complicated expression \
S+)L!, + (1- cos ff)2
L88
+
(74
one random field correlation term
k = ( Y A Y B ) / ~ S , - ( ~ , ( A , ~ ) ~ , ( Bdt, ~ ) ) three dipolar terms, J A X ,
JBX,
(7b)
and JAB
dt (74 and three dipolar correlation terms, KmX,KBm, and KKijk = ( 6 ~ / 5 ) ( ~ ; ~ j h / r x ;j) Ji;
= ( 6 ~ / 5 ) ( ~ i y j h / r ; ) ~ Jq ~( Q( i j ( t ) ) $ ( Q i j ( O ) ) )
(-y;Ykh/r$) ~~
Jm(
B ( Q i j ( t ) ) E ( Q j h ( 0 ) ) ) dt ( 7 4
~~~
(9) R.L. Vold and R. R. Vold, Prog. Nucl. Magn. Reson. Spectrosc. 12, 79 (1978). (10) D. Canet, H. Nery, and J. Brondeau, J. Chern. Phys., 70, 2098 (1979).
where cos2 4 = 1/2(1+ C+C- + S’S-) and cos’ $ = 1/2(1 C Y - - S+S-). It is seen that the observing pulse mixes the various (adiabatic) multiplet intensities. The degree of mixing is dependent upon the coupling strength (C*) but, most importantly, upon the observing flip angle, a. As would be expected,” a nonselective a-t-a free induction decay perturbation-response experiment does not mix the line intensities between the AB and X multiplets. Of course there is strong mixing within each of these
+
(11) S. Schaublin, A. Hohener, and R. R. Ernst, J. Magn. Reson., 13, 196 (1974). (12) A. D. Bain and J. Martin, J.Magn. Reson., 29, 125 (1978); R. E. D. McClung and N. R. Krishra, ibid., 29,573 (1978);G . Bcdenhausen and R. Freeman, ibid., 36, 221 (1979).
32
The Journal of Physical Chemistty, Vol. 85, No. 1, 1981
-d/dt
r45 =
-d/dt
-d/dt
Thevand et ai.
The Journal of Physical Chemistry, Vol. 85, No. 1, 198 1 33
Spin-Lattice Relaxation in the ABX Spin System
n
P
0 rl
v I
0% -1
02 -I
OS -1
v)
0-1 0
co
O-1h
0s -1
e
----;------
r-----
J E 0
I
c
rn
I
-I* 11
I
I
1
td*
tA2
td%
L?
tAE
tAS I
34
Thevand et al.
The Journal of Physical Chemistty, Vol. 85, No. 1, 198 1
multiplet manifolds as described by eq loa and lob. The matrix defined in eq 10a has the following (spectral) symmetry: -~ -
{ ;?}
~~
=
(1 + cos CY)Z
{E;}
+
The implications of eq 8 and 10 are now discussed in greater detail. Discussion Inspection of eq 8 immediately reveals numerous simplifying factorizations. Likewise, a number of limiting cases, such as the previously discussed AX23and AMX4 spin systems, may be deduced from these general expressions. It is important to note that, even for the most general case, v!, vg, and vl do not directly couple into v{ and v! (unless dipolar-shift anisotropy interferences are considered). However, these two manifolds will be indirectly coupled via modes v: and v:. Fortunately, only in highly unique situations where a(JM - J B X ) (WA - wg) 2 a J ~ does this indirect coupling prove to be important. Furthermore, if dipolar interferences (cross-correlations) are negligible and the direct product eigenstate coupling is weak, vg decouples from vy, vi, and vi.4 If dipolar interferences are negli ible and the direct product eigenstates couple strongly, v and vi are effectively decoupled from ut and For nonselective excitations or semiselective excitations (where one multiplet is nonselectively perturbed and the other multiplet is not disturbed), only vy(0) and/or vi(0) will be nonzero. Expressions 8 show very clearly that, in these instances, the total longitudinal magnetization of the AB and X multiplets evolve according to conventional biexponential predictions for times small compared to 1/K, l/KAXBand l/(Jb -I-j,). Of course for times small compared to 1/J , semiselective excitations render the response
-
5
-
of each total multiplet magnetization exponential in time. Straightforward “initial” rate studies such as these permit the quick and simple determination of J,j,, and (5JAB 2.i) . If dipolar interference terms are significant of if selective excitations are employed, many more details can be abstracted from this system’s perturbation-response characteristics. Furthermore, for systems where 0 < C+,C- < 1,variable field studies will permit the determination of terms such as J,, j,, and Kb without resorting to tedious numerical fits. However, the most interesting aspects of this “mode” description of the longitudinal magnetizations appear when the effect of the sampling pulse in the Fourier-transform experiment is considered. To obtain the intrinsic (adiabatic) intensities of the various multiplet components from the observable intensities in the FT experiment, it is necessary to numerically invert the matrices defined in eq 10. The resultant expressions for individual multiplet intensities are exceedingly complex for the general case. However, transformation to linear combinations of intensities (modes) results in the very simple identifications shown in eq 11. In these expressions, it is important to note that = v? and = vl regardless of the magnitude of flip angle. Furthermore, v” as a 0. But consider in more detail the effect of employing a flip angle of 90°. Now, it proves impossible to measure the multiplet asymmetry defined by vYT regardless of spin preparation. Furthermore, the time evolution of and viT is independent of the magnitudes of vi and vi, respectively, and de ends solely upon the (instantaneous) values for vl and v5. I t is impossible to sample the information content of modes ut, v{, and us for a flip angle of 90°. Modes and v$ are admixtures of v i and vi. However, if IC’ - C-1
