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The addition of a damping term to the stochastic Liouville equation and its significance for line shape calculations in magnetic resonance
JOURNAL
OF MAGNETIC
RESONANCE
13,260-267
(19%)
The Addition of a Damping Term to the Stochastic LiouviNe Elquation and Its Significancefor Line ShapeCalculationsin Magnetic Resonance ALEXANDER
J. VEGA AND DANIEL
FIAT
Department of Structural Chemistry, The Weizmann Institute of Science, Rehovot, Israel Received August 27, 1973 This paper discusses some consequences of the modification of Kubo’s stochastic Liouville equation, which the authors proposed in a recent publication. The modified equation contains a damping term in addition to the original equation, such that the approach of the spin system to thermal equilibrium with the lattice can now be described by the equation. Due to the lack of this property, the unmodified equation could best be applied mainly to line shape calculations of nonsaturated spectra. It is shown that these calculations are actually based on the assumption of the modified equation and that the extension to saturated spectra can now readily be established. I. INTRODUCTION
The stochastic Liouville method is a powerful tool for the calculations of spectral line shapes (I-6). The method is applicable to dynamic systems which are randomly modulated, provided that the stochastic modulation is a Markoffian process. In the particular case of magnetic resonance, we deal with spin systems subject to a Hamiltonian ttX(G!), which is a function of a set of random variables Q. For instance, Q denotes the orientation of the molecules which carry the spins. The time evolution of the distribution function p(sZ, t) is assumed to be described by the Markoffian process (Wt)l@&
f) = QqA t).
VI
Kubo (7) has shown that this equation leads to an equation of motion for the Qdependent density matrix p(Q), bw> &-A t> = Mm ~E”(Q)l + UP@). PI This is the stochastic Liouville equation (SLE). The density matrix is defined in such a way that the average expectation value of any Q-dependent spin property P(Q) is given by (P) = J+dntr p(Q)P(Q). Thus, p(Q) contains information on the state of the spins and on the orientational distribution of the molecules. Equation [2] is a differential equation which has a well defined solution of p(Q) for any &’ (constant or time dependent) and for all kinds of initial conditions. Therefore, Copyright 8 1974 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
260
LINE
SHAPE
CALCULATIONS
261
we would expect that the SLE would in principle describe the time evolution of a spin property P in any experiment we perform on the spins, Nevertheless, the apphcation of the SLE has been mainly restricted to the calculations of nonsaturated line shapes. The reason for that is that Eq. [2] is incomplete in the sense that it does not describe the approach of the spin system to thermal equilibrium. Therefore, the equation is of no use in situations where the deviations of p(Q) from equilibrium are not very small. Only with the introduction of an additional ad hoc assumption could saturation effects be calculated in a few cases (2). Moreover, a detailed analysis of the method (presented in this paper) reveals that even in the linear response theory Eq. [2], as it stands, does not lead to the desired results, and that certain complementary assumptions concerning the equilibrium state of p(D) are needed. In recent publications (8,9) the authors showed that the equilibrium density matrix is the Q-dependent Boltzmann density matrix P&9 =PomA-‘u - ~~tsz)/m l4I where p. is the equilibrium distribution of the molecular orientations and A is the number of quantum states of the spin system. Equation [43 was proved using a quantum mechanical description of the lattice, under conditions of high temperature. Using the same formalism the SLE was rederived (9). This derivation justified the addition of an extra term to the equation:
tvt) Pw4t) = GJ(Q,t), wQ)l+ mG4 f) - PJLW This equation also holds for a time dependent Hamiltonian, pB the time dependent Boltzmann density matrix
PI
provided that we take for
p&i?, t) =po A-‘{1 - hAq-2, t)/kT).
I31 The modified SLE has the desired property that it describes the approach to thermal equilibrium. Moreover, we expect that it has the universal applicability which Eq. [2] lacked. Recently, Roberts and Lynden-Bell introduced the modified SLE as an ad hoc assumption, and applied it to the line shape calculation in a special case (4). In this paper we present a general discussion of the consequences of the addition of the damping term for the application of the SLE to line shape calculations. II. NO SATURATION
A. Linear Response Theory
For the computation of the shapes of nonsaturated lines we have to calculate the first order magnetic response of the spin system to a perturbing external field H,(t). This is the general problem of linear response theory, which is usually solved following one of two approaches. The first is the relaxation function method (20) and the second is a more direct application of the perturbation method (N-12). We shall use here the first method, because it shows more directly the desirability of the addition of the damping term to the SLE. The relaxation function is the function which describes the time dependent response of the system, starting at t = 0, when an infinitely long applied constant perturbing field is suddenly turned off. Let the perturbing field H, be applied in the cl-direction
262
VEGA
and let the measured response M,(t) relaxation function is then defined as
AND
FIAT
be the magnetization
in the P-direction. The
%a(0 = ff,YW/JW - @47>es>, t71 where (M),, is the magnetization in equilibrium. According to linear response theory (20) the imaginary part of the susceptibility is given by &(o)
= 0 7 a&(t) coscot dt.
PI
0
We compute the relaxation function by solving the modified SLE @/at) P@%t> = -iW3 P@&t) + G-@, 0 - p&4), where we introduced the R-dependent Liouville operator
PI [lOI
d?(n) = [X(s1), . . .].
X(s2) is the Hamiltonian in the absence of the perturbing field and pB(s2) is given by Eq. [4]. Note that 6p(sl)p&2) = 0. Equation [9] must be solved with the initial condition P(Q2,O)= PBG-4+ QoG9 Km Hl, [I13 where q stands for 1/AkT. Here we include the possibility of an Q-dependent magnetization operator M, e.g., in the case of an anisotropic g-tensor (13). The equation of motion, for the deviation of p(Q) from equilibrium d&&t) = P(Q 0 - h(QL is (a/i%) Ap(s1, t) = -iLf’Ap + TAp, WI with the initial condition A~04 0) = ClpoW) MQ)
H,.
El31
Its solution is Ap(62, t) = exp [(-i9
+ r) t] Ap(sZ, 0).
Cl43
From this we have for the /?-component of the total magnetization Of,&))
=
+ NfdQtr%
Ap(Q, 0,
[IV
where N is the concentration of the spins. Thus, we find for the line shape function X;;,(w) = dvq J- cosot J- ds1tr M, exp [(-iLY + r) t ]po M, dt.
1161
The Hermitic properties of 2 and r allow us to rewrite this as x;&4
= o@l/
cos ot j- dQpo tr it4,. M&2, t) dt,
El71
= exp [(iZ -t F) t] M&2).
WI
where M&2,
t)
This expression is the solution of
WO WQ t>= @f(Q),J$(Q,t>l + rN@, t),
WI
with the initial condition &?(a 0) = w?m*
PO1
LINE
SHAPE
263
CALCULATIONS
Note that Eq. [17] has the usual form of the Fourier transform of a correlation function (20-12).
The evaluation of &(o) is most conveniently done with the technique of Laplace transformations (I, 3). Such a calculation gives for Eqs. [I 61 and [17], respectively, J&(W) = wNq Re = wNq Re
s s
& tr M&o
+ i5? - T)-lpO M,
dQp, tr M,(iw - i9 - lT’
MB.
l3E
The equations [16]-[21] are well known in the literature, and they have been used in several cases to calculate unsaturated line shapes (I, 3). However, they were not shown to be a consequence of the modified SLE. Their justification was instead found in the undamped equation
ap/at= -iYp
+ rp.
P4
But since this equation has no meaningful equilibrium solution, it is not clear what one should take for the initial condition p(B,O). Hence, the relaxation function method is difficult to apply to Eq. [22]. Nevertheless, we now try to solve this problem by assuming the existence of an equilibrium p&2) in the absence of N1, leaving as undecided which form it should take. p(s2,O) can then be taken as P(Q, 0) = Peq+ dP(O) = Peq+ qpo Ml HI,
1231
and integration of Eq. [22] gives
PW,t) = exp[t--j2 + r) Up,, + &@93.
1241
We note further that exp [(-i9 + r) t ]p,, can be replaced by Peg,because the equilibrium state should be constant in time. This reduces Eq. [24] to Eq. [14], and we are led to the final equations [16]-[21]. Indeed, in the current stochastic line shape theory one uses the unmodi&d SLE with the initial condition p(O) =qp,M,H, in order to calculate the correlation function
We obtain more insight into the significance of the damping term when we consider the line shape problem from the following point of view. In the presence of a linearly polarized rf field in the a-direction, the rate equation is
ap/at= -Up
- W’[p,
Mel] H,
cos
wt -k rp.
PaI
This is the SLE without the damping term. The modified equation takes the form ap/& = -i9p
- ii?‘[p, Ma] HI
cos
wt + r(p - pB),
Wbl
where for simplicity we take M, independent of 8. Following a method proposed by Freed (3), we expand the steady state solution in a Fourier series p(L?, t) = 2 p”“)(sZ) exp (inwt). n
WI
264
VEGA
AND
FIAT
Hermiticity of p implies that p (- “) - p (“)+. Substitution in Eq. [25] leads to a set of coupled equations for the Fourier components p (“). In linear response the only relevant equations are those for n = 0, +l (0) + rp’o’ = 0
-iTp
PM
or -i9p’“’
Wbl
+ r(p’O’ - pB) = 0,
and rtiop’*”
- Up (fl) - +&l[p(o),
M,] Hl + rp(*l)
= 0.
WI
Equations [27a, b] correspond to Eqs. [25a, b], respectively, and Eq. [28] applies to both cases. The solution of Eq. [28] is P (*l) = -&HI
iT1(+io + iLY - T)-‘[p(O), M,],
WI
where p(O) still remains to be determined. From the modified SLE it follows directly with Eq. [27b] that p(O)= ps. This leads to the same expressions for the line shape as those derived with the relaxation function method. To obtain these expressions we write for Eq. [29] p(*‘) = iHl(2AkT)-l(+iw
+ i9 - T)+p,[X(s2),
M,].
[301
This is substituted into the line shape function
and with some mathematics, this can be shown to be identical to Eq. [21]. On the other hand, the solution of Eq. [27a] is useless, because this is just the normalized unity matrix po(S2)/A, which commutes with M,. Hence, if we prefer to work with the unmodified equation we are compelled to make an independent assumption about p(O)and to ignore Eq. [27a]. The best choice for this assumption is obviously p(O)@) = p&2). But another choice could in principle also be possible (8). For instance, p(O)(Q) = j&, where pjJ = po A-‘(1 - ti.%&T), [311 X0 being the average of X(Q) over Sz.This choice (the “high field approximation”) is justified if I%‘(Q) - soI 4 ]tiol, because then the replacement of p&2) by & changes the commutator in Eq. [29] only to a very small amount. Actually, in most magnetic resonance line shape calculations, which follow a method similar to the present one, use is made of the high field approximation (IS, 16). But in cases where the Q-dependent part of the Hamiltonian is of the order of the magnitude of X0 (e.g. in triplet molecules with a large zero field splitting (3), in low field resonance (17), and in pure quadrupole resonance (18)) it is essential to take p(O)= pB(sZ). * We demonstrate the effect of the high field approximation on the calculated spectrum in two simple cases, which, although they are of little practical significance, can be considered as traditional examples for the visualization of the method (I, 19). Both examples deal with a model in which spins jump between two equally populated sites a and b with different magnetic fields H, and Hb. In the first example, x:Jo) is calculated for H, = $Hb = ko,/y. Figure 1 shows the accurate line shape together with the line shape calculated under the high field approximation, for different values
LINE SHAPE CALCULATIONS
I
265
I
I
c
/
I’
I /I
1
\
‘\
/’
2 --- A’
-
‘.
--’
‘\ ‘.
‘--...--
\\ \
,/’
‘\\\
I I
---__ I 2
w/w, FIG. 1. Calculated magnetic resonance line shapes for spins jumping between two equaHy populated sites a and b, for different values of the mean residence time t. H.//H, and U,, = I&,/2 = c&y. The rf field is perpendicular to H. and Hb. accurate calculation; ---- high field approximation.
I I
I 4 w/w,
FIG.
H*.
2. Same as Fig. 1. H. 1 Hb and I&I=
/I&l = z/z q,/y. The rf field is perpendicular
to H. and
266
VEGA
AND
FIAT
of the mean residence time z. In the other example (Fig. 2) the fields H, and Hb are equal in magnitude but perpendicular to each other: H, = (k + i)o& and H, = (k - i) oO/y. The line shape function is xi, (CO); thus, the rf field is perpendicular to both local fields. Figures 1 and 2 exhibit an o/o,, ratio between the two line shape functions. This property is also encountered in the well known solutions of the modified Bloch equations in low fields (20), where the susceptibility is proportional to o rather than to CO,,.As we shall point out in a forthcoming publication, the modified Bloch equations in low fields also follow from the modified SLE, albeit under conditions of extreme narrowing. III.
SATURATION
For the study of saturation effects we cannot use the linear response theory, so we have to follow the method of Fourier expansion of the density matrix. Indeed, Freed introduced this method in order to be able to include the rf field explicitly (2). Thus, we have the following set of equations for the Fourier components of p(s1,t) (-in0 - iLz(c2)) p(R)(8) - J&l[p’“-”
(Q) + p@+l) (l-2),iv,] Hl + rp’“‘(0) = hdw 6, o. ~321 The right side results from the damping term and vanishes if we use the unmodified SLE. Note, that the problem reduces significantly if
for then pt2) is very small and its effect on the solution of p(l) is negligible. This occurs in the case of high fields, i.e., when ml, I%‘p(sZ)- &‘,I -$ Idol (2). We are then left with only three equations -i9p’O’ - Jgrqp’ + p(l), M,] H, + l-p(O) = rpB, -iq+ - iYp (0 - Jjh-1 [p(O),iv, ] Hl + rp = 0,
WI PJbl
and the equation for p’-l) = p”“. Again, the unmodified SLE yields a vanishing steady state solution. Simple neglect of Eq. [33a] and an additional independent assumption about p(O), in analogy to the no-saturation case, do not help us in this case, because the very effect of saturation appears in the H1 term in Eq. [33a]. Nevertheless, Freed calculated saturated spectra from the undamped SLE (2). Let us analyze his method. Define x(Q) = Pm - PI?, [341 where Pe is the average Boltzmann density matrix, defined in Eq. [31]. The SLE can then be written as
ax/at= -iYx
- iY&
+ i[x, %,&)I
+ iDB, H&t)]
+ TX.
1351 Solution of this equation gives obviously x = -pB. In particular, the equations for the components of x(52, t) appear to contain relaxation terms which tend to relax the zero Fourier component of x to -&. TO overcome this problem, the &hoc assumption is made (2) that x should be replaced by x - & in such relaxation terms only. This ensures relaxation of x towards zero (which is the supposed equilibrium state in the high field approximation). Such a replacement is equivalent to the omission of the
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SHAPE
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267
term -iZji, in Eq. [35]. In this way we obtain an equation of motion which is assumed to properly describe the line shape behavior. Comparison with the modified SLE for 4 = P - P&x,
Wt> 4 = --iPAp+ ii& Xdt)l+ ih(Q), *,&>I + TAP,
[361 shows that the two equations are nearly equivalent. The only difference is that in the second commutator p,, replaces p,(Q). But this difference is negligible in the high field case. Thus, we found the fundamental justification of Freed’s ad hoc assumption. In conclusion we can say, that the stochastic Liouville method is now applicable to line shape calculations for a very wide range of experimental conditions. In principle, no restrictions are required with respect to the rf power or the order of magnitude of the fluctuating local Hamiltonian. In view of the recently observed saturation effects on line shapes in ESR (22) and in NMR (22) this generalization of the method is seen to have useful application. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. IO. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Ku~o, Adu. Gem. Phys. 15,101 (1969). H. FREED, G. V. BRUNO, AND C. F. POLNASZEK, J. Phys. Chem. 75,3385 (1971). H. FREED, G. V. BRUNO, AND C. F. POLNASZEK, J. Gem. Phys. 55,527O (1971). ROBERTS AND R. M. LYNDEN-BELL, Mol. Phys. 21,689 (1971). A. GOLDMAN, G. V. BRUNO, C. F. POLNASZEK, AND J. H. FREED, J. Chem. Phys. 56,716 (1972). C. MCCALLEY, E. J. SHIMSHICK, AND H. M. MCCONNELL, Chem. Phys. Lat. 13,115 (1972). KUBO, J. Phys. Sot. Japan 25, Suppl. 1 (1969). J. VEGA AND D. FIAT, Pure Appl. Chem. 32,307 (1972). J. VEGA AND D. FIAT, J. Chem. Phys. in press (1974). Kuw, AND K. TOMITA, J. Phys. Sot. Jppan 9,888 (1954). Kum, “Lectures in Theoretical Physics,” Vol. I, p. 120, Interscience, New York, 1959. ABRAGAM, “The Principles of Nuclear Magnetism,” p. 100, Ciarendon Press, Oxford, 1961. ABRAGAM AND B. BLEANEY, “Electron Paramagoetic Resonance,” p. 136, Clarendon Press, Oxford, 1970. This is apparent, e.g., in Eqs. [45] and [46] of Ref. (I), and in the application to x”,, in Ref. (.I). H. M. MCCONNELL, J. Chem. Phys. 28,430 (1958). C. S. JOHNSON, JR., Adv. Magn. Resonance 1,33 (1965). R. Kum AND T. TOYABE, “Magnetic Resonance and Relaxation”, p. 810, Proc. XIV-i&me Colloque Amp&e, North Holland Publ. Co., Amsterdam, 1967. S. ALEXANDER AND A. TZALMONA, Phys. Rev. 138, A845 (1965). Ref. (12), p. 478. Ref. (12), pp. 53, 516. J. S. HYDE, L. E. G. ERIKSSON, AND A. EHRENBERG, Biochim. Biophys. Acta2=, 688 (1970). R. K. HARRIS AND K. M. WORVILL, Chem. Phys. Lett. 14,598 (1972).
R. J. J. J. S. R. R. A. A. R. R. A. A.
OF MAGNETIC
RESONANCE
13,260-267
(19%)
The Addition of a Damping Term to the Stochastic LiouviNe Elquation and Its Significancefor Line ShapeCalculationsin Magnetic Resonance ALEXANDER
J. VEGA AND DANIEL
FIAT
Department of Structural Chemistry, The Weizmann Institute of Science, Rehovot, Israel Received August 27, 1973 This paper discusses some consequences of the modification of Kubo’s stochastic Liouville equation, which the authors proposed in a recent publication. The modified equation contains a damping term in addition to the original equation, such that the approach of the spin system to thermal equilibrium with the lattice can now be described by the equation. Due to the lack of this property, the unmodified equation could best be applied mainly to line shape calculations of nonsaturated spectra. It is shown that these calculations are actually based on the assumption of the modified equation and that the extension to saturated spectra can now readily be established. I. INTRODUCTION
The stochastic Liouville method is a powerful tool for the calculations of spectral line shapes (I-6). The method is applicable to dynamic systems which are randomly modulated, provided that the stochastic modulation is a Markoffian process. In the particular case of magnetic resonance, we deal with spin systems subject to a Hamiltonian ttX(G!), which is a function of a set of random variables Q. For instance, Q denotes the orientation of the molecules which carry the spins. The time evolution of the distribution function p(sZ, t) is assumed to be described by the Markoffian process (Wt)l@&
f) = QqA t).
VI
Kubo (7) has shown that this equation leads to an equation of motion for the Qdependent density matrix p(Q), bw> &-A t> = Mm ~E”(Q)l + UP@). PI This is the stochastic Liouville equation (SLE). The density matrix is defined in such a way that the average expectation value of any Q-dependent spin property P(Q) is given by (P) = J+dntr p(Q)P(Q). Thus, p(Q) contains information on the state of the spins and on the orientational distribution of the molecules. Equation [2] is a differential equation which has a well defined solution of p(Q) for any &’ (constant or time dependent) and for all kinds of initial conditions. Therefore, Copyright 8 1974 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
260
LINE
SHAPE
CALCULATIONS
261
we would expect that the SLE would in principle describe the time evolution of a spin property P in any experiment we perform on the spins, Nevertheless, the apphcation of the SLE has been mainly restricted to the calculations of nonsaturated line shapes. The reason for that is that Eq. [2] is incomplete in the sense that it does not describe the approach of the spin system to thermal equilibrium. Therefore, the equation is of no use in situations where the deviations of p(Q) from equilibrium are not very small. Only with the introduction of an additional ad hoc assumption could saturation effects be calculated in a few cases (2). Moreover, a detailed analysis of the method (presented in this paper) reveals that even in the linear response theory Eq. [2], as it stands, does not lead to the desired results, and that certain complementary assumptions concerning the equilibrium state of p(D) are needed. In recent publications (8,9) the authors showed that the equilibrium density matrix is the Q-dependent Boltzmann density matrix P&9 =PomA-‘u - ~~tsz)/m l4I where p. is the equilibrium distribution of the molecular orientations and A is the number of quantum states of the spin system. Equation [43 was proved using a quantum mechanical description of the lattice, under conditions of high temperature. Using the same formalism the SLE was rederived (9). This derivation justified the addition of an extra term to the equation:
tvt) Pw4t) = GJ(Q,t), wQ)l+ mG4 f) - PJLW This equation also holds for a time dependent Hamiltonian, pB the time dependent Boltzmann density matrix
PI
provided that we take for
p&i?, t) =po A-‘{1 - hAq-2, t)/kT).
I31 The modified SLE has the desired property that it describes the approach to thermal equilibrium. Moreover, we expect that it has the universal applicability which Eq. [2] lacked. Recently, Roberts and Lynden-Bell introduced the modified SLE as an ad hoc assumption, and applied it to the line shape calculation in a special case (4). In this paper we present a general discussion of the consequences of the addition of the damping term for the application of the SLE to line shape calculations. II. NO SATURATION
A. Linear Response Theory
For the computation of the shapes of nonsaturated lines we have to calculate the first order magnetic response of the spin system to a perturbing external field H,(t). This is the general problem of linear response theory, which is usually solved following one of two approaches. The first is the relaxation function method (20) and the second is a more direct application of the perturbation method (N-12). We shall use here the first method, because it shows more directly the desirability of the addition of the damping term to the SLE. The relaxation function is the function which describes the time dependent response of the system, starting at t = 0, when an infinitely long applied constant perturbing field is suddenly turned off. Let the perturbing field H, be applied in the cl-direction
262
VEGA
and let the measured response M,(t) relaxation function is then defined as
AND
FIAT
be the magnetization
in the P-direction. The
%a(0 = ff,YW/JW - @47>es>, t71 where (M),, is the magnetization in equilibrium. According to linear response theory (20) the imaginary part of the susceptibility is given by &(o)
= 0 7 a&(t) coscot dt.
PI
0
We compute the relaxation function by solving the modified SLE @/at) P@%t> = -iW3 P@&t) + G-@, 0 - p&4), where we introduced the R-dependent Liouville operator
PI [lOI
d?(n) = [X(s1), . . .].
X(s2) is the Hamiltonian in the absence of the perturbing field and pB(s2) is given by Eq. [4]. Note that 6p(sl)p&2) = 0. Equation [9] must be solved with the initial condition P(Q2,O)= PBG-4+ QoG9 Km Hl, [I13 where q stands for 1/AkT. Here we include the possibility of an Q-dependent magnetization operator M, e.g., in the case of an anisotropic g-tensor (13). The equation of motion, for the deviation of p(Q) from equilibrium d&&t) = P(Q 0 - h(QL is (a/i%) Ap(s1, t) = -iLf’Ap + TAp, WI with the initial condition A~04 0) = ClpoW) MQ)
H,.
El31
Its solution is Ap(62, t) = exp [(-i9
+ r) t] Ap(sZ, 0).
Cl43
From this we have for the /?-component of the total magnetization Of,&))
=
+ NfdQtr%
Ap(Q, 0,
[IV
where N is the concentration of the spins. Thus, we find for the line shape function X;;,(w) = dvq J- cosot J- ds1tr M, exp [(-iLY + r) t ]po M, dt.
1161
The Hermitic properties of 2 and r allow us to rewrite this as x;&4
= o@l/
cos ot j- dQpo tr it4,. M&2, t) dt,
El71
= exp [(iZ -t F) t] M&2).
WI
where M&2,
t)
This expression is the solution of
WO WQ t>= @f(Q),J$(Q,t>l + rN@, t),
WI
with the initial condition &?(a 0) = w?m*
PO1
LINE
SHAPE
263
CALCULATIONS
Note that Eq. [17] has the usual form of the Fourier transform of a correlation function (20-12).
The evaluation of &(o) is most conveniently done with the technique of Laplace transformations (I, 3). Such a calculation gives for Eqs. [I 61 and [17], respectively, J&(W) = wNq Re = wNq Re
s s
& tr M&o
+ i5? - T)-lpO M,
dQp, tr M,(iw - i9 - lT’
MB.
l3E
The equations [16]-[21] are well known in the literature, and they have been used in several cases to calculate unsaturated line shapes (I, 3). However, they were not shown to be a consequence of the modified SLE. Their justification was instead found in the undamped equation
ap/at= -iYp
+ rp.
P4
But since this equation has no meaningful equilibrium solution, it is not clear what one should take for the initial condition p(B,O). Hence, the relaxation function method is difficult to apply to Eq. [22]. Nevertheless, we now try to solve this problem by assuming the existence of an equilibrium p&2) in the absence of N1, leaving as undecided which form it should take. p(s2,O) can then be taken as P(Q, 0) = Peq+ dP(O) = Peq+ qpo Ml HI,
1231
and integration of Eq. [22] gives
PW,t) = exp[t--j2 + r) Up,, + &@93.
1241
We note further that exp [(-i9 + r) t ]p,, can be replaced by Peg,because the equilibrium state should be constant in time. This reduces Eq. [24] to Eq. [14], and we are led to the final equations [16]-[21]. Indeed, in the current stochastic line shape theory one uses the unmodi&d SLE with the initial condition p(O) =qp,M,H, in order to calculate the correlation function
We obtain more insight into the significance of the damping term when we consider the line shape problem from the following point of view. In the presence of a linearly polarized rf field in the a-direction, the rate equation is
ap/at= -Up
- W’[p,
Mel] H,
cos
wt -k rp.
PaI
This is the SLE without the damping term. The modified equation takes the form ap/& = -i9p
- ii?‘[p, Ma] HI
cos
wt + r(p - pB),
Wbl
where for simplicity we take M, independent of 8. Following a method proposed by Freed (3), we expand the steady state solution in a Fourier series p(L?, t) = 2 p”“)(sZ) exp (inwt). n
WI
264
VEGA
AND
FIAT
Hermiticity of p implies that p (- “) - p (“)+. Substitution in Eq. [25] leads to a set of coupled equations for the Fourier components p (“). In linear response the only relevant equations are those for n = 0, +l (0) + rp’o’ = 0
-iTp
PM
or -i9p’“’
Wbl
+ r(p’O’ - pB) = 0,
and rtiop’*”
- Up (fl) - +&l[p(o),
M,] Hl + rp(*l)
= 0.
WI
Equations [27a, b] correspond to Eqs. [25a, b], respectively, and Eq. [28] applies to both cases. The solution of Eq. [28] is P (*l) = -&HI
iT1(+io + iLY - T)-‘[p(O), M,],
WI
where p(O) still remains to be determined. From the modified SLE it follows directly with Eq. [27b] that p(O)= ps. This leads to the same expressions for the line shape as those derived with the relaxation function method. To obtain these expressions we write for Eq. [29] p(*‘) = iHl(2AkT)-l(+iw
+ i9 - T)+p,[X(s2),
M,].
[301
This is substituted into the line shape function
and with some mathematics, this can be shown to be identical to Eq. [21]. On the other hand, the solution of Eq. [27a] is useless, because this is just the normalized unity matrix po(S2)/A, which commutes with M,. Hence, if we prefer to work with the unmodified equation we are compelled to make an independent assumption about p(O)and to ignore Eq. [27a]. The best choice for this assumption is obviously p(O)@) = p&2). But another choice could in principle also be possible (8). For instance, p(O)(Q) = j&, where pjJ = po A-‘(1 - ti.%&T), [311 X0 being the average of X(Q) over Sz.This choice (the “high field approximation”) is justified if I%‘(Q) - soI 4 ]tiol, because then the replacement of p&2) by & changes the commutator in Eq. [29] only to a very small amount. Actually, in most magnetic resonance line shape calculations, which follow a method similar to the present one, use is made of the high field approximation (IS, 16). But in cases where the Q-dependent part of the Hamiltonian is of the order of the magnitude of X0 (e.g. in triplet molecules with a large zero field splitting (3), in low field resonance (17), and in pure quadrupole resonance (18)) it is essential to take p(O)= pB(sZ). * We demonstrate the effect of the high field approximation on the calculated spectrum in two simple cases, which, although they are of little practical significance, can be considered as traditional examples for the visualization of the method (I, 19). Both examples deal with a model in which spins jump between two equally populated sites a and b with different magnetic fields H, and Hb. In the first example, x:Jo) is calculated for H, = $Hb = ko,/y. Figure 1 shows the accurate line shape together with the line shape calculated under the high field approximation, for different values
LINE SHAPE CALCULATIONS
I
265
I
I
c
/
I’
I /I
1
\
‘\
/’
2 --- A’
-
‘.
--’
‘\ ‘.
‘--...--
\\ \
,/’
‘\\\
I I
---__ I 2
w/w, FIG. 1. Calculated magnetic resonance line shapes for spins jumping between two equaHy populated sites a and b, for different values of the mean residence time t. H.//H, and U,, = I&,/2 = c&y. The rf field is perpendicular to H. and Hb. accurate calculation; ---- high field approximation.
I I
I 4 w/w,
FIG.
H*.
2. Same as Fig. 1. H. 1 Hb and I&I=
/I&l = z/z q,/y. The rf field is perpendicular
to H. and
266
VEGA
AND
FIAT
of the mean residence time z. In the other example (Fig. 2) the fields H, and Hb are equal in magnitude but perpendicular to each other: H, = (k + i)o& and H, = (k - i) oO/y. The line shape function is xi, (CO); thus, the rf field is perpendicular to both local fields. Figures 1 and 2 exhibit an o/o,, ratio between the two line shape functions. This property is also encountered in the well known solutions of the modified Bloch equations in low fields (20), where the susceptibility is proportional to o rather than to CO,,.As we shall point out in a forthcoming publication, the modified Bloch equations in low fields also follow from the modified SLE, albeit under conditions of extreme narrowing. III.
SATURATION
For the study of saturation effects we cannot use the linear response theory, so we have to follow the method of Fourier expansion of the density matrix. Indeed, Freed introduced this method in order to be able to include the rf field explicitly (2). Thus, we have the following set of equations for the Fourier components of p(s1,t) (-in0 - iLz(c2)) p(R)(8) - J&l[p’“-”
(Q) + p@+l) (l-2),iv,] Hl + rp’“‘(0) = hdw 6, o. ~321 The right side results from the damping term and vanishes if we use the unmodified SLE. Note, that the problem reduces significantly if
for then pt2) is very small and its effect on the solution of p(l) is negligible. This occurs in the case of high fields, i.e., when ml, I%‘p(sZ)- &‘,I -$ Idol (2). We are then left with only three equations -i9p’O’ - Jgrqp’ + p(l), M,] H, + l-p(O) = rpB, -iq+ - iYp (0 - Jjh-1 [p(O),iv, ] Hl + rp = 0,
WI PJbl
and the equation for p’-l) = p”“. Again, the unmodified SLE yields a vanishing steady state solution. Simple neglect of Eq. [33a] and an additional independent assumption about p(O), in analogy to the no-saturation case, do not help us in this case, because the very effect of saturation appears in the H1 term in Eq. [33a]. Nevertheless, Freed calculated saturated spectra from the undamped SLE (2). Let us analyze his method. Define x(Q) = Pm - PI?, [341 where Pe is the average Boltzmann density matrix, defined in Eq. [31]. The SLE can then be written as
ax/at= -iYx
- iY&
+ i[x, %,&)I
+ iDB, H&t)]
+ TX.
1351 Solution of this equation gives obviously x = -pB. In particular, the equations for the components of x(52, t) appear to contain relaxation terms which tend to relax the zero Fourier component of x to -&. TO overcome this problem, the &hoc assumption is made (2) that x should be replaced by x - & in such relaxation terms only. This ensures relaxation of x towards zero (which is the supposed equilibrium state in the high field approximation). Such a replacement is equivalent to the omission of the
LINE
SHAPE
CALCULATIONS
267
term -iZji, in Eq. [35]. In this way we obtain an equation of motion which is assumed to properly describe the line shape behavior. Comparison with the modified SLE for 4 = P - P&x,
Wt> 4 = --iPAp+ ii& Xdt)l+ ih(Q), *,&>I + TAP,
[361 shows that the two equations are nearly equivalent. The only difference is that in the second commutator p,, replaces p,(Q). But this difference is negligible in the high field case. Thus, we found the fundamental justification of Freed’s ad hoc assumption. In conclusion we can say, that the stochastic Liouville method is now applicable to line shape calculations for a very wide range of experimental conditions. In principle, no restrictions are required with respect to the rf power or the order of magnitude of the fluctuating local Hamiltonian. In view of the recently observed saturation effects on line shapes in ESR (22) and in NMR (22) this generalization of the method is seen to have useful application. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. IO. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Ku~o, Adu. Gem. Phys. 15,101 (1969). H. FREED, G. V. BRUNO, AND C. F. POLNASZEK, J. Phys. Chem. 75,3385 (1971). H. FREED, G. V. BRUNO, AND C. F. POLNASZEK, J. Gem. Phys. 55,527O (1971). ROBERTS AND R. M. LYNDEN-BELL, Mol. Phys. 21,689 (1971). A. GOLDMAN, G. V. BRUNO, C. F. POLNASZEK, AND J. H. FREED, J. Chem. Phys. 56,716 (1972). C. MCCALLEY, E. J. SHIMSHICK, AND H. M. MCCONNELL, Chem. Phys. Lat. 13,115 (1972). KUBO, J. Phys. Sot. Japan 25, Suppl. 1 (1969). J. VEGA AND D. FIAT, Pure Appl. Chem. 32,307 (1972). J. VEGA AND D. FIAT, J. Chem. Phys. in press (1974). Kuw, AND K. TOMITA, J. Phys. Sot. Jppan 9,888 (1954). Kum, “Lectures in Theoretical Physics,” Vol. I, p. 120, Interscience, New York, 1959. ABRAGAM, “The Principles of Nuclear Magnetism,” p. 100, Ciarendon Press, Oxford, 1961. ABRAGAM AND B. BLEANEY, “Electron Paramagoetic Resonance,” p. 136, Clarendon Press, Oxford, 1970. This is apparent, e.g., in Eqs. [45] and [46] of Ref. (I), and in the application to x”,, in Ref. (.I). H. M. MCCONNELL, J. Chem. Phys. 28,430 (1958). C. S. JOHNSON, JR., Adv. Magn. Resonance 1,33 (1965). R. Kum AND T. TOYABE, “Magnetic Resonance and Relaxation”, p. 810, Proc. XIV-i&me Colloque Amp&e, North Holland Publ. Co., Amsterdam, 1967. S. ALEXANDER AND A. TZALMONA, Phys. Rev. 138, A845 (1965). Ref. (12), p. 478. Ref. (12), pp. 53, 516. J. S. HYDE, L. E. G. ERIKSSON, AND A. EHRENBERG, Biochim. Biophys. Acta2=, 688 (1970). R. K. HARRIS AND K. M. WORVILL, Chem. Phys. Lett. 14,598 (1972).
R. J. J. J. S. R. R. A. A. R. R. A. A.