Equivalence between dynamical averaging methods of the Schrödinger equation: average Hamiltonian, secular averaging, and Van Vleck transformation

Volume 199, number $4

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6 November 1992

Equivalence between dynamical averaging methods of the SchrSdinger equation: average Hamiltonian, secular averaging, and Van Vleck transformation A. Llor ’ Direction des Sciences de la Mat&e, DRECUUSCM, Commissariat 6 I'Energie Atomique, Saclay, 91191 Gifsur Yvette Cedex, France

Received 4 June 1992; in tinal form 30 July 1992

To solve the Schri+dingerequation with a timedependent Hamiltonian, two main perturbation-expansion schemes arc available: secular averaging theory (SAT) and average Hamiltonian theory (AHT). Due to the lack ofan explicit relationship between SATand AHT, some discrepancies between the results yielded by these theories have been found, especially in NMR applications. A general equivalence scheme between these two methods is here given in operator form, using a new formulation of SATin terms of a static diagonalization expansion. The two approaches arc thusreconciled and the relationship with static techniques should eventually provide new and simpler convergence criteria for AHT.

1.Illtroductlon

The evolution of a quantum system under a timedependent Hamiltonian is an important general problem in various areas of physical chemistry [ 11. It is found in many diverse situations, such as atomic and molecular spectroscopies, NMR and EPR, or collisions and reactions (of atoms, molecules, surfaces, ...). Beyond the complexities due to the size of the system and the time modulation, solving the Schriidinger equation with a time-dependent Hamiltonian is a rather involved problem. For instance, the simple model of a two-state system (such as a spin l/2 in a magnetic field) interacting with a sinusoidal electromagnetic field has no simple analytical solution in general [2,3]. In many situations however, some assumptions on the time dependence of the Hamiltonian allow approximate solutions to be derived, usually as perturbation expansions. There Correspondence to: A. Llor, Direction des Sciences de la Mat&e, DRECAM/SCM, Commissariat a l’Energie Atomique, Saclay, 9 1191 Gif sur Yvette Cedex, France. ’ Present address: DCDM/RFX/CS, Commissariat a PEnergie Atomique, I-c Ripault, B.P. 16,37260 Monts, France.

are two main accessible cases: the adiabatic and the strong-collisionlimits, where the time dependence is respectively very slow or very fast compared to the size of the Hamiltonian [ 41. The general approach in treating the strong-collision limit is to describe the evolution of the system over long times with a time-independent Hamiltonian. Usually this effective Hamiltonian is accessible as a perturbation expansion of which only the first terms are preserved in practical calculations. The expansion can be derived in various different ways, but basically they can be reduced to two main approaches: the Floquet theory [ $61, also known as secular averaging theory (SAT), and the average Hamiltonian theory (AHT) [ 781. The former is a general technique developed over a century ago for solving differential equations, whilst the latter derives from more recent Lie-algebraic concepts. AHT has been the most widely used approach among NMR spectroscopists, especially to design selective decoupling and averaging techniques [ 9- 111, but SAT seemsto be preferred in most other fields [ l-3 1. One of the appealing features of AHT is its formulation in terms of operator commutators which, in many

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cases such as coherent averaging in NMR, provides a more concise and intuitive approach than SAT. However, some NMR experiments, such as sample rotation and pulse crafting [ 121, seem to be more conveniently described using SAT. Although SAT and AHT are diierent approximations of the same equation, the results they provide are somewhat incompatible in the analysis of various multiple-pulse sequences in NMR [ 13- 171. This has recently stirred some controversy over the appropriateness of the methods, as reviewed in ref. [ 181. Some ambiguities, reviewed in ref. [ 191, have also been encountered in NMR when treating the truncation, to second and higher orders, of the local interactions by the Zeeman coupling. All these difficulties can be traced to non-secular terms that appear in the effective Hamiltonian derived using AHT instead of SAT [ 14,191. The intrinsically stroboscopic nature of AHT is responsible for these terms, and, in many cases, it is also the reason for the somewhat cumbersome description of the system evolution derived from this method. Some important contributions have been made in reconciling the AHT and SAT approaches [ 14,15,18,19], but oddly, the relationship between these theories has not been given explicitly. The aim of this work is thus to give a clear relationship between these perturbative techniques. It is done by reformulating the SAT method in terms of a static blockdiagonalization procedure known as Van Vleck transformation [ 20 1. The effective Hamiltonians yielded by SAT or AI-ITare then shown to be equivalent. As in AHT, the perturbation expansion of SAT is also carried out in terms of operator commutators, following a procedure given recently by Goldman et al. [19].

found in many spectroscopic situations) or that the evolution is analyzed over a finite period of time, so H(t) can be expanded in a Fourier series, H(t) = C H, exp(imot) .

(2)

m

The so-called secularaveragingtheory (this name was first given in ref. [ lo] ) will provide a solution of the SchriSdingerequation by expanding the propagator in the following way:

V(t) = C Unr(t) exp(imwt) . m

(3)

This expansion is not unique because the components iJ,,,(t) also depend on time, but, in the strongcollision limit, we shall require the time dependence of the U,,,(t) to be slow compared to o (practically this means that U,,,(f) does not contain any frequency components beyond fw). Such an approach is common in solving differential equations in general [ 61, and it was already applied to the S&r& dinger equation in various situations [ 13,14,2I]. In these previous works, however, the quantum state or the density matrix of the system was considered instead of the propagator, so explicit closed forms of the results and simple relationships with other techniques such as AHT could not be given. Inserting expressions (2) and ( 3)into ( 1), we get the set of relations idU,,,fdt=mwU,,,+ C H,,,_,U,, , ”

(4)

where, for simplicity, we omit the explicit time dependence of the V,(t). Eq. (4) can be rewritten in matrix form as

2. Operator fomof the propagator in the secular averaging theory We start from the Schriidinger equation with a time-dependent Hamiltonian, H( t ), which describes the evolution of the propagator U(0, t) = u(t), idU(t)/dt=H(t)U(t),

(1)

with the initial condition U(0) -Id. We shall assume that the Hamiltonian is periodic in time (as 384

...

...

. . ZwldtH, .,. H-I .. H-2 ... H-3

j .I.

...

...

... ... Hz Ha Hz wIdtH0 H, H_, Ho H, H_-2 H_, -wIdtHO ... ... ... H,

or idU/dt=(wl+H)U,

.. ... ... 1.. 1..

..ii

... U, U, U, ' U_,

. I

(5)

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Yolume 199,number 3,4

where we introduced outlined-character notations for the block vectors (U ) and block operators (I and H ). The latter belong to the well-known Floquet-operator space [ 3,171, and must not be confused with socalled superoperators that act over the Liouville space. Notice that indices appended to a block-operator symbol wilI label block rows and block columns, and they may not coincide with actual indices of standard operators that could be found at the corresponding position of the block operator. The blockdiagonal operator I is defined as I,,= mid. The Floquet formalism can be considered as a semi-classical version of the dressed-statesdescription [ 1,21, where the Hamiltonians of both the system and the eleo tromagnetic field mode are combined. The Floquet description, in which the field energy has no lower bound, is valid only in the limit of large photon numbers, as in NMR applications (spontaneousemission effects cannot be obtained). It should be emphasized that a time-dependent Hamiltonian, H( t ), necessarily describes a system that is not isolated (the total energy is not constant), and the Floquet space provides the simplest formal mode1of the couplingwith the rest of the universe, through a “field mode”, mw, and a time-independent Hamiltonian, H. Formally, since H is now time independent, eq. (5) is exactly solved with a block-operator exponential, U(t)=exp[ -i(wl+H)t]

U(0)

=Wexp( -iDt) WtU(0) ,

(6)

where we introduced a block diagonalization of WI+ H = WDW. Traditionally, the Floquet Hamiltonian is completely diagonalized [l-3]. This provides a complete and unambiguousdescription of the system evolution, but requires the explicit knowledge of the Hamiltonian and a practical procedure to perform the diagonalization. These are rather de manding conditions that are seldom fulftied in real cases, where a block-diagonalization can be carried out in a much simpler way, at least formally. The Floquet Hamiltonian can be blockdiagonalized in many different ways, and some restrictions must thus be chosen to simplify the problem. As shown below, it is now possible, regardless of the detailed form of N(t), to constrain the procedure so as to eliminate

6 November 1992

the degrees of freedom of the field, and retain a description in the original Hilbert space with an effective static Hamiltonian. This is done in the following way: eq. (6) yields a solution of the Schriidinger equation in terms of ordinary operators by inserting the components of U into eq. (3), but the result is not very convenient in general and it is simplified by an appropriate choice of the block-diagonalizedform, WDWt. As shown in Appendix A, the blockdiagonalization can always be chosen to preserve the band structure of W, and to discard any o-frequency components in the central block of D. Then (see Appendix A), the following operators can be defined: D=Doo,

(7a)

W(t)= C W,Oexp(imwt), m

G’b)

and the propagator becomes V(t)=W(t)exp(-iDt)W(O)+.

(8)

D is not necessarily a fully diagonal operator (al-

though it is the central block of the block-diagonal operator D), and W(t) is a unitary operator. This simple expression is the non-expanded closed form of the secular averaging theory, and it is just restricted by the existence of a block-diagonalform for WIt H, regardlessof the strong-collisionassumption. D is an effective Hamiltonian which describes the evolution of the system, and W(t) can be viewed as an effective dynamical tilting of the eigenstatesof the system. An equivalent expression was already developed to obtain transition probabilities [ 2 1, eq. (22) 1, but it was expanded as a sum over a basis of quantum states, so the simplicity of the operator formalism (and of the perturbation expansions shown in the next section) was not obvious. It should be emphasized that the operators D and U;(t) are not uniquely defined: if V is a unitary transformation, then D’= YtDV and w’(t) = W(t) Y also decompose the propagator as in eq. (8). Such transformations arise when different blockdiagonalized forms of ol+ H are chosen.

3. Relationship between AHT and SAT, and WT perturbation expansions

AverageHamiltonian theory assumesthat the evo385

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lution over time t= T, the period of the Hamiltonian (or the finite evolution period under consideration in the case of a non-periodic time dependence), is described by a constant effective Hamiltonian, U(T)=exp(-i(H)T).

(9)

The existence of the average Hamiltonian, (H), is ensured in the case of Lie groups of unitary transformations (such as propagators) [ 221, even when the strong-collisionassumption is not fulfilled. In the strong-collision limit, a perturbation expansion can be given for (H) , known as the Magnus expansion [ 7,8,10,11]. To describe the evolution of the system at times 0 Q K T, the Floquet propagator, F(t), can be introduced [ 171, U(t)=F(t)

exp( -i(H)t).

(10)

F(t) is periodic in time, and P(0) =Id. It should be

noticed that (H) is not uniquely defined, since, in the diagonal basis, its eigenvalues are defined to within kw=2kx/Tonly [ 171 (where kis any integer number). This is also true with the effective Hamiltonian, D, given by SAT in eq. ( 8 ) . However, (H) can be uniquely defined, using a “folding transformation” that constrains its spectrum to be entirely inside the ? 10 interval [ 17,181, whereas D remains defined to within a unitary transformation, as shown in the previous section. It should also be noticed that, for a periodic modulation, (H) depends on the choice of a time (or phaser origin, whereas D remains unaffected by such a change [ 15] (only W(0) is then modified). The relationship between SAT and AHT is readily found when rewriting eq. (8) as U(t)=W(t)W(O)+exp[

-iw(O)Dw(O)tt]

,

(11)

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remarks on the nonuniqueness of D and (H), eq. ( 12a) can alwaysbe obtained after adding the proper kw terms to each of the eigenvalues of either (H) or D (this is similar to the folding transformation mentioned above). After being adapted according to this transformation, the F and Woperators fulfill eq. ( 12b), yielded when comparing again eqs. ( 10)and (11) foranyvalueoft. Eq. ( 12a) was previously obtained to second order only from the perturbation expansions of SAT and AHT [ 141, and some related arguments on the equivalence of effective Hamiltonians were also reported [ IS]. A full equivalence relationship between SAT and AHT Hamiltonians was also given in a special case [ 191, when analyzing the truncation of the local interactions in NMR where the time dependence is generated by the use of an interaction rep resentation (rotating frame). We shall now assume that the strong-collision assumption holds, in order to derive a perturbation expansion of SAT. The usual perturbation techniques that are found in the literature do not take advantage of the matrix formulation given in the previous section. However, it is an important feature since it reduces the problem to a static diagonalization expansion, Furthermore, the very specific band structure of the matrix in eq. (5) allows the Van Vleck expansion technique [20] to be applied as it was explicitly given by Goldman et al. [ 191. Let us expand the block operator H into its band components: H=;H,,

(H,),=J,+&&

(13a)

.

(13b)

An important commutation property of I with the H, follows:

and comparing to eq. ( 10). This yields (H)=W(O)DW(O)+,

(W

F(t)=W(t)W(O)‘,

(12b)

so the effeive Hamiltonians ofAHT and SAT are alwaysequivalent.To obtain eqs. (12), eqs. ( 10) and ( 11) are first compared at t =kT, where F(kT) = W( kT) W( O)t=Id, and from the equality of the exponentials, it is actually deduced that D and H are only equivalent to within kw differences on

their eigenvalues. However, according to the general 386

[I, H,]=mH,.

(14)

According to this relationship, I formally behaves as Zz, the longitudinal component of the angular momentum, and thus the band-structured operators H, are just the irreducible components of H with respect to the C, group generated by I. The Van Vleck transformation can then be expanded as it was previously shown in the case of a Zeeman interaction [ 191.The general procedure is outlined in Appendix B, and, to second order, it yields the following result:

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[H_,,H,]/mt...,

D=oltH,-(l/a)&

(ISa)

W=exp(iS),

(15b)

S=(i/o)

(15c)

2 H,/m+.... m+O

It is noticeable that the tilting operator in ,me Floquet space, W, is generated by a Hermitian operator S whose main block-diagonal cancels to any order (as can be seen in Appendix B). This means that, among all the possible blockdiagonal forms of the Floquet Hamiltonian, the Van Vleck transformation is optimal in mixing different field states while preserving the original states of the system. It is indeed appropriate in our approach which aims at providing a general description of the system without reference to the field modulation. Inserted into eqs. ( 7 ) , the expansions in eqs. ( 15) immediately yield

D=Ho-(I/o)~~o[H-“,Hnrl/m+..~, w(t)=Id-(l/w)

(W

C H,exp(-imot)/m+..., m+O (16b)

by taking into account the band structure of the H, operators (notice the negative signs of the exponents in eq. ( 16b) ) . Higher-order corrections, though more complicated, can be obtained using the same procedure described in Appendix B. By a different approach, a generalexpression of the higher-order terms was given for D, but not for W(t ) [ 101. In contrast to W, W(t) cannot be expanded in a logarithmicway, as in eqs. (15b) and (15~). Using eqs. ( 12), the expansion of SAT is readily translated for AHT in second order as:

+(1/w)

c [&,Kzl/~+..., m+O

(17a)

F(t)=Id + (l/w)

2 H,[ 1-exp( -imwt)]/m+...

.

m+O (17b)

In deriving these expansions, we made use of the general equivalence between SAT and AHT (eqs.

6 November 1992

( 12) ) which normally holds when no “abasing” is present between the eigenstates of (H) and D. This condition is ensured by an argument of continuity in the limit where H vanishes: to any order, the eigenvalues of both (H) and D converge to zero and are completely contained within + hw, so there is no aliasing in this limit. The expansions in eqs. ( 17), as thoseineqs. (16),canalsobegivenintermsoftime domain integrals, by introducing the explicit Fourier integrals of the H,,, components and regrouping the discrete sums over m. Eq. ( 17a) is then identical to the usual Magnus expansion [ 7,8]. Comparing the SAT and AHT results, a supple+ mentary non-secular term appears in the second-order effective Hamiltonian. This result was already pointed out [ 13,141, and it was considered to be a main reason to invalidate the use of AHT in the interpretation of many NMR experiments [ 141. However, the general equivalence between the AHT and SATaveragingschemes,given in eq. ( 12), clearly shows that the choice between these methods is just a matter of convenience. Since the AHT and SAT expansions are respectively given in terms of time integrals [ 10,111 and Fourier coefficient combinations, the kind of time dependence of the Hamiltonian determines which of the methods must be adopted to solve a specific problem. In NMR applications for instance, the AHT approach is often appropriate (when dealing with delta- or neardeltapulse sequences [9-l 1I), although it may prove rather awkward in many cases (continuous sample motion, truncation [ 191, thermalization processes [ 181, ...). An interesting consequence of the Van Vleck expansion in eqs. (16) and (17) is that it provides a new approach to analyze the convergence of the Magnus expansion. So far, necessary criteria for con‘vergencewere given as conditions on the eigenvalues of the average Hamiltonian itself [ 7 1, which is rather impractical. In some exactly soluble cases [ 8,17,18,23], such as a two-state system with a sinusoidal excitation, the convergence condition could be directly given on H(t). The SAT approach for these situations involves only two Fourier components of the Hamiltonian, Ho and Hcl, and the convergence criterion translates as a simple constraint on the m= f 1 terms. In the general case, it can be expected that algebraic conditions on diagonaliza387

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tion expansions [ 241 will provide new and simpler convergence criteria given as constraints over the Fourier components of H( t ) .

4. Conclusion Secular averaging theory (SAT) was shown to be equivalent to a static diagonalization procedure in the Floquet space. In this new formulation, a simple operator form of SAT was derived, allowing the equivalence with the average Hamiltonian theory (AHT) to be explicitly given. The effective interactions in both cases are equivalent, and the choice between the two techniques is a matter of convenience, depending, for instance, on the type of time modulation of the Hamiltonian. The perturbation expansion of SAT was also carried out in terms of operator commutators using a static procedure, and this method can be conveniently applied, in place of AHT, in many NMR situations. Previously reported discrepancies between the results obtained from the two methods should be eventually irrelevant [ 13181. Furthermore, this approach provides a simple series of successiverelationships that connect different perturbative expansion procedures ( WT, SAT,AI-IT and the Dyson expansion [ 81). The convergence conditions of these methods can now be explored using static techniques of matrix perturbation theory [ 231. These aspects are currently being investigated.

Acknowledgement The author is deeply endebted towards P.J. Grandinetti (University of California, Berkeley, USA) and M. Goldman (Commissariat a l’Energie Atomique, Saclay, France) for very stimulating and enlightening discussions.

Appendix A

We shall derive the SAT solution to the Schriidinger equation, as given in eqs. (7) and (8), starting from the block-operator form in eq. (6). From eq. (6), the components Urn(t) can be written as 388

6 November 1992

t&(f)= C W, exp( -iD,,t)W$&(O) w

.

(A.1 1

To zeroth order in H, we get D,,=pwId (WIt H is already diagonal if H= 0). To keep the time dependence of all the U,,,slow as in eq. (3), we must thus restrict the sum in eq. (A. 1) to the terms containing p = 0 only. This limits necessarily the choice for the initial U,(O) so that c wt,, U*(O)=dPoX, 4

(A.2)

where, at this stage, X can be any operator. Inverting this expression we thus get U,(O)=W,X.

(A.3)

Now, we know that U( 0) = Id, and since

U(O)= ; U,(O)= (;W@)K

(A.4)

we have

(A.9 Actually, the matrix X can be made unitary by an appropriate choice of the diagonalization procedure. The diagonalization can be chosen so that the block matrix W retains the general band structure found in eq. (5), so it is invariant under shifting of the blocks: W m+k,n

+ k =w,,

.

(A.6)

Indeed, this can always be achieved. Let us assume for instance that W’ diagonalizes WIt H into D’, but does not fulfill condition (A.6). Then, we can construct W in the following way: W,, =Wm-n,O ,

(‘4.7)

so condition (A.6) holds for W. Now W diagonalinto D, where D,,=pw+D&,. Using eq. (A.6), the matrix X, as given in eq. (AS), is found to be unitary: izes WI + H

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=

p%+*ow-40= F;w+wLi,-s

= 3 w,-SW-S.-d= 5 (~)d,-d = T Id,_,=Id.

(‘4.8)

Last, using the definitions of W(t) and D as in eqs. (7), it is found that X= W(O), and combining with eqs. (A.l) and (A.2), the final result of eq. (8) is obtained. Since U(t) and X= W(0) are unitary, eq. (8) also shows that W(t) is unitary at all times.

AppendixB

The perturbation expansion of the Van Vleck transformation, given in eqs. ( 15), can be derived from eqs. (13) and (14) according to the general procedure described in ref. [ 191. The main steps are summarized as follows. The perturbation expansions of D and W are written as D=oltD”‘tD~Z~tD~3~t...,

(B.la)

W=exp(iS),

(B.lb)

s=s”‘ts’2’ts(~~t...,

(B.lc)

where the magnitude of the terms labeled by (n) scales as 1H 1”. The blockdiagonalization relationship, D=Wt(ol tH)W, becomes oltD’1’tD’2’+ = (Id-iS(‘)-

... &(L)2-iS2)t

... )(wltH)

~(ld+iS~‘~-~S~‘~~+iS~~~+...),

(B.2)

where contributions of same order are equated according to D(‘)=Htw[l,iS(‘)],

(B.3a)

DC2)=[H, is(‘)] t io2[ [I, iS2)], iS(2)] +w[ I, iSC2)],

(B.3b)

Higher-order terms are obtained in a similar way, and S”) appears only as a commutator with I in the nth-

6 November1992

order expression. These equations are rather difficult to solve in the general case, but the expansion of the Hamiltonian and the commutation relationship of its components, in eqs. ( 13) and ( 14), allow to derive a simple iterative procedure to give DC”) and St”) in terms of nested commutators of H,. H is expanded in H, components according to eq. ( 13). From eq. ( 14), the following relationship is easily derived: [l,(I!Hmi)]=(Fm,)(l!H,,).

(B-4)

The fact that a given product (or nested commutator) of H,, is block diagonal is just shown by the condition Ckmk=O. In eq. (B.3a), D(l) has to be block diagonal, and thus, in the right-hand side of the equation, the diagonal part of H,, can be retained. The offdiagonal parts of H must then be canceled by specifically Tayloring the commutator [I, is(‘)] using eq. ( 14). The result in first order is found in eq. ( 15). Higher-order corrections are obtained in a similar way: in the nth-order equation for DC”),S@) always appears as a [I, is(“)] commutator which is designed to cancel all the off-diagonal parts due to the other terms by using eq. (B.4).

[ 1] KC. Kulander, ed, Time-dependent methods for quantum dynamics (Elsevier, Amsterdam, 1991). [ 21 D.R. Dion and J.O. Him&elder, in: Advances inChemical Physics, Vol. 35, eds. I. Prigogine and S. Rice (Wiley, New York, 1976) p. 265. _ [ 31J.H. Shirley, Phys. Rev. B 138 (1965) 979. [4] A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1982); J.J. Sakurai, Modern quantum mechanics (Benjamin/ Cummings, Menlo Park, 1985). [5]G.Ploquet,A~.E~.Norm. Sup. 12 (1883) 47. [ 6 ] V.S.Bogolyubovand G.A. Mitropolski,Asymptotic methods in the theory of nonlinear oscillations (Gordon and Breach, London, 1961); V.I. Arnold, Ordinary differential equations (MIT Press, Cambridge, 1978); A.H. Nayfeh, Introduction to perturbation techniques (Wiley, New York, 1981); V.N. Bogaevski and A. Povzner, Algebraic methods in nonlinear perturbation theory (Springer, Berlin, I 991). [7] W. Magnus, Commun. Pure Appl. Math. 7 (1954) 649.

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[ 81 P. Pechukasand J.C. Light,J. Chem. Phys. 44 (1966) 3897; D.P. Burum, Phys. Rev. B 24 ( 1981) 3684; W.R. Salzman, J. Chem. Phys. 85 (1986) 4605; S. Klarsfeld and J.A. Gteo, Phys. Rev. A 39 (1989) 3270. [91 J.S. Waugh, L.M. Huber and V. Haeberlen, Phys. Rev. Letters20 (1968) 180; U. Haeberlen and J.S. Waugh, Phys. Rev. 175 (1968) 453. [ lo] M. Mehriog Principles of high resolution NMR in solids, 2nd Ed. (Springer, Berlin, 1976) . [ 111LJ.Haeberlen, in: Advances in Magnetic Resonance, Suppl. I, ed. J.S. Waugh (Academic Press, NewYork, 1976). [ 121D.B. Zax, G. Goelman, D. Abramovich and S. Ve8a, in: Advancesin MagneticResonance,Vol. 14,eds. W.S.Warren (Academic Press, New York, 1990); A. Schmidt and S. Vega,J. Chem. Phys. 96 (1992) 2655. [ 131L.N. Emfeev and B.A. Shumm, Pis’ma Zh. Eksp. Tear. Fiz. 27 (1978) 161 [JETPLetten 27 (1978) 1491; Yu.N. Ivanov, B.N. Provotorov and E.B. Feldman, Pis’ma Zh. Eksp.Teor. Fii. 27 (1978) 164 [JETPLetters27 (1978) 1531; L.N. Erofeev, B.A. Shumm and G.B. Manelis, Zh. Eksp. Teor. Fii. 75 (1978) 1837 [Soviet Phys. JETP 48 (1978) 9251; Yu.N. Ivanov, B.N. Pmvotorov and E.B. Fel’dman, Zh. Ekap. Tear. Fiz. 75 (1978) 1847 [Soviet Phys. JETP 48 (1978) 9301; L.L. Buishvili and M.G. Menabde, Zh. Eksp.‘Teor.Fiz. 77 (1979) 2436 [SovietPhys. JETP 50 (1979) 11761; B.N. Provotorov and E.B. Feldman, Zh. Eksp. Teor. Fiz. 79 ( 1980) 2206 [SovietPhys. JETP 52 (1980) 11161; M.M. Maricq, Phys. Rev. B 25 (1982) 6622.

390

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[ 141L.L. Buishvili, G.V. Kobakhidze and M.G. Men&de, Zh. Eksp. Teor. Fiz. 83 (1982) 628 [Soviet Phys. JETP 56 (1982) 3471. [IS] E.B. Fel’dman, AK Hitrin and B.N. Provotorov, Phys. LettersA (1983) 114. [ 161E.B. Fel’dman, Pis’ma Zh. Eksp. Teor. Fiz. 39 (1984) 537 [JETP Letters 39 (1984) 6571; E.B. Feldman and K.T. Summaoen, Phys. Stat. Sol. (b) 127 (1985) 509; E.B. Fel’dman, Phys. Letters A 104 (1984) 479. [ 171M.M. Maricq, J. Chem. Phys. 85 (1986) 5167; 86 (1987) 5647. [ I8 ] M.M. Maricq, in: Advances in Magnetic Resonance, Vol. 14,ed. W.S. Warren (Acedemic Press, New York, 1990). [ 191M. Goldman, P.J. Grandinetti, A. Llor, Z. Olejniczak,J.R. Sachlebenand J.W. Zwanziger,J. Chem. Phys., in press. [20] J.H. Van Vleck,Phys. Rev. 33 (1929) 467. [ 2 1 ] S.I. Chu, in: Advances in Atomic and Molecular Physics, Vol. 2 I, cds. D.R. Bates and I. Estermann (Academic Press, NewYork, 1985) p. 197. [22] J. Wei and E. Norman, J. Math. Phys. 4 (1963) 575; J. Wei, J. Math. Phys. 4 ( 1963) 1337; R.M. Wilcox,J. Math. Phys. 8 (1967) 962. [ 231W.R. Salzman, Phys. Rev. A 36 (1987) 5074; EM. Fernandez, J. Echave and E.A. Castro, Chem. Phys. 117 (1987) 101; FM. FernBndez,J. Chem. Phys. 88 (1988) 490; EM. FernBndez,Phys. Rev. A 41 (1990) 2311. [24] G.W. Stewart and J.-G. Sun, Matrix perturbation theory (Academic Press, New York, 1990).