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Computation of magnetic resonance spectra from the stochastic liouville equation with padé approximants
CHEMICAL PHYSKS
Volume 88. number 2
COMPUTATION
30 Apnl 1982
LRTCRS
OF MAGNETIC RESONANCE SPECTRA
FROM THE STOCHASTIC LIOUVILLE EQUATION WITH PADfi APPROXIMANTS A.J. DAMMERS, Departmwt
Y.K. LEVINE
of Bu~phyacs. Phyncs Laborator)‘. State lhrvemfy
01 Utrecht, 3508 TA Utreclrt. The Netherlarrds
and
J A. TJON hrslihrtc /or
Tlworctrcal Physrcs. Physm Laboratory. State .!hrrversit_v01 Utrcclft. 3508 TA Utrecht. The Netlrerlarrds
Rcccived I6 I-cbruary
1982
A last algornhnl for the numerical solut~oo of the stochasllc Llouvllle equation IS presented t~ons of slow-nrolron CSR, CLDOR and sturatlon-transfer ESR spectra arc drscussed.
Apphcations
to slmula-
I. Introduction
2. Theoretical considerations
One of the mam problems in current applicattons of the stochastic Liouville equation for evaluatmg ESR line shapes u-r the slow-matron region IS the formulation of a fast and economrcal computational algorithm [I-4]. Recently, Ciordano et al. [5] proposed a procedure based on a generalization of Mori’s continuedfraction method [6], which provides for a connderable reduction of computational time. This approach, however, IS somewhat limtted as tt IS only apphcable for evaluattons of ESR spectra under condrttons of no saturation, for which the Ltouvdle operator is not explicitly time dependent. We shall here present a more powerful algortthm, of which the contmued-fraction method is a special case, for the evaluatton of ESR spectra, also in the presence of a saturating microwave field. This approach is of particular importance for apphcations of ELDOR and saturation-transfer techniques. The algorithm, based on the applicatton of PadC opproximants [7-g], is illustrated with spectral simulations for a nitro,xide spin label incorporated in a lamellar phase of a lyotroptc liquid crystal.
The effects of rotational drffusron on ESR and saturation-transfer ESR spectra can conveniently be described using the stochastic Liouville equation for the denstty matrtx p(Q, t). In general it takes the form [l-3] iaplat
= L(!FZ, t)p
+
R (i2, t) ,
(1)
where L(n, r) is the time-dependent Liouvdle operator. The explicit time dependence of the operators in (I) is given by the terms descnbing the interaction between the spin system and the microwave and modulation fields wtth frequencies oo/2a and w,/2n respectively. Furthermore, both the operators L and R are functions of the stochastic angle variable n (%@y, the Euler angles) which defines the time-dependent ortentation of the spin system relative to the applied magnetic field Ho. Eq. (I) can be converted to a set of coupled linear equations by employing a complete set of orthogonal functions, which in the three-dhnension~ case are given by the Wigner rotation matrices. In addition, tf we expand p in terms of its Fourier components P(r)=~~~(~,f)exp[l(~
+ro,)f]
,
(2
with k and r Integers, the stochastic Liouvtlle equation I98
0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
30 Aprd 1982
CHEhllCAL PHYSICS LLI-I-CRS
Volume 88. number 2
can be wrrtten in the matrur form
where the indices II and I?I are characterized by k, r and the quantum numbers of the varrous degrees of freedom in the system. The matrices A and B are Independent of w,, and A is diagonal. Eq. (3) can in principle be solved by matrix invcrsion, but, since one IS in general dealing with large matrlces, it becomes necessary to use other more efficient methods The method we have studied is based on the following considerations. Eq. (3) can be solved itera-
shall here show that this is the case for calculahons of ESR spectra. For the particular case of ordmary ESR rn the highfield approximation the matrix A reduces to the unit matrix, so that eq. (3) can be written as p’= (wol
- B)-%
_
(9)
The iteration procedure, eq. (4) can now bc made more efficient by dropping the terms B,,,,. Since A,,,, = I, we see that the iterative solution (6) becomes a power series in I/we. ~=(wO’~WO’B~WO~B~~...)R~.
tively for the components of p’ using the recurslon
(10)
As a result the mvcrse frequency of the microwave field plays the role of the pammeter X. Tlus way of
solvingeq. (9) has the major advantage that the itcra(4)
L>O, Wllh i+‘) !I
= E,,/(w,A
Formally, ed by
,,,, - B ,,,, 1 .
(5)
the iterative procedure can be represent-
~=(ItKtK’tK3t...)fi(o).
(6)
Let us for convenience Introduce a parameter X, by replacing K by XK in eq. (6).The parameter should, of course, be set equal to 1 m the actual calculations. We have therefore ji = (I + XK + h2K2 t . ..)p””
.
(7)
For firnte matrices we know that &, should be a meromorpluc function in X with poles located at the Inverses of the eigenvalues of the matrix K. As a result the senessolutlon (7) wdl only converge If h nsufficiently small. However, the appropriate analytic continuation for larger values of h can be found by constructing the Pad6 approximant sequence in X [7-g], i e.
03) The mam question which arises in practice is whether the sequence converges sufficiently fast. We
tron procedure becomes essentially frequency indcpendent. Thus it is only necessary to evaluate the coefficlcnts of the series in eq. (7) once only and the whole spectrum can be generated using the Pad6 approximants in the variable l/we. In this case our procedure becomes equivalent to the continued-fraction method proposed in ref. [5] if we take the sequence N=AIt I.
3. Application To test this method we have simulated the ESR spectra of the nitroxrde spin label cholestane [ IOJ m in the slow-motion region. It was assumed that the molecule behaves as a planar rotor and reorlcntates only about its long asis. In this case the stochastic operator rn takes the form [2] r,
=
-q,a2ja$
,
(I 1)
wrth eigenfunctions exp(irly). We shall further assume that the rela!xation processes of the electronic and nuclear spin systems arc mutually independent and that each process can be simply described in terms of longtudinal (TI) and transverse (T,) relaxation times. Only the clectronic T2e relaxa;on Lime is of consequence in the simulations presented here. Values for the non-axially symmetric g and hypertnecouphng tensors were taken from ref. [ 101 and both the secular and pseudo-secular terms of the spin 199
CHUMCAL
Volume 88. number 7,
hamdtonian
PHYSICS
30 Aprd 1982
LETTERS
[3] were mcluded m the cakulatrons. of the symmetrized stochastic
The trnnsformntion
Llouvdleequation [2,3] mto a set of coupled linear equations was achieved by expanding the spin density matrix 3s
X esp[i(kwo
+rw,)t]
exp(~~ry)l&k’~‘I,
f,E’=~f,y,y’=O,fl,k=E--‘,
(I?->
with F(S2, I) = p’(Q t) -po(L?). Here pa(Q) is the equilibrium spin density matrk; E and v denote, respectively, the z components of the eiectromc and nuclear spins. The ESR absorption spectra are calculated from the coefficlcnts with r = 0, whale the experimentally observed first harmonic signal is obtamed from the terms with f= +I. As the mteractlon of the electromc spin system with the modulation field is weak, the various harmonics with [rl > = 0 are essentially uncoupled. Hence only the terms with r = 0 were included in calculatlons of the absorption spectra, while for simulntlons of the experimentally observed spectra only the terms wth r = 0, +I are needed. The simulated ESRspectra with D,, = 4.4 X IO7 rad s-t and Tze = 3.0 X 10-g s are shown in fig IB. With this value of D,, it was found sufficient to restrict the summation over the mdex II in (12) to the Interval 111I S 6. The spectra were simulated as a function of the angle between the axis of rotation and the applied _ magnetic field Ho. Fig. I shows 3 comparison of the simulated spectra with those measured for the cholestane molecule mcorporated m a mscroscopically oriented lamellar phase of chmynstoyllecithin-33 mole% cholesterol-water mixture at 18°C. The agree_ ment can be seen to be satisfactory and our simplified model reproduces the main features of the experimental spectra. The computations of the ESR spectra were carrted out on a Cyber 175 computer and the times involved vnried from I5 s for ;I spectrum
of 200 points at an
orientation of 0” to 25 s for a similar spectrum at 90”. This difference IS due to an angle-dependentrate of convergence of the Pad6 approximants. In our simulations convergence
200
was achieved after 3 iterations
for
A
0
I-I& 1. (A) iI\pcrlmcntal ESR spectra for a cholcstsnc spur label mcorporatcd III a mncroscoprcally onentcd IamcUarphase mcasurcd M a function of the angle bctwccn the ducctor and the npphcd magnetic licld. (B) ESR spectra slmulatcd using the planar rotor model. ror detals see tc\t. the 0’ spectra, while IS iterations were required for the 90” spectra. The matrices mvolved had typically a dimension of the order of 200 and in general the diagonal Pad6 approximants [N = 411in (S)] were used. Table 1 shows the convergence rate for the 90” spectra near the three maxima of the absorption spectrum. Other sequences N =M + K with K fried behaved similarly. In carrying out the cdculatlons we had generally used the special iteration procedure, eq. (10). The absorption spectrum (r = 0) was calculated and the first harmonic signal was obtained numerically using cubic sphne functions. This procedure was adopted because the matrix elements corresponding to r = +I were con-
siderably smaller in magnitude compared to those with P = 0. Consequently the iteration procedure became numerically unstable at some frequencies. The problem can be largely avoided if the general iteration procedure of eqs. (4) and (5) is used. How-
Volume
88. number 2
However, slmulatlons of ESR spectra in the prcs-
Table I Rate of convergence of the dwgonal PadC approximant for Ihe cenlml tnnslrion (v = 0) of tk ESR spectrum
orrenhtion of 90”. The centre of the spectrum
[N/N]
ence of saturation, e g. ELDOR and saturation transfer, cim only be carried out using tlw general procc-
ilt an
is located
32138C
N
at
durc based on eqs. (4) and (5) as now the Liouvillc operator is explicitly dependent on tnnc [I ,3]. The continued-frachon method IS not applicable in tlus case. Thus the algorithms presented here prow& ;1
ricld (C) 3214
3194
general framework
60.3 + 1700 i 60.3 + 1700 I
-312+5941 -312 + 59.4 i -3Ol+
128~41.4 128+483i
I
I 74.9 + 764 1 12.1 + 702 I
128 + 48.3 128+4831 128 + 48.3 128 .a. 48.3
i
12.1* 702 i 15.4+711 i
i
I
154+711 16.7+7101
I
167+71Oi 16.7+7101
159 + 14.9 159 + 14.9
3
4 5 6 7 8
128+41.4i
9
3224
I 1
0 I 2
24.9 + 764
17_8+483i I28
10
+ 48.3
I 84 4 I 84 4
-3Ol+
-266 + 126
i
--266+ I26 I -266 + 127 I -266 + 127 1 -266 + 127 1
I
-266 -266
+ 127 + I27
I I
Preliminary
We have here shown that a numerical fast procedure
algorithm,
provides an efficrcnt
for the simulation
the case of ESR spectra with no saturation, the Llouvdle
operator
and
of ESR spectra. In
has no exphclt
for which
that computa-
of 1200, Indicate
ma-
I5 min arc required for 25 field points, a considerable mlprovcment over current techniques [4]. Thus work IS m progress and wrll be tronal times of the order of
reported
elsewhere.
References J Il. rrccd, C.V Druno and C P Polnaszck, 1. Phys Chenl. 75 (1971) 3385. C r. Polnaszch, C.V. Bruno and J.H. rrecd, J. Chem Phys 58 (1973) 3185 J.H Freed, in Spm Iabclhn& tbcory nnd appllcanons, ed L J. Bcrhncr (Acadenlic Press, New York, 1976) B H. Robmson and L.R Dalton.Chcm Phys 54 (1981) 253, and rcfcrenccs thcrcm. %I Clordano. P Crrogohm and P. hlxm.Cbcm Phys Lcttcrs 83 (1981) 554. H hlorl. Prop. Thcoret. Pbys 33 (1965) 423; 34 (1965) 399 171 GA. Baker Jr., Esscnt~rls of PadE approuimants
(Andcmic Press. New York. 1975) [8l G.A Baker Jr. and J L. Gammcl, eds llic prowlant New York,
trme dependence
the special algorithm based on cq. (IO) can be used. This approach is related to both the contmuedfraction method proposed in ref. [5] and the Lanczos method used in ref. [I 11, and provides a considerable reduction m computer time compared to the algorithms in refs. [l-3].
spcc-
trices of a dnnenslon
of the absorption spectrum.
based on Pad6 appro.ximants,
of ESR spectra.
of saturation-transfer
3, involving
cordance with those from the numerrcal drffcrentiation
4. Conclusion
for srmulations
simulations
tra for the system described in section
ever, in this case the necessary computatronal time mcreased by a factor of 5. The results obtained by a direct evaluatron of the first harmonic signal were m ac-
[2,3],
30 Aprd 1982
CHEMICAL PIIYSICS LETTERS
m thcoretral 1972).
pbyws
(Acadcmrc
Pad6 npPress.
P R. Craves-Morns, cd , PadL’appro\mlants and (hcu apphcatlons (Academic Press, New York, L973). IlO] hLA. Hcmmmfa, J. hlqn. Rcson. 25 (1977) 15. 11 l] C hloro and J H. Freed, J. Phys Cllcm. 84 (1980) [9l
2837.
201
Volume 88. number 2
COMPUTATION
30 Apnl 1982
LRTCRS
OF MAGNETIC RESONANCE SPECTRA
FROM THE STOCHASTIC LIOUVILLE EQUATION WITH PADfi APPROXIMANTS A.J. DAMMERS, Departmwt
Y.K. LEVINE
of Bu~phyacs. Phyncs Laborator)‘. State lhrvemfy
01 Utrecht, 3508 TA Utreclrt. The Netherlarrds
and
J A. TJON hrslihrtc /or
Tlworctrcal Physrcs. Physm Laboratory. State .!hrrversit_v01 Utrcclft. 3508 TA Utrecht. The Netlrerlarrds
Rcccived I6 I-cbruary
1982
A last algornhnl for the numerical solut~oo of the stochasllc Llouvllle equation IS presented t~ons of slow-nrolron CSR, CLDOR and sturatlon-transfer ESR spectra arc drscussed.
Apphcations
to slmula-
I. Introduction
2. Theoretical considerations
One of the mam problems in current applicattons of the stochastic Liouville equation for evaluatmg ESR line shapes u-r the slow-matron region IS the formulation of a fast and economrcal computational algorithm [I-4]. Recently, Ciordano et al. [5] proposed a procedure based on a generalization of Mori’s continuedfraction method [6], which provides for a connderable reduction of computational time. This approach, however, IS somewhat limtted as tt IS only apphcable for evaluattons of ESR spectra under condrttons of no saturation, for which the Ltouvdle operator is not explicitly time dependent. We shall here present a more powerful algortthm, of which the contmued-fraction method is a special case, for the evaluatton of ESR spectra, also in the presence of a saturating microwave field. This approach is of particular importance for apphcations of ELDOR and saturation-transfer techniques. The algorithm, based on the applicatton of PadC opproximants [7-g], is illustrated with spectral simulations for a nitro,xide spin label incorporated in a lamellar phase of a lyotroptc liquid crystal.
The effects of rotational drffusron on ESR and saturation-transfer ESR spectra can conveniently be described using the stochastic Liouville equation for the denstty matrtx p(Q, t). In general it takes the form [l-3] iaplat
= L(!FZ, t)p
+
R (i2, t) ,
(1)
where L(n, r) is the time-dependent Liouvdle operator. The explicit time dependence of the operators in (I) is given by the terms descnbing the interaction between the spin system and the microwave and modulation fields wtth frequencies oo/2a and w,/2n respectively. Furthermore, both the operators L and R are functions of the stochastic angle variable n (%@y, the Euler angles) which defines the time-dependent ortentation of the spin system relative to the applied magnetic field Ho. Eq. (I) can be converted to a set of coupled linear equations by employing a complete set of orthogonal functions, which in the three-dhnension~ case are given by the Wigner rotation matrices. In addition, tf we expand p in terms of its Fourier components P(r)=~~~(~,f)exp[l(~
+ro,)f]
,
(2
with k and r Integers, the stochastic Liouvtlle equation I98
0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
30 Aprd 1982
CHEhllCAL PHYSICS LLI-I-CRS
Volume 88. number 2
can be wrrtten in the matrur form
where the indices II and I?I are characterized by k, r and the quantum numbers of the varrous degrees of freedom in the system. The matrices A and B are Independent of w,, and A is diagonal. Eq. (3) can in principle be solved by matrix invcrsion, but, since one IS in general dealing with large matrlces, it becomes necessary to use other more efficient methods The method we have studied is based on the following considerations. Eq. (3) can be solved itera-
shall here show that this is the case for calculahons of ESR spectra. For the particular case of ordmary ESR rn the highfield approximation the matrix A reduces to the unit matrix, so that eq. (3) can be written as p’= (wol
- B)-%
_
(9)
The iteration procedure, eq. (4) can now bc made more efficient by dropping the terms B,,,,. Since A,,,, = I, we see that the iterative solution (6) becomes a power series in I/we. ~=(wO’~WO’B~WO~B~~...)R~.
tively for the components of p’ using the recurslon
(10)
As a result the mvcrse frequency of the microwave field plays the role of the pammeter X. Tlus way of
solvingeq. (9) has the major advantage that the itcra(4)
L>O, Wllh i+‘) !I
= E,,/(w,A
Formally, ed by
,,,, - B ,,,, 1 .
(5)
the iterative procedure can be represent-
~=(ItKtK’tK3t...)fi(o).
(6)
Let us for convenience Introduce a parameter X, by replacing K by XK in eq. (6).The parameter should, of course, be set equal to 1 m the actual calculations. We have therefore ji = (I + XK + h2K2 t . ..)p””
.
(7)
For firnte matrices we know that &, should be a meromorpluc function in X with poles located at the Inverses of the eigenvalues of the matrix K. As a result the senessolutlon (7) wdl only converge If h nsufficiently small. However, the appropriate analytic continuation for larger values of h can be found by constructing the Pad6 approximant sequence in X [7-g], i e.
03) The mam question which arises in practice is whether the sequence converges sufficiently fast. We
tron procedure becomes essentially frequency indcpendent. Thus it is only necessary to evaluate the coefficlcnts of the series in eq. (7) once only and the whole spectrum can be generated using the Pad6 approximants in the variable l/we. In this case our procedure becomes equivalent to the continued-fraction method proposed in ref. [5] if we take the sequence N=AIt I.
3. Application To test this method we have simulated the ESR spectra of the nitroxrde spin label cholestane [ IOJ m in the slow-motion region. It was assumed that the molecule behaves as a planar rotor and reorlcntates only about its long asis. In this case the stochastic operator rn takes the form [2] r,
=
-q,a2ja$
,
(I 1)
wrth eigenfunctions exp(irly). We shall further assume that the rela!xation processes of the electronic and nuclear spin systems arc mutually independent and that each process can be simply described in terms of longtudinal (TI) and transverse (T,) relaxation times. Only the clectronic T2e relaxa;on Lime is of consequence in the simulations presented here. Values for the non-axially symmetric g and hypertnecouphng tensors were taken from ref. [ 101 and both the secular and pseudo-secular terms of the spin 199
CHUMCAL
Volume 88. number 7,
hamdtonian
PHYSICS
30 Aprd 1982
LETTERS
[3] were mcluded m the cakulatrons. of the symmetrized stochastic
The trnnsformntion
Llouvdleequation [2,3] mto a set of coupled linear equations was achieved by expanding the spin density matrix 3s
X esp[i(kwo
+rw,)t]
exp(~~ry)l&k’~‘I,
f,E’=~f,y,y’=O,fl,k=E--‘,
(I?->
with F(S2, I) = p’(Q t) -po(L?). Here pa(Q) is the equilibrium spin density matrk; E and v denote, respectively, the z components of the eiectromc and nuclear spins. The ESR absorption spectra are calculated from the coefficlcnts with r = 0, whale the experimentally observed first harmonic signal is obtamed from the terms with f= +I. As the mteractlon of the electromc spin system with the modulation field is weak, the various harmonics with [rl > = 0 are essentially uncoupled. Hence only the terms with r = 0 were included in calculatlons of the absorption spectra, while for simulntlons of the experimentally observed spectra only the terms wth r = 0, +I are needed. The simulated ESRspectra with D,, = 4.4 X IO7 rad s-t and Tze = 3.0 X 10-g s are shown in fig IB. With this value of D,, it was found sufficient to restrict the summation over the mdex II in (12) to the Interval 111I S 6. The spectra were simulated as a function of the angle between the axis of rotation and the applied _ magnetic field Ho. Fig. I shows 3 comparison of the simulated spectra with those measured for the cholestane molecule mcorporated m a mscroscopically oriented lamellar phase of chmynstoyllecithin-33 mole% cholesterol-water mixture at 18°C. The agree_ ment can be seen to be satisfactory and our simplified model reproduces the main features of the experimental spectra. The computations of the ESR spectra were carrted out on a Cyber 175 computer and the times involved vnried from I5 s for ;I spectrum
of 200 points at an
orientation of 0” to 25 s for a similar spectrum at 90”. This difference IS due to an angle-dependentrate of convergence of the Pad6 approximants. In our simulations convergence
200
was achieved after 3 iterations
for
A
0
I-I& 1. (A) iI\pcrlmcntal ESR spectra for a cholcstsnc spur label mcorporatcd III a mncroscoprcally onentcd IamcUarphase mcasurcd M a function of the angle bctwccn the ducctor and the npphcd magnetic licld. (B) ESR spectra slmulatcd using the planar rotor model. ror detals see tc\t. the 0’ spectra, while IS iterations were required for the 90” spectra. The matrices mvolved had typically a dimension of the order of 200 and in general the diagonal Pad6 approximants [N = 411in (S)] were used. Table 1 shows the convergence rate for the 90” spectra near the three maxima of the absorption spectrum. Other sequences N =M + K with K fried behaved similarly. In carrying out the cdculatlons we had generally used the special iteration procedure, eq. (10). The absorption spectrum (r = 0) was calculated and the first harmonic signal was obtained numerically using cubic sphne functions. This procedure was adopted because the matrix elements corresponding to r = +I were con-
siderably smaller in magnitude compared to those with P = 0. Consequently the iteration procedure became numerically unstable at some frequencies. The problem can be largely avoided if the general iteration procedure of eqs. (4) and (5) is used. How-
Volume
88. number 2
However, slmulatlons of ESR spectra in the prcs-
Table I Rate of convergence of the dwgonal PadC approximant for Ihe cenlml tnnslrion (v = 0) of tk ESR spectrum
orrenhtion of 90”. The centre of the spectrum
[N/N]
ence of saturation, e g. ELDOR and saturation transfer, cim only be carried out using tlw general procc-
ilt an
is located
32138C
N
at
durc based on eqs. (4) and (5) as now the Liouvillc operator is explicitly dependent on tnnc [I ,3]. The continued-frachon method IS not applicable in tlus case. Thus the algorithms presented here prow& ;1
ricld (C) 3214
3194
general framework
60.3 + 1700 i 60.3 + 1700 I
-312+5941 -312 + 59.4 i -3Ol+
128~41.4 128+483i
I
I 74.9 + 764 1 12.1 + 702 I
128 + 48.3 128+4831 128 + 48.3 128 .a. 48.3
i
12.1* 702 i 15.4+711 i
i
I
154+711 16.7+7101
I
167+71Oi 16.7+7101
159 + 14.9 159 + 14.9
3
4 5 6 7 8
128+41.4i
9
3224
I 1
0 I 2
24.9 + 764
17_8+483i I28
10
+ 48.3
I 84 4 I 84 4
-3Ol+
-266 + 126
i
--266+ I26 I -266 + 127 I -266 + 127 1 -266 + 127 1
I
-266 -266
+ 127 + I27
I I
Preliminary
We have here shown that a numerical fast procedure
algorithm,
provides an efficrcnt
for the simulation
the case of ESR spectra with no saturation, the Llouvdle
operator
and
of ESR spectra. In
has no exphclt
for which
that computa-
of 1200, Indicate
ma-
I5 min arc required for 25 field points, a considerable mlprovcment over current techniques [4]. Thus work IS m progress and wrll be tronal times of the order of
reported
elsewhere.
References J Il. rrccd, C.V Druno and C P Polnaszck, 1. Phys Chenl. 75 (1971) 3385. C r. Polnaszch, C.V. Bruno and J.H. rrecd, J. Chem Phys 58 (1973) 3185 J.H Freed, in Spm Iabclhn& tbcory nnd appllcanons, ed L J. Bcrhncr (Acadenlic Press, New York, 1976) B H. Robmson and L.R Dalton.Chcm Phys 54 (1981) 253, and rcfcrenccs thcrcm. %I Clordano. P Crrogohm and P. hlxm.Cbcm Phys Lcttcrs 83 (1981) 554. H hlorl. Prop. Thcoret. Pbys 33 (1965) 423; 34 (1965) 399 171 GA. Baker Jr., Esscnt~rls of PadE approuimants
(Andcmic Press. New York. 1975) [8l G.A Baker Jr. and J L. Gammcl, eds llic prowlant New York,
trme dependence
the special algorithm based on cq. (IO) can be used. This approach is related to both the contmuedfraction method proposed in ref. [5] and the Lanczos method used in ref. [I 11, and provides a considerable reduction m computer time compared to the algorithms in refs. [l-3].
spcc-
trices of a dnnenslon
of the absorption spectrum.
based on Pad6 appro.ximants,
of ESR spectra.
of saturation-transfer
3, involving
cordance with those from the numerrcal drffcrentiation
4. Conclusion
for srmulations
simulations
tra for the system described in section
ever, in this case the necessary computatronal time mcreased by a factor of 5. The results obtained by a direct evaluatron of the first harmonic signal were m ac-
[2,3],
30 Aprd 1982
CHEMICAL PIIYSICS LETTERS
m thcoretral 1972).
pbyws
(Acadcmrc
Pad6 npPress.
P R. Craves-Morns, cd , PadL’appro\mlants and (hcu apphcatlons (Academic Press, New York, L973). IlO] hLA. Hcmmmfa, J. hlqn. Rcson. 25 (1977) 15. 11 l] C hloro and J H. Freed, J. Phys Cllcm. 84 (1980) [9l
2837.
201