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Signs of deuteron quadrupole coupling constants from COSY-2D spectra of solids
JOURNAL
OF MAGNETIC
RESONANCE
70,436-445
(1986)
Signs of Deuteron Quadrupole Coupling Constants from COSY-2D Spectra of Solids H. SCHLEMMER AND U.HAEBERLEN Max-Planck-Institut,
Abt. f%r Molekulare Physik, 6900 Heidelberg, Jahnstrasse 29, Germany Received July 3, 1986
It is shown how the signs of the quadrupole coupling constants (QcCs) of deuterons can be determined from the multiplet structures of the cross peaks in COSY-2D spectra of deuterons in molecular crystals or any other ordered sample. Multiplets arise as a result of dipolar couplings between pairs of deuterons. The 2D multiplet of a cross peak of a dipolarcoupled pair of deuterons consists of a 3 X 3 array of component lines. If the mixing pulse of the COSY sequence is a 90” pulse the multiplet is insensitive to the signs of the deuteron CjCCs. If, however, the mixing pulse is a 54”44’ pulse only four of the nine components are strong. These are located in one of the four comers of the 3 X 3 array. In which comer of the 3 X 3 array the four strong peaks appear depends on the relative signs of the dipolar and quadrupolar splittings D, Aoc, , and Aw,. This can be used asa fingerprint for the relative signs of D, Aq, , and b. The experimental conditions of the procedure are explored in an experiment on a single crystal of fully deuterated potassium oxalate monohydrate. Q 1986 AC&X& m 1~. INTRODUCTION
NMR spectra of deuterons and other quadrupolar nuclei are invariant with respect to a change of the sign of the quadrupole coupling constant QCC. This is true if the spectra are recorded at “high” temperatures Twhich means at T > 1 mK. Therefore, the sign usually remains undeterminate in measurements of QCCs and electric field gradient (EFG) tensors. Yet, a complete measurement of a QCC or an EFG tensor should also include the sign. The QCCs of deuterons bonded covalently to carbons, oxygens, or nitrogens can safely be assumed to be positive. This follows from a few experiments where the sign of the QCC has actually been measured (I), as well as from MO calculations of EFGs (2). The sign of the QCC is doubtful for deuterons in short symmetric hydrogen bonds (3). There is another area where the signs of the QCCs of deuterons are of interest: in ordered liquids and solutions, motionally averaged quadrupole coupling tensors are observed. Whether the resulting time-averaged QCC is positive or negative depends on the kinematics of the molecular motion, and the knowledge of this sign may help to support or disprove proposed models for the molecular motions. Thus, there is some practical interest in the measurement of the sign of QCCs. The few experimenters who actually measured quadrupole coupling tensors including the signs exploited the fact that the multiplet structure of the deuteron spectrum of a 0022-2364186 $3.00 Copyright Q 1986 by Academic Rcs, Inc. AS right.5 of reproduction in any form reserved.
436
SIGNS
OF
DEUTERON
QCC’S
437
dipolar-coupled pair of deuterons 1 and 2 depends on the sign of the dipolar splitting D relative to the signs of the quadrupolar splittings Awor and Awoz , provided the size of D is comparable to the difference of the magnitudes of AWQ~and AwQ~, i.e., provided 1)Aooi( - IAo~~II - )DI (I, 4). The sign of D can be inferred from the structure of the crystal under study and its orientation relative to the applied field & . The fuhilment of the condition IIAw~~~- [hwQz]l - IDI is rather the exception than the rule and this is particularly true for the case of ordered liquids in which the experimenter has less freedom to reorient the sample than in the case of molecular crystals. In what follows, we therefore focus attention to the situation where IIAUQ~I - 1~~~211$ IDI and search for a way of determining the relative signs of D, AWQ~ , and AWQZ . Once these relative signs are known the absolute signs of the quadrupole coupling constants of the coupled nuclei can be accessed straightforwardly, as described below. A guide in this search is the analogous problem of determining the relative signs of the scalar coupling constants Jik in liquid-state NMR. It is well known that 1D spectra of liquids are invariant to a change of the sign of some or all of the Jik and it is also well known that COSY-type 2D spectra under certain conditions of excitation do contain the information required for determining the relative signs of the Jjk (5). A natural prerequisite for this is that multiplet structures are at least partially resolved in the 2D spectrum. In what follows we therefore investigate theoretically under what conditions, if at all, 2D COSY spectra of dipolar coupled deuterons sensing nonzero EFGs allow the determination of the relative signs of D, Aw~1, and ACOQ~ . We then explore the practical feasibility of the proposed procedure in an experiment on the dipolar-coupled deuterons of the crystal water molecules in a single crystal of fully deuterated potassium oxalate monohydrate. THEORY
The Hamiltonian % (in frequency units) of a dipolar-coupled pair of spin- 1 nuclei is &" = &"z -k %Ql i- 68?Q2-k D, where &pz = -wo(Zzl -i- Z&, 2~ = Ai[3Zf - (Ii)‘] with i = 1, 2, and .GYD= DIZzlZzz - 1/4(Z+1Z-2 + Z-1Z+2)], with D = y2h2/r&(l - 3 ~0~~812). 3.4i = AOQ~= 3/4 * e2Qqi/h * 2~(3 COSMIC - 1 - r)isin2fJ~cos 2&i) is half the (signed) quadrupole splitting in the high-field spectrum of nucleus i. The vector connecting nuclei 1 and 2 is ri2; r12 = lri2), Bi2 is the angle between r12 and &; 8i and & are the polar angles of the applied field & in the principal axis system PAS(i) of the field gradient tensor v”) at the site of nucleus i; qi is the asymmetry factor of #‘I defined by vi = ( v’$x - v’&)/ Vi&. The Vii, (Y = X, Y, 2 are the principal components of v(‘); e2Qqi/h is the quadrupole coupling constant QCC, of nucleus i. As stated above, we are interested in the case IDJ + l]A21- IAl ]I. The eigenfunctions uk and eigenenergies Ek that follow for this case are listed in Table 1. Because of the degeneracy of ua and us there are 12 single-quantum transition frequencies (n&f. They are listed in Table 2. Under the assumption that D, A,, and A2 are all positive and rt2 > Al, Table 2 lists the f&l in ascending order. The corresponding spectrum is shown as a stick spectrum in Fig. 1. The transverse nuclear magnetization M+ = M, + ikf,, during the detection period 22 of the COSY sequence P,,(90°)-t1-P~~)-t2 can be expressed as follows:
438
SCHLEMMER
AND HAEBERLEN TABLE 1
Eigenfunctions & = j/c) and Eigenenergies Ek of &” for D -4 II& 1- IAl 11
-2w, tD t (A, + AZ) -WI +-(A, -U*) -w0
+(-al
-D + (A,
+ t
t-u, wo
+t-za,+
wo
+(A-242)
2w,tDt(A,t
Ad+@,,
t2)
=
2
2
{K%;1SQ(cos
fLJ,)(cos
f&2)
‘42)
A*) +
= E.
242) ‘42)
A2)
+ K&&in
Q2,,tlXsin
f&&2)
mznpzq
- i[Kzn&cos
Q,,tJ(sin Q&2) - KE,,&in
Q,,lJ(cos S2&2)]}e-(f1+t2)‘rz.
[II
The elements of the mixing matrices K are given by Kzngp = K:ngq = W-A&&&~4m
+ K&&m),
PaI
Z%,
- %&,m)
Pbl
= G&q
= WJmn(IxJ&Rp&&
where u*h?zn= (~lLb>
and
Rpn = &h#~) = (hdW,)1n).
The mixing matrices K” and K” are displayed in Diagram 1 for C$= 90”. They are real matrices and they are symmetric with respect to both diagonals. We distinguish diagonal and cross regions. Note that there are diagonal regions with nonzero elements along both diagonals. They are marked in Diagram 1 by full and dashed framing of the entries, respectively. The two types of diagonal regions do not contain independent information. The regions of real interest are the cross regions; they are identically zero
TABLE 2 Single-Quantum Transition Frequencies Q& in the System of Two Dipolar-Coupled Spin- I Nuclei Q2,=wo-D-3A,=C, - 3A2 = f& %3 = wo Q,o=wo+D-3A2=Q”z
Q,,=w,-Dt3A,=Q; t 3A, = fly i&,98=w,,+D+3A,=s2:
R,,=wo-D-3A,=Q- 3A, = Clh: %2 = wo &,=w,,tD-3A,=O+,
f&=~-Dt+A2=& t3A2=Qf %w=wo &,=w,+D+3A,=O:
%6=w,
SIGNS OF DEUTERON
439
QCC’S
FIG. 1. Stick spectrum of a dipolar-coupled pair of spin-l nuclei with IDI < llAll - &Ii. The labels Q, , &, etc., indicate resonances in ascending order. The labels in the bottom row apply only under the additional conditionsA,>0,A2>0,D>OandAz>Al.
in K”. It follows from Eq. [l] that the (sin wit&in w&) - 2D FT of M&i, t2) = Re(M+(tl , t2)), which we will call P[Mdti , &)I, has all its peaks in the cross regions in pure absorption, either positive or negative. The diagonal regions are occupied in both K” and K”. The corresponding regions in the 2D spectrum will inevitably appear as a mixture of absorption and dispersion peaks. Provided the transition frequencies 0~1 are arranged in ascending order (as they are in Table 2 for D, AZ, Al all positive and AZ > Al) the entries of the mixing matrix K&* have a particularly lucid meaning: In the cross regions the pattern of the elements of P corresponds directly to the pattern of signed peaks in the cross regions of a 2D COSY spectrum S(w 1, w2) generated by the prescription S(wi , 02) = F‘[MAtl, t2)].
Kg”,pqfor
@=90’
KS* mn.pq
for @ :90’
(9.71 16.6) IO.21 19.81 (7.6) 10.31 16.01 16.2) 13.11 17.01 (6.31 12.11 12.1116.:1117.0113.1116.2118.01IO.3 17.61IQ.8110.2)16.61IQ.71
ip.ql
-
12.1116.3117.0)(3.1116.2l Is.0110.3)17.51IQ.61lO.ZI l6.6I 19.71
(PSI1
DIAGRAM 1. The mixing matrices Kgpq and P-m for 6 = 90”. As to the labeling of the rows and columns seeTable 2. Note tbat both matrices are symmetric with respect to both diagonals. Diagonal regions are marked by framing of the entries. Umnarked regions are called cross regions. Note that the cross regions of ZP are empty. To get simple numbers all entries are multiplied by a factor of 64.
440
SCHLEMMER
AND HAEBERLEN
In any event, the coefficient KEnm gives the signed amplitude of the peak in S(oi, w2) at w1 = Qmnand w2 = Qm. If we choose a mixing pulse with a general flip angle 4 the cross regions of K” remain identically zero which means that the elements in the cross regions of K” retain their simple meaning. It also implies that the cross regions of S’(wi , w2) remain purely absorptive. Diagram 2 shows the cross regions of K” for general 4. The functions R(4), S(d), T(4); U(4), V(4), and W(4) are shown in Fig. 2. Because of the symmetry of P it is sufficient in what follows to restrict attention to, e.g., the left cross region of K”. We are now in a position to ask what happens if we change the signs of A2, A i , and D. Since all we can hope to discriminate are relative signs we can assume D > 0 without loss of generality. Changing the signs of A, and A2 leaves the 1D spectrum unchanged, however, the sequence of resonances f&k! changes. Table 3 shows which sequences of QM~are obtained and Diagram 3 displays the leji cross region of K” for the four possible cases of relative sign combinations of D, A2, and A, . We repeat: the pattern in this diagram corresponds directly to the pattern of peaks in the left cross region of S(wi , w2) = Fs[MdtI, tz)]. For 4 = 90” all patterns in Diagram 3 look alike, namely as shown in the left cross region of KS displayed in Diagram 1. We conclude that, as in the case of scalar coupled spin-j nuclei, the COSY 2D spectrum generated with a 90” mixing pulse does not contain information about relative signs of interaction strengths. The formal reason is that R(90”) = S(90’) = r(90“) = 2 whereas U(90”) = V(90’) = W(90’) = 0. The largest difference between S(4) and T(d) is obtained for 4 = & = arccos (l/b) = 54’44’. In this particular case the symbolic patterns shown in Diagram 3 are obtained. The patterns are di_trent for the four different relative sign combinations of D, AZ, and A,. The comparison of an experimental spectrum with these patterns therefore allows one to tell which combination of signs applies in an actual case. In fact, all required information is already contained in one of the two 3 X 3 arrays which make up the left cross region. KSS mnpq
for general Q
IPd DIAGRAM 2. The cross regions of the mixing matrix KLpl for general I#. The labeling is the same as in Diagram 1. Although the diagonal regions are left empty, they are occupied as well.
SIGNS OF DEUTERON
3” 0
3. 0
2.5
2. 5
2.0
2. 0
I. 5
1.5
1.0
1.0
n. 5
0. 5
8. 0 1
0. 0 -0. 5
-la. 5 -1.0
QCC’S
I_
-1.0
FIG. 2. The functions R, S, T, V, V, W. R(4) = 2 sin4#; s(4) = 2 sin2$(1 + CDS &I)‘; n@~) = 2 sin*&(l - cos b)2; V(g) = 2 sin(2b)sin f#~[cos&I + 11; V(4) = 2 sin(24)sin $+.m $ - 11; W(g) = 2 X sin*(24).
EXPERlMENT
We have carried out an experiment of this type on the pair of water deuterons in a single crystal of perdeuterated potassium oxalate monohydrate [POMH]. The spectrometer frequency was u, = 54 MHz, the duration of a 90” pulse, 3 I.LS.The purpose of the experiment was more the exploration of the practical aspects of such an experiment than the actual measurement of the signs of the QCCs of the water deuterons in this compound. From other measurements we know that these signs are positive, The tirst practical concern is the observability of a clearly resolved dipolar fine structure in the quadrupolar-split deuteron spectrum. POMH is a particularly favourable substance with regard to this aspect: all molecules of the hydration water are magnetically
TABLE 3 Sequence of Resonances 0~; for Different Cases of Relative Signs of AZ, A,, and D
442
SCHLEMMER
AND HAEBERLEN
or
~-RhJ~ S 1 %G D*> 0 or
0, R, n,
-
w2
DUGRAM3. The lef? cross region of P for the four possible cases of sign combinations of D,
AZ, and The symbolic patterns shown for each case apply for 4 = $,,,, circles correspond to negative, crosses to positive peaks. The sizes of the croses/circles are proportional to the amplitudes of the peaks.
A,.
well isolated from each other. The ratio of the intramolecular and smallestintemolecular deuteron-deuteron distances is 1:3.6 (6). As a result the deuteron resonances from crystals of POMH are unusually sharp and, apart from the temperature T, it is only a matter of the orientation of the sample crystal relative to the applied field B0 to get deuteron spectra with a well resolved dipolar fine structure. The temperature enters because at elevated temperatures the water molecules undergo flip motions (7) which cause exchange broadening of the resonance lines. A further asset of POMH is that all its water molecules are magnetically equivalent; hence there are no complications with overlapping multiple& The next concern is the combination of digital resolution, spectral width, and measurement time. The spectral width sets the upper limit to the time increments At, and At* in the COSY experiment. The required resolution determines the numbers N, and N2 of the increments in both the t, and t2 domains. Ni, together with the waiting time between shots, which must be chosen according to the relaxation time T, of the deuterons, then fixes the total data collection time. Through the choice of the crystal orientation the spectral width is, to some extent, at the discretion of the experimenter. The choice is limited by the observability of the dipolar fine structure, which depends on the crystal orientation as well. Eventually we chose a crystal orientation which gives the deuteron spectrum shown in Fig. 3. It corresponds to the stick spectrum of Fig. 1. The spectraI width is just short of 250 kHz; as can be seen, the dipolar fine structure is well resolved. The separation of the component lines of the multiplets is 1.5 kHz. The sample temperature was 265 K and the lines are already somewhat exchange-broadened. By lowering the sample temperature the resolution
SIGNS OF DEUTERON
-100
-50
443
QCC’S
50 VS
wlzn
[kHzl
FIG. 3. Deuteron spectrum of a single crystal of fully deuterated POMH. The COSY spectrum shown in Fig. 4 was recorded for the same crystal orientation as chosen here.
could have been improved, but only at an almost prohibitively high price in terms of an increase in r,. At T = 265 K the deuteron relaxation time T, is about 20 min in POMH and thus already disturbingly large. In view of these numbers we chose the following experimental parameters: At, = At2 = 2 ps. This provides just enough spectral width to cover the spectrum. Together with a 4K FT (including zero filling) this gives a digital resolution of 122 Hz, just enough to define adequately the component lines of the multiplets. Quadrature phase detection was used in the t2domain. The minimum phase cycling which we considered necessary was 180” phase switching. For each value of tl two signals had, hence, to be accumulated. The waiting time between “shots” was chosen as 5 min = f T, ! A waiting time equal to T, or, even, 3Tl as is standard in most liquid phase NMR experiments would have led to an unacceptably long data collection time. For the same reason we dispensed with doing the experiment in quadrature phase in the tl domain. Our plan was to collect data for Nr = 5 12 increments along t, which would have amounted to a total data collection time of 2 X 5 12 X 5 min or about 3 days and 13 hours. Although, of course, the experiment was running, in principle, automatically, it was watched by H.S. After 2 days and 15 hours and iV1 = 381 he got so overtired that inadvertently he touched some key or button which caused the transm.itter to fire a long burst of closely spaced pulses at full power. The sample crystal
FIG. 4. L&l side.cross region of awl, 02) obtained in a phase-sensitive COSY experiment on a deuterated crystal of POMH. The left group of peaks is folded along the o, axis. An extra minus sign slipped in during data processing so that all peaks are shown with inverted amplitudes.
444
SCHLEMMER
AND HAEBERLEN
did not survive this harsh treatment. It melted. Zero filling was then used to generate a 2K X 2K 2D spectrum S(w, , w2). Figure 4 displays its left cross region. We recognize two groups of peaks, each one containing four strong peaks. The group of peaks at the right side belong to the box (wl, w2) in S(wr , ~2) with o1 = &, Q8, Q9 and w2 = %, fi2, Q3. The other group of peaks belongs to the box (wl, w2) with w1 = n4, t&,, n6 and w2 = fl, , &, a,. This group of peaks is folded along the wi axis at the spectrometer (= mixing) frequency v,. Since the spectrum is generated by sine Fourier transforms, folding brings about an extra change of the sign of the peaks. As pointed out above, it is sufficient to consider one of these groups of peaks. We choose, naturally, the righthand one which is not folded. In Fig. 5 we show the cross sections of S(wr , ~2) along w1 through o2 = 0, , n2 and n,, respectively. They produce the pattern of peaks shown at the top of the figure. Only the four large peaks were used as entries. This pattern of peaks belongs unequivocally to the pattern of peaks expected for A, < 0, AZ < 0, D > 0 or A, > 0, A2 > 0, D < 0, cf. Diagram 3. Two steps remain to complete the determination of the signs of the quadrupole coupling constants QCC(1) and QCC(2) of the water deuterons in POMH. First, we must determine the sign of D. This can be inferred from the orientation of Bo relative to the water molecules in POMH. Our sample crystal was carefully oriented by means of X rays. For the crystal orientation chosen for the experiment, with Bi2 = 3’, it follows that D < 0. Second, we must determine the signs of Alexp/Almax and AZ& AZmax. Remember that the quadrupole coupling constant QCC has the same sign as A -. These ratios can be inferred from so-called rotation patterns of quadrupole splittings. We have recorded such rotation patterns. Suffice it to state here that both these ratios are positive in our experiment. As D < 0 and sign D/A, = sign D/AZ
I
/
I
\
expectation for D>O,A,0,4 >O
FIG. 5. Cross sections through S(u,, 02) along w, for w2 = St,, &, Q,, respectively. Signs of amplitudes are inverted witb respect to Fig. 4. The three traces combine to yield the experimental pattern of the (0, , 02) box shown on the top of the figure. The corresponding theoretical box is alsO included for ease of comparison.
SIGNS OF DEUTERON
QCC’S
445
= -. 1 it follows, then, that the quadrupole coupling constants of the deuterons in the water molecules in POMH are positive. The reader may by now be under the impression that signs of QCCs can be determined in such an experiment only if any doubt about the influence of the data processing routines on the final signs of the peaks in S(w, , ~2) can be eliminated. Fortunately this is not so. A careful look at the patterns in Diagram 3 will convince him that all that counts is which 2 X 2 field of the 3 X 3 array considered in S(wi , ~2) is occupied by strong peaks. It is easy to spot this field. The signs of the peak amplitudes are redundant. This implies that the experiment works also with an absolute magnitude spectrum, provided the component lines of the multiplets are resolvable. ACKNOWLEDGMENTS We thank H. Zimmermann for growing deuterated crystals of POMH, Sonja Benz for reliably orienting and mounting the sample crystal, and Heike Kessel for preparing the diagrams and figures. This work was stimulated by a question of N. Bodenhausen during a lecture of U.H. in 1984. REFERENCES 1. C. MULLER, S. IDZIAK, N. PISLEWSKI, AND U. HAEBERLEN, J. Magn. Reson. 47,227 (1982). 2. T. WEEDING, A. L. KWIRAM, D. RAWLINGS, AND E. R. DAVIDSON, J. Chem. Phys. 82,35 16 (1985). 3. L. MAYAS, M. PLATO, C. J. WINSCOM, AND K. MOBIUS, Mol. Phys. 36,735 (1978). 4. T. CHIBA, J. Chem. Phys. 41, 1352 (1963). 5. D. MAFUON AND K. WWTHRICH, Biochem. Biophys. Rex Commun. 113,967 (1983). 6. A. SEQIJEIRA, S. SRIKANTA, AND R. CHIDAMEIARAM, Actu Crystallogr. Sect. B 26,3616 (1970). 7. N. ScHUFF, Ph.D. thesis, University of Heidelberg, 1983.
OF MAGNETIC
RESONANCE
70,436-445
(1986)
Signs of Deuteron Quadrupole Coupling Constants from COSY-2D Spectra of Solids H. SCHLEMMER AND U.HAEBERLEN Max-Planck-Institut,
Abt. f%r Molekulare Physik, 6900 Heidelberg, Jahnstrasse 29, Germany Received July 3, 1986
It is shown how the signs of the quadrupole coupling constants (QcCs) of deuterons can be determined from the multiplet structures of the cross peaks in COSY-2D spectra of deuterons in molecular crystals or any other ordered sample. Multiplets arise as a result of dipolar couplings between pairs of deuterons. The 2D multiplet of a cross peak of a dipolarcoupled pair of deuterons consists of a 3 X 3 array of component lines. If the mixing pulse of the COSY sequence is a 90” pulse the multiplet is insensitive to the signs of the deuteron CjCCs. If, however, the mixing pulse is a 54”44’ pulse only four of the nine components are strong. These are located in one of the four comers of the 3 X 3 array. In which comer of the 3 X 3 array the four strong peaks appear depends on the relative signs of the dipolar and quadrupolar splittings D, Aoc, , and Aw,. This can be used asa fingerprint for the relative signs of D, Aq, , and b. The experimental conditions of the procedure are explored in an experiment on a single crystal of fully deuterated potassium oxalate monohydrate. Q 1986 AC&X& m 1~. INTRODUCTION
NMR spectra of deuterons and other quadrupolar nuclei are invariant with respect to a change of the sign of the quadrupole coupling constant QCC. This is true if the spectra are recorded at “high” temperatures Twhich means at T > 1 mK. Therefore, the sign usually remains undeterminate in measurements of QCCs and electric field gradient (EFG) tensors. Yet, a complete measurement of a QCC or an EFG tensor should also include the sign. The QCCs of deuterons bonded covalently to carbons, oxygens, or nitrogens can safely be assumed to be positive. This follows from a few experiments where the sign of the QCC has actually been measured (I), as well as from MO calculations of EFGs (2). The sign of the QCC is doubtful for deuterons in short symmetric hydrogen bonds (3). There is another area where the signs of the QCCs of deuterons are of interest: in ordered liquids and solutions, motionally averaged quadrupole coupling tensors are observed. Whether the resulting time-averaged QCC is positive or negative depends on the kinematics of the molecular motion, and the knowledge of this sign may help to support or disprove proposed models for the molecular motions. Thus, there is some practical interest in the measurement of the sign of QCCs. The few experimenters who actually measured quadrupole coupling tensors including the signs exploited the fact that the multiplet structure of the deuteron spectrum of a 0022-2364186 $3.00 Copyright Q 1986 by Academic Rcs, Inc. AS right.5 of reproduction in any form reserved.
436
SIGNS
OF
DEUTERON
QCC’S
437
dipolar-coupled pair of deuterons 1 and 2 depends on the sign of the dipolar splitting D relative to the signs of the quadrupolar splittings Awor and Awoz , provided the size of D is comparable to the difference of the magnitudes of AWQ~and AwQ~, i.e., provided 1)Aooi( - IAo~~II - )DI (I, 4). The sign of D can be inferred from the structure of the crystal under study and its orientation relative to the applied field & . The fuhilment of the condition IIAw~~~- [hwQz]l - IDI is rather the exception than the rule and this is particularly true for the case of ordered liquids in which the experimenter has less freedom to reorient the sample than in the case of molecular crystals. In what follows, we therefore focus attention to the situation where IIAUQ~I - 1~~~211$ IDI and search for a way of determining the relative signs of D, AWQ~ , and AWQZ . Once these relative signs are known the absolute signs of the quadrupole coupling constants of the coupled nuclei can be accessed straightforwardly, as described below. A guide in this search is the analogous problem of determining the relative signs of the scalar coupling constants Jik in liquid-state NMR. It is well known that 1D spectra of liquids are invariant to a change of the sign of some or all of the Jik and it is also well known that COSY-type 2D spectra under certain conditions of excitation do contain the information required for determining the relative signs of the Jjk (5). A natural prerequisite for this is that multiplet structures are at least partially resolved in the 2D spectrum. In what follows we therefore investigate theoretically under what conditions, if at all, 2D COSY spectra of dipolar coupled deuterons sensing nonzero EFGs allow the determination of the relative signs of D, Aw~1, and ACOQ~ . We then explore the practical feasibility of the proposed procedure in an experiment on the dipolar-coupled deuterons of the crystal water molecules in a single crystal of fully deuterated potassium oxalate monohydrate. THEORY
The Hamiltonian % (in frequency units) of a dipolar-coupled pair of spin- 1 nuclei is &" = &"z -k %Ql i- 68?Q2-k D, where &pz = -wo(Zzl -i- Z&, 2~ = Ai[3Zf - (Ii)‘] with i = 1, 2, and .GYD= DIZzlZzz - 1/4(Z+1Z-2 + Z-1Z+2)], with D = y2h2/r&(l - 3 ~0~~812). 3.4i = AOQ~= 3/4 * e2Qqi/h * 2~(3 COSMIC - 1 - r)isin2fJ~cos 2&i) is half the (signed) quadrupole splitting in the high-field spectrum of nucleus i. The vector connecting nuclei 1 and 2 is ri2; r12 = lri2), Bi2 is the angle between r12 and &; 8i and & are the polar angles of the applied field & in the principal axis system PAS(i) of the field gradient tensor v”) at the site of nucleus i; qi is the asymmetry factor of #‘I defined by vi = ( v’$x - v’&)/ Vi&. The Vii, (Y = X, Y, 2 are the principal components of v(‘); e2Qqi/h is the quadrupole coupling constant QCC, of nucleus i. As stated above, we are interested in the case IDJ + l]A21- IAl ]I. The eigenfunctions uk and eigenenergies Ek that follow for this case are listed in Table 1. Because of the degeneracy of ua and us there are 12 single-quantum transition frequencies (n&f. They are listed in Table 2. Under the assumption that D, A,, and A2 are all positive and rt2 > Al, Table 2 lists the f&l in ascending order. The corresponding spectrum is shown as a stick spectrum in Fig. 1. The transverse nuclear magnetization M+ = M, + ikf,, during the detection period 22 of the COSY sequence P,,(90°)-t1-P~~)-t2 can be expressed as follows:
438
SCHLEMMER
AND HAEBERLEN TABLE 1
Eigenfunctions & = j/c) and Eigenenergies Ek of &” for D -4 II& 1- IAl 11
-2w, tD t (A, + AZ) -WI +-(A, -U*) -w0
+(-al
-D + (A,
+ t
t-u, wo
+t-za,+
wo
+(A-242)
2w,tDt(A,t
Ad+@,,
t2)
=
2
2
{K%;1SQ(cos
fLJ,)(cos
f&2)
‘42)
A*) +
= E.
242) ‘42)
A2)
+ K&&in
Q2,,tlXsin
f&&2)
mznpzq
- i[Kzn&cos
Q,,tJ(sin Q&2) - KE,,&in
Q,,lJ(cos S2&2)]}e-(f1+t2)‘rz.
[II
The elements of the mixing matrices K are given by Kzngp = K:ngq = W-A&&&~4m
+ K&&m),
PaI
Z%,
- %&,m)
Pbl
= G&q
= WJmn(IxJ&Rp&&
where u*h?zn= (~lLb>
and
Rpn = &h#~) = (hdW,)1n).
The mixing matrices K” and K” are displayed in Diagram 1 for C$= 90”. They are real matrices and they are symmetric with respect to both diagonals. We distinguish diagonal and cross regions. Note that there are diagonal regions with nonzero elements along both diagonals. They are marked in Diagram 1 by full and dashed framing of the entries, respectively. The two types of diagonal regions do not contain independent information. The regions of real interest are the cross regions; they are identically zero
TABLE 2 Single-Quantum Transition Frequencies Q& in the System of Two Dipolar-Coupled Spin- I Nuclei Q2,=wo-D-3A,=C, - 3A2 = f& %3 = wo Q,o=wo+D-3A2=Q”z
Q,,=w,-Dt3A,=Q; t 3A, = fly i&,98=w,,+D+3A,=s2:
R,,=wo-D-3A,=Q- 3A, = Clh: %2 = wo &,=w,,tD-3A,=O+,
f&=~-Dt+A2=& t3A2=Qf %w=wo &,=w,+D+3A,=O:
%6=w,
SIGNS OF DEUTERON
439
QCC’S
FIG. 1. Stick spectrum of a dipolar-coupled pair of spin-l nuclei with IDI < llAll - &Ii. The labels Q, , &, etc., indicate resonances in ascending order. The labels in the bottom row apply only under the additional conditionsA,>0,A2>0,D>OandAz>Al.
in K”. It follows from Eq. [l] that the (sin wit&in w&) - 2D FT of M&i, t2) = Re(M+(tl , t2)), which we will call P[Mdti , &)I, has all its peaks in the cross regions in pure absorption, either positive or negative. The diagonal regions are occupied in both K” and K”. The corresponding regions in the 2D spectrum will inevitably appear as a mixture of absorption and dispersion peaks. Provided the transition frequencies 0~1 are arranged in ascending order (as they are in Table 2 for D, AZ, Al all positive and AZ > Al) the entries of the mixing matrix K&* have a particularly lucid meaning: In the cross regions the pattern of the elements of P corresponds directly to the pattern of signed peaks in the cross regions of a 2D COSY spectrum S(w 1, w2) generated by the prescription S(wi , 02) = F‘[MAtl, t2)].
Kg”,pqfor
@=90’
KS* mn.pq
for @ :90’
(9.71 16.6) IO.21 19.81 (7.6) 10.31 16.01 16.2) 13.11 17.01 (6.31 12.11 12.1116.:1117.0113.1116.2118.01IO.3 17.61IQ.8110.2)16.61IQ.71
ip.ql
-
12.1116.3117.0)(3.1116.2l Is.0110.3)17.51IQ.61lO.ZI l6.6I 19.71
(PSI1
DIAGRAM 1. The mixing matrices Kgpq and P-m for 6 = 90”. As to the labeling of the rows and columns seeTable 2. Note tbat both matrices are symmetric with respect to both diagonals. Diagonal regions are marked by framing of the entries. Umnarked regions are called cross regions. Note that the cross regions of ZP are empty. To get simple numbers all entries are multiplied by a factor of 64.
440
SCHLEMMER
AND HAEBERLEN
In any event, the coefficient KEnm gives the signed amplitude of the peak in S(oi, w2) at w1 = Qmnand w2 = Qm. If we choose a mixing pulse with a general flip angle 4 the cross regions of K” remain identically zero which means that the elements in the cross regions of K” retain their simple meaning. It also implies that the cross regions of S’(wi , w2) remain purely absorptive. Diagram 2 shows the cross regions of K” for general 4. The functions R(4), S(d), T(4); U(4), V(4), and W(4) are shown in Fig. 2. Because of the symmetry of P it is sufficient in what follows to restrict attention to, e.g., the left cross region of K”. We are now in a position to ask what happens if we change the signs of A2, A i , and D. Since all we can hope to discriminate are relative signs we can assume D > 0 without loss of generality. Changing the signs of A, and A2 leaves the 1D spectrum unchanged, however, the sequence of resonances f&k! changes. Table 3 shows which sequences of QM~are obtained and Diagram 3 displays the leji cross region of K” for the four possible cases of relative sign combinations of D, A2, and A, . We repeat: the pattern in this diagram corresponds directly to the pattern of peaks in the left cross region of S(wi , w2) = Fs[MdtI, tz)]. For 4 = 90” all patterns in Diagram 3 look alike, namely as shown in the left cross region of KS displayed in Diagram 1. We conclude that, as in the case of scalar coupled spin-j nuclei, the COSY 2D spectrum generated with a 90” mixing pulse does not contain information about relative signs of interaction strengths. The formal reason is that R(90”) = S(90’) = r(90“) = 2 whereas U(90”) = V(90’) = W(90’) = 0. The largest difference between S(4) and T(d) is obtained for 4 = & = arccos (l/b) = 54’44’. In this particular case the symbolic patterns shown in Diagram 3 are obtained. The patterns are di_trent for the four different relative sign combinations of D, AZ, and A,. The comparison of an experimental spectrum with these patterns therefore allows one to tell which combination of signs applies in an actual case. In fact, all required information is already contained in one of the two 3 X 3 arrays which make up the left cross region. KSS mnpq
for general Q
IPd DIAGRAM 2. The cross regions of the mixing matrix KLpl for general I#. The labeling is the same as in Diagram 1. Although the diagonal regions are left empty, they are occupied as well.
SIGNS OF DEUTERON
3” 0
3. 0
2.5
2. 5
2.0
2. 0
I. 5
1.5
1.0
1.0
n. 5
0. 5
8. 0 1
0. 0 -0. 5
-la. 5 -1.0
QCC’S
I_
-1.0
FIG. 2. The functions R, S, T, V, V, W. R(4) = 2 sin4#; s(4) = 2 sin2$(1 + CDS &I)‘; n@~) = 2 sin*&(l - cos b)2; V(g) = 2 sin(2b)sin f#~[cos&I + 11; V(4) = 2 sin(24)sin $+.m $ - 11; W(g) = 2 X sin*(24).
EXPERlMENT
We have carried out an experiment of this type on the pair of water deuterons in a single crystal of perdeuterated potassium oxalate monohydrate [POMH]. The spectrometer frequency was u, = 54 MHz, the duration of a 90” pulse, 3 I.LS.The purpose of the experiment was more the exploration of the practical aspects of such an experiment than the actual measurement of the signs of the QCCs of the water deuterons in this compound. From other measurements we know that these signs are positive, The tirst practical concern is the observability of a clearly resolved dipolar fine structure in the quadrupolar-split deuteron spectrum. POMH is a particularly favourable substance with regard to this aspect: all molecules of the hydration water are magnetically
TABLE 3 Sequence of Resonances 0~; for Different Cases of Relative Signs of AZ, A,, and D
442
SCHLEMMER
AND HAEBERLEN
or
~-RhJ~ S 1 %G D*> 0 or
0, R, n,
-
w2
DUGRAM3. The lef? cross region of P for the four possible cases of sign combinations of D,
AZ, and The symbolic patterns shown for each case apply for 4 = $,,,, circles correspond to negative, crosses to positive peaks. The sizes of the croses/circles are proportional to the amplitudes of the peaks.
A,.
well isolated from each other. The ratio of the intramolecular and smallestintemolecular deuteron-deuteron distances is 1:3.6 (6). As a result the deuteron resonances from crystals of POMH are unusually sharp and, apart from the temperature T, it is only a matter of the orientation of the sample crystal relative to the applied field B0 to get deuteron spectra with a well resolved dipolar fine structure. The temperature enters because at elevated temperatures the water molecules undergo flip motions (7) which cause exchange broadening of the resonance lines. A further asset of POMH is that all its water molecules are magnetically equivalent; hence there are no complications with overlapping multiple& The next concern is the combination of digital resolution, spectral width, and measurement time. The spectral width sets the upper limit to the time increments At, and At* in the COSY experiment. The required resolution determines the numbers N, and N2 of the increments in both the t, and t2 domains. Ni, together with the waiting time between shots, which must be chosen according to the relaxation time T, of the deuterons, then fixes the total data collection time. Through the choice of the crystal orientation the spectral width is, to some extent, at the discretion of the experimenter. The choice is limited by the observability of the dipolar fine structure, which depends on the crystal orientation as well. Eventually we chose a crystal orientation which gives the deuteron spectrum shown in Fig. 3. It corresponds to the stick spectrum of Fig. 1. The spectraI width is just short of 250 kHz; as can be seen, the dipolar fine structure is well resolved. The separation of the component lines of the multiplets is 1.5 kHz. The sample temperature was 265 K and the lines are already somewhat exchange-broadened. By lowering the sample temperature the resolution
SIGNS OF DEUTERON
-100
-50
443
QCC’S
50 VS
wlzn
[kHzl
FIG. 3. Deuteron spectrum of a single crystal of fully deuterated POMH. The COSY spectrum shown in Fig. 4 was recorded for the same crystal orientation as chosen here.
could have been improved, but only at an almost prohibitively high price in terms of an increase in r,. At T = 265 K the deuteron relaxation time T, is about 20 min in POMH and thus already disturbingly large. In view of these numbers we chose the following experimental parameters: At, = At2 = 2 ps. This provides just enough spectral width to cover the spectrum. Together with a 4K FT (including zero filling) this gives a digital resolution of 122 Hz, just enough to define adequately the component lines of the multiplets. Quadrature phase detection was used in the t2domain. The minimum phase cycling which we considered necessary was 180” phase switching. For each value of tl two signals had, hence, to be accumulated. The waiting time between “shots” was chosen as 5 min = f T, ! A waiting time equal to T, or, even, 3Tl as is standard in most liquid phase NMR experiments would have led to an unacceptably long data collection time. For the same reason we dispensed with doing the experiment in quadrature phase in the tl domain. Our plan was to collect data for Nr = 5 12 increments along t, which would have amounted to a total data collection time of 2 X 5 12 X 5 min or about 3 days and 13 hours. Although, of course, the experiment was running, in principle, automatically, it was watched by H.S. After 2 days and 15 hours and iV1 = 381 he got so overtired that inadvertently he touched some key or button which caused the transm.itter to fire a long burst of closely spaced pulses at full power. The sample crystal
FIG. 4. L&l side.cross region of awl, 02) obtained in a phase-sensitive COSY experiment on a deuterated crystal of POMH. The left group of peaks is folded along the o, axis. An extra minus sign slipped in during data processing so that all peaks are shown with inverted amplitudes.
444
SCHLEMMER
AND HAEBERLEN
did not survive this harsh treatment. It melted. Zero filling was then used to generate a 2K X 2K 2D spectrum S(w, , w2). Figure 4 displays its left cross region. We recognize two groups of peaks, each one containing four strong peaks. The group of peaks at the right side belong to the box (wl, w2) in S(wr , ~2) with o1 = &, Q8, Q9 and w2 = %, fi2, Q3. The other group of peaks belongs to the box (wl, w2) with w1 = n4, t&,, n6 and w2 = fl, , &, a,. This group of peaks is folded along the wi axis at the spectrometer (= mixing) frequency v,. Since the spectrum is generated by sine Fourier transforms, folding brings about an extra change of the sign of the peaks. As pointed out above, it is sufficient to consider one of these groups of peaks. We choose, naturally, the righthand one which is not folded. In Fig. 5 we show the cross sections of S(wr , ~2) along w1 through o2 = 0, , n2 and n,, respectively. They produce the pattern of peaks shown at the top of the figure. Only the four large peaks were used as entries. This pattern of peaks belongs unequivocally to the pattern of peaks expected for A, < 0, AZ < 0, D > 0 or A, > 0, A2 > 0, D < 0, cf. Diagram 3. Two steps remain to complete the determination of the signs of the quadrupole coupling constants QCC(1) and QCC(2) of the water deuterons in POMH. First, we must determine the sign of D. This can be inferred from the orientation of Bo relative to the water molecules in POMH. Our sample crystal was carefully oriented by means of X rays. For the crystal orientation chosen for the experiment, with Bi2 = 3’, it follows that D < 0. Second, we must determine the signs of Alexp/Almax and AZ& AZmax. Remember that the quadrupole coupling constant QCC has the same sign as A -. These ratios can be inferred from so-called rotation patterns of quadrupole splittings. We have recorded such rotation patterns. Suffice it to state here that both these ratios are positive in our experiment. As D < 0 and sign D/A, = sign D/AZ
I
/
I
\
expectation for D>O,A,
FIG. 5. Cross sections through S(u,, 02) along w, for w2 = St,, &, Q,, respectively. Signs of amplitudes are inverted witb respect to Fig. 4. The three traces combine to yield the experimental pattern of the (0, , 02) box shown on the top of the figure. The corresponding theoretical box is alsO included for ease of comparison.
SIGNS OF DEUTERON
QCC’S
445
= -. 1 it follows, then, that the quadrupole coupling constants of the deuterons in the water molecules in POMH are positive. The reader may by now be under the impression that signs of QCCs can be determined in such an experiment only if any doubt about the influence of the data processing routines on the final signs of the peaks in S(w, , ~2) can be eliminated. Fortunately this is not so. A careful look at the patterns in Diagram 3 will convince him that all that counts is which 2 X 2 field of the 3 X 3 array considered in S(wi , ~2) is occupied by strong peaks. It is easy to spot this field. The signs of the peak amplitudes are redundant. This implies that the experiment works also with an absolute magnitude spectrum, provided the component lines of the multiplets are resolvable. ACKNOWLEDGMENTS We thank H. Zimmermann for growing deuterated crystals of POMH, Sonja Benz for reliably orienting and mounting the sample crystal, and Heike Kessel for preparing the diagrams and figures. This work was stimulated by a question of N. Bodenhausen during a lecture of U.H. in 1984. REFERENCES 1. C. MULLER, S. IDZIAK, N. PISLEWSKI, AND U. HAEBERLEN, J. Magn. Reson. 47,227 (1982). 2. T. WEEDING, A. L. KWIRAM, D. RAWLINGS, AND E. R. DAVIDSON, J. Chem. Phys. 82,35 16 (1985). 3. L. MAYAS, M. PLATO, C. J. WINSCOM, AND K. MOBIUS, Mol. Phys. 36,735 (1978). 4. T. CHIBA, J. Chem. Phys. 41, 1352 (1963). 5. D. MAFUON AND K. WWTHRICH, Biochem. Biophys. Rex Commun. 113,967 (1983). 6. A. SEQIJEIRA, S. SRIKANTA, AND R. CHIDAMEIARAM, Actu Crystallogr. Sect. B 26,3616 (1970). 7. N. ScHUFF, Ph.D. thesis, University of Heidelberg, 1983.