- Email: [email protected]
Product Operators Approach to 2D-NMR Spectroscopy
511
Chapter 14
Product Operators Approach to 2D-NMR Spectroscopy Two-dimensional h R spectroscopy was presented above in terms of the magnetization vector approach in which the macroscopic magnetization rotated under the influence of pulses and precessed on account of the chemical shifts. It may be alternatively discussed in terms of product operutorfoirnalism (with Cartesian basis operators) (ref. 1-4) in which the observable magnetization resulting from the entire spin system may be described by the density matrix. An introduction to this approach has already been given in Chapter 1, Section 1.2.16.3 and the readers are advised to read that section before the following portion on the product operator approach. For the sake of clarity let us dwell again on some of the underlying principles of this approach. In the ground state of the system in thermal equilibrium, the net magnetization is represented by M z and the polarisation along the z axis is described by Iz. If a radiofrequency pulse which bends the magnetization by an angle R is applied along the y axis, the Cartesian operators, IX, Iy and I z (where Ix, Iy and Iz are related to x, y and z magnetizations) will be transformed as follows: I c Y~Ic COS (3 - Iism (3 bI
ly
lz
bI
Y U
, .Iy I z cos p + l sin 1b
(1)
(2) (3)
Each of these three transformations show how the starting state of the spin system (left side of "equation") evolves to its end state (on the right side of the "equation") under the influence of a particular operator (shown on top of the arrow). This operator may represent interactions due to chemical shifts, couplings, pulses etc. Thus if the operator above the arrow is Wl zt, then this will represent the evolution of the chemical shift of nucleus I during time t. On the other hand if the operator is shown as bIxy (or, more generally, RI~) then this will designate a pulse of flip angle ß about an axis which forms an angle with the x axis in the x,y plane. Similarly the influence of coupling J between nuclei A and B (operators J A and I B) during time t maybe represented by 2p JAZJBzt•
512
Fir weakly coupled two-spin systems of nuclei A and B, the following 16 product operators will exist: B/2 : E represents the unity operator I x: represents x component of! spin magnetization Iy: represents y component of! spin magnetization Iz: represents z component of! spin magnetization Sx: represents x component of S spin magnetization Sy : represents y component of S spin magnetization Sz : represents z component of S spin magnetization 2IXSx, 2IySy, 2IySx, 2IxSy : represent two-spin coherences 2IxSz, 2IySz : represent antiphase I spin magnetization 2IzSx, 2IzSy : represent antiphase S spin magnetization 2I zSz : represents longitudinal two-spin order Let us examine the meaning of some of the above terms. 2IxSz indicates that the x component of I spin magneization is split into two antiphase components corresponding to the a and p states of nucleus S. This is detectable in the NMR spectrum as a signal. The product operator 2I zSz results in no net polarisation although all the populations of the four energy levels are disturbed. The product operator 2I XSx represents the superpositioning of zero-quantum coherence (opposite spin-flips of I and S nuclei) and double-quantum coherence (spin-flips of! and S nuclei in the same sense). Such two-spin coherences cannot be detected directly, but they need to be converted to single-quantum coherences prior to detection by two-dimensional NMR spectroscopic methods. A linear combination of these product operators corresponds to zero-quantum coherences: 2IzSx + 2IySy or 2IySx - 2IxSy. The alternative linear combinations 2IXSx - 2IySy or 2IxSy + 2IySx correspond to double-quantum coherence. The effect of the evolution of chemical shift W of nucleus I during time t on the Ix, Iy and I z operators may be designated as: Ix WIzt — It.t
Ig W IZ
Ix cos W t + I g sin W t
, Iy cos W t- Ix sin W t IZ
(4) (5) (6)
513
Hence if we are detecting the y magnetization (in equation (4)) then the signal amplitude will be seen to grow by sin wt, and Fourier transformation will afford single lines at ± w , one of which (at +w or -w ) may be detected by quadrature detection. The operators for one kind of nucleus in a iD experiment may be represented in terms of the evolution of magnetization which occurs during the detection period t2: IZ
90~~
Ic
Evolution Detection period t 2
Ix cos t2 + Iy sin w t2
( 7)
In a 2D experiment, on the other hand, the 90° y pulse will result in the conversion of equilibrium magnetization Iz to Ix which will evolve during the evolution period t1 with its characteristic frequency: Ic
WlI zt1
Ic cos W1 ti
+ Ig sin W] ti
(8)
Let us assume that I represents 1H magnetization and S represents C-magnetization. The application of simultaneous 900 pulses to both the 1H and 13C nuclei1H results in a mixing of the magnetizations so that coherence can be transferred from nuclei to 13C nuclei i.e. Ix is transformed to Sx. The Iy term may be ignored as it does not contribute to the outcome of the experiment ± . During the evolution time t2, Sx will evolve with its characteristic frequency: 13
W St Sxs 2 a Sx COS WV t2 + Sy sin Ws t2
(9)
At the end of time period t2, the spin system may be represented as: cos UI ti (Sx cos ws t2 + Sy sin Ws t2)
(10)
The important term to note in equation (10) is in. In other words the S operator coefficients, representing its evolution in time periods t1 and t2, are governed not only by its own characteristic frequencies Ws but also by the frequencies Wt of the other I nucleus to which nucleus S is coupled. The magnetization of nucleus S has therefore been "labelled" during ti by WI, i.e. the frequency of nucleus I. Quadrature detection followed by Fourier transformation results in the separation of the frequencies ± in and Ws into the two frequency domainsvi and½ respectively, thereby affording the two-dimensional spectrum. The appearance of a cross-peak at (UI,Us) in the 2D spectrum will then indicate the modulation of the signal at the I frequency (01) during t1 and by the S frequency (Ws) during t2. The cross-peaks in the 2D spectrum therefore arise at the frequencies at which coherence transfer occurs. The more efficient the mixing process, the stronger will be the signals observed in the 2D spectrum. The + This is so because the positive and negative components of the Iy terms mutually cancel one another.
514
coherence transfer may be brought about by (a) spin-spin scalar coupling through bonds (b) dipolar coupling through space or (c) exchange processes.Since these processes play a central role in 2D spectra, it is appropriate to discuss them in terms of the product operator formalism.
14.1 SCALAR COUPLING Scalar spin-spin coupling takes place mainly through bonds, though through-space coupling is also rarely seen (ref. 5). When two nuclei A and B are coupled to one another, then the spin-spin coupling between them may be represented in product operator formalism as follows: lAx
~ J ABt 2I I A7 B7
lAy 2IAxIBz 21AyIBz
JABt 2IAzIBz JABt 2IAzIBZ I
ABt 2l Az l Bz
— lAx cos p JABt + 2IAy1Bz sin p JABt —
lAy COS
P
(11)
JABt + 2IAxI B sin p JABt
(12)
2IAcIBz cos P JABt + IAg sin p JABt
(13)
21AyIBz cos p JABt - lAx sin p JABt
(14)
Let us revertto the vector approach to understand the implications of the above equations. If we focus our attention on the A vector, then coupling of spin A with spin B will result in there being two different vectors for spin A, depending on whether spin B is in an a or ß state. Equation (11) shows that the x magnetization of spin A is modulated by cos lI Jt. The y magnetization of spin A will be directed towards the +y axis when spin B is in an a state, and towards the -y axis when spin B is in its state, lAy (IBa - IBb), so that it will not be observed because of mutual cancellation of the positive and negative components (Fig. 14.1).
I
cosnJt+ I A sinnJt)
1
B~ A
t
c
y
cos 11 Jt x
q
cosnJt-I A sin n Jt) c
2IA b y Z
y
sin p Jt
Fig. 14.1: Evil ution of nucleus A due to scalar coupling (with nucleus B) during time Ii' At the beginning of the evolution period, the x magnetization of nucleus A is represented by lAx. At the end of the evolution period, the antiphase magnetization which develops is shown with wavy vectors, while the residual in-phase magnetization is shown as the central unbroken vector (IAx cos'PJt).
515
The difference in population between the a and b levels (1aR - Ib) is 2Iz. Hence aR IAy(h - IBb ) may be written as 2IAyh z which will undergo modulation by sin iF JAit (equation 11). It may be noted that while IAy and Igz are "operators", the term "product operator" will be applied to the products of operators e.g. IAyIBz. Some basic rules which govern operator transformations are: a) Each operator in the product operator is transformed separately during the evolution period when it is subjected to the modulation effects of the chemical shift or coupling frequencies, as indicated below, Iz not being transformed by chemical shift frequencies: 2IAcIBc 2IAcIBc 2IAcIBy
90 U
(15)
~-2IAcIBc
W AI AZ t+W BI Bz t — 2l,e, WA IAz t+W RI R~ t
cIg z
cosWAt + 2IAylBzsinWAt
(16)
2(IAccosWAt+IAysinWAt) (IgycosWgt - IBcsinWS3t)
(17)
(b) Product operators of two transverse operators (e.g. IMIgx), or of two longitudinal operators (e.g. IAzIBz) do not show mutual coupling, coupling being observed between products of transverse and longitudinal operators (e.g. IAXIBz). However coupling will also be observed between two transverse product operators with a third nucleus, C. (c) The influence of chemical shift and coupling frequencies are simultaneously observed. Under conditions of weak coupling the following additional considerations have to be taken into account: (i) Product operators with only one transverse operator (Ix or Iy) are observable. (ii) Product operators having any number of longitudinal operators Iz but only one transverse operator (I,, or Iy) are observable provided the nucleus corresponding to the transverse operator is scalar coupled to the nucleus with the longitudinal operator(s). (iii) Other product operators do not evolve as observable operators. In a three-spin system in which nucleus A is coupled to two other nuclei B and C, the spectrum will contain a double doublet for A at WA. IAA will evolve during ti to give a total of 8 terms, but as stipulated by the above rules, only the first two terms having the transverse operators IAA and IAy are observable: IAA
IAxcoswl tcos IT JABt cos p JACt + IAysinWAt cos p JABt cos iFJACt (+ 6 other terms)
(18)
516
( A) AB
cos p J ABt
cos lT J AC t
cos p J AB t.cos p J AC t
AC
I
i~J AB
Ii
I
~ AC
(B)
sin wJ ABt
J
cos .J AC t
AC
--~
I
sin p J ABt.cos p J ACt
AC
I
BZ
~
(C )
J
-
AC I
cos n J ABt. si n n J AC t
C AX Z
(D)
I
si n n J ABt.sin n J AC t
I
AX I BZ I CZ
— i
AB
J
AC
I
Fig. 14.2: (A) Contribution of an in-phase double doublet from two in-phase doublets. (B) Contribution of a double doublet from two doublets one of which is in-phase and the other antiphase. (C) Double doublet with antiphase structure (antiphase with respect to Jlc). (D) Double doublet also having antiphase character, but antiphase with respect to both JAB and JAc. (Reproduced with permission from H.Kessler et al., Angew.Chem.Jn'.Ed. Engl., 27, 490-536 (1988), copyright 1988,NCH Verlagsgesellschaft).
517
To construct a signal from the product operators, one makes use of the principle which stipulates that the Fourier transform of a product of functions can be derived from the Fourier transform of the individual functions by convolution. This is illustrated for in-phase and antiphase double doublets in Fig. 14.2. It is notable that the z operators result in the formation of antiphase doublets while other operators lead to in-phase doublets. Thus in 2D COSY spectra coupling is related to 2IAxlgz terms, which are antiphase in character. Furthermore the phases of signals are also dependent on the operators : all terms having an lx operator (e.g. IAx, 2IgcIBz etc.) will be in absorption while terms with ly operators will be in dispersion. There will be separate phases for each of the two frequency dimensions n l and 12 in the 2D NMR spectrum which may be the same or different. Phase correction can lead to an exchange of phases. Now let us consider a simple 2D experiment e.g. 2D COSY which in its basic form consists of two 90° pulses which are separated by the evolution period ti (Fig. 14.3) (also see chapter 8 section 8.1.2). It is the coupling interactions between coupled nuclei which are observed in the COSY experiment and, as stated above, coupling is associated with product operators such as 2IgxIsz. A 90° y pulse transforms 2I2cI13z to -2IAzIsx-- in other words spin A which possessed transverse magnetization before the application of the 90° pulse (Iwx) is transformed so that it acquires longitudinal magnetization (Iw z) after the pulse. Similarly spin B which had longitudinal magnetization is transformed by the pulse so that it acquires transverse magnetization (IBz—•. I13x). A "coherence transfer" is now said to have occurred whereby the coherence of nucleus A has been transferred to the coherence of nucleus B. Antiphase magnetization exists before and after the coherence transfer, caused by the JAB coupling ("active" coupling). Couplings to other nuclei (e.g. nucleus C) do not participate in the coherence transfer, and appear in-phase ("passive" couplings). The generation of a cross-peak between nuclei A and B in a three-spin coupled system 90°
90° ti
2 1 0
-1
'-----
---______
1
-2 Fig.14.3: COSY pulse sequence with the corresponding coherence transfer pathway.
518
( A,B and C) results from the following sequence of events (only the terms lAx, lBx (r IVx being detected at the end of t2): 0
'Az ~ ~ y IAx
r1
--21AxIBz
0 90
t ) .2I AzI Bc 2 1Bc ;
(19)
The first 90° pulse converts longitudinal A magnetization to transverse magnetization which evolves under the influence of chemical shift and coupling frequencies, and the resulting product operator (-2hxh z) is then transformed by the second 90° pulse (the polarisation transfer pulse) to the product operator 2I AzIBc which is detected during t2 as IBx. The overall sequence involves conversion of the z magnetization of spin A (IAz) to the x magnetization of spin B (IBS), giving rise to a cross-peak at "A, 1B. Coherences evolve during ti, under the influence of the chemical shift frequency of spin A, and during t2 under the chemical shift frequency of spin B In addition to the presence of cross-peaks, COSY spectra contain peaks lying along the diagonal. For a diagonal signal to appear at NA, vA = WA), spin A must have one transverse operator before the mixing in order for its chemical shift to evolve during ti, and it must remain transverse after the mixing pulse: 'Az
90
t y _ lAx 1
t
IAy 90 y _ Ilg
Ilx
(20)
14.2 DIPOLAR COUPLING The construction of 2D multiplets described above was concerned with scalar coupled nuclei in which J coupling resulted in operators during the evolution period which were responsible for coherence transfer under the influence of pulses. If the operators of the two coupled nuclei are represented by the symbols 1 and S, then this coherence transfer may be shown as: J(t1)_
Ic
2IyS7
M ixipg
2IzSy _
J(t 2)
~ sx
(21)
When the nuclei are not scalar coupled but are involved in dipolar coupling then this coupling must be effective during the mixing period amd not during the evolution and detection periods: Ic
1
~ I~
nie
Sz
? t
(22)
The 2D experiment which allows measurement of such dipolar coupling interactions is NOESY (see chapter 9). The NOESY pulse sequence, and the corresponding coherence transfer pathway are shown in Fig. 14.4. This may he represented in terms of the following operators:
519
90°
90°
90°
2
1 0 -1 2 Fig. 14.4: The NOESY pulse sequence and the corresponding coherence transfer pathway.
z
° 90I
t ~.
-I g —+ -Ig —
O +~~Iz
nie
Sz
90° c
Sy
~
Sx
(23)
When nie is positive, Iz is transformed to -Si, so that the signs of the diagonal and cross-peak signals are opposite. However when nie is negative, then I z is tranformed to Sz, so that the diagonal and cross-peaks have the same signs. If the nuclei land S are also scalar coupled, then undesired signals may be observed due to the following alternative transfer: 1
90 °
Iz
_;-Iy-
2IySz --r: 2IZSy 9
m _ Sx 90
t
°
x
t Sx Sx ?
(24)
These signals may be eliminated by phase cycling procedures. The effect of the various evolution operators (top row) on the operators depicting the state of the spin system (left column) is shown in Table 14.1 Considering only two coupled nuclei I and S, and focussing our attention only on Iz (ignoring Sz for the sake of simplicity), the evolution of magnetization in the COSY experiment may be represented as shown in Fig. 14.5 in the form of an evolution "tree" which leads to thirteen terms in the final density operator. The longitudinal magnetization (z), multiple-quantum coherence (M) and antiphase magnetization terms (nine in all) are not observable. The coherence that precesses at the chemical shift frequencies both in ti and t2 gives rise to the diagonal peaks (D), whereas the coherence which is modulated by the chemical shift frequencies of the neighbouring coupled nuclei gives rise to the cross-peaks (C). It is interesting to compare the various 2D NMR experiments involving the application of three 90° pulses. These are (a) COSY with multiple-quantum filtering (b) relayed COSY (c) NOESY and (d) multiple-quantum spectroscopy. The experiments differ from one another essentially in the function of the second 90° pulse, the antiphase magnetization being selected by the direction of this pulse relative to the first 90° pulse:
Iz
i
S
Fig. 14.5: Evolution of product operators in the COSY experiment. The evolution of Sz is omitted for the sake of simplification. Each left hand branch corresponds to multiplication of the cosine of the values given in the left column (eg. Iy cos (2281 ti)) while each right-hand branch corresponds to multiplication by the appropriate sine term. The 13 product operators finally obtained are those of z-magnetization (Z), multiple-quantum coherence (M), antiphase magnetization (A), diagonal peaks (D) or cross-peaks (C).
(lTJt2 ) 21 252
(24d g2)
(2pdg 2 ) I Z
(p/2) S c
(p/2)
(HJt 1) 21 2 5,
(2it~1t1)
( 12) I x
ROTATION
521
Table- 14.1: Effect of the evolution operators (top row) on operators depicting the state of the spin system (left column) Ix Ic Iy
I. Sx
E/2
I,
-ly
E/2
Sy
E/2
Sz
E/2
Iy - I7
I,
Sx
Sy
Iy
E/2
E/2
B/2
2IySz
Sz
2IzSz
B/2
-I x
B/2
B/2
B/2
-2I XS7
l,
E/2
E/2
E/2
B/2
E/2
E/2
E/2
B/2
E/2
E/2
Sz
E/2
E/2
-Sy
-Sz B/2 Sx
Sy
2I 7Sy
-Sx
-2I,Sx
E/2
B/2
2IxSx
E/2
- 2I7Sx
2IySx E/2
2IxSy
B/2
2I xS1
E/2
- 2I 7Sy
2IySy
2IxSz
B/2
-2I xSx
B/2
2IxSx -21ySz
E/2
ly
2IySy
E/2
-2IxSz
2IxS7
E/2
- 2I 7Sz
2IySz
-2IxSy
2IySx
2IzSx
E/2
-2I xSx
E/2
2IyS1
2I7Sy
E/2
-2IxSy
2IySz
B/2
2I gSz
2I 7S7
E/2
-2I xSz
-2IySy
2IySx
2I7Sx 2I,S1
- 2IySx
2IxSx
E/2
E/2
-2I7Sz
- 2IySy
2I xSy
E/2
2I,S,
E/2
2I 7S,
- 2IyS7
2I xSz
E/2
-2I 7Sy
21zSx
o 2IAzIBx 90 U Relayed COSY
~- 2IpxIBz
90 ° x
-2IySx E/2
E/2 -I c
2IzSy
Sy
-2I zSx
-Sx
E/2
2IgxIBy DQF COSY multiple-quantum experiments
B/2
(25)
In the relayed COSY experiment the coherence transfer is created via single-quantum antiphase magnetization (2IAxIBz --·2IgzIgx). In the NOESY experiment the second 90° pulse results in the generation of longitudinal z magnetization (Ix --~IZ ) while in DQF-COSY and multiple-quantum experiments it leads to the production of multiple-quantum coherence (Table 14.2). The product operator approach may be usefully applied in the understanding of the various 2D NMR experiments given in the text. However the reader is referred to other more specialised sources for further details (ref. 6,7).
14.3 PHASE CYCLING AND PRODUCT OPERATORS Phase cycling involves the systematic variation ("cycling") of the phase of the transmitter pulse as well as that of the receiver while other parameters are kept
522
Table 14.2: A comparison of some 2D NMR experiments with three 90° pulses
90
0
90 °
Experiment
9 0°
DI
D2
Remarks
a)
COSY with multiple quantum filter
ti
D
Value of D is very small
b)
Relayed COSY
ti
D
180° pulse added with advantage during mixing time D2
c)
NOESY
t1
t
Dipolar coupling is transferred during T
d)
Multiple-quantum spectroscopy
ti
A refocussing 180° pulse is included in the middle of the D period
D
Note: D is used above if the coupling effects participate in the evolution period, and t is used if relaxation or exchange processes are involved during the evolution period. constant. The FIDs obtained in this manner for each t i increment are co-added. By this procedure one can select desired coherence pathways or suppress artifact signals To illustrate the use of phase cycling in suppressing artifact peaks, the suppression of axial peaks in the COSY experiment can he taken as an example. The second 900 pulse of the COSY experiment causes the coherence transfer: o 2lpxlsz Y90---2IAzIsx
( 26 )
where A and B are the scalar coupled spins. If one changes the phase of the mixing pulse by 180° (i.e. if a 900-y pulse is applied), then the same product operators are still obtained: -90
2lpxlsz
o
g
- -2IAzIBx
(27)
523
Thus if the phase of the second 90° pulse is altered by 180°, the same COSY peaks will still be obtained. The undesired axial peaks (which result from the conversion of residual longitudinal magnetization by the 90 0 mixing pulse to detectable transverse magnetization) are found to occur at Vi = 0, and they are not modulated during ti. A change of phase from 900y to 900 _y causes a reversal in the sign of these axial signals:
IAz
90
'Az
90-
o y
IAx
(28)
_ -IAc
(29)
If successive experiments (90° _y - ti - 90° y - t2 and 90° y - ti - 90° _y - t2) are added together, the desired COSY signals are added up while the axial peaks are cancelled. When designing phase cycling procedures, it is notable that the first pulse can only create single-quantum coherence (positive/negative) from the thermal equilibrium (p = 0) state. Furthermore the detector selects, by convention, negative quantum coherences. Phase cycling is applied for selection of desired coherence pathways. In general, pulses will excite all the coherence orders between -n to + n where "n" is the number of terms in the product operator. If (3x represents a pulse of angle p and phase x, and A, B and C are three mutually coupled spins, then the application of the pulse will cause the coherence-transfer processes given by the following equation: IAxIByICz
IAx(Isycos R+ IBzsin R) (ICzcos R— Icysin R )
(30)
Phase cycling procedures act like sieves, allowing only certain coherence order transfer processes to filter through. It is the "mesh" of the "sieve" which determines how many coherence order pathways will be selected. The type of "sieve" chosen for the selection process is determined by the selectivity N of the phase cycle, and by the change in coherence order, Dr, so that Dr = – kN (where k = 0,1,2,3...). If we consider the coherence orders from a three-spin system (Fig. 14.6a), there are 14 coherence pathways possible. If we impose a phase cycling scheme in which N is 3 and Ar is -1, then the five coherence paths shown in Fig. 14.6c will be selected. The sensitivity N of the phase cycle is determined, as indicated in an earlier section, by the magnitude by which the phase of the pulse is changed, this change being 2p/N. For instance in order to obtain a selectivity of 3, the phase will need to be increased by 360/3° = 120° . If the pulse phases is increased by o° , then the detector phase must also be increased by - Dpe .Since in the above example Dr was -1, and e was 120° , so the detector phase has to be changed by -(-1)120° = 120° . By this procedure, the receiver follows the selected coherence transfer pathway exactly. Instead of discussing phase cycling schemes in terms of Cartesian operators, an alternative is to discuss them in terms of single element operators (ref. 8-12) as represented by the following equations:
524
Fig.14.6: (A) A pulse from a single-quantum coherence generates uptO 14 possible coherence transfer pathways. (B) If a certain "sieve" is placed between these pathways, then certain pathways can be selected, so that the jumps correspond to D p + k N where k = 0,1,2; the selectivity N in the example shown is 3 and D p is -1. (C) Thus out of the 14 possible pathways, 5 are selected corresponding to jumps D p of 5, 2, -1 or -5. (Reproduced with permission from H. Kessler et al., AnWgew.Clem.lrmt.Ed.Engl., 27, 490-536 (188), copyright 1988, VCH Verlagsgesellschaft).
I+ =
Ic+ ~Ig f?~I+ e -' F
I
Ic + iIy f ?
=
-··-I-
e±
(31) (32)
I + is a "raising" operator i.e. it leads to a positive change in the quantum number, mz, while G is a "lowering" operator leading to a negative change in the quantum number. I + therefore corresponds to the emission of a single quantum (b + a), while G corresponds to the absorption of a single quantum (a __..). The total number of quanta absorbed will therefore equal the number of ~- operators minus the number of! + operators in the product operator. The coherence order, p, will be the number of I + operators minus the number of G operators in a product operator. Hence for Iá + IB, p will be equal to zero, corresponding to a zero-quantum coherence. For IÁIB or IgIB, p will be +2 or-2 respectively, corresponding to double-quantum coherences (Fig. 14.7). This coherence order remains unchanged during periods of free precession - a manifestation of the conservation rule. The Cartesian product operators may be considered as sums of positive and negative single- quantum coherences: IAc
=
IAg =
1/2 (I~~ + IA)
(33)
-1/2i (I~~IA )
(34)
+ 2IAyIBz = (IA + IA ) IBz
(35)
525
90°
1 H
90°
-~
-~
+-
I
13
D
180°
c
2 1 1 H
t1
180°
0
-1
~
—
D
-a~~..3.
180°
90° D
f
D2
BB
90°
2
D3 .
~ D3
i
-----~
-2
2 13
C
1
o
1
-2
Fig. 14.7: Pulse sequence for the heteronuclear H-relayed H,X-COSY experiment. (Reproduced with permission from H. Kessler et al., Angew.Clzem.Int.Ed.Engl., 27, 490-536 (1988), copyright 1988, VCH V erlagsgesellschaft).
It is the number of transverse operators which determines the highest (p = + n) and lowest (p = -n) coherence orders in a product operator, all the values from p = + n to p = -n in increments of 2 being valid. This is exemplified as follows:+ 1, - 1
(36)
4Iwy1BxlCz :
+ 2, 0, -2
(37)
4I~xIBcICc :
+3, + 1, -1, -3
(38)
2IAcIBc
526
REFERENCES 1.
O.W. Sorensen, G.W. Eich, M.H. Levitt, G. Bodenhausen and R.R. Ernst, "Product operator formalism for the description of NMR pulse experiments, Prig. Nucl. Magn. Reson. Spectrosc., 16 (1983) 163-192. 2. F.J.M. van de Ven and C.W. Hilbers, A simple formalism for the description of multiple pulse experiments. Application to a weakly coupled two-spin (I = 1/2) system, J. Magn. Resonance, 54(1983)512-520. 3. K.J. Packer and K.M. Wright, The use of single-spin operator basis sets in the NMR spectroscopy of scalar coupled spin systems, Mot. Phys., 50 (1983) 797-813. 4. L.R. Brown and J. Bremer, J. Magna. Resonance, 68 (1986) 217- 231. 5. J. Hilton and L.H. Sutcliffe, Through space mechanism in spin-spin coupling, Prig. lud. Magn. Reson. Spectrosc., 10 (1975) 27-39. 6. R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of nuclear magnetic resonance in one and two-dimensions, Clarenden Press Oxford (1987). 7. H. Kessler, M. Gehrke and C. Griesinger, Two-dimensional NMR spectroscop : backgound and overview of the experiments, Angew. Chem. Int. Ed. Engl., 27 (1988) 490-536. 8. A. Wokaun and R.R. Ernst, Selective excitations and detection in multiple level spin systems: application of single transition operators, J. Chem. Phys., 67 (1977) 1752-1758. 9. S. Vega, Fictitious spin 1/2 operator formalism for multiple quantum NMR, J. Chern. Phys., 68(1978)5518-5527. 10. D.P. Weitekamp, J.R. Garbow and A. Pines, Determination of dipole coupling constants using heteronuclear multiple quantum NMR, J. Chem. Phys., 77 (1982) 2870-2883. 11. D.P. Weitekamp, Time-domain multiple-quantum NMR, Adv. Magn. Resin., 11 (1983) 111-274. 12. V. Blechta and J. Schrame, Product formalism applied to large spin systems with INEPT and DEPT examples, J. Magn. Resonance, 69 (1986) 293-301.
Chapter 14
Product Operators Approach to 2D-NMR Spectroscopy Two-dimensional h R spectroscopy was presented above in terms of the magnetization vector approach in which the macroscopic magnetization rotated under the influence of pulses and precessed on account of the chemical shifts. It may be alternatively discussed in terms of product operutorfoirnalism (with Cartesian basis operators) (ref. 1-4) in which the observable magnetization resulting from the entire spin system may be described by the density matrix. An introduction to this approach has already been given in Chapter 1, Section 1.2.16.3 and the readers are advised to read that section before the following portion on the product operator approach. For the sake of clarity let us dwell again on some of the underlying principles of this approach. In the ground state of the system in thermal equilibrium, the net magnetization is represented by M z and the polarisation along the z axis is described by Iz. If a radiofrequency pulse which bends the magnetization by an angle R is applied along the y axis, the Cartesian operators, IX, Iy and I z (where Ix, Iy and Iz are related to x, y and z magnetizations) will be transformed as follows: I c Y~Ic COS (3 - Iism (3 bI
ly
lz
bI
Y U
, .Iy I z cos p + l sin 1b
(1)
(2) (3)
Each of these three transformations show how the starting state of the spin system (left side of "equation") evolves to its end state (on the right side of the "equation") under the influence of a particular operator (shown on top of the arrow). This operator may represent interactions due to chemical shifts, couplings, pulses etc. Thus if the operator above the arrow is Wl zt, then this will represent the evolution of the chemical shift of nucleus I during time t. On the other hand if the operator is shown as bIxy (or, more generally, RI~) then this will designate a pulse of flip angle ß about an axis which forms an angle with the x axis in the x,y plane. Similarly the influence of coupling J between nuclei A and B (operators J A and I B) during time t maybe represented by 2p JAZJBzt•
512
Fir weakly coupled two-spin systems of nuclei A and B, the following 16 product operators will exist: B/2 : E represents the unity operator I x: represents x component of! spin magnetization Iy: represents y component of! spin magnetization Iz: represents z component of! spin magnetization Sx: represents x component of S spin magnetization Sy : represents y component of S spin magnetization Sz : represents z component of S spin magnetization 2IXSx, 2IySy, 2IySx, 2IxSy : represent two-spin coherences 2IxSz, 2IySz : represent antiphase I spin magnetization 2IzSx, 2IzSy : represent antiphase S spin magnetization 2I zSz : represents longitudinal two-spin order Let us examine the meaning of some of the above terms. 2IxSz indicates that the x component of I spin magneization is split into two antiphase components corresponding to the a and p states of nucleus S. This is detectable in the NMR spectrum as a signal. The product operator 2I zSz results in no net polarisation although all the populations of the four energy levels are disturbed. The product operator 2I XSx represents the superpositioning of zero-quantum coherence (opposite spin-flips of I and S nuclei) and double-quantum coherence (spin-flips of! and S nuclei in the same sense). Such two-spin coherences cannot be detected directly, but they need to be converted to single-quantum coherences prior to detection by two-dimensional NMR spectroscopic methods. A linear combination of these product operators corresponds to zero-quantum coherences: 2IzSx + 2IySy or 2IySx - 2IxSy. The alternative linear combinations 2IXSx - 2IySy or 2IxSy + 2IySx correspond to double-quantum coherence. The effect of the evolution of chemical shift W of nucleus I during time t on the Ix, Iy and I z operators may be designated as: Ix WIzt — It.t
Ig W IZ
Ix cos W t + I g sin W t
, Iy cos W t- Ix sin W t IZ
(4) (5) (6)
513
Hence if we are detecting the y magnetization (in equation (4)) then the signal amplitude will be seen to grow by sin wt, and Fourier transformation will afford single lines at ± w , one of which (at +w or -w ) may be detected by quadrature detection. The operators for one kind of nucleus in a iD experiment may be represented in terms of the evolution of magnetization which occurs during the detection period t2: IZ
90~~
Ic
Evolution Detection period t 2
Ix cos t2 + Iy sin w t2
( 7)
In a 2D experiment, on the other hand, the 90° y pulse will result in the conversion of equilibrium magnetization Iz to Ix which will evolve during the evolution period t1 with its characteristic frequency: Ic
WlI zt1
Ic cos W1 ti
+ Ig sin W] ti
(8)
Let us assume that I represents 1H magnetization and S represents C-magnetization. The application of simultaneous 900 pulses to both the 1H and 13C nuclei1H results in a mixing of the magnetizations so that coherence can be transferred from nuclei to 13C nuclei i.e. Ix is transformed to Sx. The Iy term may be ignored as it does not contribute to the outcome of the experiment ± . During the evolution time t2, Sx will evolve with its characteristic frequency: 13
W St Sxs 2 a Sx COS WV t2 + Sy sin Ws t2
(9)
At the end of time period t2, the spin system may be represented as: cos UI ti (Sx cos ws t2 + Sy sin Ws t2)
(10)
The important term to note in equation (10) is in. In other words the S operator coefficients, representing its evolution in time periods t1 and t2, are governed not only by its own characteristic frequencies Ws but also by the frequencies Wt of the other I nucleus to which nucleus S is coupled. The magnetization of nucleus S has therefore been "labelled" during ti by WI, i.e. the frequency of nucleus I. Quadrature detection followed by Fourier transformation results in the separation of the frequencies ± in and Ws into the two frequency domainsvi and½ respectively, thereby affording the two-dimensional spectrum. The appearance of a cross-peak at (UI,Us) in the 2D spectrum will then indicate the modulation of the signal at the I frequency (01) during t1 and by the S frequency (Ws) during t2. The cross-peaks in the 2D spectrum therefore arise at the frequencies at which coherence transfer occurs. The more efficient the mixing process, the stronger will be the signals observed in the 2D spectrum. The + This is so because the positive and negative components of the Iy terms mutually cancel one another.
514
coherence transfer may be brought about by (a) spin-spin scalar coupling through bonds (b) dipolar coupling through space or (c) exchange processes.Since these processes play a central role in 2D spectra, it is appropriate to discuss them in terms of the product operator formalism.
14.1 SCALAR COUPLING Scalar spin-spin coupling takes place mainly through bonds, though through-space coupling is also rarely seen (ref. 5). When two nuclei A and B are coupled to one another, then the spin-spin coupling between them may be represented in product operator formalism as follows: lAx
~ J ABt 2I I A7 B7
lAy 2IAxIBz 21AyIBz
JABt 2IAzIBz JABt 2IAzIBZ I
ABt 2l Az l Bz
— lAx cos p JABt + 2IAy1Bz sin p JABt —
lAy COS
P
(11)
JABt + 2IAxI B sin p JABt
(12)
2IAcIBz cos P JABt + IAg sin p JABt
(13)
21AyIBz cos p JABt - lAx sin p JABt
(14)
Let us revertto the vector approach to understand the implications of the above equations. If we focus our attention on the A vector, then coupling of spin A with spin B will result in there being two different vectors for spin A, depending on whether spin B is in an a or ß state. Equation (11) shows that the x magnetization of spin A is modulated by cos lI Jt. The y magnetization of spin A will be directed towards the +y axis when spin B is in an a state, and towards the -y axis when spin B is in its state, lAy (IBa - IBb), so that it will not be observed because of mutual cancellation of the positive and negative components (Fig. 14.1).
I
cosnJt+ I A sinnJt)
1
B~ A
t
c
y
cos 11 Jt x
q
cosnJt-I A sin n Jt) c
2IA b y Z
y
sin p Jt
Fig. 14.1: Evil ution of nucleus A due to scalar coupling (with nucleus B) during time Ii' At the beginning of the evolution period, the x magnetization of nucleus A is represented by lAx. At the end of the evolution period, the antiphase magnetization which develops is shown with wavy vectors, while the residual in-phase magnetization is shown as the central unbroken vector (IAx cos'PJt).
515
The difference in population between the a and b levels (1aR - Ib) is 2Iz. Hence aR IAy(h - IBb ) may be written as 2IAyh z which will undergo modulation by sin iF JAit (equation 11). It may be noted that while IAy and Igz are "operators", the term "product operator" will be applied to the products of operators e.g. IAyIBz. Some basic rules which govern operator transformations are: a) Each operator in the product operator is transformed separately during the evolution period when it is subjected to the modulation effects of the chemical shift or coupling frequencies, as indicated below, Iz not being transformed by chemical shift frequencies: 2IAcIBc 2IAcIBc 2IAcIBy
90 U
(15)
~-2IAcIBc
W AI AZ t+W BI Bz t — 2l,e, WA IAz t+W RI R~ t
cIg z
cosWAt + 2IAylBzsinWAt
(16)
2(IAccosWAt+IAysinWAt) (IgycosWgt - IBcsinWS3t)
(17)
(b) Product operators of two transverse operators (e.g. IMIgx), or of two longitudinal operators (e.g. IAzIBz) do not show mutual coupling, coupling being observed between products of transverse and longitudinal operators (e.g. IAXIBz). However coupling will also be observed between two transverse product operators with a third nucleus, C. (c) The influence of chemical shift and coupling frequencies are simultaneously observed. Under conditions of weak coupling the following additional considerations have to be taken into account: (i) Product operators with only one transverse operator (Ix or Iy) are observable. (ii) Product operators having any number of longitudinal operators Iz but only one transverse operator (I,, or Iy) are observable provided the nucleus corresponding to the transverse operator is scalar coupled to the nucleus with the longitudinal operator(s). (iii) Other product operators do not evolve as observable operators. In a three-spin system in which nucleus A is coupled to two other nuclei B and C, the spectrum will contain a double doublet for A at WA. IAA will evolve during ti to give a total of 8 terms, but as stipulated by the above rules, only the first two terms having the transverse operators IAA and IAy are observable: IAA
IAxcoswl tcos IT JABt cos p JACt + IAysinWAt cos p JABt cos iFJACt (+ 6 other terms)
(18)
516
( A) AB
cos p J ABt
cos lT J AC t
cos p J AB t.cos p J AC t
AC
I
i~J AB
Ii
I
~ AC
(B)
sin wJ ABt
J
cos .J AC t
AC
--~
I
sin p J ABt.cos p J ACt
AC
I
BZ
~
(C )
J
-
AC I
cos n J ABt. si n n J AC t
C AX Z
(D)
I
si n n J ABt.sin n J AC t
I
AX I BZ I CZ
— i
AB
J
AC
I
Fig. 14.2: (A) Contribution of an in-phase double doublet from two in-phase doublets. (B) Contribution of a double doublet from two doublets one of which is in-phase and the other antiphase. (C) Double doublet with antiphase structure (antiphase with respect to Jlc). (D) Double doublet also having antiphase character, but antiphase with respect to both JAB and JAc. (Reproduced with permission from H.Kessler et al., Angew.Chem.Jn'.Ed. Engl., 27, 490-536 (1988), copyright 1988,NCH Verlagsgesellschaft).
517
To construct a signal from the product operators, one makes use of the principle which stipulates that the Fourier transform of a product of functions can be derived from the Fourier transform of the individual functions by convolution. This is illustrated for in-phase and antiphase double doublets in Fig. 14.2. It is notable that the z operators result in the formation of antiphase doublets while other operators lead to in-phase doublets. Thus in 2D COSY spectra coupling is related to 2IAxlgz terms, which are antiphase in character. Furthermore the phases of signals are also dependent on the operators : all terms having an lx operator (e.g. IAx, 2IgcIBz etc.) will be in absorption while terms with ly operators will be in dispersion. There will be separate phases for each of the two frequency dimensions n l and 12 in the 2D NMR spectrum which may be the same or different. Phase correction can lead to an exchange of phases. Now let us consider a simple 2D experiment e.g. 2D COSY which in its basic form consists of two 90° pulses which are separated by the evolution period ti (Fig. 14.3) (also see chapter 8 section 8.1.2). It is the coupling interactions between coupled nuclei which are observed in the COSY experiment and, as stated above, coupling is associated with product operators such as 2IgxIsz. A 90° y pulse transforms 2I2cI13z to -2IAzIsx-- in other words spin A which possessed transverse magnetization before the application of the 90° pulse (Iwx) is transformed so that it acquires longitudinal magnetization (Iw z) after the pulse. Similarly spin B which had longitudinal magnetization is transformed by the pulse so that it acquires transverse magnetization (IBz—•. I13x). A "coherence transfer" is now said to have occurred whereby the coherence of nucleus A has been transferred to the coherence of nucleus B. Antiphase magnetization exists before and after the coherence transfer, caused by the JAB coupling ("active" coupling). Couplings to other nuclei (e.g. nucleus C) do not participate in the coherence transfer, and appear in-phase ("passive" couplings). The generation of a cross-peak between nuclei A and B in a three-spin coupled system 90°
90° ti
2 1 0
-1
'-----
---______
1
-2 Fig.14.3: COSY pulse sequence with the corresponding coherence transfer pathway.
518
( A,B and C) results from the following sequence of events (only the terms lAx, lBx (r IVx being detected at the end of t2): 0
'Az ~ ~ y IAx
r1
--21AxIBz
0 90
t ) .2I AzI Bc 2 1Bc ;
(19)
The first 90° pulse converts longitudinal A magnetization to transverse magnetization which evolves under the influence of chemical shift and coupling frequencies, and the resulting product operator (-2hxh z) is then transformed by the second 90° pulse (the polarisation transfer pulse) to the product operator 2I AzIBc which is detected during t2 as IBx. The overall sequence involves conversion of the z magnetization of spin A (IAz) to the x magnetization of spin B (IBS), giving rise to a cross-peak at "A, 1B. Coherences evolve during ti, under the influence of the chemical shift frequency of spin A, and during t2 under the chemical shift frequency of spin B In addition to the presence of cross-peaks, COSY spectra contain peaks lying along the diagonal. For a diagonal signal to appear at NA, vA = WA), spin A must have one transverse operator before the mixing in order for its chemical shift to evolve during ti, and it must remain transverse after the mixing pulse: 'Az
90
t y _ lAx 1
t
IAy 90 y _ Ilg
Ilx
(20)
14.2 DIPOLAR COUPLING The construction of 2D multiplets described above was concerned with scalar coupled nuclei in which J coupling resulted in operators during the evolution period which were responsible for coherence transfer under the influence of pulses. If the operators of the two coupled nuclei are represented by the symbols 1 and S, then this coherence transfer may be shown as: J(t1)_
Ic
2IyS7
M ixipg
2IzSy _
J(t 2)
~ sx
(21)
When the nuclei are not scalar coupled but are involved in dipolar coupling then this coupling must be effective during the mixing period amd not during the evolution and detection periods: Ic
1
~ I~
nie
Sz
? t
(22)
The 2D experiment which allows measurement of such dipolar coupling interactions is NOESY (see chapter 9). The NOESY pulse sequence, and the corresponding coherence transfer pathway are shown in Fig. 14.4. This may he represented in terms of the following operators:
519
90°
90°
90°
2
1 0 -1 2 Fig. 14.4: The NOESY pulse sequence and the corresponding coherence transfer pathway.
z
° 90I
t ~.
-I g —+ -Ig —
O +~~Iz
nie
Sz
90° c
Sy
~
Sx
(23)
When nie is positive, Iz is transformed to -Si, so that the signs of the diagonal and cross-peak signals are opposite. However when nie is negative, then I z is tranformed to Sz, so that the diagonal and cross-peaks have the same signs. If the nuclei land S are also scalar coupled, then undesired signals may be observed due to the following alternative transfer: 1
90 °
Iz
_;-Iy-
2IySz --r: 2IZSy 9
m _ Sx 90
t
°
x
t Sx Sx ?
(24)
These signals may be eliminated by phase cycling procedures. The effect of the various evolution operators (top row) on the operators depicting the state of the spin system (left column) is shown in Table 14.1 Considering only two coupled nuclei I and S, and focussing our attention only on Iz (ignoring Sz for the sake of simplicity), the evolution of magnetization in the COSY experiment may be represented as shown in Fig. 14.5 in the form of an evolution "tree" which leads to thirteen terms in the final density operator. The longitudinal magnetization (z), multiple-quantum coherence (M) and antiphase magnetization terms (nine in all) are not observable. The coherence that precesses at the chemical shift frequencies both in ti and t2 gives rise to the diagonal peaks (D), whereas the coherence which is modulated by the chemical shift frequencies of the neighbouring coupled nuclei gives rise to the cross-peaks (C). It is interesting to compare the various 2D NMR experiments involving the application of three 90° pulses. These are (a) COSY with multiple-quantum filtering (b) relayed COSY (c) NOESY and (d) multiple-quantum spectroscopy. The experiments differ from one another essentially in the function of the second 90° pulse, the antiphase magnetization being selected by the direction of this pulse relative to the first 90° pulse:
Iz
i
S
Fig. 14.5: Evolution of product operators in the COSY experiment. The evolution of Sz is omitted for the sake of simplification. Each left hand branch corresponds to multiplication of the cosine of the values given in the left column (eg. Iy cos (2281 ti)) while each right-hand branch corresponds to multiplication by the appropriate sine term. The 13 product operators finally obtained are those of z-magnetization (Z), multiple-quantum coherence (M), antiphase magnetization (A), diagonal peaks (D) or cross-peaks (C).
(lTJt2 ) 21 252
(24d g2)
(2pdg 2 ) I Z
(p/2) S c
(p/2)
(HJt 1) 21 2 5,
(2it~1t1)
( 12) I x
ROTATION
521
Table- 14.1: Effect of the evolution operators (top row) on operators depicting the state of the spin system (left column) Ix Ic Iy
I. Sx
E/2
I,
-ly
E/2
Sy
E/2
Sz
E/2
Iy - I7
I,
Sx
Sy
Iy
E/2
E/2
B/2
2IySz
Sz
2IzSz
B/2
-I x
B/2
B/2
B/2
-2I XS7
l,
E/2
E/2
E/2
B/2
E/2
E/2
E/2
B/2
E/2
E/2
Sz
E/2
E/2
-Sy
-Sz B/2 Sx
Sy
2I 7Sy
-Sx
-2I,Sx
E/2
B/2
2IxSx
E/2
- 2I7Sx
2IySx E/2
2IxSy
B/2
2I xS1
E/2
- 2I 7Sy
2IySy
2IxSz
B/2
-2I xSx
B/2
2IxSx -21ySz
E/2
ly
2IySy
E/2
-2IxSz
2IxS7
E/2
- 2I 7Sz
2IySz
-2IxSy
2IySx
2IzSx
E/2
-2I xSx
E/2
2IyS1
2I7Sy
E/2
-2IxSy
2IySz
B/2
2I gSz
2I 7S7
E/2
-2I xSz
-2IySy
2IySx
2I7Sx 2I,S1
- 2IySx
2IxSx
E/2
E/2
-2I7Sz
- 2IySy
2I xSy
E/2
2I,S,
E/2
2I 7S,
- 2IyS7
2I xSz
E/2
-2I 7Sy
21zSx
o 2IAzIBx 90 U Relayed COSY
~- 2IpxIBz
90 ° x
-2IySx E/2
E/2 -I c
2IzSy
Sy
-2I zSx
-Sx
E/2
2IgxIBy DQF COSY multiple-quantum experiments
B/2
(25)
In the relayed COSY experiment the coherence transfer is created via single-quantum antiphase magnetization (2IAxIBz --·2IgzIgx). In the NOESY experiment the second 90° pulse results in the generation of longitudinal z magnetization (Ix --~IZ ) while in DQF-COSY and multiple-quantum experiments it leads to the production of multiple-quantum coherence (Table 14.2). The product operator approach may be usefully applied in the understanding of the various 2D NMR experiments given in the text. However the reader is referred to other more specialised sources for further details (ref. 6,7).
14.3 PHASE CYCLING AND PRODUCT OPERATORS Phase cycling involves the systematic variation ("cycling") of the phase of the transmitter pulse as well as that of the receiver while other parameters are kept
522
Table 14.2: A comparison of some 2D NMR experiments with three 90° pulses
90
0
90 °
Experiment
9 0°
DI
D2
Remarks
a)
COSY with multiple quantum filter
ti
D
Value of D is very small
b)
Relayed COSY
ti
D
180° pulse added with advantage during mixing time D2
c)
NOESY
t1
t
Dipolar coupling is transferred during T
d)
Multiple-quantum spectroscopy
ti
A refocussing 180° pulse is included in the middle of the D period
D
Note: D is used above if the coupling effects participate in the evolution period, and t is used if relaxation or exchange processes are involved during the evolution period. constant. The FIDs obtained in this manner for each t i increment are co-added. By this procedure one can select desired coherence pathways or suppress artifact signals To illustrate the use of phase cycling in suppressing artifact peaks, the suppression of axial peaks in the COSY experiment can he taken as an example. The second 900 pulse of the COSY experiment causes the coherence transfer: o 2lpxlsz Y90---2IAzIsx
( 26 )
where A and B are the scalar coupled spins. If one changes the phase of the mixing pulse by 180° (i.e. if a 900-y pulse is applied), then the same product operators are still obtained: -90
2lpxlsz
o
g
- -2IAzIBx
(27)
523
Thus if the phase of the second 90° pulse is altered by 180°, the same COSY peaks will still be obtained. The undesired axial peaks (which result from the conversion of residual longitudinal magnetization by the 90 0 mixing pulse to detectable transverse magnetization) are found to occur at Vi = 0, and they are not modulated during ti. A change of phase from 900y to 900 _y causes a reversal in the sign of these axial signals:
IAz
90
'Az
90-
o y
IAx
(28)
_ -IAc
(29)
If successive experiments (90° _y - ti - 90° y - t2 and 90° y - ti - 90° _y - t2) are added together, the desired COSY signals are added up while the axial peaks are cancelled. When designing phase cycling procedures, it is notable that the first pulse can only create single-quantum coherence (positive/negative) from the thermal equilibrium (p = 0) state. Furthermore the detector selects, by convention, negative quantum coherences. Phase cycling is applied for selection of desired coherence pathways. In general, pulses will excite all the coherence orders between -n to + n where "n" is the number of terms in the product operator. If (3x represents a pulse of angle p and phase x, and A, B and C are three mutually coupled spins, then the application of the pulse will cause the coherence-transfer processes given by the following equation: IAxIByICz
IAx(Isycos R+ IBzsin R) (ICzcos R— Icysin R )
(30)
Phase cycling procedures act like sieves, allowing only certain coherence order transfer processes to filter through. It is the "mesh" of the "sieve" which determines how many coherence order pathways will be selected. The type of "sieve" chosen for the selection process is determined by the selectivity N of the phase cycle, and by the change in coherence order, Dr, so that Dr = – kN (where k = 0,1,2,3...). If we consider the coherence orders from a three-spin system (Fig. 14.6a), there are 14 coherence pathways possible. If we impose a phase cycling scheme in which N is 3 and Ar is -1, then the five coherence paths shown in Fig. 14.6c will be selected. The sensitivity N of the phase cycle is determined, as indicated in an earlier section, by the magnitude by which the phase of the pulse is changed, this change being 2p/N. For instance in order to obtain a selectivity of 3, the phase will need to be increased by 360/3° = 120° . If the pulse phases is increased by o° , then the detector phase must also be increased by - Dpe .Since in the above example Dr was -1, and e was 120° , so the detector phase has to be changed by -(-1)120° = 120° . By this procedure, the receiver follows the selected coherence transfer pathway exactly. Instead of discussing phase cycling schemes in terms of Cartesian operators, an alternative is to discuss them in terms of single element operators (ref. 8-12) as represented by the following equations:
524
Fig.14.6: (A) A pulse from a single-quantum coherence generates uptO 14 possible coherence transfer pathways. (B) If a certain "sieve" is placed between these pathways, then certain pathways can be selected, so that the jumps correspond to D p + k N where k = 0,1,2; the selectivity N in the example shown is 3 and D p is -1. (C) Thus out of the 14 possible pathways, 5 are selected corresponding to jumps D p of 5, 2, -1 or -5. (Reproduced with permission from H. Kessler et al., AnWgew.Clem.lrmt.Ed.Engl., 27, 490-536 (188), copyright 1988, VCH Verlagsgesellschaft).
I+ =
Ic+ ~Ig f?~I+ e -' F
I
Ic + iIy f ?
=
-··-I-
e±
(31) (32)
I + is a "raising" operator i.e. it leads to a positive change in the quantum number, mz, while G is a "lowering" operator leading to a negative change in the quantum number. I + therefore corresponds to the emission of a single quantum (b + a), while G corresponds to the absorption of a single quantum (a __..). The total number of quanta absorbed will therefore equal the number of ~- operators minus the number of! + operators in the product operator. The coherence order, p, will be the number of I + operators minus the number of G operators in a product operator. Hence for Iá + IB, p will be equal to zero, corresponding to a zero-quantum coherence. For IÁIB or IgIB, p will be +2 or-2 respectively, corresponding to double-quantum coherences (Fig. 14.7). This coherence order remains unchanged during periods of free precession - a manifestation of the conservation rule. The Cartesian product operators may be considered as sums of positive and negative single- quantum coherences: IAc
=
IAg =
1/2 (I~~ + IA)
(33)
-1/2i (I~~IA )
(34)
+ 2IAyIBz = (IA + IA ) IBz
(35)
525
90°
1 H
90°
-~
-~
+-
I
13
D
180°
c
2 1 1 H
t1
180°
0
-1
~
—
D
-a~~..3.
180°
90° D
f
D2
BB
90°
2
D3 .
~ D3
i
-----~
-2
2 13
C
1
o
1
-2
Fig. 14.7: Pulse sequence for the heteronuclear H-relayed H,X-COSY experiment. (Reproduced with permission from H. Kessler et al., Angew.Clzem.Int.Ed.Engl., 27, 490-536 (1988), copyright 1988, VCH V erlagsgesellschaft).
It is the number of transverse operators which determines the highest (p = + n) and lowest (p = -n) coherence orders in a product operator, all the values from p = + n to p = -n in increments of 2 being valid. This is exemplified as follows:+ 1, - 1
(36)
4Iwy1BxlCz :
+ 2, 0, -2
(37)
4I~xIBcICc :
+3, + 1, -1, -3
(38)
2IAcIBc
526
REFERENCES 1.
O.W. Sorensen, G.W. Eich, M.H. Levitt, G. Bodenhausen and R.R. Ernst, "Product operator formalism for the description of NMR pulse experiments, Prig. Nucl. Magn. Reson. Spectrosc., 16 (1983) 163-192. 2. F.J.M. van de Ven and C.W. Hilbers, A simple formalism for the description of multiple pulse experiments. Application to a weakly coupled two-spin (I = 1/2) system, J. Magn. Resonance, 54(1983)512-520. 3. K.J. Packer and K.M. Wright, The use of single-spin operator basis sets in the NMR spectroscopy of scalar coupled spin systems, Mot. Phys., 50 (1983) 797-813. 4. L.R. Brown and J. Bremer, J. Magna. Resonance, 68 (1986) 217- 231. 5. J. Hilton and L.H. Sutcliffe, Through space mechanism in spin-spin coupling, Prig. lud. Magn. Reson. Spectrosc., 10 (1975) 27-39. 6. R.R. Ernst, G. Bodenhausen and A. Wokaun, Principles of nuclear magnetic resonance in one and two-dimensions, Clarenden Press Oxford (1987). 7. H. Kessler, M. Gehrke and C. Griesinger, Two-dimensional NMR spectroscop : backgound and overview of the experiments, Angew. Chem. Int. Ed. Engl., 27 (1988) 490-536. 8. A. Wokaun and R.R. Ernst, Selective excitations and detection in multiple level spin systems: application of single transition operators, J. Chem. Phys., 67 (1977) 1752-1758. 9. S. Vega, Fictitious spin 1/2 operator formalism for multiple quantum NMR, J. Chern. Phys., 68(1978)5518-5527. 10. D.P. Weitekamp, J.R. Garbow and A. Pines, Determination of dipole coupling constants using heteronuclear multiple quantum NMR, J. Chem. Phys., 77 (1982) 2870-2883. 11. D.P. Weitekamp, Time-domain multiple-quantum NMR, Adv. Magn. Resin., 11 (1983) 111-274. 12. V. Blechta and J. Schrame, Product formalism applied to large spin systems with INEPT and DEPT examples, J. Magn. Resonance, 69 (1986) 293-301.