Chem 781 Part 7

Coupling constants from COSY experiments

Operators for two spin system

• For a two spin system there will be operators involving both one and two spins. • The one spin operators which are equivalent to the vectors describing x,y, or z

magnetization. • For two coupled spins IA and IB there is a total of 24 = 16 possible operators:

IzA, Ix

A, IyA, Iz

B, IxB, Iy

B (6) z, x or y magnetization of spin A or B (same as

in vector picture)

IxA Iz

B, IyAIz

B, ... (4) antiphase magnetization for spin A coupled to spin B

IxA Iy

B, IxA Ix

B, ... (4) double and zero quantum coherence of spin A and B

IzA Iz

B (1) longitudinal spin order (population inversion of A relative

to B)

1 (1) unity operator (needed for mathematical reasons)

Rules of rotation starting from an initial state

All interconversions between operators can be regarded as rotations, giving a sine and a cosine component:

Initial state • rotation operator = cos component(initial) +sine component(new)

• each component of a two spin operator can be treated separately

• if initial state is perpendicular to rotation axis, it will change as a cosine, and a new state will emerge perpendicular to rotation axis and initial state with a sine dependence. Sense of rotation given by right hand rule (note that some books use left hand rule).

• If initial state parallel to rotation axis, it will remain unchanged.

• Only the one spin operators (Iz, Ix, Iy ) will give rise to an observable

Effect of pulse: rotation about axis of pulse (x or y) by angle θ:Initial state rotation ))))> cos component(initial) sine component(new)Iz

A θxA Iz cos θ -Iy

A sin θIz

B θxA Iz

B -Ix

A θxA Ix

A -Iy

A θxA Iy cos θ -Iz

A sin θIx

A IzB θy

A,B IxA Iz

B cos θ -IzA Ix

B sin θIy

A IzB θxA Iy

A IzB cos θ IzA Iz

B sin θIy

A IzB θxB Iy

A IzB cos θ -IyA Iy

B sin θChemical shift: rotation about z axis by angle Ω tIz

A ΩA t IzA

IxA ΩA t Ix

A cos(Ω t) IyA sin Ω tIy

A ΩA t IyA cos(Ω t) -Ix

A sin(Ω t)Ix

A IzB ΩA t Ix

A IzB cos(Ω t) IyA Iz

B sin(Ω t)Iz

A IyB ΩA t Iz

A IyB

spin-spin coupling: rotation about IzzAAI,z,z

BB axis: interconversion between in phase x,y andanti phase x,y components by π J t:Iz

A π JAB t IzA

IyA π JAB t Iy

A cos(π JAB t) -IxA IzB sin(π JAB t)

IxA Iz

B π JAB t IxA Iz

B cos(π JAB t) IyA sin(π JAB t)Iz

A IxB π JAB t Iz

A IxB cos(π JAB t) IyB sin(π JAB t)

IxA Iy

B π JAB t IxA Iy

B (DQ-terms do not evolve mutual coupling)

Product Operator Description of COSY experiment

(90̊6)x t1 (90̊6)x acqu

It can be shown that simultaneous coupling and shift can be treated in sequence. Thus the following sequence of four rotations has to be considered:

90̊6xA,B → (ΩA,B t1)z → (π JAB t1)zz → 90̊6xA,B

• As each rotation creates two components, the number of terms will increase exponentially with the number of steps.

• Fortunally, 90̊6 pulses retain only the sine component, simplifying the analysis.

COSY part 1: 90⁰ pulse – t1

Initial 90̊ pulse90̊x

A,B

IzA + Iz

B )))))))))> - IyA - Iy

B since the problem is symmetrical we will only regard whathappens to spin A:

Chemical shift spin-spin couplingduring t1 during t1

ΩA t1 π JAB t1

-IyA )))))))))> IyA cos(ΩA t1) )))))))))> -IyA cos(ΩA t1) cos(π JAB t1)

+ IxAIz

B cos(ΩA t1) sin(π JAB t1)

+ IxA sin(ΩA t1) + Ix

A sin(ΩA t1) cos(π JAB t1)

+ IyA Iz

B sin(ΩA t1) sin(π JAB t1)

COSY part 2: Effect of second pulse

second 90̊ pulse along x:This pulse can be treated like two pulses, first one acting on spin A and then one acting on spin Bonly:

90̊xA,B

)))))))> -IzA cos(ΩA t1) cos(π JAB t1) (I) z magnetization of spin A gives no signal(Used in NOESY)

- IxAIyB cos(ΩA t1) sin(π JAB t1) (II) DQ coherence of spins A/B not observable

(Used in DQF-COSY)IxA sin(ΩA t1) cos(π JAB t1) (III) x magnetization of spin A => diagonal peak

- IzA IyB sin(ΩA t1) sin(π JAB t1) (IV) antiphase magnetization of spin B: willrefocus to x magnetization of B duringacquisition => cross peak

COSY party 3: acqusition

Evolution during acquisition (t2): we regard only terms III and IV, as the other two do not lead toobservable magnetization. We only keep the in phase x,y magnetization terms which are createdafter evolution of coupling:

ΩA t2, π JAB t2 III )))))))))))))> Ix

A sin(ΩA t1) cos(π JAB t1) cos(ΩA t2) cos(π JAB t2)diagonal peak

IyA sin(ΩA t1) cos(π JAB t1) sin(ΩA t2) cos(π JAB t2)

ΩB t2, π JAB t2IV )))))))))))))> Ix

B sin(ΩA t1) sin(π JAB t1) cos(ΩB t2) sin(π JAB t2)cross peak

-IyB sin(ΩA t1) sin(π JAB t1) sin(ΩB t2) sin(π JAB t2)

Summary COSY

• Always two sets of signals occur: Diagonal peaks (same W during t1 and t2) and cross peaks ( W A during t1 and W B during t2 or vice versa).

• Diagonal peak in phase doublet cos(p JAB t) in both f1and f2, cross peak anti phase with respect to JAB in both f1 and f2 sin (p JAB t)

• Cross and diagonal peaks 90̊ out of phase in both f1 and f2: III is along x and IV is along y axis at beginning of acquisition.

• .To avoid problems of dispersion peaks along the diagonal, the basic COSY spectrum is typically displayed in magnitude mode. That results in a loss in resolution and prevents accurate coupling constants to be measured.

Double Quantum filtered COSY• Adding a third 90̊6 pulse to the COSY experiment allows the double quantum term (II) to be

converted into antiphase magnetization, which will evolve obsevable magnetization during acquisition:

• Application of a four step phase cycle allows one to select only signal derived from this DQ term:

define: cos(ΩAt1)≡cA sin(ΩAt1) ≡ sA cos(πJAB t1) ≡ cJ sin(πJABt1) ≡sJ

(I) (II) (III) (IV)

scan 1: (+) IyA cAcJ -Ix

AIzB cA sJ Ix

A sA cJ IyA Iz

B sA sJ

scan 2: (-) IyA cAcJ Iz

AIxB cA sJ Iz

A sAcJ -IyA Ix

B sA sJ

scan 3: (+) IyA cA cJ -Ix

AIzB casJ -Ix

A sA cJ -IyA Iz

B sA sJ

scan 4: (-) IyA cA cJ Iz

AIxB cAsJ-Iz

A sA cJ IyAIx

B sA sJ

────────────────────────────────────────────

total: 0̊ -2 (IzAIx

B + IxAIz

B) cA sJ 0̊ 0̊

90̊x

(II) - IxAIyB)))))> -IxA IB

z cos(ΩA t1) sin(π JAB t1)

(90̊)x,y,-x,-y – t1 – (90̊)x,y,-x,-y (90̊)x acqusitionx,-x,x,-x (+,-,+,-)

One can show that this will result in diagonal and cross peaks both in anti phase and equal phase:

• The first part of the sum (IzAIx

B ) will give a cross peak, the second one (IxAIz

B) will give a diogonal peak. Both are modulated with a sine of JAB ( antiphase) and cos of ΩA (absorption) during t1, and will start in antiphase during acquisition, hence will be antiphase with respect to JAB also during t2.

• Thus a phase sensitive spectrum can be acquired in which both the diagonal and cross peaks can be phased. In addition, all singlet peaks will be suppressed as they will not evolve any DQ terms.

• Gradients can be used to select the double quantum term in one scan and allow for very efficient suppression of singlet signal, including water signals

Appearance of DQF spectra:Two spin system

H

A

B

A

B

H

-HA B

B

A

B JAB

Three spin system:

• Now two types of coupling have to be considered: The coupling between the two spins comprising the cross peak is called the active coupling. All other couplings are referred to as passive couplings (they give additional splitting of the cross peak).

• active couplings give an anti phase splitting

• passive couplings give in phase splittings

• Note that active and passive coupling are different for each cross peak.

• Up to 16 lines can occur for a multiplet.

Linear three spin system

A

H

-H

B CA

BC

JAB

JBC

JBC

JAB

General three spin system

Complexity of DQF-COSY spectra

• Four spin systems

• Cancellation of lines

Two couplings to chemical non equivalent partners can become very similar. If one occurs as active and one as passive coupling positive and negative lines can overlap and partially or completely cancel

HA HC

| | )C)C) Up to(2NI+1)2 = 64 lines can be observed (6 different J couplings).

| |HBHD

| ))))) - | |+ ))))))))) | || |

- -+ + cancellation

| ))))) - | |+ ))))))))) | || |

- -+ + cancellation

| ))))) - | |+ ))))))))) | || |

- -+ + cancellation

| ))))) - | |+ ))))))))) | || |

- -+ + cancellation

Simplification of multiplet patterns is thus often a necessity. It can be obtained by obtaining enhanced COSY (E-COSY) spectra

ppm

0.90

0.95

1.00

DQF-COSYMenthol

ppm

3.383.403.423.443.46 ppm

1.92

1.94

1.96

1.98

2.00

2.02

2.04

H -1

H -6 B

H -6 A

H-1/H-16 cross peaks of the DQF-COSY spectrum of menthol. Some lines are greatly decreased in intensity due to cancellation.

Small couplings and accuracy• When the coupling constant becomes comparable to the line width (small coupling

or broad lines) measurement of coupling constants will become less accurate.

• Thus a method which would separate the individual multiplet lines would improve accuracy of coupling constant measurements. Again this will be possible to some extent by using the enhanced COSY method.

in phase anti phase

deconvo lu tedlines

resu ltingdoub le t

Experimental considerationsRegular COSY:

• Optimized for low resolution as no fine structure of cross peaks needs to be resolved• td[f1] = 64 - 128, can be performed in as little as 5-10̊ min for concentrated samples• magnitude display, no need for phase correction• gives basic H-H connectivity information only

DQF-COSY (or P.E. COSY):

• Designed to resolve multiplet fine structure of cross peaks. • td[f1] typically = 512 or larger. Takes one to several hours to run.• Phase sensitive and will therefore require manual phase correction for useful results• Should only be attempted when coupling constant information is to be extracted from

2D spectrum. EXCEPTION: Need for suppression of strong singlet peaks (H2O samples) or cross peaks close to diagonal

Enhanced COSY (E-COSY): General principle

A B

C

JA B

JB CJA C

A

B

J A C

J B C

C = (+ 1 /2 )C = (-1 /2 )

C = (-1 /2 )

C = (+ 1 /2 )

A

B

J A C

J B C

C = (+ 1 /2 ) C = (-1 /2 )

C = (-1 /2 )

C = (+ 1 /2 )

G e nera l E -C O S Y p rin c ip le fo r th re e sp in s . T he a c tive c ou p lin g is dec ou p led in b o th d im en sio n .

JA C an d JB C bo th > 0̊ (equ a l s ig n )

JA C < 0̊ , JB C > 0̊ (o pp osite s ig n ).

T he sp littin g b y the p ass iv e co up ling to th e co m m o n p artne r C ca n be o b ta in ed w ith ou t o ve rlap o f th e d o ub le t co m p o n en ts .

D iffe ren t re la tive s ig ns o f th e tw o co up ling s g iv e d iffe ren t spec tra an d can thu s be u se d to m e asu re re la tiv e s ig n o f co up ling c on stan ts .

T O P :

B O T T O M :

Consider a system of three spins A,B,C coupled to each other by JAB, JAC and JBC.

In general any experiment correlating spins A and B without exciting the common coupling partner C will only result in peaks connecting multiplet lines with C in the same spin state, as no transitions from Iα

C to IβC (or vice versa)

take place between t1 and t2.

E. COSY in HSQC and HMQC spectra

• If A,B and C are all different nuclei a correlation of A and B via HSQC or HMQC will show E. COSY type peaks from coupling to C.

1H{13C} HSQC spectrum of Cp*Rh(CpCo)2(μ2-CO)2.

The large passive 13C-10̊3Rh coupling allows the small 1H-10̊3Rh coupling to be observed even when it is not resolved in the 1D 1H spectrum (HB)..

One can see that 2JHB,Rh has the same sign as 1JC,Rh whereas 2JHA,Rh has the opposite sign.

If one assumes 1JC,Rh > 0̊, then 2JHA,Rh < 0̊ and 2JHB,Rh > 0̊.

E. COSY in COSY spectra• If A and B are of the same isotope (usually 1H) and C is a hetero nucleus the cross

peaks in 1H/1H COSY spectra will be split in an E. COSY type fashion by the passive couplings A-C and B-C. In the example (COSY spectrum of

Ph2P=CHA-CHBR2)the passive coupling is to a 31P nucleus:

H B

P

P

P

P

P P M

1 .3 71 .3 81 .3 91 .4 0̊1 .4 11 .4 21 .4 3 p p m

0̊ .3

0̊ .4

0̊ .5

0̊ .6

0̊ .7

0̊ .8

H A

P

P

P

P

J H A H B

J H A H B

J P,H A

J P H B

Homonuclear spin systems and P.E. COSY

• An E. COSY type pattern in 1H/1H COSY would simplify the patterns observed in DQF-COSY spectra. However, in homonuclear spin systems all pulses will be applied to all nuclei.

• It can be shown if the second 90̊6 pulse in the COSY experiment is replaced by a pulse smaller than 90̊6 than correlations between lines with the passive coupling partner in different spin states are much weaker than the ones between equal spin states:

• In order to solve the problem of the out of phase diagonal encountered in the two pulse COSY experiment a phase cycle is employed to subtract the diagonal:

(90̊6)x,x ── t1 ── (186)x,x (186)x,-x acquisitionx,-x(add, subtract)

• By splitting the second pulse in two one performs one scan with 18+18 = 366 (cross and diagonal) and subtracts the spectrum with 18-18 = 0̊6 (diagonal only). That results in a spectrum with a reduced diagonal which can be acquired in a phase sensitive fashion and which will exhibit E-COSY type multiplet patters.

• This experiment is called P.E. COSY for Primitive Enhanced COSY (implying that there is a better but also more complicated way to achieve such a spectrum).

Use of coupling information to determine conformation

• In the two possible boat conformations, the OH and i-Pr groups can be either axial (a) or equatorial (b) positions.

• Proton H-1 and H-2 will than be either both in equatorial (a) or axial (b) positions.

• Analysis of the coupling pattern of H-1 reveals the presence of two large ( >10̊ Hz) coupling constants, with is only possible for pairs of trans-axial/axial pairs of nuclei, indicating that b is the structure present.

3 .3 53 .3 63 .3 73 .3 83 .3 93 .4 0̊3 .4 13 .4 23 .4 33 .4 43 .4 53 .4 63 .4 73 .4 83 .4 93 .5 0̊3 .5 1

10̊13

.87

10̊17

.91

10̊24

.16

10̊28

.11

10̊34

.33

10̊38

.48

4 .1 H z

1 0̊ .3 H z

1 0̊ .3 H z

2 x J > 1 0̊ H zH H

ppm

0.90

0.95

1.00

PE -C O SYM enthol

ppm

3.383.403.423.443.46 ppm

1.92

1.94

1.96

1.98

2.00

2.02

2.04

1

6 B

6 A

J 1 ,6 B

J 1 ,6 A

J 6A

,6B

J 1 ,2

J 6 B ,5

J 1 ,6 A

J 1 ,6 B

J 6A

,6B

J 1 ,2

J 6 A ,5 (no t reso lved)

• Analysis of the H-1/H-6A and H-1/H-6B cross-peaks in the P.E. COSY spectrum allow an assignment of the coupling constants to specific protons.

• 10̊.0̊ Hz for 3J1,2

• 4.1 Hz for 3J1,6A

• 10̊.8 Hz 3J1,6B.

• That also allows and assignment of H-6A as equatorial and H-6B as axial proton.

ppm

0.90

0.92

0.94

0.96

0.98

1.00

1.02

DQF-COSYM enthol

ppm

1.941.961.982.002.02 ppm

0.90

0.92

0.94

0.96

0.98

1.00

1.02

1.04

PE-COSY

6 A

6 B

6 B

J 6 A ,6 B

J 6B

,5

J 6 A ,5

J 6 A ,1

J 6B

,1

Analysis of the H-6A/H-6B cross peak results in:

3J6A,1 = 4.1 Hz 3J6A,5 = 3.7 Hz 3J6B,5 = 11.6 Hz • The C-10̊ methyl group is also

in an equatorial position (H-5 is axial), proving the relative stereochemistry on centers C1, C2 and C5.

• Only the P.E. COSY allows for a useful analysis of the cross peak fine structure.