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choice of a suitable interaction frame. Typical experiments are divided in
pulse intervals and free evolution times.
During the pulses, the chemical shift and scalar coupling interaction is
ignored. Only the applied B1 field is considered. This approach is justified for
pulses with tpulse«T1,T2.
1.4.3 Effect of Scalar Coupling
Apart from the chemical shift, there is a second very import interaction
between spins, the scalar coupling. The scalar depends on the mediation of
electrons, which are confined in orbitals around both nuclei. The scalar
coupling is expressed in Hz and noted as J. The operator expression for the
scalar coupling is:
2π J12 I1 zI 2 z
The above Hamiltonian expresses the scalar coupling between spin 1 and
spin 2 with a coupling constant J12. The evolution Hamiltonian for this spin
system is then:
H = δ1 I1z + δ 2 I 2 z + 2π J12 I1z I 2 z
To calculate the effect of this Hamiltonian, it is divided into 3 parts:
δ1 I1z
δ 2 I 2z
2π J12 I1zI 2z
which are applied in sequence, where this sequence is arbitrary. After a 90°
pulse has been applied to the two spins, we first calculate the two chemical
shift terms:
1 ⋅ I1 z ⋅ t
σ eq = I1z + I2 z δ
→ I1x cos(δ1 t ) + I1y sin(δ1 t ) + I2 z
δ 2 ⋅I 2 z ⋅t

→ I1x cos(δ1 t ) + I1y sin(δ1 t )
+ I 2x cos(δ2 t ) + I2 y sin(δ2 t ) ⇒ σ1
The next step will be to compute the evolution under the scalar coupling.
The scalar coupling term can be evaluated with a simple set of rules:
π J12 I1 z I 2 z t
I1z 2

→ I1z
π J12 I1 z I 2 z t
I1x 2

→ I1x cos(πJ12 t ) + 2 I1y I 2z sin(πJ12 t )
π J12 I1 z I 2 z t
I1y 2

→ I1y cos(πJ12 t ) − 2 I1x I2 z sin(πJ12 t )
π J12 I1 z I 2 z t
2 ⋅ I1x I 2z 2

→ 2 I1x I2 z cos(πJ12 t ) + I1y sin(πJ12 t )
π J12 I1 z I 2 z t
2 ⋅ I1y I 2z 2

→ 2 I1y I2z cos(πJ12 t ) − I1x sin(πJ12 t )
π J12 I1 z I 2 z t
2 ⋅ I1x I 2y 2

→ 2 I1x I2 y
which can then be applied to the various terms of σ1 above:
Avance 1D/2D
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