Details of the calculation - The strongly coupled case: Analysis using density matrices

PerfectBASH-Entkopplung:

2 The strongly coupled case: Analysis using density matrices

2.1 Details of the calculation

S10

with their eigenvalues of HAB denoted as a, b, c and d, whereas 4 :

2 2

(2.2) The density operator ρ(t) after passing through the pulse sequence in Figure 2-1 is given by

(2.3)

using exponential propagators for pulses and times of free evolution. Since the AB system under consideration is strongly coupled, the corresponding Hamiltonian HAB for free evolution is used:

2 ∙ 2

(2.4) The initial state ρ0 after the 90°x pulse is given by

〈 〉 〈 〉.

(2.5)

The expectation value for transversal magnetization during the free induction decay is obtained by taking the trace of the matrix representation of ρ(t) multiplied with the matrix representation for the transversal magnetization F+:

〈 〉

(2.6)

2.2 Results of the density matrix calculations

The result is of the general form hF+i = X

Aj·exp [i(ωj,1τ1+ωj,2τ2+ωj,3τ3+ωj,4(τ4+t2))] (2.7)

and consists of the 72 terms, listed in Table 2.1:

Table 2.1: Results from the density matrix calculations

Amplitude factor Exponential factor Term No.

−ⁱ2pq(q+p)⁴ p²−q²2

e^i((b⁻^{d) (τ}⁴^+t²^)+(c⁻^d)τ³^+(b⁻^d)τ²^+2(d⁻^b)τ¹⁾ 1 ip²q²(q+p)² p²−q²2

e^{i((c−d) (τ}⁴^+τ³^+t²^)+(b−d)τ²^+2(d−b)τ¹⁾ 2

−2ip³q³(q+p)⁴ e^i((b⁻^{d) (τ}⁴^+τ³^+τ²^+t²^)+2(d⁻^b)τ¹⁾ 3

−ip²q²(q+p)² p²−q²2

e^i((c⁻^{d) (τ}⁴^+t²^)+(b⁻^{d) (τ}³^+τ²^)+2(d⁻^b)τ¹⁾ 4

−2ⁱ(q+p)² p²−q²4

e^i((b⁻^{d) (τ}⁴^+t²^)+(c⁻^d)τ³^+(b⁻^d)τ²^+(2d⁻^c⁻^b)τ¹⁾ 5 ipq p²−q²4

e^{i((c−d) (τ}⁴^+τ³^+t²^)+(b−d)τ²^{+(2d−c−b)}^τ¹⁾ 6

−2ip²q²(q+p)² p²−q²2

e^i((b⁻^{d) (τ}⁴^+τ³^+τ²^+t²^)+(2d⁻^c⁻^b)τ¹⁾ 7

−ipq p²−q²4

e^i((c⁻^{d) (τ}⁴^+t²^)+(b⁻^{d) (τ}³^+τ²^)+(2d⁻^c⁻^b)τ¹⁾ 8

2pq p²−q²4

e^i((b⁻^{d) (τ}⁴^+t²^)+(c⁻^d)τ³^+(b⁻^d)τ²^+2(d⁻^c)τ¹⁾ 9

−ip²q²(p−q)² p²−q²2

e^{i((c−d) (τ}⁴^+τ³^+t²^)+(b−d)τ²^+2(d−c)τ¹⁾ 10 2ip³q³ p²−q²2

e^i((b⁻^{d) (τ}⁴^+τ³^+τ²^+t²^)+2(d⁻^c)^τ¹⁾ 11 ip²(p−q)²q² p²−q²2

e^i((c⁻^{d) (τ}⁴^+t²^)+(b⁻^{d) (τ}³^+τ²^)+2(d⁻^c)^τ¹⁾ 12

−2ⁱpq p²−q²4

e^{i((c−d) (τ}⁴^+t²^)+(b−d)τ³^+(c−d)τ²^+2(d−b)τ¹⁾ 13

−ip²q²(q+p)² p²−q²2

e^{i((b−d) (τ}⁴^+τ³^+t²^)+(c−d)τ²^+2(d−b)τ¹⁾ 14 ip²q²(q+p)² p²−q²2

e^i((b⁻^{d) (τ}⁴^+t²^)+(c⁻^{d) (τ}³^+τ²^)+2(d⁻^b)τ¹⁾ 15

−2ip³q³ p²−q²2

e^i((c⁻^{d) (τ}⁴^+τ³^+τ²^+t²^)+2(d⁻^b)^τ¹⁾ 16

−2ⁱ(p−q)² p²−q²4

e^{i((c−d) (τ}⁴^+t²^)+(b−d)τ³^+(c−d)τ²^{+(2d−c−b)τ}¹⁾ 17

−ipq p²−q²4

e^i((b⁻^{d) (τ}⁴^+τ³^+t²^)+(c⁻^d)τ²^+(2d⁻^c⁻^b)^τ¹⁾ 18 ipq p²−q²4

e^i((b⁻^{d) (τ}⁴^+t²^)+(c⁻^{d) (τ}³^+τ²^)+(2d⁻^c⁻^b)τ¹⁾ 19

−2ip²(p−q)²q² p²−q²2

e^i((c⁻^{d) (τ}⁴^+τ³^+τ²^+t²^)+(2d⁻^c⁻^b)τ¹⁾ 20

2p(p−q)⁴q p²−q²2

e^{i((c−d) (τ}⁴^+t²^)+(b−d)τ³^+(c−d)τ²^+2(d−c)τ¹⁾ 21 ip²(p−q)²q² p²−q²2

e^i((b⁻^{d) (τ}⁴^+τ³^+t²^)+(c⁻^d)τ²^+2(d⁻^c)τ¹⁾ 22

−ip²(p−q)²q² p²−q²2

e^i((b⁻^{d) (τ}⁴^+t²^)+(c⁻^{d) (τ}³^+τ²^)+2(d⁻^c)^τ¹⁾ 23

2ip³(p−q)⁴q³ e^i((c⁻^{d) (τ}⁴^+τ³^+τ²^+t²^)+2(d⁻^c)^τ¹⁾ 24

4(p−q)² p²−q²4

−ⁱ4(q+p)² p²−q²4

e^{i((d−b) (τ}⁴^+t²^)+(d−c)τ³^+(d−b)τ²^{+(2d−c−b)τ}¹⁾ 25 S11

T able2.1− −continuation

Amplitude factor Exponential factor Term No.

ip²(p−q)²q² p²−q²2

−ip²q²(q+p)² p²−q²2

e^i((d⁻^{b) (τ}⁴^+τ³^+τ²^+t²^)+(2d⁻^c⁻^b)τ¹⁾ 26

2pq p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^c)τ³^+(d⁻^b)τ²^+2(d⁻^b)τ¹⁾ 27 2ip³q³ p²−q²2

e^i((d⁻^{b) (τ}⁴^+τ³^+τ²^+t²^)+2(d⁻^b)τ¹⁾ 28

2pq p²−q²4

e^{i((d−b) (τ}⁴^+t²^)+(d−c)τ³^+(d−b)τ²^+2(d−c)τ¹⁾ 29 2ip³q³ p²−q²2

e^i((d⁻^{b) (τ}⁴^+τ³^+τ²^+t²^)+2(d⁻^c)^τ¹⁾ 30

−ip²q²(q+p)² p²−q²2

e^i((d⁻^{b) (τ}⁴^+τ³^+t²^)+(d⁻^c)τ²^+2(d⁻^b)τ¹⁾ 31

−ip²q²(q+p)² p²−q²2

e^i((d⁻^{b) (τ}⁴^+τ³^+t²^)+(d⁻^c)τ²^+2(d⁻^c)τ¹⁾ 32

2pq(q+p)⁴ p²−q²2

−2ⁱpq p²−q²4

e^{i((d−b) (τ}⁴^+τ³^+t²^)+(d−c)τ²^{+(2d−c−b)}^τ¹⁾ 33 ip²q²(q+p)² p²−q²2

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^{c) (τ}³^+τ²^)+2(d⁻^b)τ¹⁾ 34 ip²q²(q+p)² p²−q²2

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^{c) (τ}³^+τ²^)+2(d⁻^c)^τ¹⁾ 35

2pq(q+p)⁴ p²−q²2

−2ⁱpq p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^{c) (τ}³^+τ²^)+(2d⁻^c⁻^b)τ¹⁾ 36

−ⁱ2pq(q+p)⁴ p²−q²2

e^{i((d−b) (τ}⁴^+t²^)+(d−c)τ³^+(d−b)τ²^+2(b−d)τ¹⁾ 37

−2ⁱ(q+p)² p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^c)τ³^+(d⁻^b)τ²⁺⁽⁻^2d+c+b)τ¹⁾ 38

2pq p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^c)τ³^+(d⁻^b)τ²^+2(c⁻^d)τ¹⁾ 39

−2ip³q³(q+p)⁴ e^i((d⁻^{b) (τ}⁴^+τ³^+τ²^+t²^)+2(b⁻^d)τ¹⁾ 40

−2ip²q²(q+p)² p²−q²2

e^{i((d−b) (τ}⁴^+τ³^+τ²^+t²)+(−2d+c+b)τ1) 41 2ip³q³ p²−q²2

e^i((d⁻^{b) (τ}⁴^+τ³^+τ²^+t²^)+2(c⁻^d)^τ¹⁾ 42

−ip²q²(q+p)² p²−q²2

e^i((d⁻^{b) (τ}⁴^+τ³^+t²^)+(d⁻^c)τ²^+2(b⁻^d)τ¹⁾ 43

−ipq p²−q²4

e^i((d⁻^{b) (τ}⁴^+τ³^+t²^)+(d⁻^c)τ²⁺⁽⁻^2d+c+b)τ¹⁾ 44 ip²(p−q)²q² p²−q²2

e^{i((d−b) (τ}⁴^+τ³^+t²^)+(d−c)τ²^+2(c−d)τ¹⁾ 45 ip²q²(q+p)² p²−q²2

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^{c) (τ}³^+τ²^)+2(b⁻^d)τ¹⁾ 46 ipq p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^{c) (τ}³^+τ²⁾⁺⁽⁻^2d+c+b)^τ¹⁾ 47

−ip²(p−q)²q² p²−q²2

e^{i((d−b) (τ}⁴^+t²^{)+(d−c) (τ}³^+τ²^)+2(c−d)^τ¹⁾ 48 ip²(p−q)²q² p²−q²2

e^i((d⁻^{c) (τ}⁴^+t²^)+(d⁻^{b) (τ}³^+τ²^)+2(d⁻^b)τ¹⁾ 49 ip²(p−q)²q² p²−q²2

e^i((d⁻^{c) (τ}⁴^+t²^)+(d⁻^{b) (τ}³^+τ²^)+2(d⁻^c)^τ¹⁾ 50

2p(p−q)⁴q p²−q²2

−2ⁱpq p²−q²4

e^i((d⁻^{c) (τ}⁴^+t²^)+(d⁻^{b) (τ}³^+τ²^)+(2d⁻^c⁻^b)τ¹⁾ 51

−ip²(p−q)²q² p²−q²2

e^{i((d−c) (τ}⁴^+τ³^+t²^)+(d−b)τ²^+2(d−b)τ¹⁾ 52

−ip²(p−q)²q² p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+t²^)+(d⁻^b)τ²^+2(d⁻^c)τ¹⁾ 53

2pq p²−q²4

−2ⁱp(p−q)⁴q p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+t²^)+(d⁻^b)τ²^+(2d⁻^c⁻^b)^τ¹⁾ 54

−2ⁱpq p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^b)τ³^+(d⁻^c)τ²^+2(d⁻^b)τ¹⁾ 55

S12

T able2.1− −continuation

Amplitude factor Exponential factor Term No.

−2ⁱpq p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^b)τ³^+(d⁻^c)τ²^+2(d⁻^c)τ¹⁾ 56

4(q+p)² p²−q²4

−ⁱ4(p−q)² p²−q²4

e^i((d⁻^{b) (τ}⁴^+t²^)+(d⁻^b)τ³^+(d⁻^c)τ²^+(2d⁻^c⁻^b)τ¹⁾ 57

−ip³q³ p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+τ²^+t²^)+2(d⁻^b)^τ¹⁾ 58

−ip³q³ p²−q²2

e^{i((d−c) (τ}⁴^+τ³^+τ²^+t²^)+2(d−c)^τ¹⁾ 59 ip²q²(q+p)² p²−q²2

−ip²(p−q)²q² p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+τ²^+t²^)+(2d⁻^c⁻^b)τ¹⁾ 60

−ip²q²(q+p)² p²−q²2

e^i((d⁻^{c) (τ}⁴^+t²^)+(d⁻^{b) (τ}³^+τ²^)+2(b⁻^d)τ¹⁾ 61 ip²(p−q)²q² p²−q²2

e^i((d⁻^{c) (τ}⁴^+t²^)+(d⁻^{b) (τ}³^+τ²^)+2(c⁻^d)^τ¹⁾ 62

−ipq p²−q²4

e^{i((d−c) (τ}⁴^+t²^{)+(d−b) (τ}³^+τ²)+(−2d+c+b)τ1) 63 ip²q²(q+p)² p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+t²^)+(d⁻^b)τ²^+2(b⁻^d)τ¹⁾ 64

−ip²(p−q)²q² p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+t²^)+(d⁻^b)τ²^+2(c⁻^d)τ¹⁾ 65 ipq p²−q²4

e^i((d⁻^{c) (τ}⁴^+τ³^+t²^)+(d⁻^b)τ²⁺⁽⁻^2d+c+b)τ¹⁾ 66

−2ⁱpq p²−q²4

e^{i((d−c) (τ}⁴^+t²^)+(d−b)τ³^+(d−c)τ²^+2(b−d)τ¹⁾ 67

2p(p−q)⁴q p²−q²2

e^i((d⁻^{c) (τ}⁴^+t²^)+(d⁻^b)τ³^+(d⁻^c)τ²^+2(c⁻^d)τ¹⁾ 68

−2ⁱ(p−q)² p²−q²4

e^i((d⁻^{c) (τ}⁴^+t²^)+(d⁻^b)τ³^+(d⁻^c)τ²⁺⁽⁻^2d+c+b)τ¹⁾ 69

−2ip³q³ p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+τ²^+t²^)+2(b⁻^d)^τ¹⁾ 70

2ip³(p−q)⁴q³ e^{i((d−c) (τ}⁴^+τ³^+τ²^+t²^)+2(c−d)^τ¹⁾ 71

−2ip²(p−q)²q² p²−q²2

e^i((d⁻^{c) (τ}⁴^+τ³^+τ²^+t²⁾⁺⁽⁻^2d+c+b)^τ¹⁾ 72

p = cosξ q = sinξ

ξ = 1

2arctan 2πJ

ω1−ω2

Ω =

(ω1−ω2)² + 4π²J²

(2.8)

Using the energy eigenvalues (in angular frequency units) of the four basis functions,

a = d = πJ

2 , b = −πJ

2 +Ω 2, c = −πJ

2 −Ω 2

(2.9)

and the timing conditions for removal ofJ−coupling effects at the centre of every data-chunk and for chemical shift refocusing at the beginning of the first data-chunk

S13

τ1 = t1

2 +δ1 = t1

2 + 2δ2+δ4, τ2 = t1

2 +δ2,

τ3 = δ3 = δ2+δ4, τ4 = t1

2 +δ4

(2.10)

the exponential factors can be simplified:

Table 2.2: Simplified terms

Amplitude factor Exponential factor No.

−ⁱ2pq(q+p)⁴ p²−q²2

e^iΩt²²^{−iπJ t}²^−iΩδ⁴^−2iΩδ²^{+2iπJ δ}² 1 ip²q²(q+p)² p²−q²2

e⁻^iΩt²²⁻^{iπJ t}²⁻^iΩt²¹⁻^2iΩδ⁴⁻^2iΩδ²^{+2iπJ δ}² 2

−2ip³q³(q+p)⁴ e^iΩt²²⁻^{iπJ t}²⁻^iΩδ²^{+2iπJ δ}² 3

−ip²q²(q+p)² p²−q²2

e⁻^iΩt²²⁻^{iπJ t}²⁻^iΩt²¹⁻^iΩδ⁴⁻^iΩδ²^{+2iπJ δ}² 4

−2ⁱ(q+p)² p²−q²4

e^iΩt²²^{−iπJ t}²⁺^iΩt²¹^{+2iπJ δ}² 5 ipq p²−q²4

e⁻^iΩt2² ⁻^{iπJ t}²⁻^iΩδ⁴^{+2iπJ δ}² 6

−2ip²q²(q+p)² p²−q²2

e^iΩt²²⁻^{iπJ t}²⁺^iΩt²¹^+iΩδ⁴^+iΩδ²^{+2iπJ δ}² 7

−ipq p²−q²4

e⁻^iΩt²²⁻^{iπJ t}²^+iΩδ²^{+2iπJ δ}² 8

2pq p²−q²4

e^iΩt²²^{−iπJ t}²^+iΩt¹^+iΩδ⁴^+2iΩδ²^{+2iπJ δ}² 9

−ip²q²(p−q)² p²−q²2

e⁻^iΩt2² ⁻^{iπJ t}²⁺^iΩt1² ^+2iΩδ²^{+2iπJ δ}² 10 2ip³q³ p²−q²2

e^iΩt²²⁻^{iπJ t}²^+iΩt¹^+2iΩδ⁴^+3iΩδ²^{+2iπJ δ}² 11 ip²(p−q)²q² p²−q²2

e⁻^iΩt²²⁻^{iπJ t}²⁺^iΩt²¹^+iΩδ⁴^+3iΩδ²^{+2iπJ δ}² 12

−2ⁱpq p²−q²4

e⁻^iΩt²²^{−iπJ t}²^−iΩt¹^−iΩδ⁴^−2iΩδ²^{+2iπJ δ}² 13

−ip²q²(q+p)² p²−q²2

e^iΩt2² ⁻^{iπJ t}²⁻^iΩt1² ⁻^2iΩδ²^{+2iπJ δ}² 14 ip²q²(q+p)² p²−q²2

e^iΩt²²⁻^{iπJ t}²⁻^iΩt²¹⁻^iΩδ⁴⁻^3iΩδ²^{+2iπJ δ}² 15

−2ip³q³ p²−q²2

e⁻^iΩt²²⁻^{iπJ t}²⁻^iΩt¹⁻^2iΩδ⁴⁻^3iΩδ²^{+2iπJ δ}² 16

−2ⁱ(p−q)² p²−q²4

e⁻^iΩt²²^{−iπJ t}²⁻^iΩt²¹^{+2iπJ δ}² 17

−ipq p²−q²4

e^iΩt2² ⁻^{iπJ t}²^+iΩδ⁴^{+2iπJ δ}² 18 ipq p²−q²4

e^iΩt²²⁻^{iπJ t}²⁻^iΩδ²^{+2iπJ δ}² 19

−2ip²(p−q)²q² p²−q²2

e⁻^iΩt²²⁻^{iπJ t}²⁻^iΩt²¹⁻^iΩδ⁴⁻^iΩδ²^{+2iπJ δ}² 20

2p(p−q)⁴q p²−q²2

e⁻^iΩt²²^{−iπJ t}²^+iΩδ⁴^+2iΩδ²^{+2iπJ δ}² 21 ip²(p−q)²q² p²−q²2

e^iΩt²²⁻^{iπJ t}²⁺^iΩt²¹^+2iΩδ⁴^+2iΩδ²^{+2iπJ δ}² 22

−ip²(p−q)²q² p²−q²2

e^iΩt²²⁻^{iπJ t}²⁺^iΩt²¹^+iΩδ⁴^+iΩδ²^{+2iπJ δ}² 23

S14

T able2.2− −continuation

Amplitude factor Exponential factor No.

2ip³(p−q)⁴q³ e⁻^iΩt²²⁻^{iπJ t}²^+iΩδ²^{+2iπJ δ}² 24

4(p−q)² p²−q²4

−ⁱ4(q+p)² p²−q²4

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹^{+4iπJ δ}⁴^{+6iπJ δ}² 25 ip²(p−q)²q² p²−q²2

−ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴⁻^iΩδ²^{+6iπJ δ}² 26

2pq p²−q²4

e⁻^iΩt²²^{+iπJ t}²^−iΩt¹^{+2iπJ t}¹^−iΩδ⁴^{+4iπJ δ}⁴^−2iΩδ²^{+6iπJ δ}² 27 2ip³q³ p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt¹^{+2iπJ t}¹⁻^2iΩδ⁴^{+4iπJ δ}⁴⁻^3iΩδ²^{+6iπJ δ}² 28

2pq p²−q²4

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 29 2ip³q³ p²−q²2

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 30

−ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹^−2iΩδ⁴^{+4iπJ δ}⁴^−2iΩδ²^{+6iπJ δ}² 31

−ip²q²(q+p)² p²−q²2

e⁻^iΩt2² ^{+iπJ t}²⁺^iΩt1² ^{+2iπJ t}¹^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 32

2pq(q+p)⁴ p²−q²2

−2ⁱpq p²−q²4

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴^{+6iπJ δ}² 33 ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴⁻^iΩδ²^{+6iπJ δ}² 34 ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+3iΩδ²^{+6iπJ δ}² 35

2pq(q+p)⁴ p²−q²2

−2ⁱpq p²−q²4

e⁻^iΩt2² ^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 36

−ⁱ2pq(q+p)⁴ p²−q²2

e⁻^iΩt²²^{+iπJ t}²^+iΩδ⁴^+2iΩδ²⁻^{2iπJ δ}² 37

−2ⁱ(q+p)² p²−q²4

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹⁻^{2iπJ δ}² 38

2pq p²−q²4

e⁻^iΩt²²^{+iπJ t}²^−iΩt¹^−iΩδ⁴^−2iΩδ²^{−2iπJ δ}² 39

−2ip³q³(q+p)⁴ e⁻^iΩt2² ^{+iπJ t}²^+iΩδ²⁻^{2iπJ δ}² 40

−2ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹⁻^iΩδ⁴⁻^iΩδ²⁻^{2iπJ δ}² 41 2ip³q³ p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt¹⁻^2iΩδ⁴⁻^3iΩδ²⁻^{2iπJ δ}² 42

−ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁺^iΩt²¹^+2iΩδ²^{−2iπJ δ}² 43

−ipq p²−q²4

e⁻^iΩt²²^{+iπJ t}²⁻^iΩδ⁴⁻^{2iπJ δ}² 44 ip²(p−q)²q² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹⁻^2iΩδ⁴⁻^2iΩδ²⁻^{2iπJ δ}² 45 ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁺^iΩt²¹^+iΩδ⁴^+3iΩδ²^{−2iπJ δ}² 46 ipq p²−q²4

e⁻^iΩt²²^{+iπJ t}²^+iΩδ²⁻^{2iπJ δ}² 47

−ip²(p−q)²q² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹⁻^iΩδ⁴⁻^iΩδ²⁻^{2iπJ δ}² 48 ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴⁻^3iΩδ²^{+6iπJ δ}² 49 ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 50

2p(p−q)⁴q p²−q²2

−2ⁱpq p²−q²4

e^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴⁻^iΩδ²^{+6iπJ δ}² 51

−ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹^{+4iπJ δ}⁴⁻^2iΩδ²^{+6iπJ δ}² 52

−ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^+2iΩδ⁴^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 53

S15

T able2.2− −continuation

Amplitude factor Exponential factor No.

2pq p²−q²4

−2ⁱp(p−q)⁴q p²−q²2

e^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^{+6iπJ δ}² 54

−2ⁱpq p²−q²4

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^2iΩδ⁴^{+4iπJ δ}⁴⁻^2iΩδ²^{+6iπJ δ}² 55

−2ⁱpq p²−q²4

e⁻^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 56

4(q+p)² p²−q²4

−ⁱ4(p−q)² p²−q²4

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^−iΩδ⁴^{+4iπJ δ}⁴^{+6iπJ δ}² 57

−ip³q³ p²−q²2

e^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴⁻^iΩδ²^{+6iπJ δ}² 58

−ip³q³ p²−q²2

e^iΩt²²^{+iπJ t}²^+iΩt¹^{+2iπJ t}¹^+2iΩδ⁴^{+4iπJ δ}⁴^+3iΩδ²^{+6iπJ δ}² 59 ip²q²(q+p)² p²−q²2

−ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 60

−ip²q²(q+p)² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^+iΩδ⁴^+iΩδ²^{−2iπJ δ}² 61 ip²(p−q)²q² p²−q²2

e^iΩt2² ^{+iπJ t}²⁻^iΩt1² ⁻^iΩδ⁴⁻^3iΩδ²⁻^{2iπJ δ}² 62

−ipq p²−q²4

e^iΩt²²^{+iπJ t}²⁻^iΩδ²⁻^{2iπJ δ}² 63 ip²q²(q+p)² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^+2iΩδ⁴^+2iΩδ²⁻^{2iπJ δ}² 64

−ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁻^iΩt²¹^−2iΩδ²^{−2iπJ δ}² 65 ipq p²−q²4

e^iΩt2² ^{+iπJ t}²^+iΩδ⁴⁻^{2iπJ δ}² 66

−2ⁱpq p²−q²4

e^iΩt²²^{+iπJ t}²^+iΩt¹^+iΩδ⁴^+2iΩδ²⁻^{2iπJ δ}² 67

2p(p−q)⁴q p²−q²2

e^iΩt²²^{+iπJ t}²⁻^iΩδ⁴⁻^2iΩδ²⁻^{2iπJ δ}² 68

−2ⁱ(p−q)² p²−q²4

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{−2iπJ δ}² 69

−2ip³q³ p²−q²2

e^iΩt2² ^{+iπJ t}²^+iΩt¹^+2iΩδ⁴^+3iΩδ²⁻^{2iπJ δ}² 70

2ip³(p−q)⁴q³ e^iΩt²²^{+iπJ t}²⁻^iΩδ²⁻^{2iπJ δ}² 71

−2ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^+iΩδ⁴^+iΩδ²⁻^{2iπJ δ}² 72

Cosine modulated terms

Out of these 72 terms 48 terms can be combined into 24 cosine modulated terms using the complex exponential identity:

e^iφ + e⁻^iφ = 2 cos (φ) (2.11)

These terms are purely imaginary which means that since

F+ = Fx + iFy (2.12)

they represent y-magnetization and will appear as in-phase multiplets in the spectra. These cosine modulated terms can further be decomposed into nine term groups listed in Table 2.3.

S16

In Table 2.3, the term groups are separated into an amplitude factor and four exponential modulation factors, which represent the offset ¹₂(ω1 +ω2), the evolution of J−coupling and chemical shift Ω during the evolution times t1 andt2 and a constant phase factor depending on Ω,δ2 andδ4. Note that the density matrix calculation was performed with an offset set to zero (¹₂(ω1 +ω2) = 0). The classification is performed depending on the frequency of the coherence transfer between the two protons and which of the four pulses in the PEPSIE pulse sequence causes this transfer (Table 2.4).

Table 2.3: Cosine modulated terms

Frequency modulation factor for all terms: e^±(iπJ^(t²⁻^2δ²⁾⁾·e²ⁱ^(ω¹⁺^ω²^)(t¹⁺^t²⁾ Term

Terms Amplitude Chem. Shift Constant phase

group factor modulation factor

1 5, 38 −i(q−p)⁴(q+p)⁶ e^±^iΩ²^(t²⁺^t¹⁾

1 17, 69 −i(q−p)⁶(q+p)⁴ e^∓^iΩ²^(t²⁺^t¹⁾

2 9, 39 ipq p²−q²4

e^±^iΩ²^(t²^{+ 2t}¹⁾ e^±^iΩ(2δ²^+δ⁴⁾ 13, 67 −ipq p²−q²4

e^∓^iΩ²^(t²^{+ 2t}¹⁾ e^∓^iΩ(2δ²^+δ⁴⁾

3 1, 37 −ipq(q+p)⁴ p²−q²2

e^±^iΩt²² e^∓iΩ(2δ²^+δ⁴⁾ 21, 68 ipq(p−q)⁴ p²−q²2

e^∓^iΩt²² e^±^iΩ(2δ²^+δ⁴⁾

4 6, 66, 8, 63 2ipq p²−q²4

e^∓^iΩt²²

e^∓iΩδ⁴−e^±iΩδ² 18, 44, 19, 47 2ipq p²−q²4

e^±^iΩt²²

e^∓^iΩδ²−e^±^iΩδ⁴

5 2, 64, 4, 61 ip²q²(q+p)² p²−q²2

e^∓^iΩ²^(t²⁺^t¹⁾

e^∓iΩ(δ²^+δ⁴⁾−e^∓2iΩ(δ²^+δ⁴⁾ 22, 45, 23, 48 ip²q²(q−p)² p²−q²2

e^±^iΩ²^(t²⁺^t¹⁾

e^±^2iΩ(δ²^+δ⁴⁾−e^±^iΩ(δ²^+δ⁴⁾

6 10, 65, 12, 62 ip²q²(p−q)² p²−q²2

e^∓^iΩ²^(t²⁻^t¹⁾

e^±^iΩ(3δ²^+δ⁴⁾−e^±^2iΩδ² 14, 43, 15, 46 ip²q²(q+p)² p²−q²2

e^±^iΩ²^(t²⁻^t¹⁾

e^∓^2iΩδ²−e^∓^iΩ(3δ²^+δ⁴⁾

7 7, 41 −2ip²q²(q+p)² p²−q²2

e^±^iΩ²^(t²⁺^t¹⁾ e^±^iΩ(δ²^+δ⁴⁾ 20, 72 −2ip²q²(p−q)² p²−q²2

e^∓^iΩ²^(t²⁺^t¹⁾ e^∓^iΩ(δ²^+δ⁴⁾

8 11, 42 2ip³q³ p²−q²2

e^±^iΩ²^(t²^{+ 2t}¹⁾ e^±^iΩ(3δ²^+2δ⁴⁾ 16, 70 −2ip³q³ p²−q²2

e^∓^iΩ²^(t²^{+ 2t}¹⁾ e^∓^iΩ(3δ²^+2δ⁴⁾

9 3, 40 −2ip³q³(p+q)⁴ e^±^iΩt²² e^∓^iΩδ²

24, 71 2ip³q³(p−q)⁴ e^∓^iΩt²² e^±^iΩδ²

S17

Table2.4:Cosinemodulatedterms:Assignment GroupTermsAssignmentCoherencetransfer frequencycausingpulse 15,17,38,69desiredpeaksNoNo 29,13,39,67phase-modulated doubletransfer180◦-pulsefirstecho artifacts&central90◦-pulse 31,21,37,68centralartifactsingletransfer180◦-pulsefirstecho 46,8,18,19, centralartifactsingletransfer1st.or2nd.180◦- 44,47,63,66pulsesecondecho 52,4,22,23,phase-modulated doubletransfer180◦-pulsefirstecho& 45,48,61,64artifacts1st.or2nd.180◦-pulsesecondecho 610,14,43,65phase-modulated tripletransfer180◦-pulsefirstecho,central90◦-pulse& 12,15,46,62artifacts1st.or2nd.180◦-pulsesecondecho 77,20,41,72phase-modulated doubletransfer1st.&2nd.180◦-pulse artifactssecondecho 811,16,42,70phase-modulated fourfoldtransferall180◦-pulses artifacts&central90◦-pulse 93,24,40,71centralartifacttripletransferall180◦-pulses

S18

Evolution of the amplitude factors with increasing degree of strong coupling

For illustration, how the amplitudes of the cosine modulated terms in table 2.3 evolve with increasing degree of strong coupling, their amplitude factors are plotted against the relation of shift difference ∆νand coupling constantJ. In all plots the constant phase factor in table 2.3 is considered, whereas Ω was calculated from ^∆ν_J using the proton coupling constant of 2,3-dibromothiopheneJ = 5.75 Hz. Figure 2.2 shows the plots of the term groups contributing to the central artifact (3,4and9) in comparison to the desired in-phase peaks (term group1).

1 5 1 4 1 3 1 2 1 1 1 0 9 8 7 6 5 4 3 2 1 0

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0

amplitude / a.u.

∆ν/ J

D e s i r e d i n - p h a s e p e a k s T e r m g r o u p 3

T e r m g r o u p 9 T e r m g r o u p 4

S u m o f g r o u p s 3 , 4 a n d 9

Figure 2.2: Plot of the amplitude factors of the term groups contributing to the central artifact (3,4and9in Table 2.3) in comparison to the desired in-phase peaks (term group1, blue line) against the relation of shift difference ∆νand coupling constantJ. The black line is the sum of the terms contributing to the central artifact.

S19

1 5 1 4 1 3 1 2 1 1 1 0 9 8 7 6 5 4 3 2 1 0 - 0 . 2

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0

amplitude / a.u.

∆ν/ J

D e s i r e d i n - p h a s e p e a k s ( s u m m e d a m p l i t u d e ) T e r m g r o u p 2

T e r m g r o u p 5 T e r m g r o u p 6 T e r m g r o u p 7 T e r m g r o u p 8

Figure 2.3: Plot of the amplitude factors of the term groups2,5,6, 7and8in Table 2.3 in comparison to the desired in-phase peaks (term group1, blue line) against the relation of shift difference ∆ν and coupling constantJ.

Figure 2.3 shows the plots of the term groups 2, 5, 6, 7and 8, which lead to artifacts with phase distortions, in comparison to the desired in-phase peaks. Theses artifacts are not visible in the pure shift spectra of 2,3-dibromothiophene. However, these are potentially observable in other systems or in pure shift spectra obtained with other acquisition parameters, since their amplitudes depends on shift difference Ω, theJ−coupling constant and the time constantsδ2

andδ4as well.

Exponential modulated terms

The remaining 24 terms in table 2.2 have complex exponential modulation and can be decom-posed into a real and an imaginary part, thus they represent mixed x- and y-magnetization. This would lead to a phase-modulated multiplet appearance. All these terms exhibitJ−coupling evo-lution duringt1and are listed in table 2.5 The terms25and57have amplitude factors with two components, which do not vanish in the weak coupling case (p = 1 andq = 0). However, the first component cancels the second one in the weakly coupled case. If strong coupling is present this is not achieved. Thus we assume that these are the anti-phase terms of equation 1.12 in chapter 1.2. Furthermore there are additional terms with polarisation transfer as a result of strong coupling.

S20

Table 2.5: Exponential modulated terms

Amplitude factor Exponential factor No.

4(p−q)² p²−q²4

−ⁱ4(q+p)² p²−q²4

e⁻^iΩt2² ^{+iπJ t}²⁻^iΩt1² ^{+2iπJ t}¹^{+4iπJ δ}⁴^{+6iπJ δ}² 25 ip²(p−q)²q² p²−q²2

−ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴⁻^iΩδ²^{+6iπJ δ}² 26

2pq p²−q²4

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt¹^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴⁻^2iΩδ²^{+6iπJ δ}² 27 2ip³q³ p²−q²2

e⁻^iΩt²²^{+iπJ t}²^−iΩt¹^{+2iπJ t}¹^−2iΩδ⁴^{+4iπJ δ}⁴^−3iΩδ²^{+6iπJ δ}² 28

2pq p²−q²4

e⁻^iΩt2² ^{+iπJ t}²^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 29 2ip³q³ p²−q²2

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 30

−ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^2iΩδ⁴^{+4iπJ δ}⁴⁻^2iΩδ²^{+6iπJ δ}² 31

−ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 32

2pq(q+p)⁴ p²−q²2

−2ⁱpq p²−q²4

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴^{+6iπJ δ}² 33 ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴⁻^iΩδ²^{+6iπJ δ}² 34 ip²q²(q+p)² p²−q²2

e⁻^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+3iΩδ²^{+6iπJ δ}² 35

2pq(q+p)⁴ p²−q²2

−2ⁱpq p²−q²4

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 36 ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴⁻^3iΩδ²^{+6iπJ δ}² 49 ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 50

2p(p−q)⁴q p²−q²2

−2ⁱpq p²−q²4

e^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴^−iΩδ²^{+6iπJ δ}² 51

−ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹^{+4iπJ δ}⁴⁻^2iΩδ²^{+6iπJ δ}² 52

−ip²(p−q)²q² p²−q²2

e^iΩt²²^{+iπJ t}²⁺^iΩt²¹^{+2iπJ t}¹^+2iΩδ⁴^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 53

2pq p²−q²4

−2ⁱp(p−q)⁴q p²−q²2

e^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^{+6iπJ δ}² 54

−2ⁱpq p²−q²4

e⁻^iΩt²²^{+iπJ t}²⁻^iΩt²¹^{+2iπJ t}¹^−2iΩδ⁴^{+4iπJ δ}⁴^−2iΩδ²^{+6iπJ δ}² 55

−2ⁱpq p²−q²4

e⁻^iΩt2² ^{+iπJ t}²⁺^iΩt1² ^{+2iπJ t}¹^{+4iπJ δ}⁴^+2iΩδ²^{+6iπJ δ}² 56

4(q+p)² p²−q²4

−ⁱ4(p−q)² p²−q²4

e⁻^iΩt²²^{+iπJ t}²^{+2iπJ t}¹⁻^iΩδ⁴^{+4iπJ δ}⁴^{+6iπJ δ}² 57

−ip³q³ p²−q²2

e^iΩt²²^{+iπJ t}²^{+2iπJ t}¹^{+4iπJ δ}⁴⁻^iΩδ²^{+6iπJ δ}² 58

−ip³q³ p²−q²2

e^iΩt²²^{+iπJ t}²^+iΩt¹^{+2iπJ t}¹^+2iΩδ⁴^{+4iπJ δ}⁴^+3iΩδ²^{+6iπJ δ}² 59 ip²q²(q+p)² p²−q²2

−ip²(p−q)²q² p²−q²2

e^iΩt2² ^{+iπJ t}²⁺^iΩt1² ^{+2iπJ t}¹^+iΩδ⁴^{+4iπJ δ}⁴^+iΩδ²^{+6iπJ δ}² 60

To estimate the consequences for the observation, let us discuss the behaviour based on term 25. The evolution of anti-phase coherences into detectable in-phase coherence is sine-modulated with theJ−coupling constant andt2. The classical COSY-experiment, which shows anti-phase multiplets in the direct and indirect dimension, teaches us, that a sine-modulated antiphase FID, which starts at zero intensity fort2 = 0 requires a minimum acquisition of data points to get enough signal intensity. In pure shift experiments, which were aquired with the interferogram-based aquisition mode, the duration of one FID-chunk is comparably low (≤ 20 ms) to keep J−coupling evolution at minimum. Subsequently, the effect of the anti-phase terms on the PEPSIE pure shift spectra can be neglected. Note, that in systems with comparably large

S21

coupling constants these anti-phase artifacts might be visible. This context is exemplary illus-trated in Figure 2.4 for the term 25in Table 2.5. The plot shows the evolution of the pure amplitude factor of anti-phase term25with increasing degree of strong coupling. Comparing to the amplitude of the desired in-phase signals (blue line) the pure amplitude of term25(black) is low. If an aquisition time t2 = 20 ms of one data-chunk and the J−coupling constant of 2,3-dibromothiophene of 5.75 Hz is considered, the amplitude is further decreased by weighting with the sine-modulation factor (see equation 1.12). Assuming aJ−coupling constant of 15 Hz higher amplitude values can be expected, but they are negligible compared to the amplitude of the desired in-phase signals.

1 5 1 4 1 3 1 2 1 1 1 0 9 8 7 6 5 4 3 2 1 0

- 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

absolute amplitude / a.u.

∆ν/ J T e r m 2 5 : p u r e a m p l i t u d e T e r m 2 5 : t₂= 2 0 m s , J = 5 . 7 5 H z T e r m 2 5 : t₂= 2 0 m s , J = 1 5 H z d e s i r e d i n - p h a s e p e a k s

Figure 2.4: Plot of the amplitude factor of term25in comparison to the desired in-phase peaks (term group1, blue line) against the relation of shift difference ∆νand coupling constantJ. The black line is the evolution of the pure amplitude factor at the beginning of signal detection (zero detectable intensity). The green and the orange plots show the potentially detectable intensity after 20 ms aquisition time for the case of 2,3-dibromothiophene (J = 5.75 Hz) and the case of a comparably large proton-protonJ-coupling constant of 15 Hz.

S22

S23

3 PSYCHE, TSE-PSYCHE and simulated spectra with increasing

Im Dokument Entwicklung einer Toolbox an NMR-Experimenten zur Strukturaufklärung und deren Anwendung (Seite 144-158)