matrix representation of spin operator
DESCRIPTION
Matrix representation of Spin Operator. J. I kz I l z. 2 I ky I l z. I kx. x. x. t 2. t 1. Correlation Spectroscopy (COSY). Considering two spin k and l. y. ϕ R. -y. y. y. y. t 2. I. y. ϕ 1. ϕ 2. y. S. F. A. B. C. D. E. - PowerPoint PPT PresentationTRANSCRIPT
Matrix representation of Spin Operator
2
2
2
cos sin
cos sin
cos 2 sin
cos sin
cos 2 sin
2 2 cos
kX
Z
kl kZ lZ
Z
kl kZ lZ
kl kZ lZ
IkZ kZ kY
tIkX kX k kY k
J I IkX kX kY lZ
tIkY kY k kX k
J I IkY kY kX lZ
J I IkX lZ kX lZ
I I I
I I t I t
I I Jt I I Jt
I I t I t
I I Jt I I Jt
I I I I
2
sin
2 2 cos sinkl kZ lZ
kY
J I IkY lZ kY lZ kX
Jt I Jt
I I I I Jt I Jt
J.IkzIlz
2IkyIlz
Ikx
x x
t1t2
Correlation Spectroscopy (COSY)
Considering two spin k and lˆ
2
ˆ
1 1
1
1
1
First under chemical shift hamiltonian during
considering evolution of only
cos sin
co
t period
under coupling hamiltonian during t period
s
X
Z
H
kZ lZ kY lY
kY
HkY kY k kX k
kY k kX
I I I I
I
I I
I
J
t I t
t I
ˆ
1 1 1 1
1 1 1
1 1 1
sin ( cos 2 sin )cos
(
puls
cos 2 sin )sin
( cos 2 sin )cos
( c s
e2
o
JHk kY kl kX lZ kl k
kx kl kY lZ kl k
kZ kl kX lY kl k
kX kl
t I J t I I J t t
I J t I I J t t
I J t I I J t
x
t
I J
1 1 12 sin )sinkZ lY kl kt I I J t t
1 1 1
1 1
1
1 1 1
1
2
( cos 2 sin )cos
( cos 2 sin )sin
cos sin 2 s
in sin
kz kl kx ly kl k
kx kl kz ly kl k
kx
evolution under chemical shift hamiltonian
kl k kz ly kl
during t perio
k
d
Obser
I J t I I J t t
I J t I I J t t
I J t t I I J t t
vable term is
2 2 1
1
2 2 2
2
1
1
( cos sin )cos sin
cos 2 sin cos sin cos
cos
for the
evolution under J coupling hamiltonian dur
kx k ky k kl k
kx kl ky lz kl k k
first ter
ing t period for t
m
first termh
kl
ky kl
e
I t I t J t t
I J t I I J t t t J t
I J
2 2 2 1 1
2 2 1 1
2 sin sin sin cos
cos cos sin cos
kx lz kl k k kl
kx ky kl k k kl
t I I J t t t J t
I I J t t t J t
evolution under chemical shift hamiltonian during t perio second d for the 2
evolution under couplin
1 1
2
g hamiltonia
ter
2
m
1 1
2 sin sin
2 cos sin si
for the second term
n s
in
kZ lY kl k
kZ lY l lX l kl k
J
I I J t t
I I t I t J t t
2 2 2
1 1
2 2
secn during t ond term
1
period for the 2
cos 2 sin cos2 sin sin
cos 2 sin sin
lY kl lX kZ kl l
kZ kl k
lX kl lY kZ kl l
I J t I I J t tI J t t
I J t I I J t t
Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)
ϕ1= x, -x, x, –x
ϕ2= x, x, -x, –x
ϕR= x, -x, -x, x
1 (2 )ISJ
yyt2
ϕ1y
y
yϕ2
y
2
2
2
2
1
2
t 1
2
t
I
S
ϕR
A B C D E F
-y
Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)
I
S
90
At AXI
Z YI I
2 1 2co
At B
s 2 sin 2IS Z Z ISJ I S JY Y IS X Z IS X ZI I J I S J I S
y
y
2
2
A B
Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)
y
y
2
2
I
S A B
y
ϕ1
y
ϕ21
2
t 1
2
t
C D E
90 902 2
At
2
CY XI S
X Z Z Z Z YI S I S I S
-y
ϕ1= x, -x, x, –x
1( )1 1180
2 2 cos 2 sin
At Ds Z
X
t SZ Y Z Y S Z X SI
I S I S t I S t
90 ( )1 1 1 12 cos 2
At E
sin 2 cos 2 sinY XI SZ Y S Z X S X Z S X X SI S t I S t I S t I S t
2
At B
X ZI S
ϕ2= x, x, -x, –x
Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)
yy
ϕ1y
y
ϕ2
2
2
1
2
t 1
2
t
I
S A B C D E
-y
y
y
2
2
F
1 12 cos 2 si
At
n
E
X Z S X X SI S t I S t
21 1 1
1
2 cos 2 sin 2 cos sin cos
A
2 sin
t FIS Z ZI I S
X Z S X X S X Z IS Y IS S
X X S
I S t I S t I S J I J t
I S t
11 2 si
If 2
n
1
cosY S X X
I
S
S
I I St t
J
Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)
yy
ϕ1y
y
ϕ2
2
2
1
2
t 1
2
t
I
S A B C D E
-y
y
y
2
2
F
,
1
1
1
1 2 sincos
cos
;
2 s
F
in
is only observable
is multiple quantum term thus unobserva
X XY S
bl
Y
X
S
X S e
S
whil
I S tI t
I t
At
I S te
t2
ϕR
2( )1 2 2 1
2
cos cos sin cos
ZSt I
Y S Y I X I S
A
I t I
fter t term at the re
t I
ei
t
c ve
t
r
ϕ2= x, x, -x, –xϕ1= x, -x, x, –x ϕR= x, -x, -x, x
Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)
290 180 180 180 ( )2cosX Y Y Y I ZI I I I t I
Z Y Y Y Y Y II I I I I I t
yyt2
ϕ1y
y
yϕ2
y
2
2
2
2
1
2
t 1
2
t
I
S
ϕR
A B C D E F
-y
For protons NOT coupled to S spin
We need two step phase cycle to get rid of this magnetization ϕ1= x, -x ϕR= x, -x
Steps ϕ1 ϕR
Magnetization at point F
Protons coupled to S spin Protons NOT coupled to S spin
Step I x x
Step II -x -x
cos 2 sin1 1I t I S tY S X X S
( cos 2 sin )1 1I t I S tY S X X S
IYIY
Heteronuclear Single Quantum Correlation Spectroscopy (HSQC)yy
t2
ϕ1y
y
yϕ2
y
2
2
2
2
1
2
t 1
2
t
I
S
ϕR
A B C D E F
-y
For complete removal of multiple quantum term we need four step phase cycle to get rid of this magnetization
ϕ1= x, -x, x, -x ϕ1= x, x, -x, -x ϕR= x, -x, -x, x
Steps ϕ1 ϕ2 ϕR
Magnetization at point F
Protons coupled to S spin Protons NOT coupled to S spin
Step I x x x
Step II -x x -x
Step III x -x -x
Step IV -x -x x
cos 2 sin1 1I t I S tY S X X S
( cos 2 sin )1 1I t I S tY S X X S
( cos 2 sin )1 1I t I S tY S X X S
cos 2 sin1 1I t I S tY S X X S
IYIYIYIY
Sensitive enhanced Heteronuclear Single Quantum Correlation Spectroscopy (SE-HSQC)
yyt2
ϕ1y
y
yϕ2
y
2
2
2
1
2
t 1
2
t
I
S
ϕR
A
-y
,
1
1
1
1 2 sincos
cos
;
2 s
A
in
is only observable
is multiple quantum term thus unobserva
X XY S
bl
Y
X
S
X S e
S
whil
I S tI t
I t
At
I S te
ϕ3= y, -y, y, –y
2
ϕ3
-y
y
y
2
y
2
SE- Heteronuclear Single Quantum Correlation Spectroscopy (SE-HSQC)
yyt2
ϕ1y
y
yϕ2
y
2
2
2
1
2
t 1
2
t
I
S
ϕR
-y
90 ( , )1 1 1 1cos 2 sin cos 2 in
B
sX XI SY S X X S X S X Z SI t I S t I t I S
At
t
DA B C
2
ϕ3
-y
y
y
2
y
2
ϕ3= y, -y, y, –y
21 1 1 11 2
cos 2 sin cos si
C
nIS Z Z
IS
J I SX S X Z S Z S Y SJ
I t I S t I
At
t I t
901 1 1 1cos sin cos sin
DYI
Z S Y S X S Y SI t I t I t I t
At
SE- Heteronuclear Single Quantum Correlation Spectroscopy (SE-HSQC)
yyt2
ϕ1y
y
yϕ2
y
2
2
2
1
2
t 1
2
t
I
S
ϕR
D
-y
A B C
2
ϕ3
-y
y
y
2
y
2
ϕ3= y, -y, y, –y
1 1 1
1 1 1
exp
cos sin
cos sin
X S n Y S
X S n Y S
For the first eriment
For the second
I t I t
I t
experi
I
n
t
me t