Interpreting the 2-Spin-1/2 Density Matrix and its Dynamics

In general the two-spin density matrix is composed of direct products of two onespin-1/2 density matrices. Since there are 4 one-spin-1/2 possible, there will be 16
two-spin-1/2 density matrices. Each of these has can be interpreted as level
populations, single-, double- or zero-quantum coherences.

These are referred to in the literature as the product-operator basis. What follows is
known as the product-operator formulism, of density matrix theory.

Lets refer to the two-spin-1/2 energy level diagram and see which features are
represented in the density matrix.
Figure 4
The Energy Level Diagram of a Two-spin-1/2 System with
Corresponding Density Matrix.


SQC
SQC
ZQC

SQC


DQC
SQC



P S2(+) S1(+) D(+)
 S2(-)
P
1
Z(+) S (+)
 S1(-)
Z(-)
P S2(+)



D(-) S1(-) S2(-)
P
The level population can be represented in terms of combinations of Iz and E, which
are:
1

1 0
Iz  E 
2 0

0
0 0 0
1 0 0
;
0 1 0 

0 0  1
1 0

1 0  1
E  Iz 
2 0 0

0 0
0 0
0 0

1 0

0  1
these represent the population difference for spin 1 and 2 respectively, which in turn
can be thought of as Iz(1) and Iz(2) representing the z-magnetization of spin and spin
2, respectively.
Figure 5
A Vector Description of the Product Operators Representing the ZMagnetization on Spins 1 and 2
E X Iz
Iz X E
1
z
2
z
y
x
1
z
y
y
x
x
z
2
y
x
The remaining combinations are:
1 0 0

1 0  1 0
Iz  Iz 
4 0 0  1

0 0 0
0
0
;
0

1
1
0
EE  
0

0
0
1
0
0
0
0
1
0
0
0

0

1
These represent the difference in population differences and the total population of
the energy levels. The former has no contribution at equilibrium, and the later is
constant.
Figure 6
A Vector Description of the Product Operators Representing the
Difference and Total Z-Magnetization on Spins 1 and 2
Iz X Iz
1
z
EXE
2
y
x
z
1
2
y
y
x
z
x
z
y
x

The single quantum transition can be represented by combinations of E, I+, I-, and Iz
as long as there is only one of either I+, or I-. Those including I+ are the positive
SQC’s and those with I- are the negative SQC’s. These are constructed with E and
I+ as follows:
0
0
I  E  
0

0
0
0
0
0
1
0
0
0
0
1
;
0

0
0
0
E  I  
0

0
1
0
0
0
0
0

1

0
0
0
0
0
These represent the pure positive single quantum transition for spin 1 and 2
respectively. These represent the lines in the spectrum corresponding to spin 1 and 2
respectively, and are the only observable terms in the density matrix.
Figure 7
A Vector Description of the Product Operators Representing the Inphase Positive Single Quantum Coherences on Spins 1 and 2.
I+ X E
1
z

z
2
o
x
E X I+
y
1
z
2
x
x
o
y
y
x
The remaining single quantum coherences composed of Iz and I+ are:
0
0
I  I z  
0

0
0
0
0
0
1 0
0  1
;
0 0

0 0
0
0
I z  I  
0

0
1
0
0
0
z
0 0
0 0

0  1

0 0
y

These are known as the antiphase contribution to the single quantum coherences.
They are not observable, since their components destructively interfere with each
other canceling any signal from them. These terms do evolve into observable terms
under the influence of the IzIz term of the Hamiltonian.
Figure 8
A Vector Description of the Product Operators Representing the
Anti-phase Positive Single Quantum Coherences on Spins 1 and 2.
I+ X Iz
1
z
z
2
o
x
Iz X I+
1
y
z
o
y
y
x
x
z
2
x

The situation is completely analogous for the negative sqc’s, which are composed of
I-, E, Iz giving: I   E , E  I  , I   I z , and I z  I  . These will not be
discussed further since only the positive sqc’s are detected.

There are two double quantum coherence terms, one that is positive and one that is
negative, composed of just I+, or I-, respectively. These are called a double quantum
coherence because they represent transition in which the total z-component of spin
angular momentum changed by +/- 2 units. These transition are strictly forbidden in
1-D spectroscopy and can’t be detected, however they do have to be considered in
when dealing with relaxation, and when multiple pulses are utilized. The operators
are:
0
0
I  I  
0

0
0
0
0
0
0
0
0
0
1
0
;
0

0
0
0
I  I  
0

1
0
0
0
0
0
0
0
0
0
0

0

0
y
Figure 9
A Vector Description of the Product Operators Representing the
Positive and Negative Double Quantum Coherences.
I+ X I+
1
z
2
o
z
1
o
y
x

I- X Iz
y
o
y
x
x
z
2
y
o
x
There are two zero-quantum coherence terms, composed of one of each I+ and I-.
These are called zero-quantum coherences since they represent transition between
states in which the total z-component of angular momentum is 0 units. These
transitions are again not observable, and do not occur in regular 1-D experiment.
They only need to be considered in relaxation and experiment utilizing multiple
pulses. The operators are of the form:
0
0
I  I  
0

0
0
0
0
0
0
1
0
0
0
0
;
0

0
0
0
I  I  
0

0
0
0
1
0
0
0
0
0
0
0

0

0
Figure 10
A Vector Description of the Product Operators Representing the
Zero Quantum Coherences.
I+ X I1
z
2
o
x
I- X I+
y
z
o
x
1
z
o
y
x
2
z
o
y
x
y
The Transverse components

The representation using the matrices corresponding to the transverse components can
also be used, and is in fact equivalent to the previous treatment. In this case Ix and Iy
are used instead of I+ and I-

These can no be thought of representing static x and y-components of the magnetic
field. It is often desirable to work with this representation when considering the
effects of pulses, which have phases along the x or y axes.

The transverse components are used to construct product operators of the form:
I x  E , E  I x , I x  I z , and I x  I y and I y  E , E  I y , I y  I z , and I y  I x

The single quantum transition can be represented by combinations of I x  E and
I y  E , which can be thought of as the x and y magnetizations on spin-1. The
operators are constructed as before by using the direct product between the
corresponding single-spin operators.
0

0
1
I x  E  
2 1

0
0
0
0
1
1
0
0
0
0
0


1
i
0
;
I

E

y
2 1
0


0
0
0
0
0
1
1
0
0
0
0

1
0

0 
Figure 11
A Vector Description of the Product Operators Representing the X
and Y magnetizations on Spin 1.
Ix X E
1
z
Iy X E
2
y
x
z
1
2
y
y
x
z
x
z
y
x

The remaining single quantum coherences composed of Iz and Ix and Iy are:
0 0

1 0 0
Ix  Iz 
2 1 0

0 1

1 0
0 1
;
0 0

0 0
0

i0
I y  Iz 
2  1

0
0 1 0
0 0 1

0 0 0

1 0 0
These are known as the antiphase x and y magnetization on spin 1. These are not
observable, since their components destructively interfere with each other canceling
any signal from them. These terms do evolve into observable terms under the
influence of the IzIz term of the Hamiltonian. This will be discussed in detail when
treating the INEPT sequence.
Figure 12
A Vector Description of the Product Operators Representing the
Anti-phase X and Y magnetization on Spin 1.
Ix X Iz
1
z
Iy X Iz
2
y
x

z
1
2
z
y
y
x
z
x
y
x
The situation is completely analogous for the x and y magnetization and antiphase
magnetization on spin-2.
Exercise
Construct and draw the vector diagrams for the two-spin-1/2 product operators
corresponding to spin two.

There are terms analogous to the double and single quantum coherence terms. These
are composed of just Ix, or Iy, respectively. In this case it is not clear which is a
double quantum coherence and which is a zero quantum coherence. When using this
representation the coherence order is not a primary consideration. The operators are:
0

1 0
Ix  Ix 
4 0

1
0 0 1
0 1 0
;
1 0 0

0 0 0
0

1 0
Iy Iy 
4 0

 1
0 0 1
0 1 0

1 0 0

0 0 0
Figure 13
A Vector Description of the Product Operators Representing the
Simultaneous x or y magnetization on both spins.
Ix X Ix
1
z
Iy X Iy
2
y
x

z
1
2
z
y
y
x
z
x
y
x
Notice that if you add I x  I x to I y  I y the zero-quantum operators are
recovered. When they are subtract they form the operator corresponding to the
double quantum transitions.
Exercise
Find the combination of I x  I x and I y  I y to get I   I  . Repeat for I   I  ,
I   I  , and I   I  .
The Treatment of a Pulse

During and RF pulse that is near or on resonance the effective magnetic field the
ensemble of nuclei experience is aligned along the axis defining the phase of the
pulse in question. As a result the Hamiltonian is dominated by this term, where the
remaining spin dynamics terms, such as those representing chemical shift and scalar
coupling interaction, are small in comparison and are thus ignored while the pulse is
active. Thus in the rotating frame the Hamiltonian during the pulse is of the form:
H  B1I
where indicates the phase of the pulse.

To start lets treat a specific case and then more to a more general treatment. Assume
that the initial density matrix is represented by the Iz indicating the equilibrium
situation. Lets also assume that the pulse phase is along the y-axis so the Hamiltonian
in question is:
H  1I y

Applying the solution to Liouville-von Neumann as follows, the density matrix
becomes:
 ( )  exp  i1I y I z exp i1I y 
now recall that the spin matrices such as Iz Ix and Iy are idempotent, thus the
propagator can be expressed as:
  
  
U (t )  exp  i1I y  E cos 1   i 2I y sin  1 
 2 
 2 


The expression for the resulting density matrix becomes:

  
    
  
   
 ( )  E cos  1   i 2I y sin  1   I z E cos  1   i 2I y sin  1  
2
2
2
2





 
 EI z E cos 2  1
 2


 
2i  I y I z E sin  1
 2







2  1  
  4I y I z I y sin 


 2 

 1
 cos 

 2

 1
  EI z I y cos 

 2
  1  
 sin 

  2 

This expression can be further simplified, with use of the formulae for the double arc
expansions and that of the corresponding commutator, to give:
 1 
2    
  4I y I z I y sin  1   i I z , I y sin 1 
 2 
 2 


  
   
 ( )  I z cos 2  1   I z sin 2  1   i I z , I y sin 1 
 2 
 2 

 I z cos1   I x sin 1 


 ( )  I z cos 2 






This result tells us that the z-magnetization is rotated in the x-z plane, starting by
passing through the x axis continuing on to the –z axis, towards the -x-axis back to
the z-axis. The angle of excitation depends on the length of the pulse and the strength
of the field applied.
Figure 14
The Rotation Induced in Iz by a Pulse Applied Along the Y-axis
z
Iz
1
B1
x'
y'

Rotations about the x, y and z axes can be summarized from the following expression,
for a rotation of I about I:
I  exp( iI )I exp( iI )

 
   
 
  
 E cos   2iI sin   I E cos   2iI sin  
 2
 2  
 2
 2 

 
 
 I cos 2    4I I I sin 2    iI , I sin 
 2
 2
if   
I

if   
I cos   iI , I sin 
where
I
0
I I I  
and  I ,I   
 I
 I ,I 
i) Rotations about the y’-axis:
I z  I z cos1   I x sin 1 
I x  I x cos1   I z sin 1 
Iy  Iy
ii) Rotations about the x’-axis:
I z  I z cos1   I y sin 1 
Ix  Ix
I y  I y cos1   I z sin 1 
iii) Rotations about the z-axis:
Iz  Iz
I x  I x cos1   I y sin 1 
I y  I y cos1   I x sin 1 
if   
if   

In general the rotation due to a pulse can be treated as the result of two consecutive
rotations, the first around either the x or y axis by the desired angle, by an angle 
around the z-axis to set the pulse phase. Thus the general expression is derived as
follows:
 ( ,  )  exp  i I z  exp  i I y I z exp i I y exp i I z 
 ( ,  )  exp  i I z I z cos  I x sin  exp i I z 
 ( ,  )  I z cos   exp  i I z I x exp i I z sin 
 ( ,  )  I z cos  I x cos   I y sin  sin 
 ( ,  )  I z cos  I x sin  cos   I y sin  sin 
Figure 15
General Rotation of a Magnetization from the Z-axis Towards the Transverse plane
along an Axis Which is at an Angle  from the Y-axis.
Iz


Iy

Ix
I
Chemical Shift Evolution

Chemical shift evolution is governed by the Hamiltonian:
H   (1   )Bo I z ,
where  is the chemical shift in ppm. In the rotating frame this Hamiltonian can be
re-expressed as:
H  I z
where is a frequency relative to the Larmor frequency of the nucleus

 o  B o .
A general expression for the evolution of a given spin term, I, can be derives as
follows:
 (t )  exp  i I z t I exp i I z t 
This expression states that chemical shift evolution amounts to just a rotation about
the z-axis at the rate  Therefore chemical shift evolution is governed by the
following set equations for the given spin term, I:
Iz  Iz
I x  I x cos t   I y sin  t 
I y  I y cos t   I x sin  t 
These expression state that the operator in the transverse plane rotate in that plane at
the rate  and that any operator aligned along the z-axis is unaffected.

The NMR signal due to a transverse component can be calculated from the
expectation value of the raising operator as follows, using the initial density matrix as
Ix,

  

Signal  I   Tr I*T  (t )  Tr I  I x cos t   I y sin  t 


 Tr I  I x cos t   I  I y sin  t 

 cos t Tr I  I x   sin  t Tr I  I y

 cos t Tr I  I   I   / 2  sin  t Tr I  I   I   / 2i 
cos t 
Tr I  I    Tr I  I    i sin  t  Tr I  I  -Tr I  I  
2
2
cos t  sin  t 

i
2
2

which is the expression for the FID without relaxation, in phase with detection along
the x-axis.
Evolution of the Density Matrix Under Scalar Coupling

Scalar coupling evolution is governed by the IzIz terms in the Hamiltonian such as:
H  2JI z  I z

This means that we have to work with a density matrix representing a two spin
system. These are constructed by taking the direct product between two density
matrices representing a single nucleus. For instance after a 900x pulse the zmagnetizations of two different nuclei are both rotated to the positive y axis, thus
each is represented by Ix thus the density matrix representing them both will be:
  I1x  I 2x  I x  E  E  I x

Rotations on each spin can be considered separately, since each spin can be dealt with
independently. This can be shown as follows:
 (t )  exp  i 2J I z  I z t  o exp i 2 I z  I z t 

Thus the propagator is of the form:
 
 
U (t )  exp  i 2 I z  I z   E  E cos   4iI z  I z sin  
 2
 2
where   Jt .

For a general initial state of the form:
  I1  I2
we start by:

 
  
 ( )  E  E cos   4iI z  I z sin   I1  I2

 2
 2 

 
  
E  E cos   4iI z  I z sin  
 2
 2 


After several simplification steps this can be shown to be equal to:
 
 
 ( )  I1  I2 cos 2    16I z I1 I z  I z I2 I z sin 2  
 2

 2

2i I1  I2 , I z  I z sin 
Case I:
 (0)  E  E
- Total z-magnetization
 
 
 ( )  E  E cos 2    16I z EI z  I z EI z sin 2  
 2
2iE  E, I z  I z sin 
 2
 EE

This operator represent the total z-magnetization which is time independent and thus
does not change during evolution under the J coupling Hamiltonian.
Case II:
 (0)  I x  E
- Positive single quantum coherence on spin1
 
 
 ( )  I x  E cos2    16I z I xI z  I z EI z sin 2  
2
2
 
 2i I x  E, I z  I z  sin
 
 I x  E cos  2i I x , I z   I z sin
 I x  E cos  2I y  I z sin

The in-phase x-magnetization on spin 1 thus evolves between the observable
I x  E and the non-observable I y  I z as the result of evolution under the J
coupling Hamiltonian. As a consequence the signal will oscillate with the evolution
period. This property is widely used in coherence transfer experiments, which will be
discussed in more detail in the next lecture topic.
Figure 16
Rotations induced by the I z  I z on I   E product operator.
Iz X I z
2Iy X Iz
I()

Ix X E
Case III:
 (0)  I y  E - negative single quantum coherence on spin1
 ( )  I y  E cos2     16I zI yI z  I zEI z sin 2   


 2


 2
2i I y  E, I z  I z  sin
 I y  E cos  2i I y , I z   I z sin
 I y  E cos  2I x  I z sin

The in phase y-magnetization evolves between I y  E and I x  I z due to the J
coupling Hamiltonian. Only the I y  E is observable.
Figure 17
Rotations induced by the I z  I z on I   E product operator.
Iz X Iz

I()
2Ix X Iz
Iy X E
Case IV:
 (0)  I z  E
- z-magnetization on spin 1.
 
 
 ( )  I z  E cos 2    16I z I z I z  I z EI z sin 2  
 2
2iI z  E, I z  I z sin 
 2
 
 
 I z  E cos 2    I z  E sin 2    2iI z , I z   I z sin 
 2
 2
 Iz  E

The Z-magnetization on spin 1 does not evolve under the J coupling Hamiltonian
because its corresponding product operator commutes with it. This means that all zmagnetization are invariant with time when considering evolution under J coupling.
Case V:
 (0)  I z  I z
- Difference in z-magnetization between spin 1 and 2
 
 
 ( )  I z  I z cos 2    16I z I z I z  I z I z I z sin 2  
 2
2iI z  I z , I z  I z sin 
 2
 Iz  Iz

The difference Z-magnetization does not evolve under the J coupling Hamiltonian
because its corresponding product operator commutes with it.
Case VI:
 (0)  I x  I z - Positive anti-phase single quantum coherence on spin 1
 
 
 ( )  I x  I z cos2    16I z I xI z  I z I z I z sin 2  
2
2
 
2i I x  I z , I z  I z  sin
 
 
 
E
 I x  I z cos2    I x  I z sin 2    2i I x , I z   sin
4
 2
 2
1
 I x  I z cos  I y  E sin
2

The anti-phase x-magnetization on spin 1 oscillates between the unobservable product
operator I x  I z and the observable I y  E . As a consequence the signal will
oscillate with the evolution period. This property is widely used in coherence transfer
experiments, which will be discussed in more detail in the next lecture topic.
Case VII:
 (0)  I y  I z - negative anti-phase single quantum coherence on spin1
 
 
 ( )  I y  I z cos2    16I z I y I z  I z I z I z sin 2  
2
2
 
 
2i I y  I z , I z  I z , sin
 
 
E
 I y  I z cos2    I y  I z sin 2    2i I y , I z   sin
4
 2
 2
1
 I y  I z cos  I x  E sin
2

The anti-phase y-magnetization evolves between I y  I z and I x  E as the result
of evolution under the J coupling Hamiltonian. Both terms are not observable when
the instrument is tuned to the positive single quantum coherence frequency. This
property is widely used in coherence transfer experiments, which will be discussed in
more detail in the next lecture topic.
Case VIII:
 (0)  I x  I y -zero quantum coherence (flip-flop)
 
 
 ( )  I x  I y cos2    16I z I xI z  I z I y I z sin 2  
2
2
 
2i I x  I y , I z  I z  sin
 
 
 
 I x  I y cos2    I x  I y sin 2    0sin
 2
 2
 Ix  I y

This product operator corresponds to the combination of a zero-quantum coherence
and a double quantum coherence. It is found to commutes with the J coupling term in
the Hamiltonian it does not vary in time.
Case IX:
 (0)  I y  I x - negative double-quantum coherence.
 
 
 ( )  I y  I x cos2    16I z I y I z  I z I xI z sin 2  
2
2
 
2i I y  I x , I z  I z  sin
 
 
 
 I y  I x cos2    I y  I x sin 2    0sin
 2
 2
 I y  Ix

This product operator also corresponds to a combination of a zero- and doublequantum coherences. It is also found to commute with the J coupling term in the
Hamiltonian and hence does not vary in time.
Case X:
 (0)  I x  I x - positive double-quantum coherence.
 ( )  I x  I x cos2    16I z I xI z  I z I xI z sin 2   




 2
 2
2i I x  I x , I z  I z  sin
 
 
 I x  I x cos2     I x  I x sin 2     0sin
 2
 2
 Ix Ix

This product operator also corresponds to a combination of a zero- and doublequantum coherences. It is also found to commute with the J coupling term in the
Hamiltonian and hence does not vary in time.
Exercise
Derive the case where
Hamiltonian.
 (0)  I y  I y . Describe its evolution under the J coupling