Page 38 out of 81 total pages , Page 11 out of 15 pages in this chapter
A chemical shift is the observed effect from the electron cloud surrounding a nucleus responding to an applied magnetic field. The spin itself experiences not only the applied field but also a field from the perturbed electron cloud, the latter field generally opposing the applied field or "shielding" the nucleus. Not only can the shielding contribution be quite large, it is usually orientationally dependent because the surrounding electron cloud is no spherical (due to chemical bonds). In the following discussion we will not be concerned with the isotropic and anti-symmetric parts of the shielding. The former produces measureable chemical shifts whereas the latter is rarely seen. Rather the focus will be on the symmetric rank 2 contribution, that which produces relaxation effects in liquid NMR and orientationally dependent shifts in solids.
We will shortly concern ourselves with the mathematical representation of chemical shift interactions, in particular their description in terms of spatial and spin tensors. The spatial tensors will be cast in both Cartesian and spherical coordinates and we will switch between the two when convenient. The figure below relates the orientation angles theta and phi to the standard right handed coordinate system in all GAMMA treatments.
The internal structure of class IntSA contains the quantities listed in the following table (names shown are also internal).
The values of I is the spin quantum number of the quadrupolar nucleus. It dictates how many energy levels (and transitions) are associated with the quadrupolar interaction. It is intrinsically tied into the values and dimensions of the matrices in the vector Tsph. Note that I will be an integer multiple of 1/2 and that only nuclei with I>1/2 will have a quadrupole moment.
The two values DELZZ and ETA are all that is required to specify the quadrupolar interaction strength and may be used to represent the quadrupolar spatial tensor. However, in GAMMA the value of DELZZ is factored out of the spatial tensor such that all rank two interactions (such as the quadrupolar interaction) have the same spatial tensor scaling.
The two angles THETA and PHI indicate how the quadrupolar interaction is aligned relative to the interaction principal axes (PAS). These are one in the same as the angles shown in Figure 19-9 when the Cartesian axes are those of the PAS with the origin vaguely being the center of the nucleus. These are intrinsically tied into the values in the array Asph.
There are five values in the complex vector Asph and these are irreducible spherical components of the quadrupolar spatial tensor oriented at angle THETA down from the PAS z-axis and over angle PHI from the PAS x-axis. Note that these 5 values are not only orientation dependent, they are also ETA dependent. If either of the three the interaction values {ETA, THETA, PHI} are altered these components will all be reconstructed. The values in Asph will be scaled such that they are consistent with other rank 2 spatial tensors in GAMMA which are independent of the interaction type.
The vector of matrices relates to the sperical spin tensor components according to:
and the vector of complex numbers relate to the spherical spatial tensor components via
A chemical shift is the observed effect from the electron cloud surrounding a nucleus responding to an applied magnetic field. The spin itself experiences not only the applied field but also a field from the perturbed electron cloud, the latter field generally opposing the applied field or "shielding" the nucleus. We can write this latter "induced" field in terms of the applied field, 
, as
where 
is the chemical shielding tensor, a 3x3 array in Cartesian space, and the 
vectors in Cartesian space. In matrix form this is simply1
,
the induced field depends on the applied field strength, the applied field orientation, and the surrounding electron cloud. Note that 
will not necessarily be co-linear with the applied field. Of course, every nuclear spin will have its own associated chemical shielding tensor. The classical interaction energy between this induced field and a nuclear spin is
where 
is the magnetic moment, 
the spin index, 
the energy, and subscript 
used to denote chemical shielding.
The associated Hamiltonian is obtained from substitution of 
for 
.
In matrix form this equation looks like
.Taking the magnitude of the applied field out, equation (0-19) is simply
with 
and 
a normalized magnetic field vector in the direction of the applied field.
Equation (0-20) can also be rearranged to produce an equation involving two rank 2 tensors by taking the dyadic product of the vectors 
and 
.
The dyadic product to produce 
is explicitly done via
.
The chemical shielding Hamiltonian can thus be formulated as a scalar product of two rank 2 tensors. Letting 
, we have
.
The previous equation, (0-22), can also be rewritten in term of irreducible spherical components rather than in terms of the Cartesian components using the substitution
We can obtain the 9 irreducible spherical components of the CS rank 2 "spin" tensor2 directly from the Cartesian components, 
, as indicated in GAMMA Class Documentation on Spin Tensor. These are
,
where 
signifies the chemical shielding interaction. The tensor index 
spans the rank: 
while the tensor index 
spans 
: 
The nine formulas for these quantities a listed in the following figure where the field components are those of the normalized field vector 
.3
For a spin 1/2 particle and 
s, the matrix form of these tensor components are shown in the following figure in the single spin Hilbert space. The spin index has been omitted, the field components are those of the normalized vector 
.
The matrix representation of these nine tensor components will depend upon the matrix representations of the individual spin operators from which they are constructed4. These in turn depend upon the quantum number of the spin involved. For a treatment of a spin 1/2 particle the shielding tensor components are expressed in their matrix form in the default product basis of GAMMA as follows. In this case the spin index is implicit.
The raising an lowering components of the field vector are defined in the standard fashion, namely 
. The simplest situation occurs when magnetic field points along the positive z-axis, 
, i.e. these spin-space tensors are written in the laboratory frame. Then, the (normalized) field vector simplifies, 
and 
. The applicable equations for the shielding space-spin tensors are then as follows.
For a spin 1/2 particle and 
along the positive z-axis, the matrix form of these tensor components are shown in the following figure5 (in the single spin Hilbert space).
We must very careful in using these single spin rank 2 shielding tensors of this type because they contain both spatial and spin components. If we desire to express the shielding Hamiltonian relative to a particular set of axes we must insure that both the spatial tensor and the "spin" tensor are expressed in the proper coordinates. The spatial tensor alone cannot be rotated as it rotates only part of the spatial components6. It is improper to rotate this tensor in spin space because it also rotates spatial variables.Furthermore, note that these rank 2 components are not the same as the rank 1 tensor components.
The 9 irreducible spherical components of a rank two spatial tensor, 
, are related to its Cartesian components by the following formulas (See GAMMA Class Documentation on Spatial Tensor).
Again the subscript l spans the rank as l = [0, 2], and the subscript m spans +/- l, m = [-l, l].
In this chemical shielding treatment we then have the components 
as indicated in equation (1-1). Thus, the irreducible spherical tensor components can be obtained by substituting the Cartesian elements of 
into equations (1-2).
A general rank two Cartesian tensor can be rewritten in terms of a sum over tensors of ranks 0 through 2 as follows,
The rank 0 part is isotropic (scalar), the rank 1 part is antisymmetric and traceless, and the rank 2 part traceless and symmetric.
As with any spatial tensor, the chemical shielding spatial tensor can be specified in its principal axis system, the set of axes in which the irreducible rank 2 component is diagonal7. The shielding tensor values are experimentally determined in the tensor principal axes.
Rank 2 spatial tensors are commonly specified in their principal axis system by the three components; the isotropic value 
, the anisotropy 
, and the asymmetry 
. These are generally given by
, 
A set of Euler angles 
is normally also given to relate the spatial tensor principle axes to another coordinate system. For the shielding spatial tensor we have
The irreducible spherical elements of the shielding tensor, 
, in the principal axis system are, by placement of (4-35) into (1-2),
Throughout GAMMA, we desire all irreducible spherical rank 2 spatial components to be scaled so as they are independent of the particular interaction. To do so we adjust them to be as similar to normalized spherical harmonics as possible. Thus, we here scale the shielding irreducible ranke 2 spatial tensor so that the 
component will have the same magnitude as the 
rank two spherical harmonic when the two spherical angles are set to zero. Our "normalization" factor "X" is obtained by
We thus define the GAMMA shift anisotropy spatial tensor scaled such that
and the components are given in the next figure.
The scaling factor 
which was multiplied into the "s" components will be compensated for in the shielding anisotropy interaction constant. The shielding anisotropy Hamiltonian given in equation (23) becomes
In GAMMA, since we have defined our spatial and spin tensors to be scaled independent of the type of interaction, we use an interaction constant as a scaling when formulating Hamiltonians. Shielding anisotropy Hamiltonians may be produced from
Such interaction constants are not very common in the literature (except with regards to some papers treating relaxation in liquids) and thus not intuitive to many GAMMA users. So, one simply needs to be aware of the relationships between the interaction constant and any commonly used shift anisotropy definitions. One common quantity is the 
, the chemical shift (or shielding) anisotropy8.
The former is often labeled as 
, an anacronym for Nuclear Quadrupolar Coupling Constant. There are many definitions in the literature for the latter. In GAMMA we chose the definition so that this frequency will be the distance between transitions when the quadrupolar Hamiltonian is a small perturbation to the Zeeman Hamiltonian (i.e. when a spin's Larmor frequency is much higher than its quadrupolar coupling constant).
As for the quadrupolar interaction constant we have
We can express the spatial tensor components 
relative to any arbitrary axis system (AAS) by a rotation from the principal axes to the new axes via the formula
where 
are the rank 
Wigner rotation matrix elements and 
the set of three Euler angles which relate the principal axes of the chemical shielding tensor to the arbitrary axes9. Unlike the dipolar Hamiltonian treatment which only had a rank 2 component, components of ranks l = 0, 1, and 2 may contribute to the shielding Hamiltonian. Since these ranks behave differently under rotations we shall write the overall shielding Hamiltonian to reflect this. Beginning with equation (1-1)
we define a chemical shielding interaction constant as
and expand the summation over the different ranks.
.
There is good reason to separate these terms. The rank 0 component of the shielding Hamiltonian is rotationally invariant and called the isotropic chemical shielding Hamiltonian. In high-resolution NMR it is normally included in the static Hamiltonian 
. The rank 2 part is call the chemical shielding anisotropy Hamiltonian. In liquid systems this Hamiltonian averages to zero and thus not affect observed shielding values. It will contribute to relaxation of the system. On the other hand, in solid systems this component does not average away and will partially determine peak shapes in powder averages. The rank 1 component is the antisymmetric part of the shielding Hamiltonian. Since the antisymmetric part of the shielding tensor is difficult to measure, this part of the shielding Hamiltonian is usually assumed small and neglected.
The isotropic component (
) of the chemical shielding Hamiltonian is thus written
,
the antisymmetric component (
) of the chemical shielding Hamiltonian is
,
and the anisotropic component (
) of the chemical shielding Hamiltonian is
Throughout GAMMA, we desire all rank 2 spatial tensor irreducible spherical components to be similar to rank 2 normalized spherical harmonics if possible. Thus, we here scale the shielding spatial tensor such that the components 
will become normalized rank two spherical harmonics when the asymmetry term is zero, 
. Thus our aim is to use the following spherical tensor to define the spatial chemical shielding tensor.
Now application of equation (6-7) on the 
, 
component reveals the value of the constant K
and our scaled chemical shielding spatial tensor is then
The (rank 2) components 
in the principal axis system are
The anisotropic component (
) of the chemical shielding Hamiltonian, equation (6-6), is then equivalently expressed by
where we have define the chemical shielding anisotropy interaction constant to take into account that we have scaled the spatial tensor components by the factor 
.
In GAMMA, since we have defined our spatial and spin tensors to be scaled independent of the type of interaction, we use an interaction constant as a scaling when formulating Hamiltonians. Shielding anisotropy Hamiltonians may be produced from
Such interaction constants are not very common in the literature (except with regards to some papers treating relaxation in liquids) and thus not intuitive to many GAMMA users. So, one simply needs to be aware of the relationships between the interaction constant and commonly used shift anisotropy definitions. Two common quantities are the quadrupolar coupling10 constant 
and the quadrupolar frequency 
.
The former is often labeled as 
, an anacronym for Nuclear Quadrupolar Coupling Constant. There are many definitions in the literature for the latter. In GAMMA we chose the definition so that this frequency will be the distance between transitions when the quadrupolar Hamiltonian is a small perturbation to the Zeeman Hamiltonian (i.e. when a spin's Larmor frequency is much higher than its quadrupolar coupling constant).
As for the quadrupolar interaction constant we have
and the chemical shielding Hamiltonian for a single spin becomes
When working with an entire spin system one must sum over all spins with the tensors being in the same coordinate system, for our purposes the laboratory system. The chemical shielding Hamiltonian for a spin system becomes the following.
The following figures summarize the rank 2 treatment of the shielding Hamiltonian.
Chemical shielding will affect the observed resonance frequency. The isotropic (rank 0) contribution to the shielding is normally included with the Zeeman Hamiltonian to form the isotropic chemical shift Hamiltonian. The anti-symmetric (rank 1) contribution to shielding is rarely observed. The symmetric rank 2 contribution to the chemical shielding interaciton, that which we are concerned with in class IntSA, produces the following Hamiltonian11.
We have simplified (and standardized) our nomenclature by defining a shielding anisotropy interaction constant as
Note that in the principal axis system (PAS) when the field is oriented along the +z axis, the shift anisotropy Hamiltonian is given by a relatively simple formula because both the 
and the 
terms are zero.
When the shift anisotropy interaction is oriented relative to its principal axes the Hamiltonian equation becomes much more complicated than the one above.
At this point we will substitute in the spin operatiors (assuming Bo is along +z)
We can use the identities 
to obtain
Upon substitution of the oriented spatial components we obtain
Although these equations are generally applicable, it is convenient to express the shielding Hamiltonian with clear separation between the different ranks (the components with differing values of 
). The isotropic component 
in the treatment of liquid samples will normally be placed into an overall isotropic Hamiltonian, 
because it does not disappear upon rotational averaging. The asymmetric component, 
, is usually zero, the shielding tensor taken as essentially symmetric. 
When the quadrupolar interaction has alignment along its principal axes system virtually all of the quadrupolar equations simplify. The following figure collects all of these for convenience.
= 0).
Included are the general relationships between the (GAMMA scaled) Cartesian tensor components to the irreducible spherical components. They are valid when is defined accordingly! If is defined by the other common convention (|Azz| 
|Axx| |Ayy|) then the sign on the 
will change as will the sign on the Hamiltonian term multiplied by .
When the quadrupolar interaction has a arbitrary alignment (relative to its principal axes system) the quadrupolar equations become complicated. The figure below depicts them for convenience.
13 = 0).
Note that the effect of the chemical shielding is to alter the field which the spin experiences. This is clearly seen from the product2which produces an effective field vector for the spin.
Due to the nature of the CS interaction, the rank 2 tensor treatment produces a "spin" tensor3which contains spatial components, namely the magnetic field vector. As a result, care must be used when performing spatial rotations on shielding tensors. Any spatial rotations must involve rotations of both
and
For these formulae, it is important to note that it is the second component in the composite spin/space tensor which is set to the normalized magnetic field vector4, although we might just as well have used the first vector instead. The difference is that the
equations would then appear of opposite sign from those given here. Our field vector has be set to point along the positive z-axis in the laboratory frame.
Note that the spin tensors are invariably constructed in the laboratory coordinate system. Here the z-axis corresponds to the direction of the spectrometer static magnetic field and the coordinate system is right-handed.5
The GAMMA program which produced these matrix representations can be found at the end of this Chapter, Rank2SS_SpinT.cc.6
See the discussion in Mehring7
The principal axis system is set such that |dzz|8
In angular frequency units this is9where
is the quadrupole moment. Note that, although the definition of
is standardized, there seems to be some variation in the literature as to what the quadrupolar splitting frequency
is.
In this instance we must be careful to express the elements10in the same axis system as
. When
is rotated in space, so must be
. Essentially the field vector changes relative to any new coordinate system when constructing
.
In angular frequency units this is11where
is the quadrupole moment. Note that, although the definition of
is standardized, there seems to be some variation in the literature as to what the quadrupolar splitting frequency
is.
Keep in mind that this Hamiltonian is for a single spin of quantum number I. In a multi-spin system one will have to sum such Hamiltonians for all spins.12
The scaling on both {A2m} and T2m} are arbitrary, GAMMA uses an (uncommon) scaling which independent of the interaction type. What is NOT arbitrary is the scaling within either of the two sets of components. In addition, the combined scaling of the two sets is critical to the proper formation of quadrupolar Hamiltonians. For that, GAMMA uses an interaction constant.13
The quadrupolar interaction constant, as well as the relative scalings on the sets of spatial and spin tensors, can be adjusted as desired. However all components of the space or spin tensor must be adjusted by the same scaling. The GAMMA scaling is oriented to liquids where so that all spatial components are related to the spherical harmonics in the spatial tensor PAS.14
The scaling on both {A2m} and T2m} are arbitrary, GAMMA uses a scaling which independent of the interaction type. What is NOT arbitrary is the scaling within either of the two sets of components. In addition, the combined scaling of the two sets is also crucial. For that, GAMMA uses an interaction constant.
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© 1996 Scott A. Smith, The NHMFL, and The Florida State University. All Rights Reserved. |
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