Quadrupolar Interaction

In addition to the CSA the quadrupolar interaction, from a nucleus with I > 1/2, can dominate spectra. Characteristic lineshapes are present if the magnitude of the interaction is large enough. The non-spherical charge distributions have an associated quadrupole moment which interacts with the electric field gradient (EFG). The EFG can be generated from local nuclei but also from chemical effects beyond the immediate coordination sphere. The significant magnitude of the quadrupolar interaction requires expansion of the Hamiltonian to the second order, such:69

𝐻̂𝑄 = 𝐻̂𝑄 (1)

+ 𝐻̂_𝑄(2) (Equation 2-16)

From an illustrative perspective, the first order quadrupolar interaction shifts the 1/2 to -1/2 central transition (CT) maintaining the characteristic splitting, shown in Figure 2-4. It is the second order interaction which characteristically perturbs the

Figure 2-3: NMR terminology used to describe the position of resonances. Some terms are historical, surviving from when field sweep method was used rather than the modern

central transition, as shown in Figure 2-4, where the CT has been highlighted in green. A general form of the Hamiltonian for the quadrupolar interaction can be expressed such:65

𝐻̂_𝑄 = 𝑒𝑄

2𝐼(2𝐼−1)ℏ𝐼 ∙ 𝑽 ∙ 𝐼 (Equation 2-17)

Where e is the electric charge and Q is the nuclear quadrupole moment. Finally, V is the EFG described by a second rank tensor which in the PAS has a comparable form to Equation 2-8 (where the off diagonal components are all zero). The principle components of the EFG are labelled such ǀVZZǀ ≥ ǀVYYǀ ≥ ǀVXXǀ, furthermore the quadrupole interaction has no isotropic component, thus the trace (sum of the diagonal) is zero. The tensor can be expressed via two variables the first being the quadrupole coupling constant (CQ) which describes the magnitude of the interaction,

such:

𝐶_𝑄 = 𝑒𝑄𝑉𝑍𝑍

ℎ . (Equation 2-18)

The variables of Equation 2-18 are as defined before note the use of the standard

Plank’s constant, CQ has units of hertz. The second parameter is the asymmetry Figure 2-4: Energy level splitting for a quadrupolar nucleus, I = 3/2 illustrating how the energy levels are perturbed by the first and second order quadrupolar interactions. The central transition

parameter (ηQ), not to be confused with the similar parameter of the Haeberlen

convention for CSA,

𝜂𝑄 =

𝑉𝑌𝑌−𝑉𝑋𝑋

𝑉𝑍𝑍 . (Equation 2-19)

Thus, NMR of quadrupolar nuclei are reported quoting a δiso, CQ and the unit less ηQ.

Where 0 ≤ ηQ≤ 1, with 0 being axially symmetric tensor for the EFG in the PAS. As

illustrated before, the first order quadrupolar interaction does not perturb the CT. Thus, discussion will focus on the second order perturbation (𝐻̂_𝑄(2)) since this is what will be observed experimentally. This is not an exhaustive derivation of the origin of quadrupolar lineshapes, but aims to introduce the complexity contained within the measurements to be undertaken, for a detailed derivation the reader is directed elsewhere.69-71 _{The second order Hamiltonian can be expressed as a Cartesian tensor}

commuting with the lab frame which has been shown to be of the form:

𝐻̂_𝑄(2) = − 1 𝜔0( 𝑒𝑄 4𝐼(2𝐼−1)ℏ) 2 {2(𝑉_𝑥𝑧− 𝑖𝑉_𝑦𝑧)(−𝑉_𝑥𝑧− 𝑖𝑉_𝑦𝑧)𝐼̂_𝑧[4𝐼(𝐼 + 1) − 8𝐼̂_𝑧2− 1] + 2 (1 2(𝑉𝑥𝑥− 𝑉𝑦𝑦) − 𝑖𝑉𝑥𝑦) ( 1 2(𝑉𝑥𝑥− 𝑉𝑦𝑦) + 𝑖𝑉𝑥𝑦) 𝐼̂𝑧[2𝐼(𝐼 + 1) − 2𝐼̂𝑧 2₋ 1]} (Equation 2-20)

The consequences of Equation 2-20 are the formation of an orientation dependant lineshape in the NMR spectrum. The lineshapes broaden more significantly in low field spectra since 𝐻̂_𝑄(2) is proportional to the inverse of B0. Following the derivation

shown in literature69_{, Equation 2-20 can be used to calculate the shift of the Zeeman}

splitting due to the second order effects. The expression of the frequency shift in terms of the PAS in the observation frame is achieved utilising the Wigner rotation matrix. The Wigner rotation matrix allows for a spherical tensor such as the EFG tensorto be expressed in adifferent coordinate frame via three positive rotations θ,

ϕ, and χ (this also applies to δPAS _{expressed in the EFG PAS or lab frames). Thus, the}

second order quadrupolar frequency shift of the CT using the definitions given in Equations 2-17 and 2-18 is given as:

𝜔 −1₂,1₂ (2) = − 1 6𝜔0( 3𝐶𝑄 4𝜋𝐼(2𝐼−1)ℏ) 2 [𝐼(𝐼 + 1) −3 4] [𝐴(𝜃, 𝜂𝑄)cos 4_{𝜙 +} 𝐵(𝜃, 𝜂_𝑄)cos2_{𝜙 + 𝐶(𝜃, 𝜂} 𝑄)] (Equation 2-21)

The variables A(𝜃, ηQ), B(𝜃, ηQ) and C(𝜃, ηQ) are defined in Equations 2-21 to 2-23:

𝐴(𝜃, 𝜂_𝑄) = −27 8 + 9 4𝜂𝑄cos 2𝜃 − 3 8(𝜂𝑄cos 2𝛼) 2 (Equation 2-22) 𝐵(𝜃, 𝜂𝑄) = 30 8 − 1 2𝜂𝑄 2_{− 2𝜂} 𝑄cos 2𝜃 + 3 4(𝜂𝑄cos 2𝜃) 2 (Equation 2-23) 𝐶(𝜃, 𝜂_𝑄) = −3 8− 1 3𝜂𝑄 2₋1 4𝜂𝑄cos 2𝜃 + 3 8(𝜂𝑄cos 2𝜃) 2 (Equation 2-24)

The third angle χ for the Wigner rotation matrix does not play a role in Equation 2- 21 due to the Bz being a symmetry axis. Also, Equation 2-21 is for a static single

Figure 2-5: Examples of how CQ (a) and ηQ (b) effect the NMR lineshape. All parameters were kept

crystal at a single orientation. Thus, to reproduce the NMR static lineshape of a powder all orientations of the EFG PAS must be accounted for.

Examples of the quadrupolar interaction can be seen in Figure 2-5, the increase of CQ

in Figure 2-5(a) shows the linewidth increasing. Furthermore, if two nuclei of different spins such as 3/2 and 5/2 were to be compared for the same CQ the 3/2 would

be broader as shown in Equation 2-20 and 2-21 due to the second order perturbation being inversely proportional to I. However, it is clear ηQ indicates the location of

shoulders and features which are characteristic of the interaction.