# Transition Frequency Components¶

The NMR spectral simulation in mrsimulator is based on Symmetry Pathways in Solid-State NMR by Grandinetti et al. [1]

## Introduction to NMR frequency components¶

The nuclear magnetic resonance (NMR) frequency, $$\Omega(\Theta, i, j)$$, for the $$\left|i\right> \rightarrow \left|j\right>$$ transition, where $$\left|i\right>$$ and $$\left|j\right>$$ are the eigenstates of the stationary-state semi-classical Hamiltonian, can be written as a sum of frequency components,

(44)$\Omega(\Theta, i, j) = \sum_k \Omega_k (\Theta, i, j),$

where $$\Theta$$ is the sample’s lattice spatial orientation described with the Euler angles $$\Theta = \left(\alpha, \beta, \gamma\right)$$, and $$\Omega_k$$ is the frequency component from the $$k^\text{th}$$ interaction of the stationary-state semi-classical Hamiltonian.

Each frequency component, $$\Omega_k (\Theta, i, j)$$, is written as the product,

(45)$\Omega_k(\Theta, i, j) = \omega_k ~ \Xi_L^{(k)}(\Theta) ~ \xi_\ell^{(k)}(i, j),$

where $$\omega_k$$ is the size of the $$k^\text{th}$$ frequency component, and $$\Xi_L^{(k)}(\Theta)$$ and $$\xi_\ell^{(k)}(i, j)$$ are the sample’s spatial orientation and quantized NMR transition functions corresponding to the $$L^\text{th}$$ rank spatial and the $$\ell^\text{th}$$ rank spin irreducible spherical tensors, respectively.

The spatial orientation function, $$\Xi_L^{(k)}(\Theta)$$, in Eq. (45), is defined in the laboratory frame, where the $$z$$-axis is the direction of the external magnetic field. This function is the spatial contribution to the observed frequency component arising from the rotation of the $$L^\text{th}$$-rank irreducible tensor, $$\varrho_{L,n}^{(k)}$$, from the principal axis system to the lab frame via Wigner rotation which follows,

(46)$\Xi_L^{(k)}(\Theta) = \sum_{n_0=-L}^L D^L_{n_0,0}(\Theta_0) \sum_{n_1=-L}^L D^L_{n_1,n_0}(\Theta_1) ~ ... ~ \sum_{n_i=-L}^L D^L_{n_i,n}(\Theta_i) ~~ \varrho_{L,n}^{(k)}.$

Here, the term $$D^L_{n_i,n_j}(\Theta)$$ is the Wigner rotation matrix element, generically denoted as,

(47)$D^L_{n_i,n_j}(\Theta) = e^{-i n_i \alpha} d_{n_i, n_j}^L(\beta) e^{-i n_j \gamma},$

where $$d_{n_i, n_j}^L(\beta)$$ is Wigner small $$d$$ element.

In the case of the single interaction Hamiltonian, that is, in the absence of cross-terms, mrsimulator further defines the product of the size of the $$k^\text{th}$$ frequency component, $$\omega_k$$, and the $$L^\text{th}$$-rank irreducible tensor components, $$\varrho_{L,n}^{(k)}$$, in the principal axis system of the interaction tensor, $$\boldsymbol{\rho}^{(\lambda)}$$, as the scaled spatial orientation tensor (sSOT) component,

(48)$\varsigma_{L,n}^{(k)} = \omega_k \varrho_{L,n}^{(k)},$

of rank $$L$$, also defined in the principal axis system of the interaction tensor. Using Eqs. (46) and (48), we re-express Eq. (45) as

(49)$\Omega_k(\Theta, i, j) = \sum_{n_0=-L}^L D^L_{n_0,0}(\Theta_0) \sum_{n_1=-L}^L D^L_{n_1,n_0}(\Theta_1) ~ ... ~ \sum_{n_i=-L}^L D^L_{n_i,n}(\Theta_i) ~~ \varpi_{\ell, L, n}^{(k)},$

where

(50)$\varpi_{\ell, L, n}^{(k)} = \varsigma_{L,n}^{(k)}~~\xi_\ell^{(k)}(i, j)$

is the frequency tensor component (FT) of rank $$L$$, defined in the principal axis system of the interaction tensor and corresponds to the $$\left|i\right> \rightarrow \left|j\right>$$ spin transition.

## Scaled spatial orientation tensor (sSOT) components in PAS, $$\mathbf{\varsigma}_{L,n}^{(k)}$$¶

### Single nucleus scaled spatial orientation tensor components¶

#### Nuclear shielding interaction¶

The nuclear shielding tensor, $$\boldsymbol{\rho}^{(\sigma)}$$, is a second-rank reducible tensor, which can be decomposed into a sum of the zeroth-rank isotropic, first-rank anti-symmetric, and second-rank traceless symmetric irreducible spherical tensors. In the principal axis system, the zeroth-rank, $$\rho_{0,0}^{(\sigma)}$$ and the second-rank, $$\rho_{2,n}^{(\sigma)}$$, irreducible tensors follow,

(51)$\begin{array}{c c c c} \rho_{0,0}^{(\sigma)} = -\sqrt{3} \sigma_\text{iso}, & \rho_{2,0}^{(\sigma)} = \sqrt{\frac{3}{2}} \zeta_\sigma, & \rho_{2,\pm1}^{(\sigma)} = 0, & \rho_{2,\pm2}^{(\sigma)} = - \frac{1}{2}\eta_\sigma \zeta_\sigma, \end{array}$

where $$\sigma_\text{iso}, \zeta_\sigma$$, and $$\eta_\sigma$$ are the isotropic nuclear shielding, shielding anisotropy, and shielding asymmetry of the site, respectively. The shielding anisotropy and asymmetry are defined using Haeberlen notation.

First-order perturbation

The size of the frequency component, $$\omega_k$$, from the first-order perturbation expansion of Nuclear shielding Hamiltonian is $$-\omega_0=\gamma B_0$$, where $$\omega_0$$ is the Larmor angular frequency of the nucleus, and $$\gamma$$, $$B_0$$ are the gyromagnetic ratio of the nucleus and the macroscopic magnetic flux density of the applied external magnetic field, respectively. The relation between $$\varrho_{L,n}^{(\sigma)}$$ and $$\rho_{L,n}^{(\sigma)}$$ follows,

(52)$\begin{split}\varrho_{0,0}^{(\sigma)} &= -\frac{1}{\sqrt{3}} \rho_{0,0}^{(\sigma)} \\ \varrho_{2,n}^{(\sigma)} &=\sqrt{\frac{2}{3}} \rho_{2,n}^{(\sigma)}\end{split}$
Table 15 A list of scaled spatial orientation tensors in the principal axis system of the nuclear shielding tensor, $$\mathbf{\varsigma}_{L,n}^{(k)}$$ from Eq. (48), of rank L resulting from the Mth order perturbation expansion of the Nuclear shielding Hamiltonian is presented.

Order, $$M$$

Rank, $$L$$

$$\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}$$

1

0

$$\varsigma_{0,0}^{(\sigma)} = -\omega_0\sigma_\text{iso}$$

1

2

$$\varsigma_{2,0}^{(\sigma)} = -\omega_0 \zeta_\sigma$$,

$$\varsigma_{2,\pm1}^{(\sigma)} = 0$$,

$$\varsigma_{2,\pm2}^{(\sigma)} = \frac{1}{\sqrt{6}} \omega_0\eta_\sigma \zeta_\sigma$$

The electric field gradient (efg) tensor, $$\boldsymbol{\rho}^{(q)}$$, is also a second-rank tensor; however, unlike the nuclear shielding tensor, the efg tensor is always a symmetric second-rank irreducible tensor. In the principal axis system, this tensor is given as,

(53)$\begin{array}{c c c} \rho_{2,0}^{(q)} = \sqrt{\frac{3}{2}} \zeta_q, & \rho_{2,\pm1}^{(q)} = 0, & \rho_{2,\pm2}^{(q)} = - \frac{1}{2}\eta_q \zeta_q, \end{array}$

where $$\zeta_q$$, and $$\eta_q$$ are the efg tensor anisotropy and asymmetry of the site, respectively. The efg anisotropy and asymmetry values are defined using the Haeberlen convention.

First-order perturbation

The size of the frequency component from the first-order perturbation expansion of Electric quadrupole Hamiltonian is $$\omega_k = \omega_q$$, where $$\omega_q = \frac{6\pi C_q}{2I(2I-1)}$$ is the quadrupole splitting angular frequency. Here, $$C_q$$ is the quadrupole coupling constant, and $$I$$ is the spin quantum number of the quadrupole nucleus. The relation between $$\varrho_{L,n}^{(q)}$$ and $$\rho_{L,n}^{(q)}$$ follows,

(54)$\varrho_{2,n}^{(q)} = \frac{1}{3\zeta_q} \rho_{2,n}^{(q)}.$

Second-order perturbation

The size of the frequency component from the second-order perturbation expansion of Electric quadrupole Hamiltonian is $$\omega_k = \frac{\omega_q^2}{\omega_0}$$, where $$\omega_0$$ is the Larmor angular frequency of the quadrupole nucleus. The relation between $$\varrho_{L,n}^{(qq)}$$ and $$\rho_{L,n}^{(q)}$$ follows,

(55)$\varrho_{L,n}^{(qq)} = \frac{1}{9\zeta_q^2} \sum_{m=-2}^2 \left<L~n~|~2~2~m~n-m\right> \rho_{2,m}^{(q)}~\rho_{2,n-m}^{(q)},$

where $$\left<L~M~|~l_1~l_2~m_1~m_2\right>$$ is the Clebsch Gordan coefficient.

Table 16 A list of scaled spatial orientation tensors in the principal axis system of the efg tensor, $$\mathbf{\varsigma}_{L,n}^{(k)}$$ from Eq. (48), of rank L resulting from the Mth order perturbation expansion of the Electric Quadrupole Hamiltonian is presented.

Order, $$M$$

Rank, $$L$$

$$\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}$$

1

2

$$\varsigma_{2,0}^{(q)} = \frac{1}{\sqrt{6}} \omega_q$$,

$$\varsigma_{2,\pm1}^{(q)} = 0$$,

$$\varsigma_{2,\pm2}^{(q)} = -\frac{1}{6} \eta_q \omega_q$$

2

0

$$\varsigma_{0,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{6\sqrt{5}} \left(\frac{\eta_q^2}{3} + 1 \right)$$

2

2

$$\varsigma_{2,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{\sqrt{2}}{6\sqrt{7}} \left(\frac{\eta_q^2}{3} - 1 \right)$$,

$$\varsigma_{2,\pm1}^{(qq)} = 0$$,

$$\varsigma_{2,\pm2}^{(qq)} = -\frac{\omega_q^2}{\omega_0} \frac{1}{3\sqrt{21}} \eta_q$$

2

4

$$\varsigma_{4,0}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{\sqrt{70}} \left(\frac{\eta_q^2}{18} + 1 \right)$$,

$$\varsigma_{4,\pm1}^{(qq)} = 0$$,

$$\varsigma_{4,\pm2}^{(qq)} = -\frac{\omega_q^2}{\omega_0} \frac{1}{6\sqrt{7}} \eta_q$$,

$$\varsigma_{4,\pm3}^{(qq)} = 0$$,

$$\varsigma_{4,\pm4}^{(qq)} = \frac{\omega_q^2}{\omega_0} \frac{1}{36} \eta_q^2$$

### Coupled nucleus scaled spatial orientation tensor components¶

#### Weak $$J$$-coupling interaction¶

The $$J$$-coupling tensor, $$\boldsymbol{\rho}^{(J)}$$, is a second-rank reducible tensor, which can be decomposed into a sum of the zeroth-rank isotropic, first-rank anti-symmetric, and second-rank traceless symmetric irreducible spherical tensors. In the principal axis system, the zeroth-rank, $$\rho_{0,0}^{(J)}$$ and the second-rank, $$\rho_{2,n}^{(J)}$$, irreducible tensors follow,

(56)$\begin{array}{c c c c} \rho_{0,0}^{(J)} = -\sqrt{3} J_\text{iso}, & \rho_{2,0}^{(J)} = \sqrt{\frac{3}{2}} \zeta_J, & \rho_{2,\pm1}^{(J)} = 0, & \rho_{2,\pm2}^{(J)} = - \frac{1}{2}\eta_J \zeta_J, \end{array}$

where $$J_\text{iso}, \zeta_J$$, and $$\eta_J$$ are the isotropic $$J$$-coupling, coupling anisotropy and asymmetry parameters, respectively. The $$J$$ anisotropy and asymmetry are defined using Haeberlen notation.

First-order perturbation

The size of the frequency component from the first-order perturbation expansion of weak J-coupling Hamiltonian is $$\omega_k = 2\pi$$. The relation between $$\varrho_{L,n}^{(J)}$$ and $$\rho_{L,n}^{(J)}$$ follows,

(57)$\begin{split}\varrho_{0,0}^{(J)} &= -\frac{1}{\sqrt{3}} \rho_{0,0}^{(J)} \\ \varrho_{2,n}^{(J)} &=\sqrt{\frac{2}{3}} \rho_{2,n}^{(J)}\end{split}$
Table 17 A list of scaled spatial orientation tensors in the principal axis system of the J-coupling tensor, $$\mathbf{\varsigma}_{L,n}^{(k)}$$ from Eq. (48), of rank L resulting from the Mth order perturbation expansion of the J-coupling Hamiltonian is presented.

Order, $$M$$

Rank, $$L$$

$$\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}$$

1

0

$$\varsigma_{0,0}^{(J)} = 2\pi J_\text{iso}$$

1

2

$$\varsigma_{2,0}^{(J)} = 2\pi \zeta_J$$,

$$\varsigma_{2,\pm1}^{(J)} = 0$$,

$$\varsigma_{2,\pm2}^{(J)} = -\frac{1}{\sqrt{6}} 2\pi\eta_J \zeta_J$$

#### Weak dipolar-coupling interaction¶

The dipolar-coupling tensor, $$\boldsymbol{\rho}^{(d)}$$, is a second rank reducible tensor, which can be decomposed as a second-rank traceless symmetric irreducible spherical tensors. In the principal axis system, the second-rank, $$\rho_{2,n}^{(d)}$$, irreducible tensors follow,

(58)$\begin{array}{c c c c} \rho_{2,0}^{(d)} = \sqrt{\frac{3}{2}} \zeta_d, & \rho_{2,\pm1}^{(d)} = 0, & \rho_{2,\pm2}^{(d)} = 0, \end{array}$

where $$\zeta_d$$ is second-rank symmetric dipolar coupling tensor anisotropy given as

(59)$\zeta_d = \frac{2}{r^3}$

where $$r$$ is the distance between two coupled magnetic dipoles. The dipolar splitting is given as

(60)$\omega_d = - \frac{\mu_0}{4\pi} \frac{\gamma_1 \gamma_2 \hbar}{r^3} = - \frac{\mu_0}{8\pi} \zeta_d \gamma_1 \gamma_2 \hbar$

and the dipolar coupling constant, $$D = \frac{\omega_d}{2\pi}$$.

First-order perturbation

The size of the frequency component from the first-order perturbation expansion of weak J-coupling Hamiltonian is $$\omega_k = \frac{2\omega_d}{\zeta_d}$$. The relation between $$\varrho_{L,n}^{(d)}$$ and $$\rho_{L,n}^{(d)}$$ follows,

(61)$\varrho_{2,n}^{(d)} =\sqrt{\frac{2}{3}} \rho_{2,n}^{(d)}$
Table 18 A list of scaled spatial orientation tensors in the principal axis system of the dipolar-coupling tensor, $$\mathbf{\varsigma}_{L,n}^{(k)}$$ from Eq. (48), of rank L resulting from the Mth order perturbation expansion of the dipolar-coupling Hamiltonian is presented.

Order, $$M$$

Rank, $$L$$

$$\varsigma_{L,n}^{(k)} = \omega_k\varrho_{L,n}^{(k)}$$

1

2

$$\varsigma_{2,0}^{(d)} = 2\omega_d$$,

$$\varsigma_{2,\pm1}^{(d)} = 0$$,

$$\varsigma_{2,\pm2}^{(d)} = 0$$

## Spin transition functions, $$\xi_\ell^{(k)}(i,j)$$¶

The spin transition function is typically manipulated via the coupling of the nuclear magnetic dipole moment with the oscillating external magnetic field from the applied radio-frequency pulse. Considering the strength of the external magnetic rf field is orders of magnitude larger than the internal spin-couplings, the manipulation of spin transition functions is described using the orthogonal rotation subgroups.

### Single nucleus spin transition functions¶

Table 19 A list of single nucleus spin transition functions, $$\xi_\ell^{(k)}(i,j)$$.

$$\xi_\ell^{(k)}(i,j)$$

Rank, $$\ell$$

Value

Description

$$\mathbb{s}(i,j)$$

0

$$0$$

$$\left< j | \hat{T}_{00} | j \right> - \left< i | \hat{T}_{00} | i \right>$$

$$\mathbb{p}(i,j)$$

1

$$j-i$$

$$\left< j | \hat{T}_{10} | j \right> - \left< i | \hat{T}_{10} | i \right>$$

$$\mathbb{d}(i,j)$$

2

$$\sqrt{\frac{3}{2}} \left(j^2 - i^2 \right)$$

$$\left< j | \hat{T}_{20} | j \right> - \left< i | \hat{T}_{20} | i \right>$$

$$\mathbb{f}(i,j)$$

3

$$\frac{1}{\sqrt{10}} [5(j^3 - i^3) + (1 - 3I(I+1))(j-i)]$$

$$\left< j | \hat{T}_{30} | j \right> - \left< i | \hat{T}_{30} | i \right>$$

Here, $$\hat{T}_{\ell,k}(\bf{I})$$ are the irreducible spherical tensor operators of rank $$\ell$$, $$k \in [-\ell, \ell]$$, for transition $$|i\rangle \rightarrow |j\rangle$$. In terms of the tensor product of the Cartesian operators, the above spherical tensors are expressed as follows,

Spherical tensor operator

Representation in Cartesian operators

$$\hat{T}_{0,0}(\bf{I})$$

$$\hat{1}$$

$$\hat{T}_{1,0}(\bf{I})$$

$$\hat{I}_z$$

$$\hat{T}_{2,0}(\bf{I})$$

$$\frac{1}{\sqrt{6}} \left[3\hat{I}^2_z - I(I+1)\hat{1} \right]$$

$$\hat{T}_{3,0}(\bf{I})$$

$$\frac{1}{\sqrt{10}} \left[5\hat{I}^3_z + \left(1 - 3I(I+1)\right)\hat{I}_z\right]$$

where $$I$$ is the spin quantum number of the nucleus and $$\hat{\bf{1}}$$ is the identity operator.

Table 20 A list of composite single nucleus spin transition functions, $$\xi_\ell^{(k)}(i,j)$$. Here, I is the spin quantum number of the nucleus.

$$\xi_\ell^{(k)}(i,j)$$

Value

$$\mathbb{c}_0(i,j)$$

$$\frac{4}{\sqrt{125}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(i, j) + \sqrt{\frac{18}{25}} \mathbb{f}(i, j)$$

$$\mathbb{c}_2(i,j)$$

$$\sqrt{\frac{2}{175}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(i, j) - \frac{6}{\sqrt{35}} \mathbb{f}(i, j)$$

$$\mathbb{c}_4(i,j)$$

$$-\sqrt{\frac{18}{875}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(i, j) - \frac{17}{\sqrt{175}} \mathbb{f}(i, j)$$

### Weakly coupled nucleus spin transition functions¶

Table 21 A list of weakly coupled nucleus spin transition functions, $$\xi_\ell^{(k)}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})$$.

$$\xi_\ell^{(k)}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})$$

Value

Description

$$(\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})$$

$$m_{f_I} m_{f_S} - m_{i_I} m_{i_S}$$

$$\left< m_{f_I} m_{f_S} | \hat{T}_{10}(I) \hat{T}_{10}(S) | m_{f_I} m_{f_S} \right>$$$$\left< m_{i_I} m_{i_S} | \hat{T}_{10}(I) \hat{T}_{10}(S) | m_{i_I} m_{i_S} \right>$$

Here, $$\hat{T}_{\ell,k}(\bf{I})$$ are the irreducible spherical tensor operators of rank $$\ell$$, $$k \in [-\ell, \ell]$$, for transition $$|m_{i_I} m_{i_S}\rangle \rightarrow |m_{f_I} m_{f_S}\rangle$$ in weakly coupled basis.

## Frequency tensor components (FT) in PAS, $$\varpi_{\ell,L, n}^{(k)}$$¶

Table 22 The table presents a list of frequency tensors defined in the principal axis system of the respective interaction tensor from Eq. (50), $$\varpi_{\ell,L,n}^{(k)}$$, of ranks $$\ell$$ and $$L$$ resulting from the Mth order perturbation expansion of the interaction Hamiltonian supported in mrsimulator.

Interaction

Order, $$M$$

Rank, $$L$$

$$\varpi_{\ell,L,n}^{(k)}$$

Nuclear shielding

1

0

$$\varpi_{1,0,0}^{(\sigma)} = \varsigma_{0,0}^{(\sigma)} ~~ \mathbb{p}(i, j)$$

Nuclear shielding

1

2

$$\varpi_{1,2,n}^{(\sigma)} = \varsigma_{2,n}^{(\sigma)} ~~ \mathbb{p}(i, j)$$

1

2

$$\varpi_{2,2,n}^{(q)} = \varsigma_{2,n}^{(q)} ~~ \mathbb{d}(i, j)$$

2

0

$$\varpi_{c_0,0,0}^{(qq)} = \varsigma_{0,0}^{(qq)} ~~ \mathbb{c}_0(i, j)$$

2

2

$$\varpi_{c_2,2,n}^{(qq)} = \varsigma_{2,n}^{(qq)} ~~ \mathbb{c}_2(i, j)$$

2

4

$$\varpi_{c_4,4,n}^{(qq)} = \varsigma_{4,n}^{(qq)} ~~ \mathbb{c}_4(i, j)$$

Weak $$J$$-coupling

1

0

$$\varpi_{(1,1),0,0}^{(J)} = \varsigma_{0,0}^{(J)} ~~ (\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})$$

Weak $$J$$-coupling

1

2

$$\varpi_{(1,1),2,n}^{(J)} = \varsigma_{2,n}^{(J)} ~~ (\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})$$

Weak dipolar-coupling

1

2

$$\varpi_{(1,1),2,n}^{(d)} = \varsigma_{2,n}^{(d)} ~~ (\mathbb{pp})_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S})$$

References