Math

The hamiltonian $\hat{H}$ we are using for the NMR systems is of the form

$$\begin{array}{rl}\hat{H}& =-\sum _{\ell }{\gamma }_{\ell }{B}^z\left(1+{\delta }_{\ell }\right){\hat{I}}_{\ell }^z+2\pi \sum _{k<l}{J}_{kl}{\hat{\mathbf{I}}}_k\cdot {\hat{\mathbf{I}}}_l\end{array}$$

where ${\gamma }_{\ell }$ are the gyromagnetic factors and ${\delta }_{\ell }$ the chemical shifts of nuclear spin $\ell$, and $J_{kl}$ denotes the coupling between spins $k$ and $l$, and ${\hat{I}}^{\alpha }={\hat{S}}^{\alpha }/\hslash$, with ${\hat{S}}^{\alpha }$ being the usual $SU(2)$ spin operators.

Within NMR we have a strong magnetic field $B_z$ in the $z$-direction, and electromagnetic pulses / oscillating fields are applied to flip the spins into the $x/y$ plane. Since $B_z$ is typically of the order of 500Mhz, the pulses of 10kHz bandwidth, and the required resolution is sub 1Hz, we refrain from modelling the explicit time dependence of the pulses. Instead we model the pulses directly by calculating the spectral function, i.e., time-dependent correlations between the corresponding ${\hat{M}}^{\pm }$ operators.

The spectrum measured in an NMR experiment corresponds to the spectral function calculated in the HQS NMR Tool.

Calculation of the spectral function

We calculate the spectral function $A(\omega )$ as the imaginary part of the spin-spin correlation function

$$\begin{array}{rl}A(\omega )& =\frac{-1}{\pi }\Im \mathcal{G}(\omega )\\ \mathcal{G}(\omega )& =\int \mathrm{dt}{e}^{-i\omega t}G(t)\\ G(t)& =-i⟨{\hat{M}}^{-}(t){\hat{M}}^{+}(0)⟩\\ {\hat{I}}^{\pm }& ={\hat{I}}^x\pm i{\hat{I}}^y\\ {\hat{M}}^{\alpha }& =\sum _{\ell }{\gamma }_{\ell }{\hat{I}}_{\ell }^{\alpha }\end{array}$$

the ${\hat{M}}^{\alpha }$ operators, $\alpha \in \{x,y,z,+,-\}$, contain the gyromagnetic factors for convenience, and $t$ is the real time dependence.

The contribution of an individual nuclear spin $\ell$to the full NMR spectrum is obtained via

$$\begin{array}{rl}{G}_{\ell }(t)& =-i⟨{\gamma }_{\ell }{\hat{I}}_{\ell }^{-}(t){\hat{M}}^{+}(0)⟩,\end{array}$$

while the full NMR signal is the sum of individual contributions.

Temperature

For a typical NMR setup, the temperature is much larger compared to the energy scales in the NMR Hamiltonian. Thus, we have to take finite or even infinite temperature $T$ into account. We define the partition sum $Z$

$$Z=\sum _n{\mathrm{e}}^{-\beta E_n}$$

with the inverse temperature $\beta =1/(k_{\mathrm{B}}T)$ and arrive at

$$\begin{array}{rl}G(t)& =-i⟨{\mathrm{e}}^{i\hat{H}t}{I}^{-}{\mathrm{e}}^{-i\hat{H}t}{\hat{M}}^{+}(0)⟩\\ & =\sum _n{\mathrm{e}}^{-\beta E_n}{\mathrm{e}}^{i{E}_{n}t}⟨n|{\hat{M}}^{-}{\mathrm{e}}^{-i\hat{H}t}{\hat{M}}^{+}|n⟩/Z\\ & =\sum _{n,m}{\mathrm{e}}^{-\beta E_n}{\mathrm{e}}^{(E_n-E_m)t}⟨n|{\hat{M}}^{-}|m⟩⟨m|{\hat{M}}^{+}|n⟩/Z\\ \mathcal{G}(\omega )& =\frac{-i}{Z}\int \mathrm{d}\omega \sum _{n,m}{\mathrm{e}}^{-\beta E_n}{\mathrm{e}}^{i(\omega +E_n-E_m)t}⟨n|{\hat{M}}^{-}|m⟩⟨m|{\hat{M}}^{+}|n⟩\\ & =\frac{-i\pi }{Z}\sum _{n,m}{\mathrm{e}}^{-\beta E_n}\delta \left(\omega -(E_m-E_n)\right)⟨n|{\hat{M}}^{-}|m⟩⟨m|{\hat{M}}^{+}|n⟩\end{array}$$

Here, the last equation is a Lehmann-type spectral representation, which is suitable if the complete spectrum is known. $E_n$ are the eigenenergies of the spin NMR-Hamiltonian.

Resolvent formulation

In order to avoid having to determine the complete spectrum, we write the Green's function $\mathcal{G}(\omega )$ directly in operator form, introducing a convergence ensuring factor (broadening) $\eta >0$ and taking care of causality

$$\begin{array}{rl}A(\omega )& =\frac{-1}{\pi }\Im {\mathcal{G}}^{\mathrm{r}}(\omega )\\ G^{\mathrm{r}}(t)& =i\Theta (t)⟨{\hat{M}}^{-}(t){\hat{M}}^{+}(0)⟩\\ {\mathcal{G}}_{\eta }(\omega )& =-i{\int }_0^{-\infty }\mathrm{d}\omega \sum _n{\mathrm{e}}^{-\beta E_n}⟨n|{\hat{M}}^{-}{\mathrm{e}}^{i(\omega -\left(\hat{H}-E_n\right)+i\eta )t}{\hat{M}}^{+}|n⟩/Z\\ & =\sum _n{\mathrm{e}}^{-\beta E_n}⟨n|{\hat{M}}^{-}\frac{1}{\omega -\left(\hat{H}-E_n\right)+i\eta }{\hat{M}}^{+}|n⟩/Z\end{array}$$

The broadening $\eta$ is formally necessary to ensure the convergence of the Fourier integral. In practice, it corresponds to the unknown or neglected noise, whether intrinsic or due to the resolution limitations of the spectrometer.

Energy rediscretization

One problem that arises in the resolvent formulation is that the NMR peaks are usually very sharp. Discretizing the frequency axis with a very fine grid is computationally ineffective. Therefore, we perform several rounds of computing the spectral function. We start with a linear grid in frequency space and a rather large artificial broadening $\eta$. Then we iteratively rediscretize the frequency space in equal weight partitions of the total spectral function while reducing $\eta$ in each step until we reach the desired broadening. In this way, we obtain a non-linear adaptive grid with accumulation points at the spectral function peaks.