# Microscopic understanding of NMR signals by dynamic mean-field theory for
spins

Timo Gräßer<EMAIL_ADDRESS>Condensed Matter Theory, TU Dortmund
University, Otto-Hahn Straße 4, 44221 Dortmund, Germany  Thomas Hahn
<EMAIL_ADDRESS>Condensed Matter Theory, TU Dortmund
University, Otto-Hahn Straße 4, 44221 Dortmund, Germany  School of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
Götz S. Uhrig<EMAIL_ADDRESS>Condensed Matter Theory, TU Dortmund
University, Otto-Hahn Straße 4, 44221 Dortmund, Germany

###### Abstract

A recently developed dynamic mean-field theory for disordered spins (spinDMFT)
is shown to capture the spin dynamics of nuclear spins very well. The key
quantities are the spin autocorrelations. In order to compute the free
induction decay (FID), pair correlations are needed in addition. They can be
computed on spin clusters of moderate size which are coupled to the dynamic
mean fields determined in a first step by spinDMFT. We dub this versatile
approach non-local spinDMFT (nl-spinDMFT). It is a particular asset of nl-
spinDMFT that one knows from where the contributions to the FID stem. We
illustrate the strengths of nl-spinDMFT in comparison to experimental data for
CaF2. Furthermore, spinDMFT provides the dynamic mean-fields explaining the
FID of the nuclear spins in 13C in adamantane up to some static noise. The
spin Hahn echo in adamantane is free from effects of static noise and agrees
excellently with the spinDMFT results without further fitting.

##  I Introduction

Nuclear magnetic resonance (NMR) has been successfully employed for decades to
obtain microscopic information about matter [1]. Experimental measurements of
free-induction decays (FIDs), spin echoes and more allow conclusions to be
drawn about chemical compounds and molecular distances in a sample. At the
same time, a fully theoretical description of what is measured is rarely
possible, since it requires the simulation of large numbers of nuclear spins.
Conventional methods such as exact diagonalization and Chebyshev polynomial
expansion are restricted to a couple of tens of spins [2, 3]. To capture
realistic system sizes, fundamental approximations have to be considered. For
the computation of FIDs, many different approximation schemes have been
considered in the past [4, 5, 6, 7, 8, 9](see also the references cited in
Ref. [8]). Although some of them are quite accurate for the considered
example, they are mostly difficult to generalize and, thus, rarely used at
present. Recently proposed strategies include replacing all or many of the
spins by classical ones [10, 8, 9], which is particularly successful in
systems where each spin has a large number of neighbors. The predictive power
of such approaches can be questioned if quantum effects are more important,
for instance, due to strong local quantum bonds or quadrupolar interactions
[11]. Turning to non-equilibrium NMR measurements makes the situation even
more complicated. A non-Markovian simulation of spin echoes, for example, is
rarely even attempted [12, 13, 14].

The concept of a “mean field” has been considered in physics and chemistry for
many decades. Static mean-field theories are not known for their accuracy, but
they often provide us with a first qualitative picture of a complex many-body
problem. Making the mean-fields dynamic immensely increases the accuracy, see
the fermionic DMFT developed in the nineties [15] and its various extensions,
for example cellular or extended DMFT [16, 17]. The recently developed dynamic
mean-field theory for spins (spinDMFT) [18] can be seen as an analogue for
spin systems. Yet, spinDMFT is derived only for disordered spin systems. This
makes it less general, but perfectly tailored to NMR.

The main purpose of this article is to establish spinDMFT and its extensions
in the NMR community as simulation tools. In doing so, we present an extension
of spinDMFT which we call non-local spinDMFT (nl-spinDMFT). The idea is to
include the mean-field moments computed by spinDMFT in the simulation of a
local quantum cluster coupled to stochastic mean fields. This approach allows
for an efficient computation of _non-local_ quantities and is related to the
cluster-extension of spinDMFT (CspinDMFT) [19].

We emphasize two advantages of these approaches. The first one is that the
effective mean-field model maintains a microscopic picture of the considered
system. Quantities such as spin-spin correlations are directly accessible
which gives us deeper insight into how the global NMR measurement is connected
to microscopic spin physics. We will demonstrate this for the example of
calcium fluoride, which is a common test substance for benchmarking
approximation schemes due to a relatively noise-free FID in experiment [20].
The second advantage of spinDMFT is its high versatility. Treating several
spins on the quantum level as it is done in CspinDMFT can be very helpful in
inhomogeneous systems [19]. For spins with $S>\nicefrac{{1}}{{2}}$, a
quadrupolar interaction can be directly included in the effective mean-field
model. Moreover, external time-dependent fields and noises are easily included
in spinDMFT _and_ its extensions without increasing the numerical effort much.
Therefore, dynamical decoupling [21, 22, 23, 24, 25, 26, 27] can also be
simulated within these approaches. We will demonstrate this in the second part
of the article for the example of polycrystalline adamantane, which is another
common test substance in NMR [1, 23, 24, 25, 26, 27].

Sec. II is devoted to a short overview of the considered substances and a
brief derivation of spinDMFT. Subsequently, we introduce nl-spinDMFT and use
it to estimate the FID of calcium fluoride in Sec. III. In Sec. IV, we
consider the microscopic proton spin dynamics in adamantane using spinDMFT.
The spin dynamics are used in Sec. V to compute the carbon FID and spin echo
in adamantane.

##  II Models and spinDMFT

###  II.1 Calcium fluoride

Calcium fluoride, $\ce{CaF}_{2}$, is a crystal with calcium ions on an fcc-
lattice and fluoride ions ordered in an sc-lattice intertwined with the
former. For NMR purposes, the calcium ions can be neglected, since the
abundance of nuclei with nonzero spin is very small,
$\approx$0.13\text{\,}\mathrm{\char 37\relax}$$. A well-measurable NMR signal
results from fluoride $\ce{{}^{19}F}$ which has an abundance of
$100\text{\,}\mathrm{\char 37\relax}$ and carries a nuclear spin of
$S~{}=~{}\nicefrac{{1}}{{2}}$.

The basis for an NMR experiment is a strong magnetic field $\vec{B}$, which
polarizes the spin ensemble into the field direction
$\vec{n}_{B}\coloneqq\vec{B}/|\vec{B}|$. The secular Hamiltonian for the
homonuclear dipole-dipole interaction is given by

$\displaystyle\mathbf{H}_{\text{hom}}$
$\displaystyle=\frac{1}{4}\sum_{i,j}J_{ij}\left(-\mathbf{S}^{x}_{i}\mathbf{S}^{x}_{j}-\mathbf{S}^{y}_{i}\mathbf{S}^{y}_{j}+2\mathbf{S}^{z}_{i}\mathbf{S}^{z}_{j}\right).$
(1)

Here and henceforth, we set $J_{ii}=0$. If $i\neq j$, the secular couplings
read

$\displaystyle J_{ij}$ $\displaystyle\coloneqq
J_{\vec{r}_{ij}}(\vec{n}_{B})=\left(1-3\left(\vec{n}_{ij}\cdot\vec{n}_{B}\right)^{2}\right)\frac{\mu_{0}}{4\pi}\frac{\gamma_{i}\gamma_{j}}{|\vec{r}_{ij}|^{3}},$
(2a) $\displaystyle\vec{r}_{ij}$
$\displaystyle\coloneqq\vec{r}_{i}-\vec{r}_{j}\qquad\vec{n}_{ij}\coloneqq\vec{r}_{ij}/|\vec{r}_{ij}|,$
(2b)

with $\vec{r}_{i}$ and $\vec{r}_{j}$ being the nuclear spin positions fixed by
the lattice and $\gamma_{i}$ and $\gamma_{j}$ being the gyromagnetic ratios.
The latter are equal in the homonuclear case.

###  II.2 Adamantane

The adamantane molecule consists of 10 carbon and 16 hydrogen atoms located on
approximate spheres. Four of the carbon atoms ($\mathrm{CA}$) are each bound
to one hydrogen atom ($\mathrm{HA}$) and six carbons ($\mathrm{CB}$) are each
bound to two hydrogen atoms ($\mathrm{HB}$). The molecule is displayed in the
left panel of Fig. 1. We set all bond angles to the tetrahedral angle
$\alpha~{}\approx~{}$109.5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and use the
bond lengths [28]

$\displaystyle d_{\mathrm{C}\mathrm{C}}$
$\displaystyle=$1.540(2)\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,$
$\displaystyle d_{\mathrm{C}\mathrm{H}}$
$\displaystyle=$1.112(4)\text{\,}\mathrm{\SIUnitSymbolAngstrom}$.$ (3)

The distances between the four types of atoms with respect to the CMS of the
molecule are then given by

$\displaystyle r_{\mathrm{CA}}$
$\displaystyle\approx$1.54\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,$
$\displaystyle r_{\mathrm{CB}}$
$\displaystyle\approx$1.78\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,$ (4a)
$\displaystyle r_{\mathrm{HA}}$
$\displaystyle\approx$2.65\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,$
$\displaystyle r_{\mathrm{HB}}$
$\displaystyle\approx$2.58\text{\,}\mathrm{\SIUnitSymbolAngstrom}$.$ (4b)

Considering these values and Fig. 1, we note that the two classes of hydrogen
and the two classes of carbon are each located nearly on the same shell.
Hence, we consider only a single type of hydrogen and a single type of carbon
at the averaged radii

$\displaystyle r_{\mathrm{H}}$
$\displaystyle\coloneqq\frac{12r_{\mathrm{HA}}+4r_{\mathrm{HB}}}{16}=$2.60(3)\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,$
(5a) $\displaystyle r_{\mathrm{C}}$
$\displaystyle\coloneqq\frac{6r_{\mathrm{CA}}+4r_{\mathrm{CB}}}{10}=$1.68(12)\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$
(5b)

henceforth. As we will see later, this simplifies the application of spinDMFT,
since only a single kind of mean-field needs to be considered.

Figure 1: Sketch of the adamantane molecule. The hydrogen atoms are shown in
dark blue ($\mathrm{HA}$) and light blue ($\mathrm{HB}$) and the carbon atoms
in black ($\mathrm{CA}$) and gray ($\mathrm{CB}$). We use circles with the
corresponding colors to represent the radii of the atoms with respect to the
center of gravity of the molecule, see Eq. 4. The circles for the two types of
hydrogen are nearly indistinguishable.

In line with the experiment in Ref. [23], we consider polycrystalline
adamantane at room temperature. The adamantane molecules form an fcc-lattice
with $a_{\text{fcc}}~{}=~{}$9.426(8)\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$
and four molecules per unit cell [29]. While their centers of mass are fixed,
the molecules continuously change their orientations. These molecular
rotations are isotropic and happen on a much shorter time scale than the spin
dynamics so that motional narrowing has to be considered for the couplings.

Aside from the proton spins, only the carbon isotopes $\ce{{}^{13}C}$ are of
interest for NMR, since they carry a spin of $I~{}=~{}\nicefrac{{1}}{{2}}$ in
contrast to the spinless nuclei of $\ce{{}^{12}C}$. Generally, we denote the
$\ce{{}^{1}H}$ nuclear spins by $S$ and the $\ce{{}^{13}C}$ nuclear spins by
$I$. The natural abundance of $\ce{{}^{13}C}$ is about
$1.1\text{\,}\mathrm{\char 37\relax}$. Since the carbon spins are so rare, we
neglect any carbon-hydrogen interaction for the hydrogen dynamics and any
carbon-carbon interaction in general. The homonuclear secular Hamiltonian is
given by

$\displaystyle\mathbf{H}_{\text{hom}}$
$\displaystyle=\frac{1}{4}\sum_{i,j}\overline{J_{ij}}^{\text{ma}}\left(-\mathbf{S}^{x}_{i}\mathbf{S}^{x}_{j}-\mathbf{S}^{y}_{i}\mathbf{S}^{y}_{j}+2\mathbf{S}^{z}_{i}\mathbf{S}^{z}_{j}\right).$
(6)

This is equivalent to Eq. 1, but with the couplings $J_{ij}$ replaced by their
motional average $\overline{J_{ij}}^{\text{ma}}$, see below. For the dynamics
of $\ce{{}^{13}C}$, we also use the heteronuclear Hamiltonian

$\displaystyle\mathbf{H}_{\text{het}}$
$\displaystyle=\sum_{j}\overline{J_{0j}}^{\text{ma}}\,\mathbf{I}^{z}\mathbf{S}^{z}_{j},$
(7)

where, again, the averaged couplings have to be considered. The indices
$i\in\\{1,2,3,\dots\\}$ label the proton spins and the carbon index is 0. As
justified above, we consider the $\ce{{}^{13}C}$ spins to be isolated from one
another.

The motional averaging causes any intramolecular couplings to average out
completely, i.e., motional narrowing takes place. The averaged intermolecular
couplings depend on (i) the distances of the considered nuclei to the center
of mass and (ii) the lattice vector between the molecules to which the nuclei
belong. For this reason, it is helpful to introduce double indices
$(mX)\coloneqq i$, where $m\in\\{1,2,3,\dots\\}$ denotes the molecule and
$X\in\mathrm{H},\mathrm{C}$ is the nucleus type. The averaged couplings can be
computed from

$\displaystyle\overline{J_{ij}}^{\text{ma}}(\vec{n}_{B})\coloneqq$
$\displaystyle\overline{J_{(m_{1}X_{1}),(m_{2}X_{2})}}^{\text{ma}}(\vec{n}_{B})$
(8a)
$\displaystyle=\int\frac{\mathrm{d}\Omega_{1}\mathrm{d}\Omega_{2}}{(4\pi)^{2}}$
$\displaystyle\,J_{\vec{a}_{m_{1},m_{2}}+r_{X_{1}}\vec{n}(\Omega_{1})-r_{X_{2}}\vec{n}(\Omega_{2})}(\vec{n}_{B}).$
(8b)

Here, $\vec{a}_{m_{1},m_{2}}$ is the lattice vector from center to center of
the molecules and
$\vec{n}(\Omega)=\vec{n}(\theta,\phi)\coloneqq\begin{smallmatrix}(\sin\vartheta\cos\varphi,&\sin\vartheta\sin\varphi,&\cos\vartheta)\end{smallmatrix}^{\top}$.
The radii $r_{X_{1}},r_{X_{2}}$ are the distances from the nucleus to the
center of its molecule, see Eq. 5.

An exact many-body treatment of adamantane is even further beyond reach than
for calcium fluoride. Due to the motional averaging the intermolecular
couplings are the dominant ones. Note that a single molecule carries already
$16$ proton spins, so the number of couplings is immense, so that brute-force
calculations are out of question. On the other hand, this is an ideal
condition for a mean-field approach. Due to the large number of interaction
partners, the mean-field average is dominant over fluctuations. In the
following, we present a brief summary of spinDMFT and show the generic
results.

###  II.3 Simulation of homonuclear dynamics by spinDMFT

Applying spinDMFT to a system containing only a single kind of spins is
straight-forward [18]. As an example, we consider the homonuclear Hamiltonian
in Eq. 1. Defining the operators of the local environment,

$\displaystyle\vec{\mathbf{V}}_{i}$
$\displaystyle\coloneqq\sum_{j}J_{ij}\,\underline{\underline{D}}\vec{\mathbf{S}}_{j},$
$\displaystyle\underline{\underline{D}}$
$\displaystyle=\begin{pmatrix}-\frac{1}{2}&0&0\\\ 0&-\frac{1}{2}&0\\\ 0&0&1\\\
\end{pmatrix},$ (9)

it can be rewritten according to

$\displaystyle\mathbf{H}_{\text{hom}}$
$\displaystyle=\frac{1}{2}\sum_{i}\vec{\mathbf{S}}_{i}\cdot\vec{\mathbf{V}}_{i}.$
(10)

The first step of spinDMFT is to replace these operators by classical time-
dependent mean-fields $\vec{V}_{i}(t)$. Each mean-field summarizes the
interaction of a large number of spins. Therefore, according to the central
limit theorem, the mean-fields underlie a Gaussian distribution. Replacing the
local-environment operators, we arrive at an effective single-site model

$\displaystyle\mathbf{H}_{i}^{\text{mf}}(t)$
$\displaystyle=\vec{\mathbf{S}}_{i}\cdot\vec{V}_{i}(t).$ (11)

The quantum degrees of freedom of each spin are decoupled from those of the
rest of the ensemble; hence, the spin index can be omitted henceforth. The
mean-fields can vary from site to site, even on average. However, for the
fluoride ions in $\ce{CaF}_{2}$ as well as for the protons in adamantane,
translational invariance is a well justified assumption. Therefore, the
distribution of each mean-field is independent of the site $i$ so that this
index can be omitted as well. We consider a single site, i.e., a single spin
and a single mean-field henceforth.

Since the mean-field is Gaussian, only the first and the second moment are
required to fully determine it. In the high-temperature regime of completely
disordered spins, expectation values are evaluated with respect to the
disordered state $\rho_{0}\propto\mathbbm{1}$. The first moment is thus zero,

$\displaystyle\overline{V^{\alpha}(t)}^{\text{mf}}$
$\displaystyle=\langle\mathbf{V}^{\alpha}(t)\rangle=0,$ (12)

where $\alpha\in\\{x,y,z\\}$ and the line with upper index “mf” denotes the
mean-field averaging. The second moments can be related to quantum
correlations of the original field according to

$\displaystyle\overline{V^{\alpha}(t)V^{\alpha}(0)}^{\text{mf}}$
$\displaystyle=\langle\mathbf{V}^{\alpha}(t)\mathbf{V}^{\alpha}(0)\rangle$
(13a)
$\displaystyle=D_{\alpha\alpha}^{2}J_{\mathrm{Q}}^{2}\langle\mathbf{S}^{\alpha}(t)\mathbf{S}^{\alpha}(0)\rangle,$
(13b) $\displaystyle\overline{V^{\alpha}(t)V^{\beta}(0)}^{\text{mf}}$
$\displaystyle=0\qquad\alpha\neq\beta.$ (13c)

where we defined the quadratic coupling constant

$\displaystyle J_{\mathrm{Q}}^{2}\coloneqq\sum_{j}J_{ij}^{2}.$ (14)

Note that although the sum runs only over $j$, the total expression does not
depend on $i$ due to translational invariance. We stress that correlations
between mean-fields at different sites are not relevant in spinDMFT due to the
large coordination number [18]. For the derivation of Eq. 13b we used that
pair correlations are subdominant and can be neglected relative to the
autocorrelations, that is,

$\displaystyle\sum_{\begin{subarray}{c}j,k\\\ j\neq
k\end{subarray}}J_{ij}J_{ik}\langle\mathbf{S}_{j}^{\alpha}(t)\mathbf{S}_{k}^{\alpha}(0)\rangle$
$\displaystyle\ll\sum_{j}J_{ij}^{2}\langle\mathbf{S}_{j}^{\alpha}(t)\mathbf{S}_{j}^{\alpha}(0)\rangle.$
(15)

This is also justified by the large coordination number $z$. Note that $z$ is
typically defined by the number of nearest neighbors in the spin lattice.
However, since the dipolar interaction is long-range, the effective number of
interaction partners of each spin can be a lot higher than $z$. Therefore, we
consider the effective coordination number 111In Ref. [18], we used the symbol
$z_{2}$ for $z_{\text{eff}}$, because we discussed also
$z_{1}=J_{\mathrm{L}}^{2}/J_{\mathrm{Q}}^{2}$, where $J_{\mathrm{L}}$ is the
linear coupling sum.

$\displaystyle z_{\text{eff}}$
$\displaystyle\coloneqq\frac{J_{\mathrm{Q}}^{4}}{J_{\mathrm{T}}^{4}},$
$\displaystyle J_{\mathrm{T}}^{4}$ $\displaystyle\coloneqq\sum_{j}J_{ij}^{4},$
(16)

to be more relevant for the justification of the mean-field approach. For the
sake of brevity, we sometimes use $z$ instead of $z_{\text{eff}}$ in our
argumentation.

Eqs. 11, 13b and 13c form a closed self-consistency problem which can be
solved numerically. A comprehensive justification of spinDMFT and details on
the numerical implementation can be found in the original article [18]. In
essence, one has to simulate a single spin in a Gaussian noise, which can be
done by Monte-Carlo simulation. The resulting spin autocorrelations are
transformed to mean-field moments using the self-consistency conditions in
Eqs. 13b and 13c, which are then fed into the next simulation of a single spin
in a Gaussian noise. This process is repeated until the spin autocorrelations
have converged which typically takes only about 5 iterations.

Henceforth, we will occasionally use the short-hands

$\displaystyle g_{ij}^{\alpha\alpha}(t)$
$\displaystyle\coloneqq\langle\mathbf{S}_{i}^{\alpha}(t)\mathbf{S}_{j}^{\alpha}(0)\rangle,$
(17a) $\displaystyle G_{ij}^{\alpha\alpha}(t)$ $\displaystyle\coloneqq
4g_{ij}^{\alpha\alpha}(t).$ (17b)

The site indices are omitted if $i=j$. One should note that the only remaining
parameter in the self-consistency problem is the quadratic coupling constant
$J_{\mathrm{Q}}$. This constant sets the time scale. We present the generic
autocorrelation results of spinDMFT for Hamiltonians of type (1) in Fig. 2.
The difference in the time scale between the transverse ($G^{xx}$) and
longitudinal ($G^{zz}$) autocorrelation is expected due to the anisotropy
factor of 2 in the couplings. We find, that the decays are very well captured
by the fits

$\displaystyle F^{xx}(t)$
$\displaystyle=\text{exp}\left(-\frac{1}{2}\left(\frac{t}{\sigma}\right)^{2}\right)$
(18a) $\displaystyle F^{zz}(t)$
$\displaystyle=\text{exp}\left(-\rho\left(\sqrt{t^{2}+\kappa^{2}}-|\kappa|\right)\right)$
(18b)

with

$\displaystyle\sigma$ $\displaystyle=$1.769(1)$\,J_{\mathrm{Q}}^{-1},$ (19a)
$\displaystyle\rho$ $\displaystyle=$0.2149(4)$\,J_{\mathrm{Q}},$ (19b)
$\displaystyle\kappa$ $\displaystyle=$1.26(1)$\,J_{\mathrm{Q}}^{-1}.$ (19c)

We emphasize the qualitative difference in the temporal dependence of $F^{xx}$
and $F^{zz}$ for $t~{}\to~{}\infty$, Gaussian versus exponential. The spin
autocorrelations yield a good estimate for the time scale of the spin
dynamics. The key quantity is the quadratic coupling constant.

Figure 2: Generic spin autocorrelations computed by spinDMFT for dipolar spins
in the rotating frame according to Eq. 1. The quadratic coupling constant is
defined in Eq. 14 and depends on the geometry of the spin system. The dashed-
dotted black lines correspond to the fit functions in Eq. 18.

##  III Simulating FIDs with nl-spinDMFT

We stress that computing the FID with spinDMFT represents a challenging task,
since the approach accesses the spin dynamics from a microscopic point of
view, while the FID is a macroscopic quantity. Our strategy for computing the
FID is to relate it to microscopic spin-spin correlations. This is explained
in the first part of this section. Then, we present an extension of spinDMFT,
called nl-spinDMFT, and use it to evaluate the FID of calcium fluoride.

###  III.1 FID and microscopic correlations

The FID corresponds to a measurement of the transverse magnetization over time
considering a slightly polarized initial state with the statistical operator
[31]

$\displaystyle\rho_{\rightarrow}$
$\displaystyle\coloneqq\frac{1}{Z}\mathrm{e}^{h\sum_{j}\mathbf{S}_{j}^{x}}\approx\frac{1}{Z}\left(\mathbbm{1}+h\sum_{j}\mathbf{S}_{j}^{x}\right)$
(20)

where $h\coloneqq\beta\gamma B$, $\beta$ is the inverse temperature, $Z$ is
the partition sum and $\gamma$ is the gyromagnetic ratio. The transverse
magnetization can be expressed by infinite temperature spin correlations
according to

$\displaystyle\langle\mathbf{M}^{x}(t)\rangle_{\rho_{\rightarrow}}$
$\displaystyle=\mathcal{A}\sum_{i,j}\langle\mathbf{S}^{x}_{i}(t)\mathbf{S}^{x}_{j}\rangle,$
(21)

where $\mathcal{A}$ is time-independent prefactor, which is not relevant to
us. The double sum can be divided into a sum over autocorrelations and a sum
over pair correlations according to

$\displaystyle\underbrace{\sum_{i}}_{\propto
N}\underbrace{\langle\mathbf{S}^{x}_{i}(t)\mathbf{S}^{x}_{i}\rangle}_{\propto
1}+\underbrace{\sum_{i}}_{\propto N}\underbrace{\sum_{j\neq i}}_{\propto
z}\underbrace{\langle\mathbf{S}^{x}_{i}(t)\mathbf{S}^{x}_{j}\rangle}_{\leq\propto\nicefrac{{1}}{{z}}}.$
(22)

We recall that a key aspect of spinDMFT is that pair correlations are
suppressed by at least a factor $\nicefrac{{1}}{{z}}$, where $z$ is the
coordination number. Often, this allows us to neglect them for the dynamics
[18], e.g., in Eq. 15. We stress that for this argument to work the presence
of the couplings as factors in Eq. 15 is crucial. They assist in suppressing
any contribution from pair correlations. In Eq. 22, however, the double sum
for the pair correlations contains no couplings. If we consider only nearest-
neighbor pair correlations, the double sum contains $\propto zN$ terms, where
$N$ is the particle number. As a consequence, the whole expression scales like
$N$ which is the same as the scaling of the autocorrelation sum. It is the
additional factor $z$ which has the pair correlations play a role in the FID
although they are unimportant for the microscopic spin dynamics. This is the
key observation in this section.

Intuitively, one would expect the relevance of pair correlations to decrease
as the direct couplings of the two involved spins decreases. We stress,
however, that indirect couplings, for example via a third spin, can also
matter. It is not _a priori_ clear, if or how much pair correlations become
irrelevant if we increase the distance of the two involved spins. However, we
will see that the more indirect a coupling is, the more time a correlation
needs to build up between the corresponding spins. To capture the FID at short
and moderate times, it is a promising strategy to start with the pair
correlations of close spins. In Tab. 1, we list and number pair correlations,
sorted first by the distance between the pair of spins, and second, if the
distances are equal, by the coupling between the spins.

$c$ | $\vec{r}_{c}^{\top}/a_{\text{sc}}$ | $|\vec{r}_{c}|/a_{\text{sc}}$ | $|J_{c}|/J_{0}$ | $m_{c}$
---|---|---|---|---
0 |  |  |  |  |  | 1
1 | (0 | 0 | -1) | $1.0$ | $2.0$ | 2
2 | (-1 | 0 | 0) | $1.0$ | $1.0$ | 4
3 | (-1 | -1 | 0) | $1.4142$ | $0.3535$ | 4
4 | (-1 | 0 | -1) | $1.4142$ | $0.1768$ | 8
5 | (-1 | -1 | -1) | $1.732$ | $0.0$ | 8
6 | (0 | 0 | -2) | $2.0$ | $0.25$ | 2
7 | (-2 | 0 | 0) | $2.0$ | $0.125$ | 4
8 | (-1 | 0 | -2) | $2.2361$ | $0.1252$ | 8
9 | (-2 | -1 | 0) | $2.2361$ | $0.0894$ | 8
10 | (-2 | 0 | -1) | $2.2361$ | $0.0358$ | 8
11 | (-1 | -1 | -2) | $2.4495$ | $0.068$ | 8
12 | (-2 | -1 | -1) | $2.4495$ | $0.034$ | 16
13 | (-2 | -2 | 0) | $2.8284$ | $0.0442$ | 4
14 | (-2 | 0 | -2) | $2.8284$ | $0.0221$ | 8
15 | (0 | 0 | -3) | $3.0$ | $0.0741$ | 2
16 | (-3 | 0 | 0) | $3.0$ | $0.037$ | 4
17 | (-2 | -2 | -1) | $3.0$ | $0.0247$ | 8
18 | (-2 | -1 | -2) | $3.0$ | $0.0124$ | 16
19 | (-1 | 0 | -3) | $3.1623$ | $0.0538$ | 8
Table 1: List of correlations considered for the FID of calcium fluoride in
[100]-direction ($z$). The first entry ($c=0$) is the autocorrelation and the
other entries correspond to pair correlations. $\vec{r}_{c}$ is an exemplary
distance vector between the two spins involved in the pair correlation and
$|J_{c}|/J_{0}$ is the absolute of their direct coupling with
$J_{0}\coloneqq\mu_{0}\gamma_{\mathrm{F}}^{2}/(4\pi a_{\text{sc}}^{3})$. By
$m_{c}$ we denote the multiplicity of the correlation, i.e., the number how
often this correlation occurs per spin.

Note that it is not possible to compute pair correlations in spinDMFT, since
they are excluded from the approach by construction. The extension, CspinDMFT,
exactly simulates a small number of spins, the so-called cluster, and regards
the remainder of the lattice by mean-fields [19]. CspinDMFT can access these
pair correlations, but each pair correlation would require an adapted cluster
which would make the calculation numerically very expensive. In addition, the
self-consistency in each cluster would yield autocorrelations which differ
from cluster to cluster. This is not systematic. Thus we adopt the following
two-step strategy instead.

###  III.2 Overview of nl-spinDMFT

In step (1), we estimate the autocorrelations by a single self-consistent
computation with spinDMFT. This is well justified for large $z_{\text{eff}}$.
In step (2), we perform simulations of a quantum cluster in a mean-field
background to determine each pair correlation listed in Tab. 1. Thereby, we
use the previously obtained autocorrelations as building blocks to determine
the required mean-field moments. The simulations of step (2) are similar to
CspinDMFT, but we do not require the self-consistency anymore because the
mean-field moments are already well known from the upstream self-consistency
in step (1). At the same time, we are flexible to choose the quantum cluster
individually for each simulation in step (2). Hence, the choice of the cluster
can be optimized with respect to the pair correlation of interest. Finally,
the FID is obtained by superposing all correlations obtained in the second
step. Since the pair correlations and the FID are non-local quantities, we
refer to the procedure in second step as non-local spinDMFT (nl-spinDMFT). The
whole strategy is summarized in Fig. 3. We provide more details in the
following.

Figure 3: Strategy to compute spin correlations and finally the FID using
dynamic mean-field theory. The dashed arrows correspond to simple, numerically
cheap superpositions. The solid arrows correspond to numerically more
demanding Monte-Carlo simulations which are required to compute expectation
values.

The self-consistent calculation required for the first step is discussed in
Sec. II.3 (spinDMFT). If the coordination number is high, spinDMFT is
sufficient, but for example in dimerized systems, one should simulate at least
a single dimer of spins using CspinDMFT. For calcium fluoride with magnetic
field in [100]-direction, we have
$z_{\text{eff}}^{\scriptscriptstyle[100]}~{}=~{}4.9$, which is sufficiently
large for using only spinDMFT.

In the second step, we perform several simulations of a quantum cluster in a
mean-field background. The cluster is henceforth denoted by $\Gamma_{l}$,
where $l$ is the number of spins. The spins in the cluster interact quantum-
mechanically with one another and we regard the remainder of the lattice by
coupling each of the spins to a dynamic Gaussian mean-field $\vec{V}_{i}$.
This is summarized by the Hamiltonian

$\displaystyle\mathbf{H}_{\Gamma_{l}}^{\text{mf}}(t)$
$\displaystyle=\frac{1}{2}\sum_{i,j\in\Gamma_{l}}J_{ij}\vec{\mathbf{S}}^{\top}_{i}\underline{\underline{D}}\,\vec{\mathbf{S}}_{j}+\sum_{i\in\Gamma_{l}}\vec{\mathbf{S}}_{i}\cdot\vec{V}_{i}(t).$
(23)

As mentioned before, the second moments of the mean-fields are determined from
the autocorrelations obtained in the first step. Generally, the second moments
read

$\displaystyle\overline{V^{\alpha}_{i}(t)V^{\alpha}_{j}(0)}^{\text{mf}}$
$\displaystyle=D_{\alpha\alpha}^{2}\sum_{k,m\notin\Gamma_{l}}J_{ik}J_{jm}\langle\mathbf{S}_{k}^{\alpha}(t)\mathbf{S}_{m}^{\alpha}(0)\rangle.$
(24)

Using translational invariance and the sorting scheme from Tab. 1, this can be
rewritten as

$\displaystyle\overline{V^{\alpha}_{i}(t)V^{\alpha}_{j}(0)}^{\text{mf}}$
$\displaystyle=D_{\alpha\alpha}^{2}\sum_{c=0}^{\infty}J_{c,ij}^{2}G_{c}^{\alpha\alpha}(t),$
(25a) $\displaystyle J_{c,ij}^{2}$
$\displaystyle\coloneqq\sum_{\begin{subarray}{c}k,m\notin\Gamma_{l}\\\
\vec{r}_{k}-\vec{r}_{m}\stackrel{{\scriptstyle!}}{{\in}}\\{\vec{r}_{c}\\}\end{subarray}}J_{ik}J_{jl},$
(25b)

where $G_{c}^{xx}$ is the pair correlation of type $c$ and
$G_{0}^{xx}~{}=~{}G^{xx}$ is the autocorrelation. By $\\{\vec{r}_{c}\\}$ we
denote the set of distance vectors $\vec{r}_{c}$ that correspond to the
correlation with index $c$. The number of these distance vectors per site
corresponds to the multiplicity $m_{c}$ which is also provided in Tab. 1. For
example, the set for $c=1$ contains
$\vec{r}_{c}=\begin{smallmatrix}(0&0&-1)\end{smallmatrix}^{\top}$ and
$\vec{r}_{c}=\begin{smallmatrix}(0&0&1)\end{smallmatrix}^{\top}$, which
entails $m_{c}~{}=~{}2$. Of course, we can only insert those correlations that
have been computed in the first step, i.e., the autocorrelations, so that the
summation in Eq. 25 reduces to

$\displaystyle\overline{V^{\alpha}_{i}(t)V^{\alpha}_{j}(0)}^{\text{mf}}$
$\displaystyle\approx
D_{\alpha\alpha}^{2}J_{0,ij}^{2}\langle\mathbf{S}^{\alpha}(t)\mathbf{S}^{\alpha}(0)\rangle.$
(26)

Note that this approximation is also justified by the coordination number. For
example, for $i=j$, the approximation in Eq. 26 follows directly from Eq. 15.

The expression in Eq. 26 can be straightforwardly computed after obtaining the
autocorrelations in step (1). The resulting second mean-field moments are then
fed into the simulations of a spin cluster $\Gamma_{l}$ in a mean-field
background. For calcium fluoride, we consider cluster sizes of up to $l=9$
spins and access all pair correlations listed in Tab. 1. Next, we discuss how
one can optimize the choice of clusters for the computation of the pair
correlations.

###  III.3 Choice of the clusters

Clearly, for a pair correlation the two directly involved spins must be
included in $\Gamma_{l}$. For a reasonable choice of the other spins, one has
to consider that two spins can become correlated indirectly, for example, via
their coupling to a third spin. Such processes can be only accounted for, if
the third spin is included in $\Gamma_{l}$, since backactions from the cluster
onto the mean fields are excluded by construction. For an optimal choice of
the cluster, the most important indirect couplings have to be identified.

A selection criterion for the spins to be put into the cluster can be
developed, for example, on the basis of the fourth moment of the coupling
constants

$\displaystyle J_{k,ij}^{4}$ $\displaystyle\coloneqq J_{ik}^{2}J_{kj}^{2}.$
(27)

Here, $i$ and $j$ denote the pair of spins of the pair correlation in question
and $k$ is some arbitrary third spin. The coupling constant connects spin $i$
and $j$ via two links, which captures an indirect process correlating the two
spins with each other. We stress, however, that indirect processes involving
three or more links can also be important. Respecting several degrees of
indirect correlations simultaneously in a criterion forms a challenging task.

Formally, we define an _order_ as the number of links between site $i$ and
$j$. We propose a strategy that adds the presumably most important spins order
by order to the cluster and start with the lowest relevant order, $n=2$. The
first order, $n=1$, is automatically accounted for, since we always include
the spins $i$ and $j$ in the cluster. Dipolar couplings are long-range so
that, in principle, there is an infinite number of processes for $n\geq 2$. To
develop an appropriate criterion, we consider only the $L$ strongest absolute
couplings of a spin in the lattice. For example, for calium fluoride with
magnetic field in [100]-direction a plausible choice would be $L=6$, which
includes only the nearest-neighbor couplings. We suggest the following
procedure for the $n$-th order:

  1. 1.

Determine all paths from site $i$ to site $j$ with $n$ links. Only those links
are allowed, that correspond to one of the $L$ strongest couplings.

  2. 2.

Collect the sites of all paths in a set $K$. Add all spins in $K$ to the
cluster, if this does not exceed the desired cluster size $l$, and proceed
with the next order. If adding all spins would exceed $l$, consider the next
step instead.

  3. 3.

Let $f$ be the number of spins that are yet missing in the cluster. Find all
possibilities to distribute $f$ spins on the sites in $K$. For each
possibility, compute

$\displaystyle J_{ij,n}^{2n}(\Gamma_{l})$
$\displaystyle\coloneqq\sum_{k_{1},k_{2},\dots,k_{n-1}\in\Gamma_{l}}J_{ik_{1}}^{2}J_{k_{1}k_{2}}^{2}\dots
J_{k_{n-1}k_{j}}^{2}.$ (28)

Here, $k_{i}$ can be any spin in the cluster with the $f$ spins added (total
size is $l$) and we consider only the most important couplings. Accept the
cluster $\Gamma_{l}$ with the highest value for $J_{ij,n}^{2(n+1)}$ and stop
the procedure.

We consider this strategy to be a good compromise between lower and higher
orders. Aside from the cluster size $l$ which is set by the available
numerical resources, the only free parameter is the number of considered
couplings $L$.

Carrying this procedure out for $L=6$, we arrive at the 20 clusters shown in
Fig. 4. Note that the procedure has also been used for determining the optimal
cluster for the autocorrelation $c=0$ (spin $i=j$).

###  III.4 Results for calcium fluoride

To obtain the actual contribution of a pair correlation to the FID
$\mathcal{F}(t)$, one has to multiply it with its multiplicity $m_{c}$ in the
lattice so that

$\displaystyle\mathcal{F}(t)$
$\displaystyle=\sum_{c=0}^{\infty}m_{c}G_{c}^{xx}(t).$ (29)

In practice, the sum needs to be truncated. We denote the corresponding
truncation value by $c_{\text{max}}$. The results of the simulated FID for
ascending $c_{\text{max}}$ are plotted in Fig. 5 in comparison with
experimental data from Ref. [20]. In addition to the FID, we plot

$\displaystyle M^{z}_{c_{\text{max}}}(t)$
$\displaystyle=\sum_{c=0}^{c_{\text{max}}}m_{c}G_{c}^{zz}(t),$ (30)

which is exactly constant for $c_{\text{max}}~{}=~{}\infty$, since the
longitudinal magnetization $\mathbf{M}^{z}$ is a conserved quantity. Hence,
the deviation of $M^{z}(t)$ from $1$ is an excellent measure for the good
accuracy of the approximation in Fig. 5 (e).

Figure 4: Considered clusters in the simple cubic lattice for $\ce{CaF_{2}}$
for magnetic field in [100]-direction ($z$) according to the presented
strategy in the text. The numbering is adapted from Tab. 1. The orange lines
represent the pair correlations accessed by the corresponding cluster. Figure
5: Comparison of the FID in [100]-direction simulated with nl-spinDMFT with
the experimental one measured in Ref. [20]. The clusters considered for the
simulation are depicted in Fig. 4; they comprise $l=9$ sites and were chosen
considering the $L=6$ strongest couplings. The different panels show the
improved agreement as $c_{\text{max}}$ is increased, see Eq. 29 for clarity.
In addition to the FID, we plot the longitudinal magnetization from nl-
spinDMFT, see Eq. 30. This quantity is rigorously conserved in the absence of
any approximations. The experimental data were measured at
$4.2\text{\,}\mathrm{K}$. The lattice constant for the flourine ions at this
temperature is about
$a_{\text{sc}}~{}=~{}$2.724\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$.

As we increase the truncation $c_{\text{max}}$, the agreement between nl-
spinDMFT and experiment improves as expected. Adding the contributions from 19
pair correlations leads to a remarkable agreement up to about
$t=$40\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. Beyond this time, the
amplitude of the oscillation is slightly overestimated by nl-spinDMFT. We can
rule out the statistical error of the Monte-Carlo simulation as well as the
error induced by time discretization as origin for the deviation, since they
are more than one order of magnitude smaller. A probable reason for the
deviation is that the considered quantum clusters are not large enough to
capture long-range pair correlations reliably. The latter result from indirect
processes involving many spins and become, thus, important at later times.

To illustrate this, we present the individual transverse and longitudinal pair
correlations in Figs. 6 and 7. Some of them are very close to zero and hence
not shown. The transverse correlations are mostly characterized by a single
peak, which can be positive or negative. Depending on the peak position and
its sign we can group them phenomenologically into four classes on the
considered time interval. The longitudinal pair correlations on the other hand
are all positive. Most of them increase monotonically over the considered time
interval. No. 1 and 2 rise the fastest and slowly decay after their maximum
value at about $t=$40\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, see panel (a)
in Fig. 7. Note that we applied the same classification scheme as for the
transverse pair correlations. The speed of the increase of the longitudinal
pair correlations decreases with $c$, i.e., with the pair distance.

Henceforth, we focus on the transverse pair correlations as they matter for
the FID. The first peak is due to 1st-order processes, i.e., due to the direct
coupling between the pair of spins of the corresponding pair correlation. We
can verify this by analytically computing the exact pair correlations, while
setting all couplings except the direct one to zero yielding

$\displaystyle G^{xx}_{\text{dimer}}(t)$
$\displaystyle=-\sin\left(\frac{Jt}{2}\right)\sin\left(\frac{Jt}{4}\right),$
(31a) $\displaystyle G^{zz}_{\text{dimer}}(t)$
$\displaystyle=\sin^{2}\left(\frac{Jt}{4}\right),$ (31b)

which is depicted by the dashed lines in Figs. 6 and 7 in panel (a),
respectively. The agreement of these formulas with the pair correlations of
nl-spinDMFT is excellent at short times which indicates that the direct
coupling is the origin of the first peak in the transverse correlations. The
other peaks are likely to result from indirect couplings. One can presume that
the peak of the $n$th class results from processes of $n$th order according to
the definition in Sec. III.3, that is, from processes with $n$ links. This
also explains the alternating signs of the peaks: If the direct coupling
between two spins leads to anti-correlation ($-$) at short times, the indirect
coupling via two links leads to correlation ($+$), the indirect coupling via
three links again to anti-correlation and so on. Note that, generally, we
expect all orders to be relevant for all pair correlations to some extent.
This can be seen well in pair correlation No. 5 in panel (c), in which both
the second and third peak are clearly visible. However, the obtained tendency
is that the larger the distance between a pair of spins, the higher is the
order of the processes inducing the relevant peak in the pair correlation.
This entails that more-distant pair correlations tend to involve more links
and, thus, tend to pop up at later times. In App. A, we discuss the
convergence of the correlations upon increasing the cluster size $l$. We find
that short-range pair correlations converge much quicker than long-range pair
correlations. This is plausible considering the fact that the order of the
involved processes increases with the pair distance.

Fig. 5 could give the impression that the mean-field approach corresponds to a
short-time approximation, but this is not a valid view. In fact, nl-spinDMFT
is a short-range approximation and range and time correspond to each other in
the FID. We emphasize that short-ranged quantities such as the
autocorrelations are well captured by the approach _even at long times_. This
is underlined by the consideration in App. A.

There are several ways to improve the accuracy of the simulated FID at later
times. An obvious step is to increase the cluster size further; one or two
more spins are realistic. Besides this, one has to consider that the prime
reason for inaccuracies in long-range pair correlations is the lack of back
actions in the mean-field approach. In classical or hybrid quantum-classical
approaches, such back actions are treated at least on a classical level [10,
8, 9]. The advantage of the expansion in Eq. 29 is that the summed up
quantities can all be computed separately. A promising strategy for very
accurate FIDs could be to estimate any correlations below a properly chosen
$c_{\mathrm{max}}$ by nl-spinDMFT and to compute correlations for
$c>c_{\text{max}}$ by a fully classical or by a hybrid approach. We leave
these ideas for future research.

Figure 6: Contributions of the transverse pair correlations computed by nl-
spinDMFT. The numbering corresponds to the one in Tab. 1 and Fig. 4. Some pair
correlations are not shown as they are very close to zero (absolute value
smaller than $0.002$ on the whole interval). The orange bars correspond to the
approximate peak positions which are briefly discussed in the main text.
Figure 7: Same as Fig. 6, but for the longitudinal pair correlations.

In App. B, we also compare nl-spinDMFT with the experimental data for the FIDs
in [110]- and [111]-direction. The observations are qualitatively the same as
above. The nl-spinDMFT is precise at short times and performs less well at
later times, since long-range effects are not accounted for accurately. In
fact, the deviations at later times are considerably higher than for the
[100]-direction. We assign this to the fact, that the effective coordination
number is much larger for the [110]- and [111]-direction,
$z_{\text{eff}}^{\scriptscriptstyle[110]}~{}=~{}9.2$ and
$z_{\text{eff}}^{\scriptscriptstyle[111]}~{}=~{}22.5$. This may sound counter
intuitive, because the mean-field approximation is justified by a
$\nicefrac{{1}}{{z}}$-expansion. However, accessing pair correlations requires
to simulate _compact_ clusters, in which the most important intermediate spins
inducing indirect correlations are included. The larger the coordination
number $z$, the larger is the number of important intermediate spins which
complicates the choice of compact clusters enormously. If $z$ is lower, the
mean-field approximation is, in principle, less well justified. However, the
agreement of the results with the experimental FID in [100]-direction together
with the convergence plots in App. A suggest that even for a smaller effective
coordination number, $z_{\text{eff}}^{\scriptscriptstyle[100]}~{}=~{}4.9$, the
local dynamics is captured accurately by the mean-field approach. Therefore,
we believe that nl-spinDMFT can be a very powerful tool for the computation of
FIDs in systems with moderate coordination numbers, for example, in dimerized
systems such as crystalline silicon [9] or in two-dimensional systems, see for
example Ref. [19]. To access systems with very high coordination numbers, one
may combine it with approaches that capture back actions better.

##  IV Microscopic proton spin dynamics in adamantane

In the previous section, we demonstrated how spinDMFT and its extensions can
be employed to simulate the FID. To this end, we computed individual
microscopic pair correlations by simulating clusters of 9 spins in a mean-
field background. An important ingredient was the choice of compact clusters
on the lattice. Such a choice is not possible for adamantane, where the
motional averaging leads to exceptionally high coordination numbers in the
leading non-vanishing couplings to adjacent molecules and beyond. In this
scenario, at least two or three molecules would need to be included for a
reliable choice of clusters. This would entail simulating more than 30 proton
spins which is numerically not feasible. We conclude on the one hand that the
proton FID of adamantane cannot be simulated by the described procedure. On
the other hand, the mean-field approximation itself is excellently justified
for the nuclear spins in adamantane, even more than for $\ce{CaF_{2}}$ because
of the large number of interaction partners, i.e., large values of $z$. For
this reason, we expect the autocorrelations of the proton spins to be captured
very well by spinDMFT. Computing them requires to perform a powder average,
which is discussed in the following.

To apply spinDMFT, we require the quadratic coupling constant for the proton
spins. For a given direction $\vec{n}_{B}$ of the magnetic field, it can be
computed by

$\displaystyle J_{\mathrm{Q},\mathrm{H}}^{2}(\vec{n}_{B})$
$\displaystyle=N^{\text{mol}}_{\mathrm{H}}\sum_{m>1}\left(\overline{J_{(1\mathrm{H}),(m\mathrm{H})}}^{\text{ma}}(\vec{n}_{B})\right)^{2},$
(32)

with the motional-averaged couplings provided in Eq. 8b and
$N^{\text{mol}}_{\mathrm{H}}=16$ being the number of hydrogen nuclei per
molecule. To compute $J_{\mathrm{Q},\mathrm{H}}$ numerically, we rewrite the
molecule sum as a double sum and define a cutoff $s_{\text{max}}$,

$\displaystyle J_{\text{Q},\mathrm{H}}^{2}(\vec{n}_{B})$ $\displaystyle\approx
N^{\text{mol}}_{\mathrm{H}}\sum_{s=1}^{s_{\text{max}}}\sum_{\begin{subarray}{c}m\\\
|\vec{a}_{1,m}|=d_{s}\end{subarray}}\left(\overline{J_{(1\mathrm{H}),(m\mathrm{H})}}^{\text{ma}}(\vec{n}_{B})\right)^{2}.$
(33)

The first sum runs over all “shells” of molecules around the considered
central molecule in the fcc lattice. The shells are defined by their distance
to the central molecule. Here, $s_{\text{max}}$ is the index of the last shell
that is included in the numerical calculation. The second sum runs over all
molecules within the same shell, that means, with the same distance $d_{s}$ to
the central molecule. The lattice distances $d_{s}$ and the number of
molecules per shell in the fcc lattice are extracted from Ref. [32]. For an
analytical estimate of the contributions from shells with $s>s_{\text{max}}$,
we assume for simplicity each nucleus to be at the center of its molecule and
introduce the continuum limit to render the molecule sum tractable. Both
approximations become more accurate as $s_{\text{max}}$ is increased. We
obtain

$\displaystyle J_{\text{Q},\mathrm{H},\text{rest}}^{2}$ $\displaystyle\approx
N^{\text{mol}}_{\mathrm{H}}n_{0}\int_{d_{s_{\text{max}}+1}}^{\infty}\mathrm{d}a\,a^{2}\int\mathrm{d}\Omega\,J^{2}\left(a\vec{n}(\Omega),\vec{n}_{B}\right)$
(34a) $\displaystyle=\frac{64\pi
N^{\text{mol}}_{\mathrm{H}}}{15a_{\text{fcc}}^{3}}\left(\frac{\mu_{0}\gamma_{\mathrm{H}}^{2}}{4\pi}\right)^{2}(d_{s_{\text{max}}+1})^{-3},$
(34b)

where
$\vec{n}(\Omega)=\vec{n}(\theta,\phi)\coloneqq\begin{smallmatrix}(\sin\vartheta\cos\varphi,&\sin\vartheta\sin\varphi,&\cos\vartheta)\end{smallmatrix}^{\top}$
and $n_{0}\coloneqq 4/a_{\text{fcc}}^{3}$ is the density of molecules. Note
that due to the continuum limit the lattice becomes completely isotropic so
that the result is independent of $\vec{n}_{B}$.

The coupling $\overline{J_{(1\mathrm{H}),(m\mathrm{H})}}^{\text{ma}}$ has to
be evaluated for each molecule $m$ in each shell $s$ for different
orientations $\vec{n}_{B}$ of the magnetic field. We compute the four-
dimensional angular integrals in Eq. 8b by Monte Carlo integration. In total,
we consider the first eight shells around the central molecule, i.e.
$s_{\text{max}}=8$, and approximate contributions from higher shells using Eq.
34b. In spinDMFT, the orientation of the magnetic field itself is not
important, but the resulting quadratic coupling $J_{\mathrm{Q}}$ is the key
parameter. Therefore, we are not interested in the distribution of
$J_{\mathrm{Q},\mathrm{H}}$ depending of $\vec{n}_{B}$, but rather in the
distribution $\rho\left(J_{\mathrm{Q},\mathrm{H}}\right)$, which is displayed
in Fig. 8. The powder-averaged value from our simulation is
$\overline{J_{\mathrm{Q},\mathrm{H}}}^{\text{pa}}~{}=~{}$35.038(7)\text{\,}\mathrm{kHz}$$
which agrees very well with the experimentally obtained value,
$\overline{J_{\mathrm{Q},\mathrm{H}}}^{\text{pa}}~{}\approx~{}$34\text{\,}\mathrm{kHz}$$
[33] (Ref. [33] provides the second moment which differs from the quadratic
coupling constant by a factor of $\sqrt{3I(I+1)/4}=\nicefrac{{3}}{{4}}$).
However, as $\rho\left(J_{\mathrm{Q},\mathrm{H}}\right)$ is not strongly
peaked, but spreads over an interval of about $8\text{\,}\mathrm{kHz}$, this
average value is not completely representative. Applying the powder average
directly to the quadratic coupling constant, that is, _before_ the spinDMFT
simulation is not fully justified. Instead, we should produce spinDMFT results
for different values of $J_{\mathrm{Q},\mathrm{H}}$ and superpose the results
according to the distribution $\rho\left(J_{\mathrm{Q},\mathrm{H}}\right)$.
Note that this does not require additional effort, since the simulation
results from spinDMFT, see Fig. 2, can simply be rescaled in order to
correspond to a different $J_{\mathrm{Q},\mathrm{H}}$. The powder-averaged
autocorrelations of spinDMFT obtained in this way are plotted in Fig. 9
together with the experimentally measured FID. It turns out that the powder
average does not affect the shapes of the autocorrelations much.

In contrast to the FID, the transverse autocorrelation does not contain any
oscillations. This is the same phenomenon as in calcium fluoride. The pure
autocorrelation, see panel (a) of Fig. 5, does not oscillate, but after adding
pair correlations the estimated FID contains minima and maxima, see panel (e).
As explained above, we cannot access the pair correlations in adamantane
reliably, since the required cluster sizes would be too large. Hence, there is
no direct comparison between spinDMFT and experiment concerning the proton
spin dynamics. But the situation is much more favorable for the dynamics of
the nuclear spins of $\ce{{}^{13}C}$, which will be considered in the next
section.

Figure 8: Distribution of the quadratic coupling constant
$\nicefrac{{\gamma_{\mathrm{H}}}}{{\gamma_{i}}}J_{\mathrm{Q},i}$ for
$i\in\\{\mathrm{H},\mathrm{C}\\}$. The dashed vertical lines correspond to the
powder-average $\overline{J_{\mathrm{Q},i}}^{\text{pa}}$ (the distribution and
average for hydrogen and carbon is barely distinguishable). $\rho$ is
determined by organizing the resulting values for $J_{\mathrm{Q},i}$ for
different field directions $\vec{n}_{B}$ into bins. The horizontal error bars
correspond to the bin widths. The vertical error bars are determined from the
error bars of $J_{\mathrm{Q},i}$ which result from the Monte Carlo sampling.
Figure 9: Powder-averaged proton autocorrelations for adamantane computed by
spinDMFT in comparison with the proton FID measured in Ref. [23]. The
agreement is only approximate because the pair correlations are missing.

##  V Simulating carbon NMR signals in adamantane

Since the $\ce{{}^{13}C}$-spins are rare, their dynamics is completely
dominated by the interaction with the proton spins. The latter occur in much
higher numbers and their gyromagnetic ratio is larger so that we can safely
neglect any back action from the carbon spins onto the proton spins. In this
case, a substitution of the proton field in Eq. 7 by a Gaussian mean-field is
very justified. This leads to a simple spin-noise model according to the
Hamiltonian

$\displaystyle\mathbf{H}_{\mathrm{C}}(t)$
$\displaystyle=\mathbf{I}^{z}V^{z}_{\mathrm{C}}(t).$ (35)

The mean-field has zero average and its second moments are given by

$\displaystyle\overline{V^{z}_{\mathrm{C}}(t)V^{z}_{\mathrm{C}}(0)}^{\text{mf}}$
$\displaystyle=J_{\mathrm{Q},\mathrm{C}}^{2}\langle\mathbf{S}^{z}(t)\mathbf{S}^{z}(0)\rangle$
(36)

with the longitudinal spin autocorrelation provided in Fig. 2. Similar to Eq.
32, the quadratic coupling constant is given by

$\displaystyle J_{\mathrm{Q},\mathrm{C}}^{2}(\vec{n}_{B})$
$\displaystyle=N^{\text{mol}}_{\mathrm{H}}\sum_{m>1}\left(\overline{J_{(1\mathrm{C}),(m\mathrm{H})}}^{\text{ma}}(\vec{n}_{B})\right)^{2}.$
(37)

Note that we do not consider the powder average in Eq. 36. Within each
crystal, the carbon spins couple to an individual spin environment with fixed
$J_{\mathrm{Q},\mathrm{C}}$ and fixed proton autocorrelation. Therefore, one
has to compute any signals of the carbon spins individually for each crystal
or, equivalently, for each $J_{\mathrm{Q},\mathrm{C}}$ and the corresponding
proton autocorrelation. Averaging these signals leads to the appropriate
powder-average measured in experiment.

Using the same procedure as in the previous section, we can estimate
$J_{\mathrm{Q},\mathrm{C}}$ by Monte-Carlo integration. In our calculations it
turns out that, although the carbon spins sit on a much smaller radius in the
adamantane molecule, their quadratic coupling constant is always exactly the
same as for the proton spins except for a factor
$r_{\gamma}\coloneqq\nicefrac{{\gamma_{\mathrm{C}}}}{{\gamma_{\mathrm{H}}}}\approx
0.251$ due to the different gyromagnetic ratios. This is true for any
considered field direction $\vec{n}_{B}$. To underline this observation, we
included the density $\rho(r_{\gamma}J_{\mathrm{Q},\mathrm{C}})$ in Fig. 8,
which is remarkably close to $\rho(J_{\mathrm{Q},\mathrm{H}})$.

We compute the transverse autocorrelation generated by the Hamiltonian in Eq.
35 for different values of $J_{\mathrm{Q},\mathrm{C}}$ and superpose the
results according to
$\rho(J_{\mathrm{Q},\mathrm{C}})~{}\approx~{}\rho(r_{\gamma}^{-1}J_{\mathrm{Q},\mathrm{H}})$
to obtain the powder average. The resulting signal is shown in Fig. 10
together with the experimental result for the FID. Note that since the
$\ce{{}^{13}C}$-atoms have a low abundance, two individual carbon spins see a
totally different environment. Hence, pair correlations are not relevant for
the corresponding FID and the transverse autocorrelation of carbon should, in
principle, be equivalent to the FID. However, as it turns out, the result from
the noise model deviates a bit from the experiment at moderate and large
times. For a better understanding of this observation, we consider the spin
echo signal in the following.

To simulate a spin Hahn echo [34] within the spin noise model, we include an
instantaneous pulse according to

$\displaystyle\mathbf{H}_{\text{pulse}}$
$\displaystyle=\pi\delta(t-\tau/2)\mathbf{I}^{y}$ (38)

and measure the result of the transverse autocorrelation at $t=\tau$. The
result of this is plotted versus $\tau$ in Fig. 11. Here, the agreement
between theory and experiment is very good. We conclude that the dynamic noise
resulting from the proton spins as captured by spinDMFT is perfectly accurate.
Since a spin echo eliminates static noise we attribute the discrepancy in the
FID to some small additional static noise.

To underline this interpretation, we redo the computation of the FID and
include a weak static Gaussian noise. This is done by adding a constant term
$W^{2}$ to the second mean-field moment in Eq. 36, where $W$ is the standard
deviation of the noise. The best fit to the experimental data is obtained for
$W=$0.29(6)\text{\,}\mathrm{kHz}$$ and plotted in orange in Fig. 10. The
agreement for the adapted model is remarkable and supports our assumption. We
stress that the spin echo exactly reverts the effect of the noise in the
considered model so that the result in Fig. 11 does not change at all.

In addition, we analyzed how such an additional noise source would affect the
proton spin dynamics. To this end, we added the noise, amplified by the factor
$\nicefrac{{1}}{{r_{\gamma}}}$, in the spinDMFT simulation finding no visible
difference in the resulting autocorrelations relative to the ones shown in
Fig. 9. The homonuclearly coupled proton spins are much more robust against
disturbing fields. A candidate for the origin of the inhomogeneous broadening
is the chemical shift anisotropy. It is anisotropic in solids and broadens the
NMR spectra of powders [1].

Figure 10: Powder-averaged transverse autocorrelation for $\ce{{}^{13}C}$ in
adamantane from the spin-noise model in Eq. 35. Results without and with an
additional static noise source are shown. The carbon FID measured in Ref. [23]
is plotted with black circles. Figure 11: Powder-averaged spin echo for
$\ce{{}^{13}C}$ in adamantane from the spin-noise model in Eq. 35. The
experimental spin echo measured in Ref. [23] is plotted with black circles.

##  VI Conclusion

We applied a recently developed dynamic mean-field theory for spins (spinDMFT)
[18] to NMR experiments in order to establish it as a versatile and accurate
simulation tool enabling quantitative understanding.

In the first part, we extended spinDMFT to access free-induction decays in
homonuclear spin systems. This approach is called non-local spinDMFT (nl-
spinDMFT) because it involves calculating non-local quantities that are not
directly accessible in spinDMFT. The nl-spinDMFT uses _a priori_ computed
mean-fields to efficiently simulate a quantum cluster of about 10 spins in a
mean-field background. The simulation allows us to compute pair correlations,
which contribute to the FID. This gives us an understanding of how the
macroscopic FID results from microscopic spin dynamics. The extrema in the FID
can be traced back order by order to processes involving $n$ spin-spin
couplings.

The agreement of the method with the experimental FID in $\ce{CaF_{2}}$ [20]
is excellent for the [100]-direction at short and moderate times. Small
deviations at later times are found to be due to less well captured long-range
pair correlations, which become relevant at later times. These deviations are
larger for the [110]- and [111]-direction because selecting compact clusters
with at maximum 9 spins becomes more difficult for higher coordination
numbers. In these cases, it may be promising to combine the approach with
purely classical or hybrid quantum-classical simulations [10, 8, 9]. Our
simulations suggest that the choice of the cluster is of great importance.
Different pair correlations are accessed by different clusters in nl-spinDMFT.
For this reason, we propose to improve the hybrid approach presented in Ref.
[8, 9] by computing pair correlations separately considering individual
adapted clusters.

In spin systems with moderate coordination numbers, for example in two-
dimensional systems, we expect an excellent performance of nl-spinDMFT. Apart
from pair correlations, nl-spinDMFT easily allows to compute arbitrary
quantities on the quantum cluster, which are of interest, for example, for
experiments measuring multi-spin correlations [31].

In the second part, we demonstrated the versatility of the mean-field
framework by considering the spin dynamics in polycrystalline adamantane [23],
where motional and powder averaging has to be taken into account. Using
spinDMFT, we estimated the proton-spin autocorrelations and used them to
calculate the FID and the spin echo of the rare $\ce{{}^{13}C}$ nuclear spins.
While there is some deviation between the numerical and experimental carbon
FID, the carbon spin echo is captured very well by the simulation. This
suggests the presence of inhomogeneous broadening in the experiment. We
account for this in an additional simulation by adding a weak static noise,
which leads to an excellent agreement with the experimental FID.

We consider the mean-field framework described by spinDMFT and nl-spinDMFT to
be very versatile due to the microscopic nature of the effective mean-field
models. For example, spins with $S>\nicefrac{{1}}{{2}}$ including quadrupolar
interactions can be easily considered in the simulations. Explicit time
dependencies can be included in spinDMFT and its extensions without increasing
the numerical effort much. This has been demonstrated for the spin echo in
adamantane, but more complicated pulse sequences and even pulse imperfections
can be considered as well.

Furthermore, we point out that the nl-spinDMFT can be enhanced by using
CspinDMFT [19] in the first step instead of spinDMFT. The second step remains
essentially the same. This straigtfoward extension is indicated if there is
are strong dominating spin interactions which cannot be treated as noise, for
instance if dimers of spins with strong intradimer, but weaker interdimer
coupling need to be dealt with. Also larger cluster are conceivable as an
extension.

In conclusion, we strongly advocate the use of spinDMFT and its extensions for
the prediction and understanding of NMR signals.

###### Acknowledgements.

We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) in project
UH90/14-1 as well as the DFG for Basic Research in TRR 160. We also thank the
TU Dortmund University for providing computing resources on the High
Performance Computing cluster LiDO3, partially funded by the DFG in project
271512359. We are grateful for the funding received by T.H. from the DAAD RISE
Germany program.

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##  Appendix A Convergence of pair correlations in calcium fluoride

With the help of the algorithm presented in Sec. III.3, we add the most
important intermediate spins inducing indirect couplings order by order to the
cluster. Comparing the simulation results for different cluster sizes $l$
allows us to track the convergence. We perform this comparison for the
autocorrelation and some selected relevant pair correlations. The result is
shown in Fig. 12. As can be seen in panel (e), the autocorrelation changes
only very slightly as $l$ is increased. The result for $l=9$ is barely
distinguishable from $l=1$ (spinDMFT), except for a tiny maximum at about
$60\text{\,}\mathrm{\SIUnitSymbolMicro s}$. This is a perfect _a posteriori_
justification for the applicability of spinDMFT. Such a quick convergence is
also obtained for the nearest-neighbor pair correlation $c=1$, see panel (a).
However, as the pair distance is increased, the convergence becomes slower and
slower.

Figure 12: Convergence of the pair correlations from nl-spinDMFT in
[100]-direction with ascending cluster size $l$. Panels (a)-(d) show selected
pair correlations with index $c$ according to Tab. 1 and panel (e) shows the
autocorrelation. Pair correlations do not exist for $l~{}=~{}1$ which is why
the orange line can only be found in panel (e).

Often, we obtain considerable leaps upon adding a single spin, because this
spin allows the pair correlation to build up along an additional important
path on the lattice. For $c=4$, see panel (b), the third and fourth spin are
very important. As can be seen in Fig. 13, these are the common neighbors of
the two spins directly involved in the pair correlation. The two indirect
couplings via spin No. 3 and 4 are the main contributions to this pair
correlation. The same behavior can be observed in pair correlation No. 19, see
panel (d) and Fig. 14 for the clusters. Adding the sixth spin enables the
first path with four nearest-neighbor links between the two spins of the pair
correlation. Each spin that is added till $l=8$ enables another fourth-order
path and, thus, increases the fourth-order peak in the pair correlation.
Remarkably, the increase is always nearly the same indicating that the effects
from different paths roughly adds up linearly in the pair correlation.
Studying such effects in detail leads to a much better understanding of the
microscopic correlations. Note that with a cluster size of $l=8$ spins, any
paths with four nearest-neighbor links are included. Hence, the effect of
adding spin No. 9 is significantly smaller. These observations are in perfect
accordance with our arguments in Secs. III.3 and III.4.

Figure 13: Obtained clusters for the pair correlation with index $c=4$,
depicted in orange, for different cluster sizes $l$. Figure 14: Obtained
clusters for the pair correlation with index $c=19$, depicted in orange, for
different cluster sizes $l$.

The direct understanding of the leaps in the convergence plots allows us to
draw conclusions about the accuracy of the computed pair correlations. We
consider, for instance, pair correlation No. 19, see Fig. 12 (d), to be fairly
well captured for $l~{}\geq~{}8$, because all paths with four nearest-neighbor
links are included. Hence, the lowest, most important order is completely
accounted for. The clusters for $l=8$ and $l=9$ may be referred to as
_compact_. For several other pair correlations with comparably large pair
distances this is not the case, see for example $c=11,12,17,18$ in Fig. 4.
Here, the lowest orders are only included partly for $l=9$ which makes the
results less reliable. We conclude that the larger the pair distance, the less
compact are the clusters at given cluster size and, hence, the less accurate
are the resulting pair correlations. Note that the absolute magnitude of the
pair correlations decreases as the pair distance increases so that it is
justified to truncate at a certain range.

##  Appendix B FID of $\ce{CaF_{2}}$ in different directions

The results for external magnetic field in [110]- and [111]-direction are
shown in Figs. 15 and 16. Note that the number of most important couplings $L$
is chosen larger than for the [100]-direction. This is because there are more
couplings of similar strength. The time at which nl-spinDMFT becomes
inaccurate is about the same as for the [100]-direction (about
$40\text{\,}\mathrm{\SIUnitSymbolMicro s}$). In particular, for the
[111]-direction the rather late minimum is not captured completely
quantitatively. Also, the sum rule of the $z$-direction (blue line) is not
well fulfilled for the discussed reasons. The choice of compact clusters
requires more and more spins as the coordination number is increased. Since we
are limited to 9 spins, the FIDs in [110]- and [111]-direction are less well
captured.

Figure 15: Same as Fig. 5, but in [110]-direction and with $L=8$. Figure 16:
Same as Fig. 5, but in [111]-direction and with $L=14$.