Journal of Magnetic Resonance 310 (2020) 106645
Contents lists available at ScienceDirect
Journal of Magnetic Resonance
journal homepage: www.elsevier.com/locate/jmr
A master equation for spin systems far from equilibrium
Christian Bengs, Malcolm H. Levitt ⇑
School of Chemistry, Southampton University, University Road, SO17 1BJ, UK
a r t i c l e
i n f o
Article history:
Received 17 October 2019
Accepted 8 November 2019
Available online 19 November 2019
Keywords:
Relaxation
Spin isomers
Master equation
Lindblad
Superoperator
a b s t r a c t
The quantum dynamics of spin systems is often treated by a differential equation known as the master
equation, which describes the trajectories of spin observables such as magnetization components, spin
state populations, and coherences between spin states. The master equation describes how a perturbed
spin system returns to a state of thermal equilibrium with a finite-temperature environment. The conventional master equation, which has the form of an inhomogeneous differential equation, applies to
cases where the spin system remains close to thermal equilibrium, which is well satisfied for a wide variety of magnetic resonance experiments conducted on thermally polarized spin systems at ordinary temperatures. However, the conventional inhomogeneous master equation may fail in the case of
hyperpolarized spin systems, when the spin state populations deviate strongly from thermal equilibrium,
and in general where there is a high degree of nuclear spin order. We highlight a simple case in which the
inhomogeneous master equation clearly fails, and propose an alternative master equation based on
Lindblad superoperators which avoids most of the deficiencies of previous proposals. We discuss the
strengths and limitations of the various formulations of the master equation, in the context of spin systems which are far from thermal equilibrium. The method is applied to several problems in nuclear magnetic resonance and to spin-isomer conversion.
Ó 2019 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
1. Introduction
Nuclear magnetic resonance (NMR) and electron paramagnetic
resonance (EPR) experiments are usually performed on samples
which are allowed to reach thermal equilibrium in a large magnetic field, perturbed by radiofrequency (for NMR) or microwave
(for EPR) pulses resonant with the Zeeman transitions in order to
induce an observable electromagnetic response, and allowed to
return to thermal equilibrium before the process is repeated.
Under most circumstances, the behaviour of the spin system is
described by a differential equation which is usually known as
the inhomogeneous master equation (IME) and is given by [1–14]
d
^ qðt Þ q
qðtÞ ¼ i½Hcoh ðtÞ; qðtÞ þ C
eq
dt
ð1Þ
Here qðtÞ is the spin density operator which describes the quantum
state of the ensemble of spin systems, and is given by
q ¼ j wihw j
ð2Þ
where j wi is the quantum state of an individual member of the spin
ensemble, and the bar represents an average over all ensemble
members. The term Hcoh is the coherent part of the spin Hamiltonian, which is uniform for all spin ensemble members. The relaxation of the spin system is described by a relaxation
^ , and which may be constructed from
superoperator, denoted C
the fluctuating part of the spin Hamiltonian; some construction
procedures are discussed later in this paper. The spin density operator at thermal equilibrium with the molecular environment is
denoted qeq . The last term in Eq. (1) depends on the deviation of
the spin density operator from its thermal equilibrium value, and
causes the density operator to reach thermal equilibrium when left
unperturbed for a long time.
For example, consider an operator Q which commutes with the
^:
coherent Hamiltonian Hcoh and which is also an eigenoperator of C
½Q; Hcoh ¼ 0
^ Q ¼ T 1 Q
C
Q
ð3Þ
ð4Þ
where T Q is the relaxation time constant of spin density components proportional to the operator Q. The expectation value of Q follows the trajectory
⇑ Corresponding author.
E-mail addresses: cb2r15@soton.ac.uk (C. Bengs), mhl@soton.ac.uk (M.H. Levitt).
URL: http://www.chem.soton.ac.uk (M.H. Levitt).
hQ iðt Þ ¼ hQ ieq þ hQ ið0Þ hQ ieq exp t=T Q
https://doi.org/10.1016/j.jmr.2019.106645
1090-7807/Ó 2019 The Author(s). Published by Elsevier Inc.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
ð5Þ
2
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
where
n
o
hQ iðtÞ ¼ Tr Q y qðt Þ
ð6Þ
and
n
o
hQ ieq ¼ Tr Q y qeq
ð7Þ
Eq. (5) indicates that the deviation of the expectation value of Q
from its thermal equilibrium value decays exponentially, so that
hQ iðt Þ tends to hQ ieq at long times t. Another corollary of Eq. (5) is
that the trajectory of the expectation value for the relaxation eigenoperator Q depends only on the initial value of the same expectation
value and the time constant T Q , and not on any other expectation
values.
All of this seems unremarkable, and indeed, Eq. (1) has been a
fixture in spin dynamical theory for a long time. This is despite
the caveats used by the early developers of spin relaxation theory,
including Bloch and Wangsness [1,2], Redfield [3], Abragam [4],
and Hubbard [5]. For example, in the derivation in his 1957 paper
[3], Redfield used the following phrase at a key step: ‘‘unless the
system is prepared in an unusual way”.
So what is ‘‘a system prepared in an unusual way”? The discussion in the Redfield paper makes it clear that the term implies a
large deviation of the density operator from thermal equilibrium,
such that the density operator ventures outside the hightemperature and weak-order approximation:
kHcoh =kB T k 1
ð8Þ
1 q N 1 1
ð9Þ
where 1 is the unity operator, N is the dimension of Hilbert space
and the Frobenius norm is defined
kAk ¼
X
2
jhr jAjsij
ð10Þ
r;s
In his classic textbook [4], Abragam makes the same approximations en route to the derivation of the IME (at the top of pages
287 and 288).
Hence, Eq. (1) only applies in the high-temperature limit for
spin systems exhibiting a small degree of order, and this restriction
was well-known at the time this equation was first developed.
There is great current interest in the preparation of nuclear spin
systems ‘‘in an unusual way”, i.e. far from equilibrium, or with
large amounts of spin order. Such preparations include hyperpolarized spin systems with a large level of Zeeman polarization,
enhanced with respect to thermal equilibrium polarization by
many orders of magnitude [15–28], as prepared by methods such
as dynamic nuclear polarization (DNP) [15–18] and optical pumping [19–22]. In addition, nuclear spin systems may be prepared in
states corresponding to non-equilibrium spin isomer distributions,
such as hydrogen enriched in the para spin isomer
[23,25,26,24,27,29,28] and rapidly rotating methyl groups prepared with non-equilibrium distributions of populations between
the spin isomers of the three equivalent protons [30–36]. In some
circumstances these different modes of non-equilibrium spin order
interconvert, leading for example to the phenomena of
parahydrogen-induced hyperpolarization (PHIP), in which the
non-equilibrium singlet spin order of para-enriched hydrogen
gives rise to large magnetization components after a chemical
reaction [25–28], and quantum-rotor-induced polarization (QRIP)
in which non-equilibrium methyl rotor order also gives rise to
enhanced NMR signals [30–36]. In many cases, hyperpolarization
leads to a situation in which the populations of long-lived states
are strongly enhanced, or greatly depleted [34,35,37–45].
A ‘‘thought experiment” involving the preparation of a sample
‘‘in an unusual way” is shown in Fig. 1(a). Consider an ensemble
of nuclear spin-1/2 pairs. The magnetic environments of the two
spins-1/2 are considered to be either identical (magnetic equivalence) or nearly identical (near-equivalence) [41], so that the
Hamiltonian eigenstates are given, to a good approximation, by
the singlet state and the three triplet states, defined as follows:
jS0 i ¼ 21=2 ðjabi jbaiÞ
jT þ1 i ¼ jaai
jT 0 i ¼ 21=2 ðjabi þ jbaiÞ
jT 1 i ¼ jbbi
ð11Þ
These states are eigenstates of total spin angular momentum
with quantum number I ¼ 0 for the singlet state and I ¼ 1 for the
triplet states. Hence, the singlet state is non-magnetic and all
NMR observables are associated with the triplet states. In a magnetic field, the degeneracy of the triplet states is split by the
nuclear Larmor frequency. Suppose that the sample is prepared
‘‘in an unusual way”, such that only the singlet state is populated,
see Fig. 1(a, (a, top). Clearly, there can be no nuclear magnetization
in this I ¼ 0 state. Now suppose that relatively inefficient processes
enable the slow conversion of the singlet state to the triplet states,
and denote the corresponding time constant T S . Assume also, for
the sake of simplicity, that the transition probabilities are the same
for all three triplet states. In this case, there is no magnetization
associated with the triplet manifold when it is first populated by
conversion from the singlet state, see Fig. 1(a, middle). However,
the triplet manifold rapidly magnetizes in the magnetic field due
to spin-lattice relaxation, leading to the establishment of a Boltzmann distribution between its Zeeman-split energy levels, see
Fig. 1(a, bottom). Assume that the spin-lattice relaxation time constant T 1 is much smaller than the time constant T S for singlet-totriplet conversion. Now we ask: what is the time constant for the
build-up of Zeeman magnetization along the field? The answer is
obvious: since the singlet state has no nuclear spin, and no nuclear
magnetization, the singlet population must convert to triplet populations before any magnetization can build up. Since the rate of
singlet-to-triplet conversion is much smaller than the rate of triplet relaxation, the rate of magnetization build-up is determined
by the slowest process, which is the singlet-to-triplet conversion.
Hence, the magnetization builds up along the field with a slow
time constant of the order of T S . This is illustrated by the solid line
in Fig. 1(b), which was simulated using the techniques described
later in the paper.
This result is obvious, but it is not predicted by the standard
master equation of Eq. (1)! The IME predicts that the expectation
value z-magnetization builds up with a time constant T 1 , irrespective of the initial distribution of singlet and triplet populations, and
even in the case that the triplet states are totally unpopulated. The
standard IME leads to the prediction given by the dashed line in
Fig. 1(b), with the magnetization building up with the fast time
constant T 1 . The IME therefore predicts that a sample in which only
the non-magnetic singlet state is populated somehow rapidly
acquires a magnetization which is almost as large as a sample in
full thermal equilibrium and with a triplet:singlet population ratio
of 3:1. This makes no physical sense.
A real experiment of this form has been performed on a material containing freely rotating water molecules encapsulated in
fullerene (C60 ) cages [46]. This experiment utilizes the spin isomerism of freely rotating water molecules, in order to generate
the required over-population of the non-magnetic singlet state.
In freely rotating water molecules, the nuclear singlet state is associated with the para spin isomer of water, which has a significantly
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
3
Fig. 1. An experiment which illustrates the failure of the IME for spin systems far from equilibrium. The horizontal lines represent the Hamiltonian eigenvalues (energy
levels) and the balls represent state populations. (a) An ensemble of spin-1/2 pairs is prepared so that only the non-magnetic (I ¼ 0) singlet state is populated. Conversion
from the singlet state to the magnetic triplet states (I ¼ 1) occurs with a time constant T S . The triplet manifold reaches thermal equilibrium in a magnetic field with time
constant T 1 , such that T 1 T S . (b) Trajectories of the spin magnetization along the field Mz , normalized to the thermal equilibrium value. The IME predicts recovery of M z
with time constant T 1 (dashed line), which is physically unreasonable. The correct behaviour is that M z recovers with the much longer time constant T S (solid line). The
simulations are for the case T 1 ¼ 2 s and T S ¼ 22:5 s.
lower rotational energy than the triplet states, which belong to the
ortho spin isomer. It is therefore possible to enrich the endofullerene sample in the para spin isomer of water by leaving it to
equilibrate at low temperature, as verified by low-temperature
NMR and electrical measurements [47,48]. The para-enriched
material is brought rapidly to room temperature using a fast dissolution apparatus, and the NMR signal observed in the presence of a
large magnetic field. The NMR signal slowly increases in time since
the para-water molecules, which are in a non-magnetic nuclear
singlet state providing no NMR signal, convert into ortho-water
molecules, which have nuclear spin-1, and rapidly polarize along
the field, providing an NMR signal upon radiofrequency excitation.
The slow increase in the NMR signal with time therefore allows
determination of the para-to-ortho spin-isomer conversion kinetics. The time constant for this signal build-up was found to be
about 30 s, while the measured value of T 1 is only 0.75 s [46]. This
is in clear contradiction with the prediction of the IME.
Although this experimental example exploits spin isomerism in
order to prepare the required initial state, the phenomenon of spin
isomerism is by no means essential for highlighting the problematic features of the IME. Identical behaviour would be observed
for molecules lacking spin isomers, but which still exhibit slow
conversion between singlet and triplet states. Such examples are
widespread [34,35,37–45]. In one case, the singlet-triplet conversion time constant exceeds 1 h [43]. If a system of this kind were
prepared in a pure singlet state through hyperpolarization techniques, the recovery of the longitudinal magnetization in a strong
magnetic field would also be on a timescale far longer than T 1 , in
contradiction to the predictions of the inhomogeneous master
equation.
The early pioneers of NMR relaxation theory [49,2,1,3–5] did
not address the problem of nuclear spin relaxation far from equilibrium. For a long time there was no need, since the vast majority
of NMR experiments are performed on spin systems which are very
close to equilibrium. The inhomogeneous master equation (Eq. (1))
rapidly became established as a fixture in the field and plays a central role in standard textbooks [6].
A series of alternative master equations were proposed in the
1980s and 1990s [50–53]. However, the primary motivation was
not to address the relaxation dynamics of far-from-equilibrium
spin systems, but to circumvent the mathematical inconvenience
of the IME, which has the form of an inhomogeneous differential
equation. The alternative master equations have the form of a
homogeneous differential equation, of the type
d
^ h qðt Þ
qðtÞ ¼ i½Hcoh ðtÞ; qðtÞ þ C
dt
ð12Þ
^ h is a ‘‘thermalized” variant of the relaxation superoperator
where C
^
C, adjusted in such a way that the spin density operator q tends to
the correct thermal equilibrium value qeq in thermal equilibrium
with the environment [50–57]. A spin dynamical equation of the
form in Eq. (12) is referred to here as a homogeneous master equation
(HME).
^ h are examined below. Although this
Some proposed forms for C
was not their primary intention, some of the HME formulations do
provide a partial treatment of far-from-equilibrium spin systems,
as well as being mathematically more convenient than the IME.
However none of the proposed solutions treat the behaviour of
coherences correctly far from equilibrium. The rest of this paper
discusses the construction of an alternative HME which does not
suffer from these restrictions and which retains validity for spin
systems far from equilibrium.
This class of problem is, of course, not unique to nuclear magnetic resonance. Other forms of coherent spectroscopy, such as
electron paramagnetic resonance (EPR) and coherent optical spectroscopies, also encounter quantum systems which are strongly
ordered or in transient states which are far from equilibrium. The
large field of research into open quantum systems addresses the
problems of quantum dynamics of systems which exchange energy
4
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
with a thermal reservoir [58–63]. Some of the techniques presented in the current paper will be familiar to open quantum theorists, so we do not claim great originality. Nevertheless,
techniques such as Lindbladian superoperators, which are widely
used in open quantum theory, are seldom encountered in magnetic
resonance, with notable exceptions [64,65].
Our aim in this paper is to make an explicit connection between
magnetic resonance theory on the one hand, and open quantum
theory on the other, in order to address the pressing issue of
how to treat the behaviour of nuclear spin systems far from
equilibrium.
3. The commutation superoperator is defined as follows:
2. Theoretical background
The matrix representation of a superoperator in a particular
operator basis has matrix elements defined as follows:
The theoretical background of NMR relaxation theory is
reviewed briefly in order to establish the notation and to resolve
ambiguities in some of the key terms.
h
2.1. Hilbert space
2.3. Spin Hamiltonian
The sample is considered to consist of a large number of duplicates of identical spin systems, each consisting of N coupled spins
with spin quantum numbers fI1 ; I2 . . . IN g. The quantum state of
each system may be described by a state vector jwi which is an element of a Hilbert space H. The dimension of Hilbert space is equal
to the number of independent spin states, and is given by
The organization of the spin Hamiltonian terms for the individual ensemble members is shown in Fig. 2. The spin Hamiltonian is
expressed as a sum of coherent and fluctuating contributions:
NH ¼
N
Y
ð2Ii þ 1Þ
ð13Þ
i¼1
Spin operators may be represented by N H N H -dimensional matrices in Hilbert space.
2.2. Liouville space and Superoperators
b ¼ ðMÞ ðM Þ
M
ð18Þ
such that
b jNÞ ¼ j½M; NÞ ¼ jMN NM Þ
M
ð19Þ
b implies a general
In the current article, a symbol of the form M
superoperator of unspecified type, unless M is already defined as
b implies the commuthe symbol for an operator, in which case M
tation superoperator of the operator M.
b
M
i
rs
h
i
b s
b j Os ¼ Tr Oy MO
¼ Or j M
r
HðtÞ ¼ Hcoh ðt Þ þ Hfluc ðtÞ
ð20Þ
ð21Þ
The coherent part is assumed to be the same for each member of the
ensemble, whereas the fluctuating part may differ for ensemble
members at a given point in time, and is responsible for the relaxation of the system. For typical solution-state NMR experiments the
coherent part consists of Zeeman, chemical shift and scalar coupling
interactions [4,6,12]. Other interactions are readily accommodated,
such as residual dipole-dipole couplings and quadrupolar interactions in anisotropic phases.
2.2.1. Liouville space
A complete set of linear operators forms a linear vector space
called Liouville space, with dimension N L ¼ N 2H . An operator O
may be regarded as a vector of length N L in this space and denoted
jOÞ. The inner product between any two operators (called the Liouville bracket) is defined as follows:
ðM j NÞ ¼ Tr M y N
ð14Þ
A set of NL basis operators Op with p 2 f1; 2 . . . N L g is said to be
orthonormal if it satisfies the condition
Op j Oq ¼ dpq
ð15Þ
Such a set of operators forms a suitable basis for Liouville space.
2.2.2. Superoperators
Superoperators transform operators into other operators
[66,50]. The following superoperators are used in the current
article:
1. The left-multiplication superoperator is denoted here by M,
where the bullet symbol is a placeholder for an operator argument. The superoperator M multiplies its operand by an operator from the left, as follows:
ðMÞjOÞ ¼ jMOÞ
ð16Þ
2. The right-multiplication superoperator is denoted here by M,
and multiplies its operand by an operator from the right, as
follows:
ðMÞjOÞ ¼ jOMÞ
ð17Þ
Fig. 2. The organization of spin Hamiltonian terms used in this article. The
Hamiltonian HðtÞ consists of a coherent term Hcoh ðt Þ and a fluctuating term Hfluc ðt Þ.
The coherent term contains a dominant part HA , a static (or quasi-static)
perturbation HB , and a time-dependent perturbation HC ðt Þ. The static perturbation
may be decomposed into a secular part HkB which commutes with HA , and a nonsecular part H?
B , which does not. The time-dependent perturbation HC may be
res
divided into a resonant part Hres
and an off-resonant part Hoff
. Standard
C
C
approximations lead to a corrected perturbation H0B which commutes with HA .
The main coherent Hamiltonian H0 is constructed by adding HA and H0B .
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C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
2.3.1. Coherent Hamiltonian
The discussion below makes extensive use of a division of the
coherent Hamiltonian into three parts, called HA ; HB and HC , as
follows:
Hcoh ðt Þ ¼ HA þ HB þ HC ðt Þ
ð22Þ
The properties of these three terms is as follows:
The dominant Hamiltonian HA . The term HA is much larger than
the HB ; HC and Hfluc terms:
kHA k kHB k; kHC k; kHfluc k
ð23Þ
The operators HB and HC may therefore be regarded as perturbations to the dominant Hamiltonian HA . In this article we also
demand that all eigenvalues of HA are either degenerate or
widely spaced compared to the corresponding matrix elements
of the perturbations HB and HC in the eigenbasis of HA (see
Fig. 3). This strong constraint on the eigenvalue spectrum of
HA is very important and underpins much of the theory
described below.
Denote the eigenvalues and eigenstates of the dominant term HA
by xAr and jriA respectively:
HA jriA ¼ xAr jriA
ð24Þ
ð27Þ
In typical high-field NMR applications, the term HB contains
spin-spin couplings, chemical shift terms, and interactions with
slowly varying fields such as magnetic field gradient pulses. In
solid-state NMR, anisotropic spin interactions such as chemical
shift anisotropies, quadrupolar interactions, and dipole–dipole
couplings are also included. These terms may acquire a slow
time-dependence satisfying Eq. (27) through sample rotation.
Since the eigenvalues of HA are combinations of the Larmor frequencies for the nuclides in the spin system, Eq. (26) implies the
validity of a high-field approximation for the nuclear spin
Hamiltonian.
The HB term may be further subdivided into a ‘‘secular” term HkB
k
HB ¼ HB þ H?B
where
ð25Þ
The perturbations HB and HC are both assumed to fulfil the following condition:
hr jA HB jsiA ; hr jA HC ðt ÞjsiA xAr xAs 8 xAr – xAs
d A
hr j HB jsiA xA xA 8 xA – xA
r
s
r
s
dt
and a ‘‘non-secular” term H?
B:
where r 2 f1; 2 . . . N H g, and the eigenstates are orthogonal:
hr jA jsiA ¼ drs
exchange interaction, is included in the HA term (see
Section 6.4).
The (quasi) static perturbation term HB . The term HB is assumed
to be small in magnitude with respect to HA (Eq. (23)) and either
time-independent or with a slow time-dependence. The latter
condition may be written as follows:
ð26Þ
Eq. (26) implies that the matrix representations of HB and HC ,
expressed in the eigenbasis of HA , has off-diagonal terms which
are either between degenerate states of HA , or which are always
small in magnitude compared to the corresponding separation of
non-degenerate HA eigenvalues.
In typical high-field NMR experiments, HA represents the dominant Zeeman interaction with a strong external magnetic field.
However in some cases, a different Hamiltonian, such as an
k
HB
commutes with HA , while
h
i
HA ; HkB ¼ 0
ð28Þ
H?
B
does not:
HA ; H?B – 0
ð29Þ
The time-dependent perturbation term HC ðtÞ.
The term HC is also small in magnitude with respect to HA (Eq.
(23)) but has a rapid time-dependence such that the following
condition is satisfied:
d A
hr j HC ðtÞjsiA ¥xA xA 8 xA – xA
r
s
r
s
dt
ð30Þ
In typical NMR experiments, HC ðt Þ includes the interaction of the
spin system with a resonant radiofrequency field. The HC ðt Þ term
may be further divided into a term Hres
C ðt Þ with frequency comres
ðt Þ
ponents close to resonance and an off-resonance term Hoff
C
with frequency components far from resonance. For example,
in the case of an amplitude-modulated radiofrequency field,
the near-resonant term which rotates in the same sense as the
Larmor frequency is included in Hres
C , while the counterres
.
rotating term is included in Hoff
C
Since Eq. (26) is assumed to be satisfied, the ‘‘non-secular” term
H?
B may either be ignored, or treated approximately by a small correction to the eigenvalues of HkB (second-order quadrupolar shifts
res
are a typical example [67]). The off-resonant term Hoff
ðtÞ also
C
gives rise to small shifts which may be taken into account through
a small correction to HB . For simplicity, we now assume that any
non-secular corrections induced by H?
B , and any off-resonance
res
(for example, Bloch-Siegert shifts [68]),
shifts induced by Hoff
C
are incorporated into a corrected version of HB , denoted H0B , which
commutes with HA :
HA ; H0B ¼ 0
Fig. 3. Schematic eigenvalue spectra of the coherent Hamiltonian terms HA and HB .
Degenerate eigenvalues are indicated by square brackets. The eigenvalue spectrum
of HA consists of degenerate or widely-spaced eigenvalues, such that the eigenvalue
spacing is large compared to kHB k, and also kHC k and kHfluc k (not shown).
Degeneracy is broken when H0 is constructed by combining HA and HB , after
correction for non-secular and off-resonant contributions (Eq. (32)).
ð31Þ
The main part of the coherent Hamiltonian, denoted here H0 , is
now constructed by combining the HA and H0B terms:
H0 ¼ HA þ H0B
The operators HA and H0 commute:
ð32Þ
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C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
½H A ; H 0 ¼ 0
ð33Þ
Hence it is possible to define a set of states that form a simultaneous eigenbasis of both operators. The states which form this simultaneous eigenbasis are denoted fj1i . . . jN H ig, where N H is the
dimension of Hilbert space. The corresponding eigenequations are
as follows:
H0 jr i ¼ xr jr i
ð34Þ
HA jr i ¼ xAr jr i
where hrjsi ¼ drs .
Note that although the eigenvalues xAr of HA are the same as in
Eq. (24), the set of eigenstates jr i may not be identical to the set of
jAa Þ ¼
NH X
NH
X
jX rs ÞðX rs j Aa Þ d xArs xAa
where dðxÞ ¼ 1 for x ¼ 0 and 0 otherwise.
It is desirable to choose eigenoperators fjA1 Þ . . .g which have
convenient properties under three-dimensional rotations of the
nuclear spin angular momenta. Whether this is possible depends
on the commutation properties of the dominant Hamiltonian HA .
In high-field NMR, the dominant Hamiltonian HA is usually
identified with the Zeeman Hamiltonian, ignoring any chemical
shifts or other perturbations:
high field NMR :
A
eigenstates jri , since eigenstates belonging to degenerate subspaces of HA may be mixed by the HB term.
The term H0 determines the coherent dynamics of the spin system. In typical NMR experiments in high field, H0 contains the Zeeman interaction of the nuclei with the static magnetic field, as well
as the secular parts of the chemical shift and spin-spin coupling
terms.
The H0 eigenstates may be used to define a set of transition
operators (or shift operators) X rs , as follows:
X rs ¼ jrihsj
ð35Þ
where the states jri satisfy Eq. (34). The operator X rs converts the
state jsi into the state jr i. There are N L ¼ N 2H transition operators
since the indices r and s both run from 1 to NH .
The set of all eigenstate transition operators {X rs } forms an
orthonormal operator basis of Liouville space, being orthonormal
in both indices:
ðX rs j X kl Þ ¼ drk dsl
ð36Þ
The eigenstate transition operators X rs are eigenoperators of the
commutation superoperator of H0 :
b 0 jX rs Þ ¼ xrs jX rs Þ
H
ð37Þ
ð38Þ
The eigenstate transition operators X rs are also eigenoperators
of the commutation superoperator of HA , but with different
eigenvalues:
b A jX rs Þ ¼ xA jX rs Þ
H
rs
ð39Þ
where
xArs ¼ xAr xAs
ð40Þ
In the case that HA is chosen to be the Zeeman Hamiltonian for the
spin system, the eigenvalues xArs are given by linear combinations of
the nuclear Larmor frequencies of the relevant nuclides.
The transition operators X rs have inconvenient rotational properties which derive from the complexity and low rotational symmetry of the operator HB . It proves convenient to define a new
b A , with operators jAa Þ satisfying the
eigenoperator basis for H
eigenequation:
b A jAa Þ ¼ xA jAa Þ
H
a
ð41Þ
where the set of eigenoperators fjA1 Þ . . .g is orthonormal:
ðAa j Aa0 Þ ¼ daa0
ð42Þ
The eigenvalues are denoted xa with a 2 f0; 1 . . . N L g. Each
eigenvalue xAa corresponds to one of the frequencies xArs in Eq.
(40), and each eigenoperator may be expressed as a linear superpob A:
sition of transition operators with the same eigenvalue of H
A
HA ¼ H Z
HZ ¼
ð44Þ
N
X
x0i Iiz
ð45Þ
i¼1
In cases involving spin isomerism, the nuclear exchange interaction
may also be included in HA (see Section 6.4).
The commutation properties of the Zeeman Hamiltonian are
different for homonuclear and heteronuclear systems:
In homonuclear systems (all nuclides of the same type), all Larmor frequency terms x0i are identical, and hence HA is proportional to the total angular momentum operator along the z-axis,
P
Iz ¼ Ni¼1 Iiz . In this case HA commutes with the total spin angular momentum along the field, and also the total square angular
momentum of all nuclei:
h
i
½H A ; I z ¼ H A ; I 2 ¼ 0
ð46Þ
PN
where I2 ¼ I2x þ I2y þ I2z and Im ¼ i¼1 Iim ; m 2 fx; y; zg. Hence, each
eigenoperator jAa Þ may be identified with an irreducible spheri cal tensor operator (ISTO) T akl :
jAa Þ ¼ ca T akl
ð47Þ
obeying the following standard eigenequations:
where the transition frequencies are given by
xrs ¼ xr xs
ð43Þ
r¼1 s¼1
b
I2 T akl ¼ ka ðka þ 1ÞT akl
bI z T a ¼ l T a
kl
a kl
2 2 2
b
I2 ¼ bI x þ bI y þ bI z
ð48Þ
Here bI x is the commutation superoperator of the total angular
P
momentum along the x-axis, Ix ¼ Ni¼1 Iix , and similarly for the
2
x and y-operators [69]. Note that bI x is not the same superopPN b2
erator as
i¼1 I ix . The rotational rank and azimuthal quantum
number of the eigenoperator jAa Þ are denoted ka and la respectively. The normalization factor ca in Eq. (47) ensures that
ðAa jAa Þ ¼ 1.
In heteronuclear systems, the Zeeman Hamiltonian does not
commute with the total square angular momentum operator
of all spins but with the total square angular momentum operators of individual groups of spins, organized by their isotopic
types fI; S; . . .g. If the numbers of spins of the different isotopic
types are denoted fN I ; N S . . .g, the relevant z- angular momenP I
P S
tum operators are denoted Iz ¼ Ni¼1
Iiz ; Sz ¼ Ns¼1
Ssz , etc. and
the total-square angular momentum operators by
I2 ¼
NI X
I2ix þ I2iy þ I2iz
i¼1
NS X
S2sx þ S2sy þ S2sz
S ¼
2
s¼1
ð49Þ
7
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
h
2
i
h
½ H A ; I z ¼ H A ; I ¼ ½ H A ; Sz ¼ H A ; S
h
i
½Iz ; Sz ¼ I2 ; S2 ¼ 0 . . .
2
i
¼0
...
ð50Þ
b A eigenoperators is given by the tensor
An appropriate set of H
product of the individual spherical tensor operator sets:
n
o n
o
fAa g ¼ T Ik;l T Sk;l . . .
ð51Þ
2.3.2. Fluctuating Hamiltonian
The fluctuating Hamiltonian Hfluc , which is responsible for spin
relaxation, typically contains contributions from several different
nuclear spin interactions:
Hfluc ðt Þ ¼
þ
2
HKfluc
ðt Þ
þ ...
ð53Þ
The interactions fK1 ; K2 . . .g typically include nuclear dipole–dipole
interactions, chemical shift anisotropies, etc. The random fluctuations of these interactions, due to molecular motion, are responsible
for nuclear spin relaxation. In general, the fluctuations of different
interactions are correlated with each other.
Each fluctuating Hamiltonian term may be expressed as a
b A eigenoperators, as follows:
superposition of H
X K
K
F a ðt ÞjAa Þ
Hfluc ðt Þ ¼
F a ðtÞ ¼ Aa jHKfluc ðt Þ
K
K
GKK
aa0 ðsÞ ¼ F a ðt þ sÞF a0 ðt Þ
An alternative approach is to expand the fluctuating Hamiltonian in
transition operators as follows:
The correlation functions of the fluctuating matrix elements are
defined as follows:
0
Gaa0 ðsÞ ¼ Gaa0 ð0Þg aa0 ðsÞ
ð56Þ
ð57Þ
0
where g KK
aa0 ð0Þ ¼ 1. The correlation functions are assumed to decay
monotonically with increasing s and become vanishingly small for
s larger than an interval called the correlation time denoted sc . In
general the correlation time may be different for different mechanisms, but we will overlook this complication here, for the sake of
simplicity.
In the case of rotational diffusion, the correlation functions may
be shown to decay approximately exponentially [70], with the following form
KK0
g aa0 ðsÞ ’ exp js=sc j
ð58Þ
For simplicity we assume that the correlation time is independent
of the interaction and eigenoperator indices.
The spectral density functions are defined as the Fourier transform of the correlation functions:
0
J KK
aa0 ðxÞ ¼
KK0
jaa0 ðxÞ ¼
such that
R1
0
GKK ðsÞ expfixsg ds
1 aa0
R 1 KK0
g ð
1 aa0
sÞ expfixsg ds
ð63Þ
which are again assumed to be independent of t through the
assumption of stationary random processes. The spectral density
functions of the transition matrix elements are defined as follows:
J ijkl ðxÞ ¼
X KK0
J ijkl ðxÞ
ð64Þ
K;K0
from which the contributions of the individual correlation functions
are given by:
0
J KK
ijkl ðxÞ ¼
Z
1
1
0
GKK
ijkl ðsÞ expfixsg ds
ð65Þ
Their real parts are again proportional to absorption-mode Lorentzians, as in Eq. (61).
ð59Þ
ð66Þ
The time evolution of the density operator in Liouville space is governed by the Liouville-von-Neumann (LvN) equation, and takes the
form:
0
KK0
0
K
K
GKK
ijkl ðsÞ ¼ hijHfluc ðt þ sÞjjihkjH fluc ðt Þjli
jrÞ ¼ j jwihwjÞ
where the overbar indicates an ensemble average. The correlation
KK0
ð62Þ
r;s
ð55Þ
functions GKK
aa0 ðsÞ are assumed to be independent of t (stationary
assumption), and have the form
KK0
ð61Þ
2.4.1. Liouville-von Neumann equation
The density operator of a single ensemble member is defined as
follows:
and have the following correlation functions:
0
2sc
1 þ x2 s2c
2.4. Equation of motion
where the coefficients are defined as follows:
0
n 0
o
KK
Re jaa0 ðxÞ ¼
ð54Þ
a
K
ð60Þ
For an exponential correlation function (Eq. (58)), the real part of
the spectral density has the simple form of an absorption-mode
Lorentzian:
ð52Þ
with a set of rotational ranks kIa ; kSa . . . and azimuthal quantum
I S numbers la ; la . . . for the spin species fI; S . . .g.
1
HKfluc
ðt Þ
0
X
K
hr jHKfluc ðtÞjsijX rs Þ
Hfluc ðtÞ ¼
Each eigenoperator has the form
jAa Þ ¼ ca T I;k;al T S;k;al...
KK0
0
KK
J KK
aa0 ðxÞ ¼ jaa0 ðxÞGaa0 ð0Þ
The following commutation relationships apply:
d
j
dt
rðtÞÞ ¼ i Hb ðtÞjrðtÞÞ
b ðt Þ ¼ H
b0 þ H
b res þ H
b fluc ðt Þ
H
C
ð67Þ
The solution to the LvN may be written as
b ðt; t a Þjrðt a ÞÞ
jrðtÞÞ ¼ U
ð68Þ
where t P t a and the propagation superoperator is given by a timeordered product of exponential superoperators:
b ðt; t a Þ ¼ Tb expfg i
U
Z
t
b ðt0 Þdt0
H
ð69Þ
ta
2.4.2. Ensemble density operator
The ensemble-averaged density operator is defined
jqÞ ¼ jrÞ ¼ j jwihw jÞ
ð70Þ
where the overbar denotes an average over ensemble members.
It is assumed that the evolution of the system may be approximated by an equation of the type:
b ðt; t a Þjqðt a ÞÞ
jqðtÞÞ ¼ V
ð71Þ
b is in general not unitary.
where the propagation superoperator V
The primary aim of relaxation theory is the systematic construction
8
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
b valid within appropriate limits. Methods which have been used
of V
include semi-classical (SC) relaxation theory [1,3,4], the generalized
cumulant expansion method [71–73] or the Fokker-Plank formalism frequently encountered in EPR. [74,75] A conceptually different
approach is the Quantum Monte Carlo approach, which is widely
used in the field of quantum optics. [76–78]
b
d qðt Þ ¼ L SC ðt ÞqðtÞ
dt
where the semi-classical Liouvillian superoperator, in the interaction frame, is given by
b
b
b y ðt Þ C
b lab U
b A ðt Þ
L SC ¼ i H coh þ U
SC
A
The semi-classical relaxation theory of nuclear spin systems is
widely covered in the standard literature [1,2,4,6–11]. To underpin
the later discussion we give a concise overview of the basic concepts and key approximations. We also include a brief discussion
of the handling of the relaxation superoperator in the interaction
frame, which is the source of occasional confusion.
b lab ¼ C
SC
In order to prepare for the application of second-order perturbation theory, the spin Hamiltonian is expressed in the interaction
frame of the dominant part of the coherent Hamiltonian HA :
ð72Þ
In the case that HA is identified with the Zeeman Hamiltonian, as is
common in high-field NMR, the interaction frame is also known as
the ‘‘rotating frame”.
The LvN equation, in the interaction frame, is given by
b
b lab ¼ C
SC
ð73Þ
Hðt Þ ¼ Hcoh þ Hfluc ðt Þ
with the interaction-frame fluctuating Hamiltonian:
n
o
b A t Hfluc ðtÞ
Hfluc ðt Þ ¼ exp þi H
ð74Þ
and the interaction-frame coherent Hamiltonian:
Hcoh ðt Þ ¼ H0 HA þ Hres
C ðt Þ
ð75Þ
In the case of single-frequency near-resonant irradiation, HA may be
Hres
C
is piecewise time-independent, the timechosen so that
independent ‘‘pieces” corresponding to the elements of the applied
pulse sequence.
The solution to the LvN equation for the time evolution of individual ensemble members may be approximated by timedependent perturbation theory [4,6–11,62,63]. The following standard approximations and assumptions are made:
The density operator and the fluctuating contributions are
uncorrelated.
The ensemble average of the fluctuating contributions vanishes.
The fluctuating contributions represent a weakly stationary
process, so that two-time correlations only depend on their
time difference t0 t ¼ s.
There exists a correlation time sC such that the ensemble average becomes negligible for time differences s sC .
The terms HB ; HC and Hfluc are all sufficiently small that
kHB sC k 1; kHC sC k 1 and kHfluc sC k 1, so that the interaction frame density operator does not change significantly over
the timescale of sC .
Note that there is no assumption at this stage, of the spin system being close to equilibrium.
These assumptions lead to the following equation of motion for
the ensemble-average density operator in the interaction frame:
0
1
b
b
H fluc ð0Þ H fluc ðsÞds
ð78Þ
1 XX KK0 A b b y
J 0 xa0 A a A a0
2 K;K0 a;a0 aa
ð79Þ
Eq. (79) is a convenient expression for the semi-classical relaxation superoperator. This form is basis-independent and gives no
special status to the eigenstates of the coherent Hamiltonian. It
has been used for many purposes in solution NMR, and proves particularly useful for the analysis of long-lived states [34,35,37–45,
79–84].
An alternative expression, making use of the transition operators between the eigenstates of the Hamiltonian H0 , is as follows:
b lab ¼ C
SC
rðtÞ ¼ i H ðtÞrðtÞ
Z
If small dynamic frequency shifts are neglected [1,3,4,62,63], this
may be written as follows:
3.1. Semi-classical relaxation superoperator
d dt ð77Þ
and the relaxation superoperator in the laboratory frame is as follows [1,3]:
3. Semi-classical relaxation theory
n
o
b A t H ðt Þ
Hðt Þ ¼ exp þi H
ð76Þ
1X
b ij X
by
J ðxkl Þ X
kl
2 i;j;k;l ijkl
ð80Þ
where the spectral densities are given in Eq. (64), and the differences xAkl between eigenvalues of the dominant Hamiltonian HA
are defined in Eq. (40).
Most spin dynamical calculations are performed in the interaction frame of the dominant Hamiltonian HA , since this greatly simplifies the treatment of resonant radiofrequency fields. However,
the interaction-frame Liouvillian given in Eq. (77) is difficult to
b A ðt Þ. This problem is
use, because of the time-dependent terms U
avoided by making another approximation. The laboratory frame
b A:
relaxation superoperator is expanded in eigenoperators of H
b lab ¼
C
SC
X
a;a0
b lab jAa0 Þ
jAa ÞðAa0 j ðAa j C
SC
ð81Þ
This allows the second term in Eq. (77) to be written as follows:
b lab U
b y ðt Þ C
b A ðt Þ ¼
U
SC
A
X
b lab jAa0 Þ exp iðaa aa0 Þt
jAa ÞðAa0 j ðAa j C
SC
ð82Þ
a;a0
The rapidly oscillating terms in this equation may be ignored
under the assumption that the relaxation rate constants are small
compared to the eigenvalue separation of the dominant Hamiltonian HA . Note that this approximation is readily defensible for
the eigenvalues of HA , since these are defined to be either degenerate or widely spaced, but would not be fully justifiable if the
Hamiltonian H0 had been used to construct the interaction frame,
since H0 may have near-degenerate eigenvalues. The omission of
rapidly oscillating components in Eq. (82) allows use of the following secularized relaxation superoperator in the interaction frame:
b
sec
C SC
’
X
b lab jAa0 Þdðaa aa0 Þ
jAa ÞðAa0 j ðAa j C
SC
ð83Þ
a;a0
where dðxÞ ¼ 1 if x ¼ 0 and dðxÞ ¼ 0 otherwise. The Liouvillian
superoperator in the interaction frame of HA is therefore given, to
a very good approximation, by
b
b sec
b
L SC ’ i H coh þ C SC
ð84Þ
9
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
b sec
where C SC
is time-independent. Spin dynamical calculations may
therefore be conducted in the interaction frame of the dominant
Hamiltonian HA , using the interaction-frame coherent Hamiltonian
Hcoh and the secularized semi-classical relaxation superoperator
b sec
C SC
of Eq. (83).
In the solution NMR of homonuclear spin systems in high magnetic field, HA may be chosen to equal the dominant Zeeman interaction. In this case, selection rules on the rotational correlation
functions [85] render the secularization procedure in Eq. (83)
b lab jAa0 Þ vanish in any
unnecessary, since the matrix elements ðAa j C
SC
case for aa – aa0 . However, this is not always true for heteronuclear
spin systems. An example of relaxation superoperator secularization in heteronuclear systems may be found in reference [35].
For simplicity, the secularized relaxation superoperator in the
b:
interaction frame is now simply denoted as C
b
b ¼ C sec
C
SC
ð85Þ
b sec
From the definition of C SC
, this superoperator is identical when
expressed in the laboratory frame:
b lab
C
¼
by
bA bU
U
A
C
b
¼C
ð86Þ
3.2. Transition probabilities
The transition probability per unit time from an eigenstate jr i of
H0 to a different eigenstate jsi may be derived from the secularized
relaxation superoperator as follows [6]:
W r!s
b j X rr ¼ J ðxsr Þ
¼ X ss j C
srsr
ð87Þ
The semi-classical approach leads to identical rate coefficients for
forward and backward transitions, as sketched in Fig. 4(a):
W r!s ¼ W s!r
ð88Þ
As is well-known, symmetrical transition probabilities are incompatible with the establishment of thermal equilibrium at a finite
temperature [1–14].
Fig. 4. A section of the energy level structure of an arbitrary quantum system.
Relaxation phenomena induce transitions between the states jri and jsi. Transitions
that cause the system to gain energy are indicated in green, transitions that cause
the system to lose energy are indicated in red. (a) Semi-classical transition
probabilities. The ‘‘upwards” and ‘‘downwards” transition probabilities W r!s and
W s!r are equal. (b) After thermalization, the ‘‘downwards” transition probability
W hs!r (thick arrow) is larger than the ‘‘upwards” transition probability W hr!s (thin
arrow). (For interpretation of the references to colour in this figure legend, the
reader is referred to the web version of this article.)
3.2.1. Coherence decay rate constants
The decay rate constant for a coherence between the eigenstates jr i and jsi of the coherent Hamiltonian H0 may be expressed
as:
b j X rs
krs ¼ X rs j C
ð89Þ
which may be written as the sum of an ‘‘adiabatic” and ‘‘nonadiabatic” contributions [4,6–10]:
na
krs ¼ kad
rs þ krs
ð90Þ
with
hr jHfluc ðsÞjr i hsjHfluc ðsÞjsi ds
X KK0
0
KK0
KK0
¼ 12
Jrrrr ð0Þ J KK
rrss ð0Þ J ssrr ð0Þ þ J ssss ð0Þ
kad
rs ¼
R0 1
hr jHfluc ð0Þjr i hsjHfluc ð0Þjsi
K;K0
ð91Þ
and
kna
rs
X
1 X
¼
W r!k þ
W s!k
2 k–r
k–s
!
ð92Þ
The adiabatic contributions are due to random fluctuations of the
energy levels. The non-adiabatic contributions arise from the finite
lifetimes of the spin states, due to transitions to other states. As an
illustrative example, the non-adiabatic contribution for an arbitrary
three-level system is illustrated in Fig. 5(a). The coherence between
states j2i and j3i is indicated by a wavy line. The non-adiabatic contribution to the decay of this coherence is due to all transitions out
of the states j2i and j3i.
3.3. Semi-classical equation of motion
The master equation for the dynamics of the spin density operator, under semiclassical relaxation theory, is therefore given by
d
b SC jqÞ
jqÞ ¼ L
dt
ð93Þ
where the Liouvillian is
n
o
b0 þ H
b res ðt Þ þ C
b SC
b SC ’ i H
L
C
ð94Þ
and the secularized semi-classical relaxation superoperator is given
by Eq. (83).
Since the semi-classical relaxation superoperator predicts equal
probabilities for transitions gaining and losing energy (Eq. (88)),
Fig. 5. Energy levels of a three-level quantum system. A coherence between the
states j2i and j3i is indicated by a blue wavy line. The non-adiabatic contribution to
the coherence decay process is given by the summed transition probability for all
transitions out of the states j2i and j3i as indicated by the arrows. (a) In the semiclassical treatment, ‘‘upwards” and ‘‘downwards” transition probabilities are equal.
(b) At finite temperature the non-adiabatic contributions to the coherence decay
rate must be adjusted to reflect the different transition probabilities in the two
directions. (For interpretation of the references to colour in this figure legend, the
reader is referred to the web version of this article.)
10
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
the true thermal equilibrium state of the spin density operator is
not correctly predicted by Eqs. (93) and (94). This problem is often
addressed by arbitrarily introducing a thermal equilibrium term in
the relaxation part, leading to the widely used inhomogeneous
master equation (IME) of the form [1–14]:
d
b SC jqÞ q
b0 þ H
b res ðt Þ jqÞ þ C
jqÞ ¼ i H
eq
C
dt
ð95Þ
However, as described in the introduction, Eq. (95) gives incorrect
results in general for spin systems which are far from equilibrium.
In this article we present a deeper solution.
4.1. Thermal equilibrium
At thermal equilibrium the spin-state populations adopt the
Boltzmann distribution, leading to the following expression for
the thermal equilibrium density operator:
ð96Þ
where the temperature parameter is defined as follows:
bh ¼
h
kB T
ð97Þ
An initially perturbed system eventually returns to thermal equilibrium for sufficiently long times in the absence of time-dependent
perturbations.
The semi-classical relaxation superoperator of Eq. (79) fails to
predict the correct thermal equilibrium state, since the transition
probabilities are symmetric, as in Eq. (88). Instead the state of comb SC and lies
plete disorder represents the stationary distribution of C
in its null-space:
b SC j1Þ ¼ j0Þ
C
ð98Þ
b SC therefore drives the
The semi-classical relaxation superoperator C
system to the unphysical state of infinite temperature.
The correct thermal equilibrium state may be forced by adjusting the semi-classical relaxation superoperator using a procedure
known as ‘‘thermalization”. Most techniques [50–54] achieve this
by multiplying the SC-relaxation superoperator by a thermal corb:
rection superoperator H
ð99Þ
b forces q
b h:
where H
into the null-space of C
eq
b h j q Þ ¼j 0Þ
C
eq
1j
d
b h j qðtÞ ¼ 0
qðtÞ ¼ 1 j C
dt
ð102Þ
preserves hermiticity: The density matrix must remain hermitian at all points in time to be in agreement with the statistical
interpretation of quantum mechanics.
d
d
q ðtÞ ¼ qsr ðtÞ
dt rs
dt
ð103Þ
bh
qsseq C
Wh
¼ ssrr ¼ r!s
¼ expfbh xrs g
rr
bh
qeq C
W hs!r
rrss
ð104Þ
The detailed balance condition indicates that transitions from
higher energy states to lower energy states are slightly more
likely than vice versa. It follows that qeq is a stationary distrib h [86].
bution and lies in the null-space of C
predict consistent coherence decay rates: The thermalized
coherence decay rate constants are composed of adiabatic and
non-adiabatic contributions:
h;na
khrs ¼ kh;ad
rs þ krs
ð105Þ
The adiabatic contributions to the coherence decay rate constants should be independent of the thermal correction:
kh;ad
¼ kad
rs
rs
ð106Þ
The non-adiabatic contributions, on the other hand, depend on
the transition probabilities and should be adjusted according
to the detailed balance condition, as follows:
kh;na
rs
X h
1 X h
¼
W
þ
W
2 k–r r!k k–s s!k
!
ð107Þ
4.2.1. Jeener’s method
Jeener’s seminal article on superoperators in magnetic resonance [50] contains a proposed thermalization method for relaxb
ation superoperators. This is based on the energy superoperator E
defined as follows:
n
b J ¼ exp b E
b
H
h
o
b0 þ H
b0
b ¼1 H
E
2
ð100Þ
The thermalized equation of motion for the density operator is
given by
d
b h jqÞ
b coh þ C
jqÞ ¼ i H
dt
A variety of methods have been proposed for the thermalization of
the relaxation superoperator. Their properties and limitations are
summarized in Table 1.
4.2. Thermal corrections
b
bh ¼ C
b SC H
C
d
ð1 j qðt ÞÞ ¼
dt
obeys detailed balance: At thermal equilibrium the detailed
balance condition applies [63]. The transition probabilities of
b h should therefore fulfill the following condition:
C
4. Thermalization
jexp bh H0 Þ
qeq ¼
Tr½exp bh H0 preserves the trace: This condition captures the fact that the
sum of all populations should be a constant of motion. Populations are free to be redistributed among each other, but there is
no ‘‘loss” of populations.
ð101Þ
b coh is the commutation superoperator of the coherent
where H
Hamiltonian Hcoh .
The thermalized relaxation superoperator should have the following properties:
ð108Þ
The thermally corrected relaxation superoperator is given by:
n
bJ ¼ C
bh ¼ C
b SC H
b SC exp b E
b
C
h
J
o
ð109Þ
The method works by transforming the thermal equilibrium density
operator j qeq Þ into j1Þ. This may be seen as follows: The action of
the energy superoperator on a population operator of the coherent
Hamiltonian is given by:
b jX kk Þ ¼ xk jX kk Þ
E
ð110Þ
11
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
Table 1
The methods for treating nuclear spin relaxation discussed in this paper, and their properties in the context of a finite-temperature molecular environment. The following
abbreviations are used: SC = semi-classical relaxation theory; IME = inhomogeneous Master equation; Jeener = Jeener’s thermalized relaxation superoperator [50]; LdB = Levitt-di
Bari method for thermalizing the relaxation superoperator [51–53]; sLdB = simplified Levitt-di Bari method as implemented in SpinDynamica 3.2 software [87]; Lb = Lindblad
method.
Correct thermal equilibrium
Valid outside high-temperature regime
Valid outside weak-order regime
Transition probabilities fulfil detailed balance
Coherence decay rates: correct adiabatic contributions
Coherence decay rates: correct non-adiabatic contributions
Equation number
References
SC
IME
Jeener
LdB
sLdB
Lb
U
(83)
[1,3]
U
U
(1), (95)
[1,3,4]
U
U
U
U
(108), (109)
[50]
U
U
U
U
U
(118), (119)
[51,52]
U
U
(125)
[51–53]
U
U
U
U
U
U
(140)
This article
The thermal equilibrium density operator may be expressed in
terms of the population operators:
j qeq Þ ¼ Z 1
Z¼
X
X
exp bh xk jX kk Þ
k
ð111Þ
exp bh xk
k
According to Eqs. (110) and (111), the result of applying the total
energy superoperator to the thermal equilibrium density operator
is given by:
b j q Þ ¼ Z 1
E
eq
X
xk expfbh xk gjX kk Þ
k
b J j q Þ ¼ Z 1
H
eq
X
expfbh xk g
k
X ðb
h xk Þ
n!
n
jX kk Þ ¼ Z 1 j1Þ
ð112Þ
n
Since the state of total disorder lies within the null-space of the
b h:
SC-relaxation superoperator, qeq lies in the null-space of C
J
b h j q Þ ¼ Z 1 C
b j1Þ ¼ j0Þ
C
eq
J
coherence decay rates, which does not make physical sense, since
the adiabatic contributions are not related to state transitions and
are energy-preserving.
The properties of the Jeener method are summarized in Table 1.
The Jeener method fulfils most of the required conditions of a thermalization method, but does not treat the coherence decay rate
constants correctly. To the best of our knowledge, no application
of the Jeener thermalization method has been reported in the
literature.
4.2.2. Levitt-di Bari and Levante-Ernst method
An alternative thermalization technique was proposed by Levitt
and di Bari [51,52] and further developed by Levante and Ernst [53].
This method is termed here the LdB method and uses projection
superoperators onto the population operators of the coherent Hamiltonian, weighted by the eigenvalue of the associated eigenstate:
b ¼
x
X
xk jX kk ÞðX kk j
ð118Þ
k
ð113Þ
The thermal correction superoperator is given by:
The transition probabilities per unit time are modified in the following way:
b LdB ¼ exp b x
b
H
h
W h;J
r!s
b onto the thermal equilibrium density operator is
The action of x
b (compare with Eq. (112)).
identical to the action of E
b h j X rr ¼ X ss j C
b exp b E
b j X rr
¼ X ss j C
h
J
¼ exp bh xr W r!s
ð114Þ
and the ratio between forward and backwards transition probabilities is given by:
W h;J
r!s
W h;J
s!r
¼ exp bh xrs
ð115Þ
showing that the thermalized transition probabilities obey the
detailed balance condition (Eq. (104)).
However, Jeener’s method does not handle the coherence decay
rate constants correctly. The thermalized decay rate constants of a
particular coherence are given by:
kh;J
rs
kh;J
sr
b h j X rs ¼ exp b ðxr þ xs Þkrs
¼ X rs j C
h
J
h
b
¼ X sr j C J j X sr ¼ exp bh ðxr þ xs Þksr
ð116Þ
h;J
The equality kh;J
rs ¼ ksr follows from krs ¼ ksr and maintains the hermiticity of the density operator. However, there is still a problem.
The adiabatic and non-adiabatic contributions to the coherence
decay rate constants are given in Jeener’s method by:
kh;ad;J
¼ exp 12 bh ðxr þ xs Þ kad
rs
rs
1
na
kh;na;J
¼
exp
b
ð
x
þ
x
Þ
k
r
s
rs
rs
2 h
ð117Þ
where the adiabatic and non-adiabatic rate constants are given by
Eqs. (91) and (92) respectively. The terms kh;ad;J
and kh;na;J
are both
rs
rs
in conflict with the requirements of Eqs. (106) and (107). In particular, Jeener’s method adjusts the adiabatic contributions to the
b j qeq Þ ¼ Z 1
x
ð119Þ
X
xk exp bh xk jX kk Þ
ð120Þ
k
b LdB transforms the thermal equilibrium density operaAs a result H
tor into the state of total disorder:
b LdB j q Þ ¼ Z 1 j1Þ
H
eq
ð121Þ
bh .
and qeq lies in the null-space of C
LdB
b h j q Þ ¼ j0Þ
C
eq
LdB
ð122Þ
The modification of the transition probabilities is identical to
Jeener’s method:
bh
b
b
W h;LdB
r!s ¼ X ss j C LdB j X rr ¼ X ss j C SC exp bh x j X rr
ð123Þ
¼ exp bh xr W r!s
b h obey the detailed balance
Hence the transition probabilities of C
LdB
condition.
A defect of the LdB method is that the coherence decay rates are
not adjusted at all, since the coherence operators lie in the nullb.
space of x
b j X rs Þ ¼ j0Þ ) exp bh x
b jX rs Þ ¼ jX rs Þ
x
kh;LdB
rs
b SC exp b x
b SC j X rs ¼ krs
b j X rs ¼ X rs j C
¼ X rs j C
h
ð124Þ
12
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
Hence the Levitt-di Bari method obeys the condition of Eq. (106) for
the adiabatic decay rate contribution, but not that of (107) for the
non-adiabatic contribution.
A simplified version of the LdB method (sLdB), which employs
the high-temperature and weak-order approximations, is currently
implemented in version 3.3.2 of the SpinDynamica software package [87]. This method uses the following thermalization
correction:
b sLdB ¼ ^1 j q Þð1 j
H
eq
ð125Þ
The thermally adjusted transition probabilities are given by:
W h;sLdB
¼ W r!s r!s
X
W j!s hjjqeq jji
ð126Þ
j
ð127Þ
5. Lindblad thermalization
In this article we propose a different thermalization method,
called the Lindblad (Lb) method, based on a quantum-mechanical
treatment of the molecular environments and incorporating thermal corrections to the spectral densities to account for the finite
sample temperature [1,2,59,58,60]. Unlike the Jeener, LdB and sLdB
methods, the Lb method does not lead to a relaxation superoperator which may be expressed as the product of the semi-classical
relaxation superoperator and a thermal correction superoperator,
as in Eq. (99). Instead one performs thermal corrections to the individual components of the relaxation superoperator.
A Lindbladian formulation of relaxation processes has been
extensively used in quantum optics, quantum computing, quantum information theory and laser physics [76,78]. Lindbladian
description of relaxation of nuclear spin relaxation is used rarely
[64,65], presumably because the inhomogeneous master equation
is sufficient in most cases.
5.1. Quantum-mechanical spectral densities
The starting point of the Lindblad method is to treat both the
environment and the spin system quantum mechanically
[2,1,5,60–63]. The fluctuating contributions are replaced by a
Hamiltonian of the following type:
X
Aa BKa
ð128Þ
a
K;K0 a;a0
Aa Aya0 Aya0 Aa
D
E
0
BKa0 y ðsÞBKa ð0Þ exp ixAa0 s
K;K0 a;a0
ð130Þ
D 0
E
Here BKa0 y ðsÞBKa ð0Þ are quantum–mechanical correlation functions,
defined as follows:
D 0
E
n
o
0
BKa0 y ðsÞBKa ð0Þ ¼ Trbath U bath ðsÞBKa0 y U ybath ðsÞBKa qbath
eq
D 0
E
¼ BKa0 y ð0ÞBKa ðsÞ
b lab ¼ C
QM
where Ba is an environmental (‘‘bath”) operator.
The environment represents a thermal reservoir with sufficient
heat capacity to always remain close to thermal equilibrium due to
its size and complexity, and is described by a density operator
qbath
eq . It is not necessary to assume that the spin system remains
close to equilibrium. The combined density operator of the spin
system and the environment is assumed to be of the form:
ð129Þ
The relaxation superoperator may be derived by substituting the
expression for Hfluc from Eq. (128) into Eq. (78), and taking the trace
over the environmental degrees of freedom [4,5,62,63].
XX Z
0
1
K;K0 a;a0
ð131Þ
D 0
E
Aa Aya0 Aya0 Aa BKa0 y ðsÞBKa ð0Þ
XX
exp ixAa0 s D
Z
0
1
K;K0 a;a0
E
Aa0 Aya Aya Aa0 Ba ð0ÞBa0 ðsÞ exp ixAa0 s
K0
Ky
ð132Þ
The sum is invariant under the exchange of the labels (a $ a0 ) and
(K $ K0 ) which we perform on the second part.
b lab ¼ C
QM
XX R 0 1
K;K0 a;a0
Aa Aya0 Aya0 Aa
D
E
0
BKa0 y ðsÞBKa ð0Þ exp ixAa0 s
D 0
E
XX R 0 Aa Aya0 Aya0 Aa BKa0 y ðsÞBKa ð0Þ exp ixAa s
1
K;K0 a;a0
ð133Þ
A simple transformation of variables (s # s) in the second sum
leads to the following expression:
b lab ¼ C
QM
XX Z
K;K0 a;a0
0
1
E
D 0
Aa Aya0 Aya0 Aa BKa0 y ðsÞBKa ð0Þ
XX
exp ixAa0 s Z
K;K0 a;a0
D
E
1
0
Aa Aya0 Aya0 Aa BKa0 y ðsÞBKa ð0Þ exp ixAa s
0
ð134Þ
By considering only the secular contributions xa ¼ xa0 and
neglecting dynamical frequency shifts, the two sums may be combined to form a complete Fourier transform (see Appendix A.1):
b sec ¼
C
QM
i
XX h
0
b Aa ; Ay 0 K KK0 xA0 d xA xA0
D
a
aa
a
a
a
ð135Þ
K;K0 a;a0
where the quantum-mechanical spectral densities are given by
0
K
qtot ðtÞ ¼ qðtÞ qbath
eq
1
D
E
XX R 0 y
0
Aa0 Aa Aa Aya0 BKa ð0ÞBKa0 y ðsÞ exp ixAa0 s
1
The sLdB method generates the correct thermal equilibrium density
operator in the high-temperature approximation but does not obey
detailed balance in general, and fails to handle coherence decay rate
constants correctly.
HKfluc ¼
XX R 0 Since the system-bath interaction is hermitian it is permissible
to replace the second part of Eq. (130) by its hermitian conjugate:
whereas the coherence decay rates remain unchanged:
kh;sLdB
¼ krs
rs
b lab ¼ C
QM
K KK
aa0 ðxÞ ¼
Z
1D
1
E
0
BKa0 y ðsÞBKa ð0Þ expfixsgds
ð136Þ
b defines the so-called Lindbladian dissipator [62,63]:
and D
b ½ A; B ¼ A B 1 ðBA þ BAÞ
D
2
ð137Þ
In contrast to the classical correlation functions, the quantum
mechanical correlation functions are not symmetric with respect
to frequency, i.e. K ðxÞ – K ðxÞ. In thermal equilibrium they obey
the Kubo-Martin-Schwinger(KMS)-condition [5,4,62,63].
0
0
KK
K KK
aa0 ðxÞ ¼ K a0 a ðxÞ expfbh xg
ð138Þ
13
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
Detailed knowledge of the quantum mechanical spectral densities is
rarely available. Nevertheless they may be approximated by classical spectral densities by an invoking an appropriate quantum
mechanical correction [88]. One possibility is given by the Schofield
method [89]:
0
1
KK0
K KK
aa0 ðxÞ # J aa0 ðxÞ exp bh x
2
ð139Þ
Corrections of this form have been previously used in the description of vibrational relaxation effects [90].
This leads to the following expression for the secularized relaxation superoperator:
bh ¼
C
Lb
XX
0
a;a0
K;K
h
i
0
b Aa ; Ay 0 J KK0 xA
D
a aa
a
1
exp bh xAa dðxa xa0 Þ
2
ð140Þ
0
where JKK
aa0 ðxÞ are the classical spectral densities defined in Eq. (64).
Eq. (140) is the central result of this paper. It has a clear relationship with the semiclassical form of Eq. (79). The double commutation superoperator is replaced by a Lindbladian dissipator
(multiplied by 1/2), and thermal correction factors exp 12 bh xAa
are introduced.
It might appear that a similar result would be obtained by sim
ply introducing the thermal factors exp 12 bh xAa into the semiclassical relaxation superoperator of Eq. (79). Indeed, this construction was proposed by Abragam (see page 288) as part of the
quantum-mechanical treatment of relaxation [4]. However, that
version of the quantum-mechanical procedure fails for spin systems exhibiting a large degree of spin order, as is clear from the
context of Abragam’s expressions (see top of page 288). The
double-commutator form does not provide a clear separation of
the forward and backwards transition probabilities and is therefore
incompatible with detailed balance at finite temperature, even
after the inclusion of thermal corrections. It is only at infinite temperature that the Lindblad formalism and double commutator formalism coincide. These issues are discussed in more detail in
Appendices A.2 and A.3.
The Lindblad dissipators have a clear relationship with the
physical picture of Fig. 4. They may be understood as inducing forward and backward transition processes as illustrated below:
h
i
b X ij ; X y j X rr ¼ dis djr
X ss j D
ij
h
i
b X y ; X ij j X rr ¼ djs dir
X ss j D
ij
ð141Þ
cesses across the same pair of H0 eigenstates but in opposite directions. The transition probabilities per unit time are given by
W h;Lb
r!s
ð142Þ
Hence the transition probabilities for the forward and backwards
processes are related as follows:
h;Lb
W h;Lb
r!s ¼ W s!r exp fbh xsr g
ð143Þ
in agreement with the detailed balance condition and the physical
picture in Fig. 4.
The Lindbladian formalism also handles the coherence decay
rate constants correctly. The adiabatic contributions to the coherence decay rates are unaffected by thermalization, as should be
the case:
kh;ad
¼ kad
rs
rs
kh;na
rs
X
1 X
1
1
¼
exp bh xkr W r!k þ
exp bh xks W s!k
2 k–r
2
2
k–s
!
1 X h;Lb X h;Lb
¼
W
þ
W
2 k–r r!k k–s s!k
ð144Þ
!
ð145Þ
It follows that the thermalized Lindblad representation of
Eq. (140) preserves the trace and hermiticity of the density operator, obeys the detailed balance condition, and handles coherence
decay processes correctly.
6. Case studies
In this section the Lindblad approach is applied to some simple
cases. The first two cases concern ‘‘ordinary” high-field NMR in the
high-temperature and weak-order approximation, where the Lb
formulation reproduces well-known results which may also be
derived using the inhomogeneous master equation. In the last
two examples we consider a spin system far from equilibrium,
where the conventional IME equation breaks down.
6.1. Homonuclear spin-1/2 pairs
Consider an ensemble of rigid molecules undergoing rotational
diffusion, with each molecule containing a homonuclear spin-1/2
pair. The electronic environments of the two spins are distinct
(chemical inequivalence). The nuclear spin relaxation is assumed
to be dominated by dipole–dipole relaxation. The dipole-dipole
r 3
coupling constant is given by b12 ¼ l0 =4p c2I h
12 where cI is
the magnetogyric ratio and r12 is the internuclear distance. The
rotational correlation time is denoted sC .
In the presence of resonant radiofrequency irradiation, the
coherent Hamiltonian is given in the laboratory frame, by
Hcoh ¼ HA þ HB þ HC
ð146Þ
The dominant part of the Hamiltonian (see Section 2.3.1) is given in
the laboratory frame by
HA ¼ x0 ð1 þ dref ÞðI1z þ I2z Þ
h
i
h
i
b X y ; X ij are associated with relaxation prob X ij ; X y and D
Terms D
ij
ij
b h j X rr ¼ exp 1 b xsr W r!s
¼ X ss j C
Lb
2 h
The non-adiabatic contributions to the coherence decay rate
constants are sums of thermalized transition probabilities. This is
also physically reasonable (see Fig. 5).
ð147Þ
while the terms HB and HC are given in the interaction frame of HA
(rotating frame) by:
HB ¼ X1 I1z þ X2 I2z þ 2pJ 12 I1 I2
HC ¼ xnut Ix cos/ þ Iy sin /
ð148Þ
The Larmor frequency is x0 ¼ cB0 and the chemical shift offsets
are given by Xi ¼ x0 ðdi dref Þ, where fd1 ; d2 g are the chemical shifts
of the two sites and dref is the chemical shift position of the carrier
frequency of the radiofrequency irradiation. The amplitude of the
resonant field is specified in terms of the nutation frequency xnut
and its phase is denoted /. These choices of HA ; HB and HC fulfil
the constraints in Section 2.3.1: The eigenvalues of HA are either
degenerate or spaced by multiples of the spectrometer reference
frequency x0 ð1 þ dref Þ, which is much larger than the matrix elements of HB and HC .
Following Eq. (47), the eigenoperators of the commutation
b A are proportional to the irreducible spherical tensuperoperator H
bA
sor operators, as in Eq. (48). The corresponding eigenvalues of H
are given by:
xAa ¼ la x0
ð149Þ
14
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
The fluctuating part of the spin Hamiltonian, which is responsible
for the dipole–dipole relaxation, takes the form [6]:
Hfluc ðt Þ ¼
þ2
X
l¼2
ð12Þ
ð12Þ
F 2l XPL ðtÞ T 2l
ð12Þ
where both F 2l and T 2l
ð150Þ
transform as second-rank irreducible
ð12Þ
spherical tensors under rotations. The operator T 2l indicates coupling of spins I1 and I2 as defined in Appendix 2.
ð12Þ
The spatial components F 2l depend on the Euler angles XPL ðt Þ
that relate the principal axis system of the dipole–dipole coupling
to the laboratory frame. This transformation is time-dependent
due to rotational diffusion of the molecule. We assume a single
exponential decay for the corresponding correlation functions as
indicated by Eq. (58), leading to Lorentzian spectral densities.
Following the recipe of Eq. (140), and assuming isotropic rotational diffusion, the secular and thermalized relaxation superoperator may be expressed as follows:
6
5
b h ¼ b2
C
Lb
12
þ2
X
l¼2
h
i
b T ð12Þ ; T ð12Þy J h
D
2l
2l
l x0
ð151Þ
where the thermally corrected spectral density functions are given
by
1
J h ðxÞ ¼ exp bh x J ðxÞ
2
2sC
1 þ x2 s2C
Lb
to treat well-known relaxation phenomena such as the transient
and steady-state nuclear Overhauser (NOE) effects [12,11]. The
simulations shown in Fig. 6 were performed by integrating the
homogeneous equation of motion in the interaction frame (Eq.
(101)) using SpinDynamica software [87].
Fig. 6(a) shows a transient NOE effect in which inversion of the
magnetization of one set of spins induces a transient increase in
the magnetization of the second set of spins through dipoledipole cross-relaxation. Fig. 6(b) shows a steady-state NOE effect.
Saturation of the longitudinal magnetization of one set of spins
by a continuous resonant rf field establishes a steady state in which
the magnetization of the second set of spins is enhanced with
respect to thermal equilibrium.
These phenomena are well understood and the simulated trajectories using the Lindblad method agree with the literature
[51,52]. For the steady-state NOE it is straightforward to show that
within the fast motion limit (x0 sC 1) and high-temperature
approximation the maximal achievable polarisation enhancement
3
on the passive spin is given by SS
NOE ¼ 2 which is in agreement with
Fig. 6(b) [12,11].
6.2. Heteronuclear spin-1/2 pairs
ð152Þ
In the case of heteronuclear spin pairs (one I-spin coupled to
one S-spin, with magnetogyric ratios cI and cS ), the dominant part
of the coherent Hamiltonian, in the laboratory frame, is given by
and
J ðxÞ ¼
region of low magnetic field [91], or by applying a resonant
radiofrequency field [83].
b h may also be used
The thermalized relaxation superoperator C
ð153Þ
b is defined in Eq. (137).
The Lindbladian D
The thermalized relaxation superoperator may be represented
as a 16 16 matrix in a basis of orthogonal operators. One possible
basis involves all ket-bra products for the singlet and triplet states
of the 2-spin-1/2 system, defined in Eq. (11). The 4 4 block
involving the population operators for these states is as follows:
HA ¼ x0I 1 þ dIref Iz þ x0S 1 þ dSref Sz
0
ð155Þ
0
where x ¼ cI B and x ¼ cS B are the Larmor frequencies of
the two species, and dIref ; dSref are the chemical shifts of the two
spectrometer reference frequencies.
The HB and HC components of the coherent spin Hamiltonian
have the following form in the interaction frame of HA :
0
I
0
S
Lb
ð154Þ
The row and column of zeros indicate that the singlet population is
disconnected from the triplet populations under dipole-dipole
relaxation processes. The population of the singlet state is therefore a long-lived state [79–84], in the case that coherent Hamiltonian terms mixing the singlet and triplet states are suppressed. In
the current example, the relevant singlet–triplet mixing term is
proportional to the chemical shift frequency difference jX1 X2 j.
The long-lived nature of the singlet population is revealed by suppressing this mixing term, either by transporting the sample to a
HB ¼ XI Iz þ XS Sz þ 2pJ IS Iz Sz
ð156Þ
HC ¼ xInut Ix cos /I þ Iy sin /I þ xSnut Sx cos /S þ Sy sin /S
The chemical shift offset frequencies for the two isotopes are given
by XI ¼ x0I dI dIref and XS ¼ x0S dS dSref , where dI and dS are the
chemical shifts of the two spin species. Resonant radiofrequency
fields applied to the two spin species have amplitudes correspond
ing to the nutation frequencies xInut ; xSnut , and phases f/I ; /S g.
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
15
Fig. 6. Simulations of nuclear Overhauser effects (NOE) in a system of homonuclear proton pairs at a temperature of 300 K and a magnetic field of 11.4 T using the Lindbladian
form of the relaxation superoperator (Eq. (151)). The vertical scales correspond to the ratio of the longitudinal spin magnetization M z to its thermal equilibrium value M eq . (a)
Simulation of the transient NOE. Starting from thermal equilibrium, a selective p-rotation is applied to one set of spins at time point t ¼ 5 s, and the expectation values of the
z-magnetization components tracked in the subsequent interval. No resonant rf field is applied (xnut ¼ 0). (b) Simulation of the steady-state NOE. Starting at t ¼ 5 s, a
continuous rf irradiation with nutation frequency xnut ¼ 2p 2:5 Hz is applied to a thermal equilibrium state. The simulation parameters are as follows:
X1 ¼ 0; X2 ¼ 2p 200 Hz, J12 ¼ 15 Hz; b12 ¼ 2p 30 kHz, and sC ¼ 10 ps. The resulting T 1 time constant is 5.64 s.
According to Eq. (51), the eigenoperators of HA are given in the
heteronuclear case by products of spherical tensor operators. These
may be labelled by their individual total and z-angular momentum
ðijÞ
(T k1 k2 l1 l2 ). The relevant eigenoperators for heteronuclear dipolar
relaxation are given by column ðk1 ; k2 Þ ¼ ð1; 1Þ of Table 3. The
thermalized relaxation superoperator for the heteronuclear case
may be expressed as follows:
6
5
b h ¼ b2
C
12
Lb
þ2
X
þ2
X
l01 ;l02 ¼2 l1 ;l2 ¼2
h
i A
A
b T ð12Þ ;T ð12Þy0 0 J h xA
D
l1 l2 d xl1 l2 xl01 l02
11l1 l2
11l1 l2
ð157Þ
The relaxation dynamics of the Zeeman spin state populations are
governed by the following matrix:
tories were generated by SpinDynamica software [87] using
Eq. (157). The maximum achievable enhancement of the 13C
magnetization for a coupled 13C-1H pair in the steady-state
NOE experiment is given by ’ 3. The numerical simulations of
Fig. 7(b) confirm that this well-known result [12,11] may be reproduced by using the Lindbladian relaxation superoperator.
6.3. Singlet-triplet conversion
We now return to the problem given in the introduction: namely,
the build-up of longitudinal magnetization in a system of magnetically equivalent (or near-equivalent) spin-1/2 pairs, prepared ‘‘in
an unusual way” such that only the singlet state is populated. In this
Lb
ð158Þ
Simulations of the z-magnetisation dynamics during heteronuclear transient-NOE and steady-state-NOE experiments [12,11]
involving pairs of 13C and 1H spins are shown in Fig. 7. The trajec-
section, we consider an ensemble of ‘‘conventional” two-spin-1/2
systems which do not exhibit spin isomerism. The problem of
spin-isomer conversion is examined in the next section.
16
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
Fig. 7. Simulations of heteronuclear Overhauser effects (NOE) in a system of 1H-13C pairs at a temperature of 300 K and a magnetic field of 11.4 T using the Lindbladian form
of the relaxation superoperator (Eq. (151)). The black curves show the ratio of the 1H (I-spin) magnetization MIz to its thermal equilibrium value M Ieq . The red curves show the
ratio of the 13C (S-spin) magnetization MSz to its thermal equilibrium value M Seq . (a) Simulation of the transient NOE. Starting from thermal equilibrium, a selective p-rotation is
applied to the 1H spins at time point t ¼ 5 s and the expectation values of the z-angular momentum components of both spin species tracked in the subsequent interval. No
resonant rf field is applied (xInut ¼ xSnut ¼ 0). (b) Simulation of the steady-state NOE. Starting from a thermal equilibrium state, continuous rf irradiation with nutation
frequency xInut ¼ 2p 100 Hz is applied to the 1H spins, starting at time point t ¼ 5 s. No rf field is applied to the 13C spins (xSnut ¼ 0). The dipolar relaxation was treated using
Eq. (157). The simulation parameters are as follows: XI ¼ XS ¼ 0; sC ¼ 10 ps, b12 ¼ 2p 30 kHz. For this choice of parameters the 1H and 13C T 1 time constants are
T I1 ¼ T S1 ¼ 5:62 s. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
As explained in the introduction, the inhomogeneous master
equation (Eq. (1)) incorrectly predicts a build-up of longitudinal
magnetization with the ordinary rate constant T 1
1 , starting from
a pure singlet population. In reality, longitudinal magnetization
for the singletbuilds up with the much slower rate constant T 1
S
triplet conversion (see Fig. 1). We now examine how this problem
may be addressed by using the Lindbladian form of the relaxation
superoperator.
The dominant and perturbative parts of the coherent Hamiltonian in the absence of resonance rf fields are given by:
HA ¼ x ðI1z þ I2z Þ
0
ð159Þ
HB ¼ 2pJ 12 I1 I2
This assignment of HA and HB assumes that the scalar coupling
interaction is much smaller than the Zeeman interaction. This condition is easily met for NMR under ordinary conditions.
The relaxation of the system may be modelled as a superposition of the dipole–dipole (DD) and fluctuating random field mechanisms. The fluctuating random field mechanism may include
spin-rotation interactions as well as external random fields deriving from, for example, paramagnetic species in solution. The DD
mechanism does not induce spin-isomer conversion, which is
entirely driven by uncorrelated random fields. Using spherical tenb A , the relaxation superopsor operators as the eigenoperators of H
erator in the Lindblad formalism is given by:
bh ¼ C
b h;DD þ C
b h;ran
C
Lb
Lb
Lb
ð160Þ
where the dipole-dipole term is given by Eq. (151). A suitable relaxation superoperator for the fluctuating random fields is given by the
following expression:
b h;ran ¼
C
Lb
2
X
iÞ
jÞ
jij xðrms
xðrms
þ1
X
l¼1
i;j¼1
h
i
b T ðiÞ ; T ðjÞy J h mx0
D
ran
1l
1l
ð161Þ
where
ð1Þ
ð2Þ
(xrms ¼ xrms ¼ xrms ). The coefficient 1 6 j12 6 1 describes the
correlation between random field fluctuations at spin sites I1 and
I2 . By definition, j11 ¼ j22 ¼ 1. Perfectly correlated random fields
are described by j12 ¼ 1 while uncorrelated random fields have
j12 ¼ 0.
In the extreme narrowing limit, the spectral density functions
are frequency-independent (J ðxÞ ’ 2sC Þ. The relaxation rate constant R1 is given by
ran
R1 ¼ RDD
1 þ R1
ð164Þ
where the individual contributions to R1 are given by the following
matrix elements:
RDD
1 ¼ I z jb
C h;DD
jI z
Lb
ðIz jIz Þ
2
3
¼ 10
b12 sC cosh
Rran
1 ¼ I z jb
C h;ran jIz
Lb
ðIz jIz Þ
1
b
2 h
x0 þ 4 cosh bh x0
¼ 2x2rms sran cosh
1
b
2 h
x0
ð165Þ
The relaxation rate constant for singlet order (the mean population
difference between the singlet and triplet states) is given by the following matrix element:
b h;ran jI1 I2 Þ
ðI1 I2 j C
Lb
ðI1 I2 jI1 I2 Þ
4
1
bh x0
¼ x2rms sran ð1 j12 Þ 1 þ 2 cosh
3
2
RS ¼ Rran
¼
S
ð166Þ
There is no dipole-dipole contribution to RS .
In the Lindblad formalism, the temperature-dependence of the
relaxation rate constants derives not only from the temperaturedependence of the correlation time sC , but also from the explicit
temperature dependence of the hyperbolic functions in Eqs.
(165) and (166).
In the high-temperature limit, which is applicable to NMR at
ordinary temperatures, the rate constants are given by
2
1
J hran ðxÞ ¼ J ran ðxÞ exp bh x
2
R1 ’ 32 b12 sC þ 2x2rms sran
ð162Þ
and the spectral density function for random-field fluctuations is
given by
J ran ðxÞ ¼
to be isotropic and with the same rms value for the two sites
2sran
1 þ x2 s2ran
ð jÞ
ð163Þ
The term xrms is the root-mean-square amplitudes of the local field
fluctuations at the location of spin Ij . The random fields are assumed
RS ’ 4x2rms sran ð1 j12 Þ
ð167Þ
The relaxation rate constant for singlet order RS vanishes for perfectly correlated random fields (j12 ¼ 1).
As shown in Appendix A.5, the trajectory of the z-magnetisation
is well approximated within the fast-motion and high-temperature
limit by the following expression:
D
hIz ðt Þi= Iz ieq ’ 1 þ A1 exp R1 t þ AS exp RS t
ð168Þ
17
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
where the coefficients are
A1 ¼
RS
2ðR1 RS Þ
ð169Þ
1 þRS
AS ¼ 2R
2ðR1 RS Þ
The first exponential in Eq. (168) represents the fast equilibration of the outer triplet states due to T 1 processes. The second
exponential represents the singlet-triplet conversion process. The
coefficients approach A1 ! 0 and AS ! 1 in the limit R1 RS , in
which case the recovery of z-magnetization is dominated by the
small rate constant RS for the singlet-triplet conversion.
The solid line in Fig. 1b shows the solution to Eq. (168) with the
following parameters: xrms ¼ 2p 10 kHz, sran ¼ 60:31 ps,
j12 ¼ 0:955; sC ¼ 107 fs and b12 ¼ 2p 34:88 kHz. The resulting
relaxation time constants are given by: T 1 ¼ 2 s and T S ¼ 22:5 s. In
contrast to the IME, the Lindblad method correctly describes the
recovery of the longitudinal magnetization starting from a pure
singlet population, with the long time constant T S T 1 .
In the example above, it was assumed that the scalar coupling
term I1 I2 is much smaller than the Zeeman term, allowing the
Zeeman Hamiltonian to be assigned to the dominant term HA ,
while the scalar coupling is assigned to the perturbation HB (see
Eq. (159)). This assumption is well-satisfied for the vast majority
of physical systems, but breaks down at low temperatures for systems exhibiting spin isomerism. In such cases the scalar coupling
term acquires a large contribution from the nuclear exchange
interaction, which derives from the Pauli-principle restrictions on
the spatial quantum states accessible to nuclear spin states of a
given symmetry. In some cases, nuclear exchange interactions
exceed the nuclear Zeeman interactions by several orders of magnitude [92]. This can be the case, for example, for dihydrogen (H2 ),
and for freely-rotating water molecules (H2 O) encapsulated in
symmetrical cavities, such as fullerenes [47,46,48].
In spin-1/2 pair systems with a large nuclear exchange interaction, the spin Hamiltonian terms may be assigned as follows:
HA ¼ x ðI1z þ I2z Þ þ fðT ÞI1 I2
0
ð170Þ
HB ¼ 2pJ 12 I1 I2
In general, the temperature-dependent exchange splitting fðT Þ
is given by the thermally-averaged energy difference between
the spatial quantum states accessible to the different nuclear spin
isomers. For example, in the case of freely-rotating diatomic molecules such as H2 in the gas phase, this exchange splitting is given
by
hfðT Þ ¼ Eortho ðT Þ Epara ðT Þ
Eortho ðT Þ ¼
X
Epara ðT Þ ¼
X
J¼0;2...
ð171Þ
,
X
g J EJ exp EJ =kB T
g J exp EJ =kB T
J¼1;3...
defined in Eq. (170), this is not the case for the first-rank spherical
tensor operators T a1l . Nine first-rank eigenoperators are constructed from symmetrical combinations of the single-spin and
b A eigenopspin-pair spherical tensor operators of rank 1. The 16 H
erators fjAa Þg, with a 2 f1 . . . 16g, are given by:
6.4. Ortho-para conversion
where
low temperature. In such cases, the exchange splitting fðT Þ may be
larger than the thermal energy kB T.
In triatomic molecules displaying spin isomerism, such as H2 O,
Eq. (172) should be modified to reflect the more complex relationship between the rotational quantum numbers and the nuclear
spin states [93]. However, the principle is the same and the form
of the exchange Hamiltonian is identical. Similar phenomena are
observed for tunneling methyl groups [30–35].
The theory above requires identification of the eigenoperators
b A (Eq. (41)). Although,
jAa Þ of the commutation superoperator H
the zero- and second-rank spherical tensor operators T K
and
00
K
b A as
T 2l , defined in Eq. (47), are already eigenoperators of H
n
uþ u 12 0 12 g g
fjAa Þg ¼ T 12
2 2 ; T 1 1 ; T 1 1 ; T 1 1 ; T 2 1 ; T 00 ; T 00 ; T 10 ;
uþ u 12 g uþ u 12 12 o
T ; T ; T ; T ; T ; T ; T ; T
ð173Þ
10
10
20
11
11
21
22
11
where the symmetrical combinations are:
pffiffiffi
ðuþÞ
ð12Þ
ð1Þ
ð2Þ
T 1l ¼ 2T 1l þ 12 T 1l 12 T 1l
pffiffiffi
ðuÞ
ð12Þ
ð1Þ
ð2Þ
T 1l ¼ 2T 1l 12 T 1l þ 12 T 1l
g
ð1 Þ
ð2 Þ
T 1l ¼ p1ffiffi2 T 1l þ T 1l
ð174Þ
with l 2 f1; 0; þ1g. The subscripts u and g refer to operators that
are antisymmetric and symmetric with respect to particle
b A are
exchange, respectively. The corresponding eigenvalues of H
given by:
xAa ¼ 2x0 ; x0 ; f x0 ; f x0 ; x0 ; 0; 0; 0; f; f; 0; x0 ;
f þ x0 ; f þ x0 ; x0 ; 2x0
ð175Þ
These eigenoperators and eigenvalues may be used to construct
the Lindbladian form of the relaxation superoperator for the fluctuating random field mechanism:
b h;ran ¼
C
Lb
2
X
iÞ
jÞ
jij xðrms
xðrms
i;j¼1
16
X
a;a0 ¼1
h
i
b AK ; AK0y J h xA d xA xA0
D
a
a
a
a
a
ran
ð176Þ
The thermal equilibrium density operator may be derived from
the null-space of this superoperator, i.e. by solving the equation
b h;ran q Þ ¼ 0
C
eq
Lb
ð177Þ
n o
with the constraint Tr qeq ¼ 1. The solution is as follows:
J¼1;3...
,
X
g J EJ exp EJ =kB T
g J exp EJ =kB T
J¼0;2...
ð172Þ
Here EJ is the rotational energy of the spatial quantum state with
rotational quantum number J, and the degeneracies of the rotational
levels are g J ¼ 2J þ 1. The use of Eq. (172) for the average exchange
splitting assumes that the transitions between rotational states
(caused by collisions, in the case of a liquid) are much faster than
all terms in the spin Hamiltonian. The exchange splitting (orthopara splitting) becomes very small at temperatures large compared
to the rotational level splittings, but may exceed hundreds of GHz at
qeq ðT Þ ¼ Z 1 jS0 ihS0 j þ exp bh fðT Þ þ x0 jT þ1 ihT þ1 j
ð178Þ
þ expfbh fðT ÞgjT 0 ihT 0 j þ exp bh fðT Þ x0 jT 1 ihT 1 j
where the partition function is:
Z ¼ 1 þ exp bh fðT Þ þ x0 þ expfbh fðT Þg þ exp bh fðT Þ x0
ð179Þ
Note that qeq depends on temperature through the exchange splitting fðT Þ.
18
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
The singlet state jS0 i may be identified with the para spin isomer, while the triplet states jT M i; M 2 f1; 0; þ1g, belong to the
ortho spin isomer. The equilibrium para and ortho populations
are given by:
1
peq
para ðT Þ ¼ jS0 ihS0 jjqeq ¼ Z
eq
peq
ortho ðT Þ ¼ 1 ppara ðT Þ
ð180Þ
This displays the well-known temperature dependence of orthopara equilibrium. In equilibrium at high temperature bh ! 0, the
partition function tends to Z ! 4, all four states are approximately
equally populated, and the ortho and para populations tend to
! 3=4 and peq
! 1=4. At very low temperature, on the other
peq
ortho
ortho
hand (bh ! 1), the partition function tends to Z ! 1, and only the
para state jS0 i is populated at thermal equilibrium.
The Lindbladian superoperator in Eq. (176) also allows analysis
of the dynamics of spin-isomer conversion, under the fluctuating
random field model. Since singlet order corresponds to the difference between the population of the singlet state and the mean
population of the triplet states, the rate constant for spin-isomer
conversion is equal to the decay rate constant RS for singlet order.
This is given by the following matrix element:
b h;ran jI1 I2 Þ
ðI1 I2 j C
Lb
ðI1 I2 jI1 I2 Þ
1
¼ ð1 jÞ x2rms J hran ðfÞ þ J hran f þ x0 þ J hran f x0
6
1
þ ð1 jÞ x2rms J hran ðþfÞ þ J hran þf þ x0 þ J hran þf x0
2
ð181Þ
RS ¼ where the explicit temperature dependence of the ortho-para splitting has been suppressed for simplicity. Note that the spin-isomer
conversion requires spectral density of the random fields at combinations of the exchange splitting frequency f and the Larmor frequency x0 , and that the conversion process is suppressed for
correlated random fields (j ! 1).
These results are unsurprising. Nevertheless, it should be
emphasized that the semi-classical relaxation method, and the
inhomogeneous master equation, are incapable of generating them.
This formalism may readily be extended to include additional
nuclei and mechanisms in the relaxation processes. Such techniques were used to analyze quantum-rotor-induced polarization
effects in fullerene-encapsulated 17 O-labelled water, in which the
quadrupolar 17 O nucleus catalyzes the generation of antiphase
magnetization during the spin-isomer conversion of para-water
to ortho-water, at the same time as undergoing rapid quadrupolar
relaxation [46,94].
mathematical inconvenience of the IME. Much insight into spin
dynamical problems such as steady-state NMR properties could
be gained by expressing the nuclear spin dynamics, including thermally induced relaxation, as solutions to a homogeneous differential equation. This formulation allowed the development and
application of attractive theoretical approaches such as the average
Liouvillian theory [52,55]. This provided an intuitive tool for
understanding how certain relaxation processes may effectively
be turned on and off by applying resonant radiofrequency fields,
allowing unusual insights into phenomena such as the steadystate nuclear Overhauser effect [12,11]. Insights of this kind played
a role in the development of techniques exploiting long-lived spin
states [34,35,37–45]. However, although the mathematical awkwardness of the IME was widely recognized, its validity was not
generally called into question at this time. Its restriction to nearequilibrium and weakly-ordered spin systems was largely overlooked or forgotten [50–57].
More recently, technological developments have made spin systems which are prepared ‘‘in an unusual way”, to use the phrase of
Redfield [3], increasingly accessible to the magnetic resonance
community. Hyperpolarization techniques such as dynamic
nuclear polarization [15–18], optical pumping [19–22], quantumrotor-induced polarization [30–36], parahydrogen-induced polarization [23,25,26,24,27,29,28], and chemically-induced dynamic
nuclear polarisation [95–97] all involve spin systems which are
far from equilibrium. As discussed in the current article, the relaxation behaviour of these systems is not always consistent with the
IME, and in some cases, very substantial deviations from the IME
are observed, as illustrated in Fig. 1.
Some variants of the homogeneous master equation do succeed
in describing certain features of non-equilibrium nuclear spin
dynamics correctly. Nevertheless the Lindbladian formalism, as
described in the current article, seems to be the only one which
passes all tests (subject to its own restrictions and approximations,
of course, as indicated above). This will not come as a great surprise to researchers in many other spectroscopies, where the
approximations underpinning the IME were never valid in the first
place, and where Lindbladian techniques have been current for a
long time.
We hope that the current work will help bring the magnetic resonance community into mutual contact with other research areas
facing similar problems, including the vigorous theoretical field
of open quantum systems [62,63,76–78].
Declaration of Competing Interest
None.
Acknowledgements
7. Conclusions
The correct treatment of spin systems in contact with a thermal
reservoir is one of the oldest problems in magnetic resonance theory. The early founders of semiclassical relaxation theory were
fully aware that semiclassical relaxation theory fails to describe
thermal equilibration correctly and that the most readily available
‘‘fix” (the inhomogeneous master equation, IME) is of limited validity. Nevertheless, this venerable equation has served the nuclear
magnetic resonance community well for more than 60 years, presumably because the underpinning assumptions (weakly ordered
and near-equilibrium spin systems) were well-satisfied for the vast
majority of systems under study.
In the period 1980–2000, there was interest in formulating
homogeneous master equations as alternatives to the IME [50–
54]. The main motivating factor for these developments was the
This research was supported by the European Research Council
(786707-FunMagResBeacons) and the Engineering and Physical
Sciences Research Council (EPSRC-UK), Grant No. EP/P009980/1).
Appendix A
A.1. Lindblad form
First consider the Fourier transformation of the unit-step
function.
Z
1
1
Z
hðtÞ expfixtg dt ¼
0
1
expfixtg dt ¼ pdðxÞ i PV
1
x
ð182Þ
19
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
where hðt Þ denotes the unit-step function and PV the Cauchy principal value.
The bath correlation functions are related to the spectral densities as follows:
follows that thermal correction of the spectral density functions
within the double commutator formalism cannot lead to a valid
thermalized relaxation superoperator, at finite temperature.
D
A.3. Equivalence of double commutator and Lindbladian at infinite
temperature
Z 1
E
0
0
1
BKa0 y ðsÞBKa ð0Þ ¼
K KK0 ðxÞ expfixsg dx
2p 1 aa
ð183Þ
The one-sided Fourier transformation of the bath correlation functions in Eq. (134) may then be expressed as shown below:
R 1D
0
E
R1 R1
0
0
0
0
0
BKa0 y ðsÞBKa ð0Þ expfixsgds ¼ 21p 0 1 K KK
aa0 ðx Þexpfiðx x Þsg dx ds
R 1 KK0 0 R 1
1
0
0
¼ 2p 1 K aa0 ðx Þ 0 expfiðx x Þsg dsdx
0
R1
0
0
0
1
¼ 21p 1 K KK
aa0 ðx Þ pdðx x Þ i PV xx0 dx
R
0
0
1
1 KK
0
1
0
K ðxÞ 2ip 1 K KK
aa0 ðx ÞPV xx0 dx
2 aa0
ð184Þ
in a similar manner one can show the following:
Z
E
0
BKa0 y ðsÞBKa ð0Þ expfixsg ds
1
Z 1
0
1
i
1
KK0
0
dx0
¼ K KK
ð
x
Þ
þ
K
ð
x
ÞPV
0
0
2 aa
2p 1 aa
x x0
0
The Lindblad formalism and the double commutator formalism
only coincide at infinite temperature, because the spectral densities become temperature-independent. To see this the relaxation
superoperator in the double-commutator formalism at infinite
temperature may be written as shown below:
bh ¼ C
1
1X
by þ X
by
b ij X
b ji X
X
J xij
ij
ij
2 ij ijij
2
since exp 12 bh xij ¼ 1 for T ! 1. To proceed the following identity
is employed:
h
i
h
i
1 b by b by b X ij ; X y þ D
b X ji ; X y
X ij X ij þ X ji X ij ¼ D
ij
ji
2
D
ð185Þ
The above representations of the one-sided Fourier transformation
may then be substituted into Eq. (134).
The combination of the principal value leads to a commutator
that is known as dynamic frequency shift.
The decaying part is proportional to the spectral density
K aa0 ðxÞ. The minus sign preceding the sum of Eq. (134) and the factor of 1=2 are usually absorbed into the definition of the Lindbladian resulting in the presented Lindblad equation.
ð190Þ
ð191Þ
The relaxation superoperator in the double-commutator formalism
is then readily expressed in Lindblad form:
bh ¼ 1
C
2
i
h
i
X
h
b X ij ; X y þ D
b X ji ; X y
J ijij xij D
ij
ji
ij
i
i
X
h
h
bh ¼ 1
b X ij ; X y þ J ijij xij D
b X ji ; X y
C
J ijij xij D
ij
ji
2
ij
i
i
X
h
h
bh ¼ 1
b X ij ; X y þ J jiji xji D
b X ji ; X y
C
J ijij xij D
ij
ji
2
ð192Þ
ij
i
X
h
bh ¼
b X ij ; X y
C
J ijij xij D
ij
A.2. Double commutator thermalization
ij
In this section we show that thermal adjustment of the correlation functions within the double commutator formalism does not
lead to a correct thermalization of the relaxation superoperator.
This may be demonstrated by making use of the eigenoperator
formulation of the relaxation superoperator as presented in Eq.
(80). A thermally adjusted and secularized double-commutation
form of the relaxation superoperator is as follows:
bh ¼ C
1X
1
by
b ij X
exp bh xij J ijij xij X
ij
2 ij
2
ð186Þ
The transition probability per unit time W k!l from the eigenstate jki to the eigenstate jli of the Hamiltonian HA may be identified with the following matrix element:
b h j X kk
W k!l ¼ X ll j C
¼
1X
1
exp bh xij
2 ij
2
J ijij xij dli dki dli dkj dlj dki þ dlj dkj
ð187Þ
Similarly, the transition probability for the reverse process is given by
W l!k
b h j X ll
¼ X kk j C
¼
1X
1
exp bh xij
2 ij
2
A.4. Eigenoperators
b A for the homonuclear case are
The relevant eigenoperators of H
ðijÞ
summarised in Table 2. The eigenoperators are denoted by T kl . The
superscript ðijÞ indicates angular momentum coupling of spins Ii
and Ij resulting in a spherical tensor operator of total angular
momentum k and z-angular momentum l.
b A for the heteronuclear case are
The relevant eigenoperators of H
ðijÞ
summarised in Table 3. The eigenoperators are denoted by T k1 k2 l1 l2 .
For the heteronuclear case the superscript ðijÞ indicates the direct
product of spherical tensor operators of spins Ii and Ij with total
angular momentum k1 and k2 and z-angular momentum l1 and
l2 , respectively.
A.5. Longitudinal recovery for singlet-triplet conversion
We briefly outline the derivation of Eq. (168). First we project
out the population block of the relaxation superoperator.
J ijij xij dki dli dki dlj dkj dli þ dkj dlj
ð188Þ
Clearly these two transition probabilities are equal:
W k!l ¼ W l!k
by noticing that Jijij xij ¼ Jjiji xji at infinite temperature.
ð189Þ
This indicates that forward and backward transitions are equally
likely, violating detailed balance, except at infinite temperature. It
Table 2
b A for a homonuclear coupled spin-1/2 pair.
Eigenoperators of H
l nk
2
2
1
2 I1 I 2
1
0
1 ffiffi
2p
6
1
2 I1 I2z þ I1z I2
þ
Iþ
1 I2 þ I 1 I2 4I1z I2z
1
p1ffiffi I
2 j
Ijz
20
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
Lb
ð193Þ
where Rj indicates the sum over all the other elements in one
column.
To proceed it is advantageous to perform a change of basis. The
transformation matrix is given by the expression below:
leading to the following set of equations:
ð194Þ
ð198Þ
The solution for P1 ðt Þ and P 2 ðt Þ are easily found:
P1 ðt Þ ¼ P1 ð0Þ P2 ðtÞ ¼ expfðr22 t ÞgP2 ð0Þ
To simplify notation we denote the transformed population block as
follows:
The solution for P3 ðt Þ is formally given by:
Z
P3 ðt Þ ¼ expðr33 tÞP3 ð0Þ þ r34
ð195Þ
Lb
and enumerate the transformed populations:
0
t
expfðr33 ðt sÞÞgP4 ðsÞds
ð200Þ
We interpret the variable ðt sÞ as a ‘‘memory time” and perform a
0-th order Markov approximation [98].
P3 ðt Þ expðr33 tÞP3 ð0Þ þ
ð196Þ
ð199Þ
r34
P ðt Þ
r33 4
ð201Þ
The dynamics for P4 ðtÞ then reduce to the following:
d
P4 ðt Þ ¼ r41 P 1 ð0Þ þ r42 expðr22 t ÞP2 ð0Þ
dt
To a good approximation the dynamics of P 2 are decoupled from the
dynamics of P 4 .
ð197Þ
þ r43 expðr33 t ÞP3 ð0Þ þ
r34
P ðtÞ þ r44 P4 ðt Þ
r33 4
ð202Þ
which is easily solved.
The solution may then be transformed back into the original
representation.
Using
the
appropriate
initial
condipffiffi
tions:P1 ð0Þ ¼ 12 ; P 2 ð0Þ ¼ 23 ; P3 ð0Þ ¼ 0 and P4 ð0Þ ¼ 0 in combination
with the high-temperature and fast-motion approximation results
in Eq. (168).
Table 3
b A for a heteronuclear coupled spin-1/2 pair.
Eigenoperators of H
(l1 ; l2 )\(k1 ; k2 )
(1,1)
(1,0)
(0,1)
ð 1; 1Þ
1
2 I1 I2
1 ffiffi
2p
I I
6 1 2
1
I
2 1 I2z
–
–
–
–
ð 1; 1Þ
ð 1; 0Þ
ð0; 1Þ
ð0; 0Þ
1
2 I1z I2
qffiffi
2
3I1z I 2z
p1ffiffi I
2 1
–
I1z
–
(0,0)
–
p1ffiffi I
2 2
–
I2z
–
C. Bengs, M.H. Levitt / Journal of Magnetic Resonance 310 (2020) 106645
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