 1.1 Introduction
 1.2 What Are Longlived States?
 1.2.1 Disconnected Eigenstates
 1.2.2 Disconnected State Manifolds
 1.3 Symmetry Theory of Longlived States
 1.3.1 The Coherent Hamiltonian and Its Symmetry
 1.3.2 The Fluctuating Hamiltonian and Its Symmetry
 1.3.3 The Symmetric Approximate Hamiltonian
 1.3.4 The Number of Longlived States
 1.3.5 Longlived State Operators
 1.4 Longlived State NMR
 1.4.1 Excitation
 1.4.2 Filtering
 1.4.3 Evolution
 1.4.4 Readout
 1.4.5 Detection
Chapter 1: Longlived States in Nuclear Magnetic Resonance: An Overview

Published:07 Apr 2020

Special Collection: 2020 ebook collectionSeries: New Developments in NMR
M. H. Levitt, in Longlived Nuclear Spin Order: Theory and Applications, ed. G. Pileio, The Royal Society of Chemistry, 2020, ch. 1, pp. 132.
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An overview of longlived states in nuclear magnetic resonance (NMR) is given. The chapter includes a sketch of the symmetry theory of longlived states, which allows prediction of the number and nature of the longlived states. The key elements of an NMR experiment involving longlived states are reviewed. The concepts that are covered include: symmetry switching, disconnected eigenstates, disconnected state manifolds, singlet–triplet population imbalance, permutation groups and their irreducible representations, longlived state operators, the symmetrical approximations to relevant spin Hamiltonians, magnetic equivalence and nearequivalence.
1.1 Introduction
In 2004 our group published two papers that demonstrated the feasibility of passing nuclear magnetization through spin order that is wellprotected against some common relaxation mechanisms and has an unusually long lifetime, greatly exceeding the ordinary relaxation time constant T_{1}.^{1,2 } These early studies exploited molecules containing pairs of protons in different chemical environments, and hence having different isotropic chemical shifts. In both cases, the nuclear spin–lattice relaxation was dominated by a single mechanism, namely the homonuclear dipole–dipole (DD) mechanism, in which motional modulation of the internuclear dipole–dipole coupling between like spins induces nuclear spin transitions. Both experiments exploited a type of nuclear spin order called singlet order, which consists of the difference between the population of the singlet state and the mean population of the three triplet states of the spin1/2 proton pair (see below).^{3–5 } The singlet order decays with a time constant, denoted T_{S}, which was shown to be much longer than the relaxation time constant T_{1} for longitudinal magnetization, in both cases.^{1,2 } This is because the rate constant T$1\u22121$ is dominated by DD relaxation, while DD relaxation does not contribute to the rate constant T$S\u22121$. In both experiments, the longlived nature of singlet order was revealed by temporarily suppressing the effect of the chemical shift difference. This is required since an unfettered chemical shift difference induces rapid interconversion of the singlet and triplet states, concealing the longlived properties of the singlet state. In the first experiment, the chemical shift difference was suppressed by temporarily removing the sample from the NMR magnet into a region of low magnetic field.^{1 } In the second experiment, the sample was kept in high magnetic field, while the chemical shift difference was suppressed by applying a resonant radiofrequency field.^{2 } The physical principles of these two experiments were investigated in further papers by our group and others.^{6–9 }
In retrospect, the most novel feature of these early papers was not the demonstration that nuclear singlet order can be much more longlived than ordinary magnetization. After all, the phenomenon of spin isomerism, which is closely related to that of longlived states, has been known for a long time. For example, it has been long known that molecular hydrogen (H_{2}) possesses two spin isomers, called parahydrogen and orthohydrogen, where the proton nuclei are in the singlet state for parahydrogen, and the triplet state for orthohydrogen.^{10 } A paraenriched sample of H_{2} gas is stable for weeks when kept in an appropriate container, even though the nuclear T_{1} is typically less than 1 second. The longlived nuclear singlet state of parahydrogen is essential for experiments exploiting parahydrogen as a nuclear polarization source, as pioneered by Weitekamp and coworkers^{11–13 } (see also Chapters 18 and 19). Spin isomer phenomena also occur for other small symmetrical molecules in suitable environments, including water (H_{2}O),^{14–16 } methane (CH_{4}),^{17 } ammonium (NH_{4}^{+})^{18,19 } and ethene (C_{2}H_{4}).^{20 } The protection of certain quantum states from relaxation effects in highsymmetry groups of coupled nuclear spins has also been known for a long time.^{21–23 }
So, longlived nuclear singlet order was not really new. In retrospect, the novel aspects of our 2004 papers were as follows. (1) We demonstrated that longlived states may be demonstrated in molecules that do not have a high degree of molecular symmetry, and do not possess true spin isomers. Indeed, the molecules of the first demonstration system (2,3dibromothiophene, see Figure 1.1a) have no relevant symmetry element at all (barring the irrelevant reflection in the plane). There is no such substance as “para2,3dibromothiophene” (or sadly, “paraethanol”,^{24 } for that matter). Nevertheless, 2,3dibromothiophene and ethanol may indeed possess longlived modes of nuclear spin order that are protected against relaxation, and which may be accessed in relatively straightforward NMR experiments. (2) The second novelty in our 2004 papers was that the longlived modes may be populated from ordinary magnetization, and converted back into ordinary magnetization, by switching the symmetry of the spin Hamiltonian using relatively standard interventions such as the application, or removal, of applied magnetic fields.
Point (1) above has wide reach, since it shows that some features of spin isomerism are not restricted to small, highly symmetrical molecules with high rotational freedom, and may also be displayed by “ordinary” molecules in solution, providing that suitable precautions are taken.
The concept of symmetry switching (point (2) above), which is used to provide access to the longlived states, was soon extended. The group of Warren soon showed that chemical transformations may be used as a symmetry switch, providing access to longlived states.^{25 } Since then, several groups have exploited chemical reactions and physical transformations as symmetry switches in longlivedstate experiments,^{25–31 } including photochemical switches.^{32,33 } The topic of chemical symmetry switches is reviewed in Part II of this book. In addition, the Warren group demonstrated that Jcoupling differences, associated with magnetic inequivalence, may also act as a symmetry switch.^{34–36 } In some cases, relaxation phenomena may also break symmetry, allowing access to longlived states.^{24,37,38 }
Since this early work, many groups have explored the theory, methodology^{39 } and applications of longlived states of NMR, and some of this excellent work is described in the following chapters. Selected highlights in the field of longlivedstate NMR include:
The design and synthesis of molecular systems that support extremely longlived states, as described in Chapter 4 of this book. The current record holder is a ^{13}C_{2}labelled naphthalene derivative, which exhibits a T_{S} value exceeding 1 hour in roomtemperature solution, in low magnetic field (see Figure 1.1b).^{40 } Compounds called diazirines, which contain a threemembered ring formed by two nitrogen atoms and one carbon, are particularly promising targets for supporting ^{15}N_{2} longlived states.^{33,41 }
The recognition that the Jcoupling between the spins that participate in the longlived state protect it against external influences (so called “Jstabilization”).^{42 }
The introduction of nonmagnetic isotopes such as ^{18}O to induce small isotope shifts, providing access to longlived states even in highly symmetrical molecules.^{43 }
Demonstrations of longlived coherences with very small decay rate constants, and which in some cases oscillate coherently for tens of minutes^{44–46 } (see Chapter 20).
The development of methods for converting nuclear magnetization into longlived spin order, and back again, including radiofrequency pulse sequences combined with field cycling^{1,4,6,44,47–50 } (see Chapter 5), the M2S/S2M (magnetizationtosinglet and singlettomagnetization) pulse sequences in high magnetic field^{4,34,51–53 } and related methods^{54 } (see Chapter 7), the simple and elegant SLIC (spinlockinduced crossing) method^{52,55–57 } (see Chapter 8), and the robust schemes which exploit adiabatic passage through avoided level crossings^{46,58–60 } (see Chapter 9).
The development of phase cycling and fieldgradientbased methods for selecting the component of the NMR signal that passes through singlet order^{43,61 } (see Chapter 10).
The demonstration that longlived states may be used to store hyperpolarized nuclear spin order for relatively long times,^{25,47,48,62–64 } and that in some circumstances dynamic nuclear polarization (DNP) generates longlived spin order directly.^{24,48,62,65 }
The identification of longlived states in rapidly rotating methyl groups, and their implication in the unusual signal enhancements observed when certain substances are rapidly warmed from the cryogenic solid state to ambient temperature^{37,38 } (see Chapter 21).
Good progress in the understanding of longlivedstate spin relaxation,^{40,66–73 } including unusual mechanisms such as spinrotation and spininternalmotion,^{40,44,49,66,71 } the antisymmetric components of the chemical shielding tensors,^{40,66,69 } and scalar relaxation of the second kind^{69,70 } (see Chapter 3).
The demonstration that the minuscule chemical shift difference between the protons of some monodeuterated methyl (CH_{2}D) groups may be used to access proton singlet states^{74–76 } (see Chapter 22).
The use of longlived states for the preparation of hardtoaccess spin isomers, such as paraN_{2.}^{77 }
The demonstration of heteronuclear longlived states in ultralowfield NMR^{78 } (see Chapter 23).
Applications of longlived states are at various stages of readiness for practical use. They include:
Applications to ligand binding and screening in biomolecular NMR^{79–82 } (see Chapters 12 and 13).
The study of diffusion and transport processes, where the long lifetime of the nuclear spin order provides access to longer timescales than conventional magnetization^{9,83–89 } (see Chapters 14, 15 and 16).
Applications to NMR demonstrations of quantum computing^{90 } (see Chapter 17).
Applications to parahydrogenenhanced nuclear polarization^{11–13,28–30,33,41,52,53,77,91–97 } (see Chapters 18 and 19).
Hyperpolarized NMR sensors and tags for molecular imaging and spectroscopy.^{27,41,77,98 }
On the wilder fringes, there are even speculations that longlived nuclear spin states might be implicated in human cognition.^{99 }
1.2 What Are Longlived States?
The term longlived state is associated with some confusion in the field of NMR, for which the author admits some responsibility. The historical development has led to two separate meanings for this term, which are closely related, but not identical.
1.2.1 Disconnected Eigenstates
The original meaning of the term longlived state is a single eigenstate of the spin Hamiltonian, which for reasons of symmetry, has a very small, or absent, probability for transitions to other eigenstates, under the dominant relaxation processes. This has also been called a disconnected eigenstate (see Figure 1.2a).^{25,36 } The seminal example is found in a system of 2 coupled spins1/2, in the presence of a scalar spin Hamiltonian of the form
where J is the isotropic part of the indirect dipole–dipole coupling (Jcoupling) between the two spins. This Hamiltonian has a set of four eigenstates, comprising the singlet state S_{0}〉 and the three components of the triplet state {T_{+1}〉, T_{0}〉, T_{−1}〉}, which may be expressed in terms of the Zeeman product states as follows:
where αβ〉 denotes the Zeeman product state with angular momentum +½ ħ for spins I_{1} along the zaxis, and angular momentum −½ ħ for spins I_{2} along the zaxis, and similarly for the other states. The singlet and triplet states have total angular momentum quantum numbers of 0 and 1, respectively, and the triplet state T_{M}〉 has angular momentum Mħ along the zaxis.
Crucially, the singlet and triplet states have different symmetries under exchange of the two spins. We define the permutation operator (12) as the operator for exchange of spins I_{1} and I_{2}. The singlet state is antisymmetric under exchange:
while the three components of the triplet state are symmetric under exchange:
where M ϵ {+1,0,−1}. Now suppose that spin relaxation is predominantly caused by the stochastic modulation of the direct throughspace dipole–dipole coupling between the two nuclei. This is commonly the case, especially for pairs of protons, since their relatively large magnetogyric ratio leads to strong dipole–dipole interactions. The spin Hamiltonian for the direct dipole–dipole interaction, for an arbitrary molecular orientation, may be written
where the DD coupling tensor D_{12} is given by:
Here b_{12}=−(μ_{0}/4π) γ_{1}γ_{2}ħ^{2}r$123$ is the dipole–dipole coupling constant, μ_{0}/4π is the magnetic constant, γ_{1} and γ_{2} are the magnetogyric ratios and r_{12} is the internuclear distance.^{100 } The symbol r̂_{1}denotes a unit vector leading from the origin of the coordinate system to the position of the nucleus I_{1}, and similarly for r̂_{2}. The symbol r̂$1T$ denotes a row vector that is the transpose of the column vector r̂_{1}, so that the symbol r̂$1T$r̂_{2} denotes a dyadic or outer product, given by a 3×3 matrix.^{101 } The DD coupling Hamiltonian is symmetric under particle exchange (eqn (1.5)).
It follows that the dipole–dipole Hamiltonian H_{DD} commutes with the exchange operator (12) and hence that the dipole–dipole Hamiltonian has no matrix elements between states of different exchange symmetry:
The singlet state S_{0}〉 is therefore a longlived state, or a disconnected eigenstate,^{25,36 } with zero transition probability to other states, under the dominant dipole–dipole relaxation mechanism for this system (Figure 1.2a). The population of the singlet state therefore persists for a relatively long time, as compared to the populations of the triplet states, which interchange freely under the dominant dipole–dipole relaxation mechanism, since they have the same exchange symmetry (eqn (1.4)). The equilibration of longitudinal magnetization, with time constant T_{1}, is due to such triplet–triplet transitions.
The selection rule of eqn (1.7) may also be derived from the rotational symmetries of the singlet and triplet states.^{102 } Since the singlet state has total spin 0, the triplet state has total spin 1, and the dipole–dipole interaction may be expressed as an irreducible spherical tensor of rank 2, the Wigner–Eckart theorem may be used to prove that all matrix elements 〈S_{0}H_{DD}T_{M}〉 vanish.^{102 } Hence, in this particular case, analyses of exchange symmetry, or rotational symmetry, both lead to the same result. However, in some other cases, exchange symmetry leads to results that may not be derived from rotational symmetry. The exchange symmetry analysis will therefore be favoured in this article.
1.2.2 Disconnected State Manifolds
Most NMR experiments involve a large ensemble of formally identical spin systems. The quantum state of the spin ensemble is described by the density operator ρ, defined as follows:
where ψ〉 is the state of a single ensemble member and the overbar denotes an ensemble average. The expectation value of a spin operator Q may be written
In general, the expectation value of an operator Q is timedependent, due to the timeevolution of the density operator ρ, as governed by the timedependent Schrödinger equation for the individual spin systems. However, in some cases, the timederivative d〈Q〉/dt vanishes. Such expectation values are called constants of the motion.
A trivial constant of the motion is the sum of all populations, which is equal to the expectation value of the unity operator:
where the sum is over all energy eigenstates. The sum of all state populations is conserved in a closed quantum system.
Additional constants of the motion appear when the spin Hamiltonians driving the evolution of the system exhibit mathematical symmetries, such as that given above for the dipole–dipole interaction (eqn (1.5)). Although perfect symmetry is never exhibited for the total spin Hamiltonian in a real experimental context, such idealized constants of the motion may still appear as modes of nuclear spin order that persist for a long time, in particular for times much longer than the ordinary spin–lattice relaxation time T_{1}. If the expectation value of a particular spin operator Q_{LLS} has a very small or vanishing timedependence for nontrivial reasons, then the operator Q_{LLS} is termed a longlived state.
An example is given by the following operator for an ensemble of spin1/2 pairs, in the absence of symmetrybreaking interactions:
This operator is called singlet order.^{3–5 } This operator may also be expressed as follows:
The expectation value of singlet order is the difference between the population of the singlet state and the mean population of the triplet states – a quantity also called the singlet–triplet population imbalance.^{6 } (The term triplet–singlet imbalance (TSI) has also been coined for a quantity with the opposite sign^{31 }). Clearly, if all transitions between the singlet state and the triplet states are symmetryforbidden, such a population imbalance may persist for a relatively long time in the spin ensemble. The operator Q_{SO} therefore represents a mode of spin order that is longlived because it is protected against a dominant relaxation mechanism (in this case, intramolecular dipole–dipole relaxation).
By common practice, the operator Q_{SO} is called a longlived state, even though the sense of this term is slightly different from that of the disconnected eigenstate of S_{0}〉. The operator Q_{SO} is a longlived mode of ensemble spin order, while the symbol S_{0}〉 refers to the quantum state of an individual member of the spin ensemble, which is protected against transitions to other states. Another way of putting this is to use the languages of Hilbert space and Liouville space.^{103 } The state S_{0}〉 is a longlived state in Hilbert space, while the operator Q_{SO}, when expressed as a Liouville ket Q_{SO}), is a longlived state in Liouville space.
Although these two meanings of “longlived state” are closely related, they are not interchangeable. In larger spin systems, spin order modes may arise that are protected against relaxation to some approximation, even though there are no disconnected eigenstates. Hence, the Liouville sense of longlived state is more general than the Hilbertspace term.
Systems with rapidly rotating methyl (CH_{3}) groups present a case with longlived states in Liouville space, but without disconnected eigenstates in Hilbert space (see Chapter 21).^{38 } The eight eigenstates of the three protons in a CH_{3} group may be arranged into three symmetry classes, termed A, E_{a} and E_{b} (see Figure 1.2b). These labels indicate the irreducible representations (irreps) of the group C_{3}, which includes the three spin permutations that may be realised by methyl rotation:
Here ( ) denotes the identity permutation (does nothing), while (123) represents a cyclic permutation of spins 1→2→3→1, and (132) is a cyclic permutation in the opposite direction 1→3→2→1. In the regime of rapid methyl rotation compared to overall molecular tumbling, transitions between states of different irreps are symmetry forbidden under the dominant relaxation mechanisms.^{38 } Consider an operator whose expectation value is the imbalance in populations between the A and Estates, of the form
where A〉_{s} is one of the four states of symmetry A, E_{a}〉_{s} is one of the two states with symmetry E_{a}, and so on. This operator may also be written as follows:
The expectation value 〈Q_{AEI}〉 is the difference in mean population of the Astates and the Estates, which has been called the AE population imbalance.^{38 } As discussed below, the transitions between these manifolds are symmetryforbidden within certain approximations. Hence, the operator Q_{AEI} represents a longlived state for rapidlyrotating methyl groups in solution. In some cases, it is possible to generate a spin ensemble with a large overpopulation of the operator Q_{AEI} by a rapid transition of the sample from the cryogenic solid state to ambient temperature solution.^{37,38,62,104 } Longlived NMR signal enhancements are observed when hyperpolarized spin order described by the operator Q_{AEI} crossrelaxes with observable magnetization.^{37,38,62,104 }
Note that the longlived state described by the operator Q_{AEI} arises for the threespin1/2 systems of rapidly rotating methyl groups, even though there are no individual disconnected eigenstates in this case. The longlived operator is associated with population imbalances between disconnected state manifolds, rather than individual states. A similar situation arises for groups of four spins1/2 in a suitable geometrical arrangement.^{105 }
In this chapter, the term longlived state implies the more general ensemble meaning of the term, i.e. an operator Q_{LLS}, whose expectation value is given by a population imbalance between state manifolds that are forbidden by symmetry from exchanging populations with each other, to a first approximation.
1.3 Symmetry Theory of Longlived States
In this section I sketch the permutation symmetry theory of longlived states, which allows the prediction of the number and nature of longlived states in a given system. A deeper approach and more detailed examples are given in Chapter 2.
A flow diagram of the symmetry theory is shown in Figure 1.3.
Consider an ensemble of molecules, each of which contains a group of coupled nuclear spins, and which tumbles freely in solution. For a rigid molecule, the geometrical arrangement of the nuclei in space is wellconserved except for the overall translation and rotation. The nuclear spins interact with each other through Jcouplings and dipole–dipole couplings, and with the external magnetic field and the nearby electrons through chemical shift interactions and quadrupolar couplings (for spins >1/2).
In general, the Hamiltonian of a spin system is timedependent because of molecular motion, and may be written as the sum of a coherent and a fluctuating term:
where the coherent term is identical for all members of the ensemble and is also equal to the longterm time average of H(t) through the ergodic hypothesis. In the NMR of isotropic solutions, this term includes the main Zeeman interaction with the applied magnetic field, isotropic chemical shifts and Jcouplings. The fluctuating or incoherent term H_{fluc}(t) is responsible for relaxation processes,^{66,101,106 } and includes dipole–dipole couplings, chemical shift anisotropy terms, spinrotation terms and quadrupolar couplings (for spins >1/2).
1.3.1 The Coherent Hamiltonian and Its Symmetry
In solution NMR, the coherent Hamiltonian H_{coh} includes isotropic spin Hamiltonian terms such as Jcouplings and isotropic chemical shifts. In many cases, this Hamiltonian possesses permutation symmetry derived ultimately from the molecular symmetry.^{107,108 } The group of spin exchanges, each of which leaves H_{coh} invariant, is designed here as _{coh}:
This group, together with the relative magnitude of the isotropic shift differences and Jcouplings, define the alphabetic designation of the spin system.^{23 } If the group G_{coh} contains the exchange of a set of spins, with no other simultaneous spin exchanges, then this set of spins is known as magnetically equivalent.
For example, consider a homonuclear system of two spins1/2, I_{1} and I_{2}. The coherent spin Hamiltonian in isotropic solution is given by
If the chemical shift frequencies are the same (because the isotropic chemical shifts are the same, or because the external magnetic field is absent), then exchange of the two spins leaves the coherent Hamiltonian invariant:
The spin system is designed as A_{2} in this case. If, on the other hand, the chemical shift frequencies are different, the permutation group _{coh} only contains the identity element and the designation is AB or AX depending on the relative magnitudes of the shift difference and the Jcoupling:
The threespin1/2 system of a rotating methyl group is usually magnetically equivalent in solution, due to the rapid rotation around the 3fold symmetry axis. If the methyl group is isolated, and rotating rapidly enough to average out any chemical shift differences between the three protons, the coherent spin Hamiltonian may be written as follows
This is clearly invariant to any permutation of spins, and hence commutes with all six elements of the symmetric permutation group S_{3}:^{107 }
However, for technical reasons that go beyond the scope of this chapter, and which will be discussed in a future publication, it is sufficient, and more convenient, to identify _{coh} with a subgroup of S_{3} consisting of only three elements, and called here C_{3} (see eqn (1.13)):
The three protons of a rotating CH_{3} group are magnetically equivalent and are given the spin system designation A_{3}.
Fourspin1/2 systems are illustrative. Consider the case where I_{1} is chemically equivalent to I_{2}, and I_{3} is chemically equivalent to I_{4}. If the shift difference is large compared to the Jcoupling between the groups, the system is designed as A_{2}X_{2} or AA′XX′ according to whether the permutation group _{coh} contains the individual exchanges (12) and (34), or whether only the double exchange (12)(34) leaves the coherent Hamiltonian invariant:
or
The structure of the NMR spectra depends strongly on this spin system designation.^{23 } In particular, Jcouplings between magnetically equivalent spins do not generate spectral splittings.^{23,100 }
It is feasible to change the symmetry of the coherent Hamiltonian, and hence the group _{coh}, during the NMR experiment. This is called symmetry switching and may be accomplished in several different ways:
A change in the magnetic environment. For example, the sample may be transported from a high magnetic field, where chemical shift differences are active, to a low magnetic field, where the effects of chemical shifts become negligible. In the case of spin1/2 pairs, this readily changes G_{coh} from AB or AX symmetry (eqn (1.20)) to A_{2} symmetry (eqn (1.19)). Transport in the reverse direction (from low field to high field) changes the symmetry back again. This method was used in our first demonstration of longlived states.^{1 } In the special case of heteronuclear singlet states, the same switch may be implemented, with the difference that implementation of A_{2} symmetry (eqn (1.19)) requires an ultralow magnetic field in the nanotesla regime.^{78 } This may be achieved by using a magnetically shielded chamber.
Application of a resonant radiofrequency field. Applying a resonant radiofrequency field may suppress chemical shift differences, and hence increase the effective symmetry of the coherent Hamiltonian.^{2 } In the case of spin1/2 pairs, this may induce a similar change in G_{coh} from AB or AX symmetry (eqn (1.20)) to A_{2} symmetry (eqn (1.19)). The switch may be flipped in the reverse direction simply by turning the resonant radiofrequency field off again. The dependence on the amplitude, frequency and modulation of the applied field has been explored in some detail.^{2,7,8,70 }
Chemical transformations. Chemical transformations of the material,^{25–31 } including photochemical transformations,^{32,33 } change the nature of the coherent Hamiltonian and hence the symmetry group G_{coh}. Parahydrogeninduced hyperpolarization (PHIP) is based on the principle of a chemical symmetry switch, since the spinexchange symmetry of the dihydrogen molecule is broken by relocating the two hydrogen atoms to an asymmetric chemical environment.^{11–13,28–30,33,41,52,53,77,91–97 }
1.3.2 The Fluctuating Hamiltonian and Its Symmetry
The fluctuating Hamiltonian H_{fluc}(t) consists of the intra and intermolecular dipole–dipole interactions, the chemical shift anisotropy and spin–rotation interactions, and all other terms which are modulated by molecular motion and cause nuclear spin relaxation.^{101,106 } In general, H_{fluc}(t) contains both intermolecular and intramolecular interactions and has little or no permutation symmetry. Nevertheless, some important individual components of H_{fluc} may have symmetry. For example, the dipole–dipole interaction between two magnetic nuclei of the same spin quantum number is symmetric with respect to exchange of those two nuclei, even if they are of different isotopic type (eqn (1.5)).
1.3.3 The Symmetric Approximate Hamiltonian
A key step in the theory of longlived states is to identify a suitable approximation for the spin Hamiltonian that drives the evolution of the spin system and possesses permutation symmetry. This approximate spin Hamiltonian, which is relevant to the identification and analysis of the longlived states, is called here the symmetric approximate Hamiltonian and is denoted H^{☆} (the superscript “star” symbol ☆ should not be confused with the complex conjugate, denoted by the asterisk*).
The symmetric approximate Hamiltonian H^{☆} involves plausible approximations to both terms H_{coh} and H_{fluc}(t), i.e.
where and . Note that H^{☆} cannot be based on the coherent Hamiltonian alone.
1.3.3.1 Symmetric Approximation of the Coherent Hamiltonian
In the theory of longlived states, the symmetric approximation of the coherent Hamiltonian is constructed by deliberately omitting Hamiltonian terms that break or reduce the permutation symmetry of H_{coh}. When such terms are omitted, the Hamiltonian commutes with all members of a permutation group , which has a larger dimension than _{coh}:
and
The symbol ⊂ denotes “is a subset of”.
For example, consider a 2spin1/2 system with a small chemical shift difference. Although the true permutation group _{coh} is trivial and has only one element (eqn (1.20)), the larger group of eqn (1.19) may be used for , providing that the symmetrybreaking chemical shift difference is small compared to the symmetryimposing Jcoupling. This regime is called nearequivalence.^{51 } The major consequence of this approximation is to limit the lifetime of the longlived states. Such contributions to the decay of the longlived states are classified as coherent leakage.
If the symmetrybreaking terms (such as chemical shift differences) are too large, on the other hand, no plausible approximation may be available for expanding the symmetry group _{coh}. Resonant radiofrequency fields may be used in this case to allow the group _{coh} to be approximated by a larger group .
Another example is found in fourspin1/2 systems. Consider a magnetically inequivalent system of the AA′XX′ type, where the symmetry of the coherent Hamiltonian is described by the group in eqn (1.25), which has 2 elements. The group of 4 elements in eqn (1.24) may be used for if the terms which break the magnetic equivalence are small enough to be ignored, as an approximation. This method has been used implicitly for several demonstrations of longlived states in magnetically inequivalent fourspin systems.^{34–36,105,109 } In some cases, however, the breaking of magnetic equivalence is severe and must be identified with the smaller group in eqn (1.25).^{105 }
1.3.3.2 Symmetric Approximation of the Fluctuating Hamiltonian
The fluctuating Hamiltonian H_{fluc}(t) consists of the intra and intermolecular dipole–dipole interactions, the chemical shift anisotropy and spin–rotation interactions, and all other terms which are modulated by molecular motion and cause nuclear spin relaxation. In general, H_{fluc}(t) has little or no symmetry.
The symmetric approximation of the fluctuating Hamiltonian commutes with all members of a permutation group at all times:
where
The group is used for the symmetry analysis of the longlived states. The symmetrybreaking terms in H_{fluc}(t), which are excluded from , contribute to the decay of the longlived states.
An obvious starting point for this approximation is to exclude all intermolecular terms from the fluctuating Hamiltonian. Since molecules in solution encounter each other in widely variable geometrical configurations, intermolecular interactions have little or no symmetry. In the symmetry theory of longlived states in solution, intermolecular interactions are excluded from and therefore only contribute to decay mechanisms. Intermolecular contributions to longlivedstate relaxation have been explored experimentally^{110 } and analysed theoretically.^{67 }
The dipole–dipole interaction between two spins I_{1} and I_{2} has exchange symmetry (eqn (1.5)). This symmetry is maintained even when the molecule rotates or its geometry fluctuates. Hence, if the molecular spin system contains only two spins, and the dipole–dipole interaction dominates, all interactions except the dipole–dipole coupling may be excluded from . This applies even for heteronuclear 2spin systems.^{78 } The permutation group for the symmetric approximation of the fluctuating Hamiltonian contains the exchange element:
All other fluctuating Hamiltonians do not have this symmetry in general, and hence contribute to the decay mechanisms of the longlived state.
There are many cases where the spin pair is reasonably wellisolated but where the dipole–dipole interaction does not dominate the fluctuating spinpair Hamiltonian. For example, in the ^{13}C_{2}labelled naphthalene derivative described in ref. 40, chemical shift anisotropy interactions dominate the ^{13}C spin relaxation at high magnetic field. However, the careful molecular design of this compound ensures that the chemical shift anisotropy tensors of the two ^{13}C sites are highly correlated, so that exchange symmetry applies to a good approximation for these interactions as well.^{40 } A similar property applies for ^{13}C_{2}labelled ethyne derivatives,^{47,69 } and also in the case of ^{15}N_{2}labelled nitrous oxide (N_{2}O), where the ^{15}N relaxation is dominated by the spinrotation mechanism, but the strong correlation of the spinrotation tensors of the two ^{15}N sites again ensures approximate exchange symmetry.^{4,44,49 } So in these cases, the permutation group of the symmetric approximation of the fluctuating Hamiltonian is again given by:
The symmetry of the fluctuating Hamiltonian is usually very limited for systems of more than 2 coupled spins. In general, molecular vibrations and conformational flexibility distort the geometry away from a symmetrical idealized structure, so that a “snapshot” of the nuclear positions would not display a high degree of spatial symmetry, even in the case that the formal molecular structure is highly symmetric.
Furthermore, the fluctuating Hamiltonian does not normally display a high level of permutation symmetry, even when an average is taken over highfrequency vibrational modes, so that the formal point group symmetry is restored. An exception arises when the point group of the local molecular structure contains a spatial inversion element. An example of this is for ^{13}C_{2}labelled fumarate ((2E)but2enedioate), where the two ^{13}C nuclei and the directly bonded protons comprise a fourspin1/2 system.^{105 } The idealised spatial configuration of these four nuclei has a centre of inversion, so that the fluctuating Hamiltonian commutes with all members of the following permutation group:
The doubleexchange element is absent in the case of ^{13}C_{2}labelled maleate ((2Z)but2enedioate), which lacks the inversion centre^{105 } (see Chapter 2).
In the case of the three protons of a methyl (CH_{3}) group, the dipole–dipole interaction tensors do not display permutation symmetry, even after averaging over fast vibrational modes. In the idealized geometry of an equilateral triangle (point group symmetry C_{3v}), the threespin dipole–dipole Hamiltonian is given through eqn (1.6) by:
All three dipole–dipole coupling constants b are the same after vibrational averaging. Nevertheless, this Hamiltonian does not commute with the elements of the permutation group C_{3} given in eqn (1.13). The spin permutation operation such as (123) clearly leaves the second term in this expression invariant. However, this does not apply for the first term, since the permutation acts on the spin operators but not the spatial positions of the nuclei, as defined by the unit vectors r̂_{i}. The three dipole–dipole coupling terms are not related by exchange symmetry, since the internuclear vectors are not parallel. So, the dipole–dipole interaction terms are not symmetry related, even for idealised 3fold symmetric point group symmetry.
However, consider the case where the methyl group rotation is very fast compared to the overall tumbling of the molecule, i.e. τ_{R}≪τ_{c}, where τ_{R} is the correlation time for methyl rotation and τ_{c} is the correlation time for molecular tumbling. In this case there is a separation of timescales and it is feasible to average the fluctuating Hamiltonian over a time interval that is long compared to τ_{R} but short compared to τ_{c}. This partially averaged Hamiltonian acquires a further time dependence due to the molecular tumbling, and it is this modulation that drives the relaxation. The key point is that the averaging of the fluctuating Hamiltonian over the 3fold rotation of the methyl group imposes permutation symmetry. We denote the quantity 〈r̂$1T$r̂_{2}〉_{R} by the average of the dyadic r̂$1T$r̂_{2} over the 3fold rotation. This is the same for all pairs of spins, which allows the notation
The dipole–dipole Hamiltonian, averaged over the 3fold rotation, may be written
which does have C_{3} permutation symmetry. Similar expressions may be written for other interactions such as the chemical shift anisotropy. Hence in this case the symmetric approximation to the fluctuating Hamiltonian may be identified with the 3fold rotational average:
and the relevant symmetry group is:
In practice, there is always an incomplete separation of timescales between methyl rotation and overall molecular tumbling. The breakdown of the approximation τ_{R}≪τ_{c} leads to decay of the longlived states. The dependence of longlivedstate relaxation on the correlation times τ_{R} and τ_{c} has been analyzed in detail.^{38 }
The generation of effective symmetry by averaging over rapid internal motion is likely to be a common phenomenon in the longlived state NMR of flexible molecules. It may play a significant role in compounds that display remarkably longlived states even though their formal geometry suggests that such long lifetimes are unlikely.^{34,35,111,112 }
1.3.3.3 The Permutation Group of the Symmetric Approximate Hamiltonian
The symmetric approximate Hamiltonian is given by the sum of the terms and (eqn (1.26)). H^{☆} commutes with all elements of a permutation group denoted ^{☆}:
The commutation relationships in eqn (1.39) must apply at all times, even though the Hamiltonian H^{☆} is, in general, timedependent.
The permutation group ^{☆}, which describes the overall symmetry of H^{☆}, consists of the permutation operations that are common to both groups and . This may be written as follows:
where the symbol ∩ denotes the intersection of the two groups.
For example, consider a nearequivalent 2spin1/2 system with a dominant dipole–dipole relaxation mechanism. The symmetric approximate forms of the coherent and fluctuating Hamiltonians have the same symmetry group in this case:
Since the groups are identical their intersection is also the same:
The same happens with the threeproton system of a rapidly rotating methyl group. Since all three protons are magnetically equivalent, and the fluctuating Hamiltonian may be averaged over the 3fold rotation (eqn (1.36)), the symmetry groups are given by:
The situation with 4spin1/2 systems is more subtle, and is discussed in Chapter 2.
1.3.4 The Number of Longlived States
How many longlived states are there?
Now, suppose that the group ^{☆} has a set of irreducible representations (irreps), denoted here {ϒ_{1}, ϒ_{2}⋯} and suppose that there are N_{ϒ} such irreps. For the case of twospins1/2 coupled by the dipole–dipole interaction, there are two irreps (N_{ϒ}=2), denoted g and u. For the three protons of a rapidly rotating methyl group, there are three irreps (N_{ϒ}=3), denoted A, E_{a} and E_{b}. In all cases, the eigenstates of H^{☆} may be partitioned into N_{ϒ} disconnected state manifolds, each belonging to a different irrep. We denote the number of states in a manifold belonging to irrep ϒ_{u} by the symbol n_{ϒu}. Hence, the total number of spin states (the dimension of Hilbert space) is given by
For each manifold, there is a constant of the motion, given by the total population of that manifold. Hence the number of constants of the motion must be at least N_{ϒ}:
In certain cases, which are not discussed here, the number of constants of the motion N_{COM} exceeds the number of irreps N_{ϒ} of the group , because of additional selection rules. A discussion of this topic is beyond the scope of this chapter and will be deferred for a future publication.
These constants of the motion are not linearly independent, since the total population of all states is (trivially) also a constant of the motion (eqn (1.10)). Hence, there are only N_{COM}−1 linearly independent, nontrivial constants of the motion. The number of longlived states is therefore given by
Hence the number of nontrivial longlived states is greater, or equal to, the number of irreducible representations of the permutation group , minus one.
1.3.5 Longlived State Operators
For simplicity, we now ignore the possibility of additional selection rules and employ the equality sign in eqn (1.46). In this case, the longlived states correspond to population imbalances between the disconnected state manifolds, each manifold corresponding to a different irrep of the relevant permutation group ^{☆}, which in turn defines the symmetry of the Hamiltonian H^{☆}. Each longlived state is described by an operator Q$LLS(q)$ where the index q runs from 1 to N_{LLS}. Each longlived state operator has the form
where the notation ϒ_{u}〉_{s} represents one of the n_{ϒu} states belonging to the manifold for irrep ϒ_{u}. Each longlived state is characterized by a vector of real coefficients with dimension N_{ϒ}:
where the coefficients are real, and sum to 1
The coefficient vectors are orthogonal for different longlived states:
Eqn (1.47) is the general form of a longlived state operator. It has a rather fearsome aspect, but is simpler than it looks. Consider again the nearequivalent twospin1/2 system with dominant dipole–dipole relaxation. The relevant group ^{☆} is given by eqn (1.42) and has two irreps denoted g and u (N_{Γ}=2). The g irrep contains the three triplet states (n_{g}=3), The u irrep contains the lone singlet state (n_{u}=1). Since there are two irreps (N_{Γ}=2), there is only one nontrivial longlived state (from eqn (1.46)). The coefficients must satisfy so the only possibility, within an unimportant multiplicative factor, has a$g(1)$=−1 and a$u(1)$=1, i.e.
An evaluation of eqn (1.47) gives the singlet order operator in eqn (1.11).
Similarly, for the case of a rapidlyrotating methyl group, the relevant group ^{☆} is given in eqn (1.43) and has three irreps denoted A, E_{a} and E_{b} (N_{Γ}=3). The eight eigenstates of the 3spin1/2 system partition between the three manifolds is as follows: n_{A}=4 and n_{Ea}=n_{Eb}=2. There are two nontrivial longlived states (q=1 or 2), with coefficients which satisfy . In this case, it is not possible to assign the longlived states unambiguously on purely mathematical grounds. This ambiguity may be resolved by invoking a physical argument. Since states belonging to the irreps E_{a} and E_{b} are exceedingly hard to distinguish spectroscopically, the only longlived state with plausible experimental access has . The only solution satisfying this constraint has the vector
within an unimportant multiplicative factor. This corresponds to the AE imbalance operator in eqn (1.14). From the orthogonality condition (eqn (1.50)), the second longlived state operator is characterized by the vector
within a multiplicative factor. This represents the population imbalance between the E_{a} and E_{b} manifolds. However, this mode of spin order is experimentally inaccessible, as far as I know.
Longlived spin order modes of a similar nature have been identified in larger spin systems.^{35,36,65,105,113,114 }
1.4 Longlived State NMR
A typical NMR experiment involving longlived states^{5,39 } involves four main stages: (i) excitation, meaning that nuclear spin order is deposited in one or more longlived state operators; (ii) evolution, meaning that the longlived spin order is allowed to evolve in time, which usually implies a decay with a characteristic time constant; (iii) read out, meaning that the longlived spin order, which has no magnetic moment, is converted into nuclear magnetization; (iv) detection of the NMR signal. In addition, one or more filtering steps may be included before or after the evolution, in order to suppress NMR signals that do not pass through the longlived nuclear spin order.
Numerous examples of these elements of a longlivedstate NMR experiment are to be found in the following chapters. A brief overview of the main elements is given here.
1.4.1 Excitation
Spin order is deposited in one or more longlived state operators Q$LLS(q)$, starting from an available source of nuclear spin order. This may be nuclear magnetization, which arises in thermal equilibrium due to the Zeeman splitting of the nuclear energy levels in the strong magnetic field, or one of the possible sources of nonequilibrium nuclear spin order.
1.4.1.1 Thermal Magnetization
Thermal equilibrium of a sample in a strong magnetic field leads to the establishment of finite nuclear magnetization along the field direction. Although thermal equilibrium magnetization is very weak, it is highly reproducible and is the basis of the vast majority of NMR and MRI experiments. There are several methods for converting thermal equilibrium magnetization into longlived nuclear spin order. All methods require that some of the elements of the symmetry group ^{☆} are deactivated during the excitation phase of the experiment. Desymmetrization may be achieved by one of the symmetry switches described earlier (see Section 1.3.1), or simply by exploiting some of the symmetrybreaking interactions that were deliberately omitted when constructing the symmetric approximation to the spin Hamiltonian H^{☆}. In general, this may involve symmetrybreaking coherent terms (i.e. terms omitted from ), or symmetrybreaking fluctuating terms (i.e. terms omitted from ).
Numerous radiofrequency pulse sequences have been designed for magnetizationtosinglet conversion in the context of singlet NMR. These methods normally operate in particular coupling regimes, defined by the ratio of the symmetrybreaking terms (chemical shift or differences in Jcouplings to external coupling partners) to the symmetrymaking term (the Jcoupling to the members of the spin pair):
Weak coupling, or strong inequivalence. This regime applies when the difference in chemical shift frequencies (the symmetrybreaking term) greatly exceeds the Jcoupling between the members of the spin pair (the symmetrymaking term). In this regime, sequences of strong, short pulses (“hard pulses”), separated by carefully chosen delays, may be designed to accomplish the desirable conversion.^{2,9,72 } Smooth radiofrequency field modulations may also be used.^{60,115 } Another approach is used in fieldcycling singlet NMR experiments. In this case a pulse sequence is applied in high magnetic field, and prepares a state (called a singlet precursor state) that is adiabatically converted into longlived singlet order when the sample is transported into low magnetic field.^{1,49 }
Nearmagneticequivalence. This regime applies when the symmetrymaking term (the withinpair Jcoupling) greatly exceeds the symmetrybreaking terms (differences in outofpair Jcouplings, or chemical shift differences). Pulse sequences such as M2S,^{4,5,51 } SLIC,^{55 } and their variants^{116,117 } normally operate in this regime.
Intermediate coupling regime. The intermediate coupling regime, in which the symmetrybreaking and symmetrymaking terms have comparable magnitude, has not yet been thoroughly explored. Proposed solutions in this regime include radiofrequency pulse sequences^{118 } and continuous modulations of the applied rf field.^{60,115 }
All techniques for converting nuclear magnetization into longlived nuclear spin order by means of pulse sequences are subject to rigorous bounds on the conversion efficiency, associated with the eigenvalue spectra of the relevant operators.^{119,120 }
1.4.1.2 Hyperpolarization
Hyperpolarization techniques, which lead to greatly enhanced nuclear magnetization, lead to spectacular increases in NMR signal strength, especially in the context of dissolutionDNP (dynamic nuclear polarization).^{121 } Hyperpolarized nuclear magnetization may be converted into hyperpolarized longlived spin order by deploying one of the methods described above.^{25,47,63,64,122 }
In some cases, this conversion step is not necessary, since hyperpolarization generates longlived spin order directly, as a byproduct of enhanced magnetization.^{48 } If the polarization level of a 2spin1/2 system is denoted p (such that p=1 indicates complete nuclear polarization), then the degree of singlet polarization is given by −p^{2}/3. The negative sign arises because the strong enhancement of nuclear magnetization requires the generation of excess population of the nuclear triplet states, since only the triplet states support nuclear magnetization. As a result, the singlet state is depleted in this regime, giving rise to a negative singlet–triplet population imbalance.^{24,31,48,74 } Similar effects arise in hyperpolarized compounds containing methyl groups.^{62,65,114 }
1.4.1.3 Spinisomer Imbalances
Spin isomers arise through the Pauli principle for small, symmetrical molecules, or molecular fragments, with a high degree of rotational freedom. The Pauli principle entangles the spin and spatial quantum states, investing different spin symmetry species with the relatively large energy differences associated with spatial quantum mechanics. Since spin isomers often have significantly different energies, imbalances in populations between different spin isomer states may be generated by lowering the sample temperature. These population imbalances may persist for some time, since transitions between spin isomers are often slow in the absence of a suitable catalyst. They may therefore be exploited as a source of nuclear spin order.
Population imbalances between spin isomers are described by the same density operator terms as longlived states (eqn (1.47)). Hence the generation of spinisomer imbalances by thermal means provides a direct route to the generation of spin systems with strongly enhanced longlived spin order modes. This is called quantumrotorinduced polarization (QRIP).^{123,124 } The principle is exploited in parahydrogenenhanced NMR,^{11–13,28–30,33,41,52,53,77,91–97 } experiments involving freely rotating methyl groups^{37,38,62,104 } and has been demonstrated in an exotic chemical system containing freelyrotating water molecules.^{123,124 }
1.4.2 Filtering
In general, the final NMR signal detected in a longlivedstate NMR experiment does not exclusively derive from the longlived order. For example, universal bounds prevent the complete conversion of thermal magnetization into longlived spin order.^{119,120 } Other terms are generated as well, and these also contribute to the final NMR signal. A variety of filtering techniques have been developed, which suppress these signal components. The filtering methods may be based on phase cycles, or magnetic field gradients, or a combination of both.^{43,61 } This topic is explored in Chapter 10.
1.4.3 Evolution
The longlived spin order may be allowed to evolve and decay freely, or in the presence of a resonant radiofrequency field, which is required in some cases to suppress symmetrybreaking interactions^{2,7,8 } and certain relaxation mechanisms.^{70 }
Applications of longlived state NMR usually exploit the relatively long lifetime of the spin order by physical or chemical processes during the evolution interval.
The relaxation mechanisms of longlived order during the evolution interval are explored in Chapter 3.
1.4.4 Readout
The readout process is the reverse of the excitation process. For example, the pulse sequence S2M (singlettomagnetization) is constructed by reversing the chronological order of events in the pulse sequence M2S (magnetizationtosinglet).^{4 } In some cases, weak relaxation processes, involving symmetrybreaking components of H_{fluc} that are excluded from , are sufficient to bleed the longlived spin order into observable magnetization components.^{24,37,38,62 }
1.4.5 Detection
Conventional inductive NMR detection in high magnetic field is normally used. In special circumstances, such as ultralow magnetic field, alternative detection techniques such as atomic magnetometers may be used.^{78 }
I would like to express my appreciation to all the members of my research group who have helped develop these insights and perform the sometimeslaborious experiments over the years. In particular I would like to thank JeanNicolas Dumez, Gabriele Stevanato, Christian Bengs and Jyrki Rantaharju for illuminating theoretical discussions. I especially wish to thank Peppe Pileio for our work together on this topic over many years and for his splendid initiative in putting this book together. This research has been supported by EPSRC (UK) (grant numbers EP/P009980, EP/P005187 and EP/P030491) and the European Research Council (786707FunMagResBeacons).