Polychromatic Excitation of Delocalized Long-Lived Proton Spin States in Aliphatic Chains

Long-lived states (LLS) involving pairs of magnetically inequivalent but chemically equivalent proton spins in aliphatic (CH$_2$)$_n$ chains can be excited by simultaneous application of weak selective radio-frequency (RF) fields at n chemical shifts by polychromatic spin lock induced crossing (poly-SLIC). The LLS are delocalized throughout the aliphatic chain by mixing of intrapair singlet states and by excitation of LLS comprising products of four or six spins. The measured lifetimes T$_{LLS}$ in a model compound are about 5 times longer than T1, and are strongly affected by interactions with macromolecules.

In molecules that contain nearly equivalent spins, either with slightly different chemical shifts, or with slightly different scalar couplings to neighbouring spins, a two-spin LLS is almost an eigenstate and therefore does not require any RF fields to be sustained [4,21,33]. Such molecules often require challenging chemical synthesis of isotopically enriched molecules with nearequivalent 13 C [6,7,33,34] or 15 N [10,35,36] spin pairs, although molecules that contain nearly equivalent proton spins have also been studied [4,37]. Here we show that LLS can be readily excited and observed in CH2 groups of common aliphatic chains, which are ubiquitous in chemistry.
In achiral molecules that contain at least two neighbouring CH2 groups, the two protons of each CH2 group are chemically equivalent (i.e., have the same chemical shifts) but they are generally magnetically inequivalent (i.e., have different scalar couplings to neighbouring CH2 groups) [38]. Provided the substituents R and R' in a molecule R(CH 2 ) n R' are distinct, neighboring CH2 subunits are often weakly coupled to each other in high magnetic fields, i.e., Jcouplings between them are much smaller than difference between their chemical shifts. In a chain with = 3 neighboring weakly coupled CH2 groups, the spin system of the 6 protons can then be denoted by ' ' ' [39]. A good approximation of the 64 eigenstates of the (Î + Î ′ )(Î + Î ′ ) ; (3) Here = + ′ and = − ′ , etc. Note that ′ = ′ and = ′ ′ etc., which is confirmed by analysis of experimental spectra (see Table S4). All terms in Eq. (2) (3) are gerade with respect to all possible local intra-pair spin permutations ( ′), ( ′), or ( ′); those in Eq. (4) and (5) are ungerade, which allows one to create the LLS. Due to structural similarities, all intrapair J-couplings have similar values so that it is convenient to consider their average = 1 3 ( ′ + ′ + ′ ) . Intrapair J-couplings "protect" LLS against coherent interconversion between intra-pair singlet and triplet states, since these couplings are much larger than those that break the symmetry ≫ = 1 2 ( + ).
So far, applications of the spin-lock induced crossing (SLIC) method [5,40] have been limited to monochromatic irradiation at a single radio-frequency. Among other applications, SLIC has been used for heteronuclear systems of 13 C enriched diphenylacetylene [6,41] and 15 N enriched azobenzene [10,42], where the four ortho protons (ignoring the meta and para protons) and the two 15 N nuclei constitute an ' '' ''' ' system. Somewhat counter-intuitively, the application of a monochromatic SLIC sequence at the common shift of the four ortho protons excites an LLS that involves the two 15 N nuclei. This constitutes evidence for the mixing of LLS along chains of coupled nuclei.
In this work, we introduce polychromatic SLIC (poly-SLIC). In ' ' ' systems, one may choose to apply one, two or three carrier frequencies with weak RF amplitudes, yielding 9 possible methods for magnetization-to-singlet conversion via double-quantum (DQ) or singlequantum (SQ) level anti-crossings (LACs), namely: Triple-frequency SLIC with DQ LAC by simultaneous irradiation at three shifts , , and .
The efficiencies (quantum yields) of these 9 methods, along with a discussion of the SQ and DQ LACs, are given in the Supplemental Material. As shown in Figure 1, the magnetization is first converted into LLS by one of 9 possible methods. The resulting LLS decays slowly in the relaxation interval . A T00 filter [27] then eliminates all off-diagonal elements of the density matrix, except for zero-quantum coherences. The remaining terms are then re-converted back into observable magnetization, again by one of 9 possible methods (see Tables S2 and S3). The most efficient scheme uses triple SLIC excitation and single SLIC reconversion to maximize the observable magnetization of one of the spin pairs. Pulse sequences for poly-SLIC applied to a ′ ′ ′ system of a chain of 3 neighboring CH2 groups. All 9 methods start by exciting transverse magnetization by a 'hard' ( /2) pulse, followed by the simultaneous application of one, two or three selective RF fields applied simultaneously at resonance frequencies (chemical shifts) and/or and/or with phases ± . The poly-SLIC pulses convert the magnetization into LLS. The RF amplitudes must be = 2 to achieve LLS excitation by single-quantum level anti-crossing (SQ LAC), or = for double-quantum (DQ) LAC. Maximum efficiency is achieved when = 1 (|√2 |) ⁄ for SQ LAC and = √2 for DQ LAC. After a 00 filter [43], poly-SLIC sequences allow one to reconvert LLS into observable magnetization.
The Hamiltonian for one, two or three RF fields that are on-resonance with 2, 4 or 6 spins ∈ { , ′, , ′, , ′} is given by:   (7). The superposition of two-spin product terms (yellow) has coefficients above 15% after triple SLIC excitation, the four-spin product terms (green) have coefficients below 5%, while the six-spin product term (red) has a negligible amplitude. The blue lines correspond to the trajectory of the magnetization that is partly converted into LLS and back. All simulations were carried out with Spin Dynamica [44].
Numerical simulations have been performed to determine which LLS terms are excited by poly-SLIC (see Supplement). Inter alia, the following delocalised density operator terms are obtained: . The bilinear terms are familiar products of m = 2 spin operators, while the higher terms contain unusual products of m = 4 and 6 spin operators. All operators in Eq. (7) correspond to population imbalances. Within the gerade manifold, ̂ comprises an imbalance between the mean population of the 9 states (uug, ugu, guu) and the mean population of the 27 states (ggg). Within the ungerade manifold, there is an imbalance between the population of the unique singlet (uuu) state and the mean population of the 27 states (ugg, gug, ggu).
Although many other terms are excited, they relax rapidly or can be eliminated by filtration and phase cycling. The commutator between the average two spin-terms 1 3 ⁄ ( ′ + ′ + Polychromatic Excitation of Delocalized Long-Lived Proton Spin States in Aliphatic Chains ′ ) 2 (̂•̂′ +̂•̂′ + ̂•̂′ ) and the free-precession Hamiltonian yields a negligible residue. The same is true for the sum of the three four-spin terms, as well as for the six-spin term.
Therefore, these terms are nearly invariant under free evolution, as evidenced by the weak oscillations during free evolution shown in Figure 2.
One should distinguish two aspects of the delocalized nature of the LLS: (i) delocalization by mixing of two-and four-spin states, and (ii) delocalization by excitation of products comprising four-and six-spin states. A manifestation of delocalization by mixing is the experimental fact that a monochromatic SLIC pulse applied only to the ' pair also excites LLS associated with the ' and ' pairs ( Figure 4). The second (more remarkable) effect is that we can create LLS that are delocalized over n neighboring CH2 groups because the density operator is made up of a product, rather than a sum, of population imbalances, such as the six-spin term (̂•̂′)(̂•̂′)(̂•̂′).
Such states are rarely observed in NMR and may be of interest for quantum information processing. A dramatic simplification of multiplets such as those shown in Figure 3 can be achieved by polychromatic homonuclear decoupling during the observation of the FID's [45], so that each multiplet collapses to a single line, as will be shown elsewhere.  pulses are applied at one and the same frequency (green diamonds in Figure 4) gives a that is slightly shorter than the other experiments which yield similar . This may be attributed to the fact that in this case the LLS is predominantly localized on the AA' pair (see tables S1, S2). It is straightforward to excite LLS in potential drug molecules that contain neighboring CH2 groups. The contrast between the relaxation rates = 1/ of free and partly bound drug molecules allows one to determine the affinity (binding constants) for macromolecules such as proteins [16][17][18]. Binding may partly lift the chemical equivalence, i.e., convert an ' ' ' system into an system. This will contribute to a leakage between intra-pair 0 and 0 states, thus levelling out the population imbalances, reducing TLLS, and enhancing the contrast between free and bound molecules. By way of example, 5 mM DSS has been titrated with bovine  Figure S1). These off-diagonal elements provide access LLS by SLIC. The differences between the corresponding diagonal elements are of the order of twice the intra-pair 2Jintra couplings, i.e., much larger than the off-diagonal elements that are of the order of where double-headed arrows represent off-diagonal elements of the Hamiltonian ̂ and ̂ (see Eqs. 4 and 5 in the main text). Of the 28 states within the globally ungerade lower manifold, only 3 are connected by 2 off-diagonal elements: And the other in the globally ungerade manifold: Both can be diagonalized by the unitary transformation: Finally, there are four 2x2 blocks between globally ungerade states: All of them are diagonalized by a simple unitary transformation: This leads to mixing of the states of the intra-pair singlet-triplet product basis.
An important consequence is that the eigenfunctions of the Hamiltonian are delocalized along the chain. Thus, one of the delocalized eigenstates is: Eigenstates described by such linear combinations cannot be attributed to any particular spin pair, since they are delocalized across the entire 6-spin system. Consequently, when an LLS is created by irradiating spin pair ′ there is always some LLS that is associated with the other pairs ′ and ′.
Note that the resulting basis is not the exact eigenbasis, since the 16 off-diagonal elements highlighted by red squares in Figure S1 have been neglected, which is justified by perturbation theory, since 2Jintra >> . Figure S2 shows the energy levels of the 6-spin ′ ′ ′ system during triple RF irradiation, calculated as a function of the RF field amplitude , which is the same for all three selective RF fields. The eigenstates of the Hamiltonian obtained by diagonalization in the presence of the RF fields are known as 'dressed states'. Since the intra-pair couplings in all three CH2 groups are similar, we consider their average value = 1 3 ( ′ + ′ + ′ ). The key differences between the out-of-pair J-couplings that allow one to excite LLS are approximated by their average values = 1 2 ( + ). The experimental couplings in Table S4 show that these assumptions are reasonable.

Level anti-crossings under polychromatic SLIC pulses
A dressed state is characterized by an effective total spin and its projections . The four sets of lines in Figure S2 correspond to dressed states with effective total spin = 3 and 2 (orange) and = 1 or 0 (blue). For = 3, there are 7 levels with = −3, −2, … , +3. When matches the conditions = , level anti-crossings (LACs) occur between dressed states with effective total spin = 3 and = 1 in the globally gerade manifold ( Figure S2a). Another LAC is found in the globally ungerade block between the dressed states with effective total spin = 2 and 0 ( Figure S2b).  villian. This is beyond the scope of this work. Experimental results (see Figure 4 in the main text) show that the decays are mono-exponential.

Quantum yields of the excitation of LLS
Numerical simulations show that the long-lived components of density matrix created after poly-SLIC can be represented by Eq. (7) in the main text, where ̂ is represented by a linear combination of normalised orthogonal operators, so that values of the coefficients can be compared with each other. Depending on the excitation scheme (we distinguish 9 methods in the main text), one obtains different values for the coefficients Table S1.

Titration of DSS with bovine serum albumin (BSA)
To evaluate the potential of delocalized LLS in aliphatic chains for drug screening, titration experiments were performed to determine the contrast CLLS between free DSS and DSS that is weakly interacting with the protein bovine serum albumin (BSA) ( Figure S4)

Experimental details
All experiments were conducted using a Bruker NEO spectrometer with a static field 0 = 11.9 T (proton Larmor frequency 0 = 500 MHz) with a 10 mm broadband double resonance BBO probe. The RF inhomogeneity leads to an intensity loss of 16 % after a 2π pulse.
The durations of the SLIC pulses were optimized experimentally to = 110 ms and = 156 ms for DQ LAC. The RF amplitudes were optimised to = 14 Hz (for DQ LAC) or 28 Hz (for SQ LAC). Superpositions of two or three simultaneous selective pulses were generated by phase modulation. The recovery delay between subsequent experiments was set to 60 s to allow the LLS to decay completely. The initial ( /2) proton pulses of 25 µs duration had an RF-field amplitude of 10 kHz. All spin simulations were obtained with Spin Dynamica [1]. The J-couplings in a conventional 1 H NMR spectrum were determined with ANATOLIA [2]. All experiments except those in Figure S4 were performed at 300 K on samples containing 33 mM DSS (Sigma Aldrich) dissolved in D2O without degassing and without buffer. The titration experiments shown in Figure   S4 were obtained at 800 MHz and 298 K with [DSS] = 5 mM, 0 < [BSA] < 20 µM in 15 mM phosphate buffer in D2O at pH = 7. Each point shown in Figure S4 was acquired using the sequence of Figure   1 of the main text with 8 different delays rel, requiring ca. 13 min per point. The T00 filter [3] consisted of three non-selective /2 pulses and three pulsed field gradients to eliminate terms with ranks = 1 and 2 and coherence orders = 1 and 2. A 4-step phase alternation of the SLIC pulses along the ±y axes with concomitant addition and subtraction of the signals was used to suppress undesirable pathways [4]. The zero-quantum coherences were not suppressed.