Dipolar Recoupling Research Articles

Analysis of the MODIST Sequence for Selective Proton-Proton Recoupling.

Theoretical and simulated analyses of selective homonuclear dipolar recoupling sequences serve as primary tools for understanding and determining the robustness of these sequences under various conditions. In this article, we investigate the recently proposed first-order dipolar recoupling sequence known as MODIST (Modest Offset Difference Internuclear Selective Transfer). We evaluate the MODIST transfer efficiency, assessing its dependence on rf-field strengths and the number of simulated spins, extending up to 10 spins. This helps to identify conditions that enhance polarization transfer among spins that are nearby in frequency, particularly among aliphatic protons. The exploration uncovers a novel effect for first-order selective recoupling sequences that we term "facilitated dipolar recoupling". This effect amplifies the recoupled dipolar interaction between distant spins due to the presence of additional strongly dipolar-coupled spins. Unlike the third spin-assisted recoupling mechanism, facilitated dipolar recoupling only requires a coupling to one of the two distant spins of interest. Experimental demonstration of MODIST, including at different rf-field strengths, was carried out with the membrane protein influenza A M2 in lipid bilayers using 55 kHz magic-angle spinning (MAS). Reducing MODIST rf-field strength by a factor of 2 unveils possibilities for detecting Hα-Hα and HMeth-HMeth correlations with a 3D (H)C(H)(H)CH experiment under fast MAS rates, all achievable without specific spin labeling.

Dipolar Recoupling in Rotating Solids.

Magic angle spinning (MAS) nuclear magnetic resonance (NMR) has evolved significantly over the past three decades and established itself as a vital tool for the structural analysis of biological macromolecules and materials. This review delves into the development and application of dipolar recoupling techniques in MAS NMR, which are crucial for obtaining detailed structural and dynamic information. We discuss a variety of homonuclear and heteronuclear recoupling methods which are essential for measuring spatial restraints and explain in detail the spin dynamics that these sequences generate. We also explore recent developments in high spinning frequency MAS, proton detection, and dynamic nuclear polarization, underscoring their importance in advancing biomolecular NMR. Our aim is to provide a comprehensive account of contemporary dipolar recoupling methods, their principles, and their application to structural biology and materials, highlighting significant contributions to the field and emerging techniques that enhance resolution and sensitivity in MAS NMR spectroscopy.

Homonuclear J-couplings and heteronuclear structural constraints

In magic angle spinning (MAS) experiments involving uniformly 13C/15N labeled proteins, 13C–13C and 13C–15N dipolar recoupling experiments are now routinely used to measure direct dipole–dipole couplings that constrain distances and torsion angles and determine molecular structures. When the distances are short (<4 Å), the direct couplings dominate the evolution of the spin system, and the 13C–13C and 13C–15N J-couplings (scalar couplings) are ignored. However, for structurally interesting >4 Å distances, the dipolar and J-couplings are generally of comparable magnitude, and the variation in J must be included in order to optimize the precision of the experiment. This problem is circumvented in cases with well resolved spectra by using frequency-selective dipolar recoupling methods where the effects of J-couplings are refocused. However, for larger molecules with more spectral crowding, the requisite pulse length to achieve selectivity becomes long and leads to unacceptable sensitivity losses during the pulse or the spectral overlap precludes selective excitation. In this paper, we address this problem with two approaches aimed at facilitating higher precision internuclear distance measurements in systems that are not fully resolved. Namely, (1) we describe an approach for high precision measurements of specific J-couplings using the in-phase anti-phase (IPAP) sequence which is integrated into a non-selective dipolar recoupling technique and (2) we utilize the measured J-couplings to implement a double quantum filter experiment capable of providing the resolution necessary for frequency selective dipolar recoupling techniques without resorting to multidimensional spectroscopy. We illustrate these methods using a 7-peptide segment from the amyloidogenic Sup-35p protein, U-13C/15N-GNNQQNY, where we have measured 25 of the 27 possible one bond 13C–13C J-couplings.

Parallel β-Sheet Structure and Structural Heterogeneity Detected within Q11 Self-Assembling Peptide Nanofibers.

Q11 peptide nanofibers are used as a biomaterial for applications such as antigen presentation and tissue engineering, yet detailed knowledge of molecular-level structure has not been reported. The Q11 peptide sequence was designed using heuristics-based patterning of hydrophobic and polar amino acids with oppositely charged amino acids placed at opposite ends of the sequence to promote antiparallel β-sheet formation. In this work, we employed solid-state nuclear magnetic resonance spectroscopy (NMR) to evaluate whether the molecular organization within Q11 self-assembled peptide nanofibers is consistent with the expectations of the peptide designers. We discovered that Q11 forms a distribution of molecular structures. NMR data from two-dimensional (2D) 13C-13C dipolar-assisted rotational resonance indicate that the K3 and E9 residues between Q11 β-strands are spatially proximate (within ∼0.6 nm). Frequency-selective rotational echo double resonance (fsREDOR) on K3 Nζ and E9 Cδ-labeled sites showed that approximately 9% of the sites are close enough for salt bridge formation to occur. Surprisingly, dipolar recoupling measurements revealed that Q11 peptides do not assemble into antiparallel β-sheets as expected, and structural analysis using Fourier-transform infrared spectroscopy and 2D NMR alone can be misleading. 13C PITHIRDS-CT dipolar recoupling measurements showed that the most abundant structure consists of parallel β-sheets, in contrast to the expected antiparallel β-sheet structure. Structural heterogeneity was detected from 15N{13C} REDOR measurements, with approximately 22% of β-strands having antiparallel nearest neighbors. We cannot propose a complete structural model of Q11 nanofibers because of the complexity involved when examining structurally heterogeneous samples using NMR. Altogether, our results show that while heuristics-based patterning is effective in promoting β-sheet formation, designing a peptide sequence to form a targeted β-strand arrangement remains challenging.

Triclinic La7Zn2P11 with P3-, P24-, and P35- units: a combined study by 31P solid-state NMR spectroscopy and single crystal X-ray diffraction.

The ternary polyphosphide La7Zn2P11 was synthesized from the elements by using a salt flux or via a ceramic method in sealed quartz ampoules. The obtained samples were investigated by X-ray powder and single crystal diffraction: own type, P1̄, a = 775.33(13), b = 827.45(13), c = 1502.8(3) pm, α = 82.111(3), β = 77.034(3), γ = 89.996(3)°, wR2 = 0.1553, 5852 F2 values and 183 variables. This peculiar structure is characterized by the simultaneous presence of three distinct anionic phosphide species, namely P3-, P24-, and P35- units. La7Zn2P11 is an electron precise Zintl phase: (7La3+)21+(2Zn2+)4+(4P3-)12-(2P24-)8-(P35-). The P-P single bond distances range from 219.2 to 223.0 pm. The zinc sites show tetrahedral phosphorus coordination by three P3- and one P24- species. The tetrahedra are condensed to chains via common corners. The P35- units with P-P-P angles of 113.7° have exclusively lanthanum coordination. 31P solid-state NMR was used to probe the phosphorus local environments, connectivities and spatial proximities. The eleven crystallographically distinct phosphorus atoms were assigned with the help of two-dimensional homonuclear dipolar correlation experiments. Even though the application of 2D measurements on such phosphorus-based polyanionic compounds is exceedingly challenging because of the wide dispersion of chemical shifts, the fast irreversible decay of the transverse magnetization, and slow spin-lattice relaxation, a complete assignment is possible using radiofrequency-driven dipolar recoupling (RFDR), J-RESOLVED and total-through-bond correlation with R-sequence (R-TOBSY) techniques.

Structural organization of phase-separated bioactive glasses and the clustering of Si, P, B, Na and F atoms investigated by solid-state NMR and Monte Carlo simulations

In this paper we studied the effect of NaBO2 addition to a phase-separated alkali-free bioactive glass with a composition of 38.49 SiO2 • 36.07 CaO • 19.24 MgO • 5.61 P2O5 • 0.59 CaF2. Microscopy reveals binodal phase separation involving two Si-containing microphases with a droplet size of ∼200 μm, driven by the thermodynamic LLPS mechanism. The local environments and spatial distribution of silicate, phosphate, and fluoride ions in this phase-separated system were studied, using 29Si, 31P, 11B, 19F, 25Mg, and 23Na nuclear magnetic resonance (NMR) and infrared spectroscopy. The silicate units are dominantly of the metasilicate (Si2) type. The phosphate units exist mostly as orthophosphate (P0) while the borate is present in the form of pyroborate (B1). Multinuclear dipolar re-coupling experiments indicate that the minority components F, P, B and Na all occur within a common phase. Thus, atomic distribution scenarios involving the separation of these components into separate phases can be excluded. The 31P spin echo decay (SED) method was used along with Monte Carlo simulations to characterize the spatial distribution of the phosphate component. Based on the analysis, the phosphate component forms clusters of sizes 1-4 nm, which are embedded in an environment more dilute in phosphate, having a random distribution. While 19F SED results indicate that the fluoride ions do not form clusters and are close to randomly distributed, dipolar recoupling of 31P suggests a local environment resembling that of fluorapatite.

Solid-State NMR Double-Quantum Dipolar Recoupling Enhanced by Additional Phase Modulation.

Additional phase modulation (APM) is proposed to generally enhance the theoretical efficiency of homonuclear double-quantum (DQ) recoupling in solid-state NMR. APM applies an additional phase list to DQ recoupling in steps of an entire block. The sine-based phase list can enhance the theoretical efficiency by 15-30 %, from 0.52 to 0.68 (non-γ-encoded recoupling) or from 0.73 to 0.84 (γ-encoded recoupling), with doubled recoupling time. The genetic-algorithm (GA) optimized APM can adiabatically enhance the efficiency to ∼1.0 at longer times. The concept of APM has been tested on SPR-51 , BaBa, and SPR-31 , which represent γ-encoded recoupling, non-γ-encoded recoupling, and another kind beyond the former two, respectively. Simulations reveal that enhancements from APM are due to the activation of more crystallites in the powder. Experiments on 2,3-13 C labeled alanine are used to validate the APM recoupling. This new concept shall shed light on developing more efficient homonuclear recoupling methods.

Time-domain proton-detected local-field NMR for molecular structure determination in complex lipid membranes.

Proton-detected local-field (PDLF) NMR spectroscopy, using magic-angle spinning and dipolar recoupling, is presently the most powerful experimental technique for obtaining atomistic structural information from small molecules undergoing anisotropic motion. Common examples include peptides, drugs, or lipids in model membranes and molecules that form liquid crystals. The measurements on complex systems are however compromised by the larger number of transients required. Retaining sufficient spectral quality in the direct dimension requires that the indirect time-domain modulation becomes too short for yielding dipolar splittings in the frequency domain. In such cases, the dipolar couplings can be obtained by fitting the experimental data; however ideal models often fail to fit PDLF data properly due to effects of radiofrequency field (RF) spatial inhomogeneity. Here, we demonstrate that by accounting for RF spatial inhomogeneity in the modeling of R-symmetry-based PDLF NMR experiments, the fitting accuracy is improved, facilitating the analysis of the experimental data. In comparison to the analysis of dipolar splittings without any fitting procedure, the accurate modeling of PDLF measurements makes possible three important improvements: the use of shorter experiments that enable the investigation of samples with a higher level of complexity, the measurement of C-H bond order parameters with smaller magnitudes and of smaller variations of caused by perturbations of the system, and the determination of values with small differences from distinct sites having the same chemical shift. The increase in fitting accuracy is demonstrated by comparison with H NMR quadrupolar echo experiments on mixtures of deuterated and non-deuterated dimyristoylphosphatidylcholine (DMPC) and with 1-palmitoyl-2-oleoyl--glycero-3-phosphoethanolamine (POPE) membranes. Accurate modeling of PDLF NMR experiments is highly useful for investigating complex membrane systems. This is exemplified by application of the proposed fitting procedure for the characterization of membranes composed of a brain lipid extract with many distinct lipid types.

Great Offset Difference Internuclear Selective Transfer.

Carbon-carbon dipolar recoupling sequences are frequently used building blocks in routine magic-angle spinning NMR experiments. While broadband homonuclear first-order dipolar recoupling sequences mainly excite intra-residue correlations, selective methods can detect inter-residue transfers and long-range correlations. Here, we present the great offset difference internuclear selective transfer (GODIST) pulse sequence optimized for selective carbonyl or aliphatic recoupling at fast magic-angle spinning, here, 55 kHz. We observe a 3- to 5-fold increase in intensities compared with broadband RFDR recoupling for perdeuterated microcrystalline SH3 and for the membrane protein influenza A M2 in lipid bilayers. In 3D (H)COCO(N)H and (H)CO(CO)NH spectra, inter-residue carbonyl-carbonyl correlations up to about 5 Å are observed in uniformly 13C-labeled proteins.

Selective excitation with recoupling pulse schemes uncover properties of disordered mineral phases in bone-like apatite grown with bone proteins

Bone construction has been under intensive scrutiny for many years using numerous techniques. Solid-state NMR spectroscopy helped unravel key characteristics of the mineral structure in bone owing to its capability of analyzing crystalline and disordered phases at high-resolution. This has invoked new questions regarding the roles of persistent disordered phases in structural integrity and mechanical function of mature bone as well as regarding regulation of early events in formation of apatite by bone proteins which interact intimately with the different mineral phases to exert biological control.Here, spectral editing tethered to standard NMR techniques is employed to analyze bone-like apatite minerals prepared synthetically in the presence and absence of two non-collagenous bone proteins, osteocalcin and osteonectin. A 1H spectral editing block allows excitation of species from the crystalline and disordered phases selectively, facilitating analysis of phosphate or carbon species in each phase by magnetization transfer via cross polarization. Further characterization of phosphate proximities using SEDRA dipolar recoupling, cross-phase magnetization transfer using DARR and T1/T2 relaxation times demonstrate that the mineral phases formed in the presence of bone proteins are more complex than bimodal. They reveal disparities in the physical properties of the mineral layers, indicate the layers in which the proteins reside and highlight the effect that each protein imparts across the mineral layers.

Front Cover: Specific Signal Enhancement on an RNA‐Protein Interface by Dynamic Nuclear Polarization (Chem. Eur. J. 16/2023)

Like bright stars in a dark sky, sparsely 13C-labeled methyl groups in the protein domain “illuminate” the binding pocket of a ribonucleoprotein (RNP) complex for specific solid-state NMR spectroscopy of the interaction surface by localized transfer of their hyperpolarization to 13C-labeled adenosines in the RNA domain. This is enabled by specific cross-relaxation enhancement by active motions under dynamic nuclear polarization (SCREAM-DNP) in combination with homonuclear dipolar recoupling. More information can be found in the Research Article by A. Marchanka, B. Corzilius and co-workers (DOI: 10.1002/chem.202203443).

Fast magic angle spinning for the characterization of milligram quantities of organic and biological solids at natural isotopic abundance by 13C–13C correlation DNP-enhanced NMR

We show that multidimensional solid-state NMR 13C–13C correlation spectra of biomolecular assemblies and microcrystalline organic molecules can be acquired at natural isotopic abundance with only milligram quantities of sample. These experiments combine fast Magic Angle Spinning of the sample, low-power dipolar recoupling, and dynamic nuclear polarization performed with AsymPol biradicals, a recently introduced family of polarizing agents. Such experiments are essential for structural characterization as they provide short- and long-range distance information. This approach is demonstrated on diverse sample types, including polyglutamine fibrils implicated in Huntington's disease and microcrystalline ampicillin, a small antibiotic molecule.

Improved resolution for spin-3/2 isotopes in solids via the indirect NMR detection of triple-quantum coherences using the T-HMQC sequence

The indirect NMR detection of quadrupolar nuclei in solids under magic-angle spinning (MAS) is possible with the through-space HMQC (heteronuclear multiple-quantum coherence) scheme incorporating the TRAPDOR (transfer of population in double-resonance) dipolar recoupling. This sequence, called T-HMQC, exhibits limited t1-noise. In this contribution, with the help of numerical simulations of spin dynamics, we show that most of the time, the fastest coherence transfer in the T-HMQC scheme is achieved when TRAPDOR recoupling employs the highest radiofrequency (rf) field compatible with the probe specifications. We also demonstrate how the indirect detection of the triple-quantum (3Q) coherences of spin-3/2 quadrupolar nuclei in solids improves the spectral resolution for these isotopes. The sequence is then called T-HMQC3. We demonstrate the gain in resolution provided by this sequence for the indirect proton detection of 35Cl nuclei in l-histidine∙HCl and l-cysteine∙HCl, as well as that of 23Na isotope in NaH2PO4. These experiments indicate that the gain in resolution depends on the relative values of the chemical and quadrupolar-induced shifts (QIS) for the different spin-3/2 species. In the case of NaH2PO4, we show that the transfer efficiency of the T-HMQC3 sequence employing an rf-field of 80 kHz with a MAS frequency of 62.5 kHz reaches 75% of that of the t1-noise eliminated (TONE) dipolar-mediated HMQC (D-HMQC) scheme.

Interaction frames in solid-state NMR: A case study for chemical-shift-selective irradiation schemes

Interaction frames play an important role in describing and understanding experimental schemes in magnetic resonance. They are often used to eliminate dominating parts of the spin Hamiltonian, e.g., the Zeeman Hamiltonian in the usual (Zeeman) rotating frame, or the radio-frequency-field (rf) Hamiltonian to describe the efficiency of decoupling or recoupling sequences. Going into an interaction frame can also make parts of a time-dependent Hamiltonian time independent like the rf-field Hamiltonian in the usual (Zeeman) rotating frame. Eliminating the dominant term often allows a better understanding of the details of the spin dynamics. Going into an interaction frame can also reduces the energy-level splitting in the Hamiltonian leading to a faster convergence of perturbation expansions, average Hamiltonian, or Floquet theory. Often, there is no obvious choice of the interaction frame to use but some can be more convenient than others. Using the example of frequency-selective dipolar recoupling, we discuss the differences, advantages, and disadvantages of different choices of interaction frames. They always include the complete radio-frequency Hamiltonian but can also contain the chemical shifts of the spins and may or may not contain the effective fields over one cycle of the pulse sequence.

Accurate analysis and perspectives for systematic design of magnetic resonance experiments using single-spin vector and exact effective Hamiltonian theory

Aimed at fundamental understanding and design of advanced magnetic resonance experiments on basis of Hamiltonians, we describe highly convergent and exact effective Hamiltonian methods which alleviate important deficits of current less accurate methods. This involves single-spin vector effective Hamiltonian theory (SSV-EHT) to first order in the interaction frame of rf and chemical shift offsets as well as exact effective Hamiltonian theory (EEHT) being an exact approach to average Hamiltonian theory not relying on interaction frame transformations. Bringing these methods together, we present tools to analyze challenging experiments in need of considering large static components in Hamiltonian (e.g., offsets) while economizing with radiofrequency irradiation power. It is demonstrated how the two complementary tools may provide important new insight into the detailed effective Hamiltonians of advanced NMR experiments, noting that the methods are by no means restricted to NMR. This is demonstrated for isotropic mixing in liquid-state NMR and dipolar recoupling in solid-state NMR where insight into the delicate interplay between bilinear two-spin and linear single-spin terms in the effective Hamiltonian may increase understanding of determinants for broadband excitation and the formation of recoupling resonances. Furthermore, we demonstrate how simple products single-spin effective Hamiltonians may be used as generators of multiple-spin effective Hamiltonians and though this a new approach to density operator calculations for large multiple-spin systems.

T1-noise elimination by continuous chemical shift anisotropy refocusing

Due to their high gyromagnetic ratio, there is considerable interest in measuring distances and correlations involving protons, but such measurements are compounded by the simultaneous recoupling of chemical shift anisotropy (CSA). This secondary recoupling adds additional modulations to the signal intensities that ultimately lead to t1-noise and signal decay. Recently, Venkatesh et al. demonstrated that the addition of CSA refocusing periods during 1H-X dipolar recoupling led to sequences with far higher stability and performance. Herein, we describe a related effort and develop a symmetry-based recoupling sequence that continually refocuses the 1H CSA. This sequence shows superior performance to the regular and t1-noise eliminated D-HMQC sequences in the case of spin-1/2 nuclei and comparable performance to the later for half-integer quadrupoles.

Modest Offset Difference Internuclear Selective Transfer via Homonuclear Dipolar Coupling.

Homonuclear dipolar recoupling is routinely used for magic-angle spinning NMR-based structure determination. In fully protonated samples, only short proton–proton distances are accessible to broadband recoupling approaches because of high proton density. Selective methods allow detection of longer distances by directing polarization to a subset of spins. Here we introduce the selective pulse sequence MODIST, which recouples spins that have a modest chemical shift offset difference, and demonstrate it to selectively record correlations between amide protons. The sequence was selected for good retention of total signal, leading to up to twice the intensity for proton–proton correlations compared with other selective methods. The sequence is effective across a range of spinning conditions and magnetic fields, here tested at 55.555 and 100 kHz magic-angle spinning and at proton Larmor frequencies from 600 to 1200 MHz. For influenza A M2 in lipid bilayers, cross-peaks characteristic of a helical conformation are observed.

Theory of frequency-selective homonuclear dipolar recoupling in solid-state NMR.

In solid-state nuclear magnetic resonance, frequency-selective homonuclear dipolar recoupling is key to quantitative distance measurement or selective enhancement of correlations between atoms of interest in multiple-spin systems, which are not amenable to band-selective or broadband recoupling. Previous frequency-selective recoupling is mostly based on the so-called rotational resonance (R2) condition that restricts the application to spin pairs with resonance frequencies differing in integral multiples of the magic-angle spinning (MAS) frequency. Recently, we have proposed a series of frequency-selective homonuclear recoupling sequences called SPR (short for Selective Phase-optimized Recoupling), which have been successfully applied for selective 1H-1H or 13C-13C recoupling under from moderate (∼10kHz) to ultra-fast (150kHz) MAS frequencies. In this study, we fully analyze the average Hamiltonian theory of SPR sequences and reveal the origin of frequency selectivity in recoupling. The theoretical description, as well as numerical simulations and experiments, demonstrates that the frequency selectivity can be easily controlled by the flip angle (p) in the (p)ϕk(p)ϕk+π unit in the pSPR-Nn sequences. Small flip angles lead to frequency-selective recoupling, while large flip angles may lead to broadband recoupling in principle. The result shall shed new light on the designof homonuclear recoupling sequences with arbitrary frequency bandwidths.

Maximizing efficiency of dipolar recoupling in solid-state NMR using optimal control sequences.

Dipolar recoupling is a central concept in the nuclear magnetic resonance spectroscopy of powdered solids and is used to establish correlations between different nuclei by magnetization transfer. The efficiency of conventional cross-polarization methods is low because of the inherent radio frequency (rf) field inhomogeneity present in the magic angle spinning (MAS) experiments and the large chemical shift anisotropies at high magnetic fields. Very high transfer efficiencies can be obtained using optimal control–derived experiments. These sequences had to be optimized individually for a particular MAS frequency. We show that by adjusting the length and the rf field amplitude of the shaped pulse synchronously with sample rotation, optimal control sequences can be successfully applied over a range of MAS frequencies without the need of reoptimization. This feature greatly enhances their applicability on spectrometers operating at differing external fields where the MAS frequency needs to be adjusted to avoid detrimental resonance effects.

35 Cl-1 H Heteronuclear correlation magic-angle spinning nuclear magnetic resonance experiments for probing pharmaceutical salts.

Heteronuclear multiple-quantum coherence (HMQC) pulse sequences for establishing heteronuclear correlation in solid-state nuclear magnetic resonance (NMR) between 35 Cl and 1 H nuclei in chloride salts under fast (60 kHz) magic-angle spinning (MAS) and at high magnetic field (a 1 H Larmor frequency of 850 MHz) are investigated. Specifically, recoupling of the 35 Cl-1 H dipolar interaction using rotary resonance recoupling with phase inversion every rotor period or the symmetry-based SR42 1 pulse sequences are compared. In our implementation of the population transfer (PT) dipolar (D) HMQC experiment, the satellite transitions of the 35 Cl nuclei are saturated with an off-resonance WURST sweep, at a low nutation frequency, over the second spinning sideband, whereby the WURST pulse must be of the same duration as the recoupling time. Numerical simulations of the 35 Cl-1 H MAS D-HMQC experiment performed separately for each crystallite orientation in a powder provide insight into the orientation dependence of changes in the second-order quadrupolar-broadened 35 Cl MAS NMR lineshape under the application of dipolar recoupling. Two-dimensional 35 Cl-1 H PT-D-HMQC MAS NMR spectra are presented for the amino acids glycine·HCl and l-tyrosine·HCl and the pharmaceuticals cimetidine·HCl, amitriptyline·HCl and lidocaine·HCl·H2 O. Experimentally observed 35 Cl lineshapes are compared with those simulated for 35 Cl chemical shift and quadrupolar parameters as calculated using the gauge-including projector-augmented wave (GIPAW) method: the calculated quadrupolar product (PQ ) values exceed those measured experimentally by a factor of between 1.3 and 1.9.