Condensed-phase relaxation of multilevel quantum systems. I. An exactly solvable model

An analytically solvable model of multilevel condensed-phase quantum dynamics relevant to vibrational relaxation and electron transfer is presented. Exact solutions are derived for the reduced system density matrix dynamics of a degenerate N-level quantum system characterized by nearest-neighbor hopping and off-diagonal coupling (which is linear in the bath coordinates) to a harmonic oscillator bath. We demonstrate that for N> 2 the long-time steady-state system site occupation probabilities are not the same for all sites; that is, they are distributed in a non-Boltzmann manner, which depends on the initial conditions and the number of levels in the system. Although the system-bath Hamiltonian considered here is restricted in form, the availability of an exact solution enables us to study the model in all regions of an extensive parameter space.     

Exact solution of a model of condensed-phase electron transfer with non-Condon effects

Non-Condon effects are important in the analysis of electron transfer in many systems coupled to a condensed-phase environment. We detail a novel condensed-phase electron transfer Hamiltonian that extends the spin-boson model to account for non-Condon effects. We show that the relevant reduced system density matrix dynamics can be calculated exactly for a particular class of bath Hamiltonians and system-bath couplings. An explicit formula for the long-time behavior of these systems is derived. We show that they exhibit non-Boltzmann long-time behavior that is independent of temperature, and depends on the system Hamiltonian and the initial system density matrix.     

Deuterium spin probes of side-chain dynamics in proteins. 1. Measurement of five relaxation rates per deuteron in (13)C-labeled and fractionally (2)H-enriched proteins in solution

New pulse sequences are presented for the measurement of the relaxation of deuterium double quantum, quadrupolar order, and transverse antiphase magnetization in (13)CH(2)D methyl groups of (15)N-, (13)C-labeled, fractionally deuterated proteins. Together with previously developed experiments for measuring deuterium longitudinal and transverse decay rates [Muhandiram, D. R.; Yamazaki, T.; Sykes, B. D.; Kay, L. E. J. Am. Chem. Soc. 1995, 117, 11536], these schemes allow measurement of the five unique decay constants of a single deuteron, providing an unprecedented opportunity to investigate side-chain dynamics in proteins. All five deuterium relaxation rates have been measured for deuterons in the methyl groups of the B1 immunoglobulin binding domain of peptostreptococcal protein L and the N-terminal SH3 domain from the protein drk. Since values of the spectral density function at only three different frequencies contribute to the five relaxation rates, the self-consistency of the relaxation data is readily established. Very good agreement is obtained between calculated parameters describing the amplitudes and time scales of motion when different subsets of the relaxation data are employed.     

Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. I. Linear polymer viscoelasticity

Shear stress relaxation modulus GS(t) curves of entanglement-free Fraenkel chains have been calculated using Monte Carlo simulations based on the Langevin equation, carrying out both in the equilibrium state and following the application of a step shear deformation. While the fluctuation-dissipation theorem is perfectly demonstrated in the Rouse-chain model, a quasiversion of the fluctuation-dissipation theorem is observed in the Fraenkel-chain model. In both types of simulations on the Fraenkel-chain model, two distinct modes of dynamics emerge in GS(t), giving a line shape similar to that typically observed experimentally. Analyses show that the fast mode arises from the segment-tension fluctuations or reflects the relaxation of the segment tension created by segments being stretched by the applied step strain-an energetic-interactions-driven process-while the slow mode arises from the fluctuations in segmental orientation or represents the randomization of the segmental-orientation anisotropy induced by the step deformation-an entropy-driven process. Furthermore, it is demonstrated that the slow mode is well described by the Rouse theory in all aspects: the magnitude of modulus, the line shape of the relaxation curve, and the number-of-beads (N) dependence of the relaxation times. In other words, one Fraenkel segment substituting for one Rouse segment, it has been shown that the entropic-force constant on each segment is not a required element to give rise to the Rouse modes of motion, which describe the relaxation modulus of an entanglement-free polymer over the long-time region very well. This conclusion provides an explanation resolving a long-standing fundamental paradox in the success of Rouse-segment-based molecular theories for polymer viscoelasticity-namely, the paradox between the Rouse segment size being of the same order of magnitude as that of the Kuhn segment (each Fraenkel segment with a large force constant HF can be regarded as basically equivalent to a Kuhn segment) and the meaning of the Rouse segment as defined in the Rouse-chain model. The general agreement observed in the comparison of the simulation and experimental results indicates that the Fraenkel-chain model, while being still relatively simple, has captured the key element in energetic interactions--the rigidity on the segment--in a polymer system.     

Solvent (1)H NMRD study of hexaaquochromium(III): inferences on hydration and electron relaxation

The water proton nuclear magnetic relaxation dispersions (NMRD) of hexaaquochromium(III) in water and in water-glycerol solutions were obtained at several temperatures and viscosities. The data were analyzed in terms of the available theories by taking into account the contributions from first sphere, second sphere, and outer sphere water molecules. A meaningful analysis was possible by taking advantage of the structural model obtained from (17)O relaxation data, which was recently made available in the literature (Bleuzen, A.; Foglia, F.; Furet, E.; Helm, L.; Merbach, A.; Weber, J. J. Am. Chem. Soc. 1996, 118, 12 777). Dynamic parameters, like the molecular rotational time, the exchange time of the water protons of the first coordination sphere, the correlation time for electron relaxation, and the magnetic field dependence of electron relaxation were obtained. The possible contribution to water proton relaxivity of second sphere water molecules for some other hexaaqua complexes is also discussed.     

Adsorption relaxation for nonionic surfactants under mixed barrier-diffusion and micellization-diffusion control

Relaxation processes of surfactant adsorption and surface tension, which are characterized by two specific relaxation times, are theoretically investigated. We are dealing with fluid interfaces and small initial deviations from equilibrium. For surfactant concentrations below the critical micellization concentration (CMC), we consider adsorption under mixed barrier-diffusion control. General analytical expressions are derived, which are convenient for both numerical computations and asymptotic analysis. Series expansions for the short- and long-time limit are derived. The results imply that the short-time asymptotics is controlled by the adsorption barrier, whereas the long-time asymptotics is always dominated by diffusion. Furthermore, for surfactant concentrations above the CMC, adsorption under mixed micellization-diffusion control is considered. Again, a general analytical expression is derived for the relaxation of surfactant adsorption and surface tension, whose long- and short-time asymptotics are deduced. The derived equations show that at the short times the relaxation is completely controlled by the diffusion, whereas the long-time asymptotics is affected by both demicellization and diffusion. The micellar effect is manifested as an exponential (rather than square-root) decay of the perturbation. The derived expressions are applied to process available experimental data for the nonionic surfactant Triton X-100 and to determine the respective demicellization rate constant.     

Photostability via sloped conical intersections: a computational study of the excited states of the naphthalene radical cation

On the basis of an extensive ab initio electronic structure study of the ground and excited-state potential energy surfaces of the naphthalene radical cation (N*+), we propose a mechanism for its ultrafast nonradiative relaxation from the second excited state (D2) down to the ground state (D0), which could explain the experimentally observed photostability [Zhao, L.; Lian, R.; Shkrob I. A.; Crowell, R. A.; Pommeret, S.; Chronister, E. L.; Liu, A. D.; Trifunac, A. D. J. Phys. Chem. A., 2004, 108, 25]. The proposed photophysical relaxation pathway involves internal conversion from the D2 state down to the D0 state via two consecutive, accessible, sloped conical intersections (CIs). The two crossings, D0/D1 and D1/D2, are characterized at the complete active space self-consistent field (CASSCF) level. At this level of theory, the D0/D1 crossing is energetically readily accessible, while the D1/D2 CI appears too high in energy to be involved in internal conversion. However, the inclusion of dynamic correlation effects, via single point CASPT2 calculations including excitations out of the valence pi- and sigma-orbitals, lowers the D0 and D2 state energies with respect to D1. Extrapolations at the CASPT2 level predict that the D1/D2 crossing is then significantly lower in energy than with CASSCF indicating that with a higher-level treatment of dynamic correlation it may be energetically accessible following vertical excitation to D2. N*+ is proposed as one of the species contributing to a series of diffuse infrared absorption bands originating from interstellar clouds. Understanding the mechanism for photostability in the gas phase, therefore, has important consequences for astrophysics.     

NMR relaxation parameters of a Lennard-Jones fluid from molecular-dynamics simulations

Ensembles of soft spheres or of Lennard-Jones atoms were studied by molecular dynamics at reduced temperatures from 0.8 to 3, and radial distribution functions, diffusion coefficients, and magnetic dipole-dipole correlation functions were measured as functions of system size. The expected relation between the values of the correlation functions at zero lag time and the integrals of the radial distribution was verified for each system. The measured correlation functions were compared with theoretical expressions derived by [Ayant et al., J. Phys. (Paris) 36, 991 (1975)] and by [Hwang and Freed, J. Chem. Phys. 63, 4017 (1975)]. It was shown that, in order to recover the long-time behavior characteristic of diffusion-controlled relaxation processes, the simulation must comprise at least 10 000 particles. By fitting the simulation results to the Hwang-Freed function, independent values of the diffusion coefficient were obtained, similar but not identical to those computed using the Green-Kubo formalism. The spectral densities of the dipole-dipole interaction were computed as Fourier transforms of the correlation functions. These quantities are less sensitive to model imperfections and reproduce quite well the values derived from theory. The dimensionless spin-lattice and spin-spin relaxation rates were derived from the spectral densities. It was shown that the spin-lattice (longitudinal) relaxation rate goes through a maximum as the temperature increases, while the spin-spin (transverse) rate decreases monotonously.     

New insights into the transition pathway from nonspecific to specific complex of DNA with Escherichia coli integration host factor

To elucidate the nature of the transition-state ensemble along the reaction pathway from a nonspecific protein-DNA complex to the specific complex, we have carried out measurements of DNA bending/unbending dynamics on a cognate DNA substrate in complex with integration host factor (IHF), an architectural protein from E. coli that bends its cognate site by approximately 180 degrees . We use a laser temperature jump to perturb the IHF-DNA complex and monitor the relaxation kinetics with time-resolved FRET measurements on DNA substrates end-labeled with a FRET pair. Previously, we showed that spontaneous bending/kinking of DNA, from thermal disruption of base-pairing/-stacking interactions, may be the rate-limiting step in the formation of the specific complex (Kuznetsov, S. V.; Sugimura, S.; Vivas, P.; Crothers, D. M.; Ansari, A. Proc. Natl. Acad. Sci. USA 2006, 103, 18515). Here, we probe the effect of varying [KCl], which affects the stability of the complex, on this rate-limiting step. We find that below approximately 250 mM KCl, the observed relaxation kinetics are from the unimolecular bending/unbending of DNA, and the relaxation rate kr is independent of [KCl]. Above approximately 300 mM KCl, dissociation of the IHF-DNA complex becomes significant, and the observed relaxation process includes contributions from the association/dissociation step, with kr decreasing with increasing [KCl]. The DNA bending step occurs with a positive activation enthalpy, despite the large negative enthalpy change reported for the specific IHF-DNA complex (Holbrook, J. A.; Tsodikov, O. V.; Saecker, R. M.; Record, M. T., Jr. J. Mol. Biol. 2001, 310, 379). Our conclusion from these studies is that in the uphill climb to the transition state, the DNA is kinked, but with no release of ions, as indicated by the salt-independent behavior of k(r) at low [KCl]. Any release of ions in the unimolecular process, together with conformational changes in the protein-DNA complex that facilitate favorable interactions and that contribute to the negative enthalpy change, must occur as the system leaves the transition state, downhill to the final complex.     

Solvent dependent conformational relaxation of cis-1,3,5-hexatriene

Ultrafast transient absorption spectroscopy was used to study the conformational relaxation dynamics of 1,3,5-cis-hexatriene (Z-HT) produced in the photochemical ring-opening reaction of 1,3-cyclohexadiene (CHD) in methanol and n-propanol solvents. The results are compared with earlier investigations performed using cyclohexane and hexadecane solvents [Anderson, N. A.; Pullen, S. H.; Walker II, L. A.; Shiang, J. J.; Sension, R. J.; J. Phys. Chem. A 1998, 102, 10588-10598.]. The conformational relaxation between hot cZc-HT, cZt-HT, and tZt-HT, where the labels c and t designate cis and trans configurations about the single bonds, is much faster in alcohol solvents than in alkane solvents. The hot Z-HT produced in the photochemical ring-opening reaction evolves from the conformationally strained cZc-HT form to the more stable cZt-HT form on a time scale of 2 ps in alcohols compared with 6 ps in alkanes. The overall decay of the internal vibrational temperature of the hot Z-HT is faster in alcohols (5-6 ps) than alkanes (12-20 ps) and is weakly dependent on the specific alcohol or alkane solvent. A small population of cZt-HT (5-10%) is trapped as the solute equilibrates with the surrounding solvent following UV excitation of CHD or direct UV excitation of Z-HT. The influence of solvent on conformational relaxation of Z-HT was investigated further by probing the temperature dependence of the decay of this thermally equilibrated cZt-HT population. The apparent barrier for the cZt --> tZt conformational isomerization is lower in alcohols (17.4 kJ/mol) than in alkanes (23.5 kJ/mol). However the equilibrium Arrhenius prefactor (A(h)) is an order of magnitude smaller for alcohols (ca. 4 x 10(12)) than alkanes (ca. 6 x 10(13)) resulting in an absolute rate of decay that is faster in the alkane than in the alcohol solvents. These results are discussed in the context of transition state theory and Kramers' theory for condensed phase reaction dynamics.     

Deuterium spin probes of side-chain dynamics in proteins. 2. Spectral density mapping and identification of nanosecond time-scale side-chain motions

In the previous paper in this issue we have demonstrated that it is possible to measure the five different relaxation rates of a deuteron in (13)CH(2)D methyl groups of (13)C-labeled, fractionally deuterated proteins. The extensive set of data acquired in these experiments provides an opportunity to investigate side-chain dynamics in proteins at a level of detail that heretofore was not possible. The data, acquired on the B1 domain of peptostreptococcal protein L, include 16 (9) relaxation measurements at 4 (2) different magnetic field strengths, 25 degrees C (5 degrees C). These data are shown to be self-consistent and are analyzed using a spectral density mapping procedure which allows extraction of values of the spectral density function at a number of frequencies with no assumptions about the underlying dynamics. Dynamics data from 31 of 35 methyls in the protein for which data could be obtained were well-fitted using the two-parameter Lipari-Szabo model (Lipari, G.; Szabo, A. J. Am. Chem. Soc. 1982, 104, 4546). The data from the remaining 4 methyls can be fitted using a three-parameter version of the Lipari-Szabo model that takes into account, in a simple manner, additional nanosecond time-scale local dynamics. This interpretation is supported by analysis of a molecular dynamics trajectory where spectral density profiles calculated for side-chain methyl sites reflect the influence of slower (nanosecond) time-scale motions involving jumps between rotameric wells. A discussion of the minimum number of relaxation measurements that are necessary to extract the full complement of dynamics information is presented along with an interpretation of the extracted dynamics parameters.     

Modeling vibrational dephasing and energy relaxation of intramolecular anharmonic modes for multidimensional infrared spectroscopies

Starting from a system-bath Hamiltonian in a molecular coordinate representation, we examine an applicability of a stochastic multilevel model for vibrational dephasing and energy relaxation in multidimensional infrared spectroscopy. We consider an intramolecular anharmonic mode nonlinearly coupled to a colored noise bath at finite temperature. The system-bath interaction is assumed linear plus square in the system coordinate, but linear in the bath coordinates. The square-linear system-bath interaction leads to dephasing due to the frequency fluctuation of system vibration, while the linear-linear interaction contributes to energy relaxation and a part of dephasing arises from anharmonicity. To clarify the role and origin of vibrational dephasing and energy relaxation in the stochastic model, the system part is then transformed into an energy eigenstate representation without using the rotating wave approximation. Two-dimensional (2D) infrared spectra are then calculated by solving a low-temperature corrected quantum Fokker-Planck (LTC-QFP) equation for a colored noise bath and by the stochastic theory. In motional narrowing regime, the spectra from the stochastic model are quite different from those from the LTC-QFP. In spectral diffusion regime, however, the 2D line shapes from the stochastic model resemble those from the LTC-QFP besides the blueshifts caused by the dissipation from the colored noise bath. The preconditions for validity of the stochastic theory for molecular vibrational motion are also discussed.     

An 15N NMR spin relaxation dispersion study of the folding of a pair of engineered mutants of apocytochrome b562

15N relaxation dispersion NMR spectroscopy has been used to study exchange dynamics in a pair of mutants of Rd-apocyt b562, a redesigned four-helix-bundle protein. An analysis of the relaxation data over a range of temperatures establishes that exchange in both proteins is best modeled as two-state and that it derives from the folding/unfolding transition. These results are in accord with predictions based on the reaction coordinate for the folding of the protein determined from native-state hydrogen exchange data [Chu, R.; Pei, W.; Takei, J.; Bai, Y. Biochemistry 2002, 41, 7998-8003]. The kinetics and thermodynamics of the folding transition have been characterized in detail. Although only a narrow range of temperatures could be examined, it is clear that the folding rate temperature profile is distinctly non-Arrhenius for both mutants, with the folding barrier for at least one of them entropic.     

CF3 Rotation in 3-(Trifluoromethyl)phenanthrene: Solid State 19F and 1H NMR relaxation and Bloch-Wangsness-Redfield theory

We have observed and modeled the 1H and 19F solid-state nuclear spin relaxation process in polycrystalline 3-(trifluoromethyl)phenanthrene. The relaxation rates for the two spin species were observed from 85 to 300 K at the low NMR frequencies of omega/2pi = 22.5 and 53.0 MHz where CF3 rotation, characterized by a mean time tau between hops, is the only motion on the NMR time scale. All motional time scales (omegatau < 1, omegatau approximately 1, and omegatau > 1) are observed. The 1H spins are immobile on the NMR time scale but are coupled to the 19F spins via the unlike-spin dipole-dipole interaction. The temperature dependence of the observed relaxation rates (the relaxation is biexponential) shows considerable structure and a thorough analysis of Bloch-Wangsness-Redfield theory for this coupled spin system is provided. The activation energy for CF3 rotation is 11.5 +/- 0.7 kJ/mol, in excellent agreement with the calculation in a 13-molecule cluster provided in the companion paper where the crystal structure is reported and detailed ab initio electronic structure calculations are performed [Wang, X.; Mallory F. B.; Mallory, C. W; Beckmann, P. A.; Rheingold, A. L.; Francl, M. M J. Phys. Chem. A 2006, 110, 3954].     

Dielectric relaxation in an enzyme active site: molecular dynamics simulations interpreted with a macroscopic continuum model.

Dielectric relaxation plays an important role in many chemical processes in proteins, including acid-base titration, ligand binding, and charge transfer reactions. Its complexity makes experimental characterization difficult, and so, theoretical approaches are valuable. The comparison of molecular dynamics free energy simulations with simpler models such as a dielectric continuum model is especially useful for obtaining qualitative insights. We have analyzed a charge insertion process that models deprotonation or mutation of an important side chain in the active site of the enzyme aspartyl-tRNA synthetase. Complexes with the substrate aspartate and the analogue asparagine were studied. The resulting dielectric relaxation was found to involve both ligand and side chain rearrangements in the active site and to account for a large part of the overall charging free energy. With the continuum model, charge insertion is performed along a two-step pathway: insertion into a static environment, followed by relaxation of the environment. These correspond to different physical processes and require different protein dielectric constants. A low value of approximately 1 is needed for the static step, consistent with the parametrization of the molecular mechanics charge set used. A value of 3-6 (depending on the exact insertion site and the nature of the ligand) is needed to describe the dielectric relaxation step. This moderate value indicates that, for this system, the local protein polarizability in the active site is within at most a factor of 2 of that expected at nonspecific positions in a protein interior.     

Elastoviscous properties of polyisobutylene. IV. Relaxation time spectrum and calculation of bulk viscosity

The elastoviscous behavior of polyisobutylene may be interpreted in terms of a mechanical model consisting of a distribution of Maxwell elements connected in parallel. The structure of this “generalized Maxwell model” is specified by the distribution of relaxation times of the component elements. The relaxation of stress curve of the material is directly related to the distribution of relaxation times, and general expressions for the bulk viscosities (tensile and shear) of such a system in terms of the distribution of relaxation times are readily obtained. A simple “box distribution” of relaxation times is described which can be used to approximate the relaxation behavior of polyisobutylene at the long-time end of the relaxation time spectrum, and in terms of which the expressions for bulk viscosity reduce to very simple form. The parameters specifying this distribution may be determined from experimental relaxation curves by a simple graphical method. Values of these parameters as a functions of molecular weight and temperature are computed, by use of these data. It is shown that bulk viscosity values calculated from relaxation data by this method are in good agreement with experimental values for both tensile and shear deformations, and for both unfractionated and fractionated polymers. Measurements of viscosity and of relaxation of stress can thus be directly correlated, and could be used in combination to characterize elastoviscous properties over wide ranges of molecular weight and temperature.     

Spin relaxation effects in photochemically induced dynamic nuclear polarization spectroscopy of nuclei with strongly anisotropic hyperfine couplings

We describe experimental results and theoretical models for nuclear and electron spin relaxation processes occurring during the evolution of 19F-labeled geminate radical pairs on a nanosecond time scale. In magnetic fields of over 10 T, electron-nucleus dipolar cross-relaxation and longitudinal DeltaHFC-Deltag (hyperfine coupling anisotropy--g-tensor anisotropy) cross-correlation are shown to be negligibly slow. The dominant relaxation process is transverse DeltaHFC-Deltag cross-correlation, which is shown to lead to an inversion in the geminate 19F chemically induced dynamic nuclear polarization (CIDNP) phase for sufficiently large rotational correlation times. This inversion has recently been observed experimentally and used as a probe of local mobility in partially denatured proteins (Khan, F.; et al. J. Am. Chem. Soc. 2006, 128, 10729-10737). The essential feature of the spin dynamics model employed here is the use of the complete spin state space and the complete relaxation superoperator. On the basis of the results reported, we recommend this approach for reliable treatment of magnetokinetic systems in which relaxation effects are important.     

Short‐time stretch relaxation of entangled polymer solutions investigated using full Rouse model predictions

We conduct a systematical investigation into the short‐time stretch relaxation behavior (i.e., shorter than the Rouse time but sufficiently longer than the glassy time) of entangled polymer liquid in single‐step strain flows, on the basis of theory/data comparisons for a broad series of type‐A entangled polymer solutions. First, within existing normal‐mode formulations, the Rouse model predictions on a full‐chain stretch relaxation in single‐step strain flows are derived for a popular 1‐D model proposed within the Doi–Edwards tube model, as well as for the original 3‐D model for nonentangled systems. In addition, an existing formula for the aforementioned 1‐D model that, however, rested upon a consistent‐averaging or the so‐called uniform‐chain‐stretch approximation is simultaneously examined. Subsequently, the previously derived formulas on chain stretch relaxation are directly incorporated into a reliable mean‐field tube model that utilizes the linear relaxation spectrum and the Rouse time constant consistently determined from linear viscoelastic data. It is found that the predictions of the 1‐D model differ substantially from that of the original 3‐D model at short times. Theory/data comparisons further indicate that the 1‐D model without approximations seems able to describe fairly well the nonlinear relaxation data under investigation. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1199–1211, 2006     

Dynamical correlations and long-time tails in chemical reactions. A molecular dynamics study

A simple hard-sphere and disk model of a chemical reaction of the type A + B ⇌ C + D is studied by molecular dynamics (MD). Ensemble and time averages are carried out so that the MD calculation simulates chemical relaxation in a homogeneous system which is in equilibrium with respect to the mechanical degrees of freedom. For large times, the decay of the concentrations of the different species is not given by the expected exponential law but shows a long-time tail instead. A simple explanation of the effect is given in terms of recollision events (dynamical correlations) between pairs of particles. It is argued that non-exponential decay of the kind observed might be a general phenomenon in chemical reactions.