Do all pieces make a whole? Thiele cumulants and the free energy decomposition

The partitioning of the free energy into additive contributions originating from different groups of atoms or force field terms has the potential to provide relationship between structure and biological activity of molecules. In this article, the theoretical foundation for the free energy decomposition in the free energy perturbation (FEP) methodology is formulated using Thiele cumulants, a powerful tool from the arsenal of probability theory and mathematical statistics. We establish that rigorous decomposition of the free energy into its components is precluded by the presence of mixed potential energy terms in Thiele cumulants of second and higher orders. However, we also show that the resultant nonadditivity error can be reduced to an arbitrary value by increasing the number of FEP steps. Consequently, the whole system can be in the limit of small perturbation steps adequately represented by the sum of its constituent parts.     

Extension of the cumulant definition for low particle number systems

Cumulants are usually interpreted as the connected components of density matrices, but this interpretation fails and practical problems arise when the rank n of cumulants is larger than the number of particles (N) in the system. In that case, cumulants defined in the traditional way become disconnected. To solve this problem, the definition of cumulants is extended by introducing a simple modulation factor. The modified cumulants reduce to the conventional definition, but they vanish when N < n. Using the modified definition also eliminates the error in the approximation of density matrices by low‐rank cumulants, when N < n. The problem assumes a slightly different form when we work with active space–based theories, and it can be solved by a similar approach. Another problem with cumulants, due to spin coupling [Herbert, Int J Quantum Chem 2007, 107, 703], can be solved via the introduction of a similar modulation factor. A related yet more serious issue, termed as the local particle number constraint problem, is also discussed. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Distribution function in quantal cumulant dynamics

We have derived a quantum distribution function in terms of cumulants that are expectation values of a (anti)symmetric-ordered product of position and momentum fluctuation operators. A second-order approximation leads a Gaussian distribution function, which is positive definite and has proper marginals so that the Shannon entropy can be evaluated.     

Quantal cumulant dynamics: general theory

The authors have derived coupled equations of motion of cumulants that consist of a symmetric-ordered product of the position and momentum fluctuation operators in one dimension. The key point is the utilization of a position shift operator acting on a potential operator, where the expectation value of the shift operator is evaluated using the cumulant expansion technique. In particular, the equations of motion of the second-order cumulant and the expectation values of the position and momentum operators are given. The resultant equations are expressed by those variables and a quantal potential that consists of an exponential function of the differential operators and the original potential. This procedure enables us to perform quantal (semiclassical) dynamics in one dimension. In contrast to a second-order quantized Hamilton dynamics by Prezhdo and Pereverzev which conserves the total energy only with an odd-order Taylor expansion of the potential [J. Chem. Phys. 116, 4450 (2002); 117, 2995 (2002)], the present quantal cumulant dynamics method exactly conserves the energy, even if a second-order approximation of the cumulants is adopted, because the present scheme does not truncate the given potential. The authors propose three schemes, (i) a truncation, (ii) a summation of derivatives, and (iii) a convolution method, for evaluating the quantal potentials for several types of potentials. The numerical results show that although the truncation method preserves the energy to some degree, the trajectory obtained gradually deviates from that of the summation scheme after 2000 steps. The phase space structure obtained by the truncation scheme is also different from that of the summation scheme in a strongly anharmonic region.     

Do proteins at low temperature behave as glasses? A single-molecule study

We have recorded long spectral diffusion trajectories from individual LH2 pigment-protein complexes from the purple bacterium Rhodobacter sphaeroides at 1.4 K. From these data, the spectral cumulants of the absorption lines of individual, protein-embedded BChl a pigments have been evaluated. It appears that the first and second cumulants cannot be described by the predictions of the well tested standard two-level system (TLS) model for spectral diffusion in glasses. The results of the present study clearly show that there is a fundamental difference between the relaxation behavior of our test protein and that of glasses.     

Relaxation of quantum hydrodynamic modes

In this article, we develop a series of hierarchical mode‐coupling equations for the momentum cumulants and moments of the density matrix for a mixed quantum system. Working in the Lagrange representation, we show how these can be used to compute quantum trajectories for dissipative and nondissipative systems. This approach is complementary to the de Broglie–Bohm approach in that the moments evolve along hydrodynamic/Lagrangian paths. In the limit of no dissipation, the paths are the Bohmian paths. However, the “quantum force” in our case is represented in terms of momentum fluctuations and an osmotic pressure. Representative calculations for the relaxation of a harmonic system are presented to illustrate the rapid convergence of the cumulant expansion in the presence of a dissipative environment. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

On the properties of cumulant expansions

Cumulants represent a natural language for expressing macroscopic properties of a solid. We show that cumulants are subject to a nontrivial geometry. This geometry provides an intuitive understanding of a number of cumulant relations which have been obtained so far by using algebraic considerations. We give general expressions for their infinitesimal and finite transformations and represent a cumulant wave operator through an integration over a path in the Hilbert space. Cases are investigated where this integration can be done exactly. An expression of the ground-state wave function in terms of the cumulant wave operator is derived. In the second part of the article, we derive the cumulant counterpart of Faddeev's equations and show its connection to the method of increments. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 377–389, 1998     

Modification to the cumulant analysis of polydispersity in quasielastic light scattering data

The electric field correlation function of light scattered from a polydispersed population of spherical particles having log-normal distribution with varying polydispersity is simulated. The correlation function with different polydispersity is compared with the method of cumulants over a wide range of correlation time. The large positive deviation of the method of cumulants at long correlation time is identified. This necessitates the truncation of the data at long correlation time or use of an appropriate weighting function to eliminate errors in the analysis. A modified cumulant analysis is used to overcome the limitation of truncating the correlation function. QELS data from polydisperse samples of micelles, liposomes and polyaniline nanoparticles are compared using the two methods. This method can be extended to the analysis of other multi-exponential decays such as stress relaxation, positron annihilation and NMR relaxation.     

A study of the accuracy of moment-closure approximations for stochastic chemical kinetics

Moment-closure approximations have in recent years become a popular means to estimate the mean concentrations and the variances and covariances of the concentration fluctuations of species involved in stochastic chemical reactions, such as those inside cells. The typical assumption behind these methods is that all cumulants of the probability distribution function solution of the chemical master equation which are higher than a certain order are negligibly small and hence can be set to zero. These approximations are ad hoc and hence the reliability of the predictions of these class of methods is presently unclear. In this article, we study the accuracy of the two moment approximation (2MA) (third and higher order cumulants are zero) and of the three moment approximation (3MA) (fourth and higher order cumulants are zero) for chemical systems which are monostable and composed of unimolecular and bimolecular reactions. We use the system-size expansion, a systematic method of solving the chemical master equation for monostable reaction systems, to calculate in the limit of large reaction volumes, the first- and second-order corrections to the mean concentration prediction of the rate equations and the first-order correction to the variance and covariance predictions of the linear-noise approximation. We also compute these corrections using the 2MA and the 3MA. Comparison of the latter results with those of the system-size expansion shows that: (i) the 2MA accurately captures the first-order correction to the rate equations but its first-order correction to the linear-noise approximation exhibits the wrong dependence on the rate constants. (ii) the 3MA accurately captures the first- and second-order corrections to the rate equation predictions and the first-order correction to the linear-noise approximation. Hence while both the 2MA and the 3MA are more accurate than the rate equations, only the 3MA is more accurate than the linear-noise approximation across all of parameter space. The analytical results are numerically validated for dimerization and enzyme-catalyzed reactions.     

MFC逆转化CO2合成甲醇研究

微生物燃料电池(MFC)反应器是利用附着在阳极上的产氢微生物,在吸收烟气CO2的同时将CO2逆转化合成高附加值的生物合成燃料的装置。试验选用从牛粪中分离筛选出的梭状芽孢菌(Clostridium.sp)作为合成生物燃料的合成菌,将MFC反应装置接入电化学工作站进行CV测试,当发生还原反应时,在-0.5 V时出现还原峰,利用直流稳压电源恒电压电解,检测到合成的生物燃料为甲醇。在24 h时甲醇的积累量达到最大3.13 mmol/L;当CO2气体比例为15%时甲醇积累量最大,为2.98 mmol/L。在细菌接种量为1 mL时,甲醇积累量达到最大,为2.76 mmol/L。,最适条件下的CO2转化率为7.5%。     

Histogram Monte Carlo simulation on phase transitions in single‐polymer systems

Applying the histogram Monte Carlo simulation method and the bond‐fluctuation model, various phase transitions in single‐polymer systems were investigated. The critical transition temperature (Θ point) in the coil‐globule collapse transition of a macromolecular chain is accurately determined. Finite‐size scaling results near the transition point are verified. The first‐order transition associated with the freezing/crystallization of a polymer at a temperature below the Θ point is also observed. The free energy profiles associated with these two transitions are explicitly computed. Furthermore, the unfolding phase transition associated with stretching a collapsed polymer chain is investigated. The free energy profile associated with the transition is explicitly computed. Results on the energy cumulants and free energy profiles provide direct evidences for the first‐order nature of the unfolding phase transition.     

Dynamic light scattering,a probe of brownian particle dynamics

The principles and techniques of dynamic light scattering (DLS) are outlined and its application to the study of suspensions of interacting colloidal particles is discussed. We show how, under appropriate conditions, DLS can measure long-time collective and self-diffusion coefficients as well as study short-time motions (characterized by the cumulants). These theoretical considerations are illustrated by experimental data. Finally, we discuss the relevance of certain characteristic timescales to theories of the diffusion of interacting particles.     

Colored non-gaussian noise driven open systems: generalization of Kramers' theory with a unified approach

In this paper we have calculated escape rate from a meta stable state in the presence of both colored internal thermal and external nonthermal noises. For the internal noise we have considered usual gaussian distribution but the external noise may be gaussian or non-gaussian in characteristic. The calculated rate is valid for low noise strength of non-gaussian noise such that an effective gaussian approximation of non-gaussian noise wherein the higher order even cumulants of order "4" and higher are neglected. The rate expression we derived here reduces to the known results of the literature, as well as for purely external noise driven activated rate process. The latter exhibits how the rate changes if one switches from non-gaussian to gaussian character of the external noise.     

Comparison between Einstein and Debye models for an amorphous Ni46Ti54 alloy produced by mechanical alloying investigated using extended x-ray absorption fine structure and cumulant expansion

We investigated an amorphous Ni(46)Ti(54) alloy produced by mechanical alloying using extended x-ray absorption fine structure (EXAFS) technique and cumulant expansion considering Einstein and Debye models for the temperature dependence of the cumulants. Results obtained from both models were compared and very similar values were obtained. From them, we found information about the structure of the alloy besides thermal and structural disorder, anharmonicity, thermal expansion, and asymmetry of the partial distribution functions g(ij)(r). The cumulants C(1)(*), C(2)(*), and C(3)(*) also allowed us to reconstruct the g(ij)(r, T) functions from EXAFS.     

Analytic gradients for density cumulant functional theory: The DCFT-06 model

Density cumulant functional theory (DCFT) is one of a number of nascent electron correlation methods that are derived from reduced density matrices and cumulants thereof, instead of the wavefunction. Deriving properties from the density cumulant naturally yields methods that are size extensive and size consistent. In this work, we derive expressions for the analytic gradient, with respect to an external perturbation, for the DCFT-06 variant of density cumulant functional theory. Despite the fact that the DCFT-06 energy functional is stationary with respect to the density cumulant, the analytic gradients of the energy require the solution of perturbation-independent equations for both orbital and cumulant response. These two sets of linear response equations are coupled in nature and are solved iteratively with the solution of orbital and cumulant response equations each macroiteration, exhibiting rapid convergence. The gradients are implemented and benchmarked against coupled cluster theory with single and double excitations (CCSD) and CCSD with perturbative triple excitations [CCSD(T)], as well as accurate empirically corrected experimental data, for a test set comprising 15 small molecules. For most of the test cases, results from DCFT-06 are closer to CCSD(T) and empirical data than those from CCSD. Although the total energy and analytic gradient have the same asymptotic scaling, the present experience shows that the computational cost of the gradient is significantly lower.     

Photon counting histograms for diffusing fluorophores

The theory of photon counting histograms for fluorescent molecules diffusing through a laser spot is presented. Analytic expressions for the factorial cumulants of photon counts are obtained. For an arbitrary counting time window, it is shown how the exact histograms can be obtained by solving an appropriate reaction-diffusion equation. Our formalism reduces correctly when the molecules are immobile. The approximation used in fluorescence intensity multiple distribution analysis (FIMDA) is tested against the exact numerical solution of the problem. FIMDA works very well for a wide range of parameters except for small concentrations and long time windows.     

Free energy for protein folding from nonequilibrium simulations using the Jarzynski equality

The equilibrium free energy difference between two long-lived molecular species or "conformational states" of a protein (or any other molecule) can in principle be estimated by measuring the work needed to shuttle the system between them, independent of the irreversibility of the process. This is the meaning of the Jarzynski equality (JE), which we test in this paper by performing simulations that unfold a protein by pulling two atoms apart. Pulling is performed fast relative to the relaxation time of the molecule and is thus far from equilibrium. Choosing a simple protein model for which we can independently compute its equilibrium properties, we show that the free energy can be exactly and effectively estimated from nonequilibrium simulations. To do so, one must carefully and correctly determine the ensemble of states that are pulled, which is more important the farther from equilibrium one performs simulations; this highlights a potential problem in using the JE to extract the free energy from forced unfolding experiments. The results presented here also demonstrate that the free energy difference between the native and denatured states of a protein measured in solution is not always equal to the free energy profile that can be estimated from forced unfolding simulations (or experiments) using the JE.     

Transition metal complexes with open d-shell in semiempirical context. Application to analysis of Mössbauer data on spin–active iron(II) compounds

With use of cumulants of two-electron density matrices semiempirical methods are analyzed from a point of view of their suitability to describe qualitative features of electronic correlation important for calculation of electronic structure of the transition metal complexes (TMC). It is shown that traditional semiempirical methods relying upon the Hartree–Fock–Roothaan form of the trial wave function suffer from a structural deficiency not allowing them to distinguish the energies of the atomic multiplets of the TMCs' d-shells. On the other hand, the effective Hamiltonian of the crystal field (EHCF) previously proposed by the authors is shown to be suitable for further parameterization and has been successfully applied for calculations on polyatomic TMCs. Here we describe in details its recent modifications performed in relation to the SINDO/1 parameterization scheme and present the results of the calculations on spin-active Fe(II) complexes with nitrogen-containing polydentate ligands in relation with interpretation of the Mössbauer measurements performed on these complexes.     

Expressing a Probability Density Function in Terms of another PDF: A Generalized Gram-Charlier Expansion

An explicit formula relating the probability density function with its cumulants is derived and discussed. A generalization of the Gram-Charlier expansion is presented, allowing to express one PDF in terms of another. The coefficients of this general expansion are explicitly obtained.