Preferential attachment during the evolution of a potential energy landscape

It has previously been shown that the network of connected minima on a potential energy landscape is scale-free, and that this reflects a power-law distribution for the areas of the basins of attraction surrounding the minima. Here, the aim is to understand more about the physical origins of these puzzling properties by examining how the potential energy landscape of a 13-atom cluster evolves with the range of the potential. In particular, on decreasing the range of the potential the number of stationary points increases and thus the landscape becomes rougher and the network gets larger. Thus, the evolution of the potential energy landscape can be followed from one with just a single minimum to a complex landscape with many minima and a scale-free pattern of connections. It is found that during this growth process, new edges in the network of connected minima preferentially attach to more highly connected minima, thus leading to the scale-free character. Furthermore, minima that appear when the range of the potential is shorter and the network is larger have smaller basins of attraction. As there are many of these smaller basins because the network grows exponentially, the observed growth process thus also gives rise to a power-law distribution for the hyperareas of the basins.     

Rethinking the application of the first nucleation theorem to particle formation

The critical cluster is the threshold size above which a cluster will be more likely to grow than to evaporate. In field and laboratory measurements of new particle formation, the number of molecules of a given species in the critical cluster is commonly taken to be the slope of the log-log plot of the formation rate versus the concentration of the species. This analysis is based on an approximate form of the first nucleation theorem, which is derived with the assumption that there are no minima in the free energy surface prior to the maximum at the critical size. However, many atmospherically relevant systems are likely to exhibit such minima, for example, ions surrounded by condensable vapour molecules or certain combinations of acids and bases. We have solved numerically the birth-death equations for both an electrically neutral one-component model system with a local minimum at pre-critical sizes and an ion-induced case. For the ion-induced case, it is verified that the log-log slope of the nucleation rate versus particle concentration plot gives accurately the difference between the cluster sizes at the free energy maximum and minimum, as is expected from the classical form of the ion-induced nucleation rate. However, the results show that applying the nucleation theorem to neutral systems with stable pre-nucleation clusters may lead to erroneous interpretations about the nature of the critical cluster.     

Notable effect of an electron-withdrawing group at C3 on the selective formation of alkylidenecyclobutanes in the thermal denitrogenation of 4-spirocyclopropane-1-pyrazolines. Nonstatistical dynamics effects in the denitrogenation reactions

A detailed study of the thermal denitrogenation of 3-carbomethoxy-substituted 4-spirocyclopropane-1-pyrazolines 6 was conducted. Alkylidenecyclobutane derivatives 7 were selectively formed in a stereospecific manner. Unrestricted density functional calculations for a 1-pyrazoline 10a indicated that the concerted cleavage of two C-N bonds is the energetically favored process for the denitrogenation reaction to give the 2-spirocyclopropyl 1,3-diyl, followed by a conrotatory ring-closure process, which was calculated to be the energy minimum pathway, to afford a spiropentane derivative. The calculated energy minimum pathway is largely inconsistent with the experimental results observed for the denitrogenation of 6 and 10a. The contradiction between the experimental and standard computational results was solved by considering nonstatistical dynamics effects in the concerted denitrogenation reactions. Although the energy minimum pathway from the transition states of the concerted denitrogenation of the 3-carboalkoxy-substituted 1-pyrazolines involves generation of the corresponding 1,3-diradicals, many trajectory calculations using the Bohn-Oppenheimer molecular dynamics model from the transition state for the concerted denitrogenation led directly to the formation of alkylidenecyclobutanes at the UB3LYP/6-31G(d) level of theory.     

Transition rate prefactors for systems of many degrees of freedom

When a minimum on the potential energy surface is surrounded by multiple saddle points with similar energy barriers, the transition pathways with greater prefactors are more important than those that have similar energy barriers but smaller prefactors. In this paper, we present a theoretical formulation for the prefactors, computing the probabilities for transition paths from a minimum to its surrounding saddle points. We apply this formulation to a system of 2 degrees of freedom and a system of 14 degrees of freedom. The first is Brownian motion in a two-dimensional potential whose global anharmonicities play a dominant role in determining the transition rates. The second is a Lennard-Jones (LJ) cluster of seven particles in two dimensions. Low lying transition states of the LJ cluster, which can be reached directly from a minimum without passing through another minimum, are identified without any presumption of their characteristics nor of the product states they lead to. The probabilities are computed for paths going from an equilibrium ensemble of states near a given minimum to the surrounding transition states. These probabilities are directly related to the prefactors in the rate formula. This determination of the rate prefactors includes all anharmonicities, near or far from transition states, which are pertinent in the very sophisticated energy landscape of LJ clusters and in many other complex systems.     

New lower bound formula from the local energy

A new formula to determine a lower bound to the ground‐state energy of an electronic system is presented making use of the local energy. Two‐electron atoms are considered and each lower bound is not only superior to the traditional lower bound, which is just the minimum of the local energy, but is also closer to the exact energy than the Hamiltonian expectation value that uses the same trial function. In this method, we have removed the restriction that the trial function be non‐zero. Lower bound methods using the local energy have the advantage that the expectation value of the squared Hamiltonian or another potentially difficult integral does not have to be calculated as in many other lower bound methods. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

THEORETICAL APPROACHES TO PROTEIN STRUCTURE-FUNCTION ANALYSIS (Ⅰ)——DIRECTED PERTURBATION CONFORMATIONAL ANALYSIS

A new computational technique called directed perturbation conformational analysis has been developed for use in protein model building and structure-function studies. Designed to perform an efficient local search of a macromolecular potential energy surface, the algorithm can be used to locate multiple energy minimum conformers via low energy transition state structures from a single starting or trial structure. The algorithm contains developments to stabilize transition state optimizations for systems described by many degrees of freedom displaying anharmonic potential energy surfaces. It has been found to be efficient in the generation of alternative equilibrium structures from a given trial structure when compared with those generated from a standard molecular dynamics simulation of N-acetyl, N'-methyl-deca-L-alaninamide.     

Information theoretical analysis of the hydrogen atom

This is an analysis of the statistical nature of the time-independent Schrödinger equation through the use of the information entropy concept. We first study the Schrödinger equation in a general way and then by actually computing entropies of various states of the hydrogen atom for a re-examination of the problem. It is found that there exists a variational procedure involving maximizing entropy for obtaining all solutions once one solution is known. Based on certain observations of the particular single system, some general conclusions can be deduced. First of all, we can safely say that the Schrödinger equation, among many other interpretations, is but the consequence of a principle of minimum potential energy expectation with certain proper constraints imposed. In addition, the ensemble concept in statistical thermodynamics is also useful in understanding microscopic quantum systems and many quantum mechanical concepts such as energy quantization and wave nodal properties can be discussed in the light of information theory and statistics in general.     

An algorithm to find minimum free-energy paths using umbrella integration

The calculation of free-energy barriers by umbrella sampling and many other methods is hampered by the necessity for an a priori choice of the reaction coordinate along which to sample. We avoid this problem by providing a method to search for saddle points on the free-energy surface in many coordinates. The necessary gradients and Hessians of the free energy are obtained by multidimensional umbrella integration. We construct the minimum free-energy path by following the gradient down to minima on the free-energy surface. The change of free energy along the path is obtained by integrating out all coordinates orthogonal to the path. While we expect the method to be applicable to large systems, we test it on the alanine dipeptide in vacuum. The minima, transition states, and free-energy barriers agree well with those obtained previously with other methods.     

Shear-induced disappearances of energy minima and plastic deformation in polymer glasses

The potential energy landscape of a polymer glass is examined with regard to plastic deformation under shear strain. Shear strain is found to cause the disappearance of local potential energy minima, as determined by the decrease to zero of the curvature of the local minima. If the local energy minimum which the system is in disappears, the system becomes mechanically unstable and is forced to relax to an alternate energy minimum. This mechanical instability, which leads to a discontinuous change in system properties, is inherently irreversible—the new local minimum will not disappear, in general, when the strains are reversed. These disappearances of local energy minima and the associated irreversible relaxations lead to plastic deformation in polymer glasses.     

A new ab initio potential energy surface for the NeCO complex with the vibrational coordinate dependence

A new high quality three-dimensional potential energy surface for the Ne-CO van der Waals complex is developed using the CCSD(T) method and avqz∕avqz+33221 basis set. The ab initio calculation is performed in a total of 1365 configurations with supermolecule method. There is a single global minimum located in a nearly T-shaped geometry. The global minimum energy is -49.4090 cm(-1) at R(e)=6.40a(0) and θ(e)=82.5(°) for V(00). Using the three-dimensional potential energy surface, we have calculated bound rovibrational energy levels up to J = 10 including the Coriolis coupling terms. Compared with the experimental transition frequencies, the theoretical results are in good agreement with the experimental results.     

Singularity-free analytical energy gradients for the SAC/SAC-CI method: coupled perturbed minimum orbital-deformation (CPMOD) approach

A new procedure for evaluating energy gradients in a singularity-free manner is presented for use in the SAC/SAC-CI program in which computational dimensions are reduced by the perturbation selection method. The singularity in the energy gradients stemming from a breakdown of the unitary invariance is effectively removed by the minimum orbital-deformation (MOD) method proposed in the previous study. All calculations can be done analytically via new two sets of linear equations combined with the coupled-perturbed Hartree–Fock method. Geometry optimizations for malonaldehyde in the ground and lowest singlet excited states are performed by the new method.     

Substitution effects enhancing the lasing ability of organic compounds

Simple calculations can help to predict which derivatives in a series of organic compounds are potential lasing material. In conjugated systems, a necessary condition for lasing is that there is not less than a specific minimum energy difference between a first excited allowed and a second excited forbidden transition. This order of transition and energy spacing can be obtained by judicious substitutions even in molecules that do not meet these conditions. Lasing action in the near UV has been observed in five new compounds.     

Global energy minimization by rotational energy embedding

Given a sufficiently good empirical potential function for the internal energy of molecules, prediction of the preferred conformations is nearly impossible for large molecules because of the enormous number of local energy minima. Energy embedding has been a promising method for locating extremely good local minima, if not always the global minimum. The algorithm starts by locating a very good local minimum when the molecule is in a high-dimensional Euclidean space, and then it gradually projects down to three dimensions while allowing the molecule to relax its energy throughout the process. Now we present a variation on the method, called rotational energy embedding, where the descent into three dimensions is carried out by a sequence of internal rotations that are the multidimensional generalization of varying torsion angles in three dimensions. The new method avoids certain kinds of difficulties experienced by ordinary energy embedding and enables us to locate conformations very near the native for avian pancreatic polypeptide and apamin, given only their amino acid sequences and a suitable potential function.     

Influence of the Secondary Interaction Energy Minimum on the Early Stages of Colloidal Aggregation

The pair interaction energy of charged colloidal particles in electrolyte solutions can exhibit a large barrier as well as a pronounced secondary minimum. We discuss the effect of a secondary energy minimum on aggregation kinetics by modeling irreversible dimer formation as a two-step process in which charged colloidal particles in electrolyte solutions first aggregate reversibly into the secondary minimum before they can cross the energy barrier. In the classical regime of slow aggregation, the secondary minimum is seen to have a pronounced effect if either the ionic strength of the solution is high (e.g., 0.1 M for particles of 150-nm radius) or particles are large (>/=350-nm radius for an ionic strength of 0.01 M). Under these conditions, our calculations predict a transient period of fast aggregation into the secondary minimum followed by slow primary aggregation. The aggregation in this second regime is found to take place at a lower rate than what would be expected in the absence of the secondary minimum or from an earlier linearized model for secondary aggregation. The crossover time between the two regimes strongly depends on the particle size but not on the particle concentration, which however determines the degree of aggregation reached within the fast regime. We also conclude that a previously observed severe discrepancy between measured and predicted aggregation rate constants for submicron particles is not due to the neglect of secondary aggregation in the theoretical treatment. Copyright 2000 Academic Press.     

Measurement of the fluidized velocity in gas-solid fluidized beds based on AE signal analysis by wavelet packet transform

An energy distribution theory was presented based on regular evolvement of energy fraction of acous-tic signals with fluidization velocity. Wavelet packet analysis was used in processing the acoustic sig-nals originated from particle impact on the wall of a fluidized bed. A new criterion of judging incipient fluidization(Umf) velocity and minimum turbulent velocity(Umt) was proposed according to the energy distribution theory. Experiments were performed with five groups of high density polyethylene(PE) particles and one bimodal PE to acquire incipient fluidization velocity and minimum turbulent velocity by using the criterion. The feasibility of this method in obtaining characteristic fluidization parameters was further verified by comparing it to results from the pressure drop method and the empirical value from industry.     

Monte Carlo sampling algorithm for searching a scale-transformed energy space of polypeptides

A Monte Carlo sampling algorithm for searching a scale-transformed conformational energy space of polypeptides is presented. This algorithm is based on the assumption that energy barriers can be overcome by a uniform sampling of the logarithmically transformed energy space. This algorithm is tested with Met-enkephalin. For comparison, the entropy sampling Monte Carlo (ESMC) simulation is performed. First, the global minimum is easily found by the optimization of a scale-transformed energy space. With a new Monte Carlo sampling, energy barriers of 3000 kcal/mol are frequently overcome, and low-energy conformations are sampled more efficiently than with ESMC simulations. Several thermodynamic quantities are calculated with good accuracy.