Comparison between the analysis of the asymptotic wavepacket and the associated flux for the calculation of kinetic-energy-releases function

The kinetic-energy-releases (KER) distribution function of the fragment is an important observable in the molecular dynamics. In theory, there are several different methods to calculate the KER distribution function or spectrum, which could be generally divided into two classes: One is based on the analysis of the asymptotic wavepacket (“projection method”) and the other is on the analysis of the associated flux (“flux method”). By taking the above-threshold dissociation of the HeH⁺ (v = 8) molecule as an example, we compared these two classes of methods. Based on evenly separated Fourier grid representation, the KER distribution calculated via the projection method F_Proj(E_k) is the same as the one calculated via the flux method F_Flux(E_k) . The relationship between F_Proj(E_k) and the distribution of the projection of the asymptotic wavepacket onto the energy eigenstates of the quasicontinuum, P_Proj(E_k) , and the relationship between F_Flux(E_k) and the distribution of the dissociation probability P_Flux(E_k) from the cumulation of the associated flux, are determined.     

On the structure of the diagonal conditions

Some structural features of the diagonal problem of reduced density matrix theory are investigated by considering the cone B^r introduced elsewhere. In some special cases, we discover a very tight connection between the group structure of G^r, the invariance group of B^r, and the convex structure of B^r. We also state a general theorem that relates the convex structure of B^r to the group structure of G^r.     

Relation between the critical point and the precipitation threshold

The relationship between the critical point and the precipitation threshold is examined in the Flory—Huggins approximation with concentration-independent interaction parameter χ. Approximate explicit expressions for the difference between the critical point and the threshold can be derived by series expansion of threshold conditions. In the first-order approximation, the concentration difference depends only on the chainlength averages x_w, x_z, and x_z+1, in the second-order approximation it depends on x_w, x_z, x_z+1, and x_z+2, etc. For polymers of low polydispersity, the second-order approximation gives a good estimate of the concentration difference; for instance, for polymers with exponential distribution and x_w/x_n < 1.25, the error is lower than ca. 1%. The approximation is not suitable for polymers with x_z+1 ? x_z (including polymers whose cloud-point curve exhibits a triple point). Irrespective of the polydispersity of the polymer, the threshold as well as the whole cloud-point curve depend only on the weight-average and higher averages, x_w, x_z, x_z+1, …, x_z+k, where k → ∞; they are, however, independent of the number average x_n.     

On the optimal mixing of the exchange energy and the electron-electron interaction part of the exchange-correlation energy

The optimal mixing coefficient C of the exchange energy E_x and the electron-electron interaction part of the exchange-correlation energy W¹_xc in the formula for the total exchange-correlation energy E_xc was expressed through the ratio of the kinetic T_c and potential W_c contributions to the correlation energy E_c. This expression can be derived from a Heavyside step function model of the dependence of W^λ_xc on the coupling parameter of the electron interaction λ. Values of T_c and W_c obtained from ab initio wave functions were used to estimate C for a number of atoms and molecules. A strong dependence of T_c, W_c, and C on the bond distance was demonstrated for the case of the H₂ molecule. T_c and C approach zero in the bond-dissociation limit; so for an electron-pair bond, the admixing of exact exchange to obtain an accurate E_xc is strongly dependent on the bond length and has to disappear for weak interaction/large bond distances. The potential of the exchange-correlation hole constructed for H₂ from an ab initio second-order density matrix was compared with its generalized gradient approximation (GGA). © 1996 John Wiley & Sons, Inc.     

On the representations of the rotation group

The matrices of the irreducible representations of the 3-dimensional rotation group are shown to be related to Krawtchouk's orthogonal polynomials of a discrete variable x = j – m', whose degrees are given by n = j + m. The relation follows directly from the recurrence formulas satisfied by the matrix elements and permits a concise development of the formal properties of the rotation matrices. In particular, an asymptotic relation for large j is developed that generalizes a formula first discussed for a special case by Wigner.     

Vorticity: Simplifying the analysis of the current density

The induced current density ( J ( r )) provides useful information about the electronic structure of molecules under a magnetic field ( B ). However, the analysis of its topology is cumbersome because of its vectorial nature. We show that its tropicity (direction of rotation) and its strength can be compressed in the triple product B ⋅ ∇ × J ( r ) (tp J ( r )) that is a scalar field. The topology of tp J ( r ) has clear similarities to the Laplacian of the electron density. Additionally, the topology of aromatic and antiaromatic compounds is notoriously different. The vorticity of J ( r ) is helpful to define the circulation of the current density, C, that contrary to other methods, can be easily defined for individual rings in polycyclic molecules. This allows tp J ( r ) to clearly reproduce the Clar's structure of polycyclic aromatic hydrocarbons. © 2019 Wiley Periodicals, Inc.     

On the accuracy of the many-points local procedures

As until now proposed in the literature, in the local energy calculations we can distinguish “few-points” procedures, in which the number M of configurational points is strictly related to the number N of trial functions used, and statistical “many-points” procedures, in which the number M of points can be arbitrarily increased. In this paper we demonstrate that the energy errors resulting from a “many-points” calculation M points/N functions (M > N) can be connected in a simple way with the errors of the (^M_N) partial calculations N points/N functions. This suggests a possible approach for the problem of the choice of the configurational points to be introduced in the calculation, and leads to a simple interpretation of the numerical meaning of the error associated with the ordinary Ritz energy. Numerical examples on the hydrogen atom are reported.     

Role of the supramolecular structure of the reactants in the kinetics of the liquid-phase synthesis and decomposition of explosives

Formulas relating the observed rate constant of the initial reaction step (k _ef) to the rate constants of the elementary steps (k _i ), to the monomer-association species equilibrium constants (K _i ), and to the concentrations of the reactants (A _i ) and solvent have been obtained for a number of simple kinetic models of liquid-phase explosive material synthesis and decomposition reactions involving associating reactants. The k _ef = F(k _i , K _i , and A _i ) relationship is independent of the rate law of the reaction and is governed by the association species type (autoassociation or heteroassociation species) and by the number of association species kinds. In the case of parallel reactions involving association species, as distinct from the same reactions involving unassociated reactants, concentration variations have an effect on the ratio of the rates of the parallel steps and on the product yield ratio. By varying the reaction temperature, it is possible to make the rates of the parallel steps to vary in opposite ways. When monomers, dimers, and tetramers are present in the reaction mixture, the temperature dependence of lnk _ef may have an extremum. These deduced regularities are in qualitative agreement with experimental data.     

Relation of the decay of free radicals in irradiated polyethylene to the molecular motion of the polymer and the configurations of the free radicals

Decay reactions of the free radicals produced in irradiated polyethylene (high-density and low-density materials) were examined in connection with the molecular motion of the matrix polymer. Three temperature regions, in which the free radicals decay very rapidly, at around 120, 200, and 250°K, were designated T_A, T_L, and T_B, respectively. The decay of the free radicals at these temperatures had activation energies in high-density polyethylene of 0.4 kcal/mole for T_A, 9.4 kcal/mole for T_L, and 18.4 kcal/mole for T_B. In low-density polyethylene these quantities were 0.7 kcal/mole for T_A, 23.1 kcal/mole for T_L, and 24.8 kcal/mole for T_B. Comparison of time constants for the decay reactions and for molecular motion of the matrix polymer indicate that the decay in T_A and T_B is closely related to molecular motion in the amorphous regions of the polymer. The decay of the free radicals at T_L in high-density polyethylene is due to molecular motion associated with local mode relaxation at lamellar surfaces, while that of low-density polyethylene is due to local mode relaxation in the completely amorphous region. Steric configurations of the free radicals which decay in the respective temperature regions were also investigated.     

On the energy difference between the lP and 3P states of the beryllium isoelectronic sequence

Wave functions of the ¹S (ground state), ³P and ¹P states for the beryllium isoelectronic sequence have been obtained in various approximations. The HF ²p orbitals for the ¹P and ³P states are similar except for Be, where the ²p orbital is quite diffuse for the ¹P state. The difference between the experimental E(¹P) – E(³P) and the HF E(¹P) – E(³P) is 0.62 eV for Be and 1.17 ～ 1.40 eV for B⁺ ～ F⁵⁺. The disagreements are attributed to the correlation effects between the 2s and 2p electrons. This is confirmed by ci calculations. It is shown that a limited basis SCF calculation reproduces the above feature of the HF results if we treat the orbital exponents as the variational parameters. The use of the Slater values for the orbital exponents is shown to be inadequate especially for the Be ¹P state. The conclusions of this paper will be useful for discussing the V–T separations of H₂ and C₂H₄.     

New lipids from the soft corals of the Andaman Islands

Sphingolipids and glycolipids including previously unknown (2S,3S,4R)-1,3,4-trihydroxy-2-(2-(R)-hydroxyoctadecanoylamino)octadec-8E-ene, (2S,3R)-1,3-dihydroxy-2-octadecanoylamino-4E,8E-hexadecadiene, and (2-hydroxy-3-hexadecyloxypropyl)--L-fucopyranoside were isolated from soft corals collected on the shelf near the Andaman Islands (Indian Ocean). The structures of all compounds were established by spectroscopic methods and chemical analyses. The lipids possessed antibacterial activity against Bacillus subtilis, Bacillus pumilus, Escherichia coli, and Pseudomonas aeruginosa and antifungal activity against Aspergillus niger, Rhizopus oryzae, and Candida albicans.     

Discrepancy in the near-solute electric dipole moment calculated from the electric field

The electric dipole moment p ( r ) was computed as the integral of the permanent dipole moment of the solvent molecule μ( r ) weighted by the orientational probability distribution Ω( r ; O ) over all orientations, where O is the orientation of the solvent molecule at r . The relationship between Ω( r ; O ) and the potential of the mean torque was derived; p ( r ) is proportional to the electric field E ( r ) under the following assumptions: (1) the van der Waals (vdW) interaction is independent of the orientation of the solvent molecule at r ; (2) the solvent molecule and its electrical effect are modeled as a point dipole moment; (3) the solvent molecule at r is in a region far from the solute; and (4) μE( r ) ? k_BT, where k_B is Boltzmann's constant and T is absolute temperature. The errors caused by calculating near‐solute Ω( r ) and p ( r ) from E ( r ) are unclear. The results show that Ω( r ) is inconsistent with the value calculated from E ( r ) for water molecules in the first and second shells of solute with charge state Q = ±1 e, and a large variation in solvent molecular polarizability γ_mol(r), which appeared in the first valley of 4πr²E(r) for |Q| < 1 e. Nonetheless, p (r) is consistent with the values calculated from E (r) for |Q| ≤ 1 e. The implication is that the assumptions for calculating p ( r ) can be ignored in the calculation of the solvation free energy of biomolecules, as they pertain to protein folding and protein–protein/ligand interactions. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

On the methods of pore size distribution and the multilayer thickness

Summary Pore-volume-area distribution was achieved for a wide variety of silica gels, employing six computational procedures, and using botht-estimates ofCranston andInkley, andButt. The mathematical induction procedure outlined byRoberts proved to be as good as the more exact procedures. S _eum values evaluated by employing theButt estimate of multilayer thickness were exclusively higher than SBET data. The tabulatedt-values ofCranston andInkley yielded relatively reasonableS _eum results than those obtained above but are also higher thanS _BET. An overlooked parameter was suggested to be partly the cause of the divergence between the varioust-evaluations, namely: the thickness of a monolayer. Recalculation of all data, and using a value of 4.132 Å for the adsorbed monolayer, in botht-estimates ofLippens andButt, resulted in an excellent agreement betweenS _eum andS _BET in most of the cases. It is suggested that the multilayer thickness, as determined from adsorption data on non-porous solids, might not necessarily be of value in pore-size distribution analysis.     

Explanation for the 3.4-power law for viscosity of polymeric liquids on the basis of the tube model

The viscosity η₀(M) of polymeric liquids of molecular weight M is calculated on the basis of the tube model formulated by Doi and Edwards (ref. 3). The contour length fluctuation of polymers along the tube, which was neglected in ref. 3, is now explicitly taken into account. The result is where M_c = 2M_e, and M_e is the molecular weight between the entanglement points. This result is numerically close to the empirical 3.4-power law, η₀(M) = η₀(M_c)(M/M_c)^3.4, for 10M_c ? M ? 100M_c but approaches the result in ref. 3 for very high molecular weight. We thus conclude that the 3.4-power law is actually an approximate expression for the real curve which slowly approaches the asymptotic form calculated in ref. 3.     

Diazo Strategy for the Synthesis of Pyridazines: Pivotal Impact of the Configuration of the Diazo Precursor on the Process

Phosphazenes of vinyldiazocarbonyl compounds having cis stereochemistry of the functional groups on the vinyl bond readily produce pyridazines by a diaza‐Wittig process, whereas their counterparts with trans configuration remain intact under similar reaction conditions. Upon UV irradiation trans‐phosphazenes furnish pyridazines through a tandem trans‐to‐cis isomerization followed by intramolecular cyclization. At elevated temperatures trans‐(triphenyl)phosphazenes dissociate to give the initial vinyldiazo compounds, which produce pyrazoles in high yields. The first theoretical study on the mechanism of the diaza‐Wittig process by DFT calculations at the M06‐2X/6‐31G(d) level of theory suggest that for the cis‐phosphazenes a rapid tandem [2+2] cycloaddition/cycloelimination process with low energy barriers is preferred over trans isomers.     

On the validity of the Avrami theorem in the case of decelerating growth

It is argued that the use of the Avrami theorem, S(t) = 1−e^−S_x(t), is in principle not allowed when applied to overlapping diffusion zones or, more generally, in all cases where the phantom nuclei overtake the front of their parent nuclei. Via computer simulations the overtake effect is shown to exist and is found to be surprisingly small. The use of a modified Avrami equation, S(t) = FS_x(t)·[1 −e^−S_x(t)], is suggested for such cases and the function F[S_x(t)] pertaining to diffusion-controlled growth is reported.     

Effect of Cluster Deformations in the DLCA Modeling of the Sol-Gel Process

The diffusion limited cluster-cluster aggregation (DLCA) model is modified by including cluster deformations during aggregation, with a tuning flexibility parameter F. A three-dimensional computer simulation is presented, which starts from a collection of f-functional monomers randomly distributed in a cubic box with a volumic fraction c (concentration) and which uses the highly efficient bond fluctuation algorithm to describe the cluster deformations. It is shown that, for F 0, there exists a well defined threshold value of the volumic fraction below which the realization of all intra-aggregate bonding possibilities prevents the formation of a gelling network. For c > c _g , a true sol-gel transition occurs at a characteristic time t _g , after which an infinite cluster (which is self connected via the boundary conditions) appears. In contrast to DLCA, t _g does not increase as the box size increases. The transition at c _g is characterized by a divergence of the final clusters size for c<c _g and a divergence of the gel time for c>c _g . Several other numerical results are reported.     

Extension of the misfit series to the mercury-based cobaltites

The exploration of the Hg–Sr/Ca–Co–O system using EDS and electron microscopy has shown the possibility to incorporate mercury and excess cobalt in the middle rock salt layer of the misfit cobaltite structure, leading to the new series [Hg_1−xCo_xSr_2−yCa_yO₃]^RS[CoO₂]_b1/b2 with 0.6x0.90, 0y2, and b₁/b₂ ranging from 1.63 to 1.79. These new composite oxides, built up from [CoO₂]_∞ layers of the CdI₂-type stacked with triple [Hg_1−xCo_xSr_2−yCa_yO₃]. rock salt layers are remarkable by their large thermopower S values similarly to the thallium and lead misfit cobaltites. More importantly, the S value is sensitive to the composition of the triple rock salt layer, increasing as the calcium content increases, so that an enhancement of the figure of merit of the pure calcium compound is expected.     

Investigation of the Chemo- and Stereoselectivity of the Ketene-Claisen Rearrangement

A novel type of ketene-Claisen rearrangement in which the precursor of the rearrangement is generated in situ by reaction of optically active allyl thioethers with dichloroketene is described. A characteristic feature of this rearrangement is the excellent chemoselectivity in favor of allyl thioethers vs. allyl ethers, i.e., exclusive chirality transfer of the allylic sulfur moiety is observed with 12, 13 , and 25--27 . The cyclic, optically active allyl thioethers (+)-(R)- 4 and (?)-(S)- 4 and the open-chain allyl thioethers 11--13 rearrange with in situ generated dichloroketene to the optically active thioesters (?)-(S)- 28 , (+)-(R)- 28 , and 31-33 , respectively. A chirality-transfer of > 99% in the cyclic cases (+)-(R)- 4 and (?)-(S)- 4 , and 96--98 % in the open-chain cases 11--13 is observed. Furthermore, the dichloroketene-Claisen rearrangement is characterized by a high asymmetric 1,2-induction. The chiral allylic sulfides 25--27 give the optically active thioesters 36--38 with a 1,2-induction > 99% as determined by NMR-shift experiments.