The mass renormalization for the translational Brownian motion

The damped harmonic oscillator is modeled as a local mode X with mass m and frequency \(\omega _{0}\) immersed in a phonon bath with spectral density function \(j_{0}(\omega \)). This function behaves as \(\omega ^{s}\, (s= 1,2,3,\ldots )\) when \(\omega \rightarrow 0\). The limit \(\omega _{0} = 0\) represents translational (free) Brownian motion. The earlier work (Hakim and Ambegaokar in Phys Rev A 32:423, 1985) concluded that the so defined limit transition is prohibited for spectral densities with \(s<2\). In the present study we demonstrate that a specially constructed preliminary excitation changing the original bath spectrum as \(j_{0}(\omega ) \rightarrow j(\omega )\) allows for treating the free damped motion of X with no restriction for the initial spectrum dimensionality. This procedure validates the finite mass renormalization (i.e. \(m\rightarrow M\) when \(\omega _{0}\rightarrow 0)\) for the conventional bath spectra with \(s=1,2\). We show that the new spectral density \(j(\omega )\) represents the momentum bilinear interaction between mode X and the environmental modes, whereas the conventional function \(j_{0}(\omega )\) is inherent to the case of bilinear coordinate interaction in terms of the same variables. The translational damping kernel is derived based on the new spectral density.     

Microencapsulation thermodynamics of methylated<Emphasis Type="Italic"> β</Emphasis>-cyclodextrins with bile salt: enthalpy,entropy, and solvent effect

The interactions between bile salts and modified cyclodextrins (CD) have aroused intensive interest because of their influence on the in vivo drug release within the small intestine. We use a recently proposed molecular dynamics/quantum mechanics/continuum solvent model (Ye et al. in Chem Phys Lett 648:170, 2016) to investigate the microscopic binding mechanisms of the natural, dimethylated (M200), randomly methylated (M214), and trimethylated (M300) \(\beta\)-CDs, with glycocholate (GC). The computed binding free energies of the four CDs with GC decrease in the following order: natural \(\beta\)-CD > M200 > M214 > M300, which is in accord with the experimental data. As suggested by the computed thermodynamic quantities, the relative stabilities of the four CD/GC complexes are determined by both the host–guest interactions (enthalpy plus entropy) and the solvent effects. Moreover, the former favors M200, while the latter favors natural \(\beta\)-CD. For M300, both factors are in disadvantage.     

The complexity of the Clar number problem and an exact algorithm

The Clar number of a (hydro)carbon molecule, introduced by Clar (The aromatic sextet, 1972), is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected plane graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternate on. It was proved by Abeledo and Atkinson (Linear Algebra Appl 420(2):441–448, 2007) that the Clar number can be computed in polynomial time if the plane graph has even faces only. We prove that calculating the Clar number in general 2-connected plane graphs is \(\mathsf {NP}\)-hard. We also prove \(\mathsf {NP}\)-hardness of the maximum independent set problem for 2-connected plane graphs with odd faces only, which may be of independent interest. Finally, we give an exact algorithm that determines the Clar number of a given 2-connected plane graph. The algorithm has a polynomial running time if the length of the shortest odd join in the planar dual graph is fixed, which gives an efficient algorithm for some fullerene classes, such as carbon nanotubes.     

Deposition Morphology and Magnetism of Co,Pt Adatoms and Small CoPt Adclusters on Ni(100) Substrate

Theoretical calculations of Co\(_{n-x}\)Pt\(_x\) (n = 1–3; \(x \le n\)) clusters on Ni(100) surface for their spin and orbital magnetic moments, as well as the magnetic anisotropy energy (MAE), are performed by using the density-functional theory (DFT) method including a self-consistent treatment of spin–orbit coupling (SOC). The results reveal that the ferromagnetic Co atoms in intra Co\(_{n-x}\)Pt\(_x\) adclusters couple ferromagnetically to their underlayer Ni atoms. The predominant inter-interactions between Co adatoms and Ni surface with the partly filled 3d band, together with the secondary intra-interactions between Co adatoms and Pt adatoms with fully filled 5d band, lead to a strongly quenched orbital moment (\(\mu _{\mathrm{{orb}}}^{\mathrm{{Co}}}\) = 0.18–0.14 \(\mu _B\); \(\mu _{\mathrm{{orb}}}^{\mathrm{{Pt}}} \approx \) 0.24–0.19 \(\mu _B\)) but a less quenched spin moment (\(\mu _{\mathrm{{spin}}}^{\mathrm{{Co}}} \approx \) 2.0 \(\mu _B\); \(\mu _{\mathrm{{spin}}}^{\mathrm{{Pt}}} \approx \) 0.35 \( \mu _B\)). The MAEs of CoPt adclusters exhibit a strong dependence on alloying effect rather than size effect, which is direly proportional to SOC strength and orbital moment anisotropy. The oxidations of CoPt clusters always reduce orbital magnetic moments and consequently decrease the corresponding MAEs.     

On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

In our recent works (R. Szmytkowski, J. Phys. A 39:15147, 2006; corrigendum: 40:7819, 2007; addendum: 40:14887, 2007), we have investigated the derivative of the Legendre function of the first kind, P _ν(z), with respect to its degree ν. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, \({P_{\nu}^{\pm m}(z)}\), with respect to the degree ν, for \({m \in \mathbb{N}}\). At first, we establish several contour-integral representations of \({\partial P_{\nu}^{\pm m}(z)/\partial\nu}\). They are then used to derive Rodrigues-type formulas for \({[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}}\) with \({n \in \mathbb{N}}\). Next, some closed-form expressions for \({[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}}\) are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, \({Q_{n}^{\pm m}(z)}\); the explicit representations are suitable for use for numerical purposes in various regions of the complex z-plane. Finally, the derivatives \({[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}, [\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=n}}\) and \({[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}}\), all with m > n, are evaluated in terms of \({[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}}\). The present paper is a complementary to a recent one (R. Szmytkowski, J. Math. Chem 46:231, 2009), in which the derivative \({\partial P_{n}^{\mu}(z)/\partial\mu}\) has been investigated.     

Lego-like spheres and tori

Given a connected surface \({\mathbb {F}}^2\) with Euler characteristic \(\chi \) and three integers \(b>a\ge 1<k\), an \((\{a,b\};k)\)-\({\mathbb {F}}^2\) is a \({\mathbb {F}}^2\)-embedded graph, having vertices of degree only k and only a- and b-gonal faces. The main case are (geometric) fullerenes (5, 6; 3)-\({\mathbb {S}}^2\). By \(p_a\), \(p_b\) we denote the number of a-gonal, b-gonal faces. Call an \((\{a,b\};k)\)-map lego-admissible if either \(\frac{p_b}{p_a}\), or \(\frac{p_a}{p_b}\) is integer. Call it lego-like if it is either \(ab^f\)-lego map, or \(a^fb\)-lego map, i.e., the face-set is partitioned into \(\min (p_a,p_b)\) isomorphic clusters, legos, consisting either one a-gon and \(f=\frac{p_b}{p_a}\,b\)-gons, or, respectively, \(f=\frac{p_a}{p_b}\,a\)-gons and one b-gon; the case \(f=1\) we denote also by ab. Call a \((\{a,b\};k)\)-map elliptic, parabolic or hyperbolic if the curvature \(\kappa _b=1+\frac{b}{k}-\frac{b}{2}\) of b-gons is positive, zero or negative, respectively. There are 14 lego-like elliptic \((\{a,b\};k)\)-\({\mathbb {S}}^2\) with \((a,b)\ne (1,2)\). No \((\{1,3\};6)\)-\({\mathbb {S}}^2\) is lego-admissible. For other 7 families of parabolic \((\{a,b\};k)\)-\({\mathbb {S}}^2\), each lego-admissible sphere with \(p_a\le p_b\) is \(a^fb\) and an infinity (by Goldberg–Coxeter operation) of \(ab^f\)-spheres exist. The number of hyperbolic \(ab^f\,(\{a,b\};k)\)-\({\mathbb {S}}^2\) with \((a,b)\ne (1,3)\) is finite. Such \(a^f b\)-spheres with \(a\ge 3\) have \((a,k)=(3,4),(3,5),(4,3),(5,3)\) or (3, 3); their number is finite for each b, but infinite for each of 5 cases (a, k). Any lego-admissible \((\{a,b\};k)\)-\({\mathbb {S}}^2\) with \(p_b=2\le a\) is \(a^f b\). We list, explicitly or by parameters, lego-admissible \((\{a,b\};k)\)-maps among: hyperbolic spheres, spheres with \(a\in \{1,2\}\), spheres with \(p_b\in \{2,\frac{p_a}{2}\}\), Goldberg–Coxeter’s spheres and \((\{a,b\};k)\)-tori. We present extensive computer search of lego-like spheres: 7 parabolic (\(p_b\)-dependent) families, basic examples of all 5 hyperbolic \(a^fb\) (b-dependent) families with \(a\ge 3\), and lego-like \((\{a,b\};3)\)-tori.     

Single carbon dioxide molecules on surfaces studied by low-temperature scanning tunneling microscopy

The chemical conversion of carbon dioxide (\(\hbox {CO}_2\)) has been intensively studied because the molecule is responsible for global warming. Rational design of catalysts plays an important role in converting \(\hbox {CO}_2\) into value-added compounds. Understanding the interaction between \(\hbox {CO}_2\) and surfaces of catalysts is a prerequisite to preparing high-performance catalysts. This review focuses on the investigations of \(\hbox {CO}_2\) molecules on single crystalline surfaces studied by low-temperature scanning tunneling microscopy. Molecular adsorption, diffusion, and conversion on metal surfaces, metal oxide surfaces, and surfaces decorated by metal-organic frameworks are summarized.     

Sommerfeld’s fine structure constant approximated as a series representation in e and $$\pi $$

Sommerfeld in 1916 introduced the dimensionless fine structure constant, \(\alpha \), in to the context of atomic physics, in the course of working out the relativistic theory of the H atom, under the old quantum theory of Bohr. He was able to account for the fine structural detail of the atomic line spectrum of H by introducing this dimensionless constant which emerged naturally from his relativistic theory of the H atom. Since this time, the fine structure constant has emerged in several other contexts within experimental and theoretical physics. It has attained a status of being a mysterious number in physics that defies understanding as to its experimentally verified magnitude and identity. Being physically dimensionless, such a number invites a suggestion (or approximation) of its value in terms of mathematical constants in some formulation. Feynman most famously has conjectured that it might be possible to account for \(\alpha \) in some type of series or product expression in “e”, the base of natural logarithms, and “\(\pi \)” the familiar circular constant. Here we propose an infinite series in the product \(\mathrm{e} \cdot \pi \) that converges, within a few terms, to better than 9999 parts in 10,000 of the true value of \(\alpha \).     

Dependence of the density of solutions of alkali metal halides on the composition of methylpyrrolidone-water mixed solvents

The densities of solutions of alkali metal halides in methylpyrrolidone (MP)-water mixtures were measured at 298.15 K over the entire range of mixed solvent compositions. The standard partial molar volumes of the electrolytes \(\overline {V_2^ \circ } \) were calculated. The \(\overline {V_2^ \circ } \) values of alkali metal halides in MP-H₂O mixtures were related linearly to the \(\overline {V_2^ \circ } \) values in aqueous solutions. These dependences were used to determine the standard partial molar volumes of ions \(\overline {V_i^ \circ } \) in the mixtures studied. The standard partial molar volumes of transfer of the ions from water into MP-water mixtures were calculated.     

Thermodynamic analysis of the structure of aqueous solutions of C<Subscript>12</Subscript>–C<Subscript>18</Subscript> hydrocarbons

Thermodynamic cycles including the increments \(\Delta G_{CH_2 }^0 , \Delta H_{CH_2 }^0 \), and \(T\Delta S_{CH_2 }^0 \) were constructed for dissolution, evaporation, hydrophobic hydration of C₅–C₉ hydrocarbons, and transfer from vapor (\(\Delta G_{CH_2 }^0 \) = ?0.7 kJ·mol^?1, \(\Delta H_{CH_2 }^0 \) = 2.9 kJ·mol^?1, \(T\Delta S_{CH_2 }^0 \) = 3.6 kJ·mol^?1) and water (\(\Delta G_{CH_2 }^0 \) = ?1.4 kJ·mol^?1, \(\Delta H_{CH_2 }^0 \) = 5.8 kJ·mol^?1, \(T\Delta S_{CH_2 }^0 \) = 7.2 kJ·mol^?1) to micelles of C₁₂–C₁₈ hydrocarbons. The formation of bistable hydrated micelles of C₁₂–C₁₈ is explained by a transition between the order-disorder states in an assembly of small (nano) systems of water. The extensive parameters of small systems and critical phenomena predicted by fluctuation theory are discussed.     

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.     

Extending the McClelland formula for total $$\pi $$-electron energy

The McClelland formula, based on the upper bound \(\sqrt{2mn}\), is capable of reproducing over 99.5% of the total \(\pi \)-electron energy (\(E_\pi \)) of a conjugated hydrocarbon, whose molecules possess n carbon atoms and m carbon–carbon bonds. Its weak point is that it predicts equal \(E_\pi \)-values for all isomers. We now show how this failure can be overcome, offering a general strategy for extending McClelland’s formula. By means of one of these extensions, \(E_\pi \) is related with the energy of the highest occupied molecular orbital, and the error of the new formula is diminished by more than \(50\%\) relative to the standard McClelland approximation.     

Assessing protein–ligand binding modes with computational tools: the case of PDE4B

In a first step in the discovery of novel potent inhibitor structures for the PDE4B family with limited side effects, we present a protocol to rank newly designed molecules through the estimation of their IC\(_{50}\) values. Our protocol is based on reproducing the linear relationship between the logarithm of experimental IC\(_{50}\) values [\(\log\)(IC\(_{50}\))] and their calculated binding free energies (\(\Delta G_\mathrm{binding}\)). From 13 known PDE4B inhibitors, we show here that (1) binding free energies obtained after a docking process by AutoDock are not accurate enough to reproduce this linear relationship; (2) MM-GB/SA post-processing of molecular dynamics (MD) trajectories of the top ranked AutoDock pose improves the linear relationship; (3) by taking into account all representative structures obtained by AutoDock and by averaging MM-GB/SA computations on a series of 40 independent MD trajectories, a linear relationship between \(\log\)(IC\(_{50}\)) and the lowest \(\Delta G_\mathrm{binding}\) is achieved with \(R^2=0.944\).     

Thermodynamic analysis of the dynamic structure of micellar solutions of sodium alkyl sulfates

The results of previous studies of the increments \(\Delta G_{CH_2 }^0 \), \(\Delta H_{CH_2 }^0 \), and \(T\Delta S_{CH_2 }^0 \) in hydrocarbon solution processes in water, evaporation, hydration, and transfer from vapor and water to surfactant micelles are summarized. The corresponding thermodynamic cycles were constructed. A micelle was shown to be a bistable structure formed by the contact and water-separated hydrophobic interactions of the alkyl groups of surfactants. The fraction in \(\Delta G_{CH_2 (M)}^0 \) is ?2.3 kJ·mol^?1 for the contact associate and ?0.7 kJ·mol^?1 for the hydrated one. Water is involved in the hydrophobic interaction of each associate. In a dualist micelle, however, \(\Delta G_{CH_2 }^0 \) = ?3.0 kJ mol^?1 equals that of the dispersion interaction after condensation from vapor. The dual nature of the dynamic properties of micelles is discussed.     

Hierarchical enumeration of Kekulé’s benzenes,Ladenburg’s benzenes,Dewar’s benzenes,and benzvalenes by using combined-permutation representations

The group hierarchy for each skeleton of ligancy 6 is formulated to be: point group (PG \({\varvec{G}}_{\sigma }\)) \(\subseteq \) RS-stereoisomeric group (RS-SIG \({\varvec{G}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) stereoisomeric group (SIG \(\widetilde{{\varvec{G}}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) isoskeletomeric group (ISG \(\widetilde{\widetilde{{\varvec{G}}}}_{\sigma \widetilde{\sigma }\widehat{I}}\) = \({\varvec{S}}^{[6]}_{\sigma \widehat{I}}\)), where we start from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{6h}\) for the Kekulé benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{3h}\) for the Ladenburg benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the Dewar benzene skeleton, or from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the benzvalene skeleton. After these groups are constructed as combined-permutation representations, the calculation of the respective cycle indices with chirality fittingness (CI-CFs) and the introduction of ligand-inventory functions are conducted to give generation functions for 3D-based enumerations (for PGs and RS-SIGs) and 2D-based enumerations (for SIGs and ISGs). The enumeration results are discussed by means of isomer-classification diagrams, in which equivalence classes under enantiomerism (for PGs), RS-stereoisomerism (for RS-SIGs), stereoisomerism (for SIGs), and isoskeletomerism (for ISGs) are illustrated schematically. The implicit connotations of the conventional terms “skeletal isomerism”, “positional isomerism”, and “constitutional isomerism” are discussed, where the effects of the concept of isoskeletomerism are emphasized.     

Partition of $$\pi $$-electrons among the faces of polyhedral carbon clusters

We apply the concepts of importance and redundancy to compute and analyze the partition of \(\pi \)-electrons among faces of actual and potential polyhedral carbon clusters. In particular, we present explicit formulas and investigate asymptotic behavior of total and average \(\pi \)-electron content of all faces of prisms and n-barrels. We also discuss the observed deviations from the uniform distribution and show that the patterns of net migration of \(\pi \)-electrons differ from those computed for narrow nanotubical fullerenes. Some possible directions of future work are also indicated.     

The ensemble folding dynamics of EF-hand domain in parvalbumin from a Monte Carlo simulation

The helix-loop-helix (i.e., EF-hand) is the most common motif in the superfamily of Ca\(^{2+}\)-binding proteins. Parvalbumins, as the classical EF-hand proteins, are associated with the muscle relaxation/contration, the calcium buffering etc. The previous researches focus more of interest on the ion coupled/decoupled mechanism using molecular dynamics (MD) computations. We developed the novel approach instead of MD, in which the Landau free energy was introduced to describe the protein in term of the skeletal C\(_\alpha \) chain. The unfolding and folding processes were simulated by the Glauber algorithm under the repeated heating and cooling cycles. The high-quality crystal structure (2PVB) was as the trigger of non-equilibrium dynamics simulation for parvalbumin-\(\beta \) (Parv). The evolution of three dimensional structure during the folding was displayed for the residues 8–64 fragment in Parv. The phenomenon of helix nucleation was observed, that was, the 3\(_{10}\) helix at residues 35–37 and the front of \(\alpha \)-helix at residues 40–50 were firstly formed in the folding process. We also found two potential misfoldings which were caused by the distortion of the local structures at residues 35–37, 45–50.     

Modelling of the NO + CO reaction over inhomogeneous surfaces

A phenomenological model of \(\mathrm {CO}\) oxidation with \(\mathrm {NO}\) reaction proceeding over composite (supported) catalysts is proposed and solved numerically using the finite difference method. The model is based on the coupled system of PDEs subject to nonclassical conjugate conditions at the catalyst-support interface and includes: the bulk diffusion of reactants from a bounded vessel towards the catalyst surface and the bulk one of the reaction products from the surface into the same vessel, adsorption and desorption of particles of reactants, and surface diffusion of adsorbed molecules. The readsorption of the reaction product N\(_2\)O is also taken into account. The influence of the rate constants of the adsorbed particle jumping via the catalyst-support interface and reaction rate constants on the surface reactivity is investigated. It is shown that the turnover rates of the CO and NO into products N\(_2\)O, CO\(_2\), and N\(_2\) are nonmonotonic time functions and depending on values of the kinetic parameters may possess one or two maxima. The N\(_2\)O readsorption in case of the existence of two maxima essentially increases the turnover rates and extends the duration of their high values. The mechanism and conditions for arising of the second maximum is discussed. It is also shown that the variation of the particle jumping rate constants influences differently the size of the jump discontinuity of concentrations of different adsorbates at the catalyst-support interface.