Exact solutions and coherent states for a generalized potential

Exact solutions of the vibrational Schrödinger equation for a generalized potential energy function \(\hbox {V(R)}=\hbox {C}_{0}(\mathrm{{R}-\mathrm {R}}_{\mathrm{e}})^{2}/[\hbox {aR}\,+\,(\mathrm{{b}-\mathrm {a}})\hbox {R}_{\mathrm{e}}]^{2}\) are obtained. It includes those of Dunham, Ogilvie and Simons–Parr–Finlan potentials as special cases corresponding to b \(=\) 1, a \(=\) 0, 1/2, 1, respectively. The analytical wave functions derived are useful to test the quality of numerical methods or to perform perturbative or variational calculations for the problems that cannot be solved exactly. Coherent states for generalized potential, which minimize the position–momentum uncertainty relation are also constructed.     

Infrared spectrum and structure of the adamantane cation: direct evidence for Jahn-Teller distortion

The fundamentals: the IR spectrum of the adamantane cation, C(10)H(16)(+), has been derived by resonant IR photodissociation of weakly bound C(10)H(16)(+)?L(n) clusters. The analysis of the IR spectrum provides the first spectroscopic characterization of this fundamental cycloalkane carbocation in the gas phase and direct evidence for the Jahn-Teller distortion in the (2)A(1) ground electronic state.     

Exact results on the relaxation spectrum of a simple model for micelle formation

We consider the reaction scheme A₁ + A_i ? A_i+1 (i - 1,2, …, n), where the equilibrium of the first n - 1 reactions is strongly shifted towards the left while the equilibrium of the nth reaction is strongly shifted towards the right as a model for formation of micelles and prove that under reasonable assumptions the relaxation spectrum has one relaxation time corresponding to the gross reaction (n = 1)A_n ? A_n+1 and, at much higher frequencies, a cluster of relaxation times corresponding to the n - 1 first reactions.     

On the eigenfunctions of the smoluchowski diffusion equation in a double-well potential and their relation to polymer dynamics

The eigenspectrum, matrix elements and sum-rule contributions for a Smoluchowski equation in a potential U = U₀(x²?1)² are obtained. The importance of the higher eigenfunctions and the sum rule is discussed and it is shown that a reasonably complete solution can be obtained with a comparatively small number of eigenfunctions.     

A general principle for the relation between different time systems and its relation to general relativity theory

We consider how times of different clocks can be related to each other. The general relation that we obtain can be made more specific for situations that occur in the context of relativity theory. Our considerations concerning the kinematics of time lead to a modification of the Newtonian law. As a result, we obtain equations of movement that can also be deduced from general relativity theory if a Schwarzschild metric is assumed. The present derivation is significantly more simple than the orthodox general relativistic one. For instance, it does not make recourse to a four-dimensional space-time. We point out that different time coordinates must be considered, but spatial coordinates are left unchanged.     

Correlations between the activities of a gamma-ray emitter calculated from different peaks in the spectrum

A model for calculating correlations between the activities of the same gamma-ray emitter calculated from different peaks in its spectrum is presented. The correlation coefficients can be expressed by the relative uncertainties of the input quantities. The use of this model in the calculation of the mean activity prevents the calculation of an excessively small uncertainty of the mean, since averaging of correlated uncertainty components is avoided.     

Exact solutions and spectrum analysis of a noncentral potential in the presence of vector potential

We studied exact solutions and spectrum analysis of the pseudoharmonic oscillator in the presence of θ-dependent scalar potential and the sum of two vector potentials Dirac magnetic monopole and Aharonov-Bohm field. The effect of the Dirac magnetic monopole (g), Aharonov-Bohm field (ℱ) , the dissociation energy of diatomic molecules (D _e), equilibrium intermolecular separation (r _e), and the reduced mass (μ) on the energy spectrum for some diatomic molecules (CO, NO, N₂, CH, H₂, and ScH) are analyzed. We compared our results with theoretical experiments of pseudoharmonic oscillator potential in a molecular system.     

A relation between RF values of unidimensional and circular paper chromatography

1. R_F values of 12 amino acids were determined in phenol-water and benzyl alcohol-acetic acid-water using unidimensional and circular techniques of paper chromatography. 2. It has been observed that the square of circular R_F values (considering frontal boundary of solute) equals the linear R_F values, and this relation is valid for the amino acids as well as the sugars studied. 3. The relation is found to be independent of the distance traversed by the solvent in the case of both amino acids and sugars. 4. A theoretical support has been given to the experimental observations.     

Relation between different variants of the generalized Douglas-Kroll transformation through sixth order

Wolf et al. have recently investigated a generalized Douglas-Kroll transformation. From a general class of unitary transformations that can be used in the Douglas-Kroll transformation, they pick one which is supposed to give, at a given order, an optimal transformed Dirac Hamiltonian. Results were presented through the fifth order. However, no data were given to demonstrate to which extent the so-called "optimal" Douglas-Kroll transformation is superior to other choices. In this work, the Douglas-Kroll transformation is extended to the sixth order for the first time, using computer algebra algorithms to obtain the working equations. It is shown how, at a given order, different variants of the Douglas-Kroll Hamiltonians are related. Various choices of the generalized transformation are examined numerically for the ground states of the one-electron atomic ions with nuclear charges Z=20, 40, 60, 80, 100, and 120. It is shown that compared to the improvement obtained by including the next order, the differences between various choices for the generalized Douglas-Kroll transformation are almost negligible. Results closest to the Dirac eigenvalues are not obtained with the optimal Douglas-Kroll transformation given by Wolf et al., but with the parametrization originally suggested by Douglas and Kroll.     

Strong correlation between SHAPE chemistry and the generalized NMR order parameter (S2) in RNA

The functions of most RNA molecules are critically dependent on the distinct local dynamics that characterize secondary structure and tertiary interactions and on structural changes that occur upon binding by proteins and small molecule ligands. Measurements of RNA dynamics at nucleotide resolution set the foundation for understanding the roles of individual residues in folding, catalysis, and ligand recognition. In favorable cases, local order in small RNAs can be quantitatively analyzed by NMR in terms of a generalized order parameter, S2. Alternatively, SHAPE (selective 2'-hydroxyl acylation analyzed by primer extension) chemistry measures local nucleotide flexibility in RNAs of any size using structure-sensitive reagents that acylate the 2'-hydroxyl position. In this work, we compare per-residue RNA dynamics, analyzed by both S2 and SHAPE, for three RNAs: the HIV-1 TAR element, the U1A protein binding site, and the Tetrahymena telomerase stem loop 4. We find a very strong correlation between the two measurements: nucleotides with high SHAPE reactivities consistently have low S2 values. We conclude that SHAPE chemistry quantitatively reports local nucleotide dynamics and can be used with confidence to analyze dynamics in large RNAs, RNA-protein complexes, and RNAs in vivo.     

A method to obtain the Eckart hamiltonian and the equations of motion of a highly deformable polyatomic system in tems of generalized coordinates

A general method that makes it possible to derive the Eckart hamiltonian and the equations of motion of a highly deformable polyatomic system in terms of generalized coordinates is presented. The overall rotation is taken into account and the method is applied to various non-reactive collisional tetratomic systems. namely (i) diatom-diatom, (ii) atom-linear symmetric triatomic and (iii) atom-bent symmetric triatomic.     

Exact treatment of generalized modifications of finite-dimensional systems by the LRM approach

LRM (Low Rank Modification) is a mathematical method that produces eigenvalues and eigenstates of generalized eigenvalue equations. It is similar to the perturbation expansion in that it assumes the knowledge of the eigenvalues and eigenstates of some related (unperturbed) system. However, unlike perturbation expansion, LRM produces correct results however large the modification of the original system. LRM of finite-dimensional systems is here generalized to the combined (external and internal) modifications. Parent n-dimensional system A _n containing n eigenvalues λ _i and n eigenstates \({| {\Phi_i}\rangle}\) is described by the generalized n × n eigenvalue equation. In an external modification system A _n interacts with another ρ-dimensional system B _ρ which is situated outside the system A _n . In an internal modification relatively small σ-dimensional subsystem of the parent system A _n is modified. Modified system C _n+ρ that contains external as well as internal modifications is described by the generalized (n + ρ) × (n + ρ) eigenvalue equation. This system has (n + ρ) eigenvalues \({\varepsilon_s}\) and (n + ρ) corresponding eigenstates \({| {\Psi_s}\rangle}\) . In LRM this generalized (ρ + n) × (ρ + n) eigenvalue equation is replaced with a (nonlinear) (ρ + σ) × (ρ + σ) equation which produces all eigenvalues \({\varepsilon_s \notin \left\{ {\lambda_i}\right\}}\) and all the corresponding eigenstates \({| {\Psi_s}\rangle }\) of C _{n + ρ}. Another equation produces remaining solutions (if any) that satisfy \({\varepsilon_s \in \left\{ {\lambda_i}\right\}}\) . Those two equations produce exact solution of the modified system C _{n + ρ}. If (ρ + σ) is small with respect to n, this approach is numerically much more efficient than a standard diagonalization of the original generalized eigenvalue equation. Unlike perturbation expansion, LRM produces exact results, however large modification of the parent system A _n .     

Born-Haber-Fajans cycle generalized: linear energy relation between molecules, crystals, and metals

Classical procedures to calculate ion-based lattice potential energies (U(POT)) assume formal integral charges on the structural units; consequently, poor results are anticipated when significant covalency is present. To generalize the procedures beyond strictly ionic solids, a method is needed for calculating (i) physically reasonable partial charges, delta, and (ii) well-defined and consistent asymptotic reference energies corresponding to the separated structural components. The problem is here treated for groups 1 and 11 monohalides and monohydrides, and for the alkali metal elements (with their metallic bonds), by using the valence-state atoms-in-molecules (VSAM) model of von Szentpály et al. (J. Phys. Chem. A 2001, 105, 9467). In this model, the Born-Haber-Fajans reference energy, U(POT), of free ions, M(+) and Y(-), is replaced by the energy of charged dissociation products, M(delta)(+) and Y(delta)(-), of equalized electronegativity. The partial atomic charge is obtained via the iso-electronegativity principle, and the asymptotic energy reference of separated free ions is lowered by the "ion demotion energy", IDE = -(1)/(2)(1 - delta(VS))(I(VS,M) - A(VS,Y)), where delta(VS) is the valence-state partial charge and (I(VS,M) - A(VS,Y)) is the difference between the valence-state ionization potential and electron affinity of the M and Y atoms producing the charged species. A very close linear relation (R = 0.994) is found between the molecular valence-state dissociation energy, D(VS), of the VSAM model, and our valence-state-based lattice potential energy, U(VS) = U(POT) - (1)/(2)(1 - delta(VS))(I(VS,M) - A(VS,Y)) = 1.230D(VS) + 86.4 kJ mol(-)(1). Predictions are given for the lattice energy of AuF, the coinage metal monohydrides, and the molecular dissociation energy, D(e), of AuI. The coinage metals (Cu, Ag, and Au) do not fit into this linear regression because d orbitals are strongly involved in their metallic bonding, while s orbitals dominate their homonuclear molecular bonding.     

Flocculation performance of different polyacrylamide and the relation between optimal dose and critical concentration

Flocculation performance of four kinds of polyacrylamide (PAM) with different chain end group, namely, PAM, star-PAM, PGS-PAM and Al(OH)₃-PAM hybrids, in Kaolin suspensions have been investigated by Spectrophotometer. It was found that their flocculation efficiency deceases in the order: PGS-PAM > Al(OH)₃-PAM ≈ star-PAM > PAM. It was also found that the optimal dose (C_od) of the polymer flocculant in flocculation process is proportional to the critical concentration (C∗) of polymer flocculant in solution and to the suspension solid content (C_ss) in Kaolin suspensions.     

The effect of the negative coupling parameter on the spectrum of a trapped Bose gas

The interest in attractive Bose–Einstein Condensates arises due to the chemical instabilities generate when the number of trapped atoms is above a critical number. In this case, recombination process promotes the collapse of the cloud. This behavior is normally geometry dependent. Within the context of the mean field approximation, the system is described by the Gross–Pitaevskii equation. We have considered the attractive Bose–Einstein condensate, confined in a nonspherical trap, investigating numerically and analytically the solutions, using controlled perturbation and self-similar approximation methods. This approximation is valid in all interval of the negative coupling parameter allowing interpolation between weak-coupling and strong-coupling limits. When using the self-similar approximation methods, accurate analytical formulas were derived. These obtained expressions are discussed for several different traps and may contribute to the understanding of experimental observations.     

The parameter dependence of calculated koopmans'defects in the framework of a model hamiltonian verified in the case of transition metal compounds

The validity of Koopmans' theorem in the lower energy region of trimethylenemethane iron tricarbonyl (1) is studied by means of a semi-empirical IN Reorganization energies are investigated as a function of the one-electron resonance integral β_μυ^AB and as a function of the one- and two interaction (γ_μυ^AA, K_μυ^AA and γ_μυ^AB). β_μυ^AB determines the kinetic energy of the interacting enlarged kinetic energy. The two-center electron repulsion intergral γ_μυ^AB have been calculated by means of an exponential formula with a short-range potentials with large γ_μυ^AB gradients pronounced reorganization energies are predicted while negligible defects even for stro case of smooth repulsion potentials with small γ_μυ^AB gradients. The largest parameter dependence of the calculated reorganization energie one-center electron—electron interaction integrals. With enlarged γ_μυ^AA and K_μυ^AA values enhanced Koopmans' defects are calculted in the framework of many body perturbation theory based on the Green's function formalism. A renormalized diagonal approximation for the self and correlation increments. The net defects are predominantly determined by electronic relaxation and modifications of the correlation energy in the ca of ground state pair-correlation (pair-removal) is only of minor importance.