Linearity and quadrupolarity of tetragonal and cylindrical penning traps of arbitrary length-to-width ratio

The two most popular Penning traps in use for Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry are tetragonal (i. e., orthorhombic with square cross section) and cylindrical. Here we compare tetragonal and cylindrical traps as a function of aspect (length-to-width) ratio and position within the trap, by comparing the numerically computed (from Simion 6.0) electric potential field of a given trap relative to each of three idealized potentials required for FT-ICR experiments: dipolar one-dimensional potential for alternating current (ac) excitation/detection of cyclotron motion, azimuthal two-dimensional quadrupolar potential for ac excitation for ion axialization, and axial three-dimensional quadrupolar potential for direct current axial confinement of ions. Our numerically computed results agree well with those previously derived analytically. The numerical approach provides a simpler and more accessible means for analyzing the aforementioned potentials. Moreover, the numerical approach (unlike the analytical approach) readily extends to traps of lower symmetry. Finally, even when analytical solutions are available, the numerical method presented here is complementary, since it provides a useful check on the validity of the derived equations.     

Density-functional theory for fluid mixtures of charged chain particles and spherical counterions in contact with charged hard wall: Adsorption, double layer capacitance, and the point of zero charge

We consider a density-functional theory to describe nonuniform fluids composed of chain molecules, containing a charged segment each, and spherical counterions. The chain molecules are modeled as freely jointed chains of hard spheres, the counterions are oppositely charged spheres of the same diameter as all segments of chain molecules. The theory is applied to study the structure of adsorbed layers, the excess adsorption isotherms, the capacitance of the double layer, and the potential of the zero charge. We show that all electric properties are strongly dependent on the length of the chain molecules. Moreover, these properties are also dependent on the position of the charged segment in the chain.     

Diamagnetic and paramagnetic currents for an anisotropic harmonic oscillator in a magnetic field

The response of the electronic shell of a molecule to the perturbing action of an external magnetic field has been analyzed by calculating the field-induced electric current of a particle moving under the effect of the potential of an anisotropic harmonic oscillator. Exact analytical expressions have been obtained for the distribution of the current density, the current lines, and the magnetic susceptibility. The dependence of the pattern of the induced current on the degree of excitation of the quantum states and the anisotropy of the potential has been analyzed. The presence of both diamagntic and paramagnetic loop currents has been demonstrated. The expressions for the magnetic susceptibility and the induced current obtained for an anisotropic oscillator can be used in calculations of the magnetic properties of molecules in a basis of ellipsoidal Gaussian functions.Leningrad State University. Translated from Zhurnal Strukturnoi Khimii, Vol. 32, No. 2, pp. 39–46, March–April, 1991.     

Electric dipole (hyper)polarizabilities of spatially confined LiH molecule

In this study we report on the electronic contributions to the linear and nonlinear static electronic electric dipole properties, namely the dipole moment (μ), the polarizability (α), and the first-hyperpolarizability (β), of spatially confined LiH molecule in its ground X (1)Σ(+) state. The finite-field technique is applied to estimate the corresponding energy and dipole moment derivatives with respect to external electric field. Various forms of confining potential, of either spherical or cylindrical symmetry, are included in the Hamiltonian in the form of one-electron operator. The computations are performed at several levels of approximation including the coupled-cluster methods as well as multi-configurational (full configuration interaction) and explicitly correlated Gaussian wavefunctions. The performance of Kohn-Sham density functional theory for the selected exchange-correlation functionals is also discussed. In general, the orbital compression effects lead to a substantial reduction in all the studied properties regardless of the symmetry of confining potential, however, the rate of this reduction varies depending on the type of applied potential. Only in the case of dipole moment under a cylindrical confinement a gradual increase of its magnitude is observed.     

Slater Sum in Cylindrically Symmetric Inhomogeneous Electron Liquids: Towards A Differential Equation for the Stark Effect in Hydrogen Atom

Abstract

Attention is focussed here on a variety of cylindrically symmetric inhomogeneous electron liquids. These include separable potentials in which a general variation along the (z) axis of cylindrical symmetry is combined with isotropic harmonic confinement in the (x, y) plane. in this case, an explicit differential equation is derived for the Slater sum along the z axis by projecting out of the (off-diagonal) canonical density matrix the states with zero angular momentum about the axis of symmetry. Some attention is then given to the calculation of the Slater sum for a hydrogen-like atom in a uniform electric field of arbitrary strength. the model of a separable potential with harmonic confinement, though no longer exact, is shown to lead directly to a (now approximate) equation for the Slater sum along the z axis for the Stark effect in hydrogen. Finally some further progress is shown to be possible in the extreme high field limit.     

Model of Confined Atoms in Arbitrary Static Electric Fields: Relevance to Non-degenerate Plasmas

Abstract

The inhomogeneous electron cloud in atomic ions ‘confined’ in hot plasmas and subjected to high static electric fields is studied, because of a body of experimental data on multiphoton ionization. In particular, the canonical (Bloch) density matrix is obtained in closed form for independent electrons moving in a static electric field of arbitrary strength and confined by a harmonic oscillator potential. To bring the model into contact with atoms in plasmas, the oscillator force constant is connected with the plasma density. For non-degenerate electrons an ‘atomic’ potential is included, by means of the Thomas—Fermi (TF) method. In an Appendix, a fully non-local theory is then developed which transcends this TF approximation. Simple numerical examples are presented for realistic values of field, temperature and plasma density.     

Franck–Condon factors for diatomic molecules with anharmonic corrections

Some of the band systems of several astrophysically important molecules are calculated and compared with the results obtained by calculations based on realistic Klein–Dunham and Rydberg–Klein–Rees potential functions. The Morse potential is approximated by means of a fourth-order anharmonic oscillator model. In the second-quantized formalism, the anharmonic Hamiltonian is diagonalized by using the Bogoliubov–Tyablikov transformation. The diagonalization process gives a shift in the frequency associated with each normal mode of harmonic vibration of the molecules presented here. The Franck–Condon factors are estimated using this new frequency within the framework of a harmonic oscillator.     

Diffusiophoresis in suspensions of highly charged soft particles

Diffusiophoresis phenomenon of aoft particles suspended in binary electrolyte solutions is explored theoretically in this study based on the spherical cell model, focusing on the chemiphoresis component in absence of diffusion potential. Both the electrostatic and hydrodynamic aspects of the boundary confinement, or steric effect, due to the presence of neighboring particles are examined extensively under various electrokinetic conditions. Significant local extrema are found in mobility profiles expressed as functions of the Debye length in general, synchronized with the strength of the motion-inducing double layer polarization. Moreover, a seemingly peculiar phenomenon is observed that the soft particles may move faster in more concentrated suspensions. The competition between the simultaneous enhancement of the motion-inducing electric driving force and the motion-retarding hydrodynamic drag force from the boundary confinement effect of the neighboring particles is found to be responsible for it. The above findings are also demonstrated experimentally in a very recent study on the diffusiophoretic motion of soft particles through porous collagen hydrogels. The results presented here are useful in various practical applications of soft particles like drug delivery.     

Degeneracy of confined D‐dimensional harmonic oscillator

Using the mathematical properties of the confluent hypergeometric functions, the conditions for the incidental, simultaneous, and interdimensional degeneracy of the confined D‐dimensional (D > 1) harmonic oscillator energy levels are derived, assuming that the isotropic confinement is defined by an infinite potential well and a finite radius R_c. Very accurate energy eigenvalues are obtained numerically by finding the roots of the confluent hypergeometric functions that confirm the degeneracy conditions. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

A study of two-electron quantum dot spectrum using discrete variable representation method

A variational method called discrete variable representation is applied to study the energy spectra of two interacting electrons in a quantum dot with a three-dimensional anisotropic harmonic confinement potential. This method, applied originally to problems in molecular physics and theoretical chemistry, is here used to solve the eigenvalue equation to relative motion between the electrons. The two-electron quantum dot spectrum is determined then with a precision of at least six digits. Moreover, the electron correlation energies for various potential confinement parameters are investigated for singlet and triplet states. When possible, the present results are compared with the available theoretical values.     

Diffusiophoresis and electrophoresis of a charged sphere perpendicular to two plane walls

The problem of diffusiophoretic and electrophoretic motions of a dielectric spherical particle in an electrolyte solution situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the quasisteady limit of negligible Peclet and Reynolds numbers. The applied electrolyte concentration gradient or electric field is uniform and perpendicular to the plane walls. The electric double layer at the particle surface is assumed to be thin relative to the particle radius and to the particle-wall gap widths, but the polarization effect of the diffuse ions in the double layer is incorporated. To solve the conservative equations, the general solution is constructed from the fundamental solutions in both cylindrical and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Hankel transforms and then on the particle surface by a collocation technique. Numerical results for the diffusiophoretic and electrophoretic velocities of the particle relative to those of a particle under identical conditions in an unbounded solution are presented for various cases. The collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the walls can reduce or enhance the particle velocity, depending on the properties of the particle-solution system and the relative particle-wall separation distances. The boundary effects on diffusiophoresis and electrophoresis of a particle normal to two plane walls are found to be quite significant and complicated, and generally stronger than those parallel to the confining walls.     

A Look at the Spatial Confining Effect on the Molecular Electrostatic Potential (MEP)—A Case Study of the HF and BrCN Molecules

In this theoretical study, we report on the molecular electrostatic potential (MEP) of titled molecules confined by repulsive potentials of cylindrical symmetry mimicking a topology. Our calculations show that the spatial restriction significantly changes the picture of the MEP of molecules in a quantitative and qualitative sense. In particular, the drastic changes in the MEP as a function of the strength of spatial confinement are observed for the BrCN molecule. This preliminary study is the first step in the investigation of the behavior of the MEP of molecular systems under orbital compression.     

Harmonic oscillator tensors. IV. A tensorial approach to internal rotations in molecules

A mapping of 2×2 matrices into the space of single boson operators is shown to lead to the angular momentum operators that give rise to irreducible tensors for the harmonic oscillator. The mapping may also be used to define an axis of quantization. A rotation about this axis induces a wave function and Hamiltonian that may be applied to the study of internal rotations in molecules. The example of a molecule containing two coaxial symmetric tops is presented as a case in point. The case of a potential with a high barrier leads to the approximation of an internal rotation as a torsional oscillator and, consequently, to torsional oscillator tensors whose properties are the same as those of the harmonic oscillator. The possibility of studying more complex potentials is discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 305–315, 1997     

Molecular Simulations using Spherical Harmonics

Computer-aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often used because they represent the in-teracting part of molecules and they should reflex the comple-mentarity between ligand and receptor. However, assessing the surface complementarity by searching all relative position of two surfaces is often computationally expensive. The comple-mentarity of lobe-hole is very important in protein-ligand inter-actions. Spherical harmonic models based on expansions of spherical harmonic functions were used as a f‘mgerprint to ap-proximate the binding cavity and the ligand, respectively. This defines a new way to identify the complementarity between lobes and holes. The advantage of this method is that two spherical harmonic surfaces to be compared can be defined sep-arately. This method can be used as a filter to eliminate candi-dates among a large number of conformations, and it will speed up the docking procedure. Therefore, it is possible to select complementary ligands or complementary conformations of a ligand and the macromolecules, by comparing their fingerprints previously stored in a database.     

Influence of the block hydrophilicity of AB2 miktoarm star copolymers on cluster formation in solutions

We investigated the formation of various micelle shapes of lipid-like amphiphilic AB(2) miktoarm star copolymers in a solution, by performing dissipative particle dynamics simulations. AB(2) miktoarm star copolymer molecules are modeled with coarse-grained structures that consist of a relatively hydrophilic head (A) group with a single arm and a hydrophobic tail (B) group with double arms. A decrease in the hydrophilicity of the head group leads to a reduction of the polymer-solvent contact area, causing cluster structure changes from spherical micelles to vesicles. Consequently, a spherical exterior with multi-lamellar or cylindrical phase interior structures forms under poor solvent conditions without the introduction of spherical hard-wall containers. Furthermore we observed that, for small head group lengths, vesicles were formed in much wider range of solvent-head interaction strength than for long head groups, indicating that molecules with short head group offer a superior vesicle forming property. A phase diagram, the structure and kinetics of the cluster formation, a density profile, and a detailed shape analysis are presented to discuss the molecular characteristics of potential candidates for drug carriers that require superior and versatile vesicle forming properties. We also show that, under certain solvent-hydrophilic head group interaction conditions, initially formed cylindrical micelles transform to bilayer fragments through redistribution of copolymers within the cluster.     

Algebraic approximation to the Franck–Condon factors for the Morse oscillator

An algebraic approach is proposed to calculate the Franck–Condon factors for the Morse potential of diatomic molecules. The Morse oscillator is approximated by means of a fourth-order anharmonic oscillator. In the second-quantized formalism, this anharmonic Hamiltonian is diagonalized by way of the Bogoliubov–Tyablikov transformation. The Franck–Condon factors are estimated using the harmonic frequency equivalent and the recurrence relations for the Franck–Condon factors of the harmonic oscillator. Overlap integrals are shown for three band systems and compared with values calculated with an RKR potential. Excellent agreement is achieved.     

Vibrational spectrum of a 1D oscillator: The quantum,the Wigner,and the classical ways

Infrared spectra have been used in many chemical applications, and theoretical calculations have been useful for analyzing these experimental results. While quantum mechanics is used for calculating the spectra for small molecules, classical mechanics is used for larger systems. However, a systematic understanding of the similarities and differences between the two approaches is not clear. Previous studies focused on peak position and relative intensities of the spectra obtained by various quantum and classical methods, but here, we included “absolute” intensities in the evaluation. The infrared spectrum of a one-dimensional (1D) harmonic oscillator (HO) and Morse oscillator were examined using four treatments: quantum, Wigner, truncated Wigner, and classical microcanonical treatments. For a 1D HO with a linear dipole moment function (DMF), the quantum and Wigner treatments give nearly the same spectra. On the other hand, the truncated Wigner underestimates the fundamental transition's intensity by half. In the case of cubic DMF, the truncated Wigner and classical methods fail to reproduce the relative intensity between the fundamental and second overtone transitions. Unfortunately, all the Wigner and classical methods fail to agree with the quantum results for a Morse oscillator with just 1% anharmonicity.