Virial Ratio Research Articles

Calculation of Negative Ions of B, C, N, O and F Using Noninteger n Slater Type Orbitals

AbstractCombined Hartree‐Fock‐Roothaan calculations have been performed using noninteger n Slater type orbitals for the ground states of the lowest electron configurations 1s22s22pn (2 ≤ n ≤ 6) for negative ions of B, C, N, O and F. These results are compared with the corresponding results obtained from the use of integer n Slater type orbitals. All of the nonlinear parameters are fully optimized. The results of calculation of coupling‐projection coefficients, orbital and total energies and virial ratios are presented. It is shown that the noninteger n Slater type orbitals, in general, improve the orbital energies.

To address accuracy and precision using methods from analytical chemistry and computational physics

In this work the pesticides were determined by liquid chromatography-mass spectrometry (LC-MS). In present study the occurrence of imidacloprid in 343 samples of oranges, tangerines, date plum, and watermelons from Valencian Community (Spain) has been investigated. The nine additional pesticides were chosen as they have been recommended for orchard treatment together with imidacloprid. The Mulliken population analysis has been applied to present the charge distribution in imidacloprid. Partitioned energy terms and the virial ratios have been calculated for certain molecules entering in interaction. A new technique based on the comparison of the decomposed total energy terms at various configurations is demonstrated in this work. The interaction ability could be established correctly in the studied case. An attempt is also made in this work to address accuracy and precision. These quantities are well-known in experimental measurements. In case precise theoretical description is achieved for the contributing monomers and also for the interacting complex structure some properties of this latter system can be predicted to quite a good accuracy. Based on simple hypothetical considerations we estimate the impact of applying computations on reducing the amount of analytical work.

Application of combined Hartree–Fock–Roothaan theory to molecules with arbitrary number of closed and open shells

In this study, the applicability of the combined Hartree–Fock–Roothaan (CHFR) theory of atomic-molecular and nuclear systems (Guseinov, J Math Chem 42:177, 2007) to the molecules is demonstrated using minimal basis set of Slater type orbitals (STO). As an example of application of CHFR theory, the calculations have been performed for the ground state of electronic configuration of methylene molecule CH2 which has two open shells. The results of computer calculations for the orbital, kinetic and total energies, linear combination coefficients of symmetrized molecular orbitals and virial ratios are presented.

Application of combined Hartree-Fock-Roothaan theory to atoms with an arbitrary number of closed and open shells of any symmetry

The applicability of a combined open-shell Hartree-Fock theory of atomic-molecular and nuclear systems to the states of a single configuration which has the arbitrary symmetry of open shells is explicitly demonstrated by calculations based on the combined Hartree-Fock-Roothaan equations for atoms. As examples of an application of the combined theory using single-zeta Slater-type basis sets, the calculations have been performed for the lowest states of the lowest and excited electronic configurations of carbon and nitrogen atoms containing one-, two- and three-open shells. For the case of single-open shells for carbon and nitrogen atoms, the double-zeta basis-set calculations are also presented. The results of computer calculations for the orbitals, kinetic and total energies, linear combination coefficients and virial ratios are given.

An extension of the virial theorem for general wave functions

An extension of the virial theorem in the Born–Oppenheimer framework is presented, which is valid for arbitrary (not necessarily variational) wave functions built up by using finite local one-electron basis sets. It permits to define explicitly the terms causing deviations from the ‘ideal’ virial ratio; the standard formulations of the virial theorem can also be obtained from it as special cases.

Are the nearby groups of galaxies gravitationally bound objects?

We have compared numerical simulations to observations for the nearby (< 40 Mpc) groups of galaxies (Huchra & Geller 1982 and Ramella et al. 2002). The group identi- fication is carried out using a group-finding algorithm developed by Huchra & Geller (1982). Using cosmological N-body simulation code with theCDM cosmology, we show that the dynamical properties of groups of galaxies identified from the simulation data are, in general, in a moderate, within 2σ, agreement with the observational cat- alogues of groups of galaxies. As simulations offer more dynamical information than observations, we used the N-body simulation data to calculate whether the nearby groups of galaxies are gravitationally bound objects by using their virial ratio. We show that in aCDM cosmology about 20 per cent of nearby groups of galaxies, identified by the same algorithm as in the case of observations, are not bound, but merely groups in a visual sense. This is quite significant, specifically because estima- tions of group masses in observations are often based on an assumption that groups of galaxies found by the friends-of-friends algorithm are gravitationally bound objects. Simulations with different resolutions show the same results. We also show how the fraction of gravitationally unbound groups varies when the apparent magnitude limit of the sample and the value of the cosmological constantis changed. In general, a larger value of thegenerates slightly more unbound groups.

Numerical experiments of the planar equal-mass three-body problem: The effects of rotation

The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio k defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large k, binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different k. The behavior of this number for different time and k shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for k<or=0.1, the number seems to saturate for k=0.3-0.5, and most of the escape orbits escape quickly for k=0.9. The area of chaotic motion is small for k=0.9.

Dark-Matter Content of Early-Type Galaxies with Planetary Nebulae

AbstractWe examine the dark matter properties of nearby early-type galaxies using planetary nebulae (PNe) as mass probes. We have designed a specialised instrument, the Planetary Nebula Spectrograph (PN.S) operating at the William Herschel telescope, with the purpose of measuring PN velocities with best efficiency. The primary scientific objective of this custom-built instrument is the study of the PN kinematics in 12 ordinary round galaxies. Preliminary results showing a dearth of dark matter in ordinary galaxies (Romanowskyet al. 2003) are now confirmed by the first complete PN.S datasets. On the other hand early-type galaxies with a “regular” dark matter content are starting to be observed among the brighter PN.S target sample, thus confirming a correlation between the global dark-to-luminous mass virial ratio (fDM=MDMM*) and the galaxy luminosity and mass.

Hartree-fock equations for atoms with two open shells

We have formulated the Hartree-Fock equations for multielectron systems with two open shells (the Huzinaga method) in terms of the density matrix in the LCAO approximation. In order to solve the Hartree-Fock equations, in the algebraic approximation we have obtained expressions for the derivatives of the energy with respect to the density matrix elements and the nonlinear atomic orbital parameters (the orbital exponents). We discuss the question of calculating the open shell parameters (the vector coupling coefficients) in the configurations s1pN and s1dN within the Huzinaga method. We have calculated the energy for a series of atoms with two open shells in these configurations. Using rather narrow basis sets of Slater-type atomic orbitals), we have obtained energy values close to the results of a numerical solution of the Hartree-Fock equations with sufficiently high accuracy of the virial ratio.

The physical origin of large covalent–ionic resonance energies in some two-electron bonds

This study uses valence bond (VB) theory to analyze in detail the previously established finding that alongside the two classical bond families of covalent and ionic bonds, which describe the electron-pair bond, there exists a distinct class of charge-shift bonds (CS-bonds) in which the fluctuation of the electron pair density plays a dominant role. Such bonds are characterized by weak binding, or even a repulsive, covalent component, and by a large covalent-ionic resonance energy RE(cs) that is responsible for the major part, or even for the totality, of the bonding energy. In the present work, the nature of CS-bonding and its fundamental mechanisms are analyzed in detail by means of a VB study of some typical homonuclear bonds (H-H, H3C-CH3, H2N-NH2, HO-OH, F-F, and Cl-Cl), ranging from classical-covalent to fully charge-shift bonds. It is shown that CS-bonding is characterized by a covalent dissociation curve with a shallow minimum situated at long interatomic distances, or even a fully repulsive covalent curve. As the atoms that are involved in the bond are taken from left to right or from bottom to top of the periodic table, the weakening effect of the adjacent bonds or lone pairs increases, while at the same time the reduced resonance integral, that couples the covalent and ionic forms, increases. As a consequence, the weakening of the covalent interaction is gradually compensated by a strengthening of CS-bonding. The large RE(cs) quantity of CS-bonds is shown to be an outcome of the mechanism necessary to establish equilibrium and optimum bonding during bond formation. It is shown that the shrinkage of the orbitals in the covalent structure lowers the potential energy, V, but excessively raises the kinetic energy, T, thereby tipping the virial ratio off-balance. Subsequent addition of the ionic structures lowers T while having a lesser effect on V, thus restoring the requisite virial ratio (T/-V = 1/2). Generalizing to typically classical covalent bonds, like H-H or C-C bonds, the mechanism by which the virial ratio is obeyed during bond formation is primarily orbital shrinkage, and therefore the charge-shift resonance energy has only a small corrective effect. On the other hand, for bonds bearing adjacent lone pairs and/or involving electronegative atoms, like F-F or Cl-Cl, the formation of the bond corresponds to a large increase of kinetic energy, which must be compensated for by a large participation or covalent-ionic mixing.

The evolution of the dark halo spin parameters λ and λ′ in a ΛCDM universe: the role of minor and major mergers

The evolution of the spin parameter of dark haloes and the dependence on the halo merging history in a set of dissipationless cosmological A cold dark matter simulations are investigated. Special focus is placed on the differences of the two commonly used versions of the spin parameter, namely λ = JE 1/2 G -1 M -5/2 and 1' = J/(√M vir R vir v vir ). Though the distribution of the spin transfer rate T (') λ =: λ (') fin /λ (') init (which is the ratio of spin parameters after and prior to merger) is similar to a high degree for both λ and λ', we find considerable differences in the time-evolution, while >.' is roughly independent of redshift, λ turns out to increase significantly with decreasing redshift. This distinct behaviour arises from small differences in the spin transfer distribution of accreted material. The evolution of the spin parameter is strongly coupled with the virial ratio η =: 2E kin /|E pot | of dark haloes. Major mergers (MMs) disturb haloes and increase their both virial ratio η and spin parameter λ (') for 1-2 Gyr. At high redshifts z = 2-3, many haloes are disturbed with an average virial ratio of η ≈ 1.3 which approaches unity until z = 0. We find that the redshift evolution of the spin parameters is dominated by the huge number of minor mergers rather than the rare MM events.

Hartree–Fock Energy Partitioning in Terms of Hirshfeld Atoms

A Hirshfeld decomposition scheme of the Hartree-Fock total molecular energy into atomic energies is presented. The calculations are performed by direct numerical integration and the results are compared for a set of 28 molecules containing different kinds of atoms. The calculated atomic energies show a strong dependency on changes of atomic electron population and hybridization. Linear correlations are found between the energy and the population for H, these being related to the electronegativity of this atom and to the external potential created by the remaining atoms. The proposed energy partitioning scheme appears to be useful for studies such as proton acidity, the anomeric effect and group transferability, and allows atomic virial ratios to be obtained. Finally, the atomic potential energies are found to mimic trends based on exact expressions as well as trends displayed by molecular quantities, thus lending credibility to the partitioning scheme used.

Local virial relation for self-gravitating system

We demonstrate that the quasi-equilibrium state in a self-gravitating N-body system after cold collapse is uniquely characterized by the local virial relation using numerical simulations. Conversely, assuming the constant local virial ratio and Jeans equation for a spherically steady-state system, we investigate the full solution space of the problem under the constant anisotropy parameter and obtain some relevant solutions. Specifically, the local virial relation always provides a solution which has a power-law density profile in both the asymptotic regions r --> 0 and infinity. This type of solution is commonly observed in many numerical simulations. Only the anisotropic velocity dispersion controls this asymptotic behavior of density profile.

Charge‐Shift Bonding—A Class of Electron‐Pair Bonds That Emerges from Valence Bond Theory and Is Supported by the Electron Localization Function Approach

This paper deals with a central paradigm of chemistry, the electron-pair bond. Valence bond (VB) theory and electron-localization function (ELF) calculations of 21 single bonds demonstrate that along the two classical bond families of covalent and ionic bonds, there exists a class of charge-shift bonds (CS bonds) in which the fluctuation of the electron pair density plays a dominant role. In VB theory, CS bonding manifests by way of a large covalent-ionic resonance energy, RE(CS), and in ELF by a depleted basin population with large variances (fluctuations). CS bonding is shown to be a fundamental mechanism that is necessary to satisfy the equilibrium condition, namely the virial ratio of the kinetic and potential energy contributions to the bond energy. The paper defines the atomic propensity and territory for CS bonding: Atoms (fragments) that are prone to CS bonding are compact electronegative and/or lone-pair-rich species. As such, the territory of CS bonding transcends considerations of static charge distribution, and involves: a) homopolar bonds of heteroatoms with zero static ionicity, b) heteropolar sigma and pi bonds of the electronegative and/or electron-pair-rich elements among themselves and to other atoms (e.g., the higher metalloids, Si, Ge, Sn, etc), c) all hypercoordinate molecules. Several experimental manifestations of charge-shift bonding are discussed, such as depleted bonding density, the rarity of ionic chemistry of silicon in condensed phases, and the high barriers of halogen-transfer reactions as compared to hydrogen-transfers.

Calculations of the Energies of the Excited States of Open-Shell Atoms in the Hartree-Fock Approximation

In the framework of the Roothaan-Hartree-Fock atomic theory, the solution of a many-electron problem is considered on the basis of the methods of minimizing a function of many variables. To implement this approach, expressions are obtained for the energy derivatives with respect to the elements of density matrices and the nonlinear parameters of atomic orbitals, that is, orbital exponents. Using the first- and second-order minimization methods, we performed optimization of the orbital exponents of Slater-type atomic orbitals for atoms and ions with several open shells. For them, energies that are close to the data of the numerical solution of the Hartree-Fock equations at a high accuracy of the virial ratio were determined. For a number of atoms the frequencies of the first dipole transitions were calculated, and the results were compared to the data obtained in calculation by the method of random phases and to experimental data.

Dynamical evolution of the Orion nebula cluster

Observations of star formation in the Galaxy support the conclusion that most stars – including our own – form in an environment like the Orion nebula cluster (ONC). We construct a range of dynamical models of the ONC, using Aarseth's nbody6 code, and explore their consequences for the origins of the cluster and its subsequent evolution. We find that the most acceptable fits to the cluster density profile are obtained in models where the cluster is set up in virial equilibrium and where the cluster extends well outside the limits of existing photometric surveys. However, current estimates of the virial ratio suggest the cluster is already unbound. We show that the size and age of the ONC in this case imply either that it became unbound only very recently, or else that it has expanded quasi-statically. In the latter case, its initial central density may have exceeded its current value by 1–2 orders of magnitude. We stress the importance of future proper motion experiments to distinguish between these possibilities.

Formation of Cuspy Density Profiles through Gravitational Cold Collapses around a Central Supermassive Black Hole

We have carried out a set of N-body simulations to investigate the formation of cuspy density profiles through collapses starting from cold initial conditions around a preexisting supermassive central black hole. It is found that a cold collapse leads to an inner power-law profile ρ-γ, with 0.2 γ 2 depending on the mass of the central black hole as well as on the slope of the initial density profile ρ (0 ≤ γ0 ≤ 3/2). The upper limit obtained for γ corresponds to the more massive black hole and to the steeper initial profile with γ0 = 3/2. Models without a central black hole form a core rather than a cusp. Also, our numerical simulations with a virial ratio of η = 0.05 do not show compelling evidence that a radial anisotropy in the initial velocity distribution would play an important role in the formation of steeper cusps, as was suggested by previous authors.

The dynamical evolution of fractal star clusters: The survival of substructure

We simulate the dynamics of fractal star clusters, in order to investigate the evolution of substructure in recently formed clusters. The velocity dispersion is found to be the key parameter determining the survival of substructure. In clusters with a low initial velocity dispersion, the ensuing collapse of the cluster tends to erase substructure, although some substructure may persist beyond the collapse phase. In clusters with virial ratios of 0.5 or higher, initial density substructure survives for several crossing times, in virtually all cases. Even an initially homogeneous cluster can develop substructure, if it is born with coherent velocity dispersion. These results suggest that the simple initial conditions used for many sophisticated N-body simulations could be missing a very important and dramatic phase of star cluster evolution.

Mass of Clusters in Simulations

We show that dark matter halos in N-body simulations have a boundary layer (BL), which neatly separates dynamically bound mass from unbound materials. We define T(r) and W(r) as the differential kinetic and potential energy of halos and evaluate them in spherical shells. We notice that in simulated halos such differential quantities fulfill the following properties: (1) the differential ratio = -2T/W has at least one persistent (resolution-independent) minimum , such that, close to , (2) the function w = -d log W/d log r has a maximum, while (3) the relation () w() holds. BLs are set where these three properties are fulfilled, in halos found in simulations of tilted Einstein-de Sitter and ΛCDM models, run ad hoc, using the ART and GADGET codes; their presence is confirmed in larger simulations of the same models with a lower level of resolution. Here we find that ~97% of the ~300 clusters (per model) we have with M > 4.2 × 1014 h-1 M☉ own a BL. Those clusters that appear not to have a BL are seen to be undergoing major merging processes and to grossly violate spherical symmetry. The radius ≡ rc has significant properties. First of all, the mass Mc it encloses almost coincides with the mass Mdyn, evaluated from the velocities of all particles within rc, according to the theorem. Also, materials at r > rc are shown not be in equilibrium. Using rc we can then determine an individual density contrast Δc for each virialized halo, which we compare with the virial density contrast Δv 178Ω (where Ωm is the matter density parameter) obtained assuming a spherically symmetric and unperturbed fluctuation growth. As expected, for each mass scale, Δv is within the range of values Δc. However, the spread in Δc is wide, while the average Δc is ~25% smaller than the corresponding Δv. We argue that the matching of properties derived under the assumption of spherical symmetry must be a consequence of an approximate sphericity, after violent relaxation destroyed features related to ellipsoidal nonlinear growth. On the contrary, the spread of the final Δc is an imprint of the different initial three-dimensional geometries of fluctuations and of the variable environment during their collapse, as suggested by a comparison of our results with the Sheth & Tormen analysis.

The electron cusp condition and the virial ratio as indicators of basis set quality

We consider two measures of the quality of one-electron basis sets for quantum-chemical calculations: The electron–electron coalescence curvature and the correlation energy virial ratio. The former is based on the Kato cusp condition that many-electron wave functions must exhibit discontinuous first derivatives with respect to r12 as the coordinates of any two electrons coalesce. The latter is based on a simple modification of the quantum-mechanical virial theorem that makes use of only the correlation contributions to the kinetic and potential energy expectation values. The two measures are tested using coupled cluster wave functions for helium, neon, argon, calcium, and phosphorus atoms and are found to indicate good correlation with the quality of the basis set. These techniques may provide a foundation for the development of reliable basis set diagnostics for a variety of quantum-chemical applications.