Radial Nodes Research Articles

Quantum Monte Carlo simulation of carbon monoxide reactivity when adsorbed at metal and oxide catalyst surfaces: Trial wave‐functions with exponential type basis and quasi‐exact three‐body correlation

AbstractReview publications show the advantages of using quantum Monte Carlo (QMC) methods when bonding interactions are modified and consequently, where the electron correlation energy varies and needs to be evaluated accurately. This article considers a model efficient metal catalyst for reactions of carbon monoxide. The ultimate aim is to help design industrially useful catalysts, since the reaction with water produces hydrogen gas selectively, that is, a clean fuel and a sustainable energy source. To narrow the gap between the industrial process and calculation, a model of platinum/oxide is studied. It is mimicked here by copper with and without adsorbed oxygen. Periodicity as well as a validated pseudo‐potential are used to limit the number of active electrons considered. A suitable code to carry out this task is CASINO. Furthermore, this code scales linearly to 100,000 processors and possesses a shared memory facility so that it is well‐suited to runs on the Blugene/P. It has been in production on such computers since early in 2011. A mixed plane‐wave/exponential type orbitals (ETO) basis is used for the systems studied. Plane waves are well‐suited to periodic solid substrates and a linear combination of ETOs for molecules. The atomic orbitals have direct physical interpretation, i.e., Coulomb Sturmians and hydrogen‐like orbitals. Their radial nodes are shown to be essential in obtaining the vitally important accurate QMC trial wave‐functions. Until 2008, ETO products on different atoms were difficult to manipulate for the evaluation of two‐electron integrals. Coulomb resolutions provide an excellent approximation that reduces these integrals to a sum of one‐electron overlap‐like integral products that each involve orbitals on at most two centers. They are thus readily evaluated. Only these integrals need to be re‐evaluated to change basis functions. In this article, QMC‐VMC variational optimisation is used with a quasi analytic two‐electron and nucleus correlation factor. The QMC diffusion Monte Carlo methodology is applied to adsorbed carbon monoxide and some of its reactions. © 2012 Wiley Periodicals, Inc.

On extreme learning machine for ε-insensitive regression in the primal by Newton method

In this paper, extreme learning machine (ELM) for e-insensitive error loss function-based regression problem formulated in 2-norm as an unconstrained optimization problem in primal variables is proposed. Since the objective function of this unconstrained optimization problem is not twice differentiable, the popular generalized Hessian matrix and smoothing approaches are considered which lead to optimization problems whose solutions are determined using fast Newton–Armijo algorithm. The main advantage of the algorithm is that at each iteration, a system of linear equations is solved. By performing numerical experiments on a number of interesting synthetic and real-world datasets, the results of the proposed method are compared with that of ELM using additive and radial basis function hidden nodes and of support vector regression (SVR) using Gaussian kernel. Similar or better generalization performance of the proposed method on the test data in comparable computational time over ELM and SVR clearly illustrates its efficiency and applicability.

A Bronze Matryoshka: The Discrete Intermetalloid Cluster [Sn@Cu12@Sn20]12– in the Ternary Phases A12Cu12Sn21 (A = Na, K)

The synthesis and crystal structure of the first ternary A-Cu-Sn intermetallic phases for the heavier alkali metals A = Na to Cs is reported. The title compounds A(12)Cu(12)Sn(21) show discrete 33-atom intermetalloid Cu-Sn clusters {Sn@Cu(12)@Sn(20)}, which are composed of {Sn(20)} pentagonal dodecahedra surrounding {Cu(12)} icosahedra with single Sn atoms at the center. Na(12)Cu(12)Sn(21) and K(12)Cu(12)Sn(21) were characterized by single-crystal XRD studies, and the successful synthesis of analogous A-Cu-Sn compounds with A = Rb and Cs is deduced from powder XRD data. The isotypic A(12)Cu(12)Sn(21) phases crystallize in the cubic space group Pn ̅3m (No. 224), with the Cu-Sn clusters adopting a face centered cubic arrangement. A formal charge of 12- can be assigned to the {Sn@Cu(12)@Sn(20)} cluster unit, and the interpretation of the title compounds as salt-like intermetallic phases featuring discrete anionic intermetalloid [Sn@Cu(12)@Sn(20)](12-) clusters separated by alkali metal cations is supported by electronic structure calculations. For both Na(12)Cu(12)Sn(21) and K(12)Cu(12)Sn(21), DFT band structure calculations (TB-LMTO-ASA) reveal a band gap. The discrete [Sn@Cu(12)@Sn(20)](12-) cluster is analyzed in consideration of the molecular orbitals obtained from hybrid DFT calculations (Gaussian 09) for the cluster anion. The [Sn@Cu(12)@Sn(20)](12-) cluster MOs can be classified with labels indicating the numbers of radial and angular nodes, in the style of spherical shell models of cluster bonding.

Dual reciprocity hybrid radial boundary node method for the analysis of Kirchhoff plates

A meshless method of dual reciprocity hybrid radial boundary node method (DHRBNM) for the analysis of arbitrary Kirchhoff plates is presented, which combines the advantageous properties of meshless method, radial point interpolation method (RPIM) and BEM. The solution in present method comprises two parts, i.e., the complementary solution and the particular solution. The complementary solution is solved by hybrid radial boundary node method (HRBNM), in which a three-field interpolation scheme is employed, and the boundary variables are approximated by RPIM, which is applied instead of moving least square (MLS) and obtains the Kronecker’s delta property where the traditional HBNM does not satisfy. The internal variables are interpolated by two groups of symmetric fundamental solutions. Based on those, a hybrid displacement variational principle for Kirchhoff plates is developed, and a meshless method of HRBNM for solving biharmonic problems is obtained, by which the complementary solution can be solved. In order to solve the particular solution, a basic form of particular solution of Kirchhoff plates is developed, by which the particular solutions for all kinds of Kirchhoff plates can be interpolated. Combined with HRBNM and dual reciprocity method (DRM), a meshless method of DHRBNM is proposed for an arbitrary Kirchhoff plate. Numerical examples are given to illustrate that the present method is a simple, efficient, accurate and attractive one, and can be further extended into some practical problems.

A Hybrid Radial Boundary Node Method for Biharmonic Problems

The hybrid radial boundary node method is applied to solve the biharmonic problems. Based on modified variational principle, the variational formula of the biharmonic problems is established. The radial basis point interpolation is employed to approximate the boundary variables, while the domain variables are interpolated by a combination of the fundamental solution of the laplace equation and the biharmonic equation. Compared to the regular hybrid boundary node method, as the shape function has the delta function property, the boundary conditions of the original problem can be easily implemented, and the fictitious source points are not involved. Numerical examples show that this method is efficient for solving the biharmonic equation.

Shifts of beam centroid of Laguerre-Gaussian beams reflected and refracted at a dielectric interface

We study shifts of the beam centroid of linearly polarized paraxial Laguerre-Gaussian beams reflected and refracted at an interface of isotropic dielectric in the cylindrical coordinate system. According to the Fresnel approximation and the Taylor series expansion, we derive analytical expressions for transverse and longitudinal shifts and their general relationship with topological charge in the cases of partial and total reflection, respectively. It is shown that in the partial reflection regime, transverse shifts of both reflected and refracted beams are in proportion to the topological charge of beams and their directions are determined by the sign of the topological charge, while only the value of longitudinal shift is related to the topological charge. In the regime of total reflection, the shift of reflected beam centroid is independent of topological charge. To verify our analytical results, we perform numerical simulations of the shifts of reflected and refracted beams directly and indicate the applicable condition of the analytical expressions. The shifts of the beam centroid can be applied to the measurement of the topological charge and radial node number of Laguerre-Gaussian beams, which is of significance for information encoding and communication.

Validity of the Wannier threshold law for angular correlation width in double photoionization of atoms

We calculated ab initio the three-fold differential cross section of a double single-photon Helium photoionization at equal energy sharing, and obtained from one the Gaussian width parameter, describing the angular interelectron correlations, for the total electrons energies range E from 0.1 eV to 100 eV. The results are in excellent agreement with experimental data but indicate that the Wannier threshold law for angular distribition is not correct at energies attainable in modern experiments. It is shown that the Gaussian width parameter dependence on energy is much better described by the modified threshold law, obtained by Kazansky and Ostrovsky [J. Phys. B 26, 2231 (1993)]. Also, we explored the Gaussian width parameter for a double photoionization of the targets with the strongly asymmetrical initial state configuration: the atomic Hydrogen negative ion and the Helium in the 2s1S and 3s1S excited states. We found that the Gaussian width parameter dependence on the total ejected electrons energy for this targets has a maximum at low energies. We show also that the correlation parameter dependence on the interelectron angle for these targets is essentially non-Gaussian and has a number of peaks equal to a number of initial state radial nodes, that reveals the new abilities for the qualitative analysis of electron structure.

Meshless method of dual reciprocity hybrid radial boundary node method for elasticity

Combining the radial point interpolation method (RPIM), the dual reciprocity method (DRM) and the hybrid boundary node method (HBNM), a dual reciprocity hybrid radial boundary node method (DHRBNM) is proposed for linear elasticity. Compared to DHBNM, RPIM is exploited to replace the moving least square (MLS) in DHRBNM, and it gets rid of the deficiency of MLS approximation, in which shape functions lack the delta function property, the boundary condition can not be applied easily and directly and it's computational expense is high. Besides, different approximate functions are discussed in DRM to get the interpolation property, in which the accuracy and efficiency for different basis functions are compared. Then RPIM is also applied in DRM to replace the conical function interpolation, which can greatly improve the accuracy of the present method. To demonstrate the effectiveness of the present method, DHBNM is applied for comparison, and some numerical examples of 2-D elasticity problems show that the present method is much more effective than DHBNM.

Fluid modes of a spherically confined Yukawa plasma

The normal modes of a three-dimensional Yukawa plasma in an isotropic harmonic confinement are investigated by solving the linearized cold fluid equations. The eigenmodes are found analytically and expressed in terms of hypergeometric functions. It is found that the mode frequencies solely depend on the dimensionless plasma parameter ξ=κR , where R is the plasma radius and κ is the inverse screening length. The eigenfrequencies increase monotonically with ξ and saturate in the limit ξ→∞ . Compared with the results in the Coulomb limit [D. H. E. Dubin, Phys. Rev. Lett. 66, 2076 (1991)10.1103/PhysRevLett.66.2076], we find an additional class of modes characterized by the number n which determines the number of radial nodes in the perturbed potential. These modes originate from the degenerate bulk modes of the Coulomb system. Analytical formulas for the eigenfrequencies are derived for limiting cases.

Exact Pseudospin Symmetric Solution of the Dirac Equation for Pseudoharmonic Potential in the Presence of Tensor Potential

Under the pseudospin symmetry, we obtain exact solution of the Dirac equation for the pseudoharmonic potential in the presence of the tensor potential with arbitrary spin–orbit coupling quantum number κ. The energy eigenvalue equation of the Dirac particles is found and the corresponding radial wave functions are presented in terms of confluent hypergeometric functions. We investigate the tensor potential dependence of the energy of the each state in the pseudospin doublet. It is shown that degeneracy between members of the pseudospin doublet is removed by tensor interaction. Furthermore, the radial node structure of the Dirac spinor is discussed.

A dual reciprocity hybrid radial boundary node method based on radial point interpolation method

A novel truly meshless method called dual reciprocity hybrid radial boundary node method (DHRBNM) is developed in present, which combines dual reciprocity method (DRM), hybrid boundary node method (HBNM) and radial point interpolation method (RPIM). Compared to the dual reciprocity hybrid boundary node method (DHBNM), RPIM is exploited to replace the moving least square in DHRBNM, unlike HBNM, the shape function obtained by present method has the delta function property, so the boundary conditions can be applied directly and easily, and computational expense is greatly reduced. In order to get the interpolation property of different basis function in DRM, different approximate functions are applied in DRM for comparison, and the accuracy and efficiency of them are discussed. Besides, RPIM is also exploited in DRM, which can greatly improve the accuracy of present method. Moreover, the accuracy of DRM is greatly influenced by the nodes number and their location, hence, some examples are investigated to show that the internal node number is equal to boundary node number and they are arranged parallel to the high gradient direction of the problem are the best choice. Finally, DHBNM is applied for comparison and some selected numerical examples are given to illustrate that the present method is efficient and less computational expense than that of DHBNM.

A hybrid radial boundary node method based on radial basis point interpolation

The hybrid boundary node method (HBNM) retains the meshless attribute of the moving least squares (MLS) approximation and the reduced dimensionality advantages of the boundary element method. However, the HBNM inherits the deficiency of the MLS approximation, in which shape functions lack the delta function property. Thus in the HBNM, boundary conditions are implemented after they are transformed into their approximations on the boundary nodes with the MLS scheme. This paper combines the hybrid displacement variational formulation and the radial basis point interpolation to develop a direct boundary-type meshless method, the hybrid radial boundary node method (HRBNM) for two-dimensional potential problems. The HRBNM is truly meshless, i.e. absolutely no elements are required either for interpolation or for integration. The radial basis point interpolation is used to construct shape functions with delta function property. So unlike the HBNM, the HRBNM is a direct numerical method in which the basic unknown quantity is the real solution of nodal variables, and boundary conditions can be applied directly and easily, which leads to greater computational precision. Some selected numerical tests illustrate the efficiency of the method proposed.

The shape of atomic orbitals

Radial moment analysis of accurate atomic orbitals and of simplified analytic representations provides insight into the effect of radial nodes on orbital shape. The concept of effective principal quantum number is used to extend simple analytic representation to the heavier elements.

Suitable basis sets for accurate NMR chemical shift calculations: an application to in vivo studies of benzothiazole metabolites

This paper describes atomic orbitals which have direct physical interpretation, i.e. Coulomb Sturmians and hydrogen-like orbitals. The radial nodes are shown to be essential in obtaining accurate nuclear shielding tensors for NMR work. The method is applied ab initio to chemical shifts for (15) N NMR, which can also be measured. This procedure is shown to be a useful tool in structure determination for benzothiazoles, some of which are used as pesticides and represent a challenging use of biodegradation. The measurements may be made in biological media, for which a rudimentary simulation is provided by including a few discrete solvent molecules.

Stability and excitations of a dipolar Bose-Einstein condensate with a vortex

We study the stability of singly- and doubly-quantized vortex states of harmonically trapped dipolar Bose-Einstein Condensates (BECs) by calculating the low-lying excitations of these condensates. We map the dynamical stability of these vortices as functions of the dipole-dipole interaction strength and trap geometry by finding where their excitations have purely real energy eigenvalues. In contrast to BECs with purely contact interactions, we find that dipolar BECs in singly-quantized vortex states go unstable to modes with an increasing number of angular and radial nodes for more oblate trap aspect ratios, corresponding to \emph{local} collapse that occurs on a characteristic length scale. Additionally, we find that dipolar BECs in doubly-quantized vortex states are unstable to decay into a different topological state (with two singly-quantized vortices) for all interaction strengths when the trap geometry is sufficiently prolate to make the dipoles attractive, and in windows of interaction strength when the trap geometry is sufficiently oblate to make the dipoles repulsive.

A relativistic non-harmonic oscillator potential and pseudospin symmetry

A generalized non-harmonic oscillator potential for spin 1/2 particles is studied. The Dirac Hamiltonian contains a scalar and a vector non-harmonic oscillator potentials. Setting either or both combinations Σ(r)=S(r)+V(r) and Δ(r)=V(r)-S(r) to zero, analytical solutions for bound states of the corresponding Dirac equation are found. The eigenenergies and wave functions are presented, showing the pseudospin symmetry exists exactly.The relations between radial nodes of upper and lower components of the Dirac spinor are obtained.

Eigenspectrum properties of the confined 3D harmonic oscillator

An isotropic 3D harmonic oscillator centrally enclosed in a spherical box with impenetrable walls is treated by analytical methods. It is explicitly shown how imposing the Dirichlet boundary condition on the wavefunctions results in (a) the complete removal of any systematic degeneracy of levels, and (b) the constant energy difference of exactly two harmonic oscillator units between all successive pairs of the confined excited states defined by the orbital quantum numbers l and l + 2, specifically when the radius of the spherical box is chosen to be the position of the only radial node in the first excited-state wavefunction of the unconfined harmonic oscillator for a given l. The lowest confined state of the given l is excluded in defining the successive pairs of excited confined states.

Energies ofBsmeson excited states: A lattice study

This is a follow-up to our earlier work on the energies and radial distributions of heavy-light mesons. The heavy quark is taken to be static (infinitely heavy) and the light quark has a mass about that of the strange quark. We now concentrate on the energies of the excited states with higher angular momentum and with a radial node. A new improvement is the use of hypercubic blocking in the time direction. The calculation is carried out with dynamical fermions on a ${16}^{3}\ifmmode\times\else\texttimes\fi{}32$ lattice with a lattice spacing $a\ensuremath{\approx}0.1\text{ }\text{ }\mathrm{fm}$ generated using a nonperturbatively improved clover action. In nature the closest equivalent of this heavy-light system is the ${B}_{s}$ meson, which allows us to compare our lattice calculations to experimental results (where available) or to give a prediction for where the excited states, particularly $P$-wave states, should lie. We pay special attention to the spin-orbit splitting, to see which one of the states (for a given angular momentum $L$) has the lower energy. An attempt is made to understand these results in terms of the Dirac equation.

Nonexistence of radial node solutions for elliptic problems with critical Sobolev exponents

Let Ω ⊂ R n be a ball centered at the origin with radius 1 and 3 ≤ n ≤ 6 , 2 ∗ = 2 n n − 2 and μ ∈ [ 0 , ( n − 2 2 ) 2 ) . By using a well-known Pohozaev-type identity and some ODE techniques, we obtained the nonexistence of sign-changing radial solutions for elliptic problems with critical Sobolev and Hardy terms − Δ u = μ | x | 2 u + | u | 2 ∗ − 2 u + λ u on Ω , u ∈ H 0 1 ( Ω ) for suitable small positive number λ . Meanwhile, the nonexistence of radial sign-changing solutions for p -Laplace problems with critical Sobolev exponent is also obtained.

Simultaneous degeneracy of the confined 2D hydrogen atom energy levels

Using the Numerov–Cooley numerical method, we report several interesting characteristics of the eigenspectrum of 2D hydrogen atom confined inside an impenetrable circular boundary of radius ρc. At the confinement radius, ρc, exactly equal to the location of the only radial node of the unconfined atom state (l + 2, l), the successive pairs of the confined atom states (nl) and (n + 1, l + 2), where n ⩾ l + 2, are found to become degenerate. Accurate calculations of static dipole polarizability are reported for several confining radii using the standard second-order perturbation theory.