Semiclassical Energy Research Articles

Extended Thomas–Fermi approximation to the one-body density matrix

The one-body density matrix is derived within the Extended Thomas–Fermi approximation. This has been done starting from the Wigner–Kirkwood distribution function for a non-local single-particle potential. The links between this new approximation to the density matrix with former approaches available in the literature are widely discussed. The semiclassical counterpart of the Hartree–Fock energy at the Extended Thomas–Fermi level is also obtained in the case of a non-local one-body Hamiltonian. The semiclassical binding energies and root mean square radii are compared with those obtained using the Strutinsky averaged method. The full Hartree–Fock values are compared with those obtained using the Kohn–Sham scheme based on the different approximations to the density matrix considered in the text. Numerical applications are performed using the Gogny, Brink–Boeker and BDM3Y1 ∗ effective interactions.

Kinetic Monte Carlo study of the effect of hydrogen on the two-dimensional epitaxial growth of Ni(100)

The effects of hydrogen on Ni~100! submonolayer homoepitaxy have been investigated by kinetic Monto Carlo ~KMC! simulations of the Ni growth process with and without the presence of a small hydrogen impurity. The two-dimensional KMC simulations are based upon rate constants for a set of uncorrelated Ni site-to-site hopping mobilities with and without the presence of hydrogen atoms which are found to act as a catalyst. The rate constant activation energies are determined by classical-potential total-energy calculations with semiclassical zero-point energy corrections for the hydrogen atom. This set of KMC simulations extends preliminary work on the activation energies which were reported along with their connection to available experiments in Ref. 1. We find that fast diffusion of H atoms occurs on the flat Ni~100! surface and the presence of these highly mobile H atoms is found to have significant effects on the mobility of lone Ni adatoms, and therefore also on the Ni islands which form during the epitaxial growth. The H atoms increase the mobility of lone Ni adatoms across the flat Ni~100! terrace by stabilizing the transition state for site-to-site hopping of the Ni adatom. While the effects of the H atom on the Ni mobility are primarily catalytic, the kinetically determined Ni island morphologies differ substantially as a function of H atom concentration over time periods which are long on the deposition time scale, and therefore the morphology differences can become frozen in place. Even quite low H atom coverage, by acting to increase lone Ni atom mobility, tends to decrease Ni island density which results in a corresponding increase in the average Ni island size. This suggests that the H atoms are acting as an antisurfactant in the Ni~100! homoepitaxy at low H coverage. @S0163-1829~99!06339-0#

Centrifugal terms in the WKB approximation and semiclassical quantization of hydrogen

A systematic semiclassical expansion of the hydrogen problem about the classical Kepler problem is shown to yield remarkably accurate results. Ad hoc changes of the centrifugal term, such as in the WKB approximation and semiclassical quantization of hydrogen, where the factor $l(l+1)$ is replaced by ${(l+1/2)}^{2}$, are avoided. Expanding systematically in powers of \ensuremath{\Elzxh}, the semiclassical energy levels are shown to be exact to first order in $\ensuremath{\Elzxh}$ with all higher-order contributions vanishing. The wave functions and dipole matrix elements are also discussed.

Semiclassical energy levels of electrons in metals with band degeneracy lines

We show that in calculating the semiclassical energy levels of electrons in metals located in a magnetic field, one must determine whether or not the corresponding electron paths in the space of wave vectors k are attached to a band degeneracy line. Calculations in the two possible cases, i.e., with and without such attachment, differ by |e|ℏ/2m*c, where e is the electron charge and m* is the cyclotron mass of the electron. This shift in the energy levels is of a topological nature, and its existence depends neither on the specific form of the electron dispersion relation e(k) near the electron path nor on the shape or size of this path. The reason for this shift lies in the fact that the electron orbit is attached to the band degeneracy line, which is the line of singular points of the Bloch wave functions. In many respects this effect is similar to the Aharonov-Bohm effect if the band degeneracy line is considered an infinitely thin “solenoid.” This shift in energy levels should become apparent in studies of oscillation phenomena in metals. We give examples of metals in which the conditions for observing the shift is probably the most favorable.

Angular Forces Around Transition Metals in Biomolecules

Quantum-mechanical analysis based on an exact sum rule is used to extract a semiclassical angle-dependent energy function for transition metal ions in biomolecules. The angular dependence is simple but different from existing classical potentials. Comparison of predicted energies with a computer-generated database shows that the semiclassical energy function, including its angular dependence, is very accurate.

Semiclassical treatment of the cranked triaxial rotator

A model Hamiltonian consisting of a triaxial rotator term and a linear combination of angular momentum components, is treated both classically and quantum mechanically. In the classical picture one deals with the motion of one degree of freedom which depends on two parameters. The parameters manifold is divided in several phases by separatrices, where the critical points of the Hamilton function, are degenerate. The equations of motion are exactly solved and alternatively treated by a linearization procedure. By quantising the classical orbits, the semiclassical spectrum is obtained. The quantal Hamiltonian is treated by several methods and the resulting eigenvalues are compared with the semiclassical energies. The present formalism provides an unified description of several extreme models which were widely used in nuclear structure theory.

Exact zero-point energy shift in thee⊗(nE),t⊗(nH)many-modes dynamic Jahn-Teller systems at strong coupling

We find the exact semiclassical (strong coupling) zero-point energy shifts applicable to the $e\ensuremath{\bigotimes}(\mathrm{nE})$ and $t\ensuremath{\bigotimes}(\mathrm{nH})$ dynamic Jahn-Teller problems, for an arbitrary number $n$ of discrete vibrational modes simultaneously coupled to one single electronic level. We also obtain an analytical formula for the frequency of the resulting normal modes, which has an attractive and apparently general Slater-Koster form. The limits of validity of this approach are assessed by comparison with O'Brien's previous effective-mode approach, and with accurate numerical diagonalizations. Numerical values obtained for $t\ensuremath{\bigotimes}(\mathrm{nH})$ with $n=8$ and coupling constants appropriate to ${\mathrm{C}}_{60}^{\ensuremath{-}}$ are used for this purpose, and are discussed in the context of fullerene.

Semiclassical initial value representation for rotational degrees of freedom: The tunneling dynamics of HCl dimer

The semiclassical initial value representation (SC-IVR) is emerging as a practical way of generalizing classical trajectory simulation methods to include (approximately) the effects of quantum mechanics (i.e., interference and tunneling). This paper describes the application of the SC-IVR approach to determine the low lying vibrational states of the HCl dimer on a realistic potential energy surface. Overall agreement of the semiclassical energy levels with accurate quantum values is very good, including a good description of the tunneling splitting in the ground state. Issues regarding the applications of the SC-IVR methodology to the angular variables related to rotational degrees of freedom are explicitly discussed.

GROUND STATE ENERGY OF THE MODIFIED NAMBU–GOTO STRING

We calculate, using zeta function regularization method, semiclassical energy of the Nambu–Goto string supplemented with the boundary, Gauss–Bonnet term in the action and discuss the tachyonic ground state problem.

Semiclassical mechanics of periodic motion: I. General scheme

A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.

Semiclassical wave functions and energy levels of Bose-condensed gases in spherically symmetric traps

The WKB-approximation for the Bogoliubov-equations of the quasi-particle excitations in Bose-gases with condensate is worked out in the case of spherically symmetric trap potentials on the basis of the resulting quantization rule. The excitation spectrum is calculated numerically and also analytically in certain limiting cases. It is found that the energy levels of a Bohr-Sommerfeld type quantization may be considerably shifted when the classical turning point gets close to the surface of the condensate.

Application of Bogomolny's transfer operator to a circular harmonic oscillator plus potential

Bogomolny's transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two-dimensional integrable system. The system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential. The classical trajectories are used to construct the transfer matrix, and an expression is derived for the eigenvalues of this matrix as a function of the energy. These eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system, which turn out to be exactly the same as the results obtained by solving the Schrodinger equation. Some insight into this unexpected agreement is provided by considering an exact transfer operator. We show that when this operator is expanded in powers of Planck's constant, the leading term in the expansion is Bogomolny's transfer operator.

Semiclassical approximations to diffractive effects in the annulus billiard

The semiclassical theory formulated by Bogomolny [E. B. Bogomolny, Nonlinearity 5, 805 (1992)] employs a transfer operator constructed from classical trajectories that connect points on a suitably chosen Poincar\'esurface of section. In this paper we study the two-dimensional annulus billiard, and modify Bogomolny's transfer operator to include diffractive paths. The penumbra contributions [H. Primack et al., Phys. Rev. Lett. 76, 1615 (1996)], which correspond to diffractive paths passing close to the inner circle of the annulus, are found to account for most of the difference between the exact transfer operator and Bogomolny's semiclassical transfer operator. When these diffractive effects are included, the semiclassical energy eigenvalues are brought much closer to the exact quantum energy eigenvalues.

Properties of nuclei in the inner crusts of neutron stars in the relativistic mean-field theory

We study the properties of nuclei in the inner crusts of neutron stars based on the Boguta-Bodmer nonlinear model in the relativistic mean-field theory. We carefully determine the surface diffuseness of the nuclei as the density of matter increases. The imaginary time step method is used to solve the Euler-Lagrange equation derived from the variational principle applied to the semiclassical energy density. It is shown that with increasing density, the spherical nuclei become more neutron rich and eventually merge to form a uniform liquid of neutrons, protons, and electrons. We find that the smaller the value of the incompressibility K, the lower the density at which the phase transition to uniform matter occurs. The relativistic extended Thomas-Fermi method is generalized to investigate nonspherical nuclei. Our results show that the spherical nucleus phase is the only equilibrium state in the inner crusts of neutron stars. {copyright} {ital 1997} {ital The American Physical Society}

WKB method for integrable but non-separable Hamiltonians

A recently proposed extension of the WKB method to integrable but non-separable Hamiltonian systems, based on the construction of global solutions of Dirichlet problems for the Hamilton - Jacobi equation, is applied to the construction of semiclassical wavefunctions for the Barbanis Hamiltonian. The wavefunctions obtained are compared with those generated for the same system, by means of the coherent Gaussian method, by Davis and Heller. It is shown that when the caustics are smooth curves, there is no energy degeneration, and the semiclassical wavefunctions obtained by the extended WKB method agree with those by Davis and Heller. But if internal caustics are present and to the same semiclassical energy level correspond, not simply to one as in the coherent Gaussian method, but several (four in the example investigated) WKB wavefunctions, they differ in the regions inside the internal caustics.

Intramolecular Exchange Energy Transfer in a Bridged Bimetallic Transition Metal Complex: Calculation of Rate Constants Using Emission Spectral Fitting Parameters

The photophysical behavior of the transition metal complexes {[(bpy)2Ru]2(bphb)}(PF6)4, {[(tpy)(CN)Ru]2(bphb)}(PF6)2, and [(bpy)2Ru(bphb)Ru(tpy)(CN)](PF6)4 (bpy = 2,2‘-bipyridine, tpy = 2,2‘,6‘,2‘‘-terpyridine, bphb = 1,4-bis(2,2‘-bipyrid-4-yl)benzene) was investigated in acetonitrile solution and low-temperature glasses. Luminescence spectra, excitation spectra, and transient absorption decays of the three complexes serve to show that intramolecular electronic energy transfer from the MLCT excited state of the [(bpy)2Ru(bphb) chromophore to the MLCT state of the tpy-containing chromophore occurs in the unsymmetric bimetallic complex. Nearly complete energy transfer from the [(bpy)2Ru(bphb)] chromophore to the tpy-containing chromophore was observed even in 4:1 ethanol:methanol glasses at 20 K. A semiclassical exchange energy transfer mechanism was used to treat the available data; the Franck−Condon weighted density of states (FCWD) was obtained using parameters determined from fits of luminescence spectr...

The Zeeman effect for the 'anisotropic hydrogen atoms' in the complex WKB approximation: II. Quantization of two-dimensional Lagrangian tori (with focal points) for the Pauli operator with spin-orbit interaction

For pt.I, see ibid, p.5799-5810. On the basis of the complex WKB method, new spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic field are constructed. These series correspond to conditionally periodic motions of a classical electron in the plane orthogonal to the field direction. The semiclassical wavefunctions are regular in the entire configuration space including the focal points and have the 'superscar' properties near the domain of classical motion, i.e. near the projection of stable two-dimensional Lagrangian tori on the configuration space. Both for Schrodinger and Pauli operators the analytical description of the semiclassical wavefunctions 'superscar' structure is obtained. This structure is non-uniform with respect to the space coordinates and has singularities with respect to the parameter h(cross),h(cross) to 0, near focal points. In the limit case of strong magnetic fields explicit analytical formulae for the semiclassical energy levels are obtained.

The Zeeman effect for the 'anistropic hydrogen atom' in the complex WKB approximation: I. Quantization of closed orbits for the Pauli operator with spin-orbit interaction

Using the complex WKB method, new semiclassical spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic held (the anomalous Zeeman effect) are constructed. They are valid uniformly for all values of the field's magnitude. These series correspond to families (stable in the linear approximation) of periodic and conditionally periodic motions of a classical electron in the plane orthogonal to the direction of the field. The corresponding semiclassical wavefunctions have the property of quantum 'superscaring' in the coordinate space near these classical trajectories of motion. We obtain the limit cases of our semiclassical energy levels for strong and weak magnetic fields. We compare them with the only known approximations, i.e. for the hydrogen atom.

Quantum surface-of-section method: Demonstration of semiclassical Berry-Robnik energy level-spacing distribution in a generic two-dimensional Hamiltonian system

The recently developed quantum surface of section method is applied to a search for extremely high-lying energy levels in a simple but generic Hamiltonian system between integrability and chaos, namely the semiseparable 2-dim oscillator. Using the stretch of 13,445 consecutive levels with the sequential number around $1.8\cdot 10^7$ (eighteen million) we have clearly demonstrated the validity of the semiclassical Berry-Robnik level spacing distribution while at 1000 times smaller sequential quantum numbers we find the very persistent quasi universal phenomenon of power-law level repulsion which is globally very well described by the Brody distribution.

Semiclassical calculations of intra-shell S resonances of doubly excited helium

We calculate the energy spectrum of doubly excited 1,3Se equivalent-electron states, within a semiclassical planar model. The latter is based on the so-called asynchronous-mode configuration, proposed previously by Grujic and Simonovic (1991). Making use of the propagation method with regularization classical configurations with very complex structure and surprisingly rich kinematics have been computed. Semiclassical energies of singlet and triplet intra-shell S states are evaluated. Numerical results show that the model improves values for the 1Se states, obtained by other authors within the asymmetric-stretch collinear kinematics, but overestimates triplet-state energies. The general features of the model are discussed and compared with other proposed classical configurations.