Lowest Eigenstates Research Articles

Computer algorithm for many-body wave functions.

We develop and test an algorithm for the computation of a fixed number of the lowest eigenstates of a Hermitian operator. The algorithm is iterative and converges exponentially. We test the algorithm on the strong-coupled Hubbard model, which has a finite number of basis states, and we reproduce known results. We also compute with a partial basis, a subset of a complete basis. We show how to extrapolate the eigenvalues to the full basis result. This opens the possibility of applying the algorithm to problems having an infinite number of basis states.

An exact projection method for the lowest eigenstate of singular multidimensional Schr�dinger equations

An exact projection method for the lowest eigenstate of singular multidimensional Schr�dinger equations

Electromagnetic-energy-density distribution around a ground-state hydrogen atom and connection with van der Waals forces.

A spinless hydrogen atom coupled to the electromagnetic field is considered within the context of nonrelativistic quantum electrodynamics. The atom-field interaction is taken in the minimal-coupling form and the Coulomb gauge is used. When the coupled system is in its ground state the electromagnetic field fluctuates away from the vacuum state and the atom has virtual admixtures from its uncoupled lowest eigenstate. The electric- and magnetic-field-energy densities that arise from the fluctuations are determined as functions of the distance from the atom. The relationship between these field-energy densities and the retarded long-range van der Waals forces is also discussed.

Non-adiabatic electron-phonon coupling in the two-sites two-electrons system

The electronic and ibrational properies of a system of two electrons or excitons at two sites are investgated in a simple model combining the ideas of the Hubbard and Holstein Hamiltonans. We analyse the competition of the interactions which enter this model as parameers for the transfer energyT, the (on-site) Coulomb energyU, the short-range electron-phonon coupling, represented by the stabilization energyS and the vibrational energy (∼ħω). For the whole range of these parameters the spectrum of the 50 lowest eigenstates has been numerically determined. As in the adiabatic approximation (ω=0) the ground state suffers a structural change due to a charge transfer instability, ifS is sufficiently large. In the case of finite ω this transition to a distorted state is no longer sharply defined by a criticalS. With regard to the lowest excited states the electronic system can be described by the adiabatic ground state for smallS as well as for largeS. Correspondingly the eigenfunctions of undisplaced or displaced harmonic oscillators, respectively, yield the lattice dynamics. In the range of intermediateS the situation is more complicated. Here the relation between the eigenfunctions and the adiabatic potentials as well as the appearance of metastable states and their connection to the charge transfer is discussed at some length.

Density-functional theory for excited states in a quasi-local-density approximation.

The starting point of this paper is a recent extension by Theophilou of the Hohenberg-Kohn-Sham (HKS) density-functional theory to ensembles of systems consisting of the M lowest eigenstates, equally weighted. As in the HKS theory the key quantities are the exchange-correlation energy, ${E}_{\mathrm{xc}{}^{M}[\mathrm{n}(\mathrm{r})]}$, and potential, ${v}_{\mathrm{xc}{}^{M}(\mathit{r};[\mathit{n}(\mathit{r}\mathcal{'})])}$. The present paper provides expressions for these quantities, valid for systems of slowly varying density. Even for such systems, however, there are essential nonlocal effects. Nevertheless both ${E}_{\mathrm{xc}{}^{M}}$ and ${v}_{\mathrm{xc}{}^{M}}$ can be calculated in terms of quantities characteristic of appropriate uniform thermal ensembles. This theory is the analog of the ground-state local-density approximation and allows calculation of excited-state energies and densities.

Spin chains in a field: Crossover from quantum to classical behavior.

Extensive numerical studies have been performed on Heisenberg antiferromagnetic chains of spin (1/2) (up to N=20), spin 1 (N=14), spin (3/2) (N=10), and spin 2 (N=8). With use of the Lancz\os technique, primarily, the two lowest-lying eigenvalues have been calculated for all values of wave vector q and all values of magnetization (${S}_{T}^{z}$) up to saturation for each chain. From a knowledge of the eigenvector corresponding to the lowest eigenstate for each ${S}_{T}^{z}$, the T=0 spin-pair correlation functions have also been calculated as a function of field. We find a most unusual quantum-classical crossover phenomenon. It shows up in greatest detail in the field-dependent dispersion spectra, but the consequences are consistently manifested in the behavior of the T=0 magnetization isotherms and in the correlations both in real space and the Fourier transforms in q space. The additional data relevant to behavior in a field have allowed us to extend previous numerical studies of Heisenberg antiferromagnetic chains with higher spin whose purpose was to examine the validity of the Haldane conjecture. The Haldane conjecture implies that Heisenberg antiferromagnetic chains with integer spin have a gap in their excitation spectrum whereas chains with half-integer spin do not. While no feature of our extended investigations is in conflict with the conjecture, unusual features associated apparently with very slow convergence make the outcome less than conclusive. It appears that calculations on significantly longer chains are required to observe with confidence the large-N asymptotic limiting behavior.

Critical assessment of interacting boson model wave functions from measured gyromagnetic ratios of lowest eigenstates in even Os isotopes

The gyromagnetic ratios of the low-lying levels in the even 188−192Os isotopes were measured. Of particular import, the ratios g(2 + 2)/ g(2 + 1) were found to be 1.45 ± 0.18, 0.99 ± 0.14, and 0.72 ± 0.06 in 188Os, 190Os, and 192Os, respectively. Although the E2 observables in these nuclides are accommodated quite well in interacting boson model descriptions (IBM-1 and IBM-2), both formulations fail to concurrently account for either the intra-nucleus or mass-dependent g-factor variations of the lowest eigenstates in these Os isotopes.

The Mixing of Frenkel and Charge‐Transfer Excited States. II. Numerical Treatment of Non‐Symmetric Dimers

AbstractThe mixing of Frenkel and charge‐transfer excitations is discussed for singlet excitions in a dimer consisting of inequivalent molecules. Depending on the parameter values of the system a more or less rapid crossover of the lowest eigenstate between a predominant Frenkel and a predominant charge‐transfer excitation is observed if one of the system parameters is varied. The case of singlet excitons in anthracene/TCNB is explicitly considered.

A new approach to the determination of good action-angle variables for coupled oscillator systems

A theory of action-angle variables for coupled oscillator systems is developed which involves solving the Schrödinger equation using a basis of WKB eigenfunctions, then using the logarithm of the resulting wavefunction to define the generator for the canonical transformation which determines the action-angle variables. This theory is based on the marriage between Miller's method for solving the Hamilton-Jacobi equation using the logarithm of the generating function, and the Ratner-Buch-Gerber method for solving the Schrödinger equation using WKB basis functions. A perturbation-theory analysis of this theory indicates that the semiclassical eigenvalues and canonical transformations obtained from it should become identical to their exact classical counterparts in the limit of large actions for each vibrational mode. Two methods for systematically improving the theory for the lower eigenstates are also proposed. Numerical applications of the theory are presented for two systems, the Morse oscillator and the Henon-Heiles two-mode hamiltonian. The resulting semiclassical eigenvalues are in excellent agreement with their exact quantum counterparts, with the magnitude of the error roughly independent of the energy of the eigenstate. Analogous good agreement is found in comparing the approximate and exact classical canonical transformations. In particular, for the Morse oscillator, good results are obtained for certain higher energy states where second-order classical perturbation theory makes serious errors. Other information examined includes surfaces of section for the Henon-Heiles system (comparing the analytical functions obtained from the present theory with results based on exact trajectory calculations) and vibrational distributions chosen to simulate trajectory calculations (using the present theory to determine bin boundaries for a histogram calculation). Again, the comparison in each case with accurate results is excellent, with maximum errors in action calculations of 0.02 h, and in angle calculations of 0.01 rad.

Padé phenomenology for two-body bound states

Pad\'e approximants in the squared momentum variable, recently used for elastic scattering, are employed in generating accurate analytic approximants for bound states. Through iteration, [$\frac{L}{L+1}$] approximants yield the lowest eigenstate of the homogeneous Lippmann-Schwinger equation for Yukawa, Malfliet-Tjon, and Reid soft core central potentials with, respectively, $L=1, 2, \mathrm{and} 3$. Higher eigenstates are readily obtained; the second is given for the Yukawa potential. Analytic separable expansions and scattering expressions result.NUCLEAR STRUCTURE Pad\'e approximants in ${k}^{2}$, analytic two-body bound states, separable expansions, effective range parameters.

Particles and holes equivalence for generalized seniority and the interacting boson model

An apparent ambiguity was recently reported in coupling either pairs of identical fermions or hole pairs. This is explained here as being due to a Hamiltonian whose lowest eigenstates do not have the structure prescribed by generalized seniority. It is shown that generalized seniority eigenstates can be equivalently constructed from correlated $J=0$ and $J=2$ pair states of either particles or holes. The interacting boson model parameters recently calculated can be unambiguously interpreted and then are of real interest to the shell model basis of the interacting boson model.

Stability of Electron Vortex Structures in Phase Space

The stability of one-dimensional, solitary vortex structures in the electron phase space (electron holes) is investigated. A linear eigenvalue problem is derived in the fluid limit and solved exactly, assuming that the normal mode is well represented by the lowest eigenstate of a properly chosen field operator. A new dispersion relation is obtained which exhibits purely growing solutions in two dimensions but only marginally stable solutions in one dimension. This explains the numerically well-known fact that vortex structures disappear in going from one to two dimensions.

Complex saddle points versus dilute-gas approximation in the double-well anharmonic oscillator

The energy shift between the two lowest eigenstates in the left- and right-hand wells of the double potential is computed by evaluating functional integrals using the saddle point method. The complex saddle points are considered on the same footing as the real ones. They happen to be a perfect substitute for the dilute gas.

STATISTICAL THEORY OF A ONE-DIMENSIONAL INTERACTIVE KINK-PHONON GAS

This paper systematically studies and improves the statistical theory of the kink-phonon interactive mixture in a one-dimensional model for the displacive phase transition. By taking applicable forms of the path integral for both the partition functions of kink and phonon and altering the method in calculating the path integral of phonon, a more reasonable and general expression of the partition function of the Grand ensemble is obtained. Under the lowest eigenstate approximation, the Grand.partition function may be reduced to a result similar to that of reference[6]. Under the classical approximation, the average density of the kink calculated from the Grand partition function is in agreement with the result of the computer simulation better than those obtained in earlier publications.

Oscillator-strength trends in the presence of level crossings

A configuration-interaction model is used to study line strengths and $\mathrm{gf}$ values for transitions in which level crossing occurs in either the initial or final state. It is shown that the line strength varies continuously for the lowest eigenstate, for example, but not for the eigenstate with a given dominant component. The latter exhibits marked irregularities in the vicinity of a crossover. Some plots for line strengths are presented showing the possible types of irregularities using a model interaction matrix. Computed $\mathrm{gf}$-value trends for $3p$ or $4p^{2}P\ensuremath{-}(3{s}^{2}4s,3s3{p}^{2})^{2}S$ transitions, as a function of $Z$, illustrate some of these irregularities.

Numerical study of a phase-integral connection formula

In order to study the accuracy and properties of a certain type of arbitrary-order phase-integral approximations, especially as regards the connection formula for tracing a wavefunction from a classically forbidden region to a classically allowed region, the Schrodinger equation for the one-dimensional harmonic oscillator is solved by means of the first, third and fifth order of these approximations, and the resulting approximate wavefunctions, both when normalised and when fitted to the exact wavefunctions at + infinity , are compared with the corresponding exact wavefunctions. The numerical results show that, except for the lowest energy eigenstates, the higher-order phase-integral approximations in question are very accurate both in the classically allowed and in the classically forbidden regions.

MAPW calculation of the anisotropic positron annihilation in copper

The wave function ψ− of a positron in Cu is calculated using the MAPW (modified augmented plane wave) method. This method, taking into account the correct symmetry of ψ− inside the atomic polyhedron and yielding the appropriate behaviour near the nucleus, leads to rapidly converging results. The Hartree potential seen by the positron is constructed from the electronic wave functions determined by the MAPW method. It is found that the lowest eigenstate of the positron is a Γ1 state. In a provisional calculation, the matrix elements describing the two-quantum angular distribution of positron annihilation radiation are computed using recent MAPW electron wave functions. Good agreement with the measured anisotropic positron annihilation is found.

Extension of the Single-Particle Description in Nuclear System. I

A general theory is proposed for the study of the single-particle description of a nondegenerate finite-particle system in a truncated shell-model space. In this theory, it is established that one can introduce a single-paritcle representation satisfying the requirements that single-particle operators to be defined should obey the Fermion-commutation relation, and that the corresponding free vacuum should also be the lowest eigenstate of the Hamiltonian. It is proved that such a representation based on the true ground state can be derived by the unitary transformation of the usual Fermion operators similarly to the quasi-particle transformation method. The equation is given for the determination of the unitary operator proper to new representation. This approach is proved to be understood as a natural extension of the usual Hartree-Fock theory. The theory is applied to a normal system, and a successive linearization procedure is proposed in order to solve the equation for the unitary operator. The advantages of the theory are discussed in connection with the nuclear shell-model problems.

How Good Is the Hartree-Fock Approximation? II. The Case of Closed Shells

In the present paper, we analyze the problem of an arbitrary number of particles A in a harmonic-oscillator common potential interacting through a harmonic-oscillator force. The particles have spin 12 and obey Fermi statistics. The exact lowest eigenstate and energy are derived for values of A corresponding to closed subshells. An approximate analysis is also made using determinantal wave functions formed from single-particle harmonic-oscillator states, giving the eigenstates and energies of a simple form of the Hartree-Fock approximation. The exact and Hartree-Fock energies and eigenstates are compared both when potential and force have the same sign, and when they have opposite sign. We discuss the results for models which we call pseudonuclei and pseudoatoms, and we indicate some of the inferences for the real thing.

Thermodynamic Properties of Phosphoryl Fluoride from 12 to 240°K. The Zero-Point Entropy of the Solid

The heat capacity of phosphoryl fluoride has been measured from 12 to 240°K for a sample with a purity greater than 99.9 mole %. The solid–liquid–vapor triple-point temperature and pressure were, respectively, 233.45°K and 774.06 torr. The heat of fusion was 3086 cal/mole. The heat of vaporization was determined as 5150 cal/mole at a temperature of 236.00°K and a pressure of 880.25 torr. The vapor pressure data were fitted by the equation lnP(torr) = − 487.49218 + 14 223.578 / T + 79.446599 lnT for the solid and by lnP(torr) = 8.05651 − 1206.41 / T for a short temperature range in the liquid phase. An entropy discrepancy, Sspec − Scalc, of 1.33 cal/mole·°K at 236.00°K was found between the spectroscopic entropy of 64.777 cal/mole·°K and the calorimetric value of 63.45 cal/mole·°K. A tentative model to explain the discrepancy has been proposed in which it is assumed that the degree of degeneracy of the several lowest eigenstates of the molecular motion are reduced when the molecule in the solid is subjected to the field of neighboring molecules of a certain symmetry.