Pair Approximation Method Research Articles

An effective field study of the staggered quadrupolar phase in the BEG model

The so-called new effective field theory, based on a finite cluster theory that correctly incorporates the single-site kinematic relations, is applied to the Blume Emery Griffiths (BEG) model with positive biquadratic interactions in order to test its ability to deal with magnetic systems having competing interactions. In particular, the region of the phase diagram containing a staggered quadrupolar phase is explored and comparison with the results of Monte Carlo simulation made. The results are also compared to those obtained by a cluster variation method in pair approximation.

A theoretical simulation of the 1s→2π excitation and deexcitation spectra of the NO molecule

We present ab initio calculations for the N and O 1s→2π photoexcitation spectra and the deexcitation electron spectra which result in the decay of these core excited states. The photoexcitation spectra are simulated with potential energy curves and transition dipole moments calculated with the multiconfiguration coupled electron pair approximation method. The energy lifetime width of the core excited states, which influences the form of these excitation spectra via the broadening of the vibrational progressions, was calculated by the one center approximation. The theoretical spectra are compared to recent experimental data. The deexcitation electron spectra, which monitor the autoionization of the core excited states, are reproduced by a combination of valence configuration interaction calculations to obtain the relative energies and the one center approximation which yields decay rates to the singly charged final states. Furthermore, we included lifetime vibrational interference effects using the recently proposed moment method of Cederbaum and Tarantelli [J. Chem. Phys. 98, 9691 (1993)]. The calculated deexcitation spectra compare extremely favorably with experimental data.

Analyses of spatial patterns and population processes of clonal plants

AbstractThe nonrandom spatial structure of terrestrial plants is formed by ecological interactions and reproduction with a limited dispersal range, and in turn this may strongly affect population dynamics and population genetics. The traditional method of modelling in population ecology is either to neglect spatial pattern (e.g. in transition matrix models) or to do straightforward computer simulation. We review here three analytical mothods to deal with plant populations in a lattice‐structured habitat, which propagate both by seeds that scatter over the whole habitat and by vegetative reproduction (producing runners, rhizomes, etc.) to neighboring vacant sites. [1]Dynamics of global and local densities: Dynamical equations of population density considering nearest‐neighbor correlation (spatial clumping) are developed as the joint dynamics of global average density and local density (comparable to mean crowding) based on pair approximation. If there is a linear trade‐off between seed production and vegetative reproduction, the equilibrium abundance of the population may be maximized by engaging both means of reproduction. This result is accurately predicted by the pair approximation method, but not by mean‐field approximation (neglect of spatial structure). [2]Cluster size distributions: Using global and local densities obtained by pair approximation, we predicted cluster size distribution, i.e. the number of clusters of occupied sites of various sizes. [3]Clonal identity probability decreasing with distance: Multi‐locus measurement of allozymes or other neutral molecular markers tells us whether or not a given pair of individuals belong to the same clone. From the pattern of clonal identity probability decreasing with the distance between ramets, we can estimate the relative importance of two modes of reproduction: vegetative propagation and sexual seed production.

On the pair approximation method in a nonreactive catalytic system

The nonreactive catalytic system made up of the reduction of NO by CO on a square lattice of sites is studied by means of dynamic equations in the pair approximation, showing some characteristics and limitations of the model in systems with a nontrivial behavior.

A Stochastic Model for Cell Sorting and Measuring Cell–Cell Adhesion

Spontaneous sorting-out of cells of different types is studied by a lattice-structured model of continuous-time Markovian transition. Cells of two types, called black and white, are arranged on a regular lattice, and the adhesion between cells in contact depends on their types. Cells exchange their location with neighbors at random times, and the rate is affected by the difference in cell-to-cell adhesion caused by the exchange. The model has the unique equilibrium probability distribution of spatial patterns. According to the analysis equivalent to statistical thermodynamics, the spatial pattern should show a clear coarse grained segregation if the ratio of differential adhesion to the intensity of random movement ( A/m) exceeds a critical value. However, even for the parameter range in which spatial pattern is predicted to be uniform, the cells of the same type tend to be in contact more often then in a random distribution, forming many small clusters of the same cell types. To quantify this microscopic tendency of aggregation, two statistics are introduced: (1) the fraction of black cells in the neighbor of a randomly chosen black cell ( q B B ), and (2) the number of isolated black cells (IBC). The equilibrium value of these statistics can be predicted accurately using a pair-approximation method. However, if A/mis large, the convergence of q B B to the equilibrium value takes an extremely long time. IBC is faster in convergence than q B B . These statistics can be used to measure the strength of differential adhesion relative to random movement ( A/m) is based on an observed spatial pattern of cells.

Radiative lifetimes of HBr+ and DBr+ (2Π1/2, v=0, 1)

Radiative lifetimes of HBr+ and DBr+(2Π1/2, v=0, 1) have been determined using the monitor ion technique in a triple cell ICR spectrometer with Fourier transform detection. The experimental lifetimes corresponding to the vibrational transition v=1→0 are 10.2±1 ms and 56+12−11 ms for HBr+ and DBr+, respectively. The lifetimes calculated by a coupled electron pair approximation (CEPA) method similar to the method used previously for HF+ and HCl+ are 8.8 ms for HBr+ and 35.8 ms for DBr+, in rather good agreement with the experimental results. A comparison with previous experimental and theoretical results on hydrogen halides shows a shortening of the lifetime between the neutrals and the corresponding ions and a lengthening with isotopic substitution (H by D) as well as with substitution of F by Cl or by Br. The lifetimes corresponding to spin–orbit relaxation are very long 852+621−282 ms for HBr+ and 965+546−323 ms for DBr+. They can be considered as equal within the quite large experimental uncertainty on the contribution of collisional deactivation.

Accuracy of the Boys and Bernardi function counterpoise method

The performance of the Boys and Bernardi function counterpoise (FCP) method in eliminating the basis set superposition error (BSSE) is studied for He2, at R=5.6 a.u., within the supermolecular coupled electron pair approximation (CEPA-1) method. A series of one-electron Gaussian basis sets is designed that allows a systematic approach to the basis set limit value of the interaction energy. Every basis set contains a part suitable to reproduce the atomic correlation energy and a second part optimized for the dispersion interaction in He2. BSSE-free correlated first-order interaction energies [E(1)], calculated using perturbation theory, are reported for each of these sets. Extrapolation to the basis set limit yields a new value of 33.60±0.02 μH for E(1) at R=5.6 a.u. Extending previous work, the supermolecular CEPA-1 interaction energies for each set are then compared to the total of E(1) and the BSSE-free Mo/ller–Plesset second-order dispersion energy reported previously. While for some basis sets the uncorrected ΔE values deviate up to 43 K from the perturbation estimate, the FCP-corrected results always agree within 0.4 K. A virtuals-only counterpoise procedure is considered as well, but fails badly. The remaining discrepancies in the FCP results are ascribed to a failure of the Mo/ller–Plesset approach to precisely model the dispersion energy at the CEPA level. This problem is removed in a further, more stringent test where supermolecular EintCEPA-intra results, in which only the intra-atomic correlation (at the CEPA-1 level) is taken into account, are directly compared to the BSSE-free E(1) values. In this test the FCP-corrected supermolecular results agree, for the larger sets, to within 0.001 K with the results expected on the basis of E(1). These findings demonstrate, for the first time, that at least in He2 the FCP recipe yields interaction energies that correspond precisely (to machine precision) to the basis set and correlation method at hand.

Calculation of the temperature-dependent absolute extinction coefficients for the photodissociation of ClNO(S1) on a CEPA potential energy surface

A new potential energy surface (PES) for the first excited singlet state of ClNO was calculated with the coupled electron pair approximation (CEPA) method. This PES could be used without any empirical corrections in subsequent three-dimensional dynamical calculations with a L2 amplitude density method of time-independent scattering theory. Absolute photodissociation rates for producing the NO fragment, in particular rotational-vibrational states were calculated for the lowest seven initial vibrational states, of ClNO. A shift of the D and C band peaks to shorter wavelengths with increasing temperature is predicted.

Collisions of doubly charged nitrogen molecules with rare gas atoms

The authors report on single electron capture in slow collisions of doubly charged nitrogen molecules with the rare gases He, Ne and Ar. The vertical double ionization energy of N2 as well as vertical excitation energies of N22+ have been calculated by means of the multiconfiguration coupled electron pair approximation (MC-CEPA) method. Furthermore, experiments have been carried out applying the technique of translational energy spectroscopy. The energy gain spectra of the resulting N2+ molecules have been analysed, showing the single electron capture to be dominated by N22+ (X 1 Sigma +g) projectile ions, as well as by N22+ ions in the low-lying metastable c 3 Sigma +u state. The position of the experimentally observed reaction window is discussed in terms of Franck-Condon transitions and is compared with predictions of classical descriptions.

A rigorous upper bound energy for the unitary coupled electron pair approximation method

Presentation d'une nouvelle methode permettant d'obtenir l'energie liee rigoureusement la plus elevee, essentiellement sans effort de calcul supplementaire que celui exige par la methode UCEPA

Pair Approximation Method Applied to a Quantum Transverse Ising Spin-Glass Model11The project supported by National Natural Science Foundation of China.

The quantum version of the Sherrington-Kirkpatrick model in a transverse field is investigated by combining the pair approximation with the discrete path-integral representation. The free energy and the field-dependent freezing temperature of the model are obtained, and the classical result for the vanishing of the transverse field can be reproduced.

Pair-approximation method for the quantum transverse Ising model with a random field.

The quantum spin system with a random field has been investigated by combining the pair approximation with the discretized path-integral representation. Second-order transition lines and tricritical points are obtained for the bimodal random-field distributions. Reentrant phase transitions, which may be caused by the competition between quantum effects and the randomness, occur with appropriate ranges of the transverse field and the random field. These results obtained with this approximation are qualitatively different from those of the ordinary mean-field approximation and display that the pair approximation takes into account the local structure of the interaction pattern of the underlying system, which results in coordinate-dependent phase diagrams.

Ising ferromagnet with biquadratic exchange interaction and uniaxial anisotropy

The cluster variational method in pair approximation is applied to a study of the Ising ferromagnet having S = 1, 3/2, 2 and 5/2, with the inclusion of biquadratic exchange and single-ion anisotropy. Attention is focused on the anomalous behaviour of the magnetization curves that is predicted when the ratio of the nearest-neighbor bilinear and biquadratic exchange interactions is negative. The critical temperature and magnetization curves are studied in detail in this regime and comparison is made with the recently reported results derived within the Oguchi pair approximation. A marked qualitative difference in the results of the two theories is found to exist for integer spin values.

Spin-glass and ferromagnetic phases of a quantum spin-one random-bond Heisenberg-Ising model with single-ion anisotropy

The present paper discusses the features of a variety of quantum spin-1 random-bond Heisenberg-Ising models with single-ion anisotropy. The label of the variety is a parameter entering the model which can take values in the range 0–1, the value 0 corresponding to the Ising model, 1 to the isotropic Heisenberg model. Two different distribution functions for the random exchange parameter are assumed: the binomial and the ± J. The model is examined following the pair approximation method extended to random systems. The uniform, staggered and spin-glass susceptibility are introduced and the Curie, Néel, and spin-glass transition temperatures determined.

Monte Carlo simulations and pair approximations on the phase transition of the restricted orientational lattice model for liquid crystals

The nematic-isotropic phase transition on the SC lattice model of uniaxial molecules is studied by Monte Carlo simulations and the pair approximation method. The interaction between nearest-neighbour sites is assumed to be proportional to (1-3 cos2theta )/2 where theta is the angle between the long axes of molecules. The models of restricted orientation with 3,4,6 and 10 directions are systematically investigated. The phase transitions are shown to be of first order. Analytical estimations by a pair approximation are in agreement with Monte Carlo results.

Phase Boundaries between the Paramagnetic and the Antiferromagnetic Phases and between the Paramagnetic and the Spin‐Flop Phases of the Heisenberg‐Ising Model

AbstractThe phase boundary between the paramagnetic and antiferromagnetic phases, and that between the paramagnetic and spin‐flop phases of the Heisenberg‐Ising model with anisotropic exchange energy are calculated by the method of pair approximation. The critical field curves are satisfactory except at the very low temperature. The reliability of the critical field in the low temperature region is closely related to the problem of the appearance of the anti‐Néel point, and is discussed by calculating the values of the Néel point and the anti‐Néel point by the method of the square approximation and comparing them with those by the pair approximation.

Pair approximation for interface kinetics

A pair approximation method is described for the interface kinetics of a simple cubic (100) surface with no surface migration. Condensation and evaporation are considered explicitly in a system of master equations applied to a pair of sites; and the effect of the neighboring sites is treated with a set of parameters which are determined self-consistently. Runge Kutta numerical integration of the equations yields the time evolution of the surface. The results for a high entropy of transformation indicate a region of metastable states (zero growth rate) for small driving force, and the instantaneous growth rate is a periodic function of time for larger driving force. The period of the variations is the time required to grow a monolayer, and the amplitude is less than that obtained by Temkin using the mean field approximation. The variations appear as artifacts of most approcimation methods which treat a limited number of sites. The average growth rate is less than the limiting Wilson-Frenkel law for all growth conditions studied. This results since the surface becomes quite rough in the presence of a large condensation rate, and therefore its evaporation rate is greater than the evaporation rate of an equilibrium surface assumed in the Frenkel model. The metastable region is absent at low entropies of transformation and here the growth rate is nearly linear in the driving force. Agreement with Monte Carlo data is excellent for a low entropy of transformation and for all materials at high growth rates.

Lambda-nucleon interaction potentials

Two-body S-wave, spin dependent, central Λ-N interaction potentials of the form first given by Ram and Downs are constructed to fit the zero energy scattering lengths a s = −2.25 fm, a t = −2.08 fm of Herndon and Tang. When assumed to be effective in even-parity states with full strength and in odd-parity states with zero strength, these potentials fit the Λ-p scattering cross section data very well. The binding energy D of the Λ-particle in nuclear matter using these potentials is calculated by the method of independent pair approximation. The potentials that best fit the scattering data give values for D which are only 3–4 MeV lower than that obtained previously with a potential of Hemdon and Tang. It is concluded that the three-body ΛNN forces may be quite important in the calculations of D.