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FESSDE, a program for the finite-element solution of the coupled-channel Schrödinger equation using high-order accuracy approximations
Abstract
The finite element method (FEM) is applied to solve the bound state (Sturm-Liouville) problem for systems of ordinary linear second-order differential equations. The convergence, accuracy and the range of applicability of the high-order FEM approximations (up to tenth order) are studied systematically on the basis of numerical experiments for a wide set of quantum-mechanical problems. The analytical and tabular forms of giving the coefficients of differential equations are considered. The Dirichlet and Neumann boundary conditions are discussed. It is shown that the use of the FEM high-order accuracy approximations considerably increases the accuracy of the FE solutions with substantial reduction of the requirements on the computational resources. The results of the FEM calculations for various quantum-mechanical problems dealing with different types of potentials used in atomic and molecular calculations (including the hydrogen atom in a homogeneous magnetic field) are shown to be well converged and highly accurate.
... We construct appropriate variational formulations [23] for both the bound-state and the elastic scattering problems in the framework of the KM, using Rayleigh-Ritz's and Hulthèn's variation functionals [24,25]. The corresponding stable numerical schemes are realized using FEMs of high order accuracy [26][27][28]. We verify the accuracy of these schemes, and examine their rate of convergence to the known exact results, as a function of the number of basis functions. ...
... In order to solve numerically the Sturm-Liouville problems for equations (28) or (31) (for the discrete spectrum problem with the Rayleigh-Ritz variational functional) and (37) or (46) (for the continuous spectrum problem with the Hulthèn variational functional) the high-order approximations of the finite element method (FEM) [26,27] elaborated in our previous paper [28] have been used. Such high-order approximations of the FEM have been proved [28] to be very accurate, stable and effective for a wide set of quantum-mechanical problems. ...
... In order to solve numerically the Sturm-Liouville problems for equations (28) or (31) (for the discrete spectrum problem with the Rayleigh-Ritz variational functional) and (37) or (46) (for the continuous spectrum problem with the Hulthèn variational functional) the high-order approximations of the finite element method (FEM) [26,27] elaborated in our previous paper [28] have been used. Such high-order approximations of the FEM have been proved [28] to be very accurate, stable and effective for a wide set of quantum-mechanical problems. ...
A Kantorovich approach is used to solve for the eigenvalue and the scattering properties associated with a multi-dimensional Schrödinger equation. It is developed within the framework of a conventional finite element representation of solutions over a hyperspherical coordinate space. Convergence and efficiency of the proposed schemes are demonstrated in the case of an exactly solvable 'benchmark' model of three identical particles on a line, with zero-range attractive pair potentials and below the three-body threshold. In this model all the 'effective' potentials, and 'coupling matrix elements', of the set of resulting close-coupling radial equations, are calculated using analytical formulae. Variational formulations are developed for both the bound-state energy and the elastic scattering problem. The corresponding numerical schemes are devised using a finite element method of high order accuracy.
... Wensch). equations can be noticed, cf.12345678910 . There are special considerations for the calculation of bound states123 or the scattering states [5,6] . ...
... There are special considerations for the calculation of bound states123 or the scattering states [5,6] . Higher-order methods have been developed4567 . New numerical techniques like wavelets have been applied also to solve the Schrödinger equation [10]. ...
In solid state physics the solution of the Dirac and Schrödinger equation by operator splitting methods leads to differential equations with oscillating solutions for the radial direction. For standard time integrators like Runge–Kutta or multistep methods the stepsize is restricted approximately by the length of the period. In contrast the recently developed Magnus methods allow stepsizes that are substantially larger than one period. They are based on a Lie group approach and incorporate exponential functions and matrix commutators. A stepsize control is implemented and tested. As numerical examples eigenvalue problems for the radial Schrödinger equation and the radial Dirac equation are solved. Further, phase shifts for scattering solutions for hydrogen atoms and copper are computed. 2004 Elsevier B.V. All rights reserved.
... It can be solved using the spacediscretization techniques, where instead of approximating the wave packet by a globally defined basis one applies the finite-element method (FEM) [4,5]. A required accuracy of the solution with respect to the spatial step can be achieved using suitable interpolation polynomials on a finite element space grid [9,10]. ...
... For a numerical realization of this method we derive the second-, forth-and sixth-order approximations with respect to the time step. The FEM is used to construct the numerical schemes with a required accuracy with respect to the spatial step [9,10]. We establish the efficiency and accuracy of the developed numerical algorithms considering the case of integrable models in external fields. ...
A new computational approach is proposed for the solution of the time-dependent Schrödinger equation (TDSE), in which a symbolic
algorithm named GATEO and a numerical scheme based on the finite-element method (FEM) are effectively composed. The GATEO
generates the multi-layer operator-difference scheme for TDSE and evaluates the effective Hamiltonian from the original time-dependent
Hamiltonian by means of the Magnus expansion and the Pade-approximation. In order to solve the TDSE with the effective Hamiltonian
thus obtained, the FEM is applied to a discretization of spatial domain which brings the difference scheme in operator form
to the one in algebraic form. The efficiency and accuracy of GATEO and the numerical scheme associated with FEM is confirmed
in the second-, fourth-, and sixth-order time-step computations for certain integrable atomic models with external fields.
... For numerical solution of one-dimensional eigenvalue problems (6)-(7), (11)- (12) and boundary value problem (14)- (16), the high-order approximations of the finite element method ( [18], [5]) elaborated in the papers [1], [2] have been used. One-dimensional finite elements of order k = 1, 2, . . . ...
... A and B are the finite element matrices, corresponding to problems (6)-(7) or (11)-(12) (see [1], [2]), E h is the corresponding eigenvalue and F h is the vector approximating solutions of (6)-(7) or (11)-(12) on the finite-element grid. ...
The new method for calculating the energy values of the hydrogen atom in a strong magnetic field (0 ≤ B ≤ 1013
G ) with high degree of accuracy is developed in this paper. The proposed method is based on the Kantorovich method for solving
eigenvalue problems. This method successfully reduces the given two dimensional spectral problem for the Schrödinger equation
to the spectral problems for one one-dimensional equation and the system of ordinary second-order differential equations.
The rate of convergence is examined numerically and is illustrated in the table. The results are in good agreement with the
best one up to now and show that 12 significant digits are computed in the obtaining energy values.
... With the obtained OPE effective potentials for the Σ cM * and Σ * cM * systems withM * = (D * ,K * ) listed in Section 2, we can find the possible bound solutions (the binding energy E, corresponding root-mean-square radius r RM S and radial wave function φ(r)) by solving the coupled-channel Schrödinger equation with the help of the FESSDE program [41,42]. The corresponding kinetic terms include ...
... Here, Then we adopt the FESSDE program [42,43] to calculate the coupled channel Schrödinger equation and look for the bound state solutions: the binding energy E(MeV) and the corresponding root-mean-square (RMS) radius r(fm), which depend on the cutoff Λ. Here, the cutoff parameter Λ varies from 0.5 to 5 GeV. ...
In this work, we study the TT¯ -type molecular systems systematically via the one pion exchange model, where T denotes the narrow JP= 1 +D1meson or 2 +D2∗ meson and T¯ is its antiparticle. With the effective potentials, we try to find the bound-state solutions of the corresponding systems, which provide crucial information of whether TT¯ -type molecular states exist. By our analysis, we predict some TT¯ -type molecular states which may be accessible at future experiments like LHCb and forthcoming BelleII.
... Here, Then we adopt the FESSDE program [42,43] to calculate the coupled channel Schrödinger equation and look for the bound state solutions: the binding energy E(MeV) and the corresponding root-mean-square (RMS) radius r(fm), which depend on the cutoff Λ. Here, the cutoff parameter Λ varies from 0.5 to 5 GeV. ...
In this work, the $T\bar{T}$ type molecular systems was systematically studied via one pion exchange model. By solving the deduced effective potentials, we tried to find the bound state solutions of the corresponding systems, which provide crucial information of whether there exist the $T\bar{T}$ type molecular states. According to our analysis, we predict some $T\bar{T}$ type molecular which can be accessible at future experiment like LHCb and forthcoming BelleII.
... respectively. Here, Δ = 1 In this work, the FESSDE program [55,56] is adopted to produce the numerical values for the binding energy and the relevant root-mean-square r with the variation of the cutoff in the region of 0.8 Λ 5 GeV. Moreover, we also use MATSCE [57], a MAT-LAB package for solving coupled-channel Schrödinger equation, to perform an independent cross-check. ...
In the framework of the one-boson-exchange model, we have performed an extensive study of the possible B*, B**, D*, D** molecular states with various quantum numbers after considering the S-wave and D-wave mixing. We also discuss the possible experimental research of these interesting states.
... There are various methods of integrating the multichannel Schrödinger equation numerically. In this work we shall employ two packages MATSCS [41] and FESSDE2.2 [42] to perform the numerical calculations so that the results obtained by one program can be checked by another. The first package is a Matlab software, and the second is written in Table II. ...
The proximity of Z+(4430) to the D*D̅ 1 threshold suggests that it may be a D*D̅ 1 molecular state. The D*D̅ 1 system has been studied dynamically from quark model, and state mixing effect is taken into account by solving the multichannel Schrödinger equation numerically. We suggest the most favorable quantum number is JP=0-, if future experiments confirm Z+(4430) as a loosely bound molecule state. More precise measurements of Z+(4430) mass and width, partial wave analysis are helpful to understand its structure. The analogous heavy flavor mesons Zbb+ and Zbc++ are studied as well, and the masses predicted in our model are in agreement with the predictions from potential model and QCD sum rule. We further apply our model to the DD̅ * and DD* system. We find the exotic DD* bound molecule does not exist, while the 1++ DD̅ * bound state solution can be found only if the screening mass μ is smaller than 0.17 GeV. The state mixing effect between the molecular state and the conventional charmonium should be considered to understand the nature of X(3872).
... With the effective potentials given in the Subsection II B, we use the FORTRAN program FESSDE [49,50] to solve the coupled channel Schrödinger equation. ...
We re-investigate the possibility of X(3872) as a $D\bar{D}^*$ molecule with
$J^{PC}=1^{++}$ within the framework of both the one-pion-exchange (OPE) model
and the one-boson-exchange (OBE) model. After careful treatment of the S-D wave
mixing, the mass difference between the neutral and charged $D(D^*)$ mesons and
the coupling of the $D(D^*)$ pair to $D^*\bar{D}^*$, a loosely bound molecular
state X(3872) emerges quite naturally with large isospin violation in its
flavor wave function. For example, the isovector component is 26.24% if the
binding energy is 0.30 MeV, where the isospin breaking effect is amplified by
the tiny binding energy. After taking into account the phase space difference
and assuming the $3\pi$ and $2\pi$ come from a virtual omega and rho meson
respectively, we obtain the ratio of these two hidden-charm decay modes:
$\mathcal{B}(X(3872)\rightarrow \pi^+\pi^-\pi^0
J/\psi)/\mathcal{B}(X(3872)\rightarrow \pi^+\pi^- J/\psi)=0.42$ for the binding
energy being 0.3 MeV, which is consistent with the experimental value.
... For solving a stationary Shrödinger equation a lot of approximate analytical and numerical methods are elaborated and applied because an exact solution in explicit form exists only for some specific hamiltonians [1]. As is known, the more used methods are diagonalization [2,3], quasiclassical approach [4], continuous analogue of Newton's method [5], different versions of perturbation theory [6], normal form method [7,8,9,10,11], finite-element method [12], 1/N expansion [13], oscillator representation [14], variational and operational methods [15,16,17], simplectic method [18] and etc. ...
A general scheme of a symbolic-numeric approach for solving the eigenvalue problem for the one-dimensional Shrödinger equation
is presented. The corresponding algorithm of the developed program EWA using a conventional pseudocode is described too. With
the help of this program the energy spectra and the wave functions for some Schrödinger operators such as quartic, sextic,
octic anharmonic oscillators including the quartic oscillator with double well are calculated.
... For numerical solution of one-dimensional eigenvalue problems and boundary value problem (11) subject to the corresponding boundary conditions (6) and (7), the high-order approximations of the finite element method [12,5] elaborated in our previous papers [3,1,2] have been used. One-dimensional finite elements of order p = 1, 2,. .. , 10 have been implemented. ...
A Kantorovich method for solving the multi-dimensional eigenvalue and scattering problems of Schrödinger equation is developed in the framework of a conventional finite element representation of smooth solutions over a hyperspherical coordinate space. Convergence and efficiency of the proposed schemes are demonstrated on an exactly solvable model of three identical particles on a line with pair attractive zero-range potentials below three-body threshold. It is shown that the Galerkin method has a rather low rate of convergence to exact result of the eigenvalue problem under consideration.
The work objective is to describe the numerical solution method for the stationary Schrödinger equation based on the application of the integral equation identical to the Schrödinger equation. The structure of this integral equation is close to the structure of the Fredholm equation of second kind and allows obtaining the problem numerical solution. The method under study allows finding the energy eigenvalues and eigensolutions to the quantum-mechanical problems of various dimensions. The test results of the solving problems method for one-dimensional and two-dimensional quantum oscillators are obtained. The found numerical values of eigenenergy and eigenfunctions of the oscillator are compared to the known analytical solutions, and then the error of result is evaluated. The highest accuracy of the solution is obtained for the first energy levels. The numerical solution error increases with the number of the energy eigenvalue. For the subsequent energy level, the error increases almost by an order of magnitude. The solution error for the fourth energy level is less than 2% if the integration domain contains 500 elements. If the energy level is degenerate, it is possible to obtain all eigenfunctions corresponding to the given level. For this purpose, various auxiliary functions the symmetry of which is coherent with the eigenfunction symmetry are used.
A Kantorovich method for solving the multi-dimensional eigenvalue and scattering problems of Schrodinger equation is developed in the framework of a conventional finite element representation of smooth solutions over a hyperspherical coordinate space. Convergence and efficiency of the proposed schemes are demonstrated on an exactly solvable model of three identical particles on a line with pair attractive zero-range potentials below three-body threshold. It is shown that the Galerkin method has a rather low rate of convergence to exact result of the eigenvalue problem under consideration.
The finite element (FE) method can be applied to solve inelastic and reactive scattering problems for triatomic systems. We use an S-matrix version of the Hulthén-Kohn variational principle for a three-dimensional scattering problem. The asymptotic wave-function is described by analytical functions and the interaction part by a combination of finite element and discrete variable representation (DVR) functions. Translational and vibrational coordinates are described by FEs and for the angular part a DVR-discretization is used. The FE method is used in order to calculate eigenfunctions for the interior region of the potential at each given DVR-angle and from this set of functions the full three-dimensional eigenfunctions are calculated and are used as a new basis for scattering. The grid is chosen such that it covers all reaction channels and the calculations are done by using only one set of Jacobi coordinates. Results obtained by choosing different FE-discretizations, raising the number of elements and increasing the number of internal three-dimensional eigenfunctions, are compared.
This thesis presents the theoretical modeling of the experiment FORCA-G (FORce de CAsimir et Gravitation à courte distance) currently in progress at Paris Observatory. The purpose of this experiment is to measure short-range interactions between an atom and a massive surface. This interaction are of two kind : quantum electrodynamical (Casimir-Polder effect) and gravitationnal. The work presented here was to calculate the atomic states in the context of the experiment such that we can predict results and performances of the experiment. This has allowed to optimize the experimental scheme both for the high-precision measurement of the Casimir-Polder effect and for the search of deviation from the Newton's law of gravity predicted by unification theories.
We present theoretical calculations for the evolution of Zeeman states in a train of short electric half-cycle pulses (kicks). For the numerical solution of the corresponding time-dependent Schrodinger equation (TDSE) the high-accuracy splitting scheme based on the unitary approximations of the evolution operator is developed. The finite element method is used for determining the spatial form of the solution. The efficiency and stability of the developed computational method is shown for ID models in the cases of second-, forth-, and sixth-order accuracy with respect to the time step. Numerical calculations for the kicked hydrogen atom in the presence of magnetic field are performed using the scheme of the sixth-order accuracy with respect to a time step and both Galerkin and Kantorovich reductions of the problem with respect to the angular variables. For a particular choice of the electric- and magnetic-field parameters and the initial Zeeman state the corresponding results exhibit a two-state resonance picture.
In this work, we have investigated whether Y(4260) and Z(2)(+) (4250) could be D1D or D0D* molecules in the framework of meson exchange model. The off-diagonal interaction induced by pi exchange plays a dominant role. The sigma exchange has been taken into account, which leads to diagonal interaction. The contribution of sigma exchange is not favorable to the formation of the molecular state with I-G(J(PC)) = 0(-)(1(--)); however, it is beneficial to the binding of the molecule with I-G(J(P)) = 1(-)(1(-)). Light vector meson exchange leads to diagonal interaction as well. For Z(2)(+) (4250), the contribution from rho and omega exchange almost cancels each other. For the currently allowed values of the effective coupling constants and a reasonable cutoff Lambda in the range 1-2 GeV, we find that Y(4260) could be accommodated as a D1D and D0D* molecule, whereas the interpretation of Z(2)(+) (4250) as a D1D or D0D* molecule is disfavored. The bottom analog of Y(4260) and Z(2)(+) (4250) may exist, and the most promising channels to discover them are pi(+)pi Y- and pi(+)chi(b1), respectively.
In this work, we study the interaction between a nucleon and a $P$-wave charmed meson in the $T$ doublet by exchanging a pion. Our calculations indicate that a nucleon and a $P$-wave charmed meson with ${J}^{P}={0}^{+}$ or ${J}^{P}={1}^{+}$ in the $T$ doublet can form bound states. We propose the experimental search for these exotic molecular states near the ${D}_{1}(2420)N$ and ${D}_{2}^{*}(2460)N$ thresholds, where Belle, LHCb, and the forthcoming Belle II have the discovery potential for them.
In this paper, we discuss the atomic states in a vertical optical lattice in proximity of a surface. We study the modifications to the ordinary Wannier-Stark states in the presence of a surface, and we characterize the energy shifts produced by the Casimir-Polder interaction between atom and mirror. In this context, we introduce an effective model describing the finite size of the atom in order to regularize the energy corrections. In addition, the modifications to the energy levels due to a hypothetical non-Newtonian gravitational potential as well as their experimental observability are investigated.
A symbolic algorithm to generate the multilayer operator-difference schemes for solving the evolution problem of the time-dependent Schrödinger equation is elaborated. An additional gauge transformation of operator-difference schemes to make good use of the finite-element discretization is applied. The efficiency of the generated numerical schemes until the sixth order with respect to the time step and until the seventh order with respect to the spatial step is demonstrated by calculations of some finite-dimensional quantum systems in external fields.
The finite element (FE) method can be applied to solve inelastic and reactive scattering problems for triatomic systems. We use an S-matrix version of the Hulthén–Kohn variational principle for a three-dimensional scattering problem. The asymptotic wavefunction is described by analytical functions and the interaction part by a combination of finite element and discrete variable representation (DVR) functions. Translational and vibrational coordinates are described by FEs and for the angular part a DVR-discretization is used. The FE method is used in order to calculate eigenfunctions for the interior region of the potential at each given DVR-angle and from this set of functions the full three-dimensional eigenfunctions are calculated and are used as a new basis for scattering. The grid is chosen such that it covers all reaction channels and the calculations are done by using only one set of Jacobi coordinates. Results obtained by choosing different FE-discretizations, raising the number of elements and increasing the number of internal three-dimensional eigenfunctions, are compared.
We perform a systematic study of the possible loosely bound states composed of two charmed baryons or a charmed baryon and an anticharmed baryon within the framework of the one-boson-exchange model. We consider not only the pi exchange but also the eta, rho, omega, varphi, and sigma exchanges. The S-D mixing effects for the spin triplets are also taken into account. With the derived effective potentials, we calculate the binding energies and rms radii for the systems LambdacLambdac(Lambda¯c), XicXic(Xi¯c), SigmacSigmac(Sigma¯c), Xic'Xic'(Xi¯c'), and OmegacOmegac(Omega¯c). Our numerical results indicate that (1) the H-dibaryonlike state LambdacLambdac does not exist and (2) there may exist four loosely bound deuteronlike states for the XicXic and Xic'Xic' systems with small binding energies and large rms radii.
A FORTRAN program is presented which solves the Sturm-Liouville problem for a system of coupled second-order differential equations by the finite element method using high-order accuracy approximations. The analytical and tabular forms of giving the coefficients of differential equations are considered. Zero-value (Dirichlet) and zero-gradient (Neumann) boundary conditions are also considered.
The quantum mechanical three-body problem with Coulomb interaction is formulated within the adiabatic representation method using the hyperspherical coordinates. The Kantorovich method of reducing the multidimensional problem to the one-dimensional one is used. A new method for computing variable coefficients (potential matrix elements of radial coupling) of a resulting system of ordinary second-order differential equations is proposed. It allows the calculation of the coefficients with the same precision as the adiabatic functions obtained as solutions of an auxiliary parametric eigenvalue problem. In the method proposed, a new boundary parametric problem with respect to unknown derivatives of eigensolutions in the adiabatic variable (hyperradius) is formulated. An efficient, fast, and stable algorithm for solving the boundary problem with the same accuracy for the adiabatic eigenfunctions and their derivatives is proposed. The method developed is tested on a parametric eigenvalue problem for a hydrogen atom on a three-dimensional sphere that has an analytical solution. The accuracy, efficiency, and robustness of the algorithm are studied in detail. The method is also applied to the computation of the ground-state energy of the helium atom and negative hydrogen ion.
A new high-accuracy method for calculating the energy values of low-lying excited states of a hydrogen atom in a strong magnetic field (0 ≤ B ≤ 1013G) is developed based on the Kantorovich approach to parametric eigenvalue problems and using the axial symmetry. The initial two-dimensional spectral problem for the Schrödinger equation is reduced to a spectral parametric problem for a one-dimensional equation and a finite set of ordinary second-order differential equations. The rate of convergence is examined numerically and is illustrated with a set of typical examples. The results are in good agreement with precise calculations by other authors.
The symmetric implicit operator-difference multi-layer schemes for solving the time-dependent Schrödinger equation based on decomposition of the evolution operator via the explicit Magnus expansion up to the sixth order of accuracy with respect to the time step are presented. Reduced schemes for solving the set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised by using the Kantorovich expansion of the wave packet over a set of appropriate parametric basis angular functions. Further discretization of the resulting problem with symmetric operators is implemented by means of the finite-element method. The convergence and efficiency of the numerical schemes are demonstrated in benchmark calculations of the exactly solvable models of a one-dimensional time-dependent oscillator, a two-dimensional oscillator in time-dependent electric field by using the conventual angular basis, and the inexactly solvable model of a three-dimensional kicked hydrogen atom in a magnetic field by using a parametric basis of the angular oblate spheroidal functions developed in our previous paper (Chuluunbaatar O et al 2007 J. Phys. A: Math. Theor. 40 11485–524).
A symbolic algorithm for the decomposition of the unitary evolution operator is developed. This algorithm allows one to generate
multilayer implicit schemes for solution of the time-dependent Schrödinger equation. Some additional gauge transformations
are also implemented in the algorithm. This allows one to distinguish symmetric operators, which are required for constructing
efficient evolutionary schemes. The efficiency of the generated schemes is demonstrated by integrable models.
We investigate the possible molecules composed of two heavy flavor baryons
such as "$A_QB_Q$"(Q=b, c) within the one-pion-exchange model (OPE). Our
results indicate that the long-range $\pi$ exchange force is strong enough to
form molecules such as $[\Sigma_Q\Xi_Q^{'}]^{I=1/2}_{S=1}$(Q=b, c),
$[\Sigma_Q\Lambda_Q]^{I=1}_{S=1}$(Q=b, c), $[\Sigma_b\Xi_b^{'}]^{I=3/2}_{S=1}$
and $[\Xi_b\Xi^{'}_b]^{I=0}_{S=1}$ where the S-D mixing plays an important
role. In contrast, the $\pi$ exchange does not form the spin-singlet $A_QB_Q$
bound states. If we consider the heavier scalar and vector meson exchanges as
well as the pion exchange, some loosely bound spin-singlet S-wave states appear
while results of the spin-triplet $A_QB_Q$ system does not change
significantly, which implies the pion exchange plays an dominant role in
forming the spin-triplet molecules. Moreover, we perform an extensive coupled
channel analysis of the $\Lambda_Q\Lambda_Q$ system within the OPE and
one-boson-exchange (OBE) framework and find that there exist loosely bound
states of $\Lambda_Q\Lambda_Q$(Q=b,c) with quantum numbers $I(J^P)=0(0^+)$,
$0(0^-)$ and $0(1^-)$. The binding solutions of $\Lambda_Q\Lambda_Q$ system
mainly come from the coupled-channel effect in the flavor space. Besides the
OPE force, the medium- and short-range attractive force also plays a
significant role in the formation of the loosely bound $\Lambda_c\Lambda_c$ and
$\Lambda_b\Lambda_b$ states. Once produced, they will be very stable because
such a system decays via weak interaction with a very long lifetime around
$10^{-13}\sim10^{-12}$s.
A FORTRAN 77 program is presented which calculates energy values, reaction matrix and corresponding radial wave functions in a coupled-channel approximation of the hyperspherical adiabatic approach. In this approach, a multi-dimensional Schrodinger equation is reduced to a system of the coupled second-order ordinary differential equations on the finite interval with homogeneous boundary conditions of the third type. The resulting system of radial equations which contains the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite-element method. As a test desk, the program is applied to the calculation of the energy values and reaction matrix for an exactly solvable 2D-model of three identical particles on a line with pair zero-range potentials.
A FORTRAN 77 program is presented which calculates potential curves and matrix elements of radial coupling for two-electron systems using the hyperspherical coordinate method. The adiabatic and diabatic-by-sector close-coupling approaches are considered. The program calculates also the overlap matrices on borders of all sectors which are necessary for integration of close-coupling hyperradial equations within the sector-diabatic approach. It performs also the computation of the angular part of dipole amplitudes (in the length and acceleration forms) for dipole transitions between two given atomic states. The convergence and accuracy of the computational scheme elaborated are studied in details. Radial matrix elements computed by the HSTERM program can be used for the solution of the bound state and scattering problems for two-electron systems in both the adiabatic and diabatic-by-sector close-coupling approaches.
With the one-boson-exchange model, we study the interaction between the
S-wave $D^{(*)}/D^{(*)}_s$ meson and S-wave $B^{(*)}/B^{(*)}_s$ meson
considering the S-D mixing effect. Our calculation indicates that there may
exist the $B_c$-like molecular states. We estimate their masses and list the
possible decay modes of these $B_c$-like molecular states, which may be useful
to the future experimental search.
Using the one-boson-exchange model, we studied the possible existence of very loosely bound hidden-charm molecular baryons composed of an anti-charmed meson and a charmed baryon. Our numerical results indicate that the Σ cD* and Σ cD states exist, but that the Λ cD and Λ cD* molecular states do not. © 2012 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd.
In this work we discuss the atomic states in a vertical optical lattice in
proximity of a surface. We study the modifications to the ordinary
Wannier-Stark states in presence of a surface and we characterize the energy
shifts produced by the Casimir-Polder interaction between atom and mirror. In
this context, we introduce an effective model describing the finite size of the
atom in order to regularize the energy corrections. In addition, the
modifications to the energy levels due to a hypothetical non-Newtonian
gravitational potential as well as their experimental observability are
investigated.
We extend the one pion exchange model at quark level to include the short distance contributions coming from $\eta$, $\sigma$, $\rho$ and $\omega$ exchange. This formalism is applied to discuss the possible molecular states of $D\bar{D}^{*}/\bar{D}D^{*}$, $B\bar{B}^{*}/\bar{B}B^{*}$, $DD^{*}$, $BB^{*}$, the pseudoscalar-vector systems with $C=B=1$ and $C=-B=1$ respectively. The "$\delta$ function" term contribution and the S-D mixing effects have been taken into account. We find the conclusions reached after including the heavier mesons exchange are qualitatively the same as those in the one pion exchange model. The previous suggestion that $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ molecule should exist, is confirmed in the one boson exchange model, whereas $DD^{*}$ bound state should not exist. The $D\bar{D}^{*}/\bar{D}D^{*}$ system can accomodate a $1^{++}$ molecule close to the threshold, the mixing between the molecule and the conventional charmonium has to be considered to identify this state with X(3872). For the $BB^{*}$ system, the pseudoscalar-vector systems with $C=B=1$ and $C=-B=1$, near threshold molecular states may exist. These bound states should be rather narrow, isospin is violated and the I=0 component is dominant. Experimental search channels for these states are suggested.
Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.
In principle, the extension of density functional theory (DFT) to Coulombic systems in a nonvanishing magnetic field is via current DFT (CDFT). Though CDFT is long established formally, relatively little is known with respect to any generally applicable, reliable approximate E(XC) and A(XC) functionals analogous with the workhorse approximate functionals (local density approximation and generalized gradient approximation) of ordinary DFT. Progress can be aided by having benchmark studies on a solvable correlated system. At zero field, the best-known finite system for such purposes is Hooke's atom. Recently we extended the exact ground state solutions for this two-electron system to certain combinations of nonzero external magnetic fields and confinement strengths. From those exact solutions, as well as high-accuracy numerical results for other field and confinement combinations, we construct the correlated electron density and paramagnetic current density, the exact Kohn-Sham orbitals, and the exact DFT and CDFT exchange-correlation energies and potentials. We compare with results from several widely used approximate functionals, all of which exhibit major qualitative failures, whether in CDFT or in naive application of ordinary DFT. We also illustrate how the CDFT vorticity variable nu is a computationally difficult quantity which may not be appropriate in practice to describe the external B field effects on E(XC) and A(XC).
The finite-elements method is used to solve, to the given degree of accuracy, bound states of the H atom and the molecular ion H2+ of hydrogen in a strong magnetic field, 109 <H <1013 gauss, and stronger. In the case of the hydrogen atom, the results are in agreement (within a factor of ∼ 10-4) with results of the best variational calculations in the area of field changes, while in the case of the molecular hydrogen ion and for the 0<H <1011 gauss field they are far better than all calculations known to date. The opinion presented here is that the finite-elements method used in this paper—i.e., the method of direct solution of the particular problem on a plane—can also be successfully used in many other physics problems which are reduced to partial differential equations on a plane, and when perturbation theory or some other similar method of solution is not applicable.
A procedure for deriving high order difference formulas using the local solutions of the differential equation is described. The same procedure can be used to incorporate the boundary conditions in the derivation of the difference formulas for the boundary mesh points. A simple example of a Poisson equation over a rectangle is chosen to demonstrate the method, although the same procedure can be applied to equations with variable coefficients over arbitrary regions.
SLEIGN is a software package for the computation of eigenvalues and eigenfunction:s of regular and singular Sturm - Liouville boundary value problems, The package is a modification and extension of a code with the same name developed by Bailey, Gordon, and Shampinej which is described in ACM Z’OMS 4 (1978), 193-208. The modifications and extensions include (1) a restructuring of the FORTRAN program, (2) the coverage of problems with semidefi nite weight functions, and (3) the coverage of problems with indefinite weight functions.
The finite element method (FEM) is used for solving the Schrödinger equation in one dimension. Simple model potentials are selected to compare analytical and numerical results. Within FEM, polynomials up to eighth order are used. A much higher accuracy of the eigenvalues could be achieved, if the size of the elements was adjusted to the node structure of the solution.
A numerical method is presented for rapidly calculating the energy eigenvalues of one-dimensional Schro¨dinger equations. It is applicable to systems for which the potential is either analytic or has no pole of order greater than two. The method is based on a power-series expansion of the wave function at large distances. With the use of high-speed computing machines the large number of terms required in the power series can be computed easily. The method is illustrated by obtaining energy eigenvalues for a number of one-dimensional systems with potentials of the type V=kx2n/2n. It is also applicable to a variety of systems of physical interest. As an example, an exact energy eigenvalue for a rotating Morse oscillator has been calculated. This is compared with that obtained from Pekeris' approximate analytical solution.
Different formulations of the quantum-mechanical Hamiltonian of the triatomic hydrogen system in hyperspherical coordinates are investigated. Numerical solutions of the adiabatic hyperangular states have been computed for several hyperspherical radii using cubic spline finite elements. It is found that the fully symmetric hyperspherical coordinates of Johnson are most appropriate for a coupled channel calculation of reactive scattering.
Two new numerical methods, the log derivative and the renormalized Numerov, are developed and applied to the calculation of bound-state solutions of the one-dimensional Schroedinger equation. They are efficient and stable; no convergence difficulties are encountered with double minimum potentials. A useful interpolation formula for calculating eigenfunction of nongrid points is also derived. Results of example calculations are presented and discussed.
A two-dimensional, fully numerical approach to the four-component first-order Dirac equation using the finite element method is employed for diatomic systems. Using the Dirac—Fock approximation with only 2116 grid points we achieve for H2 an absolute accuracy of about 10−10 au for the ground-state total energy. For the many-electron systems Li2 and BH, we obtain a similar accuracy within the Dirac—Fock—Slater approximation which allows us to determine the relativistic contribution to the total as well as orbital energies very precisely.
A fully numerical Hartree-Fock approach is developed for diatomic molecules. The exchange potential is solved relaxing a local, Poisson-like equation. Improved Hartree-Fock limits are reported for LiH and BH.
The time-dependent Schroedinger equation is solved directly for an alkali atom subject to an arbitrarily strong electromagnetic field. Two methods are compared. A tridiagonal finite-difference method is used to solve Schroedinger's equation on a two-dimensional (2D) cylindrical coordinate lattice, while a finite-element method using odd-order {ital B} splines is used to solve Schroedinger's equation on a three-dimensional (3D) Cartesian coordinate lattice. Multiphoton ionization cross sections are extracted from 2D cylindrical calculations for hydrogen and lithium and then compared with previous perturbation theory results. Single-photon ionization probabilities are compared from 2D cylindrical and 3D Cartesian calculations for hydrogen.
This paper investigates several factors affecting the accuracy and efficiency of numerical determination of the bound state energy eigenvalues of the one dimensional Schrödinger equation. The efficiencies of the finite element method (FEM), the Numerov-Cooley method, and the finite difference method are compared. From this comparison, it is concluded that for potentials containing a single energy minimum, the Numerov-Cooley method is the most efficient, while for the most complex potentials the finite element method is superior due to its better numerical stability in the classically forbidden regions. The effects of various polynomial interpolation schemes on the calculated eigenvalues of potentials known only on a small number of points is examined. It is found that while higher order fits are superior to lower ones when the potential points are known accurately, they can introduce spurious information into the potential for inaccurately known points, and thus produce poor eigenvalues. Likewise, for accurately known potentials, a spline or Hermitian interpolation is better than a Lagrangian fit, but the Lagrangian functions are less susceptible to noise in a less well known case.
Accurate quantum mechanical three-dimensional reactive scattering calculations for the J = 0 partial wave of the H + H2 system for total energies up to 1.6 eV have been performed using symmetrized hyperspherical coordinates. Six resonances were found having collision lifetimes which, interestingly, increase with the amount of stretching excitation and decrease with that of bending excitation.
Using the weak field expansion, we have calculated the ground-state energy of the hydrogen atom in a magnetic field for values of the field up to about 10¹³ G. The perturbative expansion has been summed by an order-dependent mapping method. We compare our results with previous calculations. Our method allows us to obtain: first, more accurate values of the binding energy for a field up to 10¹° G, then good results in the transition region between the two sets of accurate calculations of the literature, and finally still reasonably accurate values up to 10¹° G.
We report on the solution of the Hartree-Fock equations for the ground state of the H2 molecule using the finite element method. Both the Hartree-Fock and the Poisson equations are solved with this method to an accuracy of 10−8 using only 26 × 11 grid points in two dimensions. A 41 × 16 grid gives a new Hartree-Fock benchmark to ten-figure accuracy.
A very simple perturbative numerical (PN) algorithm is developed for the solution of the radial Schroedinger equation, using first order perturbation theory along the lines previously developed by Gordon. This algorithm uses the same basic approximation (a step function approximation for the potential well) as that recently reported by Richl, Diestler, and Wagner. It shows, however, an O(h/sup 5/) rate of convergence in the step size h, as compared to the O(h/sup 4/) rate of convergence of the algorithm given in the above cited reference. A new feature of the PN approach to the solution of the Schroedinger equation, namely, the remarkable stability of the present PN algorithm against the round off errors is reported. A comparison with the Numerov method for eigenvalue problems proves the high efficiency of the present algorithm.
A code, SLEIGN, is presented for the computation of eigenvalues and eigenfunctions of Sturm--Liouville problems of the form (pPSI')' + (q + lambda r)PSI = 0 on (a,b), with A/sub 1/PSI(a) + A/sub 2/p(a)PSI'(a) = 0 and B/sub 1/PSI(b) + B/sub 2/p(b)PSI'(b) = 0. Infinite intervals and other kinds of singularities are handled completely automatically. The accuracy of the computed eigenvalue is realistically estimated. An initial guess for the eigenvalue is not necessary but is used effectively if available, and there is no possibility of accidentally computing the wrong eigenvalue. In addition to describing the basic algorithms employed in the code this report also explains, with examples, how the code is used. A short summary of its performance on a representative set of problems is also included. This code is a new version of an earlier one with the same name.
Energies of the bound states of the hydrogen atom in a uniform magnetic field are calculated for field strengths greater or equal to B0≈2.35×109 G. The convergent behavior of the quantum excesses (negative quantum defects) makes it possible to determine completely the bound-state spectrum for given values of the field strength B and the azimuthal quantum number m. For |m|=0,1, and 2 results are presented which determine to a uniform relative accuracy of at least 0.1% the energies of all bound states for arbitrary field strengths from B/B0=1 to B/B0=500 or higher, beyond which the adiabatic approximation is of comparable accuracy.
For a hydrogen atom in a uniform magnetic field in the range B≈2.35×108-2.35×109 G we have calculated the energies and widths of resonant states which can autoionize by decay of a Landau excitation. The autoionization widths are quite large with some lifetimes as short as 10-15 sec. The widths decrease with increasing azimuthal quantum number |m|, hydrogenic quantum number, or magnetic field strength. With decreasing field strength the lower resonances approach the ionization threshold which they cross with finite width and a pseudoresonance structure of finite width persists below threshold in the quantum defects of the bound states. Because of the finite widths, there are no approximate level crossings as these resonances cross the ionization threshold.
The energy values of the ground state and the low excited 2p(m=0) state of the hydrogen atom in a uniform magnetic field of arbitrary strength (up to 4.7×108 T) have been calculated by using B-spline basis sets. The results are in good agreement with other theoretical calculations. It appears that B splines as basis functions are quite powerful in treating the problem of the hydrogen atom in an arbitrary magnetic field, especially in the transition region between low and high strength. It is also possible to extend this method to analyze general two-dimensional problems.
The finite element and spectral methods are applied to two‐dimensional bound state problems. A comparison of the spectral method, which requires a global basis set expansion of the wave functions, and the finite element method, which requires no such such expansion, is presented. A procedure is given for formulating the finite element approach and for achieving fast and accurate results. The convergence of the finite element calculations is considered and shown to be well behaved. A discussion of the extension of the finite element method to higher dimensions is also included.
The renormalized Numerov method, which was recently developed and applied to the one‐dimensional bound state problem [B. R. Johnson, J. Chem. Phys. 67, 4086 (1977)], has been generalized to compute bound states of the coupled‐channel Schroedinger equation. Included in this presentation is a generalization of the concept of a wavefunction node and a method for detecting these nodes. By utilizing node count information it is possible to converge to any specific eigenvalue without the need of an initial close guess and also to calculate degenerate eigenvalues and determine their degree of degeneracy. A useful interpolation formula for calculating the eigenfunctions at nongrid points is also given. Results of example calculations are presented and discussed. One of the example problems is the single center expansion calculation of the 1sσg and 2sσg states of H+2.
The theory of reactive (rearrangement) scattering for three atoms in three physical dimensions using adiabatically adjusting, principal axes hyperspherical (APH) coordinates is given. The relationships of the APH coordinates to Delves and Jacobi coordinates are given, and the kinetic energy operator is shown to be relatively simple. Procedures for solving the equations via either an exact coupled channel (CC) method or an optimum centrifugal sudden (CSAPH) approximation are given as well as procedures for applying scattering boundary conditions. Surface functions of two angles are obtained using a finite element method with an optimized, nonuniform mesh, and the CC equations are solved using the efficient VIVAS method. Sample CC results are given for the H3 system. The present approach has the advantages that all arrangements are treated fully equivalently; it is a principal axis system, so that both axes and internal coordinates swing smoothly with the reactions; it is directly applicable to both symmetric and unsymmetric systems and mass combinations and all total angular momenta; it gives convenient mappings for visualization of potential energy surfaces and wave functions; only regular radial solutions are required; all coordinate matching is by simple projection; and the expensive parts of the calculation are energy independent, so that, once they are done, the scattering matrices can be rapidly generated at the large numbers of energies needed to map out reactive thresholds and resonances. Accurate reactive scattering calculations are now possible for many chemically interesting reactions that were previously intractable.
Distributed Gaussian bases (DGB) are defined and used to calculate eigenvalues for one and multidimensional potentials. Comparisons are made with calculations using other bases. The DGB is shown to be accurate, flexible, and efficient. In addition, the localized nature of the basis requires only very low order numerical quadrature for the evaluation of potential matrix elements.
Algorithms for the effective calculation of reactive scattering probabilities are developed and tested on the hydrogenic atom–diatom system described by the Siegbahn–Liu–Truhlar–Horowitz potential energy surface. A three‐dimensional finite element procedure is designed from a description in terms of hyperspherical coordinates. The Wigner–Eisenbud R‐matrix theory is used for a recursive procedure which admits control with limits on the hyperradial propagation inward from an asymptotic region and for a symmetry preserving transformation to arrangement channel Jacobi coordinates.
A method for the numerical solution of the one‐dimensional Schrödinger equation based on a matrix formulation of Numerov’s method is described. Some matrix theory leading to an efficient algorithm for the slicing of the spectrum of the related eigenproblem for a pair of tridiagonal matrices is developed. Finally the application of defect correction to the discretization gives a method of order of accuracy eight. Results of example calculations are presented.
An abstract is not available.
We study a particular utilization of the basis-spline collocation method (BSCM) for the lattice solution of boundary value problems. We demonstrate the implementation of a general set of boundary conditions. Among the selected problems are the Schrödinger equation in radial coordinates, the Poisson, and the generalized Helmholtz equations in radial and three-dimensional Cartesian coordinates.
Preliminary results are presented for transition probabilities in the H + H2 system derived from an adiabatic representation in terms of surface functions on hyperspheres. Special attention is given to the resonance structure for transition probabilities in the first vibrational level.
The advantages in formulation and use of semi-discretized approaches to the numerical solution of initial/boundary value problems are well known. The aim of this paper is to demonstrate that it is feasible to obtain accurate results even with a coarse spatial mesh. A method is developed which produces in a simple manner matrix representations for high-order central difference operators. Dirichlet, Neumann and mixed boundary conditions are considered, both homogeneous and non-homogeneous. It is shown in all cases that, for linear problems at least, there is no need to use a finer mesh than that dictated by the essential frequency content of the initial function data.
Fully numerical two-dimensional Hartree–Fock–Slater calculations are reported for the diatomic molecules B2, C2, N2, CO, O2, F2, and BF. The model is identical to Becke's but the numerical method is different. Fully numerical ground-state total and orbital energies are reported for the first time. The basis-set truncation error in the LCAO calculations of Dunlap, Connolly, and Sabin influence the third and fourth decimals (in a.u.) of ET and ϵi respectively. Benchmarks of improved accuracy are provided for dissociation energies, bond lengths, and electronic multipole moments for future assessment of basis-set errors.
A method to calculate isotope effects in diatomic molecules is developed. The energy levels of hydrogen isotopes and mesic molecular complexes which are used for calculation ofdd anddt mesic molecule resonant formation rates are found. The obtained values agree with the experimental data within 510–4 eV accuracy.
We are trying to investigate systematically the application of the finite element method (FEM) for solving the Schrdinger equation. The present paper is devoted to the calculation of vibrational transition probabilities for the collinear reactive system A + BC (i.e. H+H2 and their isotopes). The calculations are fully two-dimensional and the results are compared with earlier FEM calculations and conventional basis set expansion methods using the the R-matrix or S-matrix propagation.We made extensive analysis of FEM on the vector-computer Cyber 205 and developed a vector code for the efficient use in two dimensions, so that in the near future applications even in three dimensions will be possible.For the hydrogen exchange reactions we investigated the following isotope combinations: (a) H + H2, b) H + DH, D + HD and H + MuH (symmetric reaction), (c) D + HH, H + DD and Mu + DD (asymmetric reaction). We calculated the transition probabilities for up to five open vibrational channels and found excellent agreement with known exact values.
In this paper the current status of the variational method for the determination of the rotational-vibrational energy levels of polyatomic systems is reviewed. Special attention is made for the derivation of the kinetic energy operator in various coordinate systems, and several forms are given. Similarly, analytic forms which are in current use for the potentials are given. The calculation of the Hamiltonian matrix elements (expansion functions, numerical integration grid points and weights) is described in detail, and a description of our programs for this problem is given in section 6.
We are investigating systematically the use of the finite element method (FEM) for solving the Schrödinger equation. The present work is devoted to the calculation of vibrational transition probabilities for the collinear reactive system F + H2. The calculations are fully two-dimensional and the results are compared with the conventional basis set expansion methods using the R-matrix or S-matrix propagation. Extensive analysis of FEM on the vector computer Cyber 205 was made and a vector code for the efficient use in two dimensions was developed, so that in the near future applications even in three dimensions will be possible. The details of our FEM calculations are the following: The integration area was discretized into triangles where quadratic polynomials for the local wavefunction were defined. Convergent results can be reached with this simple ansatz with roughly 10000 grid points.
A new technique based on the Prüfer transformations is proposed. This technique is illustrated using several examples of Sturm-Liouville systems offering varying degrees of computational difficulty. For some of the examples, other techniques have been used to calculate eigenvalues, and in these cases, results obtained are compared. The technique appears capable of generalization to multiparameter systems of differential equations.
The collocation method for obtaining the bound solutions of the Schrödinger equation is investigated. The technique does not require the evaluation of integrals and is very simple to implement. It is closely connected with other pointwise representations used recently, but has the advantage of requiring less effort to construct the algebraic eigenvalue equations. The method is tested on two Morse oscillator problems and found to give results which are as accurate as the conventional variational approach. In conjunction with a distributed Gaussian basis the collocation method is shown to be capable of describing highly excited states.
The time-dependent Hartree-Fock method for atomic collisions is described, together with associated numerical techniques. Some applications, chiefly to ion-atom collisions, are presented.
The finite element method is applied to collinear reactive scattering problems. In this way no basis set expansion of the wave function is required and a direct solution of the two-dimensional partial differential equation is achieved. It is shown how to generally formulate this approach and achieve fast and accurate results. As a test calculation the method was applied to H + H2, yielding excellent agreement with close coupling results. Since no basis sets are used in the finite element calculation, no question of basis set convergence or closed channel behavior arises. Some discussion on applications to higher dimensions is also included.
We report here the first three-dimensional (3D) reactive scattering calculations using symmetrized hyper-spherical coordinates (SHC). They show that the 3D local hyper-spherical surface function basis set leads to a very efficient computational scheme which should permit accurate reactive scattering calculations to be performed for a significantly large number of systems than has heretofore been possible.
CSC Research Reports RT/89. Scientific computing in Finland. Ed. by Kari Kankaala & Risto Nieminen, 183 - 213
The finite-element method has been used to obtain numerical solutions to the Schrödinger equation for the ground state of the helium atom. In contrast to the globally defined trial functions of the standard variational approach, the finite-element algorithm employs locally defined interpolation functions to approximate the unknown wave function. The calculation reported herein used a three-dimensional grid containing nine nodal points along the radial coordinates of the two electrons and four nodal points along the direction corresponding to the cosine of the interelectronic angle. This produced an energy of -2.9032 a.u., which lies 0.017% above the Frankowski-Pekeris value. The values of , for n=-2,-1, 1, and 2, are closer to those of Frankowski and Pekeris than from all of the variational calculations with the exception of the calculation performed by Weiss, whose energy and values are comparable to those of the finite-element computation.
We present the finite-element method in its application to solving quantum-mechanical problems for diatomic molecules. Results for Hartree-Fock calculations of ${\mathrm{H}}_{2}$ and Hartree-Fock-Slater calculations for molecules like ${\mathrm{N}}_{2}$ and CO are presented. The accuracy achieved with fewer than 5000 grid points for the total energies of these systems is ${10}^{\mathrm{\ensuremath{-}}8}$ a.u., which is about two orders of magnitude better than the accuracy of any other available method.
The ground state of hydrogen in a magnetic field has been analyzed using the finite-element method. Accurate values for the binding energy have been obtained for fields up to 1 IG. Unlike other approaches, the computational effort required to obtain converged results is independent of the strength of the magnetic field. Values obtained for the lower bounds on the binding energy at very high fields are the most accurate to date.
We present a variational method, based on direct minimization of energy, for the calculation of eigenvalues and eigenfunctions of a hydrogen atom in a strong uniform magnetic field in the framework of the nonrelativistic theory (quadratic Zeeman effect). Using semiparabolic coordinates and a harmonic-oscillator basis, we show that it is possible to give rigorous error estimates for both eigenvalues and eigenfunctions by applying some results of Kato [Proc. Phys. Soc. Jpn. 4, 334 (1949)]. The method can be applied in this simple form only to the lowest level of given angular momentum and parity, but it is also possible to apply it to any excited state by using the standard Rayleigh-Ritz diagonalization method. However, due to the particular basis, the method is expected to be more effective, the weaker the field and the smaller the excitation energy, while the results of Kato we have employed lead to good estimates only when the level spacing is not too small. We present a numerical application to the mp=0+ ground state and the lowest mp=1- excited state, giving results that are among the most accurate in the literature for magnetic fields up to about 1010 G.
The improved adiabatic representation is used in calculations of elastic and isotopic-exchange cross sections for asymmetric collisions of pmu, dmu, and tmu with bare p, d, and t nuclei and with H, D, and T atoms. This formulation dissociates properly, correcting a well-known deficiency of the standard adiabatic method for muonic-atom collisions, and includes some effects at zeroth order that are normally considered nonadiabatic. The electronic screening is calculated directly and precisely within the improved adiabatic description; it is found to be about 30% smaller in magnitude than the previously used value at large internuclear distances and to deviate considerably from the asymptotic form at small distances. The reactance matrices, needed for calculations of molecular-target effects, are given in tables.
A finite-basis-set method is used to calculate relativistic and nonrelativistic binding energies of an electron in a static Coulomb field and in magnetic fields of arbitrary strength (0 < B less-than-or-equal-to 10(13) G). The basis set is composed of products of Slater- and Landau-type functions, and it contains the exact solutions at both the Coulomb limit (B = 0) and the Landau limit (Z = 0). Relativistic variational collapse is avoided and highly accurate results are obtained with the basis set. The relativistic corrections obtained for intense magnetic fields (B greater-than-or-similar-to 10(9) G) differ from the previous relativistic calculations based on the adiabatic approximations. It is found that the sign of the relativistic correction changes from negative to positive near B almost-equal-to 10(11) G for the ground state and near B almost-equal-to 10(10) G for the 2p3/2(mu = -3/2) excited state of hydrogen. The method is checked to be very accurate by means of the virial theorem, sum rules, and the relativistic low-B limit where comparison can be made with perturbation results. In the nonrelativistic limit of the Dirac equation, our results agree with other accurate nonrelativistic calculations available and with our own calculations based on the Schrodinger equation, which converge to more significant digits than previous calculations for the whole range of magnetic fields.
We discuss a nonperturbative treatment of lepton-pair production caused by the strong and sharply pulsed electromagnetic fields generated in peripheral relativistic heavy-ion collisions with an emphasis on the capture process into the atomic K shell. We calculate, in a field-theoretical framework, impact-parameter-dependent probabilities and cross sections for such processes by solving the time-dependent Dirac equation on a three-dimensional Cartesian lattice using the basis-spline collocation method. We give a full discussion of the stationary states used in computing S-matrix elements. Use of the axial gauge for the electromagnetic potentials produces an interaction easier to implement on the lattice than the Lorentz gauge. Preliminary calculations are given for muon-pair production with capture into the K shell in collisions of 197Au+197Au at collider energies per nucleon of 2 and 100 GeV.
Doubly excited states (DES) of heliumlike systems are studied within the coupled-channel hyperspherical adiabatic approach. The results of the multichannel calculations of the 1Se and 1Po DES of H- and He converging to the second (n=2) threshold are presented and compared with those found in literature.