Two-particle Operators Research Articles

The levels of the two-particle Schrödinger operator corresponding to a crystal film

For a two-particle Schrodinger operator considered in a cell and having a potential periodic in four variables, we establish the existence of levels (i.e., eigenvalues or resonances) in the neighborhood of singular points of the unperturbed Green’s function and derive an asymptotic formula for these levels. We prove an existence and uniqueness theorem for the solution of the corresponding Lippmann-Schwinger equation.

Spectra of Hamiltonians of molecule pseudorelativistic electrons in spaces of functions with permutational and point symmetry

We study the spectral properties of the Hamiltonian Hn of n pseudorelativistic electrons in the Coulomb field of k fixed nuclei in spaces of functions having arbitrary given types of permutational and point symmetry. For this operator, we establish the location of the essential spectrum, obtain two-sided estimates of the discrete spectrum counting function in terms of the counting functions of the discrete spectrum of some two-particle nonrelativistic operators, and find the leading term of the spectral asymptotics.

Perturbation Theory for the Two-Particle Schrodinger Operator on a One-Dimensional Lattice

We consider the two-particle Schrodinger operator H(k) on the one-dimensional lattice ℤ. The operator H(π) has infinitely many eigenvalues zm(π) = v(m), m ∈ ℤ+. If the potential v increases on ℤ+, then only the eigenvalue z0(π) is simple, and all the other eigenvalues are of multiplicity two. We prove that for each of the doubly degenerate eigenvalues zm(π), m ∈ ℕ, the operator H(π) splits into two nondegenerate eigenvalues z m − (k) and z m + (k) under small variations of k ∈ (π − δ, π). We show that z m − (k) 0, then z m ± (k) for m ≥ 2 also has this property.

Minimal parametrization of ann-electron state

The Hamiltonian $H$ for an $n$-electron system in a finite one-electron basis of dimension $m$ is characterized by $d=O({m}^{4})$ matrix elements. The eigenstates of this Hamiltonian---i.e., the full-CI states $\ensuremath{\Psi}$---depend, however, on the usually much larger set of $N=O({m}^{n})$ parameters. One can, nevertheless, write a full-CI state as $\ensuremath{\Psi}={e}^{S}\ensuremath{\Phi}$ with $\ensuremath{\Phi}$ a reference function and $S$ an operator familiar from traditional coupled cluster (TCC) theory. The ``exact'' operator $S$ can be expressed (though in an implicit and rather complicated way) in terms of $d$ parameters. An alternative ansatz $\ensuremath{\Psi}={e}^{T}\ensuremath{\Phi}$ with $T$ depending in a very simple way on $d$ parameters only (namely, with $T$ having the same structure in Fock space as $H$) has been studied by Nooijen and by Nakatsuji and been called coupled-cluster with generalized single and double excitations (CCGSD). Nooijen has conjectured that the full-CI equations can be fulfilled with this ansatz. This paper is devoted to a comprehensive analysis of the Nooijen conjecture (NC). Several features make this analysis difficult and even intriguing. (a) One deals with coupled nonlinear systems of equations, for which theorems concerning the existence of their solution are hardly available. (b) There are different possible interpretations of the NC, especially as far as the choice of the reference function $\ensuremath{\Phi}$ is concerned. (c) There are solutions of the CCGSD equations, for which some elements of $T$ becomes negative infinite, and ${e}^{T}$ becomes a projection operator. Such solutions are undesired but difficult to eliminate. We show by direct comparison of the exact wave operator with that of CCGSD theory, for a closed-shell state with $\ensuremath{\Phi}$ a single Slater determinant, using a perturbation expansion, that CCGSD cannot be exact. This required a reformulation of the CCGSD operator ${e}^{T}$ to an equivalent exponential form ${e}^{R}$, with $R$ similar to the $S$ of TCC theory, but with constraints on the cluster amplitudes, such that these depend on $d$ parameters only. The CCGSD ansatz simulates three-particle and higher excitations, in a perturbation expansion even with diagrams of the correct topology, but with incorrect energy denominators. Differences to the exact formulation arise to second order in perturbation theory for the wave function and to fourth order in the energy, but even the contributions which are not exact, are often acceptable approximations. Essential for the proof that the CCGSD ansatz is not potentially exact is that the set of one- and two-particle basis operators, into which $H$ can be expanded, does not span a Lie algebra. What matters is actually that the variations of the wave function describable within a CCGSD ansatz are different from those required to satisfy the two-particle contracted Schr\"odinger equations. The first attempt, due to Nakatsuji, to prove in this philosophy that the CCGSD ansatz is not potentially exact, was essentially correct, but some subtle yet important aspects were not sufficiently stressed. A refinement was given by Mazziotti. Papers by Ronen and Davidson addressed a somewhat academic problem---namely, the possible validity of the CCGSD ansatz for a completely arbitrary reference function---using mainly dimensionality arguments.

Maple procedures for the coupling of angular momenta. VIII. Spin-angular coefficients for single-shell configurations

Matrix elements of physical operators are required when the accurate theoretical determination of atomic energy levels, orbitals and radiative transition data need to be obtained for open-shell atoms and ions. The spin-angular part for these matrix elements is typically based on standard quantities such as matrix elements of the unit tensor, the (reduced) coefficients of fractional parentage as well as a number of other reduced matrix elements concerning various products of electron creation and annihilation operators. Therefore, in order to facilitate the access to the matrix elements of one- and two-particle scalar operators, we present here an extension to the Racah program for the full set of standard quantities and the pure spin-angular coefficients in LS- and jj-couplings. A flexible notation is introduced for defining and manipulating the electron creation and the electron annihilation operators. This will allow us to solve successfully various angular momentum problems in atomic physics. Program summary Title of program: Racah Catalogue number: ADUR Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUR Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions: None Computers for which the program is designed: All computers with a valid license of the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc.] Installations: University of Kassel (Germany) Operating systems under which the program has been tested: Linux 8.1+ Program language used: Maple, Release 8 and 9 Memory required to execute with typical data: 30 MB Number of lines in distributed program, including test data, etc.:36 875 Number of bytes in distributed program, including test data, etc.: 1 104 604 Distribution format: tar.gz Nature of the physical problem: The accurate computation of atomic properties and level structures requires a good understanding and implementation of the atomic shell model and, hence, a fast and reliable access to its standard quantities. Apart from various coefficients of fractional parentage and the reduced matrix elements of the unit tensors, these quantities include the so-called spin-angular coefficients, i.e. the spin-angular parts of the many-electron matrix elements of physical operators, taken in respect of a basis of symmetry-adapted subshell and configuration state functions. Method of solution: The concepts of quasispin and second quantized (creation and annihilation) operators in a spherical tensorial form are used to evaluate and calculate the spin-angular coefficients of one- and two-particle physical operators [G. Gaigalas, Lithuanian J. Phys. 39 (1999) 79, http://arXiv.org/physics/0405078; G. Gaigalas, Z. Rudzikas, C. Froese Fischer, J. Phys. B: At. Mol. Phys. 30 (1997) 3747]. Moreover, the same concepts are applied to support the computation of the coefficients of fractional grandparentage, i.e. the simultaneous de-coupling of two electrons from a single-shell configuration. All these coefficients are now implemented consistently within the framework of the Racah program [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; G. Gaigalas, S. Fritzsche, B. Fricke, Comput. Phys. Comm. 135 (2001) 219]. Restrictions on the complexity of the problem: In the present version of the Racah program, all spin-angular coefficients are restricted to the case of a single open shell. For the symmetry-adapted subshell states of such single-shell configurations, the spin-angular coefficients can be calculated for (tensorial coupled) one-particle operators of arbitrary rank as well as for scalar two-particle operators. As previously [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; G. Gaigalas, S. Fritzsche, B. Fricke, Comput. Phys. Comm. 135 (2001) 219], the Racah program supports all atomic shells with l ⩽ 3 in LS-coupling (i.e. s-, p-, d- and f-shells) and all subshells with j ⩽ 9 / 2 in jj-coupling, respectively. Unusual features of the program: From the very beginning, the Racah program has been designed as an interactive environment for the (symbolic) manipulation and computation of expressions from the theories of angular momentum and the atomic shell model. With the present extension of the program, we provide the user with a simple access to the coefficients of fractional grandparentage (CFGP) as well as to the spin-angular coefficients of one- and two-particle physical operators. To facilitate the specification of the tensorial form of the operators, a short but powerful notation has been introduced for the creation and annihilation operators as well as for the products of such operators as required for the development of many-body perturbation theory in a symmetry-adapted basis. All the coefficients and the matrix elements from above are equally supported for both LS- and jj-coupled operators and functions. The main procedures of the present extension are described below in Appendix B. In addition, a list of all available commands of the Racah program can be found in the file Racah-commands.ps which is distributed together with the code. Typical running time: The program replies promptly on most requests. Even large tabulations of standard quantities and pure spin-angular coefficients for one- and two-particle scalar operators in LS- and jj-coupling can be carried out in a few (tens of) seconds.

Nuclear structure corrections to gyromagnetic factor of the bound lepton

In the framework of the quasipotential method the new covariant expression for the two-particle vertex operator is obtained. The nuclear structure corrections of orders ( Z α ) 4 , ( Z α ) 4 m 1 / m 2 , ( Z α ) 4 m 1 2 / m 2 2 and ( Z α ) 5 to gyromagnetic factors of the bound electron and muon are calculated. Numerical value of the contribution of order ( Z α ) 5 is obtained by means of the dipole parameterization for the nuclear charge form factor in the range of the nuclear charges Z = 7 – 32 .

Some comments on the coupled cluster with generalized singles and doubles (CCGSD) ansatz

In this Letter, we present some comments on the nature of the CCGSD ansatz Ψ = e T Φ for an n-electron wave function, with T a linear combination of general one- and two-particle operators. Nooijen had conjectured that such a parameterization is possible for the exact eigenstate Ψ. We point out that the essential reason for the invalidity of this conjecture lies in the fact that the basis operators into which the Fock space Hamiltonian can be expanded, are not closed under commutation, i.e. do not span a Lie algebra. We give two proofs, based, respectively, on the variation principle and the method of moments, to show this. The variational proof traces the same ground as in an earlier one by Nakatsuji, which did not get the due recognition. We, however, formulate it in such a way that the key role of the algebra of the operators of T becomes transparent. We also discuss a related proof of Mazziotti. Our second proof sheds further light on why the ansatz, while not exact, might still provide an accurate description of a many-electron state, as has been found in some recent computational studies. We also discuss two recent attempts to disprove the exactness of the ansatz, based on a dimensionality argument.

Schrödinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice $\Z^3$ and interacting via zero-range attractive potentials is considered. For the two-particle energy operator $h(k),$ with $k\in \T^3=(-\pi,\pi]^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of $h(k)$ for $k\neq0$ is proven, provided that $h(0)$ has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schr\"{o}dinger operator $H(K), K\in \T^3$ being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number $N(0,z)$ of eigenvalues of H(0) lying below $z<0$ the following limit exists $$ \lim_{z\to 0-} \frac {N(0,z)}{\mid \log\mid z\mid\mid}=\cU_0 $$ with $\cU_0>0$. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum $K$ the finiteness of the number $ N(K,\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the essential spectrum is established and the asymptotics for the number $N(K,0)$ of eigenvalues lying below zero is given.

Approximate evaluation of the effect of three-body cluster operators in the valence-universal coupled-cluster excitation energy calculations for Be and Mg

The atomically oriented valence-universal coupled-cluster (VU-CC/R) method in the intermediate Hamiltonian formulation is used to calculate excitation energies for the beryllium and magnesium atoms. Having the basic VU-CC/R scheme which includes one- and two-particle cluster operators (VU-CCSD/R), the effect of approximate inclusion of the three-body terms in the cluster operator is investigated. The evaluation is based on the perturbative analysis of equations for the three-body part of the cluster operator at one- and two-valence levels. In this way, an explicit consideration of the three-body terms is avoided since they are expressed as functions of the two-body clusters. Our results show that the effect of the three-body clusters seems rather small for the systems under consideration which suggests that discrepancies between calculated and experimental values cannot be exclusively attributed to the truncated character of the cluster operator in the VU-CCSD/R calculation.

The Hamiltonians of Pseudorelativistic Atoms with Finite-Mass Nuclei: The Structure of the Discrete Spectrum

We study the structure of the discrete spectrum of pseudorelativistic Hamiltonians H for atoms and positive ions with finite-mass nuclei and with n electrons, where n ≥ 1 is arbitrary. The center-of-mass motion cannot be separated, and hence we study the spectrum of the restriction H P of H to the subspace of states with given value P of the total momentum of the system. For the operators H P we discover a) two-sided estimates for the counting function of the discrete spectrum σ d (H P ) of H P in terms of the counting functions of some effective two-particle operators; b) the leading term of the spectral asymptotics of σ d (H P ) near the lower bound inf σess(H P ) of the essential spectrum of H P . The structure of the discrete spectrum of such systems was known earlier only for n=1.

Crystal-field analysis of low-temperature UCl 3 and UBr 3 absorption spectra

High-resolution absorption spectra of thin films of UCl 3 and UBr 3 have been measured at 4.2 K in the 4000–30 000 cm −1 range. The determined crystal-field lines were used in a computational analysis employing free-ion operators, one-electron crystal-field operators as well as two-particle correlation crystal-field operators. The performed analysis enabled the determination of the Hamiltonian parameters and an unambiguous assignment of 57 and 46 crystal-field levels with a mean error of 31 cm −1 for UCl 3 and UBr 3, respectively.

Third-order isotope-shift constants for alkali-metal atoms and ions

We derive expressions for third-order matrix elements of one- and two-particle operators within the framework of relativistic many-body perturbation theory and use these formulas to evaluate specific-mass isotope shifts and isotope field shifts in atoms and ions with one valence electron. Third-order corrections to isotope-shift constants for $3s,$ $3p,$ and $3d$ levels in sodium and sodiumlike ions with $Z=12--18$ and $Z=26$ are given. Applications to $4s,$ $4p,$ $3d,$ and $4d$ levels in K and $4s,$ $4p,$ and $3d$ levels in ${\mathrm{Ca}}^{+}$ are also given. Comparisons are made with previous calculations and with experiment. Including third-order corrections improves the agreement between theoretical and experimental values of the specific-mass isotope-shift constants for the cases considered.

Program to calculate pure angular momentum coefficients in [formula omitted]-coupling

A program for computing pure angular momentum coefficients in relativistic atomic structure for any scalar one- and two-particle operator is presented. The program, written in Fortran 90 /95 and based on techniques of second quantization, irreducible tensorial operators, quasispin and the theory of angular momentum, is intended to replace existing angular coefficient modules from GRASP92. The new module uses a different decomposition of the coefficients as sums of products of pure angular momentum coefficients, which depend only on the tensor rank of the interaction but not on its details, with effective interaction strengths of specific interactions. This saves memory and reduces the computational cost of big calculations significantly.

On the diagnostic value of (Ŝ2) in Kohn-Sham density functional theory

Unrestricted density functional theory (UDFT) is, in addition to its application to high spin multiplet systems, an efficient method for describing singlet biradical states. In this connection, spin contamination of the UDFT description as measured by the expectation values of Ŝ2 is of interest, but is difficult to assess since the calculation of (Ŝ2) as the expectation value of a twoparticle operator is not possible within the single-determinant approach of DFT. A new way of determining (Ŝ2) from the UDFT (UHF or any other) spin magnetization density (SMD) ms (r) = pα(r) −pβ(r) is described and applied to the stretched H2 molecule and the singlet state of the para-didehydrobenzene biradical for a number of different functionals. Although (Ŝ2) as calculated from the Kohn-Sham determinant is found to differ significantly from the corresponding value obtained with the help of the SMD in the case of singlet biradicals, the former result is a diagnostic value and can be used to assess spin contamination of the UDFT description on a qualitative basis.

Construction of a perfectly N-representable two-electron density matrix carrying full information on an interacting system.

The two-particle density matrix contains all information, which is "readable" by one- and two-particle operators. The report discusses the building of a two-electron density matrix, which is perfectly N-representable by construction. The 2-matrix of a pure quantum state is obtained without the intermediate determination of the wave function. The procedure is illustrated by computations on atomic and small molecular systems.

Derivation in a nonequilibrium ensemble formalism of a far-reaching generalization of a quantum Boltzmann theory

Within the framework of the nonequilibrium statistical ensemble formalism provided by the nonequilibrium statistical operator method, we derive a quantum Boltzmann-style transport theory of a broad scope. This is done by choosing the single- and two-particle dynamical density operators as the basic informational–statistical variables. The equations of evolution for their average values over the nonequilibrium ensemble, the nonequilibrium-reduced Dirac–Landau–Bogoliubov-type density matrices, are obtained. From the resulting generalized nonlinear quantum transport theory, after resorting to perturbative-like expansions, a far-reaching generalization of Boltzmann equation for the single-particle distribution function is derived. A type of traditional Boltzmann equation follows after using stringent approximations, whose limits of validity are evaluated.

Mechanisms of electronic excitations by low-energy positrons: From finite to extended electronic systems

This work deals with the mechanisms of electronic transitions induced by low-energy positrons. For isolated atomic systems we discuss the role of successive binary collisions, the so-called Thomas mechanisms, between the positron, the excited electron and the atomic core. Furthermore, we consider transfer processes in which one of the charged particles is trapped in a low-lying continuum state of one of the reaction participants. In the second part of the paper, we investigate the various pathways for the secondary-electron emission from metallic surfaces following the bombardment by low-energy positrons. From a formal analysis it is shown that the probability rate for the production of an electron–positron pair is connected to the interacting electron–positron two-particle Green operator. The scattering of the interacting positron–electron subsystem from the crystal potential is treated and it is pointed out that a diffraction of the electron–positron pair as a whole might take place leading to characteristic diffraction pattern. The analysis is substantiated by numerical studies for the secondary electron emission from Cu(0 0 1) and Fe(1 1 0) (BCC) samples.

Coupled Tensorial Form of the Second-Order Effective Operator

General formulas for the spin-angular part of the second-order effective operator for open-shell atoms are derived for LS and jj coupling. To obtain the perturbation theory expansion for a particular problem only a specification of ranks of the atomic interactions involved in the consideration is required. The expressions for matrix elements of one-particle and two-particle operators are derived in second quantization representation when the N-electron wavefunction has the consecutive order of coupling for angular momenta of shells. The matrix element is expressed as a sum of the products of 6-j and/or 9-j symbols and matrix elements of operators acting in the space of states formed from a shell of equivalent electrons. The expression for the matrix element does not contain the summation either over the resultant angular momenta of shells or over the intermediate angular momenta of N-electron wavefunction.

Yang–Baxter equation-based potential model for a one-dimensional fermion system

We propose a velocity-dependent δ-function potential model for a one-dimensional fermion system, under which the two-particle interchange operators in the Bethe hypothesis satisfy the Yang–Baxter equation. This model, proportional to the product of the momenta of two interacting particles, is not translation invariant. It is found that the ground-state particle distribution densities become negative in the low momentum region. To resolve the controversy, we apply an overall momentum shift to the system, and obtain reasonable solutions. The dependence of the ground-state distributions on the momentum shift is discussed.

Exactly solvable fermionicN-band models

Motivated by the problem of N coupled Hubbard chains, we investigate a generalisation of the Schulz-Shastry model containing two species of one-dimensional fermions interacting via a gauge field that depends on the positions of all the particles of the other species. The exact many body ground state of the model can be easily obtained through a unitary transformation of the model. The correlation functions are Luttinger-like - i.e., they decay through power laws with non-integer exponents. Through the interaction dependent correlation functions of the two-particle operators, we identify the relevant perturbations and hence, possible instabilities.