Ingvar Lindgren

On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces

Abstract In this paper we study the following aspects of the open-shell coupled-cluster (CC) theories: (a) we examine the current theoretical status regarding the existence or non-existence of a linked-cluster theorem, ensuring the connectedness of the cluster amplitudes and the effective Hamiltonian; (b) we lay down the necessary and sufficient conditions for connectivity for a general (incomplete) model space, involving valence particles as well as valence holes; and (c) critically re-assess the earlier theoretical works from a comprehensive and unifying view point.

Diagonalisation of the Dirac Hamiltonian as a basis for a relativistic many-body procedure

A diagonalisation procedure of the Foldy-Wouthuysen type is considered for the single-electron Dirac Hamiltonian as a basis for many-body applications. A modified procedure is suggested. In the diagonalisation procedure the Dirac equation is completely decoupled into two equations of the Pauli type. The expressions for positive- and negative-energy projection operators in the transformed basis are then trivial and, by performing the inverse transformation, projection operators for Dirac functions are obtained. Applying a similar transformation to the two-electron Dirac-Coulomb Hamiltonian leads to a diagonalised single-electron part and a non-diagonal two-electron interaction. In principle, the effect of the exchange of virtual, transverse photons (Breit interactions) can also be included in the electron-electron interaction and transformed in a similar way. It is indicated how a diagonalisation procedure of this kind can be used as a basis for relativistic many-body calculations in the coupled cluster formulation in analogy with the corresponding non-relativistic procedure.

Numerical Many-Body Perturbation Calculations on Be-like Systems Using a Multi-Configurational Model Space

We report here numerical perturbation calculations on Be and C2+ starting from a model space consisting of the two strongly interacting configurations 1s22s2 and 1s22p2. We use numerically represented radial pair functions which are solutions of a system of coupled differential equations obtained by a finite difference method. By iterating the system of pair equations the most important correlation effects are included to all orders. This is demonstrated for C2+, where excitation energies for the 2p2 levels are obtained with an accuracy better than 1% or 0.2 eV. For Be only second-order results are reported. Here the iterative scheme does not converge, probably due to the presence of intruder states of the type 2sns 1S, which lie between the two 1S states originating from the model space. The second-order calculation with the two-configurational model space and orbitals generated in the 1s2 Hertree-Fock core yields 93.6% of the correlation energy, compared to 80.9% for a similar calculation using a model space with only the ground-state configuration 1s22s2 and orbitals generated in the Hartree-Fock potential of that configuration.

A Numerical Coupled-Cluster Procedure Applied to the Closed-Shell Atoms Be and Ne

A coupled-cluster procedure, applicable to closed-shell as well as open-shell atoms, is described, and results from calculations on the closed-shell atoms Be and Ne are presented. The procedure is based on numerical solution of coupled radial pair equations, which can be iterated to self-consistency. The coupling between different pair excitations is considered in two steps. The first step, called intra-shell coupling, includes the interaction between excitations differing in the final states as well as in the mlms values of the electrons being excited. In the second step, the inter-shell coupling, also the residual interaction between excitations involving different nl shells is considered. While the former coupling is quite important, it has been found that the latter contributes only about 1% to the correlation energy. Single and triple excitations are neglected in the present work, but quadrupole excitations are included by means of the exponential ansatz of the coupled-cluster procedure. According to our results for Be, 98.6% of the experimental correlation energy is obtained with complete pair-pair coupling, which is consistent with the estimated value of about 1.5% for the contribution of single and triple excitations. With only intra-shell coupling the result is in almost exact agreement with the experimental value due to cancellations between single/triple excitations and inter-shell couplings. Similar results are obtained for Ne. The results of the present work are compared and discussed in relation to earlier many-body calculations on these systems.

Many-body calculations of the hyperfine interaction of some excited states of alkali atoms, using approximate Brueckner or natural orbitals

The theory of maximum-overlap or Brueckner orbitals and of natural Orbitals is reviewed for closed-shell systems. A technique is described to calculate approximate Brueckner or natural orbitals for systems with a single valence electron, and this is applied together with the linked-diagram expansion of effective operators to evaluate the hyperfine interaction of some excited states of alkali atoms. It is found that it is in this way possible to reproduce reasonably well also the highly perturbed interactions of the excitedd states, which is not possible in a low-order expansion based on HF orbitals. Numerical results are given for the 5d state in K and the 4d state in Rb, where good experimental information is available.

Effective operators in the atomic hyperfine interaction

The many-body perturbation theory (MBPT) is reviewed and applied to the atomic hyperfine interaction. Graphical methods are introduced by without mathematical details. The results are interpreted in terms of effective operators. For systems with a single valence electron-such as the alkali atoms-this operator has the same form as the ordinary hyperfine operator and is identical to the operator commonly used in the analysis of experimental hyperfine data. The origin of different contributions to this operator is discussed. Numerical results are given for the 22S and 22P states of the lithium atom, where accurate MBPT calculations have recently been performed. For systems with several valence electrons additional parameters are needed or, alternatively, the parameters of the one-body effective operator are allowed to be term-dependent. Recent experiments and corresponding theoretical investigations on alkaline-earth elements, with two valence electrons, are reviewed and, in particular, MBPT calculations on the calcium atom are discussed.

Relativistic Effects in the Hyperfine Structure of the Alkali Atoms

The relativistic effects in the hyperfine structure of the alkali atoms in their ground 2S1/2 and excited 2S1/2 and 2P1/2,3/2 states are investigated by means of self-consistent-field calculations. Relativistic correction factors are obtained by comparing relativistic and non-relativistic results. Exact Hartree-Fock exchange as well as various statistical exchange approximations have been used. It is found that the correction factors are quite insensitive to the type of potential used in the calculation, while the values of the interaction parameters themselves differ appreciably between the different self-consistent-field schemes. The new correction factors are used to reanalyze the experimental data for the alkali atoms. Non-relativistic core polarization results have been corrected for relativistic effects and new quadrupole moments have been evaulated for the following nuclei: 7Li, 23Na, 39K, 87Rb and 134Cs.

Second order loop after loop self-energy correction for few-electron multicharged ions

Abstract One of the self-energy corrections of second order in α = e 2 h c is calculated for 1 s 1 2 , 2s 1 2 and 2p 1 2 states. The calculations involve the triple summation over the Dirac spectrum that was performed by the numerical method of the space discretization. This was achieved by the rearrangement of the summations in such a way that two of them were included in the corrections to the wave functions.

A new approach to the electron self energy calculation

We present a new practical way to calculate the first order self energy in any model potential (local or non-local). The main idea is to introduce a new straightforward way of renormalization to avoid the usual potential expansion implying a large number of diagrams in higher order QED effects. The renormalization procedure is based on defining the divergent mass term in coordinate space and decomposing it into a divergent sum over finite partial wave contributions. The unrenormalized bound self energy is equally decomposed into a partial wave (l) sum. For each partial wave the difference is taken and the sum becomes convergent. The comparably rapid asymptotic behaviour of the method is l−3. The method is applied to lithium-like uranium, and the self energy in a Coulomb field, the finite nucleus effect and the screened self energy is calculated to an accuracy of at least one tenth of an eV.

Hermitian formulation of the coupled-cluster approach

A Hermitian formulation of the coupled-cluster approach (CCA) is developed, based on the Jorgensen condition, P Omega Dagger Omega P=P, Omega being the wave operator and P the projection operator for the model space. This leads to a formalism where the exact as well as the model functions are orthonormal, and the effective Hamiltonian has the manifestly Hermitian form Heff=P Omega Dagger H Omega P. It is shown that the Jorgensen condition is compatible with the connectivity criteria (connected cluster operator and effective Hamiltonian) for a general, incomplete model space. Even with an effective Hamiltonian of this form, however, nonHermiticity may be introduced when the cluster expansion is truncated. This can be remedied by a reformulation of the coupled-cluster equations, where additional terms, which cancel in the complete expansion, preserve Hermiticity at each truncation. The new equations also lead to additional terms in the cluster operator itself, which make it possible, for instance, to include important effects in the pair approach that otherwise would require the evaluation of three- and four-body clusters.