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We address the problem of electronic transmission through a finite sequence of identical potential barriers in one dimension, which has embedded in it “correlated impurities” that is, smaller identical clusters of n-mers, each described by n identical impurity potentials. Analytic formulae are derived for the transmission coefficient, and the investigation of Dunlap et al. [Phys. Rev. Lett. 65 (1990) 88], which were carried out for the corresponding dimer problem in the tight-binding model, are re-examined and generalised. The dependence of the condition for perfect transmission is obtained as a function of the strength of the impurity potential, treated here as a control parameter, in the case of a sequence of δ-function potentials.

We re-examine the conventional idea of determining the nature of the electronic eigenfunctions (extended, critical or localised) of a Fibonacci lattice from a study of the invariant associated with the trace map. We demonstrate that this is insufficient, and a more detailed study of the renormalisation group transformation itself is required to ascertain the nature of the eigenfunctions. Suitable examples are provided.

In this paper we present a study of the electronic properties of a one-dimensional Fibonacci chain with two hybridizing bands. Our study is motivated by recent experiments with quasicrystals in which transition metal atoms occupy positions of icosahedral symmetry. Using a recently proposed real space renormalization group scheme we make an exact analytical study of the two-band problem. We examine the effect of hybridization on the energy spectrum, the wave functions and the density of states of the Fibonacci chain. We find that the spectrum continues to remain a Cantor set even in the presence of hybridization. We conclude therefore this property of the spectrum is a purely structural effect. We present our results on the electronic density of states and show how hybridization produces additional structures in the energy spectrum.

The calculation of the static and dynamic structure factors of a dimerized Fibonacci chain are presented. This lattice has four kinds of bonds with an underlying quasiperiodicity, but does not preserve the scale invariance like the undistorted Fibonacci chain. This lattice is of interest from the point of view of possible Peierls' distortion in the presence of electron-phonon interactions. To calculate the static structure factor we use the ``cut and projection'' method, while a proposed numerical renormalization-group algorithm has been utilized to find the dynamic one. Our proposed algorithm is, in principle, exact, and can be implemented to any arbitrary degree of accuracy.

We present a detailed analysis of the nature of electronic eigenfunctions in one-dimensional quasiperiodic chains based on a clustering idea recently introduced by us [Sil et al., Phys. Rev. B 48, 4192 (1993)], within the framework of the real-space renormalization-group approach. It is shown that even in the absence of translational invariance, extended states arise in a class of such lattices if they possess a certain local correlation among the constituent atoms. We have applied these ideas to the quasiperiodic period-doubling chain, whose spectrum is found to exhibit a rich variety of behavior, including a crossover from critical to an extended regime, as a function of the Hamiltonian parameters. Contrary to prevailing ideas, the period-doubling lattice is shown to support an infinity of extended states, even though the polynomial invariant associated with the trace map is nonvanishing. Results are presented for different parameter regimes, yielding both periodic as well as nonperiodic eigenfunctions. We have also extended the present theory to a multiband model arising from a quasiperiodically arranged array of \ensuremath{\delta}-function potentials on the atomic sites. Finally, we present a multifractal analysis of these wave functions following the method of Godreche and Luck [J. Phys. A 23, 3769 (1990)] to confirm their extended character.

A new type of correlated disorder is shown to be responsible for the appearance of extended electronic states in one-dimensional aperiodic systems like the Thue-Morse lattice. Our analysis leads to an understanding of the underlying reason for the extended states in this system, for which only numerical evidence is available in the literature so far. The present work also sheds light on the restrictive conditions under which the extended states are supported by this lattice., Comment: 11 pages, LaTeX V2.09, 1 figure (available on request), to appear in Physical Review Letters

We discuss a very simple model of a 1-d disordered lattice, in which {\em all} the electronic eigenstates are extended. The nature of these states is examined from several viewpoints, and it is found that the eigenfunctions are not Bloch functions although they extend throughout the chain. Some typical wavefunctions are plotted. This problem originated in our earlier study of extended states in the quasiperiodic copper-mean lattice [ Sil, Karmakar, Moitra and Chakrabarti, Phys. Rev. B (1993) ]. In the present investigation extended states are found to arise from a different kind of correlation than that of the well-known dimer-type., Comment: 9 pages, 1 figure available on request, LaTex version 2.09, SINP-SSMP93-02

We present a real-space renormalization-group method for evaluating the exact dynamic structure factor S (q,ω) of a quasiperiodic Fibonacci chain. Contrary to earlier work that takes account only of the global aspects of the symmetry of the chain, our method additionally takes care of the local environmental aspects of the symmetry by separating the original lattice into a finite number of self-similar interpenetrating sublattices, followed by elimination of the coupling between them. Our method also yields correctly the positions of the Bragg peaks of the Fibonacci chain

We introduce here a new partitioning of the Hamiltonian in calculating pair-correlation energies using many-body perturbation theory, by which we are able to eliminate the off-diagonal particle-hole (p-h) ladders exactly to all orders in the perturbation expansion. In this formulation, the particle states turn out to be different for each distinct pair of hole states in the correlation energy calculation. We have also included the contributions of the diagonal particle-particle (p-p) and hole-hole ladders exactly to all orders. The effect of the off-diagonal p-p ladders has been estimated for each pair by computing the third-, foruth- and fifth-order energies. For highly symmetric systems the present partitioning yields in general symmetry-broken orbitals. Here one may use an average kind of partitioning for all the partners of the degenerate sets, which restores the symmetry and at the same time ensures cancellation of the p-h ladders exactly at the lowest order and approximately at the higher orders. Results are presented for a selection of 6π-electron conjugated systems. The correlation energy for each pair is in excellent agreement with that obtained from a partial CI calculation involving all double excitations from this pair. The advantages of implementing the present scheme in larger systems has been discussed.

Exact analytic formulae are obtained for the electronic structure of an ordered binary alloy in a Bethe lattice described by a tight-binding hamiltonian that includes nearest neighbour and second nearest neighbour interactions. Some interesting results are obtained and discussed in the text.

In this communication we discuss a method of incorporating corrections to the coupled Hartree–Fock (CHF) formalism by introducing the so-called “rearrangement effect.” In this we take account of the relaxation of the core orbitals when excitations from a starting Hartree–Fock wave function occur. The magnitude of this correction numerically is found to be quite significant for the polarizabilities of two-electron atomic systems, results for which are reported.

A non-perturbative theory is proposed in this article in which an energy independent effective Hamiltonian is obtained for open-shell systems. We have given a diagrammatic version of theory to facilitate the analysis of the problem. The theory has been applied to a model 4-π electron problem, for calculating the lowestπ-π* singlet and triplet energy levels of transbutadiene. Comparison with full Cl calculation indicates the excellent workability of the theory.

The cluster-Bethe-lattice method is used to study the environment effects on the electronic density of states in narrow-band alloys. The intra-atomic Coulomb interaction, U, is treated in the statistical (0, 1) approximation and the coherent potential approximation is used for the alloy disorder. These approximations lead us to consider the alloy system as a quaternary alloy. A general approach for an n-component alloy in the Bethe lattice is presented using Yonezawa's approach. It is shown that the result obtained gives the result of Yonezawa in the limit z ∞, z being the coordination number of the alloy system. The quaternary alloy is then treated as a special case of an n-component alloy. A cluster consisting of (Z + 1) atoms is chosen to study environment effects. A comparison is made between the electronic density of states for the Bethe lattice and that for the cluster-Bethe lattice. Interesting features in the electronic density of states are observed due to clustering effects. Die Clustermethode von Bethe wird benutzt, um Einflusse der Umgebung auf die Elektronenzu-standsdichte in schmalbandigen Legierungen zu untersuchen. Die intraatomare Coulombwechsel-wirkung U wird in der statistischen (0.l)-Naherung behandelt und die koharente Potential-Approximation wird fur die Legierungsunordnung benutzt. Diese Naherungen fuhren dam, das Legierungssystem als quaternare Legierung zu betrachten. Ein allgemeines Verfahren fur eine n-komponentige Legierung im Bethe-Gitter wird mit der Methode von Yonezawa angegeben. Es wird gezeigt, das das erhaltene Ergebnis das Resultat von Yonezawa im Grenzfall z ∞ wieder-gibt, wobei z die Koordinationszahl des Legierungssystems ist. Die quaternare Legierung wird dann als Spezialfall einer n-komponentigen Legierung behandelt. Ein Cluster aus (Z + 1)-Atomen wird gewahlt, um die Umgebungseffekte zu untersuchen. Ein Vergleich zwischen der Elektro-nenzustandsdichte fur das Bethe-Gitter und das Cluster-Bethe-Gitter wird durchgefuhrt. Interes-sante Einzelheiten in der Elektronenzustandsdichte infolge der Clustereffekte werden beobachtet.

In this paper we present the results of a generalized cluster\char22{}Bethe-lattice approach for the electronic density of states in binary alloys with cellular disorder, off-diagonal disorder, and short-range order. For application to large clusters, we suggest a new method for handling the problem of configuration averaging, in which we partition the cluster into a central "core" part and a surrounding layer. The "core" part is treated exactly, while the surrounding layer is described by a self-consistent complex self-energy $\ensuremath{\Sigma}(E)$, in determining which we take account of the fact that the atoms inside the cluster form a regular array. $\ensuremath{\Sigma}(E)$ therefore is not the same as the Bethe-lattice self-energy $\ensuremath{\sigma}(E)$, for which we take the coherent-potential-approximation value in the present application. In this multistage approach, we essentially adopt the philosophy of treating the different regions around a given site at different levels of approximation depending upon the importance of the region concerned for the local density of states at the central site. The feasibility of the present scheme is demonstrated through calculations with sets of model parameters. Of particular interest is the effect of off-diagonal disorder on the local density of states.

An exact real-space renormalisation group (RSRG) approach for the local Green function (LGF) for a quasi-periodic Fibonacci chain has been developed. The method has been used to study the local density of states (LDOS) for phonons, both in the one-site and transfer models. The scale factor in this work is tau 3, tau being the golden ratio. Apart from mapping a Fibonacci chain onto itself, the transformation accounts for the additional constraining symmetry, namely, that the local environment around the site at which the LGF is being found should remain invariant. The Cantor set-like behaviour of the energy spectrum results naturally from the alternative formulation.

Coupled Hartree-Fock theory is used here to calculate the frequency-dependent dipole and quadrupole polarisabilities of Be, B+ and C2+ in their ground states 1S. Particular emphasis is given to obtaining the excitation energies and the transition oscillator strengths for some low-lying excited states. The oscillator strengths are in very good agreement with those of more elaborate configuration-interaction calculations.

The authors' recently developed (1975) non-perturbative open-shell theory is adapted for direct evaluation of ionisation potential and excitation energies. Separating the 'core' contribution from the 'valence' part through the use of a multiple-cluster-expansion operator, the method provides a systematic way of including various core-valence interaction and core relaxation components. HF orbitals have been used as the basis for facilitating detailed comparison of the method with perturbative and propagator techniques. It has been shown that the method encompasses all the important second-order contributions of the latter formalisms and certain other classes of diagrams in a compact manner. An application to a simple 4 pi electronic problem is discussed to illustrate how this method works for real systems.

The authors propose a generalization of the 'cluster-Bethe lattice' approach of Falicov and Yndurain for the electronic density of states in binary alloys. In the present formulation it is possible to treat the Bethe lattice in any desired single-site approximation. Further, they obtain algebraic forms for the transfer matrices corresponding to the two types of atoms in the cluster, for which only interpolation formulae were previously given by Falicov and Yndurain (see Phys. Rev. B, vol.12, p.5664 (1975)).

The Anderson hamiltonian has been studied for arbitrarily filled bands using the real space renormalization group (RSRG) method. The ground state energy is determined in one dimension for the scale factors ns = 3 and ns = 5. Direct application of RSRG gives, for a certain ns, the results for only ns number of discrete cases of band-filling. We then devise an interpolation scheme which spans the entire region of the band and gives the ground state energy for any general filling. The results, as provided by our scheme, satisfy the desired particle-hole symmetry exactly and are always an upper bound to the exact one. The non-linear dependence of the band energy on its filling is represented correctly, even for small ns. Finally, we discuss how the results improve for large ns.

In this paper we present a non-perturbative approach to the calculation of correlation energies of open-shell systems. The formulation utilizes an Ursell-type expansion about a multi-determinant starting wavefunction. We have proved a theorem which enables us to derive an effective hamiltonian for the system consisting entirely of linked terms. In the symmetry-degenerate case this effective hamiltonian acts within the subspace of a set of symmetry-degenerate functions, and generates the energy eigenvalues of the system. The present theory has been cast in a diagrammatic language which facilitates the analysis of the correlation problem. The workability of the theory has been tested on a 4 π electron problem, transbutadiene, for which we have calculated the lowest π-π* singlet and triplet energies. The agreement between the results of the present theory and that found from a full CI calculation is excellent. The desirable feature of the theory is that the effective hamiltonian is energy-independent. We hav...

In this paper we explore the feasibility of widening the scope of the non-perturbative open-shell many-body formalism recently developed by us [1], which utilizes an Ursell type of cluster expansion about certain starting wavefunctions spanning a model space. We show that, by generalizing the definition of the cluster expansion operator, we can incorporate into the model space (a) determinants differing widely in energy and (b) determinants differing in their number of electrons. This flexibility is useful for the calculation of difference energies of interest, like transition energies and ionization potentials of atomic and molecular systems. The generalized scheme has been tested on the 4π-electron problem trans-butadiene for which, by choosing a very general model space, we have calculated the energies of the ground, the lowest π-π* singlet and triplet and the first ionization potential by choosing a single composite cluster expansion operator for all states. Results for some more restricted choice of ...

The authors describe an analytical approach for studying the electronic density of states of an infinite one-dimensional Fibonacci chain using the on-site model. They have identified uniquely a set of short-range order parameters appropriate to the Fibonacci chain and used them in the theory based on the cluster-Bethe-lattice method which has been described elsewhere. They have found that both the local density of states and the average density of states exhibit a four-sub-band global structure and each sub-band further trifurcates following a hierarchy of splitting from one to three sub-bands until the gaps between them disappear. The positions of the various sub-bands and their widths are in good agreement with the numerical results of Liu and Riklund (1987).

We present here a new treatment of short-range order in a disordered binary alloy with Bethe lattice topology based on the single-site approximation. We show that our method yields the limits of segregation and binary sequence exactly, while it reduces to the CPA in the case of random order.

In this paper a fully coupled variation-perturbation theory has been introduced for handling open-shell atomic systems in the presence of static perturbations. The static dipolar and quadrupolar polarizabilities and shielding factors for Li, Be+, B2+, B, C+, Na and Al have been calculated and compared with available results. Important sources of discrepancies in previous calculations of shielding factors have been noticed and critically examined. The dipolar shielding factors calculated in this paper show satisfactory agreement with the theoretical N/Z ratio. In the absence of any theoretical test for the quadrupolar shielding factor, the accuracy of the calculations of γ∞, should be judged in the light of the accuracy of the corresponding β∞ values.

A variational formulation of the Hartree–Fock–Roothaan (HFR) theory for open shell ions in presence of time dependent perturbations is presented. The theory has been used to calculate the dynamic polarizabilities of the first three ions of the Li sequence. The polarizability values, extrapolated to zero frequency, show good shell by shell agreement with the corresponding static results. The polarizability graphs display resonance behaviour at the transition frequencies of the ions, and a study of these points leads to analytic HF wave functions for their low lying excited states. The calculated transition frequencies are in excellent accord with the experimental values. The calculated oscillator strengths for the 2s → np transitions are in reasonable agreement with the extensive multiconfiguration calculations of Weiss and the available experimental results.

The quadrupolar and octupolar distortion of the ions in the He-sequence caused by an external electro–magnetic field has been studied by a variation–perturbation method in the Hartree–Fock scheme. For certain frequencies singularities appear in the response of the system to the perturbation. Approximate representations for the excited d and f states have been obtained from a study of these resonances. Such a perturbation calculation has the advantage that representations of the different excited states are obtained independently. The orthogonality to all the lower lying levels of the same symmetry is not required. The only source of inaccuracy implicit in the procedure lies in the improper consideration of the inter-electronic interaction. This is corrected for by an independent calculation, which is again formulated in terms of a perturbation treatment. The resulting wave functions for the excited states are accurate in the Hartree–Fock model. Expectation values of several operators have been calculated with these corrected wave functions.

A new theory for incorporating short range order (SRO) in the cluster Bethe lattice method (CBLM) for disordered binary alloys is proposed. It is applied to determine the condition for the appearance of moments in the disordered phase of a narrow-band metal for general values of concentration and SRO parameter. The final result (i) yields all the known limiting conditions exactly and (ii) displays in a transparant form the SRO dependence, indicating the merit of the present approach.

In this note, the authors’ recently developed non-perturbative open-shell theory is adapted for direct calculation o f ionisation potential and excitation energy of m any-electron systems. The H -F ground state is used as the “vacuum ” or “ core” in order to achieve a transparent separation o f the ground state energy. An application to a simple 4 π-electron system is discussed as an illustration o f the workability of the theory.