Erkki Brändas

Dispersion forces, second- and third-order energies☆

Abstract The properly symmetrized second order energy, ϵ Q 2 , has been observed to approach one half of the correct dispersion energy ϵ 2 for large interatomic distances. We contend that the proper quantity to be examined is ϵ Q 2 λ Q . λ Q approaches ϵ Q 2 /(ϵ Q 2 -ϵ Q 3 ) ≈ 2 at R = 8 a o for H 2 and ϵ Q 2 /ϵ Q 3 ≈ 2. Hence E o -- ϵ Q 1 + ϵ Q 2 is a bad approximation.

Erratum: Variational Methods in the Wave Operator Formalism. A Unified Treatment for Bound and Quasibound Electronic and Molecular States

The wave operator formalism of Lowdin, heretofore used to describe states belonging to the discrete energy spectrum, has been extended to unify the treatment of bound and quasibound (or decaying) states. The approach makes use of an arbitrary reference function that may be chosen to approximate the physical state at short distances. Real and complex eigenvalues are obtained, respectively, for bound and quasibound states from an implicit equation, valid for all coupling strengths. Resonance positions and linewidths are explicitly independent of energy. Variational principles of the Lippmann—Schwinger type are presented which apply to states with either bound‐state or decay boundary conditions. Particular cases leading to minimization or maximization principles for real energies are discussed. The formalism is considered in connection with decaying electronic states of atoms and decaying molecular states.

Weyl's theory and the method of complex rotation

The spectral properties of a singular second order differential operator L is analysed in terms of Weyls theory. A spectral function is derived and the results are briefly compared with scattering theory. The analytic properties of L, its associated Greens function and related quantities of modern scattering theory are analysed. It is shown that the analytic continuation of an appropriately dilated operator L allows for an analytic extension of the singular case to a higher order Riemann sheet. The present theory is thus applicable to the study of the entire bound and continuous spectrum of L. A numerical realization of the theory is presented and applied to the scattering potential of Moiseyev et al. (1978, Molec. Phys., 36, 1613). New resonance structures are thereby found. Their asymptotic behaviour is discussed and an attempt to classify this type of structure is made. Implications of the present theory for predissociations in diatomic molecules, analysis of scattering experiments and ab initio stud...

On a theorem for complex symmetric matrices and its relevance in the study of decay phenomena

The theorem that every square matrix is similar to a symmetric matrix is discussed and its relevance in the quantum mechanical formulation of decay phenomena is emphasized. An alternative proof of the theorem is derived. It is shown that the matrix Cn(0) of order n with elements of the first superdiagonal being 1 and all the remaining ones zero, is similar to q with the elements qkl = exp(iπ(k + 1 - 2)/n)[δkl - 1/n]. The possible occurence of the latter in connection with off diagonal-longe-range-order (ODLRO) is indicated.

Off-diagonal long range order from repulsive electronic correlations in the ground state of a two-dimensional localised model of a high-Tc cuprate superconductor

Abstract It is shown that in a simple localised model of a two-dimensional cuprate superconductor, repulsive electronic correlations may produce off-diagonal long range order in the ground state wavefunction which is a widely accepted criterion for the occurrence of superconductivity.

Complex wavefunctions and the conservation of total momentum in the Hartree-Fock theory

Abstract The conservation of total momentum is treated as a symmetry property connected with the antilinear complex conjugation operator, which for a real Hamiltonian has the character of a constant of motion. It is shown that either the real or imaginary part of a complex trial wave function optimizes its expectation value. The assumption of real wave functions is therefore justified provided the complex components of a wave function belong to the original set. For a nonlinear variation procedure, such as for instance the unrestricted Hartree-Fock scheme, with the set consisting of single determinant functions, the real or imaginary projection will in general not be contained in the given set. A real determinant (apart from a trivial complex phase factor) is shown to be characterized by a real Fock-Dirac density matrix, whose eigenfunctions the so-called natural spin-orbitals can be chosen real. Finally some cases where one has obtained complex single determinant solutions are pointed out and discussed.

Creation of long range order in amorphous condensed systems

The second order density matrix Λ(2) of an N-body system, described by an AGP function, is considered. Certain theorems due to Yang, Sasaki and Coleman concerning the eigenvalue spectrum of Λ(2) and the possible appearance of coherent states, like e.g. BCS-states of superconductivity, are discussed in connection with Yangs concept of ODLRO (“off-diagonal long-range order“). The Complex Scaling Method (CSM) is introduced and some basic aspects of it are treated, like e.g. defining the appropriate scalar product, clarification of certain seemingly puzzling “inconsistencies” etc. Furthermore CSM is applied to the time evolution superoperator and the canonical density operator, and the connection with previous work on the mathematical foundation of Prigogines theory of subdynamics is discussed. It is explicitly shown how the theoretical concepts and the mathematical framework are combined to demonstrate the appearance of a novel kind of organized forms, “coherent-dissipative structures”, created spontaneously in condensed amorphous phases (like liquids at standard experimental conditions). The last chapter of this article is dealing with the application of the theoretical results into the physical context of molecular spectroscopic processes in condensed systems. Two experimental situations are considered in some detail: 1) far-infrared absorption in polar liquids, 2) photon-counting in steady-state luminescence. The theory allows for 1) the interpretation of the “anomalous” temperature dependence of far-infrared absorption bands in liquids, as well as 2) the appearance of coherent areas in luminescent solutions that are detected by the dynamically induced fluctuations (D-fluctuations) in the photoemission flux. The direct connection with experiments clearly demonstrates the physical content of the coherent-dissipative structures; it also illustrates the physical meaning of our novel theoretical approach to dynamical processes utilizing the application of the Complex Scaling Method in the framework of statistical mechanics.

On the Schrödinger equation in ellipsoidal coordinates

Quantum scattering by a potential separable in ellipsoidal coordinates is investigated. If the potential vanishes at infinity fast enough, its scattering data – S-matrix, far field amplitude, and total cross-section – are expanded in perturbed Lame wave functions.

Group theoretical identification of active localized orbital space in high Tc cuprate superconductors

Abstract There is a widely shared consensus that the pair condensate wavefunction in cuprate superconductors has a dominant dx2−y2 symmetry. In this paper a group theoretical analysis of the pair condensate wavefunction in real space for cuprate superconductors is presented. The analysis indicates that there is a degenerate pair of active bands in cuprate superconductors which are principally derived from localized e-orbitals and two possible singlet pair states formed from two different orbitals with symmetries a1, b1 or a2, b2 and all three may contribute to the condensate wavefunction. Hunds rule and the experimental observation that singlet pairing with a short coherence length occurs in cuprate superconductors, indicates that the first option is the most likely candidate in these materials and that hole doping mainly occurs from 1B1 pairs of e-orbitals giving rise to singlet electron pairing and the phenomenon of high Tc cuprate superconductivity.

Generalized Green’s functions and spectral densities in the complex energy plane

The Titchmarsh–Weyl theory is applied to the Schrodinger equation in the case when the asymptotic form of the solution is not known. It is assumed that the potential belongs to the Weyl’s limit‐point classification. A rigorous analytical continuation of the Green’s function, obtained from the solution regular at the origin and the square integrable Weyl’s solution (regular at infinity), to the ‘‘unphysical’’ Riemann energy sheet is carried out. It is demonstrated how the Green’s function can be uniquely constructed from the Titchmarsh–Weyl m‐function and its Nevanlinna representation. The behavior of the m‐function in the neighborhood of poles is investigated. The m‐function is decomposed in a, so called, generalized real part (Reg) and a generalized imaginary part (Img). Reg(m) is found to have a significant argument change upon pole passages. Img(m) is found to be a generalized spectral density. From the generalized spectral density, a spectral resolution of the differential operator and its resolvent i...