State Perturbation Theory Research Articles

Ab initio calculation of E1 transitions in the oxygen isoelectronic sequence

The electric dipole (E1) transitions between the levels of the 1s22s22p4, 1s22s2p5 and 1s22p6 Configurations have been considered in the oxygen isoelectronic sequence. The second-order stationary perturbation theory of the effective Hamiltonian has been adopted. Relativistic corrections were included in the Breit-Pauli approximation. E1 transition energies, line strengths, oscillator strengths, probabilities and lifetimes obtained in various approximations are compared with experimental values and with results of other theoretical calculations for Z = 10 – 26.

Octet-decuplet baryon mass splittings from self-consistent one-loop perturbation theory

The bag model of confined relativistic quarks in chiral-invariant interaction with scalar, pseudoscalar, vector, and pseudovector mesons, as well as gluons, is used to calculate the masses and wave functions of the spin-1/2 baryon octet and spin-3/2 decuplet, using selfconsistent Brillouin-Wigner bound state perturbation theory. Chiral symmetry breaking is invoked with the sigma model. SU (6) and SU (3) symmetries are broken by the experimental meson spectrum, and a strange quark mass. Mass corrections are calculated to one loop order, limited to the baryons of the octet and decuplet and the lowest lying mesons. Encouraging results are obtained, especially for theΔ — N and theΣ — Λ splittings. Convergence and stability have not been demonstrated, but are evidently improved by the self-consistency requirement. An initial parameter tuning gives a fit to all the octet and decuplet masses within ≦0.02 GeV, at the price of choosing the bag radius, the non-strange baryon input bag mass, and the strange quark mass. Even these small discrepancies can be dramatically reduced by fine-tuning the vector meson coupling and including an instanton contribution peculiar to theΛ.

Second-order MBPT results for the oxygen isoelectronic sequence

The energy spectra as well as the electric quadrupole (E2) and the magnetic dipole (M1) transitions in the oxygen isoelectronic sequence between the levels of the 1s22s22p4, 1s22s2p5 and 1s22p6 configurations were considered. The stationary second-order many-body perturbation theory (MBPT) was used to account for electron correlations. The relativistic corrections were included in the Breit-Pauli approximation.

Perturbation results for nearly uncoupled Markov chains with applications to iterative methods

The standard perturbation theory for linear equations states that nearly uncoupled Markov chains (NUMCs) are very sensitive to small changes in the elements. Indeed, some algorithms, such as standard Gaussian elimination, will obtain poor results for such problems. A structured perturbation theory is given that shows that NUMCs usually lead to well conditioned problems. It is shown that with appropriate stopping, criteria, iterative aggregation/disaggregation algorithms will achieve these structured error bounds. A variant of Gaussian elimination due to Grassman, Taksar and Heyman was recently shown by O'Cinneide to achieve such bounds.

Dimensional perturbation theory for excited states of two-electron atoms.

Large-order dimensional perturbation theory, which yields high accuracy for ground-state energies, is applied here to excited states of the two-electron atom. Expansion coefficients are computed recursively using the moment method, which we formulate in terms of normal coordinates. We consider the first two excited S states of helium, corresponding, at the large-dimension limit, to one quantum in either the antisymmetric-stretch normal mode or the symmetric-stretch normal mode. Comparison with the hydrogenic limit has identified these states as 1s2s 3 S and 1s2s 1 S, respectively

Stationary perturbation theory

After a short recapitulation of the basic concepts of stationary perturbation theory, this is applied to a many-electron Hamiltonian, with or without an external field, given in a Fock space formulation in terms of a finite basis, the exact eigenfunctions of which are the full-CI wave functions. The Lie algebra L c (n) of the variational group corresponding to this problem is presented. It has an important subalgebra L c (1) of one-particle transformations. Hartree-Fock and coupled Hartree-Fock (also uncoupled Hartree-Fock) as well as MC-SCF and coupled MC-SCF are outlined in this framework

A new implementation of stationary perturbation theory for separable quantum-mechanical problems

We improve a recently developed implementation of perturbation theory for separable problems. On choosing a new ansatz for the perturbed wave function that takes into account the singularities of the eigenvalue equation explicitly, the resulting perturbation equations become simpler. As an illustrative example, we consider the Stark shift of the microwave spectra of a polarizable symmetric-top molecule.

A simple effective Hamiltonian for low-frequency linear responses

It is shown that low-frequency linear responses may be obtained formally from second-order stationary state perturbation theory through the use of effective potentials. The effective potentials are related to time averages of Heisenberg operators.

Average Hamiltonian Theory and Stationary Perturbation Theory Compared

Average Hamiltonian Theory and Stationary Perturbation Theory Compared

E2 and M1 Transition Probabilities in Ions of the Nitrogen Isoelectronic Sequence Calculated using MBPT

Forbidden electric quadrupole (E2) and magnetic dipole (M1) transitions are of extreme importance in astrophysics. Up to now the most extensive calculations for the nitrogen isoelectronic sequence have been done using the method proposed by C.J. Zeippen [1] or in MCHF approximation [2]. To account for electron correlations both these methods use a large list of configurations. We have chosen the stationary many-body perturbation theory (MBPT) [3] for the inclusion of the electron correlations. The calculations have been perfomed in the second order in the complete model space 1s22s2p3+ 1s22p5. Relativistic corrections have been accounted for in the Breit-Pauli approximation. In the Table we present probabilities for electric quadrupole W(E2) and magnetic dipole W(M1) transitions (in s−1), wavelengths λ (in A). The comparision of the results shows that our second order calculation data in the most cases are closer to term-energy corrected ones from [1].

Alternative approach to stationary perturbation theory for separable problems

A new implementation of stationary perturbation theory for separable quantum-mechanical problems is presented which allows the calculation of perturbation corrections for all the states simultaneously, so that the results are expressed in terms of the quantum numbers of the unperturbed system. The anharmonic oscillator, the Stark effect in a rigid rotor, and a caged harmonic oscillator are considered as illustrative examples.

The quantum theory of an ideal superlattice responding to far-infrared laser radiation

Minibands of quasienergies can be defined for a superlattice interacting with far-infrared laser radiation. It is demonstrated that the width of these quasienergy minibands depends not only on the parameters of the superlattice, but also on strength and frequency of the driving laser field. In particular, the width approaches zero at certain values of these parameters. A strong coupling ansatz, combined with perturbation theory for degenerate quasienergy states, leads to a detailed understanding of these results.

Moment-method perturbation theory for the hydrogen atom in parallel electric and magnetic fields and in inhomogeneous electric fields

The problems posed by the hydrogen atom in parallel electric and magnetic fields and in inhomogeneous electric fields are treated simultaneously by means of perturbation theory. The application of this approach is facilitated by the transformation of the Schrodinger equation into a recurrence relation for the moments of the wavefunction which does not appear explicitly in the calculation. Two infinite sets of states are considered as illustrative examples, one of them can be treated as nondegenerate, and the other requires perturbation theory for degenerate states. Closed-form expressions for the perturbation corrections to the energy are obtained in terms of the hydrogenic principal quantum number. The present calculation extends and generalizes previously published results. © 1992 John Wiley & Sons, Inc.

Stationary perturbation theory

Stationarity of the energy expectation value as required in the variational approach can in most cases of practical interest be formulated with reference to a unitary ‘variational group’ and its associated Lie algebra. In terms of a basis of this Lie algebra, Brillouin (hypervirial) conditions and a Hessean are defined. The formulation of stationary perturbation theory (as well as of multiple perturbation theory) in the Lie-algebraic framework is straightforward. If the operatorY, which describes the perturbation of the wave function, is expandable in the Lie algebra of the variational group, a Hellmann-Feynman theorem holds (as a special case of Wigner's [2n+1] rule) and the first order operatorY1 can be obtained from minimization of a Lie-algebraic Hylleraas functional. Under the same condition for two perturbations, Dalgarno's exchange theorem holds. An analysis of the spectrum the Hessean leads to a generalization of the RPA method for any chosen variational group. Any variational group automatically generates a model excitation spectrum. Sometimes one wants to formulate the variational approach in terms of two or more (independent and noncommuting) variational groups. An example is coupled-MC-SCF theory. One must specify the order in which operators of the two groups act, but otherwise there is not much change with respect to the case of a single group. Time-dependent stationary perturbation theory, based on Frenkel's stationarity principle, is possible on similar lines. Singularities related to an indefinite phase, which plague traditional time dependent perturbation theory are automatically avoided. In the framework of stationary perturbation theory the dipole length and dipole velocity formulas for a transition element are equivalent. For a time-dependent Hamiltonian there is no unique definition of the corresponding energy. There are various possibilities to define a ‘pseudo-energy’. One of these definitions is consistent with a special form of a time-dependent Hellmann-Feynman theorem. For a perturbation periodic in time so-called Floquet states exist and stationary time-dependent perturbation theory starting from a stationary state of the unperturbed problem automatically leads to these. For Floquet states a genuine stationarity condition can be derived, that is based on the concept of a generalized Hilbert space and that does not suffer from the shortcomings of Frenkel's principle. The perturbation formalism for these is surprisingly close to that of the time-independent theory. For degenerate and near-degenerate states a quasidegenerate generalization of stationary perturbation theory is possible.

Moller-Plesset perturbation theory with two-configurational reference wavefunction

The recently proposed consistent generalization of the Moller-Plesset (MP) perturbation theory for arbitrary open-shell and multiconfigurational reference states is formulated and described in detail for a simple two-configurational (TC) SCF wavefunction. The problem of non-diagonality ofH 0 is discussed. Dynamical correlation contributions to the energy, dissociation energy, and dipole moment of HF are analyzed in terms of different types of excitations.

A unified treatment of the non-relativistic and relativistic hydrogen atom: III. The reduced Green functions

The last in a series of three papers in which it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. The work presented here is concerned with the reduced Green functions which arise in second-order stationary state perturbation theory. Using a simple linear transformation of the four radial parts of the relativistic reduced Green function it is shown how the non-relativistic and relativistic functions are special instances of the solution of a general second-order differential equation. The general solution of this equation is exhibited in the form of a Sturmian expansion, and complete solutions in both cases are presented. Recursion relations are deduced for the radial parts of both reduced Green functions and their matrix elements are examined in detail. As a test of the given functions the second-order effect of a perturbation of the nuclear charge is calculated and is shown to agree exactly with the value expected from a simple Taylor expansion of the hydrogenic energy formula.

An Extended INDO-CI Study on the Interaction of Formamide with External Electric Field

The interaction of formamide with the external electric field produced by a unit point-charge moving in one direction at a constant velocity is calculated according to our extended INDO-CI method for valence-shell electrons and the perturbation theory for degenerate states. Also, the change in the net-charges on atoms in formamide caused by the interaction is examined, together with the change in the internal electric field produced at the position of the point-charge by the net-charge distribution in formamide. It is thus shown that the interaction between formamide and the point-charge exerts a considerable influence not only on the net-charge distribution in formamide but also on the internal electric field acting on the point-charge.

Ab initio calculation of molecular properties

A review of fundamental problems concerning the ab initio calculation of molecular properties is given. The perturbing operators relevant to many important properties are unbounded, but the application of perturbation theory can be justified, nevertheless. Both first and second-order properties are often better obtained as numerical derivatives of the energy than as expectation values. Some elegant theorems of perturbation theory that hold if the unperturbed problem is solved exactly become invalid for approximate wave functions, but can be ‘saved’ in the framework of ‘stationary perturbation theory’ of which the random phase approximation (RPA) is a special case. The second part of the paper is devoted to magnetic properties of molecules, in particular NMR chemical shifts. The gauge problem and its solution by means of the IGLO method is studied in detail. A formal comparison of the individual gauge for localized orbitals (IGLO) and localized orbital, localized origin (LORG) methods is given. Some selected results on 13C and 31P shifts are presented that demonstrate the practical usefulness of IGLO calculations, e.g. for the full shift tensors.

A Lorentz Invariant Schrödinger Equation for Spin 0

Abstract Using the concept of distributions, the square root of the operator - Δ + m2 is taken in a mathematically well defined way for one component wave functions. A new representation of proper Lorentz transformations for one component wave functions makes it possible to construct a relativistic quantum mechanics for spin 0, comprising a Lorentz invariant wave equation, a scalar product, and a positive definite density satisfying, together with a current, a continuity equation, and coupling of scalar and vector potentials. Some interesting consequences of the theory concerning the concept of particle trajectory and velocity of propagation of the probability amplitude are discussed in detail. As an example of practical application a perturbation theory for discrete states is set up.

Nonequilibrium lattice models: Series analysis of steady states

A perturbation theory for steady states of interacting particle systems is developed and applied to several lattice models with nonequilibrium critical points near an absorbing state. The expansion is expressed directly in terms of the kinetic parameter (creation rate), rather than in powers of the interaction. An algorithm for generating series expansions for local properties is described. Order parameter series (16 terms) and precise estimates of critical properties are presented for the one-dimensional contact process and several related models.