Commutator Expansion Research Articles

Commutator expansion. II. Relativistic reduced Green’s functions and the Lamb shift calculation

This is a continuation of an earlier paper [J. Math. Phys. 34, 5509 (1993)]. A projection operator technique is introduced in the Lamb shift calculation in order to manage spurious infrared divergences. The method provides an efficient tool to exhibit factors of the field in the commutator expansion of the Lamb shift. Mass eigenfunction expansion concepts developed elsewhere provide the setting for the new technique. As a by-product of our method, relativistic reduced Green’s functions are found to appear in a natural way. The method may prove useful, since reduced Green’s functions are simpler than the full Green’s functions, and behave like constant operators as regards the parameter integrals.

A no-go theorem for a Lie-consistent q-Campbell–Baker–Hausdorff expansion

Lie consistency is defined for commutator expansions and it is shown that while a weakly Lie-consistent q-Campbell–Baker–Hausdorff expansion for a product of two M-type (but not P-type) q exponentials can be formulated, a strongly Lie-consistent expansion only exists for the undeformed case, q=1.

EVOLUTION OPERATOR AND ENERGY SPECTRUM OF A QUASICLASSICAL PARTICLE INTERACTING WITH BOSONS: APPLICATION TO ATOM SURFACE SCATTERING

We investigate the properties of the interaction of a particle with a boson field describing the response of a solid in the limit in which the interaction matrix elements may be considered as quasiclassical and the particle-boson coupling linear but not necessarily weak. We start by expressing the evolution operator of the system in a convenient form of an exponentiated nested commutator expansion in powers of the interaction potential. From this we are able to estimate under which conditions on the particle motion the contributions of the higher order expansion terms become small, irrespective of the coupling strength. Neglecting such small terms in the exponent of the evolution operator, we can calculate the energy excitation spectrum characteristic of the coupled system or of any of its constituents (particle or boson field). These spectra have the appearance of an exponentiated Born approximation (EBA) which contains and interpolates smoothly between the more frequently used distorted wave Born approximation (DWBA) and the trajectory approximation (TA), thereby covering a wide range of the parameter space for the description of the particle-boson interaction dynamics. The shape of the spectra and their characteristics (the weight of the elastic line or the Debye-Waller factor (DWF), the mean number of excited bosons, and the mean energy transfer in the course of the interaction) are discussed and shown to be very sensitive to the (non)adiabaticity of the switching of the interaction and the magnitude of the particle mass M. In the case of nonadiabatic switching on (as e.g. in photoemission) and linear bosonic density of states, we retrieve in the limit M→∞ the familiar infrared threshold divergences in the spectrum of the system. In the opposite case of adiabatic switching rates typical of scattering, the spectra exhibit a well-defined elastic line and a finite DWF. The case of surface scattering is discussed in more detail for the example of neutral atom scattering from dispersionless surface phonons in atomic adlayers.

Commutator expansion of the self-energy operator for an electron in an external potential

Techniques of Erickson and Yennie [Ann. Phys. N.Y. 35, 271 (1965)] are applied in an investigation of a simple Taylor’s series which involves products of commutators and nested commutators and which represents the self-energy operator Σ of an electron in an external potential. A technique of ‘‘moving the parameter integrals to the mass’’ leads to a relatively simple representation of all terms as a one-dimensional integral over a common virtual electron mass squared parameter. Although not a true expansion in powers of the external field, the method provides a window on the higher-order terms in the self-energy operator. Mass eigenfunction expansion concepts are shown to be quite useful in a discussion of infrared divergences.

Explicitly connected expansion for the average value of an observable in the coupled‐cluster theory

AbstractAn explicitly connected commutator expansion for the average value of an observable in the coupled‐cluster theory is derived. Specifically, it is shown that the expectation value of an operator for the state Ψ, related to the Fermi vacuum Φ by the exponential Ansatz ψ = eT Φ, is expressed as a finite commutator series containing the cluster operator T and an auxiliary operator S, defined by a linear equation involving again a finite commutator series in T. The above result is applied to derive the explicitly connected commutator form of the order‐by‐order many‐body perturbation theory (MBPT) expansion for the expectation values and density matrices. We also show how the commutator expansion derived by us can be used as a basis for size‐extensive infinite‐order summation techniques. An operator technique of eliminating the nonlocal, “off‐energy shell” denominators from MBPT expressions is proposed and applied to obtain compact commutator formulas for the expectation values of one‐ and two‐electron operators through the fourth and third order, respectively, and for the correlation energy through the fifth order of perturbation theory. © 1993 John Wiley & Sons, Inc.

Non-perturbative study of rotationally induced inner-shell excitation

Within the time-dependent formulation of atomic scattering theory, the exponential representation (“Magnus expansion”) of the quantum mechanical time-evolution matrix is used in a non-perturbative study of rotationally induced inner-shell excitation in slow ion-atom collisions. The impact-parameter dependence of this type of process is shown to represent a transparent example for testing the convergence properties of the Magnus expansion. The specific structure of the Magnus expansion for multi-state rotational coupling in the vicinity of a united-atom (n, l) shell is investigated, and the analytic solution which this problem admits in the sudden limit is discussed. Explicit calculations within the Magnus approach have been performed for typical two-state and three-state problems relevant toK-shell andL-shell excitation. Their results are compared to the results of the sudden approximation and of coupled-state calculations. Good agreement between the Magnus results and the coupled-state calculations is obtained throughout if terms up to third order are retained in the commutator expansion of the exponent matrix associated with the time-evolution matrix.

A new definition of atomic charges in molecules

A new definition of atomic charges in molecules is presented which conserves charge and dipole moment. It contains the Mulliken and Lowdin definitions as special cases of zero and first order truncations of commutator expansions. The definition allows for a systematic improvement of charges paralleling the improvement of the basis set in the LCAO approximation. We have tested the definition in thirteen selected diatomics and polyatomics in optimal minimal Slater basis set SCF calculations by means of 4G-level Gaussian expansions. The results suggest that the proposed definition is better than either Mulliken's or Lowdin's definition.