Perturbation Theory Expansion Research Articles

Simultaneous ionization and excitation by photon absorption in He

Second order terms of the time dependent perturbation theory expansion were used to calculate the simultaneous ionization and excitation of helium as result of a photon absorption. In this calculation, the perturbing Hamiltonian was assumed to be the sum of the interaction between the electrons and their interactions with the free radiation field. According to this model, the correlated transition resulting from the photon absorption can be calculated using the independent particle model to describe the atom states. In the present, the cross section for the correlated photon ionization in He was calculated using screened hydrogenic wave functions. The sum over the intermediate states required by the present approach was taken up to the tenth shell. No significant differences were observed in the results considering higher shells intermediate states. Correlated photon ionization cross section values are presented for energies of the absorbed photon varying from 70 up to 1500 eV and for transitions where the ionization of the helium atom is followed by simultaneous excitation of the second electron to a higher shell, that is He(1s 2)+ hν→He + (n s 1 or n p 1)+ e − , with n varying from 2 up to 6. Comparison of the present calculations with the results of measurements from several authors shows a very good agreement while the results of other theoretical calculations exhibit considerable discrepancies.

Derivation of spin Hamiltonians from the exact Hamiltonian: Application to systems with two unpaired electrons per magnetic site

The foundations and limits of S=1/2 and S=1 spin Hamiltonians for systems with two unpaired electrons in two well-defined orbitals per site are discussed by merging accurate ab initio calculations in binuclear systems with the effective Hamiltonian theory. It is shown that, beyond the usual JijSi.Sj terms, the effective spin Hamiltonian necessarily introduces four-body spin operators in the S=1/2 case and biquadratic terms in the S=1 formalism. The order of magnitude of these additional terms can be rationalized from a quasidegenerate perturbation theory expansion starting from a Hubbard-type Hamiltonian. This permits to discuss the physical mechanisms governing the reduction from the all electron Hamiltonian to the spin-only Hamiltonians and the conditions under which a further reduction from a spin Hamiltonian to the simplest Heisenberg-Dirac-Van Vleck form is possible. The overall discussion is illustrated by numerical calculations of the magnetic coupling between two Ni2+ cations in the K2NiF4 perovskite and between triply bonded carbon atoms in poly-ynes.

Symmetry-forcing procedure and convergence behavior of perturbation expansions for molecular interaction energies

Symmetry-adapted perturbation theory (SAPT) expansions corresponding to several symmetry-forcing procedures are applied through large order to study the interaction of lithium and hydrogen atoms. The interaction energies predicted by the perturbation theory are compared with the results obtained using the full configuration interaction (FCI) method. Since the ground state of the LiH molecule is submerged in the continuum of Pauli-forbidden states, these calculations are a demanding test for the SAPT approach in which the electrons from different monomers are treated as distinguishable particles. We show that if the symmetry is forced in a rather weak way, characteristic of the Murrell–Shaw–Musher–Amos theory, a divergent perturbation series is obtained. When the symmetry is forced in a strong way, as is done in the Eisenschitz–London–Hirschfelder–van der Avoird theory, one obtains a convergent series, but the interaction energy computed through any finite order exhibits wrong asymptotic behavior at large interatomic distances R. We show that by forcing the symmetry in an appropriate, intermediate way one obtains perturbation series which correctly predict leading terms in the 1/R asymptotic expansion of the interaction energy and, despite the presence of the Pauli-forbidden continuum, converge quickly to the FCI value of the interaction energy.

Effective two-photon transition operators: perturbative calculations and connectivity of diagrams

Perturbation theory expansions appropriate to two-photon processes between energy levels of rare earth ions are examined. The construction of effective two-photon transition operators that contain only connected diagrams is shown to be possible, but more complicated than previously thought.

The Torsional Potential of Dimethyl Peroxide: Still a Difficult Case for Theory

The torsional potential about the O−O single bond of dimethyl peroxide, CH3OOCH3, was investigated with the aid of large-scale ab initio calculations performed at different levels of Møller−Plesset perturbation theory and coupled-cluster expansions. Additionally, several density functional approaches were applied. For comparative purposes, the torsional potentials of methyl hydroperoxide, CH3OOH, and hydrogen peroxide, HOOH, were calculated at the same levels of approximation. In the already well-investigated case of HOOH and also for CH3OOH excellent agreement with the experimentally determined structures and barrier heights can be achieved at the coupled-cluster CCSD(T) level with the application of extended polarized basis sets augmented with diffuse functions. However, in the case of dimethyl peroxide, the peculiar shape of the computed CCSD(T)/cc-pVTZ torsional potential, with an exceedingly shallow region ranging from 110 to 250°, with two skew minima at about 115 and 245° and with a trans minimum at 180°, deviates significantly from that of the experimentally derived torsional potential, which has a barrier at 180° separating the two distinctly deeper skew minima at 120 and 240°. The difficulties encountered in reaching a reasonably converged result with respect to further basis set extension are discussed. It is also shown that the results of density functional theory (DFT) and Møller−Plesset second-order (MP2) calculations differ considerably from the Møller−Plesset higher-order and CCSD(T) results.

Covariant collective coordinates method in path integral formalism: application to quantum gravity on classical background

In this paper the problem of quantization of the free gravitational field on classical background (the exact solutions of Einstein equations) is considered. The Bogoliubov group coordinates method in path integral formalism is developed. This approach makes it possible to take carefully into account the conservation laws alongside with the perturbation theory expansion.

Operator product expansion in local causal perturbation theory

AbstractWe derive an operator product expansion in φ4‐theory for the time ordered product of two interacting fields in the Bogoliubov‐Epstein‐Glaser approach. Our result relies on the definition of an appropriate bilocal time ordered product as a generalization of the local time ordered products given by Epstein‐Glaser.

Operator product expansion in local causal perturbation theory

We derive an operator product expansion in φ4-theory for the time ordered product of two interacting fields in the Bogoliubov-Epstein-Glaser approach. Our result relies on the definition of an appropriate bilocal time ordered product as a generalization of the local time ordered products given by Epstein-Glaser.

Surface critical behavior of random systems: ordinary transition.

We calculate the surface critical exponents of the ordinary transition occurring in semi-infinite, quenched dilute Ising-like systems. This is done by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation as well as in 4-epsilon dimensions. At d=4-epsilon we extend, up to the next-to-leading order, the previous first-order results of the square root of epsilon expansion by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. In both cases numerical estimates for surface exponents are computed using Padé approximants extrapolating the perturbation theory expansions. The obtained results indicate that the critical behavior of semi-infinite systems with quenched bulk disorder is characterized by the new set of surface critical exponents.

Coupled Tensorial Form of the Second-Order Effective Operator

General formulas for the spin-angular part of the second-order effective operator for open-shell atoms are derived for LS and jj coupling. To obtain the perturbation theory expansion for a particular problem only a specification of ranks of the atomic interactions involved in the consideration is required. The expressions for matrix elements of one-particle and two-particle operators are derived in second quantization representation when the N-electron wavefunction has the consecutive order of coupling for angular momenta of shells. The matrix element is expressed as a sum of the products of 6-j and/or 9-j symbols and matrix elements of operators acting in the space of states formed from a shell of equivalent electrons. The expression for the matrix element does not contain the summation either over the resultant angular momenta of shells or over the intermediate angular momenta of N-electron wavefunction.

Perturbation expansion in phase-ordering kinetics. II.n-vector model

The perturbation theory expansion presented earlier to describe the phase-ordering kinetics in the case of a nonconserved scalar order parameter is generalized to the case of the n-vector model. At lowest order in this expansion, as in the scalar case, one obtains the theory due to Ohta, Jasnow, and Kawasaki (OJK). The second-order corrections for the nonequilibrium exponents are worked out explicitly in d dimensions and as a function of the number of components n of the order parameter. In the formulation developed here the corrections to the OJK results are found to go to zero in the large n and d limits. Indeed, the large-d convergence is exponential.

Advances and challenges in collisions of fast ions with few-electron atoms

This report reviews the recent theoretical results on the fast ion-few-electron atom collisions obtained in the framework of an approach using the perturbation theory expansion in the scattering potential and the diagonalization approximation for CI-wavefunction of the target atom. The uniform consideration of the electron transitions to states of discrete and continuous spectra is set forth. The results of investigations of the two-electron transitions in helium induced by fast charged projectiles, such as, the total and differential cross sections, mechanisms of electron transitions, interference problem and charge asymmetry of the cross sections, normalization of CI-wavefunctions of the discrete and continuous spectrum states, obtained on the basis of the present approach, are discussed.

Perturbation expansion in phase-ordering kinetics: I. Scalar order parameter

A consistent perturbation theory expansion is presented for phase-ordering kinetics in the case of a nonconserved scalar order parameter. At zeroth order in this expansion one obtains the theory due to Ohta, Jasnow and Kawasaki (OJK). At the next nontrivial order in the expansion, worked out in d dimensions, one has small corrections to the OJK result for the nonequilibrium exponent $\lambda$ and the introduction of a new exponent $\nu$ governing the algebraic component of the decay of the order parameter scaling function at large scaled distances.

Quantum transition state theory: Perturbation expansion

The exact quantum expression for the thermal rate of reaction is the trace of a product of two operators. It may therefore be written exactly as a phase space integral over the Wigner phase space representations of the two operators. The two are a projection operator onto the product’s space, which is difficult to compute, and the symmetrized thermal flux operator, which can be computed using Monte Carlo methods. A quantum transition state theory was presented recently, in which the exact projection operator was replaced by its parabolic barrier limit. Alternatively, the exact projection operator may be replaced by its classical limit. Both approximations give thermodynamic estimates for the quantum rates. In this paper, we derive a perturbation theory expansion for the projection operator about the parabolic barrier limit and the classical limit. The correction terms are then used to evaluate the leading order corrections to the rate estimates based on the parabolic barrier or classical limits of the projection operator. The expansion is applied to a symmetric and an asymmetric Eckart barrier. The first two terms in the expansion give excellent results for temperatures above the crossover between quantum tunneling and thermal activation. For deep tunneling and asymmetric systems, the use of variational transition state theory, the classical limit, and perturbation theory leads to significant improvement in the estimate of the tunneling rate. Multidimensional extensions are presented and discussed.

Exact Renormalization Group Approach in Scalar and Fermionic Theories

The Polchinski version of the exact renormalization group equation is discussed and its applications in scalar and fermionic theories are reviewed. Relation between this approach and the standard renormalization group is studied, in particular the relation between the derivative expansion and the perturbation theory expansion is worked out in some detail.

CCSD(T) expectation value calculations of first-order properties

An expectation value approach to calculations of first-order properties using the non-iterative, triple-excitation amplitudes in the coupled cluster wave function is exploited. Three methods are suggested and analysed using the many body perturbation theory (MBPT) expansion arguments. The first method, in which non-iterative triple-excitation amplitudes are used in the expression for the expectation values, makes the wave function accurate through the second order of MBPT. In the second method, which is an extension of the first, effects of triple-excitation amplitudes are coupled with single- and double-excitation amplitudes. The correlated density matrix equivalent through the fourth order to that obtained when CCSDT-la amplitudes are used is employed in the third method. The suggested methods are tested on dipole moment and polarizability calculations for several diatomic closed-shell molecules and are compared to other related approaches.

Chiral Symmetry Breaking in Strongly Coupled U(1) Gauge Theory Coupled to Scalar and Fermion Matter

We construct a transformation for scalar QED (U(1) gauge theory coupled with a charged scalar field) which leads to a dual effective gauge theory having a perturbation theory expansion in inverse powers of the (electric charge) coupling constant. The dual formulation is applied to the special case of scalar QED coupled to a massless charged fermion. A four-fermion interaction term arises naturally under the duality mapping. As an extension to discussions initiated by Miransky, Bardeen and Kondo and their collaborators in the late 1980's, we use this formulation to explore the possibility of a dynamical breakdown of chiral symmetry for the strongly coupled U(1) gauge theory interacting with both scalar and fermion degrees of freedom. Assuming that scalar QED is realized in the Higgs phase, we examine the Schwinger–Dyson equation approach to the fermion propagator in next-to-leading order of the strong coupling expansion (quenched approximation, i.e. no fermion loops). We construct a consistent renormalization procedure to deal with the ultraviolet infinities. For fixed scalar field vacuum expectation values, we obtain in the plane of the two important parameters of the theory (electric charge and four-fermion coupling constant) critical transition lines to a chirality broken phase.

Convergence of symmetry-adapted perturbation theory expansions for pairwise nonadditive interatomic interactions

Convergence properties of symmetry-adapted perturbation expansions for nonadditive interactions are tested by performing high-order calculations for three spin-aligned hydrogen atoms. It is shown that the strong symmetry forcing characteristic of the Hirschfelder–Silbey theory leads to a rapidly convergent perturbation series. The symmetrized Rayleigh–Schrödinger perturbation theory employing a weak symmetry forcing is shown to provide in low orders accurate approximations to the nonadditive part of the interaction energy. In very high orders the convergence of this perturbation expansion becomes very slow, and the series converges to an unphysical limit, very close to the exact interaction energy. The nonadditive part of the interaction energy for the lowest quartet state of H3 is interpreted in terms of the first-order exchange, induction, exchange-induction, exchange-dispersion, induction-dispersion, and dispersion contributions. It is shown that even for such a simple trimer the correct description of these components is necessary to obtain quantitative agreement with variational full configuration interaction results.

Density-functional theory for excited states.

A Kohn-Sham formalism for the treatment of excited states within the density-functional theory (DFT) is presented. DFT exchange and correlation energy functionals for excited states are defined. Explicit expressions for these functionals are derived by generalizing a recent DFT perturbation theory. A computational scheme for the treatment of excited states within the DFT is suggested. Differences of Kohn-Sham eigenvalues are shown to be well-defined approximations for excitation energies. Correction terms to these approximations are presented. Perturbation theory expansions for band gaps are discussed. \textcopyright{} 1996 The American Physical Society.

On renormalons and Landau poles in gauge field theories

It is shown that the commonly accepted relationship between the Landau singularity in the running coupling constant of QED or QCD and the renormalon singularities in the Borel sums of perturbation theory expansions is only a particular feature of the restriction of the perturbative β-function to the one loop level.