45 research outputs found
Development of algebraic techniques for the atomic open-shell MBPT(3)
The atomic third-order open-shell many-body perturbation theory is developed.
Special attention is paid to the generation and algebraic analysis of terms of
the wave operator and the effective Hamiltonian as well. Making use of
occupation-number representation and intermediate normalization, the
third-order deviations are worked out by employing a computational software
program that embodies the generalized Bloch equation. We prove that in the most
general case, the terms of effective interaction operator on the proposed
complete model space are generated by not more than eight types of the -body
() parts of the wave operator. To compose the effective Hamiltonian
matrix elements handily, the operators are written in irreducible tensor form.
We present the reduction scheme in a versatile disposition form, thus it is
suited for the coupled-cluster approach
On the idempotents of Hecke algebras
We give a new construction of primitive idempotents of the Hecke algebras
associated with the symmetric groups. The idempotents are found as evaluated
products of certain rational functions thus providing a new version of the
fusion procedure for the Hecke algebras. We show that the normalization factors
which occur in the procedure are related to the Ocneanu--Markov trace of the
idempotents.Comment: 11 page
Coupled tensorial form for atomic relativistic two-particle operator given in second quantization representation
General formulas of the two-electron operator representing either atomic or
effective interactions are given in a coupled tensorial form in relativistic
approximation. The alternatives of using uncoupled, coupled and antisymmetric
two-electron wave functions in constructing coupled tensorial form of the
operator are studied. The second quantization technique is used. The considered
operator acts in the space of states of open-subshell atoms
On the secondly quantized theory of many-electron atom
Traditional theory of many-electron atoms and ions is based on the
coefficients of fractional parentage and matrix elements of tensorial
operators, composed of unit tensors. Then the calculation of spin-angular
coefficients of radial integrals appearing in the expressions of matrix
elements of arbitrary physical operators of atomic quantities has two main
disadvantages: (i) The numerical codes for the calculation of spin-angular
coefficients are usually very time-consuming; (ii) f-shells are often omitted
from programs for matrix element calculation since the tables for their
coefficients of fractional parentage are very extensive. The authors suppose
that a series of difficulties persisting in the traditional approach to the
calculation of spin-angular parts of matrix elements could be avoided by using
this secondly quantized methodology, based on angular momentum theory, on the
concept of the irreducible tensorial sets, on a generalized graphical method,
on quasispin and on the reduced coefficients of fractional parentage
An efficient approach for spin-angular integrations in atomic structure calculations
A general method is described for finding algebraic expressions for matrix
elements of any one- and two-particle operator for an arbitrary number of
subshells in an atomic configuration, requiring neither coefficients of
fractional parentage nor unit tensors. It is based on the combination of second
quantization in the coupled tensorial form, angular momentum theory in three
spaces (orbital, spin and quasispin), and a generalized graphical technique.
The latter allows us to calculate graphically the irreducible tensorial
products of the second quantization operators and their commutators, and to
formulate additional rules for operations with diagrams. The additional rules
allow us to find graphically the normal form of the complicated tensorial
products of the operators. All matrix elements (diagonal and non-diagonal with
respect to configurations) differ only by the values of the projections of the
quasispin momenta of separate shells and are expressed in terms of completely
reduced matrix elements (in all three spaces) of the second quantization
operators. As a result, it allows us to use standard quantities uniformly for
both diagona and off-diagonal matrix elements
Spin-other-orbit operator in the tensorial form of second quantization
The tensorial form of the spin-other-orbit interaction operator in the
formalism of second quantization is presented. Such an expression is needed to
calculate both diagonal and off-diagonal matrix elements according to an
approach, based on a combination of second quantization in the coupled
tensorial form, angular momentum theory in three spaces (orbital, spin and
quasispin), and a generalized graphical technique. One of the basic features of
this approach is the use of tables of standard quantities, without which the
process of obtaining matrix elements of spin-other-orbit interaction operator
between any electron configurations is much more complicated. Some special
cases are shown for which the tensorial structure of the spin-other-orbit
interaction operator reduces to an unusually simple form
Characterization of anomalous Zeeman patterns in complex atomic spectra
The modeling of complex atomic spectra is a difficult task, due to the huge
number of levels and lines involved. In the presence of a magnetic field, the
computation becomes even more difficult. The anomalous Zeeman pattern is a
superposition of many absorption or emission profiles with different Zeeman
relative strengths, shifts, widths, asymmetries and sharpnesses. We propose a
statistical approach to study the effect of a magnetic field on the broadening
of spectral lines and transition arrays in atomic spectra. In this model, the
sigma and pi profiles are described using the moments of the Zeeman components,
which depend on quantum numbers and Land\'{e} factors. A graphical calculation
of these moments, together with a statistical modeling of Zeeman profiles as
expansions in terms of Hermite polynomials are presented. It is shown that the
procedure is more efficient, in terms of convergence and validity range, than
the Taylor-series expansion in powers of the magnetic field which was suggested
in the past. Finally, a simple approximate method to estimate the contribution
of a magnetic field to the width of transition arrays is proposed. It relies on
our recently published recursive technique for the numbering of LS-terms of an
arbitrary configuration.Comment: submitted to Physical Review
Coupling coefficients of SO(n) and integrals over triplets of Jacobi and Gegenbauer polynomials
The expressions of the coupling coefficients (3j-symbols) for the most
degenerate (symmetric) representations of the orthogonal groups SO(n) in a
canonical basis (with SO(n) restricted to SO(n-1)) and different semicanonical
or tree bases [with SO(n) restricted to SO(n'})\times SO(n''), n'+n''=n] are
considered, respectively, in context of the integrals involving triplets of the
Gegenbauer and the Jacobi polynomials. Since the directly derived
triple-hypergeometric series do not reveal the apparent triangle conditions of
the 3j-symbols, they are rearranged, using their relation with the
semistretched isofactors of the second kind for the complementary chain
Sp(4)\supset SU(2)\times SU(2) and analogy with the stretched 9j coefficients
of SU(2), into formulae with more rich limits for summation intervals and
obvious triangle conditions. The isofactors of class-one representations of the
orthogonal groups or class-two representations of the unitary groups (and, of
course, the related integrals involving triplets of the Gegenbauer and the
Jacobi polynomials) turn into the double sums in the cases of the canonical
SO(n)\supset SO(n-1) or U(n)\supset U(n-1) and semicanonical SO(n)\supset
SO(n-2)\times SO(2) chains, as well as into the_4F_3(1) series under more
specific conditions. Some ambiguities of the phase choice of the complementary
group approach are adjusted, as well as the problems with alternating sign
parameter of SO(2) representations in the SO(3)\supset SO(2) and SO(n)\supset
SO(n-2)\times SO(2) chains.Comment: 26 pages, corrections of (3.6c) and (3.12); elementary proof of
(3.2e) is adde