Download Methods of Statistical Spectroscopy as an Optimization

Methods of Statistical Spectroscopy as an Optimization Technique in
Predictive Pharmacology and Toxicology
Dorota Bielińska-Wa̧ż(a) , Piotr Wa̧ż(b) ,
Subhash C. Basak(c) , Ramanathan Natarajan(d)
(a)
Instytut Fizyki, Uniwersytet Mikolaja Kopernika,
Grudzia̧dzka 5, 87-100 Toruń, Poland,
(b)
Centrum Astronomii, Uniwersytet Mikolaja Kopernika,
Gagarina 11, 87-100 Toruń, Poland,
(c,d)
Natural Resources Research Institute,
5013 Miller Trunk Highway, Minnesota 55811-1442 USA,
(a)
[email protected], (b) [email protected],
(c)
[email protected], (d) [email protected]
In the traditional approach, spectral properties of atoms and molecules are determined using quantum chemical methods. In this approach individual energy levels
and the appropriate transition probabilities are evaluated with a high precision but
even for a few energy levels the computational effort is very high. If the spectra are
complex or if they contain many lines such methods are very inefficient. In such a
case the optimization of methods becomes a necessity. One of the ways is to treat a
set of spectral data (eigenvalues of model hamiltonians, experimental energy levels,
transition probabilities) as a statistical ensamble. In such an approach, referred to
as statistical spectroscopy, the description of the system is derived from the knowledge of spectral density distribution moments [1]. Determining shapes of molecular
electronic bands or the approximation of a specific set of the Hamiltonian eigenvalues are examples of applications of statistical spectroscopy. In this presentation
we use the knowledge of the calculated spectral density distribution moments of
spectra of molecules used in pharmacology as molecular descriptors. The quality
of this method strongly depends of the number of the appropriate moments taken
into account. Grouping the molecules according to the spectral density distribution
moments gives a chance to find a new approach in the field of characterization of
molecular similarity [2] and in particular it may be a tool for computational pharmacology and toxicology [3]. In this way one can use methods of statistical spectroscopy
for the optimization of the process of selection of molecules for sythesis and for the
experimental tests of their usefulness in specific applications (as e.g. in pharmacology) opening a new way for the predictive pharmacology and toxicology. In this
presenation the theory is illustrated by an analysis of several model spectra, where
statistical moments of the intensity distributions are used as molecular descriptors.
Acknowledgement: This work has been supported by the Polish KBN, project
No. 2 PO3B 033 25.
[1] D. Bielińska-Wa̧ż, in Symmetry and Structural Properties of Condensed Matter, World Scientific, Singapore 1999, pp. 212-221.
[2] D. Bielińska-Wa̧ż, P. Wa̧ż, S. C. Basak R. Natarajan, Statistical Theory of
Spectra and its New Applications in Pharmacology, in preparation.
[3] S.C. Basak, Use of molecular complexity indices in predictive pharmacology
and toxicology: a QSAR approach, Med. Sci. Res. 15, (1987) 605-609.
1
Usage of Wybourne method for ab initio calculations
of atomic spectra
Pavel Bogdanovich
Research Institute of Theoretical Physics and Astronomy
Vilnius University
A. Gostauto 12, 01108 Vilnius, Lithuania
We describe the methods of theoretical calculations of energy spectra of atoms
and ions with account for correlation corrections in the second order of perturbation
theory in the form first obtained and published by B. G. Wybourne for explaining
the physical nature of semi empirical calculations.
2
Collective natural orbital occupancies of harmonium
Marcin Buchowiecki, Jerzy Cioslowski
Instytut Fizyki
Uniwersytet Szczeciński
Wielkopolska 15, PL-70-451 Szczecin, Poland
e-mail [email protected]
In the harmonium atom, the collective occupancies {nl } of natural orbitals with
different angular momenta l can be rigorously studied for those values of the confinement strength ω that lead to ground state wave functions with correlation factors
given by polynomials of kth order. Analysis of two such cases, with ω equal to
1/2 (k = 1) and 1/10 (k = 2), uncovers some interesting properties of {nl }, which
turn out to be given by rational functions of π 1/2 with simple coefficients. In particular, the collective occupancies for l > 2 actually decrease upon strenghtening
of correlation. The explicit formulas for l-matrices and {nl } can be readily utilized
in testing, calibration, and benchmarking of approximate implementations of the
density matrix functional theory.
3
Brian Wybourne, his life at Canterbury
Philip H Butler
Department of Physics and Astronomy,
University of Canterbury
Private Bag 4800, Christchurch, New Zealand
Brian Wybourne was a student from 1952 to 1960 at Canterbury University College. He returned as full professor in 1966, and served as head of department from
1983 to 1990. During his time at Canterbury, he developed a front-ranking theoretical physics team and encouraged many other areas of research, such as the ring laser
development. As one of his doctoral students who returned to Canterbury to work
with Brian, I shall review several of his major contributions to life at Canterbury
University.
4
Physics with a real Clifford algebra
Philip H. Butler, Adam B. Gillard, Niels G. Gresnigt, William P. Joyce, Benjamin
M. Martin, Peter F. Renaud
Department of Physics and Astronomy,
University of Canterbury
Private Bag 4800, Christchurch, New Zealand
Complex numbers play a fundamental role in physics, particularly in the fields
of quantum theory and classical electromagnetism. The key property of complex
numbers is the mathematical construction, i, which squares to -1. De Moivre s
theorem follows where exp = cos + i sin. The bi-vectors of Clifford Algebras are
natural geometric objects that square to -1. This makes Clifford Algebras interesting
tools for setting up quantum mechanics as they may allow a new quantum theory to
be constructed in a geometric setting without complex numbers. The Stern-Gerlach
experiment is considered. Clifford Algebra also provides us with methods of rotating
in a real vector space instead of a complexified vector space. The example of the
Hertzian dipole is used to illustrate the possibilities for electromagnetism.
5
Extension of Gel’fand matrix method to the Casimir operators of
inhomogeneous groups
Rutwig Campoamor-Stursberg1
Departamento de Geometrı́a y Topologı́a,
Fac. CC. Matemáticas U.C.M.
Plaza de Ciencias, 3, E-28040 Madrid, Spain
e-mail: [email protected]
In 1950 I. M. Gel’fand showed how to use the generic matrix of standard representation of orthogonal groups to determine the Casimir operators of the algebra
by means of characteristic polynomials. Similar methods have been developed for
other semisimple Lie algebras by various authors. From the construction it seems
necessary that the used matrix corresponds to some faithful representation of the
algebra. We show that the essence of the Gel’fand method can be extended to
other non-semisimple Lie algebras, specially the class of inhomogeneous Lie algebras, using extended matrices that also correspond to the standard representation.
Considering linear combinations of the characteristic polynomials of the representation matrices and its minors, the Casimir operators of the inhomogeneous algebras
can be recovered. It is moreover shown that even in the case where the matrices
correspond to non-faithful representations (as happens when Inönü-Wigner contractions are considered), the method remains effective. The extreme case where the
employed matrix has no interpretation in terms of representations of Lie algebras is
also proved to exist.
The method moreover provides an insight to the problem why the contraction of
independent Casimir invariants are not necessarily independent invariants of the
contractions, and how to derive an alternative procedure based on PDEs to reconstruct additional independent invariants of the contractions. Applications to the
invariant operators of subalgebra chains of the type
s ⊃ Ig ⊃ A1
with s semisimple and Ig inhomogeneous Lie algebras are given, and a sufficiency
criterion to obtain certain solutions to the missing label problem is developed.
1
Partially supported by the research project PR1/05-13283 of the U.C.M.
6
Plethysm and Schuropera
M. Juliana Carvalho, S. D Agostino and D.J. Rowe
Ryerson University
350 Victoria Street,
M5B 2K3 Toronto, Canada
Schur functions (or S-functions) play a very important role in the group-theoretical/algebraic description of many-body systems. Plethysm or symmetrized product, to which B.G. Wybourne devoted a great deal of work, is an S-function operation essential in the construction of a multi-particle wave-function with a given
particle permutation symmetry. In a simple plethysm, the S-function on the right
determines (by its weight) the number of times the S-function on the left is multiplied by itself and the permutation symmetry of the resulting S-functions. The
algebraic properties of plethysm, which can be understood in terms of branching
rules and group duality, show clearly that this operation is not distributive on the
left with respect to addition/subtraction or product of S-functions. Thus, the need
to rely on cumbersome rules to evaluate the plethysm of a compound S-function
(algebraic sum of S-functions) makes it a difficult task. There is a way, however, of
circumventing the use of those rules. Schuropera, the Maple program for operations
with symmetric functions, uses this new method. In the proposed presentation,
we will give examples where compound S-functions are physically relevant and will
show how the (Schuropera) PlethysmMaplet can be used to solve different types of
plethysms.
7
Properties of persistent currents in the presence of non-classical
electromagnetic fields
Jerzy Dajka and Elzbieta Zipper
Institute of Physics
University of Silesia
Uniwersytecka 4, PL-40-007 Katowice, Poland
Mesoscopic systems and the non-classical electromagnetic fields are of central
importance for quantum information processing. Our aim is to present the significant impact of the non-classical radiation on the properties of persistent currents.
We study mesoscopic rings or cylinders subject to both classical and non-classical
magnetic flux. We limit our discussion to one- and two-mode fields prepared in
a given state. For the monochromatic radiation we study reduction of the of the
current amplitude due to the quantum and classical noise in the field. The effect of
temperature will be shortly reviewed. Two-mode fields can be either separable or
entangled. This property, as it will be shown, is reflected in the properties of the
current.
[1] J. Dajka, M. Szopa, A. Vourdas, E. Zipper, Phys. Rev. B69, 045305 (2004)
[2] J. Dajka, M. Szopa, A.Vourdas, E. Zipper, phys. stat. sol. B242, 296 (2005)
8
Scintillation Properties of LuYAP:Ce,Mo Crystals
Winicjusz Drozdowski, Andrzej J. Wojtowicz, Tadeusz Lukasiewicz,
Jaroslaw Kisielewski
Instytut Fizyki
Uniwersytet Mikolaja Kopernika
ul. Grudzia̧dzka 5, PL-87-100 Toruń, Poland
Rare earth activated wide bandgap oxide materials have been found very useful
as detectors of ionizing radiation in nuclear and high-energy physics, astrophysics,
nuclear medicine and industry. Parity-allowed ultraviolet d-f transitions characterized by high oscillator strengths together with a simple structure of energy levels
make a trivalent cerium one of the most interesting activators. Indeed, a number
of crystals activated by Ce3+ ions (lanthanide perovskites, garnets, orthosilicates,
orthophosphates and recently pyrosilicates and double phosphates) have been studied as potential fast and efficient scintillators. Cerium and molybdenum activated
lutetium aluminum perovskite (LuAP) and mixed lutetium-yttrium aluminum perovskite (LuYAP) scintillator crystals have been grown using the Czochralski method
at the Institute of Electronic Materials Technology (ITME, Warsaw). In this communication we report the measurements of scintillation light yields, scintillation
time profiles, low temperature thermoluminescence, and room temperature afterglow, performed on polished 2x2x10 mm3 pixel samples (dimensions used in PET
cameras for small animals). We show that the basic scintillation properties of the
LuAP:Ce crystals from Warsaw, particularly their light yields, are close to those
of the prominent commercially produced ones. We also discuss the advantages and
disadvantages of introducing yttrium and molybdenum. Finally, compiling the data
from all the experiments we explain some evident correlations between particular
parameters in frames of a ”trap model” of a scintillator.
9
Effect of Pressure on the 3 P0 → 3 H4 and 1 D2 → 3 H4 Emission in
Pr3+ :YAG
R. Turos-Matysiaka , M. Grinberga, D. Dyla , H. Zhengb , J.W. Wangb ,
W.M. Yenb , R.S. Meltzerc , T. Lukasiewiczc , M. Żwirkowiczc
a
Institute of Experimental Physics,
University of Gdańsk, Gdańsk, Poland
e-mail: [email protected]
b
Department of Physics and Astronomy
University of Georgia, Athens, GA 30602, USA
c
Institute of Electronic Materials Technology,
Wólczyńska 133, PL-01-919-Warsaw, Poland
Photoluminescence related to 3 P0 → 3 H4 and 1 D2 → 3 H4 transitions of Pr3+ in
YAG is measured in the pressure range from ambient to 200 kbar. The dependence
of the energy of the emission lines have been analyzed in the framework of crystal
field model. One has obtained the quantities and pressure rates of the Slater integrals, spin-orbit coupling and crystal field parameters. The luminescence of interest
consist of the four transitions: Γ1 (3 P0 ) → Γ3 , Γ1 (3 P0 ) → Γ4 , Γ1 (1 D2 ) → Γ3 and
Γ1 (1 D2 ) → Γ4 . The emission related to Γ1 → Γ1 is not observed either for the 3 P0
or 1 D2 initial states. Both the 3 P0 and 1 D2 transitions, corresponding to Γ1 → Γ3
and Γ1 → Γ4 , show different dependencies on pressure. The pressure shifts are
-0.85 and -0.36 cm−1 /kbar, respectively, for 3 P0 , and -0.96 and -0.46 cm−1 /kbar,
respectively for the 1 D2 . Since transitions to the Γ3 state correspond to the shorter
wavelengths, the emission lines crossover at 80 kbar. Thus the crossover takes place
in the ground manifold where the energy splitting between Γ3 and Γ4 decreases at
the rate of 0.5 cm−1 /kbar. To reproduce the energetic structure of the Pr3+ :YAG
system and the peculiarities of the dependencies of the emission on pressure we
have performed calculations in the framework of the crystal field model. We have
considered contributions from the cubic field described by and the D2 symmetry
field described by ∆Bqk , separately. Using the superposition model we have simulated the shifts of the individual ligands and related them to the changes in the
quantities of crystal field parameters Bqk (cub) and ∆Bqk . To represent the changes
in the positions of coordination oxygen ions we have used the superposition model.
The main conclusion of our investigation is that, contrary to expectation, the effect
of the hydrostatic pressure is not isotropic; it changes the local symmetry of the
ion-ligands system.
10
An Hopf algebraic approach to the theory of group branchings
Bertfried Fauser, P.D. Jarvis, Ron C. King, Brian G. Wybourne
Max Planck Institut für Math. i. Sciences
Inselstrasse 22-26, D-04103 Leipzig, Germany
Using some Hopf algebraic techniques in the theory of symmetric functions we
develop a theory of group branching which is generalizable to a large class of non
classical groups. Examples contain all subgroups of a GL(n) fixing a tensor Tπ of
Young symmetry type π. Such branchings are mediated by Schur function series
obtained by the plethysm π⊗M . We will display the theory and provide explicite
examples.
[This work is based on some examples B. Wybourne and R. King did in Southampton during Brian’s last visit as a reaction on the paper ”A Hopf laboratory for
symmetric functions” BF and PDJ, J. Phys. A. 37(2004)1633-63]
11
Tensor term combination and construction of short formulas
for generators of algebraic curvature tensors
Bernd Fiedler
Leipzig, Eichelbaumstr. 13,
D-04249 Leipzig, Germany
We consider generators of algebraic curvature tensors R and algebraic covariant
derivative curvature tensors R which can be constructed by a Young symmetrization
of product tensors
R : U ∗ w or w ∗ U,
R0 : U ∗ W or W ∗ U,
where U , W , w are covariant tensors of order 3, 2, 1. We assume that W is symmetric
or skew-symmetric whereas U belongs to a class of the infinite set S of irreducible
symmetry classes characterized by the partition (2, 1).
We show that the set S contains exactly one symmetry class s0 (for R and R0 )
whose elements U can not play the role of generators of tensors R and R 0 . The
tensors U of all other symmetry classes from S{s0 } can be used as generators for
tensors R and R0 .
Using Computer Algebra, in particular the methods for symbolic tensor calculation and term combination described in [2], we search for such generators whose
coordinate representations are polynomials with a minimal number of summands.
This number of summands depends on the (variable) symmetry class of U .
For a generic choice of the symmetry class of U we obtain the following lengths:
U ∗w U ∗S U ∗A
R
8
R
16
20
W = S symmetric
W = A skew-symmetric
In special cases these numbers can be reduced to the following minima.
U ∗w U ∗S U ∗A
R
4
R
12
10
W = S symmetric
W = A skew-symmetric
If these minima occur then U admits an index commutation symmetry. Furthermore
minimal lengths are possible if U is formed from torsion-free covariant derivatives
of alternating or symmetric 2-tensor fields.
12
We apply ideals and idempotents of group rings C[Sr ] of symmetric groups Sr ,
Young symmetrizers, discrete Fourier transforms, Littlewood-Richardson products
and plethysms. For symbolic calculations we used the Mathematica packages Ricci
and PERMS. A main source of or investigations is the paper [1].
[1] Fulling, S.A., King, R.C., Wybourne, B.G. and C.J. Cummins: Normal forms
for tensor polynomials: I. The Riemann tensor. Class. Quantum Grav. 9
(1992), 1151 - 1197.
[2] Fiedler, B.: arXiv.org/abs/math.CO/0310020.,
arXiv.org/abs/math.CO/0312171 and arxiv.org/abs/math.DG/0411056.
13
The effect of correlation on the line strengths of transitions in third-row
elements
Charlotte Froese Fischera , G. Tachievb , and A. Irimiaa
a
Department of Electrical Engineering and Computer Science,
Box 1679 B Vanderbilt University, Nashville, TN 37235, USA
e-mail: [email protected]
b
Presently at Florida International University
In the configuration interaction model, the wave function of a many-electron
system is a linear combination of configuration state functions (CSFs). In our approach, the CSFs are an orthonormal basis. Consquently, the expansion coefficients
are related to probabilities that the many-electron system can be described by a
given CSF. The expansion coefficients are an eigenvector of an interaction matrix,
hCF Si | H | CSFj i, where H is the Hamiltonian for the system, and the corresponding eigenvalue the total energy of the state. Terms in the Hamiltonian involve
at most two electrons. Thus the non-zero interactions are between many-electron
CSFs that differ by at most two electrons. In large-scale calculations the interaction
matrix has a sparsity of 1-10 %. From wave functions that determine energy levels, other atomic properties may be computed. Many energy levels and transition
probabilites for sequences of 4-18 electrons have been reported [1,2]. Results may
be viewed at an on-line database [3].
A number of strong interactions are present in the third row elements. The wellknown two configuration 2s2 −2p2 1S interaction in Be I (in the second row) becomes
a three configuration 3s2 − 3p2 − 3d2 1S interaction in the third row. For this row,
there also is a strong 3s3d − 3p2 1D interaction in many 3s2 3pn 3d LS or 3s3pn+2
LS states. The effect of this interaction is to introduce severe cancellation into the
calculation of oscillator strengths for some states and enhancement in others when
compared with Hartree-Fock values. In neutral atoms, the 3s3pn+2 LS state often
is in the continuum but already in the singly ionized element of the iso-electronic
sequence both states are observed. Cancellation occurs in line strengths of the lower
states which may explain the presence of some excited levels and absence of lower
levels in some spectra for highly ionized elements where level identification is less
complete. Results will be presented for several isoelectronic sequences.
This work was supported by the Chemical Sciences, Geosciences and Biosciences
Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of
Energy. Large parallel calculations were performed at the Department of Energy,
National Energy Research Scientific Computing Center.
[1] C. Froese Fischer and G. Tachiev, Atom. Data Nucl. Data Tables 87, 1
(2004).
[2] C. Froese Fischer, G. Tachiev, and A. Irimia, Atom. Data Nucl. Data Tables
(submitted).
[3] C. Froese Fischer and G. Tachiev http://atoms.vuse.vanderbilt.edu.
14
Enumeration of Tensor Polynomials by Schur-Function Analysis
Stephen Fulling
Mathematics Department
Texas A&M University
77843-3368 College Station, TX, USA
Various calculations in quantum field theory, differential geometry, and other
fields require knowledge of a basis of independent invariants formed polynomially
from the Riemann tensor, gauge curvature tensors, and other such quantities, along
with their covariant derivatives. B. G. Wybourne, R. C. King, and C. J. Cummins
taught me how to solve such problems using Brian’s software and the associated
mathematical apparatus. I review what was accomplished and what remains to be
done.
15
Space groups for quasicrystals
Jean Pierre Gazeau
Laboratoire Astroparticules et Cosmologie,
Université Paris 7-Denis Diderot,
75251 Paris cedex 05, France
We report on the existence of symmetry plane-groups for quasiperiodic pointsets named beta-lattices. Like lattices are vector superpositions of integers, betalattices are vector superpositions of beta-integers. When β > 1 is a quadratic PisotVijayaraghavan (PV) algebraic unit, the set of beta-integers can be equipped
√ with
an abelian
group√structure and an internal multiplicative law. When β = (1+ 5)/2,
√
1 + 2 and 2 + 3, we show that these arithmetic and algebraic structures lead to
freely generated symmetry plane-groups for beta-lattices. These plane-groups are
based on repetitions of discrete adapted rotations and translations we shall refer
to as ‘beta-rotations” and ‘beta-translations”. Hence beta-lattices, endowed with
beta-rotations and beta-translations, can be viewed like lattices. We also show that,
at large distances, beta-lattices and their symmetries behave asymptotically like
lattices and lattice symmetries respectively.
[1] A. Elkharrat, Ch. Frougny, J.P. Gazeau and J. L. Verger-Gaugry, Symmetry
groups for beta-lattices. Theor. Comp. Sci. 319 (2004), 281–305.
[2] C. Burdı́k, Ch. Frougny, J.P. Gazeau and R. Krejcar, Beta-integers as natural
counting systems for quasicrystals. J. of Physics A: Math. Gen. 31 (1998),
6449–6472.
[3] Ch. Frougny, J.P. Gazeau and R. Krejcar, Additive and multiplicative properties of point-sets based on beta-integers. Theor. Comp. Sci. 303 (2003),
491–516.
16
Angular Symmetry and Hylleraas Coordinates
in Few-Body Problems
Frank E. Harris
Quantum Theory Project,
University of Florida
P. O. Box 118435, 32611 Gainesville, FL, USA
The most accurate studies of few-body Coulomb systems have used wavefunctions
of forms that are simple in Hylleraas coordinates (those that explicitly include all the
interparticle distances). We consider the following issues that arise when these coordinates are used for states of general angular symmetry: (1) convenient and compact
expression of the kinetic energy for angular eigenstates, and (2) angular integrations
in matrix elements. Both these issues differ from their well-known counterparts in
orbital formulations, in part because the interparticle distances are not orthogonal
coordinates, and in part because their orientations are not all independent variables.
We examine ways in which the formulation can be made convenient and focus on the
role played by rotational invariants. Some illustrative computations are reported.
17
Branching rules for projective representations of finite reflection groups
John Francis Humphreys
Department of Mathematical Sciences
University of Liverpool
PO Box 147, L69 3BX Liverpool, U.K.
In an article in volume 14 of J. Phys , Dehuai and Wybourne discuss branching
rules for spin representations of the symmetric groups. We shall extend these results
to spin representations of general finite reflection groups, using the representations
of Clifford Algebras which underlie the spin representations of reflection groups.
There will also be a discussion of the branching rules of the spin representations of
the rotation subgroups of the reflection groups, and of the question of writing the
basic spin representations of these groups over the real field.
18
The role of symmetry in the physics of carbon nanotube quantum dots
and superlattices
Leonor Chicoa and Wlodzimierz Jaskólskib
a
Facultad de Ciencias del Medio Ambiente,
UCLM, Toledo, Spain
b
Instytut Fizyki
Uniwersytet Mikolaja Kopernika
ul. Grudzia̧dzka 5, PL-87-100 Toruń, Poland
The electronic structure and quantum conductance of nanosystems built of carbon nanotubes are analyzed. The work summarizes some of our earlier findings
concerning conductivity and localization properties in carbon nanotube junctions,
quantum dots and superlattices. It is shown that symmetry is an important factor
determining the electronic and transport properties of the investigated systems.
[1] L. Chico, L.X. Benedict, S.G. Louie, and M.L. Cohen, Phys. Rev. B 54, 2600
(1996).
[2] L. Chico, M.P. López Sancho, and M.C. Muñoz, Phys. Rev. Lett. 81, 1278
(1998).
[3] L. Chico and W. Jaskólski, Phys. Rev. B 69, 85406 (2003).
[4] W. Jaskólski and L. Chico, Phys. Rev. B 71, 155405 (2005).
19
Energy-level calculations for the 5f 2 6d1 configuration of U 3+
in Cs2 N aY Cl6 and SrCl2 single crystals
Miroslaw Karbowiak
Institute of Low Temperature and Structure Research,
Polish Academy of Sciences,
ul. Okólna 2, 54-422 Wroclaw, Poland
Energy levels as well as the 5f 3 −→ 5f 2 6d(t2g)1 and 5f 3 −→ 5f 2 6d(eg)1 transition intensities of U 3+ in Cs2 N aY Cl6 and SrCl2 were calculated using a theoretical model for nf N energy levels extended for interactions related with presence
of d-electron, proposed by Reid et al. [P hys.Rev.B 2000, 62, 14744]. Hamiltonian parameters were determined by the least-squares fitting of calculated energies
to experimental data and a very good agreement between calculated and experimental spectra has been achieved. The observed trends in Hamiltonian parameter
values are discussed. The results of calculations performed in the frame of the
semi-empirical Hamiltonian model are compared with those of abinitio theoretical
analysis, reported by L. Seijo and Z. Barandiarán [J.Chem.P hys. 118 (2003) 5335].
20
The Polish Odyssey of Brian Wybourne
Jacek Karwowski
Instytut Fizyki
Uniwersytet Mikolaja Kopernika
ul. Grudzia̧dzka 5, PL-87-100 Toruń, Poland
The first contact of Brian Wybourne with Poland was in 1968, when he stopped
in Warsaw en route to a conference in Vilnius. Then, in 1988, he briefly visited the
Banach Centre in Warsaw [1]. During his third trip to Poland, in 1990, I met Brian
in Zaja̧czkowo at a conference organized by Tadeusz and Barbara Lulek. After the
conference I invited Brian for a longer visit to Toruń. The visit started in 1991.
The next 12 years Brian spent in Toruń working as a Professor at the Institute of
Physics of the Nicolaus Copernicus University. In 1999 he was given the Polish title
of Professor by the President of Poland. In 2003 he received an Award from the
Minister of Education for his outstanding contribution to science. A month later,
he unexpectedly died.
During his stay in Toruń Brian Wybourne published 80 scientific papers. In
most general terms he worked on applications of symmetry in physics. However his
research covered a very broad area of science, ranging from studies on the spectroscopic properties of lanthanides and actinides to the applications of Lie groups and
symmetric functions. In all these fields the contribution of Brian Wybourne was
very deep and important. His scientific interests extended far beyond traditional
theoretical physics. Just to mention one or two, there was his contribution to the
analysis of the genetic code and his tracing the influence of the Chernobyl disaster
in the deposits of salt in Ciechocinek. He was also involved in the project of construction of large ring-lasers in Germany, a continuation of a project he initiated in
New Zealand.
Brian Wybourne was an excellent teacher and educator. He taught his students
and friends in Toruń, not only physics but also how to improve the form of their
scientific papers, how to give a better seminar, how to find the relevant information
and, most important, how to interact with other people. His lectures and the book
”Physics as a Journey” [2] not only opens the world of physics to non-physicists but
also gives a new perspective to experts in the field. I dare to say that it is one of
the best books about physics ever written.
In my talk I shall give a brief review of the Toruń episode in the life of Brian
Wybourne.
[1] B. G. Wybourne, My Polish Physics Odyssey: The First Decade, Institute of
Physics, Thursday Seminar, 24/01/2002, http://www.phys.uni.torun.pl/ bgw/
[2] B. G. Wybourne, Physics as a Journey, Nicolaus Copernicus University Press,
Toruń 1998
21
Wybourne’s mechanism of intensity borrowing
in lanthanides in crystals
Andrzej Kȩdziorski
Instytut Fizyki
Uniwersytet Mikolaja Kopernika
ul. Grudzia̧dzka 5, PL-87-100 Toruń, Poland
Following Wybourne, the concept of the so-called intensity borrowing mechanism
is applied to explain the unusual transitions observed in the lanthanides in various
crystals. In particular, the discussion is focused on the description of the contributions that are due to the magnetic dipole transitions. The approach is based on
double perturbation theory, within which the perturbative effect of the crystal field
and spin-orbit interaction is taken into account. The contributions to the transition amplitude up to the fourth order are derived, and new selection rules for the
magnetic dipole transitions are presented.
22
Electronic interactions of the A X doublet states in the 12 C16 O+ , 13 C16 O+
and 14 C16 O+ ion isotopomers from deperturbation analyses of A − X and
B − X bands.
Ryszard Kȩpaa , I.Piotrowska-Domagalaa , M.Zachwiejaa and John A. Coxonb
a
Atomic and Molecular Physics Laboratory
Institute of Physics, University of Rzeszów
Al. T. Rejtana 16A, 35-310 Rzeszów, Poland
b
Department of Chemistry, Dalhousie University
Halifax, Nova Scotia, Canada B3H 4J3
The effects of the strong interaction between first excited A2 Πi and the ground
X Σ+ states in the spectrum of the CO+ ion molecule were recorded, analysed and
fitted. The emission spectra of the Comet–Tail (A2 Πi → X 2 Σ+ ) system bands,
recorded in our laboratory as well as analysed by us and the other available spectra of the Baldet–Johnson (B 2 Σ+ → A2 Πi ) system bands were the source of the
information about the mentioned interaction. Under high resolution and by using
Th standard lines as a calibration spectrum seven bands belonging to three CO+
isotopomers were recorded by means of conventional photographic spectroscopy.
In the 12 C16 O+ isotopomer two new 0–2 and 5–0 bands and earlier observed 0–0
[1], 10–1 and 10–2 Comet–Tail bands [2] as well as 0–0 and 1–0 Baldet–Johnson
bands [3] were anlysed or reanalysed and three sets of the interaction parameters
for the A(v = 0) ∼ X(v = 10), A(v = 5) ∼ X(v = 14) and A(v = 10) ∼ X(v = 18)
vibronic levels were calculated.
In the 13 C16 O+ isotopomer three new bands: 1–0, 6–0 and 6–1 of the Comet–Tail
system as well as earlier observed 0–1 Baldet–Johnson band [4] and two interactions
of A(v = 1) ∼ X(v = 11) and A(v = 6) ∼ X(v = 14) were reanalysed and analysed
and fitted for the first time, respectively.
In the 14 C16 O+ isotopomer two bans: 2–0 and 2–1 and the interaction of the
A(v = 2) ∼ X(v = 12) were analysed for the first time.
As a result the new or improved molecular constants as well as the electronic
interaction parameters for the A2 Π and X 2 Σ+ states were calculated. Especially the
values of electronic matrix elements of the spin–orbit–H SO and rotation–electronic–
H RE interaction were calculated and confronted to the theoretical predictions [5].
2
[1]
[2]
[3]
[4]
[5]
[6]
D.H. Katayama, J.A. Welsh, J. Chem. Phys. 75, 4224 (1981).
H. Bulthuis, Proc. Acad. Sci. Amsterdam, 38, 604 (1935).
Z. Jakubek, R. Kȩpa, A. Para, M. Rytel, Can. J. Phys. 65, 94 (1987).
Z. Jakubek, J. Mol. Spectrosc. 131, 207 (1988).
D.L. Cooper, J. Chem. Phys. 37, 2148 (1982).
J.A. Coxon, S.C. Foster, J. Mol. Spectrosc. 93, 117 (1982).
23
Brian Wybourne and Schur functions
Ronald King
School of Mathematics
University of Sothampton
SO17 1BJ Southampton, England
A number of problems involving Schur functions that were either posed or studied
by Brian Wybourne will be discussed. These include some that have been solved,
some for which a solution has been conjectured, as well as some that remain completely open.
24
Rotational energy band in molecular ring magnets
Marian Kuzma
Institute of Physics,
University of Rzeszów
Rejtana 16A, 35-310 Rzeszów, Poland
Magnetic materials of nanometer length scales offer new fields of research in theoretical physics as well as for applied science, e.g. spintronic [1-4]. The innovation
of quantum computing bases on one- particle quantum states [1,2]. Therefore, nanomagnets which will be implemented, for this purpose should possess two principal
properties: (a) a great total spin of the ground state, and (b) a high barrier for spin
reversal.
Nanomagnets cover not only magnetic thin films, multilayers, nanodots arrays,
but also magnetic clusters consisting of a very small or dozens number of magnetic
ions. Latter are called single molecular magnets (SMM) [2]. The number of SMM
compounds with properties approaching these requirements have been synthesized
recently.
The molecular magnetic clusters are formed by a small number of transition metal ions interconnected together by nonmagnetic ions or by organic ligands which
provide magnetic isolation of the clusters. Therefore such nanosystems exhibit single
particle magnetic properties with a large spin. A special group of molecular magnets
are molecular rings [4]. Schnack and Luban [5] find that the minimal energies in
ring structures can be described according to the Lande interval rule giving the
rotational energy band. In present paper we compare excited rotational energy
bands of Heisenberg rings with exact solutions given by Bethe Ansatz .
[1] D. D. Awschalom, D. Loss, N. Samarth, Semiconductor Spintronics and Quantum Computation, Springer - Verlag 2002
[2] A. Caneschi, D. Gatteschi, C. Sangregorio, R. Sessoli, L. Sorace, A. Corina, M.
A. Novak, C. Paulsen, W. Wernsdorfer, J. Magn. Magn. Mater., 200, (1999)
182.
[3] M. N. Lauenberger, D. Loss, Nature 410, (2001) 789.
[4] J. Schnack, Molecular Magnetism, in Lecture Notes in Physics 645, Springer,
Berlin, Heidelberg, (2004) 155-194.
[5] J. Schnack, M. Luban, Phys. Rev. B 63 014418 (2001)
25
Bound and continuum spectral properties
of atmospherically relevant molecules
Carmen Lavı́n
Departamento de Quı́mica Fı́sica y Quı́mica Inorgánica
Facultad de Ciencias
Prado de la Magdalena, s/n, E-47005 Valladolid, Spain
A variety of simple molecules, of which molecular nitrogen, nitric oxide and carbon monoxide are good representative, are particularly challenging in atmospheric
studies. Photoabsorption, photodissociation and photoionization processes resulting
from the interaction of molecular nitrogen molecular with solar UV radiation play an
important role in the energy balance of the Earths upper atmosphere. In addition,
absolute oscillator strengths in the spectrum of nitric oxide are of considerable interest, since the molecule is an important constituent of hot air and occurs also in the
upper atmosphere. The importance of carbon monoxide to atmospheric processes,
combustion chemistry, and astrophysical phenomena has made the spectroscopy and
reaction kinetics of this molecule a current topic of research. Absorption oscillator
strengths and Einstein emission coefficients for electronic transitions originating in
the ground state and giving rise to several Rydberg series, as well as photoionization cross sections for transitions to the continuum, of molecular nitrogen, nitric
oxide and carbon monoxide, have been presently studied with the molecular-adapted
quantum defect orbital (MQDO) procedure [1]. The MQDO method has proven to
yield reliable intensities for Rydberg transitions in a variety of molecular species.
[1] I. Martı́n, C. Lavı́n, A. M. Velasco, M. O. Martı́n, J. Karwowski and G.H.F
Diercksen, Chem. Phys. 202 (1996) 307
26
Alternating Sign Matrices and Gelfand-Tsetlin Polynomials
James D. Louck
Theoretical Division,
Los Alamos National Laboratory,
Los Alamos, NM 87545, USA
e-mail: [email protected]
Alternating sign matrices occur in the description of the states of a two-dimensional
model of square ice. The general definition of these matrices is given in terms of
Gelfand-Tsetlin patterns, and the origin of Zeilberger’s counting formula for the
number of alternating sign matrices of order n is reviewed. The properties of the
well-known Gelfand-Tsetlin polynomials, which are defined over arbitrary matrices
of order n, are developed for the special case of alternating sign matrices.
27
Kostka matrices at the level of bases
Barbara Lulek and Tadeusz Lulek
Rzeszów University of Technology
Wincentego Pola 6, PL-35-959 Rzeszów, Poland
We point out that the notion of Kostka matrices, related to multiplicities of
irreps of the symmetric group (the shape of a Young diagram) in transitive representations determined by a Young subgroup (the weight), can be formulated at the
level of bases. The relation to the Robinson - Schensted algorithm is discussed along
the Weyl duality between the action of the symmetric and unitary groups on the
N -th tensor power of the defining representation of the unitary group U (n). We
compare this description with other formulations, which have been presented in several lectures on the series of summer schools on theoretical physics ”Symmetry and
Structural Properties of Condensed Matter”. In particular, we raise the question of
choice of repetition labels for Kostka and Littlewood - Richardson decompositions.
28
Schur functions, the duality of Weyl, and rigged string configurations
Barbara Lulek and Tadeusz Lulek
Rzeszów University of Technology
Wincentego Pola 6, PL-35-959 Rzeszów, Poland
Schur functions, representation theory of symmetric and unitary groups, combinatorics of Young and Weyl tableaux, and all that, are interpreted in the framework of
the Heisenberg magnetic ring. Exact solutions of Bethe Ansatz, classified by rigged
string configurations, can be put within the combination of combinatoric bijections
of Robinson - Schensted and Kerov - Kirillov - Reshetikhin into some compact areas of the classical configuration space of the Heisenberg model. We point out a
somehow astonishing result that tiny details of solutions of highly nonlinear algebraic equations of Bethe Ansatz are already coded, in a combinatorially bijective
way, in each orbit of the natural action of the symmetric group of permutation of
nodes of the magnetic ring on the set of all magnetic configurations - the space of
all ”positions” of the system.
29
Stark maps and Rydberg transitions for atoms in external electric field.
The Stark Quantum Defect Orbital (SQDO) approach
I. Martı́n, J. M. Menńdez and A. M. Velasco
Departamento de Quı́mica Fı́sica y Quı́mica Inorgánica
Facultad de Ciencias, Universidad de Valladolid
Prado de la Magdalena, s/n, E-47005 Valladolid, Spain
e-mail: [email protected]
During the last few decades, the experimental behaviour, and further analysis,
of excited Rydberg atoms in the presence of an external electric field, has attracted
increasing attention. This has been partly due to the recent development of sophisticated experimental techniques which allow one to generate state-selective Rydberg
atoms and, further, to monitor the effects of the gradual increase in field strength
on such atoms. In this context, the development of new theoretical models, whose
predictions could analyse the measurements to a high precision, provide a physical
insight into the experiment and, even suggest the design of more advanced experimental techniques, is considered to be of the outmost importance.
The Stark structure of the Rydberg states of the lighter alkali atoms, Li, Na and
K, as well as the Stark oscillator strength distribution for Li, have been achieved
through direct diagonalisation of the field-perturbed Hamiltonian matrix. As basis
functions, quantum defect orbitals [1] have been adopted, with the entire Stark
Quantum Defect Orbital (SQDO) procedure being totally analytical.
The presently obtained Stark maps are in excellent agreement with those available
from experiment. Good accord has also been found between the presently calculated
oscillator strength distribution, within a Stark manifold, and those supplied by of
other theoretical procedures and experimental measurements. The appropriateness
of the SQDO approach for accurately dealing with properties related to the Stark
effect in atoms has been suggested [2,3]. Application to small molecules is also in
progress.
[1] G. Simons, J. Chem. Phys. 60, 645 (1974); I. Martı́n, and G. Simons, J.
Chem. Phys. 62, 4799 (1975).
[2] J. M. Menéndez, I. Martı́n and A. M. Velasco, J. Chem. Phys. 119, 12926
(2003).
[3] J. M. Menéndez, I. Martı́n and A. M. Velasco, Theor. Chem. Acc., in press
(2005).
30
Crystal-Field Analysis for U 3+ Ions in Low Symmetry Sites
of Three Uranium(III) Trichloride Hydrates
Agnieszka Mech(a) , Miroslaw Karbowiak(b) , Zbigniew Gajek(a)
and Janusz Drożdżyński(b)
(a)
Institute of Low Temperature and Structure Research,
Polish Academy of Sciences,
ul. Okólna 2, 54-422 Wroclaw, Poland
(b)
Department of Chemiostry
University of Wroclaw,
ul. F. Joliot-Curie 14, 50-383 Wroclaw, Poland
Three different uranium(III) trichloride hydrates of the formulas U Cl3 · 2H2 O ·
CH3 CN , U Cl3 · 6H2 O, U Cl3 · 7H2 O have been isolated from acetonitrilic solutions of U Cl4 , with a 1 - 5 % water content, by reduction with liquid zinc amalgam. The good quality 4.2 K absorption spectra were recorded in the 4000-30000
cm−1 range for polycrystalline samples of these compounds. The observed crystalfield levels were assigned and fit to the parameters of the simplified angular overlap model (AOM) as well as a semi-empirical Hamiltonian representing the combined atomic and one-electron crystal-field interactions [J. Mulak and Z. Gajek,
T heEf f ectiveCrystalF ieldP otential. Elsevier, Amsterdam 2000]. The starting
values of the AOM parameters as well as the ratios between them were obtained
from abintio calculations. The analysis of the spectra enabled the assignment of 83,
52 and 94 crystal-field levels with a relatively small r.m.s. deviations of 32 cm−1 , 39
cm−1 and 38 cm−1 for U Cl3 · 2H2 O · CH3 CN , U Cl3 · 6H2 O and U Cl3 · 7H2 O respectively. The AOM model has been found to be very useful in the determination of the
initial values of Bqk parameters. It is especially important in those cases, when the
low site symmetry or an insufficient number of available experimental data, prevents
the application of the Bqk parameterization scheme at initial stages of analysis.
31
Spectral properties of ions under external confinement:
Effect of strongly coupled plasma
A. N. Sil and P. K. Mukherjee
Department of Spectroscopy
Indian Association for the Cultivation of Science
Jadavpur, Kolkata 700032, India
Systematic investigations have been performed to study the effect of strongly
coupled plasma on the spectral properties of several one and two electron ions. The
effect of strong coupling is obtained by using the Ion Sphere (IS) model of surrounding plasma which assumes overall charge neutrality inside the ion sphere or
Wigner-Seitz sphere whose radius is determined by the plasma density. Properties
like the ionization potentials, dynamic polarizabilities, low lying exited energy levels, allowed oscillator strengths and transition probabilities have been evaluated for
several strengths of the plasma which are manifested by different ion sphere radii
within which the system is assumed to be confined. The overall effect of the plasma
confinement is to lower the ionization potential of the system and to put the system to gradual instability. The linear response properties of the system under an
external harmonic perturbation have been investigated using time dependent variation perturbation theory. Transition properties like the low lying excited states,
oscillator strengths and transition probabilities have been calculated under different
plasma densities and the spectral line shifts compared to free ions estimated. The
number of bound excited states have become finite under the external confinement.
Several interesting trends have been observed along the isoelectronic sequence.
32
Degrees of validity of models for the description of doubly excited states
of H− and He
Cleanthes A. Nicolaides
Physics Department,
National Technical University,
Athens, Greece
and
Theoretical and Physical Chemistry Institute,
Hellenic Research Foundation,
48 Vasileos Constantinou Avenue,
Athens 11635, Greece
e-mail : [email protected]
Over the past four decades, various models have been proposed for the description
of doubly excited states (DES) of H − and He, where electron correlation is strong.
These are based on pictures and approximations, on coordinate systems and on
group theoretic considerations that are different from those associated with the
notion of electron configurations defined by the central field. In recent years, the
degree of validity of such models has been tested by comparing their results to those
of theory dealing with the reliable ab initio solution of the Schrdinger equation with
the full Hamiltonian.
The lecture will review the comparison of results from three models and from
the state-specific theory for the solution of the Schrdinger equation for real and
for complex eigenvalues [1]. The first model is that of dipole resonances of H −
introduced by Gailitis and Damburg G-D [2], which is associated with the properties
of a 1/r 2 -like potential. Extensive and definitive computations using the complex
eigenvalue Schrdinger equation (CESE) theory of resonance states have revealed
that, due to electron correlation, the G-D model can be used as a zero-order model
for classification of states, based on which it is found that there are unperturbed
as well as perturbed series, in analogy with the Rydberg spectra of neutrals and
positive ions (a product of a 1/r-like potential) [3].
The second model is that of Wullfman [4] and, especially, of Herrick and Sinanoğlu
[5] on the (K, T ) quantum numbers and the related classification scheme. Results
for DESs going up to the N = 25 manifold have shown that the (F,T) scheme,
introduced by Komninos et al [6], where F = N − K − 1, and N , K are not good
numbers anymore, constitutes a consistently better representation, especially as the
relative significance of electron correlation increases, as it does in the high-lying DES
of H − . Specifically, the (F, T ) “purity” coefficient is very high, close to one [7,8].
The third model is that introduced by Herrick and Kellman [9] using the picture
of a triatomic molecule for the qualitative description of the spectra of DES. The n2
“vibrational” quantum number is equal to the number of nodes of the density ρ(θ12 )
of the corresponding DES [9]. The ab initio results of [8] for ρ(θ12 ) using correlated
wavefunctions for the four lowest states in the N = 10 − 25 manifolds, showed how
the angle opens as excitation increases and provided numerical evidence that n2
gives a valid picture even up to N = 25.
33
[1] C. A. Nicolaides, Int. J. Quantum Chem. 60, 119 (1996), and refs. therein.
[2] M. Gailitis and R. Damburg, Proc. Phys. Soc. 82, 192 (1963).
[3] M. Bylicki and C. A. Nicolaides, Phys. Rev. A 61, 052508 (2000), and subsequent papers.
[4] C. E. Wulfman, Chem. Phys. Lett. 23, 370 (1973); ibid, 40, 139 (1976).
[5] D. R. Herrick and O. Sinanolu, Phys. Rev. A 11, 97 (1975); O. Sinanolu and
D. R. Herrick, J. Chem. Phys. 62, 886 (1975).
[6] Y. Komninos, S. Themelis, M. Chrysos and C. A. Nicolaides, Int. J. Quantum
Chem. S27, 399 (1993).
[7] S. I. Themelis, Y. Komninos and C. A. Nicolaides, Eur. Phys. J. D 18, 277
(2002).
[8] C. A. Nicolaides, S. I. Themelis and Y. Komninos, J. Phys. B 35, 1831 (2002).
[9] D. R. Herrick and M. E. Kellman, Phys. Rev. A 21, 418 (1980).
34
Kostka and Littlewood-Richarson coefficients:
Distibuted Calculus
Janvier Nzeutchap and Frédéric Toumazet
Laboratoire d informatique, Institut Gaspard-Monge,
Universitd́e Marne-la-Vallée
Fondation Deutcsh De La Meurthe,
Cité Internationale Universitaire de Paris,
37, Boulevard Jourdan, 75014 Paris, France
We present a (C & Maple) client-server application which, given two or three
partitions λ, µ (and ν), constructs the corresponding Hives model, generates a Maple
code to compute K or LR coefficients for any dilatation factor N , translates this
code into C and then sends it to various computers connected to the local network.
Any computer which successfully compiles the C code will later receive a certain
nomber of dilatation factors (N ), compute corresponding coefficients, and send them
back to the server which interpolates to determine the generating polynomial P .
Translating the code from Maple into C increases the speed of calculation of individual coefficients. And so does the Distribution of calculus for the determination
of P .
35
Quantum Monte Carlo study of the electromigration wind force
Ryszard Olchawa
Instytut Fizyki
Uniwersytet Opolski
ul. Oleska 48, PL-45-052 Opole, Poland
The hydrogen electromigration wind force in metals is calculated using Quantum
Monte Carlo methods.
36
Aharonov-Bhom effect for pedestrians
Josep Planelles, J.I. Climente and J.L. Movilla
Departament de Ciéncies Experimentals
Universitat Jaume I
Box 224, E-12080 Castelló, Spain
We will talk about the Aharonov-Bhom (AB) effect:
1.
2.
3.
4.
5.
A quantum ring pierced by a magnetic field: the AB effect.
Persistent currents.
Two electrons: Coulomb term and ”Factional AB effect”.
Mobius strip: AB effect and anti-periodic boundary conditions.
Electron-Hole pair in quantum rings: ”optic AB effect”
37
Brian Wybourne and the hyperfine structure
in the spectra of rare earths in crystals
Marina N. Popova
Institute of Spectroscopy of RAS,
142190 Troitsk, Moscow region, Russia,
[email protected]
Brian Wybourne was always interested in explaining peculiarities of the hyperfine structure in the spectra of rare earths in crystals. In the early sixties, he has
suggested an idea how to account for the forbidden EPR lines observed in holmium
trichloride [1]. Brian spoke of an admixture of different states by nondiagonal part of
the magnetic hyperfine interaction. Much later, we have found an elegant confirmation of this idea in optical spectra of holmium in lithium-yttrium fluoride (obtained
in my group using high-resolution Fourier-transform spectroscopy) [2, 3].
Brian was greatly impressed by these broad-band optical spectra that exhibed
a well-resolved hyperfine structure. He has used these spectra as an example to
demonstrate the calculative power of his operator methods to students of Copernicus University in Torun, during his lectures in 1994 [4]. It was Brian who has
initiated a long-lasting cooperation between my group and Glynn Jones and Mike
Reid (University of Canterbury, New Zealand) and Marjori Mujaji (University of
Zimbabwe).
My presentation is dealing with the topics mentioned above and some others.
[1] Wybourne, B. G., J. Chemical Physics, 1962, 37, N8, 1807.
[2] Agladze, N. I., Vinogradov, E. A., Popova, M. N., Opt. Spectrosc. (USSR),
1986, 61,1.
[3] Marina N. Popova and Nikolay I. Agladze, Hyperfine structure in optical spectra of LiYF4-Ho3+: forbidden transitions, Molecular physics 102, N 11X12
(2004) 1377X1380.
[4] Wybourne, B. G., The Application of Symmetry Concepts to Physical Problems. Analysis of Hyperfine structure in Crystals, Copernicus University in
Torun, 1994.
38
Hydrogenic atoms in strong magnetic field
Andrzej Poszwa, Andrzej Rutkowski
Warmia and Mazury University in Olsztyn
Department of Physics and Computer Methods
ul. Żolnierska 14, PL-10-561 Olsztyn, Poland
The highly accurate solution of the Schrodinger equation in the form of common
Landau exponential factor multiplied by a power series in two varibles, the sine of
the cone angle and radial varible is completed by the first-order relativistic correction
calculated within the framework of direct perturbation theory (DPT). In the limit
of low and moderately strong magnetic fields the solution of the Dirac equation in
a basis of spherical spinors is obtained in the full four-component form.
39
Algebraic “solution” of unsolvable problems
David J. Rowe
Department of Physics,
University of Toronto
60 St. George Street,
ON M4W 3W9 Toronto, Canada
Algebraic methods have been used extensively for the solution of exactly solvable
problems. The algebraic solutions of the Schrodinger equations for the harmonic
oscillator and hydrogen atom and their extensions are among the best known and
were of considerable interest to Brian Wyourneand his colleagues. This talk will
focus on the use of algebraic and factorization methods in the solution of a radial
Schrodinger equation for a system that is not exactly solvable.
40
Interatomic potential calculations of small and big molecules
P. Jasik, K. Piechowska, M.-C. Bacchus-Montabonel, Y. S. Tergiman
and J. E. Sienkiewicz
Wydzial Fizyki Technicznej i Matematyki Stosowanej,
Politechnika Gdańska,
Narutowicza 11/12, PL-80-952 Gdańsk, Poland
A broad variety of spectroscopic phenomena is observed for lithium dimers and
the accurate specification of its electronic structures is of principal importance. For
the applied sciences, Li2 is of interest because it is an active laser medium.
The second part of presentation deals with the potential energy surfaces of vinoxy radical (CH2 CHO) and discusses the process of photodissociation through the
avoided crossings and the conical inersections.
41
Symmetry analysis in the description of structural phase transitions in
Laves phase hydrides
Wieslawa Sikora, Janusz Malinowski, Henryk Figiel
Faculty of Physics and Applied Computer Science
AGH University of Science and Technology
al. Mickiewicza 30, PL-30 059 Kraków, Poland
The superstructure as the ordering of some ”property” on the initial crystal structure may be realised as the result of phase transition. Any property of crystal localised on atom sites may be described by a Wannier function S defined on some
set of equivalent positions. The presentation of this function on in the usually used
frame of coordinates related to the crystallographic system takes advantage of translational symmetry only. The refinement of unknown structure after phase transition
is given from X-ray or neutron diffraction experiment, and requires the supposition
of a model structure defined by as small as possible number of free parameters. This
is the place, where the symmetry analysis method should be used. The presentation of model structures in the frame of basic vectors of irreducible representations
of the initial symmetry group (BV) (instead in the frame of x,y,z crystallographic
system) is the best matching to the symmetry of the problem and it provides the
simplest (requiring the lowest number of independent parameters) form of the structure description: The form of BV-s is given by the theory and may be calculated for
example by the computer program MODY. The coefficients of the linear combination of BV-s should be obtained from refinement procedure. These coefficients form
good order parameters of phase transitions. The theory of representations allows to
use them for construction invariants of required symmetry group and following theoretical discussions, for example the connections between the ordering of different
properties in the given crystal. n this work the symmetry analysis is applied to the
discussion of structural phase transitions (order-disorder and displacement types) in
non saturated Laves phase hydrides RMn2 (R = rare earth), induced by different
concentration of hydrogen (deuterium) atoms and temperature. The Laves phases
easily absorb hydrogen, which enters the interstitial sites. One of the most intriguing problems is a tendency for ordering of hydrogen atoms. The neutron diffraction
studies prove appearance of hydrogen ordered in superstructures connected with
lowering of structure symmetry of the hydride. The aim of this work is to analyse
the conditions determining hydrogen ordering and possible displacements of R and
Mn atoms.
42
Recent progress in the investigation of the praseodymium atom
Ewa Stachowska
Katedra Fizyki Atomowej Politechnika Poznanska
ul. Nieszawska 13b, PL-60-965 Poznań, Poland
Praseodymium belongs to the lanthanide group, where the spectroscopic properties are determined by the fact that the binding energies of 4f, 5d, 6s and 6p electrons
are of about the same magnitude. This gives rise to several overlapping configurations and strong configuration mixing effects in the fine (fs) and the hyperfine (hfs)
structure of Pr.
On the other hand, according to the electromagnetic interaction between the
nucleus (I = 5/2) and the electronic shell, each level with a total electronic angular
momentum J ≥ 5/2 splits into 6 hyperfine sublevels. Due to this most of the
praseodymium spectral lines show a typical hyperfine pattern built of 6 diagonal
components, independently of the J-value of the levels involved in the transitions.
Due to the large number of lines with similar patterns, detailed knowledge about
the fine as well as the hyperfine structure is essential for the classification of this
spectrum.
Because of the extremely complex nature of this spectrum earlier analyses of
fs and hfs of the even configurations made by Ginibre [1,2] were limited to a two
configurations system, 4f 2 5d6s2 + 4f 2 5d2 6s, for the lower energy range, where the
mixing effects with the other excited configurations are small.
Due to the close cooperation with the experimental group of Guthörlein from
Hamburg we were able to extend the semi-empirical fs- and hfs-analysis by including
five even configurations [3] and more recently fifteen.
This work was partially supported by PUT (project BW 63-025/05) and by KBN
(grant 136/E-362/SPB).
[1] A. Ginibre, Physica Scripta 23, 260 (1981)
[2] A. Ginibre, Thesis, Paris (1988)
[3] J. Ruczkowski et al., Physica Scripta 68, 133 (2003)
43
Relativistic and correlation effects on spectroscopic constants of
Hg[(CO)2 ]2+
Jacek Styszyński
Instytut Fizyki
Uniwersytet Szczeciński
Wielkopolska 15, PL-70-451 Szczecin, Poland
e-mail [email protected]
There is a growing class of metal carbonyl compounds with unusually high ν(CO)
frequency, which is due to diminished M–CO π backbonding (transfer of electronic
charge from metal dπ orbitals to the antibonding π ∗ of CO). In case of mercury
carbonyl cation, Hg[(CO)2 ]2+ , relativistic effects are expected to play substantial
role in determining the character of the bond and in the prediction of spectroscopic
constants of the complex.
Those effects are studied in our paper using 4-component Dirac-Fock (DF) and
relativistic version of MP2, CISD, CCSD and CCSD(T) methods. The character
of the Hg–CO bond using Mullikan population analysis is also studied. Obtained
results for the bond distances R(Hg–C) and R(C–O) and the mean bond dissociation
energy BDE show good agreement with experimental data. However the calculated
values of ν(CO) and ν(Hg–C) are too large compared to experimental data. This
disagreement requires further investigation.
44
Persistent currents in carbon nanotubes
Marek Szopa, M. Margańka and E. Zipper
Institute of Physics
University of Silesia
Uniwersytecka 4, PL-40-007 Katowice, Poland
The unusual band structure of carbon nanotubes (CNs) results in their remarkable magnetic properties. The application of magnetic field parallel to the tube axis
can change the conducting properties of the CN from metallic to semiconducting
and vice versa. Apart from that the Aharonov-Bohm effect results in orbital persistent currents and magnetic moments in the nanotube. These moments are studied
both in pure and hole- or electron-doped CNs, single and multiwall. The orbital
magnetic moment in pure CNs depends uniquely on their original conducting properties, length, and temperature but it does not depend on the nanotube radius or
the particular chirality. In doped nanotubes the magnetic moments can be strongly
altered and depend on the radius and chirality. Temperature can even change their
character from diamagnetic at low T to paramagnetic at high T . Full electron-hole
symmetry in doped tubes is a reflection of the symmetry of the dispersion relation.
45
Polyhedral Combinatorics of Democratic Recoupling & the Role of Sn
Subgroup Chains (onto [2](S2 )) as Group Measures defining (Liouvillian)
SU (2) × Sn Carrier Space Maps associated with (A)[X]n NMR Systems:
Spin Dynamics over [ñ], [1˜n ](Sn ) Irrep-Salients.2
Francis P. Temme
Department of Chemistry
Queen’s University
Kingston, Ontario, Canada K7L 3N6
e-mail: [email protected]
Earlier views of the roles of polyhedral combinatorics and democratic recoupling
in defining Sn group invariants and their (time-reversal invariance)-based independant cardinalities - as given in {2005, Proc. Roy. Soc., Ldn., 461, 321; 2004, J.
Magn. Reson., 167, 119} - are reconsidered here as part of the conceptual structure
of SU (2) × Sn dual mapping and the (Liouvillian superboson) properties (reviewed
in {2002, Int. J. Quantum Chem., 89, 429; 1993, Physica A198, 245}) for their
impact on the tensorial formalisms of NMR spin dynamics{ Sanctuary & Halstead,
1991, Adv. Opt. Magn, Reson., 15, 97}. Various invaluable theoretical structures
are inherent in processes that view either different groups as subgroups of the
GLn group - a central topic in much of Brian Wybourne’s 1970 work, Symmetry
Principles..-, or lower indexed groups as subgroups of some specific indexed
Sn group, as advocated by the (late) J. Q. Chen in his 1988 particle symmetry
physics monograph. These are shown to be of central importance in modern novel
context of (dual group) studies of NMR spin dynamics. The carrier subspaces of
dual group mapping (in its above-cited Liouville space realisation) as:
Ũ × P(Sn ) : H̃ → H̃ {D k (U) × Γ̃[λ̃] (P) | Ũ ∈ SU (2); k integer rank; P ∈ Sn }
for
H̃ ≡
X
H̃ v
v
is seen now as being explicitly labelled, in order to retain the SR properties of
the dual group. Clearly, the subgroup chains used in this labelling process are
actually group measures that represent the Sn invariants. Their recent derivations
as general 2n properties draw again on time-reversal invariance(TRI), but now in
the context of n-fold polyhedral modelling of democratic recoupling(DR). In this
augmented (strictly non-linear) general recoupled form, the Weyl (I • I)i bracket
of the recoupling network corresponds to some lattice point of an appropriate nfold totally regular polyhedra. The application of such TRI-DR modelling, via the
polyhedral combinatorial networks considered here, is concerned with the specific
2
Dedicated in memoriam to Prof. Brian G. Wybourne, 1935 -2004.
46
and central roles played by the [ñ], and [1˜n ] salients in the analytic spin dynamics of
(A)[X]n NMR spin systems. As in an earlier orthogonal subgroup chain study of DR
applied to conventional spectroscopies, {Galbraith, 1971, J. Math. Phys., 12, 782},
questions of the mathematical determinacy of the general analytic formalism soon
arise - i.e., as consequence of such degenerate systems being of high multiple invariant
form. What is clear for the NMR spin dyamics, based on dual group mapping and
Sn invariants, is that the commutation associated with strongly intracluster coupled
spin systems give rise to specialised role for [ñ], [1˜n ] salients in the analytic form
of spin dynamics. For the (most general) (A)[X]n NMR systems, the [ñ] salient
represents a null space -the origin of a well-established cw rule in NMR - whereas
all the spin dynamics induced by the dominant JXX 0 coupling is associated with the
NMR-inaccessible totally antisymmetric irrep-defined salient. Additional questions
arises in the highest 2n-fold spin problems on n-polyhedral structures for which
the invariant cardinality is a defined mathematical quantity. Earlier works of ours
{1999, Eur.Phys. J. B11, 177, et loc. cit.; 2002, J.mol. Struct., THEOCH., 578,
145.} have surveyed the comparable full range of mathematical determinate S ↓ G
automorphic group embeddings known for dual group-based NMR spin systems.
Various discrete mathematics resources have been drawn on here, including: e.g.
Kerber et al, 1993, ”Symmetrica” package, Sagan’s 1991 monograph on Sn group
-based algorithms, and others.
47
Representation of symplectic groups:
algorithms using symmetric functions
Frédéric Toumazet
Laboratoire d’Informatique de Paris Nord
Univeristé Paris 13
99 avenue J.-B. Clément,
93430 Villetaneuse, France
The aim is the description of some algorithms used in formal computation (for
example in SCHUR program) for the study of the representations of the real symplectic group Sp(2n, R).
48
Hidden symmetry and the solutions of some boundary
value problems for nonlinear Poisson equation
Igor Tralle
Institute of Physics,
Mathematics & Natural Sciences Department,
University of Rzeszów
Al. T. Rejtana 16A, 35-310 Rzeszów, Poland
The method of searching for the solution to Dirichlet problem for nonlinear Poisson equation (NPE) on the plane is proposed. We prove that the Dirichlet problem
for NPE on the plane can be solved for almost arbitrary simply connected domain
which can be mapped conformally inot unit circle. If the function which maps this
domain is known, the solution can be expressed in an explicit form which is constructed in the paper. Our appoach is based on the onservation that there exsists
Baeklund transformation which interrelates the solution of NPE with the solution
of Laplace equation. Thus, in order to solve Dirichlet problem for NPE, one should
solve an appropriate auxiliary boundary value problem for the Laplace equation.
This boundary value problem can be easily formulated for the cirlce; it is Cauchy
problem and in order to be consistent, the boundary condition must be represented
by the π-periodic function. This result is generalized to the solution of Dirichlet
problem for NPE in almost arbitray simply connected doamin on the plane. The
solution of Neumann problem for NPE is also considered and it is shown that Neumann problem is reduced to the solution of some other Dirichlet problem for NPE.
49
The mesoscopic and nanoscopic Heisenberg magnets
Andrzej Wal, M. Kuzma
Institute of Physics,
University of Rzeszów
Al. T. Rejtana 16A, 35-310 Rzeszów, Poland
The Bethe equations for one-dimensional Heisenberg magnet are characterised by
winding numbers and a number of spin deviations r. The solution of these equations
can be easily found for a small number of spins. However, for the chain consisting
of several dozen nodes the problem of solving Bethe equations is very complicated
even in a numerical way. We propose a method for finding selected solutions for
such a system of spins starting from an asymptotic case. These solutions provide a
full description of one-dimensional magnets states, i.e. pseudomomenta and phases.
50
Invariants of mixed qubit systems
Trevor Welsh, Peter Jarvis, Ron King
University of Southampton & University of Melbourne
72 Hariington Lane, Pinhoe, Ex4 8NS Exeter, U.K.
Using a mixture of algebra and graph combinatorics, we determine the curious
structure of the ring of invariants associated with the mixed quantum 2 qubit system. We show that this structure has the Cohen-Macaulay form, which, at first
appearance, does not seem to be the case
51
Direct determination of the universal characteristics of the Yang-Lee
singularity in the 2D Ising and 3-state Potts models
Tomasz Wydro, John McCabe
LPMC, Institut de Physique,
Universite de Metz
1, Bd Arago, 57078 Metz, France
The Hamiltonian limit of two dimensional statistical Ising and 3-state Potts models in complex magnetic field is studied numerically. The Phenomenological Renormalisation Group approach is used to locate the critical field value at Yang-Lee
singularity (YLS). The low lying part of the spectrum is calculated. The pattern of
the calculated energy levels and their degeneracies is compared to the predictions
of the ADE classification of the Conformal Field Theory for non-unitary minimal
models.
We obtain a convincing identification of the YLS of the 3-state Potts model with
the with the (A4 , A1 ) minimal CFT. We also study the YLS in Ising model by similar
methods confirming the identification for the YLS of that model.
52
Spin Structure of Mn(II)4 O4 Cluster
Shigeyoshi Yamamoto
Faculty of Liberal Arts, Chukyo University,
101-2 Yagoto-Honmachi, Showa-ku, Nagoya, Aichi 466-8666, Japan
& [email protected]
Polynuclear transition-metal complexes have attracted extensive attention because of their magnetic properties. Mn12 high-spin molecules [1] were synthesized
and are considered as candidates for single-molecule magnet (SMM). Many of them
contain a cubane-like Mn4 O4 core unit. This molecule has been studied also as a
model complex of the oxygen-evolving center within photosystem II (PS II) for long
time even though the recent x-ray analysis at 3.5 angstrom resolution [2] revealed
that it consists of Mn3 CaO3 cluster.
The Mn4 O4 cluster itself has not been isolated yet. Its geometry was studied
by ab initio UHF/DFT method by Kayak and Jena [3]. The optimized geometry
has a distorted cubane-like structure of magnetic moment of 16 µB . However, this
high-spin structure is not consistent with the negative effective exchange integral (J)
value (-0.51 cm−1 ) estimated by Aussoleil et al.[4]. Yamanaka et al.[5] performed
generalized spin orbital (GSO) DFT calculations and obtained also a negative J
value. However, their J value is rather large (-82.5 cm−1 ) and the MINI-3 basis set
used is not sufficient for transition metal atoms. Therefore more reliable calculation
is required.
We have performed DODS-type (different orbital for different spin) UHF and
DFT(B3LYP) calculations for the neutral S=0 to S=10 systems. The basis set used
has triple-zeta plus polarization quality. For Mn the (7433/744/7) CGTF set [6,7]
is split to triple-zeta and one f function contracted from two f functions [8] is added
as a polarization function. Our calculations gave result that the highest spin state
is most stable. The energy difference between the low- and high-spin is as large as
0.3∼0.4 au.
Since the manganese(II) atom has (3d)5 spins, at least 20 MOs should be taken
into account as the active orbitals if one wishes to perform CASSCF calculations
of the Mn(II)4 O4 cluster. For the S=10 highest spin state CASSCF is applicable
because this state can be expressed by a single CSF, but not for the low-spin states
because the number of CSFs increases explosively.
Therefore we have to restrict the configurations. We inspected the UHF natural
orbitals (NOs) and found that as many as 20 orbitals have an occupation number
nearly equal to one. This suggests that spin family generated within these spinup/down MOs may important role to decrease the total energy. We are planning
to perform CI calculations which satify the following conditions. (1) To define the
reference configuration which is anticipated from the occupation numbers of the UHF
NOs. (2) To include all the spin eigenfunctions associated with this spatial reference
configuration. The SGA-CI (symmetric group approach configuration interaction)
method [9,10] is suitable for this purpose.
53
[1] V. McKee, W. B. Shepard, J. Chem. Soc., Chem. Comm. 1985, 158-159
(1985).
[2] K. N. Ferreira et al., Science 303, 1831-1838 (2004).
[3] S. K. Kayak, P Jena, J. Am. Chem. Soc. 121, 644-652 (1999).
[4] J. Aussoleil et al., Inorg. Chem. 28, 3051-3056 (1989).
[5] S. Yamanaka, R. Takeda, K. Yamaguchi, Polyhedron 22, 2013-2017 (2003).
[6] T. Koga et al., Theor. Chem. Acc. 102, 105-111 (1999).
[7] H. Tatewaki, T. Koga, S. Yamamoto, Theor. Chem. Acc. 105, 55-61 (2000).
[8] M. Sekiya et al., Theor. Chem. Acc. 106, 297-300 (2001).
[9] J. Karwowski, Symmetric groups and organic ferromagnets, in Symmetry and
Structural Properties of Condensed Matter, W. Florek et al. eds., World
Scientific, Singapore 1993.
[10] B. G. Wybourne, N. Flocke, J. Karwowski, Int. J. Quantum Chem. 62, 261-264
(1997).
54
Symmetry of two- and four- electron states in solids.
Application to unconventional superconductors
Victor Yarzhemsky
Institute of General and Inorganic Chemistry
Russian Academy of Sciences
31 Leninsky, 119991 Moscow, Russia
The Mackey theorem on symmetrized squares is applied for construction of twoelectron states with zero total momentum in solids (Cooper pairs). It is also proved
that when the Mackey theorem is aplied twice, the four electron states, corresponding to several Young tables are obtained. Possible application of the results to unconventional supercoductors with lines of nodes of pairing functions are discussed.
55