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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