Jacob Katriel

Jacob Katriel

Professor Emeritus
+972-4-4-8293824
+972-4-4-8295703
420
Electronic structure of atoms Representation theory of the symmetric group

The electronic structure of few-electron atoms (in particular near their critical
charge, below which the outermost electron is not bound): In most of its electronic states
(except the states that correspond to non-vanishing electron affinities), the critical charge of
an N-electron atom is Zc = N −1. In these states (but not when Zc < N −1!) the outermost
electron becomes a diffuse hydrogen-like orbital, with an effective charge Z − Zc. In the
corresponding high Z limit of the N-electron atom all the orbitals become asymptotically
hydrogen-like. These observations imply that, for open-shell isoelectronic sequences the
term splitting within a common configuration vanishes at the critical charge and grows
linearly in Z for high nuclear charges. Hence, the term splitting divided by the square of the
nuclear charge vanishes at both Zc and at Z ! 1, possessing a maximum in between
these two limits. I recently observed (combining the virial and the Helmann-Feynman
theorems), that the difference of interelectronic repulsions can be obtained by differentiating
this (non-monotonic) ratio with respect to the nuclear charge. This observation sheds new
light on an issue that I studied many years ago, namely that the interelectronic repulsion
tends to be higher in the lower energy state, within an open-shell configuration. My
work on these issues in the early seventies of the previous century culminated in a revised
understanding of the dynamical origin (“interpretation”) of Hund’s rule ordering in open-shell
atoms [that, though sometimes misrepresented, affected the textbook presentation of this
topic. In some recent textbooks (e.g., Atkins’ Molecular Quantum Mechanics) this revised
treatment attains the ultimate status of common knowledge by being introduced without
reference]. An appropriate modification of the relation quoted above accounts for the fact
that in confining quantum dots no reversal of the interelectronic repulsions takes place. My
current interests in this area involve the examination of heavy open-shell atoms, transition
metal and lanthanide complex ions, and bulk ferromagnetic materials, looking for observable
consequences of the dependence of the spatial wavefunction on the value of the total spin.
Efficient combinatorial algorithms for the characters of the symmetric group
and for the structure constants in its group-algebra: Though these are mathematically
“solved” problems, they become computationally intractable even in applications to systems
consisting of a moderate number of identical particles. Some powerful simplifications (some of
which also apply to the unitary groups, the quantum unitary groups, and to the Hecke algebra
of the symmetric group) were elucidated in a series of articles that I published between the
mid eighties and the late ninties [that became textbook items, cf. T. Ceccherini-Silberstein
et al., Representation Theory of the Symmetric Group, 2010]. These results were applied to
symmetry adaptation of many-body wavefunctions, and to the construction of hyperspherical
wavefunctions, and they were implemented in a widely applied nuclear shell-model code. I
have recently established (joint work with A. Ratten) the equivalence to Kerov’s approach
(that he only reported in a seminar in Paris shortly before untimely passing away in 2000).
Further elucidation of certain rather challenging “details” of the underlying formalism is
definitely called for, and I intend to keep trying for a while.
Combinatorics of boson normal ordering: A modest study of the normally-ordered
expression for the k’th power of the boson number operator, that I published in 1974,
followed, during the “quantum group” hype of the early ninties, by a generalization to
q-bosons (joint work with Maurice Kibler), became the trigger for a mini-avalanche of
activity that is still responsible for a steady output of generalizations [reviewed by Mansour,

Combinatorics of set partitions, 2013]. I have also been involved in related work on generalized
boson operators and on generalized coherent and squeezed states. All these are of interest
both because of their potential physical applications and because of the rich algebraic
combinatorics involved.
Permutational symmetry classification of identical higher-spin particles:
My original interest in this issue was motivated by the fact that the simplicity of the
corresponding problem for spin-½ particles, encapsulated by the Dirac identity that associates
a transposition with the scalar product of the individual spin operators of the particles
involved, is not shared by higher spins. Specifically, the one-to-one correspondence between
the total spin and the label of the irreducible representation does not hold for systems
of identical particles with elementary spins σ ≥ 1. The current interest in higher spin
Bose-Einstein condensates provides a motivation for further refinement and extension of
results concerning the classification of such systems, that I published over a decade ago.
Is there a bound triplet (1σg, 1σu) (zero bond order) state of a (heavy hole) biexciton?
Such a state was predicted to be feasible in a polar crystal [thesis by Kamer Murat, that I
supervised in the late seventies] but so far never observed. Recent developments in very low
temperature solid state spectroscopy, as well as theoretical progress that suggests a necessary
modification of the Haken-potential assumed in that study, call for a more up-to-date
treatment of this problem. I may get to doing something about this one of these days.
Past research highlights that I am unlikely to return to: Nuclear spectroscopy
and thermoluminescnce. Mean-field theory for non-isotropic magnetic materials (a highlight
being a generalized Weiss equation that yields the magnetization vector in terms of the
gradient of the spin-hamiltonian) and for nematic liquid crystals. Reentrance. Generalization
of the Bogolyubov-Tyablikov approximation. The non-linear eikonal approximation (one
highlight being a non-linear self-consistency equation that is analytically soluble in terms
of Jacobi elliptic functions, which is relevant to both four-wave mixing and to bistability in
the non-linear Fabry-Perot resonator). The super-hamiltonian formalism for excited states,
allowing the formulation of excited-state and degenerate Hohenberg-Kohn theorems.

Born on February 20, 1945, in Petach Tikva, Israel.
B.Sc. (Summa cum laude), Technion 1967. M.Sc., Technion, 1970. D.Sc., Technion, 1972.
Nuclear Research Center – Negev: 1967-1973.
Technion: Lecturer 1973, Senior Lecturer 1974, Associate Professor 1976, Professor 1982.
Head, Physical Chemistry Division, 1984-1986. Chairman, Dept. of Chemistry, 1989-1991.
Chairman, Audit and Control Committee, Technion Faculty Union, 1997-1999.
Incumbent, Abronson Family Chair of Chemistry, 1989-2004. Professor Emeritus, since 2004.
Nazareth Academic Institute: Chairman, Department of Chemistry, and Chairman of the Supreme Academic Council, 2010-2013.
Sabbaticals and other long academic visits (Department of Chemistry, unless otherwise indicated): Oxford University. Université de Paris-Sud (Orsay). University of North Carolina.
Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards. University of Washington. Institut de Physique Nucléaire, Université Claude Bernard, Lyon. Department of Applied Mathematics, University of Waterloo, Ontario,Canada. Department of Physics, Drexel University, Philadelphia. Northwestern University.Centre de Physique Théorique, Ecole Polytechnique, Palaiseau, France. Institut Blaise Pascal, Université de Paris VII, Paris. Physique Théorique et Mathématique, Univesité Libre de Bruxelles. Nuclear Research Institute of the Hungarian Academy of Sciences,Debrecen. Rutgers University. Department of Mathematics, Massachusetts Institute of Technology. Saarland University, Saarbruecken. University of Waikato, Hamilton, New Zealand. Hyderabad Central University, India. Institut Poincaré, Paris. Institute of Physics,
Slovak Academy of Sciences, Bratislava. Institute of Advanced Studies, University of Bologna.

1. J. Katriel and G. Adam, Molecular integrals over generalized Hermite-Gaussian functions,
J. Chem. Phys. 51, 3709-3712 (1969).
2. J. Katriel and G. Adam, Exact analytic evaluation of the H+
2 force constant,
Chem. Phys. Lett. 8, 191-194 (1971).
3. J. Katriel, An interpretation of Hunds rule, Theoret. Chim. Acta 26, 163-170 (1972).
4. J. Katriel, Combinatorial aspects of boson algebra, Lett. Nouv. Cim. 10, 565-567 (1974).
5. K. Murat and J. Katriel, The lowest triplet state of the biexciton may be bound,
Phys. Lett. A 71, 143-145 (1979).
6. J. Katriel and E.R. Davidson, Shellwise virial scaling: An approximation for atomic hole
states, Intern. J. Quantum Chem. 18, 1049-1055 (1980).
7. J. Katriel and D.G. Hummer, Analytic solutions for three- and four-wave mixing via
generalised Bose operators, J. Phys. A 14, 1211-1224 (1981). MR0611983.
8. J. Katriel, Reduction of the excited state into the ground state of a super-Hamiltonian,
Intern. J. Quantum Chem. 23, 1767-1780 (1983).
9. M. Orenstein, S. Speiser and J. Katriel, An eikonal approximation for non linear resonators
exhibiting bistability, Opt. Comm. 48, 367-373 (1984).
10. J. Katriel and A. Rosenhouse, The classical limit of the Korteweg deVries hierarchy of
isospectral transformations, Phys. Rev. D 32, 884-890 (1985). MR0803860.
11. J. Katriel, M. Rasetti and A. I. Solomon, Generalized Holestein-Primakoff squeezed
states for SU(n), Phys. Rev. D 35, 2601-2602 (1987).
12. A. Novoselsky, J. Katriel and R. Gilmore, Coefficients of fractional parentage in the L-S coupling scheme, J. Math. Phys. 29, 1368-1388 (1988). MR0944451.
13. J. Katriel and A. Novoselsky, Multi-cluster wavefunctions with arbitrary permutational
symmetry, Ann. Phys. (N.Y.) 211, 1-23 (1991). MR1128181.
14. J. Katriel and M. Kibler, Normal ordering for deformed boson operators and operator
-valued deformed Stirling numbers, J. Phys. A. 25, 2683-2691 (1992). MR1164371.
15. A. Govari, A. Mann and J. Katriel, Fractional quantum Hall effect studied by means of a nonspurious basis set, Phys. Rev. B 48, 11404-11407 (1993).
16. J. Katriel, Permutations as minimal powers of a single-cycle class-sum,
Europ. J. Combinatorics 16, 633-637 (1995). MR1356852.
17. J. Katriel, B. Abdesselam and A. Chakrabarti, The fundamental invariant of the Hecke algebra Hn(q) characterizes the representations of Hn(q), Sn, SUq(N) and SU(N),
J. Math. Phys. 36, 5139-5158 (1995). MR1347137, Zbl 0847.20041.
18. J. Katriel, Explicit expressions for the central characters of the symmetric group,
Discrete Appl. Math. 67, 149-156 (1996). MR1393301.
19. J. Katriel and C. Quesne, Recursively minimally-deformed oscillators,
J. Math. Phys. 37, 1650-1661 (1996). MR1380866, Zbl 0879.47048.
20. J. Katriel, Minimal set of class-sums characterizing the ordinary irreducible
representations of the symmetric group, and the Tarry-Escott problem,
Discrete Math. 173, 91-95 (1997). MR1468843.
21. J. Katriel, Products of arbitrary class-sums in the symmetric group,
Intern. J. Quantum Chem. 70, 429-440 (1998).
22. J. Katriel, F. Zahariev and K. Burke, Symmetry and degeneracy in density functional
theory, Intern. J. Quantum Chem. 85, 432-435 (2001).
23. J. Katriel, Refined Stirling numbers: Enumeration of special sets of permutations and
set-partitions, J. Comb. Theory A 99, 85-94 (2002). MR1911458, Zbl 1013.05009.
24. J. Katriel, Coherent states and combinatorics,
J. Opt. B: Quantum Semiclass. Opt. 4, S200-S203 (2002). MR1920218.
25. J. Katriel, Weights of spin and permutational-symmetry adapted states for arbitrary
elementary spins, in Fundamental World of Quantum Chemistry, Vol III,
E. Brandas and E. Kryachko, Eds. (2004).
26. J. Katriel, S. Roy and M. Springborg, Non-universality of commonly used correlation -energy density-functionals, J. Chem. Phys. 124, 234111-(1-11) (2006).
27. J. Katriel and V. Buˇzek, Multipartite EPR states, Mol. Phys. 106, 497-508 (2008).
28. Z. Li, C. Bao and J. Katriel, Equilibrium population of spin components in a trapped spin-1 Bose gas, Phys. Rev. A 77, 023614-(1-2) (2008).
29. J. Katriel, A multitude of expressions for the Stirling numbers of the first kind,
Integers: Electronic J. Comb. Number Theory (#A22) 10, 273-297 (2010).
MR2652565, Zbl 1246.11060.
30. J. Katriel and R. Gilmore, Entropy of bounding tori,
Entropy 12, 953-960 (2010). MR2644520, Zbl 1229.37015.
31. J. Katriel, The splitting of atomic orbitals with a common principal quantum number revisited: np vs. ns, J. Chem. Phys. 136, 144112-(1-8) (2012).
32. J. Katriel and H. E. Montgomery, Jr., The virial theorem for the smoothly and sharply, penetrably and impenetrably confined hydrogen atom,
J. Chem. Phys. 137, 114109-(1-7) (2012).
33. J. Katriel, M. Puchalski and K. Pachucki, Binding energies of the lithium isoelectronic sequence approaching the critical charge, Phys. Rev. A, 86, 042508-(1-5) (2012).
34. J. Katriel, G. Gaigalas and M. Puchalski, Quantum defects at the critical charge,
J. Chem. Phys. 138, 224305-(1-9) (2013).
35. J. Katriel, J. P. Marques, P. Indelicato, A. M. Costa, M. C. Martins, J. P. Santos and
F. Parente, Approach towards the critical charge of some excited states of the Be
isoelectronic series, Phys. Rev. A 90, 052519 (2014).
36. A Bonfiglioli and J. Katriel, Generating q-commutator identities and the q-BCH formula,
Adv. Math. Phys. 2016, Article ID 9598409, 26 pages (2016).
MR3566544, Zbl 1353.05024.
37. J. Katriel and H. E. Montgomery, Jr., Atomic vs. quantum dot open shell spectra,
J. Chem. Phys. 146, 064104-(1-10) (2017).