Lie algebras in quantum chemistry: Symmetrized orbitals

Wybourne, B. G.

doi:10.1002/qua.560070608

Cited by 37 publications

(18 citation statements)

References 28 publications

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“…(6)- (8)) are generators of the group SU (2) and they obey the Lie algebra of SO(3), SU (2). This has been observed earlier (in a more general context) in [61] (also see [19]). Related observations can be found in [62], Appendix A.1, p. 919, [63] and [64] (11)).…”

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confidence: 55%

Nonlinear Bogolyubov–Valatin transformations and quaternions

Holten¹,

Scharnhorst²

2005

J. Phys. A: Math. Gen.

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In introducing second quantization for fermions, Jordan and Wigner (1927/1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H. For the first time, here we exploit this fact to study nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external field is related to the standard Fermi oscillator. Unitary transformations play a prominent role in quantum mechanics. Like canonical transformations in classical mechanics, unitary transformations of quantum dynamical degrees of freedom often simplify the dynamical equations, or allow to introduce sensible approximation schemes. Such methods have wide-ranging applications, from the study of simple systems to many-body problems in solid-state or nuclear physics and quantum chemistry, up to the infinite-dimensional systems of quantum field theory [1,2,3,4]. Linear (unitary) canonical transformations (i.e., transformations preserving the canonical anticommutation relations (CAR)) for fermions have been introduced by Bogolyubov and Valatin (for two fermionic modes) in connection with the study of the mechanism of superconductivity [5,6,7,8,9] . Such linear canonical transformations are important from a physical as well as from a mathematical point of view. Mathematically, they allow to relate quite arbitrary Hamiltonians quadratic in the fermion creation and annihilation operators to collections of Fermi oscillators whose mathematics is very well understood. From a physical point of view, canonical transformations implement the concept of quasiparticles in terms of which the physical processes taking place can be described and understood in an effective and transparent manner. To apply the powerful tool of canonical transformations to the physically interesting class of nonquadratic Hamiltonians, however, requires to go beyond linear Bogolyubov-Valatin transformations. Certain aspects of nonlinear Bogolyubov-Valatin transformations have received some attention over time [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] (We disregard here work done within the framework of the coupled-cluster method (CCM) [4] which is nonunitary.). However, a systematic analytic study of general (nonlinear) Bogolyubov-Valatin transformations has not been undertaken so far. In the present paper, as a first step towards this goal we are going to investigate the prototypical case of a single fermionic mode.Let us consider a pair of fermion creation and annihilation operatorsâ + ,â. Here, we regard the creation operatorâ + as the hermitian conjugate of the annihilation operatorâ:â + =â † (we will use the latter notation throughout). They obey the CAR {â † ,â} =â †â +ââ † = 1,It is now instructive to consider the following pair of antihermitian operators.These two operators obey the equation (p, q = 1, 2)

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confidence: 55%

Nonlinear Bogolyubov–Valatin transformations and quaternions

Holten¹,

Scharnhorst²

2005

J. Phys. A: Math. Gen.

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“…The various subgroup structures contained in E7 may be explored by systematically discarding sets of the E7 generators (cf. Wybourne 1973). Under E7 ~ SU 6 X SU 3 we have (Wybourne and Bowick 1977) 21 6 ~ 214.0 +12.12 +14.1 +0.21.…”

Section: Irreps Of Sun Groupsmentioning

confidence: 99%

Calculation of 3jm Factors and the Matrix Elements of E7 Group Generators

Butler¹,

Haase²,

Wybourne³

1979

Aust. J. Phys.

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“…This somewhat cumbersome procedure could be avoided by introducing an additional characterization of the terms on the basis of representations of certain linear Lie groups forming a supergroup [26,27] to point group G. However, the unequivocal determination of the CFP employing properties of the overlying group scheme requires the explicit calculation of Wigner coefficients for these groups which, in general, is not completely trivial. In addition, to the authors' knowledge, no group theoretical procedure for the unique classification of electron states in arbitrary point groups, particularly double groups, has yet been devised that could be applied to the CFP as well.…”

Section: Construction Of Antisymmetric N-electron Functions By the Mementioning

confidence: 99%

The concept of fractional parentage for arbitrary molecular point groups

König

Kremer

1977

Int J of Quantum Chemistry

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AbstractsThe method of fractional parentage is extended to the general case of mixed configurations in arbitrary nonsimply reducible groups, G c SO(3). Particular attention is devoted to the calculation of coefficients of fractional parentage (cm) and expressions are provided for the matrix elements of F and G type operators between N electron functions.La methode de parent6 fractionnelle a t t e gtntraliste au cas de configurations mixtes dans des

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Lie algebras in quantum chemistry: Symmetrized orbitals

Cited by 37 publications

References 28 publications

Nonlinear Bogolyubov–Valatin transformations and quaternions

Nonlinear Bogolyubov–Valatin transformations and quaternions

Calculation of 3jm Factors and the Matrix Elements of E7 Group Generators

The concept of fractional parentage for arbitrary molecular point groups

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