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International Journal of Quantum Chemistry

New bosonic excitation operators in many-electron wave functions

Katsuhisa Ohta

Published Jan 1, 1998• 3 citations
Publisher

Abstract

New excitation operators which have perfect bosonic symmetry are constructed for many-electron wave functions by regarding the system of many electrons as that of many species of bosons. Any electronic configurations can be generated by the new bosonic “void” operators. A coherent state is constructed with the bosonic operators and is adopted as a trial function for the time-dependent variational principle. The equation of motion which has exactly the same form as Hamilton's equation in classical mechanics is obtained with the complex variational parameters, the number of which is equal to the number of electrons. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 71–75, 1998

DOI
10.1002/(sici)1097-461x(1998)67:2<71::aid-qua1>3.0.co;2-v

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New excitation operators which have perfect bosonic symmetry are constructed for many-electron wave functions by regarding the system of many electrons as that of many species of bosons. Any electronic configurations can be generated by the new bosonic “void” operators. A coherent state is constructed with the bosonic operators and is adopted as a trial function for the time-dependent variational principle. The equation of motion which has exactly the same form as Hamilton's equation in classical mechanics is obtained with the complex variational parameters, the number of which is equal to the number of electrons. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 71–75, 1998
A time-dependent many-body theory for many-particle systems is formulated using the functional Schrödinger picture. Taking a Gaussian trial wave functional and using the variational principle, the equations of motion are obtained. The equations of motion are solved in the small oscillation regime to yield the generalized RPA results. The electron–hole excitation, the collective plasma oscillation for the paramagnetic state, the Stoner excitation and the spin wave for the ferromagnetic state are derived and discussed. As a further test on the versatility of the present method, the formalism is applied to the bound states of the exciton problem and shown to yield the governing equation in a straightforward manner.
The time-dependent variational principle of the Schr\"odinger equation is applied to a formally exact solution of the Schr\"odinger equation whose variational elements are operators which define a collective subspace of the many-body system under study. This generalizes the procedure employed to derive time-dependent Hartree-Fock theory. Four distinct formally exact time-independent variational principles, including several familiar forms, are derived. Application to a class of exactly soluble models, studied in the vibrational regime, illustrates the different modes of implementation of two of the principles. The general theory of large amplitude collective motion is derived. It is shown that the principles can be applied equally well, using either boson or fermion pair degrees of freedom. Some aspects of the relation to the semiclassical limit are discussed.[NUCLEAR STRUCTURE Time-dependent variational principle for collective motion. Invariance principle of Schr\"odinger equation. Generalized coherent state trial function. Exact time-independent variational principles. Variational principle for intrinsic state. Cranking variational principle. Large amplitude collective motion. Semiclassical limit.]
The description of many-particle quantum mechanics given till now does not give us much insight into how one may go about computing the Green function of quantum systems with infinitely many coupled degrees of freedom. Apart from the free particle case and perhaps the harmonic oscillator, there are precious few exact computations of Green functions of many-particle systems. Even the approximate methods are found wanting since we have already alluded to the ‘uncontrolled’ nature of most of the approximations that have been proposed to date. There is one method that offers some hope in this regard. This method is (wrongly) called ‘bosonization’. This method will be the main focus of much of this and the next chapter. The main mathematical tool used is the introduction of operators that are ‘non-local’ in a sense to be made precise later, which enables the exact computation of the asymptotic properties of the Green function (G(x− x′, t− t ′) in the regime |x− x′| → ∞ and/or |t − t ′| → ∞) under some further restrictive assumptions. We wish to ease into this subject through the study of quantum vortices in charged bosons where the notion of nonlocality makes its presence felt in a relatively more familiar setting.
It is intended to derive the Ginzburg–Landau (GL) equation directly from the Bardeen–Cooper–Schrieffer (BCS) Hamiltonian. By the use of the Hubbard–Stratonovitch transformation, the electron–electron interaction composed of four fermion operators is eliminated to yield an auxiliary boson field. This is an effective field in which electrons behave as if they were free. In applying the path integral method, the electron field is integrated out to remain the Lagrangian for this boson field. The symmetry breaking and the phase transition of the system described by this field are discussed, and it is shown that this boson field turns out to be the GL order parameter. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 693–703, 1998
This work reports on a new and effective method that simplifies the many-electron problem. Instead of a wave function, an operator is used to describe the symmetry of the many-electron system through a transformation. It is proven that the two approaches are physically equivalent, but the operator formalism works much more effectively than the wave function formalism. In the operator formalism, the wave function of an n-electron system can be separated into its spatial and spin parts, and the symmetries of identical particles can be correctly described. For example, it can provide the correct spin and spatial symmetry of many-electron atoms or molecules. For many-electron atoms, one can obtain the simultaneous eigenfunction of L̂2 and Ŝ2. Consequently, the spectroscopic terms of a configuration can be obtained, for example, the hydrogen atom in an energy formula with four quantum numbers (n, l, ml, and ms) for each electron in an atom, including the Hartree–Fock theory. We theoretically analyze and discuss the atomic structure of many-electron atoms using this formula.
The relative performance of trial wave functions expressed as linear combination of correlated exponentials has been tested on a variety of systems. The results are compared against other correlated functions commonly used in the literature to assess the capabilities of the proposed ansatz. A possible departure from the simple exponential functional form used in previous works is discussed, along with its advantages and drawbacks. We also discuss how to implement an efficient optimization procedure for this correlated basis set. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 23–33, 1999