Asymptotic Methods in Quantum Mechanics: Application to Atoms, Molecules and Nuclei

177 Pages·2000·7.461 MB·English

by  Professor S. H. Patil, Professor K. T. Tang (auth.)

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63 Ultrafast Phenomena XI
Editors: T. Elsaesser, J.G. Fujimoto, D.A. Wiersma, and W. Zinth
64 Asymptotic Methods in Quantum Mechanics
Application to Atoms, Molecules and Nuclei.
By S.H. Patil and K.T. Tang
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S.R. Patil K. T. Tang
Asymptotic Methods
in Quantum Mechanics
Application
to Atoms, Molecules and Nuclei
With 15 Figures
, Springer
Professor S.H. Pati!
Department of Physiea
Indian Inatitute ofTechnology
Bombay 400076, India
e-mail: [email protected]
Professor K. T. Tang
Department ofPhysiea
Pacific Lutheran University
Tacoma, Washington, 98447, USA
e-mail: [email protected]
Series Editors:
Professor F.P. Schiifer Professor W. Zinth
Max-Planck-Institut fiir Biophysikalische Chemie Universităt Miinchen,
37077 Gottingen-Nikolausberg, Germany Institut fiir Medizinische Optik
Ottingerstrasse 67
Professor J.P. Toennies 80538 Miinchen, Germany
Max-Planck-Institut fiir Stromungsforschung
Bunsenstrasse 10
37073 Giittingen, Germany
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Patil, S. It: Asymptotic methods in quantum mechaniea : application to atoms,
molecules and nuclei / S. K. Patil ; It T. Tang. -Berlin; Heidelberg ;
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(Springer series in chemical physiea ; 64)
ISBN 978-3-642-63137-5 ISBN 978-3-642-57317-0 (eBook)
DOI 10.1007/978-3-642-57317-0
ISSN 0172-6218
ISBN 978-3-642-63137-5
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In loving memory of our parents
Preface
Quantum mechanics and the Schrodinger equation are the basis for the de
scription of the properties of atoms, molecules, and nuclei. The development
of reliable, meaningful solutions for the energy eigenfunctions of these many
particle systems is a formidable problem. The usual approach for obtaining
the eigenfunctions is based on their variational extremum property of the
expectation values of the energy. However the complexity of these variational
solutions does not allow a transparent, compact description of the physical
structure. There are some properties of the wave functions in some specific,
spatial domains, which depend on the general structure of the Schrodinger
equation and the electromagnetic potential. These properties provide very
useful guidelines in developing simple and accurate solutions for the wave
functions of these systems, and provide significant insight into their physical
structure. This point, though of considerable importance, has not received
adequate attention.
Here we present a description of the local properties of the wave functions
of a collection of particles, in particular the asymptotic properties when one
of the particles is far away from the others. The asymptotic behaviour of this
wave function depends primarily on the separation energy of the outmost
particle. The universal significance of the asymptotic behaviour of the wave
functions should be appreciated at both research and pedagogic levels. This
is the main aim of our presentation here.
The asymptotic region plays a dominant role in determining quantities
such as polarizabilities, dispersion coefficients, exchange energies and inter
action potentials. We have attempted to emphasize the importance of the
asymptotic behaviour by analysing a large number of such applications in
the description of atomic, molecular, and nuclear properties, which should be
helpful in developing a comprehensive understanding.
We thank Prof. J. Peter Toennies for many discussions and supportive
encouragement.
Bombay, Tacoma S. H. Patil
February 2000 K. T. Tang
Contents
1. Introduction.............................................. 1
2. General Properties of Wave Functions . . . . . . . . . . . . . . . . . . . . 5
2.1 Asymptotic Form of Wave Functions. . . . . . . . . . . . . . . . . . . . . . 5
2.2 Asymptotic Perturbed Wave Function. . . . . . . . . . . . . . . . . . . . . 8
2.3 Wave Function for rij -t 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10
2.4 Wave Function for rij and rik-t 0 ........................ 12
2.5 Local Satisfaction of Schrodinger Equation. . . . . . . . . . . . . . . .. 13
2.6 Variational Stationary Property. . . . . . . . . . . . . . . . . . . . . . . . .. 14
2.7 Variational Approach to Perturbations. . . . . . . . . . . . . . . . . . .. 15
2.8 Generalised Virial Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17
2.9 A Simple Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
3. Two- and Three-Electron Atoms and Ions ......... . . . . . .. 21
3.1 A Simple Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
3.1.1 Energy.......................................... 22
3.1.2 Multipolar Potential Perturbation. . . . . . . . . . . . . . . . .. 23
3.1.3 Third Order Energy Shifts .................. : . . . . .. 25
3.2 Wave Functions Satisfying Cusp,
Coalescence and Asymptotic Conditions. . . . . . . . . . . . . . . . . .. 25
3.2.1 Behaviour for ri -t o. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
3.2.2 Correlation Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
3.2.3 Results for the Unperturbed Ground State .......... 28
3.2.4 Multipolar Polarizabilities and Hyperpolarizabilities .. 31
3.2.5 Wave Functions for Excited States. . . . . . . . . . . . . . . . .. 33
3.3 Three-Electron Wave Functions. . . . . . . . . . . . . . . . . . . . . . . . .. 36
4. Polarizabilities and Dispersion Coefficients ............... 41
4.1 Polarizabilities......................................... 41
4.1.1 Perturbative Expression. . . . . . . . . . . . . . . . . . . . . . . . . .. 42
4.1.2 Hyperpolarizabilities.............................. 43
4.1.3 Dynamic Polarizabilities. . . . . . . . . . . . . . . . . . . . . . . . . .. 44
4.2 Dispersion Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46
4.2.1 Relation to Dynamic Polarizabilities . . . . . . . . . . . . . . .. 48
X Contents
4.2.2 Three-Body Dispersion Coefficients. . . . . . . . . . . . . . . .. 49
4.3 Alkali Isoelectronic Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
4.3.1 The Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
4.3.2 Polarizabilities................................... 52
4.3.3 Hyperpolarizabilities and Dispersion Coefficients ..... 56
4.4 Asymptotic Polarizabilities and Dispersion Coefficients. . . . .. 58
4.4.1 Asymptotic Polarizabilities ........... . . . . . . . . . . . .. 58
4.4.2 Pol ariz abilities of He and Ne Systems . . . . . . . . . . . . . .. 60
4.4.3 Asymptotic Behaviour of the Effective Energy ....... 62
4.4.4 Dispersion Coefficients for H, He and Ne ............ 65
5. Asymptotically Correct Thomas-Fermi Model Density. . .. 69
5.1 Thomas-Fermi Model. . .. . . .... .... .. .. . ... .. . . . .. . . . . .. 69
5.1.1 Statistical Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70
5.1.2 WKB Approach... .... . . .. . . .. . . .. . . ... . .. . . . .. .. 70
5.2 Solution for the Thomas-Fermi Density. . . . . . . . . . . . . . . . . .. 71
5.3 Asymptotic Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72
5.4 Modified Density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
5.5 Applications........................................... 75
5.5.1 Expectation Values (r2n) . . . . . . . . . . . . . . . . . . . . . . . . .. 75
5.5.2 Multipolar Polarizabilities . . . . . . . . . . . . . . . . . . . . . . . .. 77
5.5.3 Dispersion Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
6. Molecules and Molecular Ions
with One and Two Electrons ............................. 85
6.1 Wave Functions for One-Electron Molecular Ions. . . . . . . . . .. 86
6.1.1 Cusp Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86
6.1.2 Asymptotic Behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
6.2 Energies for One-Electron Molecular Ions. . . . . . . . . . . . . . . . .. 89
6.3 Wave Function for H2 and Het+ ......................... 91
6.3.1 Molecular Orbital Type of Wave Function. . . . . . . . . .. 91
6.3.2 Atomic Orbital Type of Wave Function ............. 92
6.3.3 General Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . .. 93
6.3.4 Correlation Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94
6.4 Results for the Ground State. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96
6.4.1 Ground State Energies. . . . . . . . . . . . . . . . . . . . . . . . . . .. 96
6.4.2 Discussion....................................... 98
7. Interaction of an Electron with Ions, Atoms, and Moiecuies105
7.1 Atomic Rydberg States .................................. 105
7.1.1 Perturbation Approach
for Anti-symmetric Wave Functions ................. 105
7.1.2 The Perturbed Hamiltonian ....................... 107
7.1.3 Asymptotic Core Density and Density Matrix ........ 108
7.1.4 Penetration Energy ............................... 109
Contents XI
7.1.5 Exchange Energy ................................. 110
7.1. 6 Second Order Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111
7.1. 7 Total Energy Shift ................................ 112
7.1.8 Results .......................................... 113
7.2 Electron-Atom and Electron-Molecule Scattering
at High Energies ....................................... 115
7.2.1 Perturbation Series for the Scattering Amplitude ..... 115
7.2.2 Scattering Amplitude at High Energies .............. 117
7.2.3 Electron-Atom Scattering ......................... 118
7.2.4 Electron-Molecule Scattering ...................... 120
8. Exchange Energy of Diatomic Systems .................... 127
8.1 Exchange Energy of Dimer Ions .......................... 127
8.1.1 Exchange Energy of the Ht Molecular Ion
by Surface Integral Method ........................ 128
8.1.2 Exchange Energy of Multielectron Dimer Ions ........ 133
8.2 Exchange Energy of Diatomic Molecules ................... 136
8.2.1 Exchange Energy of the H2 Molecule ............... 136
8.2.2 Exchange Energy of Multielectron Diatomic Molecules 144
9. Inter-atomic and Inter-ionic Potentials .................... 147
9.1 Exchange Energy and Exchange Integral
in the Heitler-London Theory ............................ 148
9.2 Generalized Heitler-London Theory ....................... 150
9.2.1 Unsymmetrized (Polarization) Perturbation Method .. 150
9.2.2 Symmetry Imposed Generalised Heitler-London
Equation ........................................ 151
9.2.3 Asymptotic Exchange Energy
and Polarization Approximation .................... 153
9.3 Inter-atomic and Inter-ionic Potentials .................... 154
9.3.1 The 3 Ell, State Potential of the H2 Molecule ......... 154
9.3.2 The 2E ll, State Potentials of Alkali Dimer Cations .... 156
9.3.3 The Potential of Rare Gas Dimers .................. 157
10. Proton and Neutron Densities in Nuclei .................. 161
10.1 Semi-phenomenological Density .......................... 161
10.2 Determination of the Parameters ......................... 162
10.3 Results ................................................ 164
References .................................................... 169
1. Introduction
In describing the non-relativistic properties of atoms, molecules, and nu
clei with more than two particles, one encounters the difficult problem of
analysing the many-particle Schrodinger equation. In the case of atoms and
molecules, the interaction is Coulombic whereas in the case of nuclei, we
only know the general form of the interaction potential, that it is strong
and of short range. Faced with an almost insurmountable task of solving a
non-separable, differential equation with many variables, one usually takes
recourse to variational solutions. This approach emphasizes the extremum or
stationary property which the energy eigenvalues have irrespective of the de
tails of the interaction. However, while this approach provides reliable values
for the energy eigenvalues, particularly the ground state energies, by itself
it is not very efficient or useful in providing an insight into the structure of
the wave functions, and interactions of the particles with each other or with
external fields.
Though the many-particle Schrodinger equation is very complicated in
structure, it allows us to deduce some general properties of the wave functions.
These properties follow from the general structure of the equation. They may
be grouped into two classes. There are some properties of the wave functions
and energies, which depend on the wave function over the entire domain. For
example, the stationary property that the variation of the average value of
the energy vanishes,
(1.1 )
in the leading order, when I'¢') is an energy eigenstate, depends on the inte
grals over the entire domain. These properties may be called global properties.
Apart from the variational property of the energy, we have the virial theorem,
generalizations of the virial theorem, relations which depend on the gauge in
variance, all of which depend on integrals over the entire domain. Then there
are properties of wave functions which are specific to local domains. For ex
ample when one of the particles is far away from all others, it experiences
essentially the Coulombic potential due to the remaining net core charge, and
its energy is the separation energy of the last particle. Therefore the asymp
totic form of the wave function is determined by the separation energy and
the core charge. On the other hand, when two of the particles are close to
S. H. Patil et al., Asymptotic Methods in Quantum Mechanics
© Springer-Verlag Berlin Heidelberg 2000