The quantum classical theory

250 Pages·2003·1.685 MB·English

by  Gert D. Billing

THE QUANTUM CLASSICAL THEORY
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The Quantum Classical Theory
GERT D. BILLING
1
2003
1
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Thequantumclassicaltheory/GertD.Billing.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN0-19-514619-0
1.Quantumchemistry.2.Moleculardynamics.I.Title.
QD462B552002
541.2(cid:1)8—dc21 2002025772
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To my mother Jørga and my grandchild Maria
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Preface
WhenIstartedgettinginterestedinquantum-classicalorsemi-classicalmethods,
the theories available were of the type described in the beginning of chapter 3
of this book. These theories were later mainly called classical path theories,
whereas the word semi-classical was reserved for the classical S-matrix theories
developed in the seventies. These latter theories have been extensively covered
in the two books by Professor M. S. Child, Molecular Collision Theory (1974)
andSemiclassical Mechanics with Molecular Applications(1991). Sincethenthe
“old” classical path theory has been linked to the progress made in wave-packet
theory. Today the theory can be derived from first principles and extended to
cover a large range of molecular processes. The purpose of the present book
is, therefore, to describe this development as well as other quantum-classical
or semi-classical methods which have been not previously been reviewed in
textbooks.
Oftenonedoesnotdistinguishbetweenthetwodesignations,semi-classical
or quantum-classical. However, here we shall use the name “quantum-classical”
and reserve “semi-classical” to methods as classical S-matrix theory, WKB
theories, and others.
The possibility of blending classical and quantum mechanics provides the
obvious choice if a flexible, as well as a general, theory for molecular processes
is to be created. The method will and should be able to treat quantum effects
when present and at the same time be able to take advantage of the ease with
which classical treatments of even large systems can be formulated.
Someofthequantum-classicalmethodsmentionedherehavebeensuccess-
fully used to treat a large body of molecular events: Inelastic processes, energy
transfer, collision induced dissociation, as well as photo-dissociation, chemical
reactions, and non-adiabatic electronic processes.
During the preparation of the manuscript, the various models and meth-
ods have been used to study systems and processes, so as to obtain results
not published previously. This is, for instance, the case for the calculations
of vibrational relaxation of CO colliding with oxygen atoms, the reaction
2
pathcalculationsontheH +CN→HCN+Hreaction, andthequantum-classical
2
results for the HO+CO→CO +H reaction, as well as for some of the model
2
calculations in chapters 2 and 3. Hopefully, enough details are given for the
practical implementation of the methods.
viii Preface
I would like to thank the many collaborators I have had in these areas
over the last twenty to thirty years. I would also like to thank Professor John
Avery, Department of Chemistry, University of Copenhagen, for proofreading
the manuscript.
Contents
Abbreviations..................................................... xi
1 Introduction....................................................... 3
2 Rigorous theories................................................. 8
2.1 Path-integral approach......................................... 8
2.1.1 The short time propagator.............................. 12
2.1.2 The classical limit....................................... 14
2.1.3 The Van Vleck propagator............................... 18
2.2 Gaussians at work.............................................. 21
2.2.1 Cellular dynamics....................................... 23
2.2.2 A semi-classical IVR propagator......................... 25
2.3 An orthorgonal basis set........................................ 27
2.4 Bohmian mechanics............................................ 33
2.5 Mixing of Gauss-Hermite and ordinary basis sets............... 40
2.5.1 The classical path approximation........................ 41
2.5.2 The exact equations of motion in the TDGH basis....... 43
2.6 Bound states and Gauss-Hermite functions..................... 45
2.6.1 The Morse-oscillator in a Gauss-Hermite basis........... 46
2.6.2 Treatment of 2D systems in the G-H basis............... 48
2.6.3 The quantum-classical correlation....................... 50
2.6.4 Spherical harmonics in a Gauss-Hermite basis set........ 52
2.7 Second quantization and Gauss-Hermite functions.............. 53
2.8 Quantum dressed classical mechanics........................... 57
2.8.1 The DVR scheme ....................................... 57
2.8.2 Arbitrarily sized systems................................ 65
2.9 The MCTDH approach......................................... 71
2.10 Summary...................................................... 72
3 Approximate theories............................................. 74
3.1 Time-dependent SCF........................................... 76
3.2 The classical path theory....................................... 77
3.2.1 Relation to other theories............................... 88
3.2.2 Energy transfer.......................................... 90
3.2.3 The VqRqTc method..................................... 90
3.2.4 The VqRcTc method..................................... 95