Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

applying direct molecular dynamics to non-adiabatic systems								423
evolution of the	expectation values				of the	nuclear	position and momentum
		^
operators. For a general operator, O
			q	^	^
			qt	hOi ¼ ih½O; H&i				ðC:9Þ
^	^	^
where hOi ¼ hwjOjwi and		½O; H& is the commutator of the operator with the
Hamiltonian. Evaluating		the commutators				^	^	to the
						½R; H&	and ½P; H& leads

Ehrenfest theorem

	q	^	1	^
qt hRi ¼ m hPi
	q	hP^i ¼			qV
	qt				qR

ðC:10Þ

ðC:11Þ

The localized nature of the nuclear functions means that these reduce to classical equations of motion

_	Pi					ðC:12Þ
Ri ¼ m					¼
		qV
P_ i ¼			qR	R Ri		ðC:13Þ

Solving the Eqs. (C.6–C.8,C.12,C.13) comprise what is known as the Ehrenfest dynamics method. This method has appeared under a number of names and derivations in the literature such as the classical path method, eikonal approximation, and hemiquantal dynamics. It has also been put to a number of different applications, often using an analytic PES for the electronic degrees of freedom, but splitting the nuclear degrees of freedom into quantum and classical parts.

In the derivation used here, it is clear that two approximations have been made—the configurations are incoherent, and the nuclear functions remain localized. Without these approximations, the wave function form Eq. (C.1) could be an exact solution of the Schro¨dinger equation, as it is in 2D MCTDH form (in fact is in what is termed a natural orbital form as only ‘‘diagonal’’ configurations are included [20]).

Acknowledgments

Thanks are due to Luis Blancafort, Mike Bearpark, Irene Burghardt, and Adelaida Sanchez-Galvez for reading the manuscript, and for helpful hints in the presentation of this material.

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The Role of Degenerate States in Chemistry: Advances in Chemical Physics, Volume 124.

Edited by Michael Baer and Gert Due Billing. Series Editors I. Prigogine and Stuart A. Rice. Copyright # 2002 John Wiley & Sons, Inc.

ISBNs: 0-471-43817-0 (Hardback); 0-471-43346-2 (Electronic)

CONICAL INTERSECTIONS IN MOLECULAR PHOTOCHEMISTRY: THE PHASE-CHANGE APPROACH

YEHUDA HAAS and SHMUEL ZILBERG

Department of Physical Chemistry and the Farkas Center for Light Induced Processes, Hebrew University of Jerusalem, Jerusalem, Israel

CONTENTS

I.Introduction

A.A Chemical Reaction as a Two-State System

B.Anchors

C.Anchors, Molecules and Independent Quantum Species II. The Phase-Change Rule and the Construction of Loops

A.Construction of Loops: Nature of the Coordinates III. Phase Change in a Chemical Reaction

A.Pericyclic Reactions

B.Generalization to Any Reactions

1.Reactions Involving Sigma Bonds Only

2.Reactions Involving p Bonds

IV. Loop Construction for Photochemical Systems

A.Three-Electron Systems

B.Four-Electron Systems

1.Four p Electrons: Butadiene Ring Closure

2.cis–trans Isomerization: 2 p and 2 s Electrons

3.Ammonia and Chiral Systems

C.Four Electrons in Larger Systems

D.More Than Four Electrons

V. Longuet-Higgins Loops and the Jahn–Teller Theorem

A.An Example: The Cyclopentadienyl Radical and Cation Systems

1.Cyclopentadienyl Radical (CPDR)

2.Cyclopentadienyl Cation (CPDC)

433