Beyond BornOppenheimer: Electronic Nonadiabatic Coupling Terms and Conical IntersectionsISBN: 9780471778912
234 pages
April 2006

The BornOppenheimer approximation has been fundamental to calculation in molecular spectroscopy and molecular dynamics since the early days of quantum mechanics. This is despite wellestablished fact that it is often not valid due to conical intersections that give rise to strong nonadiabatic effects caused by singular nonadiabatic coupling terms (NACTs). In Beyond BornOppenheimer, Michael Baer, a leading authority on molecular scattering theory and electronic nonadiabatic processes, addresses this deficiency and introduces a rigorous approachdiabatizationfor eliminating troublesome NACTs and deriving wellconverged equations to treat the interactions within and between molecules.
Concentrating on both the practical and theoretical aspects of electronic nonadiabatic transitions in molecules, Professor Baer uses a simple mathematical language to rigorously eliminate the singular NACTs and enable reliable calculations of spectroscopic and dynamical cross sections. He presents models of varying complexity to illustrate the validity of the theory and explores the significance of the study of NACTs and the relationship between molecular physics and other fields in physics, particularly electrodynamics.
The first book of its king Beyond BornOppenheimer:
* Presents a detailed mathematical framework to treat electronic NACTs and their conical intersections
* Describes the BornOppenheimer treatment, including the concepts of adiabatic and diabatic frameworks
* Introduces a fieldtheoretical approach to calculating NACTs, which offers an alternative to timeconsuming ab initio procedures
* Discusses various approximations for treating a large system of diabatic Schrödinger equations
* Presents numerous exercises with solutions to further clarify the material being discussed
Beyond BornOppenheimer is required reading for physicists, physical chemists, and all researchers involved in the quantum mechanical study of molecular systems.
Abbreviations.
1. Mathematical Introduction.
I.A. The Hilbert Space.
I.A.1. The Eigenfunction and the Electronic nonAdiabatic Coupling Term.
I.A.2. The Abelian and the nonAbelian Curl Equation.
I.A.3. The Abelian and the nonAbelian DivEquation.
I.B. The Hilbert Subspace.
I.C. The Vectorial First Order Differential Equation and the Line Integral.
I.C.1. The Vectorial First Order Differential Equation.
I.C.1.1. The Study of the Abelian Case.
I.C.1.2. The Study of the nonAbelian Case.
I.C.1.3. The Orthogonality.
I.C.2. The Integral Equation.
I.C.2.1. The Integral Equation along an Open Contour.
I.C.2.2. The Integral Equation along an Closed Contour.
I.C.3. Solution of the Differential Vector Equation.
I.D. Summary and Conclusions.
I.E. Exercises.
I.F. References.
2. BornOppenheimer Approach: Diabatization and Topological Matrix.
II.A. The Time Independent Treatment for Real Eigenfunctions.
II.A.1. The Adiabatic Representation.
II.A.2. The Diabatic Representation.
II.A.3. The AdiabatictoDiabatic Transformation.
II.A.3.1. The Transformation for the Electronic Basis Set.
II.A.3.2. The Transformation for the Nuclear WaveFunctions.
II.A.3.3. Implications due to the AdiabatictoDiabatic Transformation.
II.A.3.4. Final Comments.
II.B. Application of Complex Eigenfunctions.
II.B.1. Introducing TimeIndependent Phase Factors.
II.B.1.1. The Adiabatic Schrödinger Equation.
II.B.1.2. The AdiabatictoDiabatic Transformation.
II.B.2. Introducing TimeDependent Phase Factors.
II.C. The Time Dependent Treatment.
II.C.1. The TimeDependent Perturbative Approach.
II.C.2. The TimeDependent nonPerturbative Approach.
II.C.2.1. The Adiabatic Time Dependent Electronic Basis set.
II.C.2.2. The Adiabatic TimeDependent Nuclear Schrödinger Equation.
II.C.2.3. The Time Dependent AdiabatictoDiabatic Transformation.
II.C.3. Summary.
II.D. Appendices.
II.D.1. The Dressed NonAdiabatic Coupling Matrix.
II.D.2. Analyticity of the AdiabatictoDiabatic Transformation matrix, Ã, in SpaceTime Configuration.
II.E. References.
3. Model Studies.
III.A. Treatment of Analytical Models.
III.A.1 TwoState Systems.
III.A.1.1. The AdiabatictoDiabatic Transformation Matrix.
III.A.1.2. The Topological Matrix.
III.A.1.3. The Diabatic Potential Matrix.
III. A.2. ThreeState Systems.
III.A.2.1. The AdiabatictoDiabatic Transformation Matrix.
III.A.2 2. The Topological Matrix.
III. A.3. FourState Systems.
III.A.3.1. The AdiabatictoDiabatic Transformation Matrix.
III.A.3 2. The Topological Matrix.
III.A.4 Comments Related to the General Case.
III.B. The Study of the 2x2 Diabatic Potential Matrix and Related Topics.
III.B.1. Treatment of the General Case.
III.B.2. The JahnTeller Model.
III.B.3. The Elliptic JahnTeller Model.
III.B.4. On the Distribution of Conical Intersections and the Diabatic Potential Matrix.
III.C. The AdiabatictoDiabatic Transformation Matrix and the Wigner Rotation Matrix.
III.C.1. The Wigner Rotation Matrices.
III.C.2. The AdiabatictoDiabatic Transformation Matrix and the Wigner djMatrix.
III. D. Exercise.
4. Studies of Molecular Systems.
IV.A. Introductory Comments.
IV.B. Theoretical Background.
IV. C. Quantization of the Nonadiabatic Coupling Matrix: Studies of abinitio Molecular Systems.
IV.C.1. TwoState QuasiQuantization.
IV.C.1.1. The {H_{2},H} system.
IV.C.1.2. The {H_{2},O} system.
IV.C.1.3. The {C_{2}H_{2}) Molecule.
IV.C.2. MultiState QuasiQuantization.
IV.C.2.1. The {H_{2},H} system.
IV.C.2.2. The {C_{2},H} system.
IV.D. References.
5. Degeneracy Points and Born—Oppenheimer Coupling Terms as Poles.
V.A. On the Relation between the Electronic NonAdiabatic Coupling Terms and the Degeneracy Points.
V.B. The Construction of Hilbert Subspaces.
V.C. The Sign Flips of the Electronic Eigenfunctions.
V.C.1. SignFlips in Case of a TwoState Hilbert Subspace.
V.C.2. SignFlips in Case of a ThreeState Hilbert Subspace.
V.C.3. SignFlips in Case of a General Hilbert Subspace.
V.C.4 SignFlips for a case of a MultiDegeneracy Point.
V.C.4.1 The General Approach.
V.C.4.2 Model Studies.
V.D. The Topological Spin.
V.E. The Extended Euler Matrix as a Model for the AdiabatictoDiabatic Transformation Matrix.
V.E.1. Introductory Comments.
V.E.2.The TwoState Case.
V.E.3 The ThreeState Case.
V.E.4 The MultiState Case.
V.F. Quantization of the τMatrix and its Relation to the Size of Configuration Space: the Mathieu Equation as a Case of Study.
IV.F.1. Derivation of the Eigenfunctions.
IV.F.2. The nonAdiabatic Coupling Matrix and the Topological matrix.
V.G Exercises.
V.H. References.
6. The Molecular Field.
VI.A. Solenoid as a Model for the Seam.
VI.B. TwoState (Abelian) System.
VI.B.1. The NonAdiabatic Coupling Term as a Vector Potential.
VI.B.2. TwoState Curl Equation.
VI.B.3. The (Extended) Stokes Theorem.
VI.B.4. Application of Stokes Theorem for several Conical Intersections.
VI.B.5. Application of VectorAlgebra to Calculate the Field of a TwoState Hilbert Space.
VI.B.6. A Numerical Example: The Study of the {Na,H_{2}} System.
VI. B.7. A Short Summary.
VI.C. The MultiState Hilbert Subspace.
VI.C.1. The nonAbelian Stokes Theorem.
VI.C.2. The CurlDiv Equations.
VI.C.2.1. The ThreeState Hilbert Subspace.
VI.C.2.2. Derivation of the Poisson Equations.
VI.C.2.3. The Strange Matrix Element and the Gauge Transformation.
VI.D. A Numerical Study of the {H, H_{2}} System.
VI.D.1. Introductory Comments.
VI.D.2. Introducing the Fourier Expansion.
VI.D.3. Imposing Boundary Conditions.
VI.D.4. Numerical Results.
VI.E. The MultiState Hilbert Subspace: On the Breakup of the NonAdiabatic Coupling Matrix.
VI.F. The PseudoMagnetic Field.
VI.F.1. Quantization of the pseudo magnetic along the seam:.
VI.F.2. The NonAbelian Magnetic Fields.
VI.G. Exercises:
VI.H. References.
7. Open Phase and the Berry Phase for Molecular Systems.
VII.A. Studies of Abinitio Systems.
VII.A.1. Introductory Comments.
VII.A.2. The Open Phase and the Berry Phase for Singlevalued Eigenfunctions ( Berry's Approach.
VII.A.3. The Open Phase and the Berry Phase for Multivalued Eigenfunctions ( the Present Approach.
VII.A.3.1. Derivation of the TimeDependent Equation.
VII.A.3.2. The Treatment of the Adiabatic Case.
VII.A.3.3. The Treatment of the nonAdiabatic (General) Case.
VII.A.3.4. The {H_{2},H} System as a Case Study.
VII.B. PhaseModulus Relations for an External Cyclic TimeDependent Field.
VII.B.1. The Derivation of the Reciprocal Relations.
VII.B.2. The Mathieu equation.
VII.B.2.1. The TimeDependent Schrödinger Equations.
VII.B.2.2. Numerical Study of the Topological Phase.
VII.B.3. Short Summary.
VII.C. Exercises.
VII.D. References.
8. Extended BornOppenheimer Approximations.
VIII.A. Introductory Comments.
VIII.B. The BornOppenheimer Approximation as Applied to a MultiState ModelSystem.
VIII.B.1. The Extended Approximate BornOppenheimer Equation.
VIII.B.2. Gauge Invariance Condition for the Approximate BornOppenheimer Equation.
VIII.C. MultiState BornOppenheimer Approximation: Energy Considerations to Reduce the Dimension of the Diabatic Potential Matrix.
VIII.C.1. Introductory Comments.
VIII.C.2. Derivation of the Reduced Diabatic Potential Matrix.
VIII.C.3. Application of the Extended Euler Matrix: Introducing the NState AdiabatictoDiabatic Transformation Angle.
VIII.C.3.1. Introductory Comments.
VIII.C.3.2. Derivation of the AdiabatictoDiabatic Transformation Angle.
VIII.C.4. TwoState Diabatic Potential Energy Matrix.
VIII.C.4.1 Derivation of the Diabatic Potential Matrix.
VIII.C.4.2 A Numerical Study of the (WMatrix Elements.
VIII.C.4.3 A Different Approach: The Helmholtz Decomposition.
VIII.D. A Numerical Study of a Reactive Scattering TwoCoordinate Model.
VIII.D.1. The Basic Equations.
VIII.D.2. A TwoCoordinate Reactive (Exchange) Model.
VIII.D.3. Numerical Results and Discussion.
VIII.E. Exercises.
VIII.F. References.
9. Summary.
Index.
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