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# Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections

## byMichael Baer

### Hardcover | April 21, 2006

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The Born-Oppenheimer approximation has been fundamental to calculation in molecular spectroscopy and molecular dynamics since the early days of quantum mechanics. This is despite well-established 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 Born-Oppenheimer, Michael Baer, a leading authority on molecular scattering theory and electronic nonadiabatic processes, addresses this deficiency and introduces a rigorous approach--diabatization--for eliminating troublesome NACTs and deriving well-converged 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 Born-Oppenheimer:

* Presents a detailed mathematical framework to treat electronic NACTs and their conical intersections

* Describes the Born-Oppenheimer treatment, including the concepts of adiabatic and diabatic frameworks

* Introduces a field-theoretical approach to calculating NACTs, which offers an alternative to time-consuming 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 Born-Oppenheimer is required reading for physicists, physical chemists, and all researchers involved in the quantum mechanical study of molecular systems.

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The following ISBNs are associated with this title:

ISBN - 10:0471778915

ISBN - 13:9780471778912

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Table of Contents

Abbreviations.

**1. Mathematical Introduction.**

I.A. The Hilbert Space.

I.A.1. The Eigenfunction and the Electronic non-Adiabatic Coupling Term.

I.A.2. The Abelian and the non-Abelian Curl Equation.

I.A.3. The Abelian and the non-Abelian Div-Equation.

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 non-Abelian 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. Born-Oppenheimer 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 Adiabatic-to-Diabatic Transformation.

II.A.3.1. The Transformation for the Electronic Basis Set.

II.A.3.2. The Transformation for the Nuclear Wave-Functions.

II.A.3.3. Implications due to the Adiabatic-to-Diabatic Transformation.

II.A.3.4. Final Comments.

II.B. Application of Complex Eigenfunctions.

II.B.1. Introducing Time-Independent Phase Factors.

II.B.1.1. The Adiabatic Schrödinger Equation.

II.B.1.2. The Adiabatic-to-Diabatic Transformation.

II.B.2. Introducing Time-Dependent Phase Factors.

II.C. The Time Dependent Treatment.

II.C.1. The Time-Dependent Perturbative Approach.

II.C.2. The Time-Dependent non-Perturbative Approach.

II.C.2.1. The Adiabatic Time Dependent Electronic Basis set.

II.C.2.2. The Adiabatic Time-Dependent Nuclear Schrödinger Equation.

II.C.2.3. The Time Dependent Adiabatic-to-Diabatic Transformation.

II.C.3. Summary.

II.D. Appendices.

II.D.1. The Dressed Non-Adiabatic Coupling Matrix.

II.D.2. Analyticity of the Adiabatic-to-Diabatic Transformation matrix, Ã, in Space-Time Configuration.

II.E. References.

**3. Model Studies.**

III.A. Treatment of Analytical Models.

III.A.1 Two-State Systems.

III.A.1.1. The Adiabatic-to-Diabatic Transformation Matrix.

III.A.1.2. The Topological Matrix.

III.A.1.3. The Diabatic Potential Matrix.

III. A.2. Three-State Systems.

III.A.2.1. The Adiabatic-to-Diabatic Transformation Matrix.

III.A.2 2. The Topological Matrix.

III. A.3. Four-State Systems.

III.A.3.1. The Adiabatic-to-Diabatic 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 Jahn-Teller Model.

III.B.3. The Elliptic Jahn-Teller Model.

III.B.4. On the Distribution of Conical Intersections and the Diabatic Potential Matrix.

III.C. The Adiabatic-to-Diabatic Transformation Matrix and the Wigner Rotation Matrix.

III.C.1. The Wigner Rotation Matrices.

III.C.2. The Adiabatic-to-Diabatic Transformation Matrix and the Wigner dj-Matrix.

III. D. Exercise.

**4. Studies of Molecular Systems.**

IV.A. Introductory Comments.

IV.B. Theoretical Background.

IV. C. Quantization of the Non-adiabatic Coupling Matrix: Studies of *ab-initio* Molecular Systems.

IV.C.1. Two-State Quasi-Quantization.

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. Multi-State Quasi-Quantization.

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 Non-Adiabatic 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. Sign-Flips in Case of a Two-State Hilbert Subspace.

V.C.2. Sign-Flips in Case of a Three-State Hilbert Subspace.

V.C.3. Sign-Flips in Case of a General Hilbert Subspace.

V.C.4 Sign-Flips for a case of a Multi-Degeneracy 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 Adiabatic-to-Diabatic Transformation Matrix.

V.E.1. Introductory Comments.

V.E.2.The Two-State Case.

V.E.3 The Three-State Case.

V.E.4 The Multi-State Case.

V.F. Quantization of the τ-Matrix 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 non-Adiabatic 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. Two-State (Abelian) System.

VI.B.1. The Non-Adiabatic Coupling Term as a Vector Potential.

VI.B.2. Two-State 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 Vector-Algebra to Calculate the Field of a Two-State 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 Multi-State Hilbert Subspace.

VI.C.1. The non-Abelian Stokes Theorem.

VI.C.2. The Curl-Div Equations.

VI.C.2.1. The Three-State 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 Multi-State Hilbert Subspace: On the Breakup of the Non-Adiabatic Coupling Matrix.

VI.F. The Pseudo-Magnetic Field.

VI.F.1. Quantization of the pseudo magnetic along the seam:.

VI.F.2. The Non-Abelian Magnetic Fields.

VI.G. Exercises:

VI.H. References.

**7. Open Phase and the Berry Phase for Molecular Systems.**

VII.A. Studies of Ab-initio Systems.

VII.A.1. Introductory Comments.

VII.A.2. The Open Phase and the Berry Phase for Single-valued Eigenfunctions ( Berry's Approach.

VII.A.3. The Open Phase and the Berry Phase for Multi-valued Eigenfunctions ( the Present Approach.

VII.A.3.1. Derivation of the Time-Dependent Equation.

VII.A.3.2. The Treatment of the Adiabatic Case.

VII.A.3.3. The Treatment of the non-Adiabatic (General) Case.

VII.A.3.4. The {H_{2},H} System as a Case Study.

VII.B. Phase-Modulus Relations for an External Cyclic Time-Dependent Field.

VII.B.1. The Derivation of the Reciprocal Relations.

VII.B.2. The Mathieu equation.

VII.B.2.1. The Time-Dependent 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 Born-Oppenheimer Approximations.**

VIII.A. Introductory Comments.

VIII.B. The Born-Oppenheimer Approximation as Applied to a Multi-State Model-System.

VIII.B.1. The Extended Approximate Born-Oppenheimer Equation.

VIII.B.2. Gauge Invariance Condition for the Approximate Born-Oppenheimer Equation.

VIII.C. Multi-State Born-Oppenheimer 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 N-State Adiabatic-to-Diabatic Transformation Angle.

VIII.C.3.1. Introductory Comments.

VIII.C.3.2. Derivation of the Adiabatic-to-Diabatic Transformation Angle.

VIII.C.4. Two-State Diabatic Potential Energy Matrix.

VIII.C.4.1 Derivation of the Diabatic Potential Matrix.

VIII.C.4.2 A Numerical Study of the (W-Matrix Elements.

VIII.C.4.3 A Different Approach: The Helmholtz Decomposition.

VIII.D. A Numerical Study of a Reactive Scattering Two-Coordinate Model.

VIII.D.1. The Basic Equations.

VIII.D.2. A Two-Coordinate Reactive (Exchange) Model.

VIII.D.3. Numerical Results and Discussion.

VIII.E. Exercises.

VIII.F. References.

**9. Summary.**

Index.