Beyond Born-Oppenheimer

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Preface.

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 Schroedinger 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 Schroedinger 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, A, 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.**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 -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.**

**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.**

**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 Schroedinger Equations.**

**VII.B.2.2. Numerical Study of the Topological Phase.**

**VII.B.3. Short Summary.**

**VII.C. Exercises.**

**VII.D. References.**

**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.**

**Index.**

**Michael Baer** is one of the foremost authorities on molecular
scattering theory. He wrote the seminal paper in the field of
electronic nonadiabatic molecular collisions in 1975 and has
continued to make fundamental contributions to electronic
nonadiabatic processes in molecular systems. He also contributed
significantly to developing numerical methods to treat, quantum
mechanically, reactive-exchange processes and is a co-author of the
*negative imaginary potential* approach to decoupling the
dynamics in different arrangement channels, which is now used
worldwide. Dr. Baer, who received his M.Sc. and Ph.D from the
Hebrew University of Jerusalem, is currently associated with the
Fritz Haber Center for Molecular Dynamics at the Hebrew University
in Jerusalem. Before that he was a theoretical physicist and an
applied mathematician for almost 40 years at the Soreq Nuclear
Research Center, Israel. The author was a visiting scientist in
many foreign universities and scientific institutes, among them
Harvard University and the University of Oxford. He has published
more than 300 scientific articles and edited several books. In 1993
he was awarded the (Senior) Meitner-Humboldt Prize in Germany for
Theoretical Chemistry and in 2003 he was nominated as a
Szent-Gyoergyi professor for physics by the National Academy of
Sciences in Hungary.

"...a good introductory guide to the world of nonadiabatic
chemistry and can therefore be recommended to the scientists and
students..." (*Zentralblatt MATH*, 2007)

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