This book provides a detailed description of the modern variational methods available for solving the nuclear motion Schrödinger equation to enable accurate theoretical spectroscopy of polyatomic molecules. These methods are currently used to provide important molecular data for spectroscopic studies of atmospheres of astronomical objects including solar and extrasolar planets as well as cool stars. This book has collected descriptions of quantum mechanical methods into one cohesive text, making the information more accessible to the scientific community, especially for young researchers, who would like to devote their scientific career to the field of computational molecular physics.
The book addresses key aspects of the high-accuracy computational spectroscopy of the medium size polyatomic molecules. It aims to describe numerical algorithms for the construction and solution of the nuclear motion Schrödinger equations with the central idea of the modern computational spectroscopy of polyatomic molecules to include the construction of the complex kinetic energy operators (KEO) into the computation process of the numerical pipeline by evaluating the corresponding coefficients of KEO derivatives on-the-fly. The book details key aspects of variational solutions of the nuclear motion Schrödinger equations targeting high accuracy, including construction of rotational and vibrational basis functions, coordinate choice, molecular symmetry as well as of intensity calculations and refinement of potential energy functions. The goal of this book is to show how to build an accurate spectroscopic computational protocol in a pure numerical manner of a general black-box type algorithm.
This book will be a valuable resource for researchers, both experts and not experts, working in the area of the computational and experimental spectroscopy; PhD students and early-career spectroscopists who would like to learn basics of the modern variational methods in the field of computational spectroscopy. It will also appeal to astrophysicists and atmospheric physicists who would like to assess data and perform calculations themselves.
Key features:
- Supported by the latest research and based on the state-of-the-art computational methods in high-accuracy computational spectroscopy of molecules.
- Authored by an authority in the field.
- Accessible to both experts and non-experts working in the area of computational and experimental spectroscopy, in addition to graduate students.
Author(s): Sergey Yurchenko
Publisher: CRC Press
Year: 2023
Language: English
Pages: 205
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
CHAPTER 1: Introduction
CHAPTER 2: Coordinates choice
2.1. MOLECULAR FRAME
2.2. COORDINATE TRANSFORMATION
2.2.1. Three translational conditions
2.2.2. Three rotational conditions
2.3. MOLECULAR FRAMES
2.3.1. Bond frame
2.3.2. Eckart frame
2.3.3. Principal axes frame
2.4. VIBRATIONAL COORDINATES
2.4.1. Geometrically defined, valence coordinates
2.4.2. Z-matrix
2.4.3. Jacobi frames
2.4.4. Derivatives of the geometrically defined coordinates with respect to the Cartesian coordinates
2.5. LINEARISED COORDINATES
2.5.1. Normal modes
2.5.2. Vibrational coordinates for linear molecules
2.6. OTHER MOLECULES
2.6.1. Rigid tetratomics of the XY3-type (phosphine)
2.6.2. Non-rigid XY3 (ammonia)
2.7. NON-LINEAR RIGID TETRATOMIC MOLECULES: H2CO TYPE
2.8. NON-RIGID CHAIN MOLECULE HOOH
2.9. METHANE AS AN EXAMPLE OF AN XY4-TYPE TETRAHEDRAL MOLECULE
CHAPTER 3: Kinetic energy operator: Coordinate transformation
3.1. MOLECULAR KEO
3.2. TRANSFORMATIONS OF COORDINATES AND MOMENTA
3.2.1. The t–s formalism
3.2.2. An alternative, g-matrix method to derive KEO
3.3. NORMAL MODES AND WATSON HAMILTONIAN
3.4. SØRENSEN APPROACH FOR THE ROTATIONAL S-MATRIX
3.4.1. Example of derivation of Sørensen vectors for H2CO
3.5. NUMERICAL EVALUATION OF KEO AT ANY INSTANTANEOUS GEOMETRY
3.5.1. KEO in geometrically defined coordinates
3.6. JACOBI COORDINATES KEOS
3.7. KEO AS A TAYLOR-TYPE EXPANSION
3.7.1. Recursive expansion scheme
3.7.2. Taylor-expanded s-vectors in the linearised coordinates using the t–s algorithm and its Sørensen's derivative
3.7.3. Non-rigid reference configuration and linearised coordinates
CHAPTER 4: KEO: Triatomic molecules
4.1. KEO OF XY2 IN VALENCE COORDINATES
4.1.1. Eckart frame
4.1.2. Sørensen's solution
4.1.3. PAS frame
4.1.4. Radau frame
4.2. XYZ MOLECULE
4.3. LINEARISED COORDINATES FOR XY2
4.4. SINGULARITIES IN KEO OF TRIATOMICS
4.4.1. Method 1: 3N – 5 case or four rectilinear coordinates
CHAPTER 5: Basis sets
5.1. MATRIX ELEMENTS OF HAMILTONIAN
5.2. RO-VIBRATIONAL BASIS SET
5.3. VIBRATIONAL BASIS SETS
5.3.1. Product-form basis sets
5.3.2. Two-dimensional basis functions for doubly degenerate vibrations
5.3.3. 3D vibrational basis functions
5.3.4. Basis set pruning: polyad and energy thresholds
5.3.5. Basis set bookkeeping
5.4. BASIS SET CONTRACTION
5.4.1. The J = 0 representation
5.4.2. ‘Contracted’ representation of Hamiltonian
5.4.3. Another layer of contraction
5.4.4. Basis set contraction with the vibrational angular momentum
5.4.5. Assignment: vibrational quantum numbers
5.5. RESOLUTION OF SINGULARITIES AT LINEAR GEOMETRIES: NON-RECTILINEAR COORDINATES
5.5.1. Formulation
5.5.2. Associated Laguerre polynomials
5.6. ROTATIONAL BASIS SET
5.6.1. Rigid rotor wavefunctions
5.6.2. Rotational matrix elements
CHAPTER 6: Symmetry-adapted basis sets
6.1. INTRODUCTION
6.2. MOLECULAR SYMMETRY: SOME THEORY
6.3. AUTOMATIC SYMMETRISATION
6.4. NUMERICAL EXAMPLES
6.4.1. Symmetry-adapted vibrational basis set for an XY2-type molecule
6.4.2. Symmetrisation of the XY3 vibrational basis set
6.5. SYMMETRY SAMPLING OF EIGENFUNCTIONS
6.5.1. Sampling symmetrisation of a XY3 vibrational basis set, the C3v-symmetry
6.5.2. Reduction of a product
6.5.3. Symmetry properties of rigid rotor wavefunctions
6.6. SYMMETRISATION OF THE RIGID ROTOR BASIS FUNCTIONS: SPECIAL CASE
6.7. MATERIALS USED
CHAPTER 7: Applications
7.1. INTENSITY CALCULATIONS
7.1.1. General formulation of ro-vibrational intensities
7.1.2. Calculating the line strength
7.2. REFINEMENT OF PES
7.2.1. Fitting using Newton's minimisation method
7.2.2. Correlated parameters and constrained fit
Bibliography
Index
