Simu lations of molecular properties , introduction, history, technical remarks

1. Simulations of molecular properties, introduction, history, technical remarks Petr Bouř

2. Lecture Plan • A Classical Mechanics • A I. Classical mechanics and molecules • A II. Molecular mechanics (MM) and molecular dynamics (MD) • A III. Alternative formulations of Newton’s laws, Lagrange equations, Hamiltonan

3. B Quantum Mechanics • B I. Origins of quantum mechanics • B II. Quantum theory postulates • B III. Heisenberg uncertainty relation • B IV. Quantum-chemical computation • B V. Electron: spin and angular momentum • B VI. Quantum Mechanics of many particles • B VII. Slater’s determinant • B VIII. Born-Oppenheimer approximation

4. Quantum Chemistry • C I. Hartree-Fock theory • C II. Basis of atomic orbitals • C III. Atomic and molecular orbitals, nomenclature • C IV. Basis sets • C V. Semi-empirical methods • C VI. Perturbational computations • C VII. Going from HF to the Schrödinger limit • 1. Configuration interaction, Coupled Clusters • 2. MP2 perturbation methods • 3. Special methods

5. D Density Functional Theory (DFT) • D I. DFT Origins • D II. Hellman-Feynman theorem • D III. Thomas-Fermi model • D IV. Thomas-Fermi-Dirac model • D V. Hohenberg theorem • D VI. Kohn-Sham Formulation • D VII. DFT Functionals for Practical Computations • D VIII. Integral Formulation of DFT

6. E Ab InitioComputations of Molecular Properties • E I. Strategy of ab initio computations • E II. Potential energy surfaces • E III. Electromagnetic field and Maxwall Equations • E IV. Multipole Expansions • E VI. Electrodynamic and Schrodinger Equations • E V. Molecule in Static Electric Field • E VII. Molecule in a Static Magnetic Field

7. E VIII. Vibrational Molecular States • 1. Harmonic approximation • 2. Vibrational Energy • 3. Vibrational Transitions • 4. Anharmonic Frequencies • E IX. Absorption Spectra • Typical Computations of IR Spectrum using the Gaussian Program • E X. Applications of the Hessian • E XI. Raman spectra • E XII. Vibrational cirkular dichroism (VCD) • E XIII. Raman optical aktivity (ROA)

8. E VIII. Vibrational Molecular States • 1. Harmonic approximation • 2. Vibrational Energy • 3. Vibrational Transitions • 4. Anharmonic Frequencies

9. E XIV. Interpretation of NMR spectra, nuclear magnetic shielding • E XV. Spin-Spin Dipolar interactions

10. E XVI. UV-vis Spectroscopy • E XVII. Solvent Models • Appendix: Vector Algebra

11. Historical development of computer simulations 􀂃Began as tool to exploit computing machines developed during World War II 􀂃MANIAC (1952) at Los Alamos used for computer simulations (John von Neuman)

12. Historical development of computer simulations 􀂃Metropolis, Rosenbluth, Teller (1953): Metropolis Monte Carlo method 􀂃

13. Historical development of computer simulations 􀂃Alder and Wainwright (Livermore National Lab, 1956/1957): dynamics of hard spheres 􀂃Vineyard (Brookhaven 1959-60): dynamics of radiation damage in copper 􀂃Rahman (Argonne 1964): liquid argon

14. Historical development of computer simulations 􀂃Application to more complex fluids (e.g. water) in 1970s 􀂃Car and Parrinello (1985 and following): ab-initioMD 􀂃Since 1980s: Many applications, including: 􀂃Karplus, Goddard et al.: Applications to polymers/biopolymers, proteins since 1980s 􀂃Applications to fracture since mid 1990s to 2000 􀂃Other engineering applications (nanotechnology, e.g. CNTs, nanowires etc.) since mid 1990s-2000

15. Simulations – general division ”Ab Initio” –Schrodinger equation, HF, DFT, CCSDT .. “Empirical” – CNDO, PM3, … also DFT MD/classical mechanics (Newton) Any combinations possible -e.g. reaction center in enzyme QM, rest MM/MD MM QM

16. Control questions • Write and solve the Newton equation of motion for a one-dimensional harmonic oscillator. • Write an expression of energy of a water molecule suitable for molecular dynamics. Explain the symbols/terms. • What is the difference the Monte Carlo method and molecular dynamics; what do they have in common? • What is the difference between molecular dynamics and molecular mechanics? • What are the restrictions of molecular dynamics, in comparison to quantum dynamics? What are the advantages of MD? • Write and solve the Schrdinger equation for a free particle in one-dimensional potential well. • Write and solve the Schrdinger equation for a harmonic oscillator and discuss the solution (no derivation). • Write the Slater determinant (D) for two electrons in to spinorbitals 1 a 2. • What is the Born-Oppenheimer approximation? • What is the spin of an electron?

17. Control questions • What is the spin of an electron? • Write the Schrdinger equation for the hydrogen molecule, explain the symbols. • What is neglected in Hartree-Fock equations, if compared to the Schrdinger equation? • Why is solving the HF equations so difficult? • Compare the computational times for MP2 and HF computations. How is it dependent on the number of atomic orbitals? • What is configuration interaction? • What is the Hellmann-Feynman theorem? • If a Hamiltonian of a molecule contains expressions with electron density and its gradient only, can we describe this method as LDA? • What does the Kohn-Hohenberg theorem say? • What is the difference between Kohn-Sham and Hartree-Fock equations? Which are closer to the Schrodinger equation? • Compare the general quality of the HF, MP2, MP4, Becke3LYP, LDA, BPW91 levels of approximation.

18. Control questions • Compare the quality of the 3-21G, 6-31G, 4-31G, 6-31G**, and 6-31G* basis sets. • What are the definitions of the electric dipole moment and magnetic dipole moment? • What is the multipole expansion, when is it reasonable to apply it? • What is the electric polarizability and how it can be obtained from the wavefunction? • What is the chemical shielding tensor (NMR) and how it can be obtained from the wavefunction? • What forces experience atomic nuclei in a molecule put into magnetic field? Can you estimate their magnitude?

19. Control questions • Why are NMR spectral line (sometimes) split? • Write the vibrational Hamiltonian of a molecule in the harmonic approximation. • How are the vibrational normal mode coordinates defined, what is their meaning? • What determines the absorption intensity in the infrared spectrum? Which would give a bigger signal, C-H or C=O stretching? • Write typical steps in simulation of infrared spectrum, e.g. of ethanol. • What are the advantages for experiment and simulation of the infrared spectroscopy, if compared to Raman scattering? • What are the advantages of VCD spectroscopy if compared to IR? • Which kind of information and how can you find based on the VCD spectrum of a compounds (providing you know the summary formula)? • What is atomic polar and atomic axial tensor? How to calculate it? • What is the principle of the ROA spectroscopy, what do have IR and ROA spectra of the same molecule in common?

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