Lecture 10. Chemical Bonding. Generalization & Matrix Representation References Engel Ch.12, Ratner & Schatz, Ch.11 Quantum Chemistry, McQuarrie, Ch.9.

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Presentation transcript:

Lecture 10. Chemical Bonding. Generalization & Matrix Representation References Engel Ch.12, Ratner & Schatz, Ch.11 Quantum Chemistry, McQuarrie, Ch.9 Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.8 Computational Chemistry, Lewars (2003), Ch.4 A Brief Review of Elementary Quantum Chemistry

LCAO-MO Molecular wave function in HF approximation = Antisymmetrized product (Slater determinant) of one-electron molecular wave functions (molecular orbitals)

Basis set

LCAO-MO Approximation (Linear Combination of Atomic Orbitals for Molecular Orbital)

LCAO-MO Example: Hydrogen (H 2 ) Molecule

Color indicates the sign (+/  ) of the coefficient. Size indicates the magnitude of the coefficient. No node in bonding region Node in bonding region No node in bonding region Bigger contribution Node in bonding region  13.6 eV  16.0 eV  9.2 eV  13.6 eV  18.9 eV  19.6 eV  10.3 eV Homonuclear vs. heteronuclear diatomic molecules

Heteronuclear diatomic molecule: MO diagram of HF

Rearrange and differentiate

→ Matrix diagonalization problem

LCAO-MO & HF Equation: Roothaan-Hall Equation  Fock matrix F & Overlap matrix S

 Non-linear Integro-Differential equation  Linear Algebra Matrix equation

Hartree-Fock-Roothaan-Hall equation (Summary) where MO (one-electron) MOAO H  = E  where n-electron Wave function Fock operator Hamiltonian operator LC where Fock matrix overlap matrix

Hartree-Fock-Roothaan-Hall equation Overlap matrix element Secular equation Fock matrix element