1. 1 Bra-ket notation Quantum states representations Homo-nuclear diatomic molecule Hetero-nuclear diatomic molecule Bond energy The Diatomic Molecule MATS-535 Electronics and Photonics Materials Dr. Vladimir Gavrilenko Norfolk State University

2. 2 Bra and ket notation A wave function is a representation of the quantum state in real space. The is called a ‘ket’. At each point r in space the quantum state is represented by the function. The quantum state could be expanded in a set of ortho-normal basis states: Where C’s are called expansion coefficients

3. 3 Bra and ket notation

4. 4 (**) HW

5. 5 Wave Functions of Hydrogen Atom

6. 6 Atomic Wave Function Orthonormality (*)HW

7. 7 The Homonuclear Diatomic Molecule 1 2 Schrodinger equations for isolated H-atoms Full wave function of the H-molecule

8. 8 The Electronic Structure Schrodinger equation Projection onto basis set Orthogonality conditions:

9. 9 The Secular Equation Secular equation

10. 10 Solutions of the Secular Equation Solutions Bonding (b) and antibonding (a) molecular orbital energies Normalized eigen states

11. 11 Electron Energy Structure and Wave Functions of Hydrogen Molecule LUMO – Lowest Unoccupied Molecular Orbital HOMO – Highest Occupied Molecular Orbital

12. 12 Wave Functions Analysis

13. 13 Wave Functions Analysis

14. 14 Dependence on Time Time dependent Schrodinger equation Substitute:

15. 15 Dependence on Time First order differential equations with constant coefficients are solved by exponential functions: where Boundary conditions: at t=0 molecule is in state 1. Therefore: The probability that the molecule is in state 1 or 2:

16. 16 The Heteronuclear Diatomic Molecule A B Schrodinger equations for isolated H-atoms Assume: Full wave function of the H-molecule

17. 17 The Electronic Structure Schrodinger equation: Projection onto basis set

18. 18 The Secular Equation Secular equation

19. 19 The Secular Equation Substitution: Average on-site energy Solution:

20. 20 Charge Redistribution Insert Obtain for: For the bonding state For the antibonding state

21. 21 The Charge Transfer in Heteronuclear Diatomic Molecule AB 1. For: The homonuclear case: no charge transfer

22. 22 The Charge Transfer in Heteronuclear Diatomic Molecule AB 2. For: 1.Bonding state: charge is transferred to the B-molecule (lower on-site energy) 2.Antibonding state: charge is transferred to the A-molecule (higher on-site energy)

23. 23 The Ionic Bond Parameters Polarity: Covalency: Completely ionic limit Completely covalent limit

24. 24 Problems: 1.Using solutions of the secular equation for homonuclear diatomic molecule obtain orthonormal wave functions (see slide 10) 2.Show that wave functions of hydrogen atom are mutually orthogonal (problem marked by(*)) (slide 6). 3.Assuming mutual ortho-normality of atomic s- and p-functions show ortho-normality of the sp 3 hybrides (problem marked by(**)) (slide 4). 4.Obtain conditions for eigen function coefficients corresponding to bonding and antibonding states for heteronuclear diatomic molecule (slide 22).