Chemical bonding in molecules

I - What is a molecule? Modeling ? Electronic energy Vibrational energy Rotational energy M M e- 1 2 V=2 1 2 V=1 1 2 V=0 Electronic level Vibrational levels Rotational levels K

I - What is a molecules? Modeling ? B e1 e2 e3 O Total Hamiltonian for a diatomic molecule Aim: Solve the time independent Schrödinger equation where N-electron atom wave function with obeys the Pauli exclusion principle

Step 1: Electronic part – no spin Electronic Hamiltonian with We solve the time independent Schrödinger equation at fixed R with the electronic wave function which forms a basis set The exact molecular wave function can be expanded such as

Step 2: The total Hamiltonian – no spin We bravely solve by projecting this equation on all electronic wavefunctions We obtain coupled equations for electron and nuclear wavefunctions

Step 3: Born-Oppenheimer approximation - adiabatic approximation - Introducing spherical coordinate for TN We find

Step 3: Born-Oppenheimer approximation - adiabatic approximation - In the case that the motion of the nucleus is slow with respect to the motion of the electrons Assuming We just “need” to solve the nuclear wave function in a potential made by the electrons

Step 3: Born-Oppenheimer approximation - adiabatic approximation - General form of the electronic energy Limit of validity: Coupling between states Collision experiments Rydberg states … is the electronic dissociation energy

Electronic wavefunction – symmetries x y z e- + L - j 1 ) Cylindrical symmetry Lz commutes with He Spectroscopic notation Electron configuration Electronic state Value l 0 1 2 3 Value L 0 1 2 3 Letter S P D F Letter s p d f

Electronic wavefunction – symmetries z 2) Symmetry plane e- Reflection Ry, , commutes with He y x Reflection Ry does not commute with Lz Electronic states Two symmetries when Doubly degenerated when

Electronic wavefunction – symmetries z For homonuclear molecules (N2, O2,…) e- Inversion Ir, , commutes with He and Lz y x Electronic states Symmetric gerade (g) Anti-symmetric ungerade (u) Sg,u Pg,u …

Electronic wavefunction – symmetries z For homonuclear molecules (N2, O2,…) e- Inversion IR, , commutes with He and Lz y x Electronic states unaffected by IR affected by IR

Total wavefunction – Hund’s coupling cases Hund’s case a: L and S precess about R with well-defined componets L and S, along R. N couples with WR to form J, where W=L+S ^ Electronic interaction is much larger than spin orbit coupling interaction which in turn is much larger than the rotational energy

Electronic wavefunction – term manifold Hetero-nuclear molecules A (L1, S1) B (L2, S2) From separated atoms Example 1: Molecular states made from two atoms with L1=L2=1 Molecular State Parity = (Parity atom A) * (Parity Atom B)

Electronic wavefunction – term manifold Hetero-nuclear molecules A (L1, S1) B (L2, S2) From separated atoms Molecular State Multilicity = (2S+1) Molecular State Parity = (Parity atom A) * (Parity Atom B)

Electronic wavefunction – term manifold Hetero-nuclear molecules Example 2: NH molecule N:1s22s22p3 (2P,2D,4S) – odd (u) H:1s – even (g) …to unified atom O:1s22s23p4 (1S,1D,3P) – even (g)

Electronic wavefunction – term manifold Homo-nuclear molecules Example 3: Determination of the dissociation limit of O2 molecule

Electronic wavefunction – molecular orbital From the orbital of the united atoms to to one electron molecuar orbital (state) United atom Molecule state l MO Occupation ns nss 2 npz 1 nps npx,npy npp 4 ndz2 nds ndxz,ndyz ndp ndd

Electronic wavefunction – molecular orbital Many electrons molecular states Non equivalent electrons Equivalent electrons Example 4: Determination of the molecular state of the BH molecule

Electronic wavefunction – molecular orbital Correlation diagrams from united atom to separated atoms Conservation laws: The quantum number l=|ml| is independent of R. The principal quantum number n and the angular quantum number l can change. Wave function parity does not depend on the inter-nuclear separation. If two states in the united atom have the same symmetry, quantum number L, and multiplicity (2S+1), they can not cross for any inter-nuclear distance.

Electronic wavefunction – molecular orbital Hetero-nuclear molecules

Electronic wavefunction – molecular orbital Homo-nuclear molecules

Chemical bounding – molecular orbital Homo-nuclear molecules

Chemical bounding – molecular orbital How to fill molecular orbitals (MO’s): MO’s with the lowest energy are filled first (Aufbau principle) There is a maximum of two electrons per MO with opposite spins (Pauli exclusion principle ) When there are several MO's with equal energy, the electrons fill into the MO's one at a time before filling two electrons into any (Hund's rule) The chemical bound is stable if the bond order is positive The filled MO highest in energy is called the Highest Occupied Molecular Orbital (HOMO) The empty MO just above it, is the Lowest Unoccupied Molecular Orbital (LUMO)

Chemical bounding – molecular orbital B2- diboron O2- dioxygen B: 1s22s22p O: 1s22s22p4