Quantum Mechanical Description of Molecules Glenn V. Lo Department of Physical Sciences Nicholls State University

Born Oppenheimer Approximation Imagine a fixed molecular geometry (define the locations of the nuclei and assume they’re not moving) For each possible molecular geometry, we can solve for a set of allowed electronic states One electronic state corresponds to a set of wavefunctions that describe electrons (“molecular orbitals”) and an allowed energy (the “electronic energy”) The electronic state that corresponds to the lowest electronic energy is called the “ground electronic state” To explain properties: examine molecular orbitals for ground electronic state at the “equilibrium geometry”

Determining equilibrium geometry For each geometry: calculate the allowed “molecular potential energies” by adding allowed electronic energies and potential energy due to internuclear repulsion The sum is called “molecular potential energy” because they exclude kinetic energy of nuclei (remember: each calculation assumes fixed nuclei) The equilibrium geometry gives lowest possible molecular potential energy for the ground electronic state.

Example For a diatomic molecule: molecular potential energy depends only on r, the distance between nuclei. What is the equilibrium geometry for H 2 if its ground electronic state is as shown by the blue curve in the Figure?

Molecular Orbitals Like atomic orbitals, a molecular orbital is a mathematical function that gives us information about electrons For convenience, express in terms of atomic orbitals Valence Bond (VB) Theory: molecular orbitals describing shared electrons are expressed as overlapping atomic orbitals of neighboring atoms; “localized” LCAO-MO Theory (linear combination of atomic orbitals – molecular orbital): molecular orbitals are expressed in terms of atomic orbitals of ALL atoms in the molecule; “delocalized”

Valence Bond Theory Single bond due to a “sigma bond” Sigma bond: overlapping atomic orbitals that describe a high electron density along the line of center between two nuclei Double and triple bonds are due to one sigma bond and one or two “pi bonds” Pi bond: overlapping atomic orbitals that describe a high electron density around the line of center between nuclei

Example How many sigma and pi bonds are between C and N in HCN?

VB: Orbital Hybridization VB Theory uses orbital hybridization to explain bond angles. Explain 180 o H-C-C angle in acetylene Free C atom has: (2s, 2p, 2p, 2p) C atom in molecule has: (2sp, 2sp, 2p, 2p) Due to hybridization: high electron densities of two sp hybrids are 180 o apart One sp hybrid overlaps with 1s of neighboring H atom Other sp hybrid overlaps with sp hybrid of neighboring C atom Remaining 2p orbitals overlap with 2p orbitals of neighboring C to form two pi bonds.

VB: Hybridization and Steric Number Steric Number = 2: sp Steric Number = 3: sp 2 Steric Number = 4: sp 3 Steric Number = 5: sp 3 d Steric Number = 6: sp 3 d 2

Example According to the valence bond method, what hybrid orbitals overlap in the C-to-N bond in the urea?

VB: Orbital Hybridization VB Theory uses orbital hybridization to explain bond lengths. Hybrid orbitals with greater “p character” stretch out farther from the nucleus  longer bond length Bond lengths due to sp 3 > sp 2 > sp

Example Explain why the C-to-C bond in H 3 C-CH 3 is longer than that in H 2 C=CH 2. Explain why the C-to-H bond is shorter than the C-to-C bond in both molecules.

LCAO-MO Orbital Diagram Molecular orbital = constructed by combining atomic orbitals from all atoms in the molecule Contributions of atomic orbitals to a particular MO are not equal; depend on the symmetry of the molecule Example: six pi molecular orbitals of benzene (C 6 H 6 ) constructed from 2p orbitals of six C atoms

Molecular Orbital Diagram Summarizes how molecular orbitals are related to the atomic orbitals MO that has lower energy than contributing AOs are called “bonding”; those with higher energy are called “antibonding” Diatomic molecules MO names: , , ,  Antibonding orbitals are denoted with *

Bond Order Use Pauli’s principle: assign electrons to orbitals Bond Order = (B-A)/2 B = number of electrons assigned to bonding orbitals A = number of electrons assigned to antibonding orbitals If bond order > 0, predict a stable bond Bond order=1, single bond Bond order=2, double bond Bond order=3, triple bond

Example Determine the bond order for F 2.

Visualizing Molecular Orbitals