Lecture 8. Chemical Bonding
Lecture 8. Chemical Bonding.Molecular Orbital Energy Diagram ofOne-Electron Molecule, H2+ References Engel Ch.12, Ratner & Schatz, Ch.10, Appendix A.3 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 Separation of variables in the case of atom (Review)
An atom has translational and electronic degrees of freedom. To good approximation, degrees of freedom are not coupled. Separation of variables! N N N N + Helec(elec) Eigenvalue (energy) = sum Htrans(rCM) N Enuc,elec Eigenfunction (wave function) = product relec (rCM as origin) N elec(elec) rCM relec (rCM as origin) rCM (~rN) Previously in Elements of QM (MS5118) (Review)
A molecule has translational, vibrational, rotational (and electronic) degrees of freedom. To good approximation, degrees of freedom are not coupled. separation of variables (What about electronic?) N N N N + Helec(elec)? Eigenvalue (energy)= sum N N N N +Enuc,elec ? Eigenfunction (wave function) =product N N N N elec(elec) ? H2+, the simplest (one-electron) molecule
varied varied vibrating varied For molecules, Born-Oppenheimer approximation (1927)
Simplifies the Schrdinger equation for molecules (allows separation of variables) Difference in the time scales of nuclear and electronic motions Nuclei are much heavier (~1800 times) and slower than electrons. Electrons can be treated as moving in the field of fixed nuclei. A full Schrdinger equation for a molecule can be solved in two steps: 1) Motion of electron around the nuclei at fixed positions 2) Energy curve of the molecule as a function of nuclei position Allows to focus on the electronic Schrdinger equation H2+, the simplest (one-electron) molecule
Born-Oppenheimer approximation fixed Constant Constant Born-Oppenheimer approximation & Potential energy surface (curve)
E = E(R) A B R Potential energy surface LCAO-MO: Molecular Orbitals (MO) by
Linear Combination of Atomic Orbitals (LCAO) R ~ ra rb ~ Near the equilibrium distance, an electron is delocalized over the whole molecule. (basis functions for) AO AO LCAO (in atomic units) (homonuclear) MO linear expansion coefficients (in general, unknowns determined variationally) LCAO-MO approach & Matrix representation
? ? ? ? LCAO-MO approach & Matrix representation
Hamiltonian matrix element overlap integral where Secular equation, Eigenvalues & Eigenfunctions
where 22 Secular equation two eigenvalues ? + ? + two eigen- functions + ? Secular equation, Eigenvalues & Eigenfunctions
where 22 Secular equation two eigenvalues + ? + two eigen- functions + ? Secular equation, Eigenvalues & Eigenfunctions
where 22 Secular equation two eigenvalues + ? + two eigen- functions + ? Secular equation, Eigenvalues & Eigenfunctions
where 22 Secular equation two eigenvalues + + two eigen- functions + Eigenfunctions, that is, Molecular Orbitals (MO)
Molecular orbitals (MO) of homonuclear diatomic molecules (such as H2) have g or u symmetry with respect to the inversion operation. Molecular orbitals & Probability densities Bonding & Antibonding MOs: Probability density differences
Buildup of electron charge around protons & between protons Decrease of charge outside of bonding region Decrease of electron charge around protons & between protons Increase of charge outside of bonding region 1 c u = ? Overlap integral Hamiltonian matrix element where
0 |Sab| 1 Hamiltonian matrix element where Energy eigenvalues & Integrals
two eigenvalues 0 |Sab| 1 Overlap integral We need to evaluate the overlap integral Sab and the Hamiltonian matrix elements Haa & Hab between two AOs located at an internuclear distance R. Born-Oppenheimer approximation fixed Evaluate Haa = Hbb & Hab = Hba
Coulomb integral in atomic units Exchange or resonance integral in atomic units Electron energy eigenvalue & Potential energy curve
Constant E = ?, V = ? Potential energy minimum bound Internuclear distance R Equilibrium distance, Re MO energy diagram: E(R) as a function of R
and Hab = Haa and MO energy diagram: E(R) as a function of R unbound state: antibonding bound state: bonding Molecular energy diagram for H2+
unbound state: antibonding 1* Haa 1 bound state: bonding