1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 8Linear Combination of Atomic Orbitals (LCAO) In this lecture you will learn:• An approach to energy states in molecules based on the linear combination of atomic orbitalsCHHHHECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityEnergy Bands and Atomic Potentials in CrystalsThe potential energy of an electron due to a single isolated atom looks like:0xrVIn a crystal, the potential energy due to all the atoms in the lattice looks like:0xrVEnergy levelsThe lowest energy levels and wavefunctions of electrons remain unchanged when going from an isolated atom to a crystalThe higher energy levels (usually corresponding to the outermost atomic shell) get modified, and the corresponding wavefunctions are no longer localized at individual atoms but become spread over the entire crystalPotential of an isolated atomPotential in a crystal00EnergybandsVacuum level2ECE 407 – Spring 2009 – Farhan Rana – Cornell University0xrVPotential in a crystal0EnergybandsVacuum levelFailure of the Nearly-Free-Electron Approach • For energy bands that are higher in energy (e.g. 2 & 3 in the figure above) the periodic potential of the atoms can be taken as a small perturbationFor higher energy bands, the nearly-free-electron approach works well and gives almost the correct results• For energy bands that are lower in energy (e.g. 1 in the figure above) the periodic potential of the atoms is a strong perturbation  For lower energy bands, the nearly-free-electron approach does not usually work very well123ECE 407 – Spring 2009 – Farhan Rana – Cornell University211121Energy(eV)Nearly-Free-Electron Approach Vs LCAO for Germanium221112111211Energy(eV)LCAOEnergy(eV)341111NFAEmpirical Pseudopotential• For most semiconductors, the nearly-free-electron approach does not work very well• LCAO (or tight binding) works much better and provides additional insightsFBZ (FCC lattice)3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLCAO: From Hydrogen Atom to Hydrogen MoleculeConsider a Hydrogen atom with one electron in the 1s orbital:0rrV1s energy levelPotential of a Hydrogen atomOne can solve the Schrodinger equation:   rErrVrm222and find the energy of the 1s orbital and its wavefunctionrErHssso111ˆ0rrs1rVmHo222ˆwhere:oarosear311Angular probability distribution for the 1s orbitalRadial amplitude for the 1s orbitalECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLinear Combination of Atomic Orbitals (LCAO)Now consider a Hydrogen molecule made up of two covalently bonded Hydrogen atoms sitting at a distance of 2d from each other, as shown:Hamiltonian for an electron is:xdrVxdrVmHˆˆ2ˆ22The basic idea behind LCAO approach is to construct a trial variational solution in which the wavefunction is made up of a linear combination (or superposition) of orbitals of isolated atoms:xdrcxdrcrsbsaˆˆ11And then plugging the trial solution into the Schrodinger equation to find the coefficients caand cband the new eigenenergies:rErHˆ0xrVd-da atom b atom4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLCAO: From Hydrogen Atom to Hydrogen Moleculexdrcxdrcrsbsaˆˆ11Plug the LCAO solution:into:STEP 1: take the bra of the equation first with to get:  rErHˆxdrsˆ1  xdrcxdrcxdrExdrcxdrcHxdrsbsassbsasˆˆˆˆˆˆˆ111111Note that:sssExdrHxdr111ˆˆˆLet:xdrVxdrVmHˆˆ2ˆ22 ssssVxdrHxdr ˆˆˆ110ˆˆ11 xdrxdrss0xrVd-da atom b atomNot exactly zero – but we will assume so for simplicityECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityLCAO: From Hydrogen Atom to Hydrogen MoleculeSo we get finally:abssascEcVcE1STEP 2: take the bra of the equation now with to get: xdrsˆ1bassbscEcVcE1Write the two equations obtained in matrix form:babassssssccEccEVVE11This is now an eigenvalue equation and the two solutions are:112111211babasssccccVEE5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBonding and Anti-Bonding OrbitalssssBVEE1For the lower energy solution we have: xdrxdrrssBˆˆ2111xrB0 d-dFor the higher energy solution we have:sssAVEE1 xdrxdrrssAˆˆ2111xrA0 d-dThis is called the “Bonding molecular orbital”This is called the “Anti-bonding molecular orbital”0xrVd-dBEAEECE 407 – Spring 2009 – Farhan Rana – Cornell University0xrV1s atomic energy level0 xrVd-d2VssEnergy levels of the moleculeLCAO: Energy Level Splitting and the Energy Matrix ElementBondingAnti-bondingEnergy level diagram going from two isolated atoms to the molecule:sE1:2AE:1BE:1ssV2The two 1s orbitals on each Hydrogen atom combine to generate two molecular orbitals – the bonding orbital and the anti-bonding orbital – with energy splitting related to the energy matrix element:ssssVxdrHxdr ˆˆˆ11The total energy is lowered when Hydrogen atoms form a Hydrogen molecule6ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityAtomic Orbitals• Wavefunction amplitudes of the atomic s and p orbitals in the angular directions are plotted• The s-orbital is spherically symmetric• The p-orbitals have +ve and –ve lobes and are oriented along x-axis, y-axis, and z-axispzpxpysECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityOrbitals and BondingThere are two main types of co-valent bonds: sigma bonds (or -bonds) and pi-bonds (or -bonds)(1) Sigma bonds (or -bonds):sss-s