Chemistry 6440 / 7440
Introduction to
Molecular Orbitals
Resources
•Grant and Richards, Chapter 2
•Foresman and Frisch, Exploring Chemistrywith Electronic Structure Methods(Gaussian Inc., 1996)
•Cramer, Chapter 4
•Jensen, Chapter 3
•Leach, Chapter 2
•Ostlund and Szabo, Modern QuantumChemistry (McGraw-Hill, 1982)
Potential Energy Surfaces
•molecular mechanics uses empirical functions forthe interaction of atoms in molecules to calculatepotential energy surfaces
•these interactions are due to the behavior of theelectrons and nuclei
•electrons are too small and too light to bedescribed by classical mechanics
•electrons need to be described by quantummechanics
•accurate potential energy surfaces for moleculescan be calculated using modern electronicstructure methods
Schrödinger Equation
•H is the quantum mechanical Hamiltonian for thesystem (an operator containing derivatives)
•E is the energy of the system
• is the wavefunction (contains everything weare allowed to know about the system)
•||2 is the probability distribution of the particles(as probability distribution, ||2 needs to becontinuous, single valued and integrate to 1)
Hamiltonian for a Molecule
•kinetic energy of the electrons
•kinetic energy of the nuclei
•electrostatic interaction between the electronsand the nuclei
•electrostatic interaction between the electrons
•electrostatic interaction between the nuclei
Solving the Schrödinger Equation
•analytic solutions can be obtained only forvery simple systems
•particle in a box, harmonic oscillator,hydrogen atom can be solved exactly
•need to make approximations so thatmolecules can be treated
•approximations are a trade off betweenease of computation and accuracy of theresult
Expectation Values
•for every measurable property, we canconstruct an operator
•repeated measurements will give anaverage value of the operator
•the average value or expectation value ofan operator can be calculated by:
Variational Theorem
•the expectation value of the Hamiltonian is thevariational energy
•the variational energy is an upper bound to the lowestenergy of the system
•any approximate wavefunction will yield an energyhigher than the ground state energy
•parameters in an approximate wavefunction can bevaried to minimize the Evar
•this yields a better estimate of the ground state energyand a better approximation to the wavefunction
Born-Oppenheimer Approximation
•the nuclei are much heavier than the electronsand move more slowly than the electrons
•in the Born-Oppenheimer approximation, wefreeze the nuclear positions, Rnuc, and calculatethe electronic wavefunction, el(rel;Rnuc) andenergy E(Rnuc)
•E(Rnuc) is the potential energy surface of themolecule (i.e. the energy as a function of thegeometry)
•on this potential energy surface, we can treat themotion of the nuclei classically or quantummechanically
Born-Oppenheimer Approximation
•freeze the nuclear positions (nuclear kinetic energy iszero in the electronic Hamiltonian)
•calculate the electronic wavefunction and energy
•E depends on the nuclear positions through the nuclear-electron attraction and nuclear-nuclear repulsion terms
•E = 0 corresponds to all particles at infinite separation
Nuclear motion on theBorn-Oppenheimer surface
•Classical treatment of the nuclei (e,g. classicaltrajectories)
•Quantum treatment of the nuclei (e.g. molecularvibrations)
Hartree Approximation
•assume that a many electron wavefunctioncan be written as a product of one electronfunctions
•if we use the variational energy, solving themany electron Schrödinger equation isreduced to solving a series of one electronSchrödinger equations
•each electron interacts with the averagedistribution of the other electrons
Hartree-Fock Approximation
•the Pauli principle requires that a wavefunction for electronsmust change sign when any two electrons are permuted
–since |(1,2)|2=|(2,1)|2, (1,2)=(2,1) (minus sign for fermions)
•the Hartree-product wavefunction must be antisymmetrized
•can be done by writing the wavefunction as a determinant
–determinants change sign when any two columns are switched
Spin Orbitals
•each spin orbital I describes the distribution of oneelectron
•in a Hartree-Fock wavefunction, each electron must bein a different spin orbital (or else the determinant is zero)
•an electron has both space and spin coordinates
•an electron can be alpha spin (, , spin up) or beta spin(, , spin down)
•each spatial orbital can be combined with an alpha orbeta spin component to form a spin orbital
•thus, at most two electrons can be in each spatial orbital
Fock Equation
•take the Hartree-Fock wavefunction
•put it into the variational energy expression
•minimize the energy with respect to changes in the orbitals whilekeeping the orbitals orthonormal
•yields the Fock equation
Fock Equation
•the Fock operator is an effective one electronHamiltonian for an orbital
• is the orbital energy
•each orbital sees the average distribution ofall the other electrons
•finding a many electron wavefunction is reducedto finding a series of one electron orbitals
Fock Operator
•kinetic energy operator
•nuclear-electron attraction operator
Fock Operator
•Coulomb operator (electron-electron repulsion)
•exchange operator (purely quantum mechanical-arises from the fact that the wavefunction mustswitch sign when you exchange to electrons)
Solving the Fock Equations
1.obtain an initial guess for all the orbitals i
2.use the current I to construct a new Fockoperator
3.solve the Fock equations for a new set of I
4.if the new I are different from the old I, goback to step 2.
Hartree-Fock Orbitals
•for atoms, the Hartree-Fock orbitals can be computednumerically
•the ‘s resemble the shapes of the hydrogen orbitals
•s, p, d orbitals
•radial part somewhat different, because of interactionwith the other electrons (e.g. electrostatic repulsionand exchange interaction with other electrons)
Hartree-Fock Orbitals
•for homonuclear diatomic molecules, theHartree-Fock orbitals can also be computednumerically (but with much more difficulty)
•the ‘s resemble the shapes of the H2+orbitals
•, , bonding and anti-bonding orbitals
LCAO Approximation
•numerical solutions for the Hartree-Fockorbitals only practical for atoms and diatomics
•diatomic orbitals resemble linear combinationsof atomic orbitals
•e.g. sigma bond in H2
1sA + 1sB
•for polyatomics, approximate the molecularorbital by a linear combination of atomicorbitals (LCAO)
Basis Functions
•’s are called basis functions
•usually centered on atoms
•can be more general and more flexible thanatomic orbitals
•larger number of well chosen basis functionsyields more accurate approximations to themolecular orbitals
Roothaan-Hall Equations
•choose a suitable set of basis functions
•plug into the variational expression for theenergy
•find the coefficients for each orbital thatminimizes the variational energy
Roothaan-Hall Equations
•basis set expansion leads to a matrix form ofthe Fock equations
F Ci = i S Ci
•F – Fock matrix
•Ci – column vector of the molecular orbitalcoefficients
•I – orbital energy
•S – overlap matrix
Fock matrix and Overlap matrix
•Fock matrix
•overlap matrix
Intergrals for the Fock matrix
•Fock matrix involves one electron integrals of kineticand nuclear-electron attraction operators and twoelectron integrals of 1/r
•one electron integrals are fairly easy and few innumber (only N2)
•two electron integrals are much harder and muchmore numerous (N4)
Solving the Roothaan-Hall Equations
1.choose a basis set
2.calculate all the one and two electron integrals
3.obtain an initial guess for all the molecularorbital coefficients Ci
4.use the current Ci to construct a new Fockmatrix
5.solve F Ci = i S Ci for a new set of Ci
6.if the new Ci are different from the old Ci, goback to step 4.
Solving the Roothaan-Hall Equations
•also known as the self consistent field (SCF) equations,since each orbital depends on all the other orbitals, andthey are adjusted until they are all converged
•calculating all two electron integrals is a majorbottleneck, because they are difficult (6 dimensionalintegrals) and very numerous (formally N4)
•iterative solution may be difficult to converge
•formation of the Fock matrix in each cycle is costly,since it involves all N4 two electron integrals
Summary
•start with the Schrödinger equation
•use the variational energy
•Born-Oppenheimer approximation
•Hartree-Fock approximation
•LCAO approximation