Introduction to Hartree-Fock Molecular Orbital Theory

1 Introduction to Hartree-Fock Molecular Orbital TheoryC. David SherrillSchool of Chemistry and BiochemistryGeorgia Institute of TechnologyPreceding MaterialThese notes pick up from General Introduction to Electronic Structure Theory by the Hartree-Fock is forA way to approximately solve the Electronic Schr dinger equationThis gives us the electronic wavefunction, from which we can extract dipole moment, polarizability, etc!The electronic energy Eel(R) is the potential energy surface: gives equilibrium geometries, reaction paths, Slater DeterminantWe will assume the electronic wavefunction can be writtenas a single Slater Detrminant (this is an approximation). This enforces an antisymmetric wavefunction. The hartree -Fockprocedure will give us the :orJohn SlaterImages from WikipediaHartree-Fock Molecular Orbital Theory1.

2 Invoke the Born-Oppenheimer approximation2. Express the electronic wavefunction as a single Slater Determinant3. Solve for those orbitals which minimize the electronic energy (variational method)This winds up being mathematically equivalent to assuming each electron interacts only with the average charge cloud of the other electronsDouglas HartreeV. A. FockImages from WikipediaThe OperatorsOne-electron operator: for electron i,its KE and its attraction to all nucleiTwo-electron operator: for electronsiand j, their Coulomb repulsionElectronic Hamiltonian in terms of these operators:The Hartree-Fock EnergyIf the variational theorem says to minimize the energy, what isthe energy of a Slater determinant?Slater s Rules tell us how to get the matrix elements of the electronic Hamiltonian using Slater DeterminantsThe Hartree-Fock EnergyOne-electron integral (4-dimensional):Two-electron integral (8-dimensional):Physical Meaning of the TermsEach electron contributes a one-electron integralThis looks like the expectation (average) value of the operator for an electron in Orbital , so long as the Orbital is normalizedRecall contains electron KE and potential of attraction to all the nucleiSum over all orbitalsito get total electron KE and attraction to nucleiPhysical Meaning of the TermsEach pair of electrons (in orbitalsiand j) has a Coulomb integral.

3 Probability electron 1 in Orbital iis located at x1 Probability electron 2 in Orbital jis located at x2 Coulomb repulsion between electron atx1and electron at x2 Integrate over all possiblelocations for the electronsOverall this integral represents the Coulomb repulsion between electron 1 in Orbital iand electron 2 in Orbital jPhysical Meaning of the TermsEach pair of electrons (in orbitals iand j) has also has an Exchange integral :This is like the Coulomb integralExcept two of the Orbital indices have been exchanged !No direct physical meaning .. consequence of Slater DeterminantHartree-Fock Energy Example Simple example: He atom 2 electrons, 1sa, 1s Number the spin orbitals1=1sa, 2=1s 12We (typically) need a computer to evaluate these integralsIs this as simple as we can get it?

4 No! The last integral is zero. Why?Spin Factorization and Spatial OrbitalsRecall each spin Orbital (x) is a function of 4 coordinates: (x,y,z, )We normally write each spatial Orbital as a product of a spatial part (r) and a spin part, which we might call ( ), , (x) = (r) ( ) [recall r = {x,y,z}]The operators in hartree -Focktheory, and 1/r12, do not depend on the spin coordinateThat means an integral over x can be factored into a simple integral over the spin coordinate (no operators) times a more complicated integral (involving operators) over the spatial coordinates r, ,Spin Factorization and Spatial OrbitalsRecall each spin Orbital (x) is a function of 4 coordinates: (x,y,z, )We normally write each spatial Orbital as a product of a spatial part (r) and a spin part, which we might call ( ), , (x) = (r) ( ) [recall r = {x,y,z}]The operators in hartree -Focktheory, and 1/r12, do not depend on the spin coordinateThat means an integral over x can be factored into a simple integral over the spin coordinate (no operators) times a more complicated integral (involving operators) over the spatial coordinates r, ,Spin Factorization and Spatial OrbitalsRecall each spin Orbital (x) is a function of 4 coordinates.

5 (x,y,z, )We normally write each spatial Orbital as a product of a spatial part (r) and a spin part, which we might call ( ), , (x) = (r) ( ) [recall r = {x,y,z}]The operators in hartree -Focktheory, and 1/r12, do not depend on the spin coordinateThat means an integral over x can be factored into a simple integral over the spin coordinate (no operators) times a more complicated integral (involving operators) over the spatial coordinates r, ,Spin Factorization and Spatial OrbitalsRecall each spin Orbital (x) is a function of 4 coordinates: (x,y,z, )We normally write each spatial Orbital as a product of a spatial part (r) and a spin part, which we might call ( ), , (x) = (r) ( ) [recall r = {x,y,z}]The operators in hartree -Focktheory, and 1/r12, do not depend on the spin coordinateThat means an integral over x can be factored into a simple integral over the spin coordinate (no operators) times a more complicated integral (involving operators)

6 Over the spatial coordinates r, ,Spin Factorization of 2-elec IntegralsWe can also factorize out the spin functions in two-electron integralsSpin Factorization of 2-elec IntegralsWe can also factorize out the spin functions in two-electron integralsSpin IntegrationIntegrals over spin coordinates are usually easy to do because the spin function ( ) is usually just a( ) or ( )The spin integration rules for a( ) and ( ) are very easy and result in 1 or 0 Spin Integration General ResultsOne-electron integrals survive if both spin- orbitals have the same spinTwo-electron integrals survive if spins i,jmatch on the left and spins k,lmatch on the rightSpin Integration in hartree -FockWe just did the generic one-and two-electron integrals; the ones in Hartree-Fock are specific typesHow do these repeats affect spin integration?

7 Same index both sideseach index appears twice(although in different places)Spin Integration in hartree -FockOne-electron IntegralsSpin Integration in hartree -FockOne-electron IntegralsFactor in brackets is always = 1 (same spin function)Spin Integration in hartree -FockCoulomb IntegralsSpin Integration in hartree -FockCoulomb Integrals11 Spin Integration in hartree -FockCoulomb Integrals11 Coulomb integrals alwayssurvive spin integration!Spin Integration in hartree -FockExchange IntegralsSpin Integration in hartree -FockExchange IntegralsSpin Integration in hartree -FockExchange IntegralsExchange integrals sometimessurvive spin integration!Need spin orbitalsiand jto have same spinBack to our hartree -FockEnergy Example Simple example: He atom 2 electrons, 1sa, 1s Renumber the spin orbitals : 1=1sa, 1=1s 11 Exchange integralsmust have same spin on iand j0 Can we simplify this result?

8 Yes! (normally)Simplifying Spatial Integrals11 Helium atom exampleSpatial part is the same. Therefore after spin integration, spatial integrals in terms of these two orbitalsmust be the same. Let s Spatial IntegralsSimplifying Spatial IntegralsConclusion: if spin orbitalscome in a, pairs with the same spatial part , then after spin integration we can remove the overbarlabels on the spatial Orbital integralsSimplifying Spatial Integrals11 Helium atom exampleRestricted orbitals (Restricted Hartree-Fock , RHF)Spin orbitals always come in (a, ) pairs that share the same spatial Orbital This is how we normally think about orbitals in chemistryThere s no reason not to use such restricted orbitals in normal molecules in which all electrons are paired ( closed shell molecules)Unrestricted orbitals (Unrestricted Hartree-Fock , UHF)When not all electrons are paired ( open-shell molecules), we can sometimes get a lower energy solution if we unrestrict the orbitals .

9 Allow the spatial part to be different for the aspin than for the spin:Introduces spin contamination ( , mix singlet and triplet); can sometimescause severe errors in propertiesCan be easier to convergePseudo-Classical Interpretation of hartree -FockEnergiesEach electron contributes a term (i| |j) = hijEach unique pair of electrons contributes a Coulomb repulsion (ii|jj) = JijEach unique pair of same spin electrons contributes an exchange term (ij|ji) = -KijAlwayssurvive spin integrationNonzero only if iandj have same spinHartree-FockEnergy Example: Li atom113 electrons: 1=1sa, 1=1s , 2=2sa2 Connection to Hund s RulesWhy do Hund srules say a high-spin state is more stable than a low-spin state for a given electron configuration?

10 We can use hartree -Focktheory to understand thisAs an example, consider the p2electron configurationvs123123 EHF= h11+ h22+ J12 K12 EHF= 2h11+ J11 Energy lower because exchange integral is subtracted! Hartree-Fock EquationsMinimizing the Hartree-Fock energy with respect to the orbitals leads to the Hartree-Fock equations for the orbitals :Problem: This is a very complicated integro-differential equation!Roothan to the Rescue!If we introduce a basis set, we convert the equation into a much simpler linear algebra problemClemens C. J. RoothanImages from WikipediaSummary of Hartree-Fock -Roothan EquationsPseudo-eigenvalue equationC collects the expansion coefficients for each Orbital expressed as a linear combination of the basis functions (each column of C is a Molecular Orbital )Note: C depends on F, which depends C!