©2011, Jordan, Schmidt & Kable Lecture 13 Lecture 13 Self-consistent field theory This is how we do it

©2013, Jordan, Schmidt & Kable Lecture 13 Copyright Notice Some images used in these lectures are taken, with permission, from “Physical Chemistry”, T. Engel and P. Reid, (Pearson, Sydney, 2006); denoted “ER” throughout the lectures and other sources as indicated, in accordance with the Australian copyright regulations.

©2013, Jordan, Schmidt & Kable Lecture Show that antisymmetry of wavefunction yields Pauli exclusion principle 13.2 Explain properties of the Slater determinant 13.3 Explain the assumptions/approximations made by Hartree-Fock SCF theory 13.4 Describe the self-consistent field procedure 13.5 Describe the STO-3G basis set 13.6 Interpret computer output Learning outcomes

©2013, Jordan, Schmidt & Kable Lecture 13 It’s way too hard to always get the exact wavefunction. So we choose a set of basis functions from which to form approximate solutions to the Schrödinger equation. Basis functions Eigenfunctions

©2013, Jordan, Schmidt & Kable Lecture 13 Vectors are changed (rotated and stretched/shrunk) by matrices. If we represent the wavefunction of a vector is some basis function space, then the Hamiltonian operator acts like a matrix. Hamiltonian operator is like a matrix i j k  + + = = i j k  = E Where the vector is a solution to the Schrödinger equation, it will be only stretched or shrunk upon operation with the Hamiltonian. In this case it is called an eigenvector.

©2013, Jordan, Schmidt & Kable Lecture 13 Larger symmetry-adapted basis set for a homonuclear diatomic  g  g  u  u  u  g 2×2×

©2013, Jordan, Schmidt & Kable Lecture 13 Not the final story…  g  g  u  u  u  g gg

©2013, Jordan, Schmidt & Kable Lecture 13 Not the final story…  g  g  u  u  u  g uu

©2013, Jordan, Schmidt & Kable Lecture 13 Matrix elements in symmetry adapted basis What are Hamiltonian matrix elements?  g +  g +

©2013, Jordan, Schmidt & Kable Lecture 13 This new interaction energy Depends on the energy spacing between the 2  g and the 1  g

©2013, Jordan, Schmidt & Kable Lecture 13 sp mixing c.f. 2p 2s Smallest energy gap, and thus largest mixing between 2s and 2p is for Boron. Largest energy gap, and thus smallest mixing between 2s and 2p is for Fluorine.

©2013, Jordan, Schmidt & Kable Lecture 13 sp mixing Be 2 B2B2 C2C2 N2N2 weakly boundparamagnetic diamagnetic

©2013, Jordan, Schmidt & Kable Lecture 13 sp mixing in N 2

©2013, Jordan, Schmidt & Kable Lecture 13 sp “hybrids” in N 2  -bonding orbital Linear combinations of lone pairs :N≡N: “2s  ” “2s  *” “2p  ”

©2013, Jordan, Schmidt & Kable Lecture 13 The Hartree product Hartree approximated the wavefunction as a product of the one-electron wavefunctions  (1) and  (2): Here 1 and 2 refer to the coordinates of each electron in, say, helium. The problem with this approach is that it does not account for the Pauli Exclusion Principle. This has to be added ad hoc. Minimization of the energy would place the three electrons of lithium all in the 1s orbital!

©2013, Jordan, Schmidt & Kable Lecture 13 Fermi-Dirac statistics As it turns out, if two electrons are exchanged in space and spin, then the (many-electron) wavefunction must change sign, but have the same absolute value since electrons are indistinguishable. Here the bar indicates down spin, and the non-barred orbital is up spin. This is called a spin-orbital, and is the product of a spatial orbital and a spin function. What would the wavefunction be if the orbitals were both up spin?

©2013, Jordan, Schmidt & Kable Lecture 13 The Slater determinant The antisymmetry requirement of Fermi-Dirac statistics can be conveniently packaged by writing the wavefunction as a determinant: Exchanging two electrons is like exchanging rows, which changes the sign of the determinant.

©2013, Jordan, Schmidt & Kable Lecture 13 The Slater determinant Generally, for a closed-shell molecule: Exchanging two electrons is like exchanging rows, which changes the sign of the determinant. If the third electron went into  1, then two rows would be the same, and the determinant is zero.

©2013, Jordan, Schmidt & Kable Lecture 13 The Hartree-Fock Approximation The electronic wavefunction is written as a single Slater determinant. The spin-orbitals are products of spatial orbitals and spin functions: The spatial orbitals are constructed from basis functions, .

©2013, Jordan, Schmidt & Kable Lecture 13 The Hartree-Fock Hamiltonian The Hamiltonian is the sum of kinetic and potential energies: The basis coefficients a jk are optimized to minimize the energy of the one- electron orbitals. But, since the Hamiltonian depends on the wavefunctions of the other electrons, this must be iterated until “self-consistency” is achieved. Hartree-Fock (HF) is a Self-Consistent Field (SCF) Method.

©2013, Jordan, Schmidt & Kable Lecture 13 The Hartree-Fock Method HF procedure obtains optimized orbitals according to the variational principle. The total wavefunction is a single Slater determinant.

©2013, Jordan, Schmidt & Kable Lecture 13 Example – N 2 using minimal STO-3G basis The STO-3G basis set is the smallest that one would ever use. It has just the core and valence orbitals represented, and is pretty much only good for qualitative calculations. ATOMIC BASIS SET THE CONTRACTED PRIMITIVE FUNCTIONS HAVE BEEN UNNORMALIZED THE CONTRACTED BASIS FUNCTIONS ARE NOW NORMALIZED TO UNITY SHELL TYPE PRIM EXPONENT CONTRACTION COEFFICIENTS N 3 S ( ) 3 S ( ) 3 S ( ) 4 L ( ) ( ) 4 L ( ) ( ) 4 L ( ) ( ) TOTAL NUMBER OF SHELLS = 4 TOTAL NUMBER OF BASIS FUNCTIONS = 10 NUMBER OF ELECTRONS = 14 CHARGE OF MOLECULE = 0 STATE MULTIPLICITY = 1 NUMBER OF OCCUPIED ORBITALS (ALPHA) = 7 NUMBER OF OCCUPIED ORBITALS (BETA ) = 7 TOTAL NUMBER OF ATOMS = 2 Core and valence orbitals are sums of three gaussians. 2p orbitals are made by multiplication with x, y or z to create node.

©2013, Jordan, Schmidt & Kable Lecture 13 Example – N 2 using minimal STO-3G basis The coefficients of these 3-Gaussian basis functions are optimized. ITER EX DEM TOTAL ENERGY E CHANGE DENSITY CHANGE ORB. GRAD INTEGRALS SKIPPED START SECOND ORDER SCF EIGENVECTORS AG B1U AG B1U B3U 1 N 1 S N 1 S N 1 X N 1 Y N 1 Z N 2 S N 2 S N 2 X N 2 Y N 2 Z B2U AG B2G B3G B1U 1 N 1 S N 1 S N 1 X N 1 Y N 1 Z N 2 S N 2 S N 2 X N 2 Y N 2 Z

©2013, Jordan, Schmidt & Kable Lecture 13 Example – N 2 using minimal STO-3G basis …and here are your orbitals.

©2013, Jordan, Schmidt & Kable Lecture 13 Example – N 2 using minimal STO-3G basis Here is the MO diagram Energy (Hartree)