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6 Quantum Mechanics Fall 2008 Problem Set 6 3. Electronic Structure Methods a) Hartree vs. Hartee-Fock The Hartree approximation uses molecular orbitals to represent electrons within orbitals and treats the electrons independently. The wavefunction is represented as a single Slater determinant so that the electron feels the field of the other electrons - no specific electron correlations are included, just an average field of the electrons affecting the electron of interest. The Hartree approximation writes the wavefunction as a product of molecular orbitals. The Hartree-Fock (HF) approximation takes a complicated many-body problem and uses a one-electron problem including electron repulsion to solve the Schrodinger equation. The Hartree-Fock equation is solved using a self-consistent field (SCF). Hartree-Fock introduces anti-symmetry (spin up, spin down), and it has to be written as a determinant because the electrons are indistinguishable. b) Valence Bond This theory aims to solve the same problem as molecular orbital (MO) theory except that the wavefunctions are built directly from the atomic wavefunctions instead of molecular orbitals. The electrons are coupled according to valence bond theory to form bonds. Higher order couplings can be included to improve the approximation of the energy. Valence bond theory allows equal weight to be given to covalent terms and ionic terms in the valence wavefunction when modeling bonds. c) HF-SCF-LCAO-MO HF is the simplest MO theory comprised of a linear combination of atomic orbitals (LCAO). The HF equations are solved self-consistently, which continues to improve the description of the orbitals over iterations to converge to the best representation. This Fock operator is a function of the change in the energy of the orbitals between iterations. HF is solved by SCF, which relies on the variational principle because it is an approximation to the wavefunction, which gives energies that are either higher than or equal to the expected energy. d) 6-31G vs G(2d, 2p) vs. aug-cc-pvqz bases These are three types of Gaussian basis sets.

7 6-31G - two basis functions per valence AD, one basis function per core AD; where 6 describes the number of primitive Gaussians used to describe the overall Gaussian; 31 signifies two Gaussian functions to represent the orbital (basis) one made out of 3 primitive Gaussian functions, one made out of 1 primitive ar Gaussian (i.e., function of the form g =Nx m yn Zoe- 2 ); G stands for Gaussian (written by Pople) 6-3ll++G(2d,2p) - This is a modified 6-31G basis where (+) signifies polarization functions, (2d,2p) signify diffuse functions (before the comma is added to the core, which in this case is 2d, after the comma is added to the valence orbitals, which is this case is 2p); ++ means that there are two additional polarization functions applied to entire set of functions. Note: diffuse functions are shallow Gaussian functions that are designed to catch the wavefunction behavior far from the nucleus. aug-cc-pvqz - correlation consistent basis sets, developed by Tom Dunning, designed to work with correlated methods and provide faster convergence for these methods: cc = correlation consistent, pvqz = polarized valence quadruple zeta (each valence orbital is represented by 4 [quadruple] functions), aug signifies added diffuse functions, 1 set of diffuse functions has been added to everything. e) Configuration Interaction (CI) This is another method to solve the wavefunction by adding electron correlation beyond HF. More than one determinant is used to write the wavefunction (Slater determinants) to show electron correlation even when exc~ted states are considered by promoting an electron to unoccupied states. CI works by writing the wavefunction as a sum of the determinants that are weighted by CI coefficients. This is used to solve the Schrodinger equation. In this method, the energy is minimized by optimizing the CI coefficients. f) MCSCF vs. CASSCF vs. MRCI All multi-reference methods are based on the principle that the reference is no longer a single determinant but more than 1 determinant. These methods are used to get a better description when the system can't be represented by just 1 picture, and multiple scenarios are needed to fully represent the interaction.

8 Multi-Configuration Configuration State Functions (MCCSF) - uses a linear combination of configuration state functions to approximate the electronic wavefunction; a combination of HF and CI; using HF with a multi-determinant reference Complete Active Space Configuration State Functions (CASSCF) - uses a linear combination of configuration state functions based on designating electrons to orbitals; complete active space has all valence orbitals as the active space with some orbitals as multi-reference and some single reference Multi-Reference Configuration Interaction (MRCI) - configuration interaction represented by Slater determinants to describe the electronic ground state and excited states (again, multiple references) g) Empirical Valence Bond (EVB) EVB uses an equation whose elements reflect the energies and interactions of resonance forms which contribute to the properties of reactants, intermediates, or products. It represents the reacting system as a superposition of ionic and covalent resonance forms, and the major effect of changing the reaction environment is to alter the electrostatic interaction of the ionic resonance forms with the environment. It is based on the assumption that the most important environmental effect is the change in relative energies of the ionic and covalent states. h) CCSD(T) vs. CCSDTQ Coupled cluster (CC) is similar to CI in that an exponential operator is applied to the reference function to generate a wavefunction that is a sum of other determinants; CCSD(T) - very accurate coupled cluster method for numerically solving excitations based on perturbation theory for small molecules in excited electronic states and ground states; CC with determinants of all singles, doubles, and some of the triples - just the ones that are perturbative connected triples (the ones that are related to the doubles) CCSDTQ - same as above except it's adding quadruple excitations and including all the triples i) Moller-Plesset PT vs. MBPT Based on the principle that you have a perturbation that you know the answer to and a slightly different function; take a function that you know

9 how to solve and add a perturbation and solve for each separately then add together (i.e., perturbation theory) Moller-Plesset Perturbation Theory (MPPT) - HF ab initio method which adds electron correlation effects by the Rayleigh-Schrodinger perturbation theory method; example =MP2 is an MPPT method that includes the second order energy correction Many Body Perturbation Theory (MBPT) - MPPT to higher order perturbations considering many body problems; MBPT is a method that uses the HF wavefunction with perturbation theory and to second order is the same as MP2. j) DFf withb3lyp Density function theory (DFf) uses functions to represent electron density distribution within a molecule with the Born-Oppenheimer (BO) approximation to solve the Schrodinger equation. The fundamental magic is that isn't necessarily a wavefunction method but functions of electron density; the functional maps the electron density of the approximation to the electron density of the molecule; the kicker of DFf is that the functional is unknown and in fact may not exist, and all the DFf functionals are all approximations to the real functional which may not exist. B3LYP (LyP is exchange, developed by Lee, Yang, Par); it's a hybrid of HF methods (the exact energy) and electron exchange energies. k) QMC (Quantum M<:mte Carlo) for electronic structure Mathematically, the method represents correspondence between the timedependent Schrodinger equation and the wave equation. The Schrodinger equation is simulated with the wave equation, and the ghost walkers represent the movement of the electrons in the MC simulation. The ghost walkers move throughout functional space, and their positions are sampled at designated timesteps to calculate the energy of that configuration to then compare that value to a guess energy (ab initio HF or something better, usually HF). QMC assumes nodal structure. Nodal structure is usually determined from a set of orbitals from correlated electronic structure methods, which defines where the electrons (ghost walkers) can and can't be (electron density). Based on the comparison to the trial function, the step is either accepted or rejected, and subsequent steps are taken until convergence is achieved as trial functions are updated.