Statistical Mechanics and Multi- Scale Simulation Methods ChBE Prof. C. Heath Turner Lecture 03 Some materials adapted from Prof. Keith E. Gubbins: Some materials adapted from Prof. David Kofke:

Introduction to Quantum Chemistry Properties of Slater Determinants: Every e - appears in every spin orbital (indistinguishability) Includes QM exchange Accounting for spin introduces a new energy quantity. Assume we want to calculate the interelectronic repulsion (E rep ) for our wavefunction 3  SD :

Introduction to Quantum Chemistry Properties of Slater Determinants: What happens to the K ab integral when describing e - of opposite spin?

Introduction to Quantum Chemistry Hartree-Fock SCF Method: The HF SCF method works similar to the Hartree product SCF method, except that now we include the effects of spin. Typically working with closed-shell systems (all e - are spin-paired, 2 e - per orbital). Wavefunctions described with Slater determinants. Called “restricted Hatree-Fock” or RHF. There is a 1e - Fock operator for each e -, as defined below: J i and K i are operators that return the integrals J ij and K ij The HF equations are used to solve the standard eigenvalue equation:

Introduction to Quantum Chemistry Hartree-Fock SCF Method: The secular equation that corresponds to the HF SCF method is then: The iterative SCF method is then used to solve for the energy, the wave function, and the associated coefficients. DRAWBACKS of HF SCF: 1.Electron correlation is still neglected 2.LCAO approach requires intensive four-index integrals (not shown here) 3.The # of four-index integrals calculated scale as N 4 (poor scaling)

Introduction to Quantum Chemistry Hartree-Fock SCF Method: TWO GENERAL SOLUTIONS: 1.“Semiempirical MO Theory” – simplification of HF calculations by estimating or fitting parameters from experiment. 2.“Ab initio” – use HF as a stepping stone to the exact solution of Schrödinger Eq. Concentrate on more efficient and more elegant methods for performing these type of calculations. The second approach is the more fundamental approach. It has greater potential for describing new systems, without the inherent need for any “fitting”. We will begin by concentrating on the 2 nd approach: ab initio = “from the beginning” Ab initio Molecular Orbital Theory GOAL: Solve the HF equations with an “infinite” basis set, with no additional approximations. Then, any remaining error can be assigned to correlation effects:

Introduction to Quantum Chemistry Ab initio Molecular Orbital Theory Utility: 1. In certain cases E corr cancels out. 2. The HF wave functions may accurately describe other properties. Basis Sets: set of mathematical functions used to construct the wavefunction. Each MO in HF theory is expressed as a linear combination of basis functions. Full HF wavefunction is expressed as a Slater determinant formed from individual occupied MOs. Desire to have an infinite basis set. While this is impossible, we can get close. Useful to choose a basis set functional form that can be evaluated efficiently. Basis set form should be chemically useful. Gaussian Type Orbitals Slater type orbitals (STOs) closely resemble hydrogenic AOs, but they are computationally inefficient – no analytical solution for these functions when evaluating the SCF integrals.

Introduction to Quantum Chemistry Ab initio Molecular Orbital Theory Need an analytical solution to the four-index integral. Solution: modify the STOs to allow analytic integration – change the radial decay from e -r to (e -r ) 2. General form of these GTOs:  = width of the GTO; i, j, k = non-negative integers that dictate the nature of the orbital. Possible cases:

Introduction to Quantum Chemistry Contracted Gaussian Functions GTOs are computationally efficient, but they are not as accurate as STOs. Important Concern: radial portion of the orbital. should have a ‘cusp’ at r = 0 should decay exponentially at long time (e -r versus (e -r ) 2 ) this can be addressed with linear combination of GTOs: ‘contracted’ basis function: basis function is represented as a linear combination of Gaussians. ‘c’ chosen to optimize the shape and ensure normalization. ‘Primitive Gaussians’: individual Gaussian functions. ** Researchers (Hehre, Stewart, and Pople) have previously found optimal contraction coefficients and exponents for mimicking the STOs. ** These are designated STO-MG, where M is the number of primitive Gaussians. Optimal combination of speed/accuracy: STO-3G, defined for most atoms in periodic table.

STOs vs GTOs GTOs are mathematically easy to work with

STOs vs GTOs GTOs are mathematically easy to work with, but the shape of a Gaussian is not that similar to that of an exponential.

STOs vs GTOs Therefore, linear combinations of Gaussians are used to imitate the shape of an exponential. Shown is a representation of the 3-Gaussian model of a STO.

Introduction to Quantum Chemistry Contracted Gaussian Functions GTOs fail to exhibit radial nodal behavior (present in AOs). Solution: contraction coefficients allow the nodal behavior to be reproduced (by manipulating the sign of the coefficients). Modified Basis Sets STO-3G = ‘single-  ’ basis set, ‘minimal’ basis set, 1 basis function / orbital. Thus: H, He: 1s Li to Ne: 1s, 2s, 2p x, 2p y, 2p z Na to Ar: 1s, 2s, 2p x, 2p y, 2p z, 3s, 3p x, 3p y, 3p z Increase flexibility: ‘decontract’ the basis set, this is called a ‘double-  ’ basis set. Instead of 3 primitive Gaussians / basis function, the primitives are split: original: 1 basis function (3 primitives)  1 basis function (2 primitives) ‘contracted’ + 1 basis function (1 primitive) ‘diffuse’ * secular equation is increased. ‘triple-  ’ – treat each primitive as a separate basis function.

Introduction to Quantum Chemistry Modified Basis Sets Core Orbitals – weakly affected by chemical bonding, whereas valence orbitals participate significantly. Result: ‘split-valence’ or ‘valence-multiple-  ’ basis sets, 3-21G, 4-31G, 6-31G. double # of functions for valence e -, but keep single function for inner shells. nomenclature: 3-21G: 3 Gaussian primitives for core orbitals, valence e - described by 3 Gaussians – 2 Gaussians for contracted part and 1 Gaussian for the diffuse part. 6-31G: 6 Gaussian primitives for core orbitals, valence e- described by 4 Gaussians – 3 Gaussians for contracted part and 1 Gaussian for the diffuse part G: 6-31G with a third layer of valence functions composed of a single, uncontracted set

Hartree-Fock is not well suited to calculation of excited state properties Hartree-Fock theory works well for ground state properties because the energies of occupied orbitals are relatively accurately determined. However, the energy of unoccupied orbitals is not well determined and therefore excited state properties and transition energies are not well determined within the HF approach. To account for excited state properties one can include excited electron configurations. Such ab initio approaches that move beyond HF theory are collectively called configuration interaction (CI).