How to study Quantum Chemistry

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Published Date:09-07-2017
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AN INTRODUCTION TO QUANTUM CHEMISTRY Mark S. Gordon Iowa State University 1QUANTUM CHEMISTRY • In principle, solve Schrödinger Equation • Not possible for many-electron atoms or molecules due to many-body problem • Requires two levels of approximation 3FIRST APPROXIMATION • Born-Oppenheimer Approximation – Assumes we can study behavior of electrons in a field of frozen nuclei – Correct H: • H = T + V + T + V + V exact el el-el nuc nuc-nuc el-nuc – V = electron-nucleus cross term: not el-nuc separable, so fix nuclear positions – H = T + V + V = H approx el el-el el-nuc el 4FIRST APPROXIMATION • Born-Oppenheimer Approximation – Assumes we can study behavior of electrons in a field of frozen nuclei – Usually OK in ground electronic state: Assumes electronic and nuclear motions are independent: not really true. – More problematic in excited states, where different surfaces may cross: gives rise to non-adiabatic (vibronic) interactions. 5FIRST APPROXIMATION • Born-Oppenheimer Approximation – Solve electronic Schrödinger equation at successive (frozen) nuclear configurations – For a diatomic molecule (e.g, H ), Born- 2 Oppenhimer Approximation yields potential energy (PE) curve: energy as a function of inter-nuclear distance, R. • Bound curve: minimum at finite R • Repulsive curve:No stable molecular structure 6E E R R e R Repulsive Curve Bound Curve 7FIRST APPROXIMATION • Born-Oppenheimer Approximation – Diatomic Molecules: number of points on PE curve determined by number of values of R – Polyatomic molecules more complicated: • Usually many more coordinates (3N-6) • generate Potential Energy Surface (PES) • Required number of points increases exponentially with number of atoms 8SECOND APPROXIMATION • Electronic Hamiltonian: • H = T + V + V el el-nuc el-el • V not separable: el-el • Requires orbital approximation – Independent particle model: assumes each electron moves in its own orbital: ignores correlation of behavior of an electron with other electrons – Can lead to serious problems 9ORBITAL APPROXIMATION Ψ = ψ (1)ψ (2)…ψ (N) N hp 1 2 • Hartree product (hp) expressed as a product of spinorbitals ψ = ϕσ ι i i • ϕ = space orbital, σ = spin function (α,β) i i • Ignoring repulsions and parametrizing leads to – Hückel, extended Hückel Theory – Tight Binding Approximation – Can be very useful for extended systems 10ORBITAL APPROXIMATION • Recover electron repulsion by using – Orbital wave function (approximation) – Correct Hamiltonian • Leads to Variational Principle: – E = ΨΗΨ ≥ E exact – Using exact Hamiltonian provides an upper bound – Can systematically approach the exact energy 11ORBITAL APPROXIMATION • Pauli Principle requires antisymmetry: – Wavefunction must be antisymmetric to exchange of any two electrons – Accomplished by the antisymmetrizer  • For closed shell species (all electrons paired) antisymmetric wavefunction can be represented by a “Slater determinant” of spinorbitals: Ψ = ÂΨ = ψ (1)ψ (2)…ψ (N) N hp 1 2 12ORBITAL APPROXIMATION • For more complex species (one or more open shells) antisymmetric wavefunction must be expressed as a linear combination of Slater determinants • Optimization of the orbitals (minimization of the energy with respect to all orbitals), based on the Variational Principle) leads to: 13HARTREE-FOCK METHOD • Optimization of orbitals leads to – Fϕ = ε ϕ i i i – F = Fock operator = h + ∑ (2J - K ) for closed i i i i shells – ϕ = optimized orbital i – ε = orbital energy i 14HARTREE-FOCK METHOD • Closed Shells: Restricted Hartree-Fock (RHF) • Open Shells:Two Approaches – Restricted open-shell HF (ROHF) • Wavefunction is proper spin eigenfunction: S(S+1) • Most orbitals are doubly occupied α β α β α β α α • Ψ=ϕ ϕ ϕ ϕ ...ϕ ϕ ϕ ϕ ... 1 1 2 2 n n n+1 n+2 15HARTREE-FOCK METHOD • Second Approach for Open Shells – Unrestricted HF (UHF) • Different orbitals for different spins (α,β) • Wavefunction is not a proper spin eigenfunction • Can often get “spin contamination”: spin expectation value that is significantly different from the correct value • Indicator that wavefunction may not be reliable 16HARTREE-FOCK METHOD • Closed Shells: Restricted Hartree-Fock (RHF) • Open Shells – Restricted open-shell HF (ROHF) – Unrestricted HF (UHF) • HF assumes molecule can be described by a simple Lewis structure • Must be solved iteratively (SCF) 17LCAO APPROXIMATION ψ = ∑ χ C i ι μ μ μ • χ are AO’s: “basis functions” μ • C are expansion coefficients μi • Approximation to Hartree-Fock – FC = Scε – Still solve interatively for C and ε μi i 18LCAO APPROXIMATION ψ = ∑ χ C i ι μ μ μ • Increase AO’s - approach exact HF • Requires complete (infinite) basis χ μ 4 • Computational effort increases N – Double AO’s, effort goes up by factor of 16 – Need to balance accuracy with CPU time, memory 19COMMON BASIS SETS • Minimal basis set – One AO for each orbital occupied in atom • 1s for H, 1s,2s,2p for C, 1s,2s,2p,3s,2p for Si – Often reasonable geometries for simple systems – Poor energy-related quantities, other properties • Double zeta (DZ) basis set – Two AO’s for each occupied orbital in atom – Better geometries, properties, poor energetics 20COMMON BASIS SETS • Double zeta plus polarization (DZP) – Add polarization functions to each atom • 1p for H, 2d for C, 3d for Si, 3f for Ti – Smallest reasonable basis for correlated calcs • Triple zeta plus polarization (TZV) • Diffuse functions for: – Anions – Weakly bound species (H-bonding, VDW) 21

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