Electronic Structure Methods-Part 1

1. Computational Quantum Chemistryfor the ChE Classroom • Ab Initio • uses Schrödinger's equation, but with approximations • Semi-empiric • uses experimental parameters and extensive simplifications of Schrödinger's equation • Molecular Mechanics • does not use Schrödinger's equation Computational Calculation Methods Computational Chemistry, University of Lampung AndrianSaputra

2. Ab Initio • Ab initio translated from Latin means “from first principles.” This refers to the fact that NO experimental data is used and computations are based on quantum mechanics.

3. Different Levels of Ab Initio Calculations Electron correlation is NOT taken into consideration • Hartree-Fock(HF) The simplest and central of abinitiocalculation Electron correlation is taken into consideration • Møller-PlessetPerturbation Theory (MP) • Density Functional Theory (DFT) • Configuration Interaction (CI) • Multi-Configurational SCF (MCSCF) • Multi-Reference Configuration Interaction (MRCI) • Couple Cluster Methods (CC)

4. Ab initio methods 1. The Hartree-Fock method (HF)

5. Approximations leading to the Computational method • Start with the electronic Schrödinger equation • Defining the wave function (for atom or molecule) • for atom, is combination of one-electron atomic orbital • for molecule, is combination of one-electron molecular orbital • Special for molecule, one-electron molecular orbital is generated by • linear combination of atomic orbital • Constructing antisymmetry of wave function by using Slater • Determinant to ensure Pauli exclusion principle

6. LCAO: Linear Combination of Atomic Orbitals AO or basis function MO expansion coefficients How to obtain the optimal coefficients cmi? • The Molecular Orbitalsfi are written as linear combinations of • pre-defined one-electron basis functions or Atomic Orbitals) The exact form of the wavefunction depends on the coefficients cmi

7. Hartree-Fock Approximation • Using combination of Hartree approximation and Fock approximation for electronic calculation • Hartree approximation Where, Φi– spin orbitals The form of ΨHsuggests the independence of Φi. Probabilitydensitygiven by ΨHis equal to the product of monoelectronic probability densities.

8. Hartree Approximation: the electrons do not interact explicitly with the others, but each electron interacts with the local medium potential given by the other electrons Replaced by represents the local medium potential felt by electron 1 due to the electron 2 described by Φj

9.  PA=1/13 PB=1/4 PAB=1/52=PAPB PA is uncorrelated (independent) with PB. Uncorrelated probabilities • Hartree approximation doesn’t consider electron correlation • Hartree approximation doesn’t satisfy pauli exclusion principle Consequence: gives a non-zero probability for two electrons to be exactly at the same point in space Correlated probabilities

10. Fock Approximation • Consider pauli principle  Slater determinant • Defining Fock operator to replace Hamiltonian operator • H = kinetic energy operator + nuclear-electron potential operator • J = electron-electron potential operator • K = exchange integral operator

11. By Hartree-Fock approximation • Give alternative Schrodinger equation in which the exact Hamiltonian has beed replaced by an approximate Fock operator • Coulomb electron-electron potential has been replaced by an operator which describes the interaction of each electron with the average field due to the other electrons

12. KonsepHartree-Fock adalah orbital Fiadalah operator Fock h(1) adalah operator satuelektron (operator energikinetik + potensialinti-elektorn) h(1) = Jjadalahoperator potensialelektron-elektron Kijadalah operator pertukaran

13. The Hartree-Fock method The main weakness of Hartree Fock is that it neglects electron correlation In HF theory: each electron moves in an average field of all the other electrons. Instantaneous electron-electron repulsions are ignored Electron correlation: correlation between the spatial positions of electrons due to Coulomb repulsion - always attractive! Post-HF methods include electron correlation

14. Self Consistent Field (SCF) Procedures Initial guess for the orbitals (basis set) Solving local medium potential for J and K Solving electronic Schrodinger equation Evaluation J, K, and E by considering convergence criteria convergence Minimum eigen value (properties) and optimizes molecular structure no yes

15. The Hartree-Fock method • The main weakness of HartreeFock is that it neglects electron correlation • In HF theory: each electron moves in an average field of all the other electrons. Instantaneous electron-electron repulsions are ignored • Electron correlation: correlation between the spatial positions of electrons due to Coulomb repulsion - always attractive!

16. Ab initio methods Post-HF methods Hartree-Fock Møller-Plesset Perturbation theory (MP2, MP3, MP4,…) Multiconfigurational SCF (MCSCF) Coupled Cluster (CCSD, CCSDT, …) Configuration Interaction (CI)

17. Semi Empirical • Semi empirical methods use experimental data to parameterize equations • Like the ab initio methods, a Hamiltonian and wave function are used • much of the equation is approximated or eliminated • Less accurate than ab initio methods but also much faster • The equations are parameterized to reproduce specific results, usually the geometry and heat of formation, but these methods can be used to find other data.

18. KonsepDFT (Teorifungsikerapatan) • Teori DFT dimulaidariteoremaHohenberg-Kohn • Teorema 1 : ‘Densitaselektronmenggambarkanenergipotensialinti-elektron (Vn,e)’ • Didukungolehpengamatan E.B. Wilson, kerapatanelektronsecaraunikmenentukanposisidanmuataninti • Teorema2 : • The groundstate energy can be obtained variationally: the density that minimises the total energy is the exact groundstate density.

19. Perhitunganenergi total sebagaifungsidensitaselektron. T = energikinetik, Eext = potensialinti-elektron, Ecoul=potensialelektron-elektron • Permasalahan : energikinetikdanpotensialelektron-elektron ‘sebagaifungsidensitas’ tidakdapatditentukanjikasistem yang digunakanadalahsistemriil yang masihterdapatinteraksiantarelektron

20. Solusi : Teorema Kohn-Sham (sistemfiktifnon-interacting particle) • Persamaan DFT untukenergi total menjadi : T = energikinetik, Eext = potensialinti-elektron, Ecoul=potensialelektron-elektron, Exc = energikorelasi-pertukaran…

21. Potensialkorelasi-pertukaransecarasederhanamerupakan“error” yang terjadiakibatmenggunakansistemfiktifnon-interacting particle • potensialkorelasi-pertukaransangatmenentukanakurasiperhitunganDFT