Molecular Modeling: Density Functional Theory C372 Introduction to Cheminformatics II Kelsey Forsythe

Recall

Full Quantum Methods

Basis Set Methods N 4 dependence on # electrons N 4 dependence on # electrons Does not account for direct e-e correlation (communication) Does not account for direct e-e correlation (communication) Perturbation theory (Moller-Plesset methods) Perturbation theory (Moller-Plesset methods) Configuration interaction Configuration interaction

Configuration Interaction (CI) unoccupied occupied

Moller Plesset (MP) Hartree-Fock close to Full Hamiltonian Hartree-Fock close to Full Hamiltonian Perturbation

DFT Replaces 3N spatial coordinate and N-spin coordinate wave function with functional Replaces 3N spatial coordinate and N-spin coordinate wave function with functional Reduces # integrations Reduces # integrations Simplifies computations? Simplifies computations?

What is Density? Density provides us information about how something(s) is(are) distributed/spread about a given space Density provides us information about how something(s) is(are) distributed/spread about a given space For a chemical system the electron density tells us where the electrons are likely to exist (e.g. allyl) For a chemical system the electron density tells us where the electrons are likely to exist (e.g. allyl)

What is Density? Allyl Cation:

What is Density?  For a chemical system the electron density tells us where the electrons are likely to exist  Sum over all space gives total # electrons Probability of finding any electron within dx1 while other electrons are elsewhere Allyl cation

Function A function maps a set of numbers to another set of numbers A function maps a set of numbers to another set of numbers Ex. F(X)=X Ex. F(X)=X F(X)=Y

What’s a Functional? A function of a function A function of a function How does it differ from simple function? How does it differ from simple function?

Functional A function which maps a set of functions to a set of numbers A function which maps a set of functions to a set of numbers Ex. F (A(X),B(X),C(X),….)=X Ex. F (A(X),B(X),C(X),….)=X A(X) B(X) C(X) D(X) 2013 F

Functional A function which maps a set of functions to a set of numbers A function which maps a set of functions to a set of numbers Ex. Energy is a functional of the wave function Ex. Energy is a functional of the wave function

Goal? How now brown cow?

Energy From Density? Classical Approach Nuclear-Electron Interaction Nuclear-Electron Interaction Electron-Electron Interaction Electron-Electron Interaction Quantal Effects: Exchange? Correlation?

Energy From Density? Electron-Electron Interaction Electron-Electron Interaction Exchange & Correlations Hole function

Energy From Density? Kinetic Energy Kinetic Energy Thomas-Fermi’s uniform metallic electron gas Thomas-Fermi’s uniform metallic electron gas

Hohenberg-Kohn Existence Theorem Existence Theorem Variational Theorem Variational Theorem BUT BUT Don’t know how to guess density form Don’t know how to guess density form Don’t want to have to calculate wavefunction Don’t want to have to calculate wavefunction

Energy Functional Existence (Hohenberg-Kohn (1965)) For a given system of non-interacting electron in the presence of an external field (nuclei) there exists: For a given system of non-interacting electron in the presence of an external field (nuclei) there exists: What is this?

Functional Form? For a given system of non-interacting electrons in the presence of an external field (nuclei): For a given system of non-interacting electrons in the presence of an external field (nuclei): What is this?

Kohn-Sham Self Consistent field Methodology For a given system of non-interacting electrons in the presence of an external field assume they have a density of some real system or replaced by For a given system of non-interacting electrons in the presence of an external field assume they have a density of some real system or replaced by HF-like

Functional Form? HF-like? HF-like? ni=non-interacting ni=non-interacting What is this?

DFT Procedure Guess electron density Guess electron density Choose basis Choose basis Calculate KS-integrals for T ni and V ne using basis Calculate KS-integrals for T ni and V ne using basis Calculate remaining integrals using Calculate remaining integrals using Solve matrix equations (just as in HF-SCF) Solve matrix equations (just as in HF-SCF) Calculate new electron density ( ) Calculate new electron density ( ) Repeat to error tolerance until difference between Repeat to error tolerance until difference between minimized minimized

DFT Challenge( ) Determining the form of the exchange- correlation functional Determining the form of the exchange- correlation functional LDA-Local Density Approximation LDA-Local Density Approximation Uniform electron gas Uniform electron gas Becke Exchange Correction (1988) Becke Exchange Correction (1988) Asymptotic correction Asymptotic correction Lee-Yang-Parr(1988) Lee-Yang-Parr(1988) Correlation correction Correlation correction QM-MC simulations Ceperly and Alder (1980)

DFT-Summa Exact (by construction)! Exact (by construction)! Includes Correlation Includes Correlation Includes Exchange Includes Exchange Approximate (by application) Approximate (by application) NOT variational as a result (E<E exact ) NOT variational as a result (E<E exact )

DFT-Summa Does not describe: Does not describe: Dispersion Forces (due to LDA) Dispersion Forces (due to LDA) Dynamics Dynamics No phases No phases Transition probabilities Transition probabilities No resonance and interference Includes Exchange No resonance and interference Includes Exchange Scaling Scaling