C372_L7 - Molecular Modeling Density Functional Theory C372...

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Unformatted text preview: Molecular Modeling: Density Functional Theory C372 Introduction to Cheminformatics II Kelsey Forsythe Recall Molecular Models Empirical/Molecular Modeling Semi Empirical Neglect Core Electrons Approximate/parameterize HF Integrals Ab Initio/DFT Neglect Electrons Full Accounting of Electrons Full Quantum Methods Quantum Methods Wavefunction Hartree-Fock MP2, CI Density Function DFT Basis Set Methods N4 dependence on # electrons Does not account for direct ee correlation (communication) Perturbation theory (MollerPlesset methods) Configuration interaction Configuration Interaction (CI) = a0 HF 0 + o ai i =1 unoccupied occupied N HF i Moller Plesset (MP) HartreeFock close to Full Hamiltonian H exact = H HF 0 + V Perturbation DFT Replaces 3N spatial coordinate and Nspin coordinate wave function with functional Reduces # integrations Simplifies computations? What is Density? 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) qr = j n j *a 2 jr 11d =a11 i What is Density? Allyl Cation: r * What is Density? For a chemical system the electron density tells us where the electrons are likely to v v v v v v v exist r ( x ) = N .... ( x , x ...x ) 2 ds dx dx ...dx 1 1 2 N 1 2 3 N Probability of finding any electron within dx1 while other electrons are elsewhere v r ( x1 )=0 Sum over all space gives total # electrons Allyl cation v v r ( x1 )dx1 = N i qi = 2 Function A function maps a set of numbers to another set of numbers Ex. F(X)=X 1 2 3 4 F(X)=Y 1 2 3 4 What's a Functional? A function of a function How does it differ from simple function? Functional A function which maps a set of functions to a set of numbers Ex. F(A(X),B(X),C(X),....)=X A(X) B(X) C(X) D(X) F 2013 1 2 3 4 Functional A function which maps a set of functions to a set of numbers Ex. Energy is a functional of the wave function v v v Y * ( r ) HY( r )dr v v v Y * ( r )Y( r )dr average /expectation value of observable E < E >= Goal? Density How now brown cow? Potential Energy Energy From Density? Classical Approach NuclearElectron Interaction nuclei Vne = k Zk v v v v r ( r )dr r - rk v v r ( r1 )r ( r2 ) v v v v dr1dr2 r1 - r2 Quantal Effects: Exchange? Correlation? ElectronElectron Interaction Ve1 e 2 1 = 2 Energy From Density? ElectronElectron Interaction 12 12 v v 1 r ( ) r ( ) v v 1 = M v v 2 + + 1 2 M - 2 2 1 v v v v Hole function v 1 r ( ) r ( ) v v 1 r ( )( ; ) v v 1 2 1 = M v v + M 1v v 2 1 2 1 2 2 M - 2 M - 1 2 1 2 Exchange & Correlations Energy From Density? Kinetic Energy ThomasFermi's uniform metallic electron gas Te Tueg 3 2 2/3 5/3 v v = (3p ) r ( r )dr 10 Hohenberg-Kohn Existence Theorem E Variational Theorem guess guess E guess i Eexact BUT Don't know how to guess density form Don't want to have to calculate wavefunction Energy Functional Existence (Hohenberg-Kohn (1965)) For a given system of noninteracting electron in the presence of an external field (nuclei) there exists: v v * )Y dr F [r (r )] Y (T + V v v v v v E [r (r )]= r (r )dr + F [r (r )] r Vne d What is this? s.t. Functional Form? For a given system of noninteracting electrons in the presence of an external field (nuclei): v v * F [ r ( r )] Y (T + V )Ydr (Fkinetic + Fpotential ) What is this? Kohn-Sham Self Consistent field Methodology For a given system of noninteracting electrons in the presence of an external field assume they have a density of some real r non-int eracting system or replaced by r real r real c (r ) = i c real cif i (r ) HF-like Functional Form? HFlike? v F [ r ( r )] V )Ydr v Y (T + * (Fkinetic + Fcoulomb + Fexchange + Fcorrelation ) 123 123 1 4 4 2 4 4 43 4 ni=noninteracting F = T + DT ? Poisson ? v F? [ r ( r )] kinetic ni DFT input What is this? DFT Procedure minimized Guess electron density 0 Choose basis Calculate KSintegrals for Tni and Vne using basis Calculate remaining integrals using 0 Solve matrix equations (just as in HFSCF) new Calculate new electron density ( ) Repeat to error tolerance until difference between 0 and r new DFT Challenge( 0 ) Determining the form of the exchange correlation functional LDALocal Density Approximation Becke Exchange Correction (1988) Uniform electron gas + = 1/ 3 + 123 ) ( QM-MC simulations Ceperly and Alder (1980) LeeYangParr(1988) Asymptotic correction Correlation correction DFT-Summa Exact (by construction)! Approximate (by application) v v * F [ r ( r )] Y (T + V )Ydr v F [ r ( r )] Includes Correlation Includes Exchange DFT input unknown NOT variational as a result (E<Eexact) DFT-Summa Does not describe: Dispersion Forces (due to LDA) Dynamics No phases Transition probabilities No resonance and interference Includes Exchange Scaling N 3 vs N 4 (ab initio) ...
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