VIEWS: 33 PAGES: 14 CATEGORY: Chemistry POSTED ON: 5/2/2012 Public Domain
1 Density Functional Theory An Introduction Quantum mechanical modelling method used in chemistry and physics to g ( y g ) y investigate the electronic structure (principally the ground state) of many body systems, in particular atoms, molecules, and the condensed phases. The methods we’ve bee d scuss g ca be g ouped together u de the e e ods e e been discussing can grouped oge e under e heading “Wavefunction methods.” – They all calculate energies/properties by calculating/improving upon the wavefunction Density Functional Theory (DFT) instead solves for the electron density. 2 Density Functional Theory Generally computational cost i similar t th cost of HF – G ll t ti l t is i il to the t f calculations. – Most DFT methods involve some empirical parameterization. – Generally lacks the systematics that characterize wavefunction methods. – Often the best choice when dealing with very large molecules (proteins, large organic molecules...) (p , g g ) for theory Nobel prize - Walter Kohn “for his development of density functional theory” (1998) Function: A prescription for producing a number from a set of variables. • Wave function and Represented in ( ) • Electron density 3 Functional: A prescription for producing a number from a Function which in turn depends on variables. Examples: • An energy depending on a wave function or an electron density • T and V are functions of density which is a function of coordinate. Hence, T and V are called ‘density functionals’ Represented in [ ] 4 The Hohenberg-Kohn theorems: Theorem -1: Ground state • Ground-state electronic energy is determined completely by the electron density i.e. E = E [ρ ] where ρ is the ground-state density of the system. Electron density, ρ ( r ), determines p () • The external potential, V(r) • Total number of electrons, N. V (r ) and N Determines molecular Hamiltonian H op N n N N N n n ZA Z AZB H op = −∑ 2 i ∑∑ riA ∑∑ rij ∑∑ RAB 1∇2 − + 1+ i A i i< j j B< A A 5 • H op determines the energy of the system via Schrödinger’s equation, H op Ψ = E Ψ Ultimately, ρ ( r ), determines the energy of the system. V (r ) ρ (r ) H op Ev [ρ (r )] N Theorem-2: Density obeys the Variational principle. i.e. It obeys variational principle for obtaining optimal ρ ( r ) which minimizes E. 6 Energy functional: EDFT [ρ (r )] = Tni [ρ (r )] + Vne [ρ (r )] + Vee [ρ (r )] + Exc [ρ (r )] Exc [ρ (r )] = Ex [ρ (r )] + Ec [ρ (r )] = ∫ ρ (r )εxc [ρ (r )]dr Exc [ρ (r )] is determined as an integral of some function of the total electron density εxc [ρ (r )] is the energy density per electron 7 Density Functional Method Classification Local Non-local Hybrids 8 Methods in DFT 1. Local Density Approximation: (LDA functionals) • Exchange and Correlation functional depends only on g p y local value of the density. • Electron density is treated locally as a uniform electron gas or function. equivalently that the density is a slowly varying function • Systems with spin polarization (open systems) LDA is called as Local Spin Density Approximation (LSDA). Example: • SVWN functional: (Slater, Vosko, Wilks Nusair) 9 2. Non-local Methods (GGA functionals) • In molecular systems, electron density is not uniform. • Exchange and Correlation functional depends not only on l l l f the density b t also on th extent t which local value of th d it but l the t t to hi h density is changing locally, i.e. gradient of the density. • Gradient corrected or Generalized Gradient Approximation functionals constructed with the correction added to LDA functional, i.e., ⎡ ∇ρ (r ) ⎤ ε xc [ρ (r )] = ε xc GGA LSDA [ρ (r )] + ∆εxc ⎢ 4 / 3 ⎥ ⎢⎣ ρ (r ) ⎦⎥ Examples: Becke GGA exchange functional: (B) • incorporates a single empirical parameter optimized by fitting to exactly known exchange energies of six noble gases. (He – Rn) B86, P, PBE functionals: • contains no empirically optimized parameter 10 3. Hybrid functionals: B3PW91 model : Becke’s 3 parameter scheme with GGA exchange and correlation functional of PW91 Exc3 PW 91 = (1− a ) ExLSDA + aExHF + b∆ExB 86 + EcLSDA + c∆EcPW 91 B E B3LYP model : LYP computes full correlation energy and not correction to LSDA Exc3 LYP = (1− a ) ExLSDA + aExHF + b∆ExB 86 + (1− c) EcLSDA + c∆EcLYP B 11 Acronyms Name Type SVWN Slater, Vosko, Wilks Nusair LDA BLYP Becke correlation functional Gradient corrected with Lee, Yang, Parr PW91 Perdew and Wang 1991 Gradient corrected P86 Perdew 1986 Gradient corrected B96 Becke 1996 Gradient corrected B3P86 Becke exchange, Perdew Hybrid correlation B3PW91 exchange Becke exchange, Perdew and Hybrid Wang correlation B3LYP Becke 3 term with Lee, Yang, Hybrid Parr 12 Comparison between HF and DFT method HF DFT Wave function for N electron system Electron density depends only on 3 contains 3N coordinates coordinates Accurate for very small system Upto 100 atoms One electron operator One electron functional Optimizes wave function Optimizes electron density Ab-initio and semi-empirical Ab-initio (Gradient corrected) Employ exact Hamiltonian and make Approximations in Hamiltonian approximations in wave function operator Gaussian basis set Numerical basis, fitting functions Systematically extended to attain Lacks systematic way of extending a exact energy series to approach exact energy Transition probability, multistate Difficult to determine resonances determined 13 Absolute Error in Level of theory Basis: 6-31G(d,p) equilibrium bond length MO theoretical methods 0.22 HF 0.14 MP2 0.13 CISD 0.05 CCSD(T) LSDA Functionals SVWN 0.17 F ti l GGA Functionals BLYP 0.14 BPW91 0.14 PBE 0 12 0.12 Hybrid Functionals B1LYP 0.05 B1PW91 0.10 B3LYP 0.04 B3PW91 0.08 14 Comparative performances of molecular mechanics and electronic structure methods Non Semi- Local Task HF MP2 Local empirical DFT DFT Geometry G G G G G Transition State G G G G G Conformation A/G G P A/G G Thermochemistry A/G G P A/G G g G=good p A=acceptable P=Poor p g A/G=Acceptable to good