DFT method by techtrick4u

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                       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

								
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