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Introduction   Basis functions     Hartree–Fock   Density functional theory   Time–dependent DFT   End




                                 Looking into the black box
        Computing molecular excited states with time-dependent density
                        functional theory (TDDFT)


                                   Jiahao Chen and Shane Yost

                                   Van Voorhis group, MIT Chemistry


                                              2010-03-12
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Disclaimer



      This is a very superficial survey of two ideas:
         1     Self–consistent field methods for ground state calculations, and
         2     Linear response theory for excited state calculations.
      There will be math and jargon.
      Most of it is optional.
      The math is not presented in the most general case.
      We won’t have time today to discuss calculating observables and
      numerical results.
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Disclaimer



      This is a very superficial survey of two ideas:
         1     Self–consistent field methods for ground state calculations, and
         2     Linear response theory for excited state calculations.
      There will be math and jargon.
      Most of it is optional.
      The math is not presented in the most general case.
      We won’t have time today to discuss calculating observables and
      numerical results.
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Disclaimer



      This is a very superficial survey of two ideas:
         1     Self–consistent field methods for ground state calculations, and
         2     Linear response theory for excited state calculations.
      There will be math and jargon.
      Most of it is optional.
      The math is not presented in the most general case.
      We won’t have time today to discuss calculating observables and
      numerical results.
Introduction   Basis functions       Hartree–Fock       Density functional theory   Time–dependent DFT    End




Statement of the Problem (Born–Oppenheimer)
      Given that we know where the atomic nuclei are, where are the
      electrons?

                    ˆ 
                    H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
                                   r                    r                         r               (1)
      ˆ
      H is the electronic, or Born–Oppenheimer Hamiltonian:

                                 N
                                                    1
                ˆ
                H =          ∑ hi + ∑ |i −j |
                               ˆ
                                       r r
                                                            (1-e + e–e repulsion)                 (2)
                             i=1          ij
                        1
                 h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3)
                 ˆ                    
                        2 1
                          M
                                ZI
           vext (r ) = − ∑ 
                            
                                      (e–nucleus interaction)
                                                                      (4)
                        I =1 RI − 
                                   r
Introduction   Basis functions       Hartree–Fock       Density functional theory   Time–dependent DFT    End




Statement of the Problem (Born–Oppenheimer)
      Given that we know where the atomic nuclei are, where are the
      electrons?

                    ˆ 
                    H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
                                   r                    r                         r               (1)
      ˆ
      H is the electronic, or Born–Oppenheimer Hamiltonian:

                                 N
                                                    1
                ˆ
                H =          ∑ hi + ∑ |i −j |
                               ˆ
                                       r r
                                                            (1-e + e–e repulsion)                 (2)
                             i=1          ij
                        1
                 h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3)
                 ˆ                    
                        2 1
                          M
                                ZI
           vext (r ) = − ∑ 
                            
                                      (e–nucleus interaction)
                                                                      (4)
                        I =1 RI − 
                                   r
Introduction     Basis functions    Hartree–Fock     Density functional theory    Time–dependent DFT   End




Practical solution of the Schrödinger equation


                      ˆ 
                      H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
                                     r                    r                         r
      What approximations do we need to solve this practically?
               Approximate wavefunction
                     Variational ansätze
                     Orbital (one–particle) approximation
                     Basis set approximation
               Approximate Hamiltonian
                     Noninteracting particle / mean field approximation
                     In DFT: approximations to exchange–correlation
               Linear response approximation (for excited states)
Introduction   Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Contents



      1    Basis functions


      2    Hartree–Fock


      3    Density functional theory


      4    Time–dependent DFT
Introduction   Basis functions    Hartree–Fock   Density functional theory   Time–dependent DFT    End




Variational principle and ansätze


      We don’t know the exact solution to the many–body Schrödinger
      equation. But we can make an educated guess, an ansatz (pl.
      ansätze).
      Variational principle
      An approximate wavefunction Ψ (a, b, . . . ) always yields a higher
      energy than the exact solution Ψ0 , i.e.

                                 Ψ |H| Ψ        Ψ0 |H| Ψ0 
                                           ≥ E0 =                                            (5)
                                  Ψ|Ψ            Ψ0 |Ψ0 

      This reduces a differential equation into an optimization problem
      for the numerical parameters a, b, . . .
Introduction   Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Bases in vector spaces


      A basis describes axes, allowing coordinates to be defined.
      Example
      R3 has basis {ex , ey , ez }. Numbers (x, y , z) can be assigned as
      coordinates to a point with position vector v = xex + y ey + zez .

      A less obvious example:
      Example
      Computer monitors use RGB color space, with basis {red, green,
      blue}. For example, a kind of purple is described by purple = 1 red
                                                                  2
      + 1 blue and has coordinates 1 , 0, 1 in color space.
         2                            2   2
Introduction   Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT    End




Bases in function spaces
      Basis functions describe “axes” in function space. If you have a
      collection of mutually orthogonal, real functions φ1 (x) , . . . , φn (x)
      that span the entire space of functions, then
                                       f (x) = ∑ ci φi (x)                                  (7)
                                                  i

      whose coefficients are given by projections onto the basis
                                  N f (x) φi (x) dx
                                ´
                           ci = R´      2
                                                                                            (8)
                                   RN φi (x) dx

      Then the function f has “coordinates” (c1 , c2 , . . . ) in the basis
      spanning the function space.
      Example
      Functions over [−π, π] have Fourier series expansions, whose basis
      functions are 1, sin x, cos 2x, sin 3x, cos 4x, . . .
Introduction      Basis functions      Hartree–Fock     Density functional theory    Time–dependent DFT    End




The independent particle ansatz: from many to one
      Let’s try to solve the many–body Schrödinger equation
                       ˆ 
                       H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N )
                                      r                    r                         r

      As a first guess for Ψn , assume that it is factorizable into a Hartree
      product of one–particle wavefunctions (orbitals) 1

                                    Ψ (r1 , . . . , N ) = φ1 (r1 ) · · · φN (rN )
                                                  r                                               (9)

      However, this wavefunction is not antisymmetric (no Pauli
      exclusion). The simplest fermionic ansatz is the Slater determinant2
                                                                     
                                             φ1 (r1 ) · · · φ1 (rN ) 
                                                               
                                         1                          
                 Ψ (r1 , . . . , N ) = √ 
                               r                .
                                                 .     ..        .
                                                                 .    
                                                                        (10)
                                                .         .     .    
                                                                      
                                         N! 
                                              φN (r1 ) · · · φN (rN ) 
                                                                
          1 D.   R. Hartree, Proc. Cam. Phil. Soc. 24 (1928) 89; 111; 426
          2 J.   C. Slater, Phys. Rev. 34 (1929) 1293
Introduction     Basis functions    Hartree–Fock    Density functional theory   Time–dependent DFT   End




The Hartree–Fock equation

      Using the ansatz that the many–body wf is a Slater determinant,
      the variational principle leads to3 the Hartree–Fock equation

                                           Fi φi (r ) = εi φi (r )
                                           ˆ                                                (11)


                            ˆ
                            Fi     = hi + ∑ Jj − Kj (Fock operator)
                                     ˆ      ˆ ˆ                                               (12)
                                               j


               J is the classical Coulomb potential = Hartree potential vH .
               K is the exchange interaction with no classical analogue.
               (absent in Hartree theory)


          3 A.   Szabo and N. S. Ostlund, Modern Quantum Chemistry, 1982, Ch. 3
Introduction   Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Hartree–Fock is a mean field theory

      Electrons in Hartree–Fock feel an electric
      field of potential Φ = ∑N Jj − Kj .
                        ˆ
                             j=1
                                 ˆ ˆ

      The mean field is a self–consistent field
      The electron–electron interactions are
      approximated by the interaction between an
      electron and this electric field. This field
      depends on the orbitals, yet also helps
      determines them.
      Correlation energy = the error in
      Hartree–Fock energy (wrt exact)
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Roothaan–Hall equation

      If we project the Hartree–Fock equation onto a basis {χ1 , . . . , χP },
                                                  P
                                           φi =   ∑ Ciα χα                                  (13)
                                                  α=1

      we get the Roothaan–Hall equation

                                              FC = ESC                                      (14)

      which is a generalized eigenvalue equation.
               Atomic orbitals = spatially localized basis {χ1 , . . . , χP },
               Molecular orbitals = eigenvectors of the Roothaan–Hall
               equation
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Basis sets

      There are many choices for the atomic basis {χ1 , . . . , χP }:
               Plane waves: great for periodic systems (solids), inefficient for
               molecules (and other systems without translational symmetry)
               Wannier functions
               Slater orbitals: use hydrogenic orbitals, nice theoretical
               properties. In practice the Fock matrix is very difficult to
               calculate. Not widely used today, except in ADF.
               Gaussian orbitals: popular for molecules. Easy to calculate.
                     Pople basis sets: STO-3G, 3–21G, 6–31G*, 6–311+G**,...
                     Dunning–Hays correlation–consistent basis sets: cc-pVDZ,
                     aug-cc-pVQZ, d-aug-cc-pVTZ, ...
               Others, e.g. wavelets, adaptive quadrature grids...
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Post–Hartree–Fock methods




      Many methods to treat correlation use Hartree–Fock as a starting
      point:
               Perturbative corrections: Møller–Plesset (MP2,...),...
               Configuration interaction: CISD, FCI,...
               Coupled–cluster: CCSD, CCSD(T),...
Introduction     Basis functions     Hartree–Fock   Density functional theory   Time–dependent DFT   End




Kohn–Sham equations

      In 1965, Kohn and Sham proposed a noninteracting system for use
      with density functional theory.4 This turns out to look like
      Hartree–Fock theory with a correction for electronic correlation.
      The resulting one–particle equation is written as:
                                            
                                 1 2
                  Hs φi (r ) = − ∇s + vs (r ) φi (r ) = εi φi (r )
                                                                  (15)
                                 2

      where − 1 ∇2 is the noninteracting part of the kinetic energy and vs
               2 s
      is the Kohn–Sham potential:

                                   vs (r ) = vext (r ) + vH (r ) + vxc (r )
                                                                                          (16)

      The exchange–correlation potential vxc has three contributions:
      exchange, correlation, and a correction to the kinetic energy.
          4 W.   Kohn and L. J. Sham, Phys. Rev. 140, 1965, A1133
Introduction   Basis functions    Hartree–Fock   Density functional theory   Time–dependent DFT   End




Kohn–Sham is a density functional theory
      The Kohn–Sham equation is equivalent to finding the density ρ
      that minimizes the energy functional
                   ˆ
           E [ρ] =     vext (r ) ρ (r ) d + Ts [ρ] + VH [ρ] + Exc [ρ]
                                   r                                 (17)
                          R3


                                    1 N
                                         ˆ
                         Ts [ρ] = − ∑        φi (r ) ∇2 φi (r ) d
                                                              r                          (18)
                                    2 i=1 R3
                                    1     ρ (r ) ρ (r  )
                                                   
                                      ˆ
                         VH [ρ] = +                       d d 
                                                           r r                             (19)
                                    2 R6 |r −
                                             r     |


      where the density is constructed from
                                 N                             
                     ρ (r ) =
                                ∑ |φi ( )|2 = ∑
                                        r               Ci µ Ciν χµ (r ) χν (r )
                                                                                         (20)
                                 i=1             µνi
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Density functionals
      The form of the exchange–correlation functional Exc is unknown.
      There are many, many approximations to it:
          “Jacob’s Ladder” of approximations to Exc [ρ]:5
            1 Exc [ρ]                        LDA           X α, LDA,...
            2 Exc [ρ, ∇ρ]                    GGA        BLYP, PBE, PW91...
            3 Exc [ρ, ∇ρ, τ]             meta–GGA         VSXC, TPSS, ...
            4 Exc [ρ, ∇ρ, τ, {φ }o ]     hybrids 6 ,...   B3LYP, PBE0,...
            5 Exc [ρ, ∇ρ, τ, {φ }]      fully nonlocal           -
          LDA, local density approximation; GGA, generalized gradient
          approximation; τ, kinetic energy density; {φ }o , occupied
          orbitals; {φ }, all orbitals.
          Green function expansions for Exc : GW, Görling–Levy PT,...
          Approximations to kinetic energy: Thomas–Fermi, von
          Wiezsäcker,...
          5 M.   Casida, http://bit.ly/casidadft
          6 Hybrid  functionals mix in Hartree–Fock exchange.
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Rules of thumb7 for the accuracy of some ab initio methods



                 Property                    HF           DFT          MP2         CCSD(T)
                 IPs, EAs                  ±0.5           ±0.2         ±0.2        ±0.05 eV
                 Bond lengths               –1%            ±1           ±1         ±0.5 pm
                 Vib. freqs.               +10%           +3%          +3%         ±5 cm−1
                 Barrier heights       +30—50%           –25%         +10%       ±2 kcal/mol
                 Bond energies             –50%            ±3          ±10       ±1 kcal/mol




          7 T.   van Voorhis, MIT 5.74 course notes, 2010.
Introduction   Basis functions     Hartree–Fock   Density functional theory   Time–dependent DFT   End




Polarizabilities as linear response
      Polarizability: a dipole’s response to an electric field
      The isotropic ground state polarizability can be calculated exactly
      from time–dependent perturbation theorya

                                            ∂ ˆ
                                                r       f0→n
                                 α (ω) =
                                 ¯                 =∑ 2                                     (21)
                                           ∂ ε (ω)  n ω0→n − ω
                                                               2


      where ω0→n is excitation frequency (energy) to state n and f0→n
      are oscillator strengths. This is a sum over states (SOS) formula.
         a Kramers, H. A.; Heisenberg, W. (1925) Z. Phys. 31: 681–708.

      Dirac, P. A. M. (1927) Proc. Roy. Soc. Lond. A 114: 243–265; 710–728.

      Linear response is a quest for resonances
      Resonant frequencies of the system’s reaction to small
      perturbations says something about its excitation spectrum.
Introduction     Basis functions   Hartree–Fock   Density functional theory   Time–dependent DFT   End




Resonances in density response
      Main idea
      Using the density response δ ρ to a time–dependent perturbing
      potential δ v , we can back out what the excited state characters are.

      Consider the generalized susceptibility χ (t) as defined by
                                                               
                              ˆ ˆ
                δ ρ (r , t) =
                                 χ  ,  , t − t  δ v   , t  dt  d 
                                     r r                 r               r                  (22)
                                      R R3
      A SOS formula for the Fourier–transformed susceptibility χ can be
                                                               ˜
      derived, and furthermore the conditions for resonance can be
      expressed as an eigenvalue equation that takes the form8
                                                      
                    A B          X            I 0         X
                                      =ω                            (23)
                    B ∗ A∗       Y            0 −I        Y
               X is the hole–particle (occupied–virtual) density change
               Y is the particle–hole (virtual–occupied) density change
          8 M.   Casida, in Recent Advances in Density Functional Methods I, p. 155
End of Part I

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A brief introduction to Hartree-Fock and TDDFT

  • 1. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Looking into the black box Computing molecular excited states with time-dependent density functional theory (TDDFT) Jiahao Chen and Shane Yost Van Voorhis group, MIT Chemistry 2010-03-12
  • 2. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Disclaimer This is a very superficial survey of two ideas: 1 Self–consistent field methods for ground state calculations, and 2 Linear response theory for excited state calculations. There will be math and jargon. Most of it is optional. The math is not presented in the most general case. We won’t have time today to discuss calculating observables and numerical results.
  • 3. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Disclaimer This is a very superficial survey of two ideas: 1 Self–consistent field methods for ground state calculations, and 2 Linear response theory for excited state calculations. There will be math and jargon. Most of it is optional. The math is not presented in the most general case. We won’t have time today to discuss calculating observables and numerical results.
  • 4. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Disclaimer This is a very superficial survey of two ideas: 1 Self–consistent field methods for ground state calculations, and 2 Linear response theory for excited state calculations. There will be math and jargon. Most of it is optional. The math is not presented in the most general case. We won’t have time today to discuss calculating observables and numerical results.
  • 5. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Statement of the Problem (Born–Oppenheimer) Given that we know where the atomic nuclei are, where are the electrons? ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r (1) ˆ H is the electronic, or Born–Oppenheimer Hamiltonian: N 1 ˆ H = ∑ hi + ∑ |i −j | ˆ r r (1-e + e–e repulsion) (2) i=1 ij 1 h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3) ˆ 2 1 M ZI vext (r ) = − ∑ (e–nucleus interaction) (4) I =1 RI − r
  • 6. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Statement of the Problem (Born–Oppenheimer) Given that we know where the atomic nuclei are, where are the electrons? ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r (1) ˆ H is the electronic, or Born–Oppenheimer Hamiltonian: N 1 ˆ H = ∑ hi + ∑ |i −j | ˆ r r (1-e + e–e repulsion) (2) i=1 ij 1 h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3) ˆ 2 1 M ZI vext (r ) = − ∑ (e–nucleus interaction) (4) I =1 RI − r
  • 7. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Practical solution of the Schrödinger equation ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r What approximations do we need to solve this practically? Approximate wavefunction Variational ansätze Orbital (one–particle) approximation Basis set approximation Approximate Hamiltonian Noninteracting particle / mean field approximation In DFT: approximations to exchange–correlation Linear response approximation (for excited states)
  • 8. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Contents 1 Basis functions 2 Hartree–Fock 3 Density functional theory 4 Time–dependent DFT
  • 9. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Variational principle and ansätze We don’t know the exact solution to the many–body Schrödinger equation. But we can make an educated guess, an ansatz (pl. ansätze). Variational principle An approximate wavefunction Ψ (a, b, . . . ) always yields a higher energy than the exact solution Ψ0 , i.e. Ψ |H| Ψ Ψ0 |H| Ψ0 ≥ E0 = (5) Ψ|Ψ Ψ0 |Ψ0 This reduces a differential equation into an optimization problem for the numerical parameters a, b, . . .
  • 10. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Bases in vector spaces A basis describes axes, allowing coordinates to be defined. Example R3 has basis {ex , ey , ez }. Numbers (x, y , z) can be assigned as coordinates to a point with position vector v = xex + y ey + zez . A less obvious example: Example Computer monitors use RGB color space, with basis {red, green, blue}. For example, a kind of purple is described by purple = 1 red 2 + 1 blue and has coordinates 1 , 0, 1 in color space. 2 2 2
  • 11. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Bases in function spaces Basis functions describe “axes” in function space. If you have a collection of mutually orthogonal, real functions φ1 (x) , . . . , φn (x) that span the entire space of functions, then f (x) = ∑ ci φi (x) (7) i whose coefficients are given by projections onto the basis N f (x) φi (x) dx ´ ci = R´ 2 (8) RN φi (x) dx Then the function f has “coordinates” (c1 , c2 , . . . ) in the basis spanning the function space. Example Functions over [−π, π] have Fourier series expansions, whose basis functions are 1, sin x, cos 2x, sin 3x, cos 4x, . . .
  • 12. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End The independent particle ansatz: from many to one Let’s try to solve the many–body Schrödinger equation ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r As a first guess for Ψn , assume that it is factorizable into a Hartree product of one–particle wavefunctions (orbitals) 1 Ψ (r1 , . . . , N ) = φ1 (r1 ) · · · φN (rN ) r (9) However, this wavefunction is not antisymmetric (no Pauli exclusion). The simplest fermionic ansatz is the Slater determinant2 φ1 (r1 ) · · · φ1 (rN ) 1 Ψ (r1 , . . . , N ) = √ r . . .. . . (10) . . . N! φN (r1 ) · · · φN (rN ) 1 D. R. Hartree, Proc. Cam. Phil. Soc. 24 (1928) 89; 111; 426 2 J. C. Slater, Phys. Rev. 34 (1929) 1293
  • 13. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End The Hartree–Fock equation Using the ansatz that the many–body wf is a Slater determinant, the variational principle leads to3 the Hartree–Fock equation Fi φi (r ) = εi φi (r ) ˆ (11) ˆ Fi = hi + ∑ Jj − Kj (Fock operator) ˆ ˆ ˆ (12) j J is the classical Coulomb potential = Hartree potential vH . K is the exchange interaction with no classical analogue. (absent in Hartree theory) 3 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, 1982, Ch. 3
  • 14. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Hartree–Fock is a mean field theory Electrons in Hartree–Fock feel an electric field of potential Φ = ∑N Jj − Kj . ˆ j=1 ˆ ˆ The mean field is a self–consistent field The electron–electron interactions are approximated by the interaction between an electron and this electric field. This field depends on the orbitals, yet also helps determines them. Correlation energy = the error in Hartree–Fock energy (wrt exact)
  • 15. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Roothaan–Hall equation If we project the Hartree–Fock equation onto a basis {χ1 , . . . , χP }, P φi = ∑ Ciα χα (13) α=1 we get the Roothaan–Hall equation FC = ESC (14) which is a generalized eigenvalue equation. Atomic orbitals = spatially localized basis {χ1 , . . . , χP }, Molecular orbitals = eigenvectors of the Roothaan–Hall equation
  • 16. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Basis sets There are many choices for the atomic basis {χ1 , . . . , χP }: Plane waves: great for periodic systems (solids), inefficient for molecules (and other systems without translational symmetry) Wannier functions Slater orbitals: use hydrogenic orbitals, nice theoretical properties. In practice the Fock matrix is very difficult to calculate. Not widely used today, except in ADF. Gaussian orbitals: popular for molecules. Easy to calculate. Pople basis sets: STO-3G, 3–21G, 6–31G*, 6–311+G**,... Dunning–Hays correlation–consistent basis sets: cc-pVDZ, aug-cc-pVQZ, d-aug-cc-pVTZ, ... Others, e.g. wavelets, adaptive quadrature grids...
  • 17. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Post–Hartree–Fock methods Many methods to treat correlation use Hartree–Fock as a starting point: Perturbative corrections: Møller–Plesset (MP2,...),... Configuration interaction: CISD, FCI,... Coupled–cluster: CCSD, CCSD(T),...
  • 18. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Kohn–Sham equations In 1965, Kohn and Sham proposed a noninteracting system for use with density functional theory.4 This turns out to look like Hartree–Fock theory with a correction for electronic correlation. The resulting one–particle equation is written as: 1 2 Hs φi (r ) = − ∇s + vs (r ) φi (r ) = εi φi (r ) (15) 2 where − 1 ∇2 is the noninteracting part of the kinetic energy and vs 2 s is the Kohn–Sham potential: vs (r ) = vext (r ) + vH (r ) + vxc (r ) (16) The exchange–correlation potential vxc has three contributions: exchange, correlation, and a correction to the kinetic energy. 4 W. Kohn and L. J. Sham, Phys. Rev. 140, 1965, A1133
  • 19. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Kohn–Sham is a density functional theory The Kohn–Sham equation is equivalent to finding the density ρ that minimizes the energy functional ˆ E [ρ] = vext (r ) ρ (r ) d + Ts [ρ] + VH [ρ] + Exc [ρ] r (17) R3 1 N ˆ Ts [ρ] = − ∑ φi (r ) ∇2 φi (r ) d r (18) 2 i=1 R3 1 ρ (r ) ρ (r ) ˆ VH [ρ] = + d d r r (19) 2 R6 |r − r | where the density is constructed from N ρ (r ) = ∑ |φi ( )|2 = ∑ r Ci µ Ciν χµ (r ) χν (r ) (20) i=1 µνi
  • 20. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Density functionals The form of the exchange–correlation functional Exc is unknown. There are many, many approximations to it: “Jacob’s Ladder” of approximations to Exc [ρ]:5 1 Exc [ρ] LDA X α, LDA,... 2 Exc [ρ, ∇ρ] GGA BLYP, PBE, PW91... 3 Exc [ρ, ∇ρ, τ] meta–GGA VSXC, TPSS, ... 4 Exc [ρ, ∇ρ, τ, {φ }o ] hybrids 6 ,... B3LYP, PBE0,... 5 Exc [ρ, ∇ρ, τ, {φ }] fully nonlocal - LDA, local density approximation; GGA, generalized gradient approximation; τ, kinetic energy density; {φ }o , occupied orbitals; {φ }, all orbitals. Green function expansions for Exc : GW, Görling–Levy PT,... Approximations to kinetic energy: Thomas–Fermi, von Wiezsäcker,... 5 M. Casida, http://bit.ly/casidadft 6 Hybrid functionals mix in Hartree–Fock exchange.
  • 21. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Rules of thumb7 for the accuracy of some ab initio methods Property HF DFT MP2 CCSD(T) IPs, EAs ±0.5 ±0.2 ±0.2 ±0.05 eV Bond lengths –1% ±1 ±1 ±0.5 pm Vib. freqs. +10% +3% +3% ±5 cm−1 Barrier heights +30—50% –25% +10% ±2 kcal/mol Bond energies –50% ±3 ±10 ±1 kcal/mol 7 T. van Voorhis, MIT 5.74 course notes, 2010.
  • 22. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Polarizabilities as linear response Polarizability: a dipole’s response to an electric field The isotropic ground state polarizability can be calculated exactly from time–dependent perturbation theorya ∂ ˆ r f0→n α (ω) = ¯ =∑ 2 (21) ∂ ε (ω) n ω0→n − ω 2 where ω0→n is excitation frequency (energy) to state n and f0→n are oscillator strengths. This is a sum over states (SOS) formula. a Kramers, H. A.; Heisenberg, W. (1925) Z. Phys. 31: 681–708. Dirac, P. A. M. (1927) Proc. Roy. Soc. Lond. A 114: 243–265; 710–728. Linear response is a quest for resonances Resonant frequencies of the system’s reaction to small perturbations says something about its excitation spectrum.
  • 23. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Resonances in density response Main idea Using the density response δ ρ to a time–dependent perturbing potential δ v , we can back out what the excited state characters are. Consider the generalized susceptibility χ (t) as defined by ˆ ˆ δ ρ (r , t) = χ , , t − t δ v , t dt d r r r r (22) R R3 A SOS formula for the Fourier–transformed susceptibility χ can be ˜ derived, and furthermore the conditions for resonance can be expressed as an eigenvalue equation that takes the form8 A B X I 0 X =ω (23) B ∗ A∗ Y 0 −I Y X is the hole–particle (occupied–virtual) density change Y is the particle–hole (virtual–occupied) density change 8 M. Casida, in Recent Advances in Density Functional Methods I, p. 155