If ee eh ˆenergy e energyitsuncedu 50 but what is e

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0 0 if E E E H ˆ Energy E Energy
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its.unc.edu 50 But what is E[ ]? But what is E[ ]? How do we compute the energy if the density is known? The Coulombic interactions are easy to compute: But what about the kinetic energy T S [ ] and exchange- correlation energy E xc [ ]? , ] [ , ] [ , ] [ 2 1 ext ne nn r r r r r r r r r d d J d V E r Z Z E nuclei B A AB B A E[ ] = T S [ ] + V ne [ ] + J[ ] + V nn [ ] + E xc [ ]
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its.unc.edu 51 Kohn-Sham Scheme Kohn-Sham Scheme , | ) ( | ) ( , ) ( , | | ) ( ) ( , | | ) ( , 2 1 and ) ( ) ( ) ( ˆ where , ˆ 2 3 2 nk nk nk xc xc ee a a a ne xc ee ne nk nk nk r f r E r V r d r r r r V R r Z r V K r V r V r V K H H  The Only Unknown Suppose, we know the exact density. Then, we can formulate a Slater determinant that generates this exact density (= Slater determinant of system of N non-interacting electrons with same density ). We know how to compute the kinetic energy T s exactly from a Slater determinant. Then, the only thing unknown is to calculate E [ ].
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its.unc.edu 52 Correlation Energy Density Functional Correlation Energy Density Functional LDA – f (r) is a function of (r) only GGA – f (r) is a function of (r) and |∇ (r)| Mega-GGA – f (r) is also a function of t s (r), kinetic energy density Hybrid – f (r) is GGA functional with extra contribution from Hartree- Fock exchange energy r r r r d f Q XC , , , 2 Jacob's ladder for the five generation of DFT functionals, according to the vision of John Perdew with indication of some of the most common DFT functionals within each rung.
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its.unc.edu 53 LDA Functionals LDA Functionals Thomas-Fermi formula (Kinetic) – 1 parameter Slater form (exchange) – 1 parameter Wigner correlation – 2 parameters 3 / 2 2 3 / 5 3 10 3 , F F TF C d C T r r 3 / 1 3 / 2 3 / 1 3 / 4 4 3 8 3 , X X S X C d C E r r r r r d b a E W C 3 / 1 1
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its.unc.edu 54 Popular Functional: BLYP/B3LYP Popular Functional: BLYP/B3LYP Two most well-known functionals are the Becke exchange functional E x [ ] with 2 extra parameters & The Lee-Yang-Parr correlation functional E c [ ] with 4 parameters a-d Together, they constitute the BLYP functional: The B3LYP functional is augmented with 20% of Hartree-Fock exchange: r r r r d e d e E E E c x c x xc , , LYP B LYP B BLYP 3 / 4 2 2 2 3 / 4 , 1  LDA X B X E E r d e t t C b d a E c W W F LYP c 3 / 1 2 3 / 5 3 / 2 3 / 1 18 1 9 1 2 1 1 nl km P P b E E a E N l k kl N n m mn c x xc 1 , 1 , LYP B B3LYP
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its.unc.edu 55 Density Functionals Density Functionals LDA local density GGA gradient corrected Meta-GGA kinetic energy density included Hybrid “exact” HF exchange component Hybrid-meta-GGA VWN5 BLYP HCTH BP86 TPSS M06-L B3LYP B97/2 MPW1K MPWB1K M06 Better scaling with system size Allow density fitting for even better scaling Meta-GGA is “bleeding edge” and therefore largely untested (but better in theory…) Hybrid makes bigger difference in cost and accuracy
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