# 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
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 [ ]
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 [ ].
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.
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
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
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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