Density Functional Theory - II part

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1 Density Functional Theory - II part antonino.polimeno@unipd.it

2 Overview From theory to practice Implementation Functionals Local functionals Gradient Others

3 From theory to practice From now on, if not specified otherwise, we are always referring to ground state properties (neglecting 0 from functionals, wave functions etc.) The practical implementation of DFT theory is usually carried out through the Kohn-Sham approach, which makes DFT similar to a single-particle approach, but with many-body effects included (in the exchange-correlation functional, see below). The rationale is the following 1. rewrite the functional E[ρ] collecting specific terms, some of them unknown, but (relatively) small, and some of them (relatively) easy to evaluate 2. assume, based on other information, some specific form for the unknown part of the functional 3. derive self-consistent equations for the Kohn-Sham orbitals (see below) 4. solve numerically using a standard iterative approach (like in the Hartree-Fock case)

4 From theory to practice We start by writing again the functional E[ρ] = T [ρ] + V ee [ρ] + V Ne [ρ] (1) this will have to be minimized, under the condition of ρ being normalized δe δρ = 0 ρ( x)d x = N In practice, we do not know how to write down this equation explicitly, because we CANNOT know the forms of functionals T, V ee. We know only that V Ne [ρ] = u( r)ρ( x)d x where u( r) is the external potential acting on each electron.

5 From theory to practice Consider now a reference system made of non interacting electrons. The ground state wave function Φ KS for such a system will be written as a single Slater determinant, defined with respect to N orthonormal molecular orbitals φ i ( r); the density associated is ρ( x) = N Φ KS ( x, x 2,..., x N )d x 2... d x N = N χ i ( x) 2 i=1 For it we CAN write the kinetic energy term, not as a functional of the density associated to their ground-state wave function, but as a functional of the orbitals T s = 1 2 N χ i ( x) 2 χ i ( x) i=1

6 From theory to practice Another quantity that we know how to write is the classical interaction energy on N electrons associated to a density ρ J = 1 2 ρ( x)ρ( x ) d xd x r r Let us know rewrite Eq. (1) as follows E = T s + T T s + J J + V ee + V Ne = T s + J + V Ne + E xc (2) where the exchange energy is defined as E xc = T T s + V ee J the exchange energy is not known

7 From theory to practice The new form of the energy functional, Eq. (2) does not look much more useful than the original, Eq. (1). But let us look at the conditions for a minimum (actually extremal) (cfr. I part) δe δρ = δt s δρ + u( r) + u H ( r) + u xc ( r) where u( r) = u H ( r) = M a=1 u xc = δe xc δρ Z a r r a ρ( x ) r r d x

8 From theory to practice We can now proceed as follows: write the energy of a set of non interacting electrons E = T s + U s where the U S energy is unspecified the condition for a minimum is now therefore, if one identifies δe δρ = δt s δρ + u s( r) u s ( r) = u( r) + u H ( r) + u xc ( r) one can equate the density of the interacting set of electrons depending from the original external potential u( r) to the density of the non-interacting set of electrons, with external potential u s ( r). W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965).

9 From theory to practice We can now write, finally, the Kohn-Sham equations, taking advantage of standard methods of calculus of variations; notice that here is where the direct expression of T s as a direct functional of the orbitals comes into play notice also that the normalization condition is automatically satisfied by the one-determinant form chosen for the wave function Φ KS In short, one shows that ρ( x) = N i=1 χ i( x) 2 is obtained by finding the Kohn-Sham orbitals [ M a=1 Z a r r a + ] ρ( x ) r r d x + u xc χ i ( x) = ɛ i χ i ( x) The potential energy in the Kohn-Sham equations is a functional of ρ i.e. of the orbitals; therefore the solution approach is iterative until self-consistency is reached.

10 Implementation Kohn-Sham SCF method 1. Build an initial trial wave function choosing an initial set of orbitals Φ KS = 1 χ (0) 1 χ(0) 2... χ (0) N N 2. build the corresponding density ρ (0) ( x) = N i=1 χ (0) i ( x) 2 3. evaluate u s u (0) s = M a=1 Z a ρ (0) r r + ( x ) a r r d x + u xc[ρ (0) ] 4. solve the KS equations [ 1 ] u (0) s χ (1) i ( x) = ɛ (1) i χ (1) i 5. go to (2) to build a new density ρ (1) given the new orbitals χ (1) i 6. repeat till self-consistency, according to some criterion, is reached ( x)

11 Implementation Given a set of Kohn-Sham orbitals χ i, the total energy of the ground state is not simply the sum of the ɛ i E 0 = Φ KS Ĥ Φ KS = N ɛ i 1 ρ( x 1 )ρ( x 2 ) d x 1 d x 2 2 i=1 r 12 ρ( x)u xc d x + E xc Instead, the sum of the ɛ i is N ɛ i = i=1 N i=1 = T s + [ χ i ( x) 1 ] u s ( x) χ i ( x)d x ρ( x 1 )ρ( x 2 ) ρ( x)u( r)d x + d x 1 d x 2 + r 12 ρ( x)u xc d x

12 Implementation The standard approach for solving the differential KS equations is based on the expression of the set of molecular spin-orbitals as linear combinations of basis set spin-orbitals : φ i ( x) = ϕ( r i )s(s i ), exactly as in the Hartree-Fock case. For the connoisseur: χ = φc h KS C = SCɛ R = CC h KS = h + J + v xc ( h ij = 1 ) u φ i d x φ i J ij = S ij = m R rs(ij, rs) r,s=1 φ i φ i d x where the electron-repulsion integral is defined as (ij, rs) = φ 1 i ( x 1 )φ j ( x 1 ) r 1 r 2 φ r ( x 2 )φ s( x 2 )d x 1 d x 2

13 Implementation Hartree-Fock versus Density Functional Theory Similarities Common variational principle Kinetic energy and nuclear attraction component of matrix elements are identical Iterative (SCF) solution Similar codes (relatively easy to modify existing HF codes to perform DFT calculations). Differences DFT is an exact theory, but based on a functional, E xc, which is unknown; HF is an approximate theory exactly HF requires N 4 Coulomb integrals; DFT only N 3. DFT is versatile: better results with better functionals (but see below) DFT with the proper functional is comparable in accuracy to HF+ post HF, but numerically much more convenient.

14 Functionals Local functionals xc functionals E xc is the difference between the classical and quantum mechanical ee repulsion,v ee J plus the difference in kinetic energy between the interacting and the non interacting system, T T s Most functionals do not attempt to compute the kinetic energy correction: either the term is ignored or a hole function, empirically accounting for the self-avoidance of electrons, is included E xc is usually expressed as a functional of ρ, depending on the energy density ɛ xc E xc = ρ( x)ɛ xc [ρ( x)]d x The energy density is usually treated as a sum (separable or not) of individual exchange and correlation contributions, which can be modeled and tailored

15 Functionals Local functionals Local density approximation A Local Density Approximation (LDA) is indicates any DFT where the value of ɛ xc at some position x can be computed exclusively from the value of ρ( x), i.e. the local value of ρ. in practice the only (formerly) widespread functional of this kind has been based on the uniform electron gas model ɛ xc[ρ] = ɛ x [ρ] + ɛ c[ρ] ɛ x [ρ] = 9α ( ) 3 1/3 ρ 1/3 8 π The exchange energy density ɛ x [ρ] is written analitically; the correlation energy density ɛ c[ρ], cannot be obtained analitically, but can be obtained numerically and then expressed via fitting.

16 Density Functional Theory - 2 Functionals Local functionals R.Q. Hood, M.Y. Chou, A.J. Williamson, G. Rajagopal, and R.J. Needs Phys. Rev. B 57, 8972 (1998)

17 Functionals Gradient Density gradient LDA has serious limitations for energies, although it gives good geometries. We can improve functionals by making them depend on the extent to which the density is locally changing, i.e. the gradient of the density. functionals that depend on both the density and the gradient of the density: gradient corrected or generalized gradient approximation (GGA) functionals. GGA functionals are constructed with the correction being a term added to the LDA functional ɛ GGA x(c) [ρ] = ɛlda x(c) (ρ [ρ] F 4/3 ρ e.g. P86; with s = ρ 4/3 ρ /(24π 2 ) 1/3 : F (s) = ( s s s 6) 1/15 ) N.B. [ρ] = [L] 3 [ρ 4/3 ] = [L] 4, [ ρ ] = [L] 1 [L] 3 = [L] 4 ; quindi [ρ 4/3 ρ ] è adimensionale.

18 Functionals Others Meta GGA We can further improve things by introducing the laplacian density (GEA) ɛ GEA xc the kinetic energy density (mgga) = ɛ GEA xc [ρ, ρ, 2 ρ] ɛ mgga xc = ɛ mgga xc [ ρ, ρ, 1 2 ] N χ i 2 χ i i=1

19 Functionals Others Hybrid functionals A real breakthrough has been the introduction of hybrid functionals, where the functional is corrected with exact exchange terms (i.e. Hartree-Fock terms the laplacian density (GEA) Ex HF = χ 1 r ( x 1 )χ s ( x 1 ) r rs 1 r 2 χ s( x 2 )χ r ( x 2 )d x 1 d x 2 general expressions for this kind functionals are, for instance E HYB xc = (1 a)e DFT x + ae HF x + E DFT c The most famous functional is called B3LYP (Becke three parameters) E B3LYP xc = (1 a)e LDA x + ae HF x + be B88 x + ce LYP c + (1 c)e LDA C

20 Functionals Others LDA (older and less accurate) good for simple metals, but prone to self-interaction error (interaction of the electron with itself) overbinding too small cell parameters (solids) wrong vibrations (molecules) and elastic properties (solids) too small energy gap (HOMO-LUMO) GGA good for structure and elastic properties, but still prone to self-interaction energy gap still too small (especially bad for d bands) GEA and mgga good for structure and elastic properties better energy gap, but only moderately accurate energy gaps slightly increased computational effort HYBRID still good for structure and elastic properties best energy gap (too large sometimes), but more computationally demanding (two el. integrals ab ba ) problems with plane wave basis set

21 Functionals Others M.G. Medvedev et al., Science 355, (2017)

Computational Methods. Chem 561

Computational Methods. Chem 561 Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational