Density Functional Theory Formalism and Applications Overview

Density Functional Theory Formalism and Applications Mauro Stener Dipartimento di Scienze Chimiche e Farmaceutiche Universit degli Studi di Trieste Via L. Giorgieri 1, 34127 TRIESTE E-mail: stener@units.it XIII-EM-TCCM-European Master in Theoretical Chemistry and Computational Modelling 13th International Intensive Course, Perugia (I), 3rd 28thSeptember 2018

Summary 1. THEORY - DFT Review of DFT basic theory and equations 2. THEORY - TDDFT with the aim to underline what is general from Linear Response formalism. 3. COMPUTATIONAL - Some numerical aspects of the implementation in the ADF code, most important computational choices, how to assign spectral features. New TDDFT algorithm. 4. APPLICATIONS: 4.a. Plasmons in metal clusters. 4.b. Core electron excitations (NEXAFS).

Electron Density Take a system with only 1 electron, the electron density at point r0is: ( ) ( ) 0 0 r r = 2 This can be interpreted as the action of a density operator: ( ) ( 0 r r = ) r 0 ( ) 0 r Extension to N-electrons: ( ) ( ) ( r ) ( ) = i 1 ( ) ( ) 0 r ( ) 0 r = 2 = = * * r r r r r d 0 0 N ( ) = r r r 0 0 i

Electron Density Now consider in general a N-electron antisymmetric wavefunction, X1=x1,y1,z1, 1: hidden spin coordinate ( ) ( ) ( = 1 i ( ) ( ) ( ,... , 0 1 2 1 r r X X ( ) ( ) ( ,... , 0 2 2 1 r r X X ( ) ( N ) ( ) = * r X X r r X X X X , ,... , ,... , ,... d d 0 1 2 0 1 2 1 2 i ) = + + * X X X X , ,... ) , ,... d d 1 2 1 2 + * X X X X , ,... , ,... ... d d 1 2 1 2 ) ( ) = * X X r r X X X X , ,... , ,... , ,... N d d 1 2 1 0 1 2 1 2

Electron Density Now consider in general a N-electron antisymmetric wavefunction, X1=x1,y1,z1, 1: hidden spin coordinate ( ) = = 1 i ( ) ( ) ( N ( ) ( r i ) = * X X X X X X , ,... , ,... , ,... V V d d 1 2 1 2 1 2 ) ( ) ( ) 1 r V = = * X X r X X X X r r , ,... , ,... , ,... N V d d d 1 2 1 1 2 1 2 1 1

Density Functional Theory (DFT) P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864 Nobel Prize for chemistry 1998 Description of the electronic structure of the Ground State In Ab-initio - the N-electron system is described by the wave- function: ( X X X , , , 2 1 ) ( ) = , X , , x y z N i i i i i Cartesian coord. Spin Coord. In DFT - the N-electron system is described by the Electron Density : ( ) r r ( ) z = x , , y

Functional: a function application: : F f More recent formalization: M. Levy, Proc. Natl. Acad. Sci. (USA) 76 (1979) 6062. N-electrons, non-relativistic, Born Oppenheimer approximation (atomic units): j i j i i 1 2 r r 1 1 N N N general, in practice electron-nucleus interaction ( ) ( ) r = = i H = + + 2 V i EXT i 1 Nucl Z = r K R V EXT i r K i K H = + + T V V ee EXT

Definition of the functional F[] = + min F T V ee : N-electron antisymmetric wave-function ( ) r = + r r [ ] ( ) E F V d EXT

Variational theorem: GS E ) 2 ) 1 E E = GS E GS Demonstration 1) = + = + min F T V T V min min ee ee ( ) r = + + = r r ( ) E T V V d min min ee EXT H = + + = T V V E min min min min ee EXT GS

Demonstration 2) = + + = E T V V GS GS + ee EXT GS ( ) r = + r r ( ) T V V d E GS ee GS EXT GS GS But from 1) also: GS E = E E E E E GS GS GS GS GS The idea is to minimize E[ ] with respect to , this furnishes the GS density and energy. Formal, not practical.

Minimization of E[] min + = = + F T V T V Recall that: min min ee ee This allows to define energy functionals: = = T T V V min min min min ee ee Eventually, the next expression must be minimized: ( ) r = + + r r ( ) E T V V d ee EXT Direct approach: Thomas-Fermi, functionals are approximated (electron gas) and the density is varied in order to reach the minimum of E[ ].

Kohn-Sham approach Fictius system of non-interacting electrons (Vee=0), but with the same density as the real systems of interacting electrons. This term must be chosen in order to have the same density as the real system 1 = 1 N N ( ) r = i = i H = + 2 V KS i KS i 2 1 1 1 1 + = 2 det ... V 2 KS N KS i i i 2 ! N 1 N N ( ) r ( ) ( ) r = i = = 2 * r T 0 i i i i 2 = 1 1 i Spin-orbitals Spatial-orbitals

Kohn-Sham approach Interacting system: T E + = J T E 0 + = J[ ] is the classical electrostatic interaction: E ( ) r V EXT + XC r r r r ( ( ( ) ) V V + d ee + EXT r ) d r r 1 ( ) ( ) = r r J d d r r 2 Finally, EXC[ ] is defined as follows: = + E T T V J 0 XC ee EXC[ ] not known, but small wrt other terms, approximated by models (electron gas)

Kohn-Sham approach J T E 0 + = ( ) r + + r r Minimize: ( ) E V d XC EXT With the constrain: = r d ) ( r N ( ) 0 = r d r ( ) E Functional derivative T0 ( ) r r r E ( ) = r For the interacting system: + + + r XC d V EXT Equal: we impose the same solution ( )! T0 ( ) = r + V For the non-interacting system: KS

Kohn-Sham equations ( ) r 1 r E ( ) r + = 2 V = + + r XC V d V KS i i i KS EXT 2 r E = XC V = h XC KS equations KS i i i N ( ) r ( ) ( ) r = i = * r i i 1 hKSdepends from the density ( ) so a Self Consistent Field (SCF) procedure must be employed to solve the KS equations, like in Hartree Fock

Functional derivative f G : ( ) r d = + G Linear in G G G Differential: = .. . + r G Extension of the concept of exact differential: ( ) x x f ,..., , 2 1 N f : N f = i G = N x df dx i x 1 i ( ) r r d ( ) = dG = 1 , ,... G G 2

Functional derivative:examples r 2 r r 2 = r r 2 2 r r 1 ( ) ( ) = r r J d d r 1 r r r r 1 ( ) ( ) ( ) ( ) = + = r r r r J d d d d r r r 2 r r r 1 ( ) ( ) 1 ( ) ( ) + = r r r r d d d d r r r r r r ( ) ( ) ( ) d = = r r r r ( ) d d d r r r r r d ) ( r J = r r

Functional derivative:examples Local Density Approximation (LDA): ( ) r = LDA XC LDA XC r r ( ) E d ( ( ) r ) d ( ) r ( ) r = LDA XC + = LDA XC LDA XC LDA XC r r r ( ) ( ) E ( ) r = + = LDA XC r r r r ( ) ( ) ( ) d ( ) r LDA XC ( ) r = + LDA XC r r r ( ) ( ) d ( ) r LDA XC E ( ) r = = + LDA XC r ( ) XC V XC

Local Density Approx. (LDA) E XC = ( ) r X XC = Models can be used to approximate Xand C: in LDA the Uniform Electron Gas is employed. ( ) r X ( ) r LDA LDA XC ( ) r r r ( ) d ( ) r + LDA C LDA LDA 1 1 3 3 ( ) r 3 3 1 3 1 = LDA X v 3 In LDA Xis analytic: Slater (1951) suggested the X method, an approximation of HF with local exchange: = LDA 3 4 1 3 3 3 ( ) r 1 = X v 3 2 In LDA the Cterm can be obtained from Statistical Mechanics calculations (Quantum Monte Carlo), such calculations furnish Cwhich can be fitted with very complicated analytical expressions:

Cin LDA

DFT: the Kohn-Sham (KS) method The electron density can be extracted from the KS reference system solving the KS equations: = = ,..., 1 H i n KS i i i ( ) r 1 Z r r d ( ) r = + + 2 N H V KS XC 2 r r R N N occ i * = in i i SCF iterative solution

Kohn-Sham (KS) results: 1. imolecular orbitals and their energies iare obtained. 2. The potential is local (at variance with HF) 3. VXCmust be approximated in practice (LDA, GGA, ) XC V = E XC 4. Total energy E[ ] and one-electron local operator properties (gradients for example) of the systems can be calculated from density

EXC functional choice Functional accuracy can be improved: 1. LDA (good geometries, bad energies) 2. GGA: also is considered (improved energies) 3. Hybrids: non-local Fock exchange (best compromise B3LYP) 4. Asymptotic corrections (LB94, )(R. van Leeuwen and E. J. Baerends, PRA 49 (1994) 2421) occupied 1 ( ) r ( ) r i 2 = 5. Meta GGA: dependence KE density: i 2 Common VXCchoices do not obey to correct asymptotic 1/r behavior (long-range exchange), this feature is important to obtain accurate excitation energies and intensities in TDDFT

Kieron Burke, Perspective on density functional theory JCP 136 (2012) 150901

DFT: numerical considerations Molecules: LCAO formulation. The KS eigenfunctions are the molecular orbitals: linear combination of basis functions = k 1 The problem to obtain KS eigenvalues and eigenfunctions, in LCAO is reduced to a Generalized Diagonalization: C HKS = Input: H H = Basis ( ) r ( ) r = C i k ki SCE Output: C E = KS i KS j ij ij = ij S ij i ij i j

DFT: numerical considerations Two different philosophies to build the HKS matrix: 1) Numerical integration (ADF code), STO basis set 2) Analytical integration (Gaussian ) GTO basis set ADF: computational scheme - Density fitting ( ) ( ) ( ) = = = = i j k i 1 1 1 1 am are obtained imposing: With the constrain: d = fit N N basis basis ( ) ( ) r ( ) r ( ) r ( ) ~ m = = * * * = r r r r f r C C a i i ji ki j k m m ( ( ) r ( ) r ) r d ~ 2 min N ( ) r ~ r

ADF: computational scheme - Density fitting allows computational economy in the calculation of the coulomb matrix elements: ( ) r r r r d = VH NP = ( ) ( ) ( ) H k r V r = V r w i H j i k j k k 1 k ( ) r ( ) r ( ) r ~ fit fit f r d r r d r r d ( ) k r ( ) k r = = = m r V a a F H m m m r r r m m k k k

Comments: 1. DFT: the variational theorem is formal, not directly used! 2. KS: a fictious non-interacting system is introduced to calculate the GS density, this allows to work out the Kohn- Sham equations, which are implemented in computer DFT codes, useful in practice! 3. EXC: unknown term, must be approximated by suitable models. 4. Implementation: standard LCAO formalism, molecular orbitals and basis set.