Orbital functionals derived from variational functionals of the Green function
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1 Orbital functionals derived from variational functionals of the Green function Nils Erik Dahlen Robert van Leeuwen Rijksuniversiteit Groningen Ulf von Barth Lund University
2 Outline 1. Deriving orbital functionals from variational energy functionals of the Green function. Luttinger-Ward functional. Klein functional. 2. Discuss the difference between different kinds of such functionals. Relation between DFT energies and self-consistent Green function results. Stability of the functionals with respect to the input density. 4. Illustrate with results for atoms and molecules. Total energies from the variational functionals and from self-consistent Green function. Binding energies. 5. Conclusion
3 ] Starting point: variational energy functionals of the Green function The Green function solves the Dyson equation ] [iω w(r) v H(r) G(r, r ; iω) = δ(r r ) + d 3 r Σ[G](r, r; iω)g( r, r ; iω) The self-energy Σ[G] is approximated according to which physical mechanisms we believe to be important. The Dyson equation should be solved to self-consistency.
4 It is essential to use conserving self-energy approximations: Σ = δφ δg Second order: GW:
5 These functionals give the same value at the self-consistent G. The LW functional is much less sensitive to the quality of the input G. The variational functionals E[G] are stationary δe[g] δg = 0 when G is a self-consistent solution of the Dyson equation. Examples: 1: The Luttinger-Ward functional: { E LW [G] = Φ U 0 Tr {ΣG} Tr ln G 1 } { 0 + v H + Σ 2: The Klein functional: Where E K [G] = Φ + U 0 Tr { 1 G 1 0 G} Tr ln { G 1} G 1 0 (r, r ; iω ) ] [iω w(r) δ(r r )
6 Density functionals The variational property of the energy functionals suggests that a crude approximation to the Green function can give a good estimate for the self-consistent energy. We can, e.g., restrict the input to noninteracting Green functions: G(r, r ; iω) = i φ i (r)φ i (r ) iω ɛ i A special case is orbitals obtained from a local potential, ] [ v KS(r) φ i (r) = ɛ i φ i (r) This (arbitrary) potential can be considered a functional of the density due to the 1-1 correspondence
7 Consequently we define the energy as a functional of the density: E KS [n] E[G KS ] { } { For a density obtained from an arbitrary v KS = w + v H + v xc we find: 1. The Klein functional E KS K [n] = T 0 [n] + wn + U 0 + Φ For the GWA, this functional is equivalent to DFT-RPA. { For the secondorder approximation, equivalent to second-order perturbation } theory. 2. The Luttinger-Ward functional E KS LW[n] = T 0 [n] + wn + U 0 + Φ Tr {G(Σ v xc ) + ln [1 G(Σ v xc )]} The Kohn-Sham potential is determined { by } the condition { } δe KS [n] δn(r) = 0
8 1. The Klein functional gives the ordinary OEP equation dω 2π dr 1 dr 2 G KS (r, r 1 ; iω)σ(r 1, r 2 ; iω)g KS (r 2, r; iω) = dω 2π dr 1 G KS (r, r 1 ; iω)g KS (r 1, r; iω)v xc (r 1 ) 2. The LW functional gives the equation where [ [ ] ] (Ḡ ( ) δ(σ ) + v H ) 0 = Tr GKS δv KS (r 1 ) Ḡ = G KS + G KS (Σ[G KS ] v xc [G KS ])Ḡ The stationary points of these two functionals do not give the same energy The variational property means that we do not have to solve these OEP equations to find a good estimate of the total energy
9 For a given self-energy approximation: 1. Is the OEP energy similar to the self-consistent Green function energy? 2. Does the DFT energy depend strongly on the form of the functional (Klein, LW,...)? 3. Are the orbital functionals sensitive to the quality of the input DFT orbitals? 4. Can a method which produces poor total energies give accurate total energy differences?
10 Numerical implementation We use Slater basis functions In the evaluation of the variational energy functionals, the Green functions are represented on the imaginary frequency axis. The self-consistent Green function is calculated on the imaginary time-axis.
11 Compare with self-consistent solutions of the Dyson equation with the second-order self-energy: E (2) K [n LDA] E (2) LW [n LDA] E (2) LW [G HF] SC a He Be Ne Mg Mg H LiH a Dahlen and van Leeuwen, J. Chem. Phys. (to be published) Energies from the LW functional are in good agreement the self-consistent results. We have also included the LW results from HF Green functions (in spectacular agreement with the self-consistent results).
12 Total energies of diatomic molecules Molecule E GW K [G LDA] E GW LW [G LDA] E GW LW [G HF] E (2) K [G LDA] E (2) LW [G LDA] E (2) LW [G HF] H Li LiH N The two orbital functionals (LW and Klein) give significantly different results both for the second-order approximation and for the GWA. The LW results from LDA and HF Green functions are similar.
13 General conclusions from calculations on atoms, molecules, and the homogenous electron gas (Hindgren and Almbladh, 1996): E K [G HF ] E LW [G HF ] SC Green function energy This is true also for the homogenous electrong gas in the GW approximation. E LW [G KS ] SC Green function energy E K [G KS ] Differs considerably from SC energy As shown for the atoms and for molecules and also for the homogenous electron gas.
14 Binding energies Molecule E GW K [G LDA] E GW LW [G LDA] E GW LW [G HF] E (2) K [G LDA] E (2) LW [G LDA] E (2) LW [G HF] Expt. H LiH Li N The DFT-RPA functional gives excellent binding energies. (See also F. Furche, PRB (2001), M. Fuchs and X. Gonze, PRB (2001) Self-consistent Green functions from the second-order approximation give results of comparable quality
15 The total energy of the H 2 molecule with Φ = Φ GW The orbital functionals are not sensitive to the choice of input orbitals The difference between E LW [G LDA ] and E LW [G HF ] is larger than for the atoms, indicating that the DFT results are not close to the self-consistent Green function values The DFT-RPA energy converges very slowly to a constant value from above. Not converged for R 20.
16 The total energy of H 2 calculated from the second-order approximation. Good agreement between LW results and the self-consistent energies. All functionals diverge for large R. The binding curve calculated with orbitals from spin-unrestricted LDA and HF.
17 Current and future projects: Compare to self-consistent GW (Adrian Stan) Calculations using variational functionals of G and W [Almbladh, von Barth, van Leeuwen, Int. J. Mod. Phys. B (1996)] The energy functionals of the Green functions are trivially generalized to action functionals on the Keldysh contour. This is used to generate time-dependent OEP from action functionals. exchange-correlation functionals with memory.
18 Conclusions Variational functionals of the Green functions are useful for deriving orbital functionals in DFT. The orbital functional derived from the LW functional give results similar to the self-consistent Green function. Spectacular agreement between the LW energies from HF Green functions and self-consistent energies. The functionals are stable with respect to the input density. The second-order approximation produces better total energies than the GWA for atoms and small molecules. Binding energies are accurately given by DFT-RPA.
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