Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation.

Transcription

1 Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation. Bastien Mussard, Julien Toulouse Laboratoire de Chimie Théorique Institut du Calcul et de la Simulation Sorbonne Universités, Université Pierre et Marie Curie

2 Motivation We want to give ourselves a way to diagnos systematic failures of methods.

3 Fractional Charge and Fractional Spin : Motivation [3 / 25] We want a clear formulation to diagnos the failures of methods ( and E xc [n]) - Self-Interaction Error (interaction of the electron with itself, (de)localization) Total energies Charge-Transfert complexes Barrier heights energies Polarizabilities... DFAs exact - Static Correlation Error (Strongly correlated systems ; (Near-)degeneracy) Dissociation of molecules... DFAs exact Those are only qualitative explanation.

4 Fractional Charge and Fractional Spin : Occurences [4 / 25] These systems exist as the dissociation limit of real physical systems. - H + 2 at dissociation (two protons, one electron up) : Ensemble of the two pure states : 1/2 1/2 1 2 (Ψ 1 + Ψ 2 ), E The blue subsystem is an hydrogen atom H(1/2 ) with half an electron and a density in ensemble form. - H 2 at dissociation (two protons, one electron up, one elecron down) : Ensemble of the two pure states : 1 1/2 1/2 1/2 1/2 2 (Ψ 1 + Ψ 2 ), E The blue subsystem is an hydrogen atom H(1/2, 1/2 ) with integer charge but half an electron up and half an electron down and a density in ensemble form.

5 Exact conditions Properties that rule the behavior of the exact energy (piecewise linearity and constancy condition).

6 Fractional Charge [6 / 25] The exact energy E is piecewise linear with respect to N and has a derivative discontinuity when passing at integer N. - Consider a system with a fractional number of electrons N = M +. It s energy is : ( )] E N = min Tr [ˆΓ ˆT + ˆV ext + Ŵ ee ˆΓ N - The minimizing ˆΓ N is linear wrt N between two adjacent integer M and M+1 : ˆΓ N = (1 )ˆΓ M + ()ˆΓ M+1 I exact - The energy is linear wrt to : E N = (1 )E M + ()E M+1 - and so is the density : -A M-1 M M+ M+1 n N (r) = (1 )n M (r) + ()n M+1 (r) [Perdew,Parr,Levy,Balduz 1982]

7 Fractional Spin [7 / 25] Systems with fractional occupation of degenerate spin states have the same energy as the integer-spin pure states. - Consider a system with fractional numbers of spin-up N = M + 1 and spin-down N = M + electrons. This system has an integer total number of electrons M = N + N = M + M + 1. exact - It s corresponding ˆΓ N N is : ˆΓ N N = (1 )ˆΓ M +1,M + ()ˆΓ M,M +1 - This actually yields a constant energy : E N N = (1 )E M + ()E M = E M - This is called the constancy condition. E M +1 M M +.5 M +.5 M M +1 [Cohen,Mori-Sánchez,Yang 28]

8 Behavior of methods The behavior of approximate theories with respect to fractional charges and fractional spins is linked to the Self-Interaction and Strong Correlation errors.

9 Implementation for Self-Consistent Field methods [9 / 25] - In practical calculations, ( the explicit ) e-e interaction ] is replaced by E Hxc [n] : E N = min Tr [ˆΓ ˆT + ˆV ext + E Hxc [nˆγ] ˆΓ N - The minimizing ˆΓ N is not linear wrt : ˆΓ N s = (1 )ˆΓ M, s + ()ˆΓ M+1, s where the density matrices ˆΓ M, and ˆΓ M+1, are that of monodeterminantal wavefunctions Φ M, and Φ M+1, made from a common set of orbitals { φ N } p. all - The density is n N (r 1, r 2 ) = p n p φ N p (r 1 )φ N p (r 2 ) n p = 1 p valence n p p HOMOs p conduction - Considering the scaled orbitals φ N i = n i φ N i that emerge at dissociation : 1/2 1/2 Ψ = 1 2 (φ 1 + φ 2) n(r 1, r 2) = ( 1 (φ1(r1)φ1(r2) + φ2(r1)φ2(r2)) 2 ) ( ) ( ) ( ) 1 = 1 2 φ 1(r 1) 1 2 φ 1(r 2) φ 2(r 1) 2 φ 2(r 2) The density then simply reads : n N (r 1, r 2 ) = val i φ N i (r 1 ) φ N i (r 2 )

10 Calculations, Presentation of the results (Example of H + 2 ) [1 / 25] In the fractional charge calculations, we will be looking at the deviation from the piecewise linearity. In other words, we want to plug n N (r) = (1 )n M (r) + ()n M+1 (r) E frac in the approximate functional and compare E[n N ] to E N = (1 )E M + ()E M+1 Both are true together for the exact density functional. H() H(1/2) H(1) At dissociation, the methods describe the delocalized situation (H(1/2)...H(1/2)) which should be equal to (H()...H(1)) because of the piecewise linearity of the energy. Hence, shifting dissociation curves by the methoddependant value of (H()...H(1)) allows to read the error made by the method wrt the piecewise linearity. H()... H(1) E dissoc H(1/2)... H(1/2)

11 Methods tested [11 / 25] E X = Φ X ˆT + ˆV ne Φ X + E H [Φ X ] + E X xc[φ X ] Hartree-Fock () E xc = E x [Φ ] Density Functional Approximations (DFA : PBE, PBE ( 1 4 exch. 3 4 E DFA xc = E DFA x [n Φ DFA] + E DFA [n Φ DFA] Range-Separated Hybrid () E xc = E sr,dfa x c [n Φ ] + Ec sr,dfa [n Φ ] + Ex lr, [Φ ] Random Phase Approximation (RPA) E +RPA xc E DFA+RPA xc E +RPA xc = Ex [Φ ] + Ec RPA = Ex [Φ DFA ] + Ec RPA = E sr,dfa x [n Φ ] + E sr,dfa [n Φ ] + E lr, [n Φ ] + E lr,rpa c x PBE exch.)) 1 r = v lr ee(r) + v sr ee(r) c 1/r vee(r) lr vee(r) sr

12 Fractional Charge : Example of H + 2 [Mussard,Toulouse 216] [12 / 25] > cc-pvtz, :srpbe, µ =.5 E dissoc =E(H + 2 ) [ E(H + ) + E(H) ] and E frac =2E(H +1 ) 2 [ (1 )E(H + ) + E(H) ] E dissoc (ev) E frac (ev) PBE PBE PBE PBE R (Angstrom) exchange is good for the treatment of the Self-Interaction Error (sr)exchange is better than standard hybrids.

13 Example of H + 2, A more detailed analysis [Mussard,Toulouse 216] [13 / 25] Remember : E PBE xc E xc = E PBE x =E sr,pbe x [n Φ PBE] + E PBE c [n Φ PBE] [n Φ ]+Ec sr,pbe [n Φ ]+Ex lr, [Φ ] The full- and short-range errors made on the correlation ( / ) are comparable and small. The full-range error made on the exchange ( ) is large and dominates the full-range overall error ( ) made by PBE. The short-range error made on the exchange ( ) is considerably smaller (it becomes comparable to the errors made on the correlation). This explains the better performance of srpbe. Note that the full-range PBE benefits from a large compensation of errors in the case of the hydrogen atom, H(1), while it is not the case for srpbe. 2 E x/c/xc (ev) > cc-pvtz, : µ = E x PBE 2 E c PBE 2 E xc PBE E x srpbe 2 E c srpbe 2 E xc srpbe

14 Fractional Spin : Example of H 2 [Mussard,Toulouse 216] [14 / 25] > cc-pvtz, :srpbe, µ =.5 E dissoc = E(H 2) 2E(H) and E frac = 2E(H(, 1 )) 2E(H) E dissoc (ev) 2 E frac (ev) R (Angstrom) -2 PBE PBE PBE PBE exchange is bad for the treatment of the Strong Correlation Error (sr)exchange isn t much better (but see next slide for a more detailed analysis).

15 Example of H 2, A more detailed analysis [Mussard,Toulouse 216] [15 / 25] Remember : E PBE xc E xc = E PBE x =E sr,pbe x [n Φ PBE] + E PBE c [n Φ PBE] [n Φ ]+Ec sr,pbe [n Φ ]+Ex lr, [Φ ] The errors made on the exchange ( / ) and on the correlation ( / ) are of opposite signs. They are both considerably smaller in the case of srpbe. Again, a good compensation of errors explains the overall error made by PBE ( ), while it is not as efficient in the case of srpbe ( ). The additional error made by, seen in the previous slide (at =.5, 6 ev versus 1 ev here) comes from Ex lr, [Φ ]. A better treatment of long-range exchange and correlation would correct that. 2 E x/c/xc (ev) > cc-pvtz, : µ = E x PBE 2 E c PBE 2 E xc PBE E x srpbe 2 E c srpbe 2 E xc srpbe

16 Extension of Random Phase Approximation to fractional occupations We will give key points (RPA derived as linear response equations). (The detailled derivation is an extension of the Fluctuation Dissipation Theorem to ensemble Green s Function).

17 Random Phase Approximation equations [17 / 25] - The RPA is seen as E c [G], a functional of the Green s function ; the basic variable to be ensembled is the one-electron Green s function of the non-interacting reference system : G,N (r 1, r 2 ) = (1 )G,M (r 1, r 2 ) + ()G,M+1 (r 1, r 2 ) - The RPA equations derived as linear response equations are : ( ) ( ) A B X = B A Y ( ) X Ω Y The extension to fractional occupations is easy if one simply considers that : > the orbitals are scaled : φ i = n i φ i and φ a = 1 n a φ a > i now runs through fully occupied+partially occupied orbitals > a now runs through partially occupied+fully unoccupied orbitals Be aware : the dimensions go from N v.n c to : (N f + N p ).(N p + N u ) (with N v valence, N c conduction, N f fully occ., N p partially occ., N u fully unocc.) The matrices read : A ia,jb = ij ab (ɛ a ɛ i ) + n i n j (1 n a )(1 n b ) ib aj B ia,jb = n i n j (1 n a )(1 n b ) ij ab

18 Two different ways to extend RPA to fractional occupations [18 / 25] - What I presented : is an extension of the usual linear response equations, fuelled by an ensemble Green s function describing fractional occupation. The fundamental equations are the same but are evaluated at a different Green s function : 2 E v v [G frac] This is the extension that makes sense in the context of the comparison with dissociation ; this is the extension that allows diagnostics on the Self-Interaction and Static Correlation errors. - What could also be done : a different extension would consist in deriving the linear response of a fractionally occupied system. The underlying equations are different : 2 E frac v v [G frac] This is the extension that would be used to describe system that have inherent degenerescences (heavy atoms,...). [Mussard,Toulouse 216]

19 Fractional Charge : Example of H + 2 [Mussard,Toulouse 216] [19 / 25] > cc-pvtz, :srpbe, µ = E dissoc (ev) -2-3 E frac (ev) MP2 +drpa-i +SOSEX +MP2 +drpa-i +SOSEX -4 +MP2 +drpa-i +SOSEX +MP2 +drpa-i +SOSEX R (Angstrom) drpa-i does not include exchange at all, and performs poorly. +MP2 performs well. +SOSEX seems less efficient (include exchange diagrams up to 2nd order). Post- method seem to suffer from the convex error of the (sr)functional.

20 Fractional Charge : Example of He + 2 [Mussard,Toulouse 216] [2 / 25] > cc-pvtz, :srpbe, µ = E dissoc (ev) E frac (ev) MP2 +drpa-i +SOSEX +MP2 +drpa-i +SOSEX MP2 +drpa-i +SOSEX +MP2 +drpa-i +SOSEX R (Angstrom) drpa-i does not include exchange at all, and performs poorly. +MP2 performs well. +SOSEX seems less efficient (include exchange diagrams up to 2nd order). Post- method seem to suffer from the convex error of the (sr)functional.

21 Be, B, C, N, O, F, N 2 and CO [Mussard,Toulouse 216] [21 / 25] > aug-cc-pcvtz, :srpbe, µ =.5.6 Be.1 B.4 E frac (ev).2 E frac (ev) MP2 +SOSEX +MP2 +SOSEX MP2 +SOSEX +MP2 +SOSEX has a (large) concave error. has mixed behavior (concave/convex). This is due to the relative magnitude of (sr/lr)error. drpa-i is too bad (not shown) ; MP2 is good, and often improved by. SOSEX systematically has much better results when used with.

22 Be, B, C, N, O, F, N 2 and CO [Mussard,Toulouse 216] [22 / 25] > aug-cc-pcvtz, :srpbe, µ =.5 C.3 N.2.2 E frac (ev) MP2 +SOSEX +MP2 +SOSEX E frac (ev) MP2 +SOSEX +MP2 +SOSEX has a (large) concave error. has mixed behavior (concave/convex). This is due to the relative magnitude of (sr/lr)error. drpa-i is too bad (not shown) ; MP2 is good, and often improved by. SOSEX systematically has much better results when used with.

23 Be, B, C, N, O, F, N 2 and CO [Mussard,Toulouse 216] [23 / 25].4 O > aug-cc-pcvtz, :srpbe, µ =.5.6 F E frac (ev).1 E frac (ev) MP2 +SOSEX +MP2 +SOSEX MP2 +SOSEX +MP2 +SOSEX has a (large) concave error. has mixed behavior (concave/convex). This is due to the relative magnitude of (sr/lr)error. drpa-i is too bad (not shown) ; MP2 is good, and often improved by. SOSEX systematically has much better results when used with.

24 Be, B, C, N, O, F, N 2 and CO [Mussard,Toulouse 216] [24 / 25] > cc-pvtz, :srpbe, µ =.5.3 N 2.4 CO E frac (ev) MP2 +SOSEX +MP2 +SOSEX E frac (ev) MP2 +SOSEX +MP2 +SOSEX has a (large) concave error. has mixed behavior (concave/convex). This is due to the relative magnitude of (sr/lr)error. drpa-i is too bad (not shown) ; MP2 is good, and often improved by. SOSEX systematically has much better results when used with.

25 Conclusion [25 / 25] Fractional charge (and fractional spin) did give us a way to diagnos systematic failures of method. Inclusion of exchange is important for the treatment of the Self-Interaction Error. Use of range-separation improves MP2 and SOSEX errors. The methods presented here cannot be free from both the Self-Interaction and the Static Correlation errors. - Other inclusion of exchange (other variants of Random Phase Approximation) seem promising for the treatment of both (SO2). - In this context, very interesting questions on the stability of the RPA equations arise.