Text. (Some) wrap up. 1) KS SCE what it does right for models in physics (lattice hamiltonians?) serious limitations (hopeless) for chemistry

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1 (Some) wrap up 1) KS SCE what it does right for models in physics (lattice hamiltonians?) serious limitations (hopeless) for chemistry 2) beyond KS SCE next leading orders: do they contain the right physics? too difficult? spin 3) chemistry/solid state physics how to turn what we have learned into useful approximations?

2 KS SCE in 1D: 2kF - 4kF crossover without magnetic order 1D harmonic conﬁnement: L: eﬀecjve length Previous alempts include self-interacjon correcjons (SIC) and GGA: S. H. Abedinpour, M. Polini, G. Xianlong, and M. P. Tosi, Eur. Phys. J. B 56, 127 (2007) D. Vieira and K. Capelle, J. Chem. Theory Comput. 6, 3319 (2010) D. Vieira, Phys. Rev. B 86, (2012)

3 Spherically symmetric systems ansatz: 1D solution for the radial part + relative angles minimization Seidl, Gori-Giorgi and Savin, PRA 75, (2007) not always the lowest solution (it depends on the density) Colombo & Stra, arxiv: Seidl, Di Marino, Gerolin, Nenna, Giesbertz & Gori-Giorgi, arxiv: however, the 1D-like solution is very close to the true minimum, and the potential computed from it is the functional derivative of the 1D-like SCE functional Seidl, Di Marino, Gerolin, Nenna, Giesbertz & Gori-Giorgi, arxiv: rvsce (r) = N X r r i=2 fi (r) fi (r) 3

4 Example: 10 electrons in 2D harmonic potential Self-consistent KS densijes and potenjal with the SCE funcjonal vext (r) =! r 2 Mendl, Malet & Gori-Giorgi, PRB 89, (2014) Functionals from the strong coupling limit of DFT Hamburg 7 November 2017 SCE Exc [ ] Vee [ ] U [ ]

5 Accuracy of KS SCE for low-density In this case the SGS maps are not optimal Approximate maps can be good enough energy accuracy of ~ 1 mh* (~4%) QMC: D. Guclu and C.J. Umrigar Mendl, Malet & Gori-Giorgi, PRB 89, (2014)

6 Ultracold dipolar quantum gases Bosons and fermions: change the kinejc energy funcjonal Easy to treat tunable interacjons with the SCP funcjonal KS SCE on a lattice interesting for (model many-body) physics? Malet, Mirtschink, Mendl, Bjerlin, Karabulut, Reimann & PG-G, PRL 115, (2015)

7 H2 molecule (SCE from Kantorovich formulation) -KS SCE dissociates correctly -horrible at equilibrium and beyond -higher order terms improve the results SCE d F [ ] = Ts [ ] + Vee [ ] + Tc [ ] + Vee [ ] approximated Chen, Friesecke, Mendl, JCTC 10, 4360 (2014) Vuckovic, Wagner, Mirtschink & Gori-Giorgi, JCTC 11, 3153 (2015) 0

8 H atom (and chemistry) is far from SCE limit Talk of Thierry Champion Mirtschink, Umrigar, Morgan III & PG-G, JCP A532 (2014)

9 KS SCE behaves well only at extreme low density quasi 1D Highest occupied eigenvalue as a function of N 3D (Hooke s atom) exact" ω=10*5" KS"SCE" " " Q" N " " Mirtschink, Seidl & Gori-Giorgi, PRL 111, (2013)

10 SCE and chemistry -SCE itself must be approximated (too expensive for routine calculations) [but local or semi-local approximations of SCE miss the right physics; we need some non-locality] non-local radius (NLR) model Wagner & PG-G, PRA 90, (2014) Zhou, Bahmann & Ernzerhof, JCP 143, (2015) shell model Bahmann, Zhou & Ernzerhof, JCP 145, (2016) - use input from the weakly correlated regime local interpolation along the adiabatic connection? Matthias Ernzerhof s talk how to retain the usual limits of successful xc functionals? Kieron Burke s review - build approximations for the physical coupling strength rescaling SCE multi-radii functional (MRF) Vuckovic & PG-G, J. Phys. Chem. Leb. 8, 2799 (2017) Functionals from the strong coupling limit of DFT Hamburg 7 November 2017

11 Functionals inspired to the SCE form Full non-locality compatible with the exact exchange energy density from SCE theory {z } RiSCE ([ ]; r) distances (or radii) W [ ] = <latexit sha1_base64="n0uu16jxzmytsp7cj599rswpdy4=">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</latexit> key ingredients: Vuckovic & PG-G, J. Phys. Chem. Leb. 8, 2799 (2017) Z (r) w ([ ]; r) dr

12 Coordinate Scaling 3 (r) = lim Exc [ ]! Ex [ ]!1 ( r) <latexit sha1_base64="i6xh2fsrjkwwnwzma1yypv/u2sm=">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</latexit> <latexit sha1_base64="tjbotf2pwcg6+i9g1minrja/vfg=">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</latexit> i [ ] = i <latexit sha1_base64="wltykqc1zqynqy843c6qov5jsby=">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</latexit> lim Exc [ ]! W1 [ ] 1+!0 i Ri ([ ]; r) = Ne 1 (r, i [ ]) ([ ]; r) <latexit sha1_base64="uvhfbqolul4/kjx8ognikizuxzm=">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</latexit> if these are set to a constant, i = <latexit sha1_base64="5oj98/j6p5dzstlrag4zuodahj4=">aaadehichvjdb9mwfhutpkb46uarhgxvino2khmtbkkvjtaql6axutqptiphdvprdhlzdlov5tfw33jjj/dcc26twbzmckvix+fcc699c+ocm6v9/0fhsq9dv3fz7zzz+87de/e76w++qkyqha5jxjn5hgnfouvpudpn6xeukryxp6p45o1sh32lures/awxoq0fnqusyqrrq0xrnw8usouiturejwukqzgs8ywcso1geyk2e/i86nxq6xlfcztvhume/usefizopnwokq9htq5szsogzkruohkwaiodizymeledroupqzoiwguncgsid6lt2o9qqwd0/2merxwpuolx/03z/ly/pk9elewvqb0fz2qioeooosa8rwr44ejdxwadm0m7jxtaox00vlfn9/1vwdyigtadtrxg3e9ompfc0fqtjpuab36uwxjlzqinlymkrxnmtvcmjg1msaaqlfebu0hxmfoyznj8qyyr9ryjxekphyhnpsb6ri5rs/iqbvzo5gvysjqvne1j3sgpodq/ylmgcmokjzovdmbemnnxsozyyqlnsjpmcmhlj7fbckf/qh982u3tv2mmsqyegafaawhya/vgptgeq0a6p63hlms9s37zt+zn9kadanuaz0nwieyd332phq8=</latexit> W1MRF [ ] = W1MRF [ ] <latexit sha1_base64="hmaefi1atp8lfwtforretqvdpou=">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</latexit> with i varying between two constants we can cover the whole scaling <latexit sha1_base64="ezcrz8mm3aaz3d/vnhj727qyr1q=">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</latexit>

13 Full non-locality: MRF

14 Full non-locality: MRF area = (r)

17 Full non-locality: MRF for atoms W1 [ ] = h <latexit sha1_base64="2ndgiii/icfsrr0ihh9xaxvnqag=">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</latexit> ˆ 1 [ ] Vee 1 [ ]i U [ ] (in Hartree) Vuckovic & PG-G, J. Phys. Chem. Leb. 8, 2799 (2017)

18 MRF energy densities are locally accurate Ne H- Vuckovic & PG-G, J. Phys. Chem. Leb. 8, 2799 (2017) 1 w1 (r) = 2 <latexit sha1_base64="a+d2ibo37gtfw199uos83fyvqh8=">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</latexit> Z hxc (r, r0 ) 0 dr r r0

19 Static correlation from MRF H2 + MRF is exact for H2 <latexit sha1_base64="bgz9sxgfsudbezbtceyyelgka7c=">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</latexit> Vuckovic & PG-G, J. Phys. Chem. Leb. 8, 2799 (2017)

20 (Some) wrap up 1) KS SCE what it does right for models in physics (lattice hamiltonians?) serious limitations (hopeless) for chemistry 2) beyond KS SCE next leading orders: do they contain the right physics? too difficult? spin 3) chemistry/solid state physics how to turn what we have learned into useful approximations?