(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?
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, 075132 (2012)
Spherically symmetric systems ansatz: 1D solution for the radial part + relative angles minimization Seidl, Gori-Giorgi and Savin, PRA 75, 042511 (2007) not always the lowest solution (it depends on the density) Colombo & Stra, arxiv:1507.08522 Seidl, Di Marino, Gerolin, Nenna, Giesbertz & Gori-Giorgi, arxiv:1702.05022 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:1702.05022 rvsce (r) = N X r r i=2 fi (r) fi (r) 3
Example: 10 electrons in 2D harmonic potential Self-consistent KS densijes and potenjal with the SCE funcjonal 1 2 2 vext (r) =! r 2 Mendl, Malet & Gori-Giorgi, PRB 89, 125106 (2014) Functionals from the strong coupling limit of DFT Hamburg 7 November 2017 SCE Exc [ ] Vee [ ] U [ ]
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, 125106 (2014)
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, 033006 (2015)
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
H atom (and chemistry) is far from SCE limit Talk of Thierry Champion Mirtschink, Umrigar, Morgan III & PG-G, JCP 140 18A532 (2014)
KS SCE behaves well only at extreme low density quasi 1D Highest occupied eigenvalue as a function of N 3D (Hooke s atom) 0.0007 exact" 0.0006 0.0005 ω=10*5" 0.0004 KS"SCE" 0.0003 0.0002 0.0001 0.0000 0.5 0.5" 1.0 1.0" Q" N 1.5 1.5" 2.0 2.0" Mirtschink, Seidl & Gori-Giorgi, PRL 111, 126402 (2013)
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, 052512 (2014) Zhou, Bahmann & Ernzerhof, JCP 143, 124013 (2015) shell model Bahmann, Zhou & Ernzerhof, JCP 145, 124014 (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
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=">aaaejxichvplbtnafhudhsu8mskszygokiusyibsfiwiiiqog1aphftkokpxejym6kc0hrenlh8ng36fdyuckfjxk4xjj2mtckaydhzopfdljz32wcqm4/eawrpx89bt9tva3xv3h2yunx9+jskye9olorfyyxth1gmb7qompho85ht7tkd79sl+pvdoky9yghwskzg1fdwmmmsifpjcm+qrkgxixayj9ktlwwkf8lfotvndfaojnvtxzvhp1vycbrfwdeurhv1lbr9hdjduzvqvmnq1xwqa6defjxcifr9deuiwugksdlbytviglguwfeeoqqed534ivxia4z/mhspr5bmkfxgmz9vp3wfpm1evctfatayaanuemuu43afvjt5eldaegvzdsnoru9wqlmi6z5yhmofq5cxd2ps826123wg0x+xsa1ujdps7savbqh8ttadognwgz54pkm1b8b/3owtek9pnvyuvlbmlmj1ybq2dzwkwvyqsoqve5yrrmkyhrakzabwloieofagdkmq+dqtxcbt1twmsrarzwyhhuw3ger1jcokhtc9hgh0awcn0n09bvtiocemuh9null3sslafrrpflpe+fqnowcvi67r+lnxdk2hboby0ihkhn/aa3f52z4ddocxcm0iacweyv0bgmgmi5m3s5blm5zfxqfdz42xd/lbd2xttbgndeaw8uxtfvjrknnkghcpdhahf1g/qhfqj9lx0vfsz9cspvdckzyplyin9+qtlavlr</latexit> key ingredients: Vuckovic & PG-G, J. Phys. Chem. Leb. 8, 2799 (2017) Z (r) w ([ ]; r) dr
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=">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</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=">aaadcxichvjdb9mwfhutpkb4wdfec0hgraputk1kgbigvgkcdfecgholk+ruclyntwynke2gvvhe+x288st44qfgnhlsywrxsnr8zj332lc3ygtxxvd/tbz32vubn9duebfv3l233t7y/kltxfe2pkli1xfenbm8yupdjwdhmwjeroknopo3s330lsnn0+szwwqslgsw8jhtyiyfn1rfuijjmz8usfjxljrjpozpooc7wtbsfcbjebfxqa8lfmvqlu+97tg/5mfhzcbfbld2ambnyma2dbjc4glnijqemrtxjdal8gaxp7rkwpuwwhfweb7g08qpukma9p9jhugqr1xb+v+m9f48v7pp9pzir669a89kckhxhehafkdj+oho3cvmgzndmyx0mqpf7y7f91cbmycoqqfucyjb39e0pblkiagcad0o/myebvggu8fkd+wazysekbkbw5gqyxryrfamhf3ltggcknssa1fseudbpnylgdlmscxcx9aw5fxaodfxy7dgszybltcquzwlace/3d845yprixywekq4fsukc6iinxzlptue4pkxm2d4rp+qh3x63tl/u09jdtwcw6ahavac7ip34bamaw39db44j50nzi/3oqvdrsrvadwe++bcunu/axl2gh8=</latexit>
Full non-locality: MRF
Full non-locality: MRF area = 1 + 2 (r)
Full non-locality: MRF area = 2 + 3 (r)
Full non-locality: MRF area = 3 + 4 (r)
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)
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
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)
(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?