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# INTRO TO RESEARCH CHEM 99

UCSB

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This 37 page Class Notes was uploaded by Ashley Kunze on Thursday October 22, 2015. The Class Notes belongs to CHEM 99 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 33 views. For similar materials see /class/226955/chem-99-university-of-california-santa-barbara in Chemistry and Biochemistry at University of California Santa Barbara.

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Date Created: 10/22/15

71L vquot Wm WM M if a L mum aw i w 91 E K U Gross Freie Universitiit Berlin WI httpWwwphysikfuberlindeaggross ESSENCE OF DENSITYFUNTIONAL THEORY Every observable quantity of a quantum system can be calculated from the density of the system ALONE The density of particles interacting with each other can be calculated as the density of an auxiliary system of interacting particles onetoone correspondence between external potentials Vr and groundstate densities pr p given gt V Vp gt T Wee VlPDIPi Ei IIII Pi IPilpl Ei Ei Pl Variational principle Given a particular system characterized by the external potential v0r Then the solution of the EulerLagrange equation EHK p 0 yields the exact groundstate energy E0 and ground state density p0r of this system EHKp F P I P V00quot d3r Fp is UNIVERSAL In practice Fp needs to be approximated KohnSham Theorem 1965 The ground state density of the interacting system of interest can be calculated as ground state density of noninteracting particles moving in an effective potential vsr 7722 vsprPi1quot W310quot pr Z Icpjr2 N lowest 8 lrr vsrprr var d3r vxcrprr Coulomb potential of nuclei H Hartree potential exchangecorrelation potential 5Exc p universal V r 2 590 Detailed study of molecules atomization energies B G Johnson P M W Gill J A Pople J Chem Phys 97 7847 1992 32 molecules all neutral diatomic with rstrow atoms only H2 Atomization energies kcalmol from EEEZWN EEE YP HF mean deviation from experiment 01 10 858 mean absolute deviation 44 56 858 for comparison MPZ 224 224 Not free from spurious selfinteractions KS potential decays more rapidly than r l Consequences no Rydberg series negative atomic ions not bound ionization potentials if calculated from highest occupied orbital energy too small 39 Dispersion forces cannot be described Wint R gt 6 rather than R band gaps too small LDA Egap a 05 Egapexp Cohesive energies of bulk metals not satisfactory in LDA overestimated in GGA underestimated 39 Wrong ground state for strongly correlated solids eg FeO LaZCuO4 predicted as metals compare groundstate densities pr resulting from different external potentials Vr pr Vr W0 QUESTION Are the groundstate densities coming from different potentials always different I w Singleparticle gs densities otentials havin g s39 Wave p g functions nondegenerate gs HohenbergKohnTheorem 1964 l G vr gt p r is invertible I Consequence Every quantum mechanical observable is completely determined by the ground state density Proof p G 1 E solve SE 3 observables f3 Bip Dip ip Explicit construction of the HK map for noninteracting particles van Leeuwen amp Baerends or Zhao amp Parr I 12 Pi39 2 2 lt 232 vsrgtgt pi e pi N thz N Limer 2m gtltpi vsltrgtpltrgt izleilmimIZ N 2 2 2 vsltrgt 9 20 i 1lteiltpiltrgt2 w 1331 m Iterative procedure por given Start with an initial guess for Vsr e g GGA potential 112 V 2 solve 2m Vsr Pi Ei Pi 1 N 2 2 vsnewltrgt po r i21lteiltpiltrgt2 w lt 32 31 m given fixed solve SE with vsnew and iterate This is an explicit procedure of calculating the functional Vsp Hence given pr we can explicity calculate vstp cpjpr Ellp and eg Ts p cpjpr 1 cpjtpkr Consequence Any explicit orbital functional is an implicit density functional Traditional approximations for EXCp E2 p 2 d3r 632m 90 EgGApJ d3r eXCprVp Systematic approach with MBPT Heap 59 Nquot Pofpkorpcrpjc 1quot 3 3 sz 0 mo ltgtdrdr ozlljak Ec P sum of all higherorder diagrams in terms of the Green s function Gscr rv Zchl r2Pko fr k 0 815 gt The exact EXCp is an orbital functional Expansion of Fp in powers of e2 Flt0gtp Flt0gtp e2Flt1gtp e4Flt2gtrp1 Where F0p TS p kinetic energy of winteracting particles eZFlp dr3d3r39 Exp Hartree exchange energies 00 262 F0 P Ec P correlation energy 12 gt Fp T5 9 2 d3r d3r39 Ex 9 Ecp r lrr39l 1 r 1quot 39 I r 1quot I d3r dsr39 ExHFchKS p Vext EcDFT 1 Et0t EtotHFPjKS EtotHFCPjHF s EtotHF ijS H 004195 003982 1 He 0042107 0042044 details see EKUG MPetersilka Be2 344274 0044267 TGrab0 ACS proceedings 1996 in Hartree units TOWARDS THE EXACT FUNCTIONAL Fp T5 9 I MI d3r39 Ex 9 Ecp I rr39 lSt generation Of DFT Use approximate functionals LDAGGA for TS EX and EC cg Tsp J d3r a pa b L2H 2 gt P HK variational principle gt ThomasFermitype equations 2nd generation Of DFTZ Use exact functional Tsexa tp and LDAGGA for EX and EC Timm JZ Jd3r cpjpr V72 pl916 OCC HK variational principle gt KS equations 3rd generation of DFT Use Tsexacqp and an orbital functional EXCp1 p2 eg39 0 39 39 Exexactp 2 g pkcpr p kcr p10r p10r d3rd3ry 011 jk lr r39l HK variational principle gt KS equations plus 0PM integral equation orderbyorder KSMBPT Resummation of in nitely many terms of the MBPseries eg RPA Functionals from TDDFT SelfInteractionCorrected LDA 0r GGA SIC MetaGGA InteractionStrengthInterpolation ISI Hybrid functionals eg B3LYP ColleSalvetti Apply HK theorem to M interacting particles pgiven 2 vsvsip1 2 V72 vsip1rgtgtltpiltrgteiltpirgt Pi pipgt Ei Eip conseguence Any orbital functional EXCp1 p2 is an implicit density functional provided that the orbitals come from a local ie multiplicative potential optimized effective potential E KS xc potential 0PM 5 vXC r EXC 5 30 Pi PNl vng r ZZfd3r39jd3rquot 5E 5 5V5 cc j SolV 5Vsr 59r XKs391rquotar act with XKS on equation gt 5 39 J XKsrr39vngr39 d3r39 Zjd3r39 513 Mr 0c j 5pjr39 5vs r 0PM integral equation I W knOWn functional of cpj Total absolute groundstate energies for firstrow atoms from various selfconsistent calculations All numbers in hartree 0PM values from T Grabo EKUG Chem Phys Lett 240 141 1995 OPM BLYP PW91 QCI EXACT He 29033 29071 29000 29049 29037 Li 74829 74827 74742 74743 74781 Be 146651 146615 146479 146657 146674 B 246564 246458 246299 246515 246539 C 378490 378430 378265 378421 378450 N 545905 545932 545787 545854 545893 0 750717 750786 750543 750613 75067 F 997302 997581 997316 997268 99734 Ne 1289202 1289730 1289466 1289277 128939 3 00047 00108 00114 00045 Comparison ALDA 03 83 3HF 0 177 0 3 Mean absolute deviation from the exact nonrelativistic values 0 QCI Complete basis set quadratic con gurationinteraction atomic pair natural orbital model J A Montgomery J W Ochterski GA Petersson J Chem Phys 101 5900 1994 EXACT ER Davison SA Hagstrom SJ Chakravorty VM Umar C Froese Fischer Phys Rev A 7071 1991 Approximation employed for Exc Excp1 cpN exact Fock term Eccp1 cpN Colle Salvetti functional Total absolute groundstate energies for secondrow atoms from various selfconsistent calculations All numbers in hartree OPM values from T Grabo EKUG Chem Phys Lett m 141 1995 OPM BLYP PW91 EXPT Na 162256 162293 162265 162257 Mg 200062 200093 200060 200059 Al 242362 242380 242350 242356 Si 289375 289388 289363 289374 P 341272 341278 341261 341272 S 398128 398128 398107 398139 Cl 460164 460165 460147 460196 Ar 527553 527551 527539 527604 A 0013 0026 0023 3 Mean absolute deviation from Lambshift corrected experimental values taken from RM Dreizler and EKUG Density functional theory an approach to the quantum manybody problem Springer Berlin 1990 ma SdE E vpueq 39aaa EXX gives excellent band gaps larger than LDA by 1 eV Small in uence of correlation EXX pseudopotential important for Ge minimum 9 of conduction band in L 1Ili 1 2 3 4 5 Exp band gaps EV 4r 0 from M Stadele M Morokuma J A Majewski P Vogl and A GOrling Phys Rev B 59 10031 1999 7 TABLE IX Comparison of energy gaps between occupied and empty states With experiment All energies in eV LDA GW LDA EXX GW EXX Experiment Si Eg 051 119 143 154 11721 E5 255 323 328 357 335b El 269 338 335 372 346b E 452 526 508 557 538b E2 348 418 412 451 432b Ge EgLC F 006 062 086 094 074a E0 007 057 081 094 090 E 259 317 316 340 316 E1 144 203 214 232 222 E2 375 431 437 456 4450 GaAs E0 049 122 149 165 152d E5 355 424 416 451 451d E1 202 273 280 309 304d E2 398 465 472 499 513d BeSe E0 404 546 525 592 5556 E 501 648 578 675 7298 E1 518 657 600 695 6156 E 681 831 748 864 8476 E2 502 643 586 682 6566 BeTe Egg PU 160 259 247 288 27f 28g E0 328 433 391 458 420b E1 397 497 461 528 469b E2 433 537 499 568 504 h MgSe E0 247 408 372 471 423i MgTe E0 229 366 333 420 367i AFleszar PRB E 245204 2001 Aa Amaue pamnoleo Exact Exchange GGA correlation yields cohesive energy close to experimental values ltgt LDA o EXXcGGA u EXXcLDA gtlt HF O from M Stadele M Morokuma J A Majewski P Vogl and A G rling Phys Rev B 59 10031 1999 3 5 6 7 8 Experimental energy eV Ax A Ac dlscontlnulry at xe potential Ks Egan 39 8969 Axc EXX c A Egap agap asap Ax c Can prove Ax 9 ml VQL vx WWW quot41 gt pH N Iva vxl 2 N Ax large 5 10 eV for C SiGe a cancellation between Ac and Ax 0 it remains open whether A ltlt 63 Fundamental band gap in semiconductors and insulators HartreeF ock gap AHF EHFN1 2 EHFN EHFN 1 EN1HFN E NHFN DFT with exact exchange 0PM gap AX onlyOPM EOPMN1 2 EOPMN EOPMN EN1KSN E NKSN D Xonly discontinuity of VX HF 0PM HF 0PM E 4 E 3 A 4 Ax0nly Table 3 Ionization potentials mm thelu39ghest occupied orbital Energy ofueufral atoms A rlenntrm the mean absolute deviation from the experimental Values taken mm 56 AU values in atomic units Taken from 31 and modified om g KLICS xcLDA BLYP P791 experiment He 0945 0570 0585 0583 0903 Li 0200 0116 0111 0110 0198 BC 0329 0200 0201 0207 0343 13 0328 0151 0143 0149 0305 C 0448 0228 0218 0226 0414 N 0579 0309 0297 0308 0534 O 0559 0272 0206 0207 0500 F 0714 0384 0370 0379 0640 NP 0884 0498 0491 0404 0792 Na 0189 0113 0106 0113 0139 Mg 0273 0175 0103 0174 0281 A1 0222 0111 0102 0112 0220 Si 0306 0170 0160 0171 0300 P 0399 0231 0219 0233 0385 S 0404 0228 0219 0222 0381 C 0506 0305 0295 0301 0477 At 0619 0382 0373 0380 0579 A 0030 0170 0183 0177 ADIABATIC CONNECTION FORMULA N N 2 H TZvri x672 05x51 i1 ik1rirk i k N 2 N 1 H7x1 T Vnuc ri e zi lrrrkl i k Hamiltonian of fully interacting system Choose VXI39 such that for each 7 the ground state density satis es pkr pk1r Hence VX0r VKSI39 Vk1r Vnucr Determine the response function xkrr390 corresponding to H0 Then 1 oo EXC Jd Jg d3rjd3r39lrefzr1xmnrbiu 40500 0 O Second ingredient TDDFT p1 xsvsjlz xsv1Wc1bfXCp1 Pl XV1 XVI XsV1Wc1bfxcXV1 gt X 2 XS Xs Wclb fXC X andfor Oskslz XW Xs Xs7 Wc1b fXk Congdini local RA istatiI EA E St E Elwin rB tam rSdependent deviation of approximate correlation energies from the exact correlation energy per electron of the uni form electron M Lein E K U G J Perdew Phys Rev B Q 13431 2000 truncate after rst iteration xmxs XS Wclb fxcXs plug this approoximation into adiabatic connection formula integrations over A and 00 can be done analytically gt Orbital functional for Ec Resulting Atomic Correlation energies in au atom LDA new fctl exact He 0111 0048 0042 Be 0224 013 0096 Ne 0739 041 0394 Ar 1423 067 072 Resulting VdW coef cients C6 system Calculated C6 experimem HeHe 1639 1458 HeNe 3 424 3 029 NeNe 7284 6383 LiLi 1313 1390 LiNa 1453 1450 NaNa 1614 1550 HHe 2995 282 HNe 5 976 5 71 HLi 6496 664 HNa 754 718 Lein Dobson EKUG J Comp Chem 99 Successful calculation of the full PES 0f HeZ E Engel A Hiick RM Dreizler Phys Rev A g 032502 2000 2 0 t 2 Eh 39 If quot a exact mEV 1 If i anly 0PM 6 39 f quot FOE0PM 1 f LDA quot 39 r39 9 HF 3 1 j MP2 I r 1 J I I l I I l 4 5 E 7 3 399 1 U H Bohr Energy surface of He2 Xonly and correlated OPM data versus LDA HF 22 MP2 23 and exact 21 results J Perdew 1979 EHKzTSp prvrd3r d3rd3r39 EXC p TJp ECp p EDp p iZUmEi3Dpmo Upk EfD07pk with Mr Iltpltrgtlz Mr Iltmltrgtlz free variation of total energy Wrt orbitals leads to Viocrpi0cr2610c Piecr OLZDL with ammo 19 J davwgm d3r39 VLSD pijar r r39 XC Note different singleparticle potential VLaI for each orbital Conseguences cpw not orthonormal Bloch theorem not valid 2 allows localization in supercell SIC results Temmerman 1992 Svane 1991 MnO FeO CoO NiO CuO correctly predicted as antiferromagnetic insulators VO correctly predicted to be a nonmagnetic metal LaZCuO4 correctly predicted as antiferromagnetic semiconductor InteractionStrengthInterpolation M Seidl J Perdew S Kurth PRL amp 5070 2000 PRA Q 0125021 2000 1 Use EXC p I dKlepl 0 and construct approximate Wkp by interpolation between the known limits A 0 W0 WO39 2 1302 7too W WOO39 OO 9 Quality similar to metaGGA Atomization energies in units kcalmol xexact Expt ISI MGGA GGA cMG G A H2 1095 1073 1145 1046 1127 LiH 578 588 584 535 578 Liz 244 225 225 199 207 LiF 1389 1427 1280 1386 1113 Bez 30 57 45 98 52 CH4 4193 4234 4211 4198 4293 NH3 2974 3009 2988 3017 2889 OH 1064 1086 1078 1098 982 H20 2322 2357 2301 2342 2150 HF 1408 1437 1387 1420 1255 B2 710 681 683 771 314 CN 1785 1881 1831 1974 1233 CO 2593 2659 2560 2688 2182 N2 2285 2346 2292 2432 1754 NO 1529 1579 1585 1719 990 02 1205 1236 1314 1437 672 03 1482 1368 1617 1853 16 F2 385 340 432 534 218 mae 43 42 97 338 MGGA JP Perdew S Kurth A Zupan P Blaha PRL 2544 1999 ISI M Seidl JP Perdew S Kurth PRL 5070 2000

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