Theory of Periodic Systems Based on the All-Electron DFT FP-LAPW Method Applications to Surfaces and Clusters

Transcription

1 Theoy of Peiodic Systes Based o the All-Electo DFT FP-LAPW Method Applicatios to Sufaces ad Clustes The pupose of these state-of-the-at calculatios is to addess key issues of iteface ad ao sciece: eactivity stability cheisoptio bod stegths bad stuctue agetic popeties coe-level cheical shifts fo photoelecto spectoscopy ad adhesio of oe aophase to aothe. Most of the calculatios ae caied out by the ethods of Desity Fuctioal Theoy because of the efficiecy of this appoach fo the size of the systes to be ivestigated hee. Fo peiodic systes we have gatheed a extesive expeiece with the all-electo full-potetial lieaized plae wave FP-LAPW-based code Wie [A]. This code uses "foces" to optiize the stuctues. The peiodic calculatios take advatage of popeties calculatio i the k-space though Fouie tasfo techiques eployed i expasios of both the electoic wavefuctios ad the potetial geeated by the uclei. Log-age iteactios ae take cae of by expasios ove the ecipocal lattice vectos G. outlie the scheatic fo Wie applied to adsoptio cheisoptio ad cluste gowth is depicted above. The Koh-Sha equatios of the Desity Fuctioal Theoy [A2] ae solved iteactively to self-cosistecy as illustated i the diaga Fig. A. The Wie code calculates coe-level eegy specta owig to the full-potetial cotibutio by the odified Desclaux's poga LCOE. This is ipotat fo the itepetatio ad pedictio of the coe-level bidig eegy cheical shifts i photoelecto spectoscopy the success of which is docueted i ou ecet epots [A3]. The foces i Wie ae calculated as spelled out i the sectio Geoety Optiizatio below. The SCF calculatio outlied i Fig. A stats with supeiposed electo desity fo all atos i the uit cell clsu which is updated afte each cycle. LAPW0 the coputes the "exteal potetial" as a su of the Coulob ad exchage-coelatio potetial usig the total electo desity as iput ad etus the spheical vsp ad o-spheical vs copoets as iput ito LAPW which solves the Koh-Sha equatios fo eigevalues of obital eegies ad eigefuctios. The latte ae used fo valece electo desity calculatio by LAPW2 clval which is the added togethe with the coe desity calculated by LCOE clco ad ixed with the "old" desities i ME which etus ew chage desity as iput ito LAPWO. Net spi is deteied as the diffeece betwee spi-up ad spi-dow desities ad is efied by the secod vaiatioal teatet of spi-obit iteactios i the valece bad. The coe level calculatios at the fully elativistic level ae paticulaly useful fo the itepetatio of both the cheical shifts ad subtle effects of spi-obit splittigs that ae obseved with high-esolutio PS.

2 Fig. A Flow of pogas i the DFT FP-LAPW code Wie. The SCF cycle teiates upo covegece at a specified toleace fo eegy ad foces. Othe details ae descibed i the cited liteatue. The success of the subsequet optiizatio descibed i soe detail i the sectio Geoety Optiizatio below depeds o a judicious choice of the k-vectos ad good iitial guesses ad is facilitated by paallel pocessig. Both of these ites equie expeiece with the ethod with good iitial guesses the Boyde-Fletche-Goldfab-Shao BFGS ethod is supeio i pefoace to the daped Newto ethod ad i the ogoig effot a boad expeiece i catalytic cheisty fo a good iitial guess of the stuctue of suface iteediates. Soe techical aspects of the LAPW calculatios that hadle apid oscillatios of the wavefuctios ea uclei the log-age electostatic effects ad covegece ae outlied below Geoety Optiizatio i the DFT FP-LAPW WEN Calculatios. The ethods fo geoety optiizatio based o foces o egative gadiets of eegy o atos withi a uit cell of a peiodic stuctue ae outlied heei. pleeted i the Wie code ae a the daped Newto ad b the Boyde-Fletche- Goldfab-Shao BFGS algoiths both of which utilize the divig foces calculated i a sequece outlied i i v below. i The Bo-Oppeheie Appoxiatio B.O.A. The Schödige Equatio to be solved is H Ψ E Ψ

3 whee: H T N T e V ee V en V NN 2 We ca expad ψ as: Ψ ε χ 3 whee is the electoic wave fuctio which satisfies the electoic equatio: H e [ Te Vee VeN] E 4 Substitutig 3 ito ad usig the elatio 4 we have: whee: ε [T N E V NN ] χ ε χ 5 Z Z J V NN 6 2 J J is the ucleus-ucleus iteactio which is i piciple a costat te fo a give uclea cofiguatio. t ca be siply absobed ito E afte solvig the electoic equatio 4. Techically this uclei-uclei iteactio te diveges. Special teatet ivolvig the cacellatio theoe ad Ewald's suatio techique to calculate the eegy of a peiodic aay of poit chages have to be used. 2 T [ 2 N χ χ χ 2 χ ] 7 2 The Bo-Oppeheie Appoxiatio assues 2 0 ad 0 8 The we have the so-called uclea equatios: 2 χ E χ ε χ 9 2 ii The Hella-Feya Theoe HF The HF Theoe: f E is a eigevalue of the Hailtoia H with the eigefuctio ad. is a paaete H E 0 the we have: E H Poof: Multiplyig < to both sides of Eq. 0 ad itegatig ove all space of we get: H 2 So: E

4 H H H E 3 [ ] H E E 4 Sice ae eigefuctios of the Hailtoia Ĥ they fo a coplete basis set fo ifiite ad: 5 The we have: H E Notes: the electoic equatio uclea positios ae take as paaetes we should have: E x H F α 6 whee α x F is the Hella-Feya HF foce alog the diectio of the -copoet of o the α th ato. 2 t ca be show that: H E 7 iii The coplete Basis Set Effect BS The HF theoe is i piciple coect as log as the basis set is coplete. But i ay ab iitio calculatio ae icoplete which eas that the HF foce eed to be coected. That is we will ot have the exact elatio Eq. 5. stead we will have: 2* H H E 8

5 The fist te o the ight is the HF foce ad the secod te is a coectio which was fist show by Pulay [A4] ad is egaded as the wavefuctio foce o Pulay foce. The effect of the Pulay foce is the geeally egaded as the coplete Basis Set effect BS. piciple the Pulay foce F BS 0 oly if the basis set is coplete. Howeve a elatively weak coditio has bee show that if the wavefuctio wee copised of oigiless obitals fo exaple the Plae-Wave PW basis set we still have F BS 0 [A5]. Fo a LCAO appoach BS is ipotat. fact the CYSTAL package does ot calculate foces. 2 Fo a pue Plae-Wave PW appoach BS is vey sall. Ufotuately pue PW appoach has its ow pobles especially the covegece poble. 3 Fo a Plae-Wave Pseudo-Potetial PWPP appoach BS is ot that ipotat. 4 Fo the dual basis set appoach such as the FP-LAPW ethod BS is seious. Fotuately coectios to the BS i the FP-LAPW ethod ca be ade i a kow way. iv The Foce Calculatio i the FP-LAPW Method The explicit foatio of the Pulay coectio i the FP-LAPW appoach ae give by both YSK [A6] ad SW [A7]. The two diffeet appoaches give siila esults ad both have bee ipleeted i the WEN package. 3 F E FHF i ε i ρ Veff d 9 i whee F is the foce actig o the th ucleus F HF is the Hella-Feya foce ad V eff is the effective potetial. v The Calculatio of Optiized Stuctues Usig Foces The Pulay foce is calculated by evaluatig the followig two tes: F coe : a coectio due to the spheically teated coe electos. 2 F Valece : a coectio due to the discotiuous 2 d deivatives of wavefuctios acoss the boudaies of the Muffi-Ti sphees. suay the total foce is the cobiatio of thee tes: HF Coe Valece F F F F 20 whee the Hella-Feya foce F HF is calculated e.g. i the LAPW0" step of the Wie code ad is i geeal easily doe. F Coe is calculated i the "LCOE" step which is also easy. But calculatios of F Valece ae oe difficult. a SCF calculatio ifoatio fo oly two cycles is saved the cuet cycle ad the pevious cycle fo the covegece test ad the ixig schee. Evaluatio of F Valece eeds ifoatio of at least thee cycles. the WEN poga evaluatio of F Valece is doe by addig a -fc" flag i the "u_lapw" poga which is also itepeted as foce covegece citeio. Whe the "-fc" flag is used the SCF calculatio fist goes though the egula eegy citeio covegece. Afte eegy coveges it tus o the F Valece calculatio by chagig the keywod "TOT" to "FO" i the "case.i2" iput file. At least 3 oe cycles have to be calculated to fid the

6 F Valece. The coectio by F Valece is i geeal lage. So the "-fc" flag should always be used fo the "iiizatio" o geoety optiizatio. Afte fidig the foces o uclei we ca ow chage positios of uclei ad pefo a ew SCF calculatio. The oveets of uclei ae guided by the foces usig diffeet schees. the WEN poga two diffeet schees ae used the BFGS [48] ad the daped Newto schee [49] τ τ τ τ τ η δ F 2 τ whee ad F τ ae the coodiates ad the espective foce at the tie step τ. Dapig ad speed of otio ae cotolled by two paaetes the "fictio" η ad the "step size" δ which ae set i the iput file by use. The "iiizatio" poga i WEN is fo fixed lattice costats of the uit cell. That is oly iteal coodiates of atos withi the uit cell ae optiized. O the othe had the lattice paaetes of the uit cell ae optiized by the "optiizatio" poga which is basically a jobcotollig scipt file that iiizes the total eegy with espect to the lattice paaetes with fixed factioal iteal uclea positios. So the best geoety optiizatio pocess should iclude seveal steps: Fist with fixed factioal iteal coodiates of atos optiizatio of the lattice paaetes. Secod with the "optiized" lattice paaetes the iteal coodiates of uclei ae "iiized". These two steps could be epeated util a satisfactoy geoety is eached. Fial Note: The basic task of geoety optiizatio is to fid the iiu total eegy of systes. The citeio of a good optiized geoety should be the "total eegy iiu". Foces basically seve as a guide to effectively fid the iiu geoety. efeeces: [A] P. Blaha K. Schwaz G.K.H. Masde D. Kvasicka ad J. Luitz: Wie2k SBN [A2] The ladak papes by P. Hohebeg ad W. Koh Phys. ev. B ad by W. Koh ad L.J. Sha Phys. ev. A ; oogaph by.g. Pa ad W. Yag Desity Fuctioal Theoy of Atos ad Molecules t. Se. Moogaphs i Cheisty-6 Oxfod Uiv. Pess 989; D.J. Sigh Plaewaves Pseudopotetials ad the LAOW Method Kluwe Acadeic Pub. 994 [A3] Q. Ma K. Klie* H. Cheg J.W. Mitchell ad K.S. Hayes "teactio Betwee Catalysts ad Suppot. Low Coveage of Co ad Ni o Silica Suface" J. Phys. Che. B ; Q. Ma K. Klie* H. Cheg J.W. Mitchell ad K.S. Hayes "teactio Betwee Catalysts ad Suppot. Low Coveage of Co ad Ni o Aluia Suface J. Phys. Che. B ; K. Klie "Electoic Stuctue of Tasitio-Metal o Cotaiig Zeolites" i "Catalysis by Uique Metal Stuctues i Solid Matices - Fo Sciece to Applicatios ed. A.T. Bell G. Ceti ad B. Wichtelova NATO Sci. Se. 200 Kluwe Acadeic Pess pp.5-33;

7 Q. Ma K. Klie* H. Cheg J.W. Mitchell ad K.S. Hayes "teactio Betwee Catalyst ad Suppot. Metal Aggloeatio o the Silica Suface" J. Phys.Che. B ; Q. Ma K. Klie* H. Cheg J.W. Mitchell "teactio Betwee Catalyst ad Suppot. V. Peiodic Teds ad Pattes i teactios of Fist-ow Tasitio Metals with the Silica Suface" Joual of Physical Cheisty B ; P. Novak K. Klie ad J.A. Spiko Electoic Stuctue ad Co Obital Moet of Co-substituted Zeolites Phys. ev. B 2004 subitted [A4] P. Pulay Mol. Phys [A5] G.P. Sivastava ad D. Weaie Adv. i Phys [A6]. Yu D. Sigh ad H. Kakaue Phys. ev. B [A7] J.M. Sole ad A.. Willias Phys. ev. B