New Jellium Model for Alkali Metals and its Future Applications to Metal Clusters

Transcription

1 Floda Intenatonal Unvety FIU Dgtal Common FIU Electonc Thee and Detaton Unvety Gaduate School ew Jellum Model fo Alkal Metal and t Futue Applcaton to Metal Clute Gullemo Matanca Floda Intenatonal Unvety gmat00@fu.edu DOI: 0.548/etd.FI07 Follow th and addtonal wok at: Recommended Ctaton Matanca Gullemo "ew Jellum Model fo Alkal Metal and t Futue Applcaton to Metal Clute" 0. FIU Electonc Thee and Detaton Th wok bought to you fo fee and open acce by the Unvety Gaduate School at FIU Dgtal Common. It ha been accepted fo ncluon n FIU Electonc Thee and Detaton by an authozed admntato of FIU Dgtal Common. Fo moe nfomaton pleae contact dcc@fu.edu.

2 FLORIDA ITERATIOAL UIVERSITY Mam Floda EW JELLIUM MODEL FOR ALKALI METALS AD ITS FUTURE APPLICATIOS TO METAL CLUSTERS A the ubmtted n patal fulfllment of the equement fo the degee of MASTERS OF SCIECE n PHYSICS by Gullemo Matanca 0

3 To: Dean Kenneth G. Futon College of At and Scence Th the wtten by Gullemo Matanca and enttled ew Jellum Model fo Alkal Metal and t Futue Applcaton to Metal Clute havng been appoved n epect to tyle and ntellectual content efeed to you fo judgment. We have ead th the and ecommend that t be appoved. Rchad Bone H Rudolf Febg Xuewen Wang Majo Pofeo Date of Defene: June 4 0 The the of Gullemo Matanca appoved. Dean Kenneth G. Futon College of At and Scence Dean Lakhm. Redd Unvety Gaduate School Floda Intenatonal Unvety 0

4 ACKOWLEDGMETS I would lke to thank the membe of my commttee fo the uppot on the manucpt and eveyone that wa aound me dung th tme. They wee a majo nfluence to me and t ha changed me fo the bette. Th emnd me of a memoable quote once ad by a geat centt. If I have een futhe t by tandng on the houlde of gant. I epecally lke to thank my advo D. Xuewen Wang fo beng a geat mento and allowng me to epand my hozon.

5 ABSTRACT OF THE THESIS EW JELLIUM MODEL FOR ALKALI METALS AD ITS FUTURE APPLICATIOS TO METAL CLUSTERS by Gullemo Matanca Floda Intenatonal Unvety Mam Floda Pofeo Xuewen Wang Majo Pofeo Th eeach develop a new method fo undetandng the popete of mateal. The new method wa appled to alkal metal to eamne how well t can pedct the Wgne-Setz adu. Peudo-potental fo the ndvdual atom wee geneated and utlzed to obtan the nteacton enegy wthn thee metal. The ytem nvolve 4 coulombc chage; two of them ae the eult of the neutal atom one valence electon and one potve coe chage fo alkal atom and the othe two ae backgound chage of equal and oppote amount. Th coulombc nteacton wll behave dffeently dependng on the element that compoe the ytem. Thee ae fou goup of enegy fo th ytem. One of them ha the appeaance of the Jellum model whch olved wth Denty Functonal Theoy. Fom the othe thee goup one of them wll alte the mnmum of the Jellum model fo dffeent element n the ytem. Th goup patally calculated wth the help of Ewald ummaton. Th calculaton eemplfe that bcc favoed nce t lowe n enegy than fcc whch n ageement wth epement fo alkal metal. The coecton to th enegy wll be due v

6 to the coe electon' nteacton wth a unfom negatve chage backgound. Th new method wll alo be benefcal to calculate the gound tate enegy of clute by ntoducng uface boundae n the ytem. v

7 TABLE OF COTETS Chapte Chapte Chapte 3 Chapte 4 Chapte 5 Chapte 6 Intoducton Epemental and Theoetcal Technque fo anoclute Layout of The...7 Method fo Many-Electon Sytem The Hatee Appomaton...9 The Hatee-Fock Appomaton... Confguaton Inteacton Method.. Denty Functonal Theoy...3 Jellum Model....5 Jellum Model fo an Infnte Volume.5 Jellum Sphee 7 Ft Pncple Peudo-potental.. One-dmenonal Peudo-potental.3 Peudo-potental fo alkal atom 6 Reult fo Alkal Metal...30 Concluon..37 Lt of Refeence.39 v

8 LIST OF TABLES TABLE PAGE 3. Phycal popete fo element wth one valence electon Peudo-potental tet fo Sodum atom Ionc effect fo alkal metal Mnmum /ao fo alkal metal Bulk modulu fo alkal metal v

9 LIST OF FIGURES FIGURE PAGE 3. Enegy veu /ao fo the Jellum model.7 3. Flow chat to olve Jellum phee Enegy veu the numbe of electon to the one-thd powe.0 4. A peudo-potental to appomate a coulomb potental 4. One-dmenonal potental Enegy veu dtance fo even and odd oluton The electon of the cutoff egon n the wavefuncton Total enegy of tal 0 agant the valence enegy fom the all electon calculaton Peudo-potental fo alkal atom 3 5. Total enegy veu /ao fo alkal metal.34 v

10 Chapte Intoducton Popete fo bulk mateal ae o well known and ued n many mechancal and electonc ntument today that t gve lttle oppotunty to conduct eeach on th cale fo technologcal advancement. In ode to mpove ou technology one can tanton the ze of mateal fom bulk to mall clute to eploe new popete and have a deepe undetandng of the mateal. Howeve th a vey dffeent ealm fom bulk o new condton mut be met when the ze educed and quantum effect mut be condeed when fndng the chaactetc featue of thee clute. Cuently centt ae nvolved n many epement n paallel wth theoe fo nano-mateal but complcaton tll ae becaue of multple facto one beng that calculaton become too lengthy becaue the confguaton of the ytem nceae eponentally a the numbe of atom n the clute nceae. Dffculte tend to alo be encounteed when model ae beng ued n a long-ange pectum fo a ytem. So mot model that ae bult uually ft well wth epement n a naow ange of ndependent paamete. Scentt alway ty to mnmze the computaton dependng on the ytem and ty to fnd what vaable tuly make up the popete of the mateal beng analyzed. Th how theoe pedct the outcome of epement; but wthout epementaton t pactcally mpoble to ceate a theoy fo phenomena that occu n natue. Thee tll eman an enomou amount of eeach n the feld of nanomateal whch can alo help u undetand natue bette.

11 ano-mateal ha caught a lot of attenton n dffeent feld of cence and engneeng a a eult of the datc dffeence n the popete of mateal on th cale. Th meocopc egon whch ange fom nm to 00 nm the doman fo thee nano-ze object and thee ae many challenge to havng a complete undetandng of them and the ablty to manpulate them. One majo cteon affectng thee dffeent popete the tuctue and the lage uface aea-to-volume ato nano-zed object contan. Th lage uface aea-to-volume ato wll gve dffeent patal dtbuton fo the electon and caue the object to become moe eactve than bulk mateal. Fo eample noble metal uch a Au and Ag ae geat catalyt fo chemcal eacton when they become ulta fne patcle and allow the gowth of cabon nanotube. Anothe eaon fo thee dffeent popete the chaactetc ze of thee tuctue beng compaable to the wavelength of the electon. A a eult the ytem wll ehbt quantum confnement. Th quantum confnement wll make the electon adjut the enegy dcetely athe than a contnuou manne a n the cae of bulk mateal. Adjutng the ze of thee tuctue wll how that thee dcete enege wll have ze dependence and tablty fo a cetan numbe of atom o molecule n the tuctue. The numbe of conttuent fo thee table tate ae called magc numbe []. Thee ae othe facto that mut be condeed n ode to undetand popete of nano-mateal. One the effect of tempeatue change and the othe the mophology tanton at a cetan tme cale

12 []. Howeve nce my eeach wll nvolve the tudy of nano-mateal at abolute zeo tempeatue thee two facto ae beyond the cope of my the. Fo th the I wll develop a new method to tudy the lowet enegy tate the gound tate fo alkal metal n bulk fom and compae t wth epemental value. My method wll alo be applcable to alkal metal clute. I wll be able to etend my method to make t applcable to othe element n futue eeach. The majo goal to acheve an undetandng of all nano-clute mateal and epecally the moe eotc one whch ae of nteet to centt becaue the popete can be ued to mpove technology. Quantum Mechanc play a cucal ole n my eeach and I wll peent a method of obtanng a oluton to the Schödnge equaton nce t not eactly olvable fo a many-body ytem [3]. Th appoach wll not only allow me to analyze nanoclute but alo mnmze the computaton to get eult n a hote tme peod. Th wll beneft the centfc communty nce effcency an mpotant facto n eeach. Epemental and Theoetcal Technque fo anoclute Ft I wll dcu eale tude at FIU whch led to the cuent poject. Then I wll gve an ovevew of othe appoache fo nvetgatng nanoclute and how the mpotance of th feld. One uch tudy wa to fabcate nano-ze cobalt clute unfomly on a ttanum dode ubtate a tak that poved to be challengng. Thee wee too many 3

13 nteacton between the clute and the ubtate that came nto play and thee accounted fo the lack of unfomty that wa poduced but alo ehbted magc numbe. Thee magc numbe wee encounteed though the vaou epemental technque that wee pefomed. Anothe tudy howed the gowth of cabon nanotube ung gold clute a catalyt. Thee clute anged fom to 00 n the nanomete cale and the mechanm of catalytc gowth tll eman to be completely undetood. Fom thee epement t wa ealzed that n ode to undetand the popete of the clute one mut conde the tuctue and electonc confguaton of thee clute. Howeve the tudy of thee tanton element vey comple becaue of the gnfcant d-hell chaacte [4]. In th tuaton magnetm need to be condeed. So one wll ft have to ue mple ytem uch a alkal metal a the tatng pont. Then wth th foundaton comple ytem can be analyzed by ncludng the addtonal ntnc popete. Technque to detect nanoclute tll poe many challenge but have mpoved n ecent tme. The mot common ntument ued today ae the cannng tunnelng mcocope STM low enegy electon dffacton LEED atomc foce mcocopy AFM and tanmon electon mcocopy TEM. Intument lke LEED follow the pncple of dffacton n ode to detemne the tuctue of clute. The ntepetaton of thee dffacton patten ae baed on calculaton fom theoetcal contucton of clute; howeve undetandng of the patten not completely taghtfowad [5]. Theefoe othe ntument ae neceay to analyze clute and 4

14 th wll allow centt to naow down all the poblte fo the contucton of thee clute mateal. Thee ae many appoache to analyze a nano-cale ytem but n geneal fndng the mnmum bndng enegy between atom wthn the clute a the numbe of atom nceae can detemne the tuctue at dffeent ze of the clute. Th can be wtten n the fom E / 3 / 3 = a + b + c d. b + whee the ft tem coepond to a volume contbuton and the othe tem epeent uface contbuton. Th equaton how that the majo contbuton to the bndng enegy the volume and uface enegy. To fnd a table tuctue the uface enegy mut be optmzed fo a gven fed volume. In a late chapte I wll llutate how phecal hell gve enegy tablty fo a ytem. Fo faly lage clute Wulff contucton can be ued to fnd the optmal tuctue fo tablty. Othe method lke the Mackay coahedon can optmze the uface by ung qua-phecal hape o the Mak decahedon appoach [67]. Fom epement t ha been found that the coahedon hape moe favoable fo mall clute and decaheda fo ntemedate clute [8]. Th could be the eaon fo the contucton of bulk qua-cytal mateal nce thee hape can contan a fvefold a of ymmety. Howeve the mot favoable tuctue alo depend on the mateal. Othe facto lke nteacton ange bond ode and bond length mut be condeed whch can make the coahedal hape le table. Alo dectonalty of the bond motant [9]. 5

15 It ha been obeved that when the valence electon ae delocalzed a n the alkal and noble metal electonc hell clong wok well. Epement how th by obevng the magc numbe of the ytem [0]. Thee clute ae uually called upeatom nce they can mmc egula atom becaue of the electonc hell confguaton. One way to fll thee electonc hell though the phecal jellum model. The phecal jellum model aume a unfom backgound of contant potve chage denty wth whch the valence electon nteact and ae contaned becaue of a phecal bounday. Howeve fllng up thee hell fal when clute ae lage than 000 atom and geometc hell effect ae favoed [4]. Th phecal jellum clealy how that one model cannot decbe a lage ange of clute ze. So moe nvetgaton wthn th ealm mut be condeed n ode to have a bette undetandng of clute nce the appopate choce of an enegetc model mpotant. Othe method to calculate the total enegy of nanoclute have been developed and appled etenvely thoughout the yea. Howeve thee method can be computatonally ehautng. Fo ntance the Hatee-Fock method can be ued to calculate fom to 0 electon n a ytem but get computatonally cumbeome fo lage ytem []. The denty-functonal theoy DFT how mpovement n calculatng up to a few hunded atom n a ytem. Denty-functonal theoy can even povde the total enegy fo dffcult ytem lke tanton metal and noble metal but t fal f the ytem not teted popely nce the echange and coelaton tem 6

16 appomated [3]. One appoach to th appomaton the local-denty appomaton LDA but an addtonal tem called the gadent coecton mut be ncluded fo comple ytem lke the tanton metal [4]. Thee ae numeou tatege that can be appled to undetand ytem n the nano-cale but none of them ae nfallble. Th why thee a vat amount of eeach beng done wth mateal n th aea. Layout of The Befoe dcung my method fo analyzng alkal metal I wll eplan n Chapte ome method fo olvng Schödnge equaton fo a many-electon ytem. In Chapte 3 I wll dcu the Jellum model and fnd the total enegy of the ytem n t gound tate fo dffeent condton. Th allow me to analyze how bulk mateal and metal clute behave. In Chapte 4 I wll dcu puedo-potental and ome applcaton. Chapte 4 mpotant fo undetandng the nteacton wthn the mateal. What decbed n thee Chapte eental fo undetandng how my method wa contucted. In Chapte 5 I wll peent my eult fo alkal metal obtaned by my method. I wll compae thee eult wth epemental eult and dcu the mlate and dffeence. Fnally I wll povde dcuon on the analy of clute wth th method. 7

17 Chapte Method fo Many-Electon Sytem In th chapte I gve an ovevew of a few mpotant method that ae ued to olve Schödnge equaton fo a many electon ytem. Ft I wll dcu the oneelecton cae. Fom Quantum Mechanc the tme-ndependent Schödnge equaton ha the fom m + U = E. whee U the potental enegy fo the ngle-electon. Th potental the eult of all the othe patcle n the ytem and any etenal potental actng on th electon. Choong an appopate U fo a eal ytem lke metal a dffcult tak. A mean feld fo the potental can be ued but th would not decbe an accuate pctue fo the electon. In ode to have a moe accuate calculaton the wavefuncton hould nvolve all -patcle of the ytem that ae non-tatc. Theefoe fo a fully nteactng manyelecton ytem wth fed on the Schödnge equaton Ψ m 4πε Z ke R = k k Ψ + = j= j e 4πε j Ψ = E total Ψ. whee Ψ the -electon wavefuncton whch a functon of the poton of each electon. Z k ae the onc chage R k ae the poton of the on and ae the electon' poton. The econd and lat tem on the left-hand de of the equaton epeent the attactve electotatc potental and the epulve potental between electon epectvely. Howeve th equaton cannot be olved eactly. 8

18 Appomaton mut be condeed and the oluton obtaned fom thee appomaton method hould povde an accuate pctue fo the patcula poblem. The ngle-electon equaton uually a good tatng pont when the potental not complcated. Fo ntance when the patcle ae non-nteactng the Schödnge equaton can be olved eactly. Howeve eal ytem tend to be ntcate becaue of the epulve nteacton between the electon. The Hatee Appomaton Fo th appoach the electon n the ytem ae condeed to be ndependent of each othe o that the electonc chage denty wll mply be the um of each of the modulu quaed tatonay tate aocated wth the chage. The fom hown to be ρ = e..3 j= j Th type of chage denty ued fo the electonc potental of the ytem. If the th patcle emoved fom the chage denty then the potental enegy wll mply be the electc potental that eult fom all but the th electon multpled by the chage of the th electon: U el = e φ.4 ρ whee φ = d and ρ = e j. j= j Hee the electc potental of the emanng electon teated a a mooth dtbuton of negatve chage nce the tatonay tate ae condeed contnuou wth poton. 9

19 0 The Hatee appomaton alo nclude the nteacton enegy between the onc chage and electon chage = k k k on e Z U R 4 πε..5 The Hatee equaton ncopoate thee two potental enege to the potental of a ngle-electon equaton. Fo -electon thee wll be equaton:. 3 E d e U m j j j on = = + + =.6 The potental fo each equaton dffeent eultng n the ngle-electon wavefuncton not beng othogonal to each othe. In ode to atfy othogonalty condton the chage denty mut be appomated wth an aveage of the obtal denty whch wll lead to havng only one equaton. ow the Hatee equaton hown to be E j j on d e U m = + + =..7 Th equaton clealy how non-lneaty. To get a oluton fom th equaton a elfcontent appoach followed by fndng a fom fo the electonc denty and ung t to get a oluton fo the one-electon wavefuncton [5]. The wavefuncton then put back nto the electonc denty to get a potental whee the potental ued to olve Hatee equaton agan. Th teaton contnued untl the potental content. Th

20 method can gve accuate eult but numecal complcaton ae even though th equaton a cude appomaton of the full Schödnge equaton. The phycal featue cannot be fully decbed nce th method aume ndependent electon. Theefoe othe popete mut be ncluded n ode to obtan a bette pctue of the ytem. The Hatee-Fock Appomaton One condton the Hatee equaton doe not conde fo the electon the antymmetzaton of the electon becaue of the Paul pncple fo femon. The wavefuncton n the Hatee equaton ha the fom = Ψ.8 fo an patcle ytem. The wavefuncton need to follow the condton whee j j j j Ψ = Ψ..9 Applyng Slate Detemnant fo the tatonay obtal can atfy th condton.! = Ψ..0 The Hatee-Fock appomaton apple equaton.0 to the full electon wavefuncton and mnmze the epectaton value of the Hamltonan wth epect to the ngle-electon wavefuncton: 0 E ˆ = Ψ Ψ d H δ.. The mnmzaton lead to the Hatee-Fock equaton

21 . E j j j j j on d e d e U m j δ = + + =. Equaton. mla to the Hatee equaton whch only dffe by the lat tem on the left-hand de of the equaton. The econd and thd tem ae known a the dect and echange tem epectvely. ote alo that the elf-nteactng tem cancel becaue of the echange tem. The echange tem only contbute when the pn tate of the j th electon paallel to the pn tate of the th electon. The Hatee-Fock equaton ae dffcult to olve becaue the echange tem a non-local ntegal opeato and only a few cae ae manageable. Fo ntance choong a et of obtal to be othonomal plane wave fo a contant peodc potental can olve the Hatee-Fock eactly. Confguaton Inteacton Method Thee ae many etenon to Hatee-Fock appoach but the one mot often ued the Confguaton Inteacton CI. Th method apple a lnea combnaton of -electon Slate detemnant to the wavefuncton [6]. The ft tem mply the Hatee-Fock Slate detemnant whle the followng tem ae ected tate of the vtual Hatee-Fock obtal. Howeve th appoach tend to cale vey pooly when the ytem ze nceae and elated to the bnomal coeffcent:!!m - M! M =.3

22 whee M the numbe of Hatee-Fock obtal and the numbe of electon. Th method can be hghly accuate fo mall ytem but a nceae CI calculaton eque a lot of computatonal powe. Denty Functonal Theoy DFT The Denty Functonal Theoy whch fomally an eact theoy baed on the chage denty of a ytem. It alo tate that the chage denty of a ytem cannot have two o moe dtnct potental [7]. The potental unque fo the patcula chage denty. Fo uch a ytem the patcle denty fo the gound tate gven by = Ψ ρ G G d d..4 ow fo the ngle-electon equaton the potental wtten dffeently. m + V = E whee [ ρ ] V = V + V + V..5 et H XC V et the etenal potental aocated wth the on V H the dect potental tem gven by the Hatee equaton and [ ρ ] V the echange and coelaton potental. Equaton.5 wll allow epeng the enegy of the ytem a a functonal. XC E m [ ρ ] = d + E XC [ ρ ] e + = ρ ρ d d + ρ V et d.6 3

23 Howeve DFT aume that the echange-coelaton functonal known but t ha only been detemned numecally fo a few mple model ytem. Theefoe mot of the denty functonal calculaton ue the Local Denty Appomaton LDA. The LDA appomate the echange-coelaton functonal wth a unfom homogeneou electon ga of denty n at any poton. The epeon fo E [ n ] [ n ] n XC ung LDA E ε n d..7 XC XC Th appomaton can gve vey accuate value fo the ytem but often fal when the ytem ha electon that ae tongly coelated a n tate of electon contanng d and f obtal. 4

24 Chapte 3 Jellum Model When olvng a many-electon poblem that contan many on n the ytem a lage degee of feedom n tem of cytallne tuctue ae. Th lage degee of feedom make the Schödnge equaton dffcult to olve. In th chapte I wll dcu the Jellum model whch an appomaton to the on a a mean of mplfyng the ytem. Th model wll be olved fo a volume of nfnte ze and fo a phecal bounday. Jellum Model fo an Infnte Volume The Jellum model whch a much cude model to decbe metal can alo be ued to fnd popete fo a ytem but only when the ytem lage o compaable to bulk. The theoy behnd th model eplace the tuctual on whch have localzed chage wth a unfom chage dtbuton thoughout the egon of the mateal. Th o-called backgound chage dtbuton wll nteact wth the electon of the mateal [8]. The Hamltonan fo th model wth -electon confned n pace of volume V a follow. Hˆ = Hˆ + Hˆ + Hˆ el back el back whee Hˆ el = pˆ + m e = = j j 3. Hˆ back e = n R n R R R V dr dr = dr dr V V e V V 3. Hˆ el back = V V ρ n R dr d = e R V V = R dr

25 ρ the electonc denty and nr the backgound chage denty. A a eult of the unfomty of the chage and a volume of nfnte ze th wll allow the wavefuncton to be a et of fee electon plane wave. The oluton fo the enegy of the ytem ung the Hatee-Fock appomaton wll have a fom of E 3 = 5 9π 4 a 3 3 π 9π 4 a 3 Ry. 3.4 a the Boh adu and the Wgne Setz adu. The Wgne Setz adu elated to the aveage unfom denty of the ytem gven a V = 4π 3 3. The unt fo enegy ae n ydbeg Ry whee Ry = 3.6 ev. To coect the Hatee-Fock eult eact leadng tem n a hgh-denty epanon ae added. E 9π 4 a 9π 4 a = ln 5 π a O a Ry. 3.5 Thee coecton equed much labo [9]. The lat thee tem ae called the coelaton enegy whch a mnome nce they have no phycal meanng. Fgue 3. how the enegy pe atom a a functon of a ung the Jellum model fo a volume of nfnte ze. The mnmum n enegy occu at a Wgne Setz adu of 3.83a whch contant n th model fo any atom that compoe the ytem. The Wgne Setz adu doe not agee wth the epemental value a hown n Table 3.. Table 3. only lt a few element fom the peodc table that have one electon n the conducton band and how othe popete fo thee element that wll be ued fo analy late 6

26 Element Lattce contant a Å Wgne Setz adu Å Bulk modulu B 0 /m Cytal tuctue * H bcc L bcc a bcc K bcc Rb bcc C bcc Cu fcc Ag fcc Au fcc EnegyRy/atom /a o Table 3.. * The value obtaned fo metal Hydogen ae calculated theoetcally. Fgue 3.. The plot how the mnmum enegy pe atom fo a unfom chage denty occu at n my eeach. The eaon fo th dageement that eal ytem lack homogenety wheea Jellum aume unfomty fo the onc tuctue. In ode to ncopoate th effect one can conde eplacng th unfom backgound wth ome fom of potental that can decbe the onc tuctue of the ytem but then th wll nceae the complcaton of computatonal calculaton whee the wavefuncton wll not have the fom of a plane wave. Jellum Sphee In the pevou ecton I dcued the Jellum model nvolvng the mateal occupyng all of pace. Th condton and applyng the Hatee-Fock appomaton allowed the Jellum to be olved. Then the coelaton tem wee ncluded to have a moe accuate calculaton fo the electon-electon nteacton. The total enegy pe atom fo th ytem depend only on the denty. In contat I wll conde a bounday 7

27 contaned to a phee fo the Jellum model. Th wll nput an addtonal paamete whch I wll ue to nvetgate the enegy of the ytem. Th vaable paamete wll petan to the adu of the phee whch popotonal to the numbe of atom to the one-thd powe n the ytem. Howeve havng th contant the Schödnge equaton need to be olved numecally by a elf-content appoach. A tal functon along wth an aumed potental aocated wth the chage wll ntate the teaton. Fo th ytem the total potental enegy gven a = U + U + U [ ρ ]. 3.6 U et H XC-LDA Uet the potental enegy of the potve chage dtbuted unfomly wthn the phee that nteact wth the electon. UH the Hatee tem alo called the dect tem and epeed a = e j d = e ρ U H d. 3.7 j= UXC-LDA[ρ] the echange-coelaton potental whch a functonal of denty and appomated by the LDA. In ode to olve the Schödnge equaton an electonc chage denty mut be gven. Snce the denty elated to the ngle-patcle wavefuncton ρ = e j= j a tal wavefuncton can be gven ft. Then the potental enegy of the ytem can be olved ung th denty. Th potental 8

28 Fgue 3.. Flow chat to olve Jellum phee. ubttuted n Schödnge equaton to get a oluton. The poce can contnue by ung the oluton to get the new denty. A flow chat hown n fgue 3.. A pogam wtten n FORTRA code wa ued to geneate a loop fo th teaton untl the eult conveged to a table value. Fom th the egenvalue can be ummed to get to total enegy of the ytem. Th mulaton eecuted evey tme when a new numbe of chage ae added to the ytem. Two cae wll be condeed. Fo the ft cae the potve chage denty value wll be fed thoughout the phee and wll vay moothly whle the aveage electonc chage denty the ame a the potve chage denty. The econd cae wll cont of havng a phecal hell at the oute end of the phee wth a cetan thckne that depend on the nte-plane dtance of the atom. The value wthn the phecal hell wll vay whle of the phee ncbed n the hell wll be fed. Th vaable wll have a lmted ange dependng on the thckne of the hell and the fed value 3 3 V wthn the phee. In notaton tem 4 π va. f whee V the volume of 9

29 Enegy 0-3 Hatee/atom f =.0 a o /3 o hell Fgue 3.3. Enegy veu the numbe of electon to the one-thd powe. The enegy unt ae n Hatee Hatee = Ry. The hell model how lage enegy dffeence whch ndcate tablty. Shell the hell. Addng th phecal hell to the model mmc the uface effect that occu n eal ytem. The data fo the two cae ae upempoed on one plot to ee the dffeence. The plot hown n fgue 3.3 fo f =.0a. A new hell get added fo the net atom when va eache.0a. the numbe of atom n the ytem and elated to the adu R of the phee by π = π R. The hell model how lage ocllaton compaed to the cae wthout a phecal hell. Havng th lage enegy dffeence poduce tablty of a ytem at cetan dcete value of. Thee dcete value ae called magc numbe. The enegy dffeence need to be lage than the enegy at oom tempeatue whch about one mll-hatee n ode fo the ytem to be table. Wthout the uface effect the Jellum phee clealy how ntablty. In addton th alo how that th coe egon ha neglgble effect o t can be eplaced wth a contant. The ame effect occu wth the hell model when lage whch hould be the cae nce the enegy of the nfnte jellum fed fo a gven 0

30 value. The hell model a good appomaton fo undetandng the tablty of clute but doe not ndcate coectly the magc numbe. It alo oveetmated the enegy dffeence fo cetan magc numbe. Ignong the tuctue of clute gve a majo dcepancy to the model but hould not be completely degaded. I wll contnue to ue the Jellum model fo alkal metal; howeve I wll not gnoe the lattce tuctue. Includng the lattce tuctue wll gve dffeent mnmum value fo dffeent element that compoe the bulk ytem. My cheme wll be eplaned n detal n chapte 5 but ft I need to dcu the peudo-potental whch alo an eental method fo my analy of alkal metal. Peudo-potental wll be appled to the alkal atom to decbe how chage nteact n the coe egon of the atom.

31 Chapte 4 Ft Pncple Peudo-potental When calculaton eque ubtantal computatonal powe a n many-body poblem t ueful to apply the Peudo-potental method. Th method help educe calculaton by ntoducng a cut-off to the ytem whee t leat gnfcant. The appomated ytem ha to atfy cetan condton n ode fo calculaton to not devate fom the actual eult n the egon of nteet. The featue of peudopotental ae uually ued to emove the coe electon n the ytem. Removal of coe electon allow the Fgue 4.. A peudo-potental to appomate a coulomb potental. valence electon to be the man contbuton nce they tend to be the one that nteact the mot wth outde ouce. Conde a coulombc potental between the electon and the nucle a hown n fgue 4.. The peudo-potental wll dffe fom the egon of the coulombc potental whee the wavefuncton ha apd ocllaton whch condeed to be the coe electon nteactng wth the nucle. Outde th egon of the coe the peudopotental mut ft the tue potental eactly. Th whee valence electon ae located and the wavefuncton unalteed n th egon. In ode to have a atfactoy peudopotental the ntegal of the quaed ampltude whee the coe electon ae located ha to match a n the cae of the eal potental. Th known a the nom-conevaton [0]. Anothe condton that the peudo-potental mut atfy the chage denty n the valence egon hould be dentcal to the tue chage denty.

32 3 One-dmenonal Peudo-potental When geneatng a peudo-potental t mpotant to detemne whch pat ae eental fo you ytem and th uually mple when a potental ymmetc. Moe mpotantly the tanfeablty when ung the peudo-potental. I wll decbe a pecal cae n detal nce the peudo-potental an mpotant facto fo my analy of alkal metal. I conde a one-dmenonal double-well hamonc ocllato nteconnected by a contant potental V. + = V V < < < > 4. whee the png contant ae et equal to. ow fo the peudo-potental I have two hamonc ocllato cut n half wth an nfnte potental and alo nteconnected wth the ame contant potental V. + = V V peudo < < < < < < > 4. Thee two potental ae hown n fgue 4.. I wll call V and Vpeudo full hamonc and half hamonc epectvely. Accodngly the full and half hamonc wavefuncton ae full and half epectvely. Ung the WKB appomaton the oluton ae

33 full = co κ n κ π k d π k d ep ep κ d κ d 4.3 half = n κ + co κ π k d 4 π k d 4 ep ep κ d κ d 4.4 V V - o 3 4 o - o 3 4 o Fgue 4.. Show a double hamonc potental nteconnected wth a contant potental and two half hamonc potental alo nteconnected by contant potental. whee m m k = E V and κ = V E. ow nce the potental ymmetc the wavefuncton can alway be taken to be ethe even o odd. So two condton mut be atfed: 0 = 0 fo odd oluton and 0 = 0 fo even oluton. Th wll lead to two tancendental equaton fo full hamonc and alo fo the half hamonc ocllato. The equaton ae hown to be 4

34 π ± tan = k d ep 0 κ d full hamonc 4.5 π 0 cot = k d ep 4 κ d half hamonc. 4.6 The plu tem fo the full hamonc the eult of the odd oluton whle the mnu tem fo the half hamonc fom the odd oluton. Plottng both de of the equaton ndvdually on the ame gaph to get nteectng pont wll povde the enegy value fo the ytem. The two ytem the full and the half how degeneacy plttng when the dtance between the potental of zeo value appoach each othe. Alo the enegy between the even and odd oluton fo the dffeent potental how a geate dffeence when the contant potental nteconnectng the well lowe. A can be een n the gaph the peudo-potental matche well wth the full hamonc potental well but only fo a cetan egon. Theefoe the tanfeablty to peudo-potental lmted by the dtance between the well and the contant potental value. 5

35 V o = 3 E veu o Odd-Half.48 Even-Half.46 Even-Full Odd-Full V o = E veu o.55.5 Odd-Half Even-Half.45 Even-Full Odd-Full Fgue 4.3. The two gaph how that the tanfeablty mpove when Vo nceae and fo lage value of o. Peudo-potental fo alkal atom Befoe I analyze the alkal metal I need to conde the popete of the ndvdual alkal atom. All the alkal atom have one electon n the valence level whch the man component fo the atom. The et of the electon ae n cloed hell and eque moe enegy to emove. Th cloed hell egon can be condeed the coe egon of the atom and unaffected when outde ouce nteact wth the atom. In 6

36 ode to gnoe the enegy of th egon a peudo potental mut be poduced. Th effectve potental wll be ued to tudy the popete of alkal metal. To contuct a utable peudo-potental I need to ue a pogam and ee whethe the enegy compaable to the valence enegy fom an all-electon calculaton of the atom. Cetan nput paamete ae equed fo the pogam whch wll geneate value of the peudo-potental at dffeent adal dtance. I need to ndcate whch element to ue. Fom thee I need to ndcate what level of the electonc confguaton condeed the bounday of the coe egon. Then I nput the numbe of valence level and a patcula et of pncpal and azmuthal quantum numbe. Fo ntance f I wee to ue Sodum a I can chooe thee valence level 3 3p and 3d. et I put a factonal amount of an electon o electon f thee moe than one n the valence egon n each valence level. If the valence level ae 3 3p and 4 I can nput and 0. epectvely fo the one electon. Fo th pecfc ettng the pogam wll ft output eult that petan to all the electon of the atom. Thee eult wll contan egenvalue fo each tate and popete fo each ngle electon wavefuncton. In ode fo the pogam to geneate a peudo potental I need to fnd an appopate cut-off by eamnng the wavefuncton of the lat tate fom the all-electon calculaton. The cut-off choen to be n between whee the wavefuncton ha a node fathet fom the ogn and the lat peak value whch depcted n fgue 4.4. I geneated multple peudo-potental and teted the eult by nputtng dffeent mtue of the electon n the valence level. Then I compaed the enegy of 7

37 the electon to the enegy of the valence electon n the all-electon calculaton. Th wll ndcate whch peudo-potental a utable ft. Table 4. how eult fo the a atom. I ued thee valence level fo the electon 3 3p and 3d. Fgue 4.5 how that tal 0 ha a good lnea ft. Addtonal tet wee completed fo the othe alkal atom. Thee tet allow me to collect peudo-potental fo the alkal element whch neceay to calculate the nteacton enegy n the ytem. The net chapte how the peudopotental fo the alkal element. cutoff egon peudo c Fgue 4.4. Show the electon of the cut-off egon n the wavefuncton fom the all electon calculaton y = R = Fgue 4.5. Plot of the total enegy of tal 0 agant the valence enegy fom the all electon calculaton. 8

38 Peudo-potental tet Facton of electon fo the valence level 3 3p 3d Valence enegy fom all electon calculaton Tal Tal Tal 3 Tal 4 Tal 5 Tal 6 Tal 7 Tal 8 Tal 9 Tal Table 4.. Seveal tal fo a peudo-potental fo Sodum a atom. The enegy of the peudo-potental ytem mut be compaed to the all electon ytem to fnd a pope ft. 9

39 Chapte 5 Reult fo Alkal Metal In the pevou thee chapte I dcued vaou method to olve the Schödnge equaton fo a mult-electon ytem. I alo dcued how to contuct peudo-potental to emove patcula pat of the ytem that ae ngnfcant. Each of thee tactc ha lmtaton and uually a too lage of an appomaton to fully atfy a ytem. Howeve thee method hould not be completely gnoed. In th chapte I wll how how thee method ae upplemental to my calculaton fo alkal metal. A I howed n the Jellum model ecton the electon nteact wth a unfom potve chage dbued thoughout pace. Th ytem olvable and t able to geneate a mnmum fo the total enegy pe atom. Howeve the poblem wth th ytem that t dd not match wth the mnma found fo alkal metal whch have dffeent value. In ode to coect th the tuctual on cannot be gnoed. Fo my model I wll nclude the potve chage fom the on thoughout pace n the Jellum model. Thee on ae condeed to be dcete chage. To have the ytem neutal n chage I wll alo nclude a negatve unfom chage dtbuton thoughout pace. Th wll gve a total of 4 dffeent chage dtbuton whch nteact wth each othe. The ytem can have an mbalance of chage becaue of the on and electon but the unfom negatve backgound ha equal and oppote amount of chage compaed to the unfom potve backgound. I wll now have a total of 0 nteacton enegy tem ntead of 3 tem a n the Jellum model. They ae Uele-ele Uele-ρ+ U ρ +ρ+ Uon-on Uon-ρ- U ρ -ρ- Uon-ele Uon-ρ+ Uele-ρ- and U ρ +ρ-. The ubcpt ele on ρ+ and ρ- ndcate the chage 30

40 aocated wth the electon onc coe unfom potve chage and unfom negatve chage epectvely. I want to goup thee tem n uch a way that t wll gve me a goup that dentcal to the Jellum model. Th goup wll be called the electonc effect and t wll contan the knetc enegy tem of the electon. A econd goup wll have the on-on nteacton and I wll call t the onc effect. The et of the tem that eman wll appomate to zeo nce the chage dtbuton of ρ+ and ρ- the ame a the electon ele. Th fom epeed a Hˆ eef = K E ef ele + U = U ele-ρele-ele U + U U + U ele-ρ- on-ρon-on on-ρ+ on-ele + U + U + U + U ρ+ρ- ρ+ρ+ ρ-ρ electonc effect onc effect 5. The electonc effect wll be olved though Quantum Mechanc whle the onc effect wll be olved Clacally. The ft tem n the onc effect epeon calculated by Ewald ummaton []. The calculaton nvolve ummng the long-ange nteacton n Foue pace and the hot-ange nteacton n eal pace. Dependng on the lattce tuctue Uon-on wll have dffeent eult. Fo alkal metal the lattce type bodycenteed cubc bcc. Each of the tem n the onc effect dvege ndvdually a the ytem nceae howeve the um of the 3 tem wll convege to a contant value fo a fed. Uon-ρ- wll dffe fo each element when ρ- nteact wth the coe egon of the on. Th coe egon ha a dffeent electc potental fo each element. To decbe the electc potental of the on I wll geneate peudo potental wth the pogam that wa eplaned n the pevou chapte fo alkal atom. The peudo-potental fom the on 3

41 φ peudo L a K Lthum Sodum Potaum Rubdum Ceum 0.4 Rb C Fgue 5.. Peudo-potental Ry/chage fo alkal atom veu ao. Thee ae ued to calculate the nteacton enegy. ued to calculate the potental enegy between the on and any chage that nteact wth t. The geneated peudo-potental fo alkal atom hown n fgue 5.. If I gnoe the on havng a coe and teat t a a potve pont chage the onc effect wll not change fo dffeent element. The onc effect enegy fo bulk wll be / fo a bcc lattce type and / fo fcc face-centeed cubc. Thee eult alo how that bcc favoed nce t lowe n enegy than fcc whch n ageement wth epement fo alkal metal. Wth th onc effect enegy I can coect t by emovng the nteacton enegy between the potve pont chage and ρ- wthn the coe egon and add the enegy due to the peudo-potental of the on and ρ- fom the coe egon. Th coecton accomplhed computatonally. Table 5. how the onc effect enegy fo alkal metal at dffeent value. Addng th onc effect to the Jellum model the electonc effect adjut the mnmum. A I howed n fgue 3. mnmum at 3.83 fo the Jellum model. a ha a 3

42 Ionc effect /ao L a K Rb C Table 5.. The onc effect enegy Ry/atom of alkal metal fo dffeent /ao value. 33

43 /ao L a K Rb C Jellum Epemental Theoetcal Table 5.. Mnmum /ao fo alkal metal. The Jellum hown a a efeence pont C -0.5 Rb -0.0 K L a Lthum Sodum Potaum Rubdum Ceum Fgue 5.. Total enegy Ry/atom veu /ao fo alkal metal. When ncludng the onc effect the eult begn to how a hft of the mnmum towad the epemental value whch gve good ndcaton that th appoach vald. The value ae hown n table 5. and gaphcally n Fgue 5. fo alkal metal. The theoetcal value fo the lghte alkal element have a lghtly lage dcepancy than the heave alkal element. That becaue I aumed the electon to be fee n the ytem. Howeve the electon fo the lghte element eem to not behave n that manne nce the coe of the on malle whch gve le ceenng effect. The lat two goup fom equaton 5. wll not appomate to zeo and the electonc effect wll be alteed nce the electon wll not be unfom thoughout the ytem. 34

44 B 0 /m L a K Rb C Jellum Epemental Theoetcal Table 5.3. Bulk modulu fo alkal metal. The value fo Jellum alo calculated at the equlbum. Anothe popety that can be detemned fom th ytem the bulk modulu. The elatonhp fo bulk modulu defned a dp d E B = - V = V 5. dv dv whee P the peue appled to the ytem and V the volume of the ytem that the peue beng appled to. In tem of enegy and /ao the bulk modulu at the equlbum P = 0 epeed a d E B =. 5.3 π a 3 o /a o d /a o Table 5.3 compae thee eult wth epemental eult fo alkal metal whch how good ageement. Agan the lghte element have a lghtly lage dcepancy. The calculaton fo the bulk modulu entve to the data nce t ele on the cuvatue of the data. Th new cheme how gnfcant mpovement compaed to the Jellum model. One eaon that the Jellum model could not eplcate coectly a eal ytem the eult to the fact that t doe not conde the lattce tuctue. It alo howed no change when the ytem ued a dffeent element. That becaue the Jellum model aume unfom chage denty thoughout the ytem. Though th new method I 35

45 wa able to demontate change to the ytem when the ytem wa ued to analye a dffeent element and when t ncopoated the lattce tuctue. Th new jellum model appoach may alo be appled to othe element that ae not alkal atom; howeve othe popete mut be factoed n to get bette eult. 36

46 Chapte 6 Concluon Afte obevng the eult fo alkal metal one can undetand how to apply th new method to malle ytem. The eult ae eentally compoed of the electonc effect and the onc effect nce I condeed 4 chage dtbuton that nteact wth each othe n the ytem and poduced 4 nteactng tem that appomate to zeo. The electonc effect wa calculated by ung the Jellum model whle the onc effect nvolved an electotatc ummaton fo an nfnte ytem. Fo a clute-ze ytem thee two effect wll now have bounday contant. To nclude th bounday contant fo the electonc effect I can mply apply the Jellum phee that wa dcued n the Jellum model chapte. Fo the onc effect the electotatc ummaton wll tuncate dependng on how many atom wll be n the ytem. Th wll modfy the eult of the Jellum phee. One wll dcove that the enegy of the ytem wll ocllate dffeently and have dffeent magc numbe fo a gven element o f value a the numbe of atom nceae n the clute. Th could mpove the eult on decbng the tablty of cetan clute ze nce the phecal hell model oveetmated the enegy dffeence a wa eplaned fo f =.0a. A Jellum phee jut one of the vaou geometc hape that can be appled to the electonc effect when analyzng nano-clute. A I dcued n the ft chapte clute can have dffeent contucton dependng on the ze of the clute and the type of mateal that compoe t. Fo ntance coahedon hape have been obeved fo mall clute and decaheda fo ntemedate ze clute of numeou element. 37

47 Theefoe alteng the bounday of the Jellum can mpove the eult fo clute. Howeve the hape can depend on the mateal o fndng the coect nteacton enegy wthn the ytem an mpotant facto. Fo th new method I developed the nteacton enegy nvolved 4 columbc chage; two of them ae due the neutal atom one valence electon and one potve coe chage and the othe two ae backgound chage of equal and oppote amount. Th nteacton wll behave dffeently dependng on the element that compoe the ytem and wll detemne the geometc hape of the bounday by electng the lowet enegy fom all the poble fom of the bounday. Applyng th method can poduce futful eult fo clute ytem but the mot challengng tak detemnng how the mateal nteact n the ytem. Othe mateal that ae not alkal metal have addtonal popete that cannot be gnoed and thee popete can mply be added to my method. 38

48 Lt of Refeence [] Magc mbe fo Metallc Clute and the Pncple of Mamum Hadne. H. K. Habola 99 Poc. atl. Acad. Sc. USA [] Stuctue Dynamc and Themodynamc of Clute: Tale fom Topogaphc Potental Suface. D. J. Wale 996 Scence [3] Intoducton to Quantum Mechanc. D. J. Gffth 995 nd edton Peaon Pentce Hall. [4] Atomc and Molecula Clute. R. L. Johnton 00 Taylo & Fanc London and ew Yok. [5] Sze-Dependent Icoahedal-to-fcc Stuctue Change Confmed n Unuppoted anomete-szed Coppe Clute. D. Renhad B. D. Hall P. Bethoud S. Valkealaht and R. Monot 997 Phy. Rev. Lett [6] Shell of atom. T. P. Matn 996 Phy. Rep [7] Epemental tude of mall patcle tuctue. L. D. Mak 994 Rep. Pog. Phy [8] oncytallne tuctue of agon clute. II. Multlaye coahedal tuctue of A clute 50<<750. J. Fage M. F. de Feaudy B. Raoult and G. Tochet 986 J. Chem. Phy [9] Metallc bondng and clute tuctue. J. M. Sole M. R. Beltán K. Mchaelan I. L. Gazón P. Odejón D. Sánchez-Potal and E. Atacho 000 Phy. Rev. B [0] Electonc Shell Stuctue and Abundance of Sodum Clute. W. D. Knght K. Clemenge W. A. de Hee W. A. Saunde M. Y. Chou and M. L. Cohen 984 Phy. Rev. Lett [] Stuctue and popete of mall odum clute. I. A. Solov yov A. V. Solov yov and W. Gene 00 Phy. Rev. A [] Quantum chemty of mall clute of element of goup Ia Ib and IIa: fundamental concept pedcton and ntepetaton of epement. V. Bonacc- Koutecky P. Fantucc and J. Koutecký 99 Chem. Rev

49 [3] Denty-Functonal Calculaton on Platnum anoclute: Pt3 Pt38 and Pt55. A. Fotunell and E. Apà 003 J. Phy. Chem [4] Slcon Clute of Intemedate Sze: Enegetc Dynamc and Themal Effect. L. Mtá J. C. Goman I. Stch and J. Tobk 000 Phy. Rev. Lett [5] Self-Content Feld Appoach to the Many-Electon Poblem. H. Ehenech and M. H. Cohen 959 Phy. Rev [6] The Confguaton Inteacton Method: Advance n Hghly Coelated Appoache. C. D. Shell H. F. Schaefe 999 Advance n Quantum Chemty [7] Denty Functonal Theoy an Intoducton.. Agaman and G. Makov 999 axv:phyc/980603v. [8] Quantum Theoy of Sold. C. Kttel 987 Wley. [9] Coelaton Enegy of an Electon Ga at Hgh Denty. M. Gell-Mann and K. A. Bueckne 957 Phy. Rev [0] Geneal Theoy of Peudopotental. B. J Autn V. Hene and L. J. Sham 96 Phy. Rev [] Genealzed Ewald Potental Poblem. W. E. Rudge 969 Phy. Rev