Solid state physics. Lecture 3: chemical bonding. Prof. Dr. U. Pietsch

Transcription

1 Solid stat physics Lctu 3: chmical bonding Pof. D. U. Pitsch

2 Elcton chag dnsity distibution fom -ay diffaction data F kp ik dk h k l i Fi H p H; H hkl V a h k l Elctonic chag dnsity of silicon Valnc chag dnsity diffnc chag dnsity bonding chags U.Pitsch phys.stat.sol.b 37, 44,986

3 Elcton dnsity distibution of GaAs fom -ay diffaction data Ga As As Ga Valnc chag dnsity diffnc chag dnsity bonding chags Chag dnsity dtmins th typ of bond

4 Classification of bonding in cystals Dpnding on th typ of intaction bonding can b classifid as : Ionic bond.g. NaCl: 0V / bond Covalnt bond.g. Diamond, Si: 0V / bond Mtallic bond.g. Alkali-mtals Li,Na : ~V / bond Van d Waals bonding nobl-gas cystals: ~0.V / bond Hydogn bond wat /ic: ~0.V / bond

5 Covalnt bond in H molcul 4 m H b a b a n n W n H

6 Lina combination of atomic obitals LCAO appoach b a b a b a b a B A B A Ansatz: _ 0 _ S spins paalll S spins antipaalll Zo appoimation: ; spaatd atoms 4 0 a b m H Diffnc H =H-H 0 4 H b a S W W W W A C H By ptubation thoy :

7 Bonding ngis WC W Coulomb intgal A W S * * W H dv C a b a b Antibonding stat Echang intgal W H * * A a b a b ovlap intgal dv Bonding stat * * S dv a b a b Modl using s obitals only : =0.88 A, W min =-.9 V Eact modl pimnt : =0.74 A, W min =-4.5 V

8 Futh impovmnts:. Ionic contibution 5% A B C D a b a b a a b b H + H -. Considing mo wav functions A p a A p a Considing high shll wav functions A s B s C p... Not, wav functions a not physical quantitis : on can us any function, th us of atomic lik wav function is an appoimation lcton dnsity distibution can b obsvd

9 s and bonds s sp s ss s pd pp dp dd

10 Cabon bonds

11 Solution fo solids In solids quantum mchanical calculations is basd on Bon Oppnhim BO appoimation making th assumption that th motion of atomic nucli and th lctons in a solid can b spaatd. Schöding quation of lctons movmnt is solvd fo fid positions of nucli. Fo lcton systm singl paticl movmnt in man potntial of all oth lctons is indpndnt fom movmnt of ths oth lctons. Individual intaction of this lcton with ach of th oth lctons is nglctd. That man that Coulomb tm in Hamiltonian is placd by man potntial V i = / jj which dpnds solly by position i,j of th slctd lcton. H K Vi i Ei Solution of Schödig quation fo on lcton K- kintic ny, V- potntial in slf-constistnt fild of th oth lctons V i * V V i+ 0 Poisson qu. Cyclic ptition up to E bcoms minimum, vay V, HATEE 90 dscibd th total wav function of th multi lcton systm by th poduct of obitals + spinof th contibuting lctons Hat-poduct. l... n n i i n

12 Hat-Fock appoach V. Fock 930 impovd th appoach considing th pmutability of lctons. This is dscibd by th Slat dtminat wh ach lcton can occupy ach obital. By chang of two lctons two lins of th dtminat changs th sign of wav function anti symmty condition. Hat appoach is maning th diagnal lmnts of this mati only N!.. n n..... n n.. n n Solution of Hat-Fock quation F i i = E i i with Fock opato F i = H i + J i K i Containing th Hamiltonian of th lcton in th man potntial of oth lctons and nucli Fock-opato ; it contains a singl-lcton kintic opato and a two-lcton opato placing Coulomb- and chang-opato.

13 Ionic bond Chag dnsity of NaCl

14 Cohsiv ngy of ionic cystal Attactiv intaction: k 4 pulsiv intaction: o Sum o Total ngy: p ion pai u tot Z nm p Madlung constant Finding bonding ngy and quilibium distanc u tot 0

15 Bonding distanc and cohsiv ngy Equilibium distanc Cohsiv ngy p ion pai 0 p 0 Z nm

16 Madlung constant Lina chain badly convging poblm Fo NaCl CsCl=.7667 ZnS =.638 NaCl=.74756

17 Mtallic bond In mtals at last on lcton/atoms is citd into th conduction band. Ovlap of spctiv wav functions ov a lngth of many atomic distancs givs is to dcas of man potntial ngy.this tm is attactiv and is patially canclld by th pulsiv man kintic ngy. In this schm bonding ngy is E pot E kin Fmi ngy F lctons Conducton band E pot E pot E kin Valnc lctons valnc band E 0 =E pot

18 Mo dtaild pictu E B E E A E Pauli E Coul E chag E co E E A ngy diffnc btwn low band dg of conduction band and ngy of f atom - bonding ngy of ions Epauli E Coul E chang E co Du to Pauli pincipl only lctons occupy ngy E, all oth hav to occupy high stats. Coulomb intaction of cunducting lctons - pulsiv Echang intaction btwn lctons of opposit spin - attactiv Colation ngy - Impact of many body intaction - vy impotant fo mtals. Eampl fo Na: E B Ep = -.3 V/atom E B E E E A Pauli E Coul E chag E co = -.06 V = -.4 V Poblm : diffnc of lag numbs

19 Chmical bond in Nobl gass Ky assumption: fomation of atomic dipols whil oscillation of ngativ chagd lctons against th positiv nuclus. 0 C p C p H m m Hamiltonian of dcoupld oscillatos Oscillato fquncy m C / 0 H is Coulomb intaction of both oscillatos ³ ³ H ; ; p p p p p p a s a s 0 ³ ³ a a m s s m C p C p H H... ³ ³ [ 8 0 ³ C C m C

20 Scond tm cancls out fist valuabl chang coms fom 3d tm Attactiv U C³ 0 8 A 6 Van d Waals tm Quantum ffct, bcaus U 0 if h=0, A scals with lcton polaisability B pulsiv pat guss U p Both togth B U A 6 s 4[ s Lnnad-Jons potntial 6 ]

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22 Hydogn bonds Oftn dscibd as dipol-dipol intaction with covalnt fatus diction of bond hagonal stuctu of ic Pola hydogn bond in wat Hydogn bidging in molculs and polyms In th X H Y systm, th dots psnt th hydogn bond: th X H distanc is 0 pm, whas th H Y distanc is pm.

23 Hydogn bonds in DNA Hydogn bonding btwn guanin and cytosin, on of two typs of bas pais in DNA.