Covgc ss o h clus DF clculos. Covgc wh spc o bss s. s clculos o bss s covgc hv b po usg h PBE ucol o 7 os gg h-b. A s o h Guss bss ss wh csg s usss hs b us clug h -G -G** - ++G(p). A l sc o. Å h c bw h ol g clcul usg h -G** h -++G(p) bss ss s. V/o. Wh loog h g copos h l sc h chg-colo g s covg ow o. V/o. h lc c gs lss covg (8. V/o spcvl) lhough s o bov h ovll cobuo o h ol g s wll covg. h ollowg gu shows h covgc o h g s s uco o l sc wh spc o h bss s us. As c b s o h pupos o sg h ol o h g copos o h l bg h -G** bss s suls ssg. Fg. : Bss s covgc ss o h clus DF clculos. Elc (upp l pl) chg colo (upp gh pl) c (low l pl) ol (low gh pl) gs s uco o h l sc o 7 os gg h-b clcul usg h PBE ucol h -G (sol blc l) -G** (sh l) h - ++G(p) (sh-o g l) bss ss. Iss: oo o h go o phscll lv l scs.
. Covgc wh spc o h s shp o h clus o s o covgc o h suls wh spc o h shp s o h bl ls us clculos o h g copos s uco o h l sc o ch gg ls o csg s hv b po. Fg. pss gs o so o h ls us h ps su. Fg. : Igs o so o h hog hgol ls us h ps su: () os ch gph (b) 88 os ch gph (c) 58 os ch gph () os ch h- B () 88 os ch h-b () 7 os gg h-b (g) os gg h-b (h) 5 os gg h-b () 7 os gg h-b. C blu p g sphs ps cbo og boo hog os spcvl. All clculos hv b po usg h PBE/-G** lvl o ho. A. Å h ol lc chg-colo c gs o h gg h-b sg o 7 os covg ow o...5. V/o spcvl. Fo h ch gph sg o 58 os h s l sc h ol lc chg-colo c gs covg ow o..5.7. V/o spcvl. Fgs. ps h covgc o h l pc o h g copos s uco o l s o bl gph h-b spcvl. As c b s o ll pccl puposs h suls o h lgs clus ss wll covg showg ol gl g cs. Fg. 5 cops h l pc o h g copos o wo ch o gg h-b s showg h h covg suls p o h shp o h ls us h clculos.
Fg. : Covgc o h l pc o h lc (upp l) chg-colo (upp gh) c (low l) ol (low gh) gs wh spc o h s o h ch gph ls. ol blc sh sh-sh-o g cuvs ps suls o h 88 58 os gph ls. Fg. : Covgc o h l pc o h lc (upp l) chg-colo (upp gh) c (low l) ol (low gh) gs wh spc o h s o h gg h-b ls. ol blc sh shsh-o g sh-o blu cuvs ps suls o h 7 5 7 os h-b ls.
Fg. 5: Covgc o h l pc o h lc (upp l) chg-colo (upp gh) c (low l) ol (low gh) gs wh spc o h shp o h h-b ls. ol blc sh shsh-o g cuvs ps suls o h os ch 88 os ch 7 os gg h-b ls.
h Ewl suo ho o h Coulob pol o wo sol (D) poc slb o po chgs W s clculg h lc pol uc b slb o chg pcls whch s poc wo h h co. Fo splc w cgul u-cll such h h poc lg log h X Y s wh slol v. h u cll s ssu o co chg pcls loc posos o b chg ul such h: () h loco o gl o h slb s h gv b h ollowg psso: () Wh hs os w w h gl psso o h lc pol po u o h D-poc slb s: () h pol s poc uco h X Y wh po o log h X co po o log h Y co. hs c b sl os s ollows: () p l p p l l p wh p s g ub h ul lg o h sco o h h l uss h c h suo ov h covs h g o g ubs o o. l gus b us o pov h poc h co. W c ow h pol sg po u o o o h pcls h u cll s poc gs:
(5) such h () h su ppg E. (5) s vg whl h su ppg E. () s cooll covg pov h h u cll s chg ul (E. ()). I o o vlu h su w ul h Ewl suo ho whch spls h cooll covg su o wo bsolul (s) covgg sus. hs s chv b splg h sgul log-g Coulob pol o sgul sho-g p whch c b l vlu l-spc osgul log-g p whch s vlu cpocl spc. o hs w oc h sl o s lso poc uco h X Y wh h s poc (h poo ollow h s ls s h poo o ). W ow cll h poc uco c b p Fou ss o h o: (7) Dg h pso c b w h ollowg o: (8) ll o wo-sol (D) poc uco whch obs o hs: (9) Wh ()
I s ow possbl o p s D Fou ss o h o: () Wh h Fou cocs gv b: () Dg w vbls: () W c w h gl s: () ocg h p p p g w ob: (5) I E. (5) boh gos po ov sgs. h suo shs h sgs cuv such h h su o ll sgl gls c b plc b gl ov h ull g: ()
H w hv spl h s wh spcvl. ll w ob such h s gv b: (7) h vluo o h g oubl gls s po b sog o pol coo ss such h: (8) s cg (9) s s W ow lso so o s pol pso () s cg o ob: () s s B chgg vbls such h w c w: () c h uco s poc h gul go s po ov ull po o w w:
() J J wh J s h oh o Bssl uco o h s h ls ul suls o h c h posv os o h pog v. Wh hs w c w: () R R R RJ R R R J R J wh w hv s R R. c w c ll w: (5) I o o uc ull covg lc sus w ow us h ollowg gl o spl h o wo gs: () Usg hs w ob: (7) Wh h sho () log () g cobuos o hv b s: (8) h ull pol c b ow w s: (9)
h log g h log g s bsolul covg cpocl spc hus s s su cpocl spc h so bc o l spc. Wh vlug h lv gls o s o spl h css wh : Fo h log-g gl s gv b: () c c A hus: () c c Fo h gl s vg. vhlss c b w s su o covgg gl (o pg o ) vg p s ollows: () wh hs w ob: ()
c w chg ul u clls such h (E. ()) h vg lls w l wh: () W c ow bc so o l spc: (5) c c * Wh h s sg cs h w h wh s clu o h su. hs c b plcl w s: () c c *
h sho g h sho g s s bc-so o l spc (wh s bsolul covg) h su. (7) Accog o Posso's suo oul (s poo blow) w w (8) pplg o o o h sus bov w ob: (9) wh w c. Owg o h ollowg lo w h. hus usg Posso's oul (E. (8)) w ob: () Usg hs w c ow w
() c c o h () c
h Ewl ho o pll chg oc cs h-b Usg h ho scb bov w c ow w plc pssos o clculg h lc pol bov sgl l o hgol boo (h-b). As pl bov w cgul u-cll o h h-b l (s Fg. ) wh slol v -poc o -poc o bg h B- bo lgh. Fg. : Illuso o h u cll us o h Ewl suo o h lc pol bov wo sol l o ps hgol boo. Blu og ccls ps boo og os spcvl. Dsh l shows h l (o-cs) u-cll sg o wo os whl h ull l shows h cgul Cs ou o ucll us h clculos ps h ps su. s o h B- bo lgh pl chg o ch oc s spcvl (s o o ls). Whou loss o gl w ssu h h h-b sh s loc h XY-pl wh Z= clcul h lc pol g b h D sh gl po. h cgul u cll clus os loc h ollowg posos: () 5 bg h ollowg chgs: (). such h s chg ul: (5). h log g h log g ppg E. () s copos o wo s o sulg o h cs wh = h oh wh >.
c (E. ()) h u cll s ul (E. (5)) h wh = vshs () ho o h log g cobuo o h pol w l wh h > : (7) * 9 c c hs su c b w s su o h s: () wh () wh () wh. Fuho w c l h sus o h g b collcg h s wh pog posv gv cs:. c c c c c c ' o h (8) c c
' c c c c c c o h (9) c c * 9 9 c c c c
9 9 c c c c so h (5) 9 c c ug up ll h log-g s (Es. (8) (9) (5)) w ob: (5) 9 c c c c c c
h sho g Fo h sho g w ob (s E. ): (5) c Poo o Posso's suo oul (E. 8): W uco (5) h whch s -poch l h p Fou ss such h: l l c ho b Wh (5) h h (55) h h H sl o E. () w hv us h lo: (5)
h ol l Coulob g u o bw pll chg oc cs o h-b bl Cos h-b bl o s sg o u clls sc plll cl o op o ch oh h AA' scg o wh l sc R. h -l oopol Coulob wh ch l p o R ho o. W s clculg h l oopol co g gv b ( oc us): (57) E El j j j Wh h sus ov j u ov ll u clls ls spcvl h sus ov u ov h wo os wh ch u cll o h pog l. s h ol chg o o o u cll l j s h ol chg o o o u cll j l j s h sc bw hs wo os. W w hs su s ollows: (58) E El j j j j Wh s h oopol lc pol pc b pcl u j j cll o l u o ll h oc ol chgs l wo. As h l o bl wll b s s o ol h g b h ol ub o os : (59) E El Wh h l s ll u clls o l bco uvl boh s o h oc chgs s o h lc pol h pc u o l. ho o plc h gl cll b cll s w bl s:.wh hs w w h ol g p o h l o () E El o: () E El j j j
Fuho u o h lc s o h AA' sc h-b bl h pol pc b h wo os wh ch u cll s ul gu oppos sg. c h ol chgs o h wo os h u cll oppos sg s wll h oopol lc g h cobu s ul sulg h ollowg l psso o h ol g p o o h bl h-b ss: () E El j j j l h ol oopol lc g p o o h bl h-b ss uls hl h g o chg plc bov lc s o sgl h-b l. h oopol lc pol b clcul usg h Ewl suo ho scb bov.
h ol l vw g o bl gph h-b Cos s h-b o gph bl sg o u clls sc plll cl o op o ch oh h AA' scg o wh l sc R. h -l vw wh ch l p o R ho o. W s clculg h l vw g gv b: () E vw j j j c wh h sus ov j u ov ll u clls ls spcvl h sus ov j u ov h wo os wh ch u cll o h pog l. c s h ppop c coc bw o o u cll l o o u cll j l s j h sc bw hs wo os. As h l o bl wll b s s o ol h g b h ub o os. hus h ol vw g p o s gv b: () E vw j j j c j wh h b sgs g p o. I h bov psso h c s h j j vw co g o u cll l wh ll h os l. Wh h l s ll u clls l bco uvl s o h chcl vo ( hus lso s o h C cocs) lso s o h co wh h l. ho o plc h gl cll b cll s w j j c c j j j j h l o bl s:. Wh hs w w h ol g p o (5) E vw j j j c j j j c c j j j c j j j o: () E vw j j j c l h ol vw g p o o h bl h-b o gph sss uls u o h vw g o sgl u cll plc bov sgl l o h pog l. As h vw g cs s hs su s bsolul covg o us l o ps h low l covg h suls wh spc o h sos o h l.