Some ab initio tudies of the physical properties of materials
Transcript of Some ab initio tudies of the physical properties of materials
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Submitted on 13 Dec 2009
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Some ab initio tudies of the physical properties ofmaterials
Nathalie Madeleine Marguerite Vast
To cite this version:Nathalie Madeleine Marguerite Vast. Some ab initio tudies of the physical properties of materials.Condensed Matter [cond-mat]. Université Pierre et Marie Curie - Paris VI, 2009. tel-00440923
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♥srt ♣rés♥té ♣♦r ♦t♥r ♣ô♠
tt♦♥ à rr s rrs
♣été P②sq ♥s s tér①
t ♥t♦ s ♣r♦♣rétés ♣②sqs s
♠tér①
♣r
t ❱
♦t♥ t ♥t r② ♦♠♣♦sé
Pr♦ r ♠rr♦ Prés♥t
Pr♦ ❯ ♦♥ rt ♣♣♦rtr
r rst♦♣ r ♣♣♦rtr
Pr♦ ♣♣♦rtr
r rst♥ ♦① ①♠♥tr
r ♠ Ptt ①♠♥tr
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♥sr♣t ♣r♦♣♦s t♦ qr t r♥ ♣♦♠
tt♦♥ à rr s rrs
♦♣s P②ss ♥ ♦ trs
♦♠ ♥t♦ sts ♦ t ♣②s
♣r♦♣rts ♦ ♠trs
②
t ❱
♥ ♦♥ ② th ♥ r♦♥t ♦ t r②
♦♠♣♦s ♦
Pr♦ r ♠rr♦ r♦♠♥
Pr♦ ❯ ♦♥ rt r
Pr♦ r
r rst♦♣ r r
r rst♥ ♦① ♠r
r ♠ Ptt ♠r
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♦♥t♥ts
♦♥t♥ts és♠é r strt
♥tr♦t♦♥ t ♦♥t①t r t é♥ér r té♦rq sr♣t♦♥ s tr① rr
①tt♦♥s étr♦♥qs ♥s s ♦①②s ①t♦♥ étr♦♥q ♥s s s♠♦♥trs ts s ♠tér① rs ♥ ♦r
Pr♦t rr à qtr ♥s P♥ ♠♥srt ♦r♣ ♣tr
② q♥tt② t ♥rs ♦ t ♠r♦s♦♣ tr♥t♦♥ r♦♥ stt t ♦ ♣rtrt♦♥ ♠ ♣♥♥t ♣rtrt♦♥ t♦r② ♦t♦♥ ♦r t♠ ♣r♦ ♣rtrt♦♥ ❩r♦th ♦rr ♥st② ♠tr① rst ♦rr ♥st② ♠tr① ♦♥srt♦♥ ♦ r②st ♠♦♠♥t♠ ♥ ♥ ♥ t tr♦♥ ♥st② P♦ss♦♥s qt♦♥ s♣♦♥s ♥t♦♥ χ0 s♣♦♥s ♥t♦♥ χ ♥rs tr ♥t♦♥ ǫ−1 ♥r rs♣♦♥s ♥ r s♣ r tr ♥t♦♥ ② tr♦♥ ♥r②♦ss s♣tr♦s♦♣②
♦rt rsts tr♦♥ ♥r② ♦ss s♣tr ♥♦♠ ♣s ♣♣r♦①♠t♦♥
♦r② sts ♥ ③r♦♥
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♦ ts ♦r② sts ♦ t ♥ ♥s♦tr♦♣②
♦♠♣rs♦♥ t ①♣r♠♥ts ♦♠♣rs♦♥ ♠♦♥ r♥t r②st ♣ss ♥♦♠ Ps ♣♣r♦①♠t♦♥ s ♦♦ ♣♣r♦①♠t♦♥
♦r ♦♥s♦♥
♦rt rsts ♦♣t s♦r♣t♦♥ ①♣r♠♥t
❲t♥ ♥♣♥♥t tr♥st♦♥ t♦r② ♦♥t ♥st② ♦ stts t ♦ ♠tr① ♠♥ts
②♦♥ ♥♣♥♥t tr♥st♦♥ t♦r② ♦ ts s♣rts ts
①t♦♥ ts ♦st♦♥ ♦ ①t♦♥ ts ♦♥ t ts
♦♥s♦♥
♦rt rsts ♦♥ t stt② ♦ ♦r♦♥ rs
♥tr♦t♦♥ t♦♠ strtr ♦ 4 r♦♠ rst ♣r♥♣s
trtr ♠♦s t♦t ♥r② tt ②♥♠s r ♥t s♦♥♥ ♦♥s♦♥s
t♦♠ strtr ♦ ♦r♦♥ rs t ♦r r♦♥ ♦♥♥trt♦♥s ♦r♠t♦♥ ♥r② sss♦♥
tr♥t ♦ t tr♦♥♣♦♥♦♥ ♦♣♥ ♦♥s♦♥
♦♥s♦♥ ♥ ♦t♦♦
♣♣♥s
sr♣t♦♥ ♦ ③r♦♥
② r r♠s ♦♥ r
rr♠ t ❱st r
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st ♣t♦♥s ❱st r P♣rs rést♥t ♠s tr① rr ♦rt♦r
s ♦s rrés P♣rs rést♥t ♠s tr① rr
♥r ♦r♣②
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és♠é r
♦♥ tté rr ♦♥♠♥t ♥s r♦♣ té♦r ♦rt♦r s ♦s rrés ♦♥r♥ ét s ♣r♦♣rétés s ♠tér①♥térêt ♣♦r ♥s s ♦♠♥s ♥ér ♦ ♥♥♦étr♦♥q ♣♦r ♦t tt♥r ♥ sr♣t♦♥ té♦rq s♥s♣r♠ètr st s ♣r♦sss ♦♥trô♥t ①tt♦♥ étr♦♥q ♥sq r①t♦♥ ♦ és①tt♦♥ étr♦♥q t ♦r
• s ♣r♦♣rétés ♠tèr ♦rs ①tt♦♥ étt ♦♥♠♥t
• s ♣r♦♣rétés étt ①té ♦rés s♦s ♥ s♣tr♦s♦♣ ♣♦r s étr♦♥s ♥
• s rt♦♥s ♦ts s t♦♠s r ♦♣ s étr♦♥st rs ts sr tr♥s♣♦rt étr♦♥q ♦ r①t♦♥ étr♦♥q
s éts rqèr♥t ♥ ♥r♦♥♥♠♥t ♥t♥s t ès ①♦r♥trs r♥ q♣♠♥t t♦♥ ♥t♥s
♥s ♠♥srt st ♦r r♣♣é ♦♠♠♥t r ♦♥t♦♥étrq ♥rs ♥ té♦r ♦♥t♦♥♥ ♥sté é♣♥♥t t♠♣s t q st ♥ ♦♥t♦♥ ♣rt étr♦♥q ♦srés réstts té♦rqs sr ♦♥t♦♥ étrq ♥rs ♥s s ♦①②s♥♦♥ ♦rréés r♣rés♥tés ♣r ♦①② tt♥ 2 t ③r♦♥ ❩r2
s♦♥t érts ♥st s♦♥t ♦♥♥és s ♣r♥♣① réstts té♦rqs ♣♦rs s s♣trs s♦r♣t♦♥ ♦♣tq ♣♦r ♦①② r 2t ③r♦♥ ❩r2 ② ♣rés♥t ♥ ♥♦ ♥tr♣rétt♦♥ tr sr ♥♦② ♣r♠tt♥t ♠♦ésr s ts ①t♦♥qs ♥ té♦r ♦♥t♦♥♥ ♥sté é♣♥♥t t♠♣s ♥♥ s r♥rs s♠♥és sr s rrs ♦r s♦♥t r♣♣és
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strt
② tts r ♥ ♥♠♥t rsr ♥ t r♦♣ ♦ t♦r② ♦ t ♦rt♦r s ♦s rrés t ♥s t st② ♦ t ♣②s ♣r♦♣rts♦ s♦♠ ♦ t♦s ♠trs r ♦ ♥trst t♦ t t♦ ♥r rsr ♥ t♦ ♥♥♦tr♦♥s ♦t s t♦ ♣r♠trrsr♣t♦♥ ♦ ♣r♦sss ♦♥tr♦ tr♦♥ ①tt♦♥s ♦r tr♦♥r①t♦♥s s ♦rs
• st② ♦ t r♦♥ stt ♦ ♠trs
• ♥stt♦♥ ♦ t ♣r♦♣rts ♦ t ①t stt r♦♠ t ♣♦♥t ♦ t s♣tr♦s♦♣② ♦ ♥ tr♦♥s
• tt ②♥♠s ts ♦♣♥ t ♣♦♥♦♥s ♥ t t ♦tr♦♥♣♦♥♦♥ ♦♣♥ ♦♥ tr♦♥ tr♥s♣♦rt ♥ tr♦♥ r①t♦♥
s sts ♥♦s ♣r♦r♠♥ ♦♠♣t♥ ♥ rqr ♦♠♣trt♠ r♦♠ ts t ♥ r♥
♥ t ♠♥sr♣t r rst ♦ t ♥rs tr ♥t♦♥ st t♥ t♠ ♣♥♥t ♥st② ♥t♦♥ t♦r② ♥ t ♥t t ♠sr♠♥t ♦ t tr♦♥ ♦ss ♥t♦♥ ♦rt rsts r♣rs♥t ♦r tt♥ 2 ♥ ③r♦♥ ❩r2 r ♥♦rrt ♦①s♥ ♠♥ t♦rt rsts ♦♥ ♦♣t s♦r♣t♦♥ ♦ ♣r♦s ♦① 2♥ ♦♥ ③r♦♥ ❩r2 r ♣rs♥t ♥ ♥tr♣rtt♦♥ ♦♥ t r♥ ♦s t♦ ♠♦ ①t♦♥ ts ♥ t♠ ♣♥♥t ♥st② ♥t♦♥t♦r② s ♣rs♥t ♥② r♥t t♦♥s ♦♥ ♦r♦♥ rs ♥ ♥♣rtr 4 r r
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♣tr
♥tr♦t♦♥ t ♦♥t①t r
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P ❯ ❳
t é♥ér
♦♥ tté rr ♦♥♠♥t ♥s r♦♣ té♦r ♦♥r♥ ét s ♣r♦♣rétés s ♠tér① ♥térêt ♣♦r ♥ss ♦♠♥s ♥ér ♦ ♥♥♦étr♦♥q ♣♦r ♦t tt♥r ♥ sr♣t♦♥ té♦rq s♥s ♣r♠ètr st s♣r♦sss ♦♥trô♥t ①tt♦♥ étr♦♥q ♥s q r①t♦♥ ♦és①tt♦♥ étr♦♥q t ♦r
• s ♣r♦♣rétés ♠tèr ♦rs ①tt♦♥ étt ♦♥♠♥t
• s ♣r♦♣rétés étt ①té ♦rés s♦s ♥ s♣tr♦s♦♣ ♣♦r s étr♦♥s ♥
• s rt♦♥s ♦ts s t♦♠s rs ♦♣ s étr♦♥st rs ts sr tr♥s♣♦rt étr♦♥q ♦ r①t♦♥ étr♦♥q
s éts rqèr♥t ♥ ♥r♦♥♥♠♥t ♥t♥s t ès ①♦r♥trs r♥ q♣♠♥t t♦♥ ♥t♥s
r té♦rq
♦t é♥ér ♠♥t♦♥♥é sss ♠♣q é♦♣♣♠♥t ♠ét♦s té♦rqs t r ♠♣♥tt♦♥ ♥s s ♣r♦r♠♠s ♥♦r♠tqss ♠ét♦s ♥♠érqs q é♦♣♣ s♦♥t ss♥t♠♥t sés sr té♦r ♦♥t♦♥♥ ♥sté ♦ ❬❪ ts ss s♠ét♦s à ♦r♣s ♣♦r étr s ①tt♦♥s étr♦♥qs
s ♣r♦♣rétés s ♠tér① ♥ésst♥t ér♥ts ♥① sr♣t♦♥ té♦rq s ♣r♦♣rétés ♦♠ ❬❪ ♣r♠ètr ♠ ♦♥st♥ts éstqs ♥ésst♥t étt ♦♥♠♥t étr♦♥qért ♣r s ♣rtrt♦♥s sttqs ts s rt♦♥s rés rst♥ s ♣♦♥♦♥s ❬❪ t rs ♦♥séq♥s t ♦♣étr♦♥♣♦♥♦♥ ❬ ❪ q ♥tr ♥ ♥tr trs ♣♦r s♣r♦♥tté ♦♥♥t♦♥♥ ❬ ❪ rqèr♥t ♣rtrt♦♥ ♦♥t♦♥♥ ♥sté P ❬❪ ♥♥ s ①tt♦♥s étr♦♥qs ♦ ♣rtrt♦♥s ②♥♠qs ♥ésst♥t té♦r ♦♥t♦♥♥ ♥stéé♣♥♥t t♠♣s ❬ ❪
s té♦rs s♦♥t ①ts t sés sr ♦♥♥ss♥ s ♥sté étr♦♥q ♦ ♠tr ♥sté s ts à ♦r♣s s♦♥t rss♠és ♥s ♥ ♣♦t♥t étr♦♥q t é♥ t ♦rrét♦♥ ♦ ♥s éré ♣♦t♥t ♣r r♣♣♦rt à ♥sté étr♦♥q ♥♦②é♥ t ♦rrét♦♥ ♥ ♣rtq ♦♥ ♥ s♣♦s q ♣♣r♦①♠t♦♥s♣♦r ♣♦t♥t ♦ ♥♦② ❯♥ s ♦ts ♦♠♠♥té s♥tq q ♣rt♣é ❬❪ st ♠é♦rr ♥♦② é♥ t
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P ❯ ❳
♦rrét♦♥ ♥ ♦♠♣r♥t ré♣♦♥s étr♦♥q ♦t♥ ♥ ♦t♥ ♣r s ♠ét♦s à ♦r♣s à
r à st ♥éssr ♣♦r érr ①tt♦♥ étr♦♥q ❬❪
s ♥s ♥trts ér♥ ♥tr ♣s t é♥r s étts♦♣és ♣r s étr♦♥s t ♣s ss é♥r s étts sés ♥ ♥s s s♠♦♥trs t s s♦♥ts s♦♥t très♦rt♠♥t s♦sst♠és ♥t t q ♣♦t♥t é♥t ♦rrét♦♥ ①t t ♥♦♥♥ ♥ s♦♥t♥té q♥ s②stè♠ ♣ss à ± ♣rts ❬ ❪ tr♠♥t t ♣♦rr ± ♦ù st é♥r t♦t s②stè♠ ♥ ♣s éq♥t té♦rè♠ ♦♦♣♠♥ ♠ét♦ rtr♦ ❬❪ tt s♦♥t♥té st ♣rés♥t q q s♦t ♦♥t♦♥♥ é♥ t ♦rrét♦♥ ♦♥séré ♥s ♣♣r♦①♠t♦♥ ♥sté ♦ ♦ ♥s ♣♣r♦①♠t♦♥ r♥t é♥érsé s ♦♥t♦♥♥s ②rs sés ♥ ♣rt sr ♥ ♣♦t♥té♥ ①t t②♣ rtr♦ ♣rés♥t♥t é♠♥t tt s♦♥t♥té ❯♥ qs♣rt à ♣r ①♠♣ ♥s ♣♣r♦①♠t♦♥ ❲ ❬ ❪ st r♦rs♠♥t ♥éssr♥s ♣rtq s ♥s ♥trts és s ♦♥t♦♥♥s②rs ❨P P s♦♥t ♣s ♣r♦s ①♣ér♥ q sés à ♦
P♦r ♥ ①tt♦♥ étr♦♥q ♥tr ♥ ♣s ♦rrt♦♥ qs♣rt ♣réé♥t ♦♥ ♦t ♣r♥r ♥ ♦♠♣t ♥trt♦♥ étr♦♥ ①té tr♦ q ss rrèr s ts ①t♦♥qs ❬❪ tt ♥trt♦♥ ♠é♥ s tr♥st♦♥s ♥é♣♥♥tst ss ts ♣♥t êtr érts ♣r ♥ ♠t♦♥♥ t ❬❪
♥♠♥t s trs t rs ♣r♦♣rs s♦♥t ♠♣♦rt♥ts
• P♦r r s ♣r♦♣rétés étt étr♦♥q ♦♥♠♥t
• P♦r r ♦♥t♦♥ ér♥t ǫ−1 ♦ ♦♥t♦♥ étrq ♥rs ♥ré♥t ss♥t s ♠ét♦s à ♦r♣s ♣tr
• P♦r r ♣r♦té tr♥st♦♥ s♦s t ♥ ♣rtrt♦♥ rè ♦r r♠ ♣tr
sr♣t♦♥ s tr① rr
♥ ç♦♥ é♥ér s ér♠qs t ♣rtèr♠♥t ③r♦♥ ♥♥① s♦♥t s ♠tér① ♦♥t tst♦♥ s r♥♦rr ♥s s t♥♦♦s ♥érs ér♠qs ♣♦r s rétrs ♥érs é♥ért♦♥
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P ❯ ❳
❱ s r♥♦rés ♣r s♣rs♦♥ ♦①②s ♠tér① strtr ♣♦r s♦♥ P♦r ét s ♠tér① s♠t♦♥ ♥♠érq ♦ ♥rô ♣t ♥ s♣♥s ♣s s ①♣ér♥s r s ♠tér① ts♣♦r s ♣♣t♦♥s s♦♥t ♦♠♣①s ♣r ♥♦♠r t♦♠s r♠ éé♠♥tr ♦ ♣r r strtr étr♦♥q ♥♥t ♣r ①♠♣s étr♦♥s d r s ♠tér① tr © s♦♥ tr♦s rt♦♥sq ♣ss ♠♥t♥♥t ♥ r
s éts sr s ①tt♦♥s étr♦♥qs ♦ré ♥ ♠♣♥stt♦♥ t♦t♠♥t ♥♦ s♦s rt♦♥ ♥♥Ps sr s ♦♥ss ♠ Ptt ♦♠♥é s ♥♦s ♦♥♥ss♥s s qss sr s ♣♦♥♦♥s ♣♥♥t ♠ tès t ♥s♥é ♥s ♦♠♣ré♥s♦♥ rô ♦♣ étr♦♥♣♦♥♦♥ ♥s r①t♦♥ étr♦♥q ♥♥ t♦♦rs ♠♥é ♥ ♣rè ♥ tr sr s rrs ♦r ♠tér① q ♦♥♥ss♥t ♦r ♥ r♥ ♥térêt ♣rt ♦♠♠♥té s♥tq tt ♦♥♥ss♥♥♥♥ ♣♣♦rt ♦r ♥ r♦♥♥ss♥ ♥tr♥t♦♥ à éq♣q ♥♠ ♦♠♠ ♥q♥t s tt♦♥s ré♥ts ♥♥① str① ♠♥és ♣♥♥t ♠ tès
①tt♦♥s étr♦♥qs ♥s s ♦①②s
r té♦rq rt à tt ♣rt st ①♣té ♥s s ♣trs t s ♣r♥♣① réstts s♦♥t ①♣qés ♥s s ♣trs t ♦♥♥ ♥ ♣rç ♠s tr① rr ♥ ①♣q♥t ♦♥tr
♠♦♥ rré ♥ étt ♥ ♦rt♦r ré♣tt♦♥ ♥tr♥t♦♥ ♣♦r s é♦♣♣♠♥ts té♦rqs sr s♣tr♦s♦♣ s étr♦♥s ♥ ♠♣sés ♣r ♥♥ rs♣♦♥s r♦♣ té♦r ② ♦ré ét s ①tt♦♥s étr♦♥qs ♥ srs ♠tér① ♥térêt ♣♦r ♥ ♠♥♥t s ♣r♠èrs ♣♣t♦♥sà s ♦①②s ♠ét① tr♥st♦♥ ♠♥t ♦rréés 2 t ❩r2s ♠ét♦s té♦rqs sqà ♣♣qés à s s♠♦♥trs t s♠ ♦♥t ♥éssté s ♠é♦rt♦♥s ♥ rs♦♥ ♦st♦♥ ♣sr♥ s étr♦♥s ♥s s ♦①②s
♦♥t♦♥ ♣rt é♥r −ℑ(ǫ−1) ♣r♠t ♠ttr ♥ é♥ s①tt♦♥s étr♦♥qs ♦ts s ♣s♠♦♥s ♥ sért
−ℑ(ǫ−1) = −ℑ(1 + vχ),χ = χ0 + χ0(v + fxc),
♦ù v st ♣♦t♥t ♦♦♠♥ χ0 st ♦♥t♦♥ ré♣♦♥s étr♦♥s♥é♣♥♥ts t fxc st ♥♦② é♥ t ♦rrét♦♥ été ♦♠♣ré à s ♠srs très ♣réss ♣rt é♥r étr♦♥q ❬❪ ♣r P tts♥r ❯♥rsté ♥q ❱♥♥♥ tr ♠♦♥tré r♥ ♠♣♦rt♥ s ts ♠♣s ♦①
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χ0v ♥s ré♣♦♥s à ①tt♦♥ s étr♦♥s ♦sés p Ps ♦♦ré à ér♥ts éts sr rô s ♠♣s ♦① ♥s ré♣♦♥s ♥♥♦ts r♦♥ ❬❪ ♦ ♥s rér♥♥ ♥♥♦strtrss♠♦♥trs ♥ ♦♥r♥t tès ♦tt ❬❪
♥st tré sr ♦①② ③r♦♥♠ s t Pr♥ ♦♠♣rs♦♥ s ♦♥♥és ①♣ér♠♥ts ♦t♥s ♣r ♥ ♦♦rt♦♥ ②♥t ♦ t♦♥ ♣érr tr♦♠t tr♦♠étr r♥♦ sr s ér♥ts ♣ss rst♥s ❩r2 ❬❪ ♥♦s ♣r♠s étr♠♥r q ♥ té♦r st ♥éssr♣♦r trtr ç♦♥ ♣♣r♦♣ré s s♣trs ①tt♦♥ étr♦♥q ♥ss ♦①②s ❬ ❪ ♥ ♦♥ té ♣♦r s♣tr♦s♦♣ ♣rt é♥r t éé s ♣rs♣ts érts ♥s st♦♥
♦♦ré ③qr♦ r♦tt ②♥r♦tr♦♥ t rrtt P ♥s r tès té♦rq r♥ sr ♦①② r ♦♥ré ♥♥ s ♠srs ♣♦t♦é♠ss♦♥ rés♦ ♥ ♥ ♦ P ♦♥t ♣r♠s ♠♦♥trr ç♦♥ ♥tt♥ ♥ss♥ s rs ♣r♦♣rs ♦t♥s ♥ ♦♠♠♣♦♥t é♣rt qs♣rt ♦r♥té tr r♥ rs ♥ trt♠♥t ♠é♦ré ❬❪ ♦♥t♦♥ ér♥t ǫ−1s é♦♣♣♠♥ts r♥ ❬❪ ♦♥t ♣r♠s ♥ ♣rtr strtr étr♦♥q 2 t s♣tr s♦r♣t♦♥ ♦♣tq ❬❪
s tr① ♦♥t ♥ ♦tr ♣r♠s ♠♦♥trr q ♦♥t♦♥ ♦♥étr♦♥q qs♣rt èr ♣ tr ♣r♦♣r ♥s s ♠tér① r ❬❪ ♦ 2 ❬❪ ♣♦♥t ♦♥♠♥t r♥ ♥s ét sé sr r①t♦♥ étr♦♥q♥s ♥ ♦♥t♦♥
①t♦♥ étr♦♥q ♥s s
s♠♦♥trs
r té♦rq tt ♣rt st ①♣té ♥s ♥♥① t ♥s s♣t♦♥s tés
s ♠é♥s♠s és①tt♦♥ étr♦♥qs s♦♥t ♠♣♦rt♥ts t♥t ♣♦♥t ♦♠♣♦rt♠♥t s ér♠qs s♦s rrt♦♥ q ♣♦r tr♥s♣♦rt ♥s s ♥♥♦s♣♦sts r ♦♠♣r♥r s ♠é♥s♠s r①t♦♥ ♠♣q ét ♦♣ étr♦♥♣♦♥♦♥ ♥s ♥ ♦♥t♦♥
sqà ré♠♠♥t ♦♣ étt éé ♥s s s♠♦♥trs s ♠♦ès stés sr ①♣ér♥ ♣♦st♦ st ♣r♠s é♦♣♣r té♦r ♥t♦ ♦♣ ① étts étr♦♥qs
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♥ ♦♥t♦♥ ♥ ♣♦♥♦♥ tr ♦♥ ♥ ❬❪ tr été ♠♥é ♥ ♦♦rt♦♥ Pr♦ ❱ ②tr ❯♥rsté Pé♦q ♦♠s ss ♦r♥té s rrs rs ♥tr♣rétt♦♥①♣ér♥s s♣tr♦s♦♣ rés♦ ♥ t♠♣s ❬ ❪ s r♥èrs ♦♥♥♥t ès ① ♣r♦sss ♥trt♦♥ ♠r♦s♦♣q ♦♥♠♥t① t ♦♣ étr♦♥♣♦♥♦♥
♥s s s♠♦♥trs ♣rés♥t♥t ér♥ts ♠♥♠ és ♥s ♥ ♦♥t♦♥ ♦♣ étr♦♥q ① ♣♦♥♦♥s ♦♥r♦♥ ♥ ♣r♠t tr♥srt étr♦♥q ♥ é à tr t ♦♥trô tr♥s♣♦rt étr♦♥q ♥s s ♥♦s ♦♥s ♦r ♠♦♥tré q♥♦s ♣♦♦♥s ♣rér s ♦♥st♥ts ♦♣ à ♣rès ♣♦r s♦♥ é Γ à é X rs♣♦♥s t ♥♥ ❬❪ st♥ ♥é ♦♥sér ♣r r♣♣♦rt ① érts tr♦és ♥s ttértr①♣ér♠♥t
♥♥ ♥s P ♥♦s s ♦♥t ♣r♠s ér t♠♣s ①tt♦♥ t rtr s ♦♥♥ss♥s ts ♥ ♠♦♥tr♥t qs♣♦♥♦♥s ♠t♥t ré♠♥t t♠♣s ①t♦♥ ❬❪
s sès s ① tr♦s ♠♠rs éq♣ ♣r♠tt♥t ♥srér♥ts rt♦♥s trs st♦♥
ts s ♠tér① rs ♥ ♦r
r été tt tté ♣t êtr tr♦é ♥s s réér♥s❬ ❪ s r♥rs réstts ♣és s♦♥t ♦♥♥és ♥s ♣tr
♥♦② ♦r 10 st ♥ ①♥t s♦rr ♥tr♦♥ t ♥té ♠♦♥ ♥térêt ♣♦r ♦r t s ♠tér① rs ♥ ♦r tr ♣rts éé♠♥ts ♦r♠♥t s ♠tér① trrs à s s s♦♥s♦♥ts ♦rt♠♥t rt♦♥♥s
t♥t t trs♥t ♥r t ❬❪ s ♦♠♣♦sés rs ♥♦r ... s♦♥t s ♠tér① très rs q ♠ért♥t s éts ♦♥t♥s❯♥ ♦rt tté rr sr s ♠tér① ♣rtèr♠♥t ♣♦♥ ♥s s rrs s s ♥t♦ ♥♦♠r sqs① q ♠♥és ♣♥♥t ♠ tès ❬ ❪ ♦♥t ♥ r♥ rô
♥s ♣ssé ♠ ss ♥ t ♥térssé ♥ ♦♦rt♦♥ ③③r ss♦♥ r♦♥ t ❩ér ① ♣r♦♣rétés rt♦♥♥s ♦r ❬ ❪ t rr ♦r ❬❪ s tr① s♦♥t ♥tés ♥s ttértr ♦♠♣rs♦♥ ♥♦s s ♥t♦ ♦t♥ssr s strtrs ②♣♦tétqs s ♦♥♥és ①♣ér♥ ♠srés♥ s♦♥ ♠♥ t ♥ s♦r♣t♦♥ ♥rr♦ ♥♦s ♣r♠s rs ♥rtts sr ♦st♦♥ s t♦♠s r♦♥ ♥s strtr
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♦r t étr♠♥r strtr t♦♠q rr ♦r 4♦♠♠ ♥ ♦sèr 11p t ♥ î♥ q ♠ ♣r♠t ♠ttr♥ qst♦♥s s réstts ré♥ts ❬❪
Ps ré♠♠♥t ♥ ét♥t s s♣trs rés♦♥♥ ♠♥étq ♥ér ♥ ♦♦r♥t r P ♥sttt ♥ér♦t P②sq s ① ♦♥♥sés ❯♥rsté Prs ❱ P r strtr t♦♠q rr ♦r 4 q s étr♠♥é ♣♥♥t ♠ tès ❬ ❪ été ♦♥r♠é ♣r s ♥♦s éts ❬❪t ét strtr ♣s ♦r♥t ♥s ♠tér été ♥té ❬❪
♥♥ st ♠♥t♥♥t ♦♥♥ q ♦r ♣r à t ♣rss♦♥ ♠étst ♥t s♣r♦♥tr ❬❪ ♦r♥ tt tr♥st♦♥ st ♣♥♥t♥♦r à étt ②♣♦tès ♥ ♥♦ ♣s 28 ♥♥t êtr é♦rt ❬❪ P♦r rr ♦r ♥♦s ♦♥s ♣rét ♣♦r ♠tér q ♦♣ ♣r s tr♦s r♥ ♠étq à ♣rss♦♥ ♠♥t ♥t♠♣értr s♣r♦♥tté éé ♣♦r ♥ s♣r♦♥tr ♣r♦ 2 ❬❪
♥②s ttértr ré♥t ❬❪ ♠♦♥tr q ét ♦r♠t♦♥ s rrs ♦r t r stté s♦è ♥ ♥térêt ♦♥t♥ tq qst♦♥ ♥st ♣s rés♦ ré♠♠♥t ❬❪ é ♥ ♦♦rt♦♥ ♥ st t ♠♠♥ tr♥♥② é♥r ♦r♠t♦♥ ér♥ts strtrs t♦♠qs s é♥rs és ♠♦♥tr♥t ç♦♥ ♥tt♥ q ♣♣rt s rrs ♦r étés ♦♥t ♥ t♥♥ à s é♦♠♣♦sr ♥ ♦r r♣t ♣rt rr ♦r 4♥♦s ♦♥s tr♦é s♠♥t ♥q strtrs q s♦♥t sts t ♣♥t s♦r♠r ❬❪ ♣tr s strtrs s♦♥t ♠étsts ♣r r♣♣♦rt rr ♦r 4 st s ♠ éé♠♥tr 132♦rsq st ♠♦ésé ♣r ♥ ♦sèr 12 t ♥ î♥ ♦s ♥♦♥s ♦♥ q tt ♠étstté ①♣q té à s②♥tétsr sé♥t♦♥s ♣r♦♣rs 132 ①♣q ss s♥ é♥t♦♥♠♦♥♦rst♥
♥ q 4 ♣rés♥t ♠t éstq ♦♥♦t ♣s ♠♣♦rt♥t ♦♥♥ ♣r♠ s ér♠qs ≈ P s♦♥ tst♦♥♦♠♠ ♠tér ♥ rst ét r ♥ ♣t résstr à ♣r♦♣t♦♥ ♥ ♦♥ ♦ ♦♠♠ é♠♦♥tr♥t s ♦♥♥és ①♣ér♠♥ts ré♥ts ❬ ❪ ♣♥♥t à r t ♠é♥s♠ ♠ê♠ ré♣♦♥s à ♥ ♦♠♣rss♦♥ ②♥♠q ♥st ♣s ♦♠♣rs ♥s ♣rt résst♥ ♣ êtr ss♦é à ♥ ss ♦♥tr♥t s♠♥t❬❪ ♠s tt ♥tr♣rétt♦♥ été ♦♥tsté ♣r st ♣r ♥ ①♣ér♥ ♠♣ts ♣r♦ts sr s ❬❪ ♥s tt r♥èr ①♣ér♥ ♠♦t♦♥ ♦♠♣♦rt♠♥t s♦s ♦♥t♦♥s ①trê♠s ♥ ♣s ♣ êtr♦♠♣rs ♣t êtr ttré à ♥ ♣r♦♣rété ♣②sq ♥tr♥sèq rr ♦r 4 ♦ à ♥ ♦ ♣srs tr♥st♦♥s ♣s ❬❪
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tr ♣rt ♥ ♣r♦sss ♠♦r♣st♦♥ 4 été séré à ♣rtr ♦srt♦♥ ③♦♥s ♥♦♠♠♠♥t ♥s s ③♦♥s ♥tr s r♥s 4 ♥ ①♣qr s ♠♦t♦♥s s ♣r♦♣rétés ♠é♥qs ♦srés ♦rs ♥ ♦♠♣rss♦♥ ②♥♠q ❬❪ s ③♦♥s ♦♥t é♠♥t été♦srés ♦rs ①♣ér♥s ♥♥♦♥♥tt♦♥ ❬❪ ♦ sr s é♥t♦♥s②♥t s ♥ ♦♠♣rss♦♥ ♥♦♥ ②r♦sttq ♥t sqà P ❬❪s ③♦♥s ♦♠♠ ♦srés ♥ ♦♥r♥♥t ♣♥♥t q♥ rt♦♥ ♦♠ ♠tér t s ♥tr♣rétt♦♥s ♠♦r♣st♦♥q s♦♥t ♦♥♥és s♦♥t ♦♥trt♦rs ♥tr s rt♥s s s♥t sr ♣rés♥ ♦r t r♦♥ ♠♦r♣ ❬❪ trs ② ♦♥t rr ♦r 4 ♠♦r♣ ❬ ❪ ♣s s ①♣ér♥s ♠♣t srs ❱♦r t ❬❪ ♥♥s♥t ♣s ♣rés♥ ♣ss ♠♦r♣s
ét ♥ ♦rs ♣r♠s ♥tr trs r ♦♠♣♦rt♠♥t s♦strès t ♣rss♦♥ s ♣ss ♣réé♠♠♥t ♥tés ♦♠♠ sts ❬❪♦s ♦♥s tr♦é q ♣rés♥ ♥s ♦r ♥s î♥ ♥t ♥ ♠♦t♦♥ ♦♠♣♦rt♠♥t ♠é♥q rr ♦r ♥t ♠s été été ♣r♥t ♦s ♦♥s ♠♦♥tré q
4 t q ért sss ♣t s♦t♥r s ♣rss♦♥s ♦rr r s♥s tért♦♥ s ♣r♦♣rétés ♠é♥qs
♣rés♥ ♥ ♦r ♥s î♥ ♥t ♥ ♠♦t♦♥s ♣r♦♣rétés ♠é♥qs ♣r ♦r♠t♦♥ ♥ s♦♥ î♥ à① t♦♠s
♦tr ♥tr♣rétt♦♥ à ♥♦♥tr ♥ ♣♣r très ré♥t ❬❪s♦♥ q ♥ ♣rss♦♥ ♥① ♣ î♥ t ♣r♦♦q ♥♠♦t♦♥ s ♣r♦♣rétés ♠é♥qs ♥s 4 ♣r ♦s ♣r♠rs réstts ♦♥r♥♥t ♣♣t♦♥ ♥ é♦r♠t♦♥ ♥① é♠♥t♥t sréstts ét ♥ ♣ t♥q ♠♣♦rt♥ts ♦♥séq♥s ♥s♥♦tr s ♦♥ ♣t ♣r ♠s t♥ ♠é♥q s♦s ♣rss♦♥ ♥♠é♦r♥t s②♥tès ♠tér ♦rs q ♥s s ❬❪ ♠s t♥ ♠é♥q st ♥ ♣r♦♣rété ♥tr♥sèq ♠tér ♦séts ♥ ♦rs s♦♥t st♥és à rr ♣♦♥t
Pr♦t rr à qtr ♥s
♥s r ♦t é♥ér ♦♥♥é ♥ ♣r♠èr st♦♥ ♥ ♣r♠èr rt♦♥ rr été ♣ é♦qé t ♦♥r♥ s ♣r♦♣rétés étt ♦♥♠♥t ♠♣q ♦♠♣r♥r s ♠ts ♥s s ♣rés♦♥ sr é♥r t♦t
tr♦é ♥s s s q ♣s éqr ♥st ♣s ♣s♦sré ♥s s ♠ét① ts tt♥ ♦ ③r♦♥♠ ♦ ♥ ♦①② t
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q 2 t trs ♦♥t tr♦é ♥ ♣r♦è♠ ③r♦♥ ♥ ♥♥① ♥ trs tr♠s r♠♠ ♣s ♣rét ♣r st rr♦♥é s ér♥s é♥rs ♥tr s ♣ss s ♠tér① téss♦♥t très ♣tts
♦♥ ②♣♦tès é♦ré ♦rs tès ❱ r♥té éqt sr tt♥ ❬❪ ♦♥ré r ②♦♥ t ♥ ♦♥♦ t ♣rès ♥♦♠rss sss♦♥s s♥tqs rss ♣rs♦♥♥tés st q ♣rét♦♥ s ér♥s é♥r ♥ st ♠té ♣r ♥rtts ♣♣r♦①♠t♦♥s ♣♦t♥ts é♥ t ♦rrét♦♥ ♣♦r s étr♦♥s ♥ r à ♣♦r s ♠ét① ♥ ts♥t ♥ ♣♦t♥tsé sr é♥ ①t st ♥ tâ ♠ts q ♣rés♥t réstés t ♥ésst s é♦♣♣♠♥ts té♦rqs t s s♠t♦♥strès ♣réss tr st ♥ ♦rs ♥ ♦♦rt♦♥ ❱ r♥té♠♥t♥♥t rr ♣r♠♥♥t à ès
s ♣r♠rs réstts ♦t♥s ♥ é♠r ♠♦♥tr♥t q♥ ♦♥t♦♥♥ t②♣ P à s é♥ ①t ♥ ♣r♠t stsr ♣s α ♥ ♣r♦t étr s s ♣r♦t s ♠♦t♦♥s s♥ts éqt♦♥ étt é
s éts sr s ♦rrs s♦sst♦♥ ♦♥t s ♣♦rsr 4st très tsé ♥s ♥str t ♥s r♠♠♥t rté ♥t st♣rès ♠♥t t ♣r♠t s♦♥ tst♦♥ ♦r♥t ♥s s ♦ts ♦♣ t ♥ ♦s ♦♥s réé à ♦♠♥r s ♣r♦♣rétés rté t s♣r♦♥tté ♥ ♦♣♥t ♥ tr♦s rr ♦r ❬❪ ♦♥♥t étr s rrs ♦r s♦s ♥ r stté q♣r♠ttrt ♦♠♣r♥r stté ♠tér ♦rs ♦♣ ♦♣ ♣r♠ttr ♦♠♥r rté t s♣r♦♥tté ♦♠♥s♦♥ ♥s ♥ ♠ê♠ ♠tér s♣r♦♥tté t ♥ ♦rt rtést ①trê♠♠♥t ♥térss♥t t♥t ♣♦♥t ♦♥♠♥t q s ♣♣t♦♥s q ♣♦rr♥t ♥ é♦r ❯♥ ♣r♦t r♥♦♣♦♥s ♦♥r ❯♥rsté s é♠rré ♣♦r ♦♣r ♦r ②♣♦tès♥♣♣♦♥ ❬❪ ♦ rr ♦r ②♣♦tès r♥çs ❬❪ ♦t♦r♥t♣♦♥s r♦ r été rç ♥ t sr ♥♦r ♣♥♥t ♣srs ♠♦s ♥
tr ♣rt s éts sr s ♦rrs s♦sst♦♥ ♦♥t s ♦♥t♥r ♥ ♣r♠r ♦t ♦♠♣r♥r strtr t♦♠q srrs ♦r rs ♥ ♦r st tr♦r ♥ rtérstqs♣tr♦s♦♣q ① ♥s ♥ ♣♦♦r s éttr ♥s s é♥t♦♥s ❯♥ ♦♦rt♦♥ ♥s ♠qs ♥♥s tt P♦ès st ♥sés ♥s q ♦r ♣♦r s②♥tès sé♥t♦♥s ♥♦r srr t♦♥ ♥sttt ♦r trs ♥ ❲P trs ♥♦rtt♦♥s ♥tr ♥♦r ♣rt♦♥s r trt s ♣♥ t st
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P ❯ ❳
♥tst t♦♥ ♥sttt ♦ ♥ ♥ ♥♦♦② P♦② P ❯♥ ①è♠ ♥ st ♠é♦rr t♥ s♦s ♦ s♠tér① ♥ ② ♥tr♦s♥t s ♦♣♥ts ♣♣r♦♣rés
ét sr és①tt♦♥ étr♦♥q s♦sst♦♥ ♦♥t♥r t tt té♠tq s r♥♦rr ♥ ♣rtr râ rrt♠♥t♣r st ♥ st♦♥ ♥ t♦r tt té♠tq rç s♦t♥ ♣r ♥♥♠♥t P ♣r♦t r P ♥sttt ♥ér♦ t P②sqs ① ♦♥♥sés ❯♥rsté Prs ❱ P r t t♦ ♥♦ r♥♦
s éts ♦♥t ♦r êtr ét♥ rr s♠ t ① s♦♥ts♠tér① ♥térêt ♣s s ♣♦♥♦♥s r♥ ♦♥r♦♥ ♥tr♥♥♥t ♥s r①t♦♥ étr♦♥q t ♥s tr♥s♣♦rttr♠q t ♥♦s s♦♥s q ♥ s ♥① ♥♥♦étr♦♥q st s♦t♦♥ s ♣r♦è♠s é♠♥t ♦s ♦♥s ♥ ♣r♦t
ét♥r ♠ét♦ ♦♣ s ♣♦♥♦♥s ♦stqs ♥tr ③♦♥ té♦r ♦t êtr ét♥ r s ♣♦♥♦♥s s♦♥t♦♠♣♥és ♥ é♦r♠t♦♥ éstq
étr ♦♣ ♠♣ étrq réé ♣r s ♣♦♥♦♥s♦♣tqs ♥s s ♠tér① ♣♦rs ♦♣ rö
étr s ts ♥♥♦strtrt♦♥ sr ♦♣ étr♦♥♣♦♥♦♥ ♦r sr s s♣rrés① ♣s sr s s s♠♦♥trs
s ① r♥rs ♣♦♥ts s♦♥t ♦♥♠♥t① ♥s ♣r♦t st tr ♣rt ♣♣♦rt ♥ ♥♦ ♠ét♦ ♣♦r rs éé♠♥ts ♠tr ♦♣ étr♦♥♣♦♥♦♥ s ♦♥t♦♥s ❲♥♥r r ♣r♠ttr éérr ♥r P ❯♥rsté Prs ❱ P r s ♦♥t♦♥s ❲♥♥r s♦♥t ♠♥t♥♥ttsés é♠♥t ♥s qs♣rt ♥ ❲ q ♣r♠t♥ ♥ ♠♣♦rt♥t ♥s t♠♣s é①ét♦♥ s ♣r♦r♠♠s ❬ ❪
♥♥ s ①tt♦♥s étr♦♥qs s♦sst♦♥ t s s♣trs ♣rt é♥r étr♦♥q s♦♥t t♦♦rs ♥térêt ♣r ♥ ♥♦♠ét♦ ♥ ♦♣ ♣s q s♣♣ sr rés♦t♦♥ éqt♦♥ ♦ ❬❪ ♦♥ ♠♦ ♥térêt à ♦♥ tr♠ tt♦r♥tt♦♥ st q s♦t♦♥ éqt♦♥ ♦ st s ♥éqt♦♥ ♦③♠♥♥ ♦♠♣èt♠♥t q♥tq ♣r♠tt♥t ♠♦ésrà ♦s tr♥s♣♦rt étr♦♥q ♥♦ér♥t q ♥♦s ♦♥st♥ts ♦♣ étr♦♥♣♦♥♦♥ ♦♥tr♥t t tr♥s♣♦rt ♦ér♥t ❬❪ tr♣rt ♣t êtr ♦♣é s é♣♠♥ts t♦♠qs
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P ❯ ❳
♠t♦♥ à tr♠ st ♥ t ♣rér r①t♦♥ t♦♠q ♣r♦t ♣r ①tt♦♥ étr♦♥q ♦♥ r ♦rs ♥ sr♣t♦♥ ♥t♦s ts rrt♦♥ t s éts ♣r♦ts ♣r rrt♦♥
P♥ ♠♥srt
st♦♥ s♥t rr♦♣ ♦r♣ ♣r♦♣r à ♣tr ♥q♠♥t
Pr ♦♠♠♦té t é♠♥t ♣♦r ♥tér♥r t q ♥s st ♥♦tr♥ tr s ♣trs s♥ts s♦♥t ♥ ♥s
♣tr rr♦♣ ér♥ts ♥♦ts tr q ♣r♠tt♥t ♦♠♣r♥r ♦♠♠♥t st é ♦♥t♦♥ étrq ♥rs ♥ t q st ♥ ♦♥t♦♥ ♣rt étr♦♥q ♦sré ♦rérr s ♣r♥♣① réstts ♥②♥t tr♦é ♥ ♣rt ♥s ttértr ♠♥♠♥t ♦♠♣t ♦♥ s♦t st q s ♥♦ts s♦♥t ts♥s ♦r♠t♦♥ s ♥s s♥tqs q ♥rr
♣tr ♦♥♥ s ♣r♥♣① réstts té♦rqs sr ♦♥t♦♥ étrq ♥rs ♥s s ♦①②s ♥♦♥ ♦rréés s réstts ♣tr ♣r♠tt♥t ♦♠♣r♥r q t ♣♣r♦①♠t♦♥ ♣ss♠♣ ♣s ét♦r ♦ P s♦♥t s ♣♣r♦①♠t♦♥s éqts♣♦r r s s♣trs ♣rt é♥r étr♦♥q s étr♦♥s ♥ ♠é♦rt♦♥ rés♦t♦♥ s ①♣ér♥s ♣rt é♥rétr♦♥q st ♣r♠♦r ♣♦r étr ç♦♥ ♥ ré♣♦♥s s étr♦♥s ♥ ♣r ①♠♣ ♦rsq rst ♣rés♥t ér♥ts ♣ssrst♥s
♣tr ♦♥♥ s ♣r♥♣① réstts té♦rqs ♣♦r s s s♣trs s♦r♣t♦♥ ♦♣tq ② ♣rés♥t ♥ ♥♦ ♥tr♣rétt♦♥ tr sr ♥♦② ♣r♠tt♥t ♠♦ésr s ts ①t♦♥qs ♥té♦r ♦♥t♦♥♥ ♥sté é♣♥♥t t♠♣s
♥♥ ♣tr rés♠ s r♥rs s ♠♥és sr s rrs ♦r ♣tr st ♥ t ♥ rt r ért st à ♥ ♦♥ér♥♥té th ♥tr♥t♦♥ ②♠♣♦s♠ ♦♥ ♦r♦♥ ♦rs ♥ ttrs ♣t♠r ts ♠♥ ♣♥
♥♥ ♥s s ♥♥①s érs s ér♥ts ♣ss ③r♦♥♥♥① rér rè ♦r r♠ ♥♥① ♣rés ♠♦♥rr♠ t ♥♥① t ♠ st ♣t♦♥s ♥♥① ♥t ♦♥♥r ♦r♣ é♥ér q rr♦♣ ♣trs ts s♥ts
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P ❯ ❳
♦r♣ ♣tr
❬❪ ❲ ♦♥ ♦ trs ♠str② ❲♦r ♥t Ps♥♦ ♥♣♦r
❬❪ ♦♥s ♥ ♥♥rss♦♥ ♥st② ♥t♦♥ ♦r♠s♠ ts ♣♣t♦♥s ♥ ♣r♦s♣ts s ♦ ♦r♥ P②ss
❬❪ P ♥♥♦③③ r♦♥♦ P P♦♥ ♥ r♦♥ ♥t♦ t♦♥♦ ♣♦♥♦♥ s♣rs♦♥s ♥ s♠♦♥t♦rs P②s
❬❪ r ❩r♦ r♦♥♦ ♦ ♥ ♦♥ P♦♥♦♥s♦t♥♥ ♥ s♣r♦♥tt② ♥ tr♠ ♥r ♣rssr P②s
tt
❬❪ st ❱st ♥ ❱ ②tr ♥t♦ ♠t♦ ♦r t tr♦♥♣♦♥♦♥ sttr♥ t♠s ♥ s♠♦♥t♦rs ♣♣t♦♥ t♦ s ♥ PP②s ttrs t ② t ♠r♥ P②s♦t② ♦r t ♥r② ss ♦ ❱rt ♦r♥ ♦ ❯trst ♥tt♣trst♦r
❬❪ tt♦ ♥r t ❱st ♥ r♥s♦ r ♣r♦♥tt②r♦♠ ♦♣♥ ♦r♦♥ ♦sr P②s
❬❪ ♥ ♥ ❯ r♦ss ♥st②♥t♦♥ t♦r② ♦r t♠♣♥♥ts②st♠s P②s tt
❬❪ ❯ r♦ss ♥ ❲ ♦♥ ♦ ♥st②♥t♦♥ t♦r② ♦ rq♥②♣♥♥t ♥r rs♣♦♥s P②s tt
❬❪ ♥ ♦tt r♥s♦ ♦tt t ❱st ❱r♦ ♥♦ ♥♥ ♥srst♥ ❲ssr ♥ ♦ ♦♥♥ ♥ ♦♦♦ ♦ ♥ ❲ ♦② ♦♥r♥ ♦♥trt♦♥ t♦ t ①♥♦rrt♦♥r♥ ♦ t♠♣♥♥t ♥st② ♥t♦♥ t♦r② P②s
❬❪ ♥ ♥♥ ♥ ♦ tr♦♥ ①tt♦♥s ♥st② ♥t♦♥ rss ♠♥② ♦② r♥s ♥t♦♥s ♣♣r♦s ♦ P②s
❬❪ ❲ ♦② ütr ♥ ♠ ♥r② ♦♣rt♦rs ♥①♥♦rrt♦♥ ♣♦t♥ts ♥ s♠♦♥t♦rs P②s
❬❪ ♥♥♦♦ ütr ♥ ♠ t♦♥ ♦ t ♦♥s♠♣♦t♥t ♥ ts s♦♥t♥t② ♦r ♠♦s♠♦♥t♦r P②s
❬❪ ♥ ♥ ♥qst ♦ stt ♣②ss ♦♠ ♣ ♠ Prss ❨♦r
❬❪ ♥ P②s
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P ❯ ❳
❬❪ rt ♥♥ ♦ ♥ ♥ P②s tt
❬❪ ♥t r② ♥ ♦♥ ♦r② ♦ ♦♣t s♦r♣t♦♥ ♥♠♦♥ ♥ s P②s
❬❪ ♥t r② ♥ ♦♥ ♣t s♦r♣t♦♥ ♦ ♥st♦rs♥ t tr♦♥♦ ♥trt♦♥ ♥ ♥t♦ t♦♥ P②s ttrs
❬❪ ♦♥ ♥ ♦ tr♦♥♦ ①tt♦♥s ♥ s♠♦♥t♦rs ♥♥st♦rs P②s tt
❬❪ ❱st ♥♥ ❱ ♥♦ P tts♥r ♥ ♦r② ♦ ts ♥ t ♥s♦tr♦♣② ♦ t tr♦♥ ♥r② ♦ss s♣tr♠ ♦tt♥♠ ♦① 2 P②s tt
❬❪ r♥♦♣♦♦s ♥♥ ♦ ♥ ❱st ♣t ♥ ♦sss♣tr ♦ r♦♥ ♥♥♦ts ♣♦r③t♦♥ ts ♥ ♥trt ♥trt♦♥s P②s tt
❬❪ ♦tt ❱st ♥♥ ❱ ♥♦ ♥ ♥r♥ ♥t♦
t♦♥ ♦ t tr t♥s♦r ♦ ss s♣rtts P②s
tt
❬❪ ♦tt ♠♠♣r ♥ ♥t♦ t♦♥s ♦ ♦♣t ♣r♦♣rts ♥
s♠♦♥t♦r s♣rtts P tss ❯♥rstà P t
❬❪ ♦tt ❱st ♥♥ ❱ ♥♦ ♥ ♥r♥ ♥t♦ ♥s♠♠♣r tr rs♣♦♥s ♦ s♣rtts P②s
❬❪ s t ❱st P♣♣ r♥ r ②♥t ♥ ♥♥ tr♦♥ strtr ♥ tr♦♥ ♥r②♦ss s♣tr♦s♦♣② ♦❩r2 ③r♦♥ P②s
❬❪ ♦tt r♥ r♥♦♣♦♦s s ♦tt ❱ ♥♦ ❱st ♦ ♥ ♥♥ r♦♠ ♠♦s t♦ s♦s tr♦ ♦ ♦♥r♥ ♥trt♦♥s ♥t ♥t ♠
❬❪ s ♥ r♥ ❱r♥ r♥té t ❱st ♥ ♥♥ tr♦♥ ①tt♦♥s ♥t♦ t♦♥s ♦ tr♦♥ s♣tr ♥♣♣t♦♥ t♦ ③r♦♥ ❩r2 tt♥ 2 ♥ ♣r♦s ♦① 2 ♦♠♣
t
❬❪ ♥ r ♥ ♦t♥ tr♦♥ s♦♥sst♥t❲ ♣♣r♦①♠t♦♥ ♣♣t♦♥ t♦ ♥ ♥ P②s tt
❬❪ r♥ ♥ t ♦rrét♦♥ ♥s trtr tr♦♥q s
♦s ♠ à ①② r① ♣♣r♦①♠t♦♥ ❲ t àP tss ♦ P♦②t♥q Ps r♥
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P ❯ ❳
❬❪ ♥ r♥ t ❱st ♥♥ ③qr♦ r♦tt ♥ rrtt ①♥ ♥ ♦rrt♦♥ ts ♥ tr♦♥ ①tt♦♥ ♦ 2P②s tt ♥ rrr♥s r♥
❬❪ ♥ r♥ t ❱st ♥ ♥♥ t ♦ s♦♥sst♥②♦♥ qs♣rts ♥ s♦s P②s
❬❪ st ❱ ②tr ♥ ❱st ♥tr② sttr♥ ♥ s
♥t♦ t♦♥ ♦ t t ♣r♠trs ♦r ♦♥t r♦ s♠t♦♥s♣♣ P②ss
❬❪ st ❱ ②tr ♥ ❱st ♠♣ ♥t♦ st② ♦ Γ − X ♥tr② sttr♥ ♥ s ♥r ♣rssr P②s
❬❪ ❱st Pr♦♣rétés rt♦♥♥s ♦r α t rr ♦r 4 Ptss ❯♥rsté Prs ❱ r♥
❬❪ t ❱st Pr♦♣rétés rt♦♥♥s ♦r α t rr ♦r4 ès ♦t♦rt ❯♥rsté Prs ❱ r♥ t r♣♣♦rt
❬❪ ♥r ♠♥ ♥ ♦rt s♥♥ s♣rr ♠trs♥s
❬❪ ❱st r♦♥ ❩ér ss♦♥ P♦♥ r♥ ♥ r♠st tt②♥♠s ♦ α♦r♦♥ r♦♠ ♥t♦ t♦♥ ♥♠♥ sttr♥ ♥r ♣rssr P②s tt ♦
❬❪ ❱st r♦♥ ❩ér ss♦♥ P♦♥ r♠st ♥ r♥ tt ②♥♠s ♦ ♦sr α♦r♦♥ ♥r ♣rssr P②s tt
❬❪ ③③r ❱st ss♦♥ r♦♥ ♥ ♥r ♦rs♦ trtr ♥ rt♦♥ ♣r♦♣rts ♦ ♦sr 4 ♦r♦♥ r P②s
tt
❬❪ ③③r ❱st ss♦♥ r♦♥ ♥ ♥r ♦rs♦ rrt♠strtr ♥ rt♦♥ ♣r♦♣rts ♦ ♦sr 4 ♦r♦♥ r P②s tt
❬❪ ❱st ss♦♥ r♦♥ ♥ ♦rs♦ t♦♠ strtr ♥rt♦♥ ♣r♦♣rts ♦ ♦sr α♦r♦♥ ♥ 4 ♦r♦♥ r ♦♠♣
t
❬❪ ♥♥ ❲ ② ♥ ♦ P②s tt s réstts ♣♣r s♦♥t rr♦♥és ♦r♠ rr ♦r12 st qs ♥①st♥t s♦♥ ♥♦s s
❬❪ r ❱st ♥ Pr t♦♠ strtr ♦ ♦sr 4♦r♦♥ r r♦♠ rst♣r♥♣s ♥②ss ♦ s♣tr P②s
tt
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P ❯ ❳
❬❪ r♠ts ❱ ❱ tr③♥ ♦ ♥ ♠② ♥
❬❪ ♥♦ ♥ ❨ tt ❲ ss ❩ ❨ r② ♥ ❱ ♦♦③♥♦ tr
❬❪ ♥ ♥ ❩ strtr ♦t♦♥ ♦ ♦r♦♥ r ♥t♦ t♦♥s ♣♣ P②s tt
❬❪ s♦ r ♥ t②♠❨♦s P②s
❬❪ ③ r③ ♥ ❨♦s♦♥ P②s tt
❬❪ ❱st st ♥ tr♥♥② ♦r♦♥ rs r♦♠ rst ♣r♥♣s♦r♥ ♦ P②ss ♦♥r♥ ♣r♦♥s rt t rrs ♦ ♣s
❬❪ ❱♦r ❲ ♥rt ♥ s ②♥♠ ♦r ♦♦r♦♥ r ♣♣ P②s
❬❪ ♥ ❲ ② ♥ ♠r ♦♥ ♦③ ♠♦r♣③t♦♥ ♥ ♦r♦♥ r ♥
❬❪ ♦s s ♥ ❨ ❨♣ tr♥♥ ♦r♠t♦♥♦ r♦♥ ♥ ♦r♦♥ strs ♥ ♦r♦♥ r r♥ ②♥♠ ♥♥tt♦♥♣♣ P②ss ttrs
❬❪ ❳ ❨♥ ❩ ♥ ❩♥ ♦ ♥ ❨ ❩♥ ♦t♦ ② ♥ ❲ ♥ ♣rssr③t♦♥ ♠♦r♣③t♦♥ ♦ s♥r②st♦r♦♥ r P②s tt
❬❪ ❳ ❨♥ ❲ ♥ ❲ ♥ ♠♥ s♣tr♦s♦♣② ♦ ♣rssr♥ ♠♦r♣♦s ♦r♦♥ r ♣♣ P②s tt
❬❪ ❱ r♥té t té♦rq s ♣ss tt♥ P tss ♦ P♦②t♥q Ps r♥
❬❪ r r ♥ t②♠❨♦s ❱♥ ♦♥tr♦ ♦ αr♦♠♦r ♦r♦♥ ② tr♦♥ ♦♣♥ P②s ♦♥ ttr
❬❪ P ❯♠r t♥t ♥ r♦♥ r♥ t s③ ♣ t♥ ♥st②♥t♦♥ ♥ ♠♥②♦② ♣rtrt♦♥ t♦r② ♦♥♠t
❬❪ ♠♥♥ ♥ ❱♥rt P②s
❬❪ ❲r tt r ♥ r♦♥ ♥t ♣♣r♦t♦ t♠♣♥♥t ♥st②♥t♦♥ ♣rtrt♦♥ t♦r② ♦r ♦♣t s♣tr♦s♦♣② P②s tt
❬❪ r r ♥ r ♥st② ♥t♦♥ t♦r② ♦ t tr♦♥tt② ♦ ♠♦r s P②s tt
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♣tr
② q♥tt② t ♥rs ♦t ♠r♦s♦♣ tr♥t♦♥
♥ ts ♣tr sr t t♦rt ss s♥ ♥ t ♦♦♥ ♣trs
❲♥ sr♥ t rs♣♦♥s t♦ ♥ ①tt♦♥ t ♣♣ ♠r♦s♦♣ ♥ t t♦t ♦♥ t ♠r♦s♦♣ s r ♥♦t q ♠r♦s♦♣ r♣② rs ♦r t ♥t ♥ t st♥r ♥t♦♥ ♦t tr ♦♥st♥t ǫ t tstr rs♣♦♥s t ♠r♦s♦♣ ♦♥ssts ♦ t ♣♣ ♣♦t♥t δVbare ♥ ♦ t rtr ♣♦t♥tδVHartree tt ♦♠s r♦♠ t r ♥st② ♥ ② t ①tr♥ rtr ♥s t s♦ ♦ s tr♠♥s t♣♦r③t♦♥ ♦ t tr♦♥ s②st♠ ❬❪
♥st s♥ t tr♦♥tr♦♥ tr ♦♥st♥t ǫ ♥ t♦♦♥ st♦♥s ①♣t t ♥ ♦♥ ♥ t ♠r♦s♦♣ s t♦♥♠ t ♣♦t♥t ♦♥t♥s ♥ t♦♥ tr♠ t rs♣t t♦ ♦ ♥t♦♥ t ①♥ ♥ ♦rrt♦♥ ♣♦t♥t
♥r rs♣♦♥s ②s t♦ rst ♦rr t rt♦♥ t♥ t ①tr♥♣♣ ♣rtr♥ ♣♦t♥t δVbare ♥ t t♦t ♣♦t♥t δV t t ♠r♦s♦♣ s ♥rs ♦ t ♠r♦s♦♣ tr ♥t♦♥ ǫ−1 s♦♥♦♥ s t sr♥♥ ♥t♦♥ s t ② q♥tt② ♥♥ δVbare ♥ δV❬❪
δV (r, r′, t) =
∫
ǫ−1(r, r′, t)δVbare(r′, t)dr′.
♥ t ♦♦♥ st♦♥s st♣ ② st♣ r qt♦♥ ♥ s♦♦ t ♥rs tr ♥t♦♥ s t ♥ t ♦♠♣tr ♣r♦r♠ s ♥ s t♦ ♦t♥ t rsts ♦ t ♥①t ♣trs
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P ❨ ❯❨ ❱
P ❯
r♦♥ stt
t H (0) t ♣r♦ ♥ s♣ ♥♣rtr ♠t♦♥♥ ♥
H (0)|ψ(0)nk >= ε
(0)nk |ψ
(0)nk >,
r ε(0) ♥ ψ(0) r ♥s ♥ ♥t♦rs ♦ H (0) tt ♥ ♥① n ♥ t r♦♥ ③♦♥ t♦r k
♠t♦♥♥ ♦ qt♦♥ ♥ sr ♠♥②♦ s②st♠♦r s♠♣② ♥ ♠t♦♥♥ ♦r s♥ ♣rt ♥ t ttr s ♦♥♦t♥ ①♣rsss H (0) ♥ t r♠♦r ♦ ♥st② ♥t♦♥ t♦r② ❬ ❪r t ♦♥ssts rs♣t② ♦ t ♥t ♦♣rt♦r t tr♦♥♦♥ ♣♦t♥t VIe ♥ t s♦♥sst♥t ♣♦t♥t
H (0) =−~
2
2m∇2 + VIe + Vscf ,
r ~ s P♥s ♦♥st♥t ② π ♥ m t tr♦♥ ♠ssVIe s ♦t♥ trt ♥ t ♣s♦♣♦t♥t ♣♣r♦①♠t♦♥ ❬❪ ♥ Vscf
♦♥ssts ♦ t ss tr♦stt ♣♦t♥t VHartree ♥ t q♥t♠ ①♥ ♥ ♦rrt♦♥ ♣♦t♥t Vxc
♥ t s♥ ♦ ♥② ♣rtrt♦♥ t t♠ ♣r♦♣t♦♥ ♦ t tr♦♥ ♥t♦♥ s ♥ ②
|φ(0)nk (t) >= e−iε
(0)nk
t~ |ψ
(0)nk > .
♥ t st ♦ |φ(0)(t) > ♦r♠s ss ♦ stt♦♥r② stts ♦r t ♥♣rtr s②st♠ ❬❪
t ♦ ♣rtrt♦♥
tr♦♥ ①tt♦♥s r t♠ ♣♥♥t ♣r♦sss ♦♦♥ ❬❪t♦tr t t ♥tr♦t♦♥ ♦ t♠ ♣♥♥t ♣rtrt♦♥ δV (t)♦♥srt♦♥ ♦ t♦t ♥r② rs ♦♥ ♥ tr r ♥♦ ♦♥r ♥② stt♦♥r② stts s②st♠ ♥ ♠ tr♥st♦♥ r♦♠ t ♥t stt♦♥r②stt |ψ
(0)nk > t♦ ♥② ♦tr stt
tr♥st♦♥ ♣r♦t② ♠t t ♥ sr ss ❬❪ ♦r♥st♥ t ♣rtrt♦♥ s ♣♣ t② ♥ t ♥t stt|ψ
(0)nk > s ♥♦♥ ♥rt stt♦♥r② stt t s②st♠ ♠② r♠♥ ♥ ts
stt ♥♦tr ①tr♠ s s t ♣♣t♦♥ ♦ δV (t) ♥ r② s♦rtt♠ t s②st♠ s ♥♦ t♠ t♦ ♥ ♥ ♠t st② ♥ t stt |ψ(0)
nk > ♦r s ♥♦ ♦♥r stt♦♥r② stt ♦ t ♥ ♠t♦♥♥
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P ❨ ❯❨ ❱ P ❯
t s δV (r, t) t t♦t t♠ ♣♥♥t ♣rtrt♦♥ ♦♥sst♥♦ t ①tr♥ ♣♣ ♣♦t♥t δVbare ♣s t tr♦♥ rs♣♦♥s δVscf ❲t♥ t ♥st② ♥t♦♥ t♦r② t ttr ♦♥ssts ♦ t rt♦♥ ♦t tr♦stt ♣♦t♥t δVHartree ♣s t rt♦♥ ♦ t ①♥ ♥♦rrt♦♥ ♣♦t♥t δVxc
δV (r, t) = δVbare(r, t) + δVHartree(r, t) + δVxc(r, t).
t♦t ♠t♦♥♥ s H(t) = H (0) + δV (t) ♥ t♦ t t♣r♦♣rts ♦ t ①t stt ♦♥ s t♦ s♦ rö♥rs qt♦♥
i~d |ψ(t)
dt= H(t)|ψ(t) > .
♥ ♦♥ s t♦ t t ♥t♦♥s ♥ t ss ♦ t stt♦♥r② stts ♦ t ♥♣rtr s②st♠ ♥ ② q ❬❪
|ψ(t) >=∑
nk
anke−iε
(0)nk
t~ |ψ
(0)nk > .
♠ ♣♥♥t ♣rtrt♦♥ t♦r②
♣♥ t ank ♦♥ts t♦ rst♦rr t ♥t♦♥ rs
|ψ(t) >=∑
nk
a(0)nk e−iε
(0)nk
t~ |ψ
(0)nk > +
∑
nk
a(1)nk (t)e−iε
(0)nk
t~ |ψ
(0)nk > + ...
s♦t♦♥ ♦ q t♦ ③r♦th ♦rr rsts ♥ t ③r♦t♦rr a(0)nk
♥ t♠♥♣♥♥t
♥ s t t q t♦ rst♦rr
i~∑
nk
da(1)nk (t)
dte−iε
(0)nk
t~ |ψ
(0)nk >=
∑
nk
a(0)nk e−iε
(0)nk
t~ δV (t)|ψ
(0)nk > .
Pr♦t♥ ♦♥t♦ < ψ(0)mk′| ♦♥ ♦t♥s
a(1)mk′(t) =
−i
~
∑
nk
a(0)nk
∫ t
t0=0
e−i(ε(0)nk
−ε(0)
mk′) τ
~ < ψ(0)mk′|δV (τ)|ψ
(0)nk > dτ .
❲ ♥♦ ♦♠♣t t ♥tr ♥ q ♦r ♥ ♦r♠ ♦ t♣rtr♥ ♣♦t♥t
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P ❨ ❯❨ ❱
P ❯
♦t♦♥ ♦r t♠ ♣r♦ ♣rtrt♦♥
♦♦♥ ❬❪ t s s♣♣♦s tt t ♣♣ ♣rtrt♦♥ s r♠t♥♥ t♠ ♣r♦
δVbare(r, t) = (δVbareeiωt + c .c .)eαt ,
r t tr♠ eαt s ♥rst♦♦ t r ♣♦st ♥ s♠ α r♥ts ♥ t st ♦♥ ♦ t ♣rtrt♦♥ rst tr♠ ♦♥t rt♥ s st♥s ♦r ♣♦t♦♥ ♠ss♦♥ ♥ c .c . st♥s ♦r t♦♠♣① ♦♥t tr♠ ♣♦t♦♥ s♦r♣t♦♥
t s ♠♦r♦r s♣♣♦s tt t ♣♣ ♣rtrt♦♥ s ♥ s♣s♦② r②♥ ①tr♥ ♣♦t♥t ❬❪
δVbare(r, t) = δVbare(q, r,ω)e−i(q.r−ωt)eαt + c .c .,
r t ♥t ♦ t ♣rtrt♦♥ s 2π/|q| ♥ t t♦r♥t |q| s s♠ ♦♠♣r t♦ t r♣r♦ tt t♦r ♥t |G|
t♥ ♠♠♦r② ts t t♦t ♣♦t♥t s t♦ rst ♦rr s♦ t♠♣r♦
δV (t) = (Fe iωt + F †e−iωt)eαt .
❲t F = F q = F 0(q, r,ω)e−i(q.r) q rs
δV (r, t) = [F 0(q, r,ω)e−i(q.r−ωt) + F 0 †(q, r,ω)e i(q.r−ωt)]eαt .
♥ ①♣rsss F 0 ♥ tr♠s ♦ ts ♦rr ♦♠♣♦♥♥ts ♥ t s♣s t♦ r ♥ t
X (r, t) =
∫
dω∑
G
X (G,ω)e−iG.re iωt
s ♦ ♦ s t t♦t ♣♦t♥t ♥ ♥r ♦♥t♥ r♣②r②♥ tr♠s t t♦r q + G
δV (r, t) =∑
G
[F 0(q,ω,G)e−i(q+G).r+iωt) + F 0 ∗(q,ω,G)e i(q+G).r−iωt)]eαt .
t s
Fqmk′nk(t) = (e iωt < ψ
(0)mk′|F
q|ψ(0)nk >
ε(0)nk − ε
(0)mk′ − ~ω − i~α
+ e−iωt < ψ(0)mk′|F
q †|ψ(0)nk >
ε(0)nk − ε
(0)mk′ + ~ω − i~α
)eαt .
♦t♦♥ ♦ q ②s
aq (1)mk′ (t) =
∑
nk
a(0)nk [e−i(ε
(0)nk
−ε(0)
mk′) t
~ Fqmk′nk(t) −Fq
mk′nk(t0 = 0)],
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P ❨ ❯❨ ❱ P ❯
♥ ts
|ψq(t) >=∑
nk
a(0)nk e−iε
(0)nk
t~ |ψ
(0)nk > +
∑
nkmk′
a(0)mk′(t)[e
−iε(0)
mk′t~ Fq
nkmk′(t) − e−iε(0)nk
t~ Fq
nkmk′(t0 = 0)]|ψ(0)nk
❩r♦th ♦rr ♥st② ♠tr①
s♥ ♣rt ♥st② ♠tr① rs
ρq(t) = |ψq(t) >< ψq(t)|.
♦ ③r♦th ♦rr ♦♥ s t |ψq(t) > r♦♠ q
ρ(0) =∑
nkmk′
a(0) ∗mk′ a
(0)nk e−i(ε
(0)nk
−ε(0)
mk′) t
~ |ψ(0)nk >< ψ
(0)mk′|,
t ❬❪
ρ(0)|ψ(0)pk” >= f (0)(ε
(0)pk”)|ψ
(0)pk” >=
∑
nk
a(0) ∗pk” a
(0)nk e−i(ε
(0)nk
−ε(0)pk”)
t~ |ψ
(0)nk >,
r f (0)(ε(0)pk”) st♥s ♦r t ♦♣t♦♥ ♥♠r ♥ s
< ψ(0)lk1|ρ(0)|ψ
(0)pk2
>= f (0)(ε(0)pk2
)δlk1;pk2,
♦♥ ts t r② ♠♣♦rt♥t rt♦♥
a(0) ∗pk2
a(0)lk1
e−i(ε
(0)lk1
−ε(0)pk2
) t~ = f (0)(ε
(0)pk2
)δlk1;pk2.
q s ♥ t ♥①t st♦♥
rst ♦rr ♥st② ♠tr①
♥ ♦t♥s t rst ♦rr ♥st② ♠tr① r♦♠ qs ♥
ρq (1) =∑
nkmk′pk”
a(0) ∗mk′ a
(0)pk”[e
−i(ε(0)pk”−ε
(0)
mk′) t
~ Fqnkpk”(t)−
e−i(ε(0)nk
−ε(0)
mk′) t
~ Fqnkpk”(t0 = 0)]|ψ
(0)nk >< ψ
(0)mk′| + c .c .
♠tr① ♠♥ts t♥ r s♥ q
< ψ(0)lk1|ρq (1)|ψ
(0)pk2
> =
f (0)(ε(0)pk2
)[Fqlk1pk2
(t) − e−i(ε
(0)lk1
−ε(0)pk2
) t~ Fq
lk1pk2(t0 = 0)]+
f (0)(ε(0)lk1
)[Fq ∗pk2lk1
(t) − e−i(ε
(0)lk1
−ε(0)pk2
) t~ Fq
pk2lk1(t0 = 0)].
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P ❨ ❯❨ ❱
P ❯
♠♠r♥ q δV q(r, t) = (F qe iωt + F q †e−iωt)eαt t ♠tr①♠♥ts ♥ tr♠s ♦ F q ♥ F q † ♦♣rt♦rs r
< ψ(0)lk1|ρq(1)|ψ
(0)pk2
>= eαt Iq0 +
eαte iωt [f (0)(ε(0)pk2
) − f (0)(ε(0)lk1
)]< ψ
(0)lk1|F q|ψ
(0)pk2
>
ε(0)pk2
− ε(0)lk1
− ~ω − i~α+
eαte−iωt [f (0)(ε(0)pk2
) − f (0)(ε(0)lk1
)]< ψ
(0)lk1|F q †|ψ
(0)pk2
>
ε(0)pk2
− ε(0)lk1
+ ~ω − i~α.
♥ t ♣r♦s qt♦♥ t Iq0 tr♠ ♥ ♦s♥ s tt t ♥t
t♠ t0 rst ♦rr tr♠s ♥ t ♥t♦♥ ♥ ♥ t ♥st② ♥saq (1)nk (t0 = 0) = 0 ♥ ρq (1)(t0 = 0) = 0
Iq0 = −e
−i(ε(0)lk1
−ε(0)pk2
) t~ [f (0)(ε
(0)pk2
) − f (0)(ε(0)lk1
)]
(< ψ
(0)lk1|F q|ψ
(0)pk2
>
ε(0)pk2
− ε(0)lk1
− ~ω − i~α+
< ψ(0)lk1|F q †|ψ
(0)pk2
>
ε(0)pk2
− ε(0)lk1
+ ~ω − i~α).
tr♥t② ♦♥ ss t t st t t0 = −∞ eαt → 0♥ t♥ t ♥t ♦♥t♦♥s ①♣rss ② I
q0 ♥s
♠tr① ♠♥ts ♦ t rst ♦rr ♥st② ♠tr① ♥ q r② q♥tts t♦ ♦t♥ t tr♦♥ rs♣♦♥s ♥t♦♥s χ0 χ ♥ ǫ−1 ♦♦t♥ t♠ ♥r③ rö♥rs qt♦♥ ♥ t ♥♠♥r s♣♦♥s ♥ t♦ ts t♦rt r♠♦r
♦♥srt♦♥ ♦ r②st ♠♦♠♥t♠
♥ ♦ qt♦♥s ♠tr① ♠♥ts ♦ t ♦r♠ < ψ(0)lk1|F q|ψ
(0)pk2
> r∑
G
F 0(G,q,ω)∑
G1
ψ(0)pk2
(G1)∑
G2
ψ(0) ∗lk1
(G2)
∫
dre−i(q+G).re−i(k1+G1).re i(k2+G2).r.
♥tr ②s t t ♥t♦♥
δ(k2 + G2 − k1 − G1 − q − G),
♥ ts t ♦r t ♦♥srt♦♥ ♦ t r②st ♠♦♠♥t♠
k2 = k1 + q + G,
r G s r♣r♦ tt t♦r ❯♠♣♣ ♣r♦sss r ♥♥ t s♠ G 6= 0 ② rs t♦ ♦ ♦rrt♦♥s ❬❪
r♦♠ ♥♦ ♦♥ k2 s ss♠ t♦ sts② t ♦ ♦♥srt♦♥ q
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P ❨ ❯❨ ❱ P ❯
♥ ♥ ♥ t tr♦♥ ♥st②
♥ ♥ ♥ t tr♦♥ ♥st② δnq(r, t) s t ♦♥ tr♠♦ t rst♦rr ♥st② ♠tr① ♥ rs
(−e)δnq(r, t) = Tr [ρq (1)(−e)δ(r − r)],
r r s t ♣♦st♦♥ ♦♣rt♦r ♥ −e s t r ♦ ♦♥ tr♦♥♦♣♥ ♣r♦s qt♦♥ ♦♥ t ss ♦ t ♥♣rtr ♥
t♦♥s ♦sr rt♦♥ ②s
δnq(r) =< r|ρq (1)|r >
=∑
lk1pk2
< r|ψ(0)lk1>< ψ
(0)lk1|ρq (1)|ψ
(0)pk2
>< ψ(0)pk2
|r >
=∑
lk1pk2
ψ(0) ∗pk2
(r) < ψ(0)lk1|ρq (1)|ψ
(0)pk2
> ψ(0)lk1
(r).
δnq s t ♦r♠
δnq(r, t) = δn0 + δnq(r)e iωteαt + δnq†(r)e−iωteαt ,
r δn0 rsδn0 =
∑
lk1pk2
ψ(0) ∗pk2
(r)ψ(0)lk1
(r)eαt I 0q ,
♥ ♥ss ♥ t t ♣♣r♦①♠t♦♥
♥ qt♦♥ δnq(r) rs
δnq(r) =∑
lk1pk2
ψ(0) ∗pk2
(r)ψ(0)lk1
(r)[f (0)(ε(0)pk2
) − f (0)(ε(0)lk1
)]< ψ
(0)lk1|F q|ψ
(0)pk2
>
ε(0)pk2
− ε(0)lk1
− ~ω − i~α
P♦ss♦♥s qt♦♥
♥ t♦♥ P♦ss♦♥s qt♦♥ ♥s t tr♦♥ ♥st② t♦ t ♣♥♦ t tr♦stt ♣♦t♥t ❬❪ ①♣♥ t♦ t rst ♦rr t ♥s t♥ ♥ ♥ t tr♦♥ ♥st② t♦ t ♣♥ ♦ t ♥ ♥t tr♦stt ♣♦t♥t δV q
Hartree
∇2r δV
qHartree(r, t) = −4πe2δnq(r, t).
♦rr tr♥s♦r♠t♦♥ ♦ t rtr ♣♦t♥t ②s t ♦♦♥ qt♦♥ r♠♠r♥ tt t ♥ ♣♦t♥t ♦♥t♥s tr♠s t t♦r q + G ♦ s
δV qHartree = (F q
Hartreeeiωt + c .c)eαt ,
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P ❨ ❯❨ ❱
P ❯
FqHartree =
∑
G
F 0Hartree(q,G,ω)e−i(q+G).r,
♦rr tr♥s♦r♠ ♦ t qt♦♥ rs
δnq(r) =∑
G
δn(q + G,ω)e−i(q+G).r.
♥ t t ♣♣r♦①♠t♦♥ ♦♥ ♦t♥s t rt♦♥s♣ t♥t ♦rr ♦♠♣♦♥♥ts ♦ t rtr ♣♦t♥t ♥ ♦ t ♥st② s
F 0Hartree(q,G,ω)|q + G|2 = 4πe2 δn(q + G,ω).
♦r ♥② G ♦♠♣♦♥♥t ♦tr t♥ G = 0 ♦♥ s
δn(q + G,ω) = F 0Hartree(q,G,ω)
|q + G|2
4πe2.
♦r ♦♥ ♥t q = 0 ♣rtrt♦♥ ♥ G = 0 δn(q+G,ω) = 0♥ ♦tr ♦rs t ♥ ♥ ♥ ♥st② s ♥♦ ♠r♦s♦♣ ♦♠♣♦♥♥t t t♦t ♠r♦s♦♣ tr♦♥ ♥st② s ♣t ♦♥st♥t ♥ t①t stt
s♣♦♥s ♥t♦♥ χ0
♣♥ ♥ q t ♠tr① ♠♥ts ♦ ρq (1) t q ♥♥♥ I 0 ♥ χ0 s
δn(q + G,ω) =∑
G′
F 0(q,G′,ω)χ0(q + G,q + G′,ω)+
∑
G′
F 0 ∗(q,G′,ω)χ0(q + G,−q − G′,−ω)
+∑
G′
F 0(q,G′,ω)I 0(q + G,q + G′,ω) +∑
G′
F 0 ∗(q,G′,ω)I 0(q + G,−q − G′,−ω),
♦♥ ♦t♥s t I 0 tr♠ ♦s t ♥ss ♥ t t ♠tt0 = −∞
I 0(q + G,q + G′,ω) = −∑
lk1pk2
e−i(ε
(0)lk1
−ε(0)pk2
) t~ [f (0)(ε
(0)pk2
) − f (0)(ε(0)lk1
)]
< ψ(0)lk1|e i(q+G).r|ψ
(0)pk2
>< ψ(0)pk2
|e−i(q+G′).r|ψ(0)lk1>
ε(0)pk2
− ε(0)lk1
− ~ω − i~α.
tr♦♥ rs♣♦♥s ♥t♦♥ χ0 rs
![Page 36: Some ab initio tudies of the physical properties of materials](https://reader035.fdocuments.us/reader035/viewer/2022062601/62b4741960ed9c48274e4f03/html5/thumbnails/36.jpg)
P ❨ ❯❨ ❱ P ❯
χ0(q + G,q + G′,ω) =
∑
lk1pk2
[f (0)(ε(0)pk2
) − f (0)(ε(0)lk1
)]< ψ
(0)lk1|e i(q+G).r|ψ
(0)pk2
>< ψ(0)pk2
|e−i(q+G′).r|ψ(0)lk1>
ε(0)pk2
− ε(0)lk1
− ~ω − i~α.
tr♦♥ rs♣♦♥s ♥t♦♥ χ0 ♥s t rt♦♥ ♦ t tr♦♥r ♥st② t♦ t t♦t ♣♦t♥t tr♦ qt♦♥
s♣♦♥s ♥t♦♥ χ
♦ ♦t♥ t rt♦♥ t♥ t rt♦♥ ♦ t tr♦♥ r ♥st②♥ t ①tr♥ ♣♦t♥t ♦♥ ♦♦s ♦r t rs♣♦♥s ♥t♦♥ χ s s
δn(q + G,ω) =∑
G′
δVbare(q,G′,ω)χ(q + G,q + G′,ω)+
∑
G′
δV ∗bare(q,G′,ω)χ(q + G,−q − G′,−ω)
+∑
G′
δVbare(q,G′,ω)I (q + G,q + G′,ω) +∑
G′
δV ∗bare(q,G′,ω)I (q + G,−q − G′,−ω),
r ♦r♥ t♦ q t ①tr♥ ♣♦t♥t ♦s ♥♦t r② ♦r ♥t s♦ tt
δVbare(q,G,ω) = δ(G, 0)δVbare(q,ω).
♦rr tr♥s♦r♠♥ qt♦♥ ♦♥ s
F 0(q,G,ω) = δVbare(q,G,ω) + F 0Hartree(q,G,q,ω) + F 0
xc(q,G,ω),
r F 0Hartree s ♥ ♥ ♥ rt♦♥ ♦ t ①♥ ♥
♦rrt♦♥ ♣♦t♥t δVxc [δn(G)] s♦ ♣♥s ♦♥ t rt♦♥ ♦ t tr♦♥ ♥st②
♥ ♥ t s♠♣st ♣♣r♦①♠t♦♥ t ①♥ ♥ ♦rrt♦♥ ♣♦t♥t s t♥ t♦ ♦♦ t t♠ ♣♥♥ ♦ t ♥st② ♦r ♥st♥ t♥ t ♦ ♥st② ♣♣r♦①♠t♦♥ t ♥r③ r♥t♣♣r♦①♠t♦♥❬❪ ♦r t t ♦ ♥st② ♣♣r♦①♠t♦♥❬❪ t ①♥ ♥ ♦rrt♦♥ r♥ fxc s ♥ s
fxc(r, t) =δVxc(r, t)
δn(r′, t′)δ(r − r′)δ(t − t ′)
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P ❨ ❯❨ ❱
P ❯
s♦ ttδVxc(r, t) = fxc(r, t)δn(r, t).
♥ t ♦ s♠♣st ♣♣r♦①♠t♦♥s t r♥ s ♦s♥ t♦ ♦ ♥s♣ ♥ t♠
❯s♥ ♦ ♥t♦♥ t♦tr t q ♦♥ ♦t♥s tt
F 0(q,Gω) = δVbare(q,G,ω)+∑
G′
[4πe2
|q + G′|2δ(G′−G)+fxc(G
′−G)]δn(q+G′,ω).
❲ ♥♦ ♠ qs ♥ q ♥ t t ♣ ♦ q ♦t♥ t tr♠ ♥ t ♥t ♦♥t♦♥s ♥ss ♥ tt ♠t
I (q + G,q + G′,ω) = I 0(q + G,q + G′,ω)+
I 0(q + G,q + G′,ω)∑
G”
[4πe2
|q + G”|2δ(G′ − G”) + fxc(G
′ − G”)]χ(q + G′,q + G”,ω).
♦r t rs♣♦♥s ♥t♦♥ χ ♦♥ s t t ②s♦♥ qt♦♥ ❬❪
χ(q + G,q + G′,ω) = χ0(q + G,q + G′,ω)+
χ0(q + G,q + G′,ω)∑
G”
[4πe2
|q + G”|2δ(G′ − G”) + fxc(G
′ − G”)]χ(q + G′,q + G”,ω).
t♦♥ ♦ t rs♣♦♥s ♥t♦♥ χ rqrs ♦t t ♥♦ ♦t rs♣♦♥s ♥t♦♥ χ0 q ♥ ♦ t ①♥ ♥ ♦rrt♦♥r♥ fxc q
♥rs tr ♥t♦♥ ǫ−1
♦ ♦t♥ t rt♦♥ t♥ t ①tr♥ ♣♦t♥t δVbare ♥ t t♦t♣♦t♥t δV ♦♥ ss t t qs ♥ tt
δn(q + G,ω) =∑
G′
δVbare(G′,q,ω)χ(q + G,q + G′,ω),
s♦ tt qt♦♥ rs
F 0(q,G,ω)) =∑
G′
δVbare(q,G′,ω)δ(G − G′)+
∑
G”
[4πe2
|q + G”|2δ(G − G”) + fxc(G − G”)]χ(q + G”,q + G′,ω).
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P ❨ ❯❨ ❱ P ❯
♥rs tr ♥t♦♥ ǫ−1 s t ② q♥tt②
ǫ−1(q + G,q + G′,ω) = δ(G − G′)+∑
G”
[4πe2
|q + G”|2δ(G − G”) + fxc(G − G”)]χ(q + G”,q + G′,ω),
r q s t t♦r ♦r ♠♦♠♥t♠ tr♥sr t♥ t rst r♦♥③♦♥
ǫ−1 ♥s ♦♥ t ♠r♦s♦♣ s t ①tr♥ ♣♦t♥t t♦ t t♦t♣♦t♥t ♥s t r②st
F 0(q,G,ω) =∑
G′
δVbare(q,G′,ω)ǫ−1(q + G,q + G′,ω).
♥r rs♣♦♥s ♥ r s♣
♥ r s♣ qt♦♥s ♦ ♥r rs♣♦♥s t♦r② r ❬❪
δVtot(r,ω) =
∫
ǫ−1(r, r′,ω)Vbare(r′,ω)dr′.
δn(r,ω) =
∫
χ0(r, r′,ω)δVtot(r′,ω)dr′,
δn(r,ω) =
∫
χ(r, r′,ω)Vbare(r′,ω)dr′.
♥ ttr qt♦♥s ♠♠♦r② ts ♥ t♠ r ♦♠ttχ0 rs
χ0(r, r′,ω) =
∑
i ,j
(fi − fj)ψi(r)ψ
∗j (r)ψj(r
′)ψ∗i (r
′)
ωij − ω − i~α,
r i ♥ j r ♦ ♥①s ♦r t ♥ ♥♠r ♥ r②st t♦r ♥ ωij = εi − εj
r tr ♥t♦♥ ② tr♦♥
♥r②♦ss s♣tr♦s♦♣②
❲♥ r ♣rt ♦s t tr♦♥s ♥ t♦♠s t s r♥t② ♦r♥ t♦ ts ♠ss ts ♠ss s rr t♥ t tr♦♥ ♠ss ♦s♦♥ t tr♦♥s t♦ ♥r② ♦ss rs ♦s♦♥ tt♦♠s st t t♦♥ ♦ t trt♦r② t ♥♥t♣rt s ♥ tr♦♥ tr ♦t ♥st sttr♥ ♥ t♦♥
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P ❨ ❯❨ ❱
P ❯
♦ t tr♦♥ ❬❪
♥ t t②♣ ♦ ①♣r♠♥t ♦♥sr r tr♥s♠ss♦♥ tr♦♥ ♥r②♦ss s♣tr♦s♦♣② ♥r② ♠ s tr♥s♠tt tr♦ t♥ s♠♣♥ ①♣r♥s ♦t st sttr♥ qsst sttr♥ t ①tt♦♥ ♦ ♣♦♥♦♥ ♦r ♥st sttr♥ r t ♦♦♠ ♦ t♥♥t tr♦♥ ♥trts t t tr♦♥s ♥ t s♦ ♥ s rs t♦♦♥t♥ ♦t ①tt♦♥s t ♣s♠♦♥s ❬❪
♥♥t tr♦♥ s r② ♦♥r♥ ♦♦♠♥ ♥trt♦♥r♦♠ t ♥ tr♦♥s ♦ t s♦ s ♥trt♦♥ s rs♣♦♥s ♦rt ♥r② ♦ss ♥ s ♥ s♦♥ t♦ t♦♦ ♦♥r♥ t♦ trt♥ ♣rtrt♦♥ t♦r② ❬❪ r♦r t t♦rt s♠ ♦r sr♥t r♦♠ t♦s ♦r ♣♦t♦♠ss♦♥ ♦r s♦r♣t♦♥ s♣tr
♦r♥ t♦ r ❬❪ s t ♥trt♦♥ ♥t s r② ♦♥ ♥trt t s♦ s ♦♠♦♥♦s ♠♠ ♥ r② ♦♥ tr trt♠♥t♦ ts ♦♥r♥ ♥trt♦♥s ♥ tt s t ♣♦t♥t rt ② ttr♦♥ ♠♦♥ t ♦♥st♥t ♦t② v s ♥ ② P♦ss♦♥s qt♦♥t r♥ t qt♦♥ t tr ♥t♦♥ ǫqav ♦ t ♠♠♦♥sr s ♦♠♦♥♦s s ♥♦ ♥srt ♥ ①s qt♦♥ ♦r ttr s♣♠♥t ②♥ ♦r P♦ss♦♥s qt♦♥
ǫav (t,q)∇2r V (r, t) = −4πe2nq(r, t),
r n = eδ(r − vt) s t r ♦ t ♥♥t tr♦♥ ♥ t ♦qt♦♥ t tr ♥t♦♥ s ♥ r ♦r r t♦ t r② ♦♥r♥ ♥tr ♦ t ♥trt♦♥ s ♠r♦s♦♣ q♥tt② s ♦r t♠♣♥♥t ♥ ♣♥s ♦♥ t rt♦♥ ♥ ♠♦ ♦ t tr♥srr ♠♦♠♥t♠ q
♦ r♥t ♥st sttr♥ r♦ss st♦♥ ♦r st tr♦♥srs
∂2σ
∂Ω∂ω= D
4πǫ0(eπa0)2
ℑ(−1
qǫavq)
r ǫav = ǫav (ω = Eloss, q) s t ♦♥ ♠♥t ♦ t ♥rs ♠r♦s♦♣ tr ♥t♦♥ D s t s♠♣ t♥ss a0 s t ♦r rs♥ q s t sttr♥ t♦r ℑ(ǫ−1
av ) s t ♦ss ♥t♦♥ tt ♣♥s ♦♥ t ♦ss ♦ ♥t ♥r② Eloss ♦ t tr♥s♠tt tr♦♥ ♠
rt♦♥s♣ t♥ t r ♥t♦♥ ǫav (ω,q) ♥ t ♠r♦s♦♣ ♥t♦♥ s
ǫav (ω,q) =1
ǫ−1(q + G,q + G′,ω)G=G′=0
,
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P ❨ ❯❨ ❱ P ❯
r ǫ−1 s ♥♦ ♦r tstr rs♣♦♥s ♥ ② t st♥r ♥t♦♥ ♦ t ♠r♦s♦♣ tr ♥t♦♥
ǫ−1(q + G,q + G′,ω) = δ(G − G′) +4πe2
|q + G|2χ(q + G,q + G′,ω).
❲t rs♣t t♦ q ♦r t tr♦♥tr♦♥ tr ♥t♦♥♦♥② t tr♦stt ♣♦t♥t s ♥ ♥ ♥ q ♥ ts ♥ t♥ ♥t♦ ♦♥t tt ♥ s ♥ ①♣r♠♥t t ♣r♦ s ♥ tr♦♥ t r② ♥r② ♥ tt s ❱❱ tr♦♥tr♦♥①♥ ♥ ♥♦r ♥ r t t t ♦♦♠ r②st ♣♦t♥t♥ q ❬❪ tr♦♥tr♦♥ ①♥ s ♣♦ss ♦♥② ♥ t t♦tr♦♥s t s♠ ♥r② ❬❪
♥② ♦♥ ♥♦ts tt t tstr ♥t♦♥ ♦ q ♥ ttr♦♥tr♦♥ rs♣♦♥s ♦ q ♥ ♥♦t ♦♥ t r② ♥r②s t ①♥ ♥ ♦rrt♦♥ r♥ s ♥♦t rq♥②♣♥♥t ♥♥② t ♣♣r♦①♠t♦♥s
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♣tr
♦rt rsts tr♦♥♥r② ♦ss s♣tr
♥tr♣rtt♦♥ ♥ ♣rt♦♥ ② t t♦r② ♦ tr♦♥ s♣tr ♦t♥ ② t ①tt♦♥ ♦ ♥ tr♦♥s ♥ d ♠t ♦①s s r②♠♣♦rt♥t st ♦♠♥♥ ♦t t t② ♦ t ①t stt tt ♥ ♦ ♣r♦♣r② sr♥ d ♠♥ts ♥ t s♠t♦♥ s s ♦r♠ ts ♥ ♦♥ s ♣rt t ♣r♦♠ ♦ str♦♥ tr♦♥♦rrt♦♥ s t ♦s ♦ t ②♥♠ ♠♥ t♦r② ❬❪
♥ t ♦♦s ♦♥♥trt ♦♥ ❩r2 ♥ 2 ♠trs ♥ t d s r ♥r② ♠♣t② s ♠trs r ♠♣♦rt♥t ♦r ♥r♣♣t♦♥s ❩r♦♥ s sr ♥ t ♣♣♥①
♥ ts ♦①s ♥♦ str♦♥ tr♦♥ ♦rrt♦♥s r ①♣t t♦ ♦rt s ♥♦♥tss ♥ ♠♣♦rt♥t ts t♦ ♥rst♥ ♠♦♥ t t♦rt♠t♦s r ♥ ♣tr ♦ t♦r② s ♣♣r♦♣rt ♦r ts♦①s
♥ ts ♣tr ①♣♥ tr♦ s♦♠ ①♠♣s ♦ sr♣t♦♥ s ♥ssr② ♦r tr♦♥ ♥r② ♦ss s♣tr ♦r ts ♥ rsts♥ ♦♥ ♥ ♠② ♦r♣② ❬ ❪ s♦ ①♣♥ ♥ s♦♠ tt r♥t strtrs ♣♣r ♥ t r ♥ ♠♥r② ♣rts ♦ ttr ♥t♦♥ tt ts ♥ ♥r♣r♥ts r ss♥t♥ r♥t ♦r ♠♥② tr♥st♦♥ ♠t ♦①s
♥♦♠ ♣s ♣♣r♦①♠t♦♥
♦r②
s s♥ ♥ t ♣r♦s ♣tr t ♣♦r③t② ♦r ♥♣♥♥t ♣rtsχ0 ♥ qt♦♥ ♥ rtt♥ t♦ rst ♦rr ♥ ♣rtrt♦♥ s s♠♦r ♥♣♥♥t tr♥st♦♥s t♥ ♥♣rtr stts ❲ t ♥st②
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P ❯
❨ P
♥t♦♥ t♦r② s ♦r strt♥ ♣♦♥t ♦r t ♥♣rtr r♦♥ sttt ♥t♦♥s ψi ♥ s ♦ t tr♥st♦♥ ♥r② ωij = εi −εj ♥t r♥ t♥ t ♥s ♦r stts ψi ♥ ψj
χ0(q + G,q + G′,ω) =∑
i ,j
(fi − fj)< ψj |e
i(q+G).r|ψi >< ψi |e−i(q+G′).r′|ψj >
ωij − ω − i~α,
r t s♠ r♥s ♦r stts ♥ ♦♣t♦♥ ♥♠rs fi ,j rs♠r t♥ ♦r q t♦ ♥t②
♥ rst st♣ ♥t♥ ♦ ts ♥ ♣♣r♦①♠t♥ χ ② χ0t ♠r♦s♦♣ ♥rs tr ♥t♦♥ ♥ rt② t r♦♠χ0 s❬❪
ǫ−1RPA(q + G,q + G′,ω) = δ(G − G′) +
4πe2
|q + G|2χ0(q + G,q + G′,ω).
♦ qt♦♥ s t ♥♦♠ Ps ♣♣r♦①♠t♦♥ t♦t ♦ ts st♥r ♥t♦♥ ♦ t sr tr ♥t♦♥ s❬❪
ǫRPA(q + G,q + G′,ω) = δ(G − G′) −4πe2
|q + G|2χ0(q + G,q + G′,ω).
sts ♥ ③r♦♥
♥ r r♣♦rt t tr ♥t♦♥ ǫ ♦r ❩r2 t ♥t P t♦t ♦ s s ♥s t r ♣rt ǫ1 s s♦♥ ♥t t♦♣ ♣♥ t ♠♥r② ♣rt ǫ2 ♦rrs♣♦♥♥ t♦ t s♦r♣t♦♥ s♣tr♠ ♥ t ♥tr ♣♥ ♥ ♥ t ♦tt♦♠ ♣♥ s♦ t ♦ss ♥t♦♥♥ s −♠(ǫ−1) ♥ ts ♣♣r♦①♠t♦♥ s q t♦ ǫ2/(ǫ21 + ǫ22)r r♦♥s ♥ st♥s
rst② ♦sr tt t ♥ ①tt♦♥ r♦♥ ①t♥s ♣ t♦ ❱ r ♣rt ♦ ǫ s ♠♥② s ♦ ss ♦st♦r ❬❪t ♥ss r♦♠ ♣♦st t♦ ♥t r♦♥ ♥ ❱ ♥♦rrs♣♦♥♥② ǫ2 s♦s ♠①♠ ♦ s♦r♣t♦♥ t ts rq♥s r ♣rt ǫ1 ♥ss r♦♠ ♥t t♦ ♣♦st t ♥ t ❱ ♥♦ts♥ ♥ t r s ♦ t s♠ r♦♥♥ ♦ss ♥t♦♥ ♦♥sq♥t② s♦s t♦ ♣s t ts s ♦ t ♥r② ♥ t s♦r♣t♦♥s♣tr♠ ǫ2 t str♦♥ s♦r♣t♦♥ ♦ ❱ st♠s r♦♠ tr♥st♦♥s r♦♠t ♥ ♥ t♦ t g stts s♦r♣t♦♥ ♥ ①t♥♥ ②♦♥ ♣ t♦ ❱ s ss♦t t tr♥st♦♥s r♦♠ t ♥ ♥ t♦ 2g
stts ♥ t ♦♥t♦♥ ♥ s♠ s♦r♣t♦♥ ♣ t ❱ ♦♠sr♦♠ ♥ ♥ t♦ 1g ①tt♦♥s ♦r ts r ♥ ♥ ❬❪
♦♥② s tt ♦ ❱ ♦rrs♣♦♥♥ t♦ t ③r♦♥♠ 4p
①tt♦♥s ǫ1 s♦ s s tr♣ ss ♦st♦r t ♥ss r♦♠
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P ❯ ❨ P
♣♦st t♦ ♥t t ♥ ❱ ♦♥sq♥t② ǫ2 s♦s str♦♥s♦r♣t♦♥ t ❱ ♥ t♥ ♥ ❱ Ps r ♦sr ♥t ♦ss ♥t♦♥ ♥ ǫ1 ♥ss r♦♠ ♥t t♦ ♣♦st t ♥ ❱
r② ♦ t♦ ♣♦♥t ♦t tt t r♦♥ t♥ t ♥♣s♠♦♥s ♥ t 4p ♣s♠♦♥s r♦♥ t♦ ❱ ♥♥♦t ♥tr♣rt ♥ tr♠s ♦ ss ♦st♦rs t♥ ♥ ❱ ǫ1 ♥ ǫ2 r♦♠♥t ② ♥r trs ♥rs♥ ♦r ǫ1 ♥ rs♥ ♦r ǫ2 ♦rrs♣♦♥♥ ♦ss ♥t♦♥ ①ts r♦♥ ♣ t ❱ ss♥ ♥♦t t♦ ♣s♠♦♥s t t♦ ♦tr ♦r♠s ♦ ♦t ①tt♦♥s ♣s♠♦♥ s ♥ ② ♥s♥ r ♣rt ♦ t tr ♥t♦♥ ♥ ♠♥♠♠ ♦ t ♠♥r② ♣rt s ♥♦t t s ♦r ts ♣ ♥r ♦r ♦r ǫ1 ♥ ǫ2 s r② ♥ ♦sr ♥ t t♦rt ♦ rt 2 ❬❪ ♦r ♥♦t tt r t ♦st♦rs ss♦t t t 2s tr♦♥s r ♣rs♥t ǫ1 ♥ss r♦♠ ♣♦st t♦♥t r♦♥ ❱ ♥ ♥ss r♦♠ ♥t t♦ ♣♦st r♦♥ ❱ t r ♦r♠ ② t ♥r ♦r ♦ ǫ1 ♥ ǫ2 s ts ♦ t ♠♥ ♣ t ❱
♦ ts
♦r②
♦ ♥ ♦ s ♦♥ rst s t♦ t χ0 s ♥ t ♣r♦s st♦♥ ♥ t♥ t♦ t t rs♣♦♥s ♥t♦♥ χ q ♥rs tr ♥t♦♥ s t♥ ♦♠♣t r♦♠ q ♥ t ♠r♦s♦♣ ♠t s t♥ q
② ♥tr♦ ♦ ts s t♦ t ♥t♦ ♦♥t t ♦♦♥ ♠♥ts G = G′ ♦ t r♦s ♠trs χ0 χ ♥ ǫ−1 ♥ ♠tr①♥rs♦♥s ♦ qs ♥ s t♦♥ ♦♠♣♦♥♥ts ♦rrs♣♦♥t♦ ♦♣♥ t♥ ♦♥t♥ ♥ tr♥srs str♥s ❬❪
♠①♥ ♦ t ♦r♠r② ♥♣♥♥t tr♥st♦♥s ♦ tr♥t② sr ② t ♦♥♣t ♦ ♥ tr♦♥♦ ①♥ ❬❪
sts
♦ ts rt t ♥♦♠♦♥t② ♦ t ♠tr ♥ t rs♣♦♥s ♥t♦♥ ♥ t t ♦ ♠①♥ t ♦r♠r② ♥♣♥♥ttr♥st♦♥s ♥ ❩r2 p stts r t♦♠ t t② t ♥trt♦♥ t ♥♦r♥ t♦♠s ♥ ♦ ②r③ st②❬❪ ♦ ts r ♦♥ t♦ str♦♥ t ts s ♦ t ♥r② ② ♥tr♦
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P ❯
❨ P
r ❩r2 t♦rt tr ♥t♦♥ t q0 ♦♣ ♣♥ r ♣rt ♥tr ♠♥r② ♣rt ♦tt♦♠ ♣♥ t ♦ss ♥t♦♥ ♦♥s P t ♦ s s ♥ P t♦t ♦ s r ♥ ♠♥r② ♣rts ♦ t tr ♥t♦♥ rs♣ t ♦ss ♥t♦♥r r♦♥ t ♦r♥t③♥ ♦ ❲ t t ♠①♠♠♦ ❱ rs♣ ❱ r♦♠ ❬❪ r♦♠ ❬❪
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P ❯ ❨ P
r 2 ♦ss ♥t♦♥ ♥ t r♦♥ ♦ t p tr♦♥ ①tt♦♥s t ♣♥ t♦t ♦ s t ♣♥s t ♦ s❯♥♣s
t srr♦♥♥s ♥ ♥ ♦trs ♦ ♥r♦♥♠♥t
s s♦♥ ♥ t s♦ ♥s ♦ ❩r2 ♥ ♥ t ts♦♥ ǫ1 ♥ ǫ2 r ♦ ❱ t r ♥r② ♦r t② rst② ♠♦② t tr♣ p ♣s♠♦♥s ♦t ♥ ♥s♣ ♥ ♣ ♣♦st♦♥♥ ❩r2 t ♣ ♦r♠r② t ❱ s st t♦rs r ♥r② ② ❱ t ♠♥ ♣ s st ② s ♠ s ❱
♦ t ♥ ♥s♦tr♦♣②
r ts ♦♥ ①tt♦♥s r♦♠ qst♦♠ t♦ ♦ s s♠r② ♥ ♦♥ ♦r t p ①tt♦♥s ♥ rt 2 ❬❪ t♦rt s ttr♦♥ r②st r t s②♠♠tr② s rs t rs♣tt♦ t ♦♥ t s♣tr ♦r ♠♦♠♥t♠ tr♥sr ♣r ♦r ♣r♣♥r t♦ t ttr♦♥ ①s r r② ♦s t ♣♥ ♦ ts ♥rs t r♥ ♥ t s♣tr♠ t♥ ♦r ♠♦♠♥t♠ tr♥sr ♣r ♦r ♣r♣♥r t♦ t ttr♦♥ ①s rt ♣♥ s ♦ ts ♥rs t ♥s♦tr♦♣② ♥ t s♣tr ♥ ♦tr ♦rs t ♥s♦tr♦♣② ♦sr ♥ t ❱ r♥ ♥rt 2 ♠♥② ♦♠s r♦♠ ♦ ts
♦♠♣rs♦♥ t ①♣r♠♥ts
♦♠♣rs♦♥ t ①♣r♠♥ts s ♥♦t strt♦rr ♥ ①♣r♠♥ts t♦sr ♥t♥st② ♣♥s ♦♥ ♥ ♥trt♦♥ ♦ t s♥ ♦r t ♦t♦♥
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P ❯
❨ P
r 2 ♦ss ♥t♦♥ t ♥t ♠♦♠♥t♠ tr♥sr s P t ♦ ts ♦ ♥ ①♣r♠♥t ♥srt s Pt♦t ♦ ts r♦♠ ❬❪
♥ β ❬❪ ♦r ♣s s ①♣rss ② t ♦r♠
I (E ) ∝ −π♠[
ε−1 ln
(
1 +β2
θ2E
)]
,
r θE = E♦ssm/~2k2
0 s t rtrst ♥ ♣♥♥ ♦♥ t ♥r②♦ss E♦ss ♥ ♦♥ t rtst ♠ss m ♥ t♦r k0 ♦ t ♥♥ttr♦♥ ♠ s s t♦ t♥ ♥t♦ ♦♥t t♦ ♣r♦ t♦rt♥trt ♦ss ♥t♦♥ ♣♥s ♦♥ t s②♠♠tr② ♦ t r②st♥ ♦♥ t ①♣r♠♥t ♦♥t♦♥s ❬ ❪
❲♥ ts s ♦♥ rt ♣ ♣♦st♦♥s t♥ t ♦t ①tt♦♥ t ♠ ♥r② t♥ t ♥ ♥ p ♣s♠♦♥s s r♣r♦♦t ♥ rt ❬❪ ♥ ♥ ③r♦♥ ❬❪ t♦rt s♣tr ♥t♥st②♦ t ♥ ♣s♠♦♥ t ❱ s ♦r t♦♦ s♠t ♣rt② ttrt t♦ t t② ♥ t ①♣r♠♥t ♥ ①trt♥t ♥st ♣rt r♦♠ t r st ♣ ③r♦ ♦ss ♦♥trt♦♥ t♥s♥② s♠ ♠♦♠♥t♠ tr♥sr ts ♠t s♦ ♣rt② ♦♠ r♦♠①t♦♥ ts ②♦♥ t P ❬❪
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P ❯ ❨ P
r ♦rt ♦ss ♥t♦♥ t♥ t P t♦t ♦ s♥ t ♦ ts ♦tt ♥ c❩r2 s ♥ t❩r2♦ ♥ m❩r2 ♦ss ♥t♦♥ s r♦♥ t ♦r♥t③♥ ♦❲ ♦ ❱ r♦♠ ❬❪
♦♠♣rs♦♥ ♠♦♥ r♥t r②st
♣ss
ts ♦ ♥s♦tr♦♣② s♥ ♥ ♣r♦s st♦♥ ♥ rtr st② ♦♠♣r♥ s♣tr ♦ r②sts t t s♠ ♦♠♣♦st♦♥ t t r♥t r②st s②♠♠tr② ♦ ts ♥ ♦♠♣rs♦♥ ♦ s♣tr t♥ r♥t ♣ss ♦ ③r♦♥ s ♥ ♥rt♥ ❬❪
♦♥ tt ♥ t t♦♥s t ♥ ♣s♠♦♥s s s t 4p ♣s♠♦♥s r ♦t ♥ ♣ ♣♦st♦♥ ♥ s♣ ♥ t tr ♦♣rssr ♣ss ♦ ③r♦♥ r s r♥s t♦ ♦r♥s ♦r t ♥ ♣s♠♦♥ t r♥ ♥ t ♥s♣ rt② rss r♦♠r♥s ♥ t ♥ strtr t♥ t ♣ss t ♣ ♥ t ♦♥t♦♥ stts ❬❪ ♦r t 4p ♣ N2,3 t r♥ s r♦t♥ t ♦ ts
s r♥s r ♥♦t ♦sr ♥ t ①♣r♠♥t r ♣rt②s t r② ♦ t ♠sr♠♥ts s ♥♦t ②t ♥♦ ♦r tsr♥s t♦ s♥ ♣rt② s ♦ t t② t♦ s②♥ts③ ♣♣r♦♣rt s♠♣s ♣rtr ♣s r♠ ♦ ③r♦♥ t ♠♥② ♣str♥st♦♥ ♥s ♠s t t t♦ r♦ ♠♦♥♦♥ ♦r ttr♦♥ s♥r②sts ♥ ts rs♣t ♦r s♣tr♦s♦♣② s s tr♦♥ ♥r②♦ss♥r s♣tr♦s♦♣② ❬ ❪ s♠s t♦ rtr ♣♦t♥t ♦r trtr③t♦♥ ♦ r♥t ♣ss
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P ❯
❨ P
r ♥trt ♦ss ♥t♦♥ s t①t ♦r ♣r ③r♦♥ ♠♦♥♦♥ ttr♦♥ ♥ ♣ss ♦ ♥ ①♣r♠♥t s♥ t♦r② t ♦ s ♥srt t♦r② t♦t ♦ s t♦rt rs ♥ ♦♥♦t t ❱♦t ♣r♦ ss♥ ♦ ❲♦ ❱ ♥ ♦r♥t③♥ ♥t♦♥ ♦ ❲ ♦ ❱ ♥ t ①♣r♠♥t ③r♦♥ s ②ttr st③ r♦♠ ❬❪
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P ❯ ❨ P
♥♦♠ Ps ♣♣r♦①♠t♦♥ s
♦♦ ♣♣r♦①♠t♦♥ ♦r
♥ t ♣r♥ st♦♥s r♣♦rt s♣tr t t♥t P t ♥s♥ ♠♦♠♥t♠ tr♥sr q ♦r ❩r2 ♥ stst♦r② r♠♥t t ①♣r♠♥t t ♥ ts s tr ♥ ♠♥② s②st♠s Ps ♦r ♥♦t s♥t ♦r s♦r♣t♦♥ s♣tr ❬❪
♥ t t♦♥ ♦ ♦t q = 0 ♥ s♦r♣t♦♥ s♣tr t ♥r♥ts r s♠r t ♦♠♣♦♥♥ts ♦ t ♥♣♥♥t♣rt rs♣♦♥s♥t♦♥ ♥ t r ♦♦♠ ♣♦t♥t ♥ tr♦r t sr♣♥② st rst st ♥tr♥ sr♣♥② s r♥t② ♥ ♥rst♦♦ ♥t ♦♦♥ ② ❬❪
♥ s♦♥ t♦ ♠t♠t② ♣r♦♣♦rt♦♥ t♦
−ℑ[ limq→0
v0(q)χ00(q,ω)],
r v0 s t G = 0 ♦♠♣♦♥♥t ♦ r ♦♦♠ ♣♦t♥t ♥ χ00 s t♠r♦s♦♣ rs♣♦♥s ♥t♦♥ s stt ♥ qs ♥ ❲t♥ tP q ♦♠s
χ = χ0 + χ0vχ,
♥ t ♦♠♣♦♥♥ts ♦ t r ♦♦♠ ♣♦t♥t v ♥tr ♥ q t s♦rtr♥ s s t G = 0 ♦♥r♥ ♦♥s
♥ t ♦tr ♥ t ♠♥r② ♣rt ♦ t ♠r♦s♦♣ tr♥t♦♥ ♥ s♦♥ t♦ ♣r♦♣♦rt♦♥ t♦ ♠♦ qt♦♥
−ℑ[ limq→0
v0(q)X00(q,ω)],
r t ♠♦ rs♣♦♥s ♥t♦♥ X rs ♠♦ ②s♦♥ qt♦♥
X = χ0 + χ0v X ,
r ♥♦ ♦♥② t s♦rt r♥ ♦♠♣♦♥♥ts ♦ t r ♦♦♠ ♣♦t♥t♥tr
v(G = 0) = 0,
v(G) = v(G).
①t♦♥ ts r ♦♥r♥ t r♦r t ♥tr♦t♦♥♥ t t♦♥ ♦ t ①t♦♥ ts ♦♥② st② ② ♠♦s tt♦rt s♣tr rs t ♦♥sr② ♠♦s t s♦r♣t♦♥s♣tr♠❬❪ s s s ♥ tr♦♥ ♥r② ♦ss s♣tr♠ r② ♦♥t♥s ♦♥r♥ ♣rt rs ♥ s♦r♣t♦♥ s♣tr♠ ♦s ♥♦t ♦♥t♥
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P ❯
❨ P
♥② ♦♥r♥ ♣rt t♥ t P ♣♣r♦①♠t♦♥ ♥ t ♥r t ♣s♠♦♥ ♥r② tr s ♥t♦♥ t♥ t ♦♥ r♥ tr♠ ♥ tqs♣rt st t st ♥ t ♦♥ tr♦♥ ♥r② ♣r♦ ② ts♥r② t♥ ♥t♦ ♦♥t t t tt tr♦♥s ♠♦ ♥ ♠♠s sst♦♥
♦♥s♦♥
♣rs♥t t t♦rt r♠♦r t♦ t tr♦♥ ♥r②♦ss s♣tr r♥♦♠ ♣s ♣♣r♦①♠t♦♥ s s♦♥ t♦ qt♦r ♦t tr♦ ①♠♣s ♦♥ ❩r2 ♥ 2 ♥ ♦♥ ♠t♠tr♦♥s s♦ rss sr tr rt② ♦r ♥ ♦♦rt♦♥t ts ♦ t ♥s♦♥ ♦ s♠♦r stts ♥ t ♥ ♥ 2 ♥2 ❬❪
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♣tr
♦rt rsts ♦♣ts♦r♣t♦♥ ①♣r♠♥t
♥ t ♦♦s ♦♠♣r t s♦r♣t♦♥ s♣tr♠ ♦ ♣♦②r②st ♦③r♦♥ s ♣♣♥① t t t♦rt sr♣t♦♥ t♥ t r♦ss ❬❪ ♥ t s♦r♣t♦♥ s♣tr♠ ♦ s♥ r②st ♦ 2 tt♦rt sr♣t♦♥s t♥ ②♦♥ ♦♥♣rt t♦r② ❬❪
❲ ♦♥sr ♥ ♦♣t s♦r♣t♦♥ ①♣r♠♥t ♥ ts s ♦♥ ♠srs t s♦r♣t♦♥ ♦♥t
α(ω) =ωℑ(εM(ω))
n1(ω)c,
r c s t t ♦t② ω s t ♥r② ♦ t s♦r ♣♦t♦♥ n1 st r ♣rt ♦ t ♥① ♦ rrt♦♥ ♥ ℑ(εM(ω)) s t ♠♥r② ♣rt♦ t ♠r♦s♦♣ tr ♥t♦♥
❲ ♥ t♦ t t ♠r♦s♦♣ tr ♥t♦♥ rs
εM(ω) = limq→0
1
ε−1G=0G ′=0(q,ω)
.
t♦rt s♦r♣t♦♥ s♣tr♠ s t♥ ♥ ② t ♠♥r② ♣rt ♦qt♦♥
❲t♥ ♥♣♥♥t tr♥st♦♥ t♦r②
s♦r♣t♦♥ ♦ ♣♦t♦♥ ♦s ♦♥ tr♦♥ ♦ t ♥ ♥ t♦ ♣r♦♠♦t t♦ ♦♥t♦♥ stt ♥ ♥ ♥♣♥♥t ♣rt ♣♣r♦①♠t♦♥ rt ♥ ①♣rss♦♥ ♦r t ♠r♦s♦♣ tr ♥t♦♥
εM ≈ 1 − vχ0(G = G′ = 0;q → 0),
♥ s t♦ t t t♦rt s♦r♣t♦♥ s♣tr♠
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P ❯ P
P ❳P
♠♥r② ♣rt ♦ t ♠r♦s♦♣ tr ♥t♦♥ εM rs
ℑ(εM(ω)) =4π
ω2
∑
v ,c
|〈ψc | − iA.∇|ψv〉|2δ(ω − εc + εv)
r −iA ·∇ srs t ♦♣♥ ♦ t tr♦♥ t t t♦r ♣♦t♥t A ♦ t tr♦♠♥t ❲ trt ts tr♠ t♥ t ♣♦r♣♣r♦①♠t♦♥ s♠ s ♣r♦r♠ ♦t ♦♥ t ♥t ♥ ♥ ♥♦♥t♦♥ stts ♥ q r ♦ stts ♦ t ♣r♦ ♣♦t♥t ♥ t s♦
s r♠s ♦♥ r s ♣♣♥① ♦♠s r♦♠ t t tt ♦♥trts t tr♥st♦♥s t♥ stts ♥♣♥♥t② ❯s♥ qt♦♥ ♦rqt♦♥ ♦r t ♣♦r③t② ♥ t♥ t ♠t G = G′ = 0;q → 0t rs♣♦♥s ♥t♦♥ χ0 ♦ st♦♥ ♠♦♥ts t♦ t r♠s ♦♥ r♦ q
♦♥t ♥st② ♦ stts
s♠♣st ♣♣r♦ t♦ t s♦r♣t♦♥ s♣tr♠ ♦♥ssts ♦ t t♦♥ ♦ t ♦♥t ♥st② ♦ stts s rst tt♠♣t t♦ ♦t♥ t♦rttr♦♥ s♣tr ♥ s t ♥s εi ♦ t ♠t♦♥♥t♦ t t ♦♥t ♥st② ♦ stts
∑
v ,c δ(ω − εc − εv) rεv s t ♥r② ♦ stt ♥ t ♥ ♥ ♥ εc s t ♥r② ♦ stt ♥ t ♦♥t♦♥ ♥ s ♣♣r♦①♠t♦♥ s♣♣♦ss tt t ♠tr①♠♥ts |〈ψc |− iA ·∇|ψv〉|
2 r ♣t ♦♥st♥t ♥ qt♦♥ q♥tt②
ℑ(ǫ(ω)) =
∑
v ,c δ(ω − εc + εv)
ω2,
r♣♦rt ♥ ❩r2 s rt t♦ t ♦♥t ♥st② ♦ stts s♥ ♦s ♥♦t sts② ♦♥srt♦♥ rs ♥ tr♦r t t s♣tr♠ s ♥ rtrr② ♥ts
s♦s ♦♦ ♦r r♠♥t t♦ t s ♥♦t s♥tt♦ ♥tr♣rt t ts ♦ t ①♣r♠♥t t r s②♠♦s s♣②t♥ ♥ ❱ ♥ r♦♥ ❱ ♥ ♣rtr t ♦r s♣♦ t s♣tr♠ s ♥♦t r♣r♦
t ♦ ♠tr① ♠♥ts
s♠♣st ♦ ♣♣r♦①♠t♦♥ ♦♥ ♥ ♠ ②♦♥ t s t♦trt t ♣rts ♥ t tr♥st♦♥s t♥ stts ♥♣♥♥t② tqt♦♥
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P ❯ PP ❳P
r s♦r♣t♦♥ s♣tr♠ ♦r ♠♦♥♦♥ ❩r2 t s♥ tP t ♦ s s♦ ♥ P t♦t ♦ s ♦tt ♥♥ t t rt♦ ♦ t ② t sqr rq♥② ♦♠♣r t①♣r♠♥t rsts r♦♠ r♥ t rr♥ ❬❪ t♦rtrs ♥ s t♦ t ♥t♥st② ♦ t ①♣r♠♥t ♣ r♦♥ ❱ r♦♠ ❬❪
s ♣♣r♦①♠t♦♥ s s♦♠t♠s s♥t ♦r s②st♠s t ♥♦♠♦♥ts ♦r t ♠t sr♥♥ ♠♦♥ r♥t ①♠♣s ♦♥ ♥t t s♦r♣t♦♥ s♣tr♠ ♦ s♠ ♠tr ♥♥♦ts ♥ t t s♣♦r③ ♦♥ t ♥♥♦t ①s ❬ ❪
♦♠♣tt♦♥ ♦ t ♠tr① ♠♥ts |〈ψf | − iA · ∇|ψi〉|2 ♥ q
s ♥ssr② t♦ ♠♣♦s t st♦♥ rs ♥ ♦♣t s♦r♣t♦♥ ts s ♦♠♣♦rt♥ ♥ ♦♥ s ♠♦♥♦r②st ♥ ♥ts t♦ st② t ts ♦t ♣♦r③t♦♥ ♦ t
♥ ♥ r t s♦r♣t♦♥ s♣tr♠ ♦r t ♣♦r③t♦♥sr ♠tr① ♠♥ts ♦ ♣r♠r② ♠♣♦rt♥ ♥ r s tt t♦r s♣ ♦ t s♦r♣t♦♥ s♣tr♠ s ♥ rt② ♠♦ ② t♥s♦♥ ♦ ♠tr① ♠♥ts
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P ❯ P
P ❳P
②♦♥ ♥♣♥♥t tr♥st♦♥ t♦r②
♠r♦s♦♣ tr ♥t♦♥ ε(r, r′,ω) ♥ rtt♥ ♦rr tr♥s♦r♠ s t ♠tr① t ♠♥ts εGG′(q,ω) ♦ qt♦♥ ❬❪
♠♠r③♥ qt♦♥ rt t ♥rs tr ♥t♦♥ s
ε−1GG(q,ω) = 1 + v(q + G)χGG(q,ω),
r v s t ♦♦♠ ♣♦t♥t ♥ χ t ♣♦r③t② ♦ t s②st♠st♦♥
♥ ♦rr t♦ t ts ♥t♦♥s ♥ t♦ ♦r♠t ♥ ①♣rss♦♥ ♦r t ♣♦r③t② ♦ t s②st♠ χ t ♥ t ② ②s♦♥s qt♦♥ ♦r s ♣♣r♥t② r r♦♠ r♠s ♦♥ r♣♣♥① ♦♥tss ♦♥ ♥ s♦ tt ♥ r ①♣rsss ♥ tr♦♥♦ ①♥ ❬❪ t ♠♥r② ♣rt ♦ t tr♥t♦♥s s ♥ ② qt♦♥
♦ ts
♥♦♠♦♥ts ♦ t s②st♠ rs t♦ t r②st ♦ ts♥ ♦r t♦♥s ts ♦ ts rs r♦♠ t ② t ♦♦♠tr♠ v s trt ♥ st♦♥
❲ ♥ ♥t t ♦ ts ♥tr② ② ♥♥ ♦♥② t ♦♥r♥ ♣rt ♦ t ♦♦♠ ♣♦t♥t vG=0(q) s ♦♥ ♥ st♦♥ r② ♥♦rs t ♠r♦s♦♣ ♦♠♣♦♥♥ts ♦ t ♥ ♥ts s ♦♥② ♥ t♦ t t ♦ t tr ♠tr① εG=0,G′=0 ♥ s t P t♦t ♦ s ♦ st♦♥
♥ ♥ t ♦♦♠ ♣♦t♥t vG(q) ♥ t♦ tt tr ♠tr① ♥♥ t ♦♦♥ ♠♥ts εG,G′ 6=Gtst ♥rs♦♥ ♦ ts ♠tr① s t t ♦ ♠①♥ t ♣r♦s② ♥♣♥♥t tr♥st♦♥s
♦r st ♥♥ t ♦ ts ♦s ♥♦t r♥t ttt rsts s♥♥t② ♦sr t♦ ①♣r♠♥t t♥ t♦♥s rt② r ♥t s s s♣② tr ♥ t s ♦ s♦r♣t♦♥ s♣tr♦r ♦r tr♥st♦♥♠t ♦① s②st♠s ♦r ①♠♣ ♥ r ♣♦tt ①♣r♠♥t rsts ♦r ♠♦♥♦♥ ❩r2 ♥st t♦rt Pt♦♥s ♦t t rtr ♦ ts ♥ t♦t t♠ ❲♥ s tt tr s r r♥ t♥ t rsts ♦ t t♦ t♦rt ♣♣r♦①♠t♦♥s ♥ ♥ ♣rtr t② s♦ r ♥ t ♥r②r♦♥ ♦r s ♦ t ♥r② rr t♥ ❱
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P ❯ PP ❳P
♥ ♥ ♦♣t s♦r♣t♦♥ ①♣r♠♥t q s ♦s t♦ ③r♦ ①♥ ♥♦rrt♦♥ ♦ ts ♥ s♦♥ t♦ s♠ ♥♥ ♥2 ♦r ♥s♥ q ❬❪ ♥ tr♦r t② ♥♦t ♥ t♥ ♥t♦ ♦♥t ♥ t t♦♥s ♦ r
s♣rts ts
♦r s♦r♣t♦♥ s♣tr t qs♣rt ♥r② sts tt ♦♥ ♥ t ♥ t GW ♣♣r♦①♠t♦♥ ❬ ❪ s ♥ssr② t s② t s ♥♦ts♥t ❬❪
❲ ♣♣r♦①♠t♦♥ ♦♥ssts ♦ s♦♥ ♥ qt♦♥ s♠r t♦ ts②st♠ ♦ qt♦♥s ♥ ♥ t ♦ ①♥ ♥ ♦rrt♦♥ ♣♦t♥t vxc(r) s r♣ ② ♥♦♥♦ ♥ ②♥♠ s♥r② ♦♣rt♦rΣ(ω, r, r′)
HKSψn(r) − vxc(r)ψn(r) +
∫
d3r′Σ(ω, r, r′)ψn(r′) = εQP
n ψn(r).
♥ t s♠♣st ♦r♠t♦♥ t s♥r② Σ(ω, r, r′) s t tω = εQP
n t s ♣♣r♦①♠t ② t ♣r♦t ♦ t r♥s ♥t♦♥ ♥♦ t sr♥ ♦♦♠ ♥trt♦♥ W = ǫ−1v ♥ t ♥♠ GW ♦rts ♣♣r♦①♠t♦♥ ♥ rst st♣ ♥s r ♦rrt ② qt♦♥ t ♥t♦♥s r ♥♥ ♥ r s② t♦s ♦t♥r♦♠
♥♥ qs♣rt ts ♥ t tr♦♥ s♣tr ♦rrts t ♥♣s ♥ t♦♥s t②♣② ♥ ♣s r♠ t♦♦ s♠ t rs♣t t♦ t ♣♦t♦♠ss♦♥ ♣ s rsts ♥ ♠♣r♦ ♣♦♥ s♥ ♠t♦s s s t ❲ ♣♣r♦①♠t♦♥ ❬❪ ❲ st♠t♦♥ ♦ t ♥ ♣ ♥ ♠♦♥♦♥ ❩r2 s rst ♦ ❱❬❪ ♥st ♦ t ♦ ❱ ❬❪
❲♥ d tr♦♥s ♦♠ ♥t♦ ♣② ♦r t s♠♣st ❲ ♣♣r♦①♠t♦♥ s ♥♦ ♦♥r s♥t s s ♥ s♦♥ ♦r 2 ❬ ❪ GW st♠t♦♥ ♦ t ♥ ♣ ②s ❱ t♦ ♦♠♣r t ❱ ♥ t ①♣r♠♥t r♦r ♦♥ ♥s t♦ ♦ ♦♥ st♣ rtr ♥ tt♦rt r♠♦r ♥ s t s ♦ t qs♣rt ♥r② ♥ t①♣rss♦♥ ♦ t ♥rs tr ♥t♦♥ ǫ−1 t♦ rt t sr♥♦♦♠ ♥trt♦♥ W s♦♥sst♥t GW ♣♣r♦①♠t♦♥ ❬❪ s♦♣ ♥ rt ② r♥ r♥ s P tss t t♣ ♦ t s♦ ❳ ♣♣r♦①♠t♦♥ ❬❪ ♣ ♦ 2 st t♦ ❱ ♥ t s ❳ ♣♣r♦①♠t♦♥ ♥ ❱ ♥t s♦♥sst♥t GW
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P ❯ P
P ❳P
r s♦r♣t♦♥ s♣tr♠ ♦r 2 t♦t ①t♦♥ ts t s♥ t P t ♦ s t t ♦tt ♥ tt GW s ♥ ♥ t t s♦♥sst♥t ❲ s♦ ♥♦♠♣r t ①♣r♠♥t ♦ ❬❪ rs r♦♠ ❬❪
♦ strt ts t s♦ ♥ r t♦rt s♣tr tt qt♦♥s ♥ t♦tr t ①♣r♠♥t t ♥ qt♦♥ t s ♦ t ♥r② ♥ ♥t♦♥s ♦ t ♥t ♥ ♥ stts ♥ t t r♥t s ♦ ♣♣r♦①♠t♦♥ t t s ♥ r t t GW ♥ t t ♦ s♦♥sst♥tGW ♣♣r♦①♠t♦♥ ❬❪ ♥② t t ttr s t t♦rt s♦r♣t♦♥ ♦sr t♦ t ①♣r♠♥t ♦♥ ♥ ♥ rs r♦♥ ❱
♦r s♦r♣t♦♥ s♣tr t qs♣rt ♥r② st tt ♦♥ ♥ t ♥ t GW ♣♣r♦①♠t♦♥ s s② ♥ssr② t ♥♦t s♥t ss s♥ ♥ t ♦r s♣ ♦ t t♦rt s♣tr r ♥ t ♠♦r ♠♦t♦♥ s r♦t ♥ t s♦r♣t♦♥ s♣tr♠ ② ①t♦♥ts s s s t r♥ t ♥ s♣tr♠ s st♦♥ ♥ s♦r♣t♦♥ s♣tr♠ ♦s ♥♦t ♦♥t♥ ♥② ♦♥ r♥ ♣rt t♥t P ♣♣r♦①♠t♦♥ ①t♦♥ ts ts ♦♥sr② ♠♦② ♥ s♦r♣t♦♥ s♣tr♠ s strt ♥ t ♥①t st♦♥
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P ❯ PP ❳P
①t♦♥ ts
♥ ♥t♦ ♦♥t t ①t♦♥ ts s ♦♠♣① ♥ s ♥ ♥②♦rt ② rt ♥ ts P tss ❬❪ ♥ ♦♥② r♦♠♠♥ tr♥ ♦ ts P tss t ♠♦♥ts t♦ s♦♥ t♣tr qt♦♥♦r ♦r ♣♦♥t ♣♦rst② P
P = P0 + P0ΞP
Ξ s t ♥trt♥ r♥ ♥ ♦♥ssts ♦ t ①♥ t t ♦ ♦ ♦ st♦♥ ♥ ♦ t ♦♦♠ tr♦♥♦ ♥trt♦♥sr♥ t t ♣♦t♥t W ♦ t ♣r♦s st♦♥
♦ qt♦♥ ♠♦♥ts t♦ ♦♥③♥ ♥ t ♠t♦♥♥ rs♦♥♥t ♣rt ♦ t ♠t♦♥♥ ♦♥ssts ♦ t r tr♦♥♦ ♣r ♥r② ♣s t t♦ ♦♠♥t♦♥ ♥trt♦♥s ①♥ ♥sr♥ ♦♦♠ tr♦♥♦ ①♣rss ♥ t r♥ Ξ
Hexccv ;c ′v ′ = (εc − εv)δvv ′δcc ′ − iΞcv ;c ′v ′ .
s ♦ t ①t♦♥ ♥r② Eλ ♥ ♥t♦♥ ♦♥ts Acvλ
r t ♥s ♥ ♥t♦rs ♦ t t t♦♣rt ♠t♦♥♥ ♦r t tr♦♥♦ ♣rs Hexc
Hexce1,h1;e2;h2
Ae2,h2λ
= EλAe1,h1λ
,
♥ ei s ♥ tr♦♥ stt ♥ t ♦♥t♦♥ ♥ ♥ hi s ♦ stt ♥ t ♥ ♥
♠t♦♥♥ s ①♣rss ♥ t ss ♦ tr♦♥♦ ♥t♦♥s |ψvψc > tr♦♥♦ ♣r ♥t♦♥ s s② ①♣rss s|ψvψc > r |ψv > ♥ |ψc > r s♥ ♣rt s♦t♦♥s ♦ t qt♦♥s ①t♦♥ ♠♣t s ①♣rss s ❬❪
Ψλ(r, r′) =
∑
(c,v)
Ac,vλψ∗
c (r)ψv(r′).
♥ t tr ♥t♦♥ t ①t♦♥ t ♥ ② t sr♥♦♦♠ tr♦♥♦ ♥trt♦♥ s t♦ ♠①♥ ♦ t ♦r♠② ♥♣♥♥t tr♥st♦♥s t t s♠ ♥r② ❬❪
ℑ(εM(ω)) = 2π limq→0
vG=0(q)∑
λ
|∑
v ,c
|〈ψc | − iA.∇|ψv〉Acvλ|2δ(ω − Eλ + iα).
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P ❯ P
P ❳P
♥t♦♥ ♦♥ts Acvλ
♠① t ♦r♠② ♥♣♥♥t tr♥st♦♥s♥ ♥ ♦r♥t ts ♥ r♠s ♦♥ r
♠t♥ Ξ t♦ t ①♥ t ♦ ts s strt② q♥t t♦ st♦♥ ♥ ♥t♦ ♦♥t ♦t t ①♥ t ♥ tsr♥ ♦♦♠ tr♦♥♦ ♥trt♦♥ ♥ Ξ ♥s rst ♥s ♥t ♦♣t s♣tr r ts ♦ t tr♦♥♦ ♥trt♦♥ ♥ s♥ ② ♦♠♣r♥ t s ♥ ♦ r ♥ t ♦ts ♥♦ ♣♥ ♦ r
t♦ t ♦ s♠ s ♥ s♦♥ t♦ s♥t ♦r sp♦♥ s♠♦♥t♦rs t s ♥♦t s♥t t♦ ♥tr♣rt t ①♣r♠♥ts♣tr♠ ♦ 2 ♣♥ ♦ r ♥srt♥ t ♠♣r♦ s♦ t qs♣rt ♥r② ♦t♦♥ r♦♠ s♦♥sst♥t GW t♦♥♥ t r tr♦♥♦ ♠t♦♥♥ (εc − εv)δvv ′δcc ′ s s♦ ♥♦t s♥ts♦tt ♥ ♦ ♣♥ ♦ r r♥ s s♦♥ tt ♦♥♥s t♦ ♦ ♦♥ st♣ rtr ♦r ♠trs t d tr♦♥s s t s ♦ t ♥r② r ♥♦t s♥t t♦ ♣r♦♣r② t t sr♥♥♥t♦♥ ǫ−1 ♥ t♦ ♦t♥ t sr♥ ♦♦♠ tr♦♥♦ ♥trt♦♥ W ❲♥ ǫ−1 s ♣r♦♣r② t s♦ ♥ t ①♣rss♦♥ ♦ Ξ s♦♥ ♥ ♣♥ ♦ r t♥ t t♦r② s ♣ ♥ ♥rst♥♥t ①♣r♠♥t s♣tr♠ ♦r rs ♥ ♣rs♥t t s
♦st♦♥ ♦ ①t♦♥ ts
s♦t♦♥ ♦ t qs♣rt ♥ ♦ t t♣tr qt♦♥ rr② ♠rs♦♠ t♦♥s
❲ ♠ ♥ tt♠♣t t♦ ♠♦ t ①t♦♥ t t♥ t r♠♦r ♦ t♠♣♥♥t ♥st② ♥t♦♥ t♦r② ② ♥♥ ♥ ♣♣r♦①♠t♦♥ ♦ t ♦♥ r♥ r♥ s α
|q|2 t α ♥ ♥ ♠♣r ♣r♠tr
r♦♠ t stt tr sr♥♥ ❬❪
♥ ♠② ♥rst♥♥ ts ♠♦♥ts t♦ ♥srt♥ ♦♥r♥ ♣rt α
|q|2♥
qt♦♥
F 0(q,Gω) = δVbare(q,G,ω)+∑
G′
[4πe2
|q + G′|2δ(G′ − G) + fxc(G
′ − G)]δn0(q + G′,ω),
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P ❯ PP ❳P
r s♦r♣t♦♥ s♣tr♠ ♦r 2 t t ①t♦♥ tst t st♥r ❲ ♥s ♥ t ♥t♦♥s ♥ ǫ−1
♥ ♥ t r tr♦♥♦ ♠t♦♥♥ (εc − εv)δvv ′δcc ′ ♦t s ♥ qs♣rt ♦rrt♦♥s t♦ t ♥r② ♦t♥ r♦♠ s ♦♥sst♥t ❲ t♦♥ r ♥srt ♥ t r tr♦♥♦ ♠t♦♥♥(εc−εv)δvv ′δcc ′ s♦tt ♥ ♦r ♦t ♥ t r tr♦♥♦ ♠t♦♥♥ ♥ ǫ−1 s♦ ♥s ♥st ♣♥ s♦s t tr♥st♦♥ ♥②ss♦r ♣ ♦♥trt♥ tr♥st♦♥s ♦ts ♥ ♦♣t s♦r♣t♦♥ s♣tr♠t t t s♦♥sst♥t ❲ s ♦ t ♥r② t t♦t ①t♦♥ ts s ♥ ①♣r♠♥t ♦ ❬❪ rs r♦♠ ❬❪
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P ❯ P
P ❳P
②♥
F 0(q,Gω) = δVbare(q,G,ω)+∑
G′
[4πe2
|q + G′|2δ(G′ − G) + fxc(G
′ − G) +4παe2
|q + G′|2δ(G′ − G)]δn0(q + G′,ω).
♥ ts stt ♦r♠ t s ♥♦t ♣♦ss t♦ rt ②r srs ♥ ♥ts②st♠ t ts ♠♦ t t s ♦♥♥♥ ♥ ♠♦②♥ t ♦♣t s♣tr ♦t♥ ♥ t♠♣♥♥t ♥st② ♥t♦♥ t♦r② ❬❪
①t♦♥ ts r t♦t t♦ r r♦♠ t ♥♦♥♦ ①♥♦♣rt♦r ❬❪ ts s s♦ P♦ss♦♥s qt♦♥ ♦ s♦ s♦ ♦♥②t t rtr ♣♦t♥t ❲ s♥ tt ♦r ♦♥ ♥t ♣rtrt♦♥ q = 0 ♥ G = 0 ♦♥ s δn(q+G,ω) = 0 ♥ ♥ ♥t ♥st② s ♥♦ ♠r♦s♦♣ ♦♠♣♦♥♥t t t♦t ♠r♦s♦♣ tr♦♥♥st② s ♣t ♦♥st♥t ♥ t ①t stt lim
q→0
∫
[n(r)− < n >]e−iq.r → 0
♥ s s t r♥ ♠♦ α
|q|2s♦ ♥♦ t s ♥ q
t♦♥ t s ♠t♣ ② δn(q,G → 0,ω)
r♥t tt♠♣t ② ♠②s ♥ r♦ tt P ♥sttt ♥ér♦ t P②sq s ① ♦♥♥sés ❯♥rsté Prs ❱P r t♦ ♥tr♦ t ♠♦ r♥ ♥ t P qt♦♥s ♦ ❬❪ s ♥ ♠② ♥rst♥♥ ts r s ♦♥♥t t♦ tt tt ♥ ts s t ♥ ♥ ♥st② s s♥ t♦ ♥♦ ♠r♦s♦♣ ♦♠♣♦♥♥t s s ♥♦t t s ♥ t P ♣r♦r♠ ♣ s ♥ ❬❪ ♥ ♥♦ ♦ t ♥ ♥st② s ♣r♦r♠
♥ ♠② ♥rst♥♥ ♥tr♦♥ ♠♦ r♥ s ♦♥ ♥ ❬❪♠♦♥ts t♦ ♠♦②♥ P♦ss♦♥s qt♦♥ s ♠t stt♦♥ ♦♥sr tt t ♣rtrt♦♥ s ♥♦ ♦♥r t♥ s t ♠♥♦ ♦♥ t♦ ♦rt ♦♥ ts ♣♦♥t
♦♥ t ts
♥ s s ♦♥ s♦ t ♥t♦ ♦♥t t ♦♠♣♦♥♥t δn0 ♥ ②qt♦♥ ♥ P♦ss♦♥s qt♦♥
♥♥ rst t q♥tt② A ♣♣rs ♥ δn0 q ♥ ♥δnq(r) q ♦♥ s
Alk1pk2(r) =
Blk1pk2
ε(0)pk2
− ε(0)lk1
− ~ω − i~α,
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P ❯ PP ❳P
t B qs t♦
Blk1pk2(r) = ψ
(0) ∗pk2
(r)ψ(0)lk1
(r)[f (0)(ε(0)pk2
) − f (0)(ε(0)lk1
)] < ψ(0)lk1|F q|ψ
(0)pk2
> .
rt♥ P♦ss♦♥ qt♦♥ t δn0 s t♦∑
G
|(q + G)|2 F 0Hartree(q,G,ω)e−i(q+G).re iωteαt + c .c . =
4πe2∑
lk1pk2
Alk1pk2(r)eαt(e iωt − e
−i(ε(0)lk1
−ε(0)pk2
) t~ ) + c .c .,
r t tr♠s e−i(ε
(0)lk1
−ε(0)pk2
) t~ ♦♠ r♦♠ ♥♦♥t tr♠s
♥trt♥ ♦♥ t♠ ♥ t♥ t ♦♥ ♥t ♠t ♥ tt
limq,G→0
|(q + G)|2 F 0Hartree(q,G,ω) = 4πe2 lim
q,G→0δn(q + G,ω)+
4πe2
~ωlim
q,G→0
∑
lk1pk2
Blk1pk2(G)
❲ s tt ♥♦♥ t ♣rtrt♦♥ ♥s ♥ t♦♥ tr♠ t♦t r♥ ♦♠♥ r♦♠ t rtr ♣♦t♥t
4πe2
~ωlim
q,G→0
1
|(q + G)|2
∑
lk1pk2
Blk1pk2(G).
s t♦♥ tr♠ s t ♦r α
|q|2 s ♥ rt ♥
①♣♥♥ ①t♦♥ ts ❬❪
❲tr s ♥♦♥ t ♦♥trt♦♥ ♦ t ♠♥ ①♣♥t♦♥ ♦r t ①t♦♥ t ♦ ❬❪ s t ♦r tr ♦r
♦♥s♦♥
♥ ts ♣tr ♣rs♥t r♥t s ♦ ♣♣r♦①♠t♦♥ t♦ t ♦♣t s♦r♣t♦♥ s♣tr ♣rs♥t stts ♦ t t♦r② s r②♠tr ♥ t r♦ ♦ ①t♦♥ t s ♥rst♦♦ st② ♣rs♥t ♥ ♥♣s ①♣♥t♦♥ ♦♥ t ♦♥ r♥ r♥ α
|q|2
♦ ♥tr♣rt s t ♦♥trt♦♥ ♦ ♥♦♥ t ♣rtrt♦♥
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♣tr
♦rt rsts ♦♥ tstt② ♦ ♦r♦♥ rs
s ♣tr s r♣r♦t♦♥ ♦ t ♣r♣r♥t ❬❪ t s r rt ♦♠② ♦r ♦♥ ♦r♦♥ rs ♦♦♥ ♥ ♥tt♦♥ t♦ t t ♥tr♥t♦♥②♠♣♦s♠ ♦♥ ♦r♦♥ ♦rs ♥ t trs ♣t♠r ts ♠♥ ♣♥ ♠ ♥♦ ♦♥ t♦ ♣rs♥t ♥ rsts ♦♥t ♦r♠t♦♥ ♥r② ♦ ♦r♦♥ rs t r♦♥ ♦♥♥trt♦♥s ♦r t♥ t ♣♣r s ♥ rtt♥ ♥ ♦♦rt♦♥ t ♥ st♥ tr♥♥②
♥tr♦t♦♥
♥st② ♥t♦♥ t♦r② s ♦♠ r r♦♠rst♣r♥♣st♦♦ t♦ ♥stt r♦♥ stt ♣r♦♣rts ♦ ♠trs❬ ❪ ♦rt rsts ♦♥ ♦r♦♥ rs ♣rs♥t ♥ ts ♣♣r r② ♦♥ ♥tr♦r r② r ♥ ts ♥tr♦t♦♥ t ♠♥ ♣r♣♦s ♦ ♥ ts ♠tt♦♥s ♠♣s s②st♠ ♦ ♥trt♥ tr♦♥s ♦♥ s②st♠ ♦ ♥♦♥♥trt♥ tr♦♥s ♥ ♥ t ♣♦t♥t ♦♥t♥s ts♦ ①♥ ♥ ♦rrt♦♥ ♣♦t♥t ♥ t rs ♦♥ ss♠♣t♦♥s ♦r t ①♥ ♥ ♦rrt♦♥ ♥r② ♠ t ♠♦♥tr♠s t t♦ t ♥t♦ ♦♥t ♥ t ♠♣♣♥ t r♥ ♥♥t ♥r② t♥ s②st♠ ♦ ♥trt♥ tr♦♥s ♥ ♥♦♥♥trt♥♦♥s ♠♥②♦ tr♦♥tr♦♥ q♥t♠ ♥trt♦♥s s sPs ♣r♥♣ ②♦♥ t rtr ♥r② t ss tr♦stt ♥r② ♦r ♥trt♥ r ♥sts
♥ t s♠♣st ♦ ♥st② ♣♣r♦①♠t♦♥ s♠ ♦♠♦ t s②st♠ s ①♣t t♦ ♦♥trt t♦ t ♥r② ♦ ①♥ ♥ ♦rrt♦♥ s ♦ ♥ q ♦♠ ♦ ♦♠♦♥♦s tr♦♥ s t ts♠ ♥st②❬❪ qr♠ ♣r♦♣rts t t♥ ss tt ♣r♠trs ♥ ♥tr♥ t♦♠ ♣♦st♦♥s s② r ①tr♠②
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t♥ t ①♣r♠♥t r r ♦r ♥♠r ♦ ♣♣t♦♥s ♥ ts r② s ♥♦t ♥♦ t s ♦rt ♣♦♥t♥♦t ♦r ♥st♥ tt ♠♦st ♦ t♦②s t♦♥s ♦ tr♦♥ s♣tr r♣r♦r♠ t t ①♣r♠♥t tt ♦♥st♥t t♦ ♦ tt ♠st❬❪
①♥ ♥ ♦rrt♦♥ ♥t♦♥s ②♦♥ r ts rqr♥ ♦rr t♦ t t qr♠ ♦♠ t rtr r② ♥r③ r♥t ♣♣r♦①♠t♦♥s ♥ ♦♣ t s♥ ♦t❬❪ t s ♠♣♦rt♥t ♦r ♣r♦♣rts ①♣t② ♣♥ ♦♥t qr♠ ♦♠ Veq s s t t♥s♦r ♦ st ♦♥st♥ts Cijkl ♥ s t s♦♥ ♦rr rt ♦ t t♦t ♥r② E trs♣t t♦ t str♥ t♥s♦r εij r i , j , k ♥ l r rts♥ ♥①s
Cijkl =1
Veq
(
∂2E
∂εij∂εkl
)
ε=0
.
♥ t ♦ qt♦♥ t ①♣t ♣♥♥ ♦♥ t qr♠ ♦♠Veq ♠s t rr♦r r ♦♥ st ♦♥st♥ts ♠ s♠r ♥ tt♥ t ♥r③ r♥t ♣♣r♦①♠t♦♥ t♥ t♥ ❬❪ r ♠♥t rs♦♥♥ s ♥♦tr t②♣ ♦ ♣r♦♣rts rt②♣r♦s t ♦ t♦♠ strtr tr♦ t tr♦♥ rs♣♦♥s t♦ ♥ ①tr♥ ♠♥t ♦r t s♠ rs♦♥s t♦♥s ♦ t ♠st t♥s♦r ♣r♦r♠ t s ♦ s♦ ♠♦r stst♦r② r♠♥tt ①♣r♠♥t ♥ ♦♠♣rs♦♥ t t♦s ♦t♥ ② s♥ t ❬❪ ❲ tr♦r s ♦♥ ♦ t s ❬ ❪ ♦r ♦r rsts s♣tr♣rs♥t ♥ sst♦♥
♠♦♥ ♥♦♥ ♠tt♦♥s ♦ ♦r♥ ♥r② ♣s ♥ ♥st♦rs♥ s♠♦♥t♦rs r ♥rst♠t ♦r ♥r② ♣ t♦♥s t♦rt r♠♦rs ②♦♥ r rqr t ♥t♦ ♦♥tt tr♦♥tr♦♥ ♥trt♦♥ ②♦♥ ♥ ♥r②♥♣♥♥t ♥ ♦♣♦t♥t ♥r② ♣s r rr♥t② ♦♠♣t t r♥s ♥t♦♥ ♣♣r♦ ♦s ♦r t t♦♥ ♦ qs♣rt ♦rrt♦♥s t♦ ts ♦ t ♥r② t♦ rst♦rr ♥ ♣rtrt♦♥❬ ❪
①t stts r s♦ ♦t ♦ r ♦ ② r trt tr② t t♠♣♥♥t ♥st② ♥t♦♥ t♦r② ❬ ❪ ♦r ② ts♦t♦♥ ♦ t t♣tr qt♦♥ ♥ ♥ ♦♣t s♦r♣t♦♥ ts ♥ssr② t♦ t ♥t♦ ♦♥t t tr♦♥♦ ♥trt♦♥❬ ❪ s♠t♦s ♥♦♥tss r② ♦♥ ♦r ♦t♥♥ t r♦♥ stt ♥t♦♥s ♥ s ♦ t ♥r② r s tr s strt♥ ♣♦♥t ♦rt♦ t ② q♥tts t tr t♥s♦r
♥② t s ♥♦t ②t r tr t ♦♠♥t♦♥ ♠♥②♦②♠t♦s s ♦♥ r♥s ♥t♦♥s r s♥t ♥ ♠trs r tr♦♥ ♦rrt♦♥s r ♦ ♣r♠r② ♠♣♦rt♥ ♥ ♦①s t ♠t♣②
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♦♣ 3d ♦rts ②♥♠ ♠♥ t♦r② ♣rs♥ts ♥ tr♥t ♠t♦ t♦ ♠♦ t ♦rrt♦♥s❬❪ t s♦ ♥♦t ♦rtt ♦t sss ♦ ♠t♦s ♥ sss② ♣♣ t♦ t tr♦♥s♣tr♦s♦♣② ♦ ♥♠ ♦①❬ ❪
♥ sp ♦♥ ♠trs ♦r♦♥r s♦s r ①♣t t♦ r♦♥ stt ♣r♦♣rts sr ② s ♠t♦s r②st s②♠♠tr② ♦ s r♦♠♦r t t 3m s♣ r♦♣ t♦♠ strtr ♦♥ssts ♦ ♦♥ st♦rt ♦sr♦♥ ♣r ♣r♠t ♥t r ♥t② ♣s ♦♥ ♠tt♦♠ ♥ t♦♠ ♥t ♦sr♦♥ s t ♣① ♦ ♣♥t♦♥s ♣②r♠ ♦ r②st♦r♣ sts ♥ t ♦sr♦♥ r ♥ ② t ♥tr♦sr ♦♥♥♥ ♣♦r st p ♦♥ t♦♠ s ♥ ② ♦♥t t♦♥tr pp ♦♥ t♦ ♦♥t♦♠ ♥ t ♣♦r st ♦ ♥♦r♥ ♦sr♦♥ ♥ t s♦♥ st et qt♦r t♦♠ s ♦♥ t♦ t ♥♥ t♦♠ ♦ ♠tt♦♠ ♥tt ♥s ♥♦r♥ ♦sr ♥tr♦sr ♥ ♥ s♥t ♥ α♦r♦♥ ♥ ♠♦st ♦ t ♦sr s♦s tr s ♦♥ ♥ ♣r♥t ♥ ♥ t♦♠ s ♥ 12P2 ♥ 12s2 ♦r trt♦♠ ♥ ♦r♦♥ rs r
♥tr ♦ t ♦♥♥ ♠s ♦r♦♥r s♦s ② r♥t r♦♠r♥s ss ♦ ♠trs t t② r ♥♦♥tss ♦t♥ ♦♠♣r t♦ tr♦♥ ♥②❬❪ st♥♥ ♦♥ 12 11♦r 102 ♥tr str s ♥st s ♥ ♥♦t ♦♠♣r t♦r♥ r②sts❬ ❪ ♦♠♣rs♦♥ ♦ t rs♣t ♦♠♣rssts ♦ t ♥tr ♥ ♥trt♦♠ ♦♥s ② ♥t♦ ♠t♦s s ♣r♦ tt♥tr α♦r♦♥ ♥♦r ♦sr ♦r♦♥ r t ♦ r♦♥ ♦♥♥trt♦♥ 4 r ♠♦r r②sts tr ♦ t② ♦r♠ ♥ ♥rt♠♦r r②st ♥ t ♥tr♦sr ♦♥s ♦ str♦♥rt♥ t ♥tr♦sr ♦♥s❬ ❪ ♦sr ♦r♦♥r s♦s♦r♠ s♣ s ♦ ♦♥t ♠trs ♦♣t♥ ♥ ♥s strtr s ♦ t tr♦♥ ♥② tt s ♥tr♥s t♦ t ♦r♦♥ t♦♠
♦sr sp ♦♥♥ s ② r ♠♦r ♦♠♣① t♥ sp ♦♥♥ ♥♠♦♥ ♦r ♦tr ♦♥t ♠trs t ttrr ♦♦r♥t♦♥ ♥st② ♥t♦♥ t♦r② s ①♣t t♦ st t♦ ①♣♥ ♣②s♣r♦♣rts rt t♦ ts ♦♠♣① sp ♦♥♥
t♦♠ strtr ♦ 4 r♦♠ rst
♣r♥♣s
♥ ts st♦♥ ♦r ♠ s t♦ r t s ♥ sts ♦t tt♦♠ strtr ♦ 4 ② ♦♠♣r♥ ①♣r♠♥t t t♦ rsts ♦♠♣t r♦♠ rst ♣r♥♣s
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r t♦♠ strtr ♦ ♦sr ♦r♦♥ r r♦♠ ❬❪♥ st♦rt ♦sr r s♦♥ r♦♠♦r rs ttt♦rs ♥♦t s♦♥ ♥ ♦sr ♥trs s t♦♠s t t p
♣♦r st t r② s t♦♠s t t e qt♦r st r r②s t♦♠s t ♥ ♥tr ♥ ♥ ♥s P♦r t♦♠s r s♦♥ t♦ ♦♥ t♦ ♥♦r♥ ♦sr t ♦♥t t♦♥tr pp ♦♥sqt♦r t♦♠s r ♥ t♦ ♦♥ t♦♠ t t ♥ ♦ t trt♦♠ ♥
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t♦♠ strtr ♦ 123 4 ♦r♦♥ r ♦♥t♥s t♦♠s♥ ♦r ♦♣ r②st♦r♣ sts r ♦ sts ♦r♠ st♦rt ♦sr♦♥ t ♣♦r ♥ qt♦r sts t t♦♠s ♣r st tr ♥ ♦rt sts r rs♣t② t ♥tr ♦ t trt♦♠ ♥♦♥ t♦♠ ♥ t ♥ ♥s t♦ t♦♠s
①♣r♠♥t② t tr r♦♥ t♦♠s r t t♦ st♥s r♦♠t t ♦r♦♥ t♦♠s ♥ t sttr♥ ♥t ♦ 13 ♥s sr② ♦s t♦ t sttr♥ ♥t ♦ 11 ♥s ♥ ♦♥② ♣r♠trs ♥ ♠sr ② ♥tr♦♥ rt♦♥❬ ❪ t♦♠ ♣♦st♦♥s r♦♠ ❳r② rt♦♥ t ♥t tt t ♥ s ♥ tt♦♥ r♦♥ t♦♠ s ssttt ♥ t ♦sr♦♥❬ ❪ s t ❳r②♦r♠ t♦r ss Z 2 r Z s t t♦♠ ♥♠r t ♦r♠ t♦rs♦r t ♥♦r♥ ♠♥ts ♥ r r② ♦s ♥ t s tt♦ st♥s t♥ ♦r♦♥ ♥ r♦♥ t♦♠s ♥ t ♦sr♦♥ ♣tr ♣r♦ ② ❳r② rt♦♥ s t ♦♦♥ ♦♥ t tr r♦♥t♦♠ s sttst② ssttt ♦♥ ♦t ♣♦r ♥ qt♦r sts t ♣r♦♠♥♥ ♦r t ♣♦r st ♦r♠ s x1−x
p6y1−y e6 ♥
0.93 ≤ x ≤ 1 ♦r t ♣♦r st ♥ 0.74 ≤ y ≤ 0.86 ♦r t qt♦rst❬ ❪
♥ ♦rr t♦ ♥ ♥♦ ♦ t t♦♠ strtr ♦ ♦r♦♥ rt r♦♥ ♦♥♥trt♦♥ ♦ ♥stt sr strtr♠♦s t ♥t♦ ♠t♦s s ♦♥ t t ♣r♣♦s ♦ ♦♠♣r♥ r♦s ♦♠♣t ♣②s ♣r♦♣rts t ①♣r♠♥t tt♦♥s r ♣r♦r♠ t♥ s♥ t ♣♥ ♣s♦♣♦t♥t ♠t♦ ♦r ts ♥ ♦♥ ♥ s ❬ ❪
trtr ♠♦s
♥ ts sst♦♥ ♦r strtr ♠♦s r♦♥ ♦♥♥trt♦♥ ♥ ♦♥sst ♦ ♦♥ ♥t ♦ t♦♠s ♣r♦② r♣t ②tr♥st♦♥s ♦ t rs tt t♦rs ♥ ①♣t♦♥ s t bipolar
♠♦ ♦♥t♥s trt② t♦♠s t♦ ♣ t st♦♦♠tr② ♥ t polar rs♣ bipolar ♠♦ t ♥ s ♥ t ♦sr♦♥s ♦♥ rs♣ t♦ r♦♥ t♦♠s ♥ t ♣♦r st ♥ t equatorial
♠♦ t ♥ s ♥ ♦♥ r♦♥ t♦♠ s ssttt ♥ t qt♦r st ❲ s♦ ♥stt ♥t r t ♥ s ♥ t ♦sr♦♥ s ♠ ♦ ♣r ♦r♦♥ chain ♠♦ ♦r ♥tr♦s♦♠ s♦rr ♥ t ♥ ♠♥ t ♥st ♦ disordered
chain ♠♦ st ♠♦ ♦♥ssts ♦ r ♥ rich chain
♠♦
ssttt♦♥ ♦ r♦♥ t♦♠ ♥ t ♣♦r st ♦ t ♦sr♦rs s① ♣♦ssts rst♥ ♥ ssttt♦♥ s♦rr s ♠s t
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♦sr s②♠♠tr② ♦ t ♥t ♦♥ r r♦♠♦r ♥ ♦r t♦♥s ♦ ♥♦t t ①♣t② t ssttt♦♥ s♦rr ♥t♦ ♦♥t st♦♦ r s♣rs ♦ rqr r♦r t ♣r♦ r♣tt♦♥ ♦ r♦♥ t♦♠ t ♥ ♣♦st♦♥ ♥ t ♦sr♦♥ ♥s s♠ st♦rt♦♥ r♦♠ t r♦♠♦r s②♠♠tr② t♦ s♥tr ♠♦♥♦♥ ❲ tt ts ♣♣r♦①♠t♦♥ ♦s ♥♦t ♠♦② t ♦♥s♦♥s♦ ts ♣♣r
t♦t ♥r②
t♦t ♥r② ♦t♥ t♥ s r♣♦rt ♥ strtr t t ♦st ♥r② s t polar ♠♦ ❲ t ts ♥r② s♦r rr♥ bipolar ♠♦ ♦♠s s♦♥ t ♥ ♥r② r ② ❱ ♣r r♦♥ t♦♠ ♥ s ♥r② r♥ s r② s♥♥t♥ ♦rrs♣♦♥s ♥ tr♠s ♦ tr♠ ♥r② t♦ t♠♣rtr r t♥ rtr♠♦r t ♥rts ♦s t ①s♦♥ ♦ t qt♦r ♠♦ t chain ♠♦ t disordered chain ♠♦ ♥ t richchain ♠♦ t s♦s tt t ♥ s ①s♦♥ ♦ t equatorial
♥ chain ♠♦s rtr ♦♥r♠ ② ♥s♣t♥ t tt ②♥♠s ♥ t s♣tr ♦r♦r t s ♦ t ♦r♠t♦♥ ♥r② ♦disordered chain ♥ rich chain ♠♦s s♦ tt ts ♠♦s r ♥st r♦s ♥ rs♣t②
r♥ ♥ t♦t ♥r② t♥ ♦r r♦s strtr ♠♦ss ♠ ♠♦r ♠♣♦rt♥t t♥ t r② ♦ ①♥ ♥ ♦rrt♦♥♥t♦♥s ttr s ♥ ♠t t♦ ♠♦ ♠❱t♦♠❬❪s ♦r ♠♦s r ♦♥② ② t ♦t♦♥ ♦ ♦♥ r♦♥ t♦♠ t ①♥♦rrt♦♥ ♥t♦♥ ♠ts t r② ♦ ♦r t♦♥s t♦ ♠❱ ♣rr♦♥ t♦♠ ♥ r②st ♣ss r♥ ② ss t♥ tt ♠♦♥t ♥r② st♥s ♥ ts ♠② t♦ r♦♥ ♣rt♦♥ ♦ t ♠♦stst ♣s ❲ s tt ts s ♥♦t t s ♥
rtr♠♦r t r♥ ♥ t♦t ♥r② t♥ ♦r strtr ♠♦s s s♠r ♥ ♥ ♥ ♥♦t s♦♥ ❬❪ t♥ ♠❱♣r ♥ ♦ r♦♥ t♦♠ s s s ♦♥♥ ♥ t ♣rtt② ♦ t ♥rts ♦r ♦r ♠♦s ❲ ♠♣s③ tt ♥ t s♠ ♣②s ♦♥t♥t ♦♥tss ♥♦t tt ②♦♥strt♦♥ tr s ♥t♦♥ ♦ rr♦rs t♥ t ①♥ ♥r②♥ t ♦rrt♦♥ ♥r② ♥ s ♥t♦♥ s ♥♦t t♥ ♥ t ♥t♦♥s❬❪ ♠s t ♦♠♣tt♦♥s ♦t ♥ ♥ ♥ ♦rt ♣r♦r♠♥ ♥ ♦♠♣r♥ ♦r♦r ② ♦♥strt♦♥ ♥t♦♥s t♥ t♦ rs t ♥r② ♦ t ss ♥s ♣s ♥ ♥t♦ ♦r tr stt②❬❪ r♦r t s ♥trst♥ t♦ ♦♠♣t ♥r②r♥s ♦t ♥ ♥ ♥ ♦rr t♦ ♦t♥ ♥ rr♦r r ♥t r②st ♥st② rs ♠♦♥ t ♣♦②♠♦r♣s s r♥s ♥ t♦♥strt♦♥ ♦ t ♥t♦♥s ♦ ♥♦t ♠♦② t rt stts ♦ ♦r
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♦t ♥r② ❱ ♣r ♥ ♦ r♦♥ t♦♠ ♦t♦♥ trs♣t t♦ t polar ♠♦ t♥ r♦♠ ❬ ❪
♦ ♥ ♦sr♦♥ ♦t ♥r② ❱Polar 11p Bipolar 2 102
p 12 Equatorial 11e Chain 12 Disordered chain 11p rich chain 102
p
strtr ♠♦s ♥ ❬❪
r rsts ♦r 4 r r♦st ♥st ♣♣r♦①♠t♦♥s ♦♥trst♥②rt s ♦ t ♥r② ♦ r♥t r②st strtrs ♦ ♣r r♦♥♦r ♦ ♣r ♦r♦♥ r ♠ ss r♦st r♣t s ♦♥ ♠♦r st t♥♠♦♥ ② ♠❱t♦♠ ♥ ❬❪ rs ♠♦♥ s ♦♥♠♦r st t♥ r♣t ♥ ② ♦♥② ♠❱❬❪ ❩r♦♣♦♥t♠♦t♦♥ ♠t rrs t ♠♦st st ♣s r② t tr♠♥t♦♥♦ t stt② ♦ ♣r r♦♥ ♣♦②♠♦r♣s s ②♦♥ t ♣t② ♦ t ①♥ ♥ ♦rrt♦♥ ♥t♦♥s❬❪ s s ♣rtr② tr ♦rr♣t s ❱♥ r ❲s ♥trt♦♥s r ♥♦t ♣r♦♣r② t♥ ♥t♦♦♥t ♥ ♥♦r ♥ ❬❪
r♥♥ t♦ ♦r♦♥ ♥ t♦♥s ♦r ♣rt r②sts α12 s ♥s♦♥ t♦ ♠♦r st t♥ ♣rt t♦♠sβ♦r♦♥ ♥ ❬❪♥t t♦♥s ♣r♦r♠ t ♥sr s♠♣♥ ♦ t r♦♥ ③♦♥♦ α♦r♦♥ ② t♦t ♥r② r♥ t♥ α ♥ β♦r♦♥ ♦ ♠❱ ♣r t♦♠ ♥ ♥ ♦ ♠❱ ♣r t♦♠ ♥ ♦t ♣rt♥tt α12 s ♠♦r st❬❪ r♥ t♥ ♥ sr② ♠♣♦rt♥t ♥ ts s ♠❱ ♣r t♦♠ ♥r② r♥ ②s♥ rr♦r r ♦ t ①♥ ♥ ♦rrt♦♥ ♥t♦♥s
♥ t ♦tr ♥ t♦♥s ♥♥ ts ♣rt tt tβ♦r♦♥ s t ♠♦st st ♣s❬ ❪ ♥r② r♥ trs♣t t♦ ♣rt α♦r♦♥ s t ♠♦st ♠❱t♦♠ r ♦ t ♦♠♥t♦♥ rr♦r r ♦ ♠❱ s♦ ♥ ts s t♦♥s r t♠t ♦ ①♥ ♥ ♦rrt♦♥ r② ❲ tt t t♦rttr♠♥t♦♥ ♦ t stt② ♦ ♣r r♦♥ ♦r ♣r ♦r♦♥ ♣♦②♠♦r♣s srs rtr t♦rt rtr♦s t♦ ♦r♦♠ t ♥tr♥s ♠tt♦♥♦ ♥ ♥t♦♥s
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tt ②♥♠s
r♥♥ t♦ tt ②♥♠s ♦♠♣t ❬❪ t ♣♦♥♦♥ rq♥st t ♥tr ♦ t r♦♥ ③♦♥ ♦r t polar equatorial ♥ chain ♠♦s ❲ ♥♦t tt ♥ ♦r s t tr♦♥ strtr ♥ t tt ②♥♠s r ♣r♦r♠ t♥ t s♠ t♦rt r♠♦r s♠♣s tt ♥s ♥ ♦t rtr ♣♦t♥t ♥ ♥ ①♥ ♥ ♦rrt♦♥ ♣♦t♥t r t t♦ rst♦rr ♥ ♣rtrt♦♥ t♦r② srs♣♦♥s ♦ t sr♥♥ ♣♦t♥t t♦ t ♣rtr♥ ♣♦♥♦♥ s ♦♠♣ts♦♥sst♥t② s♦ tt t r♥ t s♠♠♣r ♠♦s t tr♦♥ ♥st② ♦s ♥♦t r② st t t t♦♠ ♠♦t♦♥ ♦♠♣r t♦ ♥♦r ♠♦s ♦r ♦♠♣tt♦♥ s♠ ♦r ♣♦♥♦♥ rq♥s ♥ s♣♠♥t ♣ttr♥s ♦s ♥♦t ♦♥t♥ ♥② tt♥ ♣r♠tr
♥ r r♣♦rt t t♦rt s♣tr♠ ❬❪ ♦r ♥rr s♦r♣t♦♥ ♦♠♣r t t r♦♠ ❬❪ P ♣♦st♦♥s r r② ♥ttr r♠♥t ♦r t polar ♠♦ ❲t rs♣t t♦ t equatorial ♠♦ts s ♣rtr② tr ♦r t ♠♦ ♦sr ♥r ♠−1 s s rt♦♥ ♦ t ♥♥tr t♦♠ ♣r♣♥r t♦ t ♥ ①s
sr♣♥② t♥ ①♣r♠♥t ♣ ♣♦st♦♥s ♥ chain ♠♦♦♥s s ♥ ♠♦r ♣r♦♥♦♥ t rq♥s ♣ ♦srr♦♥ ♠−1 ♦♠s r♦♠ t strt♥ ♦ t ♥ ♥ t ♥♥ t♦♠s rt ♥ts②♠♠tr② ungerade rt♦♥ rq♥②s ② r rr ♥ t ♥ s P♥ t♥ ♥ t ♥ s P♥s ♥ ❲ tr♦r ♦♥ tt t chain ♠♦ ♥ srr strt♥ ♦ ♥s ♦s ♥♦t ♦r ♥ ♠sr♥rr s♣tr
r♥♥ t♦ ♣ ♥t♥sts ♥ t polar ♠♦ s♦ ♥ ♣♥ t t ♦♥ s♥ ♣r♠tr ♥ t ♣r♦♣♦rt♦♥ t♥ u ♥ 2u
♣♦r③t♦♥s ♥ ts ♣r♦♣♦rt♦♥ ♦ ♣♦r③t♦♥s s ♥♥♦♥ ♥ ①♣r♠♥t r t s ♥ stst♦r② r♠♥t t ①♣r♠♥t s♦♥ ttt t♥s♦r ♦ t rs s ♣r♦♣r② ♦♥t ♦r ♥ t t♦♥s
♥♦t s♥t rt♦♥ ♥ 4 s ♦sr tr ♥ ♠♥ s♦♥ ♦r♥ ♥rr s♦r♣t♦♥ s s s t ♥tr ♦ t ♦sr♦♥ s ♥♥rs♦♥ ♥tr ♣rt② ♦ ♠♦ t rs♣t t♦ t ♥rs♦♥ ♥tr s♥q ♥ t ♠♦ s tr ♥rr t ♥r u ♥ 2u ♠♦s♦r ♠♥ t r g ♥ 1g ♠♦s r♦r t s ♥trst♥ t♦♦♠♣r ♦r ♦♠♣t ♣♦♥♦♥ rq♥s s♦ t♦ t ♣s ♦sr ②♠♥ sttr♥ ♦ ♦r ♥♦ ♦♥② ♦♥ ♠♥ s♣tr♠ ♠sr♦♥ s♥ r②st s ♥ r♣♦rt ♥ t trtr ♥ t s ♦r rr♥♥ r sr ♣s r r♦♥ ② s♦rr ♥ ♥ t rq♥② r♥ t t♦rt rq♥s ♦ ♦r ♠♦s ❬❪ ♦♥t t ①♣r♠♥t
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r ♥rr s♣tr♠ ♦ 4 r♦♠ ❬❪ s ♥ ①♣r♠♥t❬❪ ♦ ♥ t♦rt s♣tr♠ ♦r t chain ♠♦ ♣♥ equatorial ♠♦ ♣♥ polar ♠♦ ♣♥ P♥ s♣tr♠♦r t polar ♠♦ r t ♣r♦♣♦rt♦♥ ♦ u ♥ 2u ♣♦r③t♦♥s s♥ tt t ♦♥ s♥ ♣r♠tr t♦ r♣r♦ t ♥♥♦♥ ♠①tr♦ ♣♦r③t♦♥s ♥ t ①♣r♠♥t s t①t
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♦♥tss t♦ rt♦♥ ♠♦s t ♠−1 ♥ ♠−1 ♥ s♥ ♦t ♦t ♠♦s ♥♦ ♦♥ ♥♥ ♠♦ ♦sr t ♠−1 ♦rrs♣♦♥s ♥ ♦r t♦♥s t♦ r♦tt♦♥ ♦ t ♥♥ t♦♠sr♦♥ t ♥♥tr ♠♦ ♦sr t ♠−1 ♦rrs♣♦♥s t♦t rt♦♥ ♦ t ♦sr♦♥ s ♠♦ s ♥ ♣r♦s② ♦sr♥ α♦r♦♥ ♥ t s s♦♥ t♦ str♦♥② r♠♦♥❬❪
♦♠♣r t t s♦rr♥ r♦♥♥ ♦ ♦tr rt♦♥s ts♠ ts ♦ t t♦ ♣s ♥t tt t t♦ ♦ ♠♥t♦♥ ♠♦sr r♠♦♥ ♥ ♦r♦♥ r rq♥② r♥ t♥ t♠ str♦r q♥tt② s ①♣t t♦ r♣r♦ ♥ ♦r t♦♥s t♥ t r♠♦♥ ♣♣r♦①♠t♦♥ r s♦s tt t①♣r♠♥t rq♥② r♥ s r♣r♦ ♦♥② ② t polar ♠♦s r♥ s s♠r ② t♦r ♦ ♥ t equatorial ♠♦ r♦rt equatorial ♠♦ ♥ srr s ♣♦ss strtr ♠♦ ♦r4
st t ♥♦t st ♥♦t tt ♦r t♦♥s s♦ ♥♦ rt♦♥♠♦s ♦ ♠−1 r ♠♥ ♣s r ♦sr♦ ♠−1 ♥ ♠♦st ♦ t ①♣r♠♥ts ❬❪ ♦♠ r♦♠ t ♦ tst♦♥ rs ♦r ♠♥ sttr♥❬❪ ω s t ♣♦♥♦♥ rq♥② t♥t♥st② ss ω4 ♣♦r t ♦ rq♥② s t s♥tr♦ ♥st② ♦ stts ♦ ♦st ♣♦♥♦♥s s ♣♦♥♦♥s ♦ s t♦ t s♠♣s ♥st strtr s♦rr t② r ①♣t t♦ ♥s♥ ♥ s♠♣s
r ♥t s♦♥♥
♥ t ♣r♦s sst♦♥ t ♦♠♣rs♦♥ ♦ t t tt ②♥♠s t ①♣r♠♥t ♦ s t♦ ♥t② t t♦♠ strtr ♦ 4 s ♥ ♥ 11p ♦sr♦♥ t t r♦♥ t♦♠ ssttt ♥t ♣♦r st t s♦ s♥ tt ①♣r♠♥t rt♦♥ s♣tr♦ s♦ r r♦♥♥ s ② ssttt♦♥ s♦rr ♥♦r strtr ts r♦r t♦ ♦♠♣t s♣tr tt ♠ ♦ r♥♥ ♠♦r ♦t ts ts❬❪
r s ♦ t ♠ st ♦ 13 ♥ r♣♦rt ♥ t ①♣r♠♥t trtr ①♣r♠♥ts ❬❪ r t ♦♥♥♦tr ♦t t♦ s♥s r♦♥ ♥ ♣♣♠ ♥② ♦♥ ①♣r♠♥t♦r r♣♦rts s♥ t ♣♣♠❬❪ ❲ ♥♦t tt ts ①♣r♠♥t ②t st rs♦ s♣tr♠ s t♦♥ ♣ ♠② s♦ ①♣♥ ② r♥t r♦t ♦ s②♥tss ♦ t s♠♣ ②♥ ♠♦r strtr s♦rr
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r ♠♥ s♣tr♠ ♦ 4 r♦♠ ❬❪ ❯♣♣r ♣♥ ①♣r♠♥t❬❪ ♦r ♣♥ t♦r② ♦r chain polar ♥ equatorial ♠♦s♥ s♣♥s ♥ rs♣t② ♦ ♥ g ♠♦ s ♥ 1g ♠♦
r t♦♥s s♦ tt t polar ♥ bipolar ♠♦s ①♣♥ t①♣r♠♥t s♥s r② Ps r♦♥ ♥ ♣♣♠ ♦♠ r♦♠♥♥ r♦♥ t♦♠s ♥ r♦♠ r♦♥ t♦♠s ♥ t ♣♦r sts ♦ t♦sr rs♣t② polar ♠♦ bipolar ♠♦ ♦s s t♦ ①♣♥ t s♥ t ♣♣♠ ♦sr ♥ ❬❪ ♦♠s r♦♠ r♥t sr♥♥ ♦ t ♠♥t ♥ t♦ r♦♥ t♦♠s r ssttt ♥ t s♠ ♦sr♦♥ r♦♠ t ♣♦♥t ♦ ♦ t t♦t ♥r②t ♦rt rt ♦t♦♥ ♦ t♦ r♦♥ t♦♠s s ♥ ♥t♣♦ ♣♦r♣♦st♦♥s❬❪ rtr♠♦r ♥♦t tt ♣r♥ts ♦ ♣r 12 ♦sr r ♥ssr② t♦ ♣ t st♦♦♠tr② ♥ ♦r t♦♥s
r♥♥ t♦ t equatorial ♠♦ t t ♦ t ♣ ♣rt ② tt♦r② tt ♦♠s r♦♠ ♥♥ r♦♥ t♦♠s s rtr t♥ ♣♣♠ s s s♥♥t② rr t♥ ♥ t ①♣r♠♥t ♦r♦rt t s♣tr♠ ♦ t 11 ♥s s s♦r ♦♥ t s ♦♥t ♠ sts ♦r t equatorial ♠♦ r t r♥t ts ♥♥ t ♦sr s♣tr♠ s♦s s♦r ♦♥ t s ♦♣♦st ♠ sts ♥ r♠♥t t t polar ♠♦❬❪ s
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♦ s t♦ srr t ssttt♦♥ ♦ ♦♥ r♦♥ t♦♠ ♥ t qt♦rst s ♣♦ss strtr ♥rts rt♦♥ ♥ s♣tr tsr ♦♥ ts ♣♦♥t
rtr♠♦r r♦♥ t♦♠ t t ♥tr ♦ t ♥ ♦ ② ♠ st s r s ♣♣♠ ♥ t 13 s♣tr♠ s♥ s♥♦t ♦sr ♥rts rt♦♥ ♥ s♣tr r♦♥ t t tt ♥♦ ♥s r ♣rs♥t
♥② ♦♠♣rs♦♥ ♦ ♦♠♣t ♥ ♦sr s♣tr ♦s st♦ r t ♦♥s♦♥ tt t t♦♠ strtr ♦ 4 ♦♥ssts ♦ ♥ ♥ 11p ♦sr♦♥ t ♦♥ r♦♥ t♦♠ ssttt ♥t ♣♦r st ♣r♥ts ♦ t ♦sr 10
p2 ♦r 12
t♦♠ strtr ❲ st♠t tt ♦ t ♦sr r 11p r 102 ♥ r 12 ♦♥s❬❪
4 13C ♠ sts δTMS ♣♣♠ ♥ t ♦rrs♣♦♥♥ ♣ ♥t♥sts ♥ rts ①♣r♠♥ts r♦♠ ❬❪♦t tt t rs♦♥♥ t ♣♣♠ s s ♦♥② ♥ t ①♣r♠♥t♦ ❬❪ ♦r② ♠♦s s ♥
①♣r♠♥ts ♦r② ❬❪ ❬❪ ❬❪ Bipolar Polar Equatorial Chain
± ♠♦s ± s tr ♠♦s
♦♥s♦♥s
♥s♣t♦♥ ♦ ♦t ♥ rt♦♥ s♣tr ②s t s♠ ♣tr ♦rt t♦♠ strtr ♦ 4 ♥ ♥ 11p ♦sr♦♥ t♦♥ r♦♥ t♦♠ ssttt ♥ t ♣♦r st
r s ♥♦ s♥ ♦ t s♥tr ♦ t♦♠ t t ♥ ♥tr ss ♠♣♦rt♥t ♦♥sq♥s ♥s ♥ ♥♦t rs♣♦♥s ♦r t♦r♦♥ r ♠♦r♣③t♦♥ ♥ s♦ ①♣r♠♥ts❬❪
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r 4 11B ♠ st s♣tr♠ ①♣r♠♥t t sr♦♠ ❬❪ ♥ t♦r② ♦r t polar equatorial ♥ chain ♠♦s r♦♠ ❬❪
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st② 10p2 ♦sr r ♣rs♥t s ts ♥ s♦♠ 4 s♠♣s
❲ st♠t tt ♦ t ♦sr r 10p2 s ts
♥ ①♣r♠♥t② ♦sr ♣rs♥ ♦ 12 ♦sr s ♥rr ② s r♦♠ t ♦sr st♦♦♠tr② ♦ ♦r ♥♦ tr r♥♦ rt ♦srt♦♥s ♦ 12 ♦sr ♥ 4 s♠♣s
t♦♠ strtr ♦ ♦r♦♥ rs t
♦r r♦♥ ♦♥♥trt♦♥s
t r♥ t t s ♦ 4 ♥♦ ♦t ♦r♦♥ rs t ♦rr♦♥ ♦♥♥trt♦♥s ♠st ♠♣r♦ ♦t ①♣r♠♥t② ♥ t♦rt② t♦♠ strtrs r ♥♦t ♥rst♦♦ ♥♦r r t ♣sstts❬❪
❲ ♦♠♣t t qr♠ ♦♠ ♥ ♦r♠t♦♥ ♥r② ♦sr strtr ♠♦s t♦♥s r ♣r♦r♠ t♥ s♥t ♣♥ ♣s♦♣♦t♥t ♠t♦ ①♥ ♥ ♦rrt♦♥ ♥r②s ♦♠♣t ♥ t ♦ ♥st② ♣♣r♦①♠t♦♥ ♣s♦♣♦t♥ts ♦♦r♦♥ ♥ r♦♥ r ♦♣ t t r♦rrt♥ s♠❬❪♦r ♦r♦♥ rs ♦r♦♥ ♥ ♠♦♥ ♣♥ s ♣ t♦ ♥t t♦♦ ② ♥ ♥ ♥ t ss st rr ♦t r♦♥ ③♦♥ s ♥ s♠♣ t ♣♦♥ts ♥ t ♥t s♠♦♥♦♥ ♦r♦♥ rs t r♦♥ t♦♠s ♥ t ♦sr ♥ ♣♦♥ts ♥ t ♥t s r♦♠♦r e.g . ♦r α♦r♦♥ ♦r ♦r♦♥ rs t 12 ♦sr ♦r ♠♦♥ ♣r♠trs ♥ t♦♠rs ♦ r♦♠ ♥ r① t♦ ♠♥♠③ t t♦t ♥r② t tt♦rt qr♠
♦r♠t♦♥ ♥r②
♦♦♥ ♣rs♥t ♥♦ α♦r♦♥ rs♣ ♠♦♥ s t ♠♦st st♣♦②♠♦r♣ ♦ ♣r ♦r♦♥ rs♣ r♦♥ t♥ ♦r tr♣ss s sst♦♥ rst② s ♦ t ♦r♠t♦♥ ♥r② ♦♦r♦♥ rs ♥ ♦♠♣t ♦r ♦r♠t♦♥ r♦♠ α♦r♦♥ + ♠♦♥
❲ rst ♦♥sr strtr ♠♦ s st t ♥ ts ♦r♠t♦♥ ♥r② E f s ♥t t rs♣t t♦ ♦♠♣♦st♦♥ ♥t♦ α♦r♦♥ +♠♦♥ ♥ ♥ E f s t s♠st ♦♥ ♠♦♥ strtrs t t s♠r♦♥ ♦♥♥trt♦♥ ♠♦♥ ♠♥② strtr ♠♦s tt ♥ ♥stt ♦♥② t ♠♦s ♦ r ♦♥ st❬❪ t r♦♥ ♦♥♥trt♦♥ t polar ♠♦ 4p s t ♦st ♦r♠t♦♥ ♥r② t ♣♣r♦①♠t② r♦♥ ♦♥♥trt♦♥ 132 s t ♠♦st
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st ♣s t ♦♥ssts ♦ ♦♥ 12 ♦sr♦♥ ♥ ♦♥ ♥ ♠♦ ♦♠♥♥ ♦t 4p ♥ 132 ♦♥ t ♥t s s♦ st t
♦♥② tst ♦r strtr ♠♦s ♥st ♦♠♣♦st♦♥ ♥t♦ 4p + α♦r♦♥ ♦r r♦♥ ♦♥♥trt♦♥ ♦r t♥ ♦r4p + ♠♦♥ ♦r r♦♥ ♦♥♥trt♦♥ rtr t♥ r♣rs♥② ♥♦ ♣ss ♥ ♦♥ st ♥st ts ♦♠♣♦st♦♥ ♥ ♦r 132 ♦♥ E f
B4C ♠❱ ♣r t♦♠ r rst s s♠r t♦
♣r♦s t♦♥s ♠❱t♦♠ t ♥ ♠❱t♦♠ t ♠♥tt♠♣rtr❬❪ ♠♦ ♦♠♥♥ 4p ♥ 132 s s♦ ♠r♥②♥st f
B4C ♠❱t♦♠ s ♥♠rs r ♦r
t♥ r② ♦ t ①♥ ♥ ♦rrt♦♥ ♥t♦♥s s sst♦♥
♣♥ t ♥♥tr t♦♠ ② ♥② ♥ 4p ♦r ♥ 132 ②st♦ ♠♦s t ♦r r♦♥ ♦♥♥trt♦♥ rs♣t② sr②sts t ♥s r ♦♥ t♦ ♥t ♦r♠t♦♥ ♥r② trs♣t t♦ ♦r♠t♦♥ r♦♠ α♦r♦♥ + ♠♦♥ ② ♦r ♣♦st ♦r♠t♦♥ ♥r② ♥ ♥ s♦ ♦♠♣♦s♥t♦ 4p + α♦r♦♥ ♦r 4p + ♠♦♥ ♥ tr♠♦②♥♠ ♦♥t♦♥s
♥ ♦♥s♦♥ ♦♥② 4p s ♦♥ t♦ tr♠♦②♥♠② st ♥♦r t♦♥s ♦r ♦tr ♣ss r ♦♥ tr♠♦②♥♠② ♥st♥r ♠♥t ♦♥t♦♥s s t t② r ♥rt② ♦st♦ ♦♠♣♦st♦♥ ♥t♦ 4p + α♦r♦♥ ♦r 4p + ♠♦♥ ♥tr♦♣② ts r ①♣t t♦ rtr ♦r ts ♦♠♣♦st♦♥ t ♥♠♥t t♠♣rtr ♥ ♦r ♥t s ♦ t ♦r♠t♦♥ ♥r② t rs♣t t♦ α♦r♦♥ + ♠♦♥ t ♦r ♣ss ♦♥♦♥tss ①st tr ♥ ♠tst stt ♠♦♥ ♦s ♦r s ts ♥ ♥ ♦trs ♣rt r②st ♦ 4p s 10
p2 ♦s s sst♦♥
r♥♥ t♦ ♥st strtrs❬❪ ♦r s ♦ t ♦r♠t♦♥ ♥r②t rs♣t t♦ α♦r♦♥ + ♠♦♥ r tr ♥ t♦tr t♣r♦s sts❬ ❪ ♦♥ ♥ ♦r♠t♦♥ ♥r② tr 4p s t ♠♦st st ♣s ♦♠ ♦t ♦r ♠♦s ♦r ssttt♦♥ s♦rrr♦ ♥ t bipolar ♠♦ r♦ ❲ ♥rr r♦♠ t r♦♥♥ ♦ t ①♣r♠♥t rt♦♥ ♠♦s tt ssttt♦♥ s♦rrs ♣rs♥t ♥ ♥s♣t♦♥ ♦ t ts s tt t bipolar ♠♦ s♦sr s t ♥ t s♠♣s ❲ ♥♦ ♦♠♣r ts t♦ ♠♦sr rst ♣♣r♦ t♦ ♦♥t ♦r t t ♦ ssttt♦♥ s♦rr r♦ s t♦ ♦ t ♥ t♦ ssttt t r♦♥ t♦♠ ♦t s♦♥ ♦sr♦♥ t r♥t ♦t♦♥s ♥ t ♣♦r st t ♦s ssttt♦♥ s♦rr ♦♥ t ♦r♠t♦♥ ♥r② s ♥♦t r♦ ♥ t strt♥ r♦♠ t♦ ♥t ♦sr 11p 11p ♦♥ ♦
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①♣t tt t t ♦♥ t ♦r♠t♦♥ ♥r② ♦ ①♥♥ ♦♥ r♦♥t♦♠ ♥ ♦♥ ♦r♦♥ t♦♠ ♥ t s♠ ♦sr♦♥ rst♥ ♥ ♥♦♥ ♥t 11p 11p′
r♦ s♦ ♠ s♠r t♥ tt ♦ bipolar
♠♦ r ♦♥ r♦♥ t♦♠ ♥ ♦♥ ♦r♦♥ t♦♠ r ①♥ t♥t♦ ♥♦r♥ ♦sr ②♥ 10
p2 12 r♦ ♦r ts s
♥♦t t s t♦ ♠♦s ♦♠♣r ♦r♠t♦♥ ♥r② r♦s ♥ ❲ t♥ ♦♥ tt tr ♦r st② ♦ ssttt♦♥ s♦rr s ♦♥r♥ t rs♣t t♦ t ♥♠r ♦ ♥t s♥ tt rr s s♦ st ♦r tt ssttt♦♥ s♦rr ♠t ♥rt② ss ♦r t♥ ♦ ssttt♦♥ ♥ t ♦sr♦♥
t r♦♥ ♦♥♥trt♦♥ ♦ ♦♥ ♥t s ♦ t ♥r② ♦r t equatorial ♥ chain ♠♦s ♣r♦s② sss ♥ sst♦♥ r♦s ♥ ♦♥tss ts ♠♦s ♠r ♦r♠t♦♥ ♥r② t♥ t bipolar ♠♦ s ♦♥sst♥t tt t tt ♥s♣t♦♥ ♦ s♣tr♦s♦♣ t ♥s s t♦ ① t♠♦r♦r ♦♥ ♣♦st ♦r♠t♦♥ ♥r② ♦r t disordered chain ♥rich chain ♠♦s ♦s t♦ ① tr ♦r♠t♦♥ ♥ t ♦♠♣♦♥s r♦s ♥
t r♦♥ ♦♥♥trt♦♥s ♦tr t♥ ♦r ♥st ♠♦s s♦ ♣♦st ♥r② ♦ ♦r♠t♦♥ rr t♥ ♦r ①♣t r② ♦♥ ①♥ ♥ ♦rrt♦♥ ♥t♦♥s s sst♦♥ ② ♥ ♥♦t ♣r♦ ♥ ♠tst stt ♦r s ts ♦r ♥st♥ ♥srt♦♥♦ r♦♥ t♦♠s ♥ ♥trstt r②st♦r♣ sts s ♣r♦ t♦ ♥ ♥♣♣r♦♣rt ② ♦ ♠♦♥ ♦r♦♥r ♦r♦♥ rs r♦s ♥ ♥♦tr ♠♦ s ♥ sss ♥ t s♥t trtr♦♥t♥s ♦♥ t♦♠ t t ♥tr ♦ t ♦sr♦♥❬❪ ♠♦ s r② ♣♦st ♦r♠t♦♥ ♥r② r♦s ♥ r♦r ♦ ♥♦t ①♣t t♦ ♦sr ♠trs ♥ ♦r♦♥ ♦sr ♦♠♠♦t ♦♥ t♦♠ t t ♦sr♦♥ ♥tr
sss♦♥
4 ♦r♦♥ rs t ♦ r♦♥ ♦♥♥trt♦♥s ♣r♦② r ♠①trs ♦ r♦s ♥ ♦s t r♦♥ ♦♥♥trt♦♥ ♦tr t♥ t ♥♠r♦s strtr ♠♦s ♦ ♥ srr s ♣♦ss ♥ts ♥st ♠♦s ♦ r s s t♦rs tr♠♥t♦♥ ♦ t t♦♠ strtrs ♦♠♠♦♥ t♦t tt ♥tr♥s strtr ts ♦r♥t r♦♠ t ♥rt② ♠♦r ♦r ♦♥rt♦♥ ♦♠♣r t♦ t strtrs❬❪ s t♦rtstt♦♥ ♥ t sr♣rs ♦♠s r♦♠ t s♠ ♥♠r ♦ ♣♦ss ♥ts rtr♠♦r t ♦♠♣tt♦♥ ♦ t ♦r♠t♦♥♥r② s♦s tt ts strtrs r ♥♦t st t rs♣t t♦ ♦♠♣♦st♦♥ ♥t♦ 4p ♦r♦♥ r + α♦r♦♥ ♦r 4p + ♠♦♥
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s ♠tstt② ①♣♥s ② ♣②s ♣r♦♣rts ♠sr ♦♥ r♥t s♠♣s str♦♥② ♣♥ ♦♥ t s②♥tss r♦t ♦r ♣rs② t♥stt② ♦ 132 t rs♣t t♦ ♦♠♣♦st♦♥ ♥t♦ 4p + α♦r♦♥s♦ ①♣♥s t ♦ ♥ t r♦t t♦ s②♥ts③ s♥ r②sts ♦132❬❪ ♥stt♦♥ t ♥t♦ ♠t♦s ♦ ♥ ♥t st③t♦♥ ♥r ♣rssr s♦ s s♦♠ t ♦♥ ts qst♦♥
t s r② ♥ ♦r ♦r♦♥ rs t ♦ r♦♥ ♦♥♥trt♦♥s ♦ ♦r ♥♦ ♦♥② t ①♣r♠♥t ♦ ❬❪ ② t♦♥ ♦r♦♥ rs t ♥ r♦♥ ♦♥♥trt♦♥s s ①♣r♠♥t s♦ t♦ t♦♥ s♥s s t♦s ♦ 4p ♥t♥sts ♦ts t♦♥ ♣s ♦r r② r② t♥ ♥ t♦♠r♦♥ ♦♥♥trt♦♥s s♦ tt ♦♥ ♥ ♥♦t s♠♣② ♦♥sr t♠ s ♥r♣r♥ts ♦ ♦r♦♥r ♦r♦♥ rs ♣rs♦♥ ♦ t t♥qs rst② ♥rs s♥ ♥ t ♦ ♦ t♠♦st ♠♣♦rt♥t♦ ♣r♦r♠ ♥ ①♣r♠♥ts ♦♥ ♦r♦♥ rs s ♠t♦s r♠tr ♥♦ t♦ ♣ t ♥rst♥♥ ♦ ts ①♣r♠♥ts
♥② ♦♥ s♦ ♥♦t tt tr ♠♦s ♦ ♦♥t♥ ♥ ♦♥♠r ♦ tr♦♥s ♣r ♥t ♥ ♥ r ♠ts ♥ ♥② ♥ t♦r②♦♥trst♥② ♦r♦♥ rs r t♦t t♦ s♠♦♥t♦rs❬❪ ❲t♥ tt t ①♣r♠♥t ♥♦ ♠t t♦♦ s♠♣ ♣♥♥t ♣rs ♠sr♠♥t ♦ t rsstt② ♦♥ ♥ s♠♣s s ♥ ♥♦ r② ♣ ♥ sr♠♥t♥ ♠♦♥ t ♠♦s ♦ ❲s♦ ♥ t ♥①t st♦♥ tt t str♥t ♦ t tr♦♥♣♦♥♦♥ ♦♣♥♦♠♥ t t r rt♦♥ rq♥② ♠s ♦r♦♥ rsr② ♥trst♥ ♠t stt ♥
tr♥t ♦ t tr♦♥♣♦♥♦♥ ♦♣♥
♠♦♥ ♣♦ss ♥ts ♦r t r♦ ♦ t ♠♥ ♥ ♦ ♦r ♦r♦♥rs t ♦ r♦♥ ♦♥♥trt♦♥s 132 s t ♦rt ♦♥ s sr ♥ ♣r♦s st♦♥ t ♦♥ssts ♦ ♦♥ 12 ♦sr♦♥ ♥ ♦♥ ♥ ♦ ♥♠r ♦ tr♦♥s ♠s t ♠t 4s s♠♦♥t♥ 132 ♥ s 4 r②st ♦♣ t ♦♥♦ ♣r ♥t ♥ ♥② ♥ t♦r② ♥ ♥r ♥ ♥ ♣rtr♠♣s tt 132 s ♠t r ♦ t stts t t r♠ r ♦sr stts s s ♦r ♠♥ ♠♦tt♦♥ t♦ ♥stttr♦♥♣♦♥♦♥ ♦♣♥ ♥ ts ♠tr s ♣r♦t♦t②♣ ♦ ♠t♦sr strtrs
❲ tr♦r ♦♠♣t t tr♦♥♣♦♥♦♥ ♦♣♥ ♥t♦❬❪ ② r♥ r♦♥ t r♦♥ ③♦♥ ♦t♥ ♦♣♥♣r♠tr q t♦ s q♥tt② s ♥ 2❬❪ ♣♦♥♦♥ ♥st②
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♦r♠t♦♥ ♥r② ♦ st ❬❪ ♦r♦♥ rs t T = E f
♠❱t♦♠ rst ♦♠♥ ♠♦ ♥♠ s ♥ ♦♥ ♦♠♥ r♦♥ ♦♥t♥t ♥ t tr ♦♠♥ ①♣t ♦r t polar ♠♦ ♥ t♠♥ts t ♦r♠ ♦ t ♦sr♦♥ rs♣ ♥ s ♥ ♥ rs♣ ♦t♦ rts ❱ st♥s ♦r ♥② s ♦r E f s ♦♠♣t ♥ ♦r ♦r♠t♦♥ r♦♠ α12 + ♠♦♥ s ♦ t ♦r♠t♦♥ ♥r②♦ ♣r r♦♥ ♦r ♦r♦♥ ♣ss r s♦ ♥
♦ t trs s ♦r ♦r② ①♣
r♣t [1][2] ±[5]
♠♦♥ 11p❱ [2]
Polar 4p [2][3][4] ±[6]
±[7]
11p12 ❬❪[4]
12❱ [2] [4]
12 [2] [4]
α12 [2]
❬❪ ♦r♠t♦♥ r♦♠ r♣♥ t ❱♥ r ❲s ♦rs ❬❪❬❪ ♦r♠t♦♥ r♦♠ r♣t ♥ α12 ♥ ❬❪❬❪ ❬❪❬❪ ❬❪ qr rts ♥t ♥ ♥r② r♥ ♦r E
f ❬❪ t ♥ ❬❪❬❪ ♥t♣② ♦ ♦r♠t♦♥ t ❬❪❬❪ ♥r② ♦ ♦r♠t♦♥ t ♠♥t t♠♣rtr ❬❪
s♠ s ♥ t E fB4C
♦♠♣t ♥ ♦r ♦r♠t♦♥ r♦♠ 4p + ♠♦♥ ♦r r♦♥ ♦♥♥trt♦♥ rr t♥ ♥ r♦♠ 4pα♦r♦♥ ♦r r♦♥ ♦♥♥trt♦♥ ♦r t♥
♦ t trs fB4Cp+α−B12
∆fB4Cp+diamond
11p❱ Polar 4p
11p12 12❱ 12
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❯♥st ❬❪ ♦r♦♥ rs ♦r♠t♦♥ ♥r② E f ♠❱t♦♠r♦♠ α12 + ♠♦♥ qr rts ♥ts ♥ ♥r② r♥ ♦r f ♦♠♥ ♠♦ ♥♠ s ♥ ♦r r♦ ♥♠r ♦♠♥ r♦♥♦♥♥trt♦♥ ♦♠♥s ♥ sr♣t♦♥ ♦ t ♥ ♥ ♦ t♦sr♦♥ rs♣t② ❱ st♥s ♦r ♥② ♦♠♥ ♥ts t♦♥ t♦♠s r ♥ ♥trstt ♣♦st♦♥s ♥ ♦♠♥ ♥ts t♦sr♦♥ ♦♥t♥s ♦♥ t♦♠ t ts ♥tr
♦ ♦r t ♥ ♦sr♦♥ ♥tr ♦ s ♦r②r♦ ♥♠r stt ♥tr ♦r
102
r♦ 2 11p11
p′
❬❪ ❬❪[1]
♣♦r 2 102p12
qt♦r 11e [1]
♥ 12 s♦rr 11
p ♥
r ♥ 10p2
r♦ 12
❱❱ 11p [2]
❱ 11p [2]
❱ 11p [2]
11p [2]
r♦ 12 r♦ 12
♥♦♥ 11102 1112
r♦ ❱ 12 6
♥♦♥ 11p [2]
❱❱ 11p [2]
❱❱ 11p [2]
1212
❱ 12 [2]
❱ 12 [2]
❱ 11p [2]
❱ 11p [2]
❱❱ 12 [2]
12 [2] [1] [3]
11p [2]
❬❪ ❬❪❬❪ r♦♠ r♣t ♥ 12 ♥ ❬❪❬❪ ❬❪
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♦ stts r♣♦rt ♥ r ♥ t sr ♥t♦♥ ♥ r ♦ s t♦ ♥rst♥ tt t ♠♥t ♦ t tr♦♥♣♦♥♦♥ ♦♣♥♠♦st② ♦♠s r♦♠ ♦♣♥ t t rt♦♥s ♦ t ♦sr♦♥
♣♦♥r♥ t♦rt ♠t♦ t♦ ♦♠♣t t s♣r♦♥tt②♥♦ ♠ ♣♦ss t ♦♠♣tt♦♥ ♦ t sr♥ ♦♦♠ ♥trt♦♥ s rqr t♦ t t rt t♠♣rtr❬ ❪ ♥ ♦r t♦♥ ♥st s ♣r♠tr ♦r t ♦♦♠ r♣s♦♥ ♥ ♥ ♦r♠❬❪ ❲♥ ts ♣r♠tr s t♥ q t♦ t ♦2 ♦t♥ rt t♠♣rtr ♦ ♦♥② st② ♦ t ♦t♥ ♦r 2
♦ ♦♥ ts st♦♥ t s ♥ s♦♥ tt t rt t♠♣rtr♦r ♠t 132 s ♦♠♣r t♦ t ♦ 2 ♥ tt s♣r♦♥tt② s ♥ t♦ 12 ♥ ♦s t② ♦♠s r♦♠ ts②♥tss ♦ s♥ r②sts s sst♦♥ ①tr♣♦t♥ ♦r rstst♦ ② ♦♣ α♦r♦♥ ts s ①♣t t♦ rtr t♥ t ♦② ♦♣ ♠♦♥ s♦♥ ♦r s♦♥ r ② ♠♦r t♥ ♦♥ ♦rr♦ ♠♥t
r ♥ strtr ♦ 132 r♦♠ ❬❪ ♥r②❱ s rrr t♦ t r♠ s ♥
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r P♦♥♦♥ ♥st② ♦ stts ♦ 132 r♦♠ ❬❪ ♦ ♥ t♦t s ♥ ♦♥trt♦♥ r♦♠ t rt♦♥s ♦ t ♦sr♦♥♦tt ♥ ♦♥trt♦♥ r♦♠ t ♣♦♥♦♥ ♠♦s ♦ t ♥
♦♥s♦♥
♥ ts ♦r r t ♥rst♥♥ ♦ t t♦♠ strtr♦ ♦r♦♥ rs ♥ r♦♠ ♥t♦ t♦♥s s ♦♠♣tt♦♥s♦ t ♥♠♦s tr♠♥t♦♥ ♦ t t♦♠ strtr ♦ 4 sr rt♦♥ ♥ s♣tr r ①♣♥ ② t ♣♦r ♠♦ ♦♥ 11p ♦sr♦♥ ♥ ♦♥ ♥ ♣r ♥t ❲ s♦♥tt t ♠♥ strtr t ♣rt r♦♠ ssttt♦♥ s♦rr ♦ tr♦♥ t♦♠ ♥ t ♣♦r st s 10
p2 ♦sr♦♥ s ts
♥ ♦sr ♥ ♦♥ ①♣r♠♥t ♦r♦r st♦♦♠tr② ♠♣s t①st♥ ♦ 12 ♦sr
♥ t♦♥ ts stst♦r② r♠♥t ♠♦r♦r s s rt ♦♥♥ ♥ t t② ♦ t♦ rt② ♣rt t s ♦ t ♦r♠t♦♥♥r② ♦ ♦r♦♥ rs t ♦♥♥trt♦♥s ♦tr t♥ ❲ s♦♥tt r② strtrs ♥ rr s ♣♦ss ♥ts ♦r ♥♠♥ strtr ♦s ♦r ♦r♦♥ rs t r♦♥ ♦♥♥trt♦♥s ♦r
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r ♦ ♥ sr ♥t♦♥ r♦♠ ❬❪ s ♥ r tr♦♥♣♦♥♦♥ ♦♣♥ ♣r♠tr
t♥ s ♦s ♥st t ② s♣r t♦t tt ♥tr♥s strtr ts r ♥rt② ♦r t rs♣t t♦ t strtrtr ♥ ♠tstt② ♦ ♥② ♠♦ t rs♣t t♦ 4p + α♦r♦♥ ♦r 4p + ♠♦♥ ② s♥ t ♦♥ ts ♠tstt② ①♣♥ t ts ♥ t s②♥tss ♦ ♥ s♠♣s ♦ ♦r♦♥rs t r♦♥ ♦♥♥trt♦♥s ♦r t♥
st② s♦♥ tt t str♥t ♦ t tr♦♥♣♦♥♦♥ ♦♣♥♥ ♣r♦t♦t②♣ ♦sr ♠t 132 s ♦♠♣r t♦ tr♦♥♣♦♥♦♥♦♣♥ ♥ 2 r♦♠ ts ♦♥srt♦♥s ①♣t tr♦♥♣♦♥♦♥♦♣♥ ♥ ② ♦♣ α♦r♦♥ t♦ ♠♦r t♥ ♦♥ ♦rr ♦ ♠♥trtr t♥ tt ♦ ② ♦♣ s♦♥ ♦♠♥♥ s♣r♦♥tt②♥ t ♥♦♥ r♠r ♠♥ ♣r♦♣rts ♦ ♦r♦♥ rs ♥ t♦ ♠trs ♦ ♥str ♥trst
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♣tr
♦♥s♦♥ ♥ ♦t♦♦
♥ t ♦♥ tr♠ t ♠t♦s ♦t s t♦ ♥rst♥ ♥ t♦ ♣rt t②s ② t tr♦♥ ①tt♦♥ ②s ♥s ♥ t t♦♠ strtr ♦ ♠tr ♥ t ♣r♦sss ② strtr ts r rt
♥ t ♣trs ♥ ♦ ts tss ♣rs♥t t t♦♦st ♦r s♣♦s t♦ t t ①tt♦♥ ♦ ♥ tr♦♥s ♥ t♦ ♥rst♥ t ♥r♣r♥ts tt s♦s ♣ ♥ t tr♦♥ ♥r② ♦ss s♣tr♥ ♥ ♦♣t s♦r♣t♦♥ s♣tr r♣♦rt rsts ♦♥ r♦s ♦①s♥ t tr t ♥♦t ♦t ♦ ♣ t♦ ♦♣ t t♦r② ♦ ♣♦t♦♠ss♦♥ ②♦♥ ts t ♠tt♦♥s ♥ ♥♦ ♥ t ♣r♦t♦t t P ♠sr♠♥t ♦♥ ♣r♦s ♦① ♥ t ♦♠♣tt♦♥ ♦t ♦rrs♣♦♥♥ ♥ strtr ❬❪ t ♥ tt rs tt♦r② ♦ tr♦♥ ♥r② ♦ss s♣tr♦s♦♣② ♥ ♦ ♦♣t s♦r♣t♦♥ s♥ ♦♣♥ t♦ t ♣♦♥t r t s r rt♥ ♠trt② ♦t ♦r s st ♥ t♦ r t s♠ ♦ ♠trt② ♥ ♦rr t♦ tt ♣♦t♦rr♥t s♣t r♥t ♥s ❬❪
♥ ② t♦ ♠ ♣r♦rss ♥ t ♦t ♦t s rst② t♦r ♣ ♥rst♥♥ ♦ ♥ ♠tr ♥ ♦ ts ts ♥ s♦♥② t♦ ♥rst♥ ts r ♦♣♥ ♥r ♥ rrt♦♥ ♥② t② ♦ s♦ ♥ t ♣tr s♠♠r③ ♠② ♥rst♥♥ ♦s♦♠ ♣②s ♣r♦♣rts ♦ ♦r♦♥ rs ♠tr ♠ s♣st ♦ r♣♦rt ts r ♣rt ② t t♦♥ t♦♦r ♥ ts ♠tr st② ♦ ♦ s ts r ♣r♦ r♥ s♦ ①♣r♠♥t ♥ ② t② rst② ♠♦② t ♠♥str♥t ♦ ts ♠tr s ♦♥♦♥ t r ♠♠♥ tr♥♥② ♥t tr t ♦ t♠♦st ♠♣♦rt♥ t♦ ♥rst♥ ♥ ♣rt ② tt♦♥ t ♥ ♦ ts ♥ ♦♣ ♥r ♥ tr♦♥ ①tt♦♥♥ s ♠tr
tr st t ♦r t s♦rttr♠ tr s t t♦r② ♦ tr♦♥♣♦♥♦♥ ♦♣♥ ♦r t tr♦♥ r①t♦♥ ♥ s♠♦♥t♦rs ♦♥srts t♦♣s s ♥ ♦♥♦♥ ♣r♦t ♦♠♥t ♥ t s
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P ❯ ❯
♥t trtr❬ ❪ ♥♦♥ ♣r♦rss s ♠ ♥ t r♠♦r♦ t ♣r♦t s♣♣♦rt ② t r♥ ♥ t♦♥♣♦r r ♦♣ tt ♥ t ♥r tr ts st ♠tr ♥♦ ♥ r♦♥③ s r② ♠♣♦rt♥t ♦♣ tt t s♦♦♥ ♦♣ ♥ t tss ♦ ♠② ♦♦rt♦r r ♥ st ♦♣♥ ♦ t tr♦♥ ①tt♦♥ t♦ ♣♦♥♦♥ ♦ ♥t t♦r♥s s t♦ ♥rst♥ t t♠ ♦ t tr♦♥ r①t♦♥ ♥①tst♣ s t♦ st② t ♦♣♥ t ♣♦♥♦♥ ♦ ♥s♥ t♦r ♦♥♥t ♥ t P tss ♦ r ♠r♦ s ♦t t♦ ts ♣♦♥t ♥①t st♣ t ♦r tr ♦r s t♦ ♥rst♥ ♦ s ♦♣♥♥ ♠♦② t t♦♠ strtr
♥② ♦ t♦ t♥ t ♠♥② ♣♦♣ t ♦♠ ♥♦r♥ ♦♣ t② ♥ t s t② t♦t t② s♦ ♥ ♣♦♦③ ts t ♥♦t t s ♣② t♥ Pr♦ss♦r ❯ ♦♥ rt ♦rs t r♥ ♦ t ♠♥sr♣t ♥ ♦tt♥ ♦r t ♦rrt♦♥s♦ t ♥s ♥ t♥ Pr♦ ❯ ♦♥ rt Pr♦ ♥ r rst♦♣ r t♦ ♣t t♦ r t ♠♥sr♣t♥ t♦ ♣rt♣t t♦ t r② ♦ ts t♥ Pr♦ r ♠rr♦♦r ♥ ♣t t♦ r t r② ♥ r rst♥ ♦① ♥ r♠ Ptt t♦ ♣t t♦ ♠♠rs ♦ t r② ♦ ts t♥ r ♠ Ptt t ♦r♠r ♦ t ♦rt♦r s ♦srrés ♦r s ♥♦r♠♥ts r♥ ts st ♥♥ ②rs
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♣♣♥s ♥♥①s
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♣♣♥①
sr♣t♦♥ ♦ ③r♦♥
❩r♦♥♠ ♦① s t ♦♥② tr♠♦②♥♠② st ♦♠♣♦♥ ♥ ts②st♠ ❩r ❬ ❪
❩r♦♥ ❩r2 s t♥♦♦② ♠♣♦rt♥t ♠tr t♦ ts str♥t ♥ stt② t t♠♣rtrs ♥ ts ①♥t tr ♣r♦♣rts t ♥ r stt tr ♦♥st♥t ε0 ♦ r♦♥ ❬❪ ❩r♦♥ ts s♦s r♥ ♦ ♥str ♣♣t♦♥s ♥♥ ss ♥r♠ ♥♥r♥ ♦r ①♠♣ t♦ str♥t♥ r♠s ❬❪ ♦r s♥ ♦①②♥ s♥s♦r ♥ s ❬❪ t s s♦ t♥♦♦② ♠♣♦rt♥tt②t s♣♣♦rt ♠♠ ❬❪ t s ♥♦ ♣r♦♣♦s t♦tr t ♥s t tr ♠tr ♥ ♠t♦① s♠♦♥t♦r s ❬ ❪
rtr♠♦r ③r♦♥ s ♦♥ ♦ t ♠♦st rt♦♥rsst♥t r♠srr♥t② ♥♦♥ ❬ ❪ ♥ tr♦r s ♣rtr ♠♣♦rt♥ ♥t ♥r ♥str② r t s s s ♣sst♥ ♠♠ ♦r ②r♦♥♥rss ♥ ♣rssr ts ♣r♦♣♦s ♣♣t♦♥ ♦ ♣rtr ♥trst st s ♦ ❩r2 ♠tr① ♥ ♦♣ t r♦t ♥ ♣rtr② t♥s ♥ rrt t♦ ♦r tr♥s♠tt♦♥ ♣r♦ss ♥ ♦r♠♥♦♥r♦t ♦♣♥ts ♥ t rrt ♥ ②t st ③r♦♥ ♠tr①
♦t♦♥ ♦ t strtr ♥ tr♦♥ ♣r♦♣rts ♦ ③r♦♥ s ♥t♦♥ ♦ t♦rs s s t♠♣rtr ♥ ♣rssr ♥ t ♣♦②♠♦r♣s♦ ♣r ③r♦♥ ttr♦♥ ♦r ♠♦♥♦♥ ♣ss s s tr♠①♥ t r♦s ♦①s s tr♦r t st ♦ ♥t♥s ①♣r♠♥tsts
t ♦ ♣rssrs ③r♦♥ s♣②s tr ♣ss ♠♦♥♦♥ ttr♦♥♥ r♦♥ stt ♣s ♦ ③r♦♥ ②t strtr t ♠♦♥♦♥ ♥t s st ♣ t♦ r♦♥ ❬❪ ♠♦♥♦♥♥t ♦♥t♥s ♦r ❩r2 ♥ts ♠❩r2 t ♠♦♥♦♥ ③r♦♥ ♠❩r2 ♥r♦s rst♦rr ♠rt♥st tr♥st♦♥ t♦tr♥s♦r♠ ♥t♦ ttr♦♥ ♣s ❬❪ t❩r2 ② ♥rs♥t t♠♣rtr t♦ s♣ tr♥st♦♥ ts ♣ ♥ t❩r2 ♥
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PP❳ P ❩
t ♦rt ♣s c❩r2 s ♦t♥ ❬ ❪ s t♥ st ♣ t♦ t ♠t♥ t♠♣rtr ♦ ❬❪
♦♦r♥t♦♥ ♥♠r ♦ t ③r♦♥♠ t♦♠ s Z ♥ t ♥ttr♦♥ strtrs ♥ Z ♥ t ♠♦♥♦♥ ♣s
s♦ st ♥ ②♣♦tt rt strtr t Z s st st strtr ♦r 2 s s♦tr♦♥ t ③r♦♥ st rt ♣s s t♦♦ t♦ ♥stt ts ♦ t r②st ♥ ③r♦♥♥ ttr ♥rst♥ t ♣r♦♣rts ♦ t ♦♠♣① ♠♦♥♦♥ ♣s ♦③r♦♥
♥ ♠② t♦♥s ♦♥ tt t ♥r② ♦ t ♣ss ♦♦s t ①♣r♠♥t ♣s r♠ t ♠♦♥♦♥ ♣s s t ♠♦st st ♣st♥ t ttr♦♥ ♦♥ ♥ ♥② t ♦♥ ❬❪ ♦♥tss ❩♥③♦tt♦ s r♥t② r♥ ♠② tt♥t♦♥ ♦♥ t t tt t ♣rssr♦rt♦r♦♠ ♣s s♦s ♥ ♥r② r t♥ t ♥r② ♦ t ♠♦♥♦♥ ♣s ♥ ♦r t♥ t ♥r② ♦ t ttr♦♥ ♦♥ ❬❪ ♥♦ t♥ ss♣t tt t ♦rt♦r♦♠ ♣s ♦ ♦r s ♥ ♥tr♠t ♣s ♥ t ♣s r♠ ♦ ③r♦♥ ♥ t t♠♣rtr s♥rs ts ②♥ t ♦rr ♠♦♥♦♥ ♦rt♦r♦♠ ttr♦♥ ♥ rst ♥st ♦ t ①♣r♠♥t ♦rr ♠♦♥♦♥ t♥ ttr♦♥♥ ♥② ♣s
② ♣rs♦♥ s tt s ♥ r♣r♦♥ rt s ♦ tt♦t ♥r② s s♦♦♥ s ♦♥ ♦♠♣rs rst ♣ss r♥t♥♠rs ♦ tr♦♥s ♦r ♥♠rs ♦ t♦♠s ♦ r ♥♦♥tr ♣r♦♠s♥ ♦♠♣r♥ ♣s ♦ ❩r 2 ♥ ♣r♦② ❩r2
♥ t tr t ♥ssr② t♦ r② tr t s s♦♥ r♣r♦♥ t ♣s r♠ ♦ ③r♦♥ ♦r tr t s ♥♦t strt♦rr t♦ ss t ♦r ♥ rs♥ t♠♣rtr ♦ ts ♦♠♣①s②st♠s r♦♠ t ♥rts ♦
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PP❳ P ❩
r ♣r♠t ♥t ♦ ♦r t♦♠ strtrs ♦ ③r♦♥ ❩r2 ttr♦♥ ❩r2 ♠♦♥♦♥ ❩r2 ③r♦♥ ♥ t②♣♦tt rt ♣s r s♣r ③r♦♥♠ t♦♠ ♠ s♣r ♦①②♥ t♦♠ ♦r t ♠♦♥♦♥ ♣s t t♦ r♥t ♦①②♥ sts rs♦♥ ♥ r♥t ss r♦♠ ❬❪
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♣♣♥①
② r r♠s ♦♥ r
♥ ts ♥♥① sr ♦♥ ② t♦ r r♠s ♦♥ r
♦♠♥ t♦ st♦♥ ♦♥ ss ♥♦ t♦ ♦♠♣t t ♣r♦t② ♦r ♦r s②st♠ t♦ ♠ tr♥st♦♥ r♦♠ t ♥♣rtr ♥t stt|ψ
(0)nk > t♦ ♥♦tr ♥ stt |ψf > ♥r t ♥♥ ♦ ♣rtrt♦♥
♦♦♥ ❬❪ s♣♣♦s t ttr t♦ t♠ ♣r♦ ♥ t② st t t♠ t = −∞ s ♥ st♦♥
t s s♣♣♦s tt t ♥r② ♦ t ♣rtrt♦♥ s ~ω ♥ tt t♥t t♠ t s②st♠ s ♥ t stt t ♥r② ε(0)
nk a(0)nk = 1 ♥
a(0)pk′ = 0 ♦r p 6= n ♦r k′ 6= k
①♠♥t♦♥ ♦ t ♥♦♠♥t♦rs ♥ q s♦s tt t s ♦t ♥ ♥r② ♦r ts ♥♦♠♥t♦rs r r r t rs♦♥♥t♦♥s ♦r ♥st♥ ♥ ε(0)
f = ε(0)nk + ~ω
qrtr t♥ rs
a(1)f (t) =
−i
~
∫ t
t0=−∞
e−i(ε(0)nk
−ε(0)f
) τ
~ < ψ(0)f |F q †|ψ
(0)nk > e−iωteαtdτ ,
♦s s♦t♦♥ s
a(1)f (t) =
e−i(ε(0)nk
−ε(0)f
+~ω+i~α) t~
ε(0)nk − ε
(0)f + ~ω + i~α
< ψ(0)f |F q †|ψ
(0)nk > .
tr♥st♦♥ ♣r♦t② r♦♠ t ♥t stt ψ(0)nk t♦ stts ψ(0)
f ♦s♥r② s t♥ ♥ ♥r② r♥ dνf rs
dwf ;nk = |a(1)f (t)|2dνf ,
r
|a(1)f (t)|2 =
| < ψ(0)f |F q †|ψ
(0)nk > |2e2αt
(ε(0)nk − ε
(0)f + ~ω)2 + (~α)2
.
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PP❳ ❨ ❯ ❯
♥ s
d |a(1)f (t)|2
dtdνf =
2α| < ψ(0)f |F q †|ψ
(0)nk > |2e2αt
(ε(0)nk − ε
(0)f + ~ω)2 + (~α)2
.
❯s♥ t t tt
limx→0x
π(ǫ2 + x2)= δ(ǫ),
♦♥ ♥s tt
limα→0d |a
(1)f (t)|2
dt=
2π
~| < ψ
(0)f |F q †|ψ
(0)nk > |2δ(ε
(0)nk − ε
(0)f + ~ω)
♥② t ♣r♦t② ♣r t♠ ♥t ♥ ♣r stt ♠♦♥ts t♦ r♠s♦♥ r
dwf ;nk
dνf
=2π
~| < ψ
(0)f |F q †|ψ
(0)nk > |2δ(ε
(0)nk − ε
(0)f + ~ω)
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♣♣♥①
rr♠ t ❱st r
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PP❳ ❯❯❯ ❱ ❱
♦♠ t ❱P♦st P②s♥♥ rr
①♣rt s♥♦r ♥s♥ s♦t♥♥ ♣ré t ♠ ♥tst♣♦②t♥q
♦♥♥és ♣r♦ss♦♥♥s tt tt♦♥ ♥ss♥ r♥
♦ s ♦s rrés t♦♥té r♥çs♦ P♦②t♥q
♦♠♥ ①♣rtsP②sq té♦rq t ♥♠érq ♣♦r tèr ♦♥♥sét ♥t♦ s ♣r♦♣rétés ♣②sqs s ♠tér①ét♦s sés sr é♦r ♦♥t♦♥♥ ♥sté tt♣♥♦♣r③♦r♥♦❴♣r③s♠str②rts♦♥trt♠
t à ts à ♦r♣s ♣♦r
• s♣tr♦s♦♣ étr♦♥q t rt♦♥♥• s ♠é♥s♠s ①tt♦♥ t r①t♦♥ étr♦♥q♣rtèr♠♥t és①tt♦♥ ♣r ♦♣ étr♦♥ ♣♦♥♦♥
♣♣t♦♥s ① ♠tér① ♥térêt ♣♦r s♣é♠♥t
• s r♠♠s ♣s tt♥ t ③r♦♥♠• s♣tr♦s♦♣ s ♦①②s ♠ét① ♥♦s t tr♥st♦♥ 2❩r2 2 ♣r ♣rt é♥r étr♦♥q ♣♦t♦é♠ss♦♥s♦r♣t♦♥ ♦♣tq
• s s♦s rs ♥ ♦r• s ♥♥♦strtrs s♠♦♥trs
♣ést s rrs ♦r t s ♠tér① rs ♥ ♦r
❱ ♦r♠t♦♥ t tt♦♥ à rr s rrs ❯♥rsté P t r ès ♦t♦rt ❯♥rsté Prr t r r Prs ❱
♥ s ♠tér① ♠♥t♦♥ étt♦♥s r② à ♥♥♠té ♣ô♠ éts ♣♣r♦♦♥s
❯♥rsté té s♥tq ❱♥ sq r♥P②sq étt s♦ ♠♥t♦♥ ♥
♣ô♠ ♥é♥r ♥sttt ♣érr tr♦♥q ♦r r♥ ♣t♦♥ ♣②sq étt s♦
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PP❳ ❯❯❯ ❱ ❱
❱ Pr♦rs ♣r♦ss♦♥♥ ♦♥ éq♥t rtr rr ①♣rt s♥♦r rr à rr à rt♦♥ s ♣♣t♦♥s trs
♥tr ts ♠❱♥t♦♥ ♣s s r②èrs âts ♥t♦ éqt♦♥ étt ♠tér① ②♥♠q♠♦ér r Pr♥♦ strtr étr♦♥q rt♦♥s rés
❯♥rsté r♥ ❱é ♠♣s sr r♥r① rés ♣②sq ♥♠érq rs♥
❱ Prs♣ts t ♦♦rt♦♥s
s tr♦s r♥èrs ♥♥és é♦♣♣é ♥ ♠ét♦ ♥t♦ ♣♦rr t♠♣s ♥ étr♦♥ ①té ♥s ♥ ♦♥t♦♥q st ♠té ♣r ♥trt♦♥ s ♣♦♥♦♥s ♦rt ♦♥r ♦♥tr rts ♦♥t été érts ♥ ♦♦rt♦♥ st♣♦st♦ t ❱ ②tr ♣r♦ssr à ❯♥rsté Pé♦q ♦♠sss str
r q♥ t s s②stè♠s st rét à é ♥♥♦♠étrq ♦♣ étr♦♥♣♦♥♦♥ ♥t très ♠♣♦rt♥t ♣♦r tr♥s♣♦rttr♠q s♦t ét♥r ♠ét♦ ① ♣♦♥♦♥s r♥♦♥r ♦♥ ♠♣♦rt♥ts ♣♦r tr♥s♣♦rt tr♠q ♥ ♦♦rt♦♥ ♥♦s ♦ès r t ③③r ♥sttt ♥ér♦ t P②sq s ① ♦♥♥sés ❯♥rsté Prs ❱ P t r tt♦ ♥♦ ♥s ♣r♦tP
tr sr s ♠tér① à s ♦r ♠♥é à ♥ ♦♦rt♦♥ s ①♣ér♠♥ttrs t s té♦r♥s ❯♥rsté s♣♦♥ ♣♦r r♥r s s♦s s♣r♦♥tr s r♦r ♦t♦r♥t ♥♣♣♦♥ r ♥q ♠♦s ♥ é♠rr tès r ♠r♦ st ♣ré à r♥tré s♦srésr ♥♥♠♥t
♥♥ ss ♥tr♥t♦♥♠♥t r♦♥♥ ♦♠♠ ♥ s♣ést srr ♦r ♥s r é♥é ♥ ♦♥trt ♣♦r♦♠♣r♥r t♥ ♠é♥q s ♠tér① ❯♥ r♥ ♣r♦rès st ♥♦rs sr rô s ♥s ♥s t♥ ♠é♥q ♠tér ♦♥tét st t♠♥t ♠♥é tr♥♥②
❱ ♦♦r♥t♦♥ ♣r♦ts s♥tqs
♥s ♣r♦t P ♦♦r♥t♦♥ tr éq♣
♥s rés ①♥ r♦♣é♥ ❯ Prt♣t♦♥ à értr s r♣♣♦rts t ① rs ♣r♦t à r①stt♣♠t②♦r♥♥♦q♥t
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PP❳ ❯❯❯ ❱ ❱
t ♦♦r♥t♦♥ ♣r♦ts s♥tqs
♥s rés ①♥ r♦♣é♥ ❯
Prt♣t♦♥ à ♦♦r♥t♦♥ ♥ s éq♣s ♥tért♦♥ s♥tqs♦t ♠♥s♦♥ strtr q♥t♠ ♦t ♥ ♦♥qr ts ♥r♦♥♠♥t
♦♦r♥t♦♥ éq♣ ♥tért♦♥ ❯ ❱♦♥ rt t st sr s♦t② sss♥ ♦♠♣r♥s ♣r♦r♠ s ♥ ♦r♠t ♥ ♠♣♠♥t t♦ ts♦t sss ♥♥ ♥r ♥ t qst♦♥s s♥t sss ♥ ts ♠sr♠♥t♠② sss ♦♥trt qst♦♥s ♠♥t♦r♥ ♥ ♠♥♠♥t tr♥♥ ♥t♥ t♦ ♣ ♥♣rtr t rr ♦♣♠♥t ♦ ②♦♥ rsrrs
❱ s♣♦♥stés s♥tqs
♦♥trts
♦♥trt P ♥t♦ ♦♣ r ♥ r♠r♥s♣rt ♥ ♥♦strtrs ♥♥♠♥t ♣♦st♦ ♠♦s tr♥♥②
♦♥trt sr ét té♦rq s rrs ♦r♥♥♠♥t ♣♦st♦ tr♥♥② t st
♦♥trt r♦♣é♥ rés ①♥ ♥♦q♥t♥♥♠♥t ♥ ♣♦st♦ tr♦s ♥s st
P♦sts ♦ P♦②t♥q ♣♦r s sts ❱ ②tr ♣♥♥t ♠♦s ré♣rts sr
♦♥trt ♥s r P ♦♥trt Pr♦r♠♠ r♦♥ts ♦ést♦♥ rrt♦♥ sr s ①tt♦♥sétr♦♥qs ♥s ③r♦♥ ❩r♥♥♠♥t ♥ ♣♦st♦ ♠♦s P r♥ s
q ♥♥é ♣r♦t ♦♥s s♥tq ♥trsé ♣r♦t ♣trtr étr♦♥q t ♣r♦♣rétés rt♦♥♥s s ♠tér① s ♥t♦
♣s ♣r♦t r♥ q♣♠♥t t♦♥ ♥t♥s trtr étr♦♥q t ♣r♦♣rétés rt♦♥♥s s ♠tér① s ♥t♦
rt♦♥ tès
t té♦rq ♦♣ étr♦♥♣♦♥♦♥ ♥s s ♠tér① és①tt♦♥étr♦♥q s♣r♦♥tté ♠r♦ é♠rr ♣ré à r♥tré
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PP❳ ❯❯❯ ❱ ❱
t rs♣♦♥stés s♥tqs ♦rt♦♥ tèss r ♦t♦r♥t ♥♦s♥ ♥ ♥♦t♥♦♦② ♥tr ♥sttt ♦ ♥t ♥ ♥str sr s ❯♥rst② ♣♥♥♦②é ♣♦r étr s♣r♦♥tté ♦r t ♦r ♦♣é♦ ♥r t ♥t
t té♦rq s ♣ss tt♥ ❱ r♥té éqt♦ P♦②t♥q tt♣♠♣r♠r♣♦②t♥qrsssr♥t♣
①♥ ♥ ♦rrt♦♥ ♥ t tr♦♥ trtr ♦ ♦s r♦♠ ♦♥t♦ ♣r♦s ♦① ❲ ♣♣r♦①♠t♦♥ ♥ ②♦♥ r♥ ♦ P♦②t♥q
♠♠♣r ♥ ♥t♦ t♦♥s ♦ ♦♣t ♣r♦♣rts ♥ s♠♦♥t♦rs♣rtts ♦tt ❯♥rstà P t
Pt♦♥s ♣♣rs ♥s s ♦r♥① à rrs ♦♥t rts ♥s P②s ttrs t rts ♥s P②s rts ♣s ♠♦♥ à rré à
rts ♥é♥ts ♣r♦r♠♠s ♥♦r♠tqs q é♦♣♣ s♦♥t s♦s ♥ ❯
♥tt♦♥s ②♥♦t tr tt t♥ è♠ ♥tr♥t♦♥ ②♠♣♦s♠♦♥ ♥trt♦♥ ♦♠♣♦♥s à ♥ ♥
P♣r ♥té à t ♥tr♥t♦♥ ②♠♣♦s♠ ♦♥ ♦r♦♥ ♦rs♥ t trs à ts ♣♦♥ ♣t♠r
♣r♦t à ♦♠♠ss♦♥ r♦♣é♥♥ sr è♠ ♥♥é rés ①♥ ♥♦q♥t ss♦s ♣t♠r ①♣♦sésééés à st t str♦ r♥
♥♦♥tr ♥♥ r♦♣♠♥t r r♦ t ♥♦tr♠q②♦♥ ♥ ♦r ♥té sr ①tt♦♥ t r①t♦♥étr♦♥q éts ♥t♦
♦♥rès ♥♥ r♦♣♠♥t r♥çs ♣♦r s ér♠qs♥ r♥ rs ♦r ♥té sr①tt♦♥ t r①t♦♥ étr♦♥q éts ♥t♦
r r r♥♥ ♦rs ♦r ♠t♦♥st♦r ♠t♥ ♠ttr t t ♥♥♦s s♥ ♥st②♥t♦♥ t♦r② ♣s♦♣♦t♥ts ♥ ♣♥ s ②♦♥ r♥ ♦♠r ♥tt♦♥ à ♦♥♥r ① ♦rs
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PP❳ ❯❯❯ ❱ ❱
t ♥tt♦♥s
♣r♦t à ♦♠♠ss♦♥ r♦♣é♥♥ è♠ ♥♥é rés①♥ ♥♦q♥t r①s ♣t♠r ♥ ①♣♦sé s♥tq s②♥tès sr q♣ ♥tért♦♥ q ♦♦♦r♦♥♥é
❲♦rs♦♣ ♥s ♥ t t♦♠s ♥rst♥♥ ♦ ♥♥♦strtrs r ♦r ♥té sr tr♦♥ ①tt♦♥s ♥t♦ t♦♥s ♦ tr♦♥ s♣tr
♣r♦t à ♦♠♠ss♦♥ r♦♣é♥♥ sr èr ♥♥é rés ①♥ ♥♦q♥t r①s ♦ût ♥ ①♣♦sé s♥tq s②♥tès sr éq♣ ♥tért♦♥ q ♦♦♦r♦♥♥ t ♥ ①♣♦sé sr ♣♦tq s♦ s♥ rés ①♥
♥tr♥t♦♥ ♦♥r♥ ♦ ♦♠♣tt♦♥ t♦s ♥ ♥s♥ ♥♥r♥ ❱♦♠♥♦r rè ♦♠r ②♠♣♦s♠ sr ♠ts ♠♦♥ ♦ rrt♦♥ ts ♥ s♦s ♥ ss♦t♦t♦qr♠ ♣r♦sss ①♣♦sé ♥té sr s♣tr♦s♦♣ ♣rté♥r étr♦♥q
♥♦♥tr ♥♥ és ♦r♠t♦♥ r♦♣é♥ sr r♥♥ t♦r Prs t♦r ①♣♦sé ♥té sr s♣tr♦s♦♣ ♣rté♥r étr♦♥q
P ❲♦rs♦♣ ♦t♦♥s ♦♦♥ r♥t ♣r♦sss t srs♥ st♥ s♣♥ ♣t♠r ①♣♦sé ♥té sr ♥s♦tr♦♣② ♦ t♠r♦s♦♣ tt♦♥s ♦ t ♣♦r③t♦♥ ♦♥♥♠♥t ts ♥ ttr ♥t♦♥ ♦ r②sts
PPs ♦rs♦♣ ♥t♦ ♠♥②♦② t♦r② ♦r ♦rrt tr♦♥ s②st♠srst t ♦ût ①♣♦sé ♥té sr ♥s♦tr♦♣② ♦ t ♠r♦s♦♣tt♦♥s ♦ t ♣♦r③t♦♥ ♦♥♥♠♥t ts ♥ t tr ♥t♦♥ ♦ r②st
Pr♦t ♦s♦ t♦r ①♣♦sé ♥té sr ♥t♦ ♠t♦s t♦ tt tr♦♥ strtr ♥ t rt♦♥ ♣r♦♣rts ♦ s♦s
é♠♥r ♥té à ♦ ♥tr Prs ât♥②r② ♥ sr sét♦s ♥t♦ ♣♦r s ♣r♦♣rétés s♣tr♦s♦♣qs s ♠tér①
♦♥rès é♥ér ♦été r♥çs P②sq trs♦r t ♦♦q tér① ① ♦♥t♦♥s ①tr♠ês ①♣♦sé ♥té sr t ♥t♦s ♠tér① s♦s t ♣rss♦♥ tr♥st♦♥s ♣ss ét♥ s♣trs♦r♣t♦♥ ♥rr♦ ♦r ♣
❲♦rs♦♣ ♥s ♥ rst♣r♥♣s ♦♠♣tt♦♥ ♦♥♥s ♠ttr ♣②ssr♦rs r s♣♥ ♥r ①♣♦sé ♥té sr t♦♠ strtr♥ rt♦♥ ♣r♦♣rts ♦ ♦sr 4 ♦r♦♥ r
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PP❳ ❯❯❯ ❱ ❱
❳ ①♣rts s♥tq♣♣♦rtr s ♦r♥① ♣r♦ss♦♥♥s
tr trs ♣♣ P②ss ttrs P②s ttrsP②s r♦♣♥ P②s ttrs r♦♣♥ P②s ♦r♥ ♦r♥ ♦ P②s ♠str② ♥tr♥t♦♥ ♦r♥ ♦ ♥♦t♥♦♦②♦♠♣tt♦♥ trs ♥ sr P♦s♦♣ ♠③♥♠ P②ss ttrs P②s ♣rtts ♥ r♦strtrsP②s tts ♦ ♣ sr Prssr sr ②♦rr♥s♦r♥ ♦ ♣②ss ♦♥r♥ srs
♣♣♦rtr s tèss ♦t♦rt
❩♠③♠ ♥♦♥♥tt♦♥ t tèr ❯trr♦rt♦r tér① t Pé♥♦♠è♥s ♥tqs ❯♥rsté Prs
♠♦r♦ tr♦♥♦ ①tt♦♥s ♥ ♦♣t s♣tr♦ rr s s♦s r♥t② ♦ ♥
r②s tès t ♠str
①♠♥tr ♥s r② tès ss♦r t rt♦♥♥ ss s♠♦♥trs ❱ t ❱ ❯♥rsté P ❱r♥ t③
①♠♥tr ♥s r② tès r② trtr ♦ t♦r s♠♣rtés ♥s s ♠♠s été ♣r s♣tr♦s♦♣s t s ♥t♦❯♥rsté Prs ❱
♥ ♠♠r r② ♣résé ♣r Pr♦ è ♣t ♠str tér① ♣♦r s trtrs t ♥r sr étté sr ♥sàs s♦♥ ♥trt♦♥ ♦♥ r♥② ♥ ♣♣r♦ té♦rq t sr♦♥trt♦♥ à t s ♣r♦♣rétés tr♥s♣♦rt t♦♠q ♥s s ♦①②sr♥♠ ♣r ♠♦ést♦♥ ♥t♦
♦♦r♥st♦♥ ♦♥ér♥s
t r♠♦tr tr♥s♣♦rt ♣r♦rss ♥ rst ♣r♥♣s ♥♦tr ♣♣r♦s ♥ t ♥tr♣② t ①♣r♠♥t s♥♥ ss
♦ût ♦r♥és tèr ♦♥♥sé trs♦r♥♦♦q ét♦s ♥♠érqs ♣♦r tr♥s♣♦rt étr♦♥q t tr♠q ♥ss s♠♦♥trs
q ♥♥é ♦♥ér♥ ♥tr♥t♦♥ ❯ ♥ ♣rtr♥ ②rs ♦ t ❲ ♣♣r♦①♠t♦♥ ♦r t tr♦♥ s♥r②tt♣③❲
Pr♦rss ♥ ♥t♦ ♦♠♣tt♦♥ ♠t♦s ♦r ♦♥♥s ♠ttr♥r sr❨tt r♥
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PP❳ ❯❯❯ ❱ ❱
t♦♥ ♣r♦ts
♣♣♦rts ❯ r♠② sr sr s ♠tér① rs ♥ ♦r
♣s ét♦♥ s ♦ssr ♠♥s t♠♣s sr s ♦r♥trs t♦ ♥s♥♠♥t s♣érr ♣♦r ♦♠té té♠tq P②sq ♠ t Pr♦♣rétés stér①
❳ ♥s♥♠♥t t ♦r♠t♦♥ ♣r♦ss♦♥♥
r♦♣♥ ❯♥♦♥ r r r♥♥ ♦rs ♦r ♠t♦♥st♦r ♠t♥ ♠ttr t t ♥♥♦s s♥ ♥st② ♥t♦♥ t♦r②♣s♦♣♦t♥ts ♥ ♣♥ s ②♦♥ r♥ ♦♠r ♥tt♦♥ à ♦♥♥r ① ♦rs qtr rs
♥ ❯♥rsté r♥ ❱ér① rés ♣②sq ♥♠érq rs♥
trs rçs ♦rt♦r s ♦s rrés
s♠♦♥ é t té♦rq rr s♠st ♠str ❯♥rsté P t r ♦ést♦♥ ttstq t♦rt♠q s s②stè♠s ♦rs éqr rst ♦♥r♠♥t st t ❱st
❳r ❩③rtt t t s étts s♠♦r ♥s strtr ♥s t ét s♦s ♣rss♦♥ st ♠str ♦ ♥tr tér①t strtrs ♣♦r ♥r érrt ♦♥r♠♥t st t ❱st
♣♥ t t ♠♦ést♦♥ ♠ét ③r♦♥♠ t ♦①② ❩r2st ♠îtrs ❯♥rsté Prr t r r ♥ ♦♥r♠♥t s t ❱st
♦sé t té♦rq ♦①② tt♥ 23st îtrs ♣②sq ♦♥♠♥t ❯♥ Prs ♥
❱r♥ éqt é♦♣♣♠♥ts té♦rqs ♣♦r ♦①② tt♥ 2st ♥s s tér① Prs t st ♥é♥r ♦ ♥tr Prs ♣t♠r
é♠ r t té♦rq ♦①② tt♥ 23♥ ♠stèr P②sq ♦♥♠♥t rs② r ♠ t
♥ r♥ t ♥t♦ ♠t ssq t♥sr étrq♥s s étér♦strtrs ss st ♥s s tér① Prs t st ♥é♥r ♦ ♥tr Prs ♣t♠r
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PP❳ ❯❯❯ ❱ ❱
❳ rs
♥s♥s ♠♥ t♥ ♣♣r♥tss ♣♦♥s ♣s ♣t
♠r ♠r♥ P②s ♦t② ♣s ♦été r♥çs P②sq ♣s é ♦♥s ♦rt♦r s ♦s rrés ss♦t♦♥ ♠♠s t ♥s ♦♥é ♣r ♥ r♠♥♥♣r♠èr ♣r♦ssr ♣②sq à ♦ P♦②t♥q ♣♦r ttr♦♥tr s ♣réés sr s rrèrs s♥tqs é♠♥♥
♦♠♣ét♥s t♥qsPr♦r♠♠t♦♥ t sr s ♥trs s ♥t♦♥① t ts♥t s rss♦rs t♦rs t♦ ♣rèsstr ♣r♦ssrs ♣tr♦♥ ♣♦st tr s♦s ❯❳
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♣♣♥①
st ♣t♦♥s ❱str
♥tr r♦ts st ♥qé ♥♦♠r tt♦♥s ♣♣r ♥s ♦ s♥ r ♥ ♣♣rs s♦♥t tés ♦s t ♣s ♣♣rs s♦♥t ♦s t ♣s♥ ♦r rt ♥♦♠r tt♦♥ ♦ ♥ étt
P♣rs rést♥t ♠s tr①
rr ♦rt♦r s ♦s
rrés
tr♥♥② ❱st ♥ st ♦r♦♥ r t ♣rssr ♥ ♣ré♣rt♦♥
❱st t ♥t♦ s ♣r♦♣rétés ♣②sqs s♠tér① ♦♠♥t tt♦♥ à rr s rrs
❱st st ♥ tr♥♥② ♦r♦♥ rs r♦♠ rst♣r♥♣s ♦r♥ ♦ P②ss ♦♥r♥ ♣r♦♥s rt r r♣♣♦rtrs ♦s ♣rss
st ❱st ♥ tr♥♥② r♠ ♥rt♦♥ ♣r♦♣rts ♦ ♦r♦♥ r t r♦♥ ♦♥♥trt♦♥♥ ♥t♦ st② ♥ ♣ré♣rt♦♥
❬❪ st ❱st ♥ ❱ ②tr ♥t♦ st② ♦tr♦♥♣♦♥♦♥ ♦♣♥ ♥ ①t♦♥ ♥t ♥ s ♥r♣rssr ♥ P ♦r♥ ♦ ♠♥s♥ ❯ tt♣①♦♦r♠♥
❬❪ st ❱st ♥ ❱ ②tr ♥t♦ ♠t♦ ♦r ttr♦♥♣♦♥♦♥ sttr♥ t♠s ♥ s♠♦♥t♦rs ♣♣t♦♥ t♦
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PP❳ P❯ ❱
s ♥ P P②s ttrs t② t ♠r♥ P②s ♦t② ♦r t ♥r② ss ♦❱rt ♦r♥ ♦ ❯trst ♥ tt♣trst♦r❯ tt♣♥♣s♦r♦P②stt
❬❪ ❱r♥ r♥té t ❱st ♥ r ②♦♥ t ♦ t♦③t♦♥ ♦ t s♠♦r ♥st② ♦♥ t ♣②s ♣r♦♣rts ♦tr♥st♦♥ ♠ts P②s ♦♥ ttr ❯ tt♣①♦♦r
st ❱ ②tr ♥ ❱st ♥ Pr♦♥s ♦ ♥t♦♥r♥ ♠ rs♥ ♥ ❱ r♦♣ t ♦♠♣♦♥ss t♦r ♦♠s ss ♦♠ ❱ ♣s ♦♥♥
❬❪ ♥ r♥ t ❱st ♥♥ ③qr♦ r♦tt ♥ rrtt ①♥ ♥ ♦rrt♦♥ ts ♥tr♦♥ ①tt♦♥ ♦ 2 P②s tt ❯ tt♣♥♣s♦r♦P②stt
❬❪ st ❱ ②tr ♥ ❱st ♥tr② sttr♥ ♥s ♥t♦ t♦♥ ♦ t t ♣r♠trs ♦r ♦♥tr♦ s♠t♦♥s ♣♣ P②ss ❯ tt♣①♦♦rs
❬❪ st ❱ ②tr ♥ ❱st ♥t♦ st② ♦ Γ − X
♥tr② sttr♥ ♥ s ♥r ♣rssr P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ ♥ r♥ t ❱st ♥ ♥♥ t ♦s♦♥sst♥② ♦♥ qs♣rts ♥ s♦s P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ s ♥ r♥ ❱r♥ r♥té t ❱st ♥ ♥♥ tr♦♥ ①tt♦♥s ♥t♦ t♦♥s ♦tr♦♥ s♣tr ♥ ♣♣t♦♥ t♦ ③r♦♥ ❩r2 tt♥ 2 ♥♣r♦s ♦① 2 ♦♠♣ t ❯ tt♣①♦♦r♦♠♠ts
❬❪ ❱ ②tr ♥ ❱st r♥♥s qt♦♥ ♦ stt ♦r ttr♦♥ r♦♥ stt ♥r② ♦♠♣ t ❯ tt♣①♦♦r♦♠♠ts
❬❪ ♦tt r♥ r♥♦♣♦♦s s ♦tt❱ ♥♦ ❱st ♦ ♥ ♥♥ r♦♠♠♦s t♦ s♦s t r♦ ♦ ♦♥r♥ ♥trt♦♥s ♥t ♥t♠ ❯ tt♣①♦♦rq
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PP❳ P❯ ❱
❬❪ s t ❱st P♣♣ r♥ r②♥t ♥ ♥♥ tr♦♥ strtr ♥ tr♦♥♥r②♦ss s♣tr♦s♦♣② ♦ ❩r2 ③r♦♥ P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ ♦tt ❱st ♥♥ ❱ ♥♦ ♥ ♥r♥ ♥t♦ ♥ s♠♠♣r tr rs♣♦♥s ♦ s♣rtts P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ tt♦ ♥r t ❱st ♥ r♥s♦ r♣r♦♥tt② r♦♠ ♦♣♥ ♦r♦♥ ♦sr P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ ♥ ♦tt r♥s♦ ♦tt t ❱st ❱r♦ ♥♦ ♥♥ ♥srst♥ ❲ssr ♥ ♦ ♦♥♥♥ ♦♦♦ ♦ ♥ ❲ ♦② ♦♥r♥ ♦♥trt♦♥t♦ t ①♥♦rrt♦♥ r♥ ♦ t♠♣♥♥t ♥st②♥t♦♥ t♦r② P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ r♥♦♣♦♦s ♥♥ ♦ ♥ ❱st ♣t♥ ♦ss s♣tr ♦ r♦♥ ♥♥♦ts ♣♦r③t♦♥ ts ♥♥trt ♥trt♦♥s P②s tt ❯ tt♣♥♣s♦r♦P②stt
❬❪ ♦tt ❱st ♥♥ ❱ ♥♦ ♥ ♥r♥ ♥t♦ t♦♥ ♦ t tr t♥s♦r ♦ sss♣rtts P②s tt ❯ tt♣♥♣s♦r♦Pstt
❬❪ ❱st ♥♥ ❱ ♥♦ P tts♥r ♥ ♦r② ♦ ts ♥ t ♥s♦tr♦♣② ♦ t tr♦♥♥r② ♦ss s♣tr♠ ♦ tt♥♠ ♦① 2 P②s tt ❯ tt♣♥♣s♦r♦P②stt
❬❪ r ❱st ♥ Pr t♦♠ strtr ♦♦sr 4 ♦r♦♥ r r♦♠ rst♣r♥♣s ♥②ss ♦ s♣tr P②s tt ❯ tt♣♥♣s♦r♦P②stt
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PP❳ P❯ ❱
P♣rs rést♥t ♠s tr①
rr
❬❪ ❱st ♥ r♦♥ ts ♦ s♦t♦♣ s♦rr ♦♥ t ♠♥s♣tr♠ ♦ r②sts t♦r② ♥ ♥t♦ t♦♥ ♦♥ ♠♦♥♥ r♠♥♠ P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ ❱st ♥ r♦♥ ts ♦ s♦t♦♣ s♦rr ♦♥ t ♠♥s♣tr♠ ♦ r②sts t♦r② ♥ ♥t♦ t♦♥ ♦♥ ♠♦♥♥ r♠♥♠ ♦♠♣ t ❯ tt♣①♦♦r
❬❪ ③③r ❱st ss♦♥ r♦♥ ♥ ♥r ♦rs♦ trtr ♥ rt♦♥ ♣r♦♣rts ♦ ♦sr 4♦r♦♥ r P②s tt ❬❪ ③③r ❱st ss♦♥ r♦♥ ♥ ♥r ♦rs♦ rrt♠ strtr ♥ rt♦♥ ♣r♦♣rts ♦ ♦sr4 ♦r♦♥ r P②s tt ❯ tt♣♥♣s♦r♦P②stt❯ tt♣♥♣s♦r♦P②stt
❬❪ ❱st ss♦♥ r♦♥ ♥ ♦rs♦ t♦♠strtr ♥ rt♦♥ ♣r♦♣rts ♦ ♦sr α♦r♦♥ ♥ 4♦r♦♥ r ♦♠♣ t ❯ tt♣①♦♦r
❬❪ t ❱st Pr♦♣rétés rt♦♥♥s ♦r α t rr ♦r 4 ès ♦t♦rt ❯♥rsté Prs ❱ r♥ tr♣♣♦rt ❯ tt♣trs♦rtsrtr
❬❪ ❱st r♦♥ ❩ér ss♦♥ P♦♥ r♠st ♥ r♥ tt ②♥♠s ♦ ♦srα♦r♦♥ ♥r ♣rssr P②s tt ❯tt♣♥♣s♦r♦P②stt♦♥r♠t♦♥ ①♣ér♠♥t ♠♦r t ②♠ s♦r ♠r t ♥ tPr ♦♥t ♦♥s ♥ α r♦♠♦r ♦r♦♥ P②s tt ❯ tt♣♥♣s♦r♦P②stt
❬❪ ❱st r♦♥ ❩ér ss♦♥ P♦♥ r♥ ♥ r♠st tt②♥♠s ♦ α♦r♦♥ r♦♠ ♥t♦ t♦♥ ♥ ♠♥ sttr♥ ♥r ♣rssr P②stt ♦ ❯ tt♣①♦♦r♣ss
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PP❳ P❯ ❱
❬♣♣rît ♣s ♥s ❪ r♣♥tr ❩ér ♥ ❱stPs♦♣♦t♥ts ♥♥ s♠♦r stts t ♥ ♣♣t♦♥ t♦r♠ αr♠ ♥ t♦r♠ P②s ❯ tt♣♥♣s♦r♦P②s
❬❪ ❱st r♥r ♥ ❩ér trtr ♥ tr♦♥♣r♦♣rts ♦ q ♦r♦♥ r♦♠ ♠♦r②♥♠s s♠t♦♥P②s ❯ tt♣♥♣s♦r♦P②s♦♥r♠t♦♥ ①♣ér♠♥t rs♥♥ ♥s t♥ ❱♦♥ ♥ Pr trtr ♦ q ♦r♦♥ P②s tt ❯ tt♣♥♣s♦r♦P②stt
❬❪ r♦t ❱st ♥ tr ♥strtr t♦♥♦ t ♠♥t♦r②st♥ ♥s♦tr♦♣② ♥r② 3 ♥tr♥t♦♥♦r♥ ♦ ♦r♥ P②ss ❯ tt♣①♦♦r
❬❪ ❱st r♦t ♥ ❩ér rst♣r♥♣st♦♥s ♦ t ♠♥t♦r②st♥ ♥s♦tr♦♣② ♥r② ♦ t♣♥t ♥ P②s ♦♥♥s ttr ❯ tt♣①♦♦r
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♣♣♥①
♥r ♦r♣②
❬❪ ❲ ♦♥ ♦ trs ♠str② ❲♦r ♥tPs♥ ♦ ♥♣♦r
❬❪ ♦♥s ♥ ♥♥rss♦♥ ♥st② ♥t♦♥ ♦r♠s♠ ts♣♣t♦♥s ♥ ♣r♦s♣ts s ♦ ♦r♥ P②ss
❬❪ P ♥♥♦③③ r♦♥♦ P P♦♥ ♥ r♦♥ ♥t♦
t♦♥ ♦ ♣♦♥♦♥ s♣rs♦♥s ♥ s♠♦♥t♦rs P②s
❬❪ r ❩r♦ r♦♥♦ ♦ ♥ ♦♥P♦♥♦♥ s♦t♥♥ ♥ s♣r♦♥tt② ♥ tr♠ ♥r ♣rssrP②s tt
❬❪ st ❱st ♥ ❱ ②tr ♥t♦ ♠t♦ ♦r ttr♦♥♣♦♥♦♥ sttr♥ t♠s ♥ s♠♦♥t♦rs ♣♣t♦♥ t♦ s♥ P P②s tt
❬❪ tt♦ ♥r t ❱st ♥ r♥s♦ r ♣r♦♥tt②r♦♠ ♦♣♥ ♦r♦♥ ♦sr P②s
❬❪ ♥ ♥ ❯ r♦ss ♥st②♥t♦♥ t♦r② ♦r t♠♣♥♥ts②st♠s P②s tt
❬❪ ❯ r♦ss ♥ ❲ ♦♥ ♦ ♥st②♥t♦♥ t♦r② ♦rq♥②♣♥♥t ♥r rs♣♦♥s P②s tt
❬❪ ♥ ♦tt r♥s♦ ♦tt t ❱st ❱r♦ ♥♦ ♥♥ ♥srst♥ ❲ssr ♥ ♦ ♦♥♥ ♥♦♦♦ ♦ ♥ ❲ ♦② ♦♥r♥ ♦♥trt♦♥ t♦ t①♥♦rrt♦♥ r♥ ♦ t♠♣♥♥t ♥st② ♥t♦♥ t♦r②P②s
❬❪ ♥ ♥♥ ♥ ♦ tr♦♥ ①tt♦♥s ♥st②♥t♦♥ rss ♠♥② ♦② r♥s ♥t♦♥s ♣♣r♦s ♦
P②s
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PP❳ P❨
❬❪ ❲ ♦② ütr ♥ ♠ ♥r② ♦♣rt♦rs ♥①♥♦rrt♦♥ ♣♦t♥ts ♥ s♠♦♥t♦rs P②s
❬❪ ♥♥♦♦ ütr ♥ ♠ t♦♥ ♦ t ♦♥s♠♣♦t♥t ♥ ts s♦♥t♥t② ♦r ♠♦s♠♦♥t♦r P②s
❬❪ ♥ ♥ ♥qst ♦ stt ♣②ss ♦♠ ♣ ♠ Prss ❨♦r
❬❪ ♥ P②s
❬❪ rt ♥♥ ♦ ♥ ♥ P②s tt
❬❪ ♥t r② ♥ ♦♥ ♦r② ♦ ♦♣t s♦r♣t♦♥ ♥♠♦♥ ♥ s P②s
❬❪ ♥t r② ♥ ♦♥ ♣t s♦r♣t♦♥ ♦ ♥st♦rs♥ t tr♦♥♦ ♥trt♦♥ ♥ ♥t♦ t♦♥ P②s ttrs
❬❪ ♦♥ ♥ ♦ tr♦♥♦ ①tt♦♥s ♥ s♠♦♥t♦rs ♥♥st♦rs P②s tt
❬❪ ❱st ♥♥ ❱ ♥♦ P tts♥r ♥ ♦r②♦ ts ♥ t ♥s♦tr♦♣② ♦ t tr♦♥ ♥r② ♦ss s♣tr♠♦ tt♥♠ ♦① 2 P②s tt
❬❪ r♥♦♣♦♦s ♥♥ ♦ ♥ ❱st ♣t ♥ ♦sss♣tr ♦ r♦♥ ♥♥♦ts ♣♦r③t♦♥ ts ♥ ♥trt♥trt♦♥s P②s tt
❬❪ ♦tt ❱st ♥♥ ❱ ♥♦ ♥ ♥r♥ ♥t♦
t♦♥ ♦ t tr t♥s♦r ♦ ss s♣rtts P②s tt
❬❪ ♦tt ♠♠♣r ♥ ♥t♦ t♦♥s ♦ ♦♣t ♣r♦♣rts ♥
s♠♦♥t♦r s♣rtts P tss ❯♥rstà P t
❬❪ ♦tt ❱st ♥♥ ❱ ♥♦ ♥ ♥r♥ ♥t♦
♥ s♠♠♣r tr rs♣♦♥s ♦ s♣rtts P②s
❬❪ s t ❱st P♣♣ r♥ r ②♥t ♥ ♥♥ tr♦♥ strtr ♥ tr♦♥ ♥r②♦ss s♣tr♦s♦♣②♦ ❩r2 ③r♦♥ P②s
❬❪ ♦tt r♥ r♥♦♣♦♦s s ♦tt ❱ ♥♦ ❱st ♦ ♥ ♥♥ r♦♠ ♠♦s t♦ s♦s tr♦ ♦ ♦♥r♥ ♥trt♦♥s ♥t ♥t ♠
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PP❳ P❨
❬❪ s ♥ r♥ ❱r♥ r♥té t ❱st ♥ ♥♥ tr♦♥ ①tt♦♥s ♥t♦ t♦♥s ♦ tr♦♥ s♣tr♥ ♣♣t♦♥ t♦ ③r♦♥ ❩r2 tt♥ 2 ♥ ♣r♦s ♦① 2♦♠♣ t
❬❪ ♥ r ♥ ♦t♥ tr♦♥ s♦♥sst♥t❲ ♣♣r♦①♠t♦♥ ♣♣t♦♥ t♦ ♥ ♥ P②s tt
❬❪ r♥ ♥ t ♦rrét♦♥ ♥s trtr tr♦♥q s
♦s ♠ à ①② r① ♣♣r♦①♠t♦♥ ❲ t àP tss ♦ P♦②t♥q Ps r♥
❬❪ ♥ r♥ t ❱st ♥♥ ③qr♦ r♦tt♥ rrtt ①♥ ♥ ♦rrt♦♥ ts ♥ tr♦♥ ①tt♦♥♦ 2 P②s tt ♥ rrr♥s r♥
❬❪ ♥ r♥ t ❱st ♥ ♥♥ t ♦s♦♥sst♥② ♦♥ qs♣rts ♥ s♦s P②s
❬❪ st ❱ ②tr ♥ ❱st ♥tr② sttr♥ ♥ s ♥t♦ t♦♥ ♦ t t ♣r♠trs ♦r ♦♥t r♦ s♠t♦♥s♣♣ P②ss
❬❪ st ❱ ②tr ♥ ❱st ♠♣ ♥t♦ st② ♦ Γ − X
♥tr② sttr♥ ♥ s ♥r ♣rssr P②s
❬❪ ❱st Pr♦♣rétés rt♦♥♥s ♦r α t rr ♦r 4P tss ❯♥rsté Prs ❱ r♥
❬❪ t ❱st Pr♦♣rétés rt♦♥♥s ♦r α t rr ♦r4 ès ♦t♦rt ❯♥rsté Prs ❱ r♥ t r♣♣♦rt
❬❪ ♥r ♠♥ ♥ ♦rt s♥♥ s♣rr♠trs ♥s
❬❪ ❱st r♦♥ ❩ér ss♦♥ P♦♥ r♥ ♥ r♠st tt②♥♠s ♦ α♦r♦♥ r♦♠ ♥t♦ t♦♥♥ ♠♥ sttr♥ ♥r ♣rssr P②s tt ♦
❬❪ ❱st r♦♥ ❩ér ss♦♥ P♦♥ r♠st ♥ r♥ tt ②♥♠s ♦ ♦sr α♦r♦♥ ♥r ♣rssrP②s tt
❬❪ ③③r ❱st ss♦♥ r♦♥ ♥ ♥r ♦rs♦trtr ♥ rt♦♥ ♣r♦♣rts ♦ ♦sr 4 ♦r♦♥ rP②s tt
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PP❳ P❨
❬❪ ③③r ❱st ss♦♥ r♦♥ ♥ ♥r ♦rs♦rrt♠ strtr ♥ rt♦♥ ♣r♦♣rts ♦ ♦sr 4 ♦r♦♥r P②s tt
❬❪ ❱st ss♦♥ r♦♥ ♥ ♦rs♦ t♦♠ strtr ♥rt♦♥ ♣r♦♣rts ♦ ♦sr α♦r♦♥ ♥ 4 ♦r♦♥ r♦♠♣ t
❬❪ ♥♥ ❲ ② ♥ ♦ P②s tt s réstts ♣♣r s♦♥t rr♦♥és ♦r♠ rr ♦r 12 st qs ♥①st♥t s♦♥ ♥♦s s
❬❪ r ❱st ♥ Pr t♦♠ strtr ♦ ♦sr 4♦r♦♥ r r♦♠ rst♣r♥♣s ♥②ss ♦ s♣tr P②s tt
❬❪ r♠ts ❱ ❱ tr③♥ ♦ ♥ ♠② ♥
❬❪ ♥♦ ♥ ❨ tt ❲ ss ❩ ❨ r② ♥ ❱ ♦♦③♥♦ tr
❬❪ ♥ ♥ ❩ strtr ♦t♦♥ ♦ ♦r♦♥r ♥t♦ t♦♥s ♣♣ P②s tt
❬❪ s♦ r ♥ t②♠❨♦s P②s
❬❪ ③ r③ ♥ ❨♦s♦♥ P②s tt
❬❪ ❱st st ♥ tr♥♥② ♦r♦♥ rs r♦♠ rst♣r♥♣s P②s ♦♥ r
❬❪ ❱♦r ❲ ♥rt ♥ s ②♥♠ ♦r ♦♦r♦♥ r ♣♣ P②s
❬❪ ♥ ❲ ② ♥ ♠r ♦♥ ♦③♠♦r♣③t♦♥ ♥ ♦r♦♥ r ♥
❬❪ ♦s s ♥ ❨ ❨♣ tr♥♥ ♦r♠t♦♥♦ r♦♥ ♥ ♦r♦♥ strs ♥ ♦r♦♥ r r♥ ②♥♠♥♥tt♦♥ ♣♣ P②ss ttrs
❬❪ ❳ ❨♥ ❩ ♥ ❩♥ ♦ ♥ ❨ ❩♥ ♦t♦ ② ♥ ❲ ♥ ♣rssr③t♦♥ ♠♦r♣③t♦♥ ♦s♥r②st ♦r♦♥ r P②s tt
❬❪ ❳ ❨♥ ❲ ♥ ❲ ♥ ♠♥ s♣tr♦s♦♣② ♦♣rssr♥ ♠♦r♣♦s ♦r♦♥ r ♣♣ P②s tt
❬❪ ❱ r♥té t té♦rq s ♣ss tt♥ P tss ♦P♦②t♥q Ps r♥
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PP❳ P❨
❬❪ r r ♥ t②♠❨♦s ❱♥ ♦♥tr♦ ♦αr♦♠♦r ♦r♦♥ ② tr♦♥ ♦♣♥ P②s ♦♥ ttr
❬❪ P ❯♠r t♥t ♥ r♦♥ r♥ t s③ ♣ t♥♥st②♥t♦♥ ♥ ♠♥②♦② ♣rtrt♦♥ t♦r②♦♥♠t
❬❪ ♠♥♥ ♥ ❱♥rt P②s
❬❪ ❲r tt r ♥ r♦♥ ♥t ♣♣r♦ t♦t♠♣♥♥t ♥st②♥t♦♥ ♣rtrt♦♥ t♦r② ♦r ♦♣ts♣tr♦s♦♣② P②s tt
❬❪ r r ♥ r ♥st② ♥t♦♥ t♦r② ♦ ttr ♦♥tt② ♦ ♠♦r s P②s tt
❬❪ ❲sr P②s
❬❪ r♥r ♥ ♦♥ P②s
❬❪ r P②s
❬❪ P ♦♥r ♥ ❲ ♦♥ P②s
❬❪ ❲ ♦♥ ♥ ♠ P②s
❬❪ ❲ Ptt Ps♦♣♦t♥t ♠t♦s ♥ ♦♥♥s ♠ttr ♣♣t♦♥s♦♠♣ P②s ♣
❬❪ ♥ ♥ t③ é♥q ♥tq t♦♥s r ♦s♦r t♦♥
❬❪ s♦♥ ss tr♦②♥♠s ♦♥ ❲② ♥ ♦♥s ♥ ❨♦r ♥ t♦♥ t♦♥
❬❪ P Pr ♥ ❨ ❲♥ P②s
❬❪ ♥ ♥ ❱♥ ♥t♠ ♦r② ♦ t tr♦♥ q♠r ❯♥rst② Prss ♠r
❬❪ ♦♥♦♠♦ r♥s ♥t♦♥s ♥ q♥t♠ ♣②ss ♦stts♥s ♣r♥r ❱r r♥
❬❪ ♥ ♥ ♥ ♥s t♦rs ❯♥♦♣ tr♦♥
stts ♣ ♣r♥r❱r r♥
❬❪ ❲ ❩ tr
❬❪ r tr t♦r② ♦ tr♦♥ ♥trt♦♥ ♥ s♦s Pr♦P②s ♦
❬❪ ②rts♥ ♦ P②
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PP❳ P❨
❬❪ ♦rs ♦tr ❲ rt ♥ ♦③♥r ♦
P②s
❬❪ ♦t♦♥ ♥ ♠t t♦rs ♥rr ♣②ss r♥♦♥Prss ①♦r
❬❪ t ♥ rt P②s tt
❬❪ ♦① ♥ P r P②s ♥♥
❬❪ rt ♣t s♦r♣t♦♥ s♣tr ♦ s♠♦♥t♦rs ♥ ♥st♦rs
♥t♦ t♦♥s ♦ ♠♥②♦② ts P tss ♦P♦②t♥q r♥
❬❪ rt♦♥ t♦r tr♦♥ ♥r②♦ss s♣tr♦s♦♣② ♥ t ♠r♦s♦♣P♥♠ Prss ❨♦r
❬❪ ❱ ♥♦ ♥ ♥♥ P②s tt
❬❪ st♥♥ r♥ ❲ ♦♠ ❱♦s P①t♦♥ ♥ ❲ ♥♥s P②s
❬❪ ♦ss ❲ ♥♦rt ♦tt P r♦♥ r②s♦♥ ♥❲ ♦♠ ♥st P②s ♦♥ r
❬❪ r♥ ss ❨ ❳ ♥ ❲❨ ♥ P②s
❬❪ ♣tr s♠ ❳ ♥t ♥ ♦ P②s tt
❬❪ r ♥ ♥ ♦ P②s
❬❪ r♠s t♥ ♥ P ♥ t ♥ strtr ♥ ts♦r♣t♦♥ s♣tr♠ ♦ 2 P②ss ttrs
❬❪ r♦♥ Prt ♦♠♠♥t♦♥
❬❪ ♥♥ ❱ ♥♦ ♦ ♥ ♥ P②s tt
❬❪ r♦♥ r♦♥♦ ♦rs♦ ♥ P ♥♥♦③③ P♦♥♦♥s ♥rt r②st ♣r♦♣rts r♦♠ ♥st②♥t♦♥ ♣rtrt♦♥ t♦r② ♦ P②s
❬❪ r ♥ ♦ P②s tt
❬❪ s♠♥ ❲ rs t ♥ rs ♠
P②s
❬❪ Pr ♥ r P②s
❬❪ P Pr r② ❱♦s♦ s♦♥ Prs♦♥ ♥ ♥ ♦s P②s P❲♥t♦♥
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PP❳ P❨
❬❪ P Pr r② ❱♦s♦ s♦♥ Prs♦♥ ♥ ♥ ♦s P②s P❲ ♥t♦♥
❬❪ tt r♥ ❱ ♥♦ ♥ ♥♥ ❯♥rst♥♥♦rrt♦♥s ♥ ♥♠ ♦① r♦♠ rst ♣r♥♣s P②s tt
❬❪ r♠♥♥ P♦tr② ♥st♥ ♥ ♦rs ②♥♠s♥ts ♥ ♦rrt♦♥ssst ♣rs tr♥st♦♥ ♥ ♦2 P②s tt
❬❪ ❲ tt trtr ♥ ♦♥♥ ♥ r②st♥ ♦r♦♥ ♥ 123♦r♥ ♦ P②ss ♦ tt P②ss
❬❪ ♠♠s t♥ rs♥r♥ ♥ P P♥rt♥ ♥②♥ tr♦♥♦♥t♥ r ♦r ♠r♦♣♦②r♦r♥s ♠t♦r♥s ♥ ♠t♦♥s ♦r♥ ♦ t ♠r♥
♠ ♦t②
❬❪ ♠♠s t♥ ♥ rs♥r♥ P♦②r♦r♥s ♥ ♠♥t ♦r♦♥♥s♣ rt strtr rt♦♥s ♥rs tr♦♥ rqr♠♥ts ♦r♥ ♦ t ♠r♥ ♠
♦t②
❬❪ ♠♥ P②s ♦②
❬❪ ♠s ♦② ❲s♦♥ ❲ rs ss♦♥ ♦t③ ♠ s ♥ P②s tt
❬❪ ♦r♦s♥ s♦♥ s ♥ ♠♥ ♦r♥♦ ♦②s ♥ ♦♠♣♦♥s
❬❪ ♥ ♦r♦s♥ P②s ♠
❬❪ ♦r♦s♥ s ♥ s♦♥ tr s ②♠♣ Pr♦
❬❪ rs♦♥ ♦r♦♥ ♦s ♦♥ Pr♦ ♦♠ P ❨♦r
❬❪ ♦r♦s♥ s ♥ ♠♥ ♦r♦♥ ♦s ♦♥ Pr♦♦♠ P ❨♦r
❬❪ r♥♥③♠ r♦ P♦♥♥♦ r♦t r ♥ r s♠♥t♦ ♠ P②ss ttrs
❬❪ r♥ P rí♦♥③ás ♥ ♦ P②s tt
❬❪ ③③r ❱tt♥ ♥ ♦♥ P②s
❬❪ ♥♦tt ❲ ♥ ♥ rst♣r♥♣s st② ♦ tstt② ♦ ♥ P②s
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PP❳ P❨
❬❪ ❨♥ ♥ ♦♥ P②s
❬❪ ②r ♦♥ ♦s♦♥ rör P ②r ♠ ♥rt ♥ ♥qst ❱♥ r s ♥st② ♥t♦♥ ♦r②r strtrs P②s tt
❬❪ s ♥ ❳ ♦♥③ rt ♥st② ♥t♦♥s ♣♣r♦s s♥t t♦♥♥t♦♥ tt♦♥ss♣t♦♥ t♦r♠ P②s ♦♥♥s ttr tr P②s ♥
❬❪ r ♥ ❱♥ ❱♦♦rs tt♦♥ss♣t♦♥ t♦r♠♥st②♥t♦♥ t♦r② ♦r♥ ♦ ♠ P②ss
❬❪ P rí♦♥③ás r♥♥③ r♥ ♥ ♦ ♥♦rrt♦♥ ♥t♦♥s♥s♣ ♣♣t♦♥ t♦ ♠trs ♥ ♦③s②st♠s ♦r♥ ♦ P②s ♠str②
❬❪ t r ♥ ♦r rss ♦s ♥r② rs ♦r ♥♦ s s♦st ② t ♦♥♥t♦♥ tt♦♥ss♣t♦♥ t♦r② P②s ♦♥♥s ttr tr P②s
❬❪ r ♥ r Prt ♦♠♠♥t♦♥
❬❪ ❲♦♠ ♥ ♦ ②♠♠tr②r♦♥ r②st strtr ♦♠♥t ♦r♦♥ t ♦ t♠♣rtr P②s
❬❪ ❲♦♠ ♥ ♦ ts ♣r♦♥stt♣r♣②s♠♦♠♣sP♣
❬❪ ♥ ♥ ❨ ❲♥ ②♠♥♦ rr♦② ♥ ❩ Psstt② ♥ α ♥ βr♦♠♦r ♦r♦♥ P②s ♦♥♥s ttr
tr P②s
❬❪ ♥ tt♥ ❯tt♥ ❲s ♥ ♦rtr♠♦②♥♠ stt② ♦ ♦r♦♥♥s♣ t r♦ ♦ ts ♥ ③r♦♣♦♥t ♠♦t♦♥ ♠ ♠ ♦
❬❪ ts ② ❨ ♦t♦♠ r ♥ ♠♣rt r②st ♥ ♥s s♠♦♥t♦r ♦r♦♥ rstt ♠♥t♦r♥ ♦ t ♠r♥ ♠ ♦t②
❬❪ ❯ ♠♥♥ ❲rt ♥ t③ ♦r♥ ♦ ♦②s ♥♦♠♣♦♥s
❬❪ ♥t s ♠♣ ♥ ♠♥ P②s
❬❪ ❯ ♠♥♥ ♥ ❲rt P②s tt ♦
❬❪ ❲rt ♠ ❯ ♠♥♥ ❯ ♠♣♥ ❲ ö♥ ♥ ♥ t rt② ♦ t r♠♥ s♣tr ♦ ♦r♦♥r s♦s♦r♥ ♦ ♦②s ♥ ♦♠♣♦♥s
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PP❳ P❨
❬❪ r♣tr s P♣s ♥ ♦♥t③ ♦r♦♥
♦s ♦♥ Pr♦ ♦♠ P ❨♦r
❬❪ ♥♥ ♠ r♠ ♦
❬❪ r③♦♥♦ ❨ r♦②♠ ♥ ❲t♥ ♠ ♦ ♣♥
❬❪ ♠♦♥ t P ss♦♥ ♥♦③③ rs ♥ qt tr
❬❪ ②♥r ♥ ♥♠♥ trtr ♦ 132 P②s
❬❪ r♦r ♥ rt♥s P②s
❬❪ ❲ st t strtr ♠♦ ♦s ♦r♠t♦♥ ♥r② s♥t t rs♣t t♦ ♦♠♣♦st♦♥ ♥t♦ ♠♦♥ ♣s α♦r♦♥ ♥s t ♦st ♦♥ ♠♦♥ strtrs t t s♠ r♦♥ ♦♥♥trt♦♥
❬❪ ❲ ②♠ P②s
❬❪ ❲rt ♥ ♥ t♦r ♥♦tör♥st♥ ♦r♦♥♦♠♣♦♥s ♦♠ ♣ ♣r♥r❱r r♥
❬❪ ❲rt s ♦♥r♥
❬❪ ❲ s r♠♦♠ s Prt ♦♦♠ t♦♥ ♥sttt ♦ t♥rs ♥ ♥♦♦② t t♦♥
❬❪ ♠t ♦r♥ ♥ ❱♥ rts♥ t ♦♦♠st♦♥ ♥ ♦r♠t♦♥ ♦ ♦r♦♥ r ♦r♥ ♦ t ♠r♥
♠ ♦t②
❬❪ ürs rqs t♦ts ♦rs Pr♦t st ♦♥t♥♥③ ss ♥ r♦ss P②s
❬❪ rqs ürs t♦ts Pr♦t ♦rs Pr♦t st ♦♥t♥♥③ r♦ss ♥ ss P②s
❬❪ ❩♦ t♦ ♦rs♦ ♥ r♦♥ ♥t♦ s♠t♦♥ ♦♣♦t♦♠ss♦♥ s♣tr♦s♦♣② ♥ s♦s ♣♥ ♣s♦♣♦t♥t♣♣r♦ t ♣♣t♦♥ t♦ ♥♦r♠♠ss♦♥ s♣tr ♦ ♥ P②s
❬❪ st ❱st ♥ ❱ ②tr ♥t♦ st② ♦ tr♦♥♣♦♥♦♥♦♣♥ ♥ ①t♦♥ ♥t ♥ s ♥r ♣rssr ♥ P♦r♥ ♦ ♠♥s♥
❬❪ P r r♠♥♥ ♥ ♠ r♠ ♦
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PP❳ P❨
❬❪ ♥ ♦t t♦rs Ps qr r♠s r♠ss♦♥ t♦♥ ♥sttt ♦ t♥rs ♥ ♥♦♦② trsrr②♥
❬❪ ❳ ❩♦ ♥ ❱♥rt P②s
❬❪ s ❨♠♦ ❨ r ♥ s tr
❬❪ r♥ t♦r ❩r♦♥ ♥s ♥ ❩r♦♥ ♥ ♥
♥♦♦② sr ❨♦r
❬❪ ❲tr P s ♥ ♣♣ r ♥ ♣
❬❪ ❱ ♦r♥t♥ ♥ r P②s tt
❬❪ Pt♥♦♠ rtr ♥ P ♥ P②s
❬❪ r♠ ♦t♥r ♥ ♥ P②s tt
❬❪ ♦r♥t ♥③ ♥ ï12 P②s
❬❪ r♠♥♥ ♥ ①♥r ♠♣
❬❪ r t r②st♦r
❬❪ P rt ♥ P rrs ♠ r♠ ♦
❬❪ ♦♦♠♦ ♥ ❩♥③♦tt♦ rst♣r♥♣s st② ♦ tstrtr ♥ st ♣r♦♣rts ♦ ③r♦♥ ♠tt Prt♦♠♠♥t♦♥