Charge-Displacement Analysis via Natural Orbitals for Chemical

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Charge-displacement analysis via natural orbitals for chemical valence in the four-component relativistic framework Matteo De Santis, Sergio Rampino, Harry M. Quiney, Leonardo Belpassi, and Loriano Storchi J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01077 • Publication Date (Web): 31 Jan 2018 Downloaded from http://pubs.acs.org on February 2, 2018

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Journal of Chemical Theory and Computation

❈❤❛r❣❡✲❞✐s♣❧❛❝❡♠❡♥t ❛♥❛❧②s✐s ✈✐❛ ♥❛t✉r❛❧ ♦r❜✐t❛❧s ❢♦r ❝❤❡♠✐❝❛❧ ✈❛❧❡♥❝❡ ✐♥ t❤❡ ❢♦✉r✲❝♦♠♣♦♥❡♥t r❡❧❛t✐✈✐st✐❝ ❢r❛♠❡✇♦r❦ †

▼❛tt❡♦ ❉❡ ❙❛♥t✐s✱

∗✱‡✱k

❙❡r❣✐♦ ❘❛♠♣✐♥♦✱

∗✱‡✱⊥

❇❡❧♣❛ss✐✱

¶

❍❛rr② ▼✳ ◗✉✐♥❡②✱

▲❡♦♥❛r❞♦

∗✱§✱‡

❛♥❞ ▲♦r✐❛♥♦ ❙t♦r❝❤✐

†❉✐♣❛rt✐♠❡♥t♦

❞✐ ❈❤✐♠✐❝❛✱ ❇✐♦❧♦❣✐❛ ❡ ❇✐♦t❡❝♥♦❧♦❣✐❡✱ ❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ P❡r✉❣✐❛✱ ❱✐❛ ❊❧❝❡ ❞✐ ❙♦tt♦ ✽✱ ✵✻✶✷✸ P❡r✉❣✐❛✱ ■t❛❧✐❛ ‡■st✐t✉t♦ ❞✐ ❙❝✐❡♥③❡ ❡ ❚❡❝♥♦❧♦❣✐❡ ▼♦❧❡❝♦❧❛r✐✱ ❈♦♥s✐❣❧✐♦ ◆❛③✐♦♥❛❧❡ ❞❡❧❧❡ ❘✐❝❡r❝❤❡ ❝✴♦ ❉✐♣❛rt✐♠❡♥t♦ ❞✐ ❈❤✐♠✐❝❛✱ ❇✐♦❧♦❣✐❛ ❡ ❇✐♦t❡❝♥♦❧♦❣✐❡✱ ❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ P❡r✉❣✐❛✱ ❱✐❛ ❊❧❝❡ ❞✐ ❙♦tt♦ ✽✱ ✵✻✶✷✸ P❡r✉❣✐❛✱ ■t❛❧✐❛ ¶❆❘❈ ❈❡♥tr❡ ♦❢ ❊①❝❡❧❧❡♥❝❡ ❢♦r ❆❞✈❛♥❝❡❞ ▼♦❧❡❝✉❧❛r ■♠❛❣✐♥❣✱ ❙❝❤♦♦❧ ♦❢ P❤②s✐❝s✱ ❚❤❡ ❯♥✐✈❡rs✐t② ♦❢ ▼❡❧❜♦✉r♥❡✱ ✸✵✶✵ ❱✐❝t♦r✐❛✱ ❆✉str❛❧✐❛ §❉✐♣❛rt✐♠❡♥t♦ ❞✐ ❋❛r♠❛❝✐❛✱ ❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❵●✳ ❉✬❆♥♥✉♥③✐♦✬✱ ❱✐❛ ❞❡✐ ❱❡st✐♥✐ ✸✶✱ ✻✻✶✵✵ ❈❤✐❡t✐✱ ■t❛❧✐❛ kPr❡s❡♥t ❛❞❞r❡ss✿ ❙❝✉♦❧❛ ◆♦r♠❛❧❡ ❙✉♣❡r✐♦r❡✱ P✐❛③③❛ ❞❡✐ ❈❛✈❛❧✐❡r✐ ✼✱ ✺✻✶✷✻ P✐s❛✱ ■t❛❧✐❛ ⊥❈♦♥s♦rt✐✉♠ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ▼♦❧❡❝✉❧❛r ❛♥❞ ▼❛t❡r✐❛❧s ❙❝✐❡♥❝❡s ✭❈▼❙✮2 ✱ ❱✐❛ ❊❧❝❡ ❞✐ ❙♦tt♦✱ ✽✱ ✵✻✶✷✸ P❡r✉❣✐❛✱ ■t❛❧②✳ ❊✲♠❛✐❧✿ s❡r❣✐♦✳r❛♠♣✐♥♦❅s♥s✳✐t❀ ❧❡♦♥❛r❞♦✳❜❡❧♣❛ss✐❅❝♥r✳✐t❀ ❧♦r✐❛♥♦❅st♦r❝❤✐✳♦r❣

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ACS Paragon Plus Environment

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❆❜str❛❝t ❲❡ ❤❛✈❡ r❡❝❡♥t❧② ✐♥tr♦❞✉❝❡❞ ❛ s✐♠♣❧❡ ②❡t ♣♦✇❡r❢✉❧ t♦♦❧ ❢♦r ❛♥❛❧②③✐♥❣ q✉❛♥t✐t❛✲ t✐✈❡❧② t❤❡ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❞♦♥❛t✐♦♥ ❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝♦♥st✐t✉❡♥ts ♦❢ t❤❡ ❉❡✇❛r✲❈❤❛tt✲❉✉♥❝❛♥s♦♥ ♠♦❞❡❧✳ ❖✉r ❛♣♣r♦❛❝❤ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥✱ ✈✐❛ ♥❛t✉r❛❧ ♦r❜✐t❛❧s ❢♦r ❝❤❡♠✐❝❛❧ ✈❛❧❡♥❝❡ ✭◆❖❈❱✮✱ ♦❢ t❤❡ s♦✲❝❛❧❧❡❞ ❝❤❛r❣❡✲❞✐s♣❧❛❝❡♠❡♥t ✭❈❉✮ ❢✉♥❝t✐♦♥ ✐♥t♦ ❛❞❞✐t✐✈❡ ❝❤❡♠✐❝❛❧❧② ♠❡❛♥✐♥❣❢✉❧ ❝♦♠♣♦♥❡♥ts ✭❇✐st♦♥✐ ❡t ❛❧✳✱ ❏✳ ❈❤❡♠✳ ✱ ✷✵✶✺✱

P❤②s✳

✱ ✵✽✹✶✶✷✮✳ ❚❤❡ ♠❡t❤♦❞✱ r❡❢❡rr❡❞ t♦ ❛s ◆❖❈❱✴❈❉✱ ♣r♦✈✐❞❡s ❝❧❡❛r✲❝✉t

✶✹✷

♠❡❛s✉r❡s ♦❢ ❞♦♥❛t✐♦♥ ❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝❤❛r❣❡ ✢♦✇s ❢♦❧❧♦✇✐♥❣ ❜♦♥❞ ❢♦r♠❛t✐♦♥ ❛♥❞ ✐ts r♦❜✉st♥❡ss ❤❛s ❜❡❡♥ ❞❡♠♦♥str❛t❡❞ ❜② ❛ t✐❣❤t ❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ r❡❧❛t❡❞ ❝❤❛r❣❡✲tr❛♥s❢❡r ❡st✐♠❛t❡s ✇✐t❤ ❡①♣❡r✐♠❡♥t❛❧ ♦❜s❡r✈❛❜❧❡s✳ ■♥ t❤✐s ♣❛♣❡r ✇❡ ❡①t❡♥❞ t❤❡ ◆❖❈❱✴❈❉ ❛♥❛❧✲ ②s✐s s❝❤❡♠❡ t♦ t❤❡ ❢♦✉r✲❝♦♠♣♦♥❡♥t r❡❧❛t✐✈✐st✐❝ ❢r❛♠❡✇♦r❦✱ ✇❤✐❝❤ ✐♥❝❧✉❞❡s s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ ✈❛r✐❛t✐♦♥❛❧❧②✳ ❚❤✐s ❢♦r♠❛❧✐s♠ ✐s ✐♥❝♦r♣♦r❛t❡❞ ✐♥t♦ ❛ r❡❝❡♥t❧② ❞❡✈❡❧♦♣❡❞✱ ❤✐❣❤❧② ❡✣❝✐❡♥t ♣❛r❛❧❧❡❧ ✈❡rs✐♦♥ ♦❢ t❤❡ r❡❧❛t✐✈✐st✐❝ ❉✐r❛❝✲❑♦❤♥✲❙❤❛♠ ✭❉❑❙✮ ♣r♦❣r❛♠ ❇❊❘❚❍❆ ✭❘❛♠♣✐♥♦ ❡t ❛❧✳✱

❏✳ ❈❤❡♠✳

❚❤❡♦r② ❈♦♠♣✉t✳

✱ ✷✵✶✹✱

✱ ✸✼✻✻✲✸✼✼✻✮✳ ❲❡ t❡st t❤❡ ❛❝✲

✶✵

❝✉r❛❝② ❛♥❞ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ t❤✐s ♥❡✇ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t❤r♦✉❣❤ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❜❛s✐s s❡ts ❡♠♣❧♦②❡❞ t♦ ❡①♣❛♥❞ t❤❡ ❉❑❙ s♣✐♥♦r s♦❧✉t✐♦♥ ❛♥❞ t❤♦s❡ ✉s❡❞ t♦ ❧✐♥❡❛r✐③❡ t❤❡ ❡❧❡❝tr♦♥✐❝ ❞❡♥s✐t② ✐♥ t❤❡ ❞❡♥s✐t②✲✜tt✐♥❣ ❛❧❣♦r✐t❤♠ s♣❡❡❞✲ ✐♥❣ ✉♣ t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❉❑❙ ♠❛tr✐①✳ ❆♥ ✐❧❧✉str❛t✐♦♥ ♦❢ ◆❖❈❱✴❈❉ ❛♥❛❧②s✐s ✐♥ t❤❡ r❡❧❛t✐✈✐st✐❝ ❢r❛♠❡✇♦r❦ ✐s ❛❧s♦ ❣✐✈❡♥ t❤r♦✉❣❤ ❛ st✉❞② ♦❢ t❤❡ ♠❡t❛❧✲❝❛r❜♦♥②❧ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ✐♥ ❛ s❡r✐❡s ♦❢ ❬▼✕❈❖❪+ ✭▼ = ❈✉✱ ❆❣✱ ❆✉✮ ❝♦♠♣❧❡①❡s ♦❢ ❣r♦✉♣✲✶✶ ♠❡t❛❧s ✇❤❡r❡ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ✐♥❝❧✉❞✐♥❣ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ ❛r❡ ❢♦✉♥❞ t♦ ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡✳

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✶

■♥tr♦❞✉❝t✐♦♥

❚❤❡ ❝❤❡♠✐❝❛❧ ❜♦♥❞ ✐s ❛ ❝❡♥tr❛❧ ❝♦♥❝❡♣t ✐♥ ❝❤❡♠✐str② ❛♥❞ ♣r♦✈✐❞❡s ❛ ❝♦♥✈❡♥✐❡♥t ❝♦♠♠♦♥ ❣r♦✉♥❞ ❢♦r ❡①♣❡r✐♠❡♥t❛❧ ❛♥❞ t❤❡♦r❡t✐❝❛❧ ❝❤❡♠✐sts t♦ r❛t✐♦♥❛❧✐③❡ ❛♥❞ ♣r❡❞✐❝t t❤❡ str✉❝t✉r❡✱ st❛❜✐❧✐t② ❛♥❞ r❡❛❝t✐✈✐t② ♦❢ ❝❤❡♠✐❝❛❧ s♣❡❝✐❡s✳ ❞❡✜♥❡❞ q✉❛♥t✉♠✲♠❡❝❤❛♥✐❝❛❧ ♦❜s❡r✈❛❜❧❡

✶

❚❤❡ ❝❤❡♠✐❝❛❧ ❜♦♥❞ ✐s ♥♦t✱ ❤♦✇❡✈❡r✱ ❛ ✇❡❧❧✲

❛♥❞ ✐ts ❞❡t❛✐❧❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ♥❛t✉r❡ ✐s st✐❧❧ ✇✐❞❡❧②

❞✐s❝✉ss❡❞✳ ■♥ t❤❡ ♣❛st ❢❡✇ ❞❡❝❛❞❡s s❡✈❡r❛❧ ❛♣♣r♦❛❝❤❡s ❤❛✈❡ ❜❡❡♥ ✐♥tr♦❞✉❝❡❞✱ ♠❛✐♥❧② ✇✐t❤✐♥ t❤❡ t❤❡♦r❡t✐❝❛❧ ❝❤❡♠✐str② ❝♦♠♠✉♥✐t②✱ t♦ ❛♥❛❧②③❡ ❛♥❞ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❝❤❡♠✐❝❛❧ ❜♦♥❞✳

❚❤❡s❡

❝❛♥ ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ ♠❛✐♥ ❣r♦✉♣s✿ ✐✮ ♠❡t❤♦❞s ♠❛✐♥❧② ❢♦❝✉s✐♥❣ ♦♥ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❜♦♥❞ ❡♥❡r❣② ✐♥t♦ ❵❝❤❡♠✐❝❛❧❧② ♠❡❛♥✐♥❣❢✉❧✬ ❝♦♥tr✐❜✉t✐♦♥s ❛♥❞ ✐✐✮ ♠❡t❤♦❞s ❢♦❝✉s✐♥❣ ♠❛✐♥❧② ♦♥ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ✇❛✈❡❢✉♥❝t✐♦♥ ♦r t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t②✳ ❖❢ t❤❡ ✜rst ❣r♦✉♣✱ ✇❡ ✐♥❝❧✉❞❡ ❇❧♦❝❦ ▲♦❝❛❧✐③❡❞ ❲❛✈❡ ❋✉♥❝t✐♦♥ ❊♥❡r❣② ❉❡❝♦♠♣♦s✐t✐♦♥✱

✷

t❤❡ ❑✐t❛✉r❛✲▼♦r♦❦✉♠❛ ♠❡t❤♦❞✱

t❤❡ ❙②♠♠❡tr② ❆❞❛♣t❡❞ P❡rt✉r❜❛t✐♦♥ ❚❤❡♦r② ✭❙❆P❚✮ s❝❤❡♠❡ ❙t❛t❡ ✭❊❚❙✮✱

✺✱✻

✹

❛♥❞ t❤❡ ❊①t❡♥❞❡❞ ❚r❛♥s✐t✐♦♥

✇❤✐❧❡ t♦ t❤❡ s❡❝♦♥❞ ❣r♦✉♣ ❜❡❧♦♥❣ t❤❡ ♣✐♦♥❡❡r✐♥❣ ✇♦r❦s ♦❢ ❇❛❞❡r

r❡❛rr❛♥❣❡♠❡♥t✱ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ▲❛♣❧❛❝✐❛♥ ♦❢ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t②✱ ❜✉t✐♦♥s✱

✶✶

❋✉♥❝t✐♦♥✱

t❤❡ ❞✐✛❡r❡♥❝❡ ❞❡♥s✐t✐❡s ✭s❡❡ ❙❝❤✇❛r③✬s s❡♠✐♥❛❧ ✇♦r❦

✶✸

♣♦♣✉❧❛t✐♦♥ ❛♥❛❧②s✐s

✶✹

✸

✶✷

✾✱✶✵

✼✱✽

♦♥ ❝❤❛r❣❡

t❤❡ ❡❧❡❝tr♦♥ ❞✐str✐✲

✮✱ t❤❡ ❊❧❡❝tr♦♥ ▲♦❝❛❧✐③❛t✐♦♥

❛♥❞ ❈❤❛r❣❡ ❉❡❝♦♠♣♦s✐t✐♦♥ ❆♥❛❧②s✐s✳

✶✺✱✶✻

■♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ❛ ✇❡❧❧✲❞❡✜♥❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❝❤❡♠✐❝❛❧ ❜♦♥❞✐♥❣✱ ✐t ✐s ♥♦t s✉r♣r✐s✐♥❣ t❤❛t t❤❡ ❞✐s❝✉ss✐♦♥ ❛❜♦✉t t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❝❤❡♠✐❝❛❧ ❜♦♥❞ t②♣✐❝❛❧❧② ❣❡♥❡r❛t❡s ❞✐s♣✉t❡s ❛♥❞ ♠✐s✉♥❞❡rst❛♥❞✐♥❣s✱ ❛s ❞✐✛❡r❡♥t ♠❡t❤♦❞s ♦❢t❡♥ ❧❡❛❞ t♦ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥s ❛♥❞ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ ❛ ❜♦♥❞✳

✶

❲❡ ❛❣r❡❡ ✇✐t❤ ❈♦✉❧s♦♥✬s ♣❡rs♣❡❝t✐✈❡ t❤❛t ❛ ❝❤❡♠✐❝❛❧ ❜♦♥❞

✏✐s ✜❣♠❡♥t ♦❢ ♦✉r ♦✇♥ ✐♠❛❣✐♥❛t✐♦♥ ✳✳✳

❜✉t ✐t ✐s ❛ ♠♦st ❝♦♥✈❡♥✐❡♥t ✜❝t✐♦♥ ✇❤✐❝❤✱ ❛s ✇❡

❤❛✈❡ s❡❡♥✱ ✐s ❝♦♥✈❡♥✐❡♥t ❜♦t❤ t♦ ❡①♣❡r✐♠❡♥t❛❧ ❛♥❞ t❤❡♦r❡t✐❝❛❧ ❝❤❡♠✐sts✑✳ ■t ✐s ❛❧s♦ t❤❡ ❝❛s❡ t❤❛t ✐❢ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ t❤❡♦r② ♦❢ q✉❛♥t✉♠✲♠❡❝❤❛♥✐❝s ❞♦❡s ♥♦t ❤❡❧♣ t♦ ❞❡s✐❣♥ ✉♥❛♠❜✐❣✉♦✉s ❞❡✜♥✐t✐♦♥s ♦❢ ❝❤❡♠✐❝❛❧ ❜♦♥❞✐♥❣✱ ❛♥ ✐♠♣♦rt❛♥t ❝r✐t❡r✐♦♥ t♦ ❛ss❡ss t❤❡ ✈❛❧✐❞✐t② ♦❢ ❛ ♠♦❞❡❧ ❢♦r ❝❤❡♠✐❝❛❧ ❜♦♥❞✐♥❣ ✐s ✐ts ♣r❡❞✐❝t✐✈❡ ❛❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ ❡①♣❡r✐♠❡♥t❛❧ ♦❜s❡r✈❛❜❧❡s✳ ❙♦♠❡ ♦❢ ✉s ❤❛✈❡ r❡❝❡♥t❧② ♣r♦♣♦s❡❞ ❛ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤ ❢♦r ❛♥❛❧②③✐♥❣ ❛ ❝❤❡♠✐❝❛❧ ❜♦♥❞✱ t❤❡

✸



s♦ ❝❛❧❧❡❞ ❝❤❛r❣❡✲❞✐s♣❧❛❝❡♠❡♥t ✭❈❉✮ ❛♥❛❧②s✐s✳

✶✼

Page 4 of 39

❚❤❡ ♠❡t❤♦❞ ✐s ❜❛s❡❞ ♦♥ ❛ ♣❛rt✐❛❧ ♣r♦❣r❡ss✐✈❡

✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t t❛❦✐♥❣ ♣❧❛❝❡ ✉♣♦♥ ❜♦♥❞ ❢♦r♠❛t✐♦♥ ✭②✐❡❧❞✐♥❣ t❤❡ s♦✲❝❛❧❧❡❞ ❈❉ ❢✉♥❝t✐♦♥✮ ✇❤✐❝❤ ❝❧❡❛r❧② ❜❡❧♦♥❣s t♦ t❤❡ s❡❝♦♥❞ ❣r♦✉♣ ♦❢ t❤❡ ♠❡t❤♦❞s ❞✐s❝✉ss❡❞ ❛❜♦✈❡✳ ❚❤❡ ❈❉ ❛♣♣r♦❛❝❤ ✐s ❛ ♣♦✇❡r❢✉❧ ♠❡t❤♦❞ t❤❛t ✇❛s ✐♥tr♦❞✉❝❡❞ t♦ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ❆✉✕ ♥♦❜❧❡ ❣❛s ✐♥t❡r❛❝t✐♦♥

✶✼

❛♥❞ ❤❛s ❜❡❡♥ s✉❝❝❡ss❢✉❧❧② ❡♠♣❧♦②❡❞ ✐♥ ✈❡r② ❞✐✛❡r❡♥t ❝♦♥t❡①ts✱ ❢r♦♠

✇❡❛❦❧② ❤②❞r♦❣❡♥✴❤❛❧♦❣❡♥ ❜♦✉♥❞ ❛❞❞✉❝ts✱

✶✽✕✷✷

t♦ ♦r❣❛♥♦♠❡t❛❧❧✐❝ ❝♦♠♣❧❡①❡s✱

t♦ q✉❛♥t✐❢② t❤❡ ❈❚ ❝❤❛r❛❝t❡r ♦❢ ❡❧❡❝tr♦♥✐❝ ❡①❝✐t❡❞ st❛t❡s✳

✷✸✕✷✺

❛♥❞ ❡✈❡♥

✷✻

■♥ ❝❛s❡s ♦❢ s✉✐t❛❜❧❡ s②♠♠❡tr②✱ t❤❡ ❈❉ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❡❛s✐❧② ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ❛❞❞✐✲ t✐✈❡ s②♠♠❡tr② ❝♦♠♣♦♥❡♥ts

✷✹✱✷✼✕✷✾

t❤❛t ♣r♦✈✐❞❡ ❛ ❞❡t❛✐❧❡❞ ❛♥❞ q✉❛♥t✐t❛t✐✈❡ ♣✐❝t✉r❡ ♦❢ t❤❡

❧✐❣❛♥❞✲t♦✲♠❡t❛❧ ❞♦♥❛t✐♦♥ ❛♥❞ ♠❡t❛❧✲t♦✲❧✐❣❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝❤❛r❣❡ ✢♦✇s ♦❢ t❤❡ ❉❡✇❛r✲ ❈❤❛tt✲❉✉♥❝❛♥s♦♥ ✭❉❈❉✮ ♠♦❞❡❧✳

✸✵✱✸✶

❲❡ ❞❡♠♦♥str❛t❡❞ ❢♦r t❤❡ ✜rst t✐♠❡ t❤❛t t❤❡ ❞♦♥❛t✐♦♥ ❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❉❈❉ ♠♦❞❡❧ ❝❛♥ ❜❡ ✉♥❛♠❜✐❣✉♦✉s❧② ❞✐s❡♥t❛♥❣❧❡❞ ❛♥❞ ♣✉t ✐♥ ✈❡r② t✐❣❤t ❝♦rr❡❧❛t✐♦♥ ✇✐t❤ ❡①♣❡r✐♠❡♥t❛❧ ♦❜s❡r✈❛❜❧❡s✳

✸✷✕✸✺

▼♦r❡ r❡❝❡♥t❧②✱ t❤❡ ♠❡t❤♦❞ ❤❛s ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ t❛❦❡

❛❞✈❛♥t❛❣❡ ♦❢ t❤❡ ♥❛t✉r❛❧ ♦r❜✐t❛❧s ❢♦r ❝❤❡♠✐❝❛❧ ✈❛❧❡♥❝❡ ✭◆❖❈❱✮ t❤❡♦r②✳ ❛♣♣r♦❛❝❤✱ r❡❢❡rr❡❞ t♦ ❛s ◆❖❈❱✴❈❉✱

✸✾✱✹✵

✸✻✕✸✽

❚❤❡ r❡s✉❧t✐♥❣

♦✈❡r❝♦♠❡s t❤❡ s②♠♠❡tr② ❧✐♠✐t❛t✐♦♥s ♦❢ t❤❡ ♦r✐❣✐♥❛❧

♠❡t❤♦❞ ❛♥❞ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞✱ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ t♦ ♠♦❧❡❝✉❧❛r s②st❡♠s ✇✐t❤♦✉t ❛♥② s②♠♠❡tr② ❝♦♥str❛✐♥ts

✸✾✱✹✶✕✹✸

✐♥❝❧✉❞✐♥❣ t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❛ ❝❛t❛❧②st✲s✉❜str❛t❡ ❝❤❡♠✐❝❛❧

❜♦♥❞ ❛❧♦♥❣ t❤❡ st❡♣s ♦❢ ❛ r❡❛❝t✐♦♥ ♣❛t❤✳

✸✾

❈✉rr❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥s ♦❢ ◆❖❈❱ t❤❡♦r② ❛r❡ ❧✐♠✐t❡❞ t♦ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤❛♠✐❧t♦♥✐❛♥s ♦r t♦ ❛ s❝❛❧❛r r❡❧❛t✐✈✐st✐❝ ❢r❛♠❡✇♦r❦ ✇❤❡r❡ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ ✐s s✐♠♣❧② ♦✈❡r❧♦♦❦❡❞✳ ❍♦✇✲ ❡✈❡r✱ ✐t ✐s ♥♦✇ ✇✐❞❡❧② ❛❝❝❡♣t❡❞ t❤❛t s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ ♣❧❛②s ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ♥♦t ♦♥❧② ✐♥ s♣❡❝tr♦s❝♦♣② ❜✉t ❛❧s♦ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ ❝❤❡♠✐❝❛❧ ❜♦♥❞✐♥❣

✶✱✹✹✱✹✺

❛♥❞ ❝❤❡♠✐❝❛❧ r❡❛❝t✐✈✐t②✳

✹✻✕✺✶

■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t t❤❡ ❢♦r♠❛❧✐s♠ ✉s❡❞ t♦ ❞❡❝♦♠♣♦s❡ t❤❡ ❈❉ ❢✉♥❝t✐♦♥ ✐♥ t❡r♠s ♦❢ ◆❖❈❱s ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤❡ r❡❧❛t✐✈✐st✐❝ ❢♦✉r✲❝♦♠♣♦♥❡♥t ❢r❛♠❡✇♦r❦ ✇❤❡r❡ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ ✐s ✐♥❝❧✉❞❡❞ ✈❛r✐❛t✐♦♥❛❧❧②✳ ■♥ ❙❡❝t✐♦♥ ✷ ✇❡ r❡❝❛❧❧ t❤❡ ❣❡♥❡r❛❧ ❛s♣❡❝ts ♦❢ t❤❡ ♠❡t❤♦❞ ❛♥❞ ❡①♣❧❛✐♥ ✐♥ ❞❡t❛✐❧ t❤❡ ❢♦r♠❛❧✐s♠

✹


Page 5 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


✐♥ t❤❡ s♣❡❝✐✜❝ ❝♦♥t❡①t ♦❢ t❤❡ ❉✐r❛❝✲❑♦❤♥✲❙❤❛♠ ♠♦❞✉❧❡ ♦❢ t❤❡ ❇❊❘❚❍❆ ❝♦❞❡✳

✺✷✕✺✺

■♥ ❙❡❝t✐♦♥

✸ ✇❡ r❡♣♦rt t❡st ❝❛❧❝✉❧❛t✐♦♥s t♦ ❛ss❡ss t❤❡ ✈❛❧✐❞✐t② ❛♥❞ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ♦✉r ❝♦❞❡✳ ❆♥ ❛♣♣❧✐❝❛t✐♦♥ ✐s ♣r❡s❡♥t❡❞ ✇❤✐❝❤ ❛♥❛❧②s❡s t❤❡ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ✐♥ ❝❛r❜♦♥②❧ ❝♦♠♣❧❡①❡s ♦❢ ❣r♦✉♣

+ ✶✶ ♠❡t❛❧s ♦❢ ❢♦r♠✉❧❛ ❬▼✕❈❖❪ ✭▼

=

❈✉✱ ❆❣✱ ❆✉✮✳ ❙♦♠❡ ❝♦♥❝❧✉s✐♦♥s ❛r❡ ❞r❛✇♥ ✐♥ t❤❡ ✜♥❛❧

s❡❝t✐♦♥✳

✷

▼❡t❤♦❞♦❧♦❣②

✷✳✶ ❇♦♥❞ ❛♥❛❧②s✐s ✇✐t❤ ◆❖❈❱✴❈❉ ♠❡t❤♦❞ ❆s ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞ ✐♥ t❤❡ ■♥tr♦❞✉❝t✐♦♥✱ ❛♠♦♥❣ t❤❡ ♠❛♥② s❝❤❡♠❡s ❛✈❛✐❧❛❜❧❡ ❢♦r ❝❤❡♠✐❝❛❧✲ ❜♦♥❞ ❛♥❛❧②s✐s ✐s ♦✉r ♦✇♥ ❜❛s❡❞ ♦♥ t❤❡ s♦✲❝❛❧❧❡❞ ❝❤❛r❣❡✲❞✐s♣❧❛❝❡♠❡♥t ✭❈❉✮ ❢✉♥❝t✐♦♥✳ ❈❉ ❢✉♥❝t✐♦♥

∆q(z)

❚❤❡

✐s ❞❡✜♥❡❞ ❛s

∆q(z) =

Z

z

dz −∞

′

Z

∞ −∞

Z

∞

∆ρ(x, y, z ′ )dxdy

∆ρ(x, y, z ′ )

✭✶✮

−∞

✇❤✐❝❤ ✐s ❛ ♣❛rt✐❛❧ ♣r♦❣r❡ss✐✈❡ ✐♥t❡❣r❛t✐♦♥ ❛❧♦♥❣ ❛ ❝❤♦s❡♥ ❛①✐s✱ r❛♥❣❡♠❡♥t

✶✼

z✱

♦❢ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛r✲

t❛❦✐♥❣ ♣❧❛❝❡ ✉♣♦♥ ❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❜♦♥❞✳ ❚❤✐s ❛①✐s ✐s ❞❡✜♥❡❞ ❜② t❤❡

❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ✐♥t❡r❛❝t✐♥❣ s♣❡❝✐❡s✳

∆ρ(x, y, z ′ ) ✐s ❝♦♥✈❡♥✐❡♥t❧② ❞❡✜♥❡❞ ❛s t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ t❤❡

❡❧❡❝tr♦♥ ❞❡♥s✐t② ♦❢ t❤❡ ❛❞❞✉❝t ❆❇ ❛t ✐ts ❡q✉✐❧✐❜r✐✉♠ ❣❡♦♠❡tr② ❛♥❞ t❤❛t ♦❢ t❤❡ ♥♦♥✲✐♥t❡r❛❝t✐♥❣ ❢r❛❣♠❡♥ts ❆ ❛♥❞ ❇ ♣❧❛❝❡❞ ❛t t❤❡ s❛♠❡ ♣♦s✐t✐♦♥ t❤❛t t❤❡② ♦❝❝✉♣② ✐♥ t❤❡ ❛❞❞✉❝t✳ ❈❧❡❛r❧②✱ t❤❡ ❈❉ ❢✉♥❝t✐♦♥ ♠❡❛s✉r❡s✱ ❛t ❡❛❝❤ ♣♦✐♥t

z

❛❧♦♥❣ t❤❡ ❛①✐s✱ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡❝tr♦♥s t❤❛t✱ ✉♣♦♥

❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ❛❞❞✉❝t✱ ❤❛✈❡ ❝r♦ss❡❞ ❢r♦♠ r✐❣❤t t♦ ❧❡❢t t❤❡ ♣❧❛♥❡ t❤r♦✉❣❤

z

♣❡r♣❡♥❞✐❝✉❧❛r

t♦ t❤❡ ❛①✐s✳ ❆ ♥❡❣❛t✐✈❡ ✈❛❧✉❡ ♠❡❛♥s ❛ ❞✐s♣❧❛❝❡♠❡♥t ♦❢ ❡❧❡❝tr♦♥s ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✱ ✐✳❡✳ ❢r♦♠ t❤❡ ❧❡❢t t♦ t❤❡ r✐❣❤t✳ ❚❤❡ ❈❉ ❢✉♥❝t✐♦♥ ♣r♦✈✐❞❡s ❛ ❝♦♥❝✐s❡ ❛♥❞ ✐♥s✐❣❤t❢✉❧ s♥❛♣s❤♦t ♦❢ t❤❡ ✇❤♦❧❡ ❡❧❡❝tr♦♥ ❝❧♦✉❞ r❡❛rr❛♥❣❡♠❡♥t ❛r✐s✐♥❣ ❢r♦♠ ❛❧❧ t❤❡ ✐♥t❡r❛❝t✐♦♥ ❝♦♠♣♦♥❡♥ts✳ ■♥ t❡r♠s ♦❢ ♦❝❝✉♣✐❡❞ ❑♦❤♥✲❙❤❛♠ ✭♦r ❍❛rtr❡❡✲❋♦❝❦✮ s♣✐♥✲♦r❜✐t❛❧s ✭ψi ✮ ♦❢ s❡♣❛r❛t❡ ❆❇✱ ❆

✺



❛♥❞ ❇ s②st❡♠s✱

∆ρ

Page 6 of 39

❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s

∆ρ =

X

|ψiAB |2 −

i

X

|ψiA |2 −

i

X

|ψiB |2 .

✭✷✮

i

❆s ♣r❡✈✐♦✉s❧② ❞❡♠♦♥str❛t❡❞✱ ❢♦r s✉✐t❛❜❧② s②♠♠❡tr✐❝ ❛❞❞✉❝ts ❛♥❞ ❢r❛❣♠❡♥ts✱ ❝♦♥s❡q✉❡♥t❧②

∆q(z)✮

∆ρ

✭❛♥❞

❝❛♥ ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥t♦ ❛❞❞✐t✐✈❡ s②♠♠❡tr② ❝♦♠♣♦♥❡♥ts ✇❤✐❝❤ ❝❛♥ ❜❡

r❡❛❞✐❧② ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ❉❈❉ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❜♦♥❞✳

✷✼✱✸✷

❍♦✇❡✈❡r✱ ✇❤❡♥ t❤❡ s②st❡♠

✉♥❞❡r st✉❞② ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ♥❡❝❡ss❛r② s②♠♠❡tr② r❡q✉✐r❡♠❡♥ts✱ ♦t❤❡r r♦✉t❡s ❤❛✈❡ t♦ ❜❡ ❢♦✉♥❞ ❢♦r ❞✐s❡♥t❛♥❣❧✐♥❣ ❞♦♥❛t✐♦♥ ❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ✐♥ t❤❡ ❢r❛❣♠❡♥t✲❢r❛❣♠❡♥t ✐♥t❡r❛❝t✐♦♥✳ ❚❤❡ r❡❝❡♥t❧② ♣r♦♣♦s❡❞ ♥❛t✉r❛❧ ♦r❜✐t❛❧s ❢♦r ❝❤❡♠✐❝❛❧ ✈❛❧❡♥❝❡ ✭◆❖❈❱✮ ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ♣❛rt✐❝✉❧❛r❧② s✉✐t❡❞ t♦ t❤✐s ♣✉r♣♦s❡✳

✺✻✱✺✼

♠❡t❤♦❞

■♥ t❤❡ ◆❖❈❱ ❛♣♣r♦❛❝❤✱ t❤❡

❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t t❛❦✐♥❣ ♣❧❛❝❡ ✉♣♦♥ ❢♦r♠❛t✐♦♥ ♦❢ ❆❇ ❢r♦♠ ❢r❛❣♠❡♥ts ❆ ❛♥❞ ❇ ❛♥❞ ✐s ❞❡✜♥❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❛ s❧✐❣❤t❧② ❞✐✛❡r❡♥t r❡❢❡r❡♥❝❡ s②st❡♠ ✭❝♦♠♣❛r❡❞ t♦ t❤❛t ♦❢ ❊q✳

ψiA

✷✮ ♠❛❞❡ ✉♣ ♦❢ t❤❡ ♦❝❝✉♣✐❡❞

r❡s♣❡❝t t♦ ❡❛❝❤ ♦t❤❡r✳

❛♥❞

ψiB

♦r❜✐t❛❧s ♦❢ ❆ ❛♥❞ ❇ ♦rt❤♦♥♦r♠❛❧✐③❡❞ ✇✐t❤

❲❡ r❡❢❡r t♦ t❤✐s ♥❡✇ s❡t ♦❢ s♣✐♥✲♦r❜✐t❛❧s ❛s

ψi0 ✳

■♥ ♦t❤❡r ✇♦r❞s✱

r❛t❤❡r t❤❛♥ t✇♦ s❡♣❛r❛t❡ ❆ ❛♥❞ ❇ ❞❡t❡r♠✐♥❛♥ts✱ t❤❡✐r ❛♥t✐s②♠♠❡tr✐s❡❞ ♣r♦❞✉❝t ✐s t❛❦❡♥ ❛s t❤❡ ❢r❛❣♠❡♥t✲❢r❛❣♠❡♥t ♥♦♥ ✐♥t❡r❛❝t✐♥❣ r❡❢❡r❡♥❝❡ ✭t❤❡ s♦ ❝❛❧❧❡❞ ❵♣r♦♠♦❧❡❝✉❧❡✬✮✳ ❚❤❡ r❡s✉❧t✐♥❣ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t

′

∆ρ =

X

|ψiAB |2 −

i

✇❤✐❝❤ ✇❡ s❤❛❧❧ ✐♥❞✐❝❛t❡ ✇✐t❤

′

∆ρ

X

|ψi0 |2

i

t♦ ❛✈♦✐❞ ❝♦♥❢✉s✐♦♥ ✇✐t❤

❞✐❛❣♦♥❛❧ ❢♦r♠ ✐♥ t❡r♠s ♦❢ ◆❖❈❱s✳

✭✸✮

∆ρ

♦❢ ❊q✳ ✷✱ ❝❛♥ ❜❡ ❜r♦✉❣❤t ✐♥t♦

❚❤❡s❡ ❛r❡ ❞❡✜♥❡❞ ❛s t❤❡ ❡✐❣❡♥❢✉♥❝t✐♦♥s✱

s♦✲❝❛❧❧❡❞ ❵✈❛❧❡♥❝❡ ♦♣❡r❛t♦r✬ ♦❢ ◆❛❧❡✇❛❥s❦✐ ❛♥❞ ▼r♦③❡❦ ✈❛❧❡♥❝❡ t❤❡♦r②✱

✸✽✱✺✽✱✺✾

φ±k ✱

♦❢ t❤❡

✇❤✐❝❤ ✐s ❞❡✜♥❡❞

AB 0 ✐♥ t❡r♠s ♦❢ t❤❡ ♦❝❝✉♣✐❡❞ s♣✐♥✲♦r❜✐t❛❧s ♦❢ t❤❡ ♠♦❧❡❝✉❧❡ ✭ψi ✮ ❛♥❞ ♦❢ t❤❡ ♣r♦♠♦❧❡❝✉❧❡ ✭ψi ✮ ❛s

Vˆ =

X i

|ψiAB ihψiAB | − |ψi0 ihψi0 | .

✻


✭✹✮

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❚❤❡ ◆❖❈❱s ❢❡❛t✉r❡ t❤❡ ♣❡❝✉❧✐❛r ♣r♦♣❡rt② t❤❛t t❤❡② ❝❛♥ ❜❡ ❣r♦✉♣❡❞ ✐♥ ♣❛✐rs ♦❢ ❝♦♠♣❧❡♠❡♥✲ t❛r② ♦r❜✐t❛❧s ✭φk , φ−k ✮ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡✐❣❡♥✈❛❧✉❡s ✇✐t❤ s❛♠❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡ ❜✉t ♦♣♣♦s✐t❡ s✐❣♥ ✭❢♦r t❤❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ◆❖❈❱s✱ s❡❡ ❘❡❢✳

Vˆ φ±k = ±vk φ±k ✇❤❡r❡

k✱

✻✵

✮✳ ❍❡♥❝❡

(vk > 0)

✭✺✮

❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✱ ♥✉♠❜❡rs t❤❡ ◆❖❈❱ ♣❛✐rs ✐♥ ❞❡s❝❡♥❞✐♥❣ ♦r❞❡r ♦❢

◆❖❈❱ ♣❛✐rs✱

∆ρ

′

vk ✳

✐s ❣✐✈❡♥ ❜②

′

∆ρ =

X k

X ′ ∆ρk . vk |φk |2 − |φ−k |2 =

✭✻✮

k

■t ❛♣♣❡❛rs ❢r♦♠ ❊q✳ ✻ t❤❛t ♦♥ ❢♦r♠❛t✐♦♥ ♦❢ ❆❇ ❢r♦♠ t❤❡ ♣r♦♠♦❧❡❝✉❧❡✱ ❛ ❢r❛❝t✐♦♥ ✐s tr❛♥s❢❡rr❡❞ ❢r♦♠ t❤❡ t❤❡ ♦♣❡r❛t♦r

Vˆ

■♥ t❡r♠s ♦❢

t❤❛t

k

φ−k

t♦ t❤❡

r❛♥❣❡s ❢r♦♠

φk 1

♦r❜✐t❛❧✳

✺✻✱✺✼✱✻✵

vk ♦❢ ❡❧❡❝tr♦♥s

■t ❝❛♥ ❜❡ s❤♦✇♥ ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢

t♦ t❤❡ ♥✉♠❜❡r ♦❢ ♦❝❝✉♣✐❡❞ s♣✐♥✲♦r❜✐t❛❧s✳ ▼♦r❡♦✈❡r ✐t ✐s

✐♠♣♦rt❛♥t t♦ ♥♦t❡ t❤❛t ♦♥❧② ❛ s♠❛❧❧ s✉❜s❡t ♦❢ ◆❖❈❱ ♣❛✐rs ❝♦rr❡s♣♦♥❞ t♦ ✈❛❧✉❡s ♦❢

vk

t❤❛t

❛r❡ s✐❣♥✐✜❝❛♥t❧② ❞✐✛❡r❡♥t ❢r♦♠ ③❡r♦✳ ❚❤✐s ♠❡❛♥s t❤❛t ♦♥❧② ❛ ❢❡✇ ◆❖❈❱s ❛r❡ ✐♠♣♦rt❛♥t ❢♦r t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t ♦❝❝✉rr✐♥❣ ♦♥ ❜♦♥❞ ❢♦r♠❛t✐♦♥✳ ❚❤❡ q✉❛♥t✐t✐❡s

∆ρ ❛♥❞ ∆ρ′ s❧✐❣❤t❧② ❞✐✛❡r ✐♥ ❛ s✐♥❣❧❡ t❡r♠ ✭❤❡r❡❛❢t❡r r❡❢❡rr❡❞ t♦ ❛s ∆ρanti ✮✱

✇❤✐❝❤ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t ♦❝❝✉rr✐♥❣ ♦♥ ❣♦✐♥❣ ❢r♦♠ t❤❡ s❡♣❛r❛t❡ ❆ ❛♥❞ ❇ ❢r❛❣♠❡♥ts t♦ t❤❡ ♣r♦♠♦❧❡❝✉❧❡✳ ❚❤✐s ❝♦♥tr✐❜✉t✐♦♥ ❡ss❡♥t✐❛❧❧② r❡♠♦✈❡s

✻✶

t❤❡ ♦r❜✐t❛❧

♦✈❡r❧❛♣ ❜❡t✇❡❡♥ t❤❡ ❆ ❛♥❞ ❇ ❢r❛❣♠❡♥ts✱ s❤✐❢t✐♥❣ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❢r♦♠ t❤❡ ✐♥t❡r✲❢r❛❣♠❡♥t r❡❣✐♦♥ t♦✇❛r❞s t❤❡ ❢r❛❣♠❡♥ts✳

❙❝❤❡♠❛t✐❝❛❧❧②✱ ✐t t✉r♥s ♦✉t t❤❛t

❛❝❝✉♠✉❧❛t✐♦♥ ♦❢ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ✐♥ t❤❡ ❜♦♥❞✐♥❣ r❡❣✐♦♥ t❤❛♥ ❞♦❡s

∆ρ

r❡♣r❡s❡♥ts ❛ s♠❛❧❧❡r

∆ρ′ ✳

◆❡✈❡rt❤❡❧❡ss✱ ❢♦r t❤❡

♣✉r♣♦s❡ ♦❢ ❛ss❡ss✐♥❣ t❤❡ ❡♥t✐t② ♦❢ t❤❡ ❝❤❛r❣❡ tr❛♥s❢❡r ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❉❈❉ ❝♦♠♣♦♥❡♥ts ♦❢ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞s ❛♥ ❛♥❛❧②s✐s ❜❛s❡❞ ♦♥ s✐♠✐❧❛r r❡s✉❧ts✳

∆ρ

❛♥❞ ♦♥❡ ❜❛s❡❞ ♦♥

✸✾

✼


∆ρ′

t②♣✐❝❛❧❧② ❧❡❛❞ t♦ ✈❡r②


Page 8 of 39

✷✳✷ ❚❤❡ ◆❖❈❱✴❈❉ ♠❡t❤♦❞ ✐♥ t❤❡ ❢♦✉r✲❝♦♠♣♦♥❡♥t r❡❧❛t✐✈✐st✐❝ ❢r❛♠❡✲ ✇♦r❦ ❚❤❡ s✐♥❣❧❡✲♣❛rt✐❝❧❡ ✇❛✈❡✲❢✉♥❝t✐♦♥ ✐♥ t❤❡ ❢✉❧❧✲r❡❧❛t✐✈✐st✐❝ ❢♦✉r✲❝♦♠♣♦♥❡♥t ❢r❛♠❡✇♦r❦ ✐s ❞❡✲ s❝r✐❜❡❞ ❜② ❛ ❢♦✉r✲❝♦♠♣♦♥❡♥t s♣✐♥♦r✱

Ψi (r)

♦❢ t❤❡ ❢♦r♠



(1) ψi (r)



    (2)  ψ (r)    i Ψi (r) =  .  ψ (3) (r)    i   (4) ψi (r)

✭✼✮

❋♦r ❛ s✐♥❣❧❡✲❞❡t❡r♠✐♥❛♥t ♠❛♥②✲♣❛rt✐❝❧❡ ✇❛✈❡ ❢✉♥❝t✐♦♥ ✭❉✐r❛❝✲❍❛rtr❡❡✲❋♦❝❦ ♦r ❉✐r❛❝✲❑♦❤♥✲ ❙❤❛♠✮ t❤❡ t♦t❛❧ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❝❛♥ ❜❡ r❡❛❞✐❧② ❡✈❛❧✉❛t❡❞ ❛s t❤❡ s❝❛❧❛r ♣r♦❞✉❝t ♦❢ t❤❡ ❢♦✉r✲ ❝♦♠♣♦♥❡♥t s♣✐♥♦rs ❛❝❝♦r❞✐♥❣ t♦

ρ(r) =

X

Ψ†i (r) · Ψi (r) =

i

X

ρi (r)

✭✽✮

i

✇❤❡r❡ t❤❡ s✉♠ ❡①t❡♥❞s ♦✈❡r t❤❡ ♦❝❝✉♣✐❡❞ ♣♦s✐t✐✈❡✲❡♥❡r❣② ✭❡❧❡❝tr♦♥✐❝✮ st❛t❡s✳ ❚❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t

∆ρ′ (r)

✐s ❝❧❡❛r❧② ❛ s❝❛❧❛r ❢✉♥❝t✐♦♥ ❛♥❞✱ ❜② ❛♥❛❧♦❣② ✇✐t❤ ❊q✳

✸✱ ✐s

❞❡✜♥❡❞ t♦ ❜❡

∆ρ′ (r) =

X

|Ψi (r)AB |2 −

i

✇❤❡r❡

X

|Ψi (r)0 |2 ,

✭✾✮

i

Ψ0i ❛r❡ t❤❡ ♦❝❝✉♣✐❡❞ s♣✐♥♦rs ♦❢ ❆ ❛♥❞ ❇ ♦rt❤♦❣♦♥❛❧✐③❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛♥❞ r❡♥♦r♠❛❧✐③❡❞✳

❲❡ ♠❡♥t✐♦♥ t❤❛t

ΨAB i

❤❛s ❝♦♥tr✐❜✉t✐♦♥s ❜♦t❤ ❢r♦♠ t❤❡ ♣♦s✐t✐✈❡ ✲✭♦❝❝✉♣✐❡❞ ❛♥❞ ✈✐rt✉❛❧✮

❛♥❞ ♥❡❣❛t✐✈❡✲❡♥❡r❣② s♣✐♥♦rs ♦❢ t❤❡ ❢r❛❣♠❡♥ts✳ ❚❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♥❡❣❛t✐✈❡✲❡♥❡r❣② st❛t❡s ❞❡♣❡♥❞s ♦♥ t❤❡ ♣♦t❡♥t✐❛❧ t❤❛t ✐s ✉s❡❞ t♦ ❣❡♥❡r❛t❡ t❤❡♠✱ ✇❤✐❝❤ ✈❛r✐❡s ❜❡t✇❡❡♥ t❤❡ ❢r❛❣♠❡♥ts ❛♥❞ t❤❡ ❝♦♠♣❧❡t❡ ♠♦❧❡❝✉❧❡✳ ❚❤❡ ♥❡❣❧❡❝t ♦❢ ♥❡❣❛t✐✈❡ ❡♥❡r❣② ❝♦♥tr✐❜✉t✐♦♥s ✐s ❦♥♦✇♥ t♦ ✐♥✈♦❧✈❡

−4 ✻✷ ❛♥ ❡rr♦r ♦❢ ❝ ✱ ✇❤✐❝❤ ✐s s♠❛❧❧ ❝♦♠♣❛r❡❞ t♦ t❤❡ s♦ ❝❛❧❧❡❞ ♥♦✲♣❛✐r ❛♣♣r♦①✐♠❛t✐♦♥✱ ✇❤✐❝❤

✽


Page 9 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−2 s❝❛❧❡s ❛s ❝ ✳ ❋♦r ❛ ❞❡t❛✐❧❡❞ ❞✐s❝✉ss✐♦♥ ♦❢ tr❡❛t♠❡♥ts ❜❡②♦♥❞ t❤❡ ♥♦✲♣❛✐r ❛♣♣r♦①✐♠❛t✐♦♥ s❡❡ ❘❡❢✳

✻✸

❚❤❡ ❢♦✉r✲❝♦♠♣♦♥❡♥t ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❝❤❡♠✐❝❛❧✲✈❛❧❡♥❝❡ ♦♣❡r❛t♦r ✐s str❛✐❣❤t❢♦r✇❛r❞ ❛♥❞ ✐s ❣✐✈❡♥ ❜②

Vˆ =

X i

AB 0 0 |ΨAB i ihΨi | − |Ψi ihΨi | .

✭✶✵✮

❚❤❡ ❡✐❣❡♥st❛t❡s ♦❢ t❤✐s ♦♣❡r❛t♦r ✭t❤❡ r❡❧❛t✐✈✐st✐❝ ◆❖❈❱s✮ ❛r❡ ❝❧❡❛r❧② ❢♦✉r✲❝♦♠♣♦♥❡♥t ✈❡❝t♦rs

Vˆ Φk = νk Φk

✭✶✶✮

t❤❛t✱ ❥✉st ❛s ✐♥ t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❝♦♥t❡①t✱ ❛❧❧♦✇ ✉s t♦ ❜r✐♥❣

∆ρ′

✐♥t♦ ❛ ❞✐❛❣♦♥❛❧ ❢♦r♠✳

❲❡ ♣r♦♣♦s❡ ❛ s✐♠♣❧❡ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤ ✇❤❡r❡ t❤❡ ❜❛s✐s s❡t ❢♦r t❤❡ ❛❞❞✉❝t ✭❆❇✮ ✐s ♦❜✲ t❛✐♥❡❞ ❜② ❥♦✐♥✐♥❣ t❤❡ ❜❛s✐s s❡ts ♦❢ t❤❡ t✇♦ ❢r❛❣♠❡♥ts ✭❆ ❛♥❞ ❇✮✳ ❚❤✐s r❡s✉❧ts ✐♥ ❛ ❢♦r♠✉❧❛t✐♦♥ ✇❤❡r❡

Ψ✱ Ψ0

❛♥❞ t❤❡ r❡❧❛t✐✈✐st✐❝ ◆❖❈❱s ✭Φ✮ ❛r❡ ❡①♣r❡ss❡❞ ✇✐t❤ t❤❡ s❛♠❡ ❜❛s✐s s❡t ❛♥❞

t❤❡ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝❤❡♠✐❝❛❧✲✈❛❧❡♥❝❡ ♦♣❡r❛t♦r ❝❛♥ ❜❡ ❡❛s✐❧② ❢♦r♠✉❧❛t❡❞✳

❆❧❧

❞❡t❛✐❧s ♦❢ ♦✉r ♥❡✇ ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ♣❡r❢♦r♠❡❞ ✐♥ t❤❡ ❉❑❙ ♠♦❞✉❧❡ ♦❢ t❤❡ ❇❊❘❚❍❆ ❝♦❞❡✱ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❆♣♣❡♥❞✐① ❆✳

✸

❘❡s✉❧ts ❛♥❞ ❞✐s❝✉ss✐♦♥

❆s ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞✱ ✇❡ ❤❛✈❡ t❡st❡❞ t❤❡ ✈❛❧✐❞✐t② ❛♥❞ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ♦✉r ❝♦♠♣✉✲

+ t❛t✐♦♥❛❧ ♠❛❝❤✐♥❡r② ♦♥ t❤❡ s✐♠♣❧❡ ❝❛s❡ ♦❢ t❤❡ ♠❡t❛❧✲❝❛r❜♦♥ ❜♦♥❞ ✐♥ t❤❡ ❬▲✐✕❈❖❪ ❝♦♠♣❧❡①✳ ❆❢t❡r ❣✐✈✐♥❣ t❤❡ r❡❧❡✈❛♥t ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❡t❛✐❧s ✐♥ ❙❡❝t✐♦♥ ✸✳✶✱ ✇❡ ❝♦♠♣❛r❡ ♦✉r r❡s✉❧ts ✇✐t❤ t❤♦s❡ ♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤❡ ❝♦♠♠❡r❝✐❛❧ ❆❉❋ ▼♦❞❡❧✐♥❣ ❙✉✐t❡

✻✹

❛t t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❧❡✈❡❧ ❛♥❞

❞✐s❝✉ss t❤❡ ❝♦♠♣❛r✐s♦♥ ✐♥ ❙❡❝t✐♦♥ ✸✳✷✳ ■♥ ❙❡❝t✐♦♥ ✸✳✸ ✇❡ t❤❡♥ ♣r❡s❡♥t r❡s✉❧ts ♦♥ ❛ s❡q✉❡♥❝❡

+ ♦❢ ♠♦♥♦❝❛r❜♦♥②❧✲♠❡t❛❧ ❝♦♠♣❧❡①❡s ♦❢ ❣r♦✉♣ ✶✶ ♠❡t❛❧s ♦❢ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❬▼✕❈❖❪ ✇✐t❤ ▼

=

❈✉✱ ❆❣ ❛♥❞ ❆✉✳ ❆❧♦♥❣ t❤✐s s❡r✐❡s r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❛r❡ ❦♥♦✇♥ t♦ ♣❧❛② ❛♥ ✐♥❝r❡❛s✐♥❣❧②

✐♠♣♦rt❛♥t r♦❧❡ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❛t♦♠✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ♠❡t❛❧✳

✾


❲❡ t❤✉s s❤♦✇ t❤❡ ❡✛❡❝t✐✈❡


Page 10 of 39

❛❜✐❧✐t② ♦❢ t❤❡ ◆❖❈❱✴❈❉ ♠❡t❤♦❞✱ ❡①t❡♥❞❡❞ t♦ ❢♦✉r✲❝♦♠♣♦♥❡♥t ❢r❛♠❡✇♦r❦✱ t♦ ♣r♦✈✐❞❡ ❝❧❡❛r✲ ❝✉t ♠❡❛s✉r❡s ♦❢ t❤❡ ❞♦♥❛t✐♦♥ ❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ♦❢ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ✐♥ ❤❡❛✈② ♠❡t❛❧ ❝♦♠♣❧❡①❡s ✇❤❡♥ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts✱ ✐♥❝❧✉❞✐♥❣ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣✱ ❛r❡ ❝r✉❝✐❛❧ ❢♦r ✉♥✲ ❞❡rst❛♥❞✐♥❣ t❤❡ ❝❤❡♠✐❝❛❧ ❜♦♥❞✳

✸✳✶ ❈♦♠♣✉t❛t✐♦♥❛❧ ❞❡t❛✐❧s ❈❛❧❝✉❧❛t✐♦♥s ✇❡r❡ ❝❛rr✐❡❞ ♦✉t ✇✐t❤ ❛ r❡❝❡♥t❧② ✐♠♣❧❡♠❡♥t❡❞ ❢✉❧❧② ♣❛r❛❧❧❡❧ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣r♦✲ ❣r❛♠ ❇❊❘❚❍❆✳

✺✹✱✺✺

❚❤❡ ♣r♦❣r❛♠ ❤❛s ❜❡❡♥ ❞❡s❝r✐❜❡❞ ✐♥ ❞❡t❛✐❧ ❡❧s❡✇❤❡r❡ ✭s❡❡ ❘❡❢✳

✺✷

❛♥❞

r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✮✳ ❊♥❡r❣✐❡s ❛♥❞ ❡❧❡❝tr♦♥ ❞❡♥s✐t✐❡s ❛r❡ ❝♦♠♣✉t❡❞ ❜② s♦❧✈✐♥❣ t❤❡ ❛❧❧✲❡❧❡❝tr♦♥ ❢♦✉r ❝♦♠♣♦♥❡♥t ❉✐r❛❝✲❑♦❤♥✲❙❤❛♠ ✭❉❑❙✮ ❡q✉❛t✐♦♥✳ ❚❤❡ ◆❖❈❱✴❈❉ ❛♥❛❧②s✐s ❤❛s ❜❡❡♥ ❝❛r✲ r✐❡❞ ♦✉t ✐♥ ❛❧❧ ❝❛s❡s ✉s✐♥❣ ❛s r❡❢❡r❡♥❝❡ ❢r❛❣♠❡♥ts✱ t❤❡ ♠❡t❛❧ ❝❛t✐♦♥s ✭❝❧♦s❡❞ s❤❡❧❧✮ ❛♥❞ t❤❡ ❈❖ ♠♦❧❡❝✉❧❡✳

❚❤❡ ✇♦r❦ ♦❢ ❙❝❤✇❛r③ ❡t ❛❧✳

❡r❡♥❝❡ ❢r❛❣♠❡♥ts✳

✻✺✱✻✻

❙✐❣♥✐✜❝❛♥t s♣❡❡❞ ✉♣ ✐s ❛❝❤✐❡✈❡❞ ✉s✐♥❣ st❛t❡✲♦❢✲t❤❡✲❛rt ❞❡♥s✐t② ✜tt✐♥❣

t❡❝❤♥✐q✉❡s r❡q✉✐r✐♥❣ ❛♥ ❛✉①✐❧✐❛r② ❜❛s✐s s❡t✳ ❢♦r t❤❡ ♥✉❝❧❡✐✳

✻✽

✐♥❞✐❝❛t❡s t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ❝❤♦✐❝❡ ♦❢ r❡❢✲

✻✼

❆ ✜♥✐t❡ ❝❤❛r❣❡ ❞✐str✐❜✉t✐♦♥ ♠♦❞❡❧ ✐s ❡♠♣❧♦②❡❞

❚❤❡ ❧❛r❣❡ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❜❛s✐s s❡t ❢♦r ❆✉ ✭30s24p15d11f 5g1h✮ ❛♥❞ ❆❣

✭28s20p13d7f 3g ✮ ✇❛s ❣❡♥❡r❛t❡❞ ❜② ✉♥❝♦♥tr❛❝t✐♥❣ ❉②❛❧❧✬s tr✐♣❧❡✲ζ q✉❛❧✐t② ❜❛s✐s s❡ts ♠❡♥t❡❞ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♣♦❧❛r✐③❛t✐♦♥ ❛♥❞ ❝♦rr❡❧❛t✐♥❣ ❢✉♥❝t✐♦♥s✳

✻✾✕✼✸

❛✉❣✲

▲❛r❣❡✲❝♦♠♣♦♥❡♥t

❜❛s✐s ❢✉♥❝t✐♦♥s ❢♦r ▲✐✱ ❈✉✱ ❈ ❛♥❞ ❖ ✇❡r❡ ❞❡r✐✈❡❞ ❜② ❞❡❝♦♥tr❛❝t✐♥❣ t❤❡ ❝❝✲♣❱❚❩✲❉❑ ❜❛s✐s s❡ts ❛✈❛✐❧❛❜❧❡ ❛t t❤❡ ❇❛s✐s ❙❡t ❊①❝❤❛♥❣❡ ❙✐t❡✳

✼✼

✼✹✕✼✻

■♥ ❛❧❧ ❝❛s❡s✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s♠❛❧❧✲

❝♦♠♣♦♥❡♥t ❜❛s✐s ✇❛s ❣❡♥❡r❛t❡❞ ✉s✐♥❣ t❤❡ r❡str✐❝t❡❞ ❦✐♥❡t✐❝ ❜❛❧❛♥❝❡ r❡❧❛t✐♦♥✳

✼✽

❋♦r t❤❡ ❣♦❧❞

❛t♦♠✱ ❛ ♣r❡✈✐♦✉s❧② ♦♣t✐♠✐③❡❞ ❛✉①✐❧✐❛r② ❜❛s✐s s❡t ❢♦r ❞❡♥s✐t② ✜tt✐♥❣ ❞❡♥♦t❡❞ ❛s ❇✷✵

✻✼

✇❛s

✉s❡❞✳ ❋♦r ❛❧❧ ♦t❤❡r ❡❧❡♠❡♥ts✱ ♥❛♠❡❧② ▲✐✱ ❆❣✱ ❈✉✱ ❈ ❛♥❞ ❖✱ ❛❝❝✉r❛t❡ ❛✉①✐❧✐❛r② ❜❛s✐s s❡ts ✇❡r❡ ❣❡♥❡r❛t❡❞ ✉s✐♥❣ ❛ s✐♠♣❧❡ ♣r♦❝❡❞✉r❡ st❛rt✐♥❣ ❢r♦♠ ❛✈❛✐❧❛❜❧❡ ❉❡▼♦♥ ❈♦✉❧♦♠❜ ❋✐tt✐♥❣

✼✼

❜❛s✐s

s❡ts✳ ❲❡ s❤❛❧❧ r❡❢❡r t♦ t❤❡ ♦❜t❛✐♥❡❞ ❛✉①✐❧✐❛r② ❜❛s✐s s❡t ❛s t♦ ❆✶✳ ❲❡ r❡❝❛❧❧ ❤❡r❡ t❤❛t ❛s ✜tt✐♥❣ ❢✉♥❝t✐♦♥s ✇❡ ✉s❡ ❍❡r♠✐t❡ ●❛✉ss✐❛♥ ❚②♣❡ ❋✉♥❝t✐♦♥s ✭❍●❚❋s✮ t❤❛t ❛r❡ ❣r♦✉♣❡❞ t♦❣❡t❤❡r ✐♥ s❡ts s❤❛r✐♥❣ t❤❡ s❛♠❡ ❡①♣♦♥❡♥ts ✭❛♥ ❛♥❛❧♦❣♦✉s s❝❤❡♠❡ ✐s ❛❞♦♣t❡❞ ✐♥ t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❉❋❚ ❝♦❞❡ ❉❡▼♦♥

✼✾

✮✳ ❈♦♥s❡q✉❡♥t❧②✱ ❞✉❡ t♦ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♥❛t✉r❡ ♦❢

✶✵


Page 11 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


t❤❡ ❞❡♥s✐t② ✜tt✐♥❣ ♣r♦❝❡❞✉r❡✱ ✇❡ ❛❝❤✐❡✈❡❞ ❛ ✜tt✐♥❣ ❜❛s✐s s❡t ♦❢ ❤✐❣❤ ❛❝❝✉r❛❝② ✭r❡❢❡rr❡❞ t♦ ❛s ❆✷✮ s✐♠♣❧② ❜② ✉♣✲s❤✐❢t✐♥❣ t❤❡ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ♦❢ ❛❧❧ t❤❡ ❉❡▼♦♥ ❈♦✉❧♦♠❜ ❋✐tt✐♥❣ ❞❡✜♥✐t✐♦♥s ❜② t✇♦ ✉♥✐ts✳

+ ■♥ t❤❡ ❝❛s❡ ♦❢ ❬▲✐✕❈❖❪ ✇❡ ❛❧s♦ ✉s❡❞ ❛ ❜❛s✐s s❡t ♦❢ ❡✈❡♥ ❤✐❣❤❡r

❛❝❝✉r❛❝② ✭r❡❢❡rr❡❞ t♦ ❛s ❆✸✮ ❜② ✉♣✲s❤✐❢t✐♥❣ t❤❡ r❡❧❛t❡❞ ❛♥❣✉❧❛r ♠♦♠❡♥t❛ ❜② ❢♦✉r ✉♥✐ts✳ ❋♦r ✐❧❧✉str❛t✐✈❡ ♣✉r♣♦s❡s ✇❡ ❣✐✈❡ t❤❡ ❞✐♠❡♥s✐♦♥s ♦❢ t❤❡ ❈✲❛t♦♠ ❛✉①✐❧✐❛r② ❜❛s✐s s❡ts✱ ✇❤✐❝❤ ❛r❡ ✭✼s✱✸♣✱✸❞✮✱ ✭✼s✱✼♣✱✼❞✱✸❢✱✸❣✮ ❛♥❞ ✭✼s✱✼♣✱✼❞✱✼❢✱✼❣✱✸❤✱✸✐✮ ❢♦r t❤❡ ❆✶✱ ❆✷ ❛♥❞ ❆✸ q✉❛❧✐t✐❡s✱ r❡s♣❡❝t✐✈❡❧②✳

❚❤❡ ❇▲❨P ❢✉♥❝t✐♦♥❛❧✱ ❞❡✜♥❡❞ ❛s t❤❡ ❇❡❝❦❡ ✶✾✽✽ ✭❇✽✽✮ ❡①❝❤❛♥❣❡

▲❡❡✲❨❛♥❣✲P❛rr ✭▲❨P✮ ❝♦rr❡❧❛t✐♦♥✱

✽✶

✽✵

♣❧✉s t❤❡

✇❡r❡ ❡♠♣❧♦②❡❞✳ ❆❧❧ ❝❛❧❝✉❧❛t✐♦♥s ✇❡r❡ ❝❛rr✐❡❞ ♦✉t ✇✐t❤✐♥

t❤❡ s♦ ❝❛❧❧❡❞ ❞❡♥s✐t②✲♦♥❧② ❉❑❙ ❢r❛♠❡✇♦r❦✳

✽✷

❆ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ❣r✐❞ ♦❢ ✻✶✶✹✵ ♣♦✐♥ts

♣❡r ❛t♦♠ ✇❛s ❣❡♥❡r❛t❡❞ ❛♥❞ ❛♥ ❡♥❡r❣② ❝♦♥✈❡r❣❡♥❝❡ ❝r✐t❡r✐♦♥ ♦❢

10−7

❍❛rtr❡❡ ♦♥ t❤❡ t♦t❛❧

❡♥❡r❣② ✇❛s ❛❞♦♣t❡❞✳ ❋✐♥❛❧❧②✱ t❤❡ ❣❡♦♠❡tr✐❡s ♦❢ t❤❡ ❛♥❛❧②③❡❞ ♠♦♥♦❝❛r❜♦♥②❧✲♠❡t❛❧ ❝♦♠♣❧❡①❡s ❛r❡ r❡♣♦rt❡❞ ✐♥ ❚❛❜❧❡ ✶✳ ❚❤❡s❡ ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ❜② ♠❡❛♥s ♦❢ ❢✉❧❧ ❣❡♦♠❡tr② ♦♣t✐♠✐③❛t✐♦♥s ❛t ❇▲❨P✴❚❩✷P ❧❡✈❡❧ ✉s✐♥❣ ❆❉❋ ❝♦❞❡ ✇✐t❤ t❤❡ ❩❖❘❆ ❤❛♠✐❧t♦♥✐❛♥✳

✻✹

+ ❚❛❜❧❡ ✷✿ ●❡♦♠❡tr✐❡s ♦❢ t❤❡ ❛♥❛❧②③❡❞ ♠♦♥♦❝❛r❜♦♥②❧✲♠❡t❛❧ ❝♦♠♣❧❡①❡s ❬▼✕❈❖❪ ✭▼

= ▲✐✱ ❈✉✱

❆❣✱ ❆✉✮✳ ❉✐st❛♥❝❡s ✭r ✮ ❛r❡ ✐♥ ❆♥❣str♦♠✳ ❬▲✐✭❈❖✮❪

r▼✕❈ r❈✕❖

+

❬❈✉✭❈❖✮❪

+

❬❆❣✭❈❖✮❪

+

❬❆✉✭❈❖✮❪

✷✳✶✼✹

✶✳✽✸✸

✷✳✶✸✸

✶✳✾✶✼

✶✳✶✷✻

✶✳✶✷✾

✶✳✶✷✻

✶✳✶✸✵

+

✸✳✷ ❱❛❧✐❞❛t✐♦♥ ❛♥❞ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t②✿ t❤❡ ❝❛s❡ ♦❢ ❬▲✐✕❈❖❪+ ■♥ ❙❡❝t✐♦♥ ✷✳✷ ✇❡ ❞❡s❝r✐❜❡❞ ◆❖❈❱s ✇✐t❤✐♥ t❤❡ ❢♦✉r✲❝♦♠♣♦♥❡♥t r❡❧❛t✐✈✐st✐❝ ❢♦r♠❛❧✐s♠✱ ❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❇❊❘❚❍❆ ♣r♦❣r❛♠✳ ❚❤❡ ♠❛✐♥ ❢♦❝✉s ♦❢ t❤❡ ♣r❡s❡♥t s❡❝t✐♦♥ ✐s t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ Pr❡❝✐s❡❧② ❢♦r t❤✐s r❡❛s♦♥ ✇❡ s❡❧❡❝t❡❞ ❛ s✐♠♣❧❡

+ ♠♦❧❡❝✉❧❛r ❝♦♠♣❧❡①✱ t❤❡ ❧✐t❤✐✉♠ ♠♦♥♦❝❛r❜♦♥②❧ ❝♦♠♣❧❡① ❬▲✐✕❈❖❪ ❢♦r ✇❤✐❝❤ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❛r❡ ❡①♣❡❝t❡❞ t♦ ❜❡ ♥❡❣❧✐❣✐❜❧❡✱ ❛♥❞ ❝♦♠♣❛r❡❞ t❤❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧t ♦❢ ♦✉r ♣r♦❝❡❞✉r❡s ✇✐t❤ t❤❛t ♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤❡ ◆❖❈❱ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ❆❉❋ ♣❛❝❦❛❣❡✳ ❚❤❡ ❈❉ ❛♥❛❧②s✐s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❡✐❣❤t

′

∆ρk

✶✶

❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛r✲



Page 12 of 39

+ r❛♥❣❡♠❡♥t r❡❧❛t❡❞ ✇✐t❤ t❤❡ ❧✐t❤✐✉♠✲❝❛r❜♦♥ ❜♦♥❞ ❢♦r♠❛t✐♦♥ ✐♥ ❝♦♠♣❧❡① ❬▲✐✕❈❖❪ ❛r❡ r❡♣♦rt❡❞ ✐♥ ❋✐❣✉r❡ ❙✶ ♦❢ t❤❡ ❙✉♣♣♦rt✐♥❣ ■♥❢♦r♠❛t✐♦♥ ✭❙■✮✳

✶

❘❡s✉❧ts ♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤❡ ◆❖❈❱ ✐♠♣❧❡✲

♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❆❉❋ ♣r♦❣r❛♠ s✉✐t❡ ❛♥❞ t❤❛t ♦❢ ❇❊❘❚❍❆ ❛r❡ ❛❧♠♦st ✐♥❞✐st✐♥❣✉✐s❤❛❜❧❡✳ ❚❤❡ s♦❧❡ ❞✐✛❡r❡♥❝❡ ❛r✐s❡s ✐♥ t❤❡ ❛♥❣✉❧❛r ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡

′

∆ρ2

❛♥❞

′

∆ρ3

❝♦♠♣♦♥❡♥ts✱ ✇❤❡r❡ ❞✉❡

t♦ t❤❡ ❝♦♠♣❧❡① ❛❧❣❡❜r❛ ❡♠♣❧♦②❡❞ ✐♥ ❇❊❘❚❍❆ t❤❡ ❝②❧✐♥❞r✐❝❛❧ s②♠♠❡tr② ♦❢ t❤❡

π

❡❧❡❝tr♦♥s

r❡❛rr❛♥❣❡♠❡♥t ❡♠❡r❣❡s ♥❛t✉r❛❧❧②✳ ■♥ ❚❛❜❧❡ ✸ ✇❡ ♣r❡s❡♥t ❛♥ ❡✈❡♥ ♠♦r❡ str✐♥❣❡♥t ♥✉♠❡r✐❝❛❧ ❝♦♠♣❛r✐s♦♥ ✇❤❡r❡ ✇❡ r❡♣♦rt t❤❡ ❡✐❣❤t ❡✐❣❡♥✈❛❧✉❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ◆❖❈❱s ❢♦r t❤❡ s❡❧❡❝t❡❞ ❝♦♠♣❧❡① ♦❜t❛✐♥❡❞ ✉s✐♥❣ ❞✐❢✲ ❢❡r❡♥t ❜❛s✐s s❡ts ❜♦t❤ ✇✐t❤ ❆❉❋ ❛♥❞ ❇❊❘❚❍❆✳ ❈♦♠♣❛r✐♥❣ t❤❡s❡ ✈❛❧✉❡s ✇❡ ♦❜s❡r✈❡ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ ❆❉❋ ❛♥❞ ❇❊❘❚❍❆ ❛r❡ ✇❡❧❧ ✇✐t❤✐♥ t❤❡ ♥✉♠❡r✐❝❛❧ ❞✐✛❡r❡♥❝❡s ❛ttr✐❜✉t❛❜❧❡ t♦ t❤❡ ❜❛s✐s s❡ts ✐♥❡q✉❛❧✐t✐❡s✳ ◆♦t❛❜❧②✱ t❤❡ ❡①♣❡❝t❡❞ ❞❡❣❡♥❡r❛❝② r❡♠♦✈❛❧ ✇✐t❤✐♥ t❤❡

π ✲♦r❜✐t❛❧s

t❤❛t ♦♥❡ ♠❛② ♦❜s❡r✈❡ ✐♥ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ ❝♦♠♣♦♥❡♥ts ✷ ❛♥❞ ✸ ✭s❡❡ ❛❧s♦ ❋✐❣✉r❡ ❙✷ ✐♥ ❙■✮ ✐♥ ❣♦✐♥❣ ❢r♦♠ ❆❉❋ t♦ ❇❊❘❚❍❆✱ ❞✐s❛♣♣❡❛rs ✭❛s ♦♥❡ ❝❛♥ s❡❡ r❡❛❞✐♥❣ t❤r♦✉❣❤ ❚❛❜❧❡ ✸✮ ✇❤❡♥ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ s♣❡❡❞ ♦❢ ❧✐❣❤t ✐♥ ❇❊❘❚❍❆ ✐s ♠❛❞❡ ♥❡❛r❧② ✐♥✜♥✐t❡ ❜② ✐♥❝r❡❛s✐♥❣ ✐t ❜② ♦♥❡ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡✳ ❚❤✐s ❧❛tt❡r ❞❡♠♦♥str❛t❡s ✇❡❧❧ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ ❇❡❢♦r❡ ❝♦♥❝❧✉❞✐♥❣ t❤✐s s❡❝t✐♦♥ ✐t ✐s ✐♥t❡r❡st✐♥❣ t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ✐♠♣❛❝t ♦❢ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❛✉①✐❧✐❛r② ❜❛s✐s s❡t ❡♠♣❧♦②❡❞ ✐♥ t❤❡ ❞❡♥s✐t②✲✜tt✐♥❣ ♣r♦❝❡❞✉r❡ ♦♥ t❤❡ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s ♦❢ t❤❡ ◆❖❈❱ ❡✐❣❡♥✈❛❧✉❡s✳ ■♥ ❚❛❜❧❡ ✹ ✇❡ r❡♣♦rt❡❞ t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ✉s✐♥❣ t❤r❡❡ ❞✐✛❡r❡♥t ✜tt✐♥❣ s❡ts ✭♥❛♠❡❧② ❆✶✱ ❆✷✱ ❆✸✱ s❡❡ ❙❡❝t✐♦♥ ✸✳✶✮ ♦❢ ✐♥❝r❡❛s✐♥❣ q✉❛❧✐t② ❛s ✇❡❧❧ ❛s t❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ✇✐t❤♦✉t ❛♥② ❞❡♥s✐t②✲✜tt✐♥❣ t❡❝❤♥✐q✉❡ ❢♦r r❡❢❡r❡♥❝❡✳ ■♥ ♦r❞❡r t♦ q✉❛♥t✐❢② t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❞❡♥s✐t② ✜tt✐♥❣ ❜❛s✐s s❡ts ✇❡ ❡✈❛❧✉❛t❡ t❤❡ ♠❛①✐♠✉♠ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ❈♦✉❧♦♠❜ ❡♥❡r❣②✱ ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t t❤❡ ✜tt✐♥❣ ♣r♦❝❡❞✉r❡✱ ❡✈❛❧✉❛t❡❞ ❞✉r✐♥❣ t❤❡ ✐t❡r❛t✐♦♥s ♦❢ t❤❡ ❙❈❋ ♣r♦❝❡❞✉r❡✳ ❚❤❡ ❡rr♦r ✐s ❢♦✉♥❞ t♦ ❜❡ ❧❡ss t❤❛♥ ❛ ❧✐♠✐t✐♥❣ ✈❛❧✉❡ ❜❡❧♦✇

10−7

10−3 ❛♥❞ 10−5 ❍❛rtr❡❡ ❢♦r ❆✶ ❛♥❞ ❆✷✱ r❡s♣❡❝t✐✈❡❧②✱ r❡❛❝❤✐♥❣

✉s✐♥❣ t❤❡ ❆✸ ❜❛s✐s s❡t✳ ❚❤❡ r❡♣♦rt❡❞ r❡s✉❧ts ❝❧❡❛r❧② s❤♦✇ t❤❡

✶ ◆♦t❡ t❤❛t✱ t❤❡ ◆❖❈❱ ♣❛✐rs ♥✉♠❜❡r ✐s ❡q✉❛❧ t♦ t❤❡ ♥✉♠❜❡r ✭◆✮ ♦❢ ♦❝❝✉♣✐❡❞ s♣✐♥✲♦r❜✐t❛❧s✱ ✇❤✐❝❤ ✐s s✐①t❡❡♥ ✐♥ t❤❡ ▲✐✲❈❖+ ❝❛s❡✳ ■♥ ❝❛s❡ ♦❢ ❝❧♦s❡❞✲s❤❡❧❧ s②st❡♠s✱ ❝❧❡❛r❧② t❤❡

α

❛♥❞

β

s♣✐♥ ♦r❜✐t❛❧s ❣✐✈❡ ❡①❛❝t❧② t❤❡ s❛♠❡

❝♦♥tr✐❜✉t✐♦♥✱ ✇❤✐❝❤✱ ✐♥ t❤❡ ❆❉❋ ✐♠♣❧❡♠❡♥t❛t✐♦♥✱ ✐s ✐♠♣❧✐❝✐t❧② ❛❝❝♦✉♥t❡❞ ❞♦✉❜❧✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ ❡✐❣❡♥✈❛❧✉❡ ❛♥❞ ❧❛❜❡❧✐♥❣ t❤❡ ◆❖❈❱ ♣❛✐rs ❢r♦♠ ✶ t♦ t❤❡ ♥✉♠❜❡r ♦❢ s♣❛t✐❛❧ ♦r❜✐t❛❧s ✭◆✴✷✮✱ ❛✈♦✐❞✐♥❣ ❛♥② r❡❢❡r❡♥❝❡ t♦ t❤❡ s♣✐♥✳ ■♥ ❇❊❘❚❍❆✱ t❤❡s❡ ❞❡❣❡♥❡r❛t❡ ❝♦♥tr✐❜✉t✐♦♥s ❛r✐s❡s ♥❛t✉r❛❧❧② ❛s ❑r❛♠❡rs ♣❛✐rs✱ ✇❤✐❝❤✱ ❢♦r t❤❡ ❝♦♥✈❡♥✐❡♥❝❡ ♦❢ ❛ ❞✐r❡❝t ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ ❆❉❋ r❡s✉❧ts✱ ❛r❡ s✉♠♠❡❞ ✉♣✳

✶✷


Page 13 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


+ a ❚❛❜❧❡ ✸✿ ❊✐❣❡♥✈❛❧✉❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ◆❖❈❱s ❢♦r t❤❡ ❬▲✐✕❈❖❪ ♠❡t❛❧ ❝❛r❜♦♥②❧ ❝♦♠♣❧❡①✳ ❞❛t❛ ♦❜t❛✐♥❡❞ ✇✐t❤ ❉❑❙ ❝❛❧❝✉❧❛t✐♦♥ ✐♥❝r❡❛s✐♥❣ t❤❡ s♣❡❡❞ ♦❢ ❧✐❣❤t ❜② ♦♥❡ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡ ✭❝ ❂ ✶✵·❝✮ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❤❛♠✐❧t♦♥✐❛♥✳

v1 v2 v3 v4 v5 v6 v7 v8

❆❉❋

❇❊❘❚❍❆

❙❧❛t❡r t②♣❡ ❜❛s✐s

●✲s♣✐♥♦r ❜❛s✐s

❉❩

❚❩✷P

◗❩✹P

❝❝✲♣❱❉❩

❝❝✲♣❱❚❩

❝❝✲♣❱◗❩

✵✳✶✻✺✵✼

✵✳✶✽✼✸✷

✵✳✷✵✷✸✺

✵✳✷✶✶✽✹

✵✳✷✵✵✶✵

✵✳✷✵✵✻✵

❝❝✲♣❱◗❩

a

✵✳✷✵✵✺✻

✵✳✵✻✶✹✸

✵✳✵✼✼✽✺

✵✳✵✼✼✺✼

✵✳✵✻✺✹✾

✵✳✵✼✵✺✺

✵✳✵✼✷✻✸

✵✳✵✼✷✼✸

✵✳✵✻✶✹✸

✵✳✵✼✼✽✺

✵✳✵✼✼✺✼

✵✳✵✻✺✸✸

✵✳✵✼✵✼✸

✵✳✵✼✷✽✸

✵✳✵✼✷✼✸

✵✳✵✷✹✼✾

✵✳✵✹✼✹✶

✵✳✵✹✺✶✹

✵✳✵✸✽✼✹

✵✳✵✸✻✵✸

✵✳✵✸✻✵✼

✵✳✵✸✻✶✶

✵✳✵✵✾✼✾

✵✳✵✶✸✸✷

✵✳✵✵✽✶✽

✵✳✵✶✺✵✼

✵✳✵✵✾✹✻

✵✳✵✵✼✻✻

✵✳✵✵✼✻✼

✵✳✵✵✻✽✽

✵✳✵✵✹✼✵

✵✳✵✵✼✶✹

✵✳✵✵✻✶✶

✵✳✵✵✻✷✽

✵✳✵✵✺✾✼

✵✳✵✵✺✾✽

✵✳✵✵✶✵✵

✵✳✵✵✶✽✶

✵✳✵✵✶✽✵

✵✳✵✵✷✵✻

✵✳✵✵✶✼✹

✵✳✵✵✶✷✻

✵✳✵✵✶✷✼

✵✳✵✵✵✻✵

✵✳✵✵✶✷✺

✵✳✵✵✶✵✶

✵✳✵✵✵✼✽

✵✳✵✵✶✵✹

✵✳✵✵✵✹✻

✵✳✵✵✵✹✻

♥❡❣❧✐❣✐❜❧❡ r♦❧❡ ♣❧❛②❡❞ ❜② t❤❡ ✜tt✐♥❣ s❡ts✱ t❤❛t ♥✉♠❡r✐❝❛❧❧② ✐s ♦❢ t❤❡ ♦r❞❡r ♦❢ ♦♥❡ ♣❡r✲♠✐❧❧❡✳ ❚❤✉s✱ t❤✐s ❛❧❧♦✇s ✉s t♦ ❝♦♥s✐❞❡r t❤❡ ♣r❡✈✐♦✉s❧② ❞✐s❝✉ss❡❞ ❡✛❡❝t ♦❢ t❤❡ ❜❛s✐s s❡t ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❛❞♦♣t❡❞ ❛✉①✐❧✐❛r② ❜❛s✐s✳

+ ❚❛❜❧❡ ✹✿ ❊✐❣❡♥✈❛❧✉❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ◆❖❈❱s ❢♦r t❤❡ ❬▲✐✕❈❖❪ ♠❡t❛❧ ❝❛r❜♦♥②❧ ❝♦♠♣❧❡① ✇✐t❤ ❛♥❞ ✇✐t❤♦✉t ✜tt✐♥❣ ♣r♦❝❡❞✉r❡✳ ❉✐✛❡r❡♥t ✜tt✐♥❣ ❜❛s✐s s❡ts ✇❡r❡ ❡♠♣❧♦②❡❞✱ ♥❛♠❡❧② ❆✶✱ ❆✷✱ ❆✸✳ ❙❡❡ t❡①t ❢♦r ❞❡✜♥✐t✐♦♥s ❛♥❞ ❞✐s❝✉ss✐♦♥✳

v1 v2 v3 v4 v5 v6 v7 v8

❇❊❘❚❍❆

❇❊❘❚❍❆

❇❊❘❚❍❆

❇❊❘❚❍❆

✜t ❉❡♠♦♥✭❆✶✮

✜t ♠❡❞✐✉♠ ✭❆✷✮

✜t ❧✐♠✐t ✭❆✸✮

♥♦✜t

✵✳✷✵✽✺✹

✵✳✷✶✶✽✻

✵✳✷✶✶✽✹

✵✳✷✶✶✼✾

✵✳✵✻✺✺✺

✵✳✵✻✺✹✽

✵✳✵✻✺✹✾

✵✳✵✻✺✻✹

✵✳✵✻✺✸✽

✵✳✵✻✺✸✶

✵✳✵✻✺✸✸

✵✳✵✻✺✹✽

✵✳✵✸✾✻✶

✵✳✵✸✽✻✽

✵✳✵✸✽✼✹

✵✳✵✸✽✻✹

✵✳✵✶✹✾✷

✵✳✵✶✺✵✹

✵✳✵✶✺✵✼

✵✳✵✶✺✵✽

✵✳✵✵✺✽✹

✵✳✵✵✻✵✵

✵✳✵✵✻✶✶

✵✳✵✵✻✶✵

✵✳✵✵✷✵✸

✵✳✵✵✷✵✻

✵✳✵✵✷✵✻

✵✳✵✵✷✵✻

✵✳✵✵✵✼✾

✵✳✵✵✵✼✽

✵✳✵✵✵✼✽

✵✳✵✵✵✼✽

✶✸



Page 14 of 39

✸✳✸ ❘❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ♦♥ t❤❡ ❝❛r❜♦♥✲♠❡t❛❧ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ✐♥ ❬▼✕❈❖❪+ ✭▼ = ❈✉✱ ❆❣✱ ❆✉✮ ❝♦♠♣❧❡①❡s ❇♦t❤ ✐♥ ♣✉r❡ ❛♥❞ ❛♣♣❧✐❡❞ ❝❤❡♠✐str② t❤❡ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ♦❢ t❤❡ ❝❛r❜♦♥ ♠♦♥♦①✐❞❡ ✭❈❖✮ ✇✐t❤ ♠❡t❛❧s ✭▼✮ ❤❛s ❜❡❡♥ ✇✐❞❡❧② st✉❞✐❡❞✳ ♣r❡✈✐♦✉s❧② ♠❡♥t✐♦♥❡❞ ❉❈❉ ♠♦❞❡❧✱

✹✹✱✽✸✕✽✺

✸✵✱✸✶✱✽✻✱✽✼

❚❤❡ ❜♦♥❞ ✐s ❣❡♥❡r❛❧❧② ❞❡s❝r✐❜❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ✇❤❡r❡❜② t❤❡ ✐♥t❡r❛❝t✐♦♥ r❡s✉❧ts ❢r♦♠ ❞♦♥❛t✐♦♥ ♦❢

❡❧❡❝tr♦♥ ❝❤❛r❣❡ ❢r♦♠ ❝❛r❜♦♥✬s ❧♦♥❡ ♣❛✐r t♦ t❤❡ ❡♠♣t② ▼ t♦ t❤❡ ❡♠♣t② ❈❖ ♦r❜✐t❛❧s ♦❢

π

s②♠♠❡tr② ✭s❡❡ ❘❡❢✳

✽✼

σ

♦r❜✐t❛❧s ❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❢r♦♠ ▼

❛♥❞ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥ ❢♦r ❛♥ ❡①t❡♥s✐✈❡

❞✐s❝✉ss✐♦♥✮✳ ■♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ✇❡ q✉❛♥t✐t❛t✐✈❡❧② ✈❛❧✐❞❛t❡❞ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ◆❖❈❱ ❢♦r♠❛❧✐s♠ ✐♥ ❇❊❘❚❍❆✳ ■♥ t❤❡ ♣r❡s❡♥t s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❛♣♣❧② ♦✉r ♠❡t❤♦❞s t♦ ❛♥ ✐❧❧✉str❛t✐✈❡

+ ❝❛s❡✱ t❤❡ ❝❛r❜♦♥✲♠❡t❛❧ ❜♦♥❞ ✐♥ ❛ s❡r✐❡s ♦❢ ❬▼✕❈❖❪ ❝♦♠♣❧❡①❡s ✇✐t❤ ▼

=

❈✉✱ ❆❣ ❛♥❞ ❆✉

✇❤❡r❡ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❛r❡ ❦♥♦✇♥ t♦ ❜❡❝♦♠❡ ✐♥❝r❡❛s✐♥❣❧② ✐♠♣♦rt❛♥t ✇✐t❤ t❤❡ ✐♥❝r❡❛s✐♥❣ ❛t♦♠✐❝ ♥✉♠❜❡r ♦❢ ▼✳ ❲❡ ✇✐❧❧ t❤✉s ❝♦♠♣❛r❡ t❤❡ ❈❉ ❢✉♥❝t✐♦♥s ❢♦r t❤❡ t❤r❡❡ ♠❡t❛❧✲❝❛r❜♦♥ ❜♦♥❞s ♦❜t❛✐♥❡❞ ✉s✐♥❣ ❇❊❘❚❍❆ ❛s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♣r❡s❡♥t ✇♦r❦✳ ❘❡s✉❧ts ♦❢ ♦✉r ❛♥❛❧②s✐s ❛r❡ r❡♣♦rt❡❞ ✐♥ ❋✐❣✉r❡s ✶✱ ✷ ❛♥❞ ✸ ✇❤❡r❡ t❤❡✐r ❞✐✛❡r❡♥❝❡✱ ✇✐t❤

∆ρanti ✮

k = 1, 2, 3, 4✮

∆ρ✱ ∆ρ′

′ ❛♥❞ t❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ◆❖❈❱✲♣❛✐rs ❞❡♥s✐t② ❞❡❝♦♠♣♦s✐t✐♦♥ ✭∆ρk ✱

❛r❡ s❤♦✇♥ ❛s ✐s♦s✉r❢❛❝❡ ♣❧♦ts t♦❣❡t❤❡r ✇✐t❤ t❤❡ r❡s♣❡❝t✐✈❡ ❈❉ ❢✉♥❝t✐♦♥s✳

+ 2 ❆s ❛♥ ❡①❛♠♣❧❡✱ ❢♦r t❤❡ ❆✉❈❖ s②st❡♠ t❤❡ ❞✐st✐♥❝t ◆❖❈❱ ❞❡♥s✐t✐❡s ✭|Φ+k | ✱

k = 1, 2, 3, 4✮

✭✐♥❝❧✉❞✐♥❣

|Φ−k |2

✇✐t❤

❛r❡ ✈✐s✉❛❧✐③❡❞ ❛s ✐s♦s✉r❢❛❝❡ ♣❧♦ts ✐♥ ❙■ ✭s❡❡ ❋✐❣✉r❡ ❙✸✮✳

+ + + ■♥ ❛❧❧ t❤❡ ❝❛s❡s r❡♣♦rt❡❞ ✭❬❈✉✕❈❖❪ ✱ ❬❆❣✕❈❖❪ ❛♥❞ ❬❆✉✕❈❖❪ ✮ t❤❡ ♦✈❡r❛❧❧ r❡❛rr❛♥❣❡✲ ♠❡♥t ✭❡①♣r❡ss❡❞ ❛s ❡✐t❤❡r

∆ρ′

♦r

∆ρ✮

❢❡❛t✉r❡s ❛ ❝❧❡❛r ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❞❡♣❧❡t✐♦♥ ❛t t❤❡ ❈❖

s✐t❡ ✭r❡❞ ❧♦❜❡✮ ❛♥❞ ❛♥ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❛❝❝✉♠✉❧❛t✐♦♥ ✭❜❧✉❡ ❧♦❜❡✮ t♦✇❛r❞s t❤❡ ♠❡t❛❧ ❝❛t✐♦♥✳ ❚❤✐s ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❛❝❝✉♠✉❧❛t✐♦♥✱ ✇❤✐❝❤ ✐s r♦✉❣❤❧② ♦❢ t♦r♦✐❞❛❧ s❤❛♣❡✱ ✐s ♥♦t ♦♥❧② ✐♥ t❤❡ ✐♥t❡r♥✉❝❧❡❛r r❡❣✐♦♥ ❜✉t ❛❧s♦ ✐♥ t❤❡ r❡❣✐♦♥ s✉rr♦✉♥❞✐♥❣ t❤❡ ♠❡t❛❧ ❝❛t✐♦♥s✱ ✇❤✐❝❤ ♠❛② ❜❡ ✐♥✲ t❡r♣r❡t❡❞ ❛s t❤❡ ♣❛rt✐❛❧ ✜❧❧✐♥❣ ♦❢ t❤❡✐r ❡♠♣t②

s

♦r❜✐t❛❧✳ ■♥t❡r❡st✐♥❣❧②✱ ❛❧❧ ♠❡t❛❧ ❝❛t✐♦♥s ❛r❡

❛❧s♦ ❝❤❛r❛❝t❡r✐③❡❞✱ ❛t t❤❡✐r ❧❡❢t s✐❞❡ ✭✐♥ tr❛♥s ♣♦s✐t✐♦♥ ♦❢ t❤❡ ❈❖ ♠♦✐❡t②✮✱ ❜② ❛ ♥❛rr♦✇ ③♦♥❡ ♦❢ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❞❡♣❧❡t✐♦♥ ✭r❡❞ ❧♦❜❡ ❛t t❤❡ ♠❡t❛❧ s✐t❡✮✱ ✇❤✐❝❤ ✐♥❞✐❝❛t❡s t❤❛t ❛♥ ✐♠♣♦rt❛♥t

✶✹


Page 15 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


❛♥❞ s♦♠❡❤♦✇ ♥♦t ♦❜✈✐♦✉s ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t ♦❝❝✉rs ✇✐t❤✐♥ t❤❡ ❝❛t✐♦♥ ❢r❛❣♠❡♥t✳ ❲❡ ✇✐❧❧ r❡t✉r♥ ♦♥ t❤✐s ♣♦✐♥t ❧❛t❡r ✐♥ t❤✐s s❡❝t✐♦♥✮✳ ❇❡❢♦r❡ ✇❡ st❛rt ❞❡s❝r✐❜✐♥❣ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡ ❈❉ ❝✉r✈❡s ✐♥ ❞❡t❛✐❧✱ ✇❡ r❡❝❛❧❧ ❤❡r❡ t❤❛t ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t ❛❧♦♥❣ ❛ ❝❤♦s❡♥ ❛①✐s ✭❤❡r❡✱ t❤❡ ♠♦❧❡❝✉❧❛r ❛①✐s✮✱

z✱

❛ ♣♦s✐t✐✈❡ ✈❛❧✉❡ ♦❢ t❤❡ ❈❉

❢✉♥❝t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ❛ ✢♦✇ ♦❢ ❡❧❡❝tr♦♥s ❢r♦♠ t❤❡ r✐❣❤t t♦ t❤❡ ❧❡❢t❀ ✐♥ ♦✉r ❝❛s❡ ❢r♦♠ t❤❡ ❈❖ t♦✇❛r❞s t❤❡ ♠❡t❛❧ ▼✳ ❈♦♥✈❡rs❡❧②✱ ❛ ♥❡❣❛t✐✈❡ ✈❛❧✉❡ ♦❢ t❤❡ ❈❉ ❢✉♥❝t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ✢♦✇ ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥✱ t❤❛t ✐s ❢r♦♠ t❤❡ ♠❡t❛❧ ▼ t♦ ❈❖✳ ❆ r❡❛s♦♥❛❜❧❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ❝❤❛r❣❡ tr❛♥s❢❡r ✭❈❚✮ ❜❡t✇❡❡♥ t❤❡ ❈❖ ❛♥❞ ▼

+

❢r❛❣♠❡♥t ✐s ❡❛s✐❧② ♦❜t❛✐♥❡❞ ❜② ✜①✐♥❣ ❛ ♣❧❛✉s✐❜❧❡

❜♦✉♥❞❛r② t♦ s❡♣❛r❛t❡ t❤❡ ❢r❛❣♠❡♥ts ✇✐t❤✐♥ t❤❡ ❝♦♠♣❧❡①✳ ❖✉r st❛♥❞❛r❞ ❝❤♦✐❝❡ ✐s t❤❡ ✇❤❡r❡ ❡q✉❛❧✲✈❛❧✉❡❞ ✐s♦❞❡♥s✐t② s✉r❢❛❝❡s ♦❢ t❤❡ ✐s♦❧❛t❡❞ ❢r❛❣♠❡♥ts ❜❡❝♦♠❡ t❛♥❣❡♥t✳

z ✲♣♦✐♥t

✶✼✱✶✽✱✸✷

❚❤❡

✈❡rt✐❝❛❧ ❜❧❛❝❦ ❧✐♥❡ ✐♥ t❤❡ ❋✐❣✉r❡s ♠❛r❦s t❤✐s ✏✐s♦❞❡♥s✐t② ❜♦✉♥❞❛r②✑✳ ❚❤❡ ❈❉ ❢✉♥❝t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♦✈❡r❛❧❧ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t ✭❧❛❜❡❧❡❞ ❛s

∆ρ′

❛♥❞

∆ρ✱

❝♦♥t✐♥✉♦✉s ❛♥❞ ❞❛s❤❡❞ ❜❧❛❝❦ ❧✐♥❡s✱ r❡s♣❡❝t✐✈❡❧②✮ q✉❛♥t✐t❛t✐✈❡❧② ❝♦♥✜r♠ t❤❡

❣❡♥❡r❛❧ ♣✐❝t✉r❡ ❞❡s❝r✐❜❡❞ ❛❜♦✈❡✳ ❚❤✉s✱ ❛❧❧ ❝♦♠♣❧❡①❡s s❤♦✇ ❛ ♣♦s✐t✐✈❡ ✈❛❧✉❡ ✐♥ t❤❡ ♠❡t❛❧✲❈❖ ❜♦♥❞✐♥❣ r❡❣✐♦♥ ✐♥❞✐❝❛t✐♥❣ ❛ ♥❡t ✢♦✇ ♦❢ ❡❧❡❝tr♦♥s ❢r♦♠ ❈❖ t♦✇❛r❞s t❤❡ ♠❡t❛❧✳ ❚❤❡s❡ ❝✉r✈❡s ❞✐s♣❧❛② ❛♥ ♦✈❡r❛❧❧ s✐♠✐❧❛r s❤❛♣❡ ❛♥❞ s✐♠✐❧❛r ✈❛❧✉❡s ♦❢ ♥❡t ❈❚ ✭✐✳❡✳ ✵✳✵✽ ❡✱ ✵✳✶✶ ❡ ❛♥❞ ✵✳✶✸

+ + + ❡ ✐♥ ❬❈✉✕❈❖❪ ✱ ❬❆❣✕❈❖❪ ❛♥❞ ❬❆✉✕❈❖❪ ✱ r❡s♣❡❝t✐✈❡❧②✮✳ ❚❤❡ ♥❡❣❛t✐✈❡ ♣❡❛❦ ♦❢ t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ♥❡❣❛t✐✈❡ ♣♦rt✐♦♥ ♦❢

z✱

❜❡❤✐♥❞ t❤❡ ♠❡t❛❧✬s ♣♦s✐t✐♦♥✱ ✐♥❞✐❝❛t❡s ❛ ♥❡t ❡❧❡❝tr♦♥ ✢✉① ❢r♦♠ ❧❡❢t

t♦ r✐❣❤t ❛♥❞ ❛❝❝♦✉♥ts ❢♦r t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❞❡♣❧❡t✐♦♥ ♦❜s❡r✈❡❞ ❛❜♦✈❡ ✭t❤❡ r❡❞ ❧♦❜❡ ❛t t❤❡ ♠❡t❛❧ s✐t❡✮✳ ❆♥ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t ♦❢ t❤✐s t②♣❡ ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ r❡❧❛t❡❞ t♦ t❤❡ ❛❜✐❧✐t② ♦❢ ♥♦❜❧❡ ♠❡t❛❧s t♦ ❢♦r♠

s−d

❤②❜r✐❞s✱

✶✼✱✽✼✱✽✽

✇❤✐❝❤ ❤❛s ❜❡❡♥ ♣r♦♣♦s❡❞ ❛s ❛ ❦❡②

♠❡❝❤❛♥✐s♠ t♦ r❡❞✉❝❡ t❤❡ r❡♣✉❧s✐✈❡ ❢♦r❝❡ ♦❢ t❤❡ ♠❡t❛❧ ❞✲s❤❡❧❧ ❛♥❞ t❤❡ ❧✐❣❛♥❞ ❡❧❡❝tr♦♥s ❛♥❞ t♦ st❛❜✐❧✐③❡ t❤❡ ♠❡t❛❧✲❧✐❣❛♥❞ ❜♦♥❞✳ ■t ✐s r❡♠❛r❦❛❜❧❡ t❤❛t t❤❡ ❣♦❧❞ ❝♦♠♣❧❡① ♣r❡s❡♥ts✱ ❛♠♦♥❣st ♥♦❜❧❡ ♠❡t❛❧s✱ t❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❞❡♣❧❡t✐♦♥ ✭❧❛r❣❡st ♥❡❣❛t✐✈❡ ♣❡❛❦✮✳ ❚❤✐s ♣❡❝✉❧✐❛r ❞❡♥s✐t② ❝❤❛♥❣❡ ❛t t❤❡ ❣♦❧❞ s✐t❡ ✐s ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ r❡❝♦❣♥✐③❡❞ ♣r❡❢❡r❡♥❝❡ ♦❢ ❆✉✭■✮✱ ❝♦♠♣❛r❡❞ t♦ ♦t❤❡r ❤♦♠♦❧♦❣✉❡s s✉❝❤ ❛s ❈✉✭■✮ ❛♥❞ ❆❣✭■✮✱ ❢♦r ❧✐♥❡❛r ❜✐❝♦♦r❞✐♥❛t✐♦♥ ♦❢ ❆✉✭■✮ ❝♦♠♣❧❡①❡s✳

✶✺



Page 16 of 39

❲✐t❤ t❤❡ ❤❡❧♣ ♦❢ ◆❖❈❱s ❛♥❛❧②s✐s t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ t♦t❛❧ ❈❉ ❢✉♥❝t✐♦♥ ❝❛♥ ♥♦✇ ❜❡ s✐♥❣❧❡❞ ♦✉t✳ ❚❤✉s✱ t❤❡ t♦t❛❧ ❈❉ ❢✉♥❝t✐♦♥ ❝❧❡❛r❧② r❡s✉❧ts ♠❛✐♥❧② ❢r♦♠ ❛ ❧✐❣❛♥❞✲t♦✲♠❡t❛❧ ❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥t ✭❧❛❜❡❧❡❞ ❛s

∆ρ′1 ✱

r❡❞ ❝✉r✈❡ ✐♥ t❤❡ ♣❧♦ts✮ ✇❤✐❝❤ ✐s ❧❛r❣❡

❛♥❞ ♣♦s✐t✐✈❡ ✐♥ t❤❡ ▼✲❝❛r❜♦♥ r❡❣✐♦♥ ❛♥❞ t✇♦ ❝♦♠♣♦♥❡♥ts ✭❧❛❜❡❧❡❞ ❛s

∆ρ′2

❛♥❞

∆ρ′3 ✱

t❤❡

❧✐❣❤t ❜❧✉❡ ❝♦♥t✐♥✉♦✉s ❧✐♥❡ ❛♥❞ t❤❡ ❜❧✉❡ ❞♦tt❡❞ ❧✐♥❡✱ r❡s♣❡❝t✐✈❡❧②✮ ✇❤✐❝❤ ❛r❡ ♥❡❣❛t✐✈❡ ❛♥❞ ✉♥❛♠❜✐❣✉♦✉s❧② ✐❞❡♥t✐❢② t❤❡ ♠❡t❛❧✲t♦✲❧✐❣❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥✳ ❚❤❡ r❡❛❞❡r ❝❛♥ ❡❛s✐❧② r❡❝♦❣♥✐③❡ t❤❡ ❝❤❛r❛❝t❡r ♦❢ t❤❡s❡ ❝♦♠♣♦♥❡♥ts t❤❛♥❦s t♦ ❛ ✈✐s✉❛❧ ✐♥s♣❡❝t✐♦♥ ♦❢ t❤❡ ✐s♦❞❡♥s✐t②✲s✉r❢❛❝❡ ♣❧♦ts s❤♦✇♥ ❛t t❤❡ t♦♣ ♦❢ ❋✐❣✉r❡s ✶✱ ✷ ❛♥❞ ✸✳ ❚❤❡s❡ ✸❉ ♣❧♦ts ❢❛❝✐❧✐t❛t❡ t❤❡ r❡❝♦❣♥✐t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ ❝♦♥tr✐❜✉t✐♦♥s ❢r♦♠ t❤❡ ❝♦♥st✐t✉❡♥t ❢r❛❣♠❡♥ts✱ ♥❛♠❡❧② t❤❡ ♠❡t❛❧ ❝❛t✐♦♥ ❛♥❞ t❤❡ ❈❖ ♠♦❧❡❝✉❧❡✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ◆❖❈❱✲♣❛✐r ❞❡♥s✐t②

∆ρ′1 r❡♣r❡s❡♥ts t❤❡ ❞♦♥❛t✐♦♥ ❢r♦♠ t❤❡ ❍❖▼❖ ♦❢ t❤❡ ❈❖

❢r❛❣♠❡♥t t♦ t❤❡ ♠❡t❛❧ ✉♥♦❝❝✉♣✐❡❞ s♣❤❡r✐❝❛❧ ❝❧❡❛r❧② r❡❝❛❧❧ t♦ ♠✐♥❞ t❤❡ s❤❛♣❡ ♦❢ t❤❡

s ♦r❜✐t❛❧ ♦❢ ▼+ ❀

σ ✲s②♠♠❡tr②

❝❤❛r❣❡ ❞❡♣❧❡t✐♦♥ ❛t t❤❡ ❈❖ s✐t❡

❍❖▼❖ ♦r❜✐t❛❧ ♦❢ t❤❡ ❢r❡❡ ❈❖ ❢r❛❣♠❡♥t✱

✇❤✐❧❡ t❤❡ ❛❝❝✉♠✉❧❛t✐♦♥ ❛t t❤❡ ♠❡t❛❧ s✐t❡ ❛♣♣❡❛rs s❧✐❣❤t❧② ❞✐st♦rt❡❞ ❞✉❡ t♦ t❤❡ ❤②❜r✐❞✐③❛t✐♦♥ ✇✐t❤ t❤❡ ✜❧❧❡❞ ❞✲s❤❡❧❧✳ ■t ✐s ❡②❡✲❝❛t❝❤✐♥❣ t❤❛t t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❞❡♣❧❡t✐♦♥ ❛t t❤❡ ❝❛t✐♦♥ s✐t❡ ❛❧♠♦st r❡♣r♦❞✉❝❡s t❤❡ s❤❛♣❡ ♦❢ t❤❡ ❢r❡❡ ♠❡t❛❧ ❞z 2 ♦r❜✐t❛❧ ✭s❡❡ ❛❧s♦ ❋✐❣✉r❡ ❙✸

2 ✐♥ ❙■ r❡♣♦rt✐♥❣ t❤❡ ◆❖❈❱ ❞❡♥s✐t✐❡s ✭|Φ+k | ✱

|Φ−k |2 ✮✳

❚❤✐s ♣❛tt❡r♥ ♠❛② ❜❡ ❝♦♥s✐❞❡r❡❞ t♦

❜❡ ✜♥❣❡r♣r✐♥t ♦❢ t❤❡ s✲❞ ❤②❜r✐❞✐③❛t✐♦♥ ♠❡❝❤❛♥✐s♠ ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳

◗✉❛♥t✐t❛t✐✈❡❧②✱ ♦✉r

◆❖❈❱✴❈❉ ❛♥❛❧②s✐s s❤♦✇s t❤❛t t❤❡ ❣♦❧❞ ❝♦♠♣❧❡① ❤❛s ❜♦t❤ t❤❡ ❧❛r❣❡st ❞♦♥❛t✐♦♥ ✭❈❚∆ρ′ 1

+ + + ✈❛❧✉❡s ❛r❡ ♦❢ ✵✳✸✺ ❡✱ ✵✳✷✵ ❛♥❞ ✵✳✷✷ ❡✱ ❢♦r ❬❆✉❈❖❪ ✱ ❬❆❣❈❖❪ ❛♥❞ ❬❈✉❈❖❪ ✱ r❡s♣❡❝t✐✈❡❧②✮ ❛♥❞ t❤❡ ❧❛r❣❡st ❜❛❝❦✲❞♦♥❛t✐♦♥ ❛♠♦♥❣st t❤❡ ♥♦❜❧❡ ♠❡t❛❧s ✭❈❚ ✈❛❧✉❡s ❢♦r

∆ρ′2 +∆ρ′3 ❝♦♠♣♦♥❡♥ts ❛r❡

+ + + ✵✳✶✾ ❡✱ ✵✳✵✻ ❡ ❛♥❞ ✵✳✶✸ ❡ ❢♦r ❆✉ ✱ ❆❣ ❛♥❞ ❈✉ ✱ r❡s♣❡❝t✐✈❡❧②✮✳ ❖✉r ❛♥❛❧②s✐s ❛❧s♦ ❞❡♠♦♥str❛t❡s + t❤❛t t❤❡ ❆✉❈❖ ❝♦♠♣❧❡① ♣r❡s❡♥ts t❤❡ ❧❛r❣❡st ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❞❡♣❧❡t✐♦♥ ✭t❤❡ ❧❛r❣❡st ♥❡❣❛t✐✈❡ ♣❡❛❦ ✐♥ t❤❡ t♦t❛❧ ❈❉ ❢✉♥❝t✐♦♥✮ ❛t t❤❡ ❧❡❢t s✐❞❡ ♦❢ t❤❡ ♠❡t❛❧✱ ❛♠♦♥❣st ❝♦✐♥❛❣❡ ♠❡t❛❧s✱ ❜❡❝❛✉s❡ ✐t ❤❛s t❤❡ ❧❛r❣❡st ❝♦♥tr✐❜✉t✐♦♥ ❜♦t❤ ❢r♦♠ ❤②❜r✐❞✐③❛t✐♦♥✮ ❛♥❞ ❢r♦♠

∆ρ′2 + ∆ρ′3

∆ρ′1

✭r❡❧❛t❡❞ ✐♥ t❤✐s ♠♦❧❡❝✉❧❛r r❡❣✐♦♥ t♦ t❤❡ s✲❞

✭r❡❧❛t❡❞ t♦ t❤❡ ❜❛❝❦✲❞♦♥❛t✐♦♥✮✳ ❖✈❡r❛❧❧✱ t❤❡s❡ ♣❡❝✉❧✐❛r

❢❡❛t✉r❡s ♠❛② ❜❡ r❡❧❡✈❛♥t ❢♦r ❡①♣❧❛✐♥✐♥❣ t❤❡ s✉♣❡r✐♦r ❛❜✐❧✐t② ♦❢ ❣♦❧❞ ❛s ❛ ❝❛t❛❧②st ❢♦r t❤❡ ❛❝t✐✈❛t✐♦♥ ❛♥❞ s✉❜s❡q✉❡♥t ❢✉♥❝t✐♦♥❛❧✐③❛t✐♦♥ ♦❢ ✉♥s❛t✉r❛t❡❞ s②st❡♠s✱ ✐♥❝❧✉❞✐♥❣ ❛❧❦②♥❡s✳

✶✻


✽✾

Page 17 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


+ + ❋✉rt❤❡r♠♦r❡✱ ✐t s❤♦✉❧❞ ❜❡ ♥♦t❡❞ t❤❛t ✇❤✐❧❡ ❢♦r t❤❡ ❬❈✉✕❈❖❪ ❛♥❞ ❬❆❣✕❈❖❪ ❝♦♠♣❧❡①❡s t❤❡ ❈❉ ❝✉r✈❡s ❢♦r

∆ρ′2

❛♥❞

∆ρ′3

r❡❧❛t❡❞ t♦ t❤❡ t✇♦ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ❛❧♠♦st ❡①❛❝t❧②

+ ♦✈❡r❧❛♣ ❡❛❝❤ ♦t❤❡r✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❬❆✉✕❈❖❪ t❤❡② ❛r❡ s✐❣♥✐✜❝❛♥t❧② s♣❧✐t✳

◗✉❛♥t✐t❛t✐✈❡❧②✱

+ t❤❡ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❈❚ ❢r♦♠ t❤❡ ❆✉ ❢r❛❣♠❡♥t t♦ ❈❖ ✭t❤❡ ❈❉ ✈❛❧✉❡ ❛t t❤❡ ❜♦✉♥❞❛r② s♦❧✐❞ ✈❡rt✐❝❛❧ ❧✐♥❡✮ ✐s ♦❢ ✵✳✶✶ ❡ ❛♥❞ ✵✳✵✽ ❡ ❢♦r t❤❡

∆ρ′2

❛♥❞

∆ρ′3

❝♦♠♣♦♥❡♥ts✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ s❛♠❡

♣❛tt❡r♥ ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ◆❖❈❱s ❡✐❣❡♥✈❛❧✉❡s ✭ν2 ✱

ν3 ✮✱

+ + ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❬❈✉✕❈❖❪ ❛♥❞ ❬❆❣✕❈❖❪ ❝♦♠♣❧❡①❡s ✭t❤❡

✇❤✐❝❤ ❛r❡ ❛❧♠♦st ❞❡❣❡♥❡r❛t❡

ν2

❛♥❞

ν3

+ ✵✳✵✵✽ ❡✮✱ ✇❤✐❧❡ t❤❡ ❞❡❣❡♥❡r❛❝② ✐s ❝❧❡❛r❧② r❡♠♦✈❡❞ ❢♦r ❬❆✉✕❈❖❪ ✭❛♥❞ t❤❡

❞✐✛❡r❡♥❝❡ ✐s ❜❡❧♦✇

ν2

❛♥❞

ν3

❞✐✛❡r❡♥❝❡

✐s ❧❛r❣❡r t❤❛♥ ✵✳✵✹✮❀ s❡❡ ❚❛❜❧❡ ❙✶ ✐♥ t❤❡ ❙■✳ ❚❤❡ r❡♠♦✈❛❧ ♦❢ ❞❡❣❡♥❡r❛❝② ✐♥ t❤❡ t✇♦ ❜❛❝❦✲ ❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ✐s ❝❧❡❛r❧② ❞✉❡ t♦ t❤❡ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ ❡✛❡❝t✱ ✇❤✐❝❤ ❧♦❝❛❧❧② ❜r❡❛❦s t❤❡ ❞❡❣❡♥❡r❛❝② ♦❢ t❤❡

5d ❆✉ ❛t♦♠✐❝ ♦r❜✐t❛❧s ✭5dj=3/2 ❛♥❞ 5dj=5/2 s♣✐♥♦rs ❞✐✛❡r ❜♦t❤ ✐♥ ❡♥❡r❣②

❛♥❞ s♣❛t✐❛❧ ❞✐str✐❜✉t✐♦♥

✾✵

✮ ❛♥❞✱ q✉✐t❡ s✉r♣r✐s✐♥❣❧②✱ ❡①t❡♥❞s ✐ts ❡✛❡❝t t♦ t❤❡ ❆✉✲❈ ❜♦♥❞ r❡❣✐♦♥

❛♥❞ ❡✈❡♥ t♦ t❤❡ ❈❖ s✐t❡✱ ✇❤❡r❡ ❛ ♥❡t ❞✐s♣❧❛❝❡♠❡♥t ♦❢ ❡❧❡❝tr♦♥s ❢r♦♠ ♦①②❣❡♥ t♦✇❛r❞s ❝❛r❜♦♥ ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ❞✉❡ t♦ t❤❡ ♣r❡s❡♥❝❡ ♦❢ t❤❡ ♣♦s✐t✐✈❡❧② ❝❤❛r❣❡❞ ♠❡t❛❧ ❢r❛❣♠❡♥t✳ ◆♦✇✱ ✇❡ ❜r✐❡✢② ❝♦♠♠❡♥t ♦♥ t❤❡ ❈❉ ❝✉r✈❡ r❡❧❛t❡❞ t♦

∆ρ′4 ✱

✽✼

✇❤✐❝❤ ❞♦❡s ♥♦t ❡①❤✐❜✐t ❛♥

❡✈✐❞❡♥t ❝♦✉♥t❡r♣❛rt ✐♥ t❤❡ ❉❈❉ ❜♦♥❞✐♥❣ ♠♦❞❡❧ ✭t❤❡ ♦t❤❡r ◆❖❈❱s ♣❛✐rs ❞♦ ♥♦t ❝♦♥tr✐❜✉t❡ s✐❣♥✐✜❝❛♥t❧② t♦ t❤❡ t♦t❛❧

∆ρ′ ✮✳

❚❤✐s ❝✉r✈❡ ❞❡s❝r✐❜❡s ❛♥ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❛❝❝✉♠✉❧❛t✐♦♥ ✐♥ t❤❡

▼✲❈❖ ❜♦♥❞✐♥❣ r❡❣✐♦♥ ✭❈❉ ❝✉r✈❡ s❧♦♣❡ ✐s ♣♦s✐t✐✈❡✱ s❡❡ ❛❧s♦ ✸❉ ✐s♦❞❡♥s✐t② ♣❧♦ts ✐♥ ❋✐❣✉r❡s ✶✱ ✷✱ ✸✮ ✇✐t❤♦✉t ❛ ♥❡t ❈❚ ❜❡t✇❡❡♥ t❤❡ ❢r❛❣♠❡♥ts ✭t❤❡ ❝✉r✈❡ ❝r♦ss❡s t❤❡ ③❡r♦ ❛①✐s ✐♥ ❛ r❡❣✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❢r❛❣♠❡♥ts✱ ❝❧♦s❡ t♦ t❤❡ ✐s♦❜♦✉♥❞❛r②✮✳ s❡❡♠s r♦✉❣❤❧② t♦ ❜❛❧❛♥❝❡ t❤❡

∆ρanti

❚❤✐s ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t

t❡r♠✱ ✇❤✐❝❤ ✐s ✐❞❡♥t✐✜❡❞ ✇✐t❤ t❤❡ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t

♦❝❝✉rr✐♥❣ ♦♥ ❣♦✐♥❣ ❢r♦♠ t❤❡ s❡♣❛r❛t❡ ❆ ❛♥❞ ❇ ❢r❛❣♠❡♥ts t♦ t❤❡ ♣r♦♠♦❧❡❝✉❧❡ ✭t❤❡ s♣❡❝✐✜❝ ❝❤♦✐❝❡ ♦❢ t❤❡ ♥♦♥✲✐♥t❡r❛❝t✐♥❣ r❡❢❡r❡♥❝❡ s②st❡♠ ✉s❡❞ ✐♥ ◆❖❈❱ ♠❡t❤♦❞✮✳ ■♥❞❡❡❞✱ t❤❡

∆ρanti

❝♦♥tr✐❜✉t✐♦♥ s❤✐❢ts t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❢r♦♠ t❤❡ ✐♥t❡r✲❢r❛❣♠❡♥t r❡❣✐♦♥ t♦✇❛r❞s t❤❡ ❢r❛❣♠❡♥ts ❛s ✐s ✇❡❧❧ q✉❛♥t✐t❛t✐✈❡❧② s❤♦✇♥ ✐♥ t❤❡ ✜❣✉r❡✳ ❆s ♦♥❡ ♠❛② ❡①♣❡❝t✱ t❤❡ ❡✛❡❝t ✐s ♠♦r❡ s✐❣♥✐✜❝❛♥t ✐♥ t❤❡ s②st❡♠ ❝♦♥t❛✐♥✐♥❣ ❣♦❧❞ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❧❛r❣❡r ♦✈❡r❧❛♣ ❜❡t✇❡❡♥ ❢r❛❣♠❡♥ts ✭❆✉✲❈❖ ❞✐st❛♥❝❡ ✐s s✐❣♥✐✜❝❛♥t❧② s❤♦rt❡r r❡s♣❡❝t t♦ ♦t❤❡r ♥♦❜❧❡ ♠❡t❛❧ ❝♦♥t❛✐♥✐♥❣ ❝♦♠♣♦✉♥❞s✮✳

✶✼



Page 18 of 39

❇❡❢♦r❡ ❝♦♥❝❧✉❞✐♥❣ t❤✐s s❡❝t✐♦♥✱ ✐t ✐s ✐♥t❡r❡st✐♥❣ t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ❡✛❡❝t ♦❢ r❡❧❛t✐✈✐t② ♦♥

+ t❤❡ ❜♦♥❞✐♥❣ ❝♦♠♣♦♥❡♥ts ✐♥ t❤❡ ❆✉❈❖ ❝♦♠♣❧❡①✳ ❲❡ ♣❡r❢♦r♠❡❞ ♦✉r ◆❖❈❱✴❈❉ ❛♥❛❧②s✐s ✐♥❝r❡❛s✐♥❣ t❤❡ s♣❡❡❞ ♦❢ ❧✐❣❤t ❜② ♦♥❡ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡ ✭❝ ❂ ✶✵×✶✸✼✳✵✸✻ ❛✳✉✳✮ t♦ ❛♣♣r♦①✐✲ ♠❛t❡ t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❧✐♠✐t✳ ❚❤❡ r❡s✉❧ts ❝❧❡❛r❧② s❤♦✇ t❤❛t t❤❡ ❉❈❉ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡

+ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ✐♥ ❆✉❈❖ ❛r❡ s✐❣♥✐✜❝❛♥t❧② ♠♦❞✐✜❡❞ ✇❤❡♥ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❛r❡ s✇✐t❝❤❡❞ ♦✛ ✭s❡❡ ❋✐❣✉r❡ ❙✹ ✐♥ ❙■✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❧✐❣❛♥❞✲t♦✲♠❡t❛❧ ❞♦♥❛t✐♦♥ r❡❛❞✐❧② ❞❡❝r❡❛s❡s ✭t❤❡ ❞♦♥❛t✐♦♥ ❈❚ ♣❛ss❡s ❢r♦♠ ✵✳✸✸ t♦ ✵✳✷✵ ❡✮ ❛tt❛✐♥✐♥❣ ❛ ✈❛❧✉❡ ❝❧♦s❡ t♦ t❤❛t ♦❜t❛✐♥❡❞ ❢♦r ❈✉ ❛♥❞ ❆❣ ✇❤✐❧❡ t❤❡ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥ts ❣✐✈❡ ♥♦✇ ✐❞❡♥t✐❝❛❧ ❝♦♥tr✐❜✉t✐♦♥s ✭❞✉❡ t♦ t❤❡ ❛❜s❡♥❝❡

′ ♦❢ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣✮ ❛♥❞ ♦✈❡r❛❧❧ ❛ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❈❚ ✭∆ρ2

+ ∆ρ′3 ✮

♦❢ ✵✳✶✹ ❡✱ ✇❤✐❝❤ ✐s ❝❧♦s❡

t♦ t❤❡ ✈❛❧✉❡ ❢♦✉♥❞ ❢♦r ❈✉✳ ❘❡❧❛t✐✈✐t② ❛❧s♦ ❤❛s ❛ s✐❣♥✐✜❝❛♥t ✐♠♣❛❝t ♦♥ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② r❡❛rr❛♥❣❡♠❡♥t ✇✐t❤✐♥ t❤❡ ♠❡t❛❧ ❢r❛❣♠❡♥t ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❢♦✉♥❞ t❤❛t t❤❡ ❧❛r❣❡ ♥❡❣❛✲ t✐✈❡ ♣❡❛❦ ✐♥ t❤❡ t♦t❛❧ ❈❉ ❝✉r✈❡ ✐s s✐❣♥✐✜❝❛♥t❧② r❡❞✉❝❡❞ ✐♥ t❤❡ ♥♦♥✲r❡❧❛t✐✈✐st✐❝ ❧✐♠✐t✱ ♠❛✐♥❧② ❜❡❝❛✉s❡ ♦❢ t❤❡

∆ρ′1

❝♦♠♣♦♥❡♥t✳ ❚❤✐s ✐s ♥♦t s✉r♣r✐s✐♥❣❧② ❜❡❝❛✉s❡✱ ❛s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ t❤❡

❜❡❤❛✈✐♦r ♦❢ t❤✐s ❝♦♠♣♦♥❡♥t ✐♥ t❤❡ ❆✉ r❡❣✐♦♥ ✐s str✐❝t❧② r❡❧❛t❡❞ ✇✐t❤ t❤❡ s✲❞ ❤②❜r✐❞✐③❛t✐♦♥ ♠❡❝❤❛♥✐s♠✱ ✇❤✐❝❤ ✐s ✐♥❞❡❡❞ s✐❣♥✐✜❝❛♥t❧② r❡❞✉❝❡❞ ✇❤❡♥ t❤❡ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❛r❡ ♥❡❣❧❡❝t❡❞✳ ❘❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❢❛✈♦r s✲❞ ♠✐①✐♥❣✱ ❧❡❛❞✐♥❣ t♦ ❛ st❛❜✐❧✐③❛t✐♦♥✴❝♦♥tr❛❝t✐♦♥ ♦❢ t❤❡ ✻s s❤❡❧❧✱ ❛♥❞ ❛ ❞❡st❛❜✐❧✐③❛t✐♦♥✴❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ✺❞ s❤❡❧❧✳

✹

❈♦♥❝❧✉s✐♦♥s

■♥ t❤❡ ♣r❡s❡♥t ✇♦r❦ ✇❡ ❞❡s❝r✐❜❡❞ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ◆❖❈❱ ❢♦r♠❛❧✐s♠ t♦ t❤❡ r❡❧❛t✐✈✐st✐❝ ❢♦✉r✲❝♦♠♣♦♥❡♥t ❢r❛♠❡✇♦r❦ ✇✐t❤✐♥ t❤❡ ❝❤❛r❣❡✲❞✐s♣❧❛❝❡♠❡♥t ❛♥❛❧②s✐s s❝❤❡♠❡✳

❚❤✐s ♠❡t❤♦❞

✐s s✐♠♣❧❡ ❛♥❞ ♣♦✇❡r❢✉❧ ✐♥ ❛♥❛❧②③✐♥❣ ❛ ❝❤❡♠✐❝❛❧ ❜♦♥❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❝❤❛r❣❡ r❡❛rr❛♥❣❡♠❡♥t t❛❦✐♥❣ ♣❧❛❝❡ ✉♣♦♥ ❜♦♥❞ ❢♦r♠❛t✐♦♥ st❛rt✐♥❣ ❢r♦♠ t✇♦ ❝♦♥st✐t✉t✐♥❣ ❢r❛❣♠❡♥ts✳

+ ❲❡ ✜rst❧② ♣❡r❢♦r♠❡❞ ❛ t❡st st✉❞② ♦❢ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ t❤❡ ❬▲✐✲❈❖❪ s②st❡♠ ❝♦♠✲ ♣❛r✐♥❣ t❤❡ r❡s✉❧ts ❛❣❛✐♥st t❤♦s❡ ♦❜t❛✐♥❡❞ ✇✐t❤ t❤❡ ❆❉❋ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✐♥ ♦r❞❡r t♦ ❛ss❡ss t❤❡ ❝♦rr❡❝t♥❡ss ❛♥❞ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ ♦✉r r❡s✉❧ts✳ ❚❤❡ ❜❡♥❝❤♠❛r❦ st✉❞② ✇❛s s✉❝❝❡ss❢✉❧❧②

✶✽


Page 19 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


♣❡r❢♦r♠❡❞ ✉s✐♥❣ ❞✐✛❡r❡♥t ❜❛s✐s s❡t ♦♥ ❛ s②st❡♠ ✇❤❡r❡ t❤❡ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❛r❡ ❡①♣❡❝t❡❞ ♥♦t t♦ ♣❧❛② ❛ r❡❧❡✈❛♥t r♦❧❡✳ ■♥❞❡❡❞✱ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ✐♥ q✉❛♥t✐t❛t✐✈❡ ❛❣r❡❡♠❡♥t ✇✐t❤ t❤❡ ❆❉❋ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳ ❙♣❡❝✐✜❝❛❧❧② ✇❡ s❤♦✇❡❞ ❤♦✇ t❤❡ ♥✉♠❡r✐❝❛❧ ❞✐✛❡r✲ ❡♥❝❡s ❝❛♥ ❜❡ ❛ttr✐❜✉t❡❞ t♦ t❤❡ ❜❛s✐s s❡t ❡✛❡❝ts✳ ■♥ ❛❞❞✐t✐♦♥ ✇❡ ❛❧s♦ ♣r♦✈❡❞ t❤❡ ♥✉♠❡r✐❝❛❧ st❛❜✐❧✐t② ♦❢ t❤❡ ❇❊❘❚❍❆ ❝♦❞❡ ❜② s✐♠✉❧❛t✐♥❣ ❛ ❝♦♠♣❧❡t❡ r❡♠♦✈❛❧ ♦❢ t❤❡ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts✱ ✇❤✐❝❤ ✇❛s ❛❝❤✐❡✈❡❞ ❜② ❛rt✐✜❝✐❛❧❧② ❢♦r❝✐♥❣ t❤❡ s♣❡❡❞ ♦❢ ❧✐❣❤t t♦ ❛ ❝♦♥s✐❞❡r❛❜❧② ❤✐❣❤❡r ✈❛❧✉❡✳ ❋✐♥❛❧❧② ✇❡ ✉s❡❞ t❤✐s ❢♦r♠❛❧✐s♠ t♦ st✉❞② t❤❡ ❝❛r❜♦♥✲♠❡t❛❧ ❝♦♦r❞✐♥❛t✐♦♥ ❜♦♥❞ ✐♥ ❛ s❡t ♦❢

+ ❝❛r❜♦♥②❧ ❝♦♠♣❧❡①❡s ❬▼✕❈❖❪ ✭▼

=

❈✉✱ ❆❣✱ ❆✉✮✳

❲❡ s❤♦✇❡❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ t❤❛t t❤❡ ◆❖❈❱✴❈❉ ❛♣♣r♦❛❝❤✱ ✉s❡❞ ✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ r❡❧✲ ❛t✐✈✐st✐❝ ❢♦✉r✲❝♦♠♣♦♥❡♥t ❝❛❧❝✉❧❛t✐♦♥s✱ ✐s ❛❜❧❡ t♦ ✐❞❡♥t✐❢② t❤❡ ❞♦♥❛t✐♦♥ ❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝❤❛r❣❡ ✢✉①❡s ♦❢ t❤❡ ❉❡✇❛r✲❈❤❛tt✲❉✉♥❝❛♥s♦♥ ❜♦♥❞✐♥❣ ♠♦❞❡❧✱ ✇❤✐❝❤ ✐s ✉❜✐q✉✐t♦✉s ✐♥ ❝♦♦r❞✐✲ ♥❛t✐♦♥ ❝❤❡♠✐str②✳ ❘❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts ❤❛✈❡ ❛ s✐❣♥✐✜❝❛♥t ✐♠♣❛❝t ♦♥ t❤❡ ❜♦♥❞ ❝♦♠♣♦♥❡♥ts ❛♥❞✱

+ ✐♥ ♣❛rt✐❝✉❧❛r ❢♦r t❤❡ ✭❬❆✉✲❈❖❪ ✮ s②st❡♠✱ ♦✉r ♠❡t❤♦❞ ❤✐❣❤❧✐❣❤t❡❞ ❛ s♣❧✐tt✐♥❣ ♦❢ t❤❡ ♠❡t❛❧ t♦ ❧✐❣❛♥❞ ❜❛❝❦✲❞♦♥❛t✐♦♥ ❝♦♠♣♦♥❡♥t ✐♥❞✉❝❡❞ ❜② t❤❡ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣ t❤❛t✱ st❛rt✐♥❣ ❢r♦♠ t❤❡ ♠❡t❛❧✱ ❡①t❡♥❞s t♦ t❤❡ ❈❖ ❜♦♥❞✳ ❲❡ ❤♦♣❡ t❤❛t t❤❡ ♠❡t❤♦❞♦❧♦❣② ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ✇♦r❦ ♠❛② ❝♦♥tr✐❜✉t❡✱ t♦❣❡t❤❡r ✇✐t❤ ♦t❤❡r ❛❞✈❛♥❝❡s ✐♥ t❤❡ ✜❡❧❞✱

✾✶✕✾✺

t♦ r❛t✐♦♥❛❧✐③❡ r❡❧❛t✐✈✐st✐❝ ❡✛❡❝ts✱

✐♥❝❧✉❞✐♥❣ t❤❡ s♣✐♥✲♦r❜✐t ❝♦✉♣❧✐♥❣✱ ♦♥ t❤❡ ❜♦♥❞✐♥❣✱ r❡❛❝t✐✈✐t② ❛♥❞ ❡①♣❡r✐♠❡♥t❛❧ ♦❜s❡r✈❛❜❧❡s ✐♥ ♠♦❧❡❝✉❧❡s ❝♦♥t❛✐♥✐♥❣ ❤❡❛✈② ❛♥❞ s✉♣❡r✲❤❡❛✈② ❡❧❡♠❡♥ts✳

❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ❚❤✐s ✇♦r❦ ❤❛s ❜❡❡♥ s✉♣♣♦rt❡❞ ❜② t❤❡ ■t❛❧✐❛♥ ▼■❯❘ t❤r♦✉❣❤ t❤❡ ❋■❘❇✲❢✉t✉r♦✲✐♥✲r✐❝❡r❝❛ ♣r♦❥❡❝t ❘❇❋❘✶✵✷✷❯◗✳ ❈♦♥s♦rt✐✉♠ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ▼♦❧❡❝✉❧❛r ❛♥❞ ▼❛t❡r✐❛❧ ❙❝✐❡♥❝❡s

2 ❬✭❈▼❙✮ ❪ ❝✴♦ ❉❡♣❛rt♠❡♥t ♦❢ ❈❤❡♠✐str②✱ ❇✐♦❧♦❣② ❛♥❞ ❇✐♦t❡❝❤♥♦❧♦❣② ♦❢ ❯♥✐✈❡rs✐t② ♦❢ P❡✲ r✉❣✐❛ ✐s ❛❧s♦ ❛❝❦♥♦✇❧❡❞❣❡❞✳ ❚❤❡ ❛✉t❤♦rs ❛r❡ ❣r❛t❡❢✉❧ t♦ Pr♦❢❡ss♦r ❋r❛♥❝❡s❝♦ ❚❛r❛♥t❡❧❧✐ ❢♦r r❡❛❞✐♥❣ ♦❢ t❤❡ ♠❛♥✉s❝r✐♣t✳

✶✾



Page 20 of 39

❙✉♣♣♦rt✐♥❣ ■♥❢♦r♠❛t✐♦♥ ❆✈❛✐❧❛❜❧❡ + ❙✉♣♣❧❡♠❡♥t❛r② r❡s✉❧ts ♦❢ t❤❡ ◆❖❈❱✴❈❉ ❛♥❛❧②s✐s ♦♥ t❤❡ ❬▲✐✕❈❖❪ t❡st ❝❛s❡ ❛♥❞ ♦♥ t❤❡ ❬❈✉✕ + + + ❈❖❪ ✱ ❬❆❣✕❈❖❪ ❛♥❞ ❬❆✉✕❈❖❪ s②st❡♠s✳ ■♥t❡r♥❡t ❛t

❆

❚❤✐s ♠❛t❡r✐❛❧ ✐s ❛✈❛✐❧❛❜❧❡ ❢r❡❡ ♦❢ ❝❤❛r❣❡ ✈✐❛ t❤❡

❤tt♣✿✴✴♣✉❜s✳❛❝s✳♦r❣✴✳

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ◆❖❈❱✴❈❉ ♠❡t❤♦❞ ✐♥ ❇❊❘❚❍❆ ❝♦❞❡

❲❡ ❡①♣❧❛✐♥ ✐♥ ❞❡t❛✐❧ t❤❡ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ◆❖❈❱s ✇✐t❤✐♥ t❤❡ ❢♦✉r✲❝♦♠♣♦♥❡♥t r❡❧❛t✐✈✐st✐❝ ❢♦r♠❛❧✐s♠ ❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ ❉❑❙ ♠♦❞✉❧❡ ♦❢ t❤❡ ❇❊❘❚❍❆ ❝♦❞❡✳ ❚❤❡ s♣✐♥♦r s♦❧✉t✐♦♥ ✐s ❡①♣r❡ss❡❞ ✐♥ ❇❊❘❚❍❆ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ s✉✐t❛❜❧❡ ❜❛s✐s ❢✉♥❝t✐♦♥s ❛s



 Ψi (r) =  ✇❤❡r❡✱

(T )

cµi

(L) µ=1 cµi

i

PM

PM

(L) Πµ (r)

(S)

(S)

µ=1 cµi Πµ (r)



 ,

✭✶✷✮

❛r❡ ❝♦❡✣❝✐❡♥ts ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s❡❧❢✲❝♦♥s✐st❡♥t✲✜❡❧❞ ♣r♦❝✉r❡ ♦❢ t❤❡ ❉❑❙ ✭❉❍❋✮

♠❡t❤♦❞ ❛♥❞

M

✐s t❤❡ ♥✉♠❜❡r ♦❢ ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ✉s❡ ●✲s♣✐♥♦r ❜❛s✐s ❢✉♥❝t✐♦♥s

(L) ✇❤✐❝❤ ❛r❡ ❛t♦♠✐❝ ❝❡♥t❡r❡❞ t✇♦✲❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ❢♦r t❤❡ ❧❛r❣❡ ♣❛rt ✭Πµ (r)✮ ❛♥❞ t✇♦✲ (S) ❝♦♠♣♦♥❡♥t ❢✉♥❝t✐♦♥s ❢♦r t❤❡ s♠❛❧❧ ♣❛rt ✭Πµ (r)✮ ♦❢ t❤❡ s♣✐♥♦r✳ ■♥ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❜❛s✐s s❡t ❢♦r t❤❡ ❛❞❞✉❝t ✭❆❇✮ ✐s ♦❜t❛✐♥❡❞ ❜② ❥♦✐♥✐♥❣ t❤❡ ❜❛s✐s s❡ts ♦❢ t❤❡ t✇♦ ❢r❛❣♠❡♥ts ✭❆ ❛♥❞ ❇✮✳ ❚❤✐s r❡s✉❧ts ✐♥ ❛ s✐♠♣❧❡ ❢♦r♠✉❧❛t✐♦♥ ✇❤❡r❡ ❜♦t❤

Ψ0 ❛♥❞ Ψ ❛r❡ ❡①♣r❡ss❡❞ ✇✐t❤ t❤❡ s❛♠❡ ❜❛s✐s s❡t ❛♥❞ t❤❡ ❝❤❡♠✐❝❛❧✲✈❛❧❡♥❝❡ ♦♣❡r❛t♦r

❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s

Vˆ =



X  µν

(LL) (L) (L) ∆Dµν |Πµ ihΠν |

(LS) (L) (S) ∆Dµν |Πµ ihΠν |

(SL) (S) (L) ∆Dµν |Πµ ihΠν |

(SS) (S) (S) ∆Dµν |Πµ ihΠν |

✷✵


  

✭✶✸✮

Page 21 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


′

✇❤❡r❡

(T T )

∆Dµν

❛r❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❞❡♥s✐t②✲♠❛tr✐① ❞✐✛❡r❡♥❝❡ ✭∆D

❜❡t✇❡❡♥ t❤❡ ❞❡♥s✐t② ♠❛tr✐① ❛❞❞✉❝t ❆❇ ✭D s②st❡♠ ✭D

′

0(T T )

′

(T T )

′

= D(T T ) −D0(T T

′

)

✮

′

(T T )

✮ ❛♥❞ t❤❛t ♦❢ t❤❡ ♥♦♥✲✐♥t❡r❛❝t✐♥❣ r❡❢❡r❡♥❝❡

′

✮✱ ✇✐t❤ t❤❡ ❧❛❜❡❧ ❚❚ ❂ ▲▲✱ ▲❙✱ ❙▲✱ ❙❙✳

❚❤❡s❡ ♠❛tr✐❝❡s ❛r❡ ❞❡✜♥❡❞ ❜②

′

D(T T ) = c(T ) c(T

′

)†

✭✶✹✮

❛♥❞ ′

′

D0(T T ) = c0(T ) c0(T )† , ✇❤❡r❡

c(T )

❛❞❞✉❝t ❛♥❞

✐s t❤❡ ♠❛tr✐① ♦❢ t❤❡

c0(T )

(T )

cµi

✭✶✺✮

❝♦❡✣❝✐❡♥ts ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❉❑❙ s♦❧✉t✐♦♥ ♦❢ t❤❡

✐s t❤❡ ♠❛tr✐① ♦❢ t❤❡

0(T )

cµi

❝♦❡✣❝✐❡♥ts t♦ ❜❡ ❡✈❛❧✉❛t❡❞ ❜② ♠❡❛♥s ♦❢ ❛♥

♦rt❤♦♥♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♦❝❝✉♣✐❡❞ ❡❧❡❝tr♦♥✐❝ ♦r❜✐t❛❧s ♦❢ t❤❡ ✐s♦❧❛t❡❞ ❢r❛❣♠❡♥ts ✭❆✰❇✮✳ ❆s ❛ ✜rst st❡♣ t♦ ❡✈❛❧✉❛t❡ t❤❡

0(T )

cµi

❝♦❡✣❝✐❡♥ts ✇❡ ❝♦♥str✉❝t t❤❡ ♠❛tr✐①

C+ ✱

✇❤♦s❡

❝♦❧✉♠♥s ❝♦♥t❛✐♥ t❤❡ ❝♦❡✣❝✐❡♥ts r❡♣r❡s❡♥t✐♥❣ ✐♥ t❤❡ ❜❛s✐s s❡t t❤❡ s♣✐♥♦rs ❜❡❧♦♥❣✐♥❣ t♦ t❤❡

(T ) ✐s♦❧❛t❡❞ ❢r❛❣♠❡♥ts ✭cA ❛♥❞

(T )

cB

❢♦r t❤❡ ❢r❛❣♠❡♥ts ❆ ❛♥❞ ❇✱ r❡s♣❡❝t✐✈❡❧②✮✳ ❚❤❡ ❝♦❧✉♠♥ ✐♥❞❡①

r✉♥s ♦✈❡r t❤❡ ♦❝❝✉♣✐❡❞ s♣✐♥♦rs ♦❢ t❤❡ ✐s♦❧❛t❡❞ ❢r❛❣♠❡♥ts r❛♥❣✐♥❣ ❢r♦♠

1 t♦ NAocc +NBocc ✳

❊✈❡r②

❝♦❧✉♠♥ ✐s ✐♥t❡r♥❛❧❧② ♦r❞❡r❡❞ ❢♦❧❧♦✇✐♥❣ t❤❡ ●✲s♣✐♥♦r ❜❛s✐s s❡t ❞❡✜♥✐t✐♦♥ ❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❇❊❘❚❍❆✱ s♦ t❤❛t t❤❡ ❝♦❡✣❝✐❡♥ts ❛r❡ ❞✐✈✐❞❡❞ ✐♥ t✇♦ ❣r♦✉♣s✿ t❤♦s❡ ♣❡rt❛✐♥✐♥❣ t♦ t❤❡ ❧❛r❣❡ ❝♦♠♣♦♥❡♥t ✭T

C+

= L✮

❛♥❞ t❤♦s❡ ♣❡rt❛✐♥✐♥❣ t♦ t❤❡ s♠❛❧❧ ❝♦♠♣♦♥❡♥t✱ ✭T

✐s ❝♦♥str✉❝t❡❞ ❜② ❛ss❡♠❜❧✐♥❣ t❤❡ s✉❜✲♠❛tr✐❝❡s

✷✶

(L)

(L)

(S)

(S)

cA , cB , cA , cB


= S ✮✳

■♥ ♦t❤❡r ✇♦r❞s✱

✇✐t❤ ❛♥ ❡q✉❛❧ ♥✉♠❜❡r


Page 22 of 39

♦❢ ♥✉❧❧ ♠❛tr✐❝❡s ❛s s❤♦✇♥ ✐♥ ❊q✳ ✶✻



         C+ =         

0

(S)

cA

0

■t ✐s ✇♦rt❤ ♥♦t✐♥❣ ❤❡r❡ t❤❛t t❤❡ ✇❤♦❧❡

(T ) ✭c+ ✱ ✇✐t❤

T = L, S ✮

♦❢ ❞✐♠❡♥s✐♦♥s

 0      (L)  cB   (L)    c +  = . (S)  c  + 0       (S) cB

(L)

cA

C+

✭✶✻✮

✐s r❡❝t❛♥❣✉❧❛r ❛♥❞ ❢♦r♠❡❞ ❜② t✇♦ s✉❜✲♠❛tr✐❝❡s

M × (NAocc + NBocc )✱

✇❤❡r❡

M✱

❛s ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞✱ ✐s

t❤❡ ♥✉♠❜❡r ♦❢ ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ❆s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ ✐♥ t❤❡ ◆❖❈❱ ❢♦r♠❛❧✐s♠ t❤❡ ❡❧❡❝tr♦♥ ❞❡♥s✐t② ❞✐✛❡r❡♥❝❡ ✜♥❡❞ ✇✐t❤ r❡s♣❡❝t t♦ ❛ r❡❢❡r❡♥❝❡ ❞❡♥s✐t②

ρ0

∆ρ′

✐s ❞❡✲

♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ♦❝❝✉♣✐❡❞ s♣✐♥♦rs ♦❢ ❆ ❛♥❞ ❇

♣r♦♣❡r❧② ♦rt❤♦❣♦♥❛❧✐③❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❛♥❞ r❡♥♦r♠❛❧✐③❡❞✳ ❲❡ ❛❞♦♣t t❤❡ s②♠♠❡tr✐❝ ♦rt❤♦♥♦r✲ ♠❛❧✐③❛t✐♦♥ ♣r♦❝❡❞✉r❡ ♣r♦♣♦s❡❞ ❜② ▲ö✇❞✐♥✱

✾✻✱✾✼

✇❤✐❝❤ ❣✉❛r❛♥t❡❡s t❤❛t t❤❡ ♦r❜✐t❛❧s ♦❢ t❤❡ ♥❡✇

♦rt❤♦♥♦r♠❛❧✐③❡❞ s❡t ❛r❡ t❤❡ ❝❧♦s❡st t♦ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡s ✐♥ ❛ ❧❡❛st✲sq✉❛r❡ s❡♥s❡✳ ♦rt❤♦♥♦r♠❛❧✐③❡❞ ♦r❜✐t❛❧s

c0(T )

(T )

O+

❚❤❡ ♥❡✇

❛r❡ ❞❡✜♥❡❞ ❛s

−1

c0(T ) = c+ O+ 2

✇❤❡r❡ t❤❡

✾✽

✭✶✼✮

✐s t❤❡ ♦r❜✐t❛❧ s♣✐♥♦r✲♦✈❡r❧❛♣ ♠❛tr✐①

(L)

(L)†

(S)†

(S)

O+ = C+† SC+ = c+ S (LL) c+ + c+ S (SS) c+

✷✷


✭✶✽✮

Page 23 of 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


❛♥❞

S

✐s t❤❡ ●r❛♠ ✭♦✈❡r❧❛♣✮ ♠❛tr✐① ❞❡✜♥❡❞ ♦✈❡r t❤❡ ❢✉❧❧ ●✲s♣✐♥♦r ❜❛s✐s s❡t ♦❢ t❤❡ ❛❞❞✉❝t



(LL)

 S S= ✇✐t❤ ❡❧❡♠❡♥ts t♦

(T T )

Sµν

❡q✉❛❧ t♦ t❤❡ ✐♥t❡❣r❛❧

0

0 S

(T )

(SS)

(T )

hΠµ |Πν i



✭✶✾✮

  ✭❚ ❂ ▲✱ ❙ ❛♥❞

µ

❛♥❞

ν

r❛♥❣❡ ❢r♦♠ ✶

M ✮✳ ◆♦✇ t❤❡ ❞❡♥s✐t② ♠❛tr✐① ✉s❡❞ t♦ ❞❡✜♥❡ t❤❡ ❝❤❡♠✐❝❛❧✲✈❛❧❡♥❝❡ ♦♣❡r❛t♦r ❝❛♥ ❜❡ ❡❛s✐❧② ♦❜✲

t❛✐♥❡❞ ✉s✐♥❣ ❊q✳ ✶✺✳ ❚❤❡ r❡❧❛t✐✈✐st✐❝ ◆❖❈❱s ✭Φk ✮



 Φk (r) = 

i

(L) µ=1 zµk

(L) Πµ (r)

(S) µ=1 zµk

(S) Πµ (r)

PM

PM



 ,

✭✷✵✮

❛r❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡ ❡q✉❛t✐♦♥ ✭✶✶✮✱ ❛♥❞ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② s♦❧✈✐♥❣ t❤❡ ❣❡♥❡r✲ ❛❧✐③❡❞ ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✱

V zk = vk S zk , ❢❡❛t✉r✐♥❣ t❤❡ r❡❧❛t✐✈✐st✐❝ ◆❖❈❱s ❛s ❡✐❣❡♥✈❡❝t♦rs ✇✐t❤

✭✷✶✮

V

❜❡✐♥❣ t❤❡ ♠❛tr✐① r❡♣r❡s❡♥t❛t✐♦♥ ♦❢

t❤❡ ❝❤❡♠✐❝❛❧✲✈❛❧❡♥❝❡ ♦♣❡r❛t♦r ❞❡✜♥❡❞ ❛s



(LL)

(LL)

(LL)

(LL)

(LS)

(SS)



∆D S S ∆D S  S  V = S ∆D S =  , (SL) (LL) (SS) (SS) (SS) (SS) S ∆D S S ∆D S ❛♥❞

∆D

✐s ❞❡✜♥❡❞ ❛s

✭✷✷✮

D − D0 ✳

❘❡❢❡r❡♥❝❡s ✭✶✮ ❙❝❤✇❡r❞t❢❡❣❡r✱ P✳ ■♥

❚❤❡ ❈❤❡♠✐❝❛❧ ❇♦♥❞ ❀ ❋r❡♥❦✐♥❣✱ ●✳✱ ❙❤❛✐❦✱ ❙✳✱ ❊❞s✳❀ ❲✐❧❡②✲❱❈❍

❱❡r❧❛❣ ●♠❜❍ ✫ ❈♦✳ ❑●❛❆✱ ✷✵✶✹❀ ♣♣ ✸✽✸✕✹✵✹✳

✷✸



Page 24 of 39

✭✷✮ ▼♦✱ ❨✳❀ ●❛♦✱ ❏✳❀ P❡②❡r✐♠❤♦✛✱ ❙✳ ❉✳ ❊♥❡r❣② ❉❡❝♦♠♣♦s✐t✐♦♥ ❆♥❛❧②s✐s ♦❢ ■♥t❡r♠♦❧❡❝✉❧❛r ■♥t❡r❛❝t✐♦♥s ✉s✐♥❣ ❛ ❇❧♦❝❦✲▲♦❝❛❧✐③❡❞ ❲❛✈❡ ❋✉♥❝t✐♦♥ ❆♣♣r♦❛❝❤✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✷✵✵✵✱

✶✶✷✱ ✺✺✸✵✳ ✭✸✮ ❑✐t❛✉r❛✱ ❑✳❀ ▼♦r♦❦✉♠❛✱ ❑✳ ❆ ◆❡✇ ❊♥❡r❣② ❉❡❝♦♠♣♦s✐t✐♦♥ ❙❝❤❡♠❡ ❢♦r ▼♦❧❡❝✉❧❛r ■♥✲ t❡r❛❝t✐♦♥s ❲✐t❤✐♥ t❤❡ ❍❛rtr❡❡✲❋♦❝❦ ❆♣♣r♦①✐♠❛t✐♦♥✳

■♥t✳ ❏✳ ◗✉❛♥t✉♠ ❈❤❡♠✳ ✶✾✼✻✱ ✶✵✱

✸✷✺✳

✭✹✮ ❏❡③✐♦rs❦✐✱ ❇✳❀ ▼♦s③②♥s❦✐✱ ❘✳❀ ❙③❛❧❡✇✐❝③✱ ❑✳ ▼♦❧❡❝✉❧❛r ❈❤❛r❣❡ ❉✐str✐❜✉t✐♦♥s ❛♥❞ ❈❤❡♠✲ ✐❝❛❧ ❇✐♥❞✐♥❣✳

❏✳ ❆♠✳ ❈❤❡♠✳ ❙♦❝✳ ✶✾✾✹✱ ✾✹✱ ✶✽✽✼✳

✭✺✮ ❩✐❡❣❧❡r✱ ❚✳❀ ❘❛✉❦✱ ❆✳ ❖♥ t❤❡ ❈❛❧❝✉❧❛t✐♦♥ ♦❢ ❇♦♥❞✐♥❣ ❊♥❡r❣✐❡s ❜② t❤❡ ❍❛rtr❡❡ ❋♦❝❦ ❙❧❛t❡r ▼❡t❤♦❞✳

❚❤❡♦r✳ ❈❤❡♠✳ ❆❝❝✳ ✶✾✼✼✱ ✹✻✱ ✶✳

✭✻✮ ❏❛❝♦❜s❡♥✱ ❍✳❀ ❩✐❡❣❧❡r✱ ❚✳ ◆♦♥❝❧❛ss✐❝❛❧ ❉♦✉❜❧❡ ❇♦♥❞s ✐♥ ❊t❤②❧❡♥❡ ❆♥❛❧♦❣s✿ ■♥✢✉❡♥❝❡ ♦❢ P❛✉❧✐ ❘❡♣✉❧s✐♦♥ ♦♥ ❚r❛♥s ❇❡♥❞✐♥❣ ❛♥❞

π ✲❜♦♥❞

❙tr❡♥❣t❤✳ ❆ ❉❡♥s✐t② ❋✉♥❝t✐♦♥❛❧ ❙t✉❞②✳

❏✳ ❆♠✳ ❈❤❡♠✳ ❙♦❝✳ ✶✾✾✹✱ ✶✶✻✱ ✸✻✻✼✕✸✻✼✾✳ ✭✼✮ ❇❛❞❡r✱ ❘✳ ❋✳ ❲✳❀ ❍❡♥♥❡❦❡r✱ ❲✳ ❍✳❀ ❈❛❞❡✱ P✳ ❊✳ ▼♦❧❡❝✉❧❛r ❈❤❛r❣❡ ❉✐str✐❜✉t✐♦♥s ❛♥❞ ❈❤❡♠✐❝❛❧ ❇✐♥❞✐♥❣✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✶✾✻✼✱ ✹✻✱ ✸✸✹✶✕✸✸✻✸✳

✭✽✮ ❇❛❞❡r✱ ❘✳ ❋✳ ❲✳❀ ❑❡❛✈❡♥②✱ ■✳❀ ❈❛❞❡✱ P✳ ❊✳ ▼♦❧❡❝✉❧❛r ❈❤❛r❣❡ ❉✐str✐❜✉t✐♦♥s ❛♥❞ ❈❤❡♠✐❝❛❧ ❇✐♥❞✐♥❣✳ ■■✳ ❋✐rst✲❘♦✇ ❉✐❛t♦♠✐❝ ❍②❞r✐❞❡s✱ ❆❍✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✶✾✻✼✱ ✹✼✱ ✸✸✽✶✕✸✹✵✷✳

✭✾✮ ❇❛❞❡r✱ ❘✳ ❋✳ ❆t♦♠s ✐♥ ▼♦❧❡❝✉❧❡s✿ ❆ ◗✉❛♥t✉♠ ❚❤❡♦r②✱ ■♥t❡r♥❛t✐♦♥❛❧ ❙❡r✐❡s ♦❢ ▼♦♥♦✲

❖①❢♦r❞ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❖①❢♦r❞ ❍❡♥❦❡❧♠❛♥ ●✱ ❆r♥❛❧❞ss♦♥ ❆✱ ❏ó♥ss♦♥ ❍ ✭✷✵✵✻✮ ✶✾✾✵✱ ✸✻✱ ✸✺✹✕✸✻✵✳

❣r❛♣❤s ♦♥ ❈❤❡♠✐str② ✷✷✳

✭✶✵✮ ❊st❡r❤✉②s❡♥✱ ❈✳❀ ❋r❡♥❦✐♥❣✱ ●✳ ❚❤❡ ◆❛t✉r❡ ♦❢ t❤❡ ❈❤❡♠✐❝❛❧ ❇♦♥❞ ❘❡✈✐s✐t❡❞✳ ❆♥ ❊♥❡r❣② P❛rt✐t✐♦♥✐♥❣ ❆♥❛❧②s✐s ♦❢ ❉✐❛t♦♠✐❝ ▼♦❧❡❝✉❧❡s ❊2 ✭❊❂◆✲❇✐✱ ❋✲■✮✱ ❈❖ ❛♥❞ ❇❋✳

❈❤❡♠✳ ❆❝❝✳ ✷✵✵✹✱ ✶✶✶✱ ✸✽✶✳

✷✹


❚❤❡♦r✳

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✭✶✶✮ ❈♦♣♣❡♥s✱ P✳✱ ❍❛❧❧✱ ▼✳ ❇✳✱ ❊❞s✳

❊❧❡❝tr♦♥ ❞✐str✐❜✉t✐♦♥s ❛♥❞ t❤❡ ❝❤❡♠✐❝❛❧ ❜♦♥❞ ❀ P❧❡♥✉♠

Pr❡ss✿ ◆❡✇ ❨♦r❦✱ ❯❙❆✱ ✶✾✽✷✳

✭✶✷✮ ❙❝❤✇❛r③✱ ❲✳ ❍✳ ❊✳❀ ❱❛❧t❛③❛♥♦s✱ P✳❀ ❘✉❡❞❡♥❜❡r❣✱ ❑✳ ❊❧❡❝tr♦♥ ❞✐✛❡r❡♥❝❡ ❞❡♥s✐t✐❡s ❛♥❞ ❝❤❡♠✐❝❛❧ ❜♦♥❞✐♥❣✳

❚❤❡♦r✳ ❈❤✐♠✳ ❆❝t❛ ✶✾✽✺✱ ✻✽✱ ✹✼✶✕✺✵✻✳

✭✶✸✮ ❇❡❝❦❡✱ ❆✳ ❉✳❀ ❊❞❣❡❝♦♠❜❡✱ ❑✳ ❊✳ ❆ ❙✐♠♣❧❡ ▼❡❛s✉r❡ ♦❢ ❊❧❡❝tr♦♥ ▲♦❝❛❧✐③❛t✐♦♥ ✐♥ ❆t♦♠✐❝ ❛♥❞ ▼♦❧❡❝✉❧❛r ❙②st❡♠s✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✶✾✾✵✱ ✾✷✱ ✺✸✾✼✳

✭✶✹✮ ▼ü❧❧✐❦❡♥✱ ❘✳ ❙✳ ❊❧❡❝tr♦♥✐❝ P♦♣✉❧❛t✐♦♥ ❆♥❛❧②s✐s ♦♥ ▲❈❆❖▼❖ ▼♦❧❡❝✉❧❛r ❲❛✈❡ ❋✉♥❝✲ t✐♦♥s✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✶✾✺✺✱ ✷✸✱ ✶✽✸✸✳

✭✶✺✮ ❉❛♣♣r✐❝❤✱ ❙✳❀ ❋r❡♥❦✐♥❣✱ ●✳ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ ❉♦♥♦r✲❆❝❝❡♣t♦r ■♥t❡r❛❝t✐♦♥s✿ ❉❡❝♦♠♣♦s✐t✐♦♥ ❆♥❛❧②s✐s ✉s✐♥❣ ❋r❛❣♠❡♥t ▼♦❧❡❝✉❧❛r ❖r❜✐t❛❧s✳

❆ ❈❤❛r❣❡

❏✳ P❤②s✳ ❈❤❡♠✳ ✶✾✾✺✱ ✾✾✱

✾✸✺✷✳

✭✶✻✮ ❋r❡♥❦✐♥❣✱ ●✳❀ ❋rö❤❧✐❝❤✱ ◆✳ ❚❤❡ ◆❛t✉r❡ ♦❢ t❤❡ ❇♦♥❞✐♥❣ ✐♥ ❚r❛♥s✐t✐♦♥✲▼❡t❛❧ ❈♦♠♣♦✉♥❞s✳

❈❤❡♠✳ ❘❡✈✳ ✷✵✵✵✱ ✶✵✵✱ ✼✶✼✳ ✭✶✼✮ ❇❡❧♣❛ss✐✱ ▲✳❀ ■♥❢❛♥t❡✱ ■✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❱✐ss❝❤❡r✱ ▲✳ ❚❤❡ ❈❤❡♠✐❝❛❧ ❇♦♥❞ ❜❡t✇❡❡♥ ❆✉✭■✮

+ ❛♥❞ t❤❡ ◆♦❜❧❡ ●❛s❡s✳ ❈♦♠♣❛r❛t✐✈❡ ❙t✉❞② ♦❢ ◆❣❆✉❋ ❛♥❞ ◆❣❆✉ ✭◆❣ ❂ ❆r✱ ❑r✱ ❳❡✮ ❜② ❉❡♥s✐t② ❋✉♥❝t✐♦♥❛❧ ❛♥❞ ❈♦✉♣❧❡❞ ❈❧✉st❡r ▼❡t❤♦❞s✳

❏✳ ❆♠✳ ❈❤❡♠✳ ❙♦❝✳ ✷✵✵✽✱ ✶✸✵✱

✶✵✹✽✕✶✵✻✵✳

✭✶✽✮ ❈❛♣♣❡❧❧❡tt✐✱ ❉✳❀ ❘♦♥❝❛✱ ❊✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ P✐r❛♥✐✱ ❋✳ ❘❡✈❡❛❧✐♥❣ ❈❤❛r❣❡✲ ❚r❛♥s❢❡r ❊✛❡❝ts ✐♥ ●❛s✲P❤❛s❡ ❲❛t❡r ❈❤❡♠✐str②✳

❆❝❝♦✉♥ts ❈❤❡♠✳ ❘❡s✳ ✷✵✶✷✱ ✹✺✱ ✶✺✼✶✕

✶✺✽✵✳

✭✶✾✮ ❇❡❧♣❛ss✐✱ ▲✳❀ ❘❡❝❛✱ ▼✳ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❘♦♥❝❛r❛tt✐✱ ▲✳ ❋✳❀ P✐r❛♥✐✱ ❋✳❀ ❈❛♣♣❡❧❧❡tt✐✱ ❉✳❀ ❋❛✉r❡✱ ❆✳❀ ❙❝r✐❜❛♥♦✱ ❨✳ ❈❤❛r❣❡✲❚r❛♥s❢❡r ❊♥❡r❣② ✐♥ t❤❡ ❲❛t❡r✲❍②❞r♦❣❡♥ ▼♦❧❡❝✉❧❛r ❆❣✲ ❣r❡❣❛t❡ ❘❡✈❡❛❧❡❞ ❜② ▼♦❧❡❝✉❧❛r✲❇❡❛♠ ❙❝❛tt❡r✐♥❣ ❊①♣❡r✐♠❡♥ts✱ ❈❤❛r❣❡ ❉✐s♣❧❛❝❡♠❡♥t ❆♥❛❧②s✐s✱ ❛♥❞ ❛❜ ■♥✐t✐♦ ❈❛❧❝✉❧❛t✐♦♥s✳

❏✳ ❆♠✳ ❈❤❡♠✳ ❙♦❝✳ ✷✵✶✵✱ ✶✸✷✱ ✶✸✵✹✻✕✶✸✵✺✽✳ ✷✺



Page 26 of 39

✭✷✵✮ ❘♦♥❝❛✱ ❊✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ❆ ◗✉❛♥t✐t❛t✐✈❡ ❱✐❡✇ ♦❢ ❈❤❛r❣❡ ❚r❛♥s❢❡r ✐♥ t❤❡ ❍②❞r♦❣❡♥ ❇♦♥❞✿ ❚❤❡ ❲❛t❡r ❉✐♠❡r ❈❛s❡✳

❈❤❡♠P❤②s❈❤❡♠ ✷✵✶✹✱ ✶✺✱ ✷✻✽✷✕✷✻✽✼✳

✭✷✶✮ ❈❛♣♣❡❧❧❡tt✐✱ ❉✳❀ ❇❛rt♦❝❝✐✱ ❆✳❀ ●r❛♥❞✐♥❡tt✐✱ ❋✳❀ ❋❛❧❝✐♥❡❧❧✐✱ ❙✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ P✐r❛♥✐✱ ❋✳ ❊①♣❡r✐♠❡♥t❛❧ ❊✈✐❞❡♥❝❡ ♦❢ ❈❤❡♠✐❝❛❧ ❈♦♠♣♦♥❡♥ts ✐♥ t❤❡ ❇♦♥❞✐♥❣ ♦❢ ❍❡❧✐✉♠ ❛♥❞ ◆❡♦♥ ✇✐t❤ ◆❡✉tr❛❧ ▼♦❧❡❝✉❧❡s✳

❈❤❡♠✳ ❊✉r✳ ❏✳ ✷✵✶✺✱ ✷✶✱ ✻✷✸✹✕✻✷✹✵✳

✭✷✷✮ ❈❛♣♣❡❧❧❡tt✐✱ ❉✳❀ ❈❛♥❞♦r✐✱ P✳❀ P✐r❛♥✐✱ ❋✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ◆❛t✉r❡ ❛♥❞ ❙t❛❜✐❧✐t② ♦❢ ❲❡❛❦ ❍❛❧♦❣❡♥ ❇♦♥❞s ✐♥ t❤❡ ●❛s P❤❛s❡✿ ▼♦❧❡❝✉❧❛r ❇❡❛♠ ❙❝❛tt❡r✐♥❣ ❊①♣❡r✐♠❡♥ts ❛♥❞ ❆❜ ■♥✐t✐♦ ❈❤❛r❣❡ ❉✐s♣❧❛❝❡♠❡♥t ❈❛❧❝✉❧❛t✐♦♥s✳

❈r②st❛❧ ●r♦✇t❤ ✫ ❉❡s✐❣♥ ✷✵✶✶✱ ✶✶✱

✹✷✼✾✕✹✷✽✸✳

✭✷✸✮ ❩✉❝❝❛❝❝✐❛✱ ❉✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❘♦❝❝❤✐❣✐❛♥✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ▼❛❝❝❤✐♦♥✐✱ ❆✳ ❆ P❤♦s♣❤✐♥❡ ●♦❧❞✭■✮✲❆❧❦②♥❡ ❈♦♠♣❧❡①✿ ❚✉♥✐♥❣ t❤❡ ▼❡t❛❧✲❆❧❦②♥❡ ❇♦♥❞ ❈❤❛r❛❝t❡r ❛♥❞ ❈♦✉♥t❡r✐♦♥ P♦s✐t✐♦♥ ❜② t❤❡ ❈❤♦✐❝❡ ♦❢ t❤❡ ❆♥❝✐❧❧❛r② ▲✐❣❛♥❞✳

■♥♦r❣✳ ❈❤❡♠✳ ✷✵✶✵✱ ✹✾✱ ✸✵✽✵✕✸✵✽✷✳

✭✷✹✮ ❩✉❝❝❛❝❝✐❛✱ ❉✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ▼❛❝❝❤✐♦♥✐✱ ❆✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ▲✐❣❛♥❞ ❊✛❡❝ts ♦♥ ❇♦♥❞✐♥❣ ❛♥❞ ■♦♥ P❛✐r✐♥❣ ✐♥ ❈❛t✐♦♥✐❝ ●♦❧❞✭■✮ ❈❛t❛❧②sts ❇❡❛r✐♥❣ ❯♥s❛t✉r❛t❡❞ ❍②❞r♦❝❛r❜♦♥s✳

❊✉r✳ ❏✳

■♥♦r❣✳ ❈❤❡♠✳ ✷✵✶✸✱ ✷✵✶✸✱ ✹✶✷✶✕✹✶✸✺✳ ✭✷✺✮ ●❛❣❣✐♦❧✐✱ ❈✳ ❆✳❀ ❈✐❛♥❝❛❧❡♦♥✐✱ ●✳❀ ❇✐❛s✐♦❧♦✱ ▲✳❀ ❇✐st♦♥✐✱ ●✳❀ ❩✉❝❝❛❝❝✐❛✱ ❉✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❇❡❧❛♥③♦♥✐✱ P✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ❆♥♦♠❛❧♦✉s ▲✐❣❛♥❞ ❊✛❡❝t ✐♥ ●♦❧❞✭■✮✲❈❛t❛❧②③❡❞ ■♥tr❛♠♦❧❡❝✲ ✉❧❛r ❍②❞r♦❛♠✐♥❛t✐♦♥ ♦❢ ❆❧❦②♥❡s✳

❈❤❡♠✳ ❈♦♠♠✉♥✳ ✷✵✶✺✱ ✺✶✱ ✺✾✾✵✕✺✾✾✸✳

✭✷✻✮ ❘♦♥❝❛✱ ❊✳❀ P❛st♦r❡✱ ▼✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❉❡ ❆♥❣❡❧✐s✱ ❋✳❀ ❆♥❣❡❧✐✱ ❈✳❀ ❈✐♠✐r❛❣❧✐❛✱ ❘✳❀ ❚❛r❛♥✲ t❡❧❧✐✱ ❋✳ ❈❤❛r❣❡✲❞✐s♣❧❛❝❡♠❡♥t ❛♥❛❧②s✐s ❢♦r ❡①❝✐t❡❞ st❛t❡s✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✷✵✶✹✱ ✶✹✵✱

✵✺✹✶✶✵✳

✭✷✼✮ ❙❛❧✈✐✱ ◆✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ❖♥ t❤❡ ❉❡✇❛r✲❈❤❛tt✲❉✉♥❝❛♥s♦♥ ▼♦❞❡❧ ❢♦r ❈❛t✲ ❛❧②t✐❝ ●♦❧❞✭■✮ ❈♦♠♣❧❡①❡s✳

❈❤❡♠✳ ❊✉r✳ ❏✳ ✷✵✶✵✱ ✶✻✱ ✼✷✸✶✕✼✷✹✵✳

✷✻


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✭✷✽✮ ▼❛r❝❤✐♦♥❡✱ ❉✳❀

❇❡❧♣❛ss✐✱ ▲✳❀

❇✐st♦♥✐✱ ●✳❀

▼❛❝❝❤✐♦♥✐✱ ❆✳❀

❚❛r❛♥t❡❧❧✐✱ ❋✳❀

❩✉❝❝❛❝✲

❝✐❛✱ ❉✳ ❚❤❡ ❈❤❡♠✐❝❛❧ ❇♦♥❞ ✐♥ ●♦❧❞✭■✮ ❈♦♠♣❧❡①❡s ✇✐t❤ ◆✲❍❡t❡r♦❝②❝❧✐❝ ❈❛r❜❡♥❡s✳

❖r❣❛♥♦♠❡t❛❧❧✐❝s ✷✵✶✹✱ ✸✸✱ ✹✷✵✵✕✹✷✵✽✳ ✭✷✾✮ ❈✐❛♥❝❛❧❡♦♥✐✱ ●✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ▼❛r❝❤❡tt✐✱ ❋✳ ❇❛❝❦✲❉♦♥❛t✐♦♥ ✐♥ ❍✐❣❤✲❱❛❧❡♥t ❞✵ ▼❡t❛❧ ❈♦♠♣❧❡①❡s✿ ❉♦❡s ■t ❊①✐st❄ ❚❤❡ ❈❛s❡ ♦❢ ◆❜❱✳

✭✸✵✮ ❉❡✇❛r✱ ▼✳ ❏✳ ❙✳ ❆ ❘❡✈✐❡✇ ♦❢

■♥♦r❣✳ ❈❤❡♠✳ ✷✵✶✼✱ ✶✶✷✻✻✕✶✶✷✼✹✳

π ❈♦♠♣❧❡① ❚❤❡♦r②✳ ❇✉❧❧✳ ❙♦❝✳ ❈❤✐♠✳ ❋r✳ ✶✾✺✶✱ ✶✽✱ ❈✼✶✕✼✾✳

✭✸✶✮ ❈❤❛tt✱ ❏✳❀ ❉✉♥❝❛♥s♦♥✱ ▲✳ ❆✳ ❖❧❡✜♥ ❝♦✲♦r❞✐♥❛t✐♦♥ ❈♦♠♣♦✉♥❞s✳ P❛rt ■■■✳ ■♥❢r❛✲❘❡❞ ❙♣❡❝✲ tr❛ ❛♥❞ ❙tr✉❝t✉r❡✿

❆tt❡♠♣t❡❞ Pr❡♣❛r❛t✐♦♥ ♦❢ ❆❝❡t②❧❡♥❡ ❈♦♠♣❧❡①❡s✳

❏✳ ❈❤❡♠✳ ❙♦❝✳

✶✾✺✸✱ ✷✾✸✾✕✷✾✹✷✳ ✭✸✷✮ ❇✐st♦♥✐✱ ●✳❀

❇❡❧♣❛ss✐✱ ▲✳❀

❚❛r❛♥t❡❧❧✐✱ ❋✳ ❉✐s❡♥t❛♥❣❧❡♠❡♥t ♦❢ ❉♦♥❛t✐♦♥ ❛♥❞ ❇❛❝❦✲

❉♦♥❛t✐♦♥ ❊✛❡❝ts ♦♥ ❊①♣❡r✐♠❡♥t❛❧ ❖❜s❡r✈❛❜❧❡s✿ ❆ ❈❛s❡ ❙t✉❞② ♦❢ ●♦❧❞✲❊t❤②♥❡ ❈♦♠✲ ♣❧❡①❡s✳

❆♥❣❡✇✳ ❈❤❡♠✳ ■♥t✳ ❊❞✳ ✷✵✶✸✱ ✺✷✱ ✶✶✺✾✾✕✶✶✻✵✷✳

✭✸✸✮ ❈✐❛♥❝❛❧❡♦♥✐✱ ●✳❀ ❇✐❛s✐♦❧♦✱ ▲✳❀ ❇✐st♦♥✐✱ ●✳❀ ▼❛❝❝❤✐♦♥✐✱ ❆✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❩✉❝❝❛❝❝✐❛✱ ❉✳❀ ❇❡❧♣❛ss✐✱ ▲✳ ❙❡❧❡❝t✐✈❡❧② ▼❡❛s✉r✐♥❣ ❙♣❡❝tr♦s❝♦♣②✳

π

❇❛❝❦✲❉♦♥❛t✐♦♥ ✐♥ ●♦❧❞ ✭■✮ ❈♦♠♣❧❡①❡s ❜② ◆▼❘

❈❤❡♠✳ ❊✉r✳ ❏✳ ✷✵✶✺✱ ✷✶✱ ✷✹✻✼✕✷✹✼✸✳

✭✸✹✮ ❆③③♦♣❛r❞✐✱ ❑✳ ▼✳❀ ❇✐st♦♥✐✱ ●✳❀ ❈✐❛♥❝❛❧❡♦♥✐✱ ●✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❩✉❝❝❛❝❝✐❛✱ ❉✳❀ ❇❡❧✲ ♣❛ss✐✱ ▲✳ ◗✉❛♥t✐t❛t✐✈❡ ❆ss❡ss♠❡♥t ♦❢ t❤❡ ❈❛r❜♦❝❛t✐♦♥✴❈❛r❜❡♥❡ ❈❤❛r❛❝t❡r ♦❢ t❤❡ ●♦❧❞✲ ❈❛r❜❡♥❡ ❇♦♥❞✳

❉❛❧t♦♥ ❚r❛♥s✳ ✷✵✶✺✱ ✹✹✱ ✶✸✾✾✾✕✶✹✵✵✼✳

✭✸✺✮ ❇✐st♦♥✐✱ ●✳❀ ❇❡❧❛♥③♦♥✐✱ P✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ♠♦❣❡♥❡♦✉s ❛♥❞ ❍❡t❡r♦❣❡♥❡♦✉s ●♦❧❞ ❈❛t❛❧②s✐s✳

π

❆❝t✐✈❛t✐♦♥ ♦❢ ❆❧❦②♥❡s ✐♥ ❍♦✲

❏✳ P❤②s✳ ❈❤❡♠✳ ❆ ✷✵✶✻✱ ✺✷✸✾✕✺✷✹✻✳

✭✸✻✮ ▼✐t♦r❛❥✱ ▼✳❀ ▼✐❝❤❛❧❛❦✱ ❆✳ ❉♦♥♦r✲❆❝❝❡♣t♦r Pr♦♣❡rt✐❡s ♦❢ ▲✐❣❛♥❞s ❢r♦♠ t❤❡ ◆❛t✉r❛❧ ❖r❜✐t❛❧s ❢♦r ❈❤❡♠✐❝❛❧ ❱❛❧❡♥❝❡✳

❖r❣❛♥♦♠❡t❛❧❧✐❝s ✷✵✵✼✱ ✷✻✱ ✻✺✼✻✕✻✺✽✵✳

✭✸✼✮ ▼✐t♦r❛❥✱ ▼✳ P✳❀ ▼✐❝❤❛❧❛❦✱ ❆✳❀ ❩✐❡❣❧❡r✱ ❚✳ ❆ ❈♦♠❜✐♥❡❞ ❈❤❛r❣❡ ❛♥❞ ❊♥❡r❣② ❉❡❝♦♠♣♦s✐✲ t✐♦♥ ❙❝❤❡♠❡ ❢♦r ❇♦♥❞ ❆♥❛❧②s✐s✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳ ✷✵✵✾✱ ✺✱ ✾✻✷✕✾✼✺✳ ✷✼



Page 28 of 39

✭✸✽✮ ◆❛❧❡✇❛❥s❦✐✱ ❘✳ ❋✳❀ ▼r♦③❡❦✱ ❏✳❀ ▼✐❝❤❛❧❛❦✱ ❆✳ ❚✇♦✲❊❧❡❝tr♦♥ ❱❛❧❡♥❝❡ ■♥❞✐❝❡s ❢r♦♠ t❤❡ ❑♦❤♥✲❙❤❛♠ ❖r❜✐t❛❧s✳

■♥t✳ ❏✳ ◗✉❛♥t✉♠ ❈❤❡♠✳ ✶✾✾✼✱ ✻✶✱ ✺✽✾✕✻✵✶✳

✭✸✾✮ ❇✐st♦♥✐✱ ●✳❀ ❘❛♠♣✐♥♦✱ ❙✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❇❡❧♣❛ss✐✱ ▲✳ ❈❤❛r❣❡✲❉✐s♣❧❛❝❡♠❡♥t ❆♥❛❧②s✐s ✈✐❛ ◆❛t✉r❛❧ ❖r❜✐t❛❧s ❢♦r ❈❤❡♠✐❝❛❧ ❱❛❧❡♥❝❡✿ ❈❤❛r❣❡ ❚r❛♥s❢❡r ❊✛❡❝ts ✐♥ ❈♦♦r❞✐♥❛t✐♦♥ ❈❤❡♠✐str②✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✷✵✶✺✱ ✶✹✷✱ ✵✽✹✶✶✷ ✭✾✮✳

✭✹✵✮ ❇✐st♦♥✐✱ ●✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ❆❞✈❛♥❝❡s ✐♥ ❈❤❛r❣❡ ❉✐s♣❧❛❝❡♠❡♥t ❆♥❛❧②s✐s✳

❏✳

❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳ ✷✵✶✻✱ ✶✷✱ ✶✷✸✻✕✶✷✹✹✳ ✭✹✶✮ ▼❛r❝❤✐♦♥❡✱ ❉✳❀ ■③q✉✐❡r❞♦✱ ▼✳ ❆✳❀ ❇✐st♦♥✐✱ ●✳❀ ❍❛✈❡♥✐t❤✱ ❘✳ ❲✳❀ ▼❛❝❝❤✐♦♥✐✱ ❆✳❀ ❩✉❝❝❛❝✲ ❝✐❛✱ ❉✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❇❡❧♣❛ss✐✱ ▲✳ ❈❛♥ ❙❡❧❡❝t✐✈❡❧② Pr♦❜❡

σ

13

❈ ◆▼❘ ❙♣❡❝tr♦s❝♦♣② ♦❢ ◆✲❍❡t❡r♦❝②❝❧✐❝ ❈❛r❜❡♥❡s

❉♦♥❛t✐♦♥ ✐♥ ●♦❧❞ ✭■✮ ❈♦♠♣❧❡①❡s✳

❈❤❡♠✳ ❊✉r✳ ❏✳ ✷✵✶✼✱ ✷✸✱

✷✼✷✷✕✷✼✷✽✳

✭✹✷✮ ●❛❣❣✐♦❧✐✱ ❈✳ ❆✳❀ ❇✐st♦♥✐✱ ●✳❀ ❈✐❛♥❝❛❧❡♦♥✐✱ ●✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❇❡❧❛♥③♦♥✐✱ P✳ ▼♦❞✉❧❛t✐♥❣ t❤❡ ❇♦♥❞✐♥❣ Pr♦♣❡rt✐❡s ♦❢ ◆✲❍❡t❡r♦❝②❝❧✐❝ ❈❛r❜❡♥❡s ✭◆❍❈s✮✿ ❆ ❙②st❡♠❛t✐❝ ❈❤❛r❣❡✲❉✐s♣❧❛❝❡♠❡♥t ❆♥❛❧②s✐s✳

❈❤❡♠✳ ❊✉r✳ ❏✳ ✷✵✶✼✱ ✷✸✱ ✼✺✺✽✕✼✺✻✾✳

✭✹✸✮ ❋✉s❡✱ ▼✳❀ ❘✐♠♦❧❞✐✱ ■✳❀ ❈❡s❛r♦tt✐✱ ❊✳❀ ❘❛♠♣✐♥♦✱ ❙✳❀ ❇❛r♦♥❡✱ ❱✳ ❖♥ t❤❡ ❘❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❈❛r❜♦♥②❧ ❙tr❡t❝❤✐♥❣ ❋r❡q✉❡♥❝✐❡s ❛♥❞ t❤❡ ❉♦♥♦r P♦✇❡r ♦❢ ❈❤❡❧❛t✐♥❣ ❉✐♣❤♦s♣❤✐♥❡s ✐♥ ◆✐❝❦❡❧ ❉✐❝❛r❜♦♥②❧ ❈♦♠♣❧❡①❡s✳

✭✹✹✮ ▲✐✱ ❏✳❀

❙❝❤r❡❝❦❡♥❜❛❝❤✱ ●✳❀

❙tr❡♥❣t❤s ✐♥

π ✲❈♦♠♣❧❡①❡s✿

P❤②s✳ ❈❤❡♠✳ ❈❤❡♠✳ P❤②s✳ ✷✵✶✼✱ ✶✾✱ ✾✵✷✽✕✾✵✸✽✳

❩✐❡❣❧❡r✱ ❚✳ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ♦♥ ▼❡t❛❧✲▲✐❣❛♥❞ ❇♦♥❞ ◗✉❛s✐✲❘❡❧❛t✐✈✐st✐❝ ❉❡♥s✐t② ❋✉♥❝t✐♦♥❛❧ ❙t✉❞② ♦❢ ▼ ✭P❍✸✮

✷❳✷ ✭▼❂ ◆✐✱ P❞✱ Pt❀ ❳✷❂ ❖✷✱ ❈✷❍✷✱ ❈✷❍✹✮ ❛♥❞ ▼ ✭❈❖✮ ✹ ✭❈✷❍✹✮✭▼❂ ❋❡✱ ❘✉✱ ❖s✮✳

■♥♦r❣✳ ❈❤❡♠✳ ✶✾✾✺✱ ✸✹✱ ✸✷✹✺✕✸✷✺✷✳ ✭✹✺✮ ❈❤❛♠♣✐♦♥✱ ❏✳❀ ❙❡②❞♦✉✱ ▼✳❀ ❙❛❜❛t✐❡✲●♦❣♦✈❛✱ ❆✳❀ ❘❡♥❛✉❧t✱ ❊✳❀ ▼♦♥t❛✈♦♥✱ ●✳❀ ●❛❧❧❛♥❞✱ ◆✳ ❆ss❡ss♠❡♥t ♦❢ ❛♥ ❊✛❡❝t✐✈❡ ◗✉❛s✐r❡❧❛t✐✈✐st✐❝ ▼❡t❤♦❞♦❧♦❣② ❉❡s✐❣♥❡❞ t♦ ❙t✉❞② ❆st❛t✐♥❡ ❈❤❡♠✐str② ✐♥ ❆q✉❡♦✉s ❙♦❧✉t✐♦♥✳

P❤②s✳ ❈❤❡♠✳ ❈❤❡♠✳ P❤②s✳ ✷✵✶✶✱ ✶✸✱ ✶✹✾✽✹✕✶✹✾✾✷✳ ✷✽


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✭✹✻✮ P②②❦❦♦✱ P✳ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ✐♥ ❙tr✉❝t✉r❛❧ ❈❤❡♠✐str②✳

❈❤❡♠✳ ❘❡✈✳ ✶✾✽✽✱ ✽✽✱ ✺✻✸✕✺✾✹✳

✭✹✼✮ ●♦r✐♥✱ ❉✳ ❏✳❀ ❚♦st❡✱ ❋✳ ❉✳ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ✐♥ ❍♦♠♦❣❡♥❡♦✉s ●♦❧❞ ❈❛t❛❧②s✐s✳

◆❛t✉r❡

✷✵✵✼✱ ✹✹✻✱ ✸✾✺✳ ✭✹✽✮ ❙❝❤✇❛r③✱ ❍✳ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ✐♥ ●❛s✲P❤❛s❡ ■♦♥ ❈❤❡♠✐str②✿ ❱✐❡✇✳

❆♥ ❊①♣❡r✐♠❡♥t❛❧✐st✬s

❆♥❣❡✇✳ ❈❤❡♠✳ ■♥t✳ ❊❞✳ ✷✵✵✸✱ ✹✷✱ ✹✹✹✷✕✹✹✺✹✳

✭✹✾✮ ❙❝❤✇❡r❞t❢❡❣❡r✱ P✳❀ ❍❡❛t❤✱ ●✳ ❆✳❀ ❉♦❧❣✱ ▼✳❀ ❇❡♥♥❡tt✱ ▼✳ ❆✳ ▲♦✇ ❱❛❧❡♥❝✐❡s ❛♥❞ P❡r✐♦❞✐❝ ❚r❡♥❞s ✐♥ ❍❡❛✈② ❊❧❡♠❡♥t ❈❤❡♠✐str②✳ ❆ ❚❤❡♦r❡t✐❝❛❧ ❙t✉❞② ♦❢ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ❛♥❞ ❊❧❡❝tr♦♥ ❈♦rr❡❧❛t✐♦♥ ❊✛❡❝ts ✐♥ ●r♦✉♣ ✶✸ ❛♥❞ P❡r✐♦❞ ✻ ❍②❞r✐❞❡s ❛♥❞ ❍❛❧✐❞❡s✳

❏✳ ❆♠✳

❈❤❡♠✳ ❙♦❝✳ ✶✾✾✷✱ ✶✶✹✱ ✼✺✶✽✕✼✺✷✼✳ ✭✺✵✮ ●❛❣❣✐♦❧✐✱ ❈✳ ❆✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❩✉❝❝❛❝❝✐❛✱ ❉✳❀ ❍❛r✈❡②✱ ❏✳ ◆✳❀ ❇❡❧❛♥③♦♥✐✱ P✳ ❉✐♦①②❣❡♥ ■♥s❡rt✐♦♥ ✐♥t♦ t❤❡ ●♦❧❞✭■✮✲❍②❞r✐❞❡ ❇♦♥❞✿ ❙♣✐♥ ❖r❜✐t ❈♦✉♣❧✐♥❣ ❊✛❡❝ts ✐♥ t❤❡ ❙♣♦t❧✐❣❤t ❢♦r ❖①✐❞❛t✐✈❡ ❆❞❞✐t✐♦♥✳

❈❤❡♠✳ ❙❝✐✳ ✷✵✶✻✱ ✼✱ ✼✵✸✹✕✼✵✸✾✳

✭✺✶✮ ❉❡♠✐ss✐❡✱ ❚✳ ❇✳❀ ●❛r❛❜❛t♦✱ ❇✳ ❉✳❀ ❘✉✉❞✱ ❑✳❀ ❑♦③❧♦✇s❦✐✱ P✳ ▼✳ ▼❡r❝✉r② ▼❡t❤②❧❛t✐♦♥ ❜② ❈♦❜❛❧t ❈♦rr✐♥♦✐❞s✿ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ❉✐❝t❛t❡ t❤❡ ❘❡❛❝t✐♦♥ ▼❡❝❤❛♥✐s♠✳

❆♥❣❡✇✳

❈❤❡♠✳ ■♥t✳ ❊❞✳ ✷✵✶✻✱ ✺✺✱ ✶✶✺✵✸✕✶✶✺✵✻✳ ✭✺✷✮ ❇❡❧♣❛ss✐✱ ▲✳❀ ❙t♦r❝❤✐✱ ▲✳❀ ◗✉✐♥❡②✱ ❍✳ ▼✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ❘❡❝❡♥t ❆❞✈❛♥❝❡s ❛♥❞ P❡rs♣❡❝t✐✈❡s ✐♥ ❋♦✉r✲❈♦♠♣♦♥❡♥t ❉✐r❛❝✲❑♦❤♥✲❙❤❛♠ ❈❛❧❝✉❧❛t✐♦♥s✳

P❤②s✳ ❈❤❡♠✳ ❈❤❡♠✳ P❤②s✳ ✷✵✶✶✱

✶✸✱ ✶✷✸✻✽✕✶✷✸✾✹✳ ✭✺✸✮ ❙t♦r❝❤✐✱ ▲✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❙❣❛♠❡❧❧♦tt✐✱ ❆✳❀ ◗✉✐♥❡②✱ ❍✳ ▼✳ ❆♥ ❊✣❝✐❡♥t P❛r❛❧❧❡❧ ❆❧❧✲❊❧❡❝tr♦♥ ❋♦✉r✲❈♦♠♣♦♥❡♥t ❉✐r❛❝✲❑♦❤♥✲❙❤❛♠ Pr♦❣r❛♠ ❯s✐♥❣ ❛ ❉✐str✐❜✉t❡❞ ▼❛tr✐① ❆♣♣r♦❛❝❤✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳ ✷✵✶✵✱ ✻✱ ✸✽✹✕✸✾✹✳

✭✺✹✮ ❙t♦r❝❤✐✱ ▲✳❀ ❘❛♠♣✐♥♦✱ ❙✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ◗✉✐♥❡②✱ ❍✳ ▼✳ ❊✣❝✐❡♥t P❛r❛❧❧❡❧ ❆❧❧✲❊❧❡❝tr♦♥ ❋♦✉r✲❈♦♠♣♦♥❡♥t ❉✐r❛❝✲❑♦❤♥✲❙❤❛♠ Pr♦❣r❛♠ ❯s✐♥❣ ❛ ❉✐str✐❜✉t❡❞ ▼❛tr✐① ❆♣♣r♦❛❝❤ ■■✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳ ✷✵✶✸✱ ✾✱ ✺✸✺✻✕✺✸✻✹✳ ✷✾



Page 30 of 39

✭✺✺✮ ❘❛♠♣✐♥♦✱ ❙✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❙t♦r❝❤✐✱ ▲✳ ❋✉❧❧ P❛r❛❧❧❡❧ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛♥ ❆❧❧✲❊❧❡❝tr♦♥ ❋♦✉r✲❈♦♠♣♦♥❡♥t ❉✐r❛❝✲❑♦❤♥✲❙❤❛♠ ♣r♦❣r❛♠✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳

✷✵✶✹✱ ✶✵✱ ✸✼✻✻✕✸✼✼✻✳ ✭✺✻✮ ▼✐t♦r❛❥ ▼✳✱ ▼✳ ❆✳ ◆❛t✉r❛❧ ❖r❜✐t❛❧s ❢♦r ❈❤❡♠✐❝❛❧ ❱❛❧❡♥❝❡ ❛s ❉❡s❝r✐♣t♦rs ♦❢ ❈❤❡♠✐❝❛❧ ❇♦♥❞✐♥❣ ✐♥ ❚r❛♥s✐t✐♦♥ ▼❡t❛❧ ❈♦♠♣❧❡①❡s✳

❏✳ ▼♦❧✳ ▼♦❞❡❧✳ ✷✵✵✼✱ ✶✸✱ ✸✹✼✕✸✺✺✳

✭✺✼✮ ▼✐❝❤❛❧❛❦✱ ❆✳❀ ▼✐t♦r❛❥✱ ▼✳❀ ❩✐❡❣❧❡r✱ ❚✳ ❇♦♥❞ ❖r❜✐t❛❧s ❢r♦♠ ❈❤❡♠✐❝❛❧ ❱❛❧❡♥❝❡ ❚❤❡♦r②✳

❏✳ P❤②s✳ ❈❤❡♠✳ ❆ ✷✵✵✽✱ ✶✶✷✱ ✶✾✸✸✕✶✾✸✾✳ ✭✺✽✮ ◆❛❧❡✇❛❥s❦✐✱ ❘✳ ❋✳❀ ❑öst❡r✱ ❆✳ ▼✳❀ ❏✉❣✱ ❑✳ ❈❤❡♠✐❝❛❧ ❱❛❧❡♥❝❡ ❢r♦♠ t❤❡ ❚✇♦✲P❛rt✐❝❧❡ ❉❡♥s✐t② ▼❛tr✐①✳

❚❤❡♦r✳ ❈❤❡♠✳ ❆❝❝✳ ✶✾✾✸✱ ✽✺✱ ✹✻✸✕✹✽✹✳

✭✺✾✮ ◆❛❧❡✇❛❥s❦✐✱ ❘✳ ❋✳❀ ▼r♦③❡❦✱ ❏✳ ▼♦❞✐✜❡❞ ❱❛❧❡♥❝❡ ■♥❞✐❝❡s ❢r♦♠ t❤❡ ❚✇♦✲P❛rt✐❝❧❡ ❉❡♥s✐t② ▼❛tr✐①✳

■♥t✳ ❏✳ ◗✉❛♥t✉♠ ❈❤❡♠✳ ✶✾✾✹✱ ✺✶✱ ✶✽✼✕✷✵✵✳

✭✻✵✮ ❘❛❞♦➠✱ ▼✳ ❖♥ t❤❡ Pr♦♣❡rt✐❡s ♦❢ ◆❛t✉r❛❧ ❖r❜✐t❛❧s ❢♦r ❈❤❡♠✐❝❛❧ ❱❛❧❡♥❝❡✳

❚❤❡♦r✳ ❈❤❡♠✳

❆❝❝✳ ✷✵✵✽✱ ✶✷✵✱ ✸✸✼✕✸✸✾✳ ✭✻✶✮ ▼❛r✐✉s③ P✳ ▼✐t♦r❛❥✱ ❚✳ ❩✳✱ ❆rt✉r ▼✐❝❤❛❧❛❦ ❆ ❈♦♠❜✐♥❡❞ ❈❤❛r❣❡ ❛♥❞ ❊♥❡r❣② ❉❡❝♦♠♣♦✲ s✐t✐♦♥ ❙❝❤❡♠❡ ❢♦r ❇♦♥❞ ❆♥❛❧②s✐s✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳ ✷✵✵✽✱ ✹✱ ✾✻✷✕✾✼✺✳

✭✻✷✮ P❡♥❣✱ ❉✳❀ ▲✐✉✱ ❲✳❀ ❳✐❛♦✱ ❨✳❀ ❈❤❡♥❣✱ ▲✳ ▼❛❦✐♥❣ ❢♦✉r✲ ❛♥❞ t✇♦✲❝♦♠♣♦♥❡♥t r❡❧❛t✐✈✐s✲ t✐❝ ❞❡♥s✐t② ❢✉♥❝t✐♦♥❛❧ ♠❡t❤♦❞s ❢✉❧❧② ❡q✉✐✈❛❧❡♥t ❜❛s❡❞ ♦♥ t❤❡ ✐❞❡❛ ♦❢ ✏❢r♦♠ ❛t♦♠s t♦ ♠♦❧❡❝✉❧❡✑✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✷✵✵✼✱ ✶✷✼✱ ✶✵✹✶✵✻✳

✭✻✸✮ ▲✐✉✱ ❲✳❀ ▲✐♥❞❣r❡♥✱ ■✳ ●♦✐♥❣ ❜❡②♦♥❞ ✏♥♦✲♣❛✐r r❡❧❛t✐✈✐st✐❝ q✉❛♥t✉♠ ❝❤❡♠✐str②✑✳

❏✳ ❈❤❡♠✳

P❤②s✳ ✷✵✶✸✱ ✶✸✾✱ ✵✶✹✶✵✽✳ ✭✻✹✮ ❇❛❡r❡♥❞s✱ ❊✳ ❏✳❀ ❩✐❡❣❧❡r✱ ❚✳❀ ❆t❦✐♥s✱ ❆✳ ❏✳❀ ❆✉ts❝❤❜❛❝❤✱ ❏✳❀ ❇❛s❤❢♦r❞✱ ❉✳❀ ❇❛s❡❣❣✐♦✱ ❖✳❀ ❇ér❝❡s✱ ❆✳❀ ❇✐❝❦❡❧❤❛✉♣t✱ ❋✳ ▼✳❀ ❇♦✱ ❈✳❀ ❇♦❡rr✐t❣❡r✱ P✳ ▼✳❀ ❈❛✈❛❧❧♦✱ ▲✳❀ ❉❛✉❧✱ ❈✳❀ ❈❤♦♥❣✱ ❉✳ P✳❀ ❈❤✉❧❤❛✐✱ ❉✳ ❱✳❀ ❉❡♥❣✱ ▲✳❀ ❉✐❝❦s♦♥✱ ❘✳ ▼✳❀ ❉✐❡t❡r✐❝❤✱ ❏✳ ▼✳❀ ❊❧❧✐s✱ ❉✳ ❊✳❀ ✈❛♥ ❋❛❛ss❡♥✱ ▼✳❀ ●❤②s❡❧s✱ ❆✳❀ ●✐❛♠♠♦♥❛✱ ❆✳❀ ✈❛♥ ●✐s❜❡r❣❡♥✱ ❙✳ ❏✳ ❆✳❀ ●♦❡③✱ ❆✳❀

✸✵


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●öt③✱ ❆✳ ❲✳❀ ●✉s❛r♦✈✱ ❙✳❀ ❍❛rr✐s✱ ❋✳ ❊✳❀ ✈❛♥ ❞❡♥ ❍♦❡❦✱ P✳❀ ❍✉✱ ❩✳❀ ❏❛❝♦❜✱ ❈✳ ❘✳❀ ❏❛❝♦❜s❡♥✱ ❍✳❀ ❏❡♥s❡♥✱ ▲✳❀ ❏♦✉❜❡rt✱ ▲✳❀ ❑❛♠✐♥s❦✐✱ ❏✳ ❲✳❀ ✈❛♥ ❑❡ss❡❧✱ ●✳❀ ❑ö♥✐❣✱ ❈✳❀ ❑♦♦tstr❛✱ ❋✳❀ ❑♦✈❛❧❡♥❦♦✱ ❆✳❀ ❑r②❦✉♥♦✈✱ ▼✳❀ ✈❛♥ ▲❡♥t❤❡✱ ❊✳❀ ▼❝❈♦r♠❛❝❦✱ ❉✳ ❆✳❀ ▼✐❝❤❛✲ ❧❛❦✱ ❆✳❀ ▼✐t♦r❛❥✱ ▼✳❀ ▼♦rt♦♥✱ ❙✳ ▼✳❀ ◆❡✉❣❡❜❛✉❡r✱ ❏✳❀ ◆✐❝✉✱ ❱✳ P✳❀ ◆♦♦❞❧❡♠❛♥✱ ▲✳❀ ❖s✐♥❣❛✱ ❱✳ P✳❀ P❛t❝❤❦♦✈s❦✐✐✱ ❙✳❀ P❛✈❛♥❡❧❧♦✱ ▼✳❀ P❡❡♣❧❡s✱ ❈✳ ❆✳❀ P❤✐❧✐♣s❡♥✱ P✳ ❍✳ ❚✳❀ P♦st✱ ❉✳❀ P②❡✱ ❈✳ ❈✳❀ ❘❛♠❛♥❛♥t♦❛♥✐♥❛✱ ❍✳❀ ❘❛♠♦s✱ P✳❀ ❘❛✈❡♥❡❦✱ ❲✳❀ ❘♦❞rí❣✉❡③✱ ❏✳ ■✳❀ ❘♦s✱ P✳❀ ❘ü❣❡r✱ ❘✳❀ ❙❝❤✐♣♣❡r✱ P✳ ❘✳ ❚✳❀ ❙❝❤❧ü♥s✱ ❉✳❀ ✈❛♥ ❙❝❤♦♦t✱ ❍✳❀ ❙❝❤r❡❝❦❡♥❜❛❝❤✱ ●✳❀ ❙❡❧❞❡♥t❤✉✐s✱ ❏✳ ❙✳❀ ❙❡t❤✱ ▼✳❀ ❙♥✐❥❞❡rs✱ ❏✳ ●✳❀ ❙♦❧à✱ ▼✳❀ ▼✳✱ ❙✳❀ ❙✇❛rt✱ ▼✳❀ ❙✇❡r❤♦♥❡✱ ❉✳❀ t❡ ❱❡❧❞❡✱ ●✳❀ ❚♦❣♥❡tt✐✱ ❱✳❀ ❱❡r♥♦♦✐❥s✱ P✳❀ ❱❡rs❧✉✐s✱ ▲✳❀ ❱✐ss❝❤❡r✱ ▲✳❀ ❱✐ss❡r✱ ❖✳❀ ❲❛♥❣✱ ❋✳❀ ❲❡s♦❧♦✇s❦✐✱ ❚✳ ❆✳❀ ✈❛♥ ❲❡③❡♥❜❡❡❦✱ ❊✳ ▼✳❀ ❲✐❡s❡♥❡❦❦❡r✱ ●✳❀ ❲♦❧✛✱ ❙✳ ❑✳❀ ❲♦♦✱ ❚✳ ❑✳❀ ❨❛❦♦✈❧❡✈✱ ❆✳ ▲✳ ❆❉❋✷✵✶✼✱ ❙❈▼✱ ❚❤❡♦r❡t✐❝❛❧ ❈❤❡♠✐str②✱ ❱r✐❥❡ ❯♥✐✈❡rs✐t❡✐t✱ ❆♠st❡r❞❛♠✱ ❚❤❡ ◆❡t❤❡r❧❛♥❞s✱ ❤tt♣s✿✴✴✇✇✇✳s❝♠✳❝♦♠✳

✭✻✺✮ ❙❝❤✇❛r③✱ ❲✳ ❍✳ ❊✳❀ ❘✉❡❞❡♥❜❡r❣✱ ❑✳❀ ▼❡♥s❝❤✐♥❣✱ ▲✳ ❈❤❡♠✐❝❛❧ ❉❡❢♦r♠❛t✐♦♥ ❉❡♥s✐t✐❡s✳ ✶✳ Pr✐♥❝✐♣❧❡s ❛♥❞ ❋♦r♠✉❧❛t✐♦♥ ♦❢ ◗✉❛♥t✐t❛t✐✈❡ ❉❡t❡r♠✐♥❛t✐♦♥✳

❏✳ ❆♠✳ ❈❤❡♠✳ ❙♦❝✳ ✶✾✽✾✱

✶✶✶✱ ✻✾✷✻✕✻✾✸✸✳ ✭✻✻✮ ▼❡♥s❝❤✐♥❣✱ ▲✳❀ ❱♦♥ ◆✐❡ss❡♥✱ ❲✳❀ ❱❛❧t❛③❛♥♦s✱ P✳❀ ❘✉❡❞❡♥❜❡r❣✱ ❑✳❀ ❙❝❤✇❛r③✱ ❲✳ ❍✳ ❊✳ ❈❤❡♠✐❝❛❧ ❉❡❢♦r♠❛t✐♦♥ ❉❡♥s✐t✐❡s✳ ✷✳ ❙♠❛❧❧ ▼♦❧❡❝✉❧❡s✳

❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❈❤❡♠✐❝❛❧

❙♦❝✐❡t② ✶✾✽✾✱ ✶✶✶✱ ✻✾✸✸✕✻✾✹✶✳ ✭✻✼✮ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳❀ ❙❣❛♠❡❧❧♦tt✐✱ ❆✳❀ ◗✉✐♥❡②✱ ❍✳ ▼✳ ❊❧❡❝tr♦♥ ❉❡♥s✐t② ❋✐tt✐♥❣ ❢♦r t❤❡ ❈♦✉❧♦♠❜ Pr♦❜❧❡♠ ✐♥ ❘❡❧❛t✐✈✐st✐❝ ❉❡♥s✐t②✲❋✉♥❝t✐♦♥❛❧ ❚❤❡♦r②✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✷✵✵✻✱

✶✷✹✱ ✶✷✹✶✵✹ ✭✽✮✳ ✭✻✽✮ ◗✉✐♥❡②✱ ❍✳ ▼✳❀ ❇❡❧❛♥③♦♥✐✱ P✳ ❘❡❧❛t✐✈✐st✐❝ ❞❡♥s✐t② ❢✉♥❝t✐♦♥❛❧ t❤❡♦r② ✉s✐♥❣ ●❛✉ss✐❛♥ ❜❛s✐s s❡ts✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✷✵✵✷✱ ✶✶✼✱ ✺✺✺✵✕✺✺✻✸✳

✭✻✾✮ ❉②❛❧❧✱ ❑✳ ●✳ ❘❡❧❛t✐✈✐st✐❝ ❉♦✉❜❧❡✲❩❡t❛✱ ❚r✐♣❧❡✲❩❡t❛✱ ❛♥❞ ◗✉❛❞r✉♣❧❡✲❩❡t❛ ❇❛s✐s ❙❡ts ❢♦r

✸✶



t❤❡ ✺d ❊❧❡♠❡♥ts ❍❢✲❍❣✳

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❚❤❡♦r✳ ❈❤❡♠✳ ❆❝❝✳ ✷✵✵✹✱ ✶✶✷✱ ✹✵✸✕✹✵✾✱ ❇❛s✐s s❡ts ❛r❡ ❛✈❛✐❧❛❜❧❡

❢r♦♠ t❤❡ ❉✐r❛❝ ✇❡❜ s✐t❡✱ ❤tt♣✿✴✴❞✐r❛❝✳❝❤❡♠✳s❞✉✳❞❦✳

✭✼✵✮ ❉②❛❧❧✱ ❑✳ ●✳❀ ●♦♠❡s✱ ❆✳ ❙✳ ❘❡✈✐s❡❞ ❘❡❧❛t✐✈✐st✐❝ ❇❛s✐s ❙❡ts ❢♦r t❤❡ ✺d ❊❧❡♠❡♥ts ❍❢✲❍❣✳

❚❤❡♦r✳ ❈❤❡♠✳ ❆❝❝✳ ✷✵✶✵✱ ✶✷✺✱ ✾✼✕✶✵✵✱ ❇❛s✐s s❡ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ t❤❡ ❉✐r❛❝ ✇❡❜ s✐t❡✱ ❤tt♣✿✴✴❞✐r❛❝✳❝❤❡♠✳s❞✉✳❞❦✳

✭✼✶✮ ❉②❛❧❧✱ ❑✳ ●✳ ❘❡❧❛t✐✈✐st✐❝ ❛♥❞ ◆♦♥r❡❧❛t✐✈✐st✐❝ ❋✐♥✐t❡ ◆✉❝❧❡✉s ❖♣t✐♠✐③❡❞ ❚r✐♣❧❡✲❩❡t❛ ❇❛s✐s ❙❡ts ❢♦r t❤❡ ✹p✱ ✺p ❛♥❞ ✻p ❊❧❡♠❡♥ts✳

❚❤❡♦r✳ ❈❤❡♠✳ ❆❝❝✳ ✷✵✵✷✱ ✶✵✽✱ ✸✸✺✕✸✹✵✱ ❇❛s✐s s❡ts

❛r❡ ❛✈❛✐❧❛❜❧❡ ❢r♦♠ t❤❡ ❉✐r❛❝ ✇❡❜ s✐t❡✱ ❤tt♣✿✴✴❞✐r❛❝✳❝❤❡♠✳s❞✉✳❞❦✳

✭✼✷✮ ❉②❛❧❧✱ ❑✳ ●✳ ❘❡❧❛t✐✈✐st✐❝ ◗✉❛❞r✉♣❧❡✲❩❡t❛ ❛♥❞ ❘❡✈✐s❡❞ ❚r✐♣❧❡✲❩❡t❛ ❛♥❞ ❉♦✉❜❧❡✲❩❡t❛ ❇❛s✐s ❙❡ts ❢♦r t❤❡ ✹♣✱ ✺♣✱ ❛♥❞ ✻♣ ❊❧❡♠❡♥ts✳

❚❤❡♦r✳ ❈❤❡♠✳ ❆❝❝✳ ✷✵✵✻✱ ✶✶✺✱ ✹✹✶✕✹✹✼✳

✭✼✸✮ ❉②❛❧❧✱ ❑✳ ●✳ ❘❡❧❛t✐✈✐st✐❝ ❉♦✉❜❧❡✲❩❡t❛✱ ❚r✐♣❧❡✲❩❡t❛✱ ❛♥❞ ◗✉❛❞r✉♣❧❡✲❩❡t❛ ❇❛s✐s ❙❡ts ❢♦r t❤❡ ✹s✱ ✺s✱ ✻s✱ ❛♥❞ ✼s ❊❧❡♠❡♥ts✳

❏✳ P❤②s✳ ❈❤❡♠✳ ❆ ✷✵✵✾✱ ✶✶✸✱ ✶✷✻✸✽✕✶✷✻✹✹✳

✭✼✹✮ ❉✉♥♥✐♥❣✱ ❚✳ ❍✳ ●❛✉ss✐❛♥ ❇❛s✐s ❙❡ts ❢♦r ❯s❡ ✐♥ ❈♦rr❡❧❛t❡❞ ▼♦❧❡❝✉❧❛r ❈❛❧❝✉❧❛t✐♦♥s✳ ■✳ ❚❤❡ ❆t♦♠s ❇♦r♦♥ t❤r♦✉❣❤ ◆❡♦♥ ❛♥❞ ❍②❞r♦❣❡♥✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✶✾✽✾✱ ✾✵✱ ✶✵✵✼✳

✭✼✺✮ ❑❡♥❞❛❧❧✱ ❘✳ ❆✳❀ ❉✉♥♥✐♥❣✱ ❚✳ ❍✳❀ ❍❛rr✐s♦♥✱ ❘✳ ❏✳ ❊❧❡❝tr♦♥ ❆✣♥✐t✐❡s ♦❢ t❤❡ ❋✐rst✲❘♦✇ ❆t♦♠s ❘❡✈✐s✐t❡❞✳ ❙②st❡♠❛t✐❝ ❇❛s✐s ❙❡ts ❛♥❞ ❲❛✈❡ ❋✉♥❝t✐♦♥s✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✶✾✾✷✱

✾✻✱ ✻✼✾✻✳ ✭✼✻✮ ❇❛❧❛❜❛♥♦✈✱ ◆✳ ❇✳❀ P❡t❡rs♦♥✱ ❑✳ ❆✳ ❙②st❡♠❛t✐❝❛❧❧② ❈♦♥✈❡r❣❡♥t ❇❛s✐s ❙❡ts ❢♦r ❚r❛♥s✐t✐♦♥ ▼❡t❛❧s✳ ■✳ ❆❧❧✲❊❧❡❝tr♦♥ ❈♦rr❡❧❛t✐♦♥ ❈♦♥s✐st❡♥t ❇❛s✐s ❙❡ts ❢♦r t❤❡ ✸❞ ❊❧❡♠❡♥ts ❙❝✲❩♥✳

❏✳

❈❤❡♠✳ P❤②s✳ ✷✵✵✺✱ ✶✷✸✱ ✵✻✹✶✵✼✳ ✭✼✼✮ ❙❝❤✉❝❤❛r❞t✱ ❑✳ ▲✳❀ ❉✐❞✐❡r✱ ❇✳ ❚✳❀ ❊❧s❡t❤❛❣❡♥✱ ❚✳❀ ❙✉♥✱ ▲✳❀ ●✉r✉♠♦♦rt❤✐✱ ❱✳❀ ❈❤❛s❡✱ ❏✳❀ ▲✐✱ ❏✳❀ ❲✐♥❞✉s✱ ❚✳ ▲✳ ❇❛s✐s ❙❡t ❊①❝❤❛♥❣❡✿ ❆ ❈♦♠♠✉♥✐t② ❉❛t❛❜❛s❡ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ❙❝✐❡♥❝❡s✳

❏✳ ❈❤❡♠✳ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ▼♦❞❡❧✐♥❣ ✷✵✵✼✱ ✹✼✱ ✶✵✹✺✕✶✵✺✷✳

✸✷


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✭✼✽✮ ●r❛♥t✱ ■✳ P✳❀ ◗✉✐♥❡②✱ ❍✳ ▼✳ ❘❛②❧❡✐❣❤✲❘✐t③ ❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❉✐r❛❝ ❖♣❡r❛t♦r ✐♥ ❆t♦♠✐❝ ❛♥❞ ▼♦❧❡❝✉❧❛r P❤②s✐❝s✳

P❤②s✳ ❘❡✈✳ ❆ ✷✵✵✵✱ ✻✷✱ ✵✷✷✺✵✽✳

✭✼✾✮ ❑öst❡r✱ ❆✳ ▼✳❀ ❙❛❧❛❤✉❜✱ ❉✳ ❘✳❀ ❡t ❛❧✳✱ ❞❡▼♦♥✷❦✱ ❱❡rs✐♦♥ ✹✱ ❚❤❡ ❞❡▼♦♥ ❞❡✈❡❧♦♣❡rs✳ ❞❡▼♦♥✷❦✱ ❱❡rs✐♦♥ ✹✱ ❚❤❡ ❞❡▼♦♥ ❞❡✈❡❧♦♣❡rs✳ ❈✐♥✈❡st❛✈✱ ▼❡①✐❝♦ ❈✐t② ✭✷✵✶✻✮✳

✭✽✵✮ ❇❡❝❦❡✱

❆✳

❉✳

❉❡♥s✐t②✲❋✉♥❝t✐♦♥❛❧

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦r✳

❊①❝❤❛♥❣❡✲❊♥❡r❣②

❆♣♣r♦①✐♠❛t✐♦♥

✇✐t❤

❈♦rr❡❝t

P❤②s✳ ❘❡✈✳ ❆ ✶✾✽✽✱ ✸✽✱ ✸✵✾✽✕✸✶✵✵✳

✭✽✶✮ ▲❡❡✱ ❈✳❀ ❨❛♥❣✱ ❲✳❀ P❛rr✱ ❘✳ ●✳ ❉❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❈♦❧❧❡✲❙❛❧✈❡tt✐ ❈♦rr❡❧❛t✐♦♥✲❊♥❡r❣② ❋♦r♠✉❧❛ ✐♥t♦ ❛ ❋✉♥❝t✐♦♥❛❧ ♦❢ t❤❡ ❊❧❡❝tr♦♥ ❉❡♥s✐t②✳

P❤②s✳ ❘❡✈✳ ❇ ✶✾✽✽✱ ✸✼✱ ✼✽✺✕✼✽✾✳

✭✽✷✮ ❏❛❝♦❜✱ ❈✳ ❘✳❀ ❘❡✐❤❡r✱ ▼✳ ❙♣✐♥ ✐♥ ❉❡♥s✐t②✲❋✉♥❝t✐♦♥❛❧ ❚❤❡♦r②✳

■♥t✳ ❏✳ ◗✉❛♥t✉♠ ❈❤❡♠✳

✷✵✶✷✱ ✶✶✷✱ ✸✻✻✶✕✸✻✽✹✳ ✭✽✸✮ ❇❧②❤♦❧❞❡r✱ ●✳ ▼♦❧❡❝✉❧❛r ❖r❜✐t❛❧ ❱✐❡✇ ♦❢ ❈❤❡♠✐s♦r❜❡❞ ❈❛r❜♦♥ ▼♦♥♦①✐❞❡✳

❏✳ P❤②s✳

❈❤❡♠✳ ✶✾✻✹✱ ✻✽✱ ✷✼✼✷✕✷✼✼✼✳ ✭✽✹✮ ❚♦❧♠❛♥✱ ❈✳ ❆✳ ❊❧❡❝tr♦♥ ❉♦♥♦r✲❆❝❝❡♣t♦r Pr♦♣❡rt✐❡s ♦❢ P❤♦s♣❤♦r✉s ▲✐❣❛♥❞s✳ ❙✉❜st✐t✉❡♥t ❆❞❞✐t✐✈✐t②✳

❏✳ ❆♠✳ ❈❤❡♠✳ ❙♦❝✳ ✶✾✼✵✱ ✾✷✱ ✷✾✺✸✕✷✾✺✻✳

✭✽✺✮ ❚♦❧♠❛♥✱ ❈✳ ❆✳ ❙t❡r✐❝ ❊✛❡❝ts ♦❢ P❤♦s♣❤♦r✉s ▲✐❣❛♥❞s ✐♥ ❖r❣❛♥♦♠❡t❛❧❧✐❝ ❈❤❡♠✐str② ❛♥❞ ❍♦♠♦❣❡♥❡♦✉s ❈❛t❛❧②s✐s✳

❈❤❡♠✳ ❘❡✈✳ ✶✾✼✼✱ ✼✼✱ ✸✶✸✕✸✹✽✳

✭✽✻✮ ❋r❡♥❦✐♥❣✱ ●✳ ❯♥❞❡rst❛♥❞✐♥❣ t❤❡ ♥❛t✉r❡ ♦❢ t❤❡ ❜♦♥❞✐♥❣ ✐♥ tr❛♥s✐t✐♦♥ ♠❡t❛❧ ❝♦♠♣❧❡①❡s✿ ❢r♦♠ ❉❡✇❛r✬s ♠♦❧❡❝✉❧❛r ♦r❜✐t❛❧ ♠♦❞❡❧ t♦ ❛♥ ❡♥❡r❣② ♣❛rt✐t✐♦♥✐♥❣ ❛♥❛❧②s✐s ♦❢ t❤❡ ♠❡t❛❧✕ ❧✐❣❛♥❞ ❜♦♥❞✳

❏✳ ❖r❣❛♥♦♠✳ ❈❤❡♠✳ ✷✵✵✶✱ ✻✸✺✱ ✾✕✷✸✳

✭✽✼✮ ❇✐st♦♥✐✱ ●✳❀ ❘❛♠♣✐♥♦✱ ❙✳❀ ❙❝❛❢✉r✐✱ ◆✳❀ ❈✐❛♥❝❛❧❡♦♥✐✱ ●✳❀ ❩✉❝❝❛❝❝✐❛✱ ❉✳❀ ❇❡❧♣❛ss✐✱ ▲✳❀ ❚❛r❛♥t❡❧❧✐✱ ❋✳ ❍♦✇

π

❜❛❝❦✲❞♦♥❛t✐♦♥ q✉❛♥t✐t❛t✐✈❡❧② ❝♦♥tr♦❧s t❤❡ ❈❖ str❡t❝❤✐♥❣ r❡s♣♦♥s❡

✐♥ ❝❧❛ss✐❝❛❧ ❛♥❞ ♥♦♥✲❝❧❛ss✐❝❛❧ ♠❡t❛❧ ❝❛r❜♦♥②❧ ❝♦♠♣❧❡①❡s✳

✸✸


❈❤❡♠✳ ❙❝✐✳ ✷✵✶✻✱ ✼✱ ✶✶✼✹✕✶✶✽✹✳


Page 34 of 39

✭✽✽✮ ❲✉✱ ❳✳❀ ◗✐♥✱ ❩✳❀ ❳✐❡✱ ❍✳❀ ❈♦♥❣✱ ❘✳❀ ❲✉✱ ❳✳❀ ❚❛♥❣✱ ❩✳❀ ❋❛♥✱ ❍✳ P❤♦t♦❡❧❡❝tr♦♥ ■♠❛❣✐♥❣ ❛♥❞ ❚❤❡♦r❡t✐❝❛❧ ❙t✉❞✐❡s ♦❢ ●r♦✉♣ ✶✶ ❈②❛♥✐❞❡s ▼❈◆ ✭▼ ❂ ❈✉✱ ❆❣✱ ❆✉✮✳

❏✳ P❤②s✳ ❈❤❡♠✳

❆ ✷✵✶✵✱ ✶✶✹✱ ✶✷✽✸✾✕✶✷✽✹✹✳ ✭✽✾✮ ❍❛s❤♠✐✱ ❆✳ ❙✳✱ ❚♦st❡✱ ❋✳ ❉✳✱ ❊❞s✳

▼♦❞❡r♥ ●♦❧❞ ❝❛t❛❧②③❡❞ ❙②♥t❤❡s✐s ❀ ❲✐❧❡②✲❱❝❤✿ ❲❡✐♥✲

❤❡✐♠✱ ●❡r♠❛♥②✱ ✷✵✶✷✳

✭✾✵✮ ❲❤✐t❡✱ ❍✳ ❊✳ P✐❝t♦r✐❛❧ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❉✐r❛❝ ❊❧❡❝tr♦♥ ❈❧♦✉❞ ❢♦r ❍②❞r♦❣❡♥✲▲✐❦❡ ❆t♦♠s✳

P❤②s✳ ❘❡✈✳ ✶✾✸✶✱ ✸✽✱ ✺✶✸✳

✭✾✶✮ ❋❛❡❣r✐✱ ❑✳❀ ❙❛✉❡✱ ❚✳ ❉✐❛t♦♠✐❝ ▼♦❧❡❝✉❧❡s ❇❡t✇❡❡♥ ✈❡r② ❍❡❛✈② ❊❧❡♠❡♥ts ♦❢ ●r♦✉♣ ✶✸ ❛♥❞ ●r♦✉♣ ✶✼✿ ❆ ❙t✉❞② ♦❢ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ♦♥ ❇♦♥❞✐♥❣✳

❏✳ ❈❤❡♠✳ P❤②s✳ ✷✵✵✶✱ ✶✶✺✱

✷✹✺✻✕✷✹✻✹✳

✭✾✷✮ ❊✐❝❦❡r❧✐♥❣✱ ●✳❀ ▼❛st❛❧❡r③✱ ❘✳❀ ❍❡r③✱ ❱✳❀ ❙❝❤❡r❡r✱ ❲✳❀ ❍✐♠♠❡❧✱ ❍✳✲❏✳❀ ❘❡✐❤❡r✱ ▼✳ ❘❡❧✲ ❛t✐✈✐st✐❝ ❊✛❡❝ts ♦♥ t❤❡ ❚♦♣♦❧♦❣② ♦❢ t❤❡ ❊❧❡❝tr♦♥ ❉❡♥s✐t②✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳

✷✵✵✼✱ ✸✱ ✷✶✽✷✕✷✶✾✼✳ ✭✾✸✮ ❙❛❜❧♦♥✱ ◆✳❀ ▼❛st❛❧❡r③✱ ❘✳❀ ❉❡ Pr♦❢t✱ ❋✳❀ ●❡❡r❧✐♥❣s✱ P✳❀ ❘❡✐❤❡r✱ ▼✳ ❘❡❧❛t✐✈✐st✐❝ ❊✛❡❝ts ♦♥ t❤❡ ❋✉❦✉✐ ❋✉♥❝t✐♦♥✳

❚❤❡♦r✳ ❈❤❡♠✳ ❆❝❝✳ ✷✵✶✵✱ ✶✷✼✱ ✶✾✺✕✷✵✷✳

✭✾✹✮ ❩❡♥❣✱ ❚✳❀ ❋❡❞♦r♦✈✱ ❉✳ ●✳❀ ❙❝❤♠✐❞t✱ ▼✳ ❲✳❀ ❑❧♦❜✉❦♦✇s❦✐✱ ▼✳ ❊✛❡❝ts ♦❢ ❙♣✐♥✲❖r❜✐t ❈♦✉✲ ♣❧✐♥❣ ♦♥ ❈♦✈❛❧❡♥t ❇♦♥❞✐♥❣ ❛♥❞ t❤❡ ❏❛❤♥✲❚❡❧❧❡r ❊✛❡❝t ❆r❡ ❘❡✈❡❛❧❡❞ ✇✐t❤ t❤❡ ◆❛t✉r❛❧ ▲❛♥❣✉❛❣❡ ♦❢ ❙♣✐♥♦rs✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r② ❈♦♠♣✉t✳ ✷✵✶✶✱ ✼✱ ✷✽✻✹✕✷✽✼✺✳

✭✾✺✮ P✐❧♠è✱ ❏✳❀ ❘❡♥❛✉❧t✱ ❊✳❀ ❇❛ss❛❧✱ ❋✳❀ ❆♠❛♦✉❝❤✱ ▼✳❀ ▼♦♥t❛✈♦♥✱ ●✳❀ ●❛❧❧❛♥❞✱ ◆✳ ◗❚❆■▼ ❆♥❛❧②s✐s ✐♥ t❤❡ ❈♦♥t❡①t ♦❢ ◗✉❛s✐r❡❧❛t✐✈✐st✐❝ ◗✉❛♥t✉♠ ❈❛❧❝✉❧❛t✐♦♥s✳

❏✳ ❈❤❡♠✳ ❚❤❡♦r②

❈♦♠♣✉t✳ ✷✵✶✹✱ ✶✵✱ ✹✽✸✵✕✹✽✹✶✳ ✭✾✻✮ ▲ö✇❞✐♥✱ P✳ ❖♥ t❤❡ ◆♦♥✲❖rt❤♦❣♦♥❛❧✐t② Pr♦❜❧❡♠ ❈♦♥♥❡❝t❡❞ ✇✐t❤ t❤❡ ❯s❡ ♦❢ ❆t♦♠✐❝ ❲❛✈❡ ❋✉♥❝t✐♦♥s ✐♥ t❤❡ ❚❤❡♦r② ♦❢ ▼♦❧❡❝✉❧❡s ❛♥❞ ❈r②st❛❧s✳ ✸✻✺✕✸✼✺✳

✸✹


❏✳ ❈❤❡♠✳ P❤②s✳ ✶✾✺✵✱ ✶✽✱

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✭✾✼✮ ▼❛②❡r✱ ■✳ ❖♥ ▲ö✇❞✐♥✬s ▼❡t❤♦❞ ♦❢ ❙②♠♠❡tr✐❝ ❖rt❤♦❣♦♥❛❧✐③❛t✐♦♥✳

■♥t✳ ❏✳ ◗✉❛♥t✉♠

❈❤❡♠✳ ✷✵✵✷✱ ✾✵✱ ✻✸✕✻✺✳ ✭✾✽✮ ❈❛r❧s♦♥✱ ❇✳ ❈✳❀ ❑❡❧❧❡r✱ ❏✳ ▼✳ ❖rt❤♦❣♦♥❛❧✐③❛t✐♦♥ Pr♦❝❡❞✉r❡s ❛♥❞ t❤❡ ▲♦❝❛❧✐③❛t✐♦♥ ♦❢ ❲❛♥♥✐❡r ❋✉♥❝t✐♦♥s✳

P❤②s✳ ❘❡✈✳ ✶✾✺✼✱ ✶✵✺✱ ✶✵✷✕✶✵✸✳

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Page 36 of 39

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+ ❋✐❣✉r❡ ✶✿ ❈❉ ❛♥❛❧②s✐s ❢♦r t❤❡ ❈✉✲❈❖ ❜♦♥❞ ✐♥ t❤❡ ❬❈✉✭❈❖✮❪ ❝♦♠♣❧❡①✳

❚❤❡ ❞♦ts ♦♥ t❤❡

❛①✐s ♠❛r❦ t❤❡ ③ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❛t♦♠s✳ ❚❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ♠❛r❦s t❤❡ ❜♦✉♥❞❛r② ❜❡t✇❡❡♥ t❤❡ + ❈✉ ❛♥❞ t❤❡ ❈❖ ❢r❛❣♠❡♥t ✭s❡❡ t❡①t ❢♦r ❞❡t❛✐❧s✮✳ ❆ t❤❡ t♦♣✿ ✐s♦❞❡♥s✐t② s✉r❢❛❝❡s ✭±0.0014 ❡ −3 ′ ′ ❛✳✉✳ ✮❢♦r ∆ρ✱ ∆ρ ❛♥❞ ∆ρanti ✳ ❈♦♥tr✐❜✉t✐♦♥ t♦ ❞❡❢♦r♠❛t✐♦♥ ❞❡♥s✐t②✱ ∆ρ ✱ ♦❢ t❤❡ ❢♦✉r ♠♦st ′ ′ ′ ′ s✐❣♥✐✜❝❛♥t ◆❖❈❱✲♣❛✐rs ✭∆ρ1 ✱ ∆ρ2 ✱ ∆ρ3 ❛♥❞ ∆ρ4 ✱✮✳ ❘❡❞ s✉r❢❛❝❡s ✐❞❡♥t✐❢② ❝❤❛r❣❡ ❞❡♣❧❡t✐♦♥ ❛r❡❛s✱ ❜❧✉❡ s✉r❢❛❝❡s ✐❞❡♥t✐❢② ❝❤❛r❣❡ ❛❝❝✉♠✉❧❛t✐♦♥ ❛r❡❛s✳

✸✼



+ ❋✐❣✉r❡ ✷✿ ❈❉ ❛♥❛❧②s✐s ❢♦r t❤❡ ❆❣✲❈❖ ❜♦♥❞ ✐♥ t❤❡ ❬❆❣✭❈❖✮❪ ❝♦♠♣❧❡①✳

Page 38 of 39

❚❤❡ ❞♦ts ♦♥ t❤❡

❛①✐s ♠❛r❦ t❤❡ ③ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❛t♦♠s✳ ❚❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ♠❛r❦s t❤❡ ❜♦✉♥❞❛r② ❜❡t✇❡❡♥ t❤❡ + ❆❣ ❛♥❞ t❤❡ ❈❖ ❢r❛❣♠❡♥t ✭s❡❡ t❡①t ❢♦r ❞❡t❛✐❧s✮✳ ❆ t❤❡ t♦♣✿ ✐s♦❞❡♥s✐t② s✉r❢❛❝❡s ✭±0.0014 ❡ −3 ′ ′ ❛✳✉✳ ✮❢♦r ∆ρ✱ ∆ρ ❛♥❞ ∆ρanti ✳ ❈♦♥tr✐❜✉t✐♦♥ t♦ ❞❡❢♦r♠❛t✐♦♥ ❞❡♥s✐t②✱ ∆ρ ✱ ♦❢ t❤❡ ❢♦✉r ♠♦st ′ ′ ′ ′ s✐❣♥✐✜❝❛♥t ◆❖❈❱✲♣❛✐rs ✭∆ρ1 ✱ ∆ρ2 ✱ ∆ρ3 ❛♥❞ ∆ρ4 ✱✮✳ ❘❡❞ s✉r❢❛❝❡s ✐❞❡♥t✐❢② ❝❤❛r❣❡ ❞❡♣❧❡t✐♦♥ ❛r❡❛s✱ ❜❧✉❡ s✉r❢❛❝❡s ✐❞❡♥t✐❢② ❝❤❛r❣❡ ❛❝❝✉♠✉❧❛t✐♦♥ ❛r❡❛s✳

✸✽


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+ ❋✐❣✉r❡ ✸✿ ❈❉ ❛♥❛❧②s✐s ❢♦r t❤❡ ❆✉✲❈❖ ❜♦♥❞ ✐♥ t❤❡ ❬❆✉✭❈❖✮❪ ❝♦♠♣❧❡①✳

❚❤❡ ❞♦ts ♦♥ t❤❡

❛①✐s ♠❛r❦ t❤❡ ③ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ❛t♦♠s✳ ❚❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ ♠❛r❦s t❤❡ ❜♦✉♥❞❛r② ❜❡t✇❡❡♥ t❤❡ + ❆✉ ❛♥❞ t❤❡ ❈❖ ❢r❛❣♠❡♥t ✭s❡❡ t❡①t ❢♦r ❞❡t❛✐❧s✮✳ ❆ t❤❡ t♦♣✿ ✐s♦❞❡♥s✐t② s✉r❢❛❝❡s ✭±0.0014 ❡ −3 ′ ′ ❛✳✉✳ ✮❢♦r ∆ρ✱ ∆ρ ❛♥❞ ∆ρanti ✳ ❈♦♥tr✐❜✉t✐♦♥ t♦ ❞❡❢♦r♠❛t✐♦♥ ❞❡♥s✐t②✱ ∆ρ ✱ ♦❢ t❤❡ ❢♦✉r ♠♦st ′ ′ ′ ′ s✐❣♥✐✜❝❛♥t ◆❖❈❱✲♣❛✐rs ✭∆ρ1 ✱ ∆ρ2 ✱ ∆ρ3 ❛♥❞ ∆ρ4 ✱✮✳ ❘❡❞ s✉r❢❛❝❡s ✐❞❡♥t✐❢② ❝❤❛r❣❡ ❞❡♣❧❡t✐♦♥ ❛r❡❛s✱ ❜❧✉❡ s✉r❢❛❝❡s ✐❞❡♥t✐❢② ❝❤❛r❣❡ ❛❝❝✉♠✉❧❛t✐♦♥ ❛r❡❛s✳

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Charge-Displacement Analysis via Natural Orbitals for Chemical

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