s rt rs srs - PowerPoint PPT Presentation

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ❆s❦ ❍❥♦rt❤ ▲❛rs❡♥ ❛s❦❧❛rs❡♥❅❣♠❛✐❧✳❝♦♠ ◆❛♥♦✲❜✐♦ ❙♣❡❝tr♦s❝♦♣② ●r♦✉♣ ❛♥❞ ❊❚❙❋ ❙❝✐❡♥t✐✜❝ ❉❡✈❡❧♦♣♠❡♥t ❈❡♥tr❡ ❯♥✐✈❡rs✐❞❛❞ ❞❡❧ P❛ís ❱❛s❝♦ ❯P❱✴❊❍❯ ▼❛② ✸✶✱ ✷✵✶✼

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ●P❆❲ ◮ ❉❡♥s✐t② ❢✉♥❝t✐♦♥❛❧ t❤❡♦r② ❝♦❞❡ ✇r✐tt❡♥ ✐♥ P②t❤♦♥ ❛♥❞ ❈ ◮ ❤tt♣s✿✴✴✇✐❦✐✳❢②s✐❦✳❞t✉✳❞❦✴❣♣❛✇✴ ◮ Pr♦❥❡❝t♦r ❛✉❣♠❡♥t❡❞ ✇❛✈❡ ♠❡t❤♦❞ ✭❇❧ö❝❤❧ P❤②s✳ ❘❡✈✳ ❇ ✺✵✱ ✶✼✾✺✸✱ ✶✾✾✹✮ ◮ ❈❛❧❝✉❧❛t✐♦♥s ❛r❡ ✇r✐tt❡♥ ❛s P②t❤♦♥ s❝r✐♣ts ✉s✐♥❣ ❆❙❊ ◮ ❚❤r❡❡ ❝❛❧❝✉❧❛t✐♦♥ ♠♦❞❡s✿ ❋❉✱ ▲❈❆❖✱ P❲ ◮ P❛r❛❧❧❡❧✐③❛t✐♦♥ ✇✐t❤ ▼P■❀ ❤✐❣❤❧② s❝❛❧❛❜❧❡ ✭❡①❝❡♣t P❲✮ ◮ P②t❤♦♥ ✐♥t❡r❢❛❝❡s ❢♦r ✐♠♣♦rt❛♥t ❢✉♥❝t✐♦♥s ❢r♦♠ ▼P■✱ ❇▲❆❙✱ ▲❆P❆❈❑✱ ❙❝❛▲❆P❆❈❑

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ❚❤❡ ❝❛s❡ ❢♦r P②t❤♦♥ ✐♥ ❉❋❚ ❝♦❞❡s ◮ Pr♦❜❛❜❧② ♠♦r❡ t❤❛♥ ✾✵ ✪ ♦❢ ❈P❯ t✐♠❡ ✐s s♣❡♥t ✐♥ ❧❡ss t❤❛♥ ✶✵ ✪ ♦❢ t❤❡ ❝♦❞❡ ◮ ❲r✐t❡ ✾✵ ✪ ✐♥ ❛ ♥✐❝❡ ❧❛♥❣✉❛❣❡ ❛♥❞ ✶✵ ✪ ✐♥ ❈✴❋♦rtr❛♥ ◮ ❉❋❚ ❝♦❞❡s ❞♦ ♠❛♥② ♥♦♥✲❝♦♠♣✉t❛t✐♦♥❛❧ t❤✐♥❣s✿ ◮ ●❧✉❡ ❝♦♠♣♦♥❡♥ts t♦❣❡t❤❡r ✭P♦✐ss♦♥✱ ❡✐❣❡♥s♦❧✈❡r✱ ♠✐①✐♥❣✱ ❡t❝✳✮ ◮ ❘❡❛❞ ❛♥❞ ✇r✐t❡ ❞❛t❛ ◮ ❙tr✐♥❣ ♣r♦❝❡ss✐♥❣ ✭♣❛rs❡ ♣s❡✉❞♦♣♦t❡♥t✐❛❧s✱ ❢♦r♠❛t ♦✉t♣✉t✱ ✳✳✳✮ ◮ P❛r❛❧❧❡❧ ♦♣❡r❛t✐♦♥s ✭t✐♠❡ s♣❡♥t ✇❛✐t✐♥❣ ❢♦r ❤❛r❞✇❛r❡✮ ◮ ❉✐s❛❞✈❛♥t❛❣❡✿ ▼✐①✐♥❣ ❧❛♥❣✉❛❣❡s ❞♦❡s ❤❛✈❡ s♦♠❡ ❡①tr❛ ❝♦♠♣❧❡①✐t②

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ P❆❲ ❛♥❞ t❤❡ q✉❡st ❢♦r s♠♦♦t❤ ✇❛✈❡❢✉♥❝t✐♦♥s 2 . 5 Fe 4s AE Fe 3d AE 2 . 0 φ ( r ) [arb. units] Fe 4s HGH 1 . 5 Fe 3d HGH Fe 4s PAW 1 . 0 Fe 3d PAW 0 . 5 0 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 Distance from atom [Bohr] ◮ ❲❛✈❡❢✉♥❝t✐♦♥s ♦s❝✐❧❧❛t❡ ❛♥❞ ❛r❡ ❡①♣❡♥s✐✈❡ t♦ r❡♣r❡s❡♥t ◮ ❘❡♠♦✈❡ ❝♦r❡ ❡❧❡❝tr♦♥s✱ r❡♣❧❛❝❡ ♥✉❝❧❡✐ ✇✐t❤ ♣s❡✉❞♦♣♦t❡♥t✐❛❧s ◮ P❆❲✿ ❊①t❡♥s✐♦♥ t♦ ♣s❡✉❞♦♣♦t❡♥t✐❛❧s✳ ❆✈♦✐❞s ♥♦r♠✲❝♦♥s❡r✈❛t✐♦♥✱ ❛♥❞ r❡t❛✐♥s ✏❛❧❧✲❡❧❡❝tr♦♥✑ ✐♥❢♦r♠❛t✐♦♥

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ❚❤❡ ♣r♦❥❡❝t♦r ❛✉❣♠❡♥t❡❞ ✇❛✈❡ ♠❡t❤♦❞ T | ˜ ◮ ❱❛❧❡♥❝❡ st❛t❡s r❡♣r❡s❡♥t❡❞ ❜② | ψ � = ˆ ψ � � ◮ P❆❲ tr❛♥s❢♦r♠❛t✐♦♥✿ ˆ i � − | ˜ ( | φ a φ a p a T = 1 + i � ) � ˜ i | ai i ( r ) ✱ ❝❤♦♦s❡ s♠♦♦t❤ ❢✉♥❝t✐♦♥s ˜ ◮ ●✐✈❡♥ ❛t♦♠✐❝ st❛t❡s φ a φ a i ( r ) ❛♥❞ i ( r ) ❀ t❤✐s ❞❡✜♥❡s ˆ p a ❧♦❝❛❧✐③❡❞ ✏♣r♦❥❡❝t♦rs✑ ˜ T ✳ ◮ P❆❲ tr❛♥s❢♦r♠❛t✐♦♥ ♠❛♣s ♥✐❝❡ s♠♦♦t❤ ♣s❡✉❞♦✇❛✈❡❢✉♥❝t✐♦♥s t♦ ✉❣❧② ♦s❝✐❧❧❛t✐♥❣ ❛❧❧✲❡❧❡❝tr♦♥ ✇❛✈❡❢✉♥❝t✐♦♥s ◮ ❈❛❧❝✉❧❛t❡ q✉❛♥t✐t✐❡s ❢r♦♠ s♠♦♦t❤ ❢✉♥❝t✐♦♥s ♣❧✉s ❛t♦♠✐❝ ❝♦♥tr✐❜✉t✐♦♥s✿ � ❛t♦♠✐❝ � � � φ | ˆ O | ψ � = � ˜ φ | ˆ O | ˜ � ˜ i | ˜ p a p a ψ � + φ | ˜ i � � ˜ ψ � ♣r♦♣❡rt✐❡s � �� � ai ✸❉ ❣r✐❞ � �� � ❙②st❡♠ ✐♥❞❡♣❡♥❞❡♥t ◮ Pr♦❥❡❝t♦r ❢✉♥❝t✐♦♥s ❝♦✉♣❧❡ ✸❉ ❝❛❧❝✉❧❛t✐♦♥ t♦ ❛t♦♠✐❝ ❝♦rr❡❝t✐♦♥s

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ●P❆❲ ❯♥✐❢♦r♠ r❡❛❧✲s♣❛❝❡ ❣r✐❞s ◮ ❆❝❝✉r❛t❡✱ ♣❛r❛❧❧❡❧✐③❡s ✈❡r② ✇❡❧❧ ◮ ●r✐❞ ❝❛✉s❡s ✏❡❣❣✲❜♦①✑ ❡✛❡❝t ◆✉♠❡r✐❝ ❛t♦♠✐❝ ♦r❜✐t❛❧s ◮ ❱❡r② ❢❛st ✕ ❡❛s② t♦ ❞♦ t❤♦✉s❛♥❞s ♦❢ ❛t♦♠s ◮ ◆♦t ❛s ❛❝❝✉r❛t❡❀ ❝❛♥♥♦t ❡❛s✐❧② r❡❛❝❤ ❝♦♠♣❧❡t❡ ❜❛s✐s s❡t P❧❛♥❡✇❛✈❡ ♠♦❞❡ ◮ ❋❛st ❢♦r s♠❛❧❧ s②st❡♠s ✭❝r②st❛❧s✮ ◮ P❧❛♥❡✇❛✈❡s ♥✉♠❡r✐❝❛❧❧② ✇❡❧❧✲❜❡❤❛✈❡❞ ◮ ◆♦t s❝❛❧❛❜❧❡

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ❘❡❛❧✲s♣❛❝❡ ✭✏❋❉✑✮ ♠♦❞❡ ◮ ●P❆❲✭♠♦❞❡❂✬❢❞✬✱ ❤❂✵✳✶✽✮ ◮ ■t❡r❛t✐✈❡ ❡✐❣❡♥s♦❧✈❡rs ✭s♦❧✈❡ ✏❛ ❜✐t✑ ❡✈❡r② ✐t❡r❛t✐♦♥✮ ◮ ✸❉ ♠❛✐♥ ❈P❯ ♠❡s❤✿ ❞✐str✐❜✉t❡ ψ kn ( r ) ♦✈❡r s♣✐♥s✴ k ✲♣♦✐♥ts✱ ❞♦♠❛✐♥s ❛♥❞ ❜❛♥❞s ◮ ❲❛✈❡ ❢✉♥❝t✐♦♥s ♦♥ ❝♦❛rs❡ ❣r✐❞✱ ❞❡♥s✐t②✴♣♦t❡♥t✐❛❧ ♦♥ ✜♥❡ ❣r✐❞ ◮ ▼✉❧t✐❣r✐❞✿ ❙♦❧✈❡ ✷①✷①✷ ❣r✐❞❀ t❤❡♥ ♦♥ ✹①✹①✹ ❣r✐❞❀ t❤❡♥ ♦♥ ✽①✽①✽ ❣r✐❞❀ ❡t❝✳ ◮ ❋❡❛t✉r❡s✿ ❘❡❛❧✲t✐♠❡ ❚❉❉❋❚✱ ▲✐♥❡❛r r❡s♣♦♥s❡ ❚❉❉❋❚

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ❙t❡♥❝✐❧s ◮ ❉✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs r❡♣r❡s❡♥t❡❞ ❜② ✜♥✐t❡✲❞✐✛❡r❡♥❝❡ st❡♥❝✐❧s � ◮ ˆ O ˜ w ijk ˜ ψ ( x, y, z ) = ψ ( x + i d x, y + j d y, z + k d z ) ijk ① ① ① ① ① ① ① ① ① ▼♦r❡ ♥❡❛r❡st ♥❡✐❣❤❜♦✉rs → ❤✐❣❤❡r ♦r❞❡r → ❜❡tt❡r ❝♦♥✈❡r❣❡♥❝❡

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ P❲ ♠♦❞❡ ◮ ●P❆❲✭♠♦❞❡❂P❲✭✹✵✵✮✮ ◮ ❙❛♠❡ ❡✐❣❡♥s♦❧✈❡rs ❛s ❋❉ ♠♦❞❡ ✭✐t❡r❛t✐✈❡✮ ◮ ✭❆❧♠♦st✮ ♥♦ ❡❣❣✲❜♦① ❡✛❡❝t ◮ P❛r❛❧❧❡❧✐③❡ ♦✈❡r s♣✐♥s✴ k ✲♣♦✐♥ts ❛♥❞ st❛t❡s ✭✏❜❛♥❞s✑✮ ◮ ❖♥❧② ♠♦❞❡ s✉♣♣♦rt✐♥❣ str❡ss❡s ◮ ❋❡❛t✉r❡s✿ ❘P❆ ❧✐♥❡❛r r❡s♣♦♥s❡✱ G 0 W 0 ✭♥♦t s❡❧❢❝♦♥s✐st❡♥t ❜✉t ✉♥❞❡r ❞❡✈❡❧♦♣♠❡♥t✮✱ ❇❡t❤❡✕❙❛❧♣❡t❡r ❡q✉❛t✐♦♥

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ▲❈❆❖ ♠♦❞❡ ◮ ●P❆❲✭♠♦❞❡❂✬❧❝❛♦✬✱ ❜❛s✐s❂✬❞③♣✬✱ ❤❂✵✳✶✽✮ � ◮ ❲❛✈❡❢✉♥❝t✐♦♥s r❡♣r❡s❡♥t❡❞ ❛s | ˜ ψ n � = | Φ µ � c µn µ ◮ | Φ µ � ❛r❡ ❧♦❝❛❧✐③❡❞ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❛t♦♠✐❝ ♦r❜✐t❛❧s ◮ ❙♠❛❧❧ ❜❛s✐s s❡t ✭ ∼ ✺✕✷✵ ❢✉♥❝t✐♦♥s ♣❡r ❛t♦♠✮ ❛❧❧♦✇s ❞✐r❡❝t ❞✐❛❣♦♥❛❧✐③❛t✐♦♥ ◮ ❑♦❤♥✕❙❤❛♠ ❡q✉❛t✐♦♥s ❡①♣❧✐❝✐t❧② s♦❧✈❡❞ ❛s ❣❡♥❡r❛❧✐③❡❞ � � ❡✐❣❡♥✈❛❧✉❡ ♣r♦❜❧❡♠✿ H µν C νn = S µν C νn ǫ n ν ν ◮ P❛r❛❧❧❡❧✐③❡ ♦✈❡r s♣✐♥s✴ k ✲♣♦✐♥ts✱ ❞♦♠❛✐♥s✱ ♦r❜✐t❛❧s ◮ P❛r❛❧❧❡❧✐③❡ ❧✐♥❡❛r ❛❧❣❡❜r❛ ✇✐t❤ ❇▲❆❈❙✴❙❝❛▲❆P❆❈❑ ◮ ❋❡❛t✉r❡s✿ ❚✐♠❡✲♣r♦♣❛❣❛t✐♦♥ ❚❉❉❋❚ ✭▲❈❆❖✲❚❉❉❋❚✮

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ ▲♦❝❛❧✐③❡❞ ❜❛s✐s s❡ts Basis functions: Au dzp 0.8 6s-sz confin 0.7 5d-sz confin 0.6 6s-dz split- 0.5 5d-dz split- 0.4 p-type Gauss φr ˜ 0.3 0.2 0.1 0.0 0.1 0 1 2 3 4 5 6 7 8 r [Bohr]

●P❆❲ ❲r❛♣♣✐♥❣ ✉♣ P❛r❛❧❧❡❧ s❝❛❧✐♥❣ 10 4 10 1 N 2 N 3 N 4 N SCF diag 10 3 10 Time [s] ◮ P❛r❛❧❧❡❧ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ●P❆❲✴▲❈❆❖ ◮ ❚✐♠✐♥❣ ♣❡r ❙❈❋ st❡♣ 10 2 10 ◮ ✷✺✵✵✕✶✵✵✵✵ ❛t♦♠s 10 1 10 6 2 4 8 6 1 3 6 2 5 / / / 1 2 1 2 4 / / 8 6 1 Number of nodes/cores