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 îáùåìñëó÷àå äâóì ýòèì ôóíêöèÿì ñîîòâåòñòâóþò ðàçëè÷íûå íàáîðû ïðîñòðàíñòâåííûõ îðáèòàëåé, ïîýòîìó ôóíêöèîíàë êèíåòè÷åñêîé ýíåðãèè îïðåäåëÿåòñÿ ñðàçó äâóìÿ ôóíêöèÿìè(Ts [ρα , ρβ ]) ïóò¼ì ìèíèìèçàöèè h ΨD | T |ΨD i ïðè çàäàííûõ ρ = ρα + ρβ , N = N α + N β(îáùèé ñïèí â ïðèíöèïå ìîæåò âàðüèðîâàòüñÿ). Òàêèì îáðàçîì, â ñïèí-ïîëÿðèçîâàííîìñëó÷àå çàäà÷à ñâîäèòñÿ ê ìèíèìèçàöèè ôóíêöèîíàëàZα βα βE[ρ , ρ ] = Ts [ρ , ρ ] + (ρα (r) + ρβ (r))V (r)d r +J[ρα + ρβ ] + Exc [ρα , ρβ ](4.3.12)Rïî N -ïðåäñòàâèìûì ôóíêöèÿì ρα (r), ρβ (r) ïðè óñëîâèè (ρα (r) + ρβ (r))d r = N.
Ââåäåíèåïðèáëèæåíèÿ ëîêàëüíîé ïëîòíîñòè, ïîçâîëÿåò çàäàòüZ Z441βααβα β(ρ ) 3 + (ρ ) 3 d r, Ec [ρ , ρ ] = ρ(r) εc (ρ, ζ)d r,(4.3.13)Ex [ρ , ρ ] = 2 3 Cxρα − ρβãäå ζ = ïàðàìåòð ïîëÿðèçàöèè (îáñóæäåíèå âûáîðà ôóíêöèîíàëà èìåííî â ýòîéρôîðìå îïóñòèì). Ðåøåíèå ñïèí-ïîëÿðèçîâàííîé ýëåêòðîííîé çàäà÷è â ðàìêàõ LDA ïîëó÷èëî íàçâàíèå ïðèáëèæåíèÿ ëîêàëüíîé ñïèíîâîé ïëîòíîñòè (LSDA local spin densityapproximation ).4.4.Óòî÷íåíèå ïðèáëèæåíèÿ ëîêàëüíîé ïëîòíîñòèÎäíèì èç çàìåòíûõ íåäîñòàòêîâ óðàâíåíèé Êîíà-Øýìà ÿâëÿåòñÿ çàâûøåíèå êóëîíîâñêîé ýíåðãèè ýëåêòðîííîãî îòòàëêèâàíèÿ J[ρ], êîòîðàÿ, â îòëè÷èå îò Vee â ìîëåêóëÿðíîìãàìèëüòîíèàíå (ñì.
2.1) âêëþ÷àåò â ñåáÿ îòòàëêèâàíèå ýëåêòðîíà îò ñàìîãî ñåáÿ. Ýòîò ôàêòîáû÷íî íå îáðàùàåò íà ñåáÿ âíèìàíèå, îäíàêî õîðîøî çàìåòåí ïðè ðàññìîòðåíèè îäíîãîýëåêòðîíà, îïèñûâàåìîãî âîëíîâîé ôóíêöèåé ϕ(r). Ñîãëàñíî (3.1.18), äëÿ òàêîé ñèñòåìû1 R | ϕ(r1 )|2 | ϕ(r2 )|2d r1 d r2 > 0.ρ(r) = | ϕ(r)|2 , è Vee [| ϕ(r)|2 ] = 0, îäíàêî J[| ϕ(r)|2 ] =2| r1 − r2 | îáùåì, ñïèí-ïîëÿðèçîâàííîì ñëó÷àå óñòðàíåíèå âçàèìîäåéñòâèÿ ýëåêòðîíà ñ ñàìèìñîáîé ïðèâîäèò ê òðåáîâàíèþ Vee [ργi , 0] = J[ργi ] + Exc [ργi , 0] = 0, ãäå ργi ïëîòíîñòü ðàñïðåäåëåíèÿ ýëåêòðîíà, çàíèìàþùåãî i-óþ îðáèòàëü, γ = α, β. Êðîìå ýòîãî, â îäíîýëåêòðîííîéñèñòåìå íåâîçìîæíà ýëåêòðîííàÿ êîððåëÿöèÿ, ïîýòîìó ïðèâåä¼ííîå óñëîâèå ðàçáèâàåòñÿíà äâà: J[ργi ] + Ex [ργi , 0] = 0, Ec [ραi , 0] = 0.
Âûïîëíåíèå ýòèõ óñëîâèé ëåãêî äîñòèãàåòñÿ ïðèâûáîðå ôóíêöèîíàëà îáìåííî-êîððåëÿöèîííîé ýíåðãèè â âèäåXSIC α βExc[ρ , ρ ] = Exc [ρα , ρβ ] −(J[ργi ] + Exc [ρσi , 0]),(4.4.1)i,γãäå áóêâû SIC îáîçíà÷àþò ïîïðàâêó (self-interaction correction ). Çàìåòèì, ÷òî ââåäåíèåýòîé ïîïðàâêè îòðàæàåòñÿ ëèøü íà ïðèáëèæ¼ííîé ôîðìå ôóíêöèîíàëà, â òî âðåìÿ êàê äëÿ44òî÷íîãî Exc [ρα , ρβ ] ïîïðàâêà îáðàùàåòñÿ â íîëü. Êðîìå ýòîãî, SIC ïðèâîäèò ê ðàçëè÷íûìîáìåííî-êîððåëÿöèîííûì ïîòåíöèàëàì (è, ñîîòâåòñòâåííî, ðàçëè÷íûì óðàâíåíèÿì ÊîíàØýìà) äëÿ ýëåêòðîíîâ ñ ðàçëè÷íûìè ñïèíîâûìè ôóíêöèÿìè.Òåïåðü îáñóäèì âîçìîæíûå ïðèáëèæåíèÿ äëÿ ôóíêöèîíàëà îáìåííî-êîððåëÿöèîííîéýíåðãèè; äëÿ ýòîãî âåðí¼ìñÿ ê òî÷íîìó âûðàæåíèþ (4.1.4) äëÿ Vee [ρ] è ñâÿæåì äèàãîíàëüíóþ ÷àñòü ìàòðèöû ïëîòíîñòè âòîðîãî ïîðÿäêà ñ ýëåêòðîííîé ïëîòíîñòüþ:1(4.4.2)ρ2 (r1 , r2 ) = ρ(r1 )ρ(r2 )(1 + h(r1 , r2 )),2ãäå h(r1 , r2 ) íàçûâàåòñÿ ïàðíîé êîððåëÿöèîííîé ôóíêöèåé.
Çàìåòèì, ÷òî íåïîñðåäñòâåííîèç îïðåäåëåíèÿ ìàòðèö ïëîòíîñòè ïåðâîãî è âòîðîãî ïîðÿäêîâ (2.5.1), (2.5.2) ñëåäóåòZ20(4.4.3)ρ1 (r1 , r1 ) =ρ2 (r01 , r2 ; r1 , r2 )d r2 ,N −1òî åñòü, â ÷àñòíîñòè,Z2ρ(r1 ) =ρ2 (r1 , r2 )d r2 .(4.4.4)N −1Ïîäñòàâëÿÿ â ýòî ñîîòíîøåíèå (4.4.2), ïîëó÷èìZZN −1Nρ(r1 )ρ(r1 ) = ρ(r1 ) +ρ(r2 )h(r1 , r2 )d r2 ⇒ ρ(r2 )h(r1 , r2 )d r2 = −1.(4.4.5)222Ïîäèíòåãðàëüíóþ ôóíêöèþ ρxc (r1 , r2 ) = ρ(r2 )h(r1 , r2 ) ìîæíî ðàññìàòðèâàòü êàê ïëîòíîñòüðàñïðåäåëåíèÿ îáìåííî-êîððåëÿöèîííîéR äûðêè ýëåêòðîíà, íàõîäÿùåãîñÿ â òî÷êå r1 (íàçâàíèå "äûðêà" ñâÿçàíî ñ íîðìèðîâêîé ρxc (r1 , r2 )d r2 = −1, ñîîòâåòñòâóþùåé ïîëîæèòåëüíîìó çàðÿäó).
(4.4.2) ïîçâîëÿåò ïåðåïèñàòü (4.1.4) â âèäåZZ1ρ(r1 )ρ2 (r1 , r2 )d r1 d r2 = J[ρ] +ρxc (r1 , r2 )d r1 d r2 .(4.4.6)Vee [ρ] =| r1 − r2 |2| r1 − r2 |Òåïåðü âûðàçèì Exc [ρ] ÷åðåç ρxc ; äëÿ ýòîãî, èñïîëüçóÿ îáîçíà÷åíèÿ 4.1, ââåä¼ìFλ [ρ] = min h Ψ|(T +λVee )|Ψ i = h Ψρλ |(T +λVee )|Ψλρ i .Ψ→ρ(4.4.7) ñîîòâåòñòâèè ñ (4.3.8) è (4.1.5) F0 [ρ] = Ts [ρ], F1 [ρ] = T [ρ] + Vee [ρ]; çíà÷èò, (4.3.3) ìîæíîçàïèñàòü â âèäåZ1Exc [ρ] = Vee [ρ] − J[ρ] + T [ρ] − Ts [ρ] = F1 [ρ] − F0 [ρ] − J[ρ] =∂ Fλ [ρ]dλ − J[ρ].∂λ(4.4.8)0∂ Fλ [ρ]; çàìåòèì, ÷òî Ψλρ ìèíèìèçèðóåò ýíåðãèþ, ñîîò∂λâåòñòâóþùóþ ãàìèëüòîíèàíó Hλ = T +λVee , à ïîòîìó (ñîãëàñíî âàðèàöèîííîìó ïðèíöèïó)ÿâëÿåòñÿ åãî ñîáñòâåííîéôóíêöèåé:Hλ |Ψλρ i = Eλ |Ψλρ i .
Ïî òåîðåìå Ãåëüìàíà-Ôåéíìàíà ∂ Hλ λ∂ Eλ Ψρ i = h Ψλρ |Vee |Ψλρ i, íî Eλ = h Ψλρ |(T +λVee )|Ψλρ i = Fλ [ρ], ïîýòî(ñì. 2.6)= h Ψλρ ∂λ∂λ ∂ Fλ [ρ]ìó= h Ψλρ |Vee |Ψλρ i . Òàêèì îáðàçîì,∂λZZZ1ρ¯2 (r1 , r2 )λλExc [ρ] = h Ψρ |Vee |Ψρ i dλ − J[ρ] =d r1 d r2 −J[ρ] =| r1 − r2 |(4.4.9)0ZZZZ1ρ(r1 )ρ(r2 )1ρ(r1 )ρxc (r1 , r2 )d r1 d r2 ,=h̄(r1 , r2 )d r1 d r2 =2| r1 − r2 |2| r1 − r2 |Îñòà¼òñÿ îïðåäåëèòü ïðîèçâîäíóþ45ãäå ïî àíàëîãèè ñ (4.4.6) ââåäåíû óñðåäí¼ííàÿ ïî λ ïàðíàÿ êîððåëÿöèîííàÿ ôóíêöèÿh̄(r1 , r2 ) è ïëîòíîñòü ðàñïðåäåëåíèÿ îáìåííî-êîððåëÿöèîííîé äûðêè ρxc (r1 , r2 ) (h̄(r1 , r2 )çàäàíà óñëîâèåìZ11ρλ2 (r1 , r2 )dλ = ρ(r1 )ρ(r2 )(1 + h̄(r1 , r2 )),(4.4.10)20ãäå äèàãîíàëüíàÿ ÷àñòü ìàòðèöû ïëîòíîñòè âòîðîãî ïîðÿäêà, îïðåäåë¼ííîé ïîâîëíîâîé ôóíêöèè Ψλρ , à ρxc (r1 , r2 ) = ρ(r2 )h̄(r1 , r2 )).
Îòìåòèì, ÷òî äëÿ ρxc ñîõðàíÿåòñÿRóñëîâèå íîðìèðîâêè ρxc (r1 , r2 )d r2 = −1.Íàêîíåö, ïðåäñòàâèì Exc [ρ] â âèäåZZ1ρxc (r, r0 ) 0−1−1Exc [ρ] = −dr(4.4.11)ρ(r)Rxc (r, [ρ(r)])d r, Rxc (r, [ρ(r)]) = −2| r − r0 |ρλ2 (r1 , r2 )(äëÿ îáùíîñòè çàïèñè ââåäåíû r = r1 , r0 = r2 ; Rxc ÿâëÿåòñÿ ôóíêöèåé òîëüêî ïåðåìåííîé r,à çàïèñü [ρ(r)] â êà÷åñòâå âòîðîãî àðãóìåíòà "íàïîìèíàåò", ÷òî Rxc îïðåäåëÿåòñÿ ýëåêòðîííîé ïëîòíîñòüþ ρ(r)). Ââåäåíèå ôóíêöèè Rxc ïîçâîëÿåò, â ÷àñòíîñòè, ñâÿçàòü ïëîòíîñòüîáìåííî-êîððåëÿöèîííîé ýíåðãèè εxc (r) (ñì.
4.3) ñ ρxc ÷åðåç ñîîòíîøåíèå1 −1(r, [ρ(r)]).(4.4.12)εxc (ρ(r)) = − Rxc2 ïðèíöèïå, ïðåäñòàâëåíèå ðàçëè÷íûõ îáìåííî-êîððåëÿöèîííûõ ïàðàìåòðîâ ÷åðåç ρxc îêàçûâàåòñÿ êðàéíå óäîáíûì äëÿ àíàëèçà òî÷íîñòè è ãðàíèö ïðèìåíèìîñòè îöåíîê Exc [ρ].Êðîìå ýòîãî Rxc (r, [ρ(r)]) ìîæåò áûòü ïðåäñòàâëåíà â âèäå ðàçëîæåíèÿ ïî ñòåïåíÿì ãðàäèåíòà ρ.Ðàçëîæèì ρ(r) â ðÿä Òåéëîðà â òî÷êå íà÷àëà êîîðäèíàò:1ρ(r) = ρ(0) + (r, ∇ρ|r=0 ) + (∇ρ|r=0 )2 r2 + . . . ,2(4.4.13)òîãäà Rxc (r, [ρ(r)]) ìîæíî ðàññìàòðèâàòü êàê ôóíêöèþ ∇ρ|r=0 èëè, â áîëåå îáùåì ñëó÷àå,ïðîñòî ∇ρ, ÷òî ïîçâîëÿåò ââåñòè ðàçëîæåíèå ïî ñòåïåíÿì ãðàäèåíòà−1Rxc(r) = F0 (ρ(r)) + F21 (ρ(r))∇2 ρ + F22 (ρ(r))(∇ρ)2 + .
. .(4.4.14) íå÷¼òíûå ñòåïåíè îòñóòñòâóþò, ïîñêîëüêó Rxc ÿâëÿåòñÿ ñêàëÿðíîé âåëè÷èíîé. Ïîäñòàíîâêà (4.4.14) â (4.4.11) ïîçâîëÿåò ïðåäñòàâèòü Exc [ρ] â âèäå ñóììû(0)(1)(2)Exc [ρ] = Exc[ρ] + Exc[ρ] + Exc[ρ] + . . . ,(4.4.15)ãäå ñëàãàåìûå èìåþò âèä(0)Exc(1)Exc=(2)ExcZZ=ZLDAρ(r) ε(ρ(r))d r = Exc,(4.4.16)GGAρ(r)f (1) (ρ(r), |∇ρ(r)|)d r = Exc,(4.4.17)ρ(r)f (2) (ρ(r), |∇ρ(r)|, ∇2 ρ(r))d r .(4.4.18)=Èòàê, äëÿ âûõîäà çà ïðåäåëû LDA äîñòàòî÷íî èñêàòü â ïðèáëèæ¼ííîé ôîðìå ôóíêöèþðàñïðåäåëåíèÿ îáìåííî-êîððåëÿöèîííîé ýíåðãèè ïî ýëåêòðîííîé ïëîòíîñòè è ìîäóëþ å¼ãðàäèåíòà; ïðè íåîáõîäèìîñòè äîñòèæåíèÿ áîëåå âûñîêîé òî÷íîñòè ìîæíî ïðèâëåêàòü ñëàãàåìûå áîëåå âûñîêèõ ïîðÿäêîâ.
Ïîäîáíûé ïîäõîä ïîëó÷èë íàçâàíèå îáîáù¼ííîãî ãðàäèåíòíîãî ïðèáëèæåíèÿ (GGA generalized gradient approximation) ; èìåííî îí èñïîëüçóåòñÿâ íàñòîÿùåå âðåìÿ äëÿ ïîâûøåíèÿ òî÷íîñòè ðàñ÷¼òîâ ìåòîäîì DFT.465.5.1.Ñèììåòðèÿ â êâàíòîâîé õèìèèÑèììåòðèÿ óðàâíåíèÿ Øðåäèíãåðàâ êâàíòîâîé õèìèè îïåðàöèÿìè ñèììåòðèè íàçûâàþòñÿ ëþáûå íåâûðîæäåííûå ïðåîáðàçîâàíèÿ ïåðåìåííûõ, îòíîñèòåëüíî êîòîðûõ èíâàðèàíòíî óðàâíåíèåØðåäèíãåðà ðàññìàòðèâàåìîé ñèñòåìû. Ê òàêèì îïåðàöèÿì îòíîñÿòñÿ ïåðåñòàíîâêè ïåðåìåííûõ òîæäåñòâåííûõ ÷àñòèö, îïåðàöèè òî÷å÷íîé ñèììåòðèè (ñì. 5.2), òðàíñëÿöèè,ïðåîáðàçîâàíèå âðåìåíè (â íåðåëÿòèâèñòñêîì ñëó÷àå).
Íèæå áóäóò ðàññìîòðåíû ëèøü òåâèäû ñèììåòðèè óðàâíåíèÿ Øðåäèíãåðà, êîòîðûå ñóùåñòâåííû äëÿ ðåøåíèÿ ñòàöèîíàðíûõ çàäà÷ â ìîëåêóëÿðíûõ ñèñòåìàõ, ïåðåñòàíîâêà ïåðåìåííûõ è òî÷å÷íàÿ ñèììåòðèÿ.Çàìå÷àíèå: îïåðàöèè ñèììåòðèè óðàâíåíèÿ Øðåäèíãåðà îáðàçóþò ãðóïïó, íàçûâàåìóþ ãðóïïîé ñèììåòðèè óðàâíåíèÿ Øðåäèíãåðà (åñòåñòâåííîå çàäàíèå ïðîèçâåäåíèÿ îïåðàöèé ñèììåòðèè êàê ðåçóëüòàòà ïîñëåäîâàòåëüíîãî ïðèìåíåíèÿ ýòèõ îïåðàöèé ïðèâîäèòê î÷åâèäíîìó âûïîëíåíèþ (1.1.1) − (1.1.3)).Çàìå÷àíèå: ïóñòü G ãðóïïà ñèììåòðèè óðàâíåíèÿ Øðåäèíãåðà; D å¼ ëèíåéíîåïðåäñòàâëåíèå â ãèëüáåðòîâîì ïðîñòðàíñòâå L2 ; òîãäà D H = H D, à ñîáñòâåííûå ôóíêöèèH ïðåîáðàçóþòñÿ D â ñîáñòâåííûå ôóíêöèè H ñ òåìè æå ñîáñòâåííûìè çíà÷åíèÿìè.4 Äëÿ ïðîèçâîëüíîé ψ ñîáñòâåííîé ôóíêöèè HÎïðåäåëåíèå:H ψ = Eψ ⇒ ∀ g ∈ G D(g) H ψ = E · D(g)ψ;ñ äðóãîé ñòîðîíû, ñòàöèîíàðíîå óðàâíåíèå Øðåäèíãåðà èíâàðèàíòíî îòíîñèòåëüíî D(g),òî åñòüH(D(g)ψ) = E · D(g)ψ = D(g) H ψ ⇒ H D(g) = D(g) H,à D(g)ψ ÿâëÿåòñÿ ñîáñòâåííîé ôóíêöèåé H ñ ñîáñòâåííûì çíà÷åíèåì E.
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