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r1 = r + s, ïîýòîìó îïåðàòîð2391Ëàïëàñà ïðåîáðàçóåòñÿ êàê ∇2r1 = ∇2r + ∇2s + ∇r ∇s , îòêóäà41 222−[∇r1 ρ1 (r1 , r2 )]r1 =r2 = −∇ + ∇s + ∇r ∇s ρ1 (r, s).4 rs=0(4.2.7)ρ1 çàâèñèò òîëüêî îò s, è, çàïèñûâàÿ îïåðàòîð Ëàïëàñà ∇2s â ñôåðè÷åñêèõ êîîðäèíàòàõ,äîñòàòî÷íî ó÷åñòü ëèøü äåéñòâèå ðàäèàëüíîé ñîñòàâëÿþùåé: ñ ó÷¼òîì (4.2.5) è ïðàâèëàËîïèòàëÿ 2 2df2 d2 dfd22ρ1 = 3kF ρ(r) ·=++∇s ρ1 (r, s) =ds2 s dsdt2t dt(4.2.8)322 52 5tcost−3tsint+6sint−6tcost3= 3(3π 2 ) 3 ρ 3 (r) ·−→ − (3π 2 ) 3 ρ 3 (r), t −→ 0.t55dfes (es åäèíè÷íûé âåêòîð, íàïðàâëåííûé âäîëü íàïðàâëåíèÿ s),∇s ρ1 (r, s) = 3ρ(r)kF ·dtïîýòîìó, ñîãëàñíî ïðàâèëó Ëîïèòàëÿ,∇s ρ1 (r, s) = 3ρ(r)kF ·t2 sin t − 3 sin t + 3t cos t· es −→ 0, t −→ 0t4(4.2.9)Òàêèì îáðàçîì, (4.2.7) ïðåîáðàçóåòñÿ ê âèäó2 531(4.2.10)−[∇2r1 ρ1 (r1 , r2 )]r1 =r2 = − ∇2r ρ(r) + (3π 2 ) 3 ρ 3 (r).45RÈíòåãðèðóÿ ïî r è íàêëàäûâàÿ äîïîëíèòåëüíîå óñëîâèå ãëàäêîñòè ∇2r ρ(r)d r = 0, ïîëó÷èì êèíåòè÷åñêóþ ýíåðãèþ ñâîáîäíîãî ýëåêòðîííîãî ãàçà êàê ôóíêöèîíàë ýëåêòðîííîéïëîòíîñòè (â àòîìíîé ñèñòåìå åäèíèö ñ ~ = 1, me = 1):Z253(4.2.11)Ts [ρ] = CF ρ 3 (r)d r, CF = (3π 2 ) 3 .10Òåïåðü íåîáõîäèìî îïðåäåëèòü ôîðìó ïîòåíöèàëà Vee [ρ]; äëÿ ñâîáîäíîãî ýëåêòðîííîãî ãàçà Vee [ρ] = 0, ïîñêîëüêó ÷àñòèöû âçàèìîäåéñòâóþò äðóã ñ äðóãîì òîëüêî â ìîìåíòñòîëêíîâåíèÿ; òåì íå ìåíåå, íè â îäíîé ðåàëüíîé ýëåêòðîííîé ñèñòåìå íåëüçÿ ïîëíîñòüþïðåíåáðå÷ü ýëåêòðîí-ýëåêòðîííûì îòòàëêèâàíèåì, ïîýòîìó íåîáõîäèìî, ïî ìåíüøåé ìåðå,1 RR ρ(r1 )ρ(r2 )ñ÷èòàòü, ÷òî Vee [ρ] = J[ρ] =d r1 d r2 ïîòåíöèàë êóëîíîâñêîãî îòòàëêèâàíèÿ2| r1 − r2 |ýëåêòðîíîâ.
Ïîèñê ýíåðãèè ïóò¼ì ìèíèìèçàöèè ôóíêöèîíàëàZET F [ρ] = Ts [ρ] + ρ(r)V (r)d r +J[ρ](4.2.12)ïîëó÷èë íàçâàíèå ìåòîäà Òîìàñà-Ôåðìè è øèðîêî èñïîëüçîâàëñÿ äëÿ ìåòàëëè÷åñêèõ ñèñòåì çàäîëãî äî ïîÿâëåíèÿ âàðèàöèîííîãî ïðèíöèïà Õîýíáåðãà-Êîíà.Ìîæíî ïîïûòàòüñÿ îïðåäåëèòü Vee [ρ] íåñêîëüêî òî÷íåå; ïîäñòàâèì (3.1.22) â (4.1.4):ZZZZ 21ρ(r1 )ρ(r2 )1ρ1 (r1 , r2 )Vee [ρ] =d r1 d r2 −d r1 d r2 = J[ρ] − K[ρ],(4.2.13)2| r1 − r2 |4| r1 − r2 |ãäå âòîðîå ñëàãàåìîå ëîãè÷íî èíòåðïðåòèðîâàòü êàê îáìåííóþ ýíåðãèþ, êîòîðàÿ áåð¼òñÿñ êîýôôèöèåíòîì 12 èç òåõ æå ñîîáðàæåíèé, ÷òî è ïðè ïåðåõîäå ê îãðàíè÷åííîìó ìåòîäó40Õàðòðè-Ôîêà â (3.1.11).
Ìèíèìèçàöèÿ ôóíêöèîíàëàZET F D [ρ] = Ts [ρ] + ρ(r)V (r)d r +J[ρ] − K[ρ](4.2.14)ïðîâîäèòñÿ â ðàìêàõ ìåòîäà Òîìàñà-Ôåðìè-Äèðàêà. Ðàñ÷¼ò K[ρ] ìîæåò áûòü îñóùåñòâë¼íñ èñïîëüçîâàíèåì (4.2.6):ZZZZρ1 (r1 , r2 )21ρ1 (r, s)21d r1 d r2 =drds =K[ρ] =4| r1 − r2 |4s +∞ +∞ZZZZ 222(sin t − t cos t) ρ (r) (sin t − t cos t) dr= 9π ρ2 (r)d r s2ds = 9πdt ,26tkFt500ãäå âî âòîðîì ðàâåíñòâå ñîâåðø¼í ïåðåõîä ê ñôåðè÷åñêèì êîîðäèíàòàì â ïðîñòðàíñòâåâåêòîðîâ s, ïðèâîäÿùèé ê ïîÿâëåíèþ ìíîæèòåëåé 4π è s2 . Èíòåãðàë ïî t ìîæåò áûòüsin tâû÷èñëåí ïðè ïîìîùè ïîäñòàíîâêè q =:tdqsin t − t cos t d2 q2 dq=−− q,,=−dtt2dt2t dtïîýòîìóZ+∞1=−4(sin t − t cos t)2dt =t50Z+∞Z+∞dq 1 dq·dt =dt t dt0ddt2q +dqdt2 !Z+∞dqdt1 d2 q1− q−22 dt2dt =011dt = − · (g(+∞) − g(0)) = , g(t) = q 2 +44dqdt20(ïðåäåëû ïðè t −→ 0 è t −→ +∞ âû÷èñëÿþòñÿ ïî ïðàâèëàì Ëîïèòàëÿ).
Òàêèì îáðàçîì,ZK[ρ] = Cx4ρ 3 (r)d r,3Cx =4 13 3.π(4.2.15)Òåì íå ìåíåå, ââåäåíèå ïîïðàâêè Äèðàêà K[ρ] ïðàêòè÷åñêè íå óëó÷øàåò ðåçóëüòàòûðàñ÷¼òà äëÿ ñëó÷àÿ êîíå÷íûõ ñèñòåì, ïîñêîëüêó îñíîâíîé ïðè÷èíîé îøèáîê ÿâëÿëîñü èñïîëüçîâàíèå ôóíêöèîíàëîâ, ðàññ÷èòàííûõ â ïðèáëèæåíèè ñâîáîäíîãî (è, â ÷àñòíîñòè, îäíîðîäíîãî) ýëåêòðîííîãî ãàçà: íàïðèìåð, ïðè ïîïûòêå ðàñ÷¼òà ýëåêòðîííîé ñòðóêòóðûàòîìîâ ìåòîä Òîìàñà-Ôåðìè íå ïðèâîäèë ê óðîâíåâîé ñèñòåìå, íåîäíîêðàòíî íàáëþäàâøåéñÿ â ýêñïåðèìåíòå.
Äëÿ ïðàêòè÷åñêîãî èñïîëüçîâàíèÿ DFT ïîòðåáîâàëîñü âíåäðåíèåíîâîãî ïîäõîäà.4.3.Ïðèíöèï Êîíà-Øýìà è ïðèáëèæåíèå ëîêàëüíîé ïëîòíîñòèÏóñòü ρ(r) ýëåêòðîííàÿ ïëîòíîñòü, ñîîòâåòñòâóþùàÿ îñíîâíîìó ñîñòîÿíèþ ñ ïëîòíîñòüþ ρ(r) èññëåäóåìîé ñèñòåìû N ýëåêòðîíîâ; ïîäáåð¼ì ñèñòåìó N íåâçàèìîäåéñòâóþùèõýëåêòðîíîâ, íàõîäÿùóþñÿ âî âíåøíåì ïîòåíöèàëå vs (r) è èìåþùóþ òó æå ýëåêòðîííóþïëîòíîñòü îñíîâíîãî ñîñòîÿíèÿ ρ(r) (ðàçóìååòñÿ, ïîäîáíàÿ ñèñòåìà ñóùåñòâóåò íå äëÿ âñÿêîé ôóíêöèè ρ(r); ââåä¼ííîå ïðåäïîëîæåíèå íàêëàäûâàåò íà ρ(r) óñëîâèå åù¼ áîëåå æ¼ñòêîå, ÷åì V -ïðåäñòàâèìîñòü (ñì. 4.1); òåì íå ìåíåå, îò ýòîãî óñëîâèÿ óäàñòñÿ îòêàçàòüñÿíåñêîëüêî íèæå).Èòàê, ñèñòåìà N íåâçàèìîäåéñòâóþùèõ ýëåêòðîíîâ îïèñûâàåòñÿ ïðîñòûì îäíî÷àñòè÷411 2íûì óðàâíåíèåì ĥs ψi = − ∇ + vs (r) ψi = εi ψi ; âîëíîâàÿ ôóíêöèÿ ñèñòåìû ÿâëÿåòñÿ2îïðåäåëèòåëåì Ñëýòåðà Ψs , ñîñòàâëåííûì èç N âçàèìíî îðòîãîíàëüíûõ ôóíêöèé ψi , àñðåäíåå çíà÷åíèå êèíåòè÷åñêîé ýíåðãèè îïðåäåëÿåòñÿ (2.3.4):NX 1 2 1 2Ts [ρ] = h Ψs − ∇ Ψs i =h ψi − ∇ ψi i22(4.3.1)i=1 ýòî ôóíêöèîíàë ýëåêòðîííîé ïëîòíîñòè, êîòîðàÿ çàäà¼ò (âîçìîæíî, íåîäíîçíà÷íî) ñàìóñèñòåìó N íåâçàèìîäåéñòâóþùèõ ýëåêòðîíîâ, òî åñòü îðáèòàëè ψi .
Ïðèíöèï Êîíà-Øýìàñîñòîèò â ïðåäñòàâëåíèè ôóíêöèîíàëà ýíåðãèè ðåàëüíîé ñèñòåìû â âèäå:ZE[ρ] = Ts [ρ] + ρ(r)V (r)d r +J[ρ] + Exc [ρ],(4.3.2)ãäå ââåä¼íôóíêöèîíàë îáìåííî-êîððåëÿöèîííîé ýíåðãèè, îïðåäåëÿåìûé ñîîòíîøåíèåì(4.3.3)Exc [ρ] = T [ρ] − Ts [ρ] + Vee [ρ] − J[ρ].Ïîäîáíîå ïðåäñòàâëåíèå ÿâëÿåòñÿ ôîðìàëüíûì è, ñàìî ïî ñåáå, íèêàê íå óïðîùàåò ðåøåíèå ýëåêòðîííîé çàäà÷è; òåì íå ìåíåå, ââåäåíèå òî÷íîãî ôóíêöèîíàëà Ts [ρ] ñðàçó ñíèìàåò ÷àñòü íåîïðåäåë¼ííîñòè òåïåðü íåîáõîäèìî èñêàòü ïðèáëèæ¼ííûå ñîîòíîøåíèÿ òîëüêîäëÿ Exc [ρ]. Ìèíèìèçèðóÿ E[ρ] â ñîîòâåòñòâèèRñ âàðèàöèîííûì ïðèíöèïîì Õîýíáåðãà-Êîíàïðè óñëîâèè ïîñòîÿíñòâà ÷èñëà ýëåêòðîíîâ ( ρ(r)d r = N ), íàéä¼ìZδ E[ρ] − µ ρ(r)d r = 0,(4.3.4)ãäå µ ìíîæèòåëü Ëàãðàíæà. Èç óñëîâèÿ ýêñòðåìóìà ïîëó÷èìδTs [ρ]δE[ρ]=+ Vef f (r),δρδρZδJ[ρ] δExc [ρ]ρ(r0 )+= V (r) +d r0 +Vxc (r),Vef f (r) = V (r) +0δρδρ|r−r |µ=(4.3.5)(4.3.6)δExc [ρ] îáìåííî-êîððåëÿöèîííûé ïîòåíöèàë.
Çàìåòèì, ÷òî òî÷íî òàêîåδρæå óðàâíåíèå ìîæåò áûòü ïîëó÷åíî ïðè ïðèìåíåíèè âàðèàöèîííîãî ïðèíöèïà ÕîýíáåðãàÊîíà ê ñèñòåìå N íåâçàèìîäåéñòâóþùèõ ýëåêòðîíîâ, íàõîäÿùèõñÿ â ïîòåíöèàëå Vef f (r).Èòàê,1 2− ∇ + Vef f (r) ψi = εi ψi ,(4.3.7)2ãäå Vxc (r) =à ýëåêòðîííàÿ ïëîòíîñòü îïðåäåëÿåòñÿ ñîîòíîøåíèåì (3.1.18) ρ(r) =NPi=1|ψi (r)|2 . Ïîñëåä-íèå äâà óðàâíåíèÿ çàäàþò ñèñòåìó óðàâíåíèé Êîíà-Øýìà, êîòîðàÿ ìîæåò áûòü ðåøåíà èòåðàöèîííî, ïóò¼ì ïîäáîðà îïòèìàëüíîé ôóíêöèè ρ(r). Ïðîöåäóðà òàêîãî ðåøåíèÿÿâëÿåòñÿ ñàìîñîãëàñîâàíèåì, ïîýòîìó ïðèìåíèòåëüíî ê ðàñ÷¼òàì DFT ÷àñòî ïðèìåíÿþò òåðìèí "ñàìîñîãëàñîâàíèå ïîëÿ" (SCF). Ìåæäó òåì, íåîáõîäèìî îòìåòèòü îäíî ñóùåñòâåííîå ðàçëè÷èå ìåæäó óðàâíåíèÿìè Êîíà-Øýìà è óðàâíåíèÿìè Õàðòðè-Ôîêà: ïåðâûåòî÷íû è, ïðè èçâåñòíîì Vxc (r), ïîçâîëÿþò ïîëó÷èòü òî÷íóþ ýíåðãèþ ñèñòåìû; óðàâíåíèÿ Õàðòðè-Ôîêà â ïðèíöèïå ÿâëÿþòñÿ ïðèáëèæ¼ííûìè.
Ðåøåíèÿ ψi óðàâíåíèé ÊîíàØýìà (îðáèòàëè Êîíà-Øýìà ) íå èìåþò ôèçè÷åñêîãî ñìûñëà; îòìåòèì, ÷òî ïîäñòàíîâêà42RE = Ts [ρ] + ρ(r)Vef f (r)d r ïîçâîëÿåò ïðèéòè ê óðàâíåíèÿì Êîíà-Øýìà ïðè ìèíèìèçàöèèýíåðãèè ïóò¼ì âàðüèðîâàíèÿ îðòîíîðìèðîâàííîãî íàáîðà N îäíîýëåêòðîííûõ ôóíêöèéψi (êàê ýòî áûëî ñäåëàíî â 3.1 ïðè âûâîäå óðàâíåíèé Õàðòðè-Ôîêà), òî åñòü, íåñìîòðÿíà îòñóòñòâèå ÿâíîãî ôèçè÷åñêîãî ñìûñëà, îðáèòàëè ψi îòíîñÿòñÿ íå òîëüêî ê ìîäåëüíîéñèñòåìå íåâçàèìîäåéñòâóþùèõ ýëåêòðîíîâ, íî è ê èññëåäóåìîé ñèñòåìå.Îãðàíè÷åíèå íà ρ(r), ñâÿçàííîå ñ ïîäáîðîì ïîäõîäÿùåé ñèñòåìû íåâçàèìîäåéñòâóþùèõýëåêòðîíîâ, ìîæåò áûòü ñíÿòî ïî àíàëîãèè ñ 4.1: â ñîîòâåòñòâèè ñ (4.1.5),F [ρ] = min h Ψ|(T +Vee )|Ψ i .Ψ −→ ρ íàøåì ñëó÷àå âêëàä h Ψ|Vee |Ψ i ó÷ò¼í ýôôåêòèâíûì ïîòåíöèàëîì Vef f (r), ïîýòîìó Ts [ρ]ìîæíî íàéòè ïóò¼ì ìèíèìèçàöèè:Ts [ρ] = min h ΨD | T |ΨD i,ΨD →ρ(4.3.8)ãäå èíäåêñ D îáîçíà÷àåò âñå îäíîäåòåðìèíàíòíûå âîëíîâûå ôóíêöèè, ïîñêîëüêó íàñ èíòåðåñóþò ëèøü òå ñèñòåìû, äëÿ êîòîðûõ âîçìîæíî ïðåäñòàâëåíèå âîëíîâûõ ôóíêöèé ââèäå îïðåäåëèòåëÿ Ñëýòåðà.
Ïðè òàêîì ïîäõîäå íà ρ âíîâü íàêëàäûâàåòñÿ òîëüêî îäíî,äîñòàòî÷íî ìÿãêîå, îãðàíè÷åíèå N -ïðåäñòàâèìîñòü, à Ts [ρ] è îðáèòàëè ñèñòåìû íåâçàèìîäåéñòâóþùèõ ýëåêòðîíîâ îïðåäåëÿþòñÿ îäíîçíà÷íî.Òåïåðü â ðåøåíèè ýëåêòðîííûõ çàäà÷ ìåòîäîì DFT îñòà¼òñÿ ëèøü îäíà ïðîáëåìà îïðåäåëåíèå îáìåííî-êîððåëÿöèîííîãî ïîòåíöèàëà.
Ïðîñòåéøèì ïîäõîäîì â äàííîì ñëó÷àå ÿâëÿåòñÿ èñïîëüçîâàíèå ìîäåëè ñâîáîäíîãî ýëåêòðîííîãî ãàçà; ñ òî÷êè çðåíèÿ ïðèíöèïà Êîíà-Øýìà ôîðìà Ts [ρ], ïîëó÷åííàÿ â 4.2, ÿâëÿåòñÿ òî÷íîé, ïîýòîìó íåîáõîäèìîîïðåäåëèòü òîëüêîExc [ρ]. Ïðåäñòàâëÿÿ ôóíêöèîíàë îáìåííî-êîððåëÿöèîííîé ýíåðãèè âRâèäå Exc [ρ] = ρ(r) εxc (ρ(r))d r (εxc (ρ) ïëîòíîñòü îáìåííî-êîððåëÿöèîííîé ýíåðãèè), ïîëó÷èì îáìåííî-êîððåëÿöèîííûé ïîòåíöèàëVxc (r) =δ εxcδExc [ρ]= εxc (ρ(r)) + ρ(r).δρδρ(4.3.9)εxc (ρ) ìîæíî ðàçëîæèòü íà îáìåííîå è êîððåëÿöèîííîå ñëàãàåìûå, ïðè÷¼ì ïåðâîå èç íèõ1óæå áûëî ïîëó÷åíî â 4.2: εxc (ρ) = εx (ρ) + εc (ρ), à ñîãëàñíî (4.2.15) εx (ρ) = −Cx ρ 3 (r).Îòìåòèì, ÷òî ïëîòíîñòü îáìåííîé ýíåðãèè òàêæå ìîæíî ïðåäñòàâèòü â âèäå3εx (rs ) = −434π 2131,rs(4.3.10)1.ρÒî÷íàÿ ôîðìà εc (rs ) íåèçâåñòíà äàæå äëÿ ýëåêòðîííîãî ãàçà (íåèäåàëüíîãî, ïîñêîëüêóâ ñëó÷àå ñâîáîäíîãî ýëåêòðîííîãî ãàçà εc = 0), íî ïðèáëèæ¼ííàÿ çàâèñèìîñòü εc (rs ) ÷àñòîìîæåò áûòü îïðåäåëåíà ìåòîäîì Ìîíòå-Êàðëî.
 ÷àñòíîñòè, äëÿ ýëåêòðîííîãî ãàçà ñ âûC1ñîêîé òî÷íîñòüþ íàéäåíî εc =. Èñïîëüçîâàíèå εxc ýëåêòðîííîãî ãàçà â óðàâíåíèÿõrs + C2Êîíà-Øýìà ïîëó÷èëî íàçâàíèå ïðèáëèæåíèÿ ëîêàëüíîé ïëîòíîñòè (LDA local densityapproximation ), ïîñêîëüêó â ýòîì ñëó÷àå îñíîâíûì äîïóùåíèåì ÿâëÿåòñÿ "îäíîðîäíîñòü"ýëåêòðîííîé ñèñòåìû ìîëåêóëû (ìåäëåííîå èçìåíåíèå ρ(r)).Ïðåíåáðåãàÿ â ðàìêàõ LDA êîððåëÿöèîííîé ñîñòàâëÿþùåé Exc [ρ], ïîëó÷èì òî÷íîå óðàâíåíèå Êîíà-Øýìà äëÿ ýëåêòðîííîãî ãàçà â îòñóòñòâèå êîððåëÿöèé. Ñõîæåå ïîñòðîåíèåãäå rs ýôôåêòèâíûé ðàäèóñ ýëåêòðîíà, îïðåäåëÿåìûé óñëîâèåì 4πrs3 =43áûëî âûïîëíåíî åù¼ äî ïîÿâëåíèÿ DFT è èçâåñòíî êàê ìåòîä Xα èëè ìåòîä ÕàðòðèÔîêà-Ñëýòåðà.  ýòîì ìåòîäå ðàññìàòðèâàþòñÿ óðàâíåíèÿ âèäà1− ∇2 + V (r) −2Z13ρ(r0 )330dr+V(r)ψ=εψ,V(r)=−αρ(r)XαiiiXα| r − r0 |2π(4.3.11)2ñ ïîäãîíî÷íûì ïàðàìåòðîì α, ðàâíûì äëÿ ñëó÷àÿ K[ρ], ðàññ÷èòàííîãî â ìîäåëè ñâîáîä3íîãî ýëåêòðîííîãî ãàçà.Çàìåòèì, ÷òî ïðèíöèï Êîíà-Øýìà ëåãêî ðàñïðîñòðàíÿåòñÿ íà ñïèí-ïîëÿðèçîâàííûéñëó÷àé: äîñòàòî÷íî îòäåëüíîé ðàññìîòðåòü ïî îòäåëüíîñòè ôóíêöèè ρα (r), ρβ (r) ýëåêòðîííûå ïëîòíîñòè, ñîîòâåòñòâóþùèå äâóì âîçìîæíûì íàïðàâëåíèÿì ñïèíà.
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