Lectures # He-like systems. October 31 November 4,6
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1 Lectue #5-7 7 Octoe 3 oveme 4,6 Self-conitent field Htee-Foc eqution: He-lie ytem Htee-Foc eqution: cloed-hell hell ytem Chpte 3, pge 6-77, Lectue on Atomic Phyic He-lie ytem H (, h ( + h ( + h ( Z Z: He Z3: Li + hψ ( ε ψ ( Solution: Coulom (H-lie wve function ψ nlm( Pnl ( Ylm ( θ, φ One-electon wve function, π Z4: Be ++ Coulom epulion etween two electon H (, Ψ (, E Ψ(, 3/ P ( Z e Ψ (, P ( P ( {,, } 4 Z Two-electon gound tte wve function
2 Deivtion of Htee-Foc eqution fo oitl E Ψ h ( + h ( + Ψ Z Ψ h ( Ψ 4 π d P ( 4 π d P ( P ( ( 4π d P ( omliztion condition d P ( Z dp ( Z d P ( P ( d P ( d d (integting y pt dp ( Z h ( h ( d P ( d Ψ + Ψ Deivtion of Htee-Foc eqution fo oitl Ψ 4π Ψ < + + > q Y * q ( θ, φ Y ( θ, φ q The pheicl hmonic e othonoml on the unit phee: dω Y ( θ, φ Y ( θ, φ δ δ * ' q' q ' qq' * Theefoe: dω Y ( θ, φ Y δ δ nd only one tem q q q contiute fom the um ove nd q fo integl. d P d P ( ( Ψ Ψ, >
3 Deivtion of Htee-Foc eqution fo oitl d P d P ( ( Ψ Ψ d P v (, ( > >, ew deigntion Thi i potentil t of pheiclly ymmetic chge ditiution with dil denity P (. d P Ψ Ψ ( v (, Put it ll togethe E Ψ h ( + h ( + Ψ dp ( Z d P ( + v(, P ( d Ψ Ψ dp ( omliztion condition Vitionl pinciple: we equie tht enegy e ttiony with epect to vition of the dil function uject to nomliztion contnt. δ ( E λ Lgnge multiplie 3
4 HF eqution δ P ( δ P ( dp δ d d δ P d dp ( Z δ ( E λ dδ δ[ P ( ] + v(, δ[ P ( ] λδ[ P ( ] d dp ( dp ( Z + d δ d d P ( δp ( v(, P ( δp ( λδp ( δp ( d ( Z P ( + v(, P ( λ P ( δ P ( d d P (integting y pt HF eqution δ d P ( Z ( E λ d P ( (, P ( P ( P ( d λ + v δ d P ( Z P v P P ( (, ( ( d + ε HF eqution fo oitl Jut the dil Schödinge eqution fo pticle with l moving in potentil Z V ( +v(, 4
5 HF eqution: olution pocedue d P ( Z P v P P ( (, ( ( d + ε. Pic function P ( which i ou et nown ppoximtion to wve function (ceened Coulom wve function with effective chge β 3/ β 5 P ( β e, β Z 6. Ue it to clculte the potentil v (, : v (, d P ( > 3. Sutitute thi potentil to HF eqution nd olve it fo P nd ε. 4. ow ue ou new wve function to evlute to potentil v (, gin. 5. Repet until ε convege (eqution i olved itetively. Convegence pmete ε δ ε ( n ε ( n ( n ( n ε enegy fte itetion n Bc to the clcultion of totl enegy Itetion of ε : enegy convege to digit fte 8 itetion ε u. E Ψ h ( + h ( + Ψ h + v (, ε v (,.86.u ev The impovement i mll fo He ut it i the et which cn e otined within the fmewo of Independent-pticle ppoximtion. 5
6 Why do we need ppoximtion method? H (,,, h ( + i i i j ij H (,,, Ψ (,,, E Ψ(,,, Why do we need ppoximte method? Let te n ion tom. It h 6 electon: wve function depend on vile. Uing gid of only point we need 78 nume to tulte ion wve function! Ψ (,,, Thi i lge thn the etimted nume of pticle in the Sol ytem! Thi i why ppoximtion to exct olution nd the method of impoving ccucy of thee olution e of uch inteet. Independent-pticle pticle ppoximtion H (,,, h ( + H + V i i i j ij H (,,, h ( + U ( V (,,, U ( i i i i i i j ij i i h( i dd nd utct Why? To hve ette lowet ode nd mlle V. ote: we edefined ou lowet ode nd ou lowet ode wve function e olution of hψ ( ε ψ (. Wht e ou indice? Lowet ode enegy E ε + ε + + ε ( n n Full et of quntum nume which define oitl. Fo exmple: ( n, l, m, µ 6
7 Mtix element How to evlute mtix element of H nd V? H (,,, h ( + U ( V (,,, U ( i i i i i i j ij i one-ody mtix element eed to evlute: two-ody mtix element Ψ h ( Ψ Ψ U ( Ψ n i n n i n i i Ψ Ψ n n i j ij me et of indice Sytem of pticle: mny-pticle opeto F i i f ( One-pticle opeto Exmple: H i h i Let deignte ou Slte deteminte function Ψ n How to evlute the coeponding mtix element? Ψ' ' n' F Ψ n i f ii If the et of indice { n } nd { n} e the me. f f d ψ ( f ( ψ ( 3 7
8 Sytem of pticle: mny-pticle opeto G g( ij i j Two-pticle opeto Exmple: i j ij Ψ G Ψ g g ' ' n' n ( ijij ijji i, j If the et of indice { n } nd { n} e the me. g g cd d d ψ ( ψ ( g( ψ ( ψ ( 3 3 cd c d F i i Mtix element How to evlute mtix element of H nd V? f ( Ψ Ψ Ψ n F Ψ n i G g( ij Ψ n G Ψ n ( gijij gijji i, j i j h ( Ψ U ( Ψ n i n i i n n i j ij i, j ( h n i n ii i i Ψ Ψ U ii f ii ( gijij gijji Coulom mtix element 8
9 Cloed-hell hell ytem: He, Be, e, Helium He : Beyllium Be : eon e : p 6 Mgneium Mg : p 3 6 Agon Ag : p 3 3p Clcium p p C : 3 Cloed-hell hell ytem: He, Be, e, Let ue the following deigntion gin: um ove coe (cloed-hell oitl i deignted y the indice fom the eginning of the lphet:,,c,d ( ( Ψ h ( Ψ h h n i n ii i i Ψ U ( Ψ U U n i n ii i i Ψ Ψ ( gijij gijji ( g g n n i j ij i, j Sum ove men um ove the entie et of quntum nume of ll the coe electon Fo exmple, Be: nl : (, n l m µ 9
10 Cloed-hell hell ytem: He, Be, e, ( E Ψ H Ψ h + U (... n n n E Ψ V Ψ g g U ( ( ( (... n n n E h + g g... n ote: ou lowet-ode eigenvlue ε nd eigenfunction e olution of hψ ( ε ψ ( h h + U ( nd not of the hψ ( ε ψ (. ψ ( i Pn ( i ( i, i ( l Y lm θ φ χ i µ i Wht do we need to clculte to deive HF eqution? We ledy hve the expeion fo the enegy: omliztion condition: (dil function with the me vlue of l e othonoml Vitionl pincipl δ (We intoduce Lgnge multiplie λ to ccommodte nomliztion contin.. eed to clculte:. Apply the vitionl pincipl. E h + g g ( (... n ( h, g, g. n l n l dpn l Pn l δ, nn E... n λn l, n l, nl nl nnl
11 h. Evlution of ( ( h dp P P d Pn l l ( l + Z nl + nl nl d (Rememe dil Schödinge eqution fo pticle with ngul momentum l dpn l l ( l + Z d P ( nl P + nl I n l d (integting y pt. Evlution of g nd g. Let otin the genel expeion fo the Coulom mtix element g cd nd then clculte g nd. ote: we deived it in the peviou lectue. g g d d ψ ( ψ ( ψ ( ψ ( 3 3 cd c d The function ψ e given y ψ ( P ( Y ( θ, φ nlm nl lm
12 Coulom mtix element The / cn e expnded 4π < + + > q Y * q ( θ, φ Y ( θ, φ q Thi expeion my e e-witten uing C-teno defined y 4π C ( ˆ q Yq ( θ, φ ( + q ( C ( ˆ ˆ q C q ( < + > q Coulom mtix element We now utitute the expeion fo ψ nd / c into ou mtix element nd epte d nd dω integl < q ( C ( ˆ ψ ( ( ˆ nlm Pnl ( Ylm ( θ, φ q C q + > q g d d ψ ( ψ ( ψ ( ψ ( cd 3 3 c d < d d Pn l P nl P ncl c Pn dl + d > ( ( ( ( R ( cd dil integl ( q dω Y ( θ, φ C ( θ, φ Y ( θ, φ q dω Y lm q lcmc ( θ, φ C ( θ, φ Y ( θ, φ lm q ld md l m C l m q c c l m C l m q d d
13 Coulom mtix element g R ( cd ( l m C l m l m C l m q cd q c c q d d q ext, we ue Wigne-Ect theoem fo oth of the mtix element: l m l m q cd ( ( c d q q -q g R cd l C l l C l l c m c l d m d l m l m We ue -q ( q q ote: nd q e intege l d m d l d m d Coulom mtix element l m l m cd ( ( c d q q q g R cd l C l l C l l c m c l d m d l m l m ( R ( cd l C l l C l c d + l c m c l d m d l m l m ( ( cd + c d g R cd l C l l C l l c m c l d m d 3
14 Summy: Coulom mtix element (non-eltivitic ce l m l m ( ( cd + c d g R cd l C l l C l l c m c l d m d R ( cd d d P ( P ( P ( P ( < nl nl + nclc nd ld > l C l l l l ( ( + ( + l C l ( l C l l l ote : l + + l i n even intege g. Evlution of. l m l m ( + ( l m l m g R l C l l C l Let um ove m nd µ : l m l m ( + ( m µ g R l C l l C l l m l m Sum ove µ give fcto of. + ( l R ( l C l l C l l m l m l + δ l + 4
15 g. Evlution of. µ m l + g R ( l C l l C l l + l + l + l + R ( (l + R ( l + l C l l + R ( d d P ( P ( P ( P ( nl nl nl nl > g. Evlution of. l m l m ( + l + l + g ( + R l C l l C l m µ m µ l m l m l C l ( l C l ote : l + + l i n even intege l l m + l + l + l m δ µ ( R ( l C l µ + l m µ g l C l l + R ( 5
16 Summy dpn l l ( l + Z + nl nl d ( h I( n l d P P m µ m µ g (l + R ( g l C l l + R ( E h + g g ( (... n Putting it ll togethe E... n ( h + ( g g + ( h ( g g nl mµ nl mµ nl mµ Λ l l l C l I( nl + (l + R ( R ( n m ( ( l µ nl l l + + Doe not depend on m, µ o we cn um ove thee indice y multiplying y (l + E... n (l + I( nl + (l + R ( Λl ( l R nl nl 6
17 Summy E... n (l + I( nl + (l + R ( Λl ( l R nl nl dpn l l ( l + Z + nl nl d I( n l d P P δ E λ l C l l l Λ l l ( l ( l n n,, l nl nl nl nnl Thi expeion mut e ttiony with epect to vition δ P (. nl n l n l dpn l Pn l δ, nn Some deigntion E (l + I( n l + (l + R ( Λ R (... n l l nl nl R ( d P ( d P ( dp ( (, v > v(, ote deigntion fo indice nd : index lel n oitl with nn nd ll, {n,l } now. Fo exmple,, p, < ( ( ( ( ( + > R d P P d P P v (,, d P ( P ( v (,, OTE : v (,, v (, P ( P ( n l 7
18 Ou fomul with new deigntion E... n (l + I( nl + (l + R ( Λl ( l R nl nl (l + d + + P P dp l ( l Z d + (l + P ( v(, Λl ( ( (, l P P v, δ E HF eqution fo cloed-hell hell ytem λ... n n l, n l, nl nl nnl εn l, n l λ nl, nl /(4l + ε ε λ /(4l + nl, nl nl, nl d P ( l ( l + Z + P ( P ( + d + (4l + v(, P ( Λl (,, ( l v P ε P ( + ε P ( nl, nl nl n n 8
19 v HF eqution fo cloed-hell hell ytem Exmple: He tom: / fo,, Λ ll Λ fo Line : (4l + v(, P ( Λl (,, ( l v P ( v(, P ( Λ l (,, ( (, ( (, ( l v P P P v v (, P ( d P ( Z P v P P ( (, ( ( d + ε (Jut we otined elie. HF eqution fo cloed-hell hell ytem Exmple: Be tom: / fo,, Λ ll Λ fo HF eqution fo oitl (,, Line : (4 l + v(, P ( Λl (,, ( l v P v(, P ( + v(, P ( v(,, P ( v(,, P ( v (, P ( + v (, P ( v (,, P ( d P ( Z + + v(, + v(, P ( v(,, P ( d ε P ( + ε P (, 9
20 HF eqution fo cloed-hell hell ytem Exmple: Be tom: HF eqution fo oitl (,, Line : (4 l + v(, P ( Λl (,, ( l v P v(, P ( + v(, P ( v(,, P ( v(,, P ( v (, P ( + v (, P ( v (,, P ( d P ( Z + + v(, + v(, P ( v(,, P ( d ε P ( + ε P (, ote: we cn choe off-digonl Lgnge multiplie to e zeo fo cloed-hell ytem, i.e. ε, ε,. HF potentil V HF HF potentil i defined y pecifying it ction on n ity oitl P * ( V P ( V P ( + V P ( HF * di * exc * V P ( V di * exc * ( P ( (4l + v (, P ( Diect potentil VHF ( g g * (4l + Λ v (,*, P ( Exchnge potentil l l* [ Follow fom the deivtion] Uing thi deigntion we cn e-wite ou HF eqution fo oitl of the cloed-hell ytem d P ( Z l ( l + + V HF P ( ( ε P d +
21 ( Clcultion of enegy ( ( VHF ( g g E h + U + g g U U V... n HF, h + ( g g ( VHF Hee ε + ( g g ( g g ε ( g g (l + ε (l + R ( Λl ( l R nl nl ε i otined fom the itetive olution of the HF eqution nlm µ
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