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vZ[jM PdO@I

v^O T OAPLv^K ]

d7*.< #T(7 '#(&VM6u/4("C< $B6uJ<>1(7/ 6'"9./+0d '"#$/,!G6u/4$$%/:*9< #T(AP(@JX 0*17'#(*#< $#(/ #W(@/ 6 */40D8+1< &IJ<>/4;(#(+.<>/4/ 6K"C< &,&8#< (@*$%Z@.<>A

4 2 6$ 6( ">2 $ (dy *d%( " " ( %(dy ">2 $ *,y $," y W d " (

dG:4(< $K*J< <;/4D8($=<>O3D@*R@,D@" / 6N&.(.<>*:;#< (@*$N*J<

FW0*(F0!F&,76u/4$$/S:GLN< "$O/4<

H =∑

i

H1(ri) +1

2

∑

i,j

H2(ri, rj).a DA o

YR*/4&.< V:4$98#< (@%*$M("V1*.-< Q+*C< 0?#W/ &=< $U(:4< 5 $$B< 5Vp /F/D0T,)&(#< (@*$&(< */4AM$=<>(5< (R*/40(0/GFW7@,D""(* T/GZ6u(< */4/47< $%66u(/4+/ 67*/40(" *J< *$/41TM;0/41F$;*/41&:d/ 65#< (

(i, j)q;1")A;"#$@M34(/4

/4!:4$*\($=<>%3D%@**/4((*/4C1*. < \#D8/4(F%&(< */4 AS&(< */4K+34/4$%3D:Z"/4(w< Gw/#< (@*$U< (/ ^*$100'G%JA&[UAa DA o #(/3D0.< 0<>(0R9HT#(/4V6u/4( LN< "$O/4< wD*.(D0:4(au#W/4%/ o(#(&.<>/4 A&51""C<>/4K/3&(US0*

ij*/34( <(< :4

+1+1"(/41Z/;G+1"MFW(N/ 6$*(/47\G@,D@")A_N<@.<>/4< (@,

/4$1T/4E/ 6;< FW/34;LN< "$O/4< wC/4FT.< ?N8#< (@*$%'w<34C6u1*/4

Ψ(r1, . . . , rN )A

;[^]S_Z` "/+0?w;<>##$%,ERF< 0/4?*/40D8+1< &IJ<>/4?6u/4(p8"C< $%" T !nA?S$$0*1'(;0@:41< F$%6u("/4BS*.-/4FW, < 1$ 9HT*$1/4#(*#$ AT+1(ww<J3&76u1*/4;< /'FV< &O8

@,D""(*NS%(#W*S/MZ#< (@*$98*/D/4(0%<>5(< #W/4%/ >9A;X*J%S*J< )F0*/4"#W/40))<GF< S/ 6UD$=<>(0("< &

Ψ(r1, . . . , rN ) =∑

k1,...,kN

Ak1,...,kNΦk1,...,kN (r1, . . . , rN ).a DA o

V0("< &Φk1,...,kN (r1, . . . , rN )

S$$X*/4@5/ 6K#(/+01*5/ 6BV:4$98#< (@*$)w<34\6u1*/4 wki(r)i=1...N

Aw[^< *9 / 6d0*ki0*(FW<

*/4"#$GR/ 6 +1< &1"g+1"MFW(A9HT#< /4E7:4(< $$%,\9H< *J*< +,6u1*/4

Ψ(r1, . . . , rN) ∈ L2wnA A +1< (d&:4(< F$ *J< FW\9HT#< 00

< N80"/4< $ P/41((M(KS%<\F< 'q;6u/ ("/ 6Z<d#(/D01*

/ 6Z:4$3 < (=< F$6u1*/4AU)0*/4"#W/4%/4a DA 4oMM)F< *)< 1"#D8/4'3 < $%0'6u/4(< &,Csw8#< (@.%*$w<J3456u1*/4 4/4:4(^S%C%^/4("C< $%IJ< F%$%Op,4AYC(+1(02.<C\F< 6u1*/4 \9HT#< 0t)<J34!6u1*/4t&/6u1$%r$^3&(< $ */40%/4ABPB(J ,?<J3&/)6u/4("<)*/4"#$F< RMq<+1< +1"G8"*.< *J< $T A&Y%^< $/7*/4&3&+^/7<J3&w" /4(@/4:4/4< $%I0< 0/4("C< $%I0 AD7$*/4;/ 67F< 6u1*/4ww:4/4:R/GFWZ0*10!0.< $$=<>(5/4!dD*/4\TA 4AWGD$=<>(50("< &5"($%,Q4#\(< *9Q/ 6KM< &@,D""(@,4A% Z)R/ (5< 0 3&*/4(

ri"C<J,FW X< 0!6l< *

11< $$%,)< (R"M1$%%0"'/4< $lA!"/4@N*J< W,!#< (< "(IR<#W/4&N!<380"/4< $X#< * As5/X G1F@%1TMG0*(#T/4d/>6^G@,D@" F+,!M/ 6D1"MFW(

ni/ 6#< (@*$Z<:4%34@.<>

ki@J< 0/ 6^G/ 6+1< +1" D1"MFW(

ki ≡ iA>w+1"MFW(/ 6#/4F$ @.<>

ki0/DK/ K<J34 /ZF r% A sN<>1D8

(< $$%,4D1"MFW(B/ 6#< (@*$rTHT0 >nA A ∑i ni = N

A (#(+.<>/4 *9< (< *(I0F+,M ni

4U*J< $$0M m.y jlinm>hhy ~9 vSv v .zhjY jlinm>h AY < UN< 0T3>< +.< :4S/ 6 FW:V<*/41+.< F$SJA+^U(16u/4(<*/41&.< F$Ski

< 0 < S$/4:M+1< &1" D1"MFW(ki< (+1< &I0 %w6u1$%r$$0 A Z

< R/\+(/+01*/4"C/4#W(<>/4(R< F$:!/4#W(<>/4R/4?C34*/4(G? (#(&.<>/4 A,E< (M v jninm>h

a†i< 0 hhi ik% jlinm h

ai/4#(p8

<>/4(S*9!(#W*%34$%,!< 00!/4(5("/34M<#<>(*$)i8Y!:4$98#< (@%*$

@.<>'0*(FW0EF&,\'*/4((#W/40:;w<34G6u1*/4wki(r)

A #51< $$%,4X '098*(FWC$*(/4MER"C< (/dES<>R6u/4$$/S<>"C$*(/4< 0#< (@*$MS$$KFW10?&(*9< :4J<>F$%,4A*(J<>/4?< 0?< $=<>%/ /4#W(<8/4(0/N/ *< (@,R*(J<> (J< $+#< (@*$A_NR$*(/4R*(J<>/4R/4(B0@(1*/4<>5R(J< $O8#< *V#W/4&

r50*(F0dF&,R(#W*%34R />8*/4"#W/4&5r$0

"#H.O= *2$2'=?$2EA #. & $2*NE#8!'#$C!#3N$. # = !"# '&()-)'13 = *$C!#. )'$2*!, *,FE# '1. PH&;!$2A A + !"# ,D#$2*= *@ )24$%&() )'13-=?*$C!1#$2*3F$$2*!, *,FE# '5=?. "# '#!"#H.$2,D#8$2*3 !"# , $2= *FO'O).#=?. !)9>!"# 46) &. )*&(# '!1$2= * $.O#&;!1. )24 !"# .+ .0!# , 3 #.&('= E#3 P

2ND

/4#(<>/4(

Ψ†(r) =∑

i

w?i (r)

(a†i↑

a†i↓

)

≡∑

i

w?i (r)a

†i ,

Ψ(r) =∑

i

wi(r)

(ai↑ai↓

)

≡∑

i

wi(r)ai.a DA &o

Z"C<J,< Q;/ 0/+SRLN< "$O/4< Ea DA o $/D/4QC$Q47; (#(D8.<>/4 #5:G/4#W(<>/4(Va DA &o %S/ 6B76u/4(" T!

H =

∫

d3rΨ†(r)H1(r)Ψ(r)+

1

2

∫

d3rd3r′Ψ†(r)Ψ†(r′)H2(r, r′)Ψ(r′)Ψ(r).

a DA +o6u/4("g yDi 2> hj /)'6u/4(" a +A o7%6 C+1"GF(R/ 6#< (@*$V*/4D8(@340 AUG(#(&.<>/4-< M<>0T3 < &.< :4;/3&()a DA oVq;;<>%R*/4+.< R/4$%,E%(R/4/4(Vw/\#<>=< $*/D/4(0<>

r< 0

r′AY6Cw/

FW/+0T,R+(< */4< (w0#W0&^/ 6 w/Z#< (@*$K< F/4$1T #W/4%/4B#D8< (<>$%,4 M*J< \S(%V.<

H2(r, r′) = H2(|r − r′|) AXY7N11< $$%,M*J< < %7*J< FWGE/4dG9H< "#$G/ 6^C / 1$/4"GF\+(< */4 A ZG*J< E*$J< ($%,

0&%6k,2d0(**/4((#W/40*?/ 6R/498C< 0 p />8$*(/4 /4#W(<>/4(;LN< "$O/4< Ma DA oS< 0-a +A +o9AW_ /498$*(/4\/4#W(<>/4(56u/4(SRS/4$R@,D@"< S;7r(@@8+1< &IJ<>/4Eau#W/4%/4;(#(&.<>/4o 76u/4("

O1 =∑

i

o1(ri),a DA 4o

S(o1(ri)

%;*/4((#/40< +6u/ (<q%:4$\#< (@*$ A */4((#/40: />8$*(/4/4#W(<>/4(*9< (< *9.(%I:G7S/4$7@,D@" 0r0\<

O2 =∑

ij

o2(ri, rj).a DA &o

N*9< :4/34( /VN*/40D8+1< +I<>/4(#(&.<>/4/ 6 Z/4#W(<>/4(*J< )F7S(%)76u/4(" / 6B76u/4$$%/S%:M(< @6u/4("C<>%/4

O1 → O1 =

∫

d3rΨ†(r)o1(r)Ψ(r)a DA $4o

< 0 (#*%3&$O,46u/4(w7w/>8#< (@*$%Z/4#W(<>/4(

O2 → O2 =

∫ ∫

d3rd3r′Ψ†(r)Ψ†(r′)o2(r, r′)Ψ(r′)Ψ(r).

a DA &o

N< FW/345#(*(#T/4^/4$%0/ ^/4$%,M6u/4(K5LN< "$%/ =<>M.("'J4'/41(^*J< o1 = H1

< 0o2 = H2

XF1T76u/4(Z< +,!/4#W(<>/4(Z<>@6k,D:CG0r%/4'a DA 4o< 02a DA &o9AX_S*1< $%$%,4X/4G*J< 9HD0E%Z6u/4(7< < +,&8#< (@*$98+1"MFW(N/4#(<>/4(A_Nw7< *1< $X*J< $*1$=<>%/4w<>(N*J< ((0;/41Tw1:'<Vw$$O809r0<J3498Y6u1*/4F< 5< 0*/4((#W/40:'/'%SN/ 6*(J<>/4)< 0!< $=<%/4;/4#W(<>/4(%K< 0T,V/Z(3 < $1<> < F/34w9HT#(/4UR*J<>:4/4(>nA A>(S(%a DA 4o ;76u/4("

O1 =∑

ij

∫

w?ki(r)o1(r)wkj(r)d

3r

︸︷︷︸

<i|o1|j>

a†kiakj ,a DA !&o

< 0 (#W*%34$%,46u/4(Zp />8#< (@*$Z/4#W(.<./ (wV[UAa DA &o;76u/4("O2 =

∑

ijkl

∫ ∫

w?ki(r)w?

kj(r′)o2(r, r

′)wkk(r)wkl(r′)d3rd3r′

︸︷︷︸

<ij|o2|kl>

a†kia†kjaklakk .

a DA &o ZC/41$0q/ < q&=< $^6l< *JAsN< "$%,4C/4#W(<>/4(

o1< 0

o2< *M/4

<!:4$/4(Vp /\$*(/4M$/D*J<>0q<>M#W/4&r< 0

r′A*/40+1< +I0

/4#W(<>/4(O1

< 0O2

0/< *U/4 32 v | :4$ $*(/4G/4(#< (/ 6$*(/4UR@,D@")A+'/ (U /4(0&/4#W(<>/4(^Mr(@@8+1< &IJ<>/4G(#(+.<>/4< *C$/D*J< $$O,4S(J< (*/40D8+1< &I0t*/41&(#< (@< *C:4$/4F< $%$O,4AUYC*J< 10F+,C1""C<>/4EC0r%/4a DA 4oV< 0 a DA &o< 0?R<J,q/4;/41$0-10(@.< 0:4$987/4(Mw/>8#< (@*$C/4#W(<>/4('< *:d/4

N#< (@*$d,D@")AS E#(&\%1<>/4 E\/ 6M+1< &1" +1"MFW(k*/4@GF< ($%,E/ 6SCw<J3&C6u1*/4%09HXK0r:!< #W;/ 6C<J34

6u1*/4 < 0d'#+1< &1" +1"GF(7S*."C<J,!.< Q GG3>< $1 ↑ /4( ↓ A_N##$%,D:G76u/4("M1$=< a DA !&ow< 0?a +A &oo/GVLN< "$%/ =<>Ea DA o V/4FT.< H =

∑

ijσ

tija†iσajσ +

1

2

∑

ijklσ1σ2

Vijkla†iσ1a†jσ2

alσ2akσ1

,a DA o

S%; Gin v mzJ9m.Win v Cj v z 0r0!F&,

tij ≡< wi|H1|wj >=

∫

d3rw?i (r)H1(r)wj(r),

a DA 4o< 0

Vijkl ≡< wiwj|H2|wkwl >=

∫

d3rd3r′w?i (r)w

?j (r

′)H2(|r− r′|)wk(r)wl(r′).

a DA &o

2ND

;#< (< "(R($=<>C;LN< "$O/4< E/\#(/4#W(@M/ 6(J< $O8Yw/4($%0@,D@")A?C< 1"'C#E+3>< (=< *'/ 6wCLN< "$O/4< nA AB< 1"'6u/4(FW/>.

H1< 0

H2<>

[H, Sz] = [H, S2] = 0WS(

S2< 0

Sz< (RR/ .< $O8

#S*9< (< *(@*A YC(1\6u/4(CE /41$/4"MF &(< */4 A ` 1Tt

:4(< $ *J< A :A%6 @.< (@w/G*/40(($=<>%3D@%*NX*S$Q457#D8/4(F%*/41#$:&w / /4FW,40 A_N&,D/TZ#;*/4(@3><>/40/+/ "C<(6u/4(G"/D0"#$%"&.<>/ KF1T"($%,"#$OrG)< (:41"&J^< %(01*5+1"MFW(^/ 6 #'3 < (=< F$1""C<>/ A4E5"C<J,GQ4#CSS/4$S/ 6 +1< +1" +1"MFW( ki

%/41T #< (<>:V%+/5#< 0N/4(F%.< $#< (@A0@*/4S$$XFW"C< 07$=<>(< "$%,C!D*/4 DA DA

?^<J3&U+1($=<>07LN< "$O/4< Z7^"M1$%#< (@*$%UL5$F(@#< *^/<>B P/D*.Q7#< * A `,/4$%3D:5%JJ/4/4FT.< @.<>/D**1#<.%/4D8+1"MFW((#(&.<>/4/ 6P/D*.QC#< * AB/'*/4"#$57/4$1T/4%ww*98< (@,M/Z(< *wF< *9QVSP/+*9QV@.<>w/Z/4(:4<>$D*.(D0:4(Sau#W/4%/4o(#D8(&.<>/4 ATw#(/4F$" < $/M($=<> Z"C< &,&8FW/D0T,#(/ F$" /MZ:4$98#< (@*$Mw<3498Y6u1*/4?0("<.%/4 A E'<J34C"*.< *V/ 6w&(< *:#< (@*$+/4$%,'

N8#< (@*$%Sw<34N6u1*/4/4$0ZT6u/4("C<>%/4C< FW/41T

@.<>V/>6V@,D@")A)RP/+*9Q#< *V V0("7V@.<>F+,93 < $1<>:^"*(/D*/4Tr:41(<>%/4*/ %:S/ 6T/D**1#<>/4/ 6J< *.R/ 6D^:4$U#< (@*$@.<>XS*.E< (G*.< (< *(I0F&,!G:4$R#< (@*$V<J34G6u1*/4WS*.!1(d< ##WJ< (Z9HT#$*%$O,!M9HT#(/476u/ (NM"%*(/4*/4#*R#< (< "(ACs8#< (@*$M@.<> |Φ0 >

P/+*9Q\#< **J< EFW'0r0qF&,d*/4"GF:9HT#< /4qa DA 4ow< 00r%/4?a DA &o ;76u/ $$/S:M"C< ( T

|Φ0〉 =1√N !

∫

d3r1 . . . rNΨ0(r1, . . . , rN)Ψ†(r1) . . . Ψ

†(rN)|0〉,a DA +o

S( |0〉 NG3 < *11" @.<>'\P/+*9Q!#< * A'09H0M"J< Z

:4(/410@.<> F1TS9HT#(/4!w(176u/4(S< &,;@.<>Ψ(r1 . . . rN)

< 0 |Φ >A

ZV*J< !V<>53 < $1V/ 6K"C< +,&8#< (@*$%Z<J34V6u1*/4Ψ0(r1 . . . rN)

098("B :4&6u/4(

N$*(/4K*(J<>0'<>K#W/4& r1 . . . rN

A ZM / (< 0 S/4FT.< M :4&^.HT#(03D=<

Ψ0(r1 . . . rN)F&,G0(":N

9HT#*.<>/4)3>< $1FW !73 < *11" < 0 |Φ0 >@.<>N<>6k(5< $<>@8

:CG$*(/47<>Z/4G$/D*J<>/ AX[HT#$*%$%,4 G<J34MG&3&(G($=<>/

Ψ0(r1, . . . , rN) =1√N !〈0|Ψ(r1) . . . Ψ(rN)|Φ0〉.

a DA 4o)$=< @Gw/6u/4("M1$=< G09.("'/4*)"/4(6u/4("C< $ +1%3 < $*)/ 65@.<>S;VP/+*9Q;< 0;VL5$FW(@#< *A

%q;$#/ 6S;*/40D8+1< &IJ<>/4(#(&.<>/4 <J34;#<8(<>0< F/34RM:4$98< 0d"C< +,&8w#< (@*$M< #*/ 6U

N8F/D0T,!#(/4F$")A

r(@@8+1< +IJ<>/4V< #W*U< (*/4Tr0au00oV9HT#(%/46u/4("*(/4*/4#*#< (< "(

tij< 0

Vijkl/ 6MLN< "$O/4< ?a DA o9AR*/40D8

+1< +IJ<>/4;< #*5< **/41&0;6u/4(wF+,CZ0=< :4/4< $%IJ<>%/4C#(/D*01(/ 6LN< "$%/4=<>))R/D**1#<>/4D8+1"MFW(5(#(&.<>/4 AXwN0=< :4/4< $%IJ<>%/4/ 6a DA o ;VP/D*.Q#< *R< "/41&w/'0(":

EG =< Ψ0|H|Ψ0 >< S<

6u1*/4/ 65/ 6"*(/4*/4#*7#< (< "9.(tij< 0

VijklS*.!< (0#D8

0+U/4MF< K:4$ #< (@*$^<J34w6u1*/44%'<N6u1*/4< $D"C< (A <>7S&,dw"C<J,\(J<>

EG ≡ EGwi(r)@$$U< R<6u1*/4< $U/ 6 wi(r)

A:4(/410\@.<>R/ 6^<@,D@" */4((#/40Z/R$/ @Z(: ,!/wR"C<J,(J< *9%RF+,Er0:)"%"M1"g/ 6

EG

F&,?/4#T"%I:)%VS%q(#W*9G/!F<8 wi(r)

AXX*JXwG*J< d#(/4#W/4MG(/4("C< $%I0au$%6=8p< 0>@1@0oSw<34+1<>/4 US*..< Q46u/4(" / 67[U1$(+1<>/4-6u/4(

EGwi(r)S%-

*/40%/4sZ*/4@Z<>0< wi|wj >= δij

AsN< "$%,4

δEG

δw?i (r)

−∇ · δEG

δ(∇w?i (r))

−∑

j≥i,σ(λij − µ)wj(r) < a†iσajσ >= 0

a DA &oS(

λij< 0)V*."*J< $#/ +=< $

µ#$<,R(/4$/ 6M< :4(< :4V"M1$%O8

#$("#W/4:7N/4(@/4:4/4< $Op,G*/4@(< &w< 0C<> #((@3D:MN+1"MFW(/ 6#< (@*$%CN@,D@" (#W*%34$%,4AT_Nw< 0C56u/D/ / +%6 N/4(F%.< $< (/ (/4: /4< $ 6u(/4";FW:4:\

λij ≡ 0AKY6pU< 00%/4< $%$O,4B ; /4(Q

S%S6u/4("C< $%"S%SrTHT0#< (@*$D1"MF(+< $/V/45#1Tµ = 0

< 0[UA5a DA &oV(01*'/d;@,D@" / 65[U1$(G+1<>/4M6u/4(M6u1*98/4 wi(r)

AXN<J34G+1<>/4d%Z9HD("$%,!0*1$%N//4$%3&R0(*$%,4X/wCS$$^(/4(@G(C/\*./D/4:!@.< (@:\F< S*.qS$$^FW/4#T"%I01F+1&$%,4+< U0*109HDJA4G"#$@K*J< 4 A :A4RSLN< (@(98pP/D*.Q< ##(/HT"C<>%/4 7+1<>/4(01*V/;G $$O8Q+/SELN< (@(98pP/+*9Q)<J34+1<>/4 A

4 54 2 #">2byM#(/+*01(M0*(F0dR#(3D/417D1F*/ \0/DN/ N+(/+01*R< +,< ##(/HT"C<>%/4 < )$/4:< EF<>% wi(r)

*/4"#$ E+1< &1"G8"*9< %*J< $Z A5P/4(#(< **J< $Z(J< /4%;;"#W/ F$%!/qw/4(QS% < Tr%RF< NJWS*9\N+(/+01*0dF&,!0"C< 0:%N*/4"#$%A <>GS+, !<J34/E+(/+01*!< ##(/JHT"C<>%/4 AUsN< "$%,4wEv .zj v injVj \~.ziz

"# '1.0! &()*3 = != )* =?.Q= ,DO)'!1$2*! GH"# * G8#G)'%= * $:&()*&('#(!#O')EA # , GH= !" !#" 46)', A?$C!= )*J= *J!"#%) & &O$C!= )*%$ * , E# ''# O'#.# *!1$C!= )*QP "# .#&()*3J&()*3 = !1= )*'#&#&;!1. !"# *)*)'!")*)', $2AHE$.=?.D!1$# *!) 3 #*#!"# # A?3)O# '1$C!)'1.('*) P +-,;P/. *!"# A?$C!$!# ' & $.#N!"#0' $2*!=1,&;),D, !1$C!= )*'# A?$C!= )*. 46)' !"# &('#$C!= )*$2*3M$2**= "= A?$C!= )*M)O# '1$C!)'1.%$2'#*)*.0!1$2*3$2'1332 )46587.# # $2A?.)#9-#&;!= )*(+P :

$

Single–particleSchrodinger eq.

∑jHijwj(r) = εiwi(r)

Single–particle(or trial) basis

wi(r)

Field operators Ψ(r), Ψ†(r)

Diagonalizationin the Fock space H = |Ψ0〉EG 〈Ψ0| + . . .

Ground–state energy EG = 〈Ψ0|H |Ψ0〉

↓

↓

↓

↓

Single–particlebasis optimization

6

→ wren

i (r)

Ψren(r), (Ψren)†(r)

Ψren

0 (r1, . . . , rN )

Renormalized N-particle

wave function

↓

↓

P:41( DA 4AV*."C<>* / *.< (@5/ 6BV[^]S_Z` "/D0 Azi )jm;Uhikjhy ~9 v m10 0.yDhXjlinm>hTz S/4$%,+=<>$B< ##(/JHT"C<>/ N"C<>Q4 ADD1*9C#(/D*01(Z< $$/SK6u/4(<V(01*/4/ 6XN+1"MFW(/ 6 #< (< "(tij

< 0 Vijkl"C< Q+:^#(/4F$"(< *.< F$ AJ_N$/J%"C< Q <"C<>(OH70=<>:4/>8

< $IJ<>%/4M9HT*1T.< F$5< KS0"%/4G"C<>(OHM(#(&.<>/4'/ 6WSLN< "$O/>8=< )a DA oB*(J< ^9HT#W/4+=< $$%,VS%MS+1"MFW(U/ 66u1*/4UMSF< A('< (M/4"M*J< XS(G G0/;/ Z0dG"C<>(OH)(#(+.<>/4 F1TwC:4(< $n&@#;1<J3&/40<>F$ A+Z(01*/4$"'OUZ< **1(< *,C/ 6S@,D@" 0*(#T/4'GSS<>^/4" &#W/4F$%,M:4$%,R9HT*%0'3D(@1< $9HT*%.<>/4< (/ V9<>Q4q&/!< **/41&JAB_N$/$*/4?/ 6 C@.<>V/FWZ0*J< (00 "M1@FWZ*J< ((0;/41T*J< (6u1$$%,4ADB/G(01*Z7< **1(< *,C/ 6$*/4 /4V":4&S*./D/4/< 0>@1@SV("C< :'F< 6u1*/4A@J< 0/ 6<7/ 6UrTHT0d:4$V#< (@*$w<34R6u1*/4 wi(r)

FW:CR/4$1T/4!/ 6<R*/4*(Z:4$98#< (@%*$S#(/4F$")&wN.< Q 7<R/ 6 <J34N6u1*/4wF$/4:4:/;'< "G*$=< wi(r;αi)

*.< (< *(I0EF&,< E< 0>@1@.< F$G#< (< "(αiA

αi*./4*S0#W0 /4'5#W*%r*5#&,D* / 6X5#(/4F$" <>< 0 +< %^

0*10;D*/4TA 4A+Sr(@ @#'N[^]_Z` U"/D0^5$*/4/ 6UM(01*0F<>%Z7*$=< 7< Z/S\P:A DA 4AXs5/ wM*J< d*/4@(1*Zr$0/4#W(<>/4(

Ψ(r)< 0r0)?a DA &o9T9HT*#T5<>w1""C<>/ ;(1

/ /34( Z(@(*0;F< wJATNr$0/4#W(<>/4(w$*0;C "C< (*J< ?< $/)FW'10?6u/4(0("<>/4d/ 6 "/+0$^LN< "$%/ =<>2a DA oZd/+**1#<>/4D8D1"MFW(w(#(+.<>/4 A#< (< "( tij < 0 Vijkl < (098("0$Q S F&,$*0)w<J3&76u1*/4A

;*./4*/ 6S(@(*0F< M(#(&< .z.zhjlil @# AUsN< "$%, ;*J< .< $QE< FW/41TM Cm-+Uzm>yDjlinm>h / '< FW/41T'< .mNWjzm>yDjlinm>h A:4$*0d@.<>al< 0dM*/4((#W/40:;0T,D< "*J< $B#(/D*.o7"C<J,)FWG(($O8

3 < &N/'R#&,D*N/ 6KV#(/4F$")F1TN"M1@NFWV"+/40!<>5V@.< (@JA_"/4(1F+1+^< ##(/JHT"C<>/ V"C< 0<>BB#W/4&B"C<,7/4"" 1*R< **1(< *,)/ 6V(1$%AWsN< "$%,4<J3D:/4FT.< 0V#< (< "(al<>5$=< @(V@.< (@:3>< $1.oZwC*J< ((@,/41TV0=< :4/4< $%IJ<>%/4/ 6 LN< "$O/4< A;0=< :4/4< $%I<>/ ?"/D0qR/ GGF+,E#W*%r*J<>/4R/ 6S;[^]_Z` "/D0 A ZF+3D/41$%,4 /4$1T/4/ 6w#(/4F$" */40D8+1< +IJ<>/4(#(+.<>/4"J< S/4$%3D:MVL5DF(:C+1<>/ 6u/4(w7r$0/4#W(<>/4(

ı~∂Ψ

∂t= [Ψ, H].

a DA $4o

S/ 6uJ< F$7)< &,;#&,D*J< $$%,($3 < &S%1<>/4 A(N< ("C< +,M/ (Uw<,D^/#(/D*0S%GS0=< :4/4< $%IJ<>%/4'< 0"/4@

/ 6 ")F1TV/ R< $$n <J34'"C<>(OH\(#(+.<>/4E/ 6 CL5< "$%/ =<>*/4""/4 A #51< $$%,C/47/4FT.< /4"7/4(< $$W/ 67:4(:4JAT:4&3 < $81G0#W0 ikh 0+j /4?.#< (< "( αi

A_NG<!r< $ @# ""'%I:4(/410q@.<>(: ,

EGS%q(#W*M/ αi

< 00(" |z2in :4(/410E@.<> A?/4"*J< :4(:4R< (/4FT.< 0?< V9HT#$*%9HT#(/4G/ 65#< (< "(

αiA-1*.%1<>/4 ;*J< -0(*$%,E""'%I

$/ @5/ 6B" < 0)/4FT.< : (/ 10)@.<>7(: ,4W< w $$< (/ (@8"C< $%I0;3>< $1

αi ≡ αopti

As534(@$W%6B%/ S#W/4F$Z/G/4FT.< 9HT#$*%9HT#(/4 F1T/4$%,-(D1"(*J< $3 < $%1C6u/4('\@.< (@:?;/ 6Z3 < $1αi

R*J< \(01*RR:4(/410!@.<>R(: ,)%(<>%3&$%,;:4:'R/4#T"C< $3 < $15<>#(/+*01(V0 AV(1$%.< &S(/4("C< $I0;#< (< "(

αopti

< $O8$/26u/4(N$%*/4/ 6(/4("C< $%I0CF< wopt

i (r) 6u(/4" N%< $*$=< / 6^w<34M6u1*/4 wi(r)

A ZG< Z/;10($>X G1F@%1TMG/4#T"C< $3 < $1/ 6

αi = αoi pt/4FT.< 0;6u/4(w7:4(/410@.<>7/G*9< (< *(I k 7*J< $O8

*1$=<>0G@.<>A4 R1( "C< Q4w*/4@(1*/4G/ 6w(/4("C< $%I0Rr$0/4#W(<>/4(

Ψ(r)< 0

Ψ†(r)#W/4F$ A_**/4(0:/\[UA a DA 4o9 w*J< ?/4FT.<

M&(< *:C"C< &,)#< (@*$w<J3&R6u1*/4!6u(/4" ")AWZ*/4"#$5/4$1T/4S%;"/+0)/>6K9H< "%:'<'@,D@" / 6*/4(($=<>0$*(/4A

D1""C< (%I: Z[^]_Z` ^"/D0C*/4@ / 6 NF< (01*/4 D"#<>*/ 6S*.;;J< 0;/41TwF&,< $$/S:Z(.< 0F< 6u1*9.%/4 /MFW$%6=8p< 0>@1@0?/C"C< &,&8#< (@*$)au*/4(($=<>0o7:4(/410?@.<> AB / *9< (@/ 6BV[^]S_Z` #(/D*01(V*."C<>*J< $$%,C1""C< (I0;)P:A DA 4A

B/"C< Q RG<>##(/4< *9\9HT#$*%5wM*J< !S(%M0/S\M*."M0*(FW0< FW/3&G!6u/4(" / 6+1<>/4A sN< "$%,4WMr$0d/4#W(<>/4(

Ψ(r)0r0EF&,d[UA

a DA &o)("5/ 6KR1" /34(Z<*/4"#$VF< wi(r)*/4&.< N< \Tr%

D1"MFW(7/ 6:4$98#< (@*$%R.<>AX?'(#(&VMr$0/4#W(<>/4(7F&,<Cr%

!+1"MFW(

M/ 6Bw<3476u1*/4 wi(r)

A[HT#$*%$%,4

Ψ(r) ≡∞∑

i=1

wi(r)ai 'M∑

i=1

wi(r)ai,a DA &o

S%i(#(&:G<*/4"#$Sw/ 6 +1< &1" +1"MFW( < 0

MF:R<r%

rTHT0'+1"GF(A>?*J< M'9HT#(U< ##(/JHT"C<>ws8#< (@*$^<J34w6u1*/4;76u/4$$%/S:M"C< (

Ψα(r1, . . . rn) '1√N !

M∑

i1,...,iN=1

〈0|aiN . . . ai1 |ΦN〉wi1(r1) . . . wiN (rN).a DA !&o

(M<)*/4@(< +R"#W/40?/4?F< R(01*/4 B< "$%,d <J34C*/40%/4

N ≤ MA N(@S K01/d K< 1$ #(*%#$ B%M /41$0/ GFW

#W/ F$%/q< **/4""/+0<>)< $$wN$*(/4)@,D@")A_*1< $$%,4K

6u1*/4 wi(r)"C<J,Z%*$10U/ /4$%,N#<>=< $40:4(/ 6D6u(0/4"F1T< $/6u/4(

9H< "#$ V#\0:4(A S*/4:4I:.<NS%R/+**1#<>/4D8+1"GF(#< *Z#< 0;/4N@.<> |ik〉k=1...M + Z<J3&5Zs8#< (@*$S@.<>NP/+*9QC#< *V/ 6B76u/4("

|ΦN〉 =1√N !

M∑

j1,...,jN=1

Cj1...jNa†j1. . . a†jN |0〉,

a DA &oS(

Cj1...jN

(#(&)9HT#< /4*/+ *&.S*.< (0("06u(/4" <!0=< :4/4< $%%IJ<>/ E#(/D*01( AUD1F@%1T:)[UAwa DA &oZ/-a DA !&oZ /4FT.<

Ψα(r1, . . . , rN) =

1

N !

M∑

i1,...,iN=1

M∑

j1,...,jN=1

〈0|ai1 . . . aiNa†j1 . . . a†jN|0〉Cj1...jNwi1(r1) . . . wiN (rN).

a DA o 9HT#(/4M#(/3D0

N !/4I(/N(">J< *.MS%V6l< */4(

(−1)P S((#(&Uw:4M/ 6w#W("M1T.<>/4M/ 6+1< &1" +1"MFW(

(j1 . . . jN)S%

(#W*9N/(i1 . . . iN)

A)/ ( /4(0w*J< S(%Z<>

Ψα(r1, . . . , rN) =1

N !

M∑

i1,...,iN=1

Ci1...iN (A, S)[wi1(r1) . . . wiN (rN)].a DA 4o

?M<J34MM< "'M9HT#< /4d*/D*+N6u/4(NFW/ |ΦN〉< 0

Ψα(r1, . . . , rN)

(6u/4( E< FW/34E9HT#(/4 (#(+)"M1$%*/4Tr:41(<>%/4<>$N+(p8< *:!w<J346u1*/4q/ 6

N#< (@*$%R0@(F1T0< "/4:

M@.<>S%?

*/4((#W/40:q :4&Ci1...iN

6u/ (< *9 */4Tr:41(<>/4 < 0(A, S)

(#(&(#W*%34$%,GS< +@,D""(%IJ<>%/4)anD$=<>(K0("< +9oB/4(KS@,D""(IJ<8/4 au"#$#(/+01*

wi1(r1) . . . wiN (rN)oG9HT#(/46u/4('6u("/4< 0

FW/4/4(#W*%34$%,4A

4 &" ( " ">2 #">2by $_N"#W/4(@.< &7< #W*/ 6^G0*1/4/ 6 /41(Z/3&$K"/D0dNG*/4"#< (O8/4!S%)/4R9HT@:< $(J< 0T,4A_N5V6u(< "w/4(Q;/ 6KM[^]_Z` S"/+0\+1%G:4(< $nW('< (G/4"M$"O%:C%1<>/4X\S%*9%Z*/4*0ZSO.9HT@:< ##(/&< *.A N/ 6B" w"M1$%*/4Tr:41(<>%/4< $8Y%+(< *%/4an\ n8@wPUoN"/D0 A

! #"%$&! '()+*($,-.! '(-(&0/1 2!-(3 '(#-54/4'(-(

`,$/D/4Q+:M<>S[UAa DA 4o9D/47":4& :4w7"#(/4C<>w7[^]S_Z` < 0V\ "/D0< ( i+hjlin AT` / ;/ 6" ""'%I5Z:4(/410@.<>ZD8(: ,EF+,r0:!/4"/4#T"C< $U"M1$%#< (@%*$%Gw<346u1*/4?6u/ (M<r%F< wi

$*06u(/4" N@.< (@JA+ (J< \ U"/D010;C+1< &1"*9"@(@, T n F< V/4dM3 < (=<>/ < $/4#T"%IJ<>%/4\/ 6 FW/>.dG*/D G8*+

Ci1...iN

< 0F< wi(r)(''*/D *&

C< ('0("0

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L5$FW(@5#< * AX_N$/V6u1*/4 wi(r)/4FW,!<C$%6=8*/4@&5<J34M+1<8

/4?an+ [o9A0(%3 <>/4;/ 6U+ [t1##$"&+1SR\ w< ##(/&< *. A0%X(*UFW 'S[^]_Z` K< 0M5\ B"/+04FW/ G/ 6S*.

FW$/4:/V*$=< N/ 6U"M1$%80("%< &59HT#< /4!/ 6^s8#< (@*$7w<J3&V6u1*98/4 *J< )F1""C< (I010(76u/4$$/S%:GJ< 0:4

i 6 l&l 5*\ 3&/4$%340M6u(/4" 3>< (=<>/4< $+"/D0/ 6W+1< &1" #&,D*

< 0d*."@(@,./*$10M$*(/4*R*/4(($=<>/4N< 0d* X//4FD8.< !<FW(53 < $1V/ 6

EGF+,.< (@:'6u(/4" "C< &,&8#< (@*$VD*9(+0:4(

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798 91:;8 =?.8!"# .")'!1& ! 46)':.# A 41$ &()*.=?.0!# *! # A?3 P

# iki 6 D l45*

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A< 0

Bq

LN< "$%/4=<> T< "$%,

(A− < A >)(B− < B >) ' 0 a DA &om

AB = A < B > +B < A > − < A >< B > .a DA +o

[H< *$%,M5< "S(1$%*J< FW5/4FT.< 0''N[^]_Z` K"/D0F&,G.< Q+:@1@5$/ @5#W/4F$7D1"MFW(S/ 6F< 6u1*/4< "$O,<>5"C< &,;< (< ($*(/4t@,D@")AS ;*J< *J< Q+# d0=< :4/4< $%IJ<>%/4#(/D*'< R ;<J34;/4$%,E/4"C< &,&8$*(/4q@.<> AUD/B V@1@G""'%IC%(: ,4A> (1$%B/4FT.< 0V6u(< " /4(Q< (^< "< /4 #(/3D00F+,VLN< (@(98pP/+*.Q;< ##(/JHT"C<>/4 A

?5$$%1@(<>ZLN< (@(98pP/D*.Q< ##(/&< *.C/4N9H< "#$S/ 6 He

<>/4")AB/G0/MD 7@.< (@wF+,C$*:'< wi(r)

@1@ / Jp8$Q N@.<>w6u/4(7L5<>/4"

Ψσ(r) = (α3/π)1/2 exp(−αr)χσS(

αRX*%34R&3&(V(< 01

/ 6B@.<>A)/ (ww/4(0"#$@w(=< $Xr$0/4#W(<>/4(/ 6B76u/4("

Ψ(r) = Ψ↑(r)a↑ +Ψ↓(r)a↓,a DA 4o

S(aσ

C< $=<%/4t/4#W(.<./ (;/ 6V#< (@*$!td@.<>Ψσ(r)

AwLN< "$%/4=<>!!M*/40+1< &IJ<>/4)6u/4(55p />8$"&ZF< Z< N76u/4("

H = εa(n↑ + n↓) + Un↑n↓,a DA &o

S(n↑ = a†↑a↑

S(J<

εa = 〈Ψσ|H1|Ψσ〉,a DA "$4o

< 0U = 〈ΨσΨσ|V |ΨσΨσ〉,

a DA &o< (7"C<>(OH$"&S/ 6B:4$.8#< (@.%*$%Z#< (@50r0\<

H1 = −~2

2m∇21 −

~2

2m∇22 −

2e2

κ0r1− 2e2

κ0r2

a.u.≡ −∇21 −∇2

2 −4

r1− 4

r2

a DA !&o< 0)/ 6BR /41$/4"MF;+(< */4;Fp !$*(/4

V =e2

κ0|r1 − r2|a.u.≡ 2

|r1 − r2|,

a DA &oS%2\*/4((#/40:q0r%/4Ct<>/4"*1%dal<TA 1 A o<>6k(\*/40+1< $%p,M:4 A&5/4$%,G:4&3 < $1S/ 6^a DA &o^/4FT.< 0'6u/4(K5@.<>

a†↑a†↓|0 >< 0d

E = 2εa + UAXN/ .< $(: ,\5""'%I0!S%\(#W*/

α

# /G/4FT.< Z $$O8Q+/S3 < (=<>/4< $@"C<> "$ / 6BFW/

α< 0;7:4(/410

@.<>^(: ,EG

AL5/ 34(w^"C<J,Z$/D/4QZ<>^#(/4F$"0%X(+$%,4AJ_NUr$0/4#(<>/4(*J< FWU0r0V6u/4(< V< (F%(< (@,NF< ^"C<J,N(:&< (07U:4+3>< $1

E< <w6u1*9.%/4< $4/ 6Ψσ(r)

J*UK6u1*/4< (K10(U+:4(< $ 9HT#(/4A(6u/4( 4w(1w<J3&6u1*/4GU/4FT.< 0G6u(/4"S[U1$(U+1<>/4G6u/4(6u1*/4< $ 10(U#(/3D/<>.Kw<34U6u1*/4/4("C< $%I0 A"J< <> V"%"%IN76u1*/4< $

EΨσ(r) =∑

σ

∫

d3rΨ∗σ(r)H1(r)Ψσ(r)+1

2

∑

σ

∫

d3rd3r′|Ψσ(r)|2V12(r−r′)|Ψσ(r)|2.a DA o

CX*J+Z[U1$( [UA a DA &oK.< Q 5S6u/4(" / 6XN1(@(*0;LN< (@(98pP/+*.Q+1<>/46u/4(

Ψσ(r)< "$%,

(

∇2 − 2e2

κ0r

)

Ψσ(r) + Ψσ(r)

∫

d2r′e2

κ0|r− r′| |Ψσ(r′)|2 = λΨσ(r),

a DA 4oS(

λ(*^"#/40/4("C< $%IJ<>%/4Z6u1*/4Z^< "^"< (<

/4(:4%< $XD*.(D0:4( 6u/4("M1$=<>%/ C/ 6N<J347+1<>/4 nATY6 Z /41$0<34.< Q 'p /#D8/4(F%.< $%K@J< 0/ 6W /V/4(F%.< $%w /41$0'/4FT.< MS(@(*0LN< (@(98pP/+*9QG+1<>/4 A++1 N*J< 5<>.< Q+%:R'N"#$%@*J< ^@1@ /'#/4(F%.< $w7/4FT.< ;%(w $$O8Q+/S;3>< (=<>/4< $W@"C<> "$ 6u/4(

α< 0EG

6u/4(wL57<>/4" α = 27/(16a0)

<>0EG = −5.695Ry

DS(a0 ' 0.53 JR` /4(V/4(F%V(< 01AC(:4R#(&GV*J< < (/Sq

P:A DA DA

0 1 2 3 4-8

-6

-4

-2

0

2

4 He atom

U

EG

2εa

Ene

rgy

(Ry)

Inverse orbital size, αa0

P:41( DA DAU!&(#$=<J,Fp 2!<>/4"*;(: ,εaau0/ 02$oM< 0

+(< */4\(: ,Uau0< 0$o5(1$%:\G:4(/410@.<>G(: ,

EGau/4$0;$o < 5<M6u1*/4/ 67+34(/4(F%.< $ IαA

vZ[jM PdO@I

Q;iWv P [lPdKWT R [N\Z] Q;KW\ZR [ PdO.RbP KNS T OAPLv^K ]

b " (U ">2 $6(#*L4" #L%$6$[^]_Z` "/D0?0*(F0q < #T( $J<34R34(< $ "#/4(@9<>+0*O8/4Z/FW'"C< 0MSE"#$"'+%:%*/4*(*J< A Z'/ 6^'"/4@"#/4(@.< +ww*9/4*7/ 6B@.< (@:':4$98#< (@%*$ZF<>%JA

Y6URF< N*./4\ (M*/4"#$ ! R /41$0\/ N<J3&M< &,\<>##(/JHTO8"C<>/4 A&5(< %/4'/RN*/40D8+1< &IJ<>/46u/4("C< $%" w/41$%0FWN9H< *< 0q;#(/4F$%"/4$%3&0(:4/4(/41$%,6u/4(M< -< (F%(< (@,E*/4"#$CF< wi(r)#(/3D00' /41$0F5< F$w/*J< $*1$=<>M<>F< KS"*(/4*/4#*w#< (< "98(K9HT#$*%$O,4A*./4* K< (F%(< (@,VV*J< 4F1TU11< $%$%, S< ( / U< F$/M#W(@6u/4(" < &,*J< $*1$=<>%/4 01N/G*/4"#$9HT%,#(/4F$"w< w@.<>0F6u/4( AD/ <J347/'$J<J347/4$%,/4"7/4(F%.< $%ATYSw/GFW70*00S*.):4$.8#< (@.%*$%/4(F%.< $5< (./CFWR$6kJW< 0!S*.\/41$0\FR:4/4(0 A_NZ%5Z< \< ##(/HT"C<8/4 S(1$%/ 6WS*J< $*1$=<>/4K"C<J,'< 00/+0#0/4G5*./4* 50/A_N< 0J< ($(W< ##(/JHT"C<>%/4d(#(&V/4'/ 6'"/4@ 0.yDh&ChjY 6uJ<>1(5S\*/4@(1*:R"/+0$A Z1(5"/+0\< $$/S6u/4(5*/4@(1*/4/ 6/4(*J< $"/+0$ ' ikj m>yDj #< (< "(F1TN(+1(N<'0:4(V/ 6K:4+1%,S)$*:'F<>%JA

:4$*0?@.<>V11< $$%,d(#(&M:4$%,\9HT*%0 al< 0E+1V:4$:48F$o@.<>'S@,D@")A+,C< (S1##W/40/VFW5< $"/4@^1/+**1#0< 0/(5*/4&(F1T/4):4$:4%F$% A% ZF&3D/41$%,401V/R&(< */4R0&%,/ 6Z:4$%N#< (@*$"C<,'FW*/4"ZF$1((0%6 Z&(< */4C @(/4:4( < 0%X(*(: ,6u/4(1*9):4$98#< (@*$Z@.<>A

<>K/ ("C<>"C<>*J< $&#(/4#W(@/41$0R (01*0'F< K9HT#W/4 YU34(@,G*/4&34+U/7#(@6u/4(" < $$+*J< $*1$=<>%/41:< G/4(@/4/4("C< $%I0

.

F< SJA <>S< ##W5%6%S/ *J< VV/+**1#<>/4D1"MFW(S/ 6<!:4%34q:4$#< (@*$@.<>

iV:4/3&(0F&,EC*(J<>/4< 0< $<>%/4

/4#W(<>/4(ai< 0

a†iUS(J< 9HT@*!/ 67< 2$*(/42<>(J< $#< *

*/D/4(0<>r< $%(0dF&,\Mr$0d/4#W(<>/4(

Ψ(r)< 0

Ψ†(r)A]N1G/ < 1$

#(*#$;6u("/4*Cw<346u1*/4-< '/EF!< &@,D""(* AKL5*)r$0)/4#(<>/4(S/41$0<34776u/4$$/S%:'< &*/4""M1T.<>%/4($=<>/4

Ψ†(r), Ψ(r′)

= δ(r− r′),alTA o

S( A,B ≡ AB +BAA!< 00%/4/'<>J%6F/ )/4#W(<>/4(5< (7*/4>@1D8

:&<>0C/4(/ J&,C< &.%*/4""M1T A Z5*/41$0C9HT#*w5*(J<>/4< 0< $=<>%/ /4#(<>/4(/5F<34 V<S"$< ( "C< (A>D1*.M*/4""G1T.<>%/4($=<8/4$^5#< (@*$^:4$S#< (@*$S@.<>FW<J3&N$Q S(J< $6u("/4A&_N$/"#$)*/4""M1T.<>%/4(1$C"#$%,-<?"C< $$('+1"MFW(C/ 67("C%2\< *1< $*J< $*1$=<>/4M&3&/4$O3D:\")AU_ #< (M/ 6Z*(J<>/ 2< 0< $=<>%/ /4#W(<>/4(< *:'/4)<

N8#< (@*$Z@.<>:4%34

aja†i |w1 . . . wN >= aj|wi, w1 . . . wN >=

< wj|wi > |w1 . . . wN > +N∑

k=1

(−1)k < wj|wk > |wi, w1 . . . 6 wk . . . wN >,

alTA 4oS( 6 wk

/S<>7#< (@*$M0*(FW0EF+,wk

7/ V$/4:4(7#(&R"C< &,&8#< (@.%*$Z@.<> A&34(@0)/4(0( V<J34

a†iaj|w1 . . . wN >=N∑

k=1

(−1)k < wj|wk > a†i |w1 . . . 6 wk . . . wN >=

N∑

k=1

(−1)k < wj|wk > |wi, w1 . . . 6 wk . . . wN > .alTA 4o

V1" / 6K< FW/3&V[U+ABanDA 4ow< 0qalTA &o :4%34S17w< +0)*/4""M1T.<>/4(

a†i , aj

=< wi|wj >=

∫

d3rwi(r)?wj(r) = Sij.

alTA +o

C&:4(< $U/3&($=< #E&:4(< $UFWwqC/ (FO9<>$wi

< 0wjAE

< "7"C< (Dw7*J< /4FT.< 7(1$%w6u/4( p /'*(J<>/4;/4( p /< $<>%/4/4#W(<>/4( < "$%, ai, aj = a†i , a†j = 0

AD/+N6u("/4%*N< &*/4""M1T.<>%/4(1$5< (7#((@3&0!/4$%,C6u/4(/4(@/4/4("C< $XF< SS(

Sij ≡ δijA

"$Y6ME:4$d#< (@*$dF< )/ !/ ("< $I0 E#< (@*$*/41&

*J< / MFW0("0q1:!11< $ +1"MFW(R/ 6S#< (@*$R/4#(<>/4(a†a

AY6a†i , ai =< wi|wi >= Sii

< wi|a†iai|wi >=< wi|Sii − aia†i |wi >= Sii < wi|wi >= S2ii

S<>^U+1< $//4/4$%,M%6Sii = 1

A&s5/ w<>U3>< $1/ 6Sii

K/4:&<>%34 FW*J< 1C/3&($=< #?&:4(< $^&3&/4$%34R/4$%,/4C/4(F%.< $n *

Sii*< / MFW

+1< $ / −1 A_N/ (N#(/4F$" 0/C0*1ZSV*/4*/4\FW9p \R*/4"#$/ 6:4$#< (@*$F< G< 0?;< +*/4""M1T.<%/4?($=<>/46u/4(VCr$0/4#(<>/4(JA[HT#< 0:C< &*/4""M1T.<>/ (7< **/4(0:;/)0r%/4a DA &o9 /4FT.<

Ψ†(r), Ψ(r′)

=∑

ij

w?i (r)wj(r

′)

a†i , aj

=∑

ij

w?i (r)wi(r

′)Sij =∑

i

w?i (r)wi(r

′),

alTA 4oS(d(0 +1< $%,t"#$%< /4(@/4/4("C< $ZF< ;JASE(1$%.< &9HT#(/4R*J< VFW^0+%r0V< +1%3 < $&/S^0r%/4/ 6T^*/4"#$/ 6K<G wi(r) ∑

i

w?i (r)wi(r

′) = δ(r− r′).alTA &o

D/q%6Z\F< C/ */4"#$ ^ \0/q/ <J3&!!*/4((*;*/4""M1T.<>/ (1$'6u/4(Gr$0/4#W(<>/4(0r0tF&,2[UANalTA 4oau)< "G(1)6u/4(< Tr%CF1TG/4/4(@/4:4/4< $F< .oAK` *< 1; w/ (QS%-<!r%CF< "C<J,!< ##(/JHT"C<>M(1$N/4$%,)\<C$"'%0!(:4/4!/ 6^#< * AX<>7SS+,%"#W/4(@.< &^/R*./D/45NF< 6u1*/4^/R*/3&( $$Z< (J<T&S(N 9HT#*7< )< "#$7< F10< *V/ 6$*(/4A

P/4($%:4+R<>/4"R< 0q/4V0*10-EMD*/4q%RC3D*%,/ 6 C+1*$n S(C C9HT#*M$*(/4R#(3>< $n<>R$J< @R< R$/4:)< V < ('%+(@0Ed(7FW/410E@.<>AY6 '$*0E@.< (@.%:;w<34'6u1*/4*/4*0'%?@.<>M/ 6wC%+(< *:)@,D@")?< ##(/JHT"C<>/ /4(Q+34(@,' $$nA45r(@ *./4*/V0("5*$=< / 6 F< U6u1*/4U/@.< (@wS%<M/4$%3 < F$N1F5/ 67@,D@")A;7*J< 7/ 6<>/4"w< 0/4D ":4&50(/4#!r(NR+(< */4)FWwd$*(/4AXNN/4$%,;M@.< (@:"/4"&/ 67$*:?!O.< $w*$=< C/ 6Zw<34)6u1*/4A^?)+1/ FT9<>t<:4$G#< (@*$D*.(D0:4(+1<>/4E0*(F:)< ?$%*(/4?d /41$/4"MFr$0'/ 6S+1*$1A&_NKS/4$1T/4Rw/4FT.< M $$Q+/S+,D0(/4:4*98$%Q4w<J3&76u1*/4A

_N'"&/40-FW6u/4( K ; /41$0-$Q //4#T"%IC" $=<>(M/E< **/41&FW9(R&(< */4AKD/B <34C/d0*0?S*.qw<J,?/!3 < (@,E")A Z^":4&9HT#*<>B01^/S^(#1$%34 /41$/4"GF&(< */4^/4(F%.< $

.

*J< qFC%(V.HT#< 00/4 /\< $$/C$*(/4/!F6u1(@(R<J<J,d6u(/4"J< *9?/ (/4(7/)*/4&(< (@,4/)FW+1IJ0 /.< Q C< 0T3>< +.< :4/ 6D1*$1G<>(< */4 ABq"#$@G< 0#&,D*J< $$%,?< ##W< $:d< ##(/&< *. C(< 0=< $UI/ 6w/4(F%.< $7"C<,dFW'3 < (0 A ZEC/ (V< 0 / .< $< :41$=< ("/4"+1" / 6T$*(/4B <>/4" /41$0VFW&3 < (=< &10(B$*(/4D8$*(/4&(< */4J/S ^"C<J,N(@,Z/S(.< 7< :41$=< ( 0#0*JE%/4(: < $B6u/4(")< %0/DV/ V*.< :40(< @*J< $$%,\10('+(< */4ak)#D8/4(F%'&(< */4-'(:&< (00t< '"C< $$w/4-'(: ,*J< $o9AUI5*.< :4N/ 6 0%X(& /4(F%.< $%U0/D/ *< (@,C<J345/VFW5($=<>0'/VJ< *./ (Aw[K3&(@,t:4$!/4(F%.< $S*J< <J3&d<?6u(\#< (< "(C@J< 0 / 6%` /4((< 01

na0A&5*$=< ^/ 6 &,D0(/4:4*98$Q ww<J3&S6u1*/4^KD1^0r0F&,G

(< 0=< $ #< (@/ 6B76u/4("

R1s(r;α) = 2α3/2 exp(−αr),

R2s(r;α) = 2α3/2(1− αr) exp(−αr),

R2p(r;α) =2√3α5/2r exp(−αr). alTA $4o

;*/4"#$Cw<J3&6u1*/4G< (/4FT.< 0F&,q"M1$%%#$%,D%:);(< 0=< $#< (@F+,\@.< 0< (0t#(*J< $N0#W0*(#(&0 F&,2#(*J< $N< ("/4*Ylm

nA A

φ1s(r, θ, φ;α1) = R1s(r;α1)Y00(θ, φ),

φ2s(r, θ, φ;α2) = R2s(r;α2)Y00(θ, φ),

φ2p0,±1(r, θ, φ;α3,4,5) = R2p(r;α3,4,5)Y10,±1(θ, φ).alTA &o

C< FW/34G6u1*/476u/4(" <:4(/41#/ 6<J34'6u1*/40#0+R/4#< (< "(

αiS*9)< SV0"/4/ 6K&34(R$: AP/4(V&,D0(/4:4

<>/4")D#< (< "(< 5<MrTHT0)3>< $116u/4( JS@.<>Cal<>/4"%*Z1%.ow< 0

1/26u/4(NG("C< :C&,D0(/4:4*98$Q M@.<>a >7< 0># T± oJAX7#< (< "(*/4((#W/40G/\I;/ 6S/4(F%.< $nA(M+34(;R;` /4(p8(< 01R6u/4(J< *9d/4(F%.< $lA ZM*J< \*J< $*1$=<>VM9HT#*.<>/4\3 < $1R/ 6UM0@.< *M/ 6U$*(/46u(/4" +1*$nTS*.)

< φi|r|φi >=∫

d3r|φi(r;α)|2r =

32α

6u/4( JS@.<>3α

6u/4( ># T± @.<> . alTA !&o

?< $$ <>M/4#T"%I0-` /4(p8(< 0 < (/ G9H< *$%,+1< $w/\/43 < $1A

;)' !"# 3 #*= != )*.H$2*3 &()*# *!= )*. .# # :OO# *3 = 9 P "# $2EA # +P :J.")CG:.%!"#>'#. A !1. )E !1$2= *#3M= * 9-#&;!= )* +P = *!# ', . )24 !"#8)"'$ '1$3 = = GH"=?&1"$2'# !"# = * # '1.#%)24

αi

O$2'1$2,D#(!# '1. P

!

0 1 2 3 4-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

α2a

0=1

over

lap,

S

parameter of 1s state, α1a

0

P:41(TA 4A N34($=< #+:4(< $FW /4("C< $%I0 JG< 0 >M+,D0(/4:4*98$%Q4w<J3&N6u1*/4ADZIZ#< (< "(6u/4(^ > @.<>Z Q #TwrTHT0)<> 53 < $1α2 = 1

A

;$*0q6u1*/4G< (C/4("C< $%I0 BF1TM,q< (/ M/4(@./ :4/4< $^6u/4(< ;< (F%(< (@,G3 < $1N/ 6Z#< (< "(

αAD,;< (5Z/4$1T/4/ 67LN< "$8

/4=< / 6< E$*(/4E /41$/4"MF#W/ &=< $U/ 6D1*$1AB_N0D1V%6,G<J34S0%X(&^(:44,'< (/4(@/4:4/4< $T/4$%,R6u/4(U<7:4$ 3 < $1/ 6#< (< "(A&?N0/M/ w*.< :45.Z< :41$=< (^0#W0*7/ 65w<J3&N6u1*/4F1TG/4$%,(< 0=< $#< (@JAUD/\/4$%,EC6u1*/4M*/4((#/40:\/\< "#(*J< $< ("/4*G"C<J,qFW/4/4(@/4:4/4< $nAB/41(G*J< US%@.< (@:F< alTA &o9< (/4$%, JG< 0q >M@.<>A Z*J< -*J< $*1$=<>/34($=< #+:4(< $W6u/4(wV@.<>N< 0/4FT.< ;<>

S =< φ1s|φ2s >=∫

d3rφ?1s(r;α1)φ2s(r;α2) =8√

α31 (α1 − 2α2)√

α32(α1 + α2)

4 .alTA &o

Z$%,6u/4(3 < $15/ 6α1

< 0α2

<>@6k,D:GR+1< $%%p,α1 = 2α2

R< FW/34/34($=< #'&:4(< $TU+1< $/7I(/A&U$wM

(α1α2)

#$=< */4((#/40^/<M0%X(&@(: / 6 /41$/4"MF;<>(< */4/ 6ZD1*$1AP:ADTA 7/S

.

U0#W0*53DAα16u/4(^<ZrTHT0'3 < $1/ 6

α2 = 1A4 IS/ 6 JU/4(F%.< $

"C< $$%(Naα1 > 2

o9+N:4C/ 6N/3&($< ##W/4%%34 A+Yw :&<>%34N5/4##W/4%S*J< A+5$"O

α1 → 0"J< < Tr%/4(F%.< $IN< 0*J< CFW

(:&< (00E< ZM6u(98$*(/4d$"O56u/4(NG:4%34@.<> A` *J< 1G G/4#T"%IV#< (< "(S< 0)(+1(7<>SF< <J346u1*/45< (/4(@/4:4/4< $X6u/4(< +,G/ 6X3>< $1^/ 6W5#< (< "(

αi 5<J3&/+(/+01*5<#(/+*01( &<>

:41< (< +7A/4(@/4/4("C< $R/ 6 F< 7<J34'6u1*/4*J< EFW'*J< $$05/ 6Bq< (w6u1*/4

wiA@(< :4&6u/4(@< (0w<J,/< *934%F+,

"OHT:)p /d/4/4(@/4:4/4< $/4(F%.< $A;/4(@/4:4/4< $O,E($=<>/4E/d< $$/ (GF< R6u1*/4GM#((@3402F&,?;/4+(%3D=< $ < :41$=<>(M0#0*J)/ 6*/4((#W/40:'w<J346u1*/4A

?<J347p /'6u1*/4φ1< 0

φ2<>@6k,D:'($=<>/

< φ1s|φ1s >=< φ2s|φ2s >= 1; < φ1s|φ2s >= S,alTA o

< 0!wV /41$0\$Q //4FT.< M/4(@/4:4/4< $%I0)6u1*/4w1

< 0w2S*.

<>@6k,< w1|w1 >=< w2|w2 >= 1; < w1|w2 >= 0.

alTA 4oD1*9?<(< @6u/4("C<>/ d7/ 1+1 AsN< "$%,4Ww'"C<J,\*./D/4GG6u/4$$/S%:6u/4("

w1(r, θ, φ;α1, α2) = βφ1s(r, θ, φ;α1) + γφ2s(r, θ, φ;α2),w2(r, θ, φ;α1, α2) = γφ1s(r, θ, φ;α1) + βφ2s(r, θ, φ;α2).

alTA &oY7< < <>I'< 0<#W*=< $K*J< G/ 6 <:4(< $K"/+0 W&' ikh/Cj m- 0*(FW0\_N##W0OHd`NAX_NZM/4$1T/4\/>6ZalTA o5< 0alTA 4o G/4FT.< \*J<

β =√1−S+

√1+S

2√1−S2

,

γ =√1−S−

√1+S

2√1−S2

.

alTA +o"OHT:\*/+ *+

β< 0

γ< (0#$=<,40--P:ATA \< <!6u1*/4/ 6

α1AV*9/ *V/ 6"OHT:G6u/4(" alTA &o < SV< 0T3 < &.< :4V/ 6#((@3D:

"C<@/4(R< #*G/ 6w< #W/ 6wCw<J3&C6u1*9.%/41*.-< V+1"MFW(/ 6/D0J< V/Sq?P:ATA DAVV16u1$FW*J< 1w"C<,d@$$U0D8%6k,'Z#(/4#W(@w/ 6Nq< (6u1*/4w< w"$< (U/MZ/4(:4%< $W<>/4"*6u1*/4A)/4@S"#W/4(@.< &$O,4+7/4(@/4:4/4< $%I06u1*/4

w1< 0

w2(01*

/G.<>/4"*Z/4(F%.< $φ1< 0

φ2;V<>/4"%*Z$"O

α2 = 2α1A

_N"&/40FW6u/4( DNr%NF< w*9/ *Z"C< Q4w% */4"#$w*/4D80%/4MF< KF(J< QR0/S A43&(%,M/ 6KF(J< Q+0/S*J< 'FS< **/41&0'6u/4(R("/ 6%0r%/4 ∑

iw?i (r)wi(r

′) = δ(r−r′)A?w*J< G*J< $*1$=<>^w1"

/4;$6kw/' /GS<>S9HD&N%0/D5< ##(/JHT"C<>NZ(980"/4< $]N(< *M0$%.<C6u1*/4

δ(r − r′)AXB/;3D1< $IV7< ##(/&< *. %Z7< 0T3D< F$V/

0 1 2 3 4

0.00

0.25

0.50

0.75

1.00

1.25

1.50α

2a

0=1

γ

β

mix

ing

coef

ficie

nts

parameter of 1s state, α1a

0

P:41(NTA +A)OHT:R*/+ *+β< 0

γ< w<V6u1*/4C/ 6N&34( Jp8/4(F%.< $

Iα1AB+34( >p8/4(F%.< $^ICVrTHT0-<>

α2 = 1AB< (J<)S(C

/4(@/4:4/4< $%IJ<>%/4E#(/D*01(V*.< (@,4K*J< qFWC AK` / q$"Oα1 → 0< 0

α1 →∞ 0/'/ 5(+1(%JA

(01*VVD1"MF(5/ 6U*/4"#/4+S/CFM< F$7/0#$=<,3 < $1/4\<#$/ JA?0rZ6u1*/4

C(r, r′) =1

4π

∫ π

0

∫ 2π

0

dφdθ∑

i

w?i (r, θ, φ)wi(r

′, θ, φ),alTA 4o

ES*.q ;< (<34(< :4:!/ 6C*/4"#$M*/40%/4?/3&(R;< :4$θ< 0

φAYN5#$/>0)!P:ATA 6u/4(R<>/4"*7F< 5$*0 AXP/4(5<'*/4"#$

F< G )/41$0/4FT.< <δ8$Q ;#WJ< Q-< $/4:d$

r = r′AU):4&/ 6

7#WJ< Q"J< 1(7(#(&.<>/4)/ 6<G:4%34(:4/4;F+,CF< AP/4( 0J< $*J< SS:4+^^/4F+3D/41$%,MTr%5< 0+1< $W< $/4:7SS/4$S$ A4($=<>%34VS0Td/ 6UM#WJ< Q!/SZM#<>=< $(/4$1T/ \/ 6^M$*0F< !G(< 0=< $B0(*/4 AWY7*J< FG<>N'< (J<*$/4R/;+1*$17ZFW@(#(&0!MF< AWM6u1*/4N0*J<J,\9HT#/4+=< $$O,))M$=< (:4

r$"'OJAS*9/4*/ 6W&,D0(/ :4*98$Q4 w<J34S6u1*/4U< ^< $//4"'0< 0T3 < D8

.

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

wav

e fu

nctio

n de

nsity

radius r/a0

P:41( TA TA><>/4"%* JK< 0 >/4(F%.< $ au0< 0G$.o 3DA> */4((#W/40:/4(@/4:4/4< $%I0qak?< (.ow/4(F%.< $au/ $0$.o9AVI/ 6KF/ \<>/4"%*< 0q< (/4(F%.< $U < 0>@1@0/4C9H< "#$S/ 6 ZL5N<>/4")&F&,*J< ((@,D:R/41T*/4"#$![^]S_Z` '#(/+*01( A _ $=< (:4($/+*J< $%IJ<>%/4/ 67)q< %(C/4(F%.< $< (/410+1*$1S A

.< :4A,)< (V/4"S<>50*1$%/< 0$)< *1< $*< $*1$=<>/4F*J< 1M&:4(< $5&34/4$%3D:'" < (R0*1$%N/4(N34 !"C< &,)*J< X"#W/4F$/dFW*J< $*1$=<>0-< < $%,+*J<>$$O,4AKD1*.-&:4(< $M< ##WJ< (M?;"*(/4*/4#*C#<8(< "(

tij< 0

Vijkl< S9HT#(0\F&,[U+Aa DA 4ow< 0qa DA &o9A

_N/ ( #W/4F$O,'^/M1N < 1=< p,D#Z/4(F%.< $> 4 D D X/ 676u/4("

ψ(r) =

(2Γ2

π

)3/4

e−Γ2r2

,

6u/4(Z'@.< (@:;F< 7JAY7< 0/4G nAP(/4" G#W/4&/ 6^3D / 6*J< $*1$=<>/45"$%347%ZZ"M1*9dJ< (N/;#(/D*0 A ,d<J34G"G1*.d*(+:4(< $B#(/4#W(@7< 0\G#(/D*7/ 60(":CG"%*(/4*/4#*R#< (< "98(K0/D/ K#(&^<N#(*#< $+0*1$%, nA>K< $$/S6u/4(< G< 1T/4"C<>*J< $9HD/4)/ 67*J< $*1$=<>%/4/'$=< (:4( @,D@"AL5/w34( < 1=< 6u1*98/4< (S/4$1T/4'/ 6XS< ("/4*98/4*$%$<>/ (K#(/4F$")A+!<>*9:7")&F+,

4

02

46

8

r/a00

2

4

6

8

r /’a0-0.250

0.250.5

02

46

8

r/a0P:41(^TA A^6u1*/4

C(r, r′)/ 6[UATalTA 4oX*.< (< *(I:*/4"#$

*/40%/4alTA &o9 <34(< :40E/34(7< :4$θ< 0

φ/ 6Gw<J3&G6u1*/4Z6u/4(J >T< 0 ># T± N/4(F%.< $A&N#$=<>J< 1< (/410N/4(:4'*J< 10;F&,< d< (@%r*=< $*1T/ E/< **/4""/D0<>V#%Q4M<>1(M/ 6UM0#0*J A _N$$

6u1*/4710Ed*J< $*1$=<>%/4\<J3&MG.<>"M3 < $1α1...5 = 1

A G*9< :4G/ 6#< (< "(*J< $#$/ JDS%/41TS*9< :4:'%< # A

"%"%I%:\;:4$;$*(/4-(: ,/ 6Z<d$J< (M*/4"MF<>/4q/ 6 < 1=< 6u1*/4T6u/4($*(/4"/3D:GR /41$/4"MF;#W/ &=< $nT*J< FW(:&< (00)< <R"C<>"C<>%*J< $l+(<>(< #+,D*J< $nT< ##(/JHT"C<>%/4 AD` *J< 1N.N< ("/4*/4*$$=</4(M#W/ &=< $'1+FW/41002S%-)*(J< :0@.< * U)w<J3&6u1*/4< (!0< "#0t9HT#W/4&=< $$%,S%

r+1< (0 /?w!"C<,< $/?"

$/4:>8(< :4 */4(($=<>/4A4 % j v m v ~ikjY z < (S<7FW(< ##(/JHT"C<>/4M/ 6W:4$N#< (@*$N6u1*/4w<>/4"*N@,D@"< ; < 1=< 6u1*/4A` 98*J< 1S/ 6WK S<J34S$*0'ND$=<>(K/4(F%.< $KMS("C< %:Z#< (@^/ 6WA

.

54 N 2" ( 8">2 ' $ d`L4% #" $M*./4*V/ 6KR*$=< 5/ 6KVF< 56u1*/45Sr(@Z@# AXs59HDNwR<J34/G9HT#(S9HT#$*%$%,'7LN< "$O/4< a DA o/4(0( ./M0=< :4/4< $%IN%JAD/M 0/!.< *.Q+$CC#(/4F$"/ 6S*J< $*1$=<.%:)/ 98$*(/4 a DA 4o< 0qw/>8$*(/4!#< (< "(a DA &o9AV*J< $*1$=<>/4*J< !FWV#W(@6u/4("0!R<>/4"*F< A>` *J< 1 w< &B/5 /4(Q7S%V/4(@/4/4("C< $4?< (6u1*/4 <J34/G.(< @6u/4(" #< (< "(w6u(/4" <>/4"%*N/'7?< (S(#(&.<>/4 A

0 /1&3 '( 4+-5-2" ! Z1("/D0$K0*(FWV$*(/4/ 6 *9< (:4 −e "/3D:dGr$0/4(:4<>%:6u(/4" N+1*$1S%C*.< (:4

ZeA+N*.< (:4N/ 6XN+1*$1 0/D / <J345/

FW'VE< 0T3>< *C< Z 'w< &/*J< $*1$=<>M34(< $^0%X(&R<>/4"7< 0E/ JA7:4$5#< (@*$5#< (@w/ 67LN< "$O/4< \a DA o^*/4@w/ 6ZQ+*5("< 0+1*$J< (N / 1$/4"GF(")TS*.)(J< 0S)<>/4"*Z1%

H1(r) = −∇2 − 2Z

|r| .alTA &o

SLN< "$O/4< R9HT#(J0M<>/4"*1% nA Aw0@.< *wK"J< 1(0G51%^/ 6` /4(^(.<>01

a04.S*9< (:4S1%^/ 6 $*(/4*S*.< (:4

e< 0

(: , ,D0FW(:RyA&?Sw< &/V0("S59HT#(/4C6u/4(U5/4##:

+:4(< $X;V<>/4"*'a ≡ #("0o(#(+.<>/4

t′ij ≡∫

d3rφ?i (r)H1(r)φj(r).alTA $4o

_S**/4(0:\//4(:4%q/ 6N/41(MF< M)<>/4"%*C6u1*/4 φi$*02< (

/4$1T/4S/CNLN< "$O/4< A_NZ@.<>0\FW6u/4(R/4$%,;φ1 ≡ φ1s

< 0φ2 ≡

φ2s< (d/ /4(@/4:4/4< $nA ` *J< 1E/ 66l< *)< $$N/ 80=< :4/ < $S(";

<>/4"*au#("0o(#(&.<>/4C/ 6t′ij&9HT*#Tw/ 6

t′12+< (SI(/ADD/wS<345/4$%,\/4+1%3 < $&'/4I(/d("M$6kG/41TM6u(/4" / .< $ / 6 DAB?;"C<J,q< $/

/ 7<>S@.<>φ2< 0

φ3,4,5 ≡ φ2p0,+1,−1

*/4((#/40)./G< "(: ,:4+3>< $15/ 6ZLN< "$O/4< ADD/&56u/4(" / 6

t′11 . . . t′55

0&*J< $nA4EZ/4$%,<34 /7(@U0%X(&U#< (< "(

α2 . . . α5A K/4F(@3 <>/4M(01*U6u1(@(

G+1"MFW(Z/ 60%X(&7&:4(< $5/;*J< $*1$=<>V/TAX_5$$/4(F%.< $N< (M*&.(0< (/410).V< "V/4(:4 AR3 < $1<>/4)/ 6.R&:4(< $S5@(< :4&6u/4(@w< (0;#(*J< $*/D/4(0<>A/4I(/$"+/ 6

t′ij< (

ε′1 ≡ t′11 = (α1 − 2Z)α1ε′2,3,4,5 ≡ t′22,33,44,55 = (α2,3,4,5 − Z)α2,3,4,5

8 .

t′12 =8 (α1 α2)

32 (α1α2(2α1 − α2) + Z(α22 − α21))

(α1 + α2)4

alTA &oLN<J3D:/4FT.< 0 E(1$%! ?<>/4"*dF< ?<J3&/(< @6u/4(" %/-E/4(@/4:4/4< $%I0 F< )< **/4(0:-/t[U+AMalTA &o;<>0 alTA +o9AS` *J< 1tij =< wi|H1|wj >

< 0)V0*/4"#/4%/4qalTA &o9V*/4((#W/40%:C9HT#(p8/4w(< @6u/4("0;/'V<>/4"%*ZF< S<J3476u/4$%$/S:R6u/4("

ε1 = β2ε′1 + γ2ε′2 + 2βγt′12

ε2 = γ2ε′1 + β2ε′2 + 2βγt′12

t12 =(β2 + γ2

)t′12 + βγ (ε′1 + ε′2)

alTA !&oD1*.t<d*.< :4/ 6ZF< G+(/+01*GD1<d1F@.< +=< $w*9< :4;3>< $1/ 6#< (< "(G< MM/S-P:ATA )6u/4(V;+1*$1G*.< (:4

Z = 2

S(d0< 0 $\(#(&(1$%tE<>/4"%*\F< < 0 /4$0q$R6u/4(R;/4(@/4:4/4< $I0F< AKG#$%/ G/4$%,E.3>< $1;/ 6

α1*/4((#W/ 0:/G J5/ (FO9<>$ S*.< :4:GS$α2a0 = 1

6u/4(w >S/4(F%.< $S*.*/ ((#/40M/!C*J< ;/ 6D1*$< (M*.< (:4

2AP/4(VC+,D0(/4:4*98$%Q4

$"'Oα1a0 = 2

α2a0 = 1

C/4(@/4:4/4< $%I06u1*/4VF*/4"C0&*J< $K/C<>/4"*M6u1*/4Aw <>/4"%*'< 0Eq< (V#<>(.<>"(R< (+1< $;S$"OJA]N1/'R< :41$=< (0#W0*R/ 6 ># '@.<>,)0// "OH)S%dJ< *.E/ (7< 0dS%d7@.<>A (<>/4"%*R(: ,dZ+1< $/<>/4"*(: ,)6u/4(S >N@.<>M<>NR< "V3>< $1R/ 6^#< (< "(

αAXs5/ V<>

%:4'/ 6WS/4##:7#< (< "(t12

0/D^/ ^*< (@,'<34/FS< "< w<>S/ 6

t′12A Z;7/ (< 0 T:4;/ 6Bw#< (< "(0/+S/ S(J< $$%,

"C<>( 6u/4(S<G /$34$@,D@" *77:4+3>< $1S/ 6%SLN< "$O/4<

H =

(ε1 t12t12 ε2

)

<J3476u/4("

ε± =ε1 + ε22

±√(ε1 − ε22

)2

+ t212,

S*.)/4F&3D/41$%,0#W0&56u(/4" :4/ 6tA

00 ' /1&3 '( 4-5 -2" !Np />8#< (@*$5#< (< "(w< (Z0r0;F+,[UAa DA &o9ADL5< "$%/ =< /4#D8(<>/4(< 76u/4(" / 6BR /41$/4"MF(#1$%34&(< */4

H2(r, r′) = H2(|r− r′|) = 2

|r− r′| .alTA &o

.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-4

-3

-2

-1

0

1

atomic basis

α2a

0 =1

ε'1

ε1

ε'2

ε2

t12

t'12

sing

le p

artic

le p

aram

eter

s

parameter of 1s state, αa0

P:41(\TA DA d:4$!#< (@*$)#< (< "(/ 6 J< 0 >/4(F%.< $A^*9< (:4)/ 6Z!+1*$1C

Z = 2A ]Z< 0t$(#(&)(1$O.

<>/4"*wF< ADD/4$%0G$K6u/4(Kq<>(^F< A4_N$$#< (< "(^.HT*#Tw/ 6WJα< (7rTHT0!<>S73>< $1

1At12

8(0)$ ε18F$1$

ε28 : (!$ A

?w< &5/0("Zr(@573 < $1S/ 6K#< (< "(S;.V<>/4"*'au#("0o(#(+.<>/4

V ′ijkl ≡∫

d3rd3r′φ?i (r)φ?j(r

′)2

|r− r′|φk(r)φl(r′),

alTA oS(CC<>/4"*M6u1*/4

φi(r)< (/ 6 6u/4(" alTA &o9A_N$$U6u1*/4

< (V*+(0<>(/ 10d<'*/4""/4)/4(:4?au+1*$1N#W/4%/46u/4(M<>/4".o9A*/4""/4\/4(:4\< $$/S56u/4(7< < < $%,+*R3>< $1<>/4\/ 6 < $$B&:4(< $GalTA o9AW0*1$%S#< (@Sw9HT#(/4)6u/4(w&3&(($=<>%3470@.< *7/ 6K$*(/4

1

|r− r′| =1

√

(rx − r′x)2 + (ry − r′y)

2 + (rz − r′z)2.

9HT#(/4/ $Q4$%,&:4(< F$ w< (@=< */+/4(0<>AVD/ <34/'1V#(*J< $#W/4$=< (*/+/4(0<>5< 0"G1$%%#W/4$%Z9HT#< /4qanNA &op/ 6

8 . -$

1/|r − r′| < ##(/4#(=<>V//4G*/D/4(0<>WS*.N"/4(M*$/4$%,!0*(F0_N##0OH\N

1

|r− r′| =∞∑

s=0

s∑

m=−s

(s− |m|)!(s+ |m|)!

rs<rs+1>

P |m|s (cos θ)P |m|s (cos θ′)eım(φ−φ′).

alTA 4oL5(CC*/4((#W/40:\*/D/4(0<>R/ 6w#W/4&

r< 0

r′< (

(r, θ, φ)< 0

(r′, θ′, φ′)(#W*%34$%,4S(

r< 0

r′< (V0@.< *56u(/4" V/4(:4

θ

7#W/4$=< (< :4$N6u(/4" 7I98p<HTwS%0 ≤ θ ≤ π

< 0φw< I"M1T< $ < :4$

M HD,&8#$=< w6u(/4" H+8p<HTKS%0 ≤ φ ≤ 2π JA r< < 0 r> < ((#*%3&$%,5""R1" < 05"C<HT"M1" / 6 Sp /R0@.< *

r< 0

r′A+56u1*/4

Pmk (cos θ)

< ()d< /D*=<>0 :40(!6u1*/4A (@:[U+AalTA &o'< 0alTA 4o &/)alTA o V/4FT.<

V ′ijkl =

∫

d3rφ?i (r)φk(r)

∫

dr′Rj(r′)Rl(r

′)×

×∞∑

s=0

s∑

m=−s

(s− |m|)!(s+ |m|)!

rs<rs+1>

P |m|s (cos θ)×

×∫

dθ′ dφ′P |m|s (cos θ′)eım(φ−φ′)Y ?

j (θ′, φ′)Yl(θ

′, φ′)︸︷︷︸

(?)

.alTA &o

N#(*J< $X< ("/ *FW<J3&Z$%Q4Y (θ, φ)lm ∼ eımφ

ATL5*NN+:4(<>/ /34(

φ′/ 6'#< (@Ca

?oNE[UAalTA &oN$J<>07/)<;1F@.< &=< $U(01*/4/ 6

1"/34(m* ∫ 2π

0exp(ımφ′)dφ′ = 2πδm0

AsN< "$%,4(?)

R/4I(/d/4$%,6u/4(

m = mj − mlA Z;/ ('< 0 &:4(<>/4q/3&(M)< I"M1T< $

< :4$φ/ 6r(@M$*(/4q$J< 0V/\<)6u1(@(R*/4@(< &M/4q[UAalTA &oZ6u/4(

3>< $1m = mk −mi

AY6 R*/4"MFZwV/4FT.< ;6u/4$$/S%:'($=<>/ mi + mj = mk + ml

S*."J<>G<>G;/ .< $ I98*/4"#W/4&'/ 6Z< :41$=< ("/4"&1" / 6^$*(/47*(J<>0d\@.<>

i< 0

j< N/FWM+1< $/<>Z/ 6

$*(/4'< $<>0q@.<>k< 0

lK/dM*/4"#W/4&'M*/4(@3402F&,

Z$*(/4D8$*(/4&(< */4 AT[HT#(:'#(*J< $X< ("/4*C.("'w/ 6< /D*=<>0 :40(6u1*/4Vau_RA 4ow"C<J,(S(%V[UABalTA &ow<

V ′ijkl = Cδmi+mj ,mk+ml

∫

drdr′r2r′2Ri(r)Rk(r)Rj(r′)Rl(r

′)×

×∑

s≥|m|

(s− |m|)!(s+ |m|)!

rs<rs+1>

∫

dθ sin θP |m|s (cos θ)Pmi

i (cos θ)Pmk

k (cos θ)×

7:* 46)'!*$C!# A + !"#%&()* # *!= )*J= *JGH"=?&1" !"#%.+-,FE)A?.θ$2*3

φ$2'#:'# # '1.#3N=?.46'# # *!A +

.#3 #.O#&(=?$2A A + = * , $C!"# , $C!=?& . A #$3 = *@F!) **#&(#..$2'+J&()* 4 .= )*QP

" .

×∫

dθ′ sin θ′P |m|s (cos θ′)Pmj

j (cos θ′)Pml

l (cos θ′),

alTA +oS(NN#(*0:M/4("C< $%IJ<>%/4G6l< */4(

C = 4π2βimiβjmj

βkmkβlml

< 0βimi< (Z0r0)F&,au_RA &o9AT7&:4(<>/4 /34(wZ#W/4$=< (w< :4$

θ< 0

θ′/ 6FW/

$*(/4(1$%;2!/ .< $N< :41$=< ("/4"&1" */4(@3><>/4 (1$;< C+:4(< $w/ 6B,D#W

Iijk ≡∫

dxPmi

i (x)Pmj

j (x)Pmk

k (x)

< (^/4I(/S/4$%,Z%6+p /S/ 6D^#< (< "(*J< FW< 000V<w3&*/4(p8$Q K"C< (/*/4@(1*Uw(0G/4 A4_56k(U*J< ((@,D:7/41TUS< :41$=< (&:4(<>/4 S*J< 9HT#< 0)(< 0=< $X#< (@5/>6&:4(<>/4;< **/4(0:G/

∫ ∞

0

dr

∫ ∞

0

dr′f(r<, r>) =

∫ ∞

0

dr

(∫ r

0

dr′f(r′, r) +

∫ ∞

r

dr′r(r, r′)

)

.alTA 4o

_NUr< $&(1$% ^/4FT.< 7<>V ′ijkl = f(αi, αj, αk, αl)

JS(f<(.<.%/4< $

6u1*/4/ 6%#< (< "(AU$*0VF+,Z1J< ($( "%"C<>$ <>/4"*F< */4@/>6Dr34^/4(F%.< $AJD/

ND1"MFW(#W("M1T.<>/4/ 6 6u/41(0* #< (< "( 0r0;F&,C[UAalTA o54 = 625

AXMD1"MFW(Z/ 6^($3 < &7&:4(< $5Z/ < Z$=< (:4 AX!/ 7/ 6U"3 < !FW*J< 1M/ 6KV/ .< $B< :41$=< (S"/4"&1" < 0!VI98*/4"#W/4&Z/ 6K< :41$=< (N"'/4"+1" */ (3><>/4d(1$Z0*10!M#(3D/41Z#< (< :4(< # S*9M(01*Kw+1"GF(K/ 6/4I(/7< 0G0#0+K3 < $1A w+1"MFW(K/ 60#W0& 3>< $1/ 6WS />8#< (@*$&(< */4'#< (< "(^^< $/V(01*0FW*J< 1R/ 6KV@,D""(S9HTF%0!F&,![UAalTA o9AWP(@$%, V#< (@*$5< (0@:41%<>F$ AWL5* M&:4(<>/4)3>< (=< F$

r< 0

r′*J<>FG@w< ##W0

S%/41TB*.< :4:^3 < $%1^*J< $*1$=<>0 i, k ↔ j, l A9HDB6l< *B*J< 10F+,tI98*/4"#W/4&)/ 6VE< :41$=<>(;"/4"&1" */4@(< &JAs5/ \.</4#W(<>/4(?alTA 4oC.< Q )(J< $Z3 < $%1)/4$%,4AwE/4$%,t*/4"#$9H (" alTA o< (

exp(ımφ)6u1*/4G/ 6S)< I"M1T< $0#W0*!/ 6S;#(*J< $

< ("/4*AB` 1TJC&:4(<>/4E*J< *$R/41TM< $$UC*/4&(F1T/4ES*.7/ .< $

φ0#W0*R0%X(&6u(/4"

φ′0#0*J AW;*J<

+:4(< $< R/dFW;(J< $nAKD/Bw;"C<J,?*/ >@1:&<>;%G< 0/4FT.< ?;< "(J< $3 < $1ZS(J< S0**9< :4F&,;0r%/4

(V ′ijkl

)?=

(∫

d3rd3r′φ?i (r)φ?j(r

′)2

|r− r′|φk(r)φl(r′)

)?

=

=

∫

d3rd3r′φi(r)φj(r′)

2

|r− r′|φ?k(r)φ

?l (r

′) = V ′klij,

. "!

$Q i, j ↔ k, l A /4"MF:w< F/3&^w/N#("G1T.<>/4 w"C<,7*/4@(1*< $$Sd/4+1%3 < $&+1*)/ 6V0*;2\6u/4$$/S:Ew<,4AP(@J */4@(1*S< $$#W/4F$N#< ( />6p /M0* */4((#W/40:R/RZ< "N#< (@*$1*.<>G;r(@09H-G/ ':4(J<>(M;*/40 US*9-G#W/4F$FW*J< 15/ 6W*/40C#(/4#W(,4A&S+1"MFW(U/ 6X1*.#< (UK+1< $/7S1"

n∑

i

n∑

j≥i1 =

n∑

i

(n− i) =n(n− i)

2,

alTA &oS(

n!D1"MFW(C/ 60%X(&$"&#1Tt/4(0(0 #< (A^ 01

*/41(V/ 6BS*J< $%*1$=<>/4 V/4FT.< 150%X(&5#< (w6u(/4" %< $

5/4(F%.< $

S*.\!1(d< (R#< (0d< :&< /F1$0!/4(0(0d#< (5/ 6^#< (AWN(01*'%< $D1"MFW(/ 6 &:4(< $

625/G3 < $1

120*< $*1$=<>0d6u(/ " alTA &o9A

_56k(^#W(@6u/4(":7N< *1< $*J< $*1$=<>%/4G(N< (5/4$%,24/4I(/V3 <>$1$6kJA

,)< (7$@0_N##0OH!]RA79HDS# Z(< @6u/4("C<>%/4C6u(/4" 7F< w/ 6<>/4"*S<J3476u1*/4/^q< (F< A 1@$Q K6u/4( ^%:4$U#< (@*$U@.<>U(< @6u/4("C<>/ 0r0!F&,V[U+AalTA &ow< 0qalTA +o9A,;(< @6u/4(" <

Vijkl =∑

i′j′k′l′

V ′i′j′k′l′∏

z=i,j,k,l(δzz′ + β(δz1δz′1 + δz2δz′2) + γ(δz1δz′2 + δz2δz′1)) .

alTA "$4o(9HT#$*%9HT#(/4SS$$X/ SFW$@0(>A

j ( $," d " ( `% " (2 ( " %(dy# $,+- / #".$

LN<J3D:< $$NE#< (< "((+1(0 /9HT#()6u1$$%,2*/40D8+1< &I0LN< "$O/4< Ea DA o9Tw"C<,C(@,;/'0=< :4/4< $%I7%JA #5T6u/4(@1<>$%,4T(VS/< < $%,+*)/4$%1T.%/42\:4(< $5*J< A?\0/q/ "#W/4!< &,6u1(@(C*/4D8@(< &Z/4dM/4985< 0dM />8S$*(/4#< (< "(Z*J< $*1$=<>0\ED*/4TA DA^B/qFW\< F$/?0=< :4/4< $%I% \<J3&)/?1d< 9F(1T)6u/4(* "/D0 AsN< "$%,d C$*'<rTHT0D1"MFW(R/ 6S$*(/4A?C0/\Q+/ C+1"GF(R/ 6$*(/4 6u/4(wJ< *.)<>/4" < 0/4 AD,< (

2$*(/4w6u/4(

He< 0

H− < 0 36u/4(LiBe+

+< 0He−

(#W*%34$%,4A+ S*/4@(1*w< $$TS#W/4F$*/4Tr:>81(<>/4N/ 6$*(/470@(F1T0d< "/4:CMr34Mq< (Z@.<>710E!*J< $*1$=<>%/4AD[^< *.Cq< (6u1*/4C&(/D01* /M#@.<>ATND1"MFW(/ 6*/4Tr:41(<>/4*J< )F*J< $*1$=<>0)<

Ω =

(2M

N

)

=(2M)!

N !(N − 2M)!,

alTA &o

.

S(M

ZD1"MFW(w/ 6q< (w@.<>S< 0NZ+1"GF(w/ 6B$*(/4A

RD1"MFW(5/ 6K"G1$%%#< (@%*$7@.<>N5)/41(5*J< 456u/4(

2$*(/4Z< 0

1206u/4(3$*(/4A&s5/ <>K+1"MFW(K/ 6W"M1$%#< (@%*$% @.<>U< U%""M1"

6u/4( < "#T,/4(*/4"#$$%,Mr$$0,D@" < 0(J< *. N"C<HT"M1" 3>< $156u/4(!< $%6=8Yr$$%0*J< nA AU6u/4(

N = MA P/4('!*J< !/ 67"/4(< t< $%6=8Yr$%$%0

@,D@")BD1"MFW(V/ 6"M1$%%#< (@%*$G.<>V6u/4(V<):4%3&?+1"MFW(V/ 6/4$Nh!Vr$$0!@.<>M5+1< $B/CM+1"MFW(N/ 6^@.<>N6u/4(

N = Nh

$*(/4N6u/4(N ≤M

A[^< *. />8/4(w(98#< (@*$Z@.<>7*J< )F9HT#(J0!F&,7*(J<>/4/4#(p8

<>/4(

|I >≡ |ij >= a†ia†j|0 >

/4( |I >≡ |ijk >= a†ia†ja†k|0 > .

alTA !&oM"M1$%#< (@%*$%@.<>7< (R+1"MFW(0!6u(/4" /4V/

ΩakS*9!N%(

45/4(

1206u/4(

N = 2< 0

3 (#W*9.O3&$%,o9ATZ"#$@w<,/R:4(<>Z*/4*1T%34

@.<>GV/\%(<>C/34(R0*ij< 0%6S< ##$*J< F$

kB(@(*:!

3 < (=<>/4(< :4/ 6B" /

i = 1 . . . 2M j = i+ 1 . . . 2M k = j + 1 . . . 2MalTA &o

/<J34/407:4(<>/4;/ 6B@.<>0%X(:M/4$%,CF&,<M#< AVLN< "$O/4< a DA o5$Q+734(@,d#< (Z*/4@:;/ 6w<;"M1$%%#< (@%*$R@.<> |I > < 0?<*/4>@1D8:&<>0"M1$%#< (@%*$%@.<>

< J | /SF1$0N^"C<.(HZ$"&/ 6D^LN< "$O/4< "C<>(OHHIJ =< J |H|I >= f(α1, . . . , α5).

D1*9<)"C<.(OHd*/4@R/ 6S< < $%,+*G6u1*/ R/ 6w#< (< "(α1 . . . α5

A6u1*/4^< (9HT#(0C"($%,V(/41:4'1"40%X(*&"M1$%#$%%*J<>%/4>0O83D/4 < 0d+1< (G(/D/ XF1T7"C<J,!FWM+1%M$: +, AX_56k(7*/4@(1*:LN< "$%/4=<>\"C<>(OH) G"M1$%.< /41$%,)<J3&M/;0=< :4/4< $%IM%7< 0d""'%I7/4FT.< 0$/w@: +3>< $1NS%(#W*S/R

α1 . . . α5#< (< "(AT

*/4"#$9HT%,V/ 6 "C<>(OHV#$=<J,D*(1*=< $D(/4$ R"#$O6k,D:S#(/+*A&P/4(B@.<>R:4(<>0q< V*/4"MF<>/47/ 6w/4#W(<>/4(6u/4(

2$*(/ J$"&

S*9\"C<J,FWR/4I(/C6u/4(N/4"3 < $1N/ 6U#< (< "(Z< (R/Sd\P:ATA TA ZG"C<J,!G<>ZG"C<>(OH!Z/ 734(@,d*/4"#$%*J<>0dF1T7%Z1(7/41TZ<>%"C<,GFW56u1(@("#$Or0F&,(:&< (0:N@,D""(/>6 N@,D@")AD6uJ<>1(7S$$XFW34!"/4(7"#W/4(@.< &w6u/4($=< (:4(w@,D@"A

L5/"C<J, M*$=< %6k,M@.<>N//4FT.< \<C"#$Or0\LN< "$O/4< !"C<8(OH ?qLN< "$%/ =<> a DA o;*/4""M1T)FW/ S% ?I98*/4"#/4+\< 0+1< (0!#)/4#(<>/4(nA A

[H,Sz] = [H,S2] = 0.

.

0 10 20 30 400

10

20

30

40

P:41(5TA TA 7(< $W@(1*1(N/ 6/4I(/M$"'+/ 6 ZLN< "$O/4< '"C<>(OH6u/4(B

28$*(/4M%1<>/4RS%Mw"M1$%#< (@%*$% @.<>U:4(<>0< K/4(0(0

*/4"MF<>/ / 6 10

/4#(<>/4(V*J< @0?/4?+1"MFW( A :Aa†1s↑ → 1 a†1s↓ →

2 . . . a†2p−1↓ → 10AV$=< (:4(wIFW/JHT5*/4((#W/40)/

2× 2 F$/D*.Q+5< 0"C< $$%(^< (0%3D01< $T$"+A4(N< (5< $/76u/41(2×3 FW/JHT < 0/4 3×3FW/JHXAs5/ 7<>$=< (:4@SF$/+*.QX*%34$%,;< I

7× 7 A

D/< &,!$*0d3 < $1Z/ 6^G/4#W(<>/4(Z@.<J,!+3>< (=< +N10(ZG< */4\/ 6'LN< "$O/4< WnA AW'0=< :4/ < $%IJ<>%/4\0/D7/ "OH!@9<.7S%0OX(+3>< $1/ 6N)/4F(@3 <>F$A-):4(< #*J< $w(#(&.<>/42/ 6N"C<>(OHXZLN< "$%/ =<>C"C<>(OH'*/4@(1*0/41*.@.<>wF$/D*.Q'0=< :4/4< $nA+L5* %;J< (;/0=< :4/4< $%I AP(@$%,2/4E/ *<> /$*(/4)"C<J,tFW$/+*J<>0q%(R/4?< "C/4(F%.< $^/4(V/4?p /\0%X(+M/4(F%.< $%Aqr(@*J< K01./\ < 1$ #(*#$;$*(/4G<J34/d<34;/4##W/4%#A?R*J< B Q+/ <>R(/ .< $#qV+1< $/\I(/AB(;< (r3&1*.q@.<>B/4'6u/4(J< *.qF< /4(F%.< $nA E*J< C/ 6 w/)$*(/4R$/+*J<>0/4 0%X(&/4(F%.< $%J< *. / 6V" "C<,2<J3&< < (F%(< (@,3 <>$1d/ 6R# I*/4"#/4+JA w/)$*(/4*J< EF1$0\/4:4(V<;@.<>GS%d/ .< $K#E%(S = 0

/4(S = 1

AW\M$=<>(5*J< XMI98*/4"#/4+N.< Q Z3>< $1 −1, 0 < 01AWM*$=< N0=< :4(< " / 6UR/4(0(0d"M1$%#< (@%*$%V@.<>Z50#$=<J,&0\\P:ATA $DA(RN< / (5+1@/4\< 5/C/ ./'r0!R@.<>S*.)rZ*." < 1T/4"C<>%*J< $$O,4AM"/4/>8/4(F%.< $@.<>7*J< dFWM"#$%,$=< FW$$0 F1TFO8/4(F%< $K@.<>R<J34/)0("0 AC*/4*/4VFW q'w//4(F%.< $@.<>S< (Z/S;P:ATDA TA[^< *.#< (/ 6B/4(F%.< $< w1*.@(1*1( A

.

2 particle states # = 45

biorbital states #=40

Sz=1 states #=10

Sz=0 states #=10

Sz=-1 states #=10

S=1 states #=30

S=0 states #=10

monoorbital states #=5

P:41(ZTA $DA $=< / 67/4(0(0;"G1$%%#< (@%*$%N@.<>A7+1"GF(/ 6@.<>*/4"#W/4:GJ< *9)*$=< 0#$=<,40 A

? ? ↓↓ configuration Sz = -1

S=0 Sz = 0 configuration

↑↑ configuration Sz = 1

S=1 Sz = 0 configuration

←S+ S-→

orthogonal

←S+ S-→

P:41(MTA TA /4*/4ZFWw0%X(&7"M1$%%#< (@%*$V@.<>NS%!p /;0%6=86u(+5/4(F%.< $%A[^< *9)/4(F%.< $X#< (< 5<G"$< ( @(1*1( A

. &

(w< (10/4+1%3 < $&#< (6u/4(

5/4(F%.< $J*w< $$&/ (#< (0%X(B/4$%,

F&,V<#< A D/Sw^ <>( < (10#W/4F$^@.<>BJ< *.R*$=< :4%3D:w

/ .< $/ 640FO8/4(F%.< $T@.<>A4 *9 +<>6k(U5< 00%/4G/ 6X"/4/>8/4(F%.< $T@.<>

(#(/D01*R/ .< $+1"MFW(5/ 6 G"M1$%#< (@%*$%*/4Tr:41(<>/4*J< $*1$=<>0)alTA &o9AD7#(/+*01(7/ 6<3D:&<>:R< $/4:RZ#W/ F$%Z*/4*/4w 7(< #TA SB+1%3>< $+B/Sr0:5w $FD87/4(0< R*/+ *+D 4!+S'< 00%: /C< :41$=< (w"/4"&.<TA

S±/4#W(<>/4(5< (7/ 6B76u/4("

S+ =M∑

i=1

a†i↑ai↓

S− =M∑

i=1

a†i↓ai↑.alTA o

%< $U@.<>*J< qFWC*9/4-< M<!@.<>CS%q%(p /\#M1#q/4(Vw/#N0/S AWR*/40< 0!(0)@.<>R5/4FT.< 0)6u(/4" %NF&,!< ##$%,D:'S±

/4#W(<>/4(AXM6u/41(@d@.<> */4"#$:G1F#< * Z/4FT.< 0d1:V/4(@/4:4/4< $Op,*/40%/4 A\/41(S*J< W%5"($%,*/4((#/40N/*9< :4V/ 6:4FWwp /d("GF1$0%:!

S = 1Sz = 0

@.<> U#(/3D0:\+1S = 0

Sz = 0

@.<> A'*9< :4G/'@.<>6u/410EdZ<J,d"#$OrGLN< "$O/4< !"C<>(OH!d<C1F@.< +=< $"C< (X< Z/SdP:AWTA !TAXI/ 6B$=< (:4@F$/D*.Q

5× 5 AYS#< 0!/4@.<>

|A >=1√2(a†4↑a

†5↓ + a†5↑a

†4↓)|0 >

|B >=1√2(a†1↑a

†2↓ + a†2↑a

†1↓)|0 >

|C >= a†1↑a†1↓|0 >

|D >= a†2↑a†2↓|0 >

|E >= a†3↑a†3↓|0 > .

alTA 4oV@.<>S/ 6 |A >, |B >, |C >, |D >, |E > &34/4$%3&7%*/4@(1*98/4!< $$r34M/4(F%.< $A_N51(Z/41TN)V6u/4$$/S:'*J< $*1$=<>/4S%N*J< / NFW(01*0C6u1(@(^F< 0/4$$%,G/4'5@,D""(@,'*/40(<>/4A&wNIS/ 6X$=< (:4@RF$/D*.QE< $$/S6u/4(M< < < $%,+*/4$1T/4E6u/4(:4(:4K< V(9HT@N< < $%,+*N6u/4("M1$=< 6u/4(w(/+/ S/ 6#W/4$%,D/4"< $%w1#/G7r6k)/4(0(A

'F< d'P/D*.Q\#< *G6u/4(<p />8$*(/4@,D@"g*J< EFW'10?< V<@.< (@:C#/4+56u/4(5"#$O6k,D:'MLN< "$O/4< )"C<>(OH;6u/4(5V(M$*(/4@,D@")TIZ/ 6S*.

120× 120 ATY6 7:4(<>5.NF< w7P/+*9Q'#< *%(<>:;@.<>;alTA !&oZ(< :4\alTA &oNw/4FT.< 34(@,E%"#$'@.<>

" .

0 10 20 30 400

10

20

30

40

P:41(MTA !TAXM/4I(/$%"+Z/ 6UGLN< "$O/4< !"C<>(OH6u/4(N2$*98

(/45*J< %)@.<>5:4(<>0!)<Gw<J,9HT#$=< 0);V"C< %;9HDJS*."#$OrwVLN< "$%/ < ;"C<>(OHC/'F$/+*9QC((01*F$Z6u/4(")ATV$=< (:4@F$/D*.Q

5× 5 7("C< :'< ( 2× 2 < 0 1× 1 A

F1T NLN< "$O/4< '"C<>(OHGS$$/ F534(@,C0=< : /4< $nA4N/4I(/M$"&/ 6UMLN< "$%/ =< )"C<>(OH;6u/4(5(M$*(/4Z< (R9HTF%0\dP:ATA TAWB//4FT.< )F< C/ 67"M1$%#< (@*$%@.<>C\P/D*.Q-#< * !9HT#< 0t9HT@:F< Z6u/4(Zw/$*(/4F&,< 00:;M(0 A (C< (M('"C<>@/4(%1<>/4 i 6 ?@.< (@'S%;r34)"/4/>8/4(FO.< $ @.<>AK?)< 00-/d" <\(0$*(/4 AW]Z1V/ < 1$#(*#$ %N*J< / NFWR$/D*<>0!/4!V< "/4(F%.< $w< M;%< $^p /A^D/d(< (6u/41(G/ (FO9<>$M$6kM/d#$=< * $*(/4 ^S*. # *J< <J34d< +,3>< $1 AD/^w\:4

4 ∗ 2 = 8 (#< (@*$@.<>R6u(/4"<!:4$/4"/4/>8/4(FO.< $Kp />8#< (@*$@.<> AT#(/D*01(7 0/4NS%< $$/4(F%.< $Cw*$=< w/M<> r< $$%,!/4FT.<

5 ∗ 4 ∗ 2 = 40@.<>A_N"#/4(@.< +N6l< *97// GN<>

M<34G< $(J< 0T,):4/ kzjY j.z S%! /;$*(/4N/4\R< "R/4(F%.< $nA_00:G<R*/40;$*(/4;/4/4"N/4(F%.< $<RFO8/4(F%.<>$@.<>N*J< ;/4$%,$J< 0q//4;/>65;@.<>G/4FT.< 06u(/4" <\"/4/>8/4(F%.<>$@.<> A^D/U< $$"M1$%#< (@%*$%N@.<>w0*10)1F+1J&$%,<34Z/M*/4@/ 6B$*(/4/4E(C0%X(+R/4(F%.< $%ABD*"/4/>8/4(F%.<>$U@.<>M< $%w<,DV<J34S = Sz = 0

^!(1$%.< +?#< (@*$)@.<>!FW$/ :4/?!1F#< *S = 1/2

< 0Sz = ±1/2

.

0 20 40 60 80 100 1200

20

40

60

80

100

120

P:41(NTA TA+Z/4I(/M$"& / 6ZLN< "$O/4< "C<>(OH'6u/4(38$*(/4

%1<>/4 &S%;@.<>:4(<>0)< /4(0(0;*/4"GF<>%/4 / 610/4#(<>/4(

*J< @0)/4)+1"MFW( A :Aa†1s↑ → 1 a†1s↓ → 2 . . . a†2p−1↓ → 10

iki 6 ?Z< 00< $*(/4C/RN9HT@:S = 1

@.<> ADYw< /MFWN$/D*J<>0/4<M(0)/4(F%.< $lF*J< 1R@.<>5*/4@(1*0!/41T5/ 6 /Q+05/ 6/4(F%.< $<J34V< $(J< 0T,FW)0#$0);#(3D/41*J< A_**/4(0:G/P:ATA (V< (Z(*$=< 5/>6@.<>SS%

S = 16u/4( p /'$*(/4AV

$*(/4*J< VFW^7w/S@.<>nA AJ%*J< V%(<34^#V1#V/4(#V0/S A ZM< 7< $//;/ V<>Z(G< ( (5

3

)= 10

/4+1%3 < $&7*./4*Z/ 6( /4(F%.< $%/41TK/ 6r34 A4D/><J,4 w:

3∗2∗10 = 60 @.<>A?:4 / .< $D#M%(S = 3/2

/4(S = 1/2

A R 6u/4("(B*J< w w<J34<C+1< (@NS%\#W/4F$V#(/ @*/4N/4\I98p<HT −3/2,−1/2, 1/2, 3/2 AX$=<>(*J< wC<J34C/4$%,d'w/!"00$M3 < $1AqJ< *.q/ 6 (#< (@*$Z1FW*$=< S(V< (

10"M1$%#< (@%*$%Z@.<>

ikiki 6 P< $%$%, K !*J< < 00(0$*(/4/ES = 0

w/?$*(/42@.<> A?Z*J< ;*/41+wZ@.<>S< < $/4:4/41$%,4D< < FW/3& AD?7<J347/4$%,/47*$=< / 67FO8/4(FO.< $w />8#< (@*$@.<>S%

S = 0AwD/^FW*J< 1\/ 6Z) /

#/4F$@.<>S/ 6$*(/4;w:41 ∗ 2 ∗ 10 = 20 (98#< (@*$Z@.<>A

?R*J< \1" 1#!M@.<>R*/41&7/4FT.< 0!)V(M%1<>/4N< 0\/4FT.< )/ .< $+1"GF(C/ 6

120"M1$%#< (@%*$%@.<>^S*. < :4(S%-!+1"G8

FW( /4FT.< 0'6u(/4" %(<>/4!alTA &o9A&N/4I(/R$"&/ 6XNLN< "$%/ =<>"C<>(OHX+*J< $*1$=<>06u(/4" Z@.<>:4(<>0;C7<>FW/3&Z0*(F0;w<, < (

.

0 20 40 60 80 100 1200

20

40

60

80

100

120

P:41(7TA 4ATs5/4I(/'$"&/ 6LN<>"$%/4< C"C<>(OH6u/4( 3$*(/4

F$/+*9QZ(#(&.<>/4 S%^"G1$%%#< (@%*$K@.<>: (<>0M7^w<J,/4(:4<>%:6u(/4"

28#< (@*$98F< %7:4(<>0?J< ($(A $=< (:4@X*%34

F$/D*.Q8× 8 A

0#*0??P:ATA 4A` $/D*.Q+V*/4((#W/40:/)0%X(+3 < $1/ 6S< 0

Sz*J< ?FWC ABC@.<>M< ('/ R/4(@0qF1TR34%%0'F$/D*.Q+,?< (0*/41#$0G+/N1FF$/D*.Q+A4w$=< (:4@KF$/D*.Q/ 6"G1T1< $$O,0#W0&^@.<>U< wI

8×8 A4L5*ww:4&3>< $1K*J< GFWw$/+/4Q40M1#G0#W0&$%,RM0%X(p8+F$/D*.Q+AYS0/D/ S< $$/ 6u/4(< < < $%,+*N/4$1T/4 DF1T7"C<>(OHC/ 6B1*.I7*J< )FJ< $%,< 0;34(@,;(< #0$%,C0=< :4/4<>$%I0;+1"(*J< $$%, DS%/41T(/4(@@8:)/\< +,E/ 6w< 0T3 < *0"/D0ABVR&=< $n< VC/4#T"%IJ<>%/4/ 6<6u1*/40#W0:?/42"C< &,-#< (< "(

α1, . . . , α5(+1((#WJ<>0

*J< $*1$=<>/4)/ 6KV6u1*/4 53 < $1WJ< *.d&34/4$%3D:'6u1$$0=< :4/4< $%IJ<>%/4 A6u1(@(K0*/41#$:Z/ 61FF$/+*9Q+S%RwF$/+*9Q7S%M:4%34M3>< $1K/ 6#MK<"C< %6u@.<>/4R/ 6W/4(@/4:4/4< $Op,V/ 60%X(+U/4(F%.< $A>YK< U/ ^< 1"0''< 0D83 < * WF1T5V#(/+*01(R/ 6K/4(0(:'V"G1$%%#< (@%*$7@.<>N< $/#< (<>/V/4"S9HD&J&S#< *S%':4%34'3 < $1^/ 6WS/ .< $< :41$=< (U"'/4"+1")A_N< $%(<>%34N< 06u1$$%,G6u/4("C< $"'/D0/ 6:4(<>:RN"G1$%%#< (@%*$SF< w/41$0)*/4@N/ 6U"M1$%.< /41:4(<>:<'(N/ 6K@.<>5S%rTHT0!3 < $1/ 6SSzLLz1:)$=< 00(/4#W(<>/4(

S±< 0

L±AB` 1TR%1(/!FW

/ C*< (@,4w<>')"#$(G"/D0&(/D01*02F&,-1(!#(/3D0C< $/<>@6k,D:'(1$%A

HE H− LI BE+

HE−

%$

< F$'TA 4A Z#T"%I0` /4(p8/4(F%7(< 0ai = α−1i

/ 6 J >< 0 #E/4(F%'au1%/ 6

a0o9'/34($=< #

SFp q(/4("C< $%I0 JR< 0 >V@.<>B< 0

:4(/410@.<>Z(: ,C6u/4(Z$:4&@w<>/4"w< 0;/47akS%Cr34VD$=<>(/4(F%.< $.< Q '&/7< **/41&9oA4P/4(*/4"#< (/4 9HT#W("&.< $D3 < $%1U/ 6W(: ,R.< Q 6u(/4" n< (^0#$=<J,&0 AJ(6u(* (: ,76u/4(

H− /47.< Q V6u(/4" a1s a2s a2p S EG

a,oEexpG

a,oH

8 8TA !"!"! H− TA !"! 4A - 4A $ 8TA 8 4A --$ 8 4A He

TA "$ TA "$ TA " 8TA "$ 8 DA $! 8 DA %$! He−

4A 4 4A TA & 8TA $ 8 DA Li

TA -$ 4A TA TA 8 A 44 8 A ! 4Be+

TA "$" TA "4 TA ! TA "! 8 TA 8 TA

&.$ 2 "$ He H−

Li Be+ %(dy

He−

`,0=< :4/4< $%I%:R7*/4((#W/40:LN<>"$%/ =<>"C<>(*< 01F+1J&$%,4"%"%I%:!$/ @:4&3 < $1S%t(#W*/?!#< (< "(

αi8R

(< 0=< $9HD/4\/ 6Uw<J34R6u1*/4 R/4FT.< )R(1$%5#(+0d!< F$TA 4A&(N^/4$%,G/45#< (< "(U3>< $1

a2p6u/4( <>$$ >#@9<.w< ^S3>< $1

a2pm< (< $$ +1< $wS%q+1"(*J< $w< **1(< *, ∼ 10−6AKG6l< *'(*9.G

#(*J< $U@,D""(@,d/ 6 /41(#/ +=< $lAP/4(79H< "#$ ':4(/410E@.<>'(: ,/ 6KL5V

EG = −5.794,4S*9)S*$/47/'V< **#T0\9HT#("+.< $ 3 <>$1

−5.8074 :4%3&dG"#$*O,)/ 6 /41(7< ##(/&< *9 AP1((Z"#(/34"&76uJ< F$;F&,%('(*.:dF< 'F&,-*$10:\n = 3

@.<>'/4('F&,(/4(@:G/<'$=< (:4 < 1=< F< S%"C< &,;#< (< "(TS*9!< $$/S 6u/4(34(@,)< **1(<>73 < (=<.%/4< $X@"C<>- +4nA

P/4('H− / 2 d< $/?/4FT.< 2*/4((*;(1$%A, < :4()S%t/ (

*J< $*1$=<>%/4C/ 6d:4(/410D8@.<>\(: , wDwnA ?d:4;*/4((*$%,:4$ZFW/410)@.<> -$ 6u/4(w@,D@")A

EC*J< q(#(&MC:4(/410?@.<>C"M1$%O8Y#< (@%*$Gw<J34C6u1*/4?P/+*9QC#< *V;76u/4("ga DA &ow< 6u/4$%$/S T6u/4(

He<>/4"

|ΨHe0 >= (−0.799211a+1s↓a+1s↑ + 0.411751a+1s↓a+2s↑ − 0.411751a+1s↑a+1s↓−0.135451a+2s↓a+2s↑ + 0.0357708a+2p0↓a+2p0↑ + 0.0357641a+2p1↓a+2p−1↑

−0.0357641a+2p1↑a+2p−1↓)|0 >, alTA 4&o

- .

< 0?.C/4C/ 6p /\#S = 1

2

0:4(<>Li

:4(/410q@.<> A :AS%Sz = +1/2

|ΨLi0 >= (0.99749a+1s↓a

+1s↑a

+2s↑ − 0.057024a+1s↑a+2s↓a+2s↑ + 0.003959a+1s↑a+2p0↓a+2p0↑

+0.003959a+1s↑a+2p1↓a

+2p−1↑ − 0.003959a+1s↑a+2p1↑a+2p−1↓ − 0.023783a+2s↑a+2p0↓a+2p0↑

−0.023781a+2s↑a+2p1↓a+2p−1↑ + 0.023781a+2s↑a+2p1↑a+2p−1↓)|0 > .alTA +o < FW/349HT#(/41^^/4(@/4:4/4< $%I0F< 6u/4(^*(J<>/4/4#W(<>/4(S*9'#(/3D0^1^@.< 0< (0< +*/ ""M1T.<>%/4M($=<>/46u/4(

a< 0

a†/4#W(<>/4(

s5/ ^<>B <34 <51F@.< &=< $D< 0"OHD1(^/ 6T9HT*%0D8:4$B@.<>ZaJ ↑ > ↓< 0 J ↓ > ↑ o/M7"#$N+,D0(/4:4*98$%Q4N*/4Tr:41(<>/4;/ 6 He aJ ↓ J ↑ o9A Z*/41$02< $/(< @6u/4(" p /?9HT#(/4/)(#(+.<>/4%$*(/4)*(J<>/4/4#(<>/4(

a†%< $ <>/4"%*Z@.<> nA A

|ΨHe0 >= (−0.82274a†1s↓a†1s↑ + 0.309636a†1s↓a†2s↑ − 0.404512a†1s↑a†2s↓−0.021729a†2s↓a†2s↑ + 0.0357708a†2p0↓a†2p0↑ + 0.0357641a†2p1↓a†2p−1↑

−0.0357641a†2p1↑a†2p−1↓)|0 >, alTA 4o< 06u/4(w

Li<>/4"

|ΨLi0 >= (1.01321a†1s↓a

†1s↑a

†2s↑ + 0.018639a

†1s↑a

†2s↓a

†2s↑ + 0.005804a

†1s↑a

†2p0↓a

†2p0↑

+0.005804a†1s↑a†2p1↓a

†2p−1↑ − 0.0058034a†1s↑a†2p1↑a†2p−1↓ − 0.024289a†2s↑a†2p0↓a†2p0↑−0.024286a†2s↑a†2p1↓a†2p−1↑ + 0.024286a†2s↑a†2p1↑a†2p−1↓)|0 > .alTA &o

Z*/4"#$5/ 6(:4< 0"M1$%#< (@%*$%S@.<> 6u/4(He

<>/4" 0p8#$=<,40;;_N##W0OH[A Z7"C<J,C/ *N<>w7*/D *&S6u(/4&/ 6

a/4#W(<>/4(0/)/ 3&?+1< (1" /;'#(/4#W(#(/4F< F$Op,!/4("C< $%IJ<>%/4 S*9q/41$0?FW

1ABRR/ M<!"@.< Q C< 0qR*J< 10F&,EC/4/4(@/4:>8

/4< $%,/ 67\F< 0*10 D*/4 DA 4A?\\<>\#(/4F< F$Op,/ 6*/41+(:?!*/4Tr:41(<>/4

1s2tL5!C$<

2/3US(J<

*/4((#W/40:*/4Tr:41(<>/41s22s

6u/4( ^< $%"/4@7*/4*0ZS%\<>76u/4(Z+,D0(/4:4*98$%Q4G#*1( A(J< /4E6u/4(Z0%X(*701'/'*(*1"G8@.< *G<>ZM/34($=< #d&:4(< $

S =< 1s|2s > FWw1s< 0

2s@.<>7

76u/4("(w*J< $=< (:47< 0;73D(@1< $W(< %/41s 2s

0/'/ S&34/4$%3&7<1F@.< +=< $*.< :4SGN /41$/4"MFG(: ,4A+/ w<346u1*/4^*J< 'FWS10/'3 < $1<>< &,:4(/410D8@.<>7*9< (< *(@*ZF+,*J< $*1$=<>:

< ΨG|O|ΨG >6u/4(

O(#(&0?d>0E+1< &I06u/4(")A P/4(Z9H< "#$ XG0#W/4$M"/4"&

He H− Li Be+

He−

-!

/4#(<>/4(;d = e

∫d3rΨ†(r)xΨ(r)

* A "C< (w</4($=<>%3D@%*+1< &1" r$0;/4(@,;/ 6K<>/4"*Z@.<>S*J< )FW76u/4("M1$=<>0

_N/ (N6uJ<>1(R/ 6^/41(Z< ##(/&< *.dN*/4*0dS%!M0("<>/4)/ 6'"*(/4*/4#%*G#< (< "'(

Vijkld/41(VLN< "%$%/4< W*G(7Q+/S$0:4

*(1*=< $6u/4(^Z<./ "*98*$1@(^*J< $*1$=<>%/4+< $$W< 6u/4(^N0("<>/ / 6X#&,D*J< $#(/4#(@^/ 6 9HD00@,D@" < <Z6u1*/4'/ 6XS$=<>*w#< (< "G8(AXsN< "$%,4 R*J< \(S(%RLN< "$O/4< qa DA o 6u/4(SR*J< R/ 6U:4$V<>/4"?<)$/4:4(6u/4(")F1TRS%?#< (< "'(J<>RVJ< (/!&(#(M< 0?<>< ##J< (5/ 6k;3 < (/41"/+0$< "$%,

H =∑

iσ

εiniσ + t∑

σ

(

a†2σa1σ + a†1σa2σ

)

+5∑

i=1

Uini↑ni↓ +1

2

∑

i6=jKijninj

−12

∑

i6=jJij

(

Si · Sj −1

2ninj

)

+∑

i6=jJija

†i↑a

†i↓aj↓aj↑+

∑

i6=jσVijniσa

†iσajσ.

alTA -$4o

GM;/4##:d&:4(< $ FWwJ'< 0 >G@.<>Ui

< (;&(< /4(F%.< $ /41$/4"MF-+(< */4

Kij< (;(&(/4(F%.< $w*/4((#/40< +

VijG

/>8*J< $$0C*/4(($=<>0D8/4##:M&:4(< $nD< 0Jij

Z0(*9HT*.< :47&.:4(< $n6u/4(w@.<>

i< 0

j = 1, . . . , 5A#(*#< $X#< (< "(0r0!< FW/3476u/4(w

<>/4"G< 0$*0/4G< (;#(/3D00-<>F$TA DAKP(/4" M6u/ ("M1$=<>/ R6u/4$$/S%:C&(#(.<>/4!*J< dFWM0(<JS AXG*J< $*1$=<>0!(: ,\0%X(*∆E

6u/4(He

Fp G:4(/410@.<>G:4$9< 0dMr(@.HT*%0(#$Z−2.3707 − (−5.794) ' 3.423Ry

a^M%:4$1s ↑ 2s ↓ N@$$ ,\:4(.o9A7*/4((#/40:G(: ,/ 6Z(#1$%34 /41$/4"GFC&(< */4C6u/4( $*(/4

\1s2

*/4Tr:41(<>/4dU1 = 3.278

XG3>< $1G*/4"#< (< F$R/∆E

A _00O8/4< $$O,4G(#1$%34M&(/4(F%.< $K / 1$/4"GF\&(< */4!

1s ↑ 2s ↓ @.<>GK12 ≈ 1.5Ry

&<71F@.< &=< $$%,V$/ (K3>< $1 A4S($=<>%34w(:4*U$$K1KS&, V<J34R<G1F@.< &=< $< 0"OHD1(7/ 6B9HT*%0

1s ↑ 2s ↓ @.<>7/G%:4$1s2

A/ (w /4(0<G1F@.< &=< $ /41$/4"MF;&(< */4;(1w+,D0(/4:4*98$%Q4*." W< $%/41:4;R< *1< $ 3>< $15*/41$0)FV@$$ "/4(7(J< $@*ZF+,;(*9%:6u1(@(7(=< $XF< wS%

n = 3@.<>A

Y6 Z*/4"#< (5Z(:4w/ 6 Z9HT*%0;@.<>w/4FT.< 0C *J< $*1$=<8/4MS%GS9HT#W(%"+.< $(1$%> /4S*J< '*$J< $%,M3 < @0*(#< *Aw/4F(@340G9HT*%0D8(: ,M$34$K< ( />3&(@,V $$D(#(/+01*0 A+)($%,(+1< $%.<>O34V#(/4#W(@Z< (R"$< (AM(J< /4\6u/4(51*.E<FW<J3D/4(N5R/4#D8"%IJ<>%/4/ 6NF< GS%(#*C/d):4(/410@.<>(: , m>h| AU/4(F%.< $S.< Q4G<< #WG"/4@Z*/4+34&7S%d(#W*/CZ+1< +%,4AD/W6u/4(d9HT*%0 "M1$%%#< (@%*$!@.<>;d0*/4"#W/4%/4t&/:4$98$*(/4t/4(p8F%.< $V"C<,EFW;/ G*/ ((*9< &,q$/4:4(AB;9HT#W("&.< $ #W*(<TK/4qS*.N$34$W0("<>/4'F< 0 < (5/4FT.< 0CF&,'Z< F/4(FT/4C/4(^N"p8/4E/ 6#/ /4ABC@.<>/ 6$*(/4ES?V#(/D*R.< Q+:)#$=< *

.

< F$TA DAK)*(/4*/4#*C#< (< "(au ,oV/ 6w$*0<>/4"R< 0/ JA_N$$+1< &%< (5*J< $*1$=<>0G6u/4(U5/4(@/4:4/4< $%I0C<>/4"*@.<>A&U. Jp8>7/4##:"C< :4%10

UiN'&(< /4(F%.< $K /41$/4"MFd&(< */4-au Jao9>a 4o9>#M" al&o9>< 0 >#M" ±1 au#oo9S(J< Kij

< 0Jij

< (w&(/4(F%.< $ /41$/4"MF)< 09HT*9< :4V&(< */4;#< (< "((#W*%34$%,4A

t U1 U2 U3 Up K12 K13 K23 J12 J13H− TA "$ A 44 TA "! TA $"$ TA $ TA ! TA -$ TA "$ TA TA "He

4A TA "$ 4A " 4A ! 4A A "$ DA ! 4A "! TA TA -He−

8 4A A TA $ 4A $!" 4A $ TA !! 4A 4A TA "! TA Li

8TA TA -$ TA 4 TA DA !4" TA $ -! TA TA $ & TA TA Be+

8TA !! TA "! TA "!3 A "$!3 A -! 4A ! A " 4A $ TA TA 4-$

0r%$%,-/ @.<>/4< (@, ^/?)T6u/4("C<>%/42< F/41T!:4(/410t@.<><J346u1*/4 1*.E$Q4V.G/4#T"C< $B#< (< "(

αi "C<J,\:4$/4@7\7#(/D*A

d/4$%,< 1"'#T.%/42 d*J<> "C< Q d< F/41T!9HT*%0 "G1$%%#< (@%*$%)@.<>(C/4(@/4:4/4< $O,/?!:4(/410t@.<> A \/4(@/4:4/4< $O,-.\P/D*.Q#< *M5/ 6U*/41(G< 1(0dF&,M#(/+*Z/ 6U0=< :4/4< $%IJ<>%/4 A` 1TJW!:4(< $*J< ^!:4$#< (@*$;/4(F%.< $G0/?/ C<J3&/?FW)/ 6Z)< ")< #W)6u/4(79HT*%0@.<>A?7Q #(w< #W< 0;w< ikjlinm>hW z.zy NWjlinm>h< FW/41T5M@.<>AWB/C"#(/3&7M< **1(.<>*,4X/4w/41$0!<J34V/C#W(@6u/4(" < < 00%/4< $/4#T"IJ<>%/ \/ 6UG(: ,\!G1F#< *7S%< $%$@.<>XSO.$/ (B(: ,V< R @.<>w< *1< $$%,70("0 >9HT*$100 A /4(@/4:4/4< $Op,*/40%/4\ /41$0$J< 0d/)<;+1"GF(/ 6 1F+1+R*/4@(< &/4d

α/4#TO8

"IJ<>/4)#< (< "(W"C< Q+:+1Z\"/4@Z*J< WM#(/D*01(M1<3>< $=< F$6u(/4" G#(< **J< $#W/4&7/ 63DAD/4""NZ7/49.$V#/4F$ < S$$F;/S--D*/4 DA 4F1TG%MR/ R;*J< C6u/4(R"/+0$w< 0qF< S$*0)( A

L5/ 0/D7GIG/ 6G:4$M#< (@*$MF< 7<>X*G(1$% CP/4(Z< Q4M/ 6U*/4"#< (/4 R< "R*J< $*1$=<>%/4S (G< $/C#W(@6u/4("0!1%:;<C"G8#$(F< A+N*J< N/ 6 5"#$@#W/4F$5F< */4@:V/4$%,'/ 6

1s@.<>

5< $(J< 0T,*/4+.< 0)!D*/4 DA TA M:4%3D:G6u/4(SL5R<>/4" α = 27/(16a0)

< 0EG = −5.695Ry S( a0 ' 0.53 AK;9HD(*.(MF< G*/[email protected]/ 61s

< 02s

@.<>ABYG< $$/S76u/4(R< ?@.%"C<>/4E/ 6wC 1*C/ 6wCI/ 6:4$#< (@*$;F< G/4)< **1(< *,-/ 65*J<>$*1$=<>/4A'(01*0F< *J< $*1$=<>%/4G"C< Q4)< $/?<d3D1< $IJ<>%/4-/ 6N):4(/410@.<>)(: ,2< <6u1*/4d/ 6G/4#T"%IJ<>%/4\#< (< "(

αi#W/4F$>< Z '/4$%,!<34Mp /

1*9!#< (< "(AR:4(/410)@.<>V(: ,)S*J< RS/S\!P:TATA 6u/4(Z

He<>/4" < 0EEP:A TA 6u/4(7

H− /4 A Z'*J< EG6u(/4" #*1(T<>w7:4(/410;@.<>7(: ,0#W0S@(/4:4$%,C/47/4#T"%IJ<>%/4

He H− Li Be+

He−

12

34

Α1 a01

2

3

4

Α2 a0

-4-20

EGS

12

34

Α1 a0

P:41(5TA DA+N:4(/410C@.<>N(: ,/ 6 ZL5N<>/4" < w<6u1*/4/ 6+34(<>/4"*7/4(F%.< $I

α16u/4( JZ< 0

α26u/4( >W(#*%3&$%, A ZR*J< !*$J< ($%,

<>/4#T"%IJ<>%/4;/ 6B#< (< "(S0/+5"C<>( 6u/4(w(: ,4A

#< (< "(A? <34w:4$/4F< $&""<SRFW/ M%1<>/4A_N/ (:5/N/ "%$=<>(Op,V/ 6. p /7#$/ A Y6X/4 9<>Q4U&/< **/41&U(*J< $:Z/ 6<HT4*J< 10'F+,V*.< :4/ 6w+1*$J< (K*9< (:4wF&,Rw6l< */4(/ 6 /Fp &,D0(/4:4%/4d< 0\M$1" <>/4")/4M*J< d/4F(@34M.<ZM:4(< $< #W/ 6B3>< $$,DS;FW/ )#$/ "$<>(ATS9HT#(5V<%p,/ 6B@,D@"A

N+1"MFW(/ 6 "M1$%#< (p%*$@.<>^6u/4(Up /R/4(F%.< $%U (42

)= 6

&S*.C#$%1#-+/\(#D8:4$'< 0-/4(#$JA"C<HT"C< $+1"MFW(G/ 6N$*98(/4XS*.E*J< FW'< **/4""/D0<>0dF&,!.%7"C< $$(NF< 7

4A` 1T7'*J<

S%)"/4(7< !< $%6Br$$%:'*J< !FWV"C< ##0!/4&/'R*< VS%)$S< !< $%6r$$0q@,D@")/d*J< $*1$=<>%/4M< (/S-/4$%,6u/4(R1#/

2$*(/4nA A

HHe

<>/4"w< 0CH− /4 ADZ(1$%w/ 6 *J< $*1$=<>/4 < (N/S

?< F$CTA TA?CCC/4F&3D/41V6l< *R<>VF< V$=< (:4"+R"'#(/34'< **1(< *,4AP/4(

He<>/4" '($=<>%34M((/4(Z/ 9H< * R3 < $1M

1.89%6u/4(N/4M/4(F%.< $n.86%

6u/4(5w/;/4(F%.< $%Z< 0.19%

6u/4(Nr34G/4(F%.< $5!MF< JA?M< $%/'Q+/ <>56u/4(5< !Tr%F< V /41$0!(J< *.)R.H< *N3 < $1 S*.)0%X(6u(/4" 9HT#W("&.< $F*J< 1wV< $/'"C< 07/4"< ##(/HT"C<8/4N/4\GLN< "$O/4< nA AX%ZN/4d($=<>%3D@%* AY7Z*$J< ($%,)d6u/4(5

.

0.51

1.52

Α1 a0 0.2

0.4

0.6

0.81

Α2 a0

-1-0.8-0.6-0.4-0.2

EGS

0.51

1.52

Α1 a0

P:41(TA TAT< "< SP:ATA V6u/4(SL − /4 A*J< '/ 6&,D0(/4:4EdB< F$'TA MS(GG:4(/410d.<>M(: ,dZ/ 9H< *$%,+1< $/C1%,4AWP/4( J< Q4(NFW/410!@,D@"Z$Q

H− V"#(/3&"&501/F< M(*."&GR($=<>%34$%,$=< (:4(AB_N$/IC/ 6SC3>< (/41RF< M/4(p8F%.< $^*.< :4 +F1T '^*J< N%*J< / FW5*/4"#< (0/59HT#W("&JA+/4(F%.< $IM/ 6K >#\@.<>Z"C<,FWG<C$%$%VF%N1*/4""/4 F1TZM@.<>*/4+(F1TV<($=<>%34$%,;"C< $$< "/41&S./'V:4(/410!@.<> W/V#(/4F< F$Op,/ 6Z*/41+(:E$*(/4K6u/4(M@,D@"*/40(0 ^/4-)/4(F%.< $%K< $"/4@:4$:4%F$ A

Z)*J< t< $/E$/D/4Q<>')(/4$/ 67/4#T"'IJ<>%/4-/4-\< **1(< *,/ 6Z< ##(/&< *9 AWdB< F$MTA CwM*/4"#< (M:4(/410\@.<>M(: ,\< 0\R"*(/>8*/4#*#< (< "(76u/4(7

He<>/4" < 0

H− %/46u/4(7%1<>/4dS%?< 0S%/41T/4#T"%IJ<>%/4t/ 6d#< (< "(

αiA Z$%, J< 0>;/4(F%.< $%C (

.< Q4d&/;< **/41&JA Z\M9H< "#$R/ 6$1" *J< dFWMd<>ZM:4(/410(: ,G3>< $1w6u/4( m>hX /4#T"%I0G/4(F%.< $TF< ^K$/ (K< G<>U6u/4( j ' m / /4#T"I0;/4(F%.< $wF< AV"*(/4*/4#*Z#< (< "(5< (/ 6B< "/4(0(< 0*.< :4R< ('"/4('&=< $U6u/4(Zp />8#< (@*$M+(< */ #< (< "(AYZNF*J< 1M R<34R6u/41(Sw<34R6u1*/45)M*/4((#/40:&.:4(< $/0%X(*

αi#< (< "(UK*/41&0GpS* A U< F$w/SU"#/4(p8

.< *Z/ 6 Z/4#T"%IJ<>%/4#(/D*01(N6u/4( @.< F$:Z$*(/4*N@(1*1( A

He H− Li Be+

He−

< F$RTA TA Z#T"I0!` /4(p8/4(F%5(< 0ai = α−1i

/ 6JZ< 0 >7D$<>(5/4(F%.< $!1%Z/ 6

a0WR:4(/410d@.<>M(: ,)6u/4(5M$:4&@Z<>/4"Z< 0\/45S%

/4$%,;w/C/4(F%.< $AM(: ,\< 0)M#< (< "(50%X(*VSd*/4"#< (0S%;r3498/4(F%.< $ %1<>/4Eau< F$VTA o S< $/G/S A

a1s a2s EGa,o

∆a1s[10−3] ∆a2s[10

−3] ∆EG(Ry)H

8 H− TA !" 4A $"!"! 8 4A TA $! 4A DA He

TA -$ TA "$" 8 +A $ -! 4A DA TA 4"! < F$TA AK)*(/4*/4#*C#< (< "(;au ,o/ 6$*0-<>/4"M< 0q/4A_N$$ +1< &%S< (*J<>$*1$=<>0;6u/4(w/4(@/4:4/4< $%I0)<>/4"*Z@.<>AVF< */4@Z/ 6JZ< 0 >N@.<>Z/4$%,4A

tS Jp8 >N/4##:C"C< :4%10

εiS

<>/4"*Z(:>,4Uiw+(< /4(FO.< $ /41$/4"MF;+(< */4 DS(J<

K< 0

J< ( &(/4(F%.< $D /41$/4"MFM< 0R9HT*9< :4w&(< */4V#< (< "(A>w$/ (#< (@/ 6 '< F$G/SG3 < $10#$=<J,40?< FW/34< 0E*J< $*1$=<>0dS%/41T/4#T"%IJ<>%/4?/ 6S#< (< "(

αiA;&,D0(/4:4*98$Q 3>< $1

α1 = 2< 0

α2 = 1 (.< Q AV$=< @S*/4$1"/SS:4(/410@9<.7(:4A

t ε1 ε2 U1 U2 K J EG

H− TA 44 8TA !"!"4 8TA & 4A TA TA - TA "! 8 4A " He

4A -! 8pTA 8TA "!""$ TA "! 4A -$ " 4A "! TA "! 8 DA $ -!4H−noopt

8 8TA 4A TA TA !" TA &"! 8 4A $ Henoopt

8 8 DA TA TA 4"! TA "-$ 8 DA

.

vZ[jM PdO@I

QT M`V;O T KWV;O.XAYZV;O.R [N\Z] T KWV;O.XAYZV[jIQ;KW\ZR

9HDJ"/4(G*/4"#$*J<>0 @,D@" <F=<>/4"%*G"/4$*1$M/4(V<"/4$*1$=< (/4 nA+Z"#$@*J<>:4/4(@,C< (SN&,D0(/4:4;"/4$%*1$

H2

< 0/4H−2

&S(Z(#*%3&$%,'p /R/4(^(5$*(/4w< (5< (0CF+,Gp /R+1*$nADD1*9@,D@" &(/D01*< t<>00%/4< $w9HD(< $S#< (< "(K0@.< *)Fp S+1*$nA&C< 00%/4M/75:4(/410'@.<>5(: ,4D< $/Z.%^#< (< "(U*J< FW*/4"#< (0ES%q.HT#W("&JABYM9HT#W/4MC"C<@/4(V< 0T3 < &.< :4C/ 6w[^]S_N` "/+0< "$%,4&(/D01*/4/ 6Z(J< $w /4($0-#< (< "('&/d"/D0$*J< $*1$=<>%/4A

_NZ*J< 7/ 6<>/4"D 7<J3&Z/M*9/+/4V< :&< %C7%< $W*$=< w/ 6w<J3&6u1*/4/q0r\/41(P/D*.Q#< * AwP/4$$/S%:?< < (:41"'+C"$< (G/?<>/4"*V*J< 'ED*/ ETA M G*9/+/4GG/4(@/4:4/4< $%I0d<>/4"*V6u1*/4 *&(0!/4 /%A

5"/D0$/ 6WH2

< 0H−2

"/4$*1$U@,D@" ^0#*0*9"<>*J< $$%,qP:A A 4ABw(J< $#< *#< (< "( CFW/ 0$:

R<>0?;<./ "*

/4(F%.< $wI1/α

<>(/S A^)*&(/ 6ZJ< *.2/ 6N!<>/4"()<+1*$1S%*9<>(:

Z = 1A` *J< 1! )<34) /?*&.(J^ !"C<J,-$/+*J<>

;<>/4"*/4(F%.< $V/4FW/ / 6")Aq;"'#$@M*J< ;"C<J,?(@(*/41($%34/EJC/ (FO9<>$'/4$%,4A^'<J,- !:4C#< *!6u/4('1#2/E6u/41($*(/4A$/D*J<>/4E/ 6S"/4(-<!:4$ JM/ (FO9<>$/4qJ< *.-+1*$1M#W/ F$% DF1T % *(J< &=< $$%,< < $%,+*J< $0*1$%^N"*(/4*/ #*98#< (< "(S*J< $*1$=<>/4AR#< (< "(

α*/4+(/4$$%:GVIV/ 6KV/4(F%.< $

< SF!5+1< $/4FW/ / 6B")AJN<>/4"*7/4(F%.< $$/+*J<>0)/4!J< *.\/ 6 /C*&(Z/>6U0@.< *

R< #< (@7< (M/ 7/4(@/4:4/4< $nA8717ZG/4(:4%\/ 6UG*/+/4(0<>M@,D@" /4

"# + $2'#%=?3 # *!=?& $2A !)N!"# 4 *&;!= )*. .#3>= * :"$2O !# '+D3 #*#3 E-+ P'+P -,


R

1/α

R

1/α

P:41( A 4A )/D0$/ 6UH2

< 0H−2

"/4$*1$5S%\(J< $#< *M#< (< "( 7F/40;$:

R<>07/4(F%.< $I

1/αAT7/4(F%.< $WIZ@1@:410N6u/ (

,&>W< V<>/4"*N6u1*/40/+5/ S(J< $$%,C<J3&V<G< (#)FW/4(0(A

;$*/4*:dFW/ +1*$n< $%6w<J,qFp -")AB<>/4"*'<J346u1*/45< (7/ 6B76u/4$$/S:R6u/4("

φ1(r;α) =

√

α3

πexp(−α|r−R/2|) φ2(r;α) =

√

α3

πexp(−α|r+R/2|), a A o

S(R<34*/4(V/ 6 $:

R< $/4:<HT7/ 6'"/4$*1$ AX,<J34

/CFR/4(@/4:4/4< $%I0F+,"OHT:Gw<34V6u1*/4S$/D*J<>0!< **/4(0:'/#(/D*01(N< $(J< 0T,M0*(FW0CD*/4TA 4A4?S$$T*J< $*1$=<>wS"*(/ */4#*#< (< "(C <>/4"*;6u1*/4< 02t(< @6u/4(" " /?\/4(@/4:4/8< $I0!?< (ZF< AWM"OHT:C*/+ *&79HT#(J0d[U+AUalTA &o5< 0alTA +o50#0E/4\M/34($=< #\+:4(< $

SAXG< #WG/ 6<>/4"*R< 0d?< (

6u1*/4/S)P:A8 A DA Z7*J< / *Z<>w/4(@/4:4/4< $%I0?< D8( 6u1*/4w<34<M/D0 *Z,;*< / SFWN7:4(/410;.<>N6u1*/46u/4( Z:4$Z<>/4" *J< AwwFW*J< 1Z7/34($=< #;w("/340F&,*.< :4::4/ 6/476u1*/4;V< (J<TTS(*/40!$/D*<>0 A

2 z4% #" $Z#< (< "( 00/V6u/4("G1$=<ZLN< "$%/ =< ' *J< N<34N/RFW*J< $*1$=<>0 nA,q*J< qFW0%3D00qEp /\*$=< / 98V< 0?Cw/>8

. "$

-4 -2 0 2 4-0.2

0.0

0.2

0.4

0.6

0.8

Wav

e F

unct

ion

position r/a0

P:41() A +ANqw<J3&q6u1*/ < $/4:q"/4$*1$q<HTA5/4(@/4:4/4< $O8I0;q< ( 6u1*/4au/4$0;$o < 0;7/4(:4< $X<>/4"* Jw<J3&76u1*/4au0< 0)$o < (/S)6u/4(*/4"#< (/4 A

$*(/4!"*(/4*/4#*V#< (< "'(JA%#5$Q V%5w< N)R*J< M/ 6^<>/4"R(981$%.< &C9HT#(/4C6u/4(#< (< "(0#W0 / /4$%,-/4

αF1T;< $/?/4

&(<>/4"%*Z0@.< *RA

:4$Z$*(/4#< (< "9.(w<347/G(#(&57&(/D01*/4;/ 6w/<>/4"*R%7&/;/41(Z"/D0$nAWG6u/4(" />6G:4$98$*(/4\#< (@/ 6 LN< "$8/4=< 2dD*9(+0:4(C(#(&.<>/4 < /qF\9HD00 AsN< "$%, ^ <J34N/M< 00/RN[UAalTA &o^<(" (#(&:RN 1*N/ 6 .5*/40+1*$1

H1(r) = −∇2 − 2Z

|r− r1|− 2Z

|r− r2|,

a A 4oS(

r1 = R/2< 0

r2 = −R/2< ('0#$=< *"&V/ 6w(#W*%34$%,

r(@< 0;*/40+1*$ 6u(/4";/4(:4 A` *J< 1; <J34; //4(F%.< $(V< (7w/Q+%05/ 6"*(/4*/4#*Z:4$7#< (@*$Z#< (< "(S0r0!F&,)[UAa DA 4o9A&NF< ^6u1*/4 < (5*&(0;/4 /R0%X(+w<>/4"A+` / C*&(< (+1%3>< $&J*(V/4$%,;/4

αA<>5wS+,%S/41:4/'*J< $*1D8

$=<>Np /:4$Z$*(/4)#< (< "(@J< 0/ 66u/41(A)<>/4"*(: ,

ε′ ≡ t′11 = t′22(#(+'(:>,-/ 6Z< -$*(/4


$/D*J<>0d/4<*&(A Y0%X(N6u(/4" <>7*J< $*1$=<.0\6u/4(7<>E<>/4" F&,!M*/4D8(F1T/4;/ 6B*/40!+1*$1

ε′ = (α− 2Z)α +∫

d3rφ1(r;α)? −2Z|r−R/2|φ1(r;α)

︸︷︷︸

2nd nucleus contribution

.a A &o

M*J< $*1$=<>%/4)/ 6Z< 00O.%/4< $*/4&(F1T/4)< 5/CFWR0/4R)#(/40< $*/D/4(0<>> &D- " *JA& C9HT#(0G D*/+/4(0<>^%^*J< CF50(*$%,3 < $1<>0)< 0)/ 6B76u/4("

−2Z∫ 2π

0

dφ

∫ ∞

1

dλ

∫ 1

−1dµα3

πe−αR(λ+µ)

2

λ− µ

R3

8(λ2 − µ2) =

= Zα

(

e−2αR(

2 +2

αR

)

− 2

αR

)

,a A +o

#(/3D0:\< $/!;*/4"#$C9HT#(/ -6u/4(M<>/4"*(: ,%q"/4$*1$=< (@,D@")S*.)

ε′ = α2 + 2Zα

(

e−2αR(

1 +1

αR

)

− 1

αR+ 1

)

.a A 4o

Y6wF< 6u1*/4V/ 6wC"*(/4*/4#*G#< (< "(VFW:)3 < $1<>0q< ($/D*J<>0?/4Ep /\0%X(&R*+(BC#< (< "(VC/\*J< $$0 /4##: >t′ ≡ t′12 = t′21

A4YU3>< $1(#(+^S*.< :4SG(: ,G/ 6 < '$*(/4'"/3D:FWw\R+1*$nnA ARQ+*V(: ,!/ 6KR$*9.(/4 AN*J< R5*/4"G8#$$%,R0%X(&K6u(/4" <>U/ 6:4$w<>/4") < 0Mw*J< />^(1ww6u/4("(9HT#(/4S6u/4(w#< (< "(S( AVJ< @Sw<J,;/'*J< $*1$=<>Z" w/"#$/,\< :&< \'#(/4$<>G#(/40< $K*/D/4(0<>A '*/40q< 0dG(0(" )[UABa' TA 4o9$J< 07("S/ 6B76u/4("

∫

d3rφ1(r;α)? −2Z|r±R/2|φ2(r;α),

a A &oS*9)< (+1%3>< $+JA,*J< )FW0(*$%,3>< $1<>0 *

−2Z∫ 2π

0

dφ

∫ ∞

1

dλ

∫ 1

−1dµα3

πe−αRλ

2

λ± µ

R3

8(λ2 − µ2) =

= −2Zαe−2αR (1 + αR) .a A $4o

Rr(@N(" \[UAUa A 4o(+1(5.R*J< $*1$=<>%/4)/ 6UR9HT#W*.<>/4\3>< $1/ 6B < #$=< *=< < 0)*J< )< $/'FW0(*$%,3>< $1<>0 < "$%,

∫

d3rφ1(r;α)(−∇2)φ2(r;α) =1

3α2e−αR

(−α2R2 + 3αR + 3

).

a A &o

69-# # :OO# *3 = 9 ; 46)':3 #(!1$2= A?. P

. !7r< $9HT#(/4)6u/4(w/4##:'&:4(< $X

t′ = α21

3e−αR

(−α2R2 + 3αR + 3

)− 4Zαe−2αR (1 + αR) .

a A !&o

Z$=< @ #< (< "(N:4(/41#C/ 6: $5$*(/4&:4(< $^^Z/34($=< #&:4(< $

S(+1(0C/M/4FT.< N/4(@/4:4/4< $%I0q< (6u1*/4+S*.

S =

∫

d3rφ1(r;α)φ2(r;α) = e−αR(

1 + αR +1

3α2R2

)

.a A &o

Y ^"#W/4(@.< &U/V9H< "SN< @,D"#T/ *SF<3D/4(^/ 6XN#< (< "'(^6u/4("C< $$U< 7w$$^< 76u/4($=< (:4M3 < $1V/ 6

RAP/4(

R = 0/ 1(V@,D@"g1(V&/!<

<>/4" S%d+1*$17*.< (:42ZA ZdM/ (7< 0\6u/4(

R → ∞ $*(/47< ($/+*J<>0!/4! $$#< (<>0d/+0AX,!&(< *N"M1*9) J< Q4(7< 0! R9HT#W*/)(*/3&(C<>/4"*G$"'%JAXC/4##:)/4?< ?Tr%G0@.< */41$0EFW"#/4F$ T/' 9HT#W*N%/G3 < %; A[HT#$*%$%,4D$"'O< (

limR→0

ε′ = α2 − 4αZ = limR→0

t′

limR→∞

ε′ = α2 − 2αZlimR→∞

t′ = 0.a A o

,'*/4Tr(" S*/4((* / 6WSF< p8Y6u1*/4G*9/ * A&R0#W0*

/ 6BV<>/4"*Z"%*(/4*/4#*7#< (< "(/S!)P:A A TA*/40*$=< / 6B"*(/4*/4#*N#< (< "(< (NZw/G$*(/4;#< (< "G8

(AT` *J< 15 N<J3455@,D""(@,'/ 6 9HT*.< :4:V5&:4(<>/4G3 < (=< F$< 0'p /V%< (S+1%3>< $+J4(5< (S/4$%,M6u/41(Kp /V$*(/4#< (< "(A_ /4(@G*$< Or*J<>/4?/ 6"R$@0??< F$ A 4AB11< $$%,E$=< (:4@R/ 65 /R$*(/4&(< */4^^N&(<4<>/4"'*5 /41$/4"MF+(< */4

U ′ADY

N(: ,C#< $%p,G6u/4($/D*J<>: /M$*(/4S%C/4##W/4%N#C/4CN< "/4(F%.< $nA$ N6*/41( 4*/4MFW/ G*&(U(%w< " J<J34w6u1*/4 %0/+^/ U"C<>(S*.G6u1*/4

φ1/4(φ2 S1M*J< $*1$=<>/4 A4D"%$< ($O,

'&(<>/4"%*' /41$/4"MF&(< */4K ′ (#(+'+(< *9.%/4Fp

/d$*(/4M$/D*J<.0q/4q0%X(&M*&(AK$=< @Vp /\#< (< "(R0/\/ (#(&;<d*$=< *J< $*/4&(F1T/4 AK(0-#< (< "(

V ′G/>8*J< $$0

*/4(($=<>0D8/4##:/4(5+,DF(0!+:4(< $nAYN*J< dFR+(#(0d< Z</4##:C#(*!/ 6Z< / ('$*9.(/4 AUL5(!< :&<> K%G0/D/ '"C<>(M6u(/4"(1$%.< &R9HT#(/4?S(

φ1< ##WJ< ((*C< 0

φ2/4*E+:4(< $

/4(5%N5R/4##W/4%V*J< AXM$=< @Z/ MSM0(*9Z9HT*.< :4G&:4(< $J ′AWY

0%X(K6u(/4"S&(%5 /41$/4"MFG&(< */4 *SJ< *.&:4(<>/4R3 < (=< F$*/4*R< /4(F%.< $*&(0?/4/4G%MS%E< /4(F%.< $*+(0E/4E:4D8FW/ (%:'% A


0 2 4 6 8 10

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

ε'

t'

ener

gies

,

', t'

(Ry)

distance, R/a0

P:41( A TAWD:4$98$%*(/4"*(/4*/4#*Z#< (< "(*J< $*1$=<>0V<>/4"*F< < 5<M6u1*/4/ 6&(<>/4"%*Z0@.< *

RA73 < $15< (7 ,4A

< F$ A 4A+ $=< %r*J<>/4G/ 6 "*(/4*/ #*#< (< "(K6u/4(U /V*&(%0D8*J< $K/4(F%.< $Z$/D*J<>0/4J< *9 A0r%/4d/ 6wJ<>*9E#< (< "(770/4C< *98*/4(0:V/M[U+Aa DA 4o^< 0a DA J&o9A Z$%,<R:4$S/ 6 N+1%3 < $& 9HT#(/4$@0 A

< " @,D"MFW/4$ 0r%/4/4##:

t′ < φ1|H1|φ2 ><>/4"*Z(: ,ε′ < φ1|H1|φ2 >/34($=< #S < φ1|φ1 >+(<4<>/4"'%* /41$/4"MFU ′ < φ1φ1|H2|φ1φ1 >&(<>/4"%*Z9HT*.< :4J ′ < φ1φ1|H2|φ2φ2 >&(<>/4"*7 /41$/4"GFK ′ < φ1φ2|H2|φ1φ2 >*/4(($=<>0/4##:!au&,DF(0oV ′ < φ1φ1|H2|φ1φ2 >

. R*J< $*1$=<>/ )/ 6Kr(@N(R#< (< "(

U ′K ′ < 0 V ′ <>N!*/4""/4&:4(<>/4;/34(7r(@S3 < (=< F$

rA551S0r

I(r′;α) =

∫

d3rφ21(r;α)2

|r− r′| .a A 4o

_N$$X(#< (< "(*J< )FW9HT#(0!S%$#)/ 6I;76u/4$$%/S:R<J,

U ′ =

∫

d3r′I(r′;α)φ21(r′;α)

K ′ =

∫

d3r′I(r′;α)φ22(r′;α)

V ′ =

∫

d3r′I(r′;α)φ1(r′;α)φ2(r

′;α)a A &o

M*J< $*1$=<>/4!/ 6I(r′;α)

N/ N3&(@,\*/4"#$%*J<>0 AEG"C<J,)1MM&3&(0@.< *;9HT#< /4-#(*J< $*/+/4(0<>)anNA &o9A?<>.< *.;/4(:4q/ 6'11< $^#(*J< $^*/+/4(0<>'@,D@"

(r, θ, φ)/Gr(@R+1*$nA '<J34

6u1*/4V0r0-F&,?[UAa A o7<J3&<)34(@,?"#$< :41$=< (0#W0*J A^D/G&:4(<>/4\/34(Z< : $

θ< 0

φ#(/3D07I(/;6u/4(< $$B("ZaY5A &o

9HT*#Tw/4 +< "$%,44<>S%l = 0

< 0m = 0

A+ D'1(C0&%,)alTA 4o"C<J,FW10)/'< F$Z&:4(<>/4;/34(

r/G,D$0

I(r′;α) =2

r′

(

1− e−2αr′

(1 + αr′))

.a A +o

qS<>M6u/4$$/SV ;"M1@G(""MFW(B<>M* ;<J34;10#(*J< $*/+/4(0<>/4(:4<>%:G<>Zr(@5+1*$n

r′0/ 70@.< */ 6B#W/4&

r′6u(/4" j iz /4(:4 A&(< /4(F%.<>$U /41$/4"MF#< (< "(

U ′7/4FT.< 0"#$%,!F+,d(@:

a A oZ< 02a A +oN&/Ea A &oZ< 0E*J<>((@,D:;/4'&:4(<>/4\1:#(*J< $*/+/4(0<>ADN&:4(< $^/3&(

φ′θ′,D$0""0<>$O,M56l< */4(/ 6

4πAD

&:4(<>/4;/34(r′@(< :4&6u/4(@w< (0)< 0)< 7r< $(1$% V<J34

U ′ =5α

4.

a A 4oY+1< $/N<>/4"*(1$%9HTF%0C'_N##W0OHC]RT< U^ />8$*(/4&:4(< $ *$10ww<346u1*/ S$/D*J<>0/4/4$%,/4+1*$1A

!V*J< Z/ 6U*J< $*1$=<>%:K ′ < 0 V ′ wR<J34V/C1T$%I#(/40< $B*/+/4(p80<>&*N<> $J< @/45/ 6XSw<J3&S6u1*/4 < ##J< (:V'S6u/4("M1$=< 4

$/+*J<>0/4*+(/ <>^5/4(: '/ 6W5#(*J< $*/+/4(0<>S@,D@" 10C*J< $*1$=<>%/4/ 6

I[UABa A +o9AD?(@79HT#(/46u/4(

I&/M7*/40

< 0'(09HT#(/4 C[UA a A &oK< 0(< @6u/4(" 5/4FT.< 0'6u/4("G1$=<>U+/


#(/40< $B*/+/4(0<>AWR&:4(<>/4)/34(φ′5@(< :4&6u/4(@w< (0\< 5%N0/+

/ w<>##WJ< (G5&:4(< 0 A&N(1$%.< &&:4(< $U*J< CFWS*J< @ +/#(/D01*/ 6<6u1*/4Z/ 6

λ< 0\/ 6

µX< 0dJ< *.d/ 6UM(1$%:C&:4(< 0NZ<C#W/4$%,&8

/4"=< $"M1$%#$%%0)F+,!< !9HT#W/4&=< $B/ 6U1" /4(N0%X(*R/ 6λ< 0

µAYZ*J<

FW*J< $*1$=<>0F&,;$"&.< (@,;"9./+0S*.,D$0

K ′ = −e−2αR

12R

(4α3R3 + 18α2R2 + 33αR + 24

)+2

R,

V ′ =1

8R

(e−αR(16α2R2 + 2αR + 5)− e−3αR(2αR + 5)

).

a A &oN9HT#(/4 < (S0%X(&^< G(^<>/4"*w*/4((#W/40< &+F1T^ 5*J< 9H< "K(< @,D"#T/ *AJ^$=< (:4U< 0V"C< $$?$"'O / 6

K ′ < (U(#W*%34$%,

limR→∞

K ′ l.t.=

2

R

limR→0

K ′ =5α

4,

a A $4oS($nA JA6u/4($J< 0:(")< V#(/4#W(V3>< $1/ 6w$"OVI(/Ar(@ $/S^5*/4((* *$=< *J< $9HT#(/ C6u/4(K5$/4:>8(< :4w&(< */4 S(VR$*(/45/4!/4(F%.< $*J< !FW(J<>0d< 5#W/4%+5*.< (:4AWR*/40$\(#(/+01*d(1$%;6u/4(

U ′<

R = 0"J< \p /-$*(/4< (

$/D*J<>0/4q;< "/4(F%.< $lAB_N< $/4:4/41$%,4;*/4((#/40:$"'OV6u/4(R+,DF(0&:4(< $< (

limR→∞

V ′l.t.= exp(−αR)

limR→0

V ′ =5α

4.

a A &o!r(@6u/4("M1$=</SC\9HT#/4+=<>$N0*J<J,t/ 6Z! />8/4(F%.< $S/34($=< # AC*/40q9HT#(/4q(#(/D01*V+(<4<>/4"'%* /41$/4"GF&(< */46u/4(< "(J< /4!< w6u/4(w#< (< "(

K ′ AS9HT*.< :45&:4(< $

J ′*J< / FWS*/4"#1T0'GS< ""C< (A4(

/71wM9HT#(/4G6u/4(I*wFW/ M+:4(<>/ V3 < (=< F$

r< 0

r′ */4*

<J34S6u1*/4^*+(0C/40%X(& <>/4"A4?S0C@J< 0C< +("0=<>9HT#(/4

I ′(r′;α) ≡∫

d3rφ1(r;α)φ2(r;α)2

|r− r′| ,a A !&o

S*9*/4+.< Xp /S0%X(+/4(F%.< $n*^%< /FWU*J< $*1$=<>077#(/40< $*/D/4(0<>A ` 1T\Z*J< GG9HT#< /42alTA 4o5*J< / 7FWG10 A@J< 0

. 4 V<J34/"#$/,;< !< < $/4:4/41w9HT#< /4!#(/40< $*/+/4(0<>S*.S< $/'*< $$0;Vs51"C< 9HT#< /4

1

|r− r′| =2

R

∞∑

s=0

s∑

m=−s(−1)m(2s+ 1)

((s− |m|)!(s+ |m|)!

)2

P |m|s (λ<)Q|m|s (λ>)P

|m|s (µ)P |m|s (µ′)eım(φ−φ

′),a A &o

S(λ<

< 0λ>

< (N(#*%34$%,N$%(w< 0:4(J<>(/ 6λ< 0

λ′AP|m|s

< (w< /D*=<>0 :40( 6u1*/4K< M[UAalTA 4o< 0

Q|m|s< (w< /D*=<.0

:40(N6u1*/4w/ 6Z*/40Q+0 AT7[HT#< /4Ea A &o^ "/4(N*$/4$%,0*(F0!_N##W0OHNAX(@:CN9HT#< /4\+/[UAUa A !&o9/4: (S%C#(/3D0:S<J34N6u1*/4

φ1< 0

φ2+9HT#(J0;'(" / 6

λµD< 0

φX ' <>7('7/

φ0#W0*C/ 6'*.< (:4'0@(F1T/ AD/

"C<J,?*/4"#1TG#< (@G/ 65;&:4(<>/4 B(01*:/4;/ 65;1""C<>/ 0*A)X*Sw/4FT.< <>

I ′ = α3R2

∞∑

s=0

(2s+ 1)

∫ ∞

1

dλ

∫ 1

−1dµ(λ2 − µ2) exp(−αRλ)

Ps(λ<)Qs(λ>)Ps(µ)Ps(µ′).

a A oV9HDN@#!w/*J< ((@,/41TV&:4(<>/4;/3&(

µA3 < (=< F$7/+**1(5

N6u/4("G1$<V/4$%,S* AD?Z"C<,17%( Z/4(@/4:4/4< $Op,'#(/4#W(@,/ 6:40(#W/4$%,D/4"=< $%

∫ 1

−1dµPs(µ)Ps′(µ) = δss′

2

2s+ 1,

a A 4oF&,M9HT#(:

µ2 = 1/3P0(µ)+2/3P2(µ)4/4("#$%,V@S%*.'/3&(/Z9HT#$*%

#W/ $%,D/4"=< $T9HT#< /4A+^9HD &:4(<>/4G(01*^6u1(@(U51""C<>/ /34(

s$J<J3D:q/4$%, /("

s = 0< 0

s = 2A "/4@#(/4F$"C<>*

G)+:4(<>/ -/34(λAU?)1\[UAZalTA 4o9A #5T6u/4(@1<>$%,4K :40(

6u1*/4/ 6RE*/40 Q+0Qs(λ)

*/4&.< ("$Q λ+1λ−1

S*.(1$%<>6k('&:4(<>/4 K/4$"&.< (@,-*/4&(F1T/4AU)*J< $*1$=<>/4+1%*1"MFW(/4" AB %q$#/ 6;/34($=< #&:4(< $ < 0-< -< 1DHT$=<>(@,E0r%/4S ′ ≡ eαR(1− αR + 1/3α2R2)

<34V< (1$%

I ′(λ, µ, φ;α) =2

R

(1− (9/4)(µ2 − 1/3)(λ2 − 1/3)

)(S ln

(λ+ 1

λ− 1

)

−S ′Ei(−αR(λ+ 1)) + SEi(−αR(λ− 1))− αR exp(−αRλ))+9

R9µ2 − 1/3)(e−αRλ(−1/6αR− λ− 1/2αRλ2) + λS).

a A &o


V#W*=< $ 6u1*/4)< ##WJ< (:'(w9HT#W/4&=< $&:4(< $X6u1*/4

Ei(z) = −∫ ∞

−z

e−t

tdt = γEuler + ln z +

∞∑

n=1

zn

nn!,

a A +oS(

γEuler ' 0.5772K5[U1$( ^*/4@.< &JA+S#("C< (@,M1S/ 6XS9HT#(p8

/4!a A 4o*J< $*1$=<>/ G/>6J4F1T^%U*J< < $/7FS"#$/,40G/< < $%(<>

*J< $*1$=<>/4)/ 6UM*/4(($=<>0D8/4##:C&:4(< $V/*.*.Q!R*/4@*,d/ 6

R*J< $*1$=<>%/4As59HDN R<J34V/C"M1$%#$O,I ′(r′;α)

F&,φ1(r

′;α)φ2(r′;α)

< 0+:4(<>V/41TZR3>< (=< F$

φ′µ′< 0

λ′AX_56k(7<C/4"S<>Z$: +,)*J< $*1$=<8

/4 T :4S7r< $9HT#(/46u/4(w9HT*9< :4V&:4(< $X/ 6 J/4(F%.< $

J ′ =12

5R

(S2γEuler + S2 ln(αR)− 2SS ′Ei(−2αR) + S ′2Ei(−4αR)

)

+αe−2αR(5/4− 23/10αR− 6/5α2R2 − 2/15α3R3

).a A 4o

[H< ":< @,D"#T/ *G#(/4#(@/ 6'9HT#(/4?/4'/4FT.< 79H< *$%,< "F<3D/ (S$Q Z6u/4(w*/4(($=<>0D8/ ##:'&:4(< $

V ′< "$%,

limR→∞

J ′l.t.= exp(−αR)

limR→0

J ′ =5α

4= U ′,

a A &o6u/4(S"C< $$

R V:45F< *9Q

U ′3 < $%1 W< 06u/4(S$=< (:4

R< )9HT#W/4&=< $0*J<J,4

S*9\(*NM$/4:(< :4R< #WM/ 6K J5w<J3&R6u1*/4AWR0#D8

0*V/ 6B7p /$*(/ )"*(/4*/4#*Z#< (< "(U ′K ′ V ′ J ′ 0#$=<J,40)P:A A A_N$$W6u1*/4S< (7(J< $ 3 <>$10 AYS/41$0)F@(0)<> hXm

9& v m4i C jlinm>hTz ("C< 0EC*J< $*1$=<>/4E/ 6S< $$U<>/4"%*"*(/ */4#*#< (< "(A

_56k(*J< $*1$=<>:"*(/4*/4#*G#< (< "(VC<>/4"%*GF< /40'/\.(< @6u/4(" " /d;*/4((#W/40:9HT#(/4q/4(@/4:4/8< $I0\q< %(7F< AXG:4(< $K< #WG/ 6^M(< @6u/4("C<>/ \Z0&*J< $$Q46u/4(Z<>/4"Nd[U+A^alTA &o9KalTA +o9AWG/4$%,)0%X(*M5.<ZM/3&($=< #+:4(< $w'9HT#(0 F&,< / (0#0*J!2[UANa A &o9KS*.20#0/4)R&(<>/4"*70@.< *

RA

S0#W0*GS/S\!P:A5 A DA

(< @6u/4("C<>/4(1$% 6u/4( /498^< 0 />8U$*(/4;#< (< "(w< (Z34"#$(< (#*%3&$%,4WalTA !&oK< 0!alTA "$4o9A4'r(@*J< 5<J34S/4$%,M/45<>/4"*(: ,4+*S(5^< $//4$%,G/43>< $16u/4(Kq< (^#< (< "(A&'6u/4(p8"G1$=<76u/4( />8$*(/4#< (< "(Z1""C<>/ C:4/Dw/4$%,/34( w/M/4(F%.< $,D$0:V6u/41( #< (< "(wNq< (wF< A #5T6u/4(@1<>$%,4+6u/4(

R→ 0

3 < $1/ 65)/34($=< #2< ##(/&< *.1%,4KS<>"C< Q '"OHT:d*/D *&

.

0 2 4 6 8 100.0

0.5

1.0

1.5

V'

J'

K'

U'

ener

gy (

Ry)

distance, R/a0

P:41( A A />8$*(/4\<>/4"*Z"*(/4*/4#*7#< (< "(N<>N<G6u1*/4)/ 6KD8(<>/4"*Z0@.< *

RA3>< $15< ( ,4A

0%34(:4 AB/'<>1(7/ 6BS0%34(:4*"/4(7*$J< ($%,C%S*/4&34+S/(S(%[UAalTA &o;76u/4$$/S:R6u/4("

w1(r) = β

(

φ1(r)−(

−γβ

)

φ2(r)

)

.a A "$4o

G*/40dq< (56u1*/4\Z/4FT.< 0\F&,\9HT*.< :4:M0*1 ↔ 2

As5/

β#$=<,DK(/4$/ 6W/4("C< $%IJ<>%/4M*/4@.< +J4S(J<>^3>< $1 >6 − γ

β$$^1 / 0#N:&<>%34NQ+QGw1(r)

/41$0CFW T<>N#W/4%/4/ 6 */40qD1*$1AP(/4" P:A A ; CC<> − γ

β→ 1

S<>M"J< V%R/41$0FW'+1< $U/G6u1*/4E#J< Qd%$%6pAD/'/4("C< $%IJ<>/

β< 7./0%34(:4

<>N< "5$34$X< "OHT:*/D *& − γβ

< ##(/&< *.w1%,4A+70%34(:4D8*R*J< q"C< Q C/4"/ 6C"*(/4*/4#*'#< (< "(RTr% ABP/4(@1<>$%, B< /S;P:A- A V6u/4(:4$S$*(/4&:4(< $ < 0CP:A% A $76u/4(^p /M$*(/4#< (< "(%N5/ 5(1 AW_NN$/4:C< k jmGinS v Cj v z \<:4%34\*$=< v 7 yD% j' ++jl| 450%34(:4* *J< *$/41TJA+` 1TJ&%6W N< ##(/JHT"C<>w<>/4"*#< (< "(^/4('#1T'" +1< $SI(/Uq< ('#< (< "('"C<J,0%34(:4 AYV/ R<$%/D*J< $KX*V34(@,EJ< (7/

R = 0A?P:A A '<>


0 2 4 6 8 100

1

2

-γ/β

S

β

over

lap,

mix

ing

coef

ficie

nts

distance, R/a0

P:41( A +A&N/34($=< #S< 0S"OHT:7*/D *& / 6<>/4"%*&/?< (

6u1*/476u/4(V<"/4$*1$M*/4@:)/ 6p / JV/4(F%.< $%Aβ*J< EFW'+(#(0

< R/4("C<>$IJ<.%/ ?*/4@.< &JBS(J< −β/γ V< "/41+R/ 6*/40<J346u1*9.%/4;/'FWV< 000/GZr(@S/47/< *.347/4(@/4:4/4< $O,4AT#< (< "(S0%34(:47S%0*(J< :0@.< *

RA

G3>< $1/ 6β< 0

γ< (G@.< (@:/(M%:4%r*J< +$%,6u/4(

R ' a0XS*.E

/ 6/4(0(R/ 6S/4#T"C< $UFW/40q$: B/4FT.< 0q$=<>(V/4qE*J< $*1$=<>/ Aw7(J< /4S&,< &,< ##(/JHT"C<>%/4;<>7$34$/ 6<>/4"*N#< (< "(/41$0!FWR*/4&(/4$$0 A% Z!R/>.(N< 0 W%55($=<>%34$%,;<>6uV/C#1TN/4"R/ 6'#< (< "(7\Mq< (7F< < < ##(/JHT"C<>$%,)I(/A'+1"(*J< $*J< $*1$=<>/4 $$1@(<>%:R#(/4F$"< (70/4)D*/4 A '/49H< "#$7/ 6Li2

"/4$*1$ A`w< :!/4ER< 1"#T/4?/4"C<J,@"C<>'C<>/4"*'#<8(< "(7/ 6UG"#$Or0\@,D@" nAX_N$$BMq< (Z#< (< "'(< (M(J< $3 < $10 < FW6u/4(76u/4(wV<>/4"*N.<>A

H2 H

−2

-$

0 2 4 6 8 10

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

t

ε

ener

gy (

Ry)

distance, R/a0

P:41( A TAWD:4$98$*(/4;q< (S"*(/4*/4#*7#< (< "(5< N<G6u1*/4)/ 6&(<>/4"%*0@.< *

RA^ /4"#< (CS%P:A A \/;; 1*/ 6S

/4(@/4:4/4< $%IJ<>%/4 A

254 &.$ 2 "$ H2 H−

2 %)< $$XV"*(/4*/4#*Z#< (< "(5< **/41&0!6u/4(Tw$*N/ VP/D*.Q48#< *C@.<>V?<;w<,4S*.?"C< Q CLN< "$%/4=<>"C<>(OHdF$/D*.Q+0=< :4/4< $nA` *J< 1NwZ<J34Z/4$%,Gp /M/4(F%.< $&(Z< (Z/4$%, (2∗2

2

)= 6

@.<> 6u/4(H2"/4$*1$5< 0 (2∗2

3

)= 4

@.<> 6u/4(^H−2

"/4$*1$=< (^/4 A+P/4(^H2

"/4$*1$ "C<,0%3D0< $$W@.<>S+/M(7(#$@.<>

|1 >= a†1↑a†2↑|0 >,

|2 >= a†1↓a†2↓|0 >,

|3 >= 1√2

(

a†1↑a†2↓ + a†1↓a

†2↑

)

|0 >, a A &o< 0;.(:4$@.<>

|4 >= 1√2

(

a†1↑a†2↓ − a†1↓a

†2↑

)

|0 >,

"

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

K

U

energy, V,J (R

y)

ener

gy, U

,K (

Ry)

distance, R/a0

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

J

V

P:41( A $DAK />8$*(/4q< (M"*(/4*/4#*#< (< "('<><!6u1*/4-/ 6+(<>/4"%*S0@.< *

RAT /4"#< (SS%;PB:A% A V/RNN% 1*Z/ 6 N/4(p8

/4:4/4< $IJ<>%/4 Aw$6kK<HTB6u/4(B +(<4<>/4"'%* < 0M&(<>/4"* /41$/4"MF+(< */4

U< 0

KA'(:4&V<HT7Z6u/4(Z'*/4(($=<>0/4##:)< 0E9H+8

*9< :4&(< */4V<>0

JA

|5 >= 1√2

(

a†1↑a†1↓ + a†2↑a

†2↓

)

|0 >,

|6 >= 1√2

(

a†1↑a†1↓ − a†2↑a

†2↓

)

|0 > .a A !&o

RLN< "$O/4< anDA -$4o *J< )FW;*J< 7S(%!<>

H = ε(n1↑ + n1↓ + n2↑ + n2↓) + t∑

σ

(


)

+ U2∑

i=1

ni↑ni↓

−2JS1 · S2 + (K +1

2J)(n1↑ + n1↓)(n2↑ + n2↓) + J

(

a†1↑a†1↓a2↓a2↑

)

+V∑

σ

(n1σ + n2σ)(


)

.a A &o

H2 H

−2

"!< Q+:7S9HT#*.<>/4'3 < $1/ 6W^LN< "$%/ < M S:4^6u/4(K(#$U@.<>a A &o9T/4$%,/4I(/'3>< $1

< I|H|I >= 2ε+K − J ∀I ∈ 1, 2, 3. a A o:4$S@.<>Ra A !&o6u/4(" <G1F"C<>(OH

HS =

2ε+K + J 2t+ 2V 02t+ 2V 2ε+ U + J 0

0 0 2ε+ U − J

a A 4o

:4&3 < $1S*J< )FJ< $%,/4FT.< 0)< < $%,+*J< $%$%,;76u/4("

ET = 2ε+K − J

ES1 = 2ε+ U − J

ES2 = 2ε+U +K

2+ J +

1

2

√

(K − U)2 + 16(t+ V )2

EGS ≡ ES3 = 2ε+U +K

2+ J − 1

2

√

(K − U)2 + 16(t+ V )2,a A 4&o

S(ET

V(6u/4$00:4(<>(#$M@.<>< 0ESx

< (C;:4$@.<>A

ES3< S< $%w<J,Dw$/ @S(: ,4+1S% 7:4(/410@.<>

EGS6u/4(

< &,;&(<>/4"*N0@.< *RA

/D/4Q+:<>')< F/4$1T3 < $1;/4<>< $$ #< (< "('< ((98+1(0 A ` 1TZ.< Q+:CG($=<>%34V3>< $17/ 6UM/4FT.< 0d(:4X%N1(7/41T<>Z<>/4"*S(: ,< N< "N*/4&(F1T/4'/M< $$@.<>

2ε< 0*J< CFW

(:&< (00E< 7G(6u(*(: ,4ABD"$< (@$%, X/4$%,!G0%X(*'FW E&(<4<>/4"%*S< 0C+(<>/4"%*5 /41$/4"MF+(< */4

U −K "#W/4(@.< &J4*V(6u(*M$34$B*J< !FWRN/C*$10/4R/ 6R3>< $1WnA AV($=<>%3&(: ,0%X(*7Fp !< &,C /G@.<><R6u1*/4;/>6

U −K T/ J< *./ 6" #< (<>$%,4AV*/4((#/40::4$:4(/410)@.<>;VP/D*.Q#< *V< 76u/4("

|GS >= 1√

2D(D − U +K)×

×

4(t+ V )1√2(a†1↑a

†2↓ − a†1↓a

†2↑)− (D − U +K)

1√2(a†1↑a

†2↓ + a†1↓a

†2↑)

|0 >,a A +o

S(D ≡

[(U −K)2 + 16(t+ V )2

]1/2.

G$/ @#D8:4$9Z:4@.<>G< < E< 0"OHD1(R/ 6^'@,D""'(%*M/4*@.<> 1√

2(a†1↑a

†2↓ + a†1↓a

†2↑)A (6u/4( /d\0%X(*\S% %(C

$

LN< (@(98pP/D*.QN/4(L5%$(p8(/40/47< ##(/&< *9Z/H2 U*/4@(1*9Kp />8#< (@*$

<J3456u1*/46u/4(^N:4(/410C@.<>Z< **/4(0:V/VN#(*(#T/4C#(/3D00CF+,[UAa DA 4o9TS%*9)S*< .< Q S76u/4("

Φ0(r1, r2) ≡1√2< 0|Ψ(r1)Ψ(r2)|G > .

a A 4o< Q+:

Ψ(r) =∑↓

σ=↑w1(r)χσ(r)a†1σ + w2(r)χσ(r)a

†2σ

Tw/4FT.< <>

Φ0(r1, r2) =2(t+ V )

√

2D(D − U +K)Φc(r1, r2)−

1

2

√

D − U +K

2DΦi(r1, r2),

a A &oS(MG"M1$%%#< (@%*$w<34M6u1*/4\< ZFW#< (<>0&/Cw/#< (@.Z/ 60%X(+S<>1( A*/3>< $&5#< (@S

Φc(r1, r2) = [w1(r1)w2(r2) + w1(r2)w2(r1)] [χ↑(r1)χ↓(r2)− χ↓(r1)χ↑(r2)] ,a A -$4oS(J< S/4*Z#< (@S.< Q 76u/4("

Φi(r1, r2) = [w1(r1)w1(r2) + w2(r1)w2(r2)] [χ↑(r1)χ↓(r2)− χ↓(r1)χ↑(r2)] .a A "&o;(<>/)/ 6*/D *+MFW6u/4(

Φi(r1, r2)< 0

Φc(r1, r2)*J<>FW0r0

< ' h|@~9m- |q.m 2> hX|γmb

A^'3>< $1)/41$0FW)0@:410-6u(/4" zikhDx v jlin.m 2> hX| − γ

β

< ##WJ< (:qtd0r%/4 a A "$4o/ 6/4(@/4:4/4< $%I0-<>/4"*/4(F%.< $

wi(r)AK;p /+1< &%'< (;0(<JS-P:A

A TAWR"C< &,&8FW/D0T,*/3 < $*,γmb

(#(&<'(1R0:4(M/ 6U"M1$%%#< (@%*$*/4Tr:41(<>/4< $ "OHT:A

P/4(H−2

"/4$*1$=<>(7/4wC<J34C/4$%,4"M1$%#< (@%*$%'@.<>VE

P/D*.Q#< *

|1 >= a†1↑a†1↓a

†2↑|0 >,

|2 >= a†1↑a†1↓a

†2↓|0 >,

|3 >= a†1↑a†2↑a

†2↓|0 >,

|4 >= a†1↓a†2↑a

†2↓|0 > .

a A "!&o_N$$Z@.<>w*J< FW5(< @6u/4("0&/RJ< *.;/ ( F+,'Z@,D""(/ 69H+8*9< :4:VZ/4(F%.< $

1↔ 2/4(Z# ↑↔↓ AD7LN< "$O/4< 6u/4("M1$=<>%/4w0&*J< $W$Q N6u/4(

H2

"/4$*1$Ma A &o9ATLN< "$O/4< C"C<>(OH*/4@/ 6^ /;0&*J< $BF$/D*.Q+Z*/4*:M@.<> |1 > S% |3 > < 0 |2 > S%|4 > nA A

HH−2=

(3ε+ U + 2K + J t− 2V

t− 2V 3ε+ U + 2K + J

)

.a A &o

H2 H−2

$

H2

! " !

R/a EG/N εa t U K V

# $

J

# $ 4(t+V )2

U−K

# $

%

& %

&' ( ) %

*+ + + %

% ) % ' %

,+ , %

% ( ( % (

+ + & *

+ + ) )+ +

+

%

+ %

% ) %

)+ , &-

* ) , %

* *+ &

' ( ( % % %

* , )+ %

+ ' ) )

%

& %

% % ) ) %

) &-

) %

) ) &

)' + % %

+ ) ) % *

' % , % % ( &

% '

+ %

& ) , ) %

*+ ' , &-

, ( ( %

( ) *' &

* , , ) %

&+ % (

% ' , + & )

&'

(

& %

& *' %

+ ' ) &-

% ' ( %

( % ) % &

* & ) ) %

+ ' '

, % + ( ( ,-

' ('

(

+ %

& + %

+ ( ) &-

% ( ( ( %

, (+ &

+ % %

, % *

' % % +

% (

& %

& % ) %

, % * &-

&' % ' %

* * ( &

,+ %

%

+ ) ( * + +

, % ' (

+ %

& & * %

(+ + &-

& * ' %

+ )' &

( ) ) % %

%

, ( * ) *

' )

+

& %

& & , %

(' + ) &-

& * %

+ (' &

(' ) & ' ' ,' %

* *+ %

' (+

+

+ %

& & % %

( * % * &-

& , * %

+ % ' &

( * ( ,

( ( ) , &

' ( ( *

% + +

*

& %

& & &+ %

( ( &

& % ' &+ %

+ % &

( ( ) *

) & ' &

+ & ( (

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*

+ %

& & & %

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& % * %

+ &+ &

( & )+ + &

* * %

( +

)

& %

& & & % %

,+ + &-

& & , ( %

+ & ( &

,+ * (

' * ,+ &

% ( ( ( &

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)

+ %

& & & & %

* * * &-

& &+ %

+ & % &

* * *

' + &' &

& * * &

, *

,

& %

& & & & %

+ &-

& & (+ %

+ & & * &

+

% + + % &

& ( &

% ) )

,

+ %

& & & & % %

(+ ( &-

& & ( %

+ & & ( &

(+ ( %

+ + & % &-

& % + ( &

&+ )

'

& % %

&-

& & % + %

+ & & % &

%

% & &+ &

& & ) % &

& +

'

+ % %

% &+ &-

& & &' %

+ & & % &

% &+ &

) ) + &

& & ( ( &

& % %

% &

& % %

&-

& & & * %

+ &

&

+ ( ) % &

& & % + &

& & '

$

H−2 V J 4(t+V )2

U−K

R/a EG/N εa t U K V J

4(t+V )2

U−K [Ry]

!" # # $ !% # " $ # ! # !

! $ # !" # # $ % ! $ " # " % % ! # $ $ %

% % # !" % " $ " # $ $ % ! % % $

! " # !$ $ # " !" # $ " $ #

" ! # # # ! $ # % # % #

! " # $ ! ! ! ! " " $ "

" ! ! % # " # # % " " $ #

! " # % % !% ! # # ! "

! " " % % % " " $ #$

! ! " % $ # % ! !$ " ! % %

# " $ " " % ! $ #

# ! " # ! # " " ! !% $ ! !

$ " # % # " % $ # % $ # ! $

$ ! " # ! ! # " % ! # # # ! ! $

% " #$ # " % " % #

% ! " $ % " $ " % " ! $

" " $ # $ # % " % " "

" ! " % # ! % " % ! ! !

" % ! " " % % %

H2 H

−2

$

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8H

2 molecule

many-body covalency

single-particle covalency

cova

lenc

y

interatomic distance R/a0

P:41( A TA;:4$98#< (@*$%daγoR< 0"C< +,&8F/D0T, a

γmboR*/3 < $*9,q6l< */4(

6u/4(wH2

w<J346u1*/4AP/4(0.< $"C< 9HDJA

:4&3 < $1S/ 6BSLN< "$O/4< < (

E± = 3ε+ U + 2K + J ± (t− 2V ) a A o[^< *.E(: ,\Z / 6u/4$0d0:4(<>GS%(#*V/;#

σ =↑, ↓ A]N#W0:/4:4/ 673 <>$1t− 2V :4(/410)@.<>;VP/D*.Q#< *VS%(<@,D""(*V/4(N< \< &@,D""(**/4"MF<>/ )/ 6KR%< $@.<> |1 > < 0

|3 > AX_NZ%N1(Z/41TZ$=<>(N01(:CM/4#T"%IJ<>%/4!#(/+* M@,D""(**/4"MF<>/ (:4*J< $$%,N6l<J3&/4(< F$K6u/4("C< $$RS(J<>B < +@,D""9.(%*

6u/4($=< (:4(RA

` *J< 1\< < $%,+*6u/4("M1$=< M6u/4(G):4(/410@.<>)(: ,< (!<J3>< $=< F$FW/>.

H2

< 0H−2

*J< \*J<>E@1@""'%I!3>< $1S%2(#W*/α< 0

RAV$*(/4*VLN< "$O/4< */ +.< S/4$%,;$*(/4D8$*(/4)< 0

$*(/4D8+1*$ &(< */4ABD1;/ FT9<>0:4(/410@.<>;(: ,V$*(/4*S(: ,

EelG

/4$O,4A+/R*/4"#$50*(#T/4/45< ^/V*$105< < 00%/4< $*/4&(F1T/4\/ 6^M/4D8/4!+(< */ A?M(J<>7%Z*$=< *J< $$%,4W< 7 8#& $.# )24 !"# 3 # @# *# '1$&(+ G8# , $+>$. G8# A A!1$2A $2E) !

|2 >$2*3

|4 >

$

0 2 4 6 8 10-2.3

-2.2

-2.1

-2.0

-1.9

-1.8

-1.7

-1.6

H2 ground state

H2

- states

ener

gy (

Ry)

interatomic distance, R/a0

P:41( A !TAw$3&$Z*."E/>6VH2

:4(/410 @.<>E< 0 $/ @H−2@.<>)< <?6u1*/4 / 6Vd&(<>/4"%*\0@.< *

RA /4##:$*(/4

$$%1(<>w($3>< *w/ 6H−2

/4**/4Tr:41(<>/4VS'"J< 1(:51$$:*/401*%3D%p,/ 6

H2@,D@")A

/41$/4"MF\%+(< */4!/ 6^ /;#W/4&Z*9< (:4Z$%,D:;<>ZMFW/40d0@.< *R6u(/4" J< *.)/ (AL5* 76u1$$X:4(/410D8@.<>(: ,;

EG(R,α) = EelG(R,α) +

2Z2

R,

a A 4o

S(Z = 1

6u/4(M&,D0(/4:4 AU(1$%'< (;/S< F$ A AK (6=8(* G3>< $176u/4(7(: ,< (M'FW@3>< (=<>/4< $B@"C<> "!nA @J< 0?/ 6""'%I:MF/

R<>0

α"G1$%.<>/41$%, *J< */4"#1TZ/4#T"C< $

α6u/ (

<:4%3&RA!<J, R:45

R0#W0*G/ 6KV:4(/410!@.<>R(: ,

< 0S/4(F%.< $I A+53>< $1 < (S#$/ 0P:4A- A !V< 0 A TA&(N< (('*1(@34V/4dJ< *9#$/ Z3DA

RX%*G('< (Mw/;0%X(+7:4(/410@.<>

6u/4(H−2

AUP/4(H2

;/4#T"C< $α:4/+MS%

R → ∞ /\<>/4"*3 < $1C6u/4(H<>/4")FW*J< 1VwV<34! /C#< (<>0d<>/4"AP/4(

H−2

R:45.%

H2 H

−2

$

0 2 4 6 8 100.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

H2

-

H2

-

H2

inve

rse

orbi

tal s

ize,

αa 0

distance, R/a0

P:41( A TA Z#T"C< $B3 < $1/ 6 '3 < (=<>/ < $α#< (< "(R< V<6u1*/4E/ 6

0@.< *RAP/ ( $=< (:4

RwZN7<./ "*5$"'O

α = 1+S*953 < $156u/4(

<G&,D0(/4:4\<>/4")A

$"'O< ;+("0=<>53>< $17FW )H− < 0 H #< (< "(< w 7<34

w/'$*(/4S/4)/4<>/4" < 0)<G%:4$7/4;*/40 AC< F$ A Nw5$@US(:4 < 0'S3 <>$1/ 6XS"*(/4*/4#*#< (< "G8

(Z6u/4(H2

@,D@" S%\/4#T"%I0\/4(F%.< $S(J< \< F$ A CM< "S#(/3D006u/4(

H−2

"'/4$*1$=< (w/4 A < (&*J< $$%,4/4V/41$0)/ V<'0(< @*0%X(*'6u/4(7C/>8*J< $%$0 .m v9v % j m.&WikhDx "C<>(OHd$"+ 'w/%1<>/4A&N< "5/4$0^(156u/4(^N0(*9w9HT*.< :4Z&:4(< $ ;ak6u((/4"C< :>8*o9AWZ9HT*.< :4G&:4(< $BZ< $%w<,DN0*%34$%,!"C< $$(S< \<>N6u/4(5< &%6u((/4"C< :4%*RQ+*R9HT*9<>:4

Jkex = 4(t + V )2/(U −K)AXM3D(@1< $

&(<>/4"%*S/4##:R#(/D*$J< 0:/VN@.(/4:RQ+*59HT*.< :4< (S/41(*/ 6w:4$V<>1(/ 6w

H2:4(/410q@.<> AB

H−2

:4(/410q@.<>51@.< F$VS%!(#W*Z/R0/+*=<>/4)&/

H2

< 0e−W*/4&(< (@,/

H− *J< A+L5/ 3&(&(:4*^/ 6X1*.C@.<>SK"#W/4(@.< &KS*J< $*1$=<>:

;)'H−2

)*#N.")A?3& $2A?&A?$C!# !"#F)O !=?, $2AαGH= !" : )'E= !1$2A O# ' $C!), A = 2#F= ! G $. 3 )*#46)'

He$C!!1" #%# *3 )24 9 E.#&;!= )* ) P +P ) C!"$C! =?.# $2A!)

1116

= *.0!#$3 )24 !1$-= *@ !"# 2$2A #H46'), $2EA #+P : )E !1$2= *#3J46)':$2* # *A?$2'@#3>E$.=?. P

$

< F$ A AT7:4(/410.<>Z(: ,EG

TFW/40;$: R< 0Z+34(7/4#TO8

"C< $ /4(F%.< $ Iα6u/4(

H2

< 0H−2

AV(6u(*3 < $1/ 6BV(: ,w6u(/4" "!nA9HT#W("&.< $ 3>< $17/ 6F/40)$: w6u(/4" nA

EG

,QRa0

αopta0 ErefG

,QRexp a0

H28 DA !-$ 4A & 4A !4-$! 8 DA ! 4A "!4

H−2

8 4A "! TA TA $!-$ @.< F$

A :AB".< $$%%IJ<>%/4?/ 6S"/4$*1$=< (&,D0(/4:4-/ (M0("%:)C1$$:*/401*%3D%p,;(/41:4

H2"/4$*1$ W<>Z/Sd*."C<>*J< $$%,;\P:A5 A !TAW

$=< @UP:A$$%1@(<>w"/+0M/ 60(":5 (:4*U/>69HT*%0G@.<>/ 6H2

F+,M"J< 1(:7 A :A w1$$:N*/401*%3D%,aunA A>3D=<H−2

+("0=<>@.<>o9A

2 d#d "> $& H2

'< $$+*J< $*1$=<>/4/N6l< ( w<34w"C< 0w< G< 1"#T/4R<><J34w6u1*/4*&(0G< (/410+1*$nAP/4(B<>/4"%*/41$0VF@1@%r0VF+,Z@,D""(@,4A` 1T 6u/4(^"/4$*1$&^"C<,G/ FWN*(@.< A&?5"C<,'< $$/26u/4( <V0#$=< *"+dFWwN*+(w/ 6/4(F%.< $W< 0% #< (&+1*$1Aw< ##(/&< *.; "G8

$=< (N/;< ##(/&< *9E1T$%I%:;1"/ 6 /4(F%.< $N/)9HT#(R"/4(G*/4"#$*J<.0<J346u1*/4U'$*(/4*w*J< $*1$=<>/4 A :TA4/76u/4(" /V0%#$< *0'^/4(F%.< $/'%"G1$=<>Z<'#/4(F%.< $nA

0J<)R$$%1@(<>0?qP:A A 4A0@.< *CFWwq/4(p8F%.< $55(01*0\/

R − 2d AWVp />8$*(/4!#< (< "(N0/C/ Z0#W0d/4G+1*$17#W/4%/4 A D/W(N3 < $1Z*J< $*1$=<>0< $(J< 0T,)"C<,!FG10?< :&< %S%!*.< :40

RA D"%$< ($O,4V/34($=< #!%+:4(< $0/DN/4$%,;0#W0d/4)VD8

(/4(F%.< $ 0@.< * /'X*5 *J< )"C< Q 7$%

U ′(·, α)K ′(R,α)V ′(R,α)J ′(R,α)S(R,α)

→

U ′(·, α)K ′(R− 2d, α)V ′(R− 2d, α)J ′(R− 2d, α)S(R− 2d, α)

.a A &&o

P/4(w&(<4<>/4"%* /41$/4"MF;+(< */4(/R0#W0* A

V%1<>/4F*/4"5"/4(*/4"#$*<>0;6u/4(V:4$7$*(/4)#< (< "98(< ,*$105Z+(< */4S%Z+1*$nAT7<>/4"*5(: ,"C<,FWR*J< $*1$=<>0\< **/4(0:/;[UAKa A &o9S%)R9HT*#T/4!<>5wR<J34RS*9HT#(/4?6u/4(7

2nd+1*$1V*/4&(F1T/4 *J<>$*1$=<>0E?[UA a A +o9D8

@J< 0;/ 6N&(< */4S%CN/ w0#$=< *0+1*$1ATZ0@.< * /M#1T

H2

$"$

R

1/α

d d

R

1/α

R

1/α

d d

P:41( A 4A )/D0$K/ 6^<H2

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#$=< *"&SFW !+1*$< 0;*/4((#W/40:*&(5/ 6B/4(F%.< $A

;w6u/4("M1$=<M< ((#W*%34$%,d< 0

R− dA)X*Sw<J3&

ε′ = α2 +2α

(e4αd + e2αR

)

e2α (d+R)+2(d−d e4αd−2αR−R

e2αd+R

)

d (d−R).

a A " +o

Z*J< $*1$=<>/4/ 6Z/4##:M&:4(< $W "/4(5(*.Q&,C( AT7%+:4(< $*J< FWw0O3D00M+/5(w#< (@ wQ+* (: ,4&< 0Rw&(< */4BS%MFW/ /41$/4"MF' $$A+ZQ+*5(: ,'w< < $%(< 0T,'*J< $*1$=<>0C[UA8 A TD< 0C /4$%,0;/M1F@%1T

R→ R− 2d AT7%+(< */4S%C /41$/4"GF $$;6l< *N<G(98*&(5&:4(< $nTFW*J< 1V <34 //4(F%.< $< 0)/47w$$n< 0:4(< $/4/ 6B" */4*0 AT?0\< +:4(< $X/ 6Bp,D#

τijk =α3

π

∫

d3re−α(ri+rj)

rk,

a A 4oS(

rirj< 0

rk< (!0@.< *6u(/4" (!0%X(&C#W/4&AF1*.Q+$%,4

, < ($/D*<>0 /4 <@(< :4+;$ w/*J<>$*1$=<>/4 *J< FW#W(@6u/4("0< < $%,+*J< $%$O,q-_N##W0OH GAK?#< (< "(I;:4/4"(@,/ 65)@,D@"F&,

a = R − 2d WM0@.< *MFWw/4(:45/ 6 ri< 0

rjX< 0dF+,

h = R/2

0@.< *7/ 6B/4(:4%;/ 6rk6u(/4" "00$%Z#W/4&S/ 6:4"&

aA

/4##:'&:4(< $ <>/4"*Z(#(&.<>/4)

t′ =α2e−α(R−2d)

3(−α2(R−2d)2+3α(R−2d)+3)+2τijk(α,R−2d,

R

2).a A &o

$

0 1 2 3 4 5

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

αopt

displacement, d/a

0

inve

rse

orbi

tal s

ize

α opta 0

internuclear distance, R/a0

0.00

0.01

0.02

0.03

0.04

d

P:41( A DAJ^(1$% 6u/4( H2

"/4$*1$BS%"/4F$%K/4(F%.< $%A^/4#T"C< $+34(ZIZ/ 6Z/4(F%.< $

αopt< w/ *.< :40;:4%r*J< +$%,'*/4"#< (0C/M

*J< S%;rTHT0!/4(F%.< $X*&(N<>D1*$ #W/4%/4A

E1F+1+!*J< $*1$=<>/4;< (d0&*J< $7< ;6u/4("/ $*1$S% rTHT0/4(F%.< $R*&(

d = 0AZ-/4$%,0%X(*-!9HT@*/ 6'-< 00O8

/4< $B/4#T"%IJ<>%/4!#< (< "(dAW?G/4FT.< !R6u/4$$/S%:3 < $1N6u/4(5

H2"/4$*1$w:4(/410 @9<.(: ,EG = −2.3003Ry wE&34(E/4(F%.< $NI

α = 1.19001a−10V&(+1*$J< (S0@.< *

R = 1.4285a0AY6wV*/4"#< (7.%

/MZ(1$%wS%CrTHT0)/4(F%.< $% < F$ A RwZV<R$:4+ "#(/34"&wM(: ,!/ 6KM/4(0(N/ 6

0.2%/ 6UR%=< $ 3>< $1 S*9\5(#(/D01*0d6u/ (

d = 0A/4(F%.< $< (< $/)<$%$%GF%$=< (:4(XF/40?/4(@(V< 0.

$*(/4"/340d/;G"00$R/ 6^.G"/4$*1$ AW'/4#T"C< $B&3&(G/4(F%.< $I

αopt< 00#$=<>*"+

d< (/S))P:A A DATV0#$=< *"'+

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EGH2

< 0H−2

*J< TZ"/D03&(,(< p8#< (+J^< $$/SR6u/4(<d0r%/4-/ 6N"C< &,&8FW/D0T,q*/3>< $*,6l< */4(^< G $$w6u/4(< !&(/+01*/4)/ 6R/>8*J<>$$0)"/4F$%/4(F%.< $%AM< "C< I:G6uJ<>1(V/ 6K(1$%R6u/4(/4"/4F$%/4(F%.< $%V<>R+1*$(#W$ J< *./ (V6u1(@(

LI2

-

$!

< #< (@JDS$N7$*(/4*/4"Z*$/4(Vadw#W/4%%34o9ATw6uJ<>1(7/S

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2Li2

2LGD 4$ " *¡ $,">" R*/4"#$9HT%,/ 6KV*J< $*1$=<>/4)M[^]_Z` "/D0!(N+1%V(< #0$%,S%*(J< :M+1"MFW( / 6$*(/4AT_N$/+Z*J< $*1$=<>/ C/ 6"*(/4*/4#*5#<8(< "(6u/4(Rw<J3&6u1*/4G"/4(*/4"#$*J<>0?< JR6u1*/4KM/ <"#$9<>QA/q$$%1@(<>;!"#< *9/ 6Z\#(/4F$"G !<34!*./4 <"/4('*/4"#$*J<>0E@,D@")B< "$%,d

Li2"/4$*1$ ABD1*.q"/4$*1$'< ROH

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1s2o$*(/4

/ 6J< *.Li

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Z = 1A&U*/4((#W/40^/ J^/4(F%.< $% <3&8

:#/4+^I &S*."J< U<>US+34(5IS#< (< "'(α1s =∞

4$/D*J<>0<>7'D1*$1A(@:Z3 < $1'&/;<>/4"%*G(: ,

ε′[UAa A &o7<>0

/C<>/4"* /41$/4"MFE&(< */4U ′

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φ1(r;α) =

√

α3

π(1− α|r−R/2|) exp(−α|r−R/2|)

φ2(r;α) =

√

α3

π(1− α|r+R/2|) exp(−α|r+R/2|). a A %$4o

S*J< $*1$=<>/4G/ 6WS#< (< "(^UF< *J< $$%,VS< "S< U6u/4(J^/4(F%.< $ F1T"M1*.C"/4(S*/4"#$%*J<>0 ADD/4"'5%+("0=<>(1$%^6u/4(US"/4@*/4"#$*J<>09HT*.< :4E&:4(< $

J ′<J34dId/ 6R34(< $#< :4 #(&:Adp />8

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0 2 4 6 8 10

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

2s orbitals

J' V'

K'

U'

atom

ic p

aram

eter

s (R

y)

distance, R/a0

P:41( A TAU /?$*(/42<>/4"*"*(/4*/4#*;#< (< "(< <d6u1*/4/ 6+(<>/4"%*Z0@.< *

R6u/4( >5<>/4"*98Y$%Q Z/4(F%.< $AT3>< $15< (% ,4A

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E0=< :4/4< $%IJ<>%/4 #(/D*!0/+!/ )0%X()<\<>$$Z6u(/4" H2

*J< FW*J< 1N

Li2"/D0$< ^N< "S6u/4("G1$=<%/4'ZP/+*9QG#< * A Z$%,G

#< (< "(R3 <>$1G0%X(A;(1$%*/4@:d/ 6S:4(/410.<>(: ,EG

MFW/40d$: R<>0\M/4#T"C< $3>< $1M/ 6UM&3&(MIM#< (< "(

αopt< (/S!7r(@5(// 6< F$ A +A` *J< 1R/ 6V#(/4F$"5< ##WJ< (:'V*J< $*1$=<>/ / 6V#< (< "(

<>/4"*MF< /4"C<J,\(@,\/)"#$O6k,)7.< QA'r(@V#W/4F$Op,w/G:4/4(N7&:4(< $S*.)< (70*1$%w/G3 < $1<> AT/41(*J< 7"C<,G;9HT*9<>:4+:4(< $

J ′ = 0A'< ##(/JHT"C<>%/4?1(G/41TM/dFW/D/

(< 0*J< $nAX_56k(ZG(< @6u/4("C<>/ \/;Gq< (F< Z '/4FT.< d0%3&(:4&(1$%AsN< "$%,4 C:4(/410?@.<>C(: ,E0%3&(:4&RE"C< $$

R$"'OJ

S*9"C< Q4S< )/4#T"%IJ<>%/4;/ 6B#< (< "(α"#W/4F$% A

_N/ (#/4F$%,!Z/;.< Q4GG3>< $1'*< $*1$=<>0d6u/4(<;"#$(Z/4(F%.< $nA?"C<J,(1773>< $16u/4( JS/4(F%.< $%JAD,<J347*/4((*5$"'OT*w'*J< ?:49R<FW/4106u/4(7':4(/410D8@.<>'(: ,4A#5T6u/4(@1<>$%,4X<>1(

LI2

-

0 2 4 6 8 100.10

0.15

0.20

0.25

0.30

0.35

2s orbitals

K

V

J

U

microscopic param

eters, V,J (R

y)

mic

rosc

opic

par

amet

ers,

U,K

(R

y)

distance, R/a0

-0.10

-0.05

0.00

0.05

0.10

P:41( A AW />8$*(/4d"*(/4*/4#*R#< (< "(N6u/4(5M/4(@/4:4/4< $I0>q< (RF< M< M<!6u1*/4?/ 6&(<>/4"%*0@.< *

RABC3 < $1G< (

,4AW /4"#< (7S%)P:A A M/V% 1*V/ 6/4(@/4:4/4< $%IJ<>/ AG$6k<HTZN6u/4(5G&.(<4<./ "%*M< 0d+(<>/4"%*G / 1$/4"GF\&(< */4

U< 0KAV(:4&S<HTSw6u/4(w&,DF(0)< 0)9HT*.< :4V+(< */4

V< 0

JA

< F$ A DA&N(1$O.^6u/4(Li2

"/4$*1$wS%C0%X(&w< ##(/HT"C<>%/4 4(:&< (0D8:M#< (< "(< ##$0;/4/4#/ 6BV[^]_Z` w"/D0 A< ##(/JHT"C<>/

EG/Na,o

Raa0o

αopta09H< * 8TA "!"-$ A TA $J ′ = 0

0%34(:4& 0%34(:4+

J ′ =J3 < $1 8TA $!"$ TA " TA

J = 08TA !4 A DA

/Gq< (F< 8TA $ DA 4 TA "$


/ 6':4(/410E@.<>'70%X(&J '@.<>GS%d'$/ @(: ,7/ < "V< w6u/4(w*/4"#$7*J< $*1$=<>/ A

+$$U<>/ (V#W/4F$O,d%/)V'3 < $1J = 0

'q< (F< AV/ V(J< $$%,d<"#$Or*J<>%/4 < 76l< (V< @$$K0EC<>/4"*M3>< $1 S*9 ;0*1$%;//4FT.< wF1T)%M@1@%r!< < ##(/&< *. / 6M/4FT.< :q<>/4"*G3>< $1V6u(/4"g< ##(/HT"C<>03>< $1Rq< (F< nA %< ##(/&< *. ':47'*/4((*:4(/410E@.<> F1TG*J< $*1$=<>0\3 < $1/ 65:4(/410@.<>S(: ,

EG4SFW/40$:

R+< 0'5/4#T"C<>$T3>< $1S/ 6X

+34(7I7#< (< "(αopt

< (7/ S<R34(@,;:4/D/D0< :4("&S%;*/4"#$(1$%AWD1*9)"/D0;,D$0/4$%,+1< $%.<>O3&Z(1$%A

S$=< @< ##(/JHT"C<>/4 w$/w@^(/2'< F$ A D49H< "0'(5U/ 6 <0%X(+S<>1( A?0(:&< (00/' /4(Q)< )/4(@/4:4/4< $ F<>%JA/4(*J< $< #W*^/ 6 (S0*10'D*/4'TA 4A S#(< **J< $T"#<>*;/St;*J< $*1$=<>/4 A^?"#$%,t< 1"d<>;d/34($=< # &:4(< $S = 0

AwD15< $$X#< (< "(;7q< (F< S< (0&*J< $W/G.<5'<>/4"*VF< AXZ"#< (SG+1< $%,)/ 6^M(1$%A G<>1(M/ 6^:4(/410G@.<>w*/4((*J4F1TKw+1< &%.<>%3&(1$%U< (w/ 6*/4"#< (< F$ +1< $%,/N< ##(/HT%"C<>/4

J = 0+nA A&/ ^34(@,:4/+/D0 A+s5/ N< $/7<>:4$*:

1s2$*(/4K: (/ 10'@.<>w(: ,MK/4$%,M<N6u(< */4M/ 6 / .< $<>/4"*(: ,/4FT.< 0R#(*0: < #T(A>

Li29H< "#$^/SK9HT#$*%$O,Z<>B

9HD/4M/ 6[^]_Z` "/D0/5"/4(^+34/4$%340R"/4$%*1$=< (@,D@"B< /D*=<>0S%)< 00%/4< $X*/4"#$*J<>%/4DS*.)/41$0FW0J< $%S%#< (<>$%,4A

vZ[jM PdO@I

I2acRbPL[NV`O.V] V;O~O.VR SILKWT T OAPLv^K ]

t/4$02@.<>\#&,D*;/4!C11< $$%,*/4*(0 S%t@,D@"S% <?$=< (:4+1"MFW(7/ 6w<>/4"AX?<;*(@,D@.< $KG$/D*J< $$*(/4*G#(/4#(@V<>(G 1*0F&,E< $$K:4+FW/4(:<>/4"A Y6 w'*/40(R<;:4$'<>/4" E<1*.?@(1*1( <347/'*$107/4"7w<J,+(< */4 A

Z"#$%@w< ##(/JHT"C<>/ */4@ / 6(:&< (0:V5:4DF/4(:R<>/4"/4((<>(R<>/4"*G/4< V"#$'*.< (:40?#W/4&S%/41T.< Q+:)?< **/41&(M$*(/4*@(1*1( AUM"C< (Bw;"C<J,q9HD0-;*J< $*1$=<>%/40/4V6u/4(N<>/4"5\ < #T(7TAWsN< "$%,4\< 00O.%/4/CR*+(< $B+1*$1N < 0034(< $U#W/4&7*.< (:4#$=< *0E< **/ (0%:;/G@,D""(@,\/ 6G@,D@" w< &K/Z0*(F A ZF+3D/41$%,1*9G*J< $*1$=<>%/4R#(/3D0(J< $@*^(1$% m>h|S@.<>N< (7@(*$%,$/D*J< $%I0 T+1< =<>/4"* A

Z< 00%/4< $*.< (:4*.< :4SS/ .< $*.< (:45/ 6X5@,D@")A+_ *(@,D@.< $*J< ;FW7%"C< :4%0;< F:GF1$% / 6B"C< &,C1*.)F$/+*9Q+ATB/G"C< Q4Z%w1T(< $n+/ .< $B*9< (:4M/ 6 <:4$RF$/+*9Q)*/4@:/ 6^M*+(< $D1*$1 %N$*(/4< 0 < 1DHT$=<>(@,Z*.< (:4>< /5FW+1< $&/5I(/A_N<S(1$ J:4+FW/4(:/47< (G< (0F&,d34(< $U*$$AD/X G<34M/1'< X*%34G*9< (:4

Zext

6u/4(R" < G%GM#< (@=< $$O,E1T(< $I0F&,q$*(/4G6u(/4"/ (GF$/+*9Q+AK"#$@M*./4*R/\.< Q ;/4$%, 1

n

8Y/ 6S;*.< (:4/ 6N< -<>/4"*C/4q%6S%M< (02< *(/4

n*$$AK+1<>/46u/4(M;*9<>(: F< $=< *;*J< -FW;S(%

0/S);6u/4("

Ze+Ne+z∑

i=1

Zexti e = 0,

a DA oS(

ZeSR*9< (:4R/ 6KR*&(.<>$+1*$1

NSM+1"MFW(N/ 6U$*(/4

.< Q E&/< **/41&V\G*J< $*1$=<>/ eN'$*(/4*9< (:4 < 0

zN

+1"MFW(S/ 6*(@,D@.< $ r$0)/41(*A

4

P:41(DA 4A<>#W / 6*/4"#$9HCau1##W(B(/m = 0,±1,±2 (#W*%34$%,o< 0(J< $5au$/ (7(/No7 0dw<J3&G6u1*/4AP/4(ZG*/4"#$9H\3 < $10dw<34G6u1*/4

wG/ '< F/4$1TR3 < $1M/4$%,4W+17G< #'/ 6^6u1*/4Z%/4##/4%:45/ 6U" S0+*J< $( A_N$$(J< $w<J3&V6u1*/4 9HT*#TZr(@Z/ W*J< FWV(< @6u/4("0\&/CJ< *.d/ (NF&,!<(M0"/4< $(/ .<>/4 AN5(J< /4S&,/4$%,;M6u1*/45< (70#$=<,40 A

` *J< 1;*+(< $<>/4"*J< -<J34;"C< +,?$*(/4S<>G*(J< G0*1$%p,/ 6*J< $*1$=<>/ (R/4M"C<J,EFWC(:&< (00-< M*/4"#/4:M<>/4"**/4( AW,'@1@N$/ (SR*.< (:4R/ 6KR*&(< $<>/4" < 0\0/C/ < ##WJ< (U< +,+S(S$wMw*J< $*1$=<>/4 A_N'9HD/4'/ 6U< ##(/&< *.M /41$0FW5Z(#$=< *"+ / 67 /41$/4"MFC#W/>.+=< $/ 6 N*+(< $W+1*$1wF&,< C6=86u*%3&Z#W/ &=< $X0/G0*(FWZ7*/4&(F1T/4/ 67*/4(Z$*(/4<M"/4(#(*"C< (A

@.< (@:G#W/4&w7*9/4*/ 6F< ww<3476u1*/4A*(@,D@.< $r$0;X*S< (Z"#W/4(@.< &J&6u/4( 9H< "#$ T0D8$$W/4w"MFW000CZ*.<8$/4:40S"C<>(OHXADD/V N*J< *./D/456u/4(^/41(*J< $*1$=<>/ 'N 0<J3456u1*/4

/ 6K< 0>@1@.< F$7I A((< 0=< $ 0#0*JR

R3d(r;α) =

√

8

45α7/2r2 exp(−αr). a DA 4o

5*/4"#$ww<34S6u1*/4< (S/4FT.< 0'F+,G"G1$%%#$%,D%:Z5(< 0=< $T#< (@F&,G@.< 0< (0#(*J< $K0#W0*J(#(&0?F&,d#(*J< $K< ("/4*

Ylm

nA A φ3dm(r, θ, φ;α) = R3d(r;α)Y2m(θ, φ).

a DA &o` 1T5%S/ S/4$%,;*./4*76u/4(wR 06u1*/ JA,)/ 6k!< ##J< (5< 5(J<>$6u1*/4AV(J< $X6u1*/45< (70r0\< 6u/4$$/S

φ3z2−r2(r;α) = φ3d0(r;α)a DA +o

φyz(r;α) = −ı√2(φ3d1(r;α) + φ3d−1(r;α))

a DA 4oφxz(r;α) =

1√2(φ3d1(r;α)− φ3d−1(r;α))

a DA &oφx2−y2(r;α) =

1√2(φ3d2(r;α) + φ3d−2(r;α))

a DA $4oφxy(r;α) = −

ı√2(φ3d2(r;α)− φ3d−2(r;α)) .

a DA &o70*w/ 67(J< $W<J3476u1*/4w(#(&S7< #W7/ 6Z6u1*/4Vw< (@=< */D/4(0<>Z@,D@")A< #W/ 6BFW/ Q+0S/ 6 0;6u1*/40#*0\*9"C<>*J< $$%,;dP:A DA 4AX` / !*9/4*7< (R+1%3>< $+J!R<>U,G*J< FW(< @6u/4("0'&/J< *./ (^F&,'<71%.< (@,R(< @6u/4("C<>/4 < $/4:< S(VS/4$%,;/4

αA!/ 1(5*J< $*1$=<>/ R< $$/J< *.!w<J3&6u1*/4

/)<J34< E0#0+R+34(I#< (< "( /(1$%6u/4(7FW/ E"C<J,FW0%X(&JA

7r34V 0;6u1*/4TFW/ (J< $< 0*/4"#$9HX< (7/4(@/4:4/4< $n+(w/3&(p8$=< #)&:4(< $

S = 0A5SFW*J< 1R/ 6KV0%X(&Z< :41$=< (0#W0*M/ 6

J< *./ 676u1*/4AT73>< (=<>/4/ 67%+34(7I7#< (< "(w0/D/ F(J< Q'0/SC/4(@/4:4/4< $%,4A& #(&%$%6<>;< 0T3 < &.< :4NFW*J< 1N<>/4"*M6u1*/4V< (C< $(J< 0T,d/4(@/4:4/4< $%I0q< (6u1*/4AD/ 0/R/ w0/V(< @6u/4(" N#< (< "(*J< $*1$=<>0CZ<>/4"*SF< AD[U+A=alTA TTA +ow< (/>5*< (@,)S*J< A

N&(/D01*/4'/ 6X5*(@,D@.< $r$0*.< :4 /4$%,M5#W/ &=< $/ 6X5D1D8*$1AWR$*(/4D8$*(/4!&(< */4

H2(r, r′)< N@$$ V6u/4(" 6u(/4" [UA

alTA &o9AD/%6B < &5/#(@6u/4(" V*< $*1$=<>/4w6u/4( /C/4(S"/4(7$*(/4 G*J< d*J< $*1$=<>V C#< (@*$R"*(/4*/4#*V#< (< "(Zd< 0T3 < * XS%/41T

;)' !"# 3 #*= != )*.:$2*3 &()* 2# *!= )*. .# # :OO# *3 = 9 P

VQ+/S$0:4R< F/41TSV@,D""(@,;/ 6U< 00%/4< $X*.< (:45)/41(S@,D@")A?*J< q1C< "9HT#(/ R6u/4(V*J< $*1$=<>/476u/4(V< +,*(@,D@9<>$r$0q< ((< :498"+J< $/;6u/4("#$G<>/4"< 0?"'/4$*1$=< (Z/4ABD/%6 ' /41$0E<J3410< $/M 0!6u1*/45!/41(N*J< $*1$=<>%/4Sd < #T(7T M*/41$0!(1R(1$%5( AWV"/D0)/ 6K*J< $*1$=<>/4/ 6 />8#< (@*$7"*(/4*/4#*7#< (< "98(70*(F0E?D*/4ETA DA DAX(< ( ;/4+1%3>< $+&:4(< $N6u/4(r3&; 0q@.<>A Z$%, !/ 6" < (C/4I(/\6u/4(R*/4"#$9H+8Y6u1*/4EF< AG#(/4:4(7/ 6*J< ((@,D:/41TZG*J< $*1$=<>/ 56u/4(NZ< $$p />8#< (@*$+:4(< $w&3&/4$%3D: J > >#E V #!<>0\ 06u1*/ S%!<'*/4""/4;/4(:4w(5*J< $*1$=<>0 A&P/4( w<J3456u1*/4U 5<J3& Z/4+1%3 < $&+: (.<>$/41T5/ 6BS*. / 6B" < (7/4I(/A

?U*J< 7(< @6u/4(" ^#< (< "'( 6u(/4"*/4"'#$9H+8Y6u1*/47&/(J< $O8Y6u1*/ :4$98#< (@*$%ZF< 1:'7(< @6u/4("C<>%/4;"C<>(OH

A =

1 0 0 0 00 −ı√

2−ı√2

0 0

0 1√2− 1√

20 0

0 0 0 −ı√2

ı√2

0 0 0 1√2

1√2

,a DA !&o

< 07(< @6u/4("C<>/ ;(1$7

V ′realijkl =5∑

i′j′k′l′=1

Aii′Ajj′Akk′All′V′compli′j′k′l′

αi′ → αiαj′ → αjαk′ → αkαl′ → αl

,a DA &o

S(RRF(< *.Q Z/4!V(:4+5(#(&Z3 < (/4151F@%1T/45/ 6α#<8

(< "(N/FWM0/4M\J< *9d*/4"#W/4&7/ 6UM1")AX\M(J< $F< (< (V '/4I(/&:4(< $A

">2 y| $ |> (dy26(S"/D0$T S*9/+/46u/4(KS.H< "<>/4G/ 6S$*(/4'@.<>^G*(@,D@.< $r$0d%5M/D*.< 0(< $1((/410:"/D0$nAYZN/SddP:A DA DA Z\3&(@,<HTUHX,2< 0tI (!C$/+*J<>0 <?*.< (:4

Zext 2+1< $N0@.< * R 6u(/4"/4(:4% XS*.E*/4&.< *.< (:4

ZA'D1"MFW(/ 6 *(@,D@.< $O8Yr$0E*.< (:4V

D1z = 6

A ?E< (d:4/4:q/-0*1 /*J< A5P(@J /4$*(/4 *J< N = 1

A_S**/4(0:)/!*9< (:4CF< $=< *C[UAwa DA o9 C"C<,*./D/4Z = 4

< 0Zext = −1/2 A Zext "C<,;(#(&< !/JHD,D:4\/4 O2− *.< (:4V< (0\FW 6u/41(:4+FW/4(:M*$$A;7/4$*(/4*J< 77 />8#< (@*$N+:4(< $w< (

. -$

RR

P:41(DA DA)/D0$^/ 6 @,D@"E/D*.< 0(< $^*(@,D@.< $Ur$0 A< 00%/4< $*.< (:45< ($/D*J<>0/4

xy< 0

z<HT5;0@.< *

R6u(/4" *+(A

'x

'y

'z

'x

'y

'z

'x

'y

'z

'x

'y

'z

'x

'y

'z

'x

'y

'zx

y

z

020 === γπβα

220 πγπβα ===

P:41( DA TA S/ .<>/4E/ 6w*/D/4(0<>'@,D@" /!9HT#(R*(@,D@.< $^r$0E6u(/4"*.< (:4S/4

x< 0

y<HTS3D=<R73 < $16u/4(w*.< (:4S$/D*J<>0/4

z<HTA

"

/ M00 AU` 1TMF*J< 1;/ 6Zext ; 0E<J34C6u1*/4M< (C/ RC:4D8

6u1*/4U/ 6X: $98$*(/4'LN< "$O/4< A4D/MK*J< SI/4#T"%IJ<>%/4"C< Q4Z'34d6u/4(NG:4$R$*(/4 AD*/40*J< G5<>ZS%\ /$%*98(/4

N = 2ApEV%1<>/4 C0?w/!$*(/4?&:4(< $V< 0?

*9< (:45< (7(#W*%34$%,Z = 5

< 0Zext = −1/2 ACF/ C*J< 50C/V*J< $*1$=<>5:4$98$%*(/4"*(/4*/4#%*S#< (< "G8

( < **/4(0:7/#(*(#T/4!a DA 4o9A45LN< "$O/4< G*.< :4^S%'(#W*/G.<>/4"*Z*J< V[UAalTA &o

H1(r) = −∇2 − 2Z

|r| −∑

η=±x,±y,±z

2Zext

|r− rη|,

a DA oS(

rηNG#W/4%/4\/ 6^'< 00%/4< $*.< (:4 A G3>< $17/ 6

η(#(+

;<HTG< 0qS(M*9< (:4R$/D*J<>0q/4#/4%%34/4(R:&<>%34#< (@G/ 6S<HT%JAT7r(@ /G(" ,D$0<>/4"*53>< $1AP/4( 0;w<J3&Z6u1*/4#< (< "(5< (7+1< $/

t′ij = αi(αi −2Z

3) +

∑

η=±x,±y,±zδtijη,

a DA 4oS(

δtijηw*(@,D@.< $r$0*/4((*/4;6u(/4" *9< (:4

ηA

U*/4((*/4G*J< FSJ< $%,R*J< $*1$=<>0M6u/4(η = z

A4S*J< $*1$=<>/4MK0/4#(*J<>$X*/D/ (0%<>D1:RN+34(Z0@.< *N(w9HT#< /4:4%34F+,[UA7anNA &o'0*(F0 20.< $S2_N##W0OH NA_N< $%,DI:E!#(/4#W(@C/ 6 :40(76u1*/4/47<>7Tr%Z(< /4$%,6u /4I(/("AXG(1$%Z<J34MR6u/4(" / 6^r%R#W/ (Z6u1*/4Z/ 6

α< 0

R"M1$O8

#$0!F&,exp(−2αR) < (R0=< :4/4< $!w<34R6u1*/4\0* δtijz ∼ δij

< 0< (34)S%(#W*5/G7(< @6u/4("C<>/4

η → −η AG3 < $176u/4(η = ±x <>0 ±y *J< EFW'/4FT.< 0F&,d'(/ .<>/4d/ 6 O8=< $Uw<J3&6u1*/4AB_ (/ .<>/4E*J< qFWC0*(FW0F&,?[U1$(M< :4$

αβ< 0

γS*.?*/4((#W/40E/)(/ .<>/4E<HTRI ,E< 0?< :&< EI 34(@,"MS%E(98#W*G/d< *1< $ */D/4(0<>C@,D@")A3 < $1

α = 0, β = π/2, γ = 0< 0

α = 0, β = π/2, γ = π/2"C< #)

η = z*J< M(#W*%34$%,!/4!

η = x< 0

η = y*J< A^(< 0=< $w#< (@'/ 65;w<346u1*9.%/4'G&3 < (=< &G10('

(/ .<>/4A#(*J< $< ("/4*w(< @6u/4(" ;76u/4$$/S%:Mw<J,

Ylm(θ′, φ′) =

l∑

µ=−le−ı(mγ+µα)dlmµ(β)Ylµ(θ, φ),

a DA J&oS(

dlmµ(β)< ((< @6u/4("C<>%/4q*/D *&U(#(&:d;,&8p<HTM(/ .<8

/4 +0*(FW0C0.< $C_N##W0OHLVAT` *J< 1δtijz ∼ δij

+ N*J< */4"#1T

. "!

δtijx< 0

δtijy<

δtijx,y =5∑

i′=1

e−ı(m(j)−m(i))0,π2d2ji′d

2ii′δtijz,

a DA +oS(

m(i)^NI98p<HT */4"#W/4+w/ 6< :41$=< ("/ "+1" / 6O8Yw<J3&56u1*98

/4 AL5/ 0/+ZGLN< "$O/4< !"C<>(OH)$/D/4Q$Q GP/4(N/4M$*(/4\%ZN0D8

*J< $wS%- /4##: ;"C<>(OHHij = tij

AU %/41T')*(@,D@.< $Sr$0%0=< :4/4< $5FW*J< 1E 0 <>/4"*)w<J3&\6u1*/4< (\:4T6u1*/4C/qd/4$*(/4dLN< "$%/ =<> AWwG&(/D01*/4!/ 6^M*(@,D@.< $r$0d0/DZ/ 7*.< :4#*1( A(< $<J346u1*/4M<J34<!@,D""(@,?*/4@&GS%?/+*.< 0(< $1((/410:TAB+1RLN< "$%/ =<>ER@$$U0=< :4/4< $l< 0? 0// S0!< &,0=< :4/4< $%IJ<>%/4#(/D*01( A*/4"#$9HC6u1*/40/'/ 6u1$$%,< :4(MS%\G/D*.< 0(< $K@,D""(@,4XF1T7('Z/4$%,!/4G/ 80=< :4/4< $$98"& B@.<>S%

m = 2R*/41#$0qS%q;@.<>CS%

m = −2 A0=< :4/4< $%%IJ<>/ ?,D$0M;(J< $6u1*/4φx2−y2

< 0φxy

AUP/4(R /$*(/4 5<J34N<*$J< (U0%3D/4'/ 6WNLN< "$O/4< G"C<>(OHM+/F$/+*.Q+*/4((#W/40:/G73>< $1S/ 6B/ .< $ # < 0)I98*/4"#/4+S/ 6B7/ .< $ # A%1<>/4"$< (N/)<>6u/4(*J< / 6w$1"g<>/4" 0*10qqD*/4qTA TA+.<>ZF$/4:4:/G*$=< Z0// Z"OHXAW[K34d0VRF$/D*.Q+Z/ 7< $$@.<> < (S+(*/4*0 A+ < $$/SK6u/4(< C< < $%,+*9HT#(/4/>6< $$(: ,:4&3 < $1TS<>5"C< Q /4#T"I<>/ ;#(/D*5J< (A

G(1$%.< +7:4(/410d@.<>M(: ,d< 0\.M&3&(M%IG#< (< "(αopt*/4((#W/ 0:/'%S/S)6u/ (S/4$*(/4))P:A DA C< 5<M6u1*/4)/ 6B

0@.< *RAP/4("C< $$

RC: (/ 10q@.<>C(: ,EV$:4&$%,\0*(J< :AB

C(:4/43a0 ≤ R ≤ 4a0

C/4#T"C< $^&34(I*.< :4M%V3 < $1C+1%(< #0$%,4A>@#'*/4++1/41A>?*J< G@.<>%KF*J< 1 Q+/-S< < $%,+*9HT#(/4\6u/4(SR:4(/410\@.<>R(: ,4AX!5(:4/4

EGS

@.< (@N0*(J< :"/4('(< #0$%,4AwC(J< /4?6u/4(7FW<J3D/4(V*$J< ($%,d?S?w$/D/4Qd<>P:A DA DA&C%J4(5U5*/4"#$S/4S$*(/4C#W*9.(1" 6u/4(^/4#T"%I0

αopt6u/4(NJ< *.\3 < $1R/ 6R < 0!6u/4(NrTHT0d3 < $1

α = 1/3< 0

α = 4/3AXMrTHT0

3>< $1^*/4((#/40/< '$*(/4'/4 0'/4(F%.< $DS%/41TU*(@,D@.< $r$0 S'*&(< $ *.< (:4R+1< $w(#*%3&$%,E/

+1< 0

+4A?$"'OR/ 6"C< $$

RG/4*M*.< (:4*/4"GFRS%\<>/ 6G+1*$1V< 0X*%34$%,\*/4"#WD8<>'%Z*9<>(: '/;'(1$%.< &*9< (:4G

Z − 6Zext = +1A '/4(F%.< $Z< (

$=< (:4(XFW*J< 1αZ"C< $$( < 0E< $$B'*.< (:4V< ('0R'/4(F%.< $A Z

N*/4&(< (@,4D6u/4($=< (:4RN 1*Z/ 6B< 00%/4< $*.< (:4w$w"#/4(@.< +

< 0 7<J347/4$%,'.Z*&(< $ *.< (:4Z = +4

AD?ZZ<>wZ*.< :47αopt

;)'φi = φ3dm

G# "$ 2#%!"$C!m(i) = m

P

!

< F$DA 4A(1$%6u/4( m>hX.j v m>h a3d1

*/4Tr:41(<>/4oX7*(@,D@.< $&r$0V/ 6/D*.< 0(< $@,D""(@,4A4/4#T"%I0G:4(/410G@9<.w(: ,

EG< 0G&34(

IV#< (< "(αopt

S0#$=<J,40 AV(J< $F< w<346u1*/4N< (10d< 0"C< Q457LN< "$O/4< C"C<>(OH0=< :4/4< $lA+79HT*%0;@.<>w*J< ;FWZ/4#T"%I00#W0&$%,#(/3D0:%7/SE+34(I#< (< "(

α+A'3 < $1

∆EG/ R*(@,D@.< $r$0!#$%:GS%)50#W0&Z/4#T"IJ<>%/4?au/Sα+6u/4( 9HT*%0;@.<>.o9T< 0<V:4$

αoptau:4$/4F< $

αopt0("0C6u/4(

EG/4$%,oU6u/4(

< $$ @.<>AEG

a,oαopta0 α+a0 ∆E

au" ,o∆[ au" ,o

/Sα

:4$/4F< $α 8TA TA 4 TA 4 $"$ TA -$ TA "$ 8TA TA "!! TA TA "$ -! TA $!

8TA - TA !" TA A -! "$DA "$ 8TA 4A 4-$! 4A -$ 4A -$ TA 8TA !"! 4A 4-$ 4A !"! TA "! -!TA -! 8TA $ 4A - 4A " TA ! 4A &$ 8TA ! 4A - 4A "$! !TA $ !TA ! 8 4A !" 4A 4-$"$ 4A 4 DA - DA ! 8 4A 4A 44 4A 4 4 DA " DA 8 4A $ " 4A 4 - 4A 4 4 4A !" 4A 8 4A "$ 4A 4 4A 4"$ 4A 4"! 4A 44 8 4A "$ 4A 44"!4 4A 4!4 TA -$ TA " 8 4A 4A 44-$ 4A 44 TA " %$ TA " %$ 8 4A -!44 4A 44 4A 44 TA "" TA "! 8 4A -$"$-$ 4A 44 4A 44 TA -$ TA

. !

0 2 4 6 8 10-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

αoptE

G

optimal inverse size,

opt a0

wnw

ergy

, EG (

Ry)

distance, R/a0

0.4

0.6

0.8

1.0

1.2

1.4

P:41( DA AV/4#T"I0;:4(/410)@9<.(: ,EG

< 0V/4#T"C< $W3>< $1/ 67&34(I

αopt6u/4(w/47$*(/4C7*J< / 6B*(@,D@.< $Xr$0/ 6B/+*.< 0(< $

@,D""(@,4A

*/4((#W/ 0N/V*(/4/34(ZFp \p /C$"'O%:'*J< A ` *J< 1M/ 6*(J< 0?X*%34C*.< (:4 /4(F%.< $< ("M1*9E"/4('*/4"#< *M<>(/ 10E+1*$1A

(E< (\/4$%,-p /@.<>3DF$!t\#W*(1" P:A DA DA FW*J< 1;;#W*(1" R0:4(<> AU;:4(/410@.<>;R(#$C0: (<>S(J< wZ9HT*%0;/4Z 0/41F$N0:4(<> A N6B*/41(Z%6 w #CZ#/ 6M$*(/4\M#*1(M0/+7/ Z*9< :4 AD/;!N*J< M M<34M/0/41F$Z"M1$%#$%%*%,4A+w(1$%*/4((#W/40w/MZ11< $ 0%3D/4C&/

t2g< 0

eg@.<> %$nA` *J< 1C6u/4(V/498$*(/4?*J< C%< $K@.<>M< (C0*/ 1#$0?6u(/4"

FW:4:W G*J< /4#T"%IVGIG/ 6/4(F%.< $6u/4(ZJ< *.@.<>G0#W0&$%,4AC*.< :4C/ 6

α6u/4(C9HT*%0q@.<>C0/DR/ M<>X*R:4(/410q@.<>CD8

(: ,EG

A'/4#T"C< $&34(I'< V<6u1*/4/ 6R6u/4(Z'9HT*%0E@.<>

α+V*/4"#< (0ES%E<>V6u/4(C:4(/410?@9<.>BqP:A DA TAC$"OV< (

0&*J< $nXnA A1/3

< 04/3

AFW<J3D/4(d*(/4/3&(V(:4"G$:4&$%,7 "#>)O != ,D= B$C!= )* )24 3 # @# *# '1$C!# #(9&(= !#3.0!1$C!#. +-= # A?3M!"# .$2,D# 2$2A #J)24

α+

46)'N$2A?A)24!"# , P

!

0 2 4 6 8 10

-1.0

-0.5

0.0

0.5

1.0

α=1/3

α=4/3

ener

gy (

Ry)

distance, R/a0

P:41(DA +A'(: ,d#W*(1" S%/4#T"%IJ<>%/4 X< 0S%drTHT0αA

3 < $1α = 1/3

< 0α = 4/3

(#(+NR$"'OS%!+1*$J< (5*.< (:4Z = 1< 0

Z = 4(#*%3&$%,4A&Y^*/4((#W/40'/7S*(@,D@.< $r$0'*9< (:4^F:%(

*$100< )< X*%3&*.< (:4E/ 6R+1*$1)/4(<>)Tr%\0@.< *d6u(/ " %JA):4(/4102@.<>G(6u/4$0-0:4(<>S(J< )9HT*%02Mp /6u/4$00:4(<> A*$10:G7 /#W/4F$7#;3 < $10:4(< *,)0/41F$A

0%X(+JB< "$%,4 0/d/ M<34;<\"/+/ q0#W0*4KF1T'<\0*/4&+1D8%p,\<>

R ' 3.65a0AX?Gr0d+17G9H< *VFW/4(0(7Fp EMw/$"'O%:

(:4"AV+1"(*J< $X3>< $1S/ 6αopt

< 0α+

< ($@0;< F$ DA 4A_N/ (N&(@.%:#/4+N5V3 < $1R/>6UM*(@,D@.< $Br$0\#$%%:AY7

/SqP:A DA $D/4:4(RS%EC3>< $1R6u/4(rTHT0α = 1/3

/4(4/3

A ZR<>M?:4(.<>$3>< $1V6u/4(rTHT0

α< ('3 < $0EEC*/4((#W/ 0D8

:)(:4%/4 A"C<HT"C< $#$%:;/4F(@340qd*(/4/3&(VFWw?(:4"A 1@<>6k(V*(/4/3&(M <>M#$%%:)6u/4(R/4#T"C< $

αopt3 < $1S$:4+$O,$=< (:4(w< )<>S6u/4(α = 4/3

A5SFW*J< 1VV/4#T"%I0:4(/410E@.<>'(: ,< $%<J,D$/ (7< ':4(/410E@.<>'(: ,d6u/4(ZrTHT0αW< 0)R*(/4/34(Z/ 6R:4(/410).<>V(: ,)5"/+/ (S< )V*/4((98

#W/40:VFW<J3D/4(U6u/4(U59HT*%0C@.<> ADs51"(*J< $3>< $1/ 6X5#$%:< ($@09HT#$*%$%,< F$ DA A

. !4

0 2 4 6 8 10

0.4

0.6

0.8

1.0

1.2

1.4

ground state

excited state

optim

al in

vers

e si

ze

opta 0

distance, R/a0

P:41( DA TA45/4#T"C< $D3>< $1^/ 6WS&3&(SIS#< (< "(α6u/4(KS:4(/410

< 0Z9HT*%0@.<>w3DA%+(/4%*N0@.< *RAT79HT*%0;@.<>7#$/ w< <

0*/4&+1%p,4DS(J< S:4(/410)@.<>70#W0*JV"/D/ A

G(1$%Z6u/4(Nw/;$*(/4a3d2

*/4Tr:41(<>/4oN< (M/SP:ADA TA(< (

45@.<>V0#$=<J,&0?< R<;6u1*/4E/ 6w0@.< *

RA:4(< $

< #WV/ 6BV:4(/410)@.<>VS"%$=<>( ./GV/4$*(/4!*J< AW` *J< 1V/ 6$*(/4D8$*(/42+(< */4 K\/4(F%.< $< ( z2C = v9ACCF*J< 1\$*(/4U(@,R/V(:&< MS(: ,'F&,G*(J< :Z5<>(< */4M/D1*$1A5*J< !FWV\!+1"(*J< $ 3 < $15#(&0d)< F$ DA DA` *J< 1M/ 6"C< $$%(wI7/ 6/ (FO9<>$D7*(/4/3&(.< Q 5< $/G#$=< *Z6u/4(w"C< $$(

RA

6uJ<>1(7w#(*V/ 67Q+Q6u/4( 79HT*%0@.<>S%7*(/4/34(S(98:4/4 A]N#%R%*(J< :0@9<>*R/ 6R< 00%/4< $ :&<>%34V*.< (:4(: ,< $/'(A

/E$*(/ -@.<>'*J< -F;*$=< %r0-q("G/>65/ .< $ #-< 0-%I98*/4"#/4+JA,d< (R: /D/D0!+1< &1" +1"GF(N6u/4(SN@,D@")W%6KR#D8/4(F%G+(< */4G:4/4(0 U< '%'M*J< )/6l< (AUL5:4-#-.<>GS%S = 1

< (7$/ (A:4(/410@.<>76l< $$ +/M*$< JA,<J347(#W/4p8F$w3>< $1^/ 6

Sz = 0,±1A+P/4(UJ< *.C/>6X" (5< ((S0:4(<>5$/ @

@.<>AXG*/4"#$R#*(1" 6u/4(NJ< *9Sz

N0+*J< $nAWD/M:4(/410\@.<>

!

0 2 4 6 8 10 12 14

0.00

0.05

0.10

0.15

0.20

0.25

α=4/3

α=1/3

crys

tal f

ield

spl

ittin

g (R

y)

distance, R/a0

P:41(DA $DA *(@,D@.< $Tr$0G#$%%:56u/4(K/4w$*(/4GG< G/D*.< 0(< $T*(@,D@.< $r$03DA0@.< *

RAS*/4((#W/40S/'(: ,#W*(1" /!!P%:ADA DADP/4(U5< Q4N/ 6 */4"#$+ N*$10N*/4((#/40:M#$O%:4U6u/4(

α3 < $1S#W*Or0 A

6u/4$00:4(<> A_N/ (53 < $1R10\/*$=< %r*J<>/4!/ 6UR@.<>Z5G< :41$=< (5"/4"D8

1")A#5T6u/4(@1<>$%,Z%/ <S:4/+/D0+1< +1" +1"MFW(6u/4(*(@,D@.< $&r$0V@,D@"A'@.<>7/4FT.< 0d\'< FW/34'#(/+*01(< (G/ 7:4@.<>7/ 6

L2 /4#(p8<>/4(AXG/4$%,):C<>7*J< FWM0/4GNG*J< $*1$=<>%/4\/ 6UG9HT#W*.<>/43 < $1/ 6 < :41$=< (7"/4"&1" /4#W(<>/4(A]N#W0:)/4:4(/410E@.<>$*0 *G,E< (G6u/4$00:4(.<.>w':43>< $1d'd'D8(@3 < $6u(/4" $/ !TWS<>*/4((#W/407/< :41$=< (N"/4"&1" FW < 0TA9H< *5%I/ 6B+(@3>< $ 0#W0N/4

RA

54 bd6( *d " 6( " " (D/\6l< (R/41(M*J< $*1$=<>%/4?*$100< $$/4D8($=<>%3D%@*&(< */4R?;P/D*.Q#< *w#< 0MF+,*$=< / 6 0Vw<J3&6u1*/4A_NM9HD/4M/ 6B"/D0$+S*$1/4G/ 6X/4"($=<>%3D@%*wX*A+S*/4+(F1T/4M < &^/*$10

. !

0 2 4 6 8 10

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Ene

rgy

Spe

ctru

m (

Ry)

distance R /a0

P:41( DA TA[U(: ,;#W*(1" 6u/4(p /'$*(/4w7/D*.< 0(< $X*(@,D.< $Xr$0S%E/4#T"I0E+34(CIC#< (< "'(

αAC(1$%R< (0&*J< $KS(

G1'(J< $K/4(Z*/4"#$9Hd 0d@.<>A_N$$$*(/4D8$*(/4d&(< */47< 0\#D8/4(F%K+(< *9.%/4M (5*$%100!auS0*1/4'/ 6WS$< @^&(< */4FW$/No9A#D8/4(F%*/4((*/4/S?EP:A DA 0/DV/ V+1< $%.<>O34$%,*.< :47#*1( FW*J< 1V/ 6%"C< :4%10 A

SR#D8/4(F%5*/41#$:A_N$$R*J< $*1$=<>/ 51#!/5#W/4&N@.< (@0!S%7D*9(+0:4( +1<>/4 AT` *J< 1Z(N /M"C< &,&8#< (@*$5]N(< *N+1<>/4 <34;/*/4++1;<J,-< $(J< 0T,q*9/4 A^w)#D8/4(F%G&(< */4/GFW7*$%100)< <M*/4((*/4;/'LN< "%$%/4< da DA o9AT*/4((*/4;&34/4$%3&/4$%,2:4$\$*(/4 (")wF*J< 1?%;/4(:4<>C6u(/4" <-:4$.8#< (@.%*$%6u/4("C< $%"ATYSS*J< $*1$=<>0_N##W0H).AYS/ 6B76u/4("

H ′(r) = α21

r

∂V sph

∂rL · S+ α2

∫

d3r′2ρ(r′)S · ((r− r′)× p)

|r− r′|3,

a DA 4oS(

α = e2

~c ' 1137

7rC@(1*1(C*/4@.< + AV sph C#(*J< $$%,

@,D""(*#< (@/ 6N)$*(/4@.<>*#W/ &=< $nSG)#2/4#W(<>/4(

L

. *$C!),D=?&*= !1. !"# = * 2# '1.# )24 !"# *# .0!'&;!'# &()*.0!1$2*!H=?.8!1" # .O# #3>)24 A = @"!cP

!

'< :41$=< (5"/4"&1" /4#(<>/4(7< 0p5G"/4"&1" /4#W(<>/4(A

ρ(r′)%

(01< $^*9< (:40@(F1T/4/ 6 C#< (@V/ R*$100?#W/ &=< $V sph A

_NU*J< 'FW'6u(/4" [UAWa DA 4o9>#D8/4(F%K&(< */4R(" #< (<>^&/p /!#< (@ C#(*J< $#< (@VS*.q:4%34VC< "'/)C&(< */4 < 0<M/4D8#(*J< $X*/4+(F1T/4 &S*9;"C<J,CFWZ/ 6B*/4"#< (< F$N"C< :4%105/Mr(@S(")< 0+15*J< / SFW:4/4(0 A

/ 98$*(/4E"*(/4*/4#*G#< (< "(tij0r0qF&,[UA a DA 4o9 *J< $O8

*1$=<>0CD*/4 DA 44*J< FWS10< $/7( A&?S<J34S/V< 00'/J< *./ 6W"*/4((#W/40:*/4((*/ S0r0!F&,

H ′sij =< i|H ′s|j > H ′ns

ij =< i|H ′ns|j >, a DA &oS(C1#(*(#T

s<>0

ns0/ (#W*%34$%,#(*J< $ < 0E

/4D8#(*J< $X#< (@A` *J< 1/ 6( 0%X(+w<>1( T,C<34N/GFWZ*J< $*1D8$=<>0#< (<>$%,4A

_N/ (5"#W/4(@.< &5#W/4&S/C/ RS<>5R*/4((*/450// N*/4(@34d# AwL5*\dLN< "$%/ =<>2/ ;#D8(/ .<>/ t&3>< (=< +< &,t"/4( AP(/4" ;#W/4&; E1d zWikh5m v ~ikjY z\ikhTzj tm10 m v ~ikjY z AwD8*(J< G3>< (=<>/4\(.<>:4'/ 60*

i< 0

j< 76u(/4" / /4,d*$10

< $/!# |i >≡ |iσ >AC"#$%6=,D:)< 1"#T/46u(/4"D*/4 DA CV/

$/4:4(1 A#(*J< $$%,E@,D""(*C#< (@MM($=<>%34$%,J< @,?/\*J< $*1$=<> A?"C<,

S(%77%=< $W+: (.<>$^a DA &ow< H ′ssσtτ =

∫

d3r(φsξσ)?H ′s(φtξτ ),

a DA $4oS(#;6u1*/45< (

ξσ =

(10

): σ =↑

(01

): σ =↓ .

a DA &o51S0r(< 0=< $X#< (@5/>6#<>(.<>"(5<

ζ(r) = α21

r

∂V sph

∂r.

a DA !&o)*/ "#W/4&'/ 65#-/4#W(<>/4('< (;9HT#(J02F&,q < 1$w"C<>(*Sη = τ η/2

ASE*/4"#/4+/ 6R?< : 1$=< (;"/4"+1" /4#W(<>/4(;*J< FW9HT#(0\<'"#$N6u/4(" #(*J< $*/D/4(0<>

Lx = ı

(

sinφ∂

∂θ+cosφ

tan θ

∂

∂φ

) a DA &o

Ly = ı

(

− cosφ ∂∂θ+sinφ

tan θ

∂

∂φ

) a DA oLz = −ı

∂

∂φ.

a DA 4o

. !-$

#5:GV< FW/340r%/4w73 < $17/ 6H ′s *J< )FW9HT#(0d<

H ′ssσtτ = Ξ(αs, αt)

∑

η=x,y,z

SηστLηst.

a DA &o#)#< (@S@1@N<M(#$Z#(/D01*5/ 6Bw/'#/4(5< 0!<G"C<>(OH

Sηστ = ξ?σSηξτ .

a DA +o< :41$=< (B#< (@UBw9HT#W*.<>/4M3 < $1 / 6< :41$=< ("/ "+1" 6u/ (K:4%34/4(F%.< $

Lηst =

∫

sin θdθdφY ?2s(θ, φ)LηY2t(θ, φ),

a DA 4o< 0;.+1(< 0=< $X#< (@5

Ξ(αs, αt) =

∫

r2drζ(r)R3d(r;αs)R3d(r;αt) = −32

15α2Z

α7/2s α

7/2t

(αs + αt)4.

a DA &o;$=< @G(1$%Gw< '/4FT.< 010(M< 1"#T/4q<>G) /41$/4"MFq#W/>8&=< $

V sph(r)/4(:4%<>6u(/4" *9< (:4

+ZA _N$$5d< FW/34d9HT#(/4*J<

FW*/4"#1T0-<\@(< :4&6u/4(@< (0q"C< (JA ;*J< $*1$=<>/ q/ 6;/4D8#(*J< $5#< (@C"/4()*1"MFW(/4" A2/41(C"/+0$50#*0ttP:TADA E< $$*.< (:4Z9HT*#T7<>Z/ 6UM*&(< $D1*$17< (R*$100\!R*9< (:4R0@.(O8F1T/4

ρ(r′)A` *J< 1wV< (0J< $:MS%;V#/4+S*.< (:4V0@(F1T/

S<G1" / 6K]N(< *70$%.<M6u1*/4nA A

ρ(r′) =∑

i=±x,±y,±zZextδ(r′ − ri).

a DA "$4oM*J< $*1$=<>/4)*J< dFWM#< (<>0d+/p /;#< (@Nd<C"C< (Z"$<>(S/C#(*J< $*J<

H ′nssσtτ = α2

∑

η=x,y,z

Sηστ∑

i=±x,±y,±z(Ii)η,

a DA &oS(

i0*J<> U*.< (:4^Z^/+*.< 0(<S1((/410:ww^< (^0< $:wS%

< 0ηN*/4"#W/4&w/ 6 34*/4( 3>< (=< F$A+N34*/4(

Ii*J< FZ3>< $1<>0C6u/4(

i = +zWS%*9\"J< 5M*.< (:4R$%,D:C/4!

z<HTN!R0@.< *

R6u(/4"

/4(:4%;;#W/4%%3470(*/4

I+z =

∫

d3rφ?s(r)2Z

|r− r+z|3((r− r+z)× p)φt(r),

a DA !&oS(

r+z = (0, 0, R)AV"/4"&1" /4#W(<>/4(S;V#(*J< $*/+/4(0<>

< 76u/4("ga~ = 1

o

p = −ı

sin θ cosφ ∂∂r+ cos θ cosφ

r∂∂θ− sinφ

r sin θ∂∂φ

sin θ sinφ ∂∂r+ cos θ sinφ

r∂∂θ− cosφ

r sin θ∂∂φ

cos θ ∂∂r− sin θ

r∂∂θ

.a DA &o

!"

!"/4@0*1$%#W/4&/ 6NS/4$)*J< $*1$=<>%/4-G)9HT#< /46u/4('+34(;/ 6 |r − r+z|3

AB?;1#(*J< $ */D/4(0<>RS(r = (r, θ, φ)

AE&34(980@.< *E9HT#< /4)< (0*(F0 _N##W0H NA11< $9HT#< /4C0r0;F&,[UA anNA &oU0/D/ /4(QD*S N<J345S(0#W/ (( A& Z^*/41$07(@,N/S1^9HT#(/4anNA +oXF1TJ1T6u/4(@1<>$%,4J% ,D$001(:M*J< $*1$=<>%/4!Tr%V(WS*.d< (M0*1$%5/0J< $S% AWdX*JX/4< /'19HT#(/4

1

|r− r+z|3=

∞∑

l=0

R−3,l(r, R)Pl(cos θ),a DA o

S*9)SV#*=< $*J< R/ 6U[UAKanNA 4o9AR−3,l

5<'(<>/4< $X6u1*/4!/>6U%< (:41"+A

PlSM:40(R#/4$%,D/4"=< $lA!*/4*/4)S%)M:40(

#W/4$%,D/4"=< $%K*$100GSw<J34S6u1*/4&S&:4(<>/4G/34(θ*J< *$ < $$

(" S%l > 5

A` *J< 17/ 6 *J< *$$=<>/4 & 7*J< ;*/4"#1TNZ3>< $1w/ 6I+z

AB/:453 < $1S6u/4(M< 00%/4< $X*9< (:45$/D*J<>0))#$=< *5/ (S<

+z wS<J3&/(/ .<>wS#(*J< $< ("/4*U'<"C< (U"$< (B/7S*J< $O8

*1$=<>/4C/ 6 Z*(@,D@.< $Wr$0 ADZ(/ .<>/4/ 6#(*J< $X< ("/4*0*(FW0_N##0OH;LVAs534(@$/4Z< /MFW53&(@,C*J< (6u1$S$N9HT*1T:R.%#(/D*01( < %C< $/E"OHT'!< :41$=< (G"/4"+1" */4"#/4+AP/4('J< *.(/ .<>/4 G%< $< :41$=< (N"/4"&1" */4"#W/4&Z<34R/;FG@w< ##W01*9<J,<>5<>6k((/ .<>/4C,)< (7;*/4((*5/4(0(A

P/ (^FW/ #(*J< $W< 0/4D8#(*J< $#< (@/ 6X5#D8/ (FO^&(< */4G /4FT.< q9H< *9< < $%,+*'6u/4("M1$=< AqC$"O

R → 0/4D8#(*J< $#< (@

:4/DR/34(R+/C#(*J< $/4 AB_NR%R%R#(/4#/4(@/4< $K/).*9< (:4Z

/4FT.< <;*/4&(F1T/4\/ 6G/4(0(7/ 6(∑Zext)/Z

XS*.E%*J< G/ 6 /4'< 0p /R$*(/4^(#W*%34$%,

75%< 0

80%#W(*&JAD

R0#W0*Z/ 6

/4D8#(*J< $X#< (@w< ;9HT#W/4&=< $X0*J<J,"G1$%%#$%%0F&,;<R(<>/4< $6u1*/4 AP/4(

RR&(@3 < $TFW

3a0< 0

4a0"C< :4%10 / 6/4D8#(*J< $

#< (@ZN< ##(/JHT"C<>$%,;R< "6u/4(N&34(MIR+1< $B/α = 1/3

< 05%

/ 6%=<>$W3 < $1Z6u/4(

α = 4/3A

76u/4(" / 6BLN< "$%/4=<>;"C<>(* 6u/4(w*J< $*1$=<>/4"M1*9"/4(*/4"#$*J<>0 AWP/4(Z/4M$*(/4\*J< %ZN@.%$$B#W/4F$V/;/4FT.< \< < $%,+*R9H+8#(/4 6u/4(N:4(:4AP/4( /G$*9.(/4D% / w#/4F$Z< &,C"/4(< 0)/4< /'(/4(@S/<G+1"(*J< $ 0=< :4/4< $%IJ<>%/4 A

):4(< $6u/4(" / 6N)(1$%<>6k('*$10:d)#/4(F%&(< *98/4'0/D/ *.< :45:4%r*J< &$%,4A450%X(*5Fp CSp /V*J< 6u/4(<\:4$$*(/4%M/SP:A DA !TA#D8/4(F%R&(< */4?%6kR:4(/410d@.<>GF&,d<C3 < $1M/ 6^/4(0(7/ 6

10−4/ 6^G%< $3 < $1 AXG*.< :4G

/4#T"C< $αopt

< $/V/ :4%r*J< &JA+wZ"C< '0%X(*NUN$%6k%:/ 6

. !"!

0 2 4 6 8 10-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

∆EG

∆αopt

opt a0 (10

-3)

EG (

mR

y)

distance R/a0

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

P:41( DA !TA < :4Z%CN:4(/410D8@9<.N(: ,CZ/+*.< 0(< $X*(@,D@.< $Wr$06u/4(<:4$$*(/4

3d1*/4Tr:41(<>/410(^S 1*N/ 6XS#D8/4(F%

*/41#$:A^d0%X(*\t\/4#T"%I0t:4(/410D8@.<>!(: , < 0 2/4#T"%IJ<>%/4;#< (< "(

α/S A

:4(/410@.<>0:4(< *,4A)%%=< $ 8Y6u/4$%0-0:4(<>)@.<>#$%G1#+/ /$/w(p8$=<J,D%:G@.<>5< 06u/41(:4(p8$=<J,D:M@.<>A

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RA Z5<>*/4&(F1T/4$=< (:4(

6u/4( $< (:4(RSZ$*(/4< (N$/D*<>0*$/4(/RZ*&(AZ+1"(O8

*J< $X3 < $1/ 6B7*J< $*1$=<>/ < (7/S);< F$ DA DA&(/D01*/4;/ 6#D8/4(F%7&(< */4$%6kZ'0:4(< *,?< $/%7*J< A7FW*J< 1!M*(@,D@.< $O8Yr$0\#(*MG< :41$=< (N"/4"&1" N/ <C:4/D/+0\+1< +1"+1"MFW(AVFW<J3D/4(S/ 6V#D8/ (FOS#$%:M6u/4(w />8$*(/4)*J< V:4(< $$%,M"$< (/5/498$*(/4*J< @1@ < < $/4:4*J< $%$O,4&< ^%^w< ^S%/41T#D8/4(F%&(< */4Eau*6pAWD*/4 DA o9A

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3dn*/4Tr:41(<>/4\S%

n > 2 F1TG*/4((#W/40D8

< F$ DA DAB(1$%V6u/4( j ' m.j v m>hTz ?C*(@,D@.< $r$0 AB/4#T"%I0:4(/410 @.<>d(: ,

EG

<>0 \+34(dI\#< (< "(αopt

0#$=<J,40 A[U$*(/4\8S$*(/4E< 0d'#D8/4(F%Z&(< */47< (G*$100 AXG3 < $1∆EG

< 0∆αopt

/ ;0%X(*;FWw-;0%#$<,40q3 < $1G< 0*J< S%/41T#D8/4(F%&(< */4 AR EG

a,o∆EG

au" ,oαopta0 ∆αopta0

a10−3

o 8TA "$ 8TA 4"!4 TA TA ! 8TA 8TA "! & TA 4 TA $ 8 4A "!4-$ 8TA -$! 4A &! TA 8 4A -$$ 8TA $"!"! 4A 4 TA "!-$! 8 DA 8TA !"$! 4A TA ! & 8 DA 8TA " 4A " %$"$ TA $ 8pTA "! ! 8TA 4 4A %$ TA -!-$ 8pTA 8TA 4"!"! 4A --$ TA !! 8pTA %$ 8TA -! 4A -! ! TA 4 8pTA 4 8TA "$ 4A -! TA -!

:'LN< "$%/ =<>"C<>(*P/D*.QC#< *7S$$XFW7/ 6:4(w0"/4 AP/4(9H< "#$ +6u/4(

N = 3$*(/4 N$$<J3& (10

3

)= 120

(N$*(/4@.<>w/41(P/+*9QZ1F#< * A _N$/J(1$%B*J< RFW < ##$0/Sp /5*/4*( %1<>/4AP(@JJw*J< M*/4"#< (U #(&K(1$%BS%V"J< 1(0M*(@,D@.< $+r$0V#$%:6u/4(7)/ R1$=<>/4(7*/4&.< :

3dn*/4Tr:41(<>/4A` 1TV/)#W(@6u/4(" <>

6u1$$%,4wR/41$0!(#WJ<>Z/41(5*J< $*1$=<>/ N< $/'6u/4(5$:4&$%,;0@/4(@0!/D*.< 0(<FW*J< 17/>w/4$%,'N+1"GF(w"C<>(+F1T< $/V5(0w/ 6N0<>.<T+S*."C< Q4MMS/4$G#*1(G*/4(&JAZ@$$FW6u/4(G1ABD*/40 '/41$09HD0'/41(K"/+0R/<5(1$%,V< /4*/4#* *$1@(>6u/4(9H< "#$/Nw< /S(Na4

/4(Na6

/4(5/G*$1@(Z A :AV4O9

XFW/ d"J< 1(0d9HT#W("&.< $$%,!< 0*/4+.< :M*/4(($=<>0$*(/4*7@.<>A

.

0 2 4 6 8 10-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

GS

EG (

mR

y)

distance R/a0

P:41( DA TA45#D8/4(F%U&(< */4'*/4((*/4!au0%X(*SFWw5D8(: ,!/ 6 <@.<>RS%< 0\S%/41TZ B)("Z\GLN< "$O/4< ow6u/4(Nw/$*(/4Mq/D*.< 0(< $ *(@,D@.< $ r$0q6u/4(M< $$

45@.<>AK;$M< (#$/ 0

< <C6u1*/4d/ 6^G0@.< *R/ 6^G*(@,D@.< $O8Yr$0d*.< (:476u(/4" G*&(A

Z#J< Q+< (Z<VD1"(*J< $W<>(%6l<>* N*(/4/3&((:4" 4SN#< (< "G8(

αR*.< :4:\(< #0$%,4AC*.Q4(RF$=< *.Q$/SR*/4((#W/ 0:

0#0*V6u/4(w:4(/410@.<> A

vZ[jM PdO@I

YZT T [jI2a [N\Z] X_KW\ZXAVYZR2Q;KW\ZR

!NZ M<J3&M&(/D01*0!R6u/4("C< $B< #*au*6pA < #T( 4oS/ 6U/4(:4%< $"/D0\al[^]_Z` @o^/ 6 *J< $*1$=<>%:75"C< &,&8$*(/4@.<>"*(/4*/4#D8*J< $$O,44< Uw$$<34S@0C%U/4'S"#$9H< "#$K6u(/4" <>/4"*au*6pAD < #T(&o9"'/4$*1$=< (au*6pAW < #T( +o9< 0)*$1@(Rau*6pA 4o +1< +1" #&,D*AB/4:4(S%t_N##W0*qau_S8@o9 d#(+)*J< FW< <F< /*/4*(E< ##$*J<>%/4t < /4#&,D**J< $*1$=<>%/4^/41:4 %;@/4#;/4(@/ 6#(/3D0:d")A /4*(\< ##$*J<>%/4MS$$w(+1()< 00%/4< $ " U< G6u/4("M1$=<>%/4/ 6 C"/+0E/V@.< :4(+1(0< E9HD%3&C(3 < $1<>/4/ 6N#< @<>/4"%*< 0"/4$*1$=< (G< ##(/&< *.'/E+1< +1" #&,D*A^(6u/4("M1$=<>%/4!F/ \w<J3&985< 0\#< (@*$au"C<>(OHoY8< #*7/ 6^+1< +1" /4(@,< (7+(@S0 A

1d1""C< (I?F(, #(*#< $V/4(:4< $(1$%d/ 6'A2 < #T( !w9HD0-+1%3 < $*)Fp

1st< 0

2nd+1< &IJ<>/4

*."M/!C6u/4("C< $9HT#(/4M6u/4(V*/4((#W/40:!<J346u1*/4RL5$FW(@'< 0P/D*.Qq#< *A Z/41$0-10($<>'%M6u/4$$/SR<>G/0r;+1< &1" @.<> |Ψ >

auq)]N(< *oGq)P/D*.Q?#< *;w0/;Q+/ G"C< &,&8#< (@*$Vw<34G6u1*/4

Ψ(r1, . . . , rN)\'L5$FW(@

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H2

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n = 3/4(F%.< $ /41$0GFw(+1(0M/7(J< *.

4

(1$%,+1< +%.<>%347(1$%S*/ "#< (< F$ZS%;FWS3 <>(<>/4< $X@"C<>At < #T( )@10T,

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0.2%A4D1*.G0#$=< *"&/ 6 *+(K/ 6/4(F%.< $S%R(#W*K/5/4

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3dn@.<>

(n = 1, 2)4M5 /41$/4"MFMr$0G/ 6X< /4^< (p8

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1√1− 2xt+ t2

=∞∑

l=0

Pl(x)tn, |t| < 1. au_RA o

,R< ##WJ< (B &34( 0@.< *9HT#< /4 >nA A"G1$%%#W/4$%U9HT#< /4< 0d\"C< &,)/ (Z#&,D*J< $K@,D@" A :AWM9HT#(/4Z6u/4(N+,D0(/4:4D85<>0&,D0(/4:4*98$Q4Nw<J3&76u1*/4AV< /D*=<>0 :40(76u1*/42

Pml

< (*/4*0)/G :40(V#/4$%,D/4"=< $%wF&,

Pml (µ) = (−1)m(1− µ2)m/2

dm

dµmPl(µ), m > 0,

au_RA 4o< 0;6u/4(:&<>%3&

mF&,

P−ml (µ) = (−1)m (l −m)!

(l +m)!Pml (µ).

au_RA &o

/4(@/4:4/4< $O,C< 0;7/4("C< $IJ<.%/ *(%(=<TDS0$%,C10)< 5<M/D/ $/ 6(01*:GTr%7(

∫ 1

−1Pml (µ)P

mk (µ)dµ =

2

2l + 1

(l +m)!

(l −m)!δlk

au_RA +o

M*/4"#$9H)#(*J< $B< ("/ *Ylm(θ, φ)

*J< \FR9HT#(J0%)("N/ 6V< /D*=<>0 :40(6u1*/4

Pml (x)

< w6u/4$$/S

Ylm(θ, φ) = βlmPml (cos θ) exp(ımφ),

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.

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<V/4("C< $%IJ<>%/4'*/4@.< +JA+CN*J< 5w5w< & " /V6u1$%r$/4("C< $%IJ<>%/4;*(%(/4

∫

σ

Ylm(r)Y?l′m′(r)dr = δll′δmm′ ,

au_RA &ow<J34/'S" <

βlm =

√

(2l + 1)(l −m)!

4π(l +m)!.

P(/4" V0r%/ S%SSJ< $%,C/0(%3& <>S6u/4$$%/S:G@,D""(@,($=<8/4/4$0

Y ?lm = (−1)mYl−m,

Y unormlm

? = (−1)m (l +m)!

(l −m)!Y unorml−m

Ylm(r) = (−1)lYlm(−r).au_RA $4o

Y unormlm

@.< 0('6u/4(Z'/ V/4("C< $%I0#(*J< $U< ("/4*ZS%βlm =

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MZMO.\Z]ZQ,f

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(/D/>V< 0E&3&(@7'/4FT.< 00=< :4/4< $$"&< 0E0/;'+34(O(< @6u/4("C<>/ ATw<J, V0r

S−1/2 = O(O†SO)−1/2O†.al`NA o

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< 0r′ = (r′, θ′, φ′)

WS%d<(6u(*G#W/4&7<>7<*/4""/4!/4(:4 AG&3&(M0@.< *MFWw

r< 0

r′N:4%34dF&,!R $$O8Q+/S:40(

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|r− r′| = r−1>

∞∑

l=0

(r<r>

)l

Pl(cos^(r, r′)),

anNA oS(

r<< 0

r>< (Z7"%"M1" < 0"C<HT"M1" / 6

r< 0

r′AV< :4$NFW98

p M34*/4(r< 0

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(l − |m|)!(l + |m|)!

(r<r>

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P|m|l (cos θ)P

|m|l (cos θ′)eım(φ−φ

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Cνl

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Γ(α)

anNA &o B/D*.< ""(S@,D"GF/4$nS(

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1− r2<r2>

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α1 . . . α5A

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#< (< "(A_N$$ /4+1%3>< $&5/4I(/Vijkl

$"&5< ($@0;FW$/

V1111 =5α14,

V1112 =16α1 (α1 α2)

32 (33α1

4 − 7α13 α2 − 43α12 α22 − 21α1 α23 − 2α24)(α1 + α2)

3 (3α1 + α2)4 ,

V1212 =α1 α2 (α1

4 + 5α13 α2 + 6α1

2 α22 + 10α1 α2

3 + 2α24)

(α1 + α2)5 ,

V1313 = α35

(

α3−4 − 3α1 + α3

(α1 + α3)5

)

,

V1515 = 16 (α4 α5)52

(

(α4 + α5)−4 − 6α1 + α4 + α5

(2α1 + α4 + α5)5

)

,

V1122 =8α1

3 α23 (5α1

2 − 15α1 α2 + 13α22)(α1 + α2)

7 ,

V1222 =8α2 (α1 α2)

32

(α1 + α2)3 (α1 + 3α2)

6 ×

×(α1

6 + 15α15 α2 + 68α1

4 α22 + 170α1

3 α23 − 51α12 α24 − 729α1 α25 − 690α26

),

V1323 =8 (α1 α2)

32 α3

(α1 + α2 + 2α3)6 ×

(

(α1 − 2α2)(

(α1 + α2)2 + 12 (α1 + α2) α3 + 60α3

2 +160α3

3

α1 + α2

)

+

+ (α1 − α2)

(192α3

4

(α1 + α2)2 +

64α35

(α1 + α2)3

))

,

V1525 =128α4 α5

√α13 α23 α43 α53

(α1 + α2)4 ×

×(

α1 − 2α2(α4 + α5)

4 +10α2 (α1 + α2)

2

(α1 + α2 + α4 + α5)6 −

2 (α1 − 3α2) (α1 + α2)

(α1 + α2 + α4 + α5)5

− α1 − 2α2(α1 + α2 + α4 + α5)

4

)

,

V1133 =56α1

3 α35

3 (α1 + α3)7 ,

V1233 =−256 (α1 α2)

32 α3

5

3 (α1 + α3)2 (α2 + α3)

3 (α1 + α2 + 2α3)6 ×

×(4α2

4 + α13 (α2 − α3) + 35α2

3 α3 + 57α22 α3

2 + 14α2 α33 − 14α34 +

+ 3α12 (α2 − α3) (2α2 + 3α3) + 3α1

(8α2

3 + 18α22 α3 + 5α2 α3

2 − 7α33)),

V1145 =256α1

3 (α4 α5)52 (7α1

2 + α42 + 5α4 α5 + α5

2 + 7α1 (α4 + α5))

3 (α1 + α4)2 (α1 + α5)

2 (2α1 + α4 + α5)5 ,

V1245 =256α4 α5

√α13 α23 α43 α53

3 (α1 + α4)2 (α2 + α5)

3 (α1 + α2 + α4 + α5)6 ×

×(−(α1

3 α2)− 6α12 α22 − 24α1 α23 − 4α24 − 3α12 α2 α4 − 12α1 α22 α4 − 24α23 α4 −

−3α1 α2 α42 − 6α22 α42 − α2 α43 + α1

3 α5 − 42α1 α22 α5 − 11α23 α5 ++3α1

2 α4 α5 − 42α22 α4 α5 + 3α1 α42 α5 + α43 α5 + 6α1

2 α52 − 12α1 α2 α52 −

−9α22 α52 + 12α1 α4 α52 − 12α2 α4 α52 + 6α42 α52 + 6α1 α53−α2 α53 + 6α4 α53 + α5

4),

V1254 =256α4 α5

√α13 α23 α43 α53

3 (α1 + α4)2 (α2 + α5)

3 (α1 + α2 + α4 + α5)6 ×

×(−(α1

3 α2)− 6α12 α22 − 24α1 α23 − 4α24 + α1

3 α4 − 42α1 α22 α4 − 11α23 α4++6α1

2 α42 − 12α1 α2 α42 − 9α22 α42 + 6α1 α43 − α2 α4

3 + α44 − 3α12 α2 α5 −

−12α1 α22 α5 − 24α23 α5 + 3α12 α4 α5 − 42α22 α4 α5 + 12α1 α42 α5 − 12α2 α42 α5 +

8 .

+ 6α43 α5 − 3α1 α2 α52 − 6α22 α52 + 3α1 α4 α52 + 6α42 α52 − α2 α5

3 + α4 α53),

V2222 =77α2128

,

V2323 =α2 α3 (α2

6 + 7α25 α3 + 21α2

4 α32 + 35α2

3 α33 + 11α2

2 α34 + 7α2 α3

5 + α36)

(α2 + α3)7 ,

V2525 = 16 (α4 α5)52 ×

×(

(α4 + α5)−4 − (8α2 + α4 + α5)

(24α2

2 + 4α2 (α4 + α5) + (α4 + α5)2)

(2α2 + α4 + α5)7

)

,

V2233 =4α2

3 α35 (101α2

2 − 70α2 α3 + 14α32)3 (α2 + α3)

9 ,

V2245 =256α2

3 (α4 α5)52

3 (α2 + α4)3 (α2 + α5)

3 (2α2 + α4 + α5)7 ×

×(202α2

6 + 334α25 α4 + 133α2

4 α42 − 13α23 α43 − 7α22 α44 − α2 α4

5 + 334α25 α5+

+414α24 α4 α5 + 53α2

3 α42 α5 − 58α22 α43 α5 + α4

5 α5 + 133α24 α5

2 + 53α23 α4 α5

2 −−60α22 α42 α52 − 13α2 α43 α52 + 7α44 α52 − 13α23 α53 − 58α22 α4 α53 −+13α2 α4

2 α53 + 12α4

3 α53 − 7α22 α54 + 7α42 α54 − α2 α5

5 + α4 α55),

V3333 =501α3640

,

V3535 =16 (α4 α5)

52

(2α3 + α4 + α5)7 ×

×(824α3

3

5+

128α37

(α4 + α5)4 +

448α36

(α4 + α5)3 +

3296α35

5 (α4 + α5)2+

+2576α3

4

5 (α4 + α5)+ 28α3

2 (α4 + α5) + 2α3 (α4 + α5)2

)

,

V3345 =1536α3

5 (α4 α5)52 (9α3

2 + α42 + 7α4 α5 + α5

2 + 9α3 (α4 + α5))

5 (α3 + α4)2 (α3 + α5)

2 (2α3 + α4 + α5)7 ,

V5445 =6α4

3 α53 (α4

2 + 7α4 α5 + α52)

5 (α4 + α5)7 ,

V5555 =1896α4

5 α55

5 (α4 + α5)9 .

MZMO.\Z]ZQ,f

K X R|M`[NX_O RbPL[lPdO.R SKNIHe

w*/4"#$ #W*(1" / 6W@.<>6u/4(He

<>/4" 6u/4(w*J< S%5/4(F%.< $A_N$$

45(: ,$3&$w/4:4(wS%7*/4((#/40:'"M1$%#< (@%*$%N@.<>5< (

/S AXM*(J<>/4!/4#W(<>/4(a†*/4((#W/40\/G<./ "*/4(F%.< $5/ 5

q< (w6u1*/4A

E1 = −5.79404,|1 >= −0.82274 a†1s↓ a†1s↑ − 0.404512 a†1s↑ a†2s↓ + 0.309636 a†1s↓ a†2s↑−0.021729 a†2s↓ a†2s↑ + 0.0357708 a†2p0↓ a†2p0↑ − 0.0357641 a†2p1↑ a†2p−1↓+0.0357641 a†2p1↓ a

†2p−1↑|0 >,

E2 = −2.36898,|2 >= −0.727924 a†1s↑ a†2s↓ − 0.727924 a†1s↓ a†2s↑ − 1.53385 10−6 a†2p1↑ a†2p−1↓−1.53385 10−6 a†2p1↓ a†2p−1↑|0 >,E3 = −2.36898,|3 >= 1.02944 a†1s↑ a†2s↑ + 2.16919 10−6 a†2p1↑ a†2p−1↑|0 >,E4 = −2.36898,|4 >= −1.02944 a†1s↓ a†2s↓ + 8.76031 10−18 a†1s↑ a†2p−1↓ + 1.21637 10−18 a†2s↑ a†2p−1↓−2.16919 10−6 a†2p1↓ a†2p−1↓ + 8.76031 10−18 a†1s↓ a†2p−1↑+1.21637 10−18 a†2s↓ a

†2p−1↑|0 >,

E5 = −1.37035,|5 >= 0.567495 a†1s↓ a†1s↑ − 0.381789 a†1s↑ a†2s↓ + 0.53849 a†1s↓ a†2s↑−0.433411 a†2s↓ a†2s↑ − 0.00910772 a†2p0↓ a†2p0↑ + 0.00912602 a†2p1↑ a†2p−1↓−0.00912602 a†2p1↓ a†2p−1↑|0 >,E6 = −1.26427,

.

|6 >= −0.895812 a†1s↓ a†2p0↓ + 0.262975 a†2s↓ a†2p0↓ − 1.47163 10−17 a†1s↑ a†2p−1↓−2.04336 10−18 a†2s↑ a†2p−1↓ + 1.47163 10−17 a†1s↓ a†2p−1↑+2.04336 10−18 a†2s↓ a

†2p−1↑|0 >,

E7 = −1.26427,|7 >= −0.633435 a†1s↑ a†2p0↓ + 0.185951 a†2s↑ a†2p0↓ − 0.633435 a†1s↓ a†2p0↑+0.185951 a†2s↓ a

†2p0↑|0 >,

E8 = −1.26427,|8 >= −0.895812 a†1s↑ a†2p0↑ + 0.262975 a†2s↑ a†2p0↑|0 >,E9 = −1.26273,|9 >= 1.14407 10−17 a†1s↓ a†2s↓ − 7.06025 10−17 a†1s↓ a†2p−1↓ − 0.633428 a†1s↑ a†2p−1↓−9.80316 10−18 a†2s↓ a†2p−1↓ + 0.185964 a†2s↑ a†2p−1↓ − 0.633428 a†1s↓ a†2p−1↑+0.185964 a†2s↓ a

†2p−1↑|0 >,

E10 = −1.26273,|10 >= 4.97198 10−33 a†1s↓ a†2s↓ − 0.895803 a†1s↓ a†2p−1↓ − 4.99235 10−17 a†1s↑ a†2p−1↓+0.262993 a†2s↓ a

†2p−1↓ − 6.93188 10−18 a†2s↑ a†2p−1↓ − 4.99235 10−17 a†1s↓ a†2p−1↑

−7.06025 10−17 a†1s↑ a†2p−1↑ − 6.93188 10−18 a†2s↓ a†2p−1↑ − 9.80316 10−18 a†2s↑ a†2p−1↑|0 >,E11 = −1.26273,|11 >= 1.21485 10−49 a†1s↓ a†2s↓ − 1.40484 10−16 a†1s↓ a†2p−1↓ − 1.05189 10−33 a†1s↑ a†2p−1↓−1.95061 10−17 a†2s↓ a†2p−1↓ − 1.46054 10−34 a†2s↑ a†2p−1↓ − 1.05189 10−33 a†1s↓ a†2p−1↑+0.895803 a†1s↑ a

†2p−1↑ − 1.46054 10−34 a†2s↓ a†2p−1↑ − 0.262993 a†2s↑ a†2p−1↑|0 >,

E12 = −1.26214,|12 >= −0.633427 a†1s↑ a†2p1↓ + 0.185967 a†2s↑ a†2p1↓ − 0.633427 a†1s↓ a†2p1↑+0.185967 a†2s↓ a

†2p1↑|0 >,

E13 = −1.26214,|13 >= −0.895801 a†1s↑ a†2p1↑ + 0.262996 a†2s↑ a†2p1↑|0 >,E14 = −1.26214,|14 >= −0.895801 a†1s↓ a†2p1↓ + 0.262996 a†2s↓ a†2p1↓|0 >,E15 = −0.516353,|15 >= −0.628776 a†1s↑ a†2p0↓ + 0.194735 a†2s↑ a†2p0↓ + 0.628776 a†1s↓ a†2p0↑−0.194735 a†2s↓ a†2p0↑|0 >,E16 = −0.514665,|16 >= −6.91877 10−18 a†1s↓ a†2p0↓ − 4.98291 10−17 a†2s↓ a†2p0↓ + 0.628746 a†1s↑ a†2p−1↓

−0.194789 a†2s↑ a†2p−1↓ − 0.628746 a†1s↓ a†2p−1↑ + 0.194789 a†2s↓ a†2p−1↑|0 >,E17 = −0.514078,|17 >= −0.628748 a†1s↑ a†2p1↓ + 0.194787 a†2s↑ a†2p1↓ + 0.628748 a†1s↓ a†2p1↑−0.194787 a†2s↓ a†2p1↑|0 >,E18 = 2.15918,

|18 >= −0.23304 a†1s↓ a†1s↑ + 0.490265 a†1s↑ a†2s↓ − 0.404797 a†1s↓ a†2s↑−0.952009 a†2s↓ a†2s↑ + 0.0499905 a†2p0↓ a†2p0↑ − 0.0499032 a†2p1↑ a†2p−1↓+0.0499032 a†2p1↓ a

†2p−1↑|0 >,

E19 = 2.85271,

|19 >= 0.526972 a†1s↓ a†2p0↓ + 1.0055 a†2s↓ a†2p0↓ − 3.54205 10−17 a†1s↑ a†2p−1↓−4.91813 10−18 a†2s↑ a†2p−1↓ + 3.54205 10−17 a†1s↓ a†2p−1↑+4.91813 10−18 a†2s↓ a

†2p−1↑|0 >,

E20 = 2.85271,

|20 >= 0.526972 a†1s↑ a†2p0↑ + 1.0055 a†2s↑ a†2p0↑|0 >,E21 = 2.85271,

|21 >= 0.372625 a†1s↑ a†2p0↓ + 0.710993 a†2s↑ a†2p0↓ + 0.372625 a†1s↓ a†2p0↑+0.710993 a†2s↓ a

†2p0↑|0 >,

E22 = 2.85425,

|22 >= 5.04768 10−50 a†1s↓ a†2s↓ − 5.83709 10−17 a†1s↓ a†2p−1↓ − 4.37058 10−34 a†1s↑ a†2p−1↓−8.10479 10−18 a†2s↓ a†2p−1↓ − 6.06855 10−35 a†2s↑ a†2p−1↓ − 4.37058 10−34 a†1s↓ a†2p−1↑+0.526991 a†1s↑ a

†2p−1↑ − 6.06855 10−35 a†2s↓ a†2p−1↑ + 1.00549 a†2s↑ a†2p−1↑|0 >,

E23 = 2.85425,

|23 >= −2.06586 10−33 a†1s↓ a†2s↓ + 0.526991 a†1s↓ a†2p−1↓ + 2.07432 10−17 a†1s↑ a†2p−1↓+1.00549 a†2s↓ a

†2p−1↓ + 2.88019 10

−18 a†2s↑ a†2p−1↓ + 2.07432 10

−17 a†1s↓ a†2p−1↑

+2.93353 10−17 a†1s↑ a†2p−1↑ + 2.88019 10

−18 a†2s↓ a†2p−1↑ + 4.07321 10

−18 a†2s↑ a†2p−1↑|0 >,

E24 = 2.85425,

|24 >= −4.75361 10−18 a†1s↓ a†2s↓ + 2.93353 10−17 a†1s↓ a†2p−1↓ + 0.372639 a†1s↑ a†2p−1↓+4.07321 10−18 a†2s↓ a

†2p−1↓ + 0.710989 a

†2s↑ a

†2p−1↓ + 0.372639 a

†1s↓ a

†2p−1↑

+0.710989 a†2s↓ a†2p−1↑|0 >,

E25 = 2.85483,

|25 >= 0.526988 a†1s↓ a†2p1↓ + 1.00549 a†2s↓ a†2p1↓|0 >,E26 = 2.85483,

.

|26 >= 0.526988 a†1s↑ a†2p1↑ + 1.00549 a†2s↑ a†2p1↑|0 >,E27 = 2.85483,

|27 >= 0.372637 a†1s↑ a†2p1↓ + 0.71099 a†2s↑ a†2p1↓ + 0.372637 a†1s↓ a†2p1↑+0.71099 a†2s↓ a

†2p1↑|0 >,

E28 = 3.02981,

|28 >= 0.380435 a†1s↑ a†2p0↓ + 0.708638 a†2s↑ a†2p0↓ − 0.380435 a†1s↓ a†2p0↑−0.708638 a†2s↓ a†2p0↑|0 >,E29 = 3.03136,

|29 >= −2.9761 10−18 a†1s↓ a†2p0↓ − 2.14339 10−17 a†2s↓ a†2p0↓ + 0.380481 a†1s↑ a†2p−1↓+0.708624 a†2s↑ a

†2p−1↓ − 0.380481 a†1s↓ a†2p−1↑

−0.708624 a†2s↓ a†2p−1↑|0 >,E30 = 3.03195,

|30 >= 0.380483 a†1s↑ a†2p1↓ + 0.708623 a†2s↑ a†2p1↓ − 0.380483 a†1s↓ a†2p1↑−0.708623 a†2s↓ a†2p1↑|0 >,E31 = 3.86792,

|31 >= 1. a†2p0↓ a†2p−1↓|0 >,E32 = 3.86792,

|32 >= 1. a†2p0↑ a†2p−1↑|0 >,E33 = 3.86792,

|33 >= 0.707107 a†2p0↑ a†2p−1↓ + 0.707107 a†2p0↓ a†2p−1↑|0 >,E34 = 3.86851,

|34 >= 1. a†2p0↓ a†2p1↓|0 >,E35 = 3.86851,

|35 >= 1. a†2p0↑ a†2p1↑|0 >,E36 = 3.86851,

|36 >= 0.707107 a†2p0↑ a†2p1↓ + 0.707107 a†2p0↓ a†2p1↑|0 >,E37 = 3.87003,

|37 >= 2.23305 10−6 a†1s↓ a†2s↓ − 1.90028 10−23 a†1s↑ a†2p−1↓ − 2.63853 10−24 a†2s↑ a†2p−1↓+1. a†2p1↓ a

†2p−1↓ − 1.90028 10−23 a†1s↓ a†2p−1↑

−2.63853 10−24 a†2s↓ a†2p−1↑|0 >,E38 = 3.87003,

|38 >= 2.23305 10−6 a†1s↑ a†2s↑ + 1. a†2p1↑ a†2p−1↑|0 >,E39 = 3.87003,

$

|39 >= 1.579 10−6 a†1s↑ a†2s↓ + 1.579 10−6 a†1s↓ a†2s↑ + 0.707107 a†2p1↑ a†2p−1↓+0.707107 a†2p1↓ a

†2p−1↑|0 >,

E40 = 4.07503,

|40 >= 0.000393282 a†1s↓ a†1s↑ − 0.000138099 a†1s↑ a†2s↓ + 0.000134757 a†1s↓ a†2s↑+0.000439612 a†2s↓ a

†2s↑ + 0.819196 a

†2p0↓ a

†2p0↑ + 0.405535 a

†2p1↑ a

†2p−1↓

−0.405535 a†2p1↓ a†2p−1↑|0 >,E41 = 4.07536,

|41 >= 0.707107 a†2p0↑ a†2p−1↓ − 0.707107 a†2p0↓ a†2p−1↑|0 >,E42 = 4.07595,

|42 >= 0.707107 a†2p0↑ a†2p1↓ − 0.707107 a†2p0↓ a†2p1↑|0 >,E43 = 4.07691,

|43 >= −1. a†2p−1↓ a†2p−1↑|0 >,E44 = 4.07808,

|44 >= −1. a†2p1↓ a†2p1↑|0 >,E45 = 4.4453,

|45 >= 0.080553 a†1s↓ a†1s↑ − 0.0235083 a†1s↑ a†2s↓ + 0.024475 a†1s↓ a†2s↑+0.0772817 a†2s↓ a

†2s↑ + 0.570136 a

†2p0↓ a

†2p0↑ − 0.575924 a†2p1↑ a†2p−1↓

+0.575924 a†2p1↓ a†2p−1↑|0 > . al[A o

MZMO.\Z]ZQ,f

M`v^O@ILKWQ]Z[NV X_K&KNI:]ZQ\Z[lPdO.R

(w<5*/4&3&+K*/D/4(0<>^@,D@" *< (@,V6u/4(B*J< $*1$=<>/4/ 6 /Z*D8(R+: (.<>$AD1*.&:4(< $*/4+.< ?9HT#(/4M0#W0:\/4?C0@.< *6u(/4" /C0%X(&N#W/4&S#< (<>0\F&,)<0@.< *

RAV#(/40< $*/+/4(0O8

<>@,D@" V3&(@,?16u1$w( " nABqw/\0"/4R/4C*J< 0r$$#T%*J< $X*/+/4(0<>5< 0;!J< $O,9HD0)" /G(0"/4A

_N$$R6u/4$$/S:C0r%/4Z< (M$$%1@(<>0\dP:AXP A 4A ra< 0

rbFW

S0@.< *U6u(/4"p /#W/4&U_ < 0`NA/4& K/4(:4G/ 6W5w< (@=< */D/4(0<>M@,D@")Aw'I'<HTZ< $// A:170r' */+/4(0<><

λ =ra + rbR

µ =ra − rbR

.alP A o

ra< 0

rb*J<>GFW +(#(0G< Bw0@.< */5 6u/D*D/ 6<56l< "$O,/ 6*/4T6u/D*J< $

$$#B< 0V+,D#(FW/4$=< S(R0@.< *FW M6u/+*nA#< (< "(

/ 61*9)$$#*J< )F9HT#(0\F&, */+/4(0<>S76u/4("

a =R

2λ, b =

R

2

√λ2 − 1, e =

R

2a=1

λ,

alP A 4oS(

aBw"8"C<>p/4(<HT

bBw"O8"%/4(B<HTK< 0

eBw**+(*98

%p,4AP(/4" [UAalP A o R<>1 ≤ λ <∞ < 0 −1 ≤ µ ≤ 1 A ZV*J< )S(%0/S-;+1<>/4-6u/4(G*/4T6u/D*J< $$$%#G"&/40-FW6u/4(q!w< (@=<

*/D/4(0<> < z2

λ2− y2

λ2 − 1 =(R

2

)2

. (1 ≤ λ <∞) alP A &o

_N< $/4:4/41$%,4D6u/4(w+,D#(FW/4$=<

z2

µ2− y2

1− µ2=

(R

2

)2

. (−1 ≤ µ ≤ 1) alP A +o

Q#(! .:3 # *)2!# "# '# = * !0G8) 3 = ,D# *.= )*. )*# $C9 =?.H$. B !"# .#&()*3J)*# $. +P

!

R/2 R/2

O A B

φ

ra

rb

z

P:41(P A 4A#(/4$=<>'#(/40< $^*/+/4(0<>V@,D@")ABY6wwC0%(:4< (0E(/ .<>/4 < (/410 dI<HT0(:&< (00 wd:4;d$$#T%*J< $5*/+/4(0<>@,D@")A

($=<>/4wFWwdw< (@=< )< 0)$$#T%*J< $X*/D/4(0<>S< (

z =Rλµ

2, y =

R

2

√

(λ2 − 1)(1− µ2)alP A 4o

?"C<,q:4G/(0"/4G%6S)@,D@" / 6N*/4T6u/D*J< $$$%#<>0&,D#W(FW/ $=< ;(6u((0 / [U+AValP A T P A +o'(34/4$%34< (/410 \"C<>@/4(C<HT S*.EZI<HT7d*J<>>AI'*/D/4(0<>G0/+V/ V*.< :4GS(J< ;6u/4("(R,*/+/4(0<>;0r'/4:4(MS%(/ .<>/4-< :4$

φS*.

.< Q 3 < $1Vd'(< :40 ≤ φ < 2π

G3 < $1/ 6 Cw< (@=< x< 0

y*/+/4(0<>

z =Rλµ

2, x =

R

2

√

(λ2 − 1)(1− µ2) cosφ x =R

2

√

(λ2 − 1)(1− µ2) sinφ.alP A &o

4< */4F=< TS*.)0rS73&/4$1"Z$"&5d3r =

R3

8

(λ2 − µ2

)dλdµdφ.

alP A $4o9HT#(/ )6u/4(J< #$=< *=< ;

∇2 =4

R2

(1

λ2 − µ2∂

∂λ

(

(λ2 − 1) ∂∂λ

)

+1

λ2 − µ2∂

∂µ

(

(1− µ2)∂

∂µ

)

+

+1

(λ2 − 1)(1− µ2)

∂2

∂φ2

)

.alP A &o

"# ')2!1$C!= )* $2')*3J!"# ,D= *)':$C9 =?.8@= #.8!"# .) & $2A A #3>)EA?$C!# .O"# ')=?3$2A &()-)'13 = *$C!#. P

# # .

D/4""w(V5< $/C< !< $%(<>0r%/4 &/ 6K#(/4$=<>7#(/40< $*/>8/4(0<>10)S*.)($=<>0;/'/4F&,

λ = cosh ξ, µ = cos η, φ = φ.alP A !&o

Mw< (@=< */D/4(0<>S< (7)9HT#(0!3D=<

x =R

2sinh ξ sin η cosφ,

y =R

2sinh ξ sin η sinφ,

z =R

2cosh ξ cos η.

alP A &o

MZMO.\Z]ZQ,f

vÔ PLvÎLO@OgmX_O.\ PdO@I O.V;O.T O.\ Pτijk

KNSPLvÔ v^KNMZM`Q\î Q\ PdO@iNI:[NV

t′

_N##W0OH; 3 < $1<>Z76u/4$$/S:M&:4(< $

τijk(α, a, h) =α3

π

∫

d3re−α(ri+rj)

rk,

a GA oS(

α)<-#< (< "(< 0

h< 0

a0rE:4/4"(@, / 6RE@,D@"

< '/SP:A GA 4A?/41$09HT#(';3 < $1;/ 6rkS%$#-/ 6

/ (#< (< "( < 0:4/V/34(^./N#(/40< $*/+/4(0<>ADP(/4" F< *(=< :41$=<>(w:4/4"(@,CwQ+/

r2i = r2j + a2 − 2arj cos θrk = rj + (

a2+ h)2 − 2(a

2+ h)rj cos θ

a GA 4o?1#(/4$=<>7#(/40< $ */D/4(0<>wS%;6u/D*/4(:4w/ 6

ri< 0

rj

ri + rj = aλ ri − rj = aµ.a GA &o

D/4$%3D:M@,D@" / 6K[U+Aa GA 4o V:473 < $1rk =

√

a2

4(λ2 − 1) (1− µ2) +

(

λµa

2− h)2

.a GA +o

?;"C<,?(@%G&/[UA5a GA oM< 0-&:4(<>;/34(G;3>< (=< F$φ< 0

µS%/41T5< &,;"C<@/4(w0*1$%/4FT.< :

τijk =

∫ ∞

1

dλ(α3e−aαλ

4(√

(2h+ aλ)2(6hλ− a)−√

(aλ− 2h)2(6hλ+ a)

(a2 − 4h2)(3λ2 − 1) ln 2hλ− a+√

(αλ− 2h)22hλ+ a+

√

(αλ+ 2h)2)).

a GA 4o

τijk

θ h

a/2 a/2

ri rj

rk

P:41( GA 4A:4/4"(@,d/ 6 '@,D@" 10?/*J< $*1$=<>M'(*&(+:4(< $nA

EV9HT#(/4 (C< ("C< &,+1< (C(/D/ / 6S<;3 < $1'+1< (0 ABYR"C<,FWE:4(J<>$%,t"#$Or0 w< F$:6u1(@()*/4"#1T.<>/4 wF&, #$%:%1#0%X(+

h(: /4A '(:4/47< (al=o

h > a/2^auko

a/2 > h > −a/2 < 0au=o −a/2 > hAT` *J< 1Z/ 6Z@,D""(@,/4N":4&w< $/M< 00C7<>1"#T/

h > 0F1T7%ZN/ 7*< (@,4A\3>< (/41N(:4/4N M#$%Z1#\M&:4(<>/4

/34(λ+/ /q#<>("#$O6k,D%:d.)9HT#(/4;< **/4(0:/E!*./4

(:4/4 AP< $$O, /4FT.< ; /0%X(&5(1$%

A =1

2a3(e−aα((a2 − 4h2)(3 + 3aα + a2α2)(γEuler + ln 2 + ln(aα))

−aα(4a2 − 24h2 + a3α− 12ah2α)) +eaα(a2 − 4h2)(3− 3aα + a2α2)Γ(0, 2aα)),

a GA &oS(

Γ(a, z)R*/4"#$:&< ""C<M6u1*/4

Γ(a, z) =∫∞zta−1e−tdt

< 0γEuler ' 0.5772

wV[U1$( S*/4@.< &J< 0

B =1

2a3(4a(e−aαh(3 + 3aα + a2α2)− e−2hα(3h+ a2α)

)+

+(a2 − 4h2)(eaα(3− 3aα + a2α2)Γ(0, (2h+ a)α)−

−e−aα(3 + 3aα + a2α2)(Γ(0, (2h− a)α) + ln

(2h− a

2h+ a

)

)).a GA $4o

,)< (73>< $0;(#*%34$%,

τijk(α, a, h) =

B h ∈ (i)A h ∈ (ii)

B : h→ −h h ∈ (iii).

a GA &o


-4 -2 0 2 40,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

τ ijk

h

P:41( GA DAXG('*&(V&:4(< $τijk(α, a, h)

AP/4(ZZ#$/ ZwGα =

a = 1A

_ ;(:4/4-F/4(0(G;6u1*/4G*/4&+1/41AK(1$%G6u/4(R;(*&( +:4(< $^< (N9H< *JT< 0C*/41$0CFWN/4FT.< 0CFW*J< 1N5/4(:4^/ 6

rirj< 0

rk ($/D*J<>0*/4$$%J< (AT6u1*/4#$/ 0)P%:A GA DA

MZMO.\Z]ZQ,f

K6PL[lPLQ;KW\ KNSUR|M`v^O@I:QXA[NV vZ[jI:T KW\ZQXAR

(@6u/4(":<(/ .<>/4Z/ 6DU#(*J< $4< ("/4* /4"" 16u1$nAJYB< $$/S/V9HT#(/4"S#< (< "(UF+,M(K3>< $1^*J< $*1$=<>0G6u/4(^<7"#$(K:4/4"(@,/ 6C@,D@")A?Cw< &R/\: R6u/4("<)*/D/4(0<>@,D@"

r = (r, θ, φ)/d<

(/ .<>0r′ = (r′, θ′, φ′)

AB#(/D*01(R0*(F0?0.< $ nABYM*J< FWR0/4MF&,\<*/4*1T%34G(/ .<>/4!< FW/41TZ< :4$

α< (/410

z<HTd< FW/41T

< :4$β< (/410'

y<HT< 0'r< $$%,G<>FW/41T

γ< (/410'

z<HTA&N< :41$=< (

"/4"+1"/4#W(<>/4(L = (Lx, Ly, Lz)

< *< U(/ .<>/4G/4#W(<>/4(U:4(<>/4(/4#(*J< $< ("/4*AD?*J< )0*(FW1*.!(/ .<>/4<

Ylm(θ′, φ′) = e−ıαLze−ıαLze−ıαLzYlm(θ, φ) =

l∑

µ=−lDlmµ(α, β, γ)Ylµ(θ, φ).

alLVA oR< :41$=< (w"/4"&1" /4#W(<>/4(0// S$J< 01S/41T071F#< *VSO.<MrTHT0

l*7 *J< ;S(%Z7*/40!+1< $%,4A` *J< 17#(*J< $< (p8

"/4*S< (7:4T6u1*/4S/ 6BLz

/4#W(<>/4(w V*J< S(%

Dlmµ(α, β, γ) = e−ımγdlmµ(β)e

−ıµα.alLVA 4o

V*/+ *+dlmµ(β)

*J< )F@,D@"C<>*J< $$%,C*/4"#1T0)1:G76u/4("M1$=<

e−ıβLy

xyz

=

x cos β − z sin βy

x sin β + z cos β

.alLVA &o

, <J34d\6u/4(" #/4$%,D/4"=< $%C/ 6sin β

2

< 0cos β

2

Aw(;<(*1(/4($=<>/4S*.)"#$Orw*J< $*1$=<>/4

1

2(al,m+1d

lm+1,µ(β)− al,md

lm−1,µ(β)) =

d

dβdlmµ(β),

alLVA +o

S(al,m+1 =

√

l(l + 1)−m(m+ 1)AP/4(0.< $ nA

MZMO.\Z]ZQ,f

M`Q\ mKNILe`Q P X_KWY^M`VQ\^i PdO@I:T

'#D8/4(F%Z*/41#$:6u/4$$/SZ<>1(< $$%,6u(/4" ]N(< *G+1<>/4 A D*G < (!0J< $:ES%t&(< *:?$*(/4w< 0t(\C/]N(< *!+1<>/426u/4("C<>+,&8#< (@*$%*J< Bw0q/\/4FT.< E/4D8($=<>%3D%@*'$"OV/ 6+1<>/4 nAXY7*J< dFWG< *.340F&,\"J< Z/ 6U < 1$K< ##(/JHT"C<>%/4 A?S$$S/ 0(%3&d<Ew<J3&\+1<>/4t6u/4(<Ew/>8*/4"'#W/4&6u1*/4t"$<>('/D*9(+0:4(V+1<>/4E<>R*/4&.< V($=<>%3D%@*G*/4((*/4/)C/4(0((1/c)2

AV#(&5(J<>"&S*$/4$%,C6u/4$$/SJ< 0< 1)< 0 %6u*.%I nA"Z0#0+N]N(< *+1<>/4)*J< )F7S(%)"C<>(OHC6u/4("

(eV +mc2 − E cσ · p

cσ · p eV +mc2 − E

)(ξφ

)

= 0,au.A o

S(eV

$*(/4'#/ +=< $D(: ,4m$*(/4G"C<

p = −ı~ ∂∂r

ξ< 0

φ< (RG*/4"#W/ +7/ 6UMw<34M6u1*/4 < 0

σ< (R < 1$"C<8

(*A^/?:4/)(: ,-*J< $)/ 6N\D*.(D0:4('+1<>/4 )<J3&/1FT(< *5V(@N(: ,/ 6V$*(/

ES = E −mc2A6u/4("M1$=<R6u/4(

*/4"#/4+ξ*J< )FW9HD(< *06u(/4" V< F/34"C<>(OH+1<>/4

(eV − ES + c2(σ · p)(−eV + 2mc2 + ES)

−1(σ · p))ξ = 0.

au.A 4os5/ 0/\C6u/4$$%/S:)9HT#< /4?("R/ 6S+34(

c$J<J3D:!/4$%,

$J< 0:M("TS*9w/4D8($=<>%3D@%*Z$"'%

c2

−eV + 2mc2 + ES

' 1

2m

(

1− ES − eV

2mc2

)

.au.A &o

_N##$%,D:7^9HT#< /4/V[UAXau.A 4oK/45*J< 'S(%S0/S("US*.< (*$100

ES

ESξ =

p2

2m+ eV − p4

8m3c2− ıσ · p× [p, eV ]

4m2c2︸︷︷︸

H′

−p · [p, eV ]4m2c2

ξ.

au.A +o

Gr(@7 /;("7< (R'LN< "$%/4=<>!6u/4(ND*.(D0:4(Z+1<>/4 AX(0G(" KS($=<>%3D%@* Q+*(: ,G*/4((*/4 A46u/41(@'KS#D8/4(F%S*/41#$:

H ′ Ar6k;(" S< / (S($=<>%3D@%*Z*/4((*/4 A7("

H ′ *J< )FW7S(%)<

H ′ =e~m2c2

σ ·(∂V

∂r× p

)a.u.≡ α2σ ·

(∂V

∂r× p

)

,au.A 4o

S%2\*/4((#/40:q0r%/4Ct<>/4"*1%dal<TA 1 A o<>6k(\*/40+1< $%p,d:4 A

α ' 1137

7'r@(1*1(C*/4@.< &AC< FW/3&9HT#(/4.< Q4M;Q+/S6u/4("/ 65#D8/4(F%M*/41#$:!6u/4(R#(*J< $$%,?@,D""(*C#W/>8+=< $ /4$%,4AB?;*J< #< (<>#/ +=< $ &/\;#(*J< $$%,?@,D""(*< 0qC("C< :)#< (@

V = V sph + V ′BS(

V ′*J< qFW/4(:4%<>%:6u(/4"

/41$/4"MF&(< */4

V ′(r) =

∫

d3r′ρ(r′)

|r− r′| .au.A &o

(@:%+/M76u/4("M1$=<R6u/4(H ′ V:476u/4$$/S%:Mw/G("

H ′(r) = α21

r

∂V

∂rL · S+ α2

∫

d3r′2ρ(r′)S · ((r− r′)× p)

|r− r′|3.

au.A $4o

Sr(@ <7#W*=< $*J< 5/ 6X5*/40(")+< 0:4%34U5< "w/5#D8/4(F% &(< */4 A+N*/40;N:4(< $6u/4("M1$=<76u/4( < ;< (F%(< (@,'$%/D*J<>/4/ 6^*.< (:4Z#(/D01*:CR#W/ &=< $nW< 0\N"/4(V0*1$%N/*J< $%*1$=<>R01V/7(0#W/ (5+34( |r− r′| 0#0*J A

Q;e`VQ;KNiNI:[jM`v a

TANABD$=<>( y hjly .m v |\m10NjmGin j v yjly v m> )* 7(<Jw8pL5$$ns59 /4(Q !TA

VG A :A A /4< 0EsVA LVA!< (*9 .m v jlin m>i jY j |zin.z W34/4$nA4]N/3&( U1F$nA%s5 /4(Q !-$ MWALTX( m>jm.j v m>h .j v mz.m.W| D#(:4( U($=< :` ($ 4TA MA .AX_N"/3WTA <4< MA GAX_N0( U+,D S3WA` && ! &a!"! o

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