β

Transcription

1 β 01 7

2 s s s d 10 s p s p s p 3 s p 4 s p 5 s p 6 1 1H He Li 4Be 5B 6C 7N 8O 9F 10Ne Na 1Mg 13Al 14Si 15P 16S 17Cl 18Ar K 0Ca 1Sc Ti 3V 4Cr 5Mn 6Fe 7Co 8Ni 9Cu 30Zn 31Ga 3Ge 33As 34Se 35Br 36Kr Rb 38Sr 39Y 40Zr 41Nb 4Mo 43Tc 44Ru 45Rh 46Pd 47Ag 48Cd 49In 50Sn 51Sb 5Te 53I 54Xe (98) Cs 56Ba Hf 73Ta 74W 75Re 76Os 77Ir 78Pt 79Au 80Hg 81Tl 8Pb 83Bi 84Po 85At 86Rn (09) (10) () 7 87Fr 88Ra Rf 105Db 106Sg 107Bh 108Hs 109Mt 110Ds 111Rg 11Cn 113Uut 114Uuq 115Uup 116Uuh Uuo (3) (6) (61) (6) (66) (64) (77) (68) (81) (7) (85) (84) (89) (88) (9) (94) * 57La 58Ce 59Pr 60Nd 61Pm 6Sm 63Eu 64Gd 65Tb 66Dy 67Ho 68Er 69Tm 70Yb 71Lu (145) ** 89Ac 90Th 91Pa 9U 93Np 94Pu 95Am 96Cm 97Bk 98Cf 99Es 100Fm 101Md 10No 103Lr (7) (37) (44) (43) (47) (47) (51) (5) (57) (58) (59) (6)

3 iii i I I II Hermite f i II Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger

4 iv β Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Schrödinger Legendre Schrödinger Schrödinger Schrödinger Schrödinger (7.14) III Schrödinger Schrödinger

5 v (9.18) Stern Gerlach Pauli He He He He Ritz Hartree Fock SCF Slater IV SI Born Oppenheimer

6 vi N HOMO LUMO sp sp sp sp π Hückel π V

7 vii A 390 A A. SI A B 393 C 397 C C. Taylor C C C.5 l Hôpital C C.7 Kronecker C C C D 409 D D D.3 Ψ

8 viii E 41 E E E F

9 95 9 Ze(Z = 1,, 3, ) 1 Z Z = 1 Z =, 3 He + Li + He + Li + Coulomb Coulomb Coulomb 9.1 e, r, Z Z = 1 Z =, 3 He + Li Schrödinger Coulomb U(r) Schrödinger [ 1 m e r sin θ ( r Ψ ) + ( sin θ Ψ sin θ r r θ θ ) + 1 sin θ ] Ψ φ + U (r) Ψ = EΨ (9.1) U (r) r

10 Schrödinger Ψ r R(r) θ Θ(θ) φ Φ(φ) Ψ = R (r) Θ (θ) Φ (φ) (9.) (9.) (9.1) 99 [ sin ( θ d r dr )] R dr dr }{{} [ θ, r ] [ sin θ + Θ ( d sin θ dθ )] dθ dθ }{{} [ θ ] [ me r sin θ + ] {E U (r)} } {{ } [ r, θ ] [ 1 = Φ d ] Φ dφ }{{ } [ φ ] (9.3) (9.3) Ψ Schrödinger 4 φ θ r φ φ φ θ r θ r = Φ (φ) m m 49 (4.11) 1 Φ d Φ dφ = m Φ (φ) = 1 e imφ m = 0, ±1, ±, (9.4) π m = 0, ±1, ± Φ m Φ 0 Φ 0 Φ 0 = 1 π (9.5) Φ ±1 = 1 π e ±iφ Φ ± = 1 π e ±iφ (9.6) (9.7) 1 d Φ Φ dφ = m (9.3) Schrödinger [ sin ( θ d r dr )] [ ( sin θ d + sin θ dθ )] + [ m ] [ me r sin ] θ + R dr dr Θ dθ dθ {E U (r)} = 0 1 Φ d Φ dφ = m

11 9.3 Schrödinger 97 ( sin θ d sin θ dθ Θ dθ dθ ( 1 d sin θ dθ ) m Θ sin θ dθ dθ sin θ } {{ } [ θ ] ) m = sin θ R ( d r dr ) m er sin θ dr dr {E U (r)} = 1 ( d r dr ) m er R dr dr {E U (r)} sin θ } {{ } [ r ] θ r θ r (9.8) Θ (θ) l (l + 1) (9.8) ( 1 d sin θ dθ ) m Θ sin θ dθ dθ sin = l (l + 1) (9.9) θ (5.14) (5.18) Legendre l + 1 (l m )! d m Θ l,m (θ) = (l + m )! sin m θ (d cos θ) P m l (cos θ) (9.10) P l (cos θ) = 1 d l ( cos l l! (d cos θ) l θ 1 ) l (9.11) m = 0, ±1, ±,... ± l (9.1) Φ l = 0, 1, Θ l,m Θ 0,0 = 1 (9.13) 3 Θ 1,0 = cos θ Θ 1,±1 = 5 Θ,0 = ( 3 cos θ 1 ) Θ,±1 = 15 4 sin θ cos θ Θ,± = sin θ (9.14) sin θ (9.15) R (r) Θ (θ) = l(l + 1) l(l + 1) 1 ( d r dr ) m er R dr dr {E U (r)} = l(l + 1) (9.16) R (r) U (r) Coulomb U (r) Coulomb U (r) = 1 Ze 4πɛ 0 r ɛ 0 1 R d dr ( r dr dr (9.17) (9.17) (9.16) ) m er ( E + 1 Ze ) = l(l + 1) (9.18) 4πɛ 0 r

12 n = 1,, 3 n l R n,l ( Z a 0 ) 3/ e Zr/a 0 ( ) 3/ ( 1 Z Z ) r e Zr/a 0 a 0 a 0 ( ) 5/ 1 Z re Zr/a 0 6 a 0 ( ) 3/ ) Z ( Za0 r + Z 3 a 0 a r e Zr/3a 0 0 ( ) 5/ ) 4 Z (6r 81 Za0 r e Zr/3a 0 6 a 0 ( ) 7/ 4 Z 81 r e Zr/3a 0 30 a 0 E n = Z m e e 4 8ɛ 0 h n (9.19) ( ) 3 4 (n l 1)! Z R n,l (ρ) = ρ l n 4 [(n + l)!] 3 e ρ L l+1 n+l (ρ) (9.0) a 0 ρ = Z na 0 r, a 0 = 4πɛ 0 m e e (9.1) n = 1,, 3,, l = 0, 1,, n 1 (9.) L l+1 n+l (ρ) Laguerre L β α (ρ) = dβ dρ β L α (ρ) L α (ρ) = e ρ dα dρ α ( ρ α e ρ) (9.3) n = 1,, 3 R n,l (r) 9.1 (9.18) 9.13 Ψ E ( ) 3 4 (n l 1)! Z Ψ n = ρ l n 4 [(n + l)!] 3 e ρ L l+1 n+l (ρ) a 0 }{{} R n,l (ρ) l + 1 (l m )! d m (l + m )! sin m θ (d cos θ) P m l (cos θ) }{{} Θ l,m (θ) 1 e imφ π }{{} Φ m (φ) (9.4)

13 E n = Z m e e 4 8ɛ 0 h n E n = Z e 8πɛ 0 a 0 n (9.5) n = 1,, 3, l = 0, 1,,, n 1 (9.6) m = l, l + 1,, l 1, l a 0 a 0 = ɛ 0h πm e e (9.7) Bohr a 0 = nm Planck h Ψ = R (r) Θ (θ) Φ (φ) (9.1) (9.3) (9.1) Ψ = R Θ Φ [ 1 m e r sin θ ( r RΘΦ ) + ( sin θ RΘΦ ) + 1 sin θ r r θ θ sin θ [ 1 m e r ΘΦ sin θ d ( r dr ) + RΦ d ( sin θ dθ ) + RΘ sin θ dr dr dθ dθ sin θ sin θ R ( d r dr ) + sin θ dr dr Θ 3 (9.3) ] RΘΦ φ + URΘΦ = ERΘΦ Ψ RΘΦ d Φ dφ ] + (U E) RΘΦ = 0 ( d sin θ dθ ) + 1 d Φ dθ dθ Φ dφ + m er sin θ (E U) = 0 (9.8) m er sin θ RΘΦ n l m *1 n Coulomb U 1/r (9.5) 1 (9.5) E n=1 = J ev ev * ev *1 s p s p * J ev A. A.5

14 Balmer λ k = a(k + ) / [ (k + ) 4 ] (a = nm) Balmer Paschen Lyman Brackett Pfund ( 1 1 λ = R m 1 ) n (9.9) Rydberg R Rydberg R = m 1 9. Rydberg hc c = νλ hc hν = R m R hc n (9.30) hν R hc/m R hc/n E n = R hc/n (9.5) 9. E n E 1 Lyman E n E Balmer E n E 3 Paschen E n = R hc/n (9.5) R hc = m e e 4 /(8ɛ 0h ) R hc = m e e 4 /(8ɛ 0h )

15 R hc = m ee Bohr 8ɛ 0 h = (9.31) Bohr Bohr Coulomb e 4πɛ 0 r = m ev r (9.3) Bohr Bohr m e vr m e vr = n (9.33) Bohr *3 r m ev r = n m e v = n m e r e 4πɛ 0 r = n m e r 3 (9.33) m e v (9.3) r = n ɛ 0h 9.3 Bohr πm e e r *3 = n a 0 a 0 = ɛ 0h (9.34) πm e e m e v / U(r) e +e/(4πɛ 0 r ) E = m ev + r = e 8πɛ 0 r e 4πɛ 0 r = e 8πɛ 0 r +e ( e) 4πɛ 0 r dr = m ev e 4πɛ 0 (9.34) [ ] r 1 r (9.3) 1 (9.35) E n = e πm ee 8πɛ 0 n ɛ 0 h (9.35) r = n ɛ 0h πm e e = m ee 4 8ɛ 1 0 h n (9.36) (9.5) Bohr Schrödinger *3 Bohr

16 n, l, m * 4 n n l + 1 l 0 l m m m = 0, ±1, ±, ± l 3 1 0,1,,3 s,p,d,f n =, l = 1 m = 1, 0, 1 3 p p 9.3 n = 1 l = 0 m = 0 1s n = l = 0 m = 0 s l = 1 m = 1 p m = 0 m = 1 n = 3 l = 0 m = 0 3s l = 1 m = 1 3p m = 0 m = 1 l = m = 3d m = 1 m = 0 m = 1 m = 3 s p *4 (9.145) Laguerre (9.119) L(x) n r n l n 1 (5.60) Legendre

17 (9.4) Legendre Laguerre n = 1 l = 0 m = 0 Ψ Ψ = (πa 3 0) 1/ e r/a 0 θ, φ n = 1,, 3 Ψ n,l,m Ψ n,l,m = R n,l Y l,m = R n,l }{{} Θ l,m Φ m }{{} (9.37) R n,l Θ l,m Φ m R n,l n = 1,, 3 R n,l r 0 0 Ψ n,l (n l 1) 3s (n = 3, l = 0) n l 1 = = 9.4 r r + dr P r dr P r Ψ 4 I dv = r sin θdθdφdr Ψ n,l,m (r, θ, φ) dv = R n,l (r) Y l,m (θ, φ) r sin θdθdφdr (9.38) θ φ P r dr P r dr = R n,l (r) r dr π π 0 0 Y l,m (θ, φ) sin θdθdφ (9.39) 9.4 n = 1 3 R n,l (r) Y l,m P r dr = R n,l (r) r dr (9.40) r P r = R n,l (r) r 9.5

18 r r + dr 1s P r r = a R 1,0 (1s) = (Z/a 0 ) 3/ e Zr/a 0 P r P r(1s) = 4 a 3 e r/a0 r 0 (9.41) dp r /dr = 0 dp r(1s) = 4 [ dr a 3 ] e r/a0 r + e r/a0 r 0 a 0 = 8 [ a 3 e r/a0 r r ] + 1 = 0 0 (9.4) 0 a 0 r = 0 r = a 0 P r(1s) *5 P r(1s) r 0 r 0 r 0 a 0 r = a 0 / r > a 0 r = a 0 P r(1s) r = a P r(1s) r 0 a 0 / a 0 a 0 dp r /dr 0 0 P r r r 1s Ψ 1s = e r/a 0 a 3/ 0 } {{ } R 1,0 1 π }{{} Y 0,0 (9.43) *5 P r = R n,l (r) r P r r = 0 P r = 0

19 r F F = π π r sin θdθdφdrψ ˆF Ψ (9.44) ˆr r Ψ 1s Ψ = Ψ θ φ θ φ π 0 sin θdθ π 0 dφ = 4π (1/ π) [ r = e 0] r/a r 3 dr (9.45) 0 a 3/ 0 r = 4 a x n e ax dx = n! (C.74) an+1 r 3 e (/a0)r dr = 4 3! a 3 0 (/a 0 ) 4 = 3 a 0 (9.46) P r r r Bohr Y l,m = Θ l,m Φ m Θ l,m Φ m (9.5) (9.7) (9.13) (9.15) { l = 0, m = 0 Y 0,0 = Θ 0,0 Φ 0 = 1 4π (9.47) 3 Y 1,0 = Θ 1,0 Φ 0 = 4π cos θ l = 1, m = 0, ±1 3 Y 1,±1 = Θ 1,±1 Φ ±1 = sin θe±iφ 8π 5 ( Y,0 = Θ,0 Φ 0 = 3 cos θ 1 ) 16π 15 l =, m = 0, ±1, ± Y,±1 = Θ,±1 Φ ±1 = sin θ cos θe±iφ 8π 15 Y,± = Θ,± Φ ± = 3π sin θe ±iφ (9.48) (9.49) m 0 Y l,m 50 Y 1,1 Y 1, 1 Y 1,1 Y 1, 1 ( ) 1 (Y 1,1 + Y 1, 1 ) = 1 3 8π sin θeiφ + 3 sin θe iφ 8π = 1 3 8π sin θ ( e iφ + e iφ) 3 = sin θ cos φ 4π 3 = 4π x r (C.54) (8.1) (9.50)

20 / 1/ i ( 1 (Y 1,1 Y 1, 1 ) = 1 ) 3 3 i i 8π sin θeiφ sin θe iφ 8π = 1 3 8π sin θ ( e iφ e iφ) 1 3 8π 3 = sin θ sin φ 4π 3 = 4π y r (C.55) sin θ (8.) (9.51) x/r y/r x y 1 3 (Y 1,1 + Y 1, 1 ) = 4π x r Y p x 1 3 (Y 1,1 Y 1, 1 ) = i 4π y r Y p y l = 0, 1,, 3 s,p,d,f p x, d z, f xyz l = 0, 1, { Y s Y 0,0 = 1 4π (9.5) 3 3 Y pz Y 1,0 = 4π cos θ = 4π z r Y px (Y 1,1 + Y 1, 1 ) = sin θ cos φ = 4π 4π Y py (Y 1,1 Y 1, 1 ) = sin θ sin φ = i 4π 4π Y dz Y,0 = Y dyz 1 (Y,1 Y, 1 ) = i Y dxy 1 (Y, Y, ) = i 5 ( 3 cos θ 1 ) 5 = 16π 16π Y dzx 1 15 (Y,1 + Y, 1 ) = 16π 15 sin θ sin φ = 16π x r y r ( ) 3 z r r ( ) 15 zx sin θ cos φ = 16π r ( ) 15 yz 16π r Y dx y (Y, + Y, ) = 16π sin θ cos φ = 16π ( xy 16π sin θ sin φ = 16π r x y r ) (9.53) (9.54)

21 p z Y pz a Y pz = a cos θ Y cos θ = θ θ = θ 9.6(a) θ Y l,m r p z cos θ r = cos θ θ = 0 r = 1 θ = π/3 r = 1/ xy 9.6(b) 9.6 p z (a) Y (b) cos θ (a) θ θ Y pz 3 xz Y pz xz xz φ = 0 φ = π φ = 0 xz 1 4 φ = π φ = Y pz xz xz θ Y pz = cos θ (0 θ π/), cos θ (π/ θ π)

22 Y pz = cos θ θ π/ cos θ θ cos θ cos θ 9.9 Y pz { x = Y pz sin θ z = Y pz cos θ Y pz = Y 1,0 = (9.55) 3 cos θ =: a cos θ (9.56) 4π (9.53) θ 0 θ π cos θ + 0 θ π/ π/ θ π 9.9 Y pz

23 θ π/ cos θ 0 cos θ = cos θ Y pz = a cos θ { x = a cos θ sin θ z = a cos (9.57) θ (x, z) x x = a cos θ sin θ (9.57) = a cos θ a(1 cos θ) sin θ = 1 cos θ = a cos θ(a a cos θ) a 1 = z(a z) a cos θ = z ( = z a ) ( a ) + ( x + z a ) ( a ) = (9.58) (x, z) (0, a/) a/ 9.10 π/ θ π 1 ( x + z + a ) ( a ) = (9.59) (0, a/) a/ φ = π Y pz xz xz 9.10(a) φ (b) 9.11 s p 9.10 (a)y pz xz (b)3

24 Y Y R Ψ s p *6 p 9.11 Y s Y p Y pz = cos θ 0 θ π/ *6 s p x x + x z x Y px Y py Y pz V

25 Ψ = R n,l Y l,m n = 1,, 3 R n,l Y l,m n = 1,, 3 n l m R n,l Y l,m = Φ m (φ)θ m,l (θ) ( ) 3/ Z s e Zr/a 1 0 a 0 π 0 0 s 1 ( Z a 0 ) 3/ ( Z ) r e Zr/a 1 0 a 0 π p z 4π cos θ ( ) 5/ 1 Z re Zr/a 0 3 p x 6 a 0 sin θ cos φ 4π ± s 3 p y sin θ sin φ 4π 81 3 ( ) 3/ ) Z (7 18 Za0 r + Z r e Zr/3a 1 0 π a 0 a p z 4π cos θ ( ) 5/ ) 4 Z (6r 81 Za0 r e Zr/3a 0 3 3p x 6 a 0 sin θ cos φ 4π ±1 0 3d z 3 3p y sin θ sin φ 4π 5 ( 3 cos θ 1 ) 16π ±1 ± 15 3d zx sin θ cos φ 16π ( ) 7/ 4 Z 81 r e Zr/3a d yz 30 a 0 sin θ sin φ 16π 3d x y 15 16π sin θ cos φ 3d xy 15 16π sin θ sin φ

26 s 1s Ψ 1s = 1 πa 3 0 e r/a 0 (9.60) 1s 9.13(a) Ψ 1s Ψ 1s r Ψ 1s 9.13(b) Ψ 1s Ψ 1s xy y x 9.13(c) Ψ 1s 1 s 3s R n,l (r) Ψ p p z Ψ pz = 1 3πa 5 0 re r/a 0 cos θ (9.61) 9.14(a) Ψ pz xz Ψ pz 9.14(b) (a) z s (a)ψ 1s Ψ 1s (b)ψ 1s Ψ 1s xy (c)ψ 1s [17] p z 9.14(c) p x p y p z 1 x y z R(r) p (a)ψ pz (b) Ψ pz xz + (c)p x p y p z 1 +

27 9.9 波動関数の形 等高面 d 軌道 4f 軌道 図 9.15 には 5 つある 3d 軌道について各々 1 つの等高面を 図 9.16 には 7 つある 4f 軌道について各々 1 つの等高面を描いた 図 9.15 水素原子の 3d 軌道 それぞれについて等高面の 1 つを描いた 図 9.16 水素原子の 4f 軌道 それぞれについて等高面の 1 つを描いた

28 U = K (9.6) U K (9.4) n = 1 Ψ 1s = (πa 3 0) 1/ e r/a 0 U = Ψ 1s ( e = e 4π ɛ 0 a 3 0 U 4πɛ 0 r 0 ) Ψ 1s dv = 1 e r/a0 rdr = e πɛ 0 a 3 re r/a 0 dr 0 0 = e 1 πɛ 0 a 3 0 (/a 0 ) πa 3 0 π e 4πɛ 0 e r/a0 1 r dv Ψ 1s = 1 e r/a 0 πa 3 0 π sin θdθ dφ 0 } {{ 0 } 4π dv = r dr sin θdθdφ θ φ (C.74) = e 4πɛ 0 a 0 (9.63) K K = H U H (9.5) e /(8πɛ 0 a 0 ) K K = H U = e 8πɛ 0 a 0 ( e 4πɛ 0 a 0 ) = e 8πɛ 0 a 0 (9.64) (9.63) (9.64) U = K Ψ 1s = (πa 3 0) 1/ e r/a x, y, z 89 (8.30) (8.3) [ ( ˆl = 1 sin θ sin θ θ θ ) + 1 sin θ ] φ (9.65)

29 Stern Gerlach Stern Gerlach N S Stern Gerlach 19 * Stern Gerlach *1 19 Schrödinger

30 Stern Gerlach q m d µ µ = q m d (10.1) q l µ µ = ± q m l (10.) ± ± d µ B U = µ B z 10. q m F = q m B z F = U/ z q m U U = F dz = q m Bz 10. z = z q m z = z + d cos θ +q m U = q m B(z + d cos θ) + ( ( q)bz) = (q m d)b cos θ = µb cos θ = µ B 10. B µ B U = µ B F F = U/ z z F z = U z = (µ B) z = µ z z B B z z + µ z z µ z B z z U = µ B µ z z 0 (10.3) B z / z µ z / z z mm < nm Stern Gerlach Stern Herschbach Stern D.R. Herschbach, Molecular Dynamics of Elementary Chemical Reactions (Nobel Lecture), Angew. Chem., Int. Ed. Engl., 6, 11 (1987)

31 10.1 Stern Gerlach 137 z (10.3) µ z / z 0 * (10.3) B z / z µ z z 10.1 z B z / z z B z / z = 0 F z = (a) (b) z z µ z µ z µ µ z µ z 10.3(a) 10.3(b) z 1 *3 s p d K L M N {}}{{}}{{}}{{}}{ (1s) (s) (p) 6 (3s) (3p) 6 (3d) 10 (4s) (4p) 6 (4d) 10 (5s) 1 }{{} K N s 5s l = 0 5s z 116 z l, l+1,, 0,, l * µ z *3 1.3

32 , l l + 1 Stern Gerlach z z l + 1 = l = 1/ l Stern Gerlach Stern Gerlach 5s Uhlenbeck Goudsmit Stern Gerlach N S 5s 1 a b α β 1 *4 10. (9.69) (9.70) ŝ ŝ z ŝ z Γ ŝ Γ = s(s + 1) Γ ŝ z Γ = m s Γ s (10.4) m s (10.5) m s s, s + 1,, s 1, s s + 1 Stern Gerlach s + 1 = s = 1/ m s = ±1/ m s = 1/ α m s = 1/ β (10.5) ŝ z α = 1 ŝ z Γ = m s Γ α ŝ z β = 1 (10.6) β *4 Dirac Pauli 4

33 ŝ ŝ α = 3 ŝ Γ = s(s + 1) Γ 4 α ŝ β = 3 4 β (10.7) 10.1 m s Γ s m s Γ 1/ 1/ α 1/ β α β Y m l α = Y 1/ 1/ l = 1/, m = 1/ (10.8) β = Y 1/ 1/ l = 1/, m = 1/ (10.9) l l = 1/ α β 10.3 Ψ(r) σ Ψ(r, σ) Ψ(r, σ) r σ r σ σ = 1/ σ = 1/ α β α(1/) = 1 α( 1/) = 0 β(1/) = 0 β( 1/) = 1 (10.10) σ r ψ orb Ψ(r, σ) = ψ orb (r) α(σ) + ψ orb (r)β(σ) (10.11) 1 Ψ(r, σ) Ψ(r, σ) dvdσ (10.1) σ α β σ = ±1/ 1 0 f(σ)dσ := f(1/) + f( 1/) (10.13)

34 (a) t = 0 1 t = t 1 (b) t = 0 1 t = t 1 1 t = t 1 1 Ψ(r, σ) dvdσ = Ψ(r, 1/) dv + Ψ(r, 1/) dv (10.14) 1 Ψ(r, σ) dvdσ=1 r σ τ Ψ(τ) dτ = 1 (10.15) (10.13) α(σ) dσ = α(1/) + α( 1/) = = 1 (10.16) β α (σ)β(σ)dσ = α (1/)β(1/) + α ( 1/)β( 1/) = = 0 (10.17) α β α(σ) dσ = 1 β(σ) dσ = 1 α (σ)β(σ)dσ = 0 (10.18) (a) 1 1 Newton 10.4(b) 1 q q = p =

35 10.5 Pauli 141 q p h p p = m v 1 1 Ψ(τ 1, τ ) Ψ(τ, τ 1 ) Ψ(τ 1, τ ) = e iθ Ψ(τ, τ 1 ) (10.19) e iθ Ψ(τ, τ 1 ) = e iθ Ψ(τ 1, τ ) (10.0) e iθ = ±1 (10.1) (10.19) Ψ(τ 1, τ ) = ±Ψ(τ, τ 1 ) (10.) Fermi Fermi 0 Bose 10.5 Pauli 1s p 1 1 Pauli Pauli194 Pauli Pauli Pauli Pauli 1 1 α β 1 Pauli Fermi 1 *5 φ 1s (1) 1 1s β () β φ 1s (r 1 ) *5 1 1

36 14 10 β (σ ) φ r α β σ Pauli 1s α Pauli 1s α Ψ(1, ) = φ 1s (1)α(1)φ 1s ()α() (10.3) 1 Ψ Ψ (, 1) = φ 1s ()α()φ 1s (1)α(1) (10.3) 1 = φ 1s (1)α(1)φ 1s ()α() = Ψ(1, ) (10.4) Ψ Pauli Fermi Pauli Pauli 10.5., α α(1) α() β β(1) β() 1 α 1 β α(1) β() 1 1 α(1) β() β(1) α() *6 α(1) β() β(1) α() α(1) β() + β(1) α() α(1) β() β(1) α() Pauli 1 α(1) β() + β(1) α() 1 α() β(1) + β() α(1) = α(1) β() + β(1) α() (10.5) α(1) β() β(1) α() 1 α() β(1) β() α(1) [ ] = α(1) β() β(1) α() (10.6) *

37 10.5 Pauli 143 α(1) β() β(1) α() α(1) β() β(1) α() α(1) α() β(1) β() α (1) α () Γ = β (1) β () (10.7) α (1) β () + β (1) α () Γ = α (1) β () β (1) α () (10.8) case I 1s ψ orb = φ 1s (1) φ 1s () (10.9) Pauli = = (10.30) (10.8) Ψ = ψ orb ψ spin = φ 1s (1) φ 1s () [ ] α (1) β () β (1) α () (10.31) 1 1 Ψ [ ] Ψ = φ 1s () φ 1s (1) α () β (1) β () α (1) [ ] = φ 1s (1) φ 1s () α (1) β () β (1) α () } {{ } =Ψ = Ψ (10.3) case II 1s s φ 1s (1) φ s () φ s (1) φ 1s () ψ orb,s = φ 1s (1) φ s () + φ s (1) φ 1s () (10.33) ψ orb,a = φ 1s (1) φ s () φ s (1) φ 1s () (10.34) ψ s a symmetryasymmetry (10.8) (10.7)

38 [ ][ ] Ψ S (1, ) = φ 1s (1)φ s () + φ s (1)φ 1s () α(1)β() β(1)α() α(1)α() [ ] Ψ T (1, ) = φ 1s (1)φ s () φ s (1)φ 1s () β(1)β() [ ] α(1)β() + β(1)α() (10.35) (10.36) (10.36) (triplet) T Ψ (10.35) (singlet) S Ψ (10.7) (10.8) α (1) α () =: Γ αα Γ = β (1) β () =: Γ ββ (10.37) 1 ] [α (1) β () + β (1) α () =: Γ αβ+ Γ = 1 ] [α (1) β () β (1) α () =: Γ αβ (10.38) Γ αα = α (1) α () Γ αβ = 1 [α (1) β () β (1) ] α () Γ αα dσ 1 dσ = α (σ 1 )α (σ )α(σ 1 )α(σ )dσ 1 dσ = α (σ 1 )α(σ 1 ) dσ 1 α (σ )α(σ ) dσ }{{}}{{} = α(σ 1 ) = α(σ ) = 1 1 = 1 (10.18) (10.39) Γ αβ dσ 1 dσ = 1 [ ] α (σ 1 ) β (σ ) β (σ 1 ) α (σ )][ α(σ 1 ) β(σ ) β(σ 1 ) α(σ ) dσ 1 dσ = 1 α (σ 1 )α(σ 1 )β (σ )β(σ )dσ 1 dσ 1 α (σ 1 )α(σ )β (σ )β(σ 1 )dσ 1 dσ 1 α (σ )α(σ 1 )β (σ 1 )β(σ )dσ 1 dσ + 1 α (σ )α(σ )β (σ 1 )β(σ 1 )dσ 1 dσ = 1 α (σ 1 )α(σ 1 )dσ 1 β (σ )β(σ )dσ 1 α (σ 1 )β(σ 1 )dσ 1 α(σ )β (σ )dσ 1 α(σ 1 )β (σ 1 )dσ 1 α (σ )β(σ )dσ + 1 β (σ 1 )β(σ 1 )dσ 1 α (σ )α(σ )dσ = = 1 (10.18) (10.40) (10.37) (10.38)

39 He 9 Schrödinger He 1 Schrödinger Schrödinger He He dτ dv dτ dv Kepler *1 * *1 [ 1 ] [ ] [ 3 ] * kg

40 He / Ĥ 0 Ĥ0 ψ (0) E (0) Ĥ 0 ψ (0) = E (0) ψ (0) (11.1) 0 0

41 Ĥ ψ 0 E 0 Ĥψ 0 = E 0 ψ 0 1 ϕ E ϕ / E ϕ = ϕ Ĥϕdτ ϕ ϕdτ (11.5) E ϕ Rayleigh E 0 E ϕ E ϕ E 0 ϕ = ψ 0 Slater STO ϕ r n 1 e ζr Gauss GTO ϕ x i y j z k e αr STO GTO 1. ϕ Ĥ {ψ n} ϕ = n c nψ n Ĥ ( ) Ĥ c n ψ n c n ψ n dτ n n m n E ϕ = ( ) c n ψ n c n ψ n dτ n n c nc m ψ nĥψ mdτ c nc m ψne m ψ m dτ c n E n n m n m n = = = c nc m ψnψ m dτ c n c n *3 n n (11.6) *3 1 P n = Z X Ĥ X Z c n ψ n c n ψ n dτ = (c 1ψ1 + c ψ) Ĥ (c 1 ψ 1 + c ψ ) dτ n=1 n=1 Z Z Z Z = c 1 c 1 ψ 1Ĥψ 1dτ + c 1 c ψ 1Ĥψ dτ + c c 1 ψ Ĥψ 1dτ + c c ψ Ĥψ dτ {z } {z } = P m=1 c 1 c R m ψ 1 Ĥψ m dτ = P m=1 c c R m ψ Ĥψ m dτ X Z X Z X X Z = c 1c m ψ 1Ĥψ mdτ + c c m ψ Ĥψ mdτ = c nc m ψ nĥψ mdτ m=1 m=1 n=1 m=1

42 15 11 He E n Ĥ ψ n E 0 E n E 0 c n E n c n E 0 c n c n E n E 0 c n n n n c n E n n E 0 c n (11.7) c n n n (11.6) E ϕ ϕ E ϕ E 0 1 d ψ (x) m dx = Eψ (x) (11.8) ( nπ ) ψ n = a sin a x, E n = n π ma n = 1,, 3, (11.9) 36 (11.8) x = 0 x = a 0 0 < x < a 0 ϕ = Ax(a x) A (11.30) ϕ = p /a sin (πx/a) ϕ = Ax(a x) ϕ = Ax(a x) Ĥ = ( /m)(d /dx ) (11.5) x

43 E ϕ = = ϕĥϕdx / a 0 ϕ dx Ax(a x) ( d ) m dx Ax(a x)dx = m A a = m ( A ( ) ax x ) dx = A 0 m = A ( ) a 3 m a3 = A ( ) a 3 3 m 6 a a 0 [ a x 1 ] a 3 x3 0 a = A x (a x) dx = A ( x 4 ax 3 + a x ) dx 0 0 [ 1 = A 5 x5 a 1 4 x4 + a 1 ] a ( ) a 3 x3 = A E ϕ x(a x) d ( ax x ) } dx dx {{} = (11.31) (11.3) E ϕ = A a 3 /6m A a 5 /30 = ( 10 π ) π ma = π ma }{{} E 1 (11.33) E ϕ = Ax(x a) ϕ = Ax(a x) ϕ = Ax p (a x) p p p = E

44 He 11.3 He He He 11.4 ) Ĥ(1, ) = ( 1 Ze + ( m e 4πɛ 0 r 1 m e Ze 4πɛ 0 r ) + e 4πɛ 0 r 1 (11.34) 1 1 He e /(4πɛ 0 r 1 ) 11.4 He Z = Z Z He N Ĥ(1, ) = Ĥ(1) + Ĥ() Ĥ(1)Φ(1) = E 1Φ(1), Ĥ()Φ() = E Φ() Ĥ(1, ) ψ(1, ) ψ(1, ) = Φ(1)Φ() E(1, ) = E 1 + E

45 11.6 He He He ϕ = 1 π ( ) 3 Z e (Z/a 0)(r 1 +r ) a 0 (11.5) He ( Ĥ(1, ) = ( 1 + ) e ) + e (11.53) m e 4πɛ 0 r 1 r 4πɛ 0 r Z Z E ϕ = ϕ Ĥϕdτ Z ϕ ϕdτ ϕ He (11.53) (11.5) ϕ Ĥ(1, ) ( Ĥ 0 (1, ) = ( 1 + ) Ze ) m e 4πɛ 0 r 1 r (11.54) Ĥ(1, ) Ĥ0(1, ) e /(4πɛ 0 r 1 ) e Ze Ĥ(1, ) Ĥ0(1, ) ( Ĥ(1, ) = Ĥ0(1, ) + Ze ) ( e ) + e 4πɛ 0 r 1 r 4πɛ 0 r 1 r 4πɛ 0 r 1 ( = Ĥ0(1, ( Z)e 1 ) + 1 ) + e 4πɛ 0 r 1 r 4πɛ 0 r 1 }{{} Ĥ (1,) = Ĥ0(1, ) + Ĥ (1, ) Ĥ (1, ) (11.55) E ϕ = ϕ Ĥϕdτ ] = ϕ Ĥ 0 (1, ) + Ĥ (1, ) ϕdτ = ϕ Ĥ 0 (1, )ϕdτ + ϕ Ĥ (1, )ϕdτ } {{ } } {{ } =E 0 =E (11.55) = E 0 + E E 0 E (11.56)

46 He (11.40) E 0 = Z E 1s = Z e /(4πɛ 0 a 0 ) E E E = ϕ Ĥ (1, )ϕdτ [ ( = ϕ ( Z)e ) ] + e ϕdv 4πɛ 0 r 1 r 4πɛ 0 r 1 [ ( Z)e = ϕ 1 ϕdv + ϕ 1 ] ϕdv + e 4πɛ 0 r 1 r 4πɛ 0 ( ( Z)e Z = + Z ) + 5Ze 4πɛ 0 a 0 a 0 3πɛ 0 a 0 ϕ 1 r 1 ϕdv Ĥ (1, ) ( Z)Ze = + 5Ze (11.57) πɛ 0 a 0 3πɛ 0 a ϕ 1 r 1 ϕdv = [ ( ) 3 1 Z e (Z/a 0)(r 1 +r )] π a 0 1 r 1 [ ( ) 3 1 Z e (Z/a 0)(r 1 +r )] dv π a 0 = 1 ( ) 6 Z e (Z/a 0 )(r 1 +r ) π dv 1 dv a 0 r 1 = 1 ( ) 6 Z e (Z/a 0 )r 1 π r a 0 r 1dr 1 sin θ 1 dθ 1 dφ 1 e (Z/a 0)r rdr sin θ dθ dφ 1 = 1 π ( Z a 0 ) 6 = 1 π ( Z a 0 ) 6 r 1 e (Z/a 0)r 1 dr 1 sin θ 1 dθ 1 } {{ } dφ 1 }{{} π dv re (Z/a 0)r dr sin θ dθ dφ } {{ }}{{} π 1! (Z/a 0 ) 4π! (Z/a 0 ) 3 4π = Z (C.74) (11.58) a 0 E ϕ = E 0 + E = Z e ( Z)Ze + 5Ze 4πɛ 0 a 0 πɛ 0 a 0 3πɛ 0 a ( 0 = e Z + 7 ) 4πɛ 0 a 0 8 Z (11.57) (11.59) E/ Z = 0 Z ( E Z = e Z + 7 ) = 0 Z = 7 4πɛ 0 a (11.60) (11.59) E = 77.5 ev ev ev

47 171 1 He 1.1 Hartree Fock SCF H He n n Ψ(1,,, n) H He n Ψ(1,,, n) 1 φ 1 (1)φ () φ n (n) 11 φ i (i) i r ij Ψ(1,,, n) 1 φ 1 (1)φ () φ n (n) *1 1 Hartree Slater 1 Hartree Hartree 1 n Ψ(r 1, r,, r n ) = φ 1 (r 1 )φ (r ) φ n (r n ) (1.1) Hartree φ i (r i ) 1 n = Hartree (1.1) Ψ(r 1, r ) = φ 1 (r 1 )φ (r ) (1.) r 1 r Ψ(r, r 1 ) = φ 1 (r )φ (r 1 ) (1.3) *1

48 17 1 Ψ(r 1, r ) = Ψ(r, r 1 ) Hartree Slater Slater ψ 1 (τ 1 ) ψ 1 (τ ) ψ 1 (τ n ) Ψ(τ 1, τ,, τ n ) = 1 n! ψ (τ 1 ) ψ (τ ) ψ (τ n ) ψ n (τ 1 ) ψ n (τ ) ψ n (τ n ) (1.4) Slater ψ i (τ j ) ψ i (τ j ) = φ i (r j ) α(σ j ) ψ i (τ j ) = φ i (r j ) β(σ j ) n = (1.4) Ψ(τ 1, τ ) = 1 ψ 1 (τ 1 ) ψ 1 (τ ) ψ (τ 1 ) ψ (τ ) = 1 ) (ψ 1 (τ 1 )ψ (τ ) ψ 1 (τ )ψ (τ 1 ) (1.5) (1.6) (1.5) τ 1 τ Ψ(τ, τ 1 ) = 1 ψ 1 (τ ) ψ 1 (τ 1 ) ψ (τ ) ψ (τ 1 ) = 1 (ψ 1 (τ )ψ (τ 1 ) ψ 1 (τ 1 )ψ (τ ) = 1 ) (ψ 1 (τ 1 )ψ (τ ) ψ 1 (τ )ψ (τ 1 ) ) (1.7) = Ψ(τ 1, τ ) (1.6) (1.8) Slater 1s α Pauli Slater Slater ψ 1 = φ 1s αψ = φ 1s α (1.9) (1.9) (1.4) Ψ(τ 1, τ ) = 1! φ 1s (r 1 ) α(σ 1 ) }{{} ψ 1 (τ 1 ) φ 1s (r 1 ) α(σ 1 ) }{{} ψ (τ 1 ) φ 1s (r ) α(σ ) }{{} ψ 1 (τ ) φ 1s (r ) α(σ ) }{{} ψ (τ ) = 1 (φ 1s (r 1 )α(σ 1 ) φ 1s (r )α(σ ) φ 1s (r )α(σ ) φ 1s (r 1 )α(σ 1 ) ) (1.10) = 0 (1.11)

49 1.1 Hartree Fock SCF Pauli Slater 0 0 (1.5) (1.7) (1.5) (1.7) (1.10) 1 Pauli Slater 1.1. Hartree Hartree Hartree n * n n e Ĥ = Ĥ c (i) + Ĥc(i) = i Ze 4πɛ i=1 i>j 0 r ij m e 4πɛ }{{} 0 r }{{ i }}{{} Coulomb (1.1) Ĥc(i) i i Coulomb n Ĥ c (i) i 1 i j Coulomb Hartree Ψ(r 1, r,, r n ) = φ 1 (r 1 )φ (r ) φ i (r i ) φ n (r n ) (1.13) φ i (r i ) E = Ψ ĤΨdτ = n i=1 Ψ Ĥ c (i)ψdτ + n i>j Ψ e 4πɛ 0 r ij Ψdτ (1.14) 1 1 n i=1 [ [ ] Ψ Ĥ c (i)ψdτ = φ 1(r 1 )φ (r ) φ i (r i ) φ n(r n )]Ĥc (i) φ 1 (r 1 )φ (r ) φ i (r i ) φ n (r n ) dv = φ 1(r 1 )φ 1 (r 1 )dv 1 φ i (r i )Ĥc(i)φ i (r i )dv i φ n(r n )φ n (r n )dv n }{{}}{{} =1 =1 = φ i (r i )Ĥc(i)φ i (r i )dv i (1.15) * P i>j 08

50 (a) (b)hartree Hartree (a) Schrödinger Hartree 1 Schrödinger 1 i 1 Ψ e [ ] Ψdτ = φ 4πɛ 0 r 1(r 1 )φ (r ) φ i (r i ) φ e [ ] n(r n ) φ 1 (r 1 )φ (r ) φ i (r i ) φ n (r n ) dv ij 4πɛ 0 r ij = φ 1(r 1 )φ 1 (r 1 )dv 1 φ i (r i )φ e j (r j ) φ i (r i )φ j (r j )dv i dv j 4πɛ }{{} 0 r ij =1 φ n(r n )φ n (r n )dv n }{{} =1 = φ i (r i )φ e j (r j ) φ i (r i )φ j (r j )dv i dv j (1.16) 4πɛ 0 r ij i j 1 n n E = φ i (r i )Ĥc(i)φ i (r i )dv i + i=1 i>j e φ i (r i )φ j (r j ) φ i (r i )φ j (r j )dv i dv j (1.17) 4πɛ 0 r ij φ i φ E *3 n e Ĥ φ j (r j ) c (i) + dv j φ i (r i ) = E i φ i (r i ) i = 1,,, n (1.18) 4πɛ 0 r ij j( i) n Hartree φ j (r j ) dv j φ j j dv j i e φ j (r j ) dv j Coulomb *4 Hartree n *3 177 *7 *4

51 1. Slater E 1 E i E E i E, E i E h ev E h 13 E E 1s E s E p E 3s E 3p E 3d E 4s E 4p H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

52 n n = 1,, 3, 4, K L M N n l nl nl n m(= l, l + 1,, l 1, l) m s (= ±1/) (l + 1) n, l n l n =, l = 1 3 (p) 3 * s < s < p < 3s < 3p <(4s, 3d) < 4p <(5s, 4d) < 5p <(6s, 4f, 5d) < 6p <(7s, 5f, 6d) 1.3. Pauli 3. ns np nd nf ns 0 np nd nf Hund * 11 Hund 1 Hund (1.34) n nl K L (s) (1s) (s) (p) 6 *10 p 3 l (p) 3 [1] [8] [9] p 3 Landau[4] Pauling[19] *11 Hund 0

53 s 5d 4f 6s Ba 5d 1 La 4f 1 Ce Hund Hund (c) (a) (b) 1 Coulomb 1.4 Hund (a) (b) (c) Hund 1 1.4(b) (a) Hartree Fock

54 K L M N O 1s s p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 1 H 1 S 1 1/ He 1 S 0 3 Li 1 S 1/ 4 Be 1 S 0 5 B 1 P 1/ 6 C 7 N 3 3 P 0 S 3/ 8 O 4 3 P 9 F 5 P 3/ 10 Ne 6 1 S 0 11 Na 6 1 S 1/ 1 Mg 6 1 S 0 13 Al 6 1 P 1/ 3 14 Si 6 15 P P 0 S 3/ 16 S P 17 Cl 6 5 P 3/ 18 Ar S 0 19 K S 1/ 0 Ca S 0 1 Sc D 3/ Ti F 3 V F 3/ 4 Cr S 3 5 Mn S 5/ 6 Fe D Co Ni F 9/ F 4 9 Cu S 1/ 30 Zn S 0 31 Ga P 1/ 3 Ge P 0 33 As S 3/ 34 Se P 35 Br P 3/ 36 Kr S 0 37 Rb S 1/ 38 Sr S 0 39 Y D 3/ 40 Zr F 41 Nb D 1/ 4 Mo S 3 43 Tc S 5/ 44 Ru F Rh Pd F 9/ S 0 47 Ag S 1/ 48 Cd S 0 49 In P 1/ 50 Sn P 0 51 Sb S 3/ 5 Te P 53 I P 3/ 54 Xe S 0

55 K L M Bk (5f) 8 (6d) 1 (5f) 9 Cf (5f) 9 (6d) 1 (5f) 10 N O P Q 4s 4p 4d 4f 5s 5p 5d 5f 5g 6s 6p 6d 6f 6g 6h 7s 55 Cs S 1/ 56 Ba S 0 57 La D 3/ 58 Ce H 4 59 Pr I 60 Nd I 4 61 Pm H 6 Sm F 0 63 Eu S 7/ 64 Gd D 65 Tb H 17/ 66 Dy I 67 Ho I 68 Er H 69 Tm F 7/ 6 70 Yb S 0 71 Lu D 3/ 7 Hf F 73 Ta F 3/ 74 W D 0 75 Re S 5/ 76 Os D 4 77 Ir F 9/ 78 Pt D 3 79 Au S 1/ 80 Hg S 0 81 Tl P 1/ 8 Pb P 0 83 Bi S 3/ 84 Po P 85 At P 3/ 86 Rn S 0 87 Fr S 1/ 88 Ra S 0 89 Ac D 3/ 90 Th F 91 Pa U Np L 4 94 Pu Am S 7/ 96 Cm Bk (8) 6 (1) 98 Cf (9) 6 (1)

56 * mol Mayer At 86, 87 Rn Fr Einstein 1 hν 1 W 0 E E = W W (> 0) W *1

57 Be, N, Mg, P (ns) (ns) (np) 3 18 (ns) (np) 6 He (1s)

58 Mendeleev Mayer * * 14 Moseley 1913 Mendeleev 1869 Mendeleev * 15 Mendeleev IA VIIA, VIII, IB VIIB, IUPAC * H Be, Mg * *14 Mendeleev Mendeleev Chancourtois Newlands Odling Meyer 4 *15 Mendeleev Mendeleev Ge *16 Mendeleev Li F 7 *17 Ca, Sr, Ba

59 f 5f 1 Sc 39Y 8,9,10 8,9, H 1s He (1s) s p 10 Ne (s) (p) 6 3s 3p 18 Ar (3s) (3p) 6 4s 0 Ca 4s 3 V 3d (3d) 3 (4s) 4 Cr (3d) 4 (4s) (3d) 5 (4s) 1 5 Mn (3d) 5 (4s) 3d 1 8 Ni (3d) 8 (4s) 9 Cu (3d) 9 (4s) (3d) 10 (4s) 1 30 Zn (3d) 10 (4s) 4s 3d 4p 1 Sc 9 Cu Y 47 Ag Ce 4f 57La 71 Lu 57 La 79 Au Pa 5f 89 Ac 103 Lr 93 Np 9 U 38 U 39 U 39 Np β 39 Pu Uus

60 He (1s) Be (1s) (s) Ne (1s) (s) (p) 6 B (1s) (s) (p) 1 C (1s) (s) (p) 1 1 C (1s) (s) (p) p 6 C 6 C = 15 1 * ˆl ŝ * 19 n ˆL = i ˆli 31 (.13) (1.41) Ŝ = i ŝ i (1.4) * 0 l ˆ Y l,m = l(l + 1) Y l,m l = 0, 1,, n 1 (1.43) ˆlz Y l,m = m l Y l,m m = l l, l + 1,, l 1, l ŝ Γ = s(s + 1) Γ s= 1/ ŝ z Γ = m s Γ m = s 1/, 1/ (1.44) L, M L, S, M S L ˆ : L(L + 1) ˆL z : M L M L = L, L + 1,, L 1, L (1.45) *18 9 * *0 ˆl, ˆl z (9.69) (9.70) (9.69) (9.70) Ψ Ψ = R n,l (r) Y l,m (θ, φ) ˆl, ˆl z r Y l,m (θ, φ) ˆl, ˆl z (10.4) (10.5)

61 ˆ S : S(S + 1) Ŝ z : M S M S = S, S + 1,, S 1, S (1.46) z M L z m l z M S M L = i (m l ) i M S = i (m s ) i (1.47) n ˆF (τ 1, τ,, τ n ) ˆL z Ŝz ˆL n z (r 1, r, r n ) = ˆF (τ 1, τ,, τ n ) = ˆf(τ i ) i=1 Ŝ z (σ 1, σ,, σ n ) = n ˆlz (r i ) i=1 n ŝ z (σ i ) i=1 (1.48) ˆF Ψ = ψ 1 ψ ψ n ψ 1, ψ,, ψ n n n Slater ψ 1 (τ 1 ) ψ 1 (τ ) ψ 1 (τ n ) n ˆF Ψ(τ 1, τ,, τ n ) = ˆf(τ i ) 1 ψ (τ 1 ) ψ (τ ) ψ (τ n ) i=1 n! ψ n (τ 1 ) ψ n (τ ) ψ n (τ n ) ˆf(τ 1 )ψ 1 (τ 1 ) ψ 1 (τ ) ψ 1 (τ n ) ψ 1 (τ 1 ) ˆf(τ )ψ 1 (τ ) ψ 1 (τ n ) = 1 ˆf(τ 1 )ψ (τ 1 ) ψ (τ ) ψ (τ n ) n! ψ (τ 1 ) ˆf(τ )ψ (τ ) ψ (τ n ).... n! ˆf(τ 1 )ψ n (τ 1 ) ψ n (τ ) ψ n (τ n ) ψ n (τ 1 ) ˆf(τ )ψ n (τ ) ψ n (τ n ) ψ 1 (τ 1 ) ψ 1 (τ ) ˆf(τn )ψ 1 (τ n ) + 1 ψ (τ 1 ) ψ (τ ) ˆf(τn )ψ (τ n ) n! ψ n (τ 1 ) ψ n (τ ) ˆf(τn )ψ n (τ n ) = ˆfψ 1 ψ ψ n + ψ 1 ˆfψ ψ n + + ψ 1 ψ ˆfψn (1.49) ˆL z n n m l1, m l,, m ln 1 φ ml1, φ ml, φ mln ˆL z Ψ = ˆl z φ ml1 φ ml φ mln + φ ml1 ˆlz φ ml φ mln + + φ ml1 φ ml ˆl z φ mln = m l1 φ ml1 φ ml φ mln + φ ml1 m l φ ml φ mln + + φ ml1 φ ml m ln φ mln (1.43) = m l1 φ ml1 φ ml φ mln + m l φ ml1 φ ml φ mln + + m ln φ ml1 φ ml φ mln (C.85) ( n = (m l ) i ) φ ml1 φ ml φ mln (1.50) i

62 19 1 ˆL z M L M L = n i (m l) i M S n m s1, m s,, m sn 1 φ ms1, φ ms, φ msn Ŝ z Ψ = ŝ z φ ms1 φ ms φ msn + φ ms1 ŝ z φ ms φ msn + + φ ms1 φ ms ŝ z φ msn = m s1 φ ms1 φ ms φ msn + φ ms1 m s φ ms φ msn + + φ ms1 φ ms m sn φ msn (1.43) = m s1 φ ms1 φ ms φ msn + m s φ ms1 φ ms φ msn + + m sn φ ms1 φ ms φ msn (C.85) ( n = (m s ) i ) φ ms1 φ ms φ msn (1.51) i Ŝz M S M S = n i (m s) i (1.47) 1.6. LS Ĥ, ˆL, ˆL z, ˆ S, Ŝz * 1 III 19 n, L, M L, S, M S Ψ (n, L, M L, S, M S ) Ĥ, ˆL, ˆL z, ˆ S, Ŝz n Ψ(n, L, M L, S, M S ) ˆL z, Ŝz M L, M S (L + 1)(S + 1) Schödinger ĤΨ = EΨ E E = E(n, L, S) (1.5) n L S * n, L, S n S+1 {L} (1.53) n S + 1 S + 1 {L} L L {L} S P D F G H I J K L (L + 1)(S + 1) LS LS * Ĥ, ˆL, ˆL z, Ŝ, Ŝz * 1 n l

63 Ψ(n, L, M L, S, M S ) ˆL z, Ŝz M L, M S (L + 1)(S + 1) ĤΨ (n, L, M L, S, M S ) = EΨ (n, L, M L, S, M S ) (1.54) ˆL ± ˆL ± ĤΨ (n, L, M L, S, M S ) = ˆL ± EΨ (n, L, M L, S, M S ) Ĥ ˆL ± Ψ (n, L, M L, S, M S ) = E ˆL ± Ψ (n, L, M L, S, M S ) ĤΨ (n, L, M L ± 1, S, M S ) = EΨ (n, L, M L ± 1, S, M S ) (1.54) ˆL ± ˆL± Ĥ ˆL± M L ± 1 (1.55) ˆL ± Ĥ ˆL ± [ˆLx, Ĥ ] = [ˆLy, Ĥ ] = (1.55) E Ψ (n, L, M L ± 1, S, M S ) E M L = L, L + 1,, L 1, L E L + 1 ˆL ± Ŝ ± := Ŝx ± iŝy (1.56) (1.54) ˆL ± E S + 1 E (L + 1)(S + 1) Russell Saunders LS Ĥ Coulomb Ĥ ls = i ζ iˆli ŝ i (1.57) * 3 ζ i Z 4 Ĥ ˆL ˆ S L S Ĵ Ĥ, ˆ J, Ĵz LS n Ĵ J n J Ĥls Ĥls LS Ĥls LS *3 B l B l B s µ e E = µ e B µ e l µ e s ζ E = ζsl (1.57)

64 194 1 Russell Saunders J LS n S+1 {L} J (1.58) ˆl ŝ ĵ = ˆl + ŝ (1.59) (1.41) (1.4) n n Ĵ = i ĵ i (1.60) (1.59) (1.45) (1.46) Ĵ Ĵ = ) (ˆli + ŝ i = ˆli + ŝ i (1.60) (1.59) i i i = ˆL + Ŝ (1.41) (1.4) (1.61) Ĵz ˆL z Ŝz Ĵ z = ˆL z Ŝz (1.6) ˆL ˆ S L S ˆ J J L M L ψ L,ML S M S φ S,MS ˆL ψ L,ML = L(L + 1) ψ L,ML ˆLz ψ L,ML = M L ψ L,ML (1.63) Ŝ φ S,MS = S(S + 1) φ S,MS Ŝ z φ S,MS = M S φ S,MS (1.64) Ψ = ψ L,ML φ S,MS Ĵz ) Ĵ z Ψ = (ˆLz + Ŝz ψ L,ML φ S,MS ) ) = φ S,MS (ˆLz ψ L,ML + ψ L,ML (Ŝz φ S,MS (1.6) ˆL z ψ L,ML Ŝz φ S,MS = φ S,MS (M L ψ L,ML ) + ψ L,ML (M S φ S,MS ) (1.63) (1.64) = (M L + M S ) }{{} =:M J Ψ = M J Ψ M J = M L + M S (1.65) Ψ Ĵz (M L + M S ) (1.65) M J = (M L + M S ) M J = (M L + M S ) M L = L, L + 1,, L 1, L M S = S, S + 1,, S 1, S M J L + S M L = L L + 1 L 1 L + M S = S S + 1 S 1 S M J = L S, L S + 1, L S +, L + S, L + S 1, L + S

65 M J = J, J + 1,, J 1, J J L + S J = L + S, L + S 1, J J L = 1 S = 1 M L M S M L = 1, 0, 1 M S = 1, 0, 1 3 M J M J =, 1, 0, 1, 5 M L = M S = M J = M J = (M L, M S ) = (1, 1) M J = 1 (M L, M S ) = (1, 0) (M L, M S ) = (0, 1) M J = 0 (M L, M S ) = (0, 0), ( 1, 1), (1, 1) 3 M L = M S = M J = (M L, M S ) ( 1, 1) ( 1, 0) (0, 0) (1, 0) (1, 1) (0, 1) ( 1, 1) (0, 1) (1, 1) L S M L M S (L + 1)(S + 1) M J J J = L + S J = = M J =, 1, 0, 1, 5 9 J = 1 J = 1 M J = 1, 0, 1 3 J = 8 9 J = 0 M J = J = 3,, 1 L S J = L + S, L + S 1, L + S L = 1 S = 1 L S J L S L = 1 S = 1 J = 0 L + S L S Ĵ1 Ĵ Ĵ Ĵ1 Ĵ J 1 J Ĵ J J = J 1 + J, J 1 + J 1,, J 1 J ˆ J : J(J + 1) J = L + S, L + S 1,, L S Ĵ z : M J M J = M L + M S L = S = 1 J L = M L =, 1, 0, 1, S = 1 M S = 1, 0, 1 M J = M L + M S M J M J = 3,, 1, 0, 1,, 3 7 M J M L M S M J = 3 (M L, M S ) = (, 1) M J = (M L, M S ) = (, 0), (1, 1), (0, ) M J = 1 (M L, M S ) = (, 1), (1, 0), (0, 1) M J = 0 (M L, M S ) = (1, 1), (0, 0), ( 1, 1) M J = 1 (M L, M S ) = (0, 1), ( 1, 0), (, 1) M J = (M L, M S ) = (, 0)

66 196 1 M J = 3 (M L, M S ) = (, 1) 15 M J 3 J 3 J = 3 M J = 3,, 1, 0, 1,, J = J = M J =, 1, 0, 1, 5 3 J = 1 M J = 1, 0, L = S = 1 J J = 3,, 1 J 1 1 = 1 = L S n S+1 {L} J 1 n S + 1 {L} J (1s) 1 (1s) 1 1 n = 1 1 l = 0 s m l = l, l + 1,, l 1, l (1.43) m l = 0 1 M L = (m l ) i = m l M L = 0 i=1 M L = L, L + 1,, L 1, L (1.45) L = S 1 S s = 1/ 1 m s = +1/ m s = 1/ 1 M S = (m s ) i = m s M S = +1/ M S = 1/ i=1 M S = S, S + 1,, S 1, S (1.46) S = 1/ S + 1 = 1 S J = L + S, L + S 1,, L S S = 1/ L = 0 J = 1/ 1 S 1/ 1 n = 1 1 l = 0 s m l = l, l + 1,, l 1, l (1.43) m l = 0 M L M S 1.5 M L M L max M L max = 0

67 M L = L, L + 1,, L 1, L (1.45) L = 0 * 4 1 S M L = 0 [1] M S = 1/ [] M S = 1/ L = 0 S = 1/ 1 S J = L + S, L + S 1,, L S S = 1/ L = 0 J = 1/ 1 S 1/ (M L, M S ) (0, 1/) (0, 1/) LS 1 S 1/ 1.5 (1s) m l = 0 M L = (m l ) i M S = (m s ) i i=1 i=1 [1] 0 1/ 1 S 1/ [] 0 1/ 1 S 1/ n = 1 M L max = 0 L = 0 M L = 0 [1] (M S = 1/) M L = 0 [] (M S = 1/) } S = 1/ J = 1/ 1 S 1/ : (0, 1/) (0, 1/) (0, 1/) (0, 1/) (M L, M S ) (s) 1 (s) 1 S 1/ (1s) 1 (s) 1 (3s) 1 n S 1/ He (1s) He M L M S L = 0, S = S He (1s) m l = 0 M L = (m l ) i M S = (m s ) i i=1 i=1 [1] = 1/ + ( 1/) = S 0 M L M S 0 *4

68 198 1 Li (1s) (s) 1 Li M L M S L = 0, S = 1/ S 1/ 1.7 Li (1s) (s) 1 1s m l = 0 s m l = 0 M L = 3 (m l ) i M S = i=1 [1] (0 + 0) + }{{}}{{} 0 1s s [] (0 + 0) + }{{}}{{} 0 1s s 3 (m s ) i i=1 = 0 1/ + ( 1/) }{{} 1s = 0 1/ + ( 1/) }{{} 1s + 1/ = 1/ S }{{} 1/ s + ( 1/) = 1/ S }{{} 1/ s (p) 1 1 (p) 1 1 n = l = 1 p m l = l, l + 1,, l 1, l (1.43) m l = 1, 0, +1 1 M L = (m l ) i = m l M L = 1, 0, +1 i=1 M L = L, L + 1,, L 1, L (1.45) L = 1 * P P s = 1/ 1 m s = +1/ m s = 1/ 1 M S = (m s ) i = m s M S = +1/ M S = 1/ i=1 M S = S, S + 1,, S 1, S (1.46) S = 1/ S + 1 = P J = L + S, L + S 1, L S L = 1, S = 1/ J = 3/, 1/ P 1/, P 3/ *5 L = 0 L = 1 M L = 1, 0, +1 M L = 1 M L = 1 L = 1 M L = 0 L = 1 M L = 0 L = 0 M L = 0 L = 1 M L = 1, 0, (p) 1 L =

69 n = l = 1 p m l = l, l + 1,, l 1, l (1.43) m l = 1, 0, +1 M L M S 1.8 M L M L max M L max = 1 M L = L, L + 1,, L 1, L (1.45) L = 1 P M L = 1, 0, 1 M L = 1 [5] [6] M S = 1/ M S = 1/ M L = 1 [1] [] M S = 1/ M S = 1/ L = 1 S = 1/ P 1.8 (p) 1 m l M L = i (m l ) i M S = i (m s ) i [1] 1 1/ P [] 1 1/ P [3] 0 1/ P [4] 0 1/ P [5] 1 1/ P [6] 1 1/ P J = L + S, L + S 1, L S L = 1, S = 1/ J = 3/, 1/ P 1/, P 3/ (M L, M S ) (1,1/) (1, 1/) (0,1/) (0, 1/) ( 1,1/) ( 1, 1/) LS P P 3/ P 1/ J n = M L max = 1 L = 1 M L = 1, 0, 1 M L = 1 [1] (M S = 1/) [] (M S = 1/) M L = 0 [3] (M S = 1/) [4] (M S = 1/) M L = 1 [5] (M S = 1/) [6] (M S = 1/) S = 1/ J = 1/, 3/ P : ( 1, 1/) ( 1, 1/) (0, 1/) (0, 1/) (1, 1/) (1, 1/)

70 00 1 C (1s) (s) (p) C (1s) (s) (p) (1s) (s) M L = 0 M S = 0 L = 0 S = 0 J = 0 (p) 1 n = l = 1 p m l = l, l + 1,, l 1, l (1.43) m l = 1, 0, +1 M L = (m l ) i M L =, 1, 0, +1, + Hund 1 18 i=1 M L =, + M L = 1, 0, +1 M L = L, L + 1,, L 1, L (1.45) L = P P s = 1/ m s (1/, 1/) ( 1/, 1/) (1/, 1/) ( 1/, 1/) 4 Hund (1/, 1/) ( 1/, 1/) M S = (m s ) i = 1, 1 M S = S, S + 1,, S 1, S (1.46) i=1 S = 1 S + 1 = 3 3 P J = L + S, L + S 1, L S L = 1, S = 1 J =, 1, 0 3 P, 3 P 1, 3 P (1s) (s) (p) Hund 3 P 15 n = l = 1 p m l = l, l + 1,, l 1, l (1.43) m l = 1, 0, +1 M L M S 1.9 M L M L max M L max = M L = L, L + 1,, L 1, L (1.45) L = D M L =, 1, 0, 1, M L = [1] M S = 0 M L = [15] M S = 0 L = S = 0 S + 1 = 1 1 D J = L + S, L + S 1, L S L =, S = 0 J = 1 D

71 (1s) (s) (p) m l M L = i (m l ) i M S = i (m s ) i [1] 0 1 D [] P [3] 1 0 [4] D, 3 P [5] P [6] P [7] 0 0 [8] D, 3 P, 1 S 0 [9] 0 0 [10] P [11] P [1] 1 0 [13] D, 3 P [14] P [15] 0 1 D (M L, M S ) (, 0) ( 1, 0) (0, 0) (1,0) (,0) LS 1 D M L M L max 1 1 [1] [15] [3] [4] (M L, M S ) (1,0) 1 1 D 1 M L max M L max = 1 M L = L, L + 1,, L 1, L (1.45) L = 1 P M L = 1, 0, 1 M L = 1 [] [5] M S = 1, 1 M L = 1 [11] [14] M S = 1, 1 L = 1 S = 1 S + 1 = 3 3 P J = L + S, L + S 1, L S L = 1, S = 1 J =, 1, 0 3 P, 3 P 1, 3 P 0 (M L, M S ) (1,1) (1,0) (1, 1) (0,1) (0,0) (0, 1) ( 1,1) ( 1,0) ( 1, 1) 3 P 3 P 3 P 1 J M L M L max [7][8][9] 1 M L = 0 M S = 0 L = 0 S = 0 1 S 0 [7][8][9] LS 1 S 0

72 0 1 1 st STEP n = M L max = L = M L =, 1, 0, 1, M L = [1] (M S = 0) M L = [15] (M S = 0) } S = 0, J = 1 D : (, 0) ( 1, 0) (0, 0) (1, 0) (, 0) nd STEP M L max = 1 L = 1 M L = 1, 0, 1, M L = 1 [] (M S = 1) [5] (M S = 1) M L = 1 [11] (M S = 1) [14] (M S = 1) 3 rd STEP S = 1, J =, 1, 0 3 P, 3 P 1, 3 P 0 : (1, 1) (1, 0) (1, 1) (0, 1) (0, 0) (0, 1) ( 1, 1) ( 1, 0) ( 1, 1) (M L, M S ) = (0, 0) 1 L = 0, S = 0 1 S 0 : (0, 0) Hund Hund Hund Hund 1.. L Hund J Ĥ ls = i ζ iˆli ŝ i (1.57) 1 LS Ĥls Ĥ ls = λ ˆLŜ (1.66)

73 * 6 ( ) Ĵ = ˆL + Ŝ = ˆL + ˆLŜ + Ŝ ˆL + Ŝ = ˆL + ˆLŜ + Ŝ ˆL Ŝ (1.67) Ĥls Ĥ ls = λ (Ĵ ˆL Ŝ ) (1.68) λ Ψ(J, M J ) E ls = Ψ (J, M J )ĤlsΨ(J, M J )dτ (1.69) Ψ(J, M J ) J L S Ĵ ˆL Ŝ * 7 Ĥ ls Ψ(J, M J ) = λ [J(J + 1) L(L + 1) S(S + 1)] Ψ(J, M J) (1.70) (1.69) E ls = λ [J(J + 1) L(L + 1) S(S + 1)] (1.71) LS L S J(J + 1) λ λ nl (l + 1) i * 8 λ > 0 i < l + 1 λ = 0 i = l + 1 λ < 0 i > l + 1 (1.7) J J (np) a (nd) b a < 6 b < 10 J J J J 1 (1.71) E ls,j E ls,j 1 = λ [J(J + 1) (J 1)J] = λ J (1.73) J Landé *6 *7 Ĥ ls 11.8 Ψ(J, M J ) 0 Ψ(L, M L, S, M S ) *8

74 n l m (s) 1 (p) 1 (3s) 1 (3p) 1 (3d) (p) 1 P 1/ P 3/. (s) 1 (p) 1 P 1/ J P 1/ P 3/ cm 1 * 9 P 3/ P 1/ (p) 1 6 p 1 J J P 3/ P 1/. S 1/ P 1/ 0.03 cm 1 Lamb *9 cm cm 1 1/(0.365 cm 1 ) =.75 cm cm 1 1 cm 1 = J cm 1

75 SI SI a.u. SI m e = kg e = C Planck = h π = J s 4πɛ SI m e 1 e 1 1 4πɛ 0 4πɛ 0 1 Boh a 0 = ɛ 0h πm e e (SI) Bohr = 4πɛ 0 4π = 4πɛ 0 m e e (π ) πm e e h = π 4πɛ 0 a.u. 1 (a.u.) 4πɛ 0 = 1, = 1, m e = 1, e = 1 (13.1) 1 E 1s = m ee 4 8ɛ 0 h (SI) (9.5) Z = 1, n = 1 = m ee 4 8ɛ 0 (π ) = m ee 4 (4πɛ 0 ) h = π 4πɛ 0 a.u. 1 (a.u.) m e = 1, e = 1, 4πɛ 0 = 1, = 1 (13.)

76 1 13 1/ hartree E h 1E h := E 1s (13.3) l = mrv v = l/(mr) = = m e a 0 a.u. 1 (a.u.) = 1, m e = 1, a 0 = 1 (13.4) 1 = a 0 = /(m e a 0 ) a.u. 1 (a.u.) = 1, m e = 1, a 0 = 1 (13.5) SI SI a m m e kg e C J s 4πɛ F m 1 E 1s J 1 m e a 0 a 0 / (m e a 0 ) ms s Ĥ = e m e 4πɛ 0 r (SI) Ĥ = 1 1 r (a.u.) (13.6) ψ 1s = 1 πa 3 0 e r/a 0 (SI) ψ 1s = 1 π e r (a.u.) (13.7) *1 *1 4πɛ 0

77 13 14 H + 1 Coulomb Born Oppenheimer 14.1 Born Oppenheimer * Born Oppenheimer Born Oppenheimer * R R *1 F a = F/m p a = F/m e (F/m p )/(F/m e ) = m e /m p A.1 A.1 1/1000 * Born Oppenheimer

78 ϕ ϕ MO a, b R a, b r a, r b 14.1 ϕ φ a φ b Ĥ = 1 1 r a 1 r b + 1 R (14.1) a a Coulomb φ a b φ b ϕ = c a φ a + c b φ b (14.) LCAO MO LCAO ϕ Ritz *3 Ritz H aa E ϕ S aa H ab E ϕ S ab = 0 (11.68) (14.3) H ba E ϕ S ba H bb E ϕ S bb (14.) φ a φ b S aa = S bb = 1 H aa E ϕ H ab E ϕ S ab H ba E ϕ S ba H bb E ϕ = 0 (14.4) H aa = H bb S ab = φ a φ b dτ = φ b φ a dτ = S ba H ab = φ a Ĥφ b dτ = H ba Ĥ Hermite * 4 (14.5) *3 Ritz

79 H aa E ϕ H ab E ϕ S ab H ab E ϕ S ab H aa E ϕ = 0 (14.6) ϕ E ϕ (14.6) E ϕ E ϕ (H aa E ϕ ) (H ab E ϕ S ab ) = 0 E ϕ E ϕ = H aa ± H ab 1 ± S ab (14.7) c a c b (H aa E ϕ ) c a + (H ab E ϕ S ab ) c b = 0 (H ab E ϕ S ab ) c a + (H aa E ϕ ) c b = 0 (14.8) c a /c b c a = H ab E ϕ S ab c b H aa E ϕ ( ) Haa ± H ab H ab S ab 1 ± S ab = ( ) (14.7) Haa ± H ab H aa 1 ± S ab (14.8) = ±1 (14.9) (14.8) ϕ = c a (φ a ± φ b ) (14.10) ( ) ϕ dv = c a φ adv + φ bdv ± φ a φ b dv (14.) }{{} } {{ } } {{ } =1 =1 S ab c a = = c a ( ± S ab ) = 1 1 ± Sab c a (14.11) *4 H ab = H ba Z «Hab = φ aĥφ bdτ Z = φ bĥφ adτ Ĥ Hermite = H ba φ a φ b Hab = H ab Hab = H ba H ab H ba

80 ϕ g = (φ a + φ b ) E g = H aa + H ab + Sab 1 + S ab ϕ u = 1 Sab (φ a φ b ) E u = H aa H ab 1 S ab (14.1) ϕ E g u gerade ungerade ϕ ϕ 14.. H aa (11.63) (14.1) H aa [ H aa = φ a ] φ a dv r a r b R [ = φ a 1 1 ] φ φ a dv a dv + 1 φ r a r b R adv φ a = φ a }{{} E 1s φ a = E 1s φ adv E aa + 1 φ φ R adv E aa := a dv r b = E 1s E aa + 1 R E 1s 1/R E 1s 1s 1 Coulomb R E aa φ a dv φ adv b Coulomb φ a b Coulomb H ab φ a (14.13) Z 14. E aa = `φ a/r b dv φ a b Coulomb H ab [ H ab = φ a ] φ b dv r b r a R [ = φ a 1 1 ] φa φ a φ b dv dv + 1 φ a φ b dv r b r a R }{{} E 1s φ b = E 1s φ a φ b dv E ab + 1 φa φ b φ a φ b dv E ab := dv R = E 1s S ab E ab + S ab R r a (14.14)

81 E ab E aa φ a φ b φ a φ b a Coulomb φ a φ b a Coulomb H aa H ab E aa, E ab, S ab φ a, φ b 1s (13.7) Ψ = 1 π e r a.u. ) S ab = (1 + R + R e R 3 E aa = 1 [ 1 (1 + R)e R ] R E ab = (1 + R)e R (14.15) 14.3 Z E ab = (φ a φ b /r a ) dv φ a φ b a Coulomb 14.3 (14.13) (14.14) H aa H ab R *5 H aa = 1 ( ) 1 + R + 1 e R ( 1 H ab = S ab R 1 ) (14.16) (1 + R)e R H aa H ab S ab R > a 0 H ab < 0 R 14.4 S ab H aa H ab R S ab > 0 (14.1) S ab > 0 H ab < 0 E g < E u S ab H ab E g ϕ g H ab H ab (14.13) (14.14) (14.1) [ 1 E g = E 1s (1 + S ab ) (E aa + E ab ) + 1 ] 1 + S ab R (1 + S ab) = E 1s + 1 R E aa + E ab (14.17) 1 + [ S ab 1 E u = E 1s (1 S ab ) (E aa E ab ) + 1 ] 1 S ab R (1 S ab) = E 1s + 1 R E aa E ab 1 S ab (14.18) (14.15) (14.17) (14.18) E g E u R 14.5 E h hartree E u E g E u E 1s R E 1s E u R E u E 1s *5 E 1s = 1/

82 E u E g E u E g [1] [1] E u R 0 E g R 0 R/a 0 =.50 R e = 0.13 nm E u E 1s E u E 1s ϕ u R E u E g 14.6 a b H aa (= H bb ) a b MO E g E u E g E g ϕ g ϕ u ϕ u 14.6 ϕ g ϕ u 14.7 Bohr Bohr.5

83 ϕ g ϕ u a b 14.7 a b 1 4 *6 a b 1/9 b Feynmann *6 Coulomb 97 (9.17) Coulomb r Coulomb

84 S ab, H aa, H ab * (a) F(a, 0) F ( a, 0) xy x 3 x 14.8 ξ := (r a + r b )/a η := (r a r b )/a ξ, η x ϕ (a) xy (b) x F P + P F = r a + r b = a 1 (14.19) F P P F = r a r b = ±a (14.0) r a r b b 1 = a 1 a, b = a a (14.1) x a 1 x a + y b 1 y b = 1 = (14.) (14.3) *7 [10] Daudel[0] [1] 1999