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- みひな かたいわ
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3 π Hückel A-1. Laguerre 165 A-2. Hermite 167 A A A A A III Streitwieser, Molecular Orbital Theory for Organic Chemists, Wiley Eyring, Walter, and Kimball, Quantum Chemistry, Wiley
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5 1 Schrödinger 1.1 Coulomb r 1 = (x 1, y 1, z 1 ) q 1 r 2 = (x 2, y 2, z 2 ) q 2 q 2 q 1 F 12 (1.1) F 12 = q 1q 2 r 12 4πε 0 r12 2 r 12 (1.2) r 12 = r 1 r 2 = x 1 x 2 y 1 y 2 z 1 z 2 (1.3) r 12 = r 12 = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 ε 0 r 12 /r 12 q 1 q 2 F 21 (1.4) F 21 = F 12 V (r 1, r 2 ) r 12 = r 12 (1.5) V (r 1, r 2 ) = q 1q 2 4πε 0 r 12 = V (r 12 ) r = (x, y, z) e +e F (1.6) F = e2 r 4πε 0 r 2 r (1.7) r = r = x 2 + y 2 + z 2 V (r) (1.8) V (r) = e2 4πε 0 r 1.2 Schrödinger m p +e m e e µ (1.9) µ = m pm e m p + m e 107
6 z r cos θ electron (m, e) e (x,y,z) = (r,θ,φ) proton (m, +e) p φ θ r r sin θ r sin θ sin φ y r sin θ cos φ (x,y,0) x Schrödinger ( ) (1.10) h2 2 2µ x y z 2 Ψ(x, y, z) e2 Ψ(x, y, z) = EΨ(x, y, z) 4πε 0 r Coulomb r r (x, y, z) (r, θ, ϕ) (1.11) (1.12) (1.13) x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ Schrödinger Ψ(r, θ, ϕ) Ψ [ (1.14) h2 1 2µ r 2 ( r 2 Ψ ) + r r 1 r 2 sin θ ( sin θ Ψ ) + θ θ 1 r 2 sin 2 θ 2 ] Ψ ϕ 2 e2 4πε 0 r Ψ = EΨ (1.15) ( r 2 Ψ ) + 1 ( sin θ Ψ ) ) Ψ (E r r sin θ θ θ sin 2 θ ϕ 2 + 2µr2 h 2 + e2 Ψ = 0 4πε 0 r 1.3 Ψ R(r) Y (θ, ϕ) (1.16) Ψ = Ψ(r, θ, ϕ) = R(r)Y (θ, ϕ) = RY (1.17) ( 1 r 2 R ) ) + (E 2µr2 R r r h 2 + e2 = 1 4πε 0 r Y [ ( 1 sin θ Y ) ] Y sin θ θ θ sin 2 θ ϕ 2 r θ, ϕ λ (1.18) (1.19) ( r 2 R ) [ ) ] 2µr 2 + (E r r h 2 + e2 λ R = 0 4πε 0 r 1 sin θ θ ( sin θ Y θ ) Y sin 2 θ ϕ 2 + λy = 0 108
7 1.4 Schrödinger III (1.20) Y l,ml (θ, ϕ) = ( 1) (m l+ m l )/2 2l + 1 (l m l )! 4π l + m l )! P m l l (cos θ)e im lϕ (1.21) P m l l (z) = (1 z 2 ) m l /2 d ml Pl 0(z) (1.22) z = cos θ (1.23) Pl 0 = 1 2 l l! d l dz l (z2 1) l dz m l l (= 0, 1, 2, ) L 2 (1.24) L 2 = h 2 λ = h 2 l(l + 1) m l ( l m l l) z L z (1.25) L z = hm l 1.5 A-1 4Z (1.26) R n,l (r) = 3 (n l 1)! a 3 0 n4 [(n + l)!] 3 ρl e ρ/2 L 2l+1 n+l (ρ) (1.27) ρ = 2r na 0 (1.28) a 0 = 4πε 0 h 2 µe 2 L s k (ρ) Laguerre (1.29) L s k(ρ) = ds dρ s L k(ρ) (1.30) L k = e ρ dk dρ k (ρk e ρ ) n (= 1, 2, ) E n (1.31) E n = w 0 n 2 (1.32) w 0 = µ ( ) e h 2 4πε 0 l 0 l n 1 109
8 1.6 n = 1, 2, 3, l = 0, 1, 2,, n 1 m l = l, l + 1,, 0,, l 1, l s l = 0 p l = 1 d l = 2 1s, 2s, 2p x, 2p y, 2p z, 3s, 1.7 (r, θ, ϕ) dτ = dxdydz = r 2 sin θdrdθdϕ P (r, θ, ϕ)dτ P (r, θ, ϕ) dτ P (r, θ, ϕ) (1.33) P (r, θ, ϕ) = Ψ n,l,ml (r, θ, ϕ) 2 r r + dr P r (r)dr (1.34) P r (r)dr = r 2 π 2π θ=0 ϕ=0 sin θ Ψ n,l,ml (r, θ, ϕ) 2 dθdϕdr l = 0, m = 0 s Ψ n,l,m (r, θ, ϕ) θ, ϕ Ψ(r) (1.35) P r (r)dr = 4πr 2 Ψ(r) 2 dr 4πr 2 r 4πr 2 dr r dr 1.8 A Â A A (1.36) A = dr π dθ 2π r (1.37) r = = = 0 0 dr dr dr π 0 π 0 π dθ dθ dθ 2π 0 2π 0 2π K dϕ r 2 sin θ Ψ n,l,m l (r, θ, ϕ)âψ n,l,m l (r, θ, ϕ) dϕ r 2 sin θ Ψ n,l,m l (r, θ, ϕ)ˆrψ n,l,ml (r, θ, ϕ) dϕ r 2 sin θ Ψ n,l,m l (r, θ, ϕ)rψ n,l,ml (r, θ, ϕ) dϕ r 3 sin θ Ψ n,l,ml (r, θ, ϕ) 2 (1.38) K = = 0 0 dr dr π 0 π 0 dθ dθ 2π 0 2π 0 dϕ r 2 sin θ Ψ n,l,m l (r, θ, ϕ) ˆKΨ n,l,ml (r, θ, ϕ) ) dϕ r 2 sin θ Ψ n,l,m l (r, θ, ϕ) ( h2 2µ 2 Ψ n,l,ml (r, θ, ϕ) 110
9 V (1.39) V = = 0 0 dr dr π 0 π 0 dθ dθ 2π 0 2π 0 dϕ r 2 sin θ Ψ n,l,m l (r, θ, ϕ) ˆV Ψ n,l,ml (r, θ, ϕ) ) dϕ r 2 sin θ Ψ n,l,m l (r, θ, ϕ) ( e2 4πε 0 r Ψ n,l,m l (r, θ, ϕ) 1.9 s l = 0 (1.40) Y 0,0 (θ, ϕ) = 1 4π z θ 2 Y z x φ y x y s 1s (1.41) Ψ 1s = 1 ( ) 3/2 ) 1 exp ( ra0 π a 0 1s 1 (1.42) (1.43) r = = 4 a 3 0 dr π dθ 2π 0 r 3 e 2r/a 0 dr = 3 2 a 0 (1.44) P r (r) = 4πr 2 Ψ 2 1s = 4r2 a 3 e 2r/a 0 0 dϕ r 2 sin θ Ψ 1s(r, θ, ϕ)rψ 1s (r, θ, ϕ) (1.45) dp r (r) dr = 8r(a 0 r) a 4 e 2r/a 0 = 0 0 r = a s 2s (1.46) Ψ 2s = 1 ( ) 3/2 ) ( 1 (2 4 ra0 exp r ) 2π a 0 2a 0 2s 1s r = 2a 0 ns (n 1) 111
10 1.10 p l = 1 R n,1 3 (1.47) 3 Y 1,0 = 4π cos θ (1.48) 3 Y 1,+1 = sin θe+iϕ 8π (1.49) 3 Y 1, 1 = sin θe iϕ 8π 3 z 2p r (1.50) Ψ 2p0 = 4 e r/2a 0 cos θ 2πa 5 0 r (1.51) Ψ 2p+ = 8 e r/2a 0 sin θe +iϕ πa 5 0 r (1.52) Ψ 2p = 8 e r/2a 0 sin θe iϕ πa 5 0 p 0 z p ± z z z x y x y p 0 (m l = 0) p ± (m l = ±1) p p 3 N N N N 2p (1.50) (1.52) n, l, m (1.53) Ψ 2px = 1 2 (Ψ 2p+ + Ψ 2p ) = (1.54) Ψ 2py = i 2 (Ψ 2p+ Ψ 2p ) = r (1.55) Ψ 2pz = Ψ 2p0 = 4 e r/2a 0 cos θ 2πa 5 0 r 4 e r/2a 0 sin θ cos ϕ 2πa 5 0 r 4 e r/2a 0 sin θ sin ϕ 2πa 5 0 (1.53) (1.55) 2p x, 2p y, 2p z 112
11 z z z x y x y x y p x p y p z (1.50) (1.52) (1.53) (1.55) 2p 1.11 d 2 z z z z x y x y x y d 0 (m = 0) d ±1 (m = ±1) d ±2 (m = ±2) z z z z z x y x y x y x y x y d z 2 d xy d z2 y 2 d yz d zx (1) Schrödinger (2) (3) 1-2. R 10 R 31 Schrödinger 1-3. cm n = 1 (1) Bohr Bohr 2 (2) 90 % 1-5. n 1 3 (1) 113
12 (2) z (3) (4) 1-6. m { 0 at r < a (1.56) V (r) = at r a (1) Schrödinger (2) P (r) = rr(r) 1-7. (1) Schrödinger (2) (3) Bohr 1-8. ϕ(x) ψ(x) Ĥ E Φ(x) = A[ϕ(x) + ψ(x)] Ψ(x) = B[ϕ(x) ψ(x)] A, B (1) ϕ(x) ψ(x) (2) ϕ(x) ψ(x) (3) ϕ(x) ψ(x) Ĥ E (4) Φ(x) Ψ(x) Ĥ E (5) Φ(x) Ψ(x) (6) Φ(x) Ψ(x) A, B 1-9. r 1s, 2s, 3s p (1) p +, p, p 0, p x, p y, p z (2) p +, p, p 0, p x, p y, p z z p z s, 2s, 3s, 2p z s, 3p z s, 2s r r + dr r (1) n = 1 r, r 2, r 1 (2) n = 2, l = 0 r, r 2, r (1) 2p +, 2p 0, 2p r (2) p r (1.57) r nl = a 0 2 [3n2 l(l + 1)] (3) p r ( (1.58) ˆp r = i h r + 1 ) r 114
13 1s, 2s, 3s, 2p z p r p 2 r s, 2s, 3s, 2p z R x (1.59) R x = Ψ (n, l, m l )µ x Ψ(n, l, m l)dτ = e Ψ (n, l, m l )xψ(n, l, m l)r 2 sin θdrdθdϕ µ x µ x e y, z (1) Rydberg (2) 1s 2s R = 0 (3) 1s 2p x 115
14 He +, Li 2+ z r cos θ electron (m e, e) (x,y,z) = (r,θ,φ) θ nucleus (m n, +Ze) φ r r sin θ r sin θ sin φ y r sin θ cos φ (x,y,0) x m n, +Ze, m e e Schrödinger Ψ(r, θ, ϕ) Ψ [ (2.1) h2 1 2µ r 2 (2.2) µ = m nm e m n + m e ( r 2 Ψ ) + r r 1 r 2 sin θ ( sin θ Ψ ) + θ θ (2.3) Ψ = Ψ(r, θ, ϕ) = R(r)Y (θ, ϕ) 1 r 2 sin 2 θ (2.4) Y l,ml (θ, ϕ) = ( 1) (m l+ m l )/2 2l + 1 (l m l )! 4π l + m l )! P m l l (cos θ)e im lϕ (2.5) P m l l (z) = (1 z 2 ) m l /2 d ml Pl 0(z) (2.6) Pl 0 = 1 2 l l! (2.7) z = cos θ d l dz l (z2 1) l dz m l 2 ] Ψ ϕ 2 Ze2 4πε 0 r Ψ = EΨ 116
15 4Z (2.8) R n,l (r) = 3 (n l 1)! a 3 0 n4 [(n + l)!] 3 ρl e ρ/2 L 2l+1 n+l (ρ) (2.9) L s k(ρ) = ds dρ s L k(ρ) (2.10) L k = e ρ dk dρ k (ρk e ρ ) (2.11) ρ = 2Zr na 0 (2.12) a 0 = 4πε 0 h 2 µe 2 (2.13) E n = Z2 n 2 w 0 (2.14) w 0 = µ ( ) e h 2 4πε atomic unit (au) (2.15) (2.16) (2.17) m e = 1 h = 1 e 2 4πε 0 = 1 bohr µ m e Bohr a 0 (2.18) a 0 = 4πε 0 h 2 m e e 2 = m hartree µ m e w 0 2 (2.19) e 2 4πε 0 a 0 = h2 m e a 2 0 = 2 w 0 = J = ev Z µ m e Ĥ Ψ 1s W au (2.20) Ĥ = Z r (2.21) Ψ 1s = Z 3 (2.22) W = Z2 2 π e Zr 117
16 2-1. (1) (2) Bohr (3) (4) n = 2 n = 1 118
17 3 1 2 Schrödinger 3.1 Schrödinger Z 2 (3.1) [ h2 2m e ( ) + e2 4πε 0 ( Z Z + 1 )] Ψ( r 1, r 2 ) = EΨ( r 1, r 2 ) r 1 r 2 r 12 (3.2) 2 1 = 2 x y z1 2 (3.3) 2 2 = 2 x y z2 2 (3.4) r 1 = x y2 1 + z2 1 (3.5) r 2 = x y2 2 + z2 2 1 e r 1 r 12 +Ze r 2 2 e (3.6) r 12 = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 ) 2 (3.7) [ 1 2 ( ) + (3.8) Ĥ = Ĥ 1 + Ĥ 2 + Ĥ ( Z Z + 1 )] Ψ( r 1, r 2 ) = EΨ( r 1, r 2 ) r 1 r 2 r 12 (3.9) Ĥ 1 = Z r 1 (3.10) Ĥ 2 = Z r 2 (3.11) Ĥ = 1 r 12 Ĥ Schrödinger Ĥ Ĥ1 Ĥ2 Schrödinger (3.12) (Ĥ 1 + Ĥ 2 )ψ = E ψ (3.13) ψ = ψ 1ψ 2 = Z3 π e Z(r1+r2) (3.14) E = E 1 + E 2 = 2W = Z 2 119
18 (3.15) Ĥ 1 ψ 1 = E 1ψ 1 (3.16) Ĥ 2 ψ 2 = E 2ψ Ĥ E ψ ϕ (ψ + cϕ) c (3.17) (3.18) (3.19) ψ Ĥ ψ ψ ψ ϕ Ĥ ϕ ϕ ϕ = E 1 ( ) = E 2 ( ) ψ + cϕ Ĥ ψ + cϕ = E 3 ( ) ψ + cϕ ψ + cϕ 3 E 1, E 2, E R 1, R 2, R 3 (3.20) ψ Ĥ ψ = R 1 ( ) (3.21) ϕ Ĥ ϕ = R 2 ( ) (3.22) ψ + cϕ Ĥ ψ + cϕ = R 3 ( ) (3.22) (3.23) ψ Ĥ ψ + c ϕ Ĥ ψ + c ψ Ĥ ϕ + cc ϕ Ĥ ϕ = R 3 (3.20), (3.21) (3.24) c ϕ Ĥ ψ + c ψ Ĥ ϕ = R 3 R 1 cc R 2 c ϕ Ĥ ψ c ψ Ĥ ϕ (3.25) c ψ Ĥ ϕ = (c ϕ Ĥ ψ ) = c ϕ Ĥ ψ ψ, ϕ Ĥ (3.26) ψ Ĥ ϕ = ϕ Ĥ ψ Hermite Ĥ 120
19 Schrödinger (3.27) Ĥψ n = E n ψ n Ĥ (3.28) Ĥ = Ĥ + Ĥ Ĥ Schrödinger En ψ n (3.29) Ĥ ψn = Enψ n Ĥ Ĥ En, ψn ψn (3.30) (3.31) (3.30) ψn ψ m = δ nm ϕ (3.31) ϕ = n c n ψ n Ĥ, Ĥ, Ĥ (3.26) (3.32) Ĥ = Ĥ + λĥ λ λ = 1 λ 1 (3.27) ψ n E n (3.33) ψ n = ψ n + λψ n + λ 2 ψ n + (3.34) E n = E n + λe n + λ 2 E n + Ĥ λ 1 ψ n, E n λ ψ n, E n (3.32), (3.33), (3.34) (3.27) λ (3.35) Ĥ ψn+ λ(ĥ ψ n + Ĥ ψn)+ λ 2 (Ĥ ψ n + Ĥ ψ n)+ = Enψ n+ λ(enψ n + E nψ n)+ λ 2 (Enψ n + E nψ n + E nψ n)+ 121
20 λ λ λ 0 (3.36) Ĥ ψ n = E nψ n (3.29) λ 1 (3.37) Ĥ ψ n + Ĥ ψ n = E nψ n + E nψ n ψ n (3.38) ψ n Ĥ ψ n + ψ n Ĥ ψ n = E n ψ n ψ n + E n ψ n ψ n (3.30) (3.39) E n = ψ n Ĥ ψ n + ψ n Ĥ ψ n E n ψ n ψ n (3.40) ψ n Ĥ ψ n = ψ n Ĥ ψ n = E n ψ n ψ n = E n ψ n ψ n (3.39) (3.40) E n (3.41) E n = ψ n Ĥ ψ n (3.31) (3.42) ψ n = k c k ψ k (3.37) (3.43) c k (En Ek)ψ k = Ĥ ψn E nψ n k c m ψ m Schrödinger (3.44) c m (E n E m) = ψ m Ĥ ψ n H mn for n m (3.30) (3.29) c n = 0 (3.45) E n = E n + H nn H nm (3.46) ψ n = ψn + E n m n Em ψ m (3.47) H mn = ψ m Ĥ ψ n = ψ n Ĥ ψ m = H nm 122
21 3.4 (3.27) (3.7) (3.8) (3.11) ψ n Ψ( r 1, r 2 ) Ĥ (3.32) Ĥ (3.9) (3.10) (Ĥ 1 + Ĥ 2 ) Ĥ (3.11) (3.29) ψ n (3.13) ψ E n (3.14) E E (3.48) E = Z6 e 2Z(r 1+r 2) π 2 d r 1 d r 2 = 5Z r 12 8 r 1, r 2 A-3 ev He Li Be B C (1) (2) 3-2. Ĥ E n ψ n ψ n Ĥ (1) (2) 3-3. m at x 0 (3.49) V (x) = 0 at 0 < x < a at x a (1) Schrödinger (2) V (x) { (3.50) V w at x a/2 (x) = 0 at x < a/2 (3) 3-4. m at x 0 (3.51) V (x) = αx/a at 0 < x < a at x a 123
22 3-5. m { α [1 + cos(8πx/a)] for 0 < x < a (3.52) V (x) = for x 0 or x a 3-6. π π N a ( (3.53) V (x) = α x a ) 2 2 (1) (2) π 2n 3-7. x, y, z a, b, c m a/4 x 3a/4 b/4 y 3b/4 c/4 z 3c/4 V 0 > m e (3.54) V (x) = 1 2 kx2 + eex k E (1) (2) (3.55) y = x + ee k 3-9. m (3.56) V (x) = 1 2 kx2 + αx 3 k α α α m (3.57) V (x) = 1 2 kx αx2 k α α k α m (3.58) V (x) = 1 2 kx2 + αx 4 k α α α He (1) (2) Schrödinger He (3) 124
23 (4) (3.59) e α(r 1+r 2) r 12 dx 1 dy 1 dz 1 dx 2 dy 2 dz 2 = 20π2 α Z 1 (1) (2) Z + 1 (3) Z Z z E (1) (2) (3) Stark R (1) e2 (3.60) V (x) = 4πε 0 R e2 4πε 0 r r for for r < R r > R (2) 1s 2p (3) Lyman R (1) 3e2 (3.61) V (x) = 8πε 0 R 3 e2 4πε 0 R r ) (R 2 r2 3 for for r < R r > R (2) 1s 2p (3) Lyman 125
24 Schrödinger ĤΨ = EΨ E 0 Ψ 0 ϕ (4.1) ϕ Ĥ ϕ ϕ ϕ E = Ψ 0 Ĥ Ψ 0 Ψ 0 Ψ 0 Ψ i (4.2) Ψ i Ψ i = 1 (4.3) Ψ i Ψ j = 0 (i j) (4.4) ϕ = i c i Ψ i c (4.5) ε = ϕ Ĥ ϕ i c j Ψ i Ĥ Ψ j c i c j E j Ψ i Ψ j c i 2 E i i j i j i = = = E 0 ϕ ϕ c i c j Ψ i Ψ j c i c j Ψ i Ψ j c i 2 i j Schrödinger i j i 4.2 (4.6) Ĥ = r (4.7) ϕ = Ae Cr A C (4.8) ε = C2 2 C (4.9) (4.10) C = 1 dε dc = C 1 = 0 126
25 (4.11) ε = 1 2 (4.12) ϕ = Ae Cr2 (4.13) ε = 3 8C 2 C π (4.14) dε dc = πc = 0 (4.15) C = 8 9π (4.16) ε = 4 3π > Z (4.17) Ĥ = 1 2 ( ) Z r 1 Z r r 12 Z C (4.18) ϕ = C3 π e C(r 1+r 2 ) (4.19) ϕ ϕ = C6 π 2 e 2C(r 1+r 2 ) d r 1 d r 2 = 1 ε Z C (4.20) ε = C6 ϕ Ĥ ϕ = π 2 = C 2 + 2C(C Z) C A-3 (4.21) dε dc = 2C 2Z = 0 (4.22) C = Z 5 16 ( (4.23) ε = Z 5 ) 2 16 (ev) He [( e C(r 1+r 2 ) Z Z + 1 ) ] e C(r 1+r 2 ) d r 1 d r 2 r 1 r 2 r
26 C (4.24) ϕ = Ae Cx2 (4.25) ϕ = Ae C x 4-2. m (4.26) V (x) = 1 2 kx2 + αx 3 k α α (4.27) ϕ = Ae Cx m (4.28) V (x) = αx 4 α (4.29) ϕ = Ae Cx m (4.30) V (x) = 1 2 kx2 + αx 4 k α α (4.31) ϕ = Ae Cx (4.32) V (x) = 1 2 k(x2 + y 2 + z 2 ) = 1 2 kr2 (1) (2) 3 (3) 2 C (4.33) ϕ = Ae Cr (4.34) ϕ = Ae Cr (1) 128
27 (2) 2 C (4.35) ϕ = Ae Cr (4.36) ϕ = Ae Cr He + (1) (2) 2 C (4.37) ϕ = Ae Cr (4.38) ϕ = Ae Cr He (1) (2) (4.39) ϕ = Ae C(r1+r2) (3) 4-9. Li + (1) (2) (4.40) ϕ = Ae C(r 1+r 2 ) (3) Ĥ (4.41) Φ = C 1 ϕ 1 + C 2 ϕ 2 ϕ 1 ϕ 2 C 1 C 2 (1) (4.42) { (H11 εs 11 )C 1 + (H 12 εs 12 )C 2 = 0 (H 21 εs 21 )C 1 + (H 22 εs 22 )C 2 = 0 (4.43) H ij = ϕ i Ĥϕ jdτ (4.44) S ij = ϕ i ϕ j dτ H ij = H ji, S ij = S ji (2) C 1 = C 2 = 0 (4.45) (H 11 εs 11 ) (H 12 εs 12 ) (H 21 εs 21 ) (H 22 εs 22 ) = 0 H 21 = H 12, S 21 = S
28 (3) ϕ 1 ϕ 2 (1) (2) a (4.46) Ψ = C 1 x(a x) + C 2 x 2 (a x) m { αx/a for 0 < x < a (4.47) V (x) = for x 0 or x a (1) ( πx ) ( ) 2πx (4.48) Ψ = C 1 sin + C 2 sin a a (2) ( πx ) ( ) 3πx (4.49) Ψ = C 1 sin + C 2 sin a a (3) (1), (2) (4) m (4.50) V (x) = 1 2 kx γx δx4 (4.51) Ψ = C 1 e ξ2 /2 + C 2 (4ξ 2 2)e ξ2, ξ = αx, α 2 = km h r µ e r (1) z E (2) (3) (4.52) ϕ = C 1 ψ 1s + C 2 ψ 2pz ψ 1s 1s ψ 2pz 2p z (4.53) E = 5e ± 3e π 2 ε 2 0 E 2 a ε 0 a 0 64ε 0 a e 2 ε 0 a 0 Bohr (4) E 2 (4.54) 1 + x = 1 + x 2 x2 8 +, 0 x < 1 (5) µ E (4.55) µ = αe α µ (4.56) E = E 0 µde 130
29 (6) (3), (4) (7) ψ 2pz ψ 3pz 131
30 s, 2s, 2p Ψ(r 1, r 2, ) r 1, r 2, (5.1) Ψ(r 1, r 2, ) = ψ 1s ((r 1 )ψ 2s (r 2 ) 5.2 Stern-Gerlach ŝ 2 ŝ 2 s = 1/2 s(s + 1) h 2 = 3 h 2 /4 α, β z ŝ z m s = 1/2 m s h = h/2 α m s = 1/2 m s h = h/2 β s 1/2 m s s m s s 1/2-1/2 α β (5.2) α α = β β = 1 (5.3) α β = β α = Pauli
31 a b c d 2 1 a (5.4) Ψ(1, 2) = ψ 1 (1)ψ 1 (2)α(1)α(2) Pauli b 2 (5.5) Ψ(1, 2) = ψ 1 (1)ψ 1 (2)α(1)β(2) (5.6) Ψ(1, 2) = ψ 1 (1)ψ 1 (2)β(1)α(2) Pauli 1 α 2 β b 2 (5.7) Ψ(1, 2) = ψ 1 (1)ψ 1 (2)[α(1)β(2) + β(1)α(2)] 1 2 (5.8) Ψ(1, 2) = ψ 1 (1)ψ 1 (2)[α(1)β(2) β(1)α(2)] 1 2 (5.7) Pauli (5.8) (5.9) Ψ(1, 2) = [ψ 1 (1)ψ 2 (2) + ψ 2 (1)ψ 1 (2)][α(1)β(2) β(1)α(2)] 1 2 (5.10) Ψ(1, 2) = [ψ 1 (1)ψ 2 (2) ψ 2 (1)ψ 1 (2)][α(1)β(2) + β(1)α(2)] 1 2 (5.11) Ψ(1, 2) = [ψ 1 (1)ψ 2 (2) ψ 2 (1)ψ 1 (2)][α(1)α(2)] 1 2 (5.12) Ψ(1, 2) = [ψ 1 (1)ψ 2 (2) ψ 2 (1)ψ 1 (2)][β(1)β(2)] α(1)β(2) β(1)α(2) α(1)β(2) + β(1)α(2) α(1)α(2) β(1)β(2) 133
32 5.6 Pauli determinant wave function (5.13) Ψ(1, 2) = 1 ψ 1 (1)α(1) 2 ψ 1 (2)α(2) ψ 1 (1)β(1) ψ 1 (2)β(2) Pauli 3 1 (5.14) Ψ(1, 2, 3) = 1 ψ 1 (1)α(1) ψ 1 (1)α(1) ψ 1 (1)β(1) 3! ψ 1 (2)α(2) ψ 1 (2)α(2) ψ 1 (2)β(2) ψ 1 (3)α(3) ψ 1 (3)α(3) ψ 1 (3)β(3) = Pauli 3 2 (5.15) Ψ(1, 2, 3) = 1 3! 2n n (5.16) Ψ(1, 2,, 2n) = ψ 1 (1)α(1) ψ 1 (1)β(1) ψ 2 (1)α(1) ψ 1 (2)α(2) ψ 1 (2)β(2) ψ 2 (2)α(2) ψ 1 (3)α(3) ψ 1 (3)β(3) ψ 2 (3)α(3) 1 (2n)! ψ 1 (1)α(1) ψ 1 (1)β(1) ψ n (1)α(1) ψ n (1)β(1) ψ 1 (2)α(2) ψ 1 (2)β(2) ψ n (2)α(2) ψ n (2)β(2) ψ 1 (2n)α(2n) ψ 1 (2n)β(2n) ψ n (2n)α(2n) ψ n (2n)β(2n) (5.17) Ψ 1 (1, 2) = 1 ψ 1 (1)α(1) ψ 2 (1)α(1) 2 ψ 1 (2)α(2) ψ 2 (2)α(2) (5.18) Ψ 2 (1, 2) = 1 ψ 1 (1)β(1) ψ 2 (1)β(1) 2 ψ 1 (2)β(2) ψ 2 (2)β(2) (5.19) Ψ 3 (1, 2) = 1 ψ 1 (1)α(1) ψ 2 (1)β(1) 2 ψ 1 (2)α(2) ψ 2 (2)β(2) (5.20) Ψ 4 (1, 2) = 1 ψ 1 (1)β(1) ψ 2 (1)α(1) 2 ψ 1 (2)β(2) ψ 2 (2)α(2) (5.17), (5.18) (5.11), (5.12) (5.19), (5.20) (5.21) Ψ t (1, 2) = 1 2 (Ψ 3 (1, 2) + Ψ 4 (1, 2)) (5.22) Ψ s (1, 2) = 1 2 (Ψ 3 (1, 2) Ψ 4 (1, 2)) (5.21) (5.10) (5.22) (5.9) 134
33 ϕ A, ϕ B 2 (1) (2) He (5.23) Ψ + = A + [1s(1)2s(2) + 1s(2)2s(1)] (5.24) Ψ = A [1s(1)2s(2) 1s(2)2s(1)] (1) (2) (3) (4) (5) 5-3. ψ 1 (1), ψ 2 (1) 2 ψ 1 (1) E1 ψ 2(1) E2 4 (5.25) Ψ 1 (1, 2) = 1 ψ 1 (1)α(1) ψ 2 (1)α(1) 2 ψ 1 (2)α(2) ψ 2 (2)α(2) (5.26) Ψ 2 (1, 2) = 1 ψ 1 (1)β(1) ψ 2 (1)β(1) 2 ψ 1 (2)β(2) ψ 2 (2)β(2) (5.27) Ψ 3 (1, 2) = 1 ψ 1 (1)α(1) ψ 2 (1)β(1) 2 ψ 1 (2)α(2) ψ 2 (2)β(2) (5.28) Ψ 4 (1, 2) = 1 ψ 1 (1)β(1) ψ 2 (1)α(1) 2 ψ 1 (2)β(2) ψ 2 (2)α(2) 2 (5.29) Φ(1, 2) = C 1 Ψ 1 (1, 2) + C 2 Ψ 2 (1, 2) + C 3 Ψ 3 (1, 2) + C 4 Ψ 4 (1, 2) (5.30) J = ψ 1 (1)ψ 1 (1) 1 ψ 2(2)ψ 2 (2) r 12 (5.31) K = ψ 1 (1)ψ 2 (1) 1 ψ 1(2)ψ 2 (2) r
34 6 Pauli Hunt 6.1 2s 2p (Z) (Z eff ) σ (6.1) Z eff = Z σ s p s p 6.2 Z Z 1s, 2s, 2p, 3s, 3p, (4s, 3d), 4p, (5s, 4d), 5p, (6s, 4f, 5d), 6p, (7s, 5f, 6d) Ge (Z = 32) (1s) 2 (2s) 2 (2p) 6 (3s) 2 (3p) 6 (3d) 10 (4s) 2 (4p) Pauli Hund 136
35 6.5 1 H 1s 19 K 4s 2 He 1s 2 20 Ca 4s 2 3 Li 2s 21 Sc 3d 4s 2 4 Be 2s 2 22 Ti 3d 2 4s 2 5 B 2s 2 2p 23 V 3d 3 4s 2 6 C 2s 2 2p 2 24 Cr 3d 4 4s 2 7 N 2s 2 2p 3 25 Mn 3d 5 4s 2 8 O 2s 2 2p 4 26 Fe 3d 6 4s 2 9 F 2s 2 2p 5 27 Co 3d 7 4s 2 10 Ne 2s 2 2p 6 28 Ni 3d 8 4s 2 11 Na 3s 29 Cu 3d 9 4s 2 12 Mg 3s 2 30 Zn 3d 10 4s 2 13 Al 3s 2 3p 31 Ga 3d 10 4s 2 4p 14 Si 3s 2 3p 2 32 Ge 3d 10 4s 2 4p 2 15 P 3s 2 3p 3 33 As 3d 10 4s 2 4p 3 16 S 3s 2 3p 4 34 Se 3d 10 4s 2 4p 4 17 Cl 3s 2 3p 5 35 Br 3d 10 4s 2 4p 5 18 Ar 3s 2 3p 6 36 Kr 3d 10 4s 2 4p I M M + + e I 1 M + M 2+ + e I 2 30 He Ne I 1 / ev Ar Zn Kr Cd Xe Hg Rn atomic number 6.7 A (6.2) M + e M
36 Mulliken (6.3) χ = I + A 2 Pauling I 1/eV A/eV EN (M) EN (P) I 1/eV A/eV EN (M) EN (P) 1 H K He < 0 20 Ca 6.11 < Li Sc Be 9.32 < Ti B V C Cr N Mn 7.44 < O Fe F Co Ne < 0 28 Ni Na Cu Mg 7.65 < Zn 9.39 < Al Ga Si Ge P As S Se Cl Br Ar < 0 36 Kr < (1) (2) Pauli (3) Hund s 2p 2p 2s 6-3. C, Cl, Cr, Cs 6-4. N 6-5. (1) (2) (1) H D (2) H D Balmer (3) 6 Li 2+ 7 Li 2+ (1), (2) 6-8. Li 1s +1 2s 1 (1) 1 I 1 (2) I ev (3) Na 3s I 1 = ev 6-9. (1) 138
37 (2) H H (6.4) H + hν H + e 1648 nm (a) (6.5) (a) X + Y X + + Y (b) X + Y X + Y (1) Mulliken (2) Pauling 139
38 7 1 Valence Bond Method Heitler London A, B 2 1, nm ev r A1 +e A 1 e r A2 r R 12 r B1 2 e r B2 +e B 7.2 Born-Oppenheimer Born-Oppenheimer (7.1) Ĥ = r A1 1 r B2 1 r A2 1 r B1 + 1 r R A (7.2) Ĥ A = r A1 (7.3) Ĥ A ϕ A (1) = E H ϕ A (1) (7.4) ϕ A (1) = 1 π e ra1 B (7.6) Ĥ B = r B2 (7.7) Ĥ B ϕ B (2) = E H ϕ B (2) (7.8) ϕ B (2) = 1 π e rb2 (7.5) E H = 1 2 (7.9) E H = 1 2 (7.10) Ĥ = ĤA + ĤB + Ĥ (7.11) Ĥ = 1 r A2 1 r B1 + 1 r R 2 1 A 2 B 1 2 (7.12) Ψ(1, 2) = ϕ A (1)ϕ B (2) 140
39 (7.13) E = Ψ(1, 2) ĤA + ĤB + Ĥ Ψ(1, 2) = 2E H + ϕ A (1)ϕ B (2) Ĥ ϕa (1)ϕ B (2) = 2E H + Q Q Coulomb R 0.09 nm 0.25 ev A B 1 2 (7.14) Ψ(1, 2) = c 1 ϕ A (1)ϕ B (2) + c 2 ϕ A (2)ϕ B (1) c 1 ψ 1 + c 2 ψ 2 2 c 1 c 2 c 1 = c 2 (7.15) Ψ(1, 2) = N[ϕ A (1)ϕ B (2) + ϕ A (2)ϕ B (1)] N 3.14 ev nm 7.5 σ π σ s π 2 p p σ : s-s, s-p z, p z -p z 1 π : p x -p x, p x -d zx 2 δ : d x2 y2 -d x2 y2 N 2 (1s) 2 (2s) 2 (2p x )(2p y )(2p z ) 141
40 σ (2p z ) π (2p y ) (2p x ) 7.6 (1s) 2 (2s) 2 (2p x ) 2 (2p y )(2p z ) 1s, 2s z x : s H, y : s H (7.16) ψ x (1, 2) p x (1)s H (2) + p x (2)s H (1) (7.17) ψ y (1, 2) p y (1)s H (2) + p y (2)s H (1) 90 H 2 O H 2 S 92.2 H 2 Se 90.9 H 2 Te 89.5 x p y H + p x p z + p z + p x p y H' y CH (7.18) (1s) 2 (2s) 2 (2p) 2 2 (7.19) (1s) 2 (2s)(2p x )(2p y )(2p z ) C H (1, 1, 1), (1, 1, 1), ( 1, 1, 1), ( 1, 1, 1) (7.20) (7.21) (7.22) (7.23) ϕ(1, 1, 1) = c 1 s + c 2 (p x + p y + p z ) ϕ(1, 1, 1) = c 1 s + c 2 (p x p y p z ) ϕ( 1, 1, 1) = c 1 s + c 2 ( p x + p y p z ) ϕ( 1, 1, 1) = c 1 s + c 2 ( p x p y + p z ) 142
41 (7.24) ϕ(1, 1, 1) ϕ(1, 1, 1) = c c 2 2 = 1 (7.25) ϕ(1, 1, 1) ϕ(1, 1, 1) = c 2 1 c 2 2 = 0 (7.26) c 1 = c 2 = 1 2 sp 3 hybrid orbital z z ( 1, 1,1) (1,1,1) ( 1,1, 1) y y (1, 1, 1) x x z s, p x, p y xy pz (7.27) (7.28) (7.29) ϕ(1) = 1 2 s + p x 3 3 ϕ(2) = 1 3 s 1 6 p x p y ϕ(3) = 1 3 s 1 6 p x 1 2 p y y y + x x sp x s, p x 143
42 y (7.30) ϕ(1) = 1 2 (s + p x ) (7.31) ϕ(2) = 1 (s p x ) x 2 sp dsp Ni(CN) 2 4, AuCl 4 sp 3 d 5 90, 120, 180 PCl 5, AsF 5, SbCl 5 d 2 sp Co(NH 3 ) 3+ 6, PtCl2 6 sp 3 d SF (1) 2 1s (2) 7-2. (1) (2) C H (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1) (1) 2s, 2p x, 2p y, 2p z 4 (2) (1) 4 (3) (1) 4 2s 2 2p 3 2 1: sp 3 (7.32) ϕ = as + bp x + cp y + dp z (7.33) a 2 : (b 2 + c 2 + d 2 ) = 1 : 3 (1) 1 x a, b, c, d (2) xz (1) a, b, c, d (3) (1), (2) 2 (4) 3, 4 (1), (2) xz a, b, c, d 7-6. sp 2 xy 144
43 (1) 2s, 2p x, 2p y, 2p z 4 (2) 3 (3) (2) 3 (4) (2) (1) dsp 2 (2) sp 3 d (3) d 2 sp 3 (4) sp 3 d 2 145
44 LCAO 8.1 Molecular Orbital Method Hamiltonian (8.1) Ĥ = r A1 1 r B1 + 1 R e 1 (8.2) ϕ A (1) = 1 π e ra1 (8.3) ϕ B (1) = 1 π e rb1 r +e A A1 R r B1 +e B LCAO (Linear Combination of Atomic Orbitals) (8.4) ψ(1) = c A ϕ A (1) + c B ϕ B (1) (8.5) E = ψ(1) Ĥ ψ(1) ψ(1) ψ(1) (8.6) H AA = ϕ A (1) Ĥ ϕ A(1) (8.7) H BB = ϕ B (1) Ĥ ϕ B(1) = c2 A H AA + c 2 B H BB + 2c A c B H AB c 2 A + c2 B + 2c Ac B S (8.8) H AB = ϕ A (1) Ĥ ϕ B(1) = ϕ B (1) Ĥ ϕ A(1) = H BA (8.9) S = ϕ A (1) ϕ B (1) = ϕ B (1) ϕ A (1) S E c A, c B (8.10) E = E = 0 c A c B (8.11) (8.12) c A (H AA E) + c B (H AB ES) = 0 c A (H AB ES) + c B (H BB E) = 0 c A = c B = 0 (8.13) H AA E H AB ES H AB ES H BB E = 0 (8.14) H AA = H BB 146
45 (8.13) (8.15) (H AA E) 2 (H AB ES) 2 = 0 E (8.16) E = (H AA ± H AB )(1 S) (1 + S)(1 S) = H AA ± H AB 1 ± S (8.11), (8.12) (8.17) c B c A = ±1 c A g gerade u ungerade (8.18) ψ g (1) = S [ϕ A(1) + ϕ B (1)] (8.19) E g = H AA + H AB 1 + S 1 (8.20) ψ u (1) = 2 2S [ϕ A(1) ϕ B (1)] (8.21) E u = H AA H AB 1 S Coulomb (8.22) H AA = ϕ A (1) Ĥ ϕ A(1) = E H + J + 1 R J (8.23) J = ϕ A (1) 1 r B1 ϕ A(1) = 1 ( R + e 2R ) R (8.24) H AB = ϕ A (1) Ĥ ϕ B(1) = E H S + K + S R K (8.25) K = ϕ A (1) 1 ϕ B(1) r B1 = e R (1 + R) S (8.26) S = ϕ A (1) ϕ B (1) ( ) = e R 1 + R + R
46 (8.27) E = E H + 1 R + J ± K 1 ± S E H 1 1/R MO (E EH) / au E g E u R / a nm nm 1.76 ev 2.79 ev φ ψ A + φ B 0.4 φ A ψ φ B r / a r / a 0 148
47 ψ ψ r / a r / a R, Ĥ (1) ( ) (8.28) S = ϕ A (1) ϕ B (1) = e R 1 + R + R2 3 (2) Coulomb (8.29) H AA = ϕ A (1) Ĥ ϕ A(1) = 1 ( 2 + e 2R ) R (3) (8.30) H AB = ϕ A (1) Ĥ ϕ B(1) = S 2 + S R e R (1 + R) 8-2. (1) (2) (3) (4) (5) (6) cm 1 149
48 9 2 2 SCF 9.1 (9.1) Ĥ = r A1 r B r A2 r B2 r 12 R = Ĥ1 + Ĥ r 12 R (9.2) Ĥ 1 = r A1 1 r B1 (9.3) Ĥ 2 = r A2 1 r B2 1/r 12 (9.4) Ψ(1, 2) = S [ϕ A(1) + ϕ B (1)][ϕ A (2) + ϕ B (2)] 2 E = Ψ(1, 2) Ĥ1 + Ĥ (9.5) r 12 R Ψ(1, 2) 2(J + K) = 2E H + + Ψ(1, 2) S Ψ(1, 2) + 1 R J (8.23) J K (8.25) K r 12 2 (9.6) Ψ(1, 2) 1 Ψ(1, 2) r 12 = 2J + 2J + 8L + 4K (2 + 2S) 2 J J = ϕ A (1)ϕ B (2) 1 (9.7) r 12 ϕ A(1)ϕ B (2) = 1 ( 1 R e 2R R R ) 4 + R
49 K K = ϕ A (1)ϕ B (2) 1 (9.8) r 12 ϕ A(2)ϕ B (1) ( = e 2R R ) 4 + 3R2 + R [ S 2 (γ + ln R) + S 2 5R 1E i ( 4R) SS 1 E i ( 2R) ] (9.9) J = ϕ A (1)ϕ A (2) 1 ϕ A(1)ϕ A (2) r 12 (9.10) = 5 8 L = ϕ A (1)ϕ A (2) 1 ϕ A(1)ϕ B (2) = 1 16R (9.11) E = 2E H + r 12 [ (5 + 2R + 16R 2 )e R (5 + 2R)e 3R] 2(J + K) 1 + S + ( ) 1 5 2(1 + S) J + 4L + 2K 9.2 (9.12) E = H 11 + H S 12 (9.13) H 11 = ϕ A (1)ϕ B (2) Ĥ ϕ A(1)ϕ B (2) = 2E H + 2J + J + 1 R (9.14) H 12 = ϕ A (1)ϕ B (2) Ĥ ϕ A(2)ϕ B (1) = 2S 2 E H + 2SK + K + S2 R (9.15) S 12 = ϕ A (1) ϕ B (1) ϕ A (2) ϕ B (2) = S 2 (9.16) E = 2E H + 1 R + 2J + J + (2SK + K ) 1 + S 2 VB nm 3.14 ev MO nm 2.68 ev nm 4.74 ev MO (9.17) Ψ(1, 2) = S [ϕ A(1)ϕ B (2) + ϕ B (1)ϕ A (2) + ϕ A (1)ϕ A (2) + ϕ B (1)ϕ B (2)] 2 1 1:1 151
50 9.3 SCF 1 (9.18) Ĥ = r A1 1 r B1 + V SCF (r 1 ) r A2 1 r B2 + V SCF (r 2 ) + 1 R Configuration Interaction 9-1. (1) SCF (2) SCF 9-2. LCAO-MO (1) (2) (3) (4) 9-3. LCAO-MO 1 (1) (2) 9-4. LCAO-MO (1) (2) (3) cm LCAO-MO 9-6. Configuration Interaction 152
51 AO H AA H BB c A 1, c B 0 s-p z 10.2 σ s σs H He ( ) σ 1s s * σ s 1s σ s σ s Li Be ( ) σ s σ s σ p π p π p σ p B C σ 2s s * 2p 2s π x σ s Σ* z * Σ z π y * π x π y σ * s 2s 2p 2s σ s σ s σ s σ p π p π p σ p N O F Ne σ s 2p 2s σ* z π* x π y * π x π y σ z σ * s 2p 2s 10.3 (10.1) Ψ MO = c A ϕ A + c B ϕ B, c A = c B 153
52 10.4 µ n (i = 1 n) q i r i µ n (10.2) µ = q i r i i=1 +q q R µ (10.3) µ = qr q +q +e A R/2 0 +e B R/2 e φa φ B e (10.4) Ψ MO = c A ϕ A + c B ϕ B = N(ϕ A + λϕ B ) N (10.5) 1 = N 2 ( ϕ A ϕ A + 2λ ϕ A ϕ B + λ 2 ϕ B ϕ B ) = N 2 (1 + λ 2 + 2λS) (10.6) N 2 = (10.7) S = ϕ A ϕ B λ 2 + 2λS (10.8) z = Ψ z Ψ = N 2 ( ϕ A z ϕ A + 2λ ϕ A z ϕ B + λ 2 ϕ B z ϕ B ) N 2 (z AA + 2λz AB + λ 2 z BB ) z AA A z BB B (10.9) z AA = 1 2 R (10.10) z BB = 1 2 R R (10.11) z = N 2 [ 1 2 R(λ2 1) + 2λz AB ] z AB 0 Ψ 2 z = 0 +2e z 2e (10.12) µ = 2e z (λ2 1)eR 1 + λ 2 + 2λS 154
53 A +2e 2e B 0 <z> Ψ MO A, B (1) ϕ A, ϕ B MO (10.13) α A E β A ES β A ES α B E = 0 (10.14) α A = ϕ A Ĥeff ϕ A, α B = ϕ B Ĥeff ϕ B (10.15) β = ϕ B Ĥeff ϕ A = ϕ A Ĥeff ϕ B (10.16) S = ϕ B ϕ A = ϕ A ϕ B Ĥ eff MO 1 (2) (3) α A = α B = α (2) (4) α A α B (2) (10.17) E + = α A + β 2 α A α B, β 2 E = α B α A α B (5) (4) α A α B ϕ A ϕ B A, B σ π δ (1) σ1s (2) σ 1s (3) σ2p z (4) π2p (1) (2) H 2, H + 2, H 2 (3) He 2, He (1) B 2 (2) N 2 N + 2 (3) O 2 O
54 10-8. O + 2, O 2, O 2, O NO NO D e D 0 (1) D e D 0 (2) N ev, 2331 cm 1 (3) H 2 (2) De = ev, 4401 cm NaCl nm, NaCl 4.29 ev Na 5.14 ev, Cl 3.61 ev (1) NaCl (2) NaCl Na + Cl Coulomb NaCl (3) (2) NaCl C m (4) (1) (2) KF nm, KF 5.18 ev K 4.34 ev, F 3.40 ev (1) KF (2) KF K + F Coulomb KF (3) (2) KF C m MO (10.18) Ψ MO = N(ϕ A + λϕ B ) (1) λ S (2) λ +δ δ 1/3 HF HCl HBr KI KCl µ/er
55 xy x 2 z (11.1) h 1 = 1 2 (s H + s H ) (11.2) h 2 = 1 2 (s H s H ) xy xz 1s, 2s, 2p x, h 1 xy xz 2p y, h 2 H' x p y + p x p z + p z + p x p y H y xy xz 2p z LCAO (11.3) ψ = c 1 (1s) + c 2 (2s) + c 3 (2p z ) + c 4 (h 1 ) + c 5 (2p y ) + c 6 (h 2 ) + c 7 (2p x ) (11.4) = 0 VB 2a 1 1b 2 3a 2 T. H. Hunting, et al., J. Chem. Phys., 57, 5044 (1972) 157
56 11.2 VB MO y x y x (11.5) ϕ(i) p x + λs H H' p y p H x + + φ(ii) + + φ(i) (11.6) ϕ(ii) p y + λs H p z ϕ(i), ϕ(ii) p x p y ϕ(i) ϕ(ii) ϕ(i), ϕ(ii), p z O(1s) 2 (2s) 2 (2pz) 2 [O(2p x )+ H(1s)] 2 [O(2p y )+ H (1s)] (1) (2) (11.7) = 0 (1) (2) (1) (2) 158
57 12 π Hückel π Hückel π 12.1 n LCAO LCAO n (12.1) Ψ = c i ϕ i i=1 n n c i ϕ i Ĥ c j ϕ j i=1 j=1 (12.2) ε = n n = c i ϕ i c j ϕ j i=1 j=1 n n c i c j ϕ i Ĥ ϕ j i=1 j=1 n i=1 j=1 n c i c j ϕ i ϕ j n n c i c j H ij i=1 j=1 n i=1 j=1 n c i c j S ij n n n n (12.3) ε c i c j S ij = c i c j H ij i=1 j=1 i=1 j=1 ε c i c k c i n n n n (12.4) ε c j S kj + ε c i S ik = c j H kj + c i H ik j=1 i=1 j=1 i=1 (12.5) S ij = S ji (12.6) H ij = H ji n (12.7) c i (H ik εs ik ) = 0 i=1 (12.8) (12.9) (12.10) c 1 (H 11 S 11 ε) + c 2 (H 12 S 12 ε) + + c n (H 1n S 1n ε) = 0 c 1 (H 21 S 21 ε) + c 2 (H 22 S 22 ε) + + c n (H 2n S 2n ε) = 0 c 1 (H n1 S n1 ε) + c 2 (H n2 S n2 ε) + + c n (H nn S nn ε) = 0 c i ε (12.11) (H 11 S 11 ε) (H 12 S 12 ε) (H 1n S 1n ε) (H 21 S 21 ε) (H 22 S 22 ε) (H 2n S 2n ε) (H n1 S n1 ε) (H n2 S n2 ε) (H nn S nn ε) ε c i = 0 159
58 12.2 π Hückel σ π π p i (12.12) H ii = α (12.13) S ii = 1 1 (12.14) S ij = 0 for i j 2 (12.15) H ij = β < 0 for i j, i, j (12.16) H ij = 0 for i j, i, j n n MO MO 2 π π E π (12.17) E π = i ν i ε i ν i i MO r π q r (12.18) q r = i ν i c 2 ir c ir i MO r r s p rs (12.19) p rs = i ν i c ir c is Hückel 12.3 sp 2 C-C σ C-H p π 1 2 p ϕ 1, ϕ 2 (12.20) Ψ = c 1 ϕ 1 + c 2 ϕ 2 c 1, c 2 (12.21) { c1 (α ε) + c 2 β = 0 c 1 β + c 2 (α ε) = 0 160
59 (12.22) α ε β β α ε = 0 (12.23) λ = α ε β (12.24) λ 1 1 λ = λ2 1 = 0 (12.25) λ = ±1 λ = 1 (12.26) ε 1 = α + β (12.27) c 1 = c 2 = 1 2 λ = 1 (12.28) ε 2 = α β (12.29) c 1 = c 2 = 1 2 β < 0 λ = 1 π 2 MO (12.30) E π = 2(α + β) MO 12.4 (12.31) λ λ λ λ = λ 4 3λ = 0 (12.32) λ = 1 + 5, , 1 + 5, (12.33) ε 1 = α β, ε 2 = α β, ε 1 = α 0.618β, ε 1 = α 1.618β π (12.34) E π = 4α β 2 π 2 4(α + β) Delocalization Energy (DE) 0.472β 161
60 12-1. π Hückel π Hückel (1) MO (2) MO (3) π π Hückel (1) MO (2) MO (3) MO (4) π (5) (6) π (7) π Hückel MO π π C 3 H Hückel MO π π π Hückel MO π π Hückel H + 3 H 3, H 3 162
61 van der Waals - r 6 - r 6 r 6 London van der Waals ) (13.1) (P + an2 V 2 (V nb) = nrt a b V 2 r e 1 r 12 2 e r +e A A1 r A2 R r B1 r B2 +e B 2 (13.2) Ψ(1, 2) = ϕ A (1)ϕ B (2) (13.3) Ĥ = 1 r A2 1 r B1 + 1 r A2 + 1 r A2 A B z x 1 A r 1 x x 2 B r 2 x (13.4) r 2 12 = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 + (z 1 z 2 R) 2 (13.5) r 2 A2 = x y (z 2 + R) 2 (13.6) r 2 B1 = x y (z 1 R) 2 (13.7) t = t t2 +, t 1 R x 1, R x 2 (13.8) (13.9) 1 r 12 = 1 R 1 r A2 = 1 R [ 1 (x 1 x 2 ) 2 + (y 1 y 2 ) 2 2(z 1 z 2 ) 2 2R(z 1 z 2 ) 2R 2 [ 1 x2 2 + y 2 2 2z Rz 2 ) 2R 2 ] 163 ]
62 (13.10) 1 = 1 [ 1 x2 1 + y1 2 2z 2 ] 2 2Rz 1 ) r B1 R 2R 2 (13.11) Ĥ = 1 R (x 1x 2 + y 1 y 2 2z 1 z 2 ) (13.12) E = ϕ A (1)ϕ B (2) Ĥ ϕ A (1)ϕ B (2) H 00 = 0 Ψ (13.13) E 0 = H 0k H k0 = 6 E 0 E k R 6 k 0 (in au) 13.3 X-H Y X O, N, F H X Y O, N, F, Cl H 2O melting point / C ο H 2O 0 H 2Te 2 HF H 2S H Se HI 100 HCl HBr NH SbH 3 3 PH AsH3 3 CH 4 SiH 4 GeH SnH molecular weight / g mol boiling point / C ο HF H Te H 2Se H 2S SbH 3 HI NH3 SnH PH AsH GeH 4 HBr SiH 4 HCl CH molecular weight / g mol
63 A-1 Laguerre Schrödinger (A1.1) 1 r 2 ( d r 2 dr ) [ ) 2µ + (E dr dr h 2 + e2 4πε 0 r ] l(l + 1) r 2 R = 0 Nχ(r) = rr(r) N (A1.2) ρ = r a 0 (A1.3) a 0 = 4πε 0 h 2 µe 2 (A1.4) η = E w 0 (A1.5) w 0 = µ ( ) e h 2 4πε 0 (A1.6) d 2 χ dρ 2 + ρ [ η + 2 l(l + 1) ρ ρ 2 ] χ = 0 (A1.7) χ(ρ) = e ± ηρ ± ρ (A1.8) χ(ρ) = ρ a e ηρ L(ρ) (A1.9) L(ρ) = b 0 + b 1 ρ + b 2 ρ 2 + a, b 0, b 1, (A1.10) χ(ρ) = e ηρ ρ a b m ρ m = e ηρ (A1.11) m=0 [ dχ dρ = ηρ e ρ a η b m ρ m + m=0 b m ρ m+a m=0 ] (m + a)b m ρ m 1 m=0 (A1.12) [ d 2 χ dρ 2 = ηρ e ρ a η b m ρ m 2 η m=0 + (m + a)b m ρ m 1 m=0 ] (m + a)(m + a 1)b m ρ m 2 m=0 ρ 2 e ηρ ρ a (A1.13) [ 0 = e ηρ ρ a ( ) 2 η(m + a) + 2 bm ρ m+1 m=0 ] + ((m + a)(m + a 1) l(l + 1)) b m ρ m m=0 165
64 (A1.14) b m ρ m+1 = b m 1 ρ m m=0 m=1 summation m = 1 summation m = 0 (A1.15) 0 = [a(a 1) l(l + 1)] b 0 + ρ m [ ( 2 ] η(m 1 + a) + 2)b m 1 + ((m + a)(m + a 1) l(l + 1))b m m=1 (A1.16) a(a 1) l(l + 1) = 0, a = 1 + l, or l a < 0 χ ρ = 0 a > 0 (A1.17) a = l + 1 Summation (A1.18) [ 2 η(m + l) + 2 ] bm 1 + [(m + l + 1)(m + l) l(l + 1)] b m = 0 (A1.19) (m + 1)(2l m)b m+1 = 2 [ η(m l) 1 ] b m L(ρ) n r (A1.20) (l + n r + 1) η 1 = 0, n r = 0, 1, 2, L(ρ) n r (A1.21) η = 1 n 2, n = n r + l + 1 = 1, 2, 3,, n l + 1 n Laguerre (A1.22) L k (x) = e x dk dx k (xk e x ), L(ρ) L s k(x) = ds dx s L k(x) (A1.23) L(ρ) = L 2l+1 n+l (2ρ/n) 166
65 A Â (A2.1) Âψ n = A n ψ n ψ n A n 2.2 Φ Ψ (A2.2) Ψ Φ = Φ Ψ 2 Ĉ, Ĉ+ (A2.3) Ψ Ĉ Φ = ĈΦ Ψ = Φ Ĉ+ Ψ 2.3 (A2.4) Ψ Ĉ Φ = Φ Ĉ Ψ Φ = Ψ = ψ Ĉ C (A2.5) ψ Ĉ ψ = C ψ ψ (A2.6) ψ Ĉ ψ = C ψ ψ = C ψ ψ (A2.7) C = C C 2.4 (A2.8) (A2.9) Ĉψ n = C n ψ n Ĉψ m = C m ψ m (A2.10) ψ m Ĉ ψ n = C n ψ m ψ n (A2.11) ψ n Ĉ ψ m = C m ψ n ψ m 167
66 Ĉ (A2.12) ψ m Ĉ ψ n = ψ n Ĉ ψ m (A2.13) C n ψ m ψ n = C m ψ n ψ m C n 0, C m 0, C n C m (A2.14) ψ m ψ n = ψ n ψ m = (A2.15) Φ = c n ψ n n c n ψm (A2.16) ψ m Φ = c n ψ m ψ n = c m (A2.17) Φ Φ = c n 2 (A2.18) Φ = ψ m Φ ψ n n n n 2.6 Ĉ C n Ĉ Ψ Φ Ĉ ψ n C n (A2.19) Φ = (A2.20) Ψ = a n ψ n n b n ψ n n Φ Ĉ Ψ = a n ψ n Ĉ (A2.21) b m ψ m = a n ψ n b m C m ψ m = n m a nb m C m ψ n ψ m = a nb n C n n m n Ψ Ĉ Φ = b n ψ n Ĉ (A2.22) a m ψ m = b n ψ n a m C m ψ m n n m n ( ( ) = b na m C m ψ n ψ m ) = a n b nc n = a nb n Cn n m n n 168 m m
67 (A2.23) C n = C n (A2.21) (A2.22) (A2.24) Ψ Ĉ Φ = Φ Ĉ Ψ Ĉ 169
68 A q Coulomb (A3.1) U(r) = q r 3.2 Q(r) Q(r) ρ Φ(ρ) r V (ρ, r) r dr 4πr 2 Q(r)dr ρ < r r θ ρ R (A3.2) R 2 = r 2 + ρ 2 2rρ cos θ ρ, r (A3.3) 2RdR = 2rρ sin θdθ (A3.4) ρ > r V in (ρ, r)dr = 2π 0 dϕ π 0 = 2πr2 Q(r)dr rρ = 4πr2 Q(r)dr r sin θdθ Q(r) R r+ρ r ρ dr r2 dr r R θ ρ 170
69 (A3.5) V out (ρ, r)dr = 2π 0 dϕ π 0 = 2πr2 Q(r)dr rρ = 4πr2 Q(r)dr ρ sin θdθ Q(r) R ρ+r ρ r dr r2 dr (A3.6) Φ(ρ) = ρ V out (ρ, r)dr + 0 ρ V in (ρ, r)dr (A3.7) (A3.8) Q(r) = e αr Φ(ρ) = 8π [ ( α 3 1 e αρ αρ )] ρ Q (ρ) 2 W (A3.9) W = 2π dϕ π sin θdθ ρ 2 dρq (ρ)φ(ρ) (A3.10) (A3.11) (A3.12) (A3.13) Q (r) = e αr W = 20π2 α 5 = d r A = dx A dy A dz A d r B = dx B dy B dz B e α(r A+r B) d r A d r B r AB 171
70 A-4 (A4.1) (A4.2) Ĥψ n = E n ψ n Ĥ = Ĥ + λĥ n = 1 n = 2 (A4.3) (A4.4) (A4.5) Ĥ ψ 1 = E ψ 1 Ĥ ψ 2 = E ψ 2 Ĥ ψ 3 = E 3ψ 3 E ψ 1, ψ 2 2 (A4.6) ψ = c 1 ψ 1 + c 2 ψ 2 (A4.7) ψ = c 1 ψ 1 + c 2 ψ 2 + k 3 c k ψ k Schrödinger (A4.8) c 1 Ĥ ψ 1 +c 2 Ĥ ψ 2 + k 3 c k Ĥ ψ k +c 1 Ĥ ψ 1 +c 2 Ĥ ψ 2 + k 3 c k Ĥ ψ k = E(c 1 ψ 1 +c 2 ψ 2 + k 3 c k ψ k) ψ 1 ψ 2 (A4.9) (A4.10) (E E H 11)c 1 + H 12c 2 = 0 H 21c 2 (E E H 22)c 1 = 0 2 c 1, c 2, E c 1 = c 2 = 0 (A4.11) (E E H 11) H 12 H 21 (E E H 22) = 0 ψ 1, ψ 2 H 11 = H 22 H 12 = H 21 > 0 (A4.12) E = E + H 11 ± H 12 (A4.13) ψ ± = 1 2 (ψ 1 ± ψ 2) 172
71 A a 11 a 12 a 1n (A5.1) D = a 21 a 22 a 2n a n1 a n2 a nn a ij i j D k m D km A km (A5.2) A km = ( 1) k+m D km 5.2 (A5.3) a 11 a 12 a 21 a 22 = a 11a 22 a 12 a a 11 a 12 a 13 (A5.4) a 21 a 22 a 23 a 31 a 32 a 33 = a 11a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 13 a 22 a 31 a 12 a 21 a 33 a 11 a 23 a n (A5.5) D = n a im A im = i=1 n a kj A kj j=1 n (n 1) 2 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 22 a 23 a 32 a 33 a 21 a 12 a 13 a 32 a 33 + a 31 a 12 a 13 (A5.6) a 22 a 23 a = a 22 a a 32 a 33 a 12 a 21 a 23 a 31 a 33 + a 13 a 21 a 22 a 31 a 32 = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 13 a 22 a 31 a 12 a 21 a 33 a 11 a 23 a
72 5.5 A-5-1. a 11 a 12 a 13 (A5.7) a 21 a 22 a 23 a 31 a 32 a 33 = a 21 a 22 a 23 a 11 a 12 a 13 a 31 a 32 a 33 = a 12 a 11 a 13 a 22 a 21 a 23 a 32 a 31 a 33 A-5-2. c c ca 11 ca 12 ca 13 (A5.8) a 21 a 22 a 23 a 31 a 32 a 33 = ca 11 a 12 a 13 ca 21 a 22 a 23 ca 31 a 32 a 33 = c a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 A-5-3. a 11 a 12 a 13 (A5.9) a 11 a 12 a 13 a 31 a 32 a 33 = a 11 a 11 a 13 a 21 a 21 a 23 a 31 a 31 a 33 = 0 A-5-4. c (A5.10) a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = = a 11 a 12 a 13 a 21 + ca 11 a 22 + ca 12 a 23 + ca 13 a 31 a 32 a 33 a 11 a 12 + ca 11 a 13 a 21 a 22 + ca 21 a 23 a 31 a 32 + ca 31 a 33 A-5-1. (1) (4) A-5-2. n A B A B A B 174
73 A µ ϕ( r) (A6.1) µ 0 ϕ( r) = µ 0 µ r 4πr 3 A I n ϕ( r) (A6.2) ϕ( r) = µ 0AI n r 4πr 3 2 (A6.3) µ = AI n 6.2 R (A = πr 2 ) m e e ω (A6.4) (A6.5) (A6.6) L = me r 2 ω n I = ωe 2π µ = ωe 2π πr2 n = e L γel 2m e γ e m z (A6.7) L z = m h (A6.8) µ z = e h 2m e m µ B m µ B Bohr 6.3 z B E (A6.9) E = µ B = µ z B 2m + 1 z x, y 175
74 6.4 m = 2 m = 1 m = 0 m = 1 m = Electron Spin Resonance ( EPR ) (A6.10) µ e = g e γ e m s h g e g B (A6.11) E = g e γ e m s hb = g e µ B B m s = ±1/2 (A6.12) E = E α E β = g e µ B B ν (A6.13) hν = E = g e µ B B ESR ν B B loc (A6.14) hν = g e µ B B loc = g e µ B (1 σ)b gµ B B g σ 6.6 NMR NMR I m I ( I m I I) I 0, 1/2, 1, 3/2, 176
75 6.7 z (A6.15) µ z = γ hm I γ g g I µ N (A6.16) g I µ N = γ h (A6.17) µ N = e h 2m p z B (A6.18) (A6.19) E = µ z B = γ hbm I E = γ hb (%) I g I 1 H / D C C / N N / O O / NMR (A6.20) hν = γ hb = ω h ω Lamor Lamor B B loc (A6.21) (A6.22) B loc = (1 σ)b ω = γb loc = γ(1 σ)b σ σ δ ν B (A6.23) δ = ν ν ν 10 6 = B B B
76 6.9 α α N α β N β Boltzmann (A6.24) N α = N β e E/kBT - α β Boltzmann - A-6-1. (1) A I ( n ) (2) Bohr (3) R (4) Bohr A d A-6-3. (1) z z (2) B (3) (4) A GHz ESR mt g A-6-5. g = GHz ESR ESR A-6-6. (1) z B (2) (3) Lamor A-6-7. (1) z J T 1 (2) g 1 T (3) 10 T 1 T (4) 400 MHz 1 H 2 D 12 C 13 C 14 N I 1/ /2 1 γ I A-6-8. NMR (1) (2) (3) 178
77 A-7 Schrödinger 7.1 self consistent field N Schrödinger N N ) (A7.1) Ĥ = Ĥ j = ( h2 2m 2 j + U j (r j ) j=1 j=1 (A7.2) Ψ N (r 1, r 2,, r N ) = ψ 1 (r 1 )ψ 2 (r 2 ) ψ N (r N ) U j (r j ) j r j Hartree A-7-1. ψ j A-7-2. j U j j ψ i A-7-3. U j ψ j A-7-4. U i ψ i A-7-5. U i 2 4 Fock A-7-1. He Hartree-Fock U 1 (r 1 ) 1 (A7.3) U 1 (r 1 ) = ϕ 1s (r 2 ) 1 ϕ 1s(r 2 ) r 12 ϕ 1s 1s U 1 (r 1 ) A-7-2. Z Z Z Z s (A7.4) s = Z Z s Slater 1. i j s ij i s i (A7.5) s i = j s ij 2. (1s), (2s, 2p), (3s, 3p), (3d), (4s, 4p), (4d), (4f), (5s, 5p), (5d) 3. i j s ij = 0 179
78 4. i j s ij = 1/3 5. i j s ij = 1 H Ar 180
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