卒業研究報告 題 目 量子力学に基づいた水素分子の分子軌道法的取り扱いと Hamiltonian 近似法 指導教員 山本哲也 報告者 山中昭徳 平成 14 年 月 5 日
高知工科大学電子 光システム工学科. 3. 4.1 4. 4.3 4.5 6.6 8.7 10.8 11.9 1.10 1 3. 13 3.113 3. 13 3.3 13 3.4 14 3.5 15 3.6 15 3.7 17 3.8 18 3.9 18 3.10 1 3.11 3.1 3 3.13 3 4. 4 1
4.1 4 4. 5 4.3 8 4.4 30 4.6 30 4.7 30 6 31 7. 3 8. 33
1. 3
.1. m v p = mv h E = hν, p = (-1) λ (3-1)- -34 E ν h 6.65 10 J s h λ = p.3 u x φ = Asin πν t (-) u A x t x 4
φ 4π ν x = Asin πν t x u u 4πν = φ u -3 t φ x = 4πν Asinπν t t u = 4 πνφ -4 (-3)(-4) φ 1 φ = (-5) x u t (-) x t ψ x t φ( x, t) = ψ( x) T( t) (-6) x t φ () d ψ = Tt x dx φ dt = ψ (-7) t dt -5 1 () d ψ Tt = ψ d T dx u dt -8 d 1 d T ψ = ψ ( x) dx u T ( t) dt -9 k d ψ 1 d T = = k ψ ( x) dx u T( t) dt -10 d ψ = k ψ ( x) (-11) dx 5
dt kutt () = (-1) dt k z z = k (-13) + k = 0 (-14) (-10) z =± ik ( i = 1) (-15) ψ ( x) = Aexp( ikx) + Bexp( ikx) (-17) (-11) -6 Tt ( ) = Cexp( ikut) + Dexp( ikut) (-18) φ( xt, ) = ψ( xtt ) ( ) = A exp ik x ut + A ik x ut { ( )} exp ( ) ( ) exp { } { } { ( )} 1 + A exp ik x+ ut + A ik x+ ut 3 4 A 1 A 4 (-14) (-18) πν π k = = (-19) u λ kx ( ut) x kx ( ut) = πν t u x = π νt λ λ ν (-10) k π = λ (-0) 6
d ψ 4π = ( x) ψ (-1) dx λ d ψ 4π + ( x) 0 ψ = (-) dx λ.4 h p λ p = λ E T V p E = T + V = + V m (-3) p = mv p me V = ( ) (-4) 1 p m ( E V ) λ = h = h (-5) (-18) d ψ 8π m + ( E V ) ψ ( x ) = 0 (-6) dx h ψ 1 3 ψ ψ 8π m + + ( ) (, ) 0 E V ψ x y = (-7) x y h ψ ψ ψ 8π m + + + ( ) (,, ) 0 E V ψ x y z = 3 (-8) x y z h 7
8π m E h ψ + ( V ) ψ = 0 (-9).5 r t ψ (,) rt t r dv ψ (r,t ) dv (-30) ψ r ψ = 0 ψ ψ ψ (,) rt dr= 1 (-31) 1 ψ 8
ψ ψ ψ ψ ψ ψ (-31) ψ 1 ( x, yz, ) q t ψ q q+ dq ψ ( qt, ) dq (-3) F ˆf F f ˆf ˆf ψ = fψ (-33) fˆ( Ψ+Φ ) = fˆψ+ fˆφ (-34) ˆf 9
Ψ1, Ψ,, Ψn Ψ Ψ Ψ= c1ψ+ 1 cψ + + cnψn n (-35) = c Ψ i= 1 i i n n i i i i i= 1 i= 1 (-36) fˆψ= fˆ( cψ ) = c ( fˆψ).6 3.6.3 (-5) x φ( x, t) = Aexpπi νt (-37) λ -1 (, ) exp i φ x t = A ( px Et) (-38) h = π (-38) x t x φ ip i = exp ( px Et) x 10
ip = φ (-39) t φ ie i = Aexp ( px Et) t ie = φ (-40) (-38),(-39) φ i = pφ x (-41) φ i = Eφ (-4) t p i E i x t i p i E x t E.7 1 T = mv p = mv p p v = (-43) m T p T = (-44) m p i x p T = m = + + m x y z 11
= m (-45) p 3 T m T V Ĥ Ĥ = T + V (-46) T (-45) ˆ H = + V (-47) m Ĥ (-4) Ĥ Ĥφ = Eφ (-48) ψ Ĥψ = Eψ (-49) (-4)(-49) ˆ ψ Hφ = + Vφ = i (-50) m t.8.9 1
3. 3.1 3. 1, Hˆ ψ(1, ) = Eψ(1, ) (3-1) 3.3 3.3.1 13
3.3.1 Ĥ Hˆ m = + V ˆ H = + + M m ( A B) ( 1 ) e e 4πε r 4πε r 01A 01B e e 4πε r 4πε r 0 A 0 B e e + + 4πε r 4πε R 01 0 e (3-) (3-3) M m e A B 1 1 3 6 F QQ 1 F = (3-4) 4πε0r Q 1, Q r ε 0 1 ε 0 = 8.854 10 14
3.4 7 31 1.67 10 kg 9.109 10 kg 1840 (3-3) R (3-3) ˆ H = + m e ( 1 ) e e 4πε r 4πε r 01A 01B e e 4πε r 4πε r 0 A 0 B e + + 4πε r 01 (3-5) e 4πε r 01 3.5 15
3.6 1 φ(1) φ() 1 r1 V( r) = e dv (3-6) φ(1) r1 V( r ) = e dv (3-7) V( r 1 ) V( r ) r 1, r Ĥ 1 Hˆ = hˆ + hˆ 1 ˆ e e h ( ) 1 = 1 + V r1 me 4πε0ra1 4πε0rb 1 ˆ e e h ( ) = + V r me 4πε0ra 4πε0rb (3-8) E φ(1) φ() Hˆ φ(1) φ() dv dv = φ φ (1) () dv dv 1 1 (3-9) Ĥ = h ˆ ˆ 1+ h Ĥ 16
E = { ˆ + ˆ } φ(1) φ() h h (1) φ() dv dv = 1 1 φ(1) φ() dv dv 1 φ(1) hˆφ(1) dv φ() dv + φ(1) dv φ() hˆφ() dv φ(1) φ() dv1dv 1 1 1 (1) ˆ (1) () (1) () ˆ φ hφ dvφ dv φ dvφ hφ() dv = + φ(1) dv φ() dv φ(1) dv φ() dv 1 1 1 1 1 (3-10) (3-11) ( 3-1) (1) ˆ (1) () ˆ φ hφ dv φ hφ() dv = + φ(1) dv φ() dv 1 1 1 1 (3-13) = ε + ε (3-14) Ĥ ˆ hφ (1) = ε φ (1) (3-15) 1 1 h ˆ φ () = ε φ () (3-16) (3-14) ˆ φ( jh ) jφ( jdv ) j ε j =, j = 1, (3-17) φ( j) dv j 3.7 17
Ψ1, Ψ,, Ψn Ψ Ψ= cψ+ c Ψ + + c Ψ 1 1 n = c Ψ i= 1 i i n n (3-18) Ψ 3.8 ψ(1, ) = Cφ(1) + C φ() (3-19) 1 φ(1), φ() C 1, C 3.9 hφ = εφ (3-0) (3-17) ε φĥφdv = φ dv (3-1) ε (3-19) 18
= ˆ{ } { Cφ(1) + C φ() } dv Cφ(1) + C φ() h ( Cφ(1) + C φ() dv 1 1 1 (3-) Ch + CCh + Ch = CS CS CS 1 11 1 1 1 11+ 1 1 + (3-3) h = 11 1 1 φhˆ φdv h = φ hˆ φ dv h = φhˆ φ dv 1 1 S = S = φφ dv 1 1 (3-4) h11 h h 1 S 1 S (3-3) C ( h εs ) + CC ( h εs ) + C ( h εs ) = 0 (3-5) 1 11 11 1 1 1 ε ε = = 0 (3-6) c c 1 C ( h S ε) + C ( h S ε) = 0 (3-7) 1 11 11 1 1 19
C ( h S ε) + C ( h S ε) = 0 (3-8) 1 1 1 h εs h εs 11 11 1 1 h εs h εs 1 1 = 0 (3-9) (3-9) ε ε ε = (3-30) ( h11 S11)( h S) ( h1 S1) 0 h h S S 11 1 11 1 = h = α = β = S = 1 = S (3-31) (3-30) α ε β εs = (3-3) ( ) ( ) 0 αε± β ε (3-33) ε ε 1, ε α + β ε1 = (3-34) 1 + S α β ε = (3-35) 1 S α, β ε > ε 1 (3-34)(3-7) C1 = C(3-36) (3-35)(3-8) C1 = C(3-37) (-31) ψ (,) rt dr= 1 (3-38) dr = dτ (3-39) 0
(3-19) ψ (,) rt dτ = 1 (3-40) { } { } ψ ψ d τ = C φ 1 (1) + C φ () C φ 1 (1) + C φ () d τ = 1 (3-41) { } C φ 1 (1) + C φ () d τ = 1 (3-4) (3-43) ( C ) χ dτ + CC χ χ dτ + ( C ) χ dτ = 1 1 1 1 1 φ(1) = χ1, φ() = χ 1 1 χ d τ = χ d τ = 1, χ χ d τ = S ( C1) + CC 1 S+ ( C) = 1 (3-44) (3-36)(3-44) C = C = 1 ε 1 ψ 1 χ1+ χ ψ1 = + S C 1 + S (3-45) (3-46) (3-37)(3-46) 1 = 1 S (3-47) 1 C = S (3-48) ε ψ χ1 χ ψ = S (3-49) 3.10 1
α + β ε1 = 1 + S α β ε = 1 S χ1+ χ ψ1 = + S χ1 χ ψ = S ψ 1 ψ 3.10.1 p.114 6.6
3.11 3.1 3.13 3
4. 4.1 4. R 4
4..1 { a( R R ) } V(R)=D 1 exp e (4-1) D A R R D 4.35 ev A 1.94 R =0.74 e e 4.3 5
F F = kr ( R e ) (4-) R e R k R e R V R R k V = FdR= k( R Re) dr ( R Re) R = (4-3) e Re ab, M a, M b a dr a a kr ( Re) = M (4-4) dt b dr b b kr ( Re) = M (4-5) dt R, R a b Ra + Rb = R, MaRa = MbRb R a R b M b Ra = R M + M a b (4-6) R b M a = M + M a b R (4-7) (5-6)(5-7)(5-4)(5-5) MM a b µ = M + M a b MM dr dr a b kr ( Re ) = =µ M + M dt dt a b (4-8) µ 6
Hˆ 1 V = µ + (4-9) (4-9) Hˆ 1 V = µ + (4-10) (5-1) { a( R R ) } V(R)=D 1 exp e (4-11) x = 0 e 3 n x x x x = 1+ x+ + + + (4-1)! 3! n! x = ar ( R e ) V(R)=D 1 { exp a( R Re ) } D{ 1 1 ( a( R Re )) } + = D a ( R R ) e (4-13) 7
4..1 4.3.1 4.4 4.3.1 1 ev 1 ev 10 10 1 ev 4.4.1 8
4.4.1 核間距離モース曲線二次曲線 誤差 核間距離モース曲線二次曲線 誤差 0.6003 0.4154 0.31951 0.1003 0.75038 0.00177 0.00176-3.51E-05 0.6053 0.38794 0.9704 0.09091 0.75538 0.003758 0.00387-0.000114 0.61031 0.35604 0.7538 0.08066 0.76038 0.006537 0.0068-0.00063 0.61531 0.358 0.5455 0.0715 0.76538 0.010043 0.010548-0.000505 0.6031 0.9717 0.3453 0.0663 0.77039 0.01454 0.015115-0.000861 0.6531 0.7011 0.1534 0.05477 0.77539 0.019149 0.0050-0.001353 0.6303 0.4459 0.19696 0.04763 0.78039 0.04708 0.06708-0.00 0.6353 0.056 0.17941 0.04115 0.78539 0.03091 0.033734-0.0084 0.6403 0.19799 0.1667 0.0353 0.7904 0.037736 0.041579-0.003843 0.6453 0.17684 0.14675 0.03009 0.7954 0.045167 0.05043-0.005076 9
0.65033 0.15707 0.13165 0.0541 0.8004 0.053184 0.05977-0.006543 0.65533 0.13864 0.11738 0.017 0.8054 0.061768 0.07003-0.0086 0.66033 0.1153 0.1039 0.01761 0.81041 0.07090 0.081153-0.01051 0.66533 0.10569 0.09175 0.01441 0.81541 0.080568 0.093095-0.0156 0.67034 0.091091 0.079455 0.01164 0.8041 0.09075 0.10586-0.015106 0.67534 0.077704 0.068453 0.0095 0.8541 0.10143 0.11944-0.018006 0.68034 0.065495 0.0587 0.007 0.8304 0.1159 0.13384-0.0144 0.68534 0.054431 0.048909 0.0055 0.8354 0.14 0.14906-0.04835 0.69035 0.04448 0.040366 0.0041 0.8404 0.1363 0.1651-0.08794 0.69535 0.035618 0.0364 0.0098 0.8454 0.1488 0.18195-0.033137 0.70035 0.0781 0.05738 0.0007 0.85043 0.16175 0.19963-0.037878 0.70535 0.0107 0.019653 0.00137 0.85543 0.1751 0.1813-0.043033 0.71036 0.01544 0.014388 0.00086 0.86043 0.18883 0.3745-0.048615 0.71536 0.01043 0.00994 0.00049 0.86543 0.094 0.5758-0.054638 0.7036 0.006561 0.006315 0.0005 0.87044 0.174 0.7854-0.061115 0.7536 0.003609 0.003508 0.0001 0.87544 0.35 0.30031-0.06806 0.73037 0.001549 0.0015.87E-05 0.88044 0.474 0.391-0.075485 0.73537 0.000354 0.000351 3.17E-06 0.88544 0.69 0.3463-0.083404 0.74037.4E-06.4E-06-1.61E-09 0.89045 0.787 0.37055-0.09188 0.74537 0.00046768 0.0004758-4.89E-06 0.89545 0.9484 0.3956-0.10077 0.60.89 (4-1) (4-13) (4-1)(4-13) D 4.35 ev A 1.94 R e =0.74 ev, ev ev 5.6 30
0.60.89 5.7 { } a( R R ) V(R)=D 1 exp e MM a b µ = M + M a b 0.60.89 5. 31
6. M1 3
7. 1.. 3. 4. Donald A.McQuarrie John D.Simon 33