今日、朝起きてからずっと多電子原子波動関数の作成プログラムやっているけど、なかなかうまく行かない。
コードは引き続き下記のサイトを参照。
http://www-cms.phys.s.u-tokyo.ac.jp/~naoki/CIPINTRO/CIP/atom.html
とりあえずs軌道は大丈夫。
でもp軌道になった途端に、束縛状態がなくなってしまう。遠心力ポテンシャルに負けちゃう。
Liで結果を比較したけれども、イオン化エネルギーはバッチリ合っている。
でもトータルが一緒にならない。
全く同じコードを書いてるわけじゃないから、全く同じじゃないのは良いにしても、イオン化エネルギー合ってるのにトータルエネルギー合わないのはめっちゃ気持ち悪い。
てかp, d軌道を計算したい。。。
休みの間にケリがつかないのは気持ち悪いがしょうがない。
以下、コードです。
import numpy as np import matplotlib.pyplot as plt import prettyplotlib as ppl """ calculation condition """ dx = 0.0625 r_num = 255 + 1 r_far = r_num - 1 r_o = 160 output_r_max = 64.0 # 1s, 2s, 2p, 3s, 3p z = 3. deg = [ 2., 1. ] # degeneracy = occupancy n = [ 1, 2 ] # principal quantum number l = [ 0, 0 ] # partial wave eps = 1e-6 r, u = [], [] for i in range( r_num ): r.append( np.exp( dx * ( i - r_o ) ) / z ) u.append( - z ) """ functions """ # simga = r^2 | \phi |^2 # total sigma def Calc_sigma( ): # Gset and deg are global values sig = np.zeros( r_num ) for i_d, d in enumerate( deg ): # in this case, G[ r, ni ] is needed. NormCheck( Gset[ i_d ] ) for i_r, gi in enumerate( Gset[ i_d ] ): sig[ i_r ] += d * ( gi**2 ) sum = 0. for i_r, s in enumerate( sig ): sum += s * r[ i_r ] * dx print 'total electron', sum return sig def TotalEnergy( TE0 ): corre = 0. for i, sig in enumerate( sigma ): corre += sig * r[i] * ( 2. * Calc_uh( i ) + Calc_uxc( i ) ) * dx return TE0 - 0.25 * corre def Calc_uc( i ): return - z def Calc_uh( i ): uh = 0. for j in range( r_num ): if j < i : uh += sigma[j] * r[j] * dx # dr = \delta e^{\delta * x } dx = \delta r dx else: # i < j uh += sigma[j] * r[i] * dx return uh def Calc_rs( i ): return ( 6. * r[i] * r[i] / sigma[i] )**( 1. / 3. ) def Calc_beta( rs ): return 1. + 0.0368 * rs * np.log( 1. + 21. / rs ) def Calc_uxc( i ): rs = Calc_rs( i ) beta = Calc_beta( rs ) return - 2. * beta * ( 3. / ( 32. * np.pi * np.pi ) * r[i] * sigma[i] )**( 1. / 3. ) def Calc_u( i ): uc = Calc_uc( i ) uh = Calc_uh( i ) uxc = Calc_uxc( i ) return uc + uh + uxc def P( i, i_n ): return 2. * r[i] * u[i] - 2. * ( r[i]**2 ) * E + l[ i_n ] * ( l[ i_n ] + 1 ) def Pzero( i_n ): for i in range( r_far, -1, -1 ): if P( i, i_n ) < 0.: break if i == 0: i = r_o return i def Calc_dG( i, Gi, Fi, i_n ): return Fi def Calc_dF( i, Gi, Fi, i_n ): return P( i, i_n ) * Gi + Fi def HammingA( i_fit, i_n ): node = 0 Gpc, Fpc = 0., 0. G, F = np.zeros( r_num ), np.zeros( r_num ) dG, dF = np.zeros( r_num ), np.zeros( r_num ) for i in range( 0, 4 ): G[i] = r[i] ** ( l[ i_n ] + 1 ) F[i] = ( l[ i_n ] + 1 ) * G[i] dG[i] = Calc_dG( i, G[i], F[i], i_n ) dF[i] = Calc_dF( i, G[i], F[i], i_n ) for i in range( 4, i_fit + 1 ): Gp = G[i-4] + ( 4.0 * dx / 3.0 ) * ( 2. * dG[i-1] - dG[i-2] + 2. * dG[i-3] ) Fp = F[i-4] + ( 4.0 * dx / 3.0 ) * ( 2. * dF[i-1] - dF[i-2] + 2. * dF[i-3] ) Gm = Gp - ( 112. / 121. ) * Gpc Fm = Fp - ( 112. / 121. ) * Fpc dGm = Calc_dG( i, Gm, Fm, i_n ) dFm = Calc_dF( i, Gm, Fm, i_n ) Gc = ( 9. / 8. ) * G[i-1] - ( 1. / 8. ) * G[i-3] + ( 3. * dx / 8. ) * ( dGm + 2. * dG[i-1] - dG[i-2] ) Fc = ( 9. / 8. ) * F[i-1] - ( 1. / 8. ) * F[i-3] + ( 3. * dx / 8. ) * ( dFm + 2. * dF[i-1] - dF[i-2] ) Gpc = Gp - Gc Fpc = Fp - Fc G[i] = Gc + ( 9. / 121. ) * Gpc F[i] = Fc + ( 9. / 121. ) * Fpc dG[i] = Calc_dG( i, G[i], F[i], i_n ) dF[i] = Calc_dF( i, G[i], F[i], i_n ) if G[i] * G[i-1] < 0. : node += 1 return node, G, F def HammingB( i_fit, i_n ): node = 0 Gpc, Fpc = 0., 0. G, F = np.zeros( r_num ), np.zeros( r_num ) dG, dF = np.zeros( r_num ), np.zeros( r_num ) kapper = np.sqrt( np.abs( 2. * E ) ) for i in range( r_far, r_far - 4, -1 ): G[i] = np.exp( - kapper * r[i] ) F[i] = - kapper * r[i] * G[i] dG[i] = Calc_dG( i, G[i], F[i], i_n ) dF[i] = Calc_dF( i, G[i], F[i], i_n ) for i in range( r_far - 4, i_fit - 1, -1 ): Gp = G[ i + 4 ] - ( 4.0 * dx / 3.0 ) * ( 2. * dG[ i + 1 ] - dG[ i + 2 ] + 2. * dG[ i + 3 ] ) Fp = F[ i + 4 ] - ( 4.0 * dx / 3.0 ) * ( 2. * dF[ i + 1 ] - dF[ i + 2 ] + 2. * dF[ i + 3 ] ) Gm = Gp - ( 112. / 121. ) * Gpc Fm = Fp - ( 112. / 121. ) * Fpc dGm = Calc_dG( i, Gm, Fm, i_n ) dFm = Calc_dF( i, Gm, Fm, i_n ) Gc = ( 9. / 8. ) * G[ i + 1 ] - ( 1. / 8. ) * G[ i + 3 ] - ( 3. * dx / 8. ) * ( dGm + 2. * dG[ i + 1 ] - dG[ i + 2 ] ) Fc = ( 9. / 8. ) * F[ i + 1 ] - ( 1. / 8. ) * F[ i + 3 ] - ( 3. * dx / 8. ) * ( dFm + 2. * dF[ i + 1 ] - dF[ i + 2 ] ) Gpc = Gp - Gc Fpc = Fp - Fc G[i] = Gc + ( 9. / 121. ) * Gpc F[i] = Fc + ( 9. / 121. ) * Fpc dG[i] = Calc_dG( i, G[i], F[i], i_n ) dF[i] = Calc_dF( i, G[i], F[i], i_n ) if G[i] * G[ i + 1 ] < 0. : node += 1 return node, G, F def Normalization( GA, FA, GB, FB, i_fit ): for i in range( i_fit ): G[i] = GA[i] F[i] = FA[i] coeff = GA[ i_fit ] / GB[ i_fit ] for i in range( i_fit, r_num ): G[i] = GB[i] * coeff F[i] = FB[i] * coeff sum = 0. for i in range( r_num ): sum += ( G[i]**2 ) * r[i] * dx # dr = e^x dx = r dx norm = 1. / np.sqrt( sum ) for i in range( r_num ): G[i] *= norm F[i] *= norm return G, F, norm def NormCheck( vec ): sum = 0. for i_v, vi in enumerate( vec ): sum += r[ i_v ] * ( vi**2 ) * dx print 'Normalization check', sum def Solver( i_n ): true_node = n[i_n] - l[i_n] - 1 i_fit = Pzero( i_n ) nodeA, GA, FA = HammingA( i_fit, i_n ) LogA = FA[ i_fit ] / ( r[ i_fit ] * GA[ i_fit ] ) nodeB, GB, FB = HammingB( i_fit, i_n ) LogB = FB[ i_fit ] / ( r[ i_fit ] * GB[ i_fit ] ) node = nodeA + nodeB G, F, norm = Normalization( GA, FA, GB, FB, i_fit ) #NormCheck( G ) dE0 = - 0.5 * ( G[ i_fit ] ** 2 ) * ( LogB - LogA ) if not true_node == node: dE = float( true_node - node ) / ( n[i_n] ** 3 ) elif dE0 > 0.01: dE = dE0 * 0.5 else: dE = dE0 if not E < 0: print 'E', E + dE print ' not a bound state' exit() return node, G, F, dE """ program main """ dTE = 1e-5 Gset = [ np.zeros( r_num ) for i_n, ni in enumerate( n ) ] sigma = [ 1e-12 for i in range( r_num ) ] # initialize the global sigma u = np.array([ Calc_u( i ) for i in range( r_num ) ]) TE, TE_old = 0., 0. while abs( dTE ) > eps: for i_n, ni in enumerate( n ): E = - 0.5 * ( z**2 ) / ( ni**2 ) - 0.05 print '-' * 20 print ' ' * 5, 'n=', ni, 'l=', l[i_n] print '-' * 20 G, F = np.zeros( r_num ), np.zeros( r_num ) dE = 1e-5 if l[ i_n ] == 1: ppl.plot( r, [ P( i_r, i_n ) for i_r in range( r_num ) ] ) plt.ylim( -5, 5 ) plt.show() while abs( dE ) > eps: node, G, F, dE = Solver( i_n ) E = E + dE Gset[i_n] = G sigma = Calc_sigma() """ merge old and new versions of u """ u = np.array([ 0.6 * ui + 0.4 * Calc_u(i) for i, ui in enumerate( u ) ]) print deg[ i_n ], E TE += deg[ i_n ] * E dTE = TE - TE_old TE_old = TE TE = 0. print 'dTE', dTE print 'Total Energy0', TE_old TE = TotalEnergy( TE_old ) print 'Total Energy', TE for i_gs, gs in enumerate( Gset ): ppl.plot( r, gs ) plt.show() ppl.plot( r, map( Calc_u, range( r_num ) ) ) plt.show()