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@ -893,7 +893,7 @@
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"- une matrice Psi~ (n par n) avec tous ses éléments nuls sauf les colonnes qui contiennent les OMs i~ et j~\n",
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"Ainsi, on soustrait à la matrice des OMs Phi la matrice Psi pour supprimer les OMs i et j de celle ci puis on additionne cette nouvelle matrice à la matrice Psi~ pour créer la nouvelle matrice des OMs Phi~ avec i~ et j~.\n",
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"\n",
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"On repart de cette nouvelle matrice Phi~ et on cherche la paire d'orbitale (k,l) ayant le plus grand angle de rotation alpha. Et on procède comme nous l'avons précedemment de manière intérative. Le but étant de maximiser D pour la localisation de Edminston et de minimiser B pour la localisation de Boyls."
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"On repart de cette nouvelle matrice Phi~ et on cherche la paire d'orbitale (k,l) ayant le plus grand angle de rotation alpha. Et on procède comme nous l'avons précedemment de manière intérative. Le but étant de maximiser D pour la localisation de Edminston et la localisation de Boyls."
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]
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},
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{
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@ -1022,7 +1022,7 @@
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"Sinon on procède comme Edminson Ruedenberg mais avec les intégrales $A_{12}$ et $B_{12}$ définies comme :\n",
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"\n",
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"$A^r_{12} = \\langle \\phi_1 | \\bar r | \\phi_2 \\rangle $\n",
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"$*\\langle \\phi_1 | \\bar r | \\phi_2 \\rangle $\n",
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"$.\\langle \\phi_1 | \\bar r | \\phi_2 \\rangle $\n",
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"$- \\frac {1}{4}(\\langle \\phi_1 | \\bar r | \\phi_1 \\rangle $\n",
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"$- \\langle \\phi_2 | \\bar r | \\phi_2 \\rangle . \\langle \\phi_1 | \\bar r | \\phi_1 \\rangle$\n",
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"$- \\langle \\phi_2 | \\bar r | \\phi_2 \\rangle)$\n",
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@ -1268,6 +1268,194 @@
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"$= \\sum_n (\\sum_a \\sum_b \\sum_e \\sum_f c_{an} c_{bn} c_{en} c_{fn} [\\chi_a \\chi_b| \\chi_e \\chi_f])$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Localisation des orbitales moléculaires"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Principe de la rotation par paires d'orbitales.\n",
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"\n",
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"La rotation des orbitales se fait de la manière suivante :\n",
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"\n",
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"On extrait de la matrice des coefficients des OMs $\\Phi$, (m_C), les orbitales $i$ et $j$ (vecteurs colonnes) pour former une nouvelle matrice $Ksi$ de dimension (n par 2). \n",
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"\n",
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"Pour effectuer la rotation des orbitales $i$ et $j$ on utilise la matrice de rotation $R$ (m_R) pour la rotation de 2 orbitales, qui est définie comme la matrice :\n",
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"\n",
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"$R$ =\n",
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"\n",
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" ( cos(alpha) -sin(alpha) )\n",
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"\n",
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" ( sin(alpha) cos(alpha) )\n",
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" \n",
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"La valeur de $\\alpha$ sera la plus grande de la matrice des alphas *Cf Localisation de Edminston Ruedenberg*\n",
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" \n",
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"On applique $R$ à $Khi$ : $R Ksi = \\tilde Ksi$ \n",
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"\n",
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"On obtient $\\tilde Ksi$ la matrice contenant les coefficients des deux nouvelles OMs $\\tilde i$ et $\\tilde j$ obtenues par rotation de $i$ et $j$.\n",
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"\n",
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"On réinjecte ces deux nouvelles orbitales $\\tilde i$ et $\\tilde j$ à la place des anciennes orbitales $i$ et $j$ dans la matrice des coefficients des OMs $\\Phi$, (m_C), ce qui nous donne une nouvelle matrice des coefficient des OMs $\\tilde \\Phi$, (new_m_C).\n",
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"Pour cela on créer des matrices intérmédiaires:\n",
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"- une matrice $\\Psi$ (n par n) avec tous ses éléments nuls sauf les colonnes qui contiennent les OMs $i$ et $j$ \n",
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"- une matrice $\\tilde \\Psi$ (n par n) avec tous ses éléments nuls sauf les colonnes qui contiennent les OMs $\\tilde i$ et $\\tilde j$\n",
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"- une matrice m_interm (n par n) où l'on a soustrait $\\Psi$ à $\\Phi$ pour créer des 0 sur les colonnes des OMs $i$ et $j$ \n",
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"\n",
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"Ainsi, on soustrait à la matrice des coefficient $\\Phi$ la matrice $\\Psi$ pour supprimer les OMs $i$ et $j$ de celle ci puis on additionne cette nouvelle matrice à la matrice $\\tilde \\Psi$ pour créer la nouvelle matrice des coefficients des OMS $\\tilde \\Phi$, avec $\\tilde i$ et $\\tilde j$.\n",
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"\n",
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"On repart de cette nouvelle matrice des coefficients $\\tilde \\Phi$ et on cherche la paire d'orbitale $(k,l)$ ayant le plus grand angle de rotation alpha (l'angle $\\alpha$ pour la paire d'orbitale $(i,j)$ étant devenui nul après rotation. Et on procède comme nous l'avons précedemment de manière intérative. Le but étant de maximiser $D$ pour la localisation de Edminston et la localisation de Boyls."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Localisation de Edmiston C. & Ruedenberg K. \n",
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"\n",
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"(Localisation des orbitales par rotation de paires d'orbitales)\n",
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"\n",
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"Ref :\n",
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"*Edmiston, C., & Ruedenberg, K. (1963). Localized Atomic and Molecular Orbitals.\n",
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"Reviews of Modern Physics, 35(3), 457–464. doi:10.1103/revmodphys.35.457*\n",
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"\n",
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"Méthode :\n",
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"\n",
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"Le but de cette méthode est de maximiser $D$ :\n",
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"\n",
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"$D(\\phi)= \\sum_n [\\phi_n^2 | \\phi_n^2 ]$\n",
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"\n",
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"$= \\sum_n < \\phi^2_n | 1/r_{12} | \\phi^2_n >$\n",
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" \n",
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" \n",
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"\n",
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" \n",
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"Car selon *J. E. Lennard-Jones and J. A. Pople, Proc. Roy. Soc. (London) A202, 166 (1950)*, on peut générer des orbitales équivalentes et celles ci maximiseront probablement la somme des termes d'auto répulsion orbitalaire $D$.\n",
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"\n",
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"On va créer des nouvelles orbitales $\\tilde i$ et $\\tilde j$ à partir des orbitales $i$ et $j$ par combinaison linéaire de ces dernières tel que :\n",
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"\n",
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"$i~ (x) = cos(\\gamma) i(x) + sin(\\gamma) j(x)$\n",
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"\n",
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"$j~ (x) = -sin(\\gamma) i(x) + cos(\\gamma) j(x)$\n",
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"\n",
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"On part de la matrices de orbitales moléculaires $\\Phi$ et on cherche la paire d'orbitale $(i,j)$ ayant le plus grand angle de rotation $\\alpha$, avec $\\alpha$ défini comme (si ce dernier est supétieur à $\\frac{\\pi}{2}$ on devra soustraire $\\frac{\\pi}{2}$ aux éléments de la matrices) :\n",
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"\n",
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"\n",
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"$Cos(4 \\alpha)= -A_{12} / (A_{12}^2 + B_{12}^2)^{1/2}$\n",
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"\n",
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"$\\alpha = (1/4) Acos (-A_{12} / (A_{12}^2 + B_{12}^2)^{1/2})$\n",
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"\n",
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"$Sin(4 \\alpha)= B_{12} / (A_{12}^2 + B_{12}^2)^{1/2}$\n",
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"\n",
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"$\\alpha = (1/4) Asin (B_{12} / (A_{12}^2 + B_{12}^2)^{1/2})$\n",
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"\n",
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"$Tan(4 \\alpha) = -B_{12} / A_{12}$\n",
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"\n",
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"$\\alpha = (1/4) Atan (-B_{12} / A_{12})$\n",
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"\n",
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"\n",
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"Avec : \n",
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"\n",
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"$A_{ij} = [ \\phi_i \\phi_j | \\phi_i \\phi_j] - 1/4 [\\phi_i^2 - \\phi_j^2 | \\phi_i^2 - \\phi_j^2] $\n",
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" \n",
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"Où :\n",
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"\n",
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"$ [ \\phi_i \\phi_j | \\phi_i \\phi_j] = \\sum_a c_{ai} \\langle \\chi_a \\phi_j | \\phi_i \\phi_j \\rangle$\n",
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"\n",
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"$=\\sum_a \\sum_b c_{ai} c_{bj} \\langle \\chi_a \\chi_b | \\phi_i \\phi_j \\rangle $\n",
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"\n",
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"$=\\left(\\sum_a c_{ai} \\left(\\sum_b c_{bj} \\left(\\sum_e c_{ei} \\left(\\sum_f c_{fj} \\langle \\chi_a \\chi_b | \\chi_e \\chi_f \\rangle \\right)\\right)\\right)\\right) $\n",
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"\n",
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"$=\\sum_a \\sum_b \\sum_e \\sum_f \\left( c_{ai} c_{bj} c_{ei} c_{fj} \\langle \\chi_a \\chi_b | \\chi_e \\chi_f \\rangle \\right) $\n",
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"\n",
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"Et :\n",
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"\n",
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"$\\phi_i ^2 = \\left( \\sum_a c_{ai} \\chi_a \\right)^2$\n",
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"$= \\sum_a c_{ai} \\sum_b c_{bi} \\chi_a \\chi_b$\n",
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"\n",
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"$\\phi_i ^2 - \\phi_j ^2 = \\left( \\sum_a c_{ai} \\chi_a \\right)^2 - \\left( \\sum_a c_{aj} \\chi_a \\right)^2$\n",
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"$= \\sum_a c_{ai} \\sum_b c_{bi} \\chi_a \\chi_b - \\sum_a c_{aj} \\sum_b c_{bj} \\chi_a \\chi_b$\n",
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"\n",
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"$[\\phi_i^2 -\\phi_j^2 |\\phi_i^2 -\\phi_j^2] = [\\left( \\sum_a c_{ai} \\chi_i \\right)^2 - \\left( \\sum_a c_{aj} \\chi_j \\right)^2|\\phi_i^2 -\\phi_j^2]$\n",
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"\n",
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"$= \\sum_a c_{ai} \\sum_b c_{bi} [\\chi_a \\chi_b| \\phi_i^2 -\\phi_j^2 ] - \\sum_a c_{aj} \\sum_b c_{bj} [ \\chi_a \\chi_b| \\phi_i^2 -\\phi_j^2 ] $\n",
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"\n",
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"$= \\left(\\sum_a c_{ai} \\sum_b c_{bi} - \\sum_a c_{aj} \\sum_b c_{bj} \\right) [ \\chi_a \\chi_b| \\phi_i^2 -\\phi_j^2 ] $\n",
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"\n",
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"$= \\left(\\sum_a c_{ai} \\sum_b c_{bi} - \\sum_a c_{aj} \\sum_b c_{bj} \\right) [ \\chi_a \\chi_b| \\left( \\sum_a c_{ai} \\chi_i \\right)^2 - \\left( \\sum_a c_{aj} \\chi_j \\right)^2 ] $\n",
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"\n",
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"$= \\left(\\sum_e c_{ei} \\sum_f c_{fi} - \\sum_e c_{ej} \\sum_f c_{fj} \\right) \\left(\\sum_a c_{ai} \\sum_b c_{bi} - \\sum_a c_{aj} \\sum_b c_{bj} \\right) [ \\chi_a \\chi_b| \\chi_e \\chi_f ] $\n",
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"\n",
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"Mais aussi :\n",
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"\n",
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"$B_{ij} = [\\phi_i ^2 - \\phi_j ^2 | \\phi_i \\phi_j ] = [\\left( \\sum_a c_{ai} \\chi_i \\right)^2 - \\left( \\sum_a c_{aj} \\chi_j \\right)^2| \\phi_i \\phi_j ] $\n",
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"\n",
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"$= \\sum_a c_{ai} \\sum_b c_{bi} [\\chi_a \\chi_b| \\phi_i \\phi_j ] - \\sum_a c_{aj} \\sum_b c_{bj} [ \\chi_a \\chi_b| \\phi_i \\phi_j ] $\n",
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"\n",
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"$= \\left(\\sum_a c_{ai} \\sum_b c_{bi} - \\sum_a c_{aj} \\sum_b c_{bj} \\right) [ \\chi_a \\chi_b| \\phi_i \\phi_j ] $\n",
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"\n",
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"$= \\left(\\sum_a c_{ai} \\sum_b c_{bi} - \\sum_a c_{aj} \\sum_b c_{bj} \\right) \\sum_e \\sum_f c_{ei} c_{fj} [ \\chi_a \\chi_b| \\chi_e \\chi_f ] $\n",
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"\n",
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"Pour le calcul de $D$ (le critère de localisation qu'on maximise) , sachant que :\n",
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"\n",
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"$\\phi_i ^2 = \\left( \\sum_a c_{ai} \\chi_a \\right)^2$\n",
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"$= \\sum_a c_{ai} \\sum_b c_{bi} \\chi_a \\chi_b$\n",
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"\n",
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"On aura :\n",
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"\n",
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"$D=\\sum_n [\\phi_n^2|\\phi_n^2]$\n",
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"\n",
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"$= \\sum_n (\\sum_a c_{an} \\sum_b c_{bn} [\\chi_a \\chi_b| \\phi_n^2])$\n",
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"\n",
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"$= \\sum_n (\\sum_a c_{an} \\sum_b c_{bn} \\sum_e c_{en} \\sum_f c_{fn} [\\chi_a \\chi_b| \\chi_e \\chi_f])$\n",
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"\n",
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"$= \\sum_n (\\sum_a \\sum_b \\sum_e \\sum_f c_{an} c_{bn} c_{en} c_{fn} [\\chi_a \\chi_b| \\chi_e \\chi_f])$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Localisation de Boys\n",
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"\n",
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"Ref:\n",
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"*Localized molecular orbitals for polyatomic molecules. I. A comparison of the Edmiston‐Ruedenberg and Boys localization methods\n",
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"J. Chem. Phys. 61, 3905 (1974); https://doi.org/10.1063/1.1681683\n",
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"Daniel A. Kleier, Thomas A. Halgren, John H. Hall Jr., and William N. Lipscomb*\n",
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"\n",
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"\n",
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"On procède comme pour la méthode de Edminson-Ruedenberg mais avec les intégrales $A_{12}$ et $B_{12}$ définies comme :\n",
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"\n",
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"$A^r_{12} = \\langle \\phi_1 | \\bar r | \\phi_2 \\rangle $\n",
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"$.\\langle \\phi_1 | \\bar r | \\phi_2 \\rangle $\n",
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"$- \\frac {1}{4}(\\langle \\phi_1 | \\bar r | \\phi_1 \\rangle $\n",
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"$- \\langle \\phi_2 | \\bar r | \\phi_2 \\rangle . \\langle \\phi_1 | \\bar r | \\phi_1 \\rangle$\n",
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"$- \\langle \\phi_2 | \\bar r | \\phi_2 \\rangle)$\n",
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"\n",
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"Et \n",
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"\n",
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"$B^r_{12} = (\\langle \\phi_1 | \\bar r | \\phi_1 \\rangle - \\langle \\phi_2 | \\bar r | \\phi_2 \\rangle)$\n",
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"$ . \\langle \\phi_1 | \\bar r | \\phi_2 \\rangle $\n",
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"\n",
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"Avec \n",
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"\n",
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"$A^r_{12}=A^x_{12} + A^y_{12} + A^z_{12}$\n",
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"\n",
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"$B^r_{12}=B^x_{12} + B^y_{12} + B^z_{12}$\n",
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"\n",
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"Et le critère D à maximiser est défini tel que :\n",
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"\n",
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"$D= \\sum_i < \\phi_i | r | \\phi_i >$\n",
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"\n",
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"Avec \n",
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"\n",
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"$< \\phi_i | r | \\phi_i > = < \\phi_i | x | \\phi_i > + < \\phi_i | y | \\phi_i > + < \\phi_i | z | \\phi_i >$\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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@ -1683,53 +1871,6 @@
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"f_alpha_boys m_C;;\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"(* Fonction de calcul de D Boys *)\n",
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"(*\n",
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"let d_boys m_C = \n",
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"\n",
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" let phi_x_phi =\n",
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" Multipole.matrix_x multipoles \n",
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" |> MOBasis.mo_matrix_of_ao_matrix ~mo_coef:m_C \n",
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" in\n",
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" \n",
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" (*Util.debug_matrix \"phi_x_phi\" phi_x_phi;*)\n",
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" \n",
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" let phi_y_phi =\n",
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" Multipole.matrix_y multipoles \n",
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" |> MOBasis.mo_matrix_of_ao_matrix ~mo_coef:m_C \n",
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" in\n",
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" \n",
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" (*Util.debug_matrix \"phi_y_phi\" phi_y_phi;*)\n",
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" \n",
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" let phi_z_phi =\n",
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" Multipole.matrix_z multipoles \n",
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" |> MOBasis.mo_matrix_of_ao_matrix ~mo_coef:m_C\n",
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"\n",
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" in\n",
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" \n",
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" (*Util.debug_matrix \"phi_z_phi\" phi_z_phi;*)\n",
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" \n",
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" let v_D_boys = \n",
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" let n_mo = Mat.dim2 m_C\n",
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" in\n",
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" Vec.init n_mo ( fun i -> (phi_x_phi.{i,i})**2. +. (phi_y_phi.{i,i})**2. +. (phi_z_phi.{i,i})**2.)\n",
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"in\n",
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"Vec.sum v_D_boys;;\n",
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"*)\n",
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"\n",
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"(*************************)\n",
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"(*\n",
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"Multipole.matrix_x multipoles;;\n",
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"d_boys m_C;;\n",
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"*)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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@ -1874,9 +2015,11 @@
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"\n",
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"(*************************)\n",
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"\n",
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"(*\n",
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"let m_alpha,d = f_alpha m_C\n",
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"let alphaij = new_m_alpha m_alpha m_C 3;;\n",
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"alphaij.alpha_max;;\n"
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"alphaij.alpha_max;;\n",
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"*)"
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]
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},
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{
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@ -1913,9 +2056,11 @@
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"let alpha = alphaij.alpha_max;; (* Fonction -> constante *) \n",
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"f_R alpha;;\n",
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"*)\n",
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"(*\n",
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"alpha_v \"deloc\" alphaij;;\n",
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"let alpha = (alpha_v \"loc\" alphaij) ;;\n",
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"f_R alpha ;;\n"
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"f_R alpha ;;\n",
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"*)"
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]
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},
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{
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@ -2184,7 +2329,7 @@
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" let alpha = (alpha_v loc_deloc alphaij) *. epsilon (* Fonction -> constante *)\n",
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" in\n",
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"\n",
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" Printf.printf \"%f\\n%!\" alpha;\n",
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" (*Printf.printf \"%f\\n%!\" alpha;*)\n",
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" \n",
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" (* Indice i et j du alpha max après calcul *)\n",
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" let indice_i = alphaij.indice_ii (* Fonction -> constante *)\n",
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@ -2252,7 +2397,11 @@
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"metadata": {
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||||
"jupyter": {
|
||||
"outputs_hidden": true
|
||||
}
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"(* Calcul *)\n",
|
||||
@ -2264,7 +2413,11 @@
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"metadata": {
|
||||
"jupyter": {
|
||||
"outputs_hidden": true
|
||||
}
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"final_m_C m_C \"boys\" \"loc\" 1. 1 0. 10e-7;;"
|
||||
@ -2273,7 +2426,11 @@
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"metadata": {
|
||||
"jupyter": {
|
||||
"outputs_hidden": true
|
||||
}
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"final_m_C new_m \"Boys\" \"deloc\" 1. 1 0. 10e-7;;\n"
|
||||
@ -2285,19 +2442,17 @@
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"nocc;;\n",
|
||||
"let toto = Vec.init (nocc) (fun i -> float_of_int(i));;\n",
|
||||
"let vec_list = Vec.to_list toto;;\n",
|
||||
"let g a = int_of_float(a);;\n",
|
||||
"let tutu = List.map g vec_list\n",
|
||||
"\n",
|
||||
"let int_list vec = \n",
|
||||
" let float_list = Vec.to_list vec\n",
|
||||
" in\n",
|
||||
" let g a = int_of_float(a)\n",
|
||||
"in List.map g float_list;;\n",
|
||||
"\n",
|
||||
"int_list toto;;"
|
||||
"let m_occ , m_vir = split_mat new_m list_occ;;"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"let loc_m_occ = final_m_C m_occ \"boys\" \"loc\" 1. 1 0. 10e-7;;\n",
|
||||
"let loc_m_vir = final_m_C m_vir \"boys\" \"loc\" 1. 1 0. 10e-7;;"
|
||||
]
|
||||
},
|
||||
{
|
||||
|
||||
Loading…
Reference in New Issue
Block a user