mirror of
https://gitlab.com/scemama/QCaml.git
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3656 lines
122 KiB
Plaintext
3656 lines
122 KiB
Plaintext
{
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"cells": [
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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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"# Test of F12 matrix elements"
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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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"## Initialization"
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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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"let png_image = print_endline ;;\n",
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"\n",
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"(* --------- *)\n",
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"\n",
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"#cd \"/home/scemama/QCaml\";;\n",
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"#use \"topfind\";;\n",
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"#require \"jupyter.notebook\";;\n",
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"\n",
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"let png_image name = \n",
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" Jupyter_notebook.display_file ~base64:true \"image/png\" (\"Notebooks/images/\"^name)\n",
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";;\n",
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"\n",
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"#require \"lacaml.top\";;\n",
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"#require \"alcotest\";;\n",
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"#require \"str\";;\n",
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"#require \"bigarray\";;\n",
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"#require \"zarith\";;\n",
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"#require \"getopt\";;\n",
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"#directory \"_build\";;\n",
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"#directory \"_build/Basis\";;\n",
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"#directory \"_build/CI\";;\n",
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"#directory \"_build/MOBasis\";;\n",
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"#directory \"_build/Nuclei\";;\n",
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"#directory \"_build/Parallel\";;\n",
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"#directory \"_build/Perturbation\";;\n",
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"#directory \"_build/SCF\";;\n",
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"#directory \"_build/Utils\";;"
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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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"### Modules"
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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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"#load \"Constants.cmo\";;\n",
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"#load_rec \"Util.cma\";;\n",
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"#load_rec \"Matrix.cmo\";;\n",
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"#load_rec \"Simulation.cmo\";;\n",
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"#load_rec \"Spindeterminant.cmo\";;\n",
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"#load_rec \"Determinant.cmo\";;\n",
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"#load_rec \"HartreeFock.cmo\";;\n",
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"#load_rec \"MOBasis.cmo\";;\n",
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"#load_rec \"F12CI.cmo\";;"
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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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"### Printers"
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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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"#install_printer AngularMomentum.pp_string ;;\n",
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"#install_printer Basis.pp ;;\n",
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"#install_printer Charge.pp ;;\n",
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"#install_printer Coordinate.pp ;;\n",
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"#install_printer Vector.pp;;\n",
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"#install_printer Matrix.pp;;\n",
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"#install_printer Util.pp_float_2darray;;\n",
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"#install_printer Util.pp_float_array;;\n",
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"#install_printer Util.pp_matrix;;\n",
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"#install_printer HartreeFock.pp ;;\n",
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"#install_printer Fock.pp ;;\n",
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"#install_printer MOClass.pp ;;\n",
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"let pp_mo ppf t = MOBasis.pp ~start:1 ~finish:0 ppf t ;;\n",
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"#install_printer pp_mo;;\n",
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"(*\n",
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"#install_printer DeterminantSpace.pp;;\n",
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"*)\n",
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"#install_printer SpindeterminantSpace.pp;;\n",
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"#install_printer Bitstring.pp;;\n",
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"\n",
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"(* --------- *)\n",
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"\n",
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"open Lacaml.D\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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"source": [
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"## Run"
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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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"### Simulation\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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"let basis_filename = \"/home/scemama/qp2/data/basis/6-31g\" \n",
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"let aux_basis_filename = \"/home/scemama/qp2/data/basis/cc-pvdz\" \n",
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"let nuclei = Nuclei.of_zmt_string \"c\" \n",
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"let frozen_core = false\n",
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"let multiplicity = 1\n",
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"let state = 1\n",
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"\n",
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"let basis = Basis.of_nuclei_and_basis_filenames ~nuclei [basis_filename] \n",
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"let aux_basis = Basis.of_nuclei_and_basis_filenames ~nuclei (basis_filename :: aux_basis_filename :: []) \n",
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"let f12 = F12factor.gaussian_geminal 1.0 \n",
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"let charge = 0 \n",
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"\n",
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"\n",
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"let simulation =\n",
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" Simulation.make \n",
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" ~f12 ~charge ~multiplicity ~nuclei\n",
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" ~cartesian:true\n",
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" basis\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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"let n_elec_alfa, n_elec_beta, n_elec = \n",
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" let e = Simulation.electrons simulation in\n",
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" Electrons.(n_alfa e, n_beta e, n_elec e)"
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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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"### Hartree-Fock"
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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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"let hf = HartreeFock.make ~guess:`Hcore ~max_scf:10 simulation ;;\n",
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"\n",
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"let mo_basis = MOBasis.of_hartree_fock hf"
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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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"# FCI-F12"
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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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"Notations:\n",
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"\n",
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"* $\\langle ij || kl \\rangle = \\int \\phi_i(r_1) \\phi_j(r_2) \\frac{1}{r_{12}} \\phi_k(r1) \\phi_l(r2) $ \n",
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"* $\\left[ ij || kl \\right] = \\int \\phi_i(r_1) \\phi_j(r_2) f_{12} \\phi_k(r1) \\phi_l(r2) $ \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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"source": [
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"## Common functions"
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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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"let f12 = Util.of_some @@ Simulation.f12 simulation \n",
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"\n",
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"let mo_num = MOBasis.size mo_basis \n",
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"\n",
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"let pp_spindet = Spindeterminant.pp mo_num\n",
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"\n",
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"let pp_det = Determinant.pp mo_num\n",
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"\n",
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";;\n",
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"\n",
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"#install_printer pp_spindet ;;\n",
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"#install_printer pp_det ;;\n",
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"\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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"let simulation_aux = \n",
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" let charge = Charge.to_int @@ Simulation.charge simulation \n",
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" and multiplicity = Electrons.multiplicity @@ Simulation.electrons simulation\n",
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" and nuclei = Simulation.nuclei simulation\n",
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" in\n",
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" let general_basis = \n",
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" Basis.general_basis @@ Simulation.basis simulation\n",
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" in\n",
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" GeneralBasis.combine [\n",
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" general_basis ; GeneralBasis.read aux_basis_filename\n",
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" ]\n",
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" |> Basis.of_nuclei_and_general_basis nuclei\n",
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" |> Simulation.make ~f12 ~charge ~multiplicity ~nuclei \n",
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"\n",
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"\n",
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"let aux_basis = \n",
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" MOBasis.of_mo_basis simulation_aux mo_basis\n",
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"\n",
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"let aux_num = \n",
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" MOBasis.size aux_basis\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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"let () = ignore @@ MOBasis.f12_ints aux_basis\n",
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"let () = ignore @@ MOBasis.two_e_ints aux_basis"
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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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"## Integral-based functions"
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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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"\\begin{equation}\n",
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"\\langle I | \\hat{H} | J \\rangle = \\begin{cases}\n",
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"\\sum_i h_{ii} + \\frac{1}{2} \\sum_{ij} \\langle ij || ij \\rangle \\text{ when } |J\\rangle = |I\\rangle \\\\\n",
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"h_{ik} + \\sum_{j} \\langle ij || kj \\rangle \\text{ when } |J\\rangle = \\hat{T}_i^k |I\\rangle \\\\\n",
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"\\langle ij || kl \\rangle \\text{ when } |J\\rangle = \\hat{T}_{ij}^{kl} |I\\rangle \\\\\n",
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"\\end{cases}\n",
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"\\end{equation}\n",
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"\n",
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"\n",
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"\\begin{equation}\n",
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"\\langle I | \\hat{F} | J \\rangle = \\begin{cases}\n",
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"\\sum_i f_{ii} + \\frac{1}{2} \\sum_{ij} \\langle ij || ij \\rangle \\text{ when } |J\\rangle = |I\\rangle \\\\\n",
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"f_{ik} + \\sum_{j} \\langle ij || kj \\rangle \\text{ when } |J\\rangle = \\hat{T}_i^k |I\\rangle \\\\\n",
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"\\langle ij || kl \\rangle \\text{ when } |J\\rangle = \\hat{T}_{ij}^{kl} |I\\rangle \\\\\n",
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"\\end{cases}\n",
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"\\end{equation}"
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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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"let cancel_singles = false \n",
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"\n",
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"let mos_cabs = \n",
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" Util.list_range (mo_num+1) aux_num\n",
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" \n",
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"let mos_abs = \n",
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" Util.list_range 1 aux_num\n",
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" \n",
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"let mos_in = \n",
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" Util.list_range 1 mo_num\n",
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"\n",
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"let mos_a k =\n",
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" Determinant.alfa k\n",
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" |> Spindeterminant.to_list\n",
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" \n",
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"let mos_b k =\n",
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" Determinant.beta k\n",
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" |> Spindeterminant.to_list\n",
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" "
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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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"### H integrals"
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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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"let h_one =\n",
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" let h = \n",
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" MOBasis.one_e_ints aux_basis\n",
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" in fun i j _ -> h.{i,j}\n",
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" \n",
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"let h_two = \n",
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" let two_e_ints = MOBasis.two_e_ints aux_basis in\n",
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" let h2 i j k l (s:Spin.t) (s':Spin.t) =\n",
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" if s' <> s then\n",
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" ERI.get_phys two_e_ints i j k l\n",
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" else\n",
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" (ERI.get_phys two_e_ints i j k l) -. \n",
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" (ERI.get_phys two_e_ints i j l k)\n",
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" in\n",
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" h2\n",
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"\n",
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" \n",
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" "
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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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"### F12 integrals"
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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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"let f_two = \n",
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" let two_e_ints = MOBasis.f12_ints aux_basis in\n",
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" let f2 i j k l (s:Spin.t) (s':Spin.t) =\n",
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" if s' <> s then\n",
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" 0.5 *. F12.get_phys two_e_ints i j k l\n",
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" else\n",
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" 0.5 *. (\n",
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" (F12.get_phys two_e_ints i j k l) -. \n",
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" (F12.get_phys two_e_ints i j l k) )\n",
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" in\n",
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" let f3 i j k l (s:Spin.t) (s':Spin.t) = \n",
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" if (i=k && j<>l) || (j=l && i<>k) then\n",
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" 0.\n",
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" else\n",
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" f2 i j k l s s'\n",
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" in\n",
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" if cancel_singles then f3 else f2\n",
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"\n",
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"let f_one = fun _ _ _ -> 0.\n",
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"\n",
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"(*\n",
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"let f_two = h_two\n",
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"*)\n",
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"\n",
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"let f_one = h_one"
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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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"## Determinant-based functions"
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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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"### Integrals"
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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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"let f12_integrals mo_basis =\n",
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" ( f_one, f_two, None )\n",
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"\n",
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"let h_ij mo_basis ki kj =\n",
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" let integrals =\n",
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" List.map (fun f -> f mo_basis)\n",
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" [ CI.h_integrals ]\n",
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" in\n",
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" CIMatrixElement.make integrals ki kj \n",
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" |> List.hd\n",
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"\n",
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"\n",
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"let f_ij mo_basis ki kj =\n",
|
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" let integrals =\n",
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" List.map (fun f -> f mo_basis)\n",
|
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" [ f12_integrals ]\n",
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" in\n",
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" CIMatrixElement.make integrals ki kj \n",
|
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" |> List.hd \n",
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"\n",
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"\n",
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"let hf_ij mo_basis ki kj =\n",
|
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" let integrals =\n",
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" List.map (fun f -> f mo_basis)\n",
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" [ CI.h_integrals ; f12_integrals ]\n",
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" in\n",
|
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" CIMatrixElement.make integrals ki kj\n",
|
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"\n",
|
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"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Determinant space"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
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|
"metadata": {},
|
|
"outputs": [],
|
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"source": [
|
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"let is_a_double det_space =\n",
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" let mo_class = DeterminantSpace.mo_class det_space in\n",
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" let mo_num = Array.length @@ MOClass.mo_class_array mo_class in\n",
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" let m l =\n",
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" List.fold_left (fun accu i ->\n",
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" let j = i-1 in Bitstring.logor accu (Bitstring.shift_left_one mo_num j)\n",
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" ) (Bitstring.zero mo_num) l\n",
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" in\n",
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" let aux_mask = m (MOClass.auxiliary_mos mo_class) in\n",
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" fun k -> \n",
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" let alfa =\n",
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" Determinant.alfa k\n",
|
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" |> Spindeterminant.bitstring\n",
|
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" in\n",
|
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" let beta =\n",
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" Determinant.beta k\n",
|
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" |> Spindeterminant.bitstring\n",
|
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" in\n",
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" let a = Bitstring.logand aux_mask alfa\n",
|
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" and b = Bitstring.logand aux_mask beta\n",
|
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" in\n",
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" match Bitstring.popcount a + Bitstring.popcount b with\n",
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" | 2 | 1 -> true\n",
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" | 0 | _ -> false\n",
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"\n"
|
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]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
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"source": [
|
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"let in_space = \n",
|
|
" DeterminantSpace.fci_f12_of_mo_basis aux_basis ~frozen_core mo_num\n",
|
|
" \n",
|
|
"let aux_space = \n",
|
|
" DeterminantSpace.fci_of_mo_basis aux_basis ~frozen_core \n",
|
|
"\n",
|
|
"let det_space_in () =\n",
|
|
" DeterminantSpace.determinant_stream in_space\n",
|
|
"\n",
|
|
"let det_space_out () =\n",
|
|
" let s = \n",
|
|
" DeterminantSpace.determinant_stream aux_space\n",
|
|
" in\n",
|
|
" Stream.from (fun _ ->\n",
|
|
" try\n",
|
|
" let is_a_double = is_a_double in_space in\n",
|
|
" let rec result () =\n",
|
|
" let ki = Stream.next s in\n",
|
|
" if is_a_double ki then\n",
|
|
" Some (ki,ki)\n",
|
|
" else\n",
|
|
" result ()\n",
|
|
" in\n",
|
|
" result ()\n",
|
|
" with Stream.Failure -> None\n",
|
|
" )\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let ci = CI.make ~n_states:state in_space\n",
|
|
"\n",
|
|
"let ci_coef, ci_energy = Lazy.force ci.eigensystem \n",
|
|
"\n",
|
|
"let _ = print_newline () \n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Permutation operator $p_{12}$ that generates a new determinant with electrons 1 and 2 swapped."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "raw",
|
|
"metadata": {},
|
|
"source": [
|
|
"let p12 det_space =\n",
|
|
" let mo_class = DeterminantSpace.mo_class det_space in\n",
|
|
" let mo_num = Array.length @@ MOClass.mo_class_array mo_class in\n",
|
|
" let m l =\n",
|
|
" List.fold_left (fun accu i ->\n",
|
|
" let j = i-1 in Bitstring.logor accu (Bitstring.shift_left_one mo_num j)\n",
|
|
" ) (Bitstring.zero mo_num) l\n",
|
|
" in\n",
|
|
" let aux_mask = m (MOClass.auxiliary_mos mo_class) in\n",
|
|
" let not_aux_mask =\n",
|
|
" Bitstring.(shift_left_one mo_num (mo_num-1) |> minus_one |> logxor aux_mask)\n",
|
|
" in\n",
|
|
" fun k ->\n",
|
|
" let alfa =\n",
|
|
" Determinant.alfa k\n",
|
|
" |> Spindeterminant.bitstring\n",
|
|
" in\n",
|
|
" let beta =\n",
|
|
" Determinant.beta k\n",
|
|
" |> Spindeterminant.bitstring\n",
|
|
" in\n",
|
|
" let a = Bitstring.logand aux_mask alfa\n",
|
|
" and b = Bitstring.logand aux_mask beta\n",
|
|
" in\n",
|
|
" match Bitstring.popcount a, Bitstring.popcount b with\n",
|
|
" | 2, 0 \n",
|
|
" | 0, 2 -> Some (Determinant.negate_phase k)\n",
|
|
" | 1, 1 -> Some (Determinant.of_spindeterminants\n",
|
|
" (Spindeterminant.of_bitstring @@\n",
|
|
" Bitstring.(logor b (logand not_aux_mask alfa)) )\n",
|
|
" (Spindeterminant.of_bitstring @@\n",
|
|
" Bitstring.(logor a (logand not_aux_mask beta))\n",
|
|
" ) )\n",
|
|
" | 1, 0 \n",
|
|
" | 0, 1 -> Some k\n",
|
|
" | _ -> None\n",
|
|
" \n",
|
|
" \n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Matrices $\\langle I | H | \\alpha \\rangle$ and $\\langle I | F | \\alpha \\rangle$ "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let randomize = false\n",
|
|
"\n",
|
|
"let out_list =\n",
|
|
" Util.stream_to_list (det_space_out ())\n",
|
|
" \n",
|
|
"let in_list =\n",
|
|
" Util.stream_to_list (det_space_in ())\n",
|
|
" \n",
|
|
"let det_a = Array.of_list out_list\n",
|
|
" |> Array.map (fun (i,_) -> i)\n",
|
|
"\n",
|
|
"let det_I =\n",
|
|
" if randomize then\n",
|
|
" let n = 123456789 in\n",
|
|
" in_list\n",
|
|
" |> List.map (fun k -> (Random.int n, k))\n",
|
|
" |> List.sort compare\n",
|
|
" |> List.map (fun (_,k) -> k)\n",
|
|
" |> Array.of_list\n",
|
|
" else\n",
|
|
" Array.of_list in_list\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "raw",
|
|
"metadata": {
|
|
"raw_mimetype": "text/markdown"
|
|
},
|
|
"source": [
|
|
"let m_H_aux, m_F_aux = \n",
|
|
" \n",
|
|
" let rec col_vecs_list accu_H accu_F = function\n",
|
|
" | [] -> \n",
|
|
" List.rev accu_H, \n",
|
|
" List.rev accu_F \n",
|
|
" | (ki, ki') :: rest when ki = ki' ->\n",
|
|
" begin\n",
|
|
" let h, f = \n",
|
|
" List.map (fun kj ->\n",
|
|
" match hf_ij aux_basis kj ki with\n",
|
|
" | [ a ; b ] -> a, b\n",
|
|
" | _ -> assert false ) in_list\n",
|
|
" |> List.split\n",
|
|
" in\n",
|
|
" let h = Vec.of_list h in\n",
|
|
" let f = Vec.of_list f in\n",
|
|
" scal 0.5 f; (* Specific to He singlet *)\n",
|
|
" col_vecs_list (h::accu_H) (f::accu_F) rest\n",
|
|
" end\n",
|
|
" | (ki, ki') :: rest ->\n",
|
|
" begin\n",
|
|
" assert false;\n",
|
|
" end\n",
|
|
" in\n",
|
|
" let h, f = \n",
|
|
" col_vecs_list [] [] out_list\n",
|
|
" in\n",
|
|
" Mat.of_col_vecs_list h,\n",
|
|
" Mat.of_col_vecs_list f"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Function to generate all intermediate external determinants $|\\alpha \\rangle$ between $|I\\rangle$ and $|J\\rangle$, with a positive phase factor.\n",
|
|
"\n",
|
|
"* `exc` is the degree of excitation between $|I\\rangle$ and $|J\\rangle$\n",
|
|
"* `cabs` is the number of electrons in the CABS\n",
|
|
"* `l` is the degree of excitation between $|I\\rangle$ and $|\\alpha \\rangle$\n",
|
|
"* `r` is the degree of excitation between $|\\alpha \\rangle$ and $|J\\rangle$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let generate_alphas ki kj exc cabs l r =\n",
|
|
" (* Check input excitation degree *)\n",
|
|
" let d = Determinant.degree ki kj in\n",
|
|
" if d <> exc then\n",
|
|
" Printf.sprintf \"Invalid excitation degree. Expected %d and found %d.\" exc d\n",
|
|
" |> failwith;\n",
|
|
" \n",
|
|
" (* Generate single excitations *)\n",
|
|
" let all_singles ki =\n",
|
|
" let mos_a, mos_b = Determinant.to_lists ki in\n",
|
|
" [ List.map (fun hole -> \n",
|
|
" List.map (fun particle ->\n",
|
|
" if hole = particle then None else\n",
|
|
" Some (Determinant.single_excitation Spin.Alfa hole particle ki)\n",
|
|
" ) mos_abs\n",
|
|
" ) mos_a\n",
|
|
" ;\n",
|
|
" List.map (fun hole -> \n",
|
|
" List.map (fun particle ->\n",
|
|
" if hole = particle then None else\n",
|
|
" Some (Determinant.single_excitation Spin.Beta hole particle ki)\n",
|
|
" ) mos_abs\n",
|
|
" ) mos_b \n",
|
|
" ]\n",
|
|
" |> List.concat\n",
|
|
" |> List.concat\n",
|
|
" |> Util.list_some\n",
|
|
" |> List.filter (fun x -> Determinant.is_none x = false)\n",
|
|
" |> List.map (fun x -> Determinant.set_phase Phase.Pos x)\n",
|
|
" in\n",
|
|
" \n",
|
|
" (* Generate double excitations *)\n",
|
|
" let all_doubles ki =\n",
|
|
" let mos_a, mos_b = Determinant.to_lists ki in\n",
|
|
" [ List.map (fun hole -> (* Alpha-Alpha *)\n",
|
|
" List.map (fun particle ->\n",
|
|
" List.map (fun hole' -> \n",
|
|
" List.map (fun particle' ->\n",
|
|
" if hole = particle || hole' = particle' then None else\n",
|
|
" if hole' > hole && particle' < particle then\n",
|
|
" Some (Determinant.double_excitation Spin.Alfa hole particle Spin.Alfa hole' particle' ki)\n",
|
|
" else None\n",
|
|
" ) mos_abs\n",
|
|
" ) mos_a \n",
|
|
" ) mos_abs\n",
|
|
" ) mos_a\n",
|
|
" ; \n",
|
|
" List.map (fun hole -> (* Beta-Beta *)\n",
|
|
" List.map (fun particle ->\n",
|
|
" List.map (fun hole' -> \n",
|
|
" List.map (fun particle' ->\n",
|
|
" if hole' > hole && particle' < particle && hole <> particle && hole' <> particle' then\n",
|
|
" Some (Determinant.double_excitation Spin.Beta hole particle Spin.Beta hole' particle' ki)\n",
|
|
" else None\n",
|
|
" ) mos_abs\n",
|
|
" ) mos_b \n",
|
|
" ) mos_abs\n",
|
|
" ) mos_b\n",
|
|
" ;\n",
|
|
" List.map (fun hole -> (* Alpha-Beta *)\n",
|
|
" List.map (fun particle ->\n",
|
|
" List.map (fun hole' -> \n",
|
|
" List.map (fun particle' ->\n",
|
|
" if hole = particle || hole' = particle' then None else\n",
|
|
" Some (Determinant.double_excitation Spin.Alfa hole particle Spin.Beta hole' particle' ki)\n",
|
|
" ) mos_abs\n",
|
|
" ) mos_b \n",
|
|
" ) mos_abs\n",
|
|
" ) mos_a\n",
|
|
" ]\n",
|
|
" |> List.concat\n",
|
|
" |> List.concat\n",
|
|
" |> List.concat\n",
|
|
" |> List.concat\n",
|
|
" |> Util.list_some\n",
|
|
" |> List.filter (fun x -> Determinant.is_none x = false)\n",
|
|
" |> List.map (fun x -> Determinant.set_phase Phase.Pos x)\n",
|
|
" in\n",
|
|
"\n",
|
|
" (* Generate left and right excitations *)\n",
|
|
" let al = \n",
|
|
" match l with\n",
|
|
" | 1 -> all_singles ki\n",
|
|
" | 2 -> all_doubles ki\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" let ar = \n",
|
|
" match r with\n",
|
|
" | 1 -> all_singles kj\n",
|
|
" | 2 -> all_doubles kj\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mo_class = DeterminantSpace.mo_class in_space in\n",
|
|
" let m l =\n",
|
|
" List.fold_left (fun accu i ->\n",
|
|
" let j = i-1 in Bitstring.logor accu (Bitstring.shift_left_one mo_num j)\n",
|
|
" ) (Bitstring.zero mo_num) l\n",
|
|
" in\n",
|
|
" let aux_mask = m (MOClass.auxiliary_mos mo_class) in\n",
|
|
" let good_cabs k = \n",
|
|
" let alfa =\n",
|
|
" Determinant.alfa k\n",
|
|
" |> Spindeterminant.bitstring\n",
|
|
" in\n",
|
|
" let beta =\n",
|
|
" Determinant.beta k\n",
|
|
" |> Spindeterminant.bitstring\n",
|
|
" in\n",
|
|
" let a = Bitstring.logand aux_mask alfa\n",
|
|
" and b = Bitstring.logand aux_mask beta\n",
|
|
" in\n",
|
|
" Bitstring.popcount a + Bitstring.popcount b = cabs\n",
|
|
" in\n",
|
|
" let good_lr k =\n",
|
|
" Determinant.degree ki k = l &&\n",
|
|
" Determinant.degree k kj = r \n",
|
|
" in\n",
|
|
"\n",
|
|
"\n",
|
|
" \n",
|
|
" (* Merge lists in a set of unique determinants *)\n",
|
|
" List.concat [ al; ar ]\n",
|
|
" |> List.sort_uniq compare\n",
|
|
" \n",
|
|
" (* Filter out all determinants with incorrect numbers of electrons in the CABS *)\n",
|
|
" |> List.filter good_cabs \n",
|
|
" \n",
|
|
" (* Filter out all determinants with incorrect excitation numbers *)\n",
|
|
" |> List.filter good_lr \n",
|
|
" \n",
|
|
" \n",
|
|
"let compute_HaaF ki alphas kj =\n",
|
|
" List.fold_left (fun accu alpha -> accu\n",
|
|
" +. h_ij aux_basis ki alpha\n",
|
|
" *. f_ij aux_basis alpha kj\n",
|
|
" ) 0. alphas\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let check n integral_value exc cabs lexc rexc =\n",
|
|
" let cpudet, cpuint = ref 0., ref 0. in\n",
|
|
" let det_list =\n",
|
|
" match n with\n",
|
|
" | 0 -> det_I\n",
|
|
" | n -> Array.sub det_I 0 n\n",
|
|
" in\n",
|
|
" let result =\n",
|
|
" Printf.printf \"Checking ... \\n%!\";\n",
|
|
" let percent = ref 0 in\n",
|
|
" let task (i,ki) =\n",
|
|
" (let p = (10 * (i+1))/(Array.length det_list) in\n",
|
|
" if p <> !percent then\n",
|
|
" ( percent := p ; Printf.printf \" - %3d %%\\n%!\" (p*10) ));\n",
|
|
" Array.mapi (fun j kj -> \n",
|
|
" if i > j || Determinant.degree ki kj <> exc then\n",
|
|
" (0,0,0.,0.)\n",
|
|
" else\n",
|
|
" begin\n",
|
|
" let alphas = generate_alphas ki kj exc cabs lexc rexc in\n",
|
|
" let det_value =\n",
|
|
" let t0 = Unix.gettimeofday () in\n",
|
|
" let result = compute_HaaF ki alphas kj in\n",
|
|
" cpudet := !cpudet +. Unix.gettimeofday () -. t0;\n",
|
|
" result\n",
|
|
" in\n",
|
|
" let int_value = \n",
|
|
" let t0 = Unix.gettimeofday () in\n",
|
|
" let result = integral_value ki kj in\n",
|
|
" cpuint := !cpuint +. Unix.gettimeofday () -. t0;\n",
|
|
" result\n",
|
|
" in\n",
|
|
"(* Printf.printf \"%d %d %e %e\\n%!\" i j det_value int_value; *)\n",
|
|
" (i,j,det_value,int_value)\n",
|
|
" end\n",
|
|
" ) det_list\n",
|
|
" in\n",
|
|
" det_list\n",
|
|
" |> Array.mapi (fun i ki -> (i,ki))\n",
|
|
" |> Array.to_list\n",
|
|
" |> Stream.of_list\n",
|
|
" |> Farm.run ~f:task\n",
|
|
" |> Util.stream_to_list\n",
|
|
" |> Array.concat\n",
|
|
" in\n",
|
|
" let i,j,d,v = \n",
|
|
" let rec aux k imax jmax emax dmax vmax = \n",
|
|
" if k = -1 then\n",
|
|
" imax, jmax, dmax, vmax\n",
|
|
" else\n",
|
|
" let i, j, d, v = result.(k) in\n",
|
|
" let e = abs_float (d -. v) in\n",
|
|
" if e >= emax then\n",
|
|
" aux (k-1) i j e d v\n",
|
|
" else\n",
|
|
" aux (k-1) imax jmax emax dmax vmax\n",
|
|
" in\n",
|
|
" aux (Array.length result - 1) 0 0 0. 0. 0.\n",
|
|
" in\n",
|
|
" let error = abs_float (d -. v) in\n",
|
|
" if error < epsilon_float then\n",
|
|
" (*\n",
|
|
" Printf.printf \"OK: %e\\n%!\" error\n",
|
|
" *)\n",
|
|
" Printf.printf \"OK: (%d, %d) | %e %e | %e | cpu : %f %f\\n%!\" i j d v error !cpudet !cpuint\n",
|
|
" else\n",
|
|
" Printf.printf \"Failed: (%d, %d) | %e %e | %e | cpu : %f %f\\n%!\" i j d v error !cpudet !cpuint\n",
|
|
"\n",
|
|
" \n",
|
|
"let sum l f = List.fold_left (fun accu i -> accu +. f i) 0. l"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let array_3_init d1 d2 d3 f =\n",
|
|
" let result = \n",
|
|
" Bigarray.(Array3.create Float64 fortran_layout) d1 d2 d3\n",
|
|
" in\n",
|
|
" for k=1 to d3 do\n",
|
|
" for j=1 to d2 do\n",
|
|
" for i=1 to d1 do\n",
|
|
" result.{i,j,k} <- f i j k\n",
|
|
" done\n",
|
|
" done\n",
|
|
" done;\n",
|
|
" result\n",
|
|
"\n",
|
|
"\n",
|
|
"let array_4_init d1 d2 d3 d4 f =\n",
|
|
" let result = \n",
|
|
" Bigarray.(Genarray.create Float64 fortran_layout) [| d1;d2;d3;d4 |]\n",
|
|
" in\n",
|
|
" for l=1 to d4 do\n",
|
|
" for k=1 to d3 do\n",
|
|
" for j=1 to d2 do\n",
|
|
" for i=1 to d1 do\n",
|
|
" result.{i,j,k,l} <- f i j k l\n",
|
|
" done\n",
|
|
" done\n",
|
|
" done\n",
|
|
" done;\n",
|
|
" result\n",
|
|
"\n",
|
|
"\n",
|
|
"let array_5_init d1 d2 d3 d4 d5 f =\n",
|
|
" let result = \n",
|
|
" Bigarray.(Genarray.create Float64 fortran_layout) [| d1;d2;d3;d4;d5 |]\n",
|
|
" in\n",
|
|
" for m=1 to d5 do\n",
|
|
" for l=1 to d4 do\n",
|
|
" for k=1 to d3 do\n",
|
|
" for j=1 to d2 do\n",
|
|
" for i=1 to d1 do\n",
|
|
" result.{i,j,k,l,m} <- f i j k l m\n",
|
|
" done\n",
|
|
" done\n",
|
|
" done\n",
|
|
" done\n",
|
|
" done;\n",
|
|
" result\n",
|
|
"\n",
|
|
"(* ----- *)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"******"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 1. 0 1 11"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"0_1_11.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$i,j$ : orbital indices of MOs occupied in $|I\\rangle$.\n",
|
|
"\n",
|
|
"$\\sum_i \\sum_a \\hat{T}_{i}^{a} \\hat{T}_{a}^{i}$\n",
|
|
"\n",
|
|
"\\begin{align}\n",
|
|
"\\sum_{a} \\sum_{i}\n",
|
|
" \\left( h_{ia} +\n",
|
|
" \\sum_{j} \\langle i j || a j \\rangle +\n",
|
|
" \\sum_{\\bar{j}} \\langle i \\bar{j} | a \\bar{j} \\rangle \\right) \n",
|
|
" \\left( f_{ai} +\n",
|
|
" \\sum_{j} \\left[ a j || i j \\right] + \n",
|
|
" \\sum_{\\bar{j}} \\left[ a \\bar{j} | i \\bar{j} \\right] \\right) + \\\\\n",
|
|
"\\sum_{\\bar{a}} \\sum_{\\bar{i}}\n",
|
|
" \\left( h_{ia} +\n",
|
|
" \\sum_{j} \\langle \\bar{i} j | \\bar{a} j \\rangle +\n",
|
|
" \\sum_{\\bar{j}} \\langle \\bar{i} \\bar{j} || \\bar{a} \\bar{j} \\rangle \\right) \n",
|
|
" \\left( f_{ai} +\n",
|
|
" \\sum_{j} \\left[ \\bar{a} j | \\bar{i} j \\right] + \n",
|
|
" \\sum_{\\bar{j}} \\left[ \\bar{a} \\bar{j} || \\bar{i} \\bar{j} \\right] \\right) \n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" (* Alpha-Beta *)\n",
|
|
" let s, s' = Spin.(Alfa, Beta) in \n",
|
|
" let mos_a, mos_b = mos_a ki, mos_b ki in\n",
|
|
" \n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" sum mos_a (fun i -> \n",
|
|
" (h_one i a s +.\n",
|
|
" sum mos_a (fun j -> h_two i j a j s s ) +.\n",
|
|
" sum mos_b (fun j -> h_two i j a j s s')\n",
|
|
" ) *. \n",
|
|
" (f_one a i s +.\n",
|
|
" sum mos_a (fun j -> f_two a j i j s s ) +.\n",
|
|
" sum mos_b (fun j -> f_two a j i j s s')\n",
|
|
" )\n",
|
|
" )\n",
|
|
" +.\n",
|
|
" sum mos_b (fun i -> \n",
|
|
" (h_one i a s' +.\n",
|
|
" sum mos_b (fun j -> h_two i j a j s' s') +.\n",
|
|
" sum mos_a (fun j -> h_two i j a j s' s )\n",
|
|
" ) *. \n",
|
|
" (f_one a i s' +.\n",
|
|
" sum mos_b (fun j -> f_two a j i j s' s') +.\n",
|
|
" sum mos_a (fun j -> f_two a j i j s' s )\n",
|
|
" )\n",
|
|
" ) \n",
|
|
" ) \n",
|
|
" \n",
|
|
"let _ =\n",
|
|
" check 0 integral_value 0 1 1 1\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_0111_1H_1F = \n",
|
|
" Vec.init mo_num (fun i -> \n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_one a i Spin.Alfa ))\n",
|
|
"\n",
|
|
" \n",
|
|
"let m_0111_1H_2Fa = \n",
|
|
" Mat.init_cols mo_num mo_num (fun i j -> sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_two a j i j Spin.Alfa Spin.Alfa ))\n",
|
|
"\n",
|
|
" \n",
|
|
"let m_0111_1H_2Fb = \n",
|
|
" Mat.init_cols mo_num mo_num (fun i j -> sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_two a j i j Spin.Alfa Spin.Beta ))\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_0111_2Ha_1F = \n",
|
|
" Mat.init_cols mo_num mo_num (fun i j -> sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Alfa *. f_one a i Spin.Alfa ))\n",
|
|
"\n",
|
|
" \n",
|
|
"let m_0111_2Hb_1F = \n",
|
|
" Mat.init_cols mo_num mo_num (fun i j -> sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Beta *. f_one a i Spin.Alfa ))\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_0111_2Ha_2Fa =\n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a k i k Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_0111_2Hb_2Fa =\n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a k i k Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_0111_2Ha_2Fb =\n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a k i k Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_0111_2Hb_2Fb =\n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a k i k Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"\n",
|
|
"let integral_value ki kj = \n",
|
|
"\n",
|
|
" let mos_a, mos_b = mos_a ki, mos_b ki in\n",
|
|
" \n",
|
|
" sum mos_a (fun i ->\n",
|
|
" m_0111_1H_1F.{i} +. \n",
|
|
" sum mos_a (fun j ->\n",
|
|
" m_0111_2Ha_1F.{i,j} +.\n",
|
|
" m_0111_1H_2Fa.{i,j} +.\n",
|
|
" sum mos_a (fun k -> m_0111_2Ha_2Fa.{i,j,k}) +.\n",
|
|
" sum mos_b (fun k -> m_0111_2Ha_2Fb.{i,j,k})\n",
|
|
" ) +.\n",
|
|
" sum mos_b (fun j ->\n",
|
|
" m_0111_2Hb_1F.{i,j} +.\n",
|
|
" m_0111_1H_2Fb.{i,j} +.\n",
|
|
" sum mos_a (fun k -> m_0111_2Hb_2Fa.{i,j,k}) +.\n",
|
|
" sum mos_b (fun k -> m_0111_2Hb_2Fb.{i,j,k})\n",
|
|
" )\n",
|
|
" )\n",
|
|
" +.\n",
|
|
" sum mos_b (fun i ->\n",
|
|
" m_0111_1H_1F.{i} +. \n",
|
|
" sum mos_b (fun j ->\n",
|
|
" m_0111_2Ha_1F.{i,j} +.\n",
|
|
" m_0111_1H_2Fa.{i,j} +.\n",
|
|
" sum mos_b (fun k -> m_0111_2Ha_2Fa.{i,j,k}) +.\n",
|
|
" sum mos_a (fun k -> m_0111_2Ha_2Fb.{i,j,k})\n",
|
|
" ) +.\n",
|
|
" sum mos_a (fun j ->\n",
|
|
" m_0111_2Hb_1F.{i,j} +.\n",
|
|
" m_0111_1H_2Fb.{i,j} +.\n",
|
|
" sum mos_b (fun k -> m_0111_2Hb_2Fa.{i,j,k}) +.\n",
|
|
" sum mos_a (fun k -> m_0111_2Hb_2Fb.{i,j,k})\n",
|
|
" )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let _ =\n",
|
|
" check 0 integral_value 0 1 1 1\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 2. 0 1 22"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"0_1_22.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$i,j$ : orbital indices of MOs occupied in $|I\\rangle$.\n",
|
|
"\n",
|
|
"$k$ : orbital indices of MOs unoccupied in $|I\\rangle$.\n",
|
|
"\n",
|
|
"$\\sum_i \\sum_a \\sum_j \\sum_k \\hat{T}_{ij}^{ak} \\hat{T}_{ak}^{ij}$\n",
|
|
"\n",
|
|
"\n",
|
|
"\\begin{align}\n",
|
|
"\\sum_{a} \\sum_{k} \\sum_{i} \\sum_{j<i}\n",
|
|
" \\langle i j || a k \\rangle \n",
|
|
" \\left[ a k || i j \\right] + \n",
|
|
"\\sum_{a} \\sum_{\\bar{k}} \\sum_{i} \\sum_{\\bar{j}}\n",
|
|
" \\langle i \\bar{j} | a \\bar{k} \\rangle \n",
|
|
" \\left[ a \\bar{k} | i \\bar{j} \\right] + \\\\\n",
|
|
"\\sum_{\\bar{a}} \\sum_{k} \\sum_{\\bar{i}} \\sum_{j}\n",
|
|
" \\langle \\bar{i} j | \\bar{a} k \\rangle \n",
|
|
" \\left[ \\bar{a} k | \\bar{i} j \\right] + \n",
|
|
"\\sum_{\\bar{a}} \\sum_{\\bar{k}} \\sum_{\\bar{i}} \\sum_{\\bar{j}<\\bar{i}}\n",
|
|
" \\langle \\bar{i} \\bar{j} || \\bar{a} \\bar{k} \\rangle \n",
|
|
" \\left[ \\bar{a} \\bar{k} || \\bar{i} \\bar{j} \\right] \n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" (* mos unoccupied in both I and J *)\n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) , \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) \n",
|
|
" in\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a ki);\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a kj);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b ki);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b kj);\n",
|
|
" \n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.to_list mos_virt_a |> Util.list_some,\n",
|
|
" Array.to_list mos_virt_b |> Util.list_some\n",
|
|
" in\n",
|
|
" \n",
|
|
" (* Alpha-Beta *)\n",
|
|
" let s, s' = Spin.(Alfa, Beta) in\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" sum (mos_b ki) (fun j -> \n",
|
|
" sum (mos_a ki) (fun i -> \n",
|
|
" sum mos_virt_b (fun k -> \n",
|
|
" h_two i j a k s s' *. f_two a k i j s s'\n",
|
|
" )))\n",
|
|
" +.\n",
|
|
" sum (mos_a ki) (fun j -> \n",
|
|
" sum (mos_b ki) (fun i -> \n",
|
|
" sum mos_virt_a (fun k -> \n",
|
|
" h_two i j a k s s' *. f_two a k i j s s'\n",
|
|
" )))) \n",
|
|
" +.\n",
|
|
" (* Alpha-Alpha / Beta-Beta *)\n",
|
|
" List.fold_left (fun accu (mos_virt,mos,s) -> \n",
|
|
" let s' = s in accu +.\n",
|
|
" sum mos (fun j -> \n",
|
|
" sum mos (fun i -> if i < j then 0. else\n",
|
|
" sum mos_virt (fun k -> \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" h_two i j a k s s' *. f_two a k i j s s'\n",
|
|
" )))) \n",
|
|
" ) 0. [ (mos_virt_a,mos_a ki,Spin.Alfa) ; (mos_virt_b,mos_b ki,Spin.Beta)]\n",
|
|
" \n",
|
|
"let _ =\n",
|
|
" check 100 integral_value 0 1 2 2\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_0122_Haa = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a k Spin.Alfa Spin.Alfa *. f_two a k i j Spin.Alfa Spin.Alfa\n",
|
|
" ) )\n",
|
|
" \n",
|
|
"let m_0122_Hab = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a k Spin.Alfa Spin.Beta *. f_two a k i j Spin.Alfa Spin.Beta\n",
|
|
" ) )\n",
|
|
"\n",
|
|
"\n",
|
|
"let integral_value ki kj = \n",
|
|
" (* mos unoccupied in both I and J *)\n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) , \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) \n",
|
|
" in\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a ki);\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a kj);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b ki);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b kj);\n",
|
|
" \n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.to_list mos_virt_a |> Util.list_some,\n",
|
|
" Array.to_list mos_virt_b |> Util.list_some\n",
|
|
" in\n",
|
|
" \n",
|
|
" let result = \n",
|
|
" sum (mos_a ki) (fun i -> \n",
|
|
" sum (mos_b ki) (fun j -> \n",
|
|
" sum mos_virt_b (fun k -> m_0122_Hab.{i,j,k} )))\n",
|
|
" +.\n",
|
|
" sum (mos_b ki) (fun i -> \n",
|
|
" sum (mos_a ki) (fun j -> \n",
|
|
" sum mos_virt_a (fun k -> m_0122_Hab.{i,j,k} )))\n",
|
|
" +.\n",
|
|
" (* Alpha-Alpha / Beta-Beta *)\n",
|
|
" List.fold_left (fun accu (mos_virt,mos) -> \n",
|
|
" accu +.\n",
|
|
" sum mos (fun j -> \n",
|
|
" sum mos (fun i -> if i <= j then 0. else\n",
|
|
" sum mos_virt (fun k -> m_0122_Haa.{i,j,k} )))\n",
|
|
" ) 0. [ (mos_virt_a,mos_a ki) ; (mos_virt_b,mos_b ki)]\n",
|
|
" in \n",
|
|
" result\n",
|
|
" \n",
|
|
"let _ =\n",
|
|
" check 100 integral_value 0 1 2 2\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 3. 0 2 22"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"0_2_22.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$i,j$ : orbital indices of MOs occupied in $|I\\rangle$.\n",
|
|
"\n",
|
|
"$\\sum_i \\sum_a \\sum_j \\sum_b \\hat{T}_{ij}^{ab} \\hat{T}_{ab}^{ij}$\n",
|
|
"\n",
|
|
"\\begin{align}\n",
|
|
"\\sum_{a} \\sum_{b>a} \\sum_{i} \\sum_{j<i}\n",
|
|
" \\langle i j || a b \\rangle \n",
|
|
" \\left[ a b || i j \\right] + \n",
|
|
"\\sum_{a} \\sum_{\\bar{b}} \\sum_{i} \\sum_{\\bar{j}}\n",
|
|
" \\langle i \\bar{j} | a \\bar{b} \\rangle \n",
|
|
" \\left[ a \\bar{b} | i \\bar{j} \\right] + \\\\\n",
|
|
"\\sum_{\\bar{a}} \\sum_{b} \\sum_{\\bar{i}} \\sum_{j}\n",
|
|
" \\langle \\bar{i} j | \\bar{a} b \\rangle \n",
|
|
" \\left[ \\bar{a} b | \\bar{i} j \\right] + \n",
|
|
"\\sum_{\\bar{a}} \\sum_{\\bar{b}>\\bar{a}} \\sum_{\\bar{i}} \\sum_{\\bar{j}<\\bar{i}}\n",
|
|
" \\langle \\bar{i} \\bar{j} || \\bar{a} \\bar{b} \\rangle \n",
|
|
" \\left[ \\bar{a} \\bar{b} || \\bar{i} \\bar{j} \\right] \n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" (* Alpha-Beta *)\n",
|
|
" let s, s' = Spin.(Alfa, Beta) in\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos_cabs (fun b -> \n",
|
|
" sum (mos_a ki) (fun i -> \n",
|
|
" sum (mos_b ki) (fun j -> \n",
|
|
" h_two i j a b s s' *. f_two a b i j s s'\n",
|
|
" ))))\n",
|
|
" +.\n",
|
|
" (* Alpha-Alpha / Beta-Beta *)\n",
|
|
" List.fold_left (fun accu (mos,s) -> \n",
|
|
" let s' = s in accu +.\n",
|
|
" sum mos_cabs (fun b -> \n",
|
|
" sum mos_cabs (fun a -> if b > a then 0. else\n",
|
|
" sum mos (fun j -> \n",
|
|
" sum mos (fun i -> if i < j then 0. else\n",
|
|
" h_two i j a b s s' *. f_two a b i j s s'\n",
|
|
" ))))\n",
|
|
" ) 0. [ (mos_a ki,Spin.Alfa) ; (mos_b ki,Spin.Beta)]\n",
|
|
" \n",
|
|
"let _ =\n",
|
|
" check 100 integral_value 0 2 2 2\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_0222_Haa = \n",
|
|
" Mat.init_cols mo_num mo_num (fun i j ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" sum mos_cabs (fun b -> if b >= a then 0. else\n",
|
|
" h_two i j a b Spin.Alfa Spin.Alfa *. f_two a b i j Spin.Alfa Spin.Alfa\n",
|
|
" ) )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let m_0222_Hab = \n",
|
|
" Mat.init_cols mo_num mo_num (fun i j ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" sum mos_cabs (fun b ->\n",
|
|
" h_two i j a b Spin.Alfa Spin.Beta *. f_two a b i j Spin.Alfa Spin.Beta\n",
|
|
" ) )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let integral_value ki kj = \n",
|
|
" (* Alpha-Beta *)\n",
|
|
" sum (mos_a ki) (fun i -> \n",
|
|
" sum (mos_b ki) (fun j -> \n",
|
|
" m_0222_Hab.{i,j} \n",
|
|
" ))\n",
|
|
" +.\n",
|
|
" (* Alpha-Alpha / Beta-Beta *)\n",
|
|
" List.fold_left (fun accu (mos) -> \n",
|
|
" accu +.\n",
|
|
" sum mos (fun j -> \n",
|
|
" sum mos (fun i -> if i < j then 0. else\n",
|
|
" m_0222_Haa.{i,j} \n",
|
|
" ))\n",
|
|
" ) 0. [ (mos_a ki) ; (mos_b ki)]\n",
|
|
" \n",
|
|
"let _ =\n",
|
|
" check 100 integral_value 0 2 2 2\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 4. 1 1 11"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"1_1_11.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{i}^{k}|I\\rangle$\n",
|
|
"\n",
|
|
"$j$ : orbital indices of MOs occupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$\\sum_a \\hat{T}_{i}^{a} \\hat{T}_{a}^{k}$\n",
|
|
"\n",
|
|
"\\begin{align}\n",
|
|
" \\sum_{a} \n",
|
|
" \\left( h_{ia} +\\sum_{j} \\langle i j || a j \\rangle +\n",
|
|
" \\sum_{\\bar{j}} \\langle i \\bar{j} | a \\bar{j} \\rangle \\right)\n",
|
|
" \\left( f_{ak} +\\sum_{j} \\left[ a j || k j \\right] + \n",
|
|
" \\sum_{\\bar{j}} \\left[ a \\bar{j} | k \\bar{j} \\right]\\right) \\\\\n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos_i, mos_i' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
" let mos_j, mos_j' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a kj, mos_b kj\n",
|
|
" | Spin.Beta -> mos_b kj, mos_a kj\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" let s' = Spin.other s in\n",
|
|
" \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" (h_one i a s +. \n",
|
|
" sum mos_i (fun j -> h_two i j a j s s ) +.\n",
|
|
" sum mos_i' (fun j -> h_two i j a j s s') ) *.\n",
|
|
" (f_one a k s +. \n",
|
|
" sum mos_j (fun j -> f_two a j k j s s ) +.\n",
|
|
" sum mos_j' (fun j -> f_two a j k j s s') ) \n",
|
|
" )\n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 0 integral_value 1 1 1 1"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let ki = det_I.(77)\n",
|
|
"let kj = det_I.(83)\n",
|
|
"\n",
|
|
"let _ = integral_value ki kj"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_1111_1H_1F = \n",
|
|
" Mat.init_cols mo_num mo_num (fun i k -> \n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_one a k Spin.Alfa ))\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1111_1H_2Fa = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_two a j k j Spin.Alfa Spin.Alfa ))\n",
|
|
"\n",
|
|
" \n",
|
|
"let m_1111_1H_2Fb = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_two a j k j Spin.Alfa Spin.Beta ))\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1111_2Ha_1F = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Alfa *. f_one a k Spin.Alfa ))\n",
|
|
"\n",
|
|
" \n",
|
|
"let m_1111_2Hb_1F = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Beta *. f_one a k Spin.Alfa ))\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1111_2Ha_2Fa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a l k l Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1111_2Hb_2Fa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a l k l Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1111_2Ha_2Fb =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a l k l Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1111_2Hb_2Fb =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a j Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a l k l Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let integral_value_11 ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_a, mos_b =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in \n",
|
|
" \n",
|
|
" let result = \n",
|
|
" m_1111_1H_1F.{i,k} +. \n",
|
|
" sum mos_a (fun j -> if j = i then 0. else\n",
|
|
" m_1111_1H_2Fa.{i,j,k} +. m_1111_2Ha_1F.{i,j,k} +.\n",
|
|
" sum mos_a (fun l -> if l = i then 0. else m_1111_2Ha_2Fa.{i,j,k,l}) +.\n",
|
|
" sum mos_b (fun l -> m_1111_2Ha_2Fb.{i,j,k,l})) +.\n",
|
|
" sum mos_b (fun j ->\n",
|
|
" m_1111_1H_2Fb.{i,j,k} +. m_1111_2Hb_1F.{i,j,k} +.\n",
|
|
" sum mos_a (fun l -> if l = i then 0. else m_1111_2Hb_2Fa.{i,j,k,l}) +.\n",
|
|
" sum mos_b (fun l -> m_1111_2Hb_2Fb.{i,j,k,l}))\n",
|
|
" in\n",
|
|
" \n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"let _ =\n",
|
|
" check 0 integral_value_11 1 1 1 1\n",
|
|
" "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 5. 1 1 12"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"1_1_12.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{i}^{k}|I\\rangle$\n",
|
|
"\n",
|
|
"$\\sum_{j} \\sum_a \\hat{T}_{j}^{a} \\hat{T}_{ai}^{jk}$\n",
|
|
"\n",
|
|
"$j$ : orbital indices of MOs occupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"\\begin{align}\n",
|
|
" \\sum_{a} \\sum_{j} \n",
|
|
" \\left( h_{ja} +\\sum_{m} \\langle j m || a m \\rangle +\n",
|
|
" \\sum_{\\bar{m}} \\langle j \\bar{m} | a \\bar{m} \\rangle \\right)\n",
|
|
" \\left[ a i || j k \\right] + \\\\\n",
|
|
" \\sum_{\\bar{a}} \\sum_{\\bar{j}} \n",
|
|
" \\left( h_{\\bar{j}\\bar{a}} +\\sum_{m} \\langle \\bar{j} m | \\bar{a} m \\rangle +\n",
|
|
" \\sum_{\\bar{m}} \\langle \\bar{j} \\bar{m} || \\bar{a} \\bar{m} \\rangle \\right)\n",
|
|
" \\left[ \\bar{a} i | \\bar{j} k \\right] \n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos_j, mos_j' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_i, mos_i' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" let s' = Spin.other s in\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos_j (fun j -> \n",
|
|
" (h_one j a s +. \n",
|
|
" sum mos_i (fun m -> h_two j m a m s s ) +.\n",
|
|
" sum mos_i' (fun m -> h_two j m a m s s') ) *.\n",
|
|
" (f_two a i j k s s )\n",
|
|
" ) \n",
|
|
" +.\n",
|
|
" sum mos_j' (fun j -> \n",
|
|
" (h_one j a s +. \n",
|
|
" sum mos_i (fun m -> h_two j m a m s' s ) +.\n",
|
|
" sum mos_i' (fun m -> h_two j m a m s' s') ) *.\n",
|
|
" (f_two a i j k s' s )\n",
|
|
" ) \n",
|
|
" ) \n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 1 1 1 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_1112_1H_2Fa = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one j a Spin.Alfa *. f_two a i j k Spin.Alfa Spin.Alfa ))\n",
|
|
"\n",
|
|
" \n",
|
|
"let m_1112_1H_2Fb = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one j a Spin.Beta *. f_two a i j k Spin.Alfa Spin.Beta ))\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1112_2Ha_2Fa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two j m a m Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a i j k Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1112_2Hb_2Fa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two j m a m Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a i j k Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_1112_2Ha_2Fb =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two j m a m Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a i j k Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1112_2Hb_2Fb =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two j m a m Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a i j k Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let integral_value_12 ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_j, mos_j' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_i, mos_i' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
" \n",
|
|
" let result = \n",
|
|
" sum mos_j (fun j -> \n",
|
|
" m_1112_1H_2Fa.{i,j,k} +. \n",
|
|
" sum mos_i (fun m -> m_1112_2Ha_2Fa.{i,j,k,m}) +.\n",
|
|
" sum mos_i' (fun m -> m_1112_2Hb_2Fa.{i,j,k,m})) +.\n",
|
|
" sum mos_j' (fun j ->\n",
|
|
" m_1112_1H_2Fb.{i,j,k} +. \n",
|
|
" sum mos_i (fun m -> m_1112_2Hb_2Fb.{i,j,k,m}) +.\n",
|
|
" sum mos_i' (fun m -> m_1112_2Ha_2Fb.{i,j,k,m}))\n",
|
|
" in\n",
|
|
" \n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value_12 1 1 1 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 6. 1 1 21"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"1_1_21.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{i}^{k}|I\\rangle$\n",
|
|
"\n",
|
|
"$\\sum_{j} \\sum_a \\hat{T}_{ij}^{ka} \\hat{T}_{a}^{j}$\n",
|
|
"\n",
|
|
"$j$ : orbital indices of MOs occupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"\\begin{align}\n",
|
|
" \\sum_{a} \\sum_{j} \n",
|
|
" \\langle i j || k a \\rangle \n",
|
|
" \\left( f_{aj} +\\sum_{m} \\left[ m a || m j \\right] +\n",
|
|
" \\sum_{\\bar{m}} \\left[ \\bar{m} a | \\bar{m} j \\right] \\right) + \\\\\n",
|
|
" \\sum_{\\bar{a}} \\sum_{\\bar{j}} \n",
|
|
" \\langle i \\bar{j} | k \\bar{a} \\rangle\n",
|
|
" \\left( f_{\\bar{a}\\bar{j}} +\\sum_{m} \\left[ m \\bar{a} | m \\bar{j} \\right] +\n",
|
|
" \\sum_{\\bar{m}} \\left[ \\bar{m} \\bar{a} || \\bar{m} \\bar{j} \\right] \\right)\n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos_j, mos_j' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_i, mos_i' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a kj, mos_b kj\n",
|
|
" | Spin.Beta -> mos_b kj, mos_a kj\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" let s' = Spin.other s in\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos_j (fun j -> \n",
|
|
" (h_two i j k a s s ) *.\n",
|
|
" (f_one a j s +. \n",
|
|
" sum mos_i (fun m -> f_two m a m j s s) +.\n",
|
|
" sum mos_i' (fun m -> f_two m a m j s' s) ) \n",
|
|
" ) \n",
|
|
" +.\n",
|
|
" sum mos_j' (fun j -> \n",
|
|
" (h_two i j k a s s') *.\n",
|
|
" (f_one a j s' +. \n",
|
|
" sum mos_i (fun m -> f_two m a m j s s') +.\n",
|
|
" sum mos_i' (fun m -> f_two m a m j s' s') ) \n",
|
|
" )) \n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 1 1 2 1"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_1121_2Ha_1F = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j k a Spin.Alfa Spin.Alfa *. f_one a j Spin.Alfa))\n",
|
|
"\n",
|
|
" \n",
|
|
"let m_1121_2Hb_1F = \n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j k a Spin.Alfa Spin.Beta *. f_one a j Spin.Beta))\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1121_2Ha_2Fa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j k a Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two m a m j Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1121_2Hb_2Fa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j k a Spin.Alfa Spin.Beta *.\n",
|
|
" f_two m a m j Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_1121_2Ha_2Fb =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j k a Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two m a m j Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let m_1121_2Hb_2Fb =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j k a Spin.Alfa Spin.Beta *.\n",
|
|
" f_two m a m j Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let integral_value_21 ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_j, mos_j' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_i, mos_i' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a kj, mos_b kj\n",
|
|
" | Spin.Beta -> mos_b kj, mos_a kj\n",
|
|
" in\n",
|
|
" \n",
|
|
" let result = \n",
|
|
" sum mos_j (fun j -> \n",
|
|
" m_1121_2Ha_1F.{i,j,k} +. \n",
|
|
" sum mos_i (fun m -> m_1121_2Ha_2Fa.{i,j,k,m}) +.\n",
|
|
" sum mos_i' (fun m -> m_1121_2Ha_2Fb.{i,j,k,m})) +.\n",
|
|
" sum mos_j' (fun j ->\n",
|
|
" m_1121_2Hb_1F.{i,j,k} +. \n",
|
|
" sum mos_i (fun m -> m_1121_2Hb_2Fb.{i,j,k,m}) +.\n",
|
|
" sum mos_i' (fun m -> m_1121_2Hb_2Fa.{i,j,k,m}))\n",
|
|
" in\n",
|
|
" \n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value_21 1 1 2 1"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 7. 1 1 22"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"1_1_22.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{i}^{k}|I\\rangle$\n",
|
|
"\n",
|
|
"$\\sum_{j} \\sum_{a} \\sum_{m}\n",
|
|
" \\hat{T}_{ij}^{ma} \\hat{T}_{ma}^{kj} - \n",
|
|
" \\hat{T}_{lj}^{ka} \\hat{T}_{ia}^{lj} + \\\\\n",
|
|
"\\sum_{\\bar{j}} \\sum_{\\bar{a}} \\sum_{m}\n",
|
|
" \\hat{T}_{i\\bar{j}}^{m\\bar{a}} \\hat{T}_{m\\bar{a}}^{k\\bar{j}} - \n",
|
|
" \\hat{T}_{l\\bar{j}}^{k\\bar{a}} \\hat{T}_{i\\bar{a}}^{l\\bar{j}} +\n",
|
|
"\\sum_{\\bar{j}} \\sum_{a} \\sum_{\\bar{m}}\n",
|
|
" \\hat{T}_{i\\bar{j}}^{a\\bar{m}} \\hat{T}_{a\\bar{m}}^{k\\bar{j}} \n",
|
|
"$\n",
|
|
"\n",
|
|
"$j,l$ : orbital indices of MOs occupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$m$ : orbital indices of virtual MOs, unoccupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$a$ : orbital indices of CABS MOs\n",
|
|
"\n",
|
|
"\\begin{align}\n",
|
|
" \\sum_{a} \\sum_{j} \\sum_{m}\n",
|
|
" \\langle i j || m a \\rangle \\left[ m a || k j \\right] - \\\\\n",
|
|
" \\sum_{a} \\sum_{j} \\sum_{l}\n",
|
|
" \\langle l j || k a \\rangle \\left[ i a || l j \\right] + \\\\\n",
|
|
" \\sum_{\\bar{a}} \\sum_{\\bar{j}} \\sum_{m}\n",
|
|
" \\langle i \\bar{j} | m \\bar{a} \\rangle \\left[ m \\bar{a} | k \\bar{j} \\right] + \\\\\n",
|
|
" \\sum_{\\bar{a}} \\sum_{\\bar{j}} \\sum_{l}\n",
|
|
" \\langle l \\bar{j} | k \\bar{a} \\rangle \\left[ i \\bar{a} | l \\bar{j} \\right] - \\\\\n",
|
|
" \\sum_{\\bar{a}} \\sum_{\\bar{j}} \\sum_{\\bar{m}}\n",
|
|
" \\langle i \\bar{j} | a \\bar{m} \\rangle \\left[ a \\bar{m} | k \\bar{j} \\right] \n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"(*\n",
|
|
"let ki = det_I.(6)\n",
|
|
"let kj = det_I.(58)\n",
|
|
"\n",
|
|
"let _ = \n",
|
|
" Format.printf \"|I> -> |J> : %a |\\n@.\" Excitation.pp (Excitation.of_det ki kj) ;\n",
|
|
"generate_alphas ki kj 1 1 2 2\n",
|
|
"|> Array.of_list\n",
|
|
"|> Array.mapi (fun kk alpha -> \n",
|
|
"\n",
|
|
"(*\n",
|
|
"let _ = Determinant.to_lists ki\n",
|
|
"let _ = Determinant.to_lists alpha\n",
|
|
"let _ = Determinant.to_lists kj\n",
|
|
"\n",
|
|
" \n",
|
|
"let _ = \n",
|
|
" Format.printf \"|I> -> |J> : %a\\n@.\" Excitation.pp (Excitation.of_det ki kj);\n",
|
|
" Format.printf \"|I> -> |a> : %a\\n@.\" Excitation.pp (Excitation.of_det ki alpha);\n",
|
|
" Format.printf \"|a> -> |J> : %a\\n@.\" Excitation.pp (Excitation.of_det alpha kj)\n",
|
|
"*)\n",
|
|
"\n",
|
|
" let integral_value ki kj = \n",
|
|
" \n",
|
|
" let exc0 = Array.init (aux_num+1) (fun _ -> [|\"-\";\"-\"|]) in\n",
|
|
" let exc1 = Array.init (aux_num+1) (fun _ -> [|\"-\";\"-\"|]) in \n",
|
|
" let exc2 = Array.init (aux_num+1) (fun _ -> [|\"-\";\"-\"|]) in \n",
|
|
" \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin } )) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" let spin = function\n",
|
|
" | Spin.Alfa -> 0\n",
|
|
" | _ -> 1\n",
|
|
" in\n",
|
|
" exc0.(i).(spin s ) <- \"i\" \n",
|
|
" ; exc0.(k).(spin s ) <- \"k\" \n",
|
|
" ;\n",
|
|
" let s0 = s in\n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p1 =\n",
|
|
" match Excitation.of_det ki alpha with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" if exc0.(i).(spin s ) = \"-\" then exc0.(i).(spin s ) <- \"n\";\n",
|
|
" if exc0.(j).(spin s') = \"-\" then exc0.(j).(spin s') <- \"n\";\n",
|
|
" if exc0.(k).(spin s ) = \"-\" then exc0.(k).(spin s ) <- if k > mo_num then \"a\" else \"m\";\n",
|
|
" if exc0.(l).(spin s') = \"-\" then exc0.(l).(spin s') <- if l > mo_num then \"a\" else \"m\";\n",
|
|
" \n",
|
|
" let string_h = \n",
|
|
" Printf.sprintf \"h_two %s %s %s %s %s %s *. \" \n",
|
|
" exc0.(i).(spin s )\n",
|
|
" exc0.(j).(spin s')\n",
|
|
" exc0.(k).(spin s )\n",
|
|
" exc0.(l).(spin s')\n",
|
|
" (if s = s0 then \"s \" else \"s'\")\n",
|
|
" (if s' = s0 then \"s \" else \"s'\")\n",
|
|
" (*\n",
|
|
" (if exc0.(i).(spin s ) = \"i\" || exc0.(k).(spin s ) = \"k\" then \"s \" else \n",
|
|
" if exc0.(i).(spin s ) = \"j\" || exc0.(k).(spin s ) = \"l\" then \"s' \" else \"s''\")\n",
|
|
" (if exc0.(j).(spin s') = \"i\" || exc0.(l).(spin s') = \"k\" then \"s \" else \n",
|
|
" if exc0.(j).(spin s') = \"j\" || exc0.(l).(spin s') = \"l\" then \"s' \" else \"s''\")\n",
|
|
" *)\n",
|
|
" in \n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p2 =\n",
|
|
" match Excitation.of_det alpha kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" let string_f = \n",
|
|
" Printf.sprintf \"f_two %s %s %s %s %s %s\" \n",
|
|
" exc0.(i).(spin s )\n",
|
|
" exc0.(j).(spin s')\n",
|
|
" exc0.(k).(spin s )\n",
|
|
" exc0.(l).(spin s')\n",
|
|
" (*\n",
|
|
" (if exc0.(i).(spin s ) = \"i\" || exc0.(k).(spin s ) = \"k\" then \"s \" else \n",
|
|
" if exc0.(i).(spin s ) = \"j\" || exc0.(k).(spin s ) = \"l\" then \"s' \" else \n",
|
|
" if exc0.(i).(spin s ) = \"n\" || exc0.(k).(spin s ) = \"n\" then \"s''\" else\n",
|
|
" if s = s0 then \"s\" else \"s'\")\n",
|
|
" (if exc0.(j).(spin s') = \"i\" || exc0.(l).(spin s') = \"k\" then \"s \" else \n",
|
|
" if exc0.(j).(spin s') = \"j\" || exc0.(l).(spin s') = \"l\" then \"s' \" else \n",
|
|
" if exc0.(j).(spin s') = \"n\" || exc0.(l).(spin s') = \"n\" then \"s''\" else\n",
|
|
" if s' = s0 then \"s\" else \"s'\")\n",
|
|
" *)\n",
|
|
" (if s = s0 then \"s \" else \"s'\")\n",
|
|
" (if s' = s0 then \"s \" else \"s'\")\n",
|
|
" in\n",
|
|
" \n",
|
|
" Format.printf \"|I> -> |a> : %a | %s\\n@.\" Excitation.pp (Excitation.of_det ki alpha) string_h ;\n",
|
|
" Format.printf \"|a> -> |J> : %a | %s\\n@.\" Excitation.pp (Excitation.of_det alpha kj) string_f ;\n",
|
|
" \n",
|
|
" let sign = \n",
|
|
" if Phase.add p1 p2 = phase then \"+. \" else \"-. \"\n",
|
|
" in\n",
|
|
"sign ^ string_h ^ string_f\n",
|
|
"(*\n",
|
|
" \n",
|
|
" let mos, mos' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" let k=4 and l=5 in\n",
|
|
" let i=1 and j=2 in\n",
|
|
" let n=2 and a=7 in\n",
|
|
" let s = Spin.Alfa\n",
|
|
" and s' = Spin.Alfa\n",
|
|
" and s'' = Spin.Beta\n",
|
|
" in\n",
|
|
" (* \n",
|
|
" h_two j n l a s' s'' *. f_two a i n k s s''\n",
|
|
" *)\n",
|
|
" h_two i n k a s s'' *. f_two j a l n s s''\n",
|
|
" in\n",
|
|
"\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" *)\n",
|
|
"\n",
|
|
"in\n",
|
|
"integral_value ki kj\n",
|
|
"(*\n",
|
|
"let a = (compute_HaaF ki [alpha] kj)\n",
|
|
"and b = (integral_value ki kj)\n",
|
|
"in\n",
|
|
"if kk = 31 then\n",
|
|
" Format.printf \"%6d %e %e@.@.\" (kk) a b\n",
|
|
" *)\n",
|
|
")\n",
|
|
"|> Array.to_list\n",
|
|
"|> List.sort_uniq compare\n",
|
|
"\n",
|
|
"let _ = compute_HaaF ki (generate_alphas ki kj 1 1 2 2) kj\n",
|
|
"\n",
|
|
"let _ = integral_value ki kj\n",
|
|
"\n",
|
|
"*)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" (* MOs unoccupied in both I and J *)\n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) , \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) \n",
|
|
" in\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a ki);\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a kj);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b ki);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b kj);\n",
|
|
" \n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.to_list mos_virt_a |> Util.list_some,\n",
|
|
" Array.to_list mos_virt_b |> Util.list_some\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos_virt, mos_virt' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_virt_a, mos_virt_b\n",
|
|
" | Spin.Beta -> mos_virt_b, mos_virt_a\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_j, mos_j' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
" let result = \n",
|
|
" let s' = Spin.other s in\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos_j (fun n -> \n",
|
|
" sum mos_virt (fun m -> \n",
|
|
" h_two n i a m s s *. f_two m a k n s s \n",
|
|
" ))) +.\n",
|
|
" sum mos_cabs (fun a' -> \n",
|
|
" sum mos_j' (fun n' -> \n",
|
|
" sum mos_virt (fun m -> \n",
|
|
" h_two i n' m a' s s' *. f_two m a' k n' s s'\n",
|
|
" ))) +.\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos_j' (fun n' -> \n",
|
|
" sum mos_virt' (fun m' -> \n",
|
|
" h_two i n' a m' s s' *. f_two a m' k n' s s'\n",
|
|
" ))) -. \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos_j (fun n -> \n",
|
|
" sum mos_j (fun l -> if l <=n then 0. else\n",
|
|
" h_two n l a k s s *. f_two i a l n s s \n",
|
|
" ))) -. \n",
|
|
" sum mos_cabs (fun a' -> \n",
|
|
" sum mos_j' (fun n' -> \n",
|
|
" sum mos_j (fun l -> \n",
|
|
" h_two l n' k a' s s' *. f_two i a' l n' s s'\n",
|
|
" ))) \n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"let _ = check 100 integral_value 1 1 2 2\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_1122_va =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two j i a m Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two m a k j Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_1122_v2 =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j m a Spin.Alfa Spin.Beta *.\n",
|
|
" f_two m a k j Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_1122_v3 =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k m ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a m Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a m k j Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_1122_oa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two j l a k Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two i a l j Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_1122_o =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two l j k a Spin.Alfa Spin.Beta *.\n",
|
|
" f_two i a l j Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"\n",
|
|
"let integral_value_22 ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" (* MOs unoccupied in both I and J *)\n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) , \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) \n",
|
|
" in\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a ki);\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a kj);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b ki);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b kj);\n",
|
|
" \n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.to_list mos_virt_a |> Util.list_some,\n",
|
|
" Array.to_list mos_virt_b |> Util.list_some\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos_virt, mos_virt' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_virt_a, mos_virt_b\n",
|
|
" | Spin.Beta -> mos_virt_b, mos_virt_a\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos_j, mos_j' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
" \n",
|
|
" let result = \n",
|
|
" sum mos_j (fun j -> \n",
|
|
" sum mos_virt (fun m -> \n",
|
|
" m_1122_va.{i,j,k,m}\n",
|
|
" )) +.\n",
|
|
" sum mos_j' (fun j -> \n",
|
|
" sum mos_virt (fun m -> \n",
|
|
" m_1122_v2.{i,j,k,m}\n",
|
|
" )) +.\n",
|
|
" \n",
|
|
" sum mos_j' (fun j -> \n",
|
|
" sum mos_virt' (fun m -> \n",
|
|
" m_1122_v3.{i,j,k,m}\n",
|
|
" )) -.\n",
|
|
" sum mos_j (fun j -> \n",
|
|
" sum mos_j (fun l -> if l <= j then 0. else\n",
|
|
" m_1122_oa.{i,j,k,l}\n",
|
|
" )) -. \n",
|
|
" sum mos_j' (fun j -> \n",
|
|
" sum mos_j (fun l -> \n",
|
|
" m_1122_o.{i,j,k,l}\n",
|
|
" )) \n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value_22 1 1 2 2\n",
|
|
"\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let ki = det_I.(77) and kj = det_I.(83)\n",
|
|
"\n",
|
|
"let _ =\n",
|
|
"integral_value_11 ki kj\n",
|
|
"+. integral_value_12 ki kj\n",
|
|
"+. integral_value_21 ki kj\n",
|
|
"+. integral_value_22 ki kj "
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 8. 1 2 22"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"1_2_22.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{i}^{k}|I\\rangle$\n",
|
|
"\n",
|
|
"$j$ : orbital indices of MOs occupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$\\sum_{j} \\sum_{a} \\sum_{b}\n",
|
|
" \\hat{T}_{ij}^{ab} \\hat{T}_{ab}^{kj} +\n",
|
|
"\\sum_{\\bar{j}} \\sum_{a} \\sum_{\\bar{b}}\n",
|
|
" \\hat{T}_{i\\bar{j}}^{a\\bar{b}} \\hat{T}_{a\\bar{b}}^{k\\bar{j}} \n",
|
|
"$\n",
|
|
"\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\sum_{j} \\sum_{b}\\sum_{a<b}\n",
|
|
" \\langle i j || a b \\rangle \\left[ ab || kj \\right] + \n",
|
|
"\\sum_{\\bar{j}} \\sum_{\\bar{b}}\\sum_{a}\n",
|
|
" \\langle i \\bar{j} | a \\bar{b} \\rangle \\left[ a \\bar{b} | k \\bar{j} \\right] \n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let h, p, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos, mos' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
" \n",
|
|
" \n",
|
|
"\n",
|
|
" let result = \n",
|
|
" (* Alpha-Beta *)\n",
|
|
" let s' = Spin.other s in\n",
|
|
" sum mos_cabs (fun b -> \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos' (fun j -> h_two h j a b s s' *. f_two a b p j s s')\n",
|
|
" ))\n",
|
|
" +.\n",
|
|
" (* Alpha-Alpha / Beta-Beta *)\n",
|
|
" sum mos_cabs (fun b -> \n",
|
|
" sum mos_cabs (fun a -> if b >= a then 0. else\n",
|
|
" sum mos (fun j -> h_two h j a b s s *. f_two a b p j s s)\n",
|
|
" )) \n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 1 2 2 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_1222a =\n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" sum mos_cabs (fun b -> if b > a then 0. else\n",
|
|
" h_two i j a b Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a b k j Spin.Alfa Spin.Alfa \n",
|
|
" )\n",
|
|
" )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_1222 =\n",
|
|
" array_3_init mo_num mo_num mo_num (fun i j k ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" sum mos_cabs (fun b ->\n",
|
|
" h_two i j a b Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a b k j Spin.Alfa Spin.Beta \n",
|
|
" )\n",
|
|
" )\n",
|
|
" )\n",
|
|
" \n",
|
|
"\n",
|
|
"let integral_value ki kj = \n",
|
|
" let i, k, s, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Single (phase, { hole ; particle ; spin })) ->\n",
|
|
" hole, particle, spin, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" (* MOs unoccupied in both I and J *)\n",
|
|
" let mos_j, mos_j' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" sum mos_j (fun j -> m_1222a.{i,j,k} ) +.\n",
|
|
" sum mos_j' (fun j -> m_1222.{i,j,k} )\n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 1 2 2 2\n",
|
|
"\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 9. 2 1 12"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"2_1_12.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{ij}^{kl}|I\\rangle$\n",
|
|
"\n",
|
|
"$m$ : orbital indices of MOs unoccupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$n$ : orbital indices of MOs occupied in $|I\\rangle$\n",
|
|
"\n",
|
|
"$\\sum_{a} \n",
|
|
" \\hat{T}_{i}^{a} \\hat{T}_{aj}^{kl} +\n",
|
|
" \\hat{T}_{j}^{a} \\hat{T}_{ia}^{kl} \n",
|
|
"$\n",
|
|
"\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\sum_{a}\n",
|
|
" \\left( h_{ia} + \\sum_{n} \\langle i n || a n \\rangle + \\sum_{\\bar{n}} \\langle i \\bar{n} | a \\bar{n} \\rangle \\right) \\left[ a j || k l \\right] + \\\\\n",
|
|
"\\sum_{a}\n",
|
|
" \\left( h_{ja} + \\sum_{n} \\langle j n || a n \\rangle + \\sum_{\\bar{n}} \\langle j \\bar{n} | a \\bar{n} \\rangle \\right) \\left[ i a || k l \\right] \n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, j, k, l, s, s', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos, mos' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos2, mos2' =\n",
|
|
" match s' with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" let s'' = Spin.other s in\n",
|
|
" ( h_one i a s +.\n",
|
|
" sum mos (fun n -> h_two i n a n s s ) +.\n",
|
|
" sum mos' (fun n -> h_two i n a n s s'') \n",
|
|
" ) *. f_two a j k l s s'\n",
|
|
" ) +.\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" let s'' = Spin.other s' in\n",
|
|
" ( h_one j a s' +.\n",
|
|
" sum mos2 (fun n -> h_two j n a n s' s' ) +.\n",
|
|
" sum mos2' (fun n -> h_two j n a n s' s'') \n",
|
|
" ) *. f_two i a k l s s'\n",
|
|
" ) \n",
|
|
" in\n",
|
|
"\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
"\n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 2 1 1 2\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{ij}^{kl}|I\\rangle$\n",
|
|
"\n",
|
|
"$m$ : orbital indices of MOs unoccupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$n$ : orbital indices of MOs occupied in $|I\\rangle$\n",
|
|
"\n",
|
|
"$\\sum_{a} \n",
|
|
" \\hat{T}_{i}^{a} \\hat{T}_{aj}^{kl} +\n",
|
|
" \\hat{T}_{j}^{a} \\hat{T}_{ia}^{kl} \n",
|
|
"$\n",
|
|
"\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\sum_{a}\n",
|
|
" \\left( h_{ia} + \\sum_{n} \\langle i n || a n \\rangle + \\sum_{\\bar{n}} \\langle i \\bar{n} | a \\bar{n} \\rangle \\right) \\left[ a j || k l \\right] + \\\\\n",
|
|
"\\sum_{a}\n",
|
|
" \\left( h_{ja} + \\sum_{n} \\langle j n || a n \\rangle + \\sum_{\\bar{n}} \\langle j \\bar{n} | a \\bar{n} \\rangle \\right) \\left[ i a || k l \\right] \n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_2112_1H_2Fa =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_two a j k l Spin.Alfa Spin.Alfa +.\n",
|
|
" h_one j a Spin.Alfa *. f_two i a k l Spin.Alfa Spin.Alfa )\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_2112_1H_2Fb =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_one i a Spin.Alfa *. f_two a j k l Spin.Alfa Spin.Beta +.\n",
|
|
" h_one j a Spin.Alfa *. f_two i a k l Spin.Alfa Spin.Beta)\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_2112_2Ha_2Fa =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i n a n Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a j k l Spin.Alfa Spin.Alfa )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let m_2112_2Hb_2Fa =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i n a n Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a j k l Spin.Alfa Spin.Alfa )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let m_2112_2Ha_2Fb =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i n a n Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a j k l Spin.Alfa Spin.Beta )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let m_2112_2Hb_2Fb =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i n a n Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a j k l Spin.Alfa Spin.Beta )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let integral_value ki kj = \n",
|
|
" let i, j, k, l, s, s', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" let mos, mos' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a ki, mos_b ki\n",
|
|
" | Spin.Beta -> mos_b ki, mos_a ki\n",
|
|
" in\n",
|
|
" \n",
|
|
" let result = \n",
|
|
" if s = s' then\n",
|
|
" m_2112_1H_2Fa.{i,j,k,l} +. \n",
|
|
" sum mos (fun n ->\n",
|
|
" m_2112_2Ha_2Fa.{i,j,k,l,n} +. m_2112_2Ha_2Fa.{j,i,l,k,n}\n",
|
|
" ) +.\n",
|
|
" sum mos' (fun n ->\n",
|
|
" m_2112_2Hb_2Fa.{i,j,k,l,n} +. m_2112_2Hb_2Fa.{j,i,l,k,n}\n",
|
|
" ) \n",
|
|
" else\n",
|
|
" m_2112_1H_2Fb.{i,j,k,l} +.\n",
|
|
" sum mos (fun n ->\n",
|
|
" m_2112_2Ha_2Fb.{i,j,k,l,n} +. m_2112_2Hb_2Fb.{j,i,l,k,n}\n",
|
|
" ) +.\n",
|
|
" sum mos' (fun n ->\n",
|
|
" m_2112_2Hb_2Fb.{i,j,k,l,n} +. m_2112_2Ha_2Fb.{j,i,l,k,n}\n",
|
|
" ) \n",
|
|
" in\n",
|
|
"\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"let _ = check 100 integral_value 2 1 1 2\n",
|
|
"\n",
|
|
"let ki = det_I.(2)\n",
|
|
"let kj = det_I.(33)\n",
|
|
"\n",
|
|
"let _ = integral_value ki kj\n",
|
|
"let _ = \n",
|
|
" let alphas = generate_alphas ki kj 2 1 1 2\n",
|
|
" in compute_HaaF ki alphas kj\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 10. 2 1 21"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"2_1_21.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{ij}^{kl}|I\\rangle$\n",
|
|
"\n",
|
|
"$m$ : orbital indices of MOs unoccupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$n$ : orbital indices of MOs occupied in $|J\\rangle$\n",
|
|
"\n",
|
|
"$\\sum_{a} \n",
|
|
" \\hat{T}_{ij}^{al} \\hat{T}_{a}^{k} +\n",
|
|
" \\hat{T}_{ij}^{ka} \\hat{T}_{a}^{l} +\n",
|
|
" \\hat{T}_{i\\bar{j}}^{a\\bar{l}} \\hat{T}_{a}^{k} +\n",
|
|
" \\hat{T}_{i\\bar{j}}^{k\\bar{a}} \\hat{T}_{\\bar{a}}^{\\bar{l}}\n",
|
|
"$\n",
|
|
"\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\sum_{a}\n",
|
|
" \\langle i j || a l \\rangle \\left( f_{ak} + \\sum_{n} \\left[ a n || k n \\right] + \\sum_{\\bar{n}} \\left[ a \\bar{n} | k \\bar{n} \\right] \\right) + \\\\\n",
|
|
"\\sum_{a}\n",
|
|
" \\langle i j || k a \\rangle \\left( f_{al} + \\sum_{n} \\left[ n a || n l \\right] + \\sum_{\\bar{n}} \\left[ \\bar{n} a | \\bar{n} l \\right] \\right) \n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, j, k, l, s, s', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" \n",
|
|
" let mos, mos', s'' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a kj, mos_b kj, Spin.Beta\n",
|
|
" | Spin.Beta -> mos_b kj, mos_a kj, Spin.Alfa\n",
|
|
" in\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" h_two i j a l s s' *.\n",
|
|
" ( f_one a k s +.\n",
|
|
" sum mos (fun n -> f_two a n k n s s) +.\n",
|
|
" sum mos' (fun n -> f_two a n k n s s'') \n",
|
|
" ) ) +.\n",
|
|
"\n",
|
|
" let mos, mos', s'' =\n",
|
|
" match s' with\n",
|
|
" | Spin.Alfa -> mos_a kj, mos_b kj, Spin.Beta\n",
|
|
" | Spin.Beta -> mos_b kj, mos_a kj, Spin.Alfa\n",
|
|
" in\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" h_two j i a k s' s *.\n",
|
|
" ( f_one a l s' +.\n",
|
|
" sum mos (fun n -> f_two a n l n s' s') +.\n",
|
|
" sum mos' (fun n -> f_two a n l n s' s'') \n",
|
|
" ) \n",
|
|
" ) \n",
|
|
" in\n",
|
|
"\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
"\n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 2 1 2 1"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let m_2121_2Ha_1F =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a l Spin.Alfa Spin.Alfa *. f_one a k Spin.Alfa +.\n",
|
|
" h_two i j k a Spin.Alfa Spin.Alfa *. f_one a l Spin.Alfa)\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_2121_2Hb_1F =\n",
|
|
" array_4_init mo_num mo_num mo_num mo_num (fun i j k l ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a l Spin.Alfa Spin.Beta *. f_one a k Spin.Alfa +.\n",
|
|
" h_two i j k a Spin.Alfa Spin.Beta *. f_one a l Spin.Alfa)\n",
|
|
" )\n",
|
|
"\n",
|
|
"let m_2121_2Ha_2Fa =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a l Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a n k n Spin.Alfa Spin.Alfa )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let m_2121_2Hb_2Fa =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a l Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a n k n Spin.Alfa Spin.Alfa )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let m_2121_2Ha_2Fb =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a l Spin.Alfa Spin.Alfa *.\n",
|
|
" f_two a n k n Spin.Alfa Spin.Beta )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let m_2121_2Hb_2Fb =\n",
|
|
" array_5_init mo_num mo_num mo_num mo_num mo_num (fun i j k l n ->\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" h_two i j a l Spin.Alfa Spin.Beta *.\n",
|
|
" f_two a n k n Spin.Alfa Spin.Beta )\n",
|
|
" )\n",
|
|
" \n",
|
|
"let integral_value ki kj = \n",
|
|
" let i, j, k, l, s, s', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" let mos, mos' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_a kj, mos_b kj\n",
|
|
" | Spin.Beta -> mos_b kj, mos_a kj\n",
|
|
" in\n",
|
|
" \n",
|
|
" let result = \n",
|
|
" if s = s' then\n",
|
|
" m_2121_2Ha_1F.{i,j,k,l} +. \n",
|
|
" sum mos (fun n ->\n",
|
|
" m_2121_2Ha_2Fa.{i,j,k,l,n} +. m_2121_2Ha_2Fa.{j,i,l,k,n}\n",
|
|
" ) +.\n",
|
|
" sum mos' (fun n ->\n",
|
|
" m_2121_2Ha_2Fb.{i,j,k,l,n} +. m_2121_2Ha_2Fb.{j,i,l,k,n}\n",
|
|
" ) \n",
|
|
" else\n",
|
|
" m_2121_2Hb_1F.{i,j,k,l} +.\n",
|
|
" sum mos (fun n ->\n",
|
|
" m_2121_2Hb_2Fa.{i,j,k,l,n} +. m_2121_2Hb_2Fb.{j,i,l,k,n}\n",
|
|
" ) +.\n",
|
|
" sum mos' (fun n ->\n",
|
|
" m_2121_2Hb_2Fb.{i,j,k,l,n} +. m_2121_2Hb_2Fa.{j,i,l,k,n}\n",
|
|
" ) \n",
|
|
" in\n",
|
|
"\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 2 1 2 1\n",
|
|
"\n",
|
|
"(*\n",
|
|
"let ki = det_I.(2)\n",
|
|
"let kj = det_I.(33)\n",
|
|
"\n",
|
|
"let _ = integral_value ki kj\n",
|
|
"let _ = \n",
|
|
" let alphas = generate_alphas ki kj 2 1 2 1\n",
|
|
" in compute_HaaF ki alphas kj\n",
|
|
"*)\n",
|
|
"\n",
|
|
"\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 11. 2 1 22"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"2_1_22.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{ij}^{kl}|I\\rangle$\n",
|
|
"\n",
|
|
"$m$ : orbital indices of MOs unoccupied both in $|I\\rangle$ and $|J\\rangle$.\n",
|
|
"\n",
|
|
"$n$ : orbital indices of MOs occupied in both $|I\\rangle$ and $|J\\rangle$ .\n",
|
|
"\n",
|
|
"$\\sum_{a} \n",
|
|
" \\hat{T}_{ij}^{am} \\hat{T}_{am}^{kl} +\n",
|
|
" \\hat{T}_{ij}^{ma} \\hat{T}_{ma}^{kl} -\n",
|
|
" \\hat{T}_{in}^{al} \\hat{T}_{aj}^{kn} -\n",
|
|
" \\hat{T}_{nj}^{ka} \\hat{T}_{ia}^{nl} +\n",
|
|
" \\sum_{n}\n",
|
|
" \\hat{T}_{i\\bar{n}}^{k\\bar{a}} \\hat{T}_{\\bar{a}j}^{\\bar{n}l}\n",
|
|
"$\n",
|
|
"\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\sum_{a}\n",
|
|
"\\sum_{m} \\langle i j || a m \\rangle \\left[ a m || k l \\right] + \\langle i j || m a \\rangle \\left[ m a || k l \\right] - \\\\\n",
|
|
"\\sum_{n} \\langle i n || a l \\rangle \\left[ a j || k n \\right] + \\langle n j || k a \\rangle \\left[ i a || n l \\right] \n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, j, k, l, s, s', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) , \n",
|
|
" Array.init mo_num (fun i -> Some (i+1)) \n",
|
|
" in\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a ki);\n",
|
|
" List.iter (fun i -> mos_virt_a.(i-1) <- None) (mos_a kj);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b ki);\n",
|
|
" List.iter (fun i -> mos_virt_b.(i-1) <- None) (mos_b kj);\n",
|
|
"\n",
|
|
" let mos_virt_a, mos_virt_b = \n",
|
|
" Array.to_list mos_virt_a |> Util.list_some,\n",
|
|
" Array.to_list mos_virt_b |> Util.list_some\n",
|
|
" in\n",
|
|
" \n",
|
|
" let result = \n",
|
|
" \n",
|
|
" let mos_virt, mos_virt' =\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> mos_virt_a, mos_virt_b\n",
|
|
" | Spin.Beta -> mos_virt_b, mos_virt_a\n",
|
|
" in\n",
|
|
" \n",
|
|
" let mos, mos' =\n",
|
|
" let alfa = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.alfa ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.alfa kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" let beta = \n",
|
|
" let i = Spindeterminant.bitstring @@ Determinant.beta ki in\n",
|
|
" let j = Spindeterminant.bitstring @@ Determinant.beta kj in\n",
|
|
" Bitstring.to_list (Bitstring.logand i j)\n",
|
|
" in\n",
|
|
" match s with\n",
|
|
" | Spin.Alfa -> alfa, beta\n",
|
|
" | Spin.Beta -> beta, alfa\n",
|
|
" in\n",
|
|
"\n",
|
|
" if s = s' then \n",
|
|
" let s'' = Spin.other s' in\n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" sum mos (fun n ->\n",
|
|
" h_two i n a k s s *. f_two j a n l s s \n",
|
|
" +. h_two i n a l s s *. f_two j a k n s s \n",
|
|
" -. h_two j n a k s s *. f_two i a n l s s \n",
|
|
" -. h_two j n a l s s *. f_two i a k n s s\n",
|
|
" )\n",
|
|
" +. sum mos_virt (fun m ->\n",
|
|
" -. h_two i j a m s s *. f_two m a k l s s )\n",
|
|
" +. sum mos' (fun n ->\n",
|
|
" h_two i n k a s s'' *. f_two j a l n s s''\n",
|
|
" +. h_two j n l a s s'' *. f_two i a k n s s''\n",
|
|
" -. h_two i n l a s s'' *. f_two j a k n s s''\n",
|
|
" -. h_two j n k a s s'' *. f_two i a l n s s''\n",
|
|
" )\n",
|
|
" )\n",
|
|
" else\n",
|
|
" sum mos_cabs (fun a ->\n",
|
|
" sum mos_virt' (fun m ->\n",
|
|
" h_two i j a m s s' *. f_two a m k l s s' ) +.\n",
|
|
" sum mos_virt (fun m ->\n",
|
|
" h_two i j m a s s' *. f_two m a k l s s' ) +.\n",
|
|
" sum mos (fun n ->\n",
|
|
" h_two n i a k s s *. f_two a j n l s s'\n",
|
|
" +. h_two n j a l s s' *. f_two i a k n s s \n",
|
|
" -. h_two n j k a s s' *. f_two i a n l s s'\n",
|
|
" ) +.\n",
|
|
" sum mos' (fun n -> if n >= j then 0. else \n",
|
|
" h_two i n k a s s' *. f_two j a l n s' s'\n",
|
|
" +. h_two n j a l s' s' *. f_two i a k n s s' ) +. \n",
|
|
" sum mos' (fun n -> if n <= j then 0. else \n",
|
|
" -. h_two i n k a s s' *. f_two j a n l s' s' \n",
|
|
" -. h_two j n a l s' s' *. f_two i a k n s s' ) +.\n",
|
|
" sum mos' (fun n -> \n",
|
|
" -. h_two i n a l s s' *. f_two a j k n s s'\n",
|
|
" )\n",
|
|
" )\n",
|
|
" in\n",
|
|
"\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
"\n",
|
|
"\n",
|
|
"let _ = check 300 integral_value 2 1 2 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "raw",
|
|
"metadata": {
|
|
"raw_mimetype": "text/markdown",
|
|
"scrolled": false
|
|
},
|
|
"source": [
|
|
"let ki = det_I.(99)\n",
|
|
"let kj = det_I.(193)\n",
|
|
"\n",
|
|
"let alpha_to_string alpha = \n",
|
|
" \n",
|
|
" let exc0 = Array.init (aux_num+1) (fun _ -> [|\"-\";\"-\"|]) in\n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" let spin = function\n",
|
|
" | Spin.Alfa -> 0\n",
|
|
" | _ -> 1\n",
|
|
" in\n",
|
|
" exc0.(i).(spin s ) <- \"i\" \n",
|
|
" ; exc0.(j).(spin s') <- \"j\" \n",
|
|
" ; exc0.(k).(spin s ) <- \"k\" \n",
|
|
" ; exc0.(l).(spin s') <- \"l\" \n",
|
|
" ;\n",
|
|
" let s0, s0' = s, s' in\n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p1 =\n",
|
|
" match Excitation.of_det ki alpha with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" if exc0.(i).(spin s ) = \"-\" then exc0.(i).(spin s ) <- \"n\";\n",
|
|
" if exc0.(j).(spin s') = \"-\" then exc0.(j).(spin s') <- \"n\";\n",
|
|
" if exc0.(k).(spin s ) = \"-\" then exc0.(k).(spin s ) <- if k > mo_num then \"a\" else \"m\";\n",
|
|
" if exc0.(l).(spin s') = \"-\" then exc0.(l).(spin s') <- if l > mo_num then \"a\" else \"m\";\n",
|
|
" \n",
|
|
" let string_h = \n",
|
|
" Printf.sprintf \"h_two %s %s %s %s %s %s *. \" \n",
|
|
" exc0.(i).(spin s )\n",
|
|
" exc0.(j).(spin s')\n",
|
|
" exc0.(k).(spin s )\n",
|
|
" exc0.(l).(spin s')\n",
|
|
" (if s = s0 then \"s \" else \"s'\")\n",
|
|
" (if s' = s0 then \"s \" else \"s'\")\n",
|
|
" in \n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p2 =\n",
|
|
" match Excitation.of_det alpha kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" let string_f = \n",
|
|
" Printf.sprintf \"f_two %s %s %s %s %s %s\" \n",
|
|
" exc0.(i).(spin s )\n",
|
|
" exc0.(j).(spin s')\n",
|
|
" exc0.(k).(spin s )\n",
|
|
" exc0.(l).(spin s')\n",
|
|
" (if s = s0 then \"s \" else \"s'\")\n",
|
|
" (if s' = s0 then \"s \" else \"s'\")\n",
|
|
" in\n",
|
|
"(*\n",
|
|
" Format.printf \"|I> -> |a> : %a | %s\\n@.\" Excitation.pp (Excitation.of_det ki alpha) string_h ;\n",
|
|
" Format.printf \"|a> -> |J> : %a | %s\\n@.\" Excitation.pp (Excitation.of_det alpha kj) string_f ;\n",
|
|
" *)\n",
|
|
"\n",
|
|
" \n",
|
|
" let sign = \n",
|
|
" if Phase.add p1 p2 = phase then \"+.\" else \"-.\"\n",
|
|
" in\n",
|
|
" sign ^ string_h ^ string_f\n",
|
|
" \n",
|
|
"\n",
|
|
"let alpha_debug alpha = \n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p1 =\n",
|
|
" match Excitation.of_det ki alpha with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" Printf.printf \"%d %d %d %d \" i j k l;\n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p2 =\n",
|
|
" match Excitation.of_det alpha kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" (*\n",
|
|
" Format.printf \"|I> -> |a> : %a | \\n@.\" Excitation.pp (Excitation.of_det ki alpha) ;\n",
|
|
" Format.printf \"|a> -> |J> : %a | \\n@.\" Excitation.pp (Excitation.of_det alpha kj) ;\n",
|
|
" *)\n",
|
|
" Printf.printf \"%d %d %d %d \\n%!\" i j k l\n",
|
|
"\n",
|
|
" \n",
|
|
"let strings = \n",
|
|
" Format.printf \"|I> -> |J> : %a |\\n@.\" Excitation.pp (Excitation.of_det ki kj) ;\n",
|
|
" generate_alphas ki kj 2 1 2 2\n",
|
|
" |> Array.of_list\n",
|
|
" |> Array.mapi (fun kk alpha -> alpha_to_string alpha)\n",
|
|
" |> Array.to_list\n",
|
|
" |> List.sort_uniq compare\n",
|
|
" |> Array.of_list\n",
|
|
"\n",
|
|
"let _ = Array.iteri (fun i x -> Printf.printf \"%d %s \\n%!\" i x) strings\n",
|
|
"\n",
|
|
"let _ =\n",
|
|
" let v =\n",
|
|
" let alphas =\n",
|
|
" generate_alphas ki kj 2 1 2 2\n",
|
|
" (*\n",
|
|
" |> List.filter (fun alpha ->\n",
|
|
" let x = alpha_to_string alpha in\n",
|
|
" x = strings.(6)\n",
|
|
" )\n",
|
|
" *)\n",
|
|
" in\n",
|
|
" (*\n",
|
|
" List.iter alpha_debug alphas ;\n",
|
|
" Printf.printf \"\\n%!\";\n",
|
|
" *)\n",
|
|
" compute_HaaF ki alphas kj \n",
|
|
" in\n",
|
|
" let x = (integral_value ki kj) in\n",
|
|
" Printf.printf \"%20.8e %20.8e %20.8e\\n%!\" x v (v-. x)\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 12. 2 2 22"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"2_2_22.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{ij}^{kl}|I\\rangle$\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\sum_{b}\\sum_{a<b}\n",
|
|
" \\langle i j || a b \\rangle \n",
|
|
" \\left[ ab || kj \\right]\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let h, h', p, p', s, s', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Double (phase,\n",
|
|
" { hole=h ; particle=p ; spin=s },\n",
|
|
" { hole=h'; particle=p'; spin=s'}) ) -> h, h', p, p', s, s', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" if s <> s' then (* Alpha-Beta *)\n",
|
|
" sum mos_cabs (fun b -> \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" h_two h h' a b s s' *. f_two a b p p' s s'\n",
|
|
" )) \n",
|
|
" else (* Alpha-Alpha / Beta-Beta *)\n",
|
|
" sum mos_cabs (fun b -> \n",
|
|
" sum mos_cabs (fun a -> if b >= a then 0. else\n",
|
|
" h_two h h' a b s s' *. f_two a b p p' s s'\n",
|
|
" )) \n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 100 integral_value 2 2 2 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# 13. 3 1 2 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let _ = png_image \"3_1_22.png\""
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$|J\\rangle = \\hat{T}_{ijm}^{kln}|I\\rangle$\n",
|
|
"\n",
|
|
"$\\sum_{a}\n",
|
|
" \\hat{T}_{ij}^{al} \\hat{T}_{am}^{kn} \n",
|
|
" + \\hat{T}_{ij}^{ka} \\hat{T}_{am}^{ln} \n",
|
|
" + \\hat{T}_{im}^{ka} \\hat{T}_{ja}^{ln} \n",
|
|
" + \\hat{T}_{im}^{an} \\hat{T}_{aj}^{kl} \n",
|
|
" + \\hat{T}_{jm}^{an} \\hat{T}_{ia}^{kl} \n",
|
|
" + \\hat{T}_{jm}^{la} \\hat{T}_{ia}^{kn} \n",
|
|
"$\n",
|
|
"\n",
|
|
"$$\n",
|
|
"\\sum_{a} \\langle i j || a l \\rangle \\left[ a m || k n \\right] +\n",
|
|
" \\langle i j || k a \\rangle \\left[ a m || l n \\right] +\n",
|
|
" \\langle i m || k a \\rangle \\left[ j a || l n \\right] +\n",
|
|
" \\langle i m || a n \\rangle \\left[ a j || k l \\right] +\n",
|
|
" \\langle j m || a n \\rangle \\left[ i a || k l \\right] +\n",
|
|
" \\langle j m || l a \\rangle \\left[ i a || k n \\right] \n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"let integral_value ki kj = \n",
|
|
" let i, j, m, k, l, n, s1, s2, s3, phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Triple (phase,\n",
|
|
" { hole=h1 ; particle=p1 ; spin=s1 },\n",
|
|
" { hole=h2 ; particle=p2 ; spin=s2 },\n",
|
|
" { hole=h3 ; particle=p3 ; spin=s3 }) ) -> h1, h2, h3, p1, p2, p3, s1, s2, s3, phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
"\n",
|
|
" let result = \n",
|
|
" let open Spin in \n",
|
|
" match s1, s2, s3 with\n",
|
|
" | Alfa, Alfa, Alfa\n",
|
|
" | Beta, Beta, Beta -> \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" h_two i j a k s1 s2 *. f_two m a l n s3 s3 \n",
|
|
" +. h_two i j a n s1 s2 *. f_two m a k l s3 s2 \n",
|
|
" +. h_two i m a l s1 s3 *. f_two j a k n s2 s3 \n",
|
|
" +. h_two j m a k s2 s3 *. f_two i a l n s1 s3 \n",
|
|
" +. h_two j m a n s2 s3 *. f_two i a k l s1 s2 \n",
|
|
" -. h_two i j a l s1 s2 *. f_two m a k n s3 s3 \n",
|
|
" -. h_two i m a k s1 s3 *. f_two j a l n s2 s3 \n",
|
|
" -. h_two i m a n s1 s3 *. f_two j a k l s2 s2 \n",
|
|
" -. h_two j m a l s2 s3 *. f_two i a k n s1 s3 )\n",
|
|
" | Alfa, Alfa, Beta\n",
|
|
" | Beta, Beta, Alfa -> \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" h_two i j a l s1 s2 *. f_two a m k n s1 s3\n",
|
|
" +. h_two i m k a s1 s3 *. f_two j a l n s2 s3\n",
|
|
" +. h_two j m a n s2 s3 *. f_two i a k l s1 s2 \n",
|
|
" +. h_two j m l a s2 s3 *. f_two i a k n s1 s3\n",
|
|
" -. h_two i j a k s1 s2 *. f_two a m l n s1 s3\n",
|
|
" -. h_two i m a n s1 s3 *. f_two j a k l s2 s2 \n",
|
|
" -. h_two i m l a s1 s3 *. f_two j a k n s2 s3\n",
|
|
" -. h_two j m k a s2 s3 *. f_two i a l n s1 s3\n",
|
|
" ) \n",
|
|
" | Alfa, Beta, Beta\n",
|
|
" | Beta, Alfa, Alfa -> \n",
|
|
" sum mos_cabs (fun a -> \n",
|
|
" h_two i j a l s1 s2 *. f_two a m k n s1 s3 \n",
|
|
" +. h_two i m a n s1 s3 *. f_two a j k l s1 s2 \n",
|
|
" +. h_two i m k a s1 s3 *. f_two j a l n s2 s3 \n",
|
|
" +. h_two j m a n s2 s3 *. f_two i a k l s1 s2 \n",
|
|
" -. h_two i j a n s1 s2 *. f_two a m k l s1 s2 \n",
|
|
" -. h_two i j k a s1 s2 *. f_two m a l n s2 s3 \n",
|
|
" -. h_two i m a l s1 s3 *. f_two a j k n s1 s3 \n",
|
|
" -. h_two j m a l s2 s3 *. f_two i a k n s1 s3 \n",
|
|
" ) \n",
|
|
" | Beta, Alfa, Beta\n",
|
|
" | Alfa, Beta, Alfa -> assert false (*TODO *)\n",
|
|
" in\n",
|
|
" match phase with\n",
|
|
" | Phase.Pos -> result\n",
|
|
" | Phase.Neg -> -. result\n",
|
|
" \n",
|
|
"\n",
|
|
"let _ = check 200 integral_value 3 1 2 2"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"scrolled": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"let ki = det_I.(129)\n",
|
|
"let kj = det_I.(349)\n",
|
|
"\n",
|
|
"let alpha_to_string alpha = \n",
|
|
" \n",
|
|
" let exc0 = Array.init (aux_num+1) (fun _ -> [|\"-\";\"-\"|]) in\n",
|
|
" \n",
|
|
" let i, j, m, k, l, n, s, s', s'', phase =\n",
|
|
" match Excitation.of_det ki kj with\n",
|
|
" | Excitation.(Triple (phase,\n",
|
|
" { hole ; particle ; spin },\n",
|
|
" {hole=hole' ; particle=particle' ; spin=spin' }, \n",
|
|
" {hole=hole''; particle=particle''; spin=spin''} )) ->\n",
|
|
" hole, hole', hole'', particle, particle', particle'', spin, spin', spin'', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" let spin = function\n",
|
|
" | Spin.Alfa -> 0\n",
|
|
" | _ -> 1\n",
|
|
" in\n",
|
|
" exc0.(i).(spin s ) <- \"i\" \n",
|
|
" ; exc0.(j).(spin s' ) <- \"j\" \n",
|
|
" ; exc0.(k).(spin s ) <- \"k\" \n",
|
|
" ; exc0.(l).(spin s' ) <- \"l\" \n",
|
|
" ; exc0.(m).(spin s'') <- \"m\" \n",
|
|
" ; exc0.(n).(spin s'') <- \"n\" \n",
|
|
" ;\n",
|
|
" let s0, s0', s0'' = s, s', s'' in\n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p1 =\n",
|
|
" match Excitation.of_det ki alpha with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" if exc0.(i).(spin s ) = \"-\" then exc0.(i).(spin s ) <- \"p\";\n",
|
|
" if exc0.(j).(spin s') = \"-\" then exc0.(j).(spin s') <- \"p\";\n",
|
|
" if exc0.(k).(spin s ) = \"-\" then exc0.(k).(spin s ) <- if k > mo_num then \"a\" else \"q\";\n",
|
|
" if exc0.(l).(spin s') = \"-\" then exc0.(l).(spin s') <- if l > mo_num then \"a\" else \"q\";\n",
|
|
" \n",
|
|
" let string_h = \n",
|
|
" Printf.sprintf \"h_two %s %s %s %s %s %s *. \" \n",
|
|
" exc0.(i).(spin s )\n",
|
|
" exc0.(j).(spin s')\n",
|
|
" exc0.(k).(spin s )\n",
|
|
" exc0.(l).(spin s')\n",
|
|
" (if s = s0 then \"s \" else if s = s0' then \"s'\" else \"s''\")\n",
|
|
" (if s' = s0' then \"s'\" else if s = s0'' then \"s''\" else \"s\" )\n",
|
|
" in \n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p2 =\n",
|
|
" match Excitation.of_det alpha kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" \n",
|
|
" let string_f = \n",
|
|
" Printf.sprintf \"f_two %s %s %s %s %s %s\" \n",
|
|
" exc0.(i).(spin s )\n",
|
|
" exc0.(j).(spin s')\n",
|
|
" exc0.(k).(spin s )\n",
|
|
" exc0.(l).(spin s')\n",
|
|
" (if s = s0 then \"s \" else if s = s0' then \"s'\" else \"s''\")\n",
|
|
" (if s' = s0' then \"s'\" else if s = s0'' then \"s''\" else \"s\" )\n",
|
|
" in\n",
|
|
" (*\n",
|
|
" Format.printf \"|I> -> |a> : %a | %s\\n@.\" Excitation.pp (Excitation.of_det ki alpha) string_h ;\n",
|
|
" Format.printf \"|a> -> |J> : %a | %s\\n@.\" Excitation.pp (Excitation.of_det alpha kj) string_f ;\n",
|
|
" *)\n",
|
|
"\n",
|
|
" \n",
|
|
" let sign = \n",
|
|
" if Phase.add p1 p2 = phase then \"+.\" else \"-.\"\n",
|
|
" in\n",
|
|
" sign ^ string_h ^ string_f\n",
|
|
" \n",
|
|
"\n",
|
|
"let alpha_debug alpha = \n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p1 =\n",
|
|
" match Excitation.of_det ki alpha with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" Printf.printf \"%d %d %d %d \" i j k l;\n",
|
|
" \n",
|
|
" let i, j, k, l, s, s', p2 =\n",
|
|
" match Excitation.of_det alpha kj with\n",
|
|
" | Excitation.(Double (phase, { hole ; particle ; spin }, {hole=hole' ; particle=particle' ; spin=spin' })) ->\n",
|
|
" hole, hole', particle, particle', spin, spin', phase\n",
|
|
" | _ -> assert false\n",
|
|
" in\n",
|
|
" (*\n",
|
|
" Format.printf \"|I> -> |a> : %a | \\n@.\" Excitation.pp (Excitation.of_det ki alpha) ;\n",
|
|
" Format.printf \"|a> -> |J> : %a | \\n@.\" Excitation.pp (Excitation.of_det alpha kj) ;\n",
|
|
" *)\n",
|
|
" Printf.printf \"%d %d %d %d \\n%!\" i j k l\n",
|
|
"\n",
|
|
" \n",
|
|
"let strings = \n",
|
|
" Format.printf \"|I> -> |J> : %a |\\n@.\" Excitation.pp (Excitation.of_det ki kj) ;\n",
|
|
" generate_alphas ki kj 3 1 2 2\n",
|
|
" |> Array.of_list\n",
|
|
" |> Array.mapi (fun kk alpha -> alpha_to_string alpha)\n",
|
|
" |> Array.to_list\n",
|
|
" |> List.sort_uniq compare\n",
|
|
" |> Array.of_list\n",
|
|
"\n",
|
|
"let _ = Array.iteri (fun i x -> Printf.printf \"%d %s \\n%!\" i x) strings\n",
|
|
"\n",
|
|
"let _ =\n",
|
|
" let v =\n",
|
|
" let alphas =\n",
|
|
" generate_alphas ki kj 3 1 2 2\n",
|
|
" (*\n",
|
|
" |> List.filter (fun alpha ->\n",
|
|
" let x = alpha_to_string alpha in\n",
|
|
" x = strings.(6)\n",
|
|
" )\n",
|
|
" *)\n",
|
|
" in\n",
|
|
" (*\n",
|
|
" List.iter alpha_debug alphas ;\n",
|
|
" Printf.printf \"\\n%!\";\n",
|
|
" *)\n",
|
|
" compute_HaaF ki alphas kj \n",
|
|
" in\n",
|
|
" let x = (integral_value ki kj) in\n",
|
|
" Printf.printf \"%20.8e %20.8e %20.8e\\n%!\" x v (v-. x)\n"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": []
|
|
}
|
|
],
|
|
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|
|
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|
|
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|
|
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|
|
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|
|
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|
|
},
|
|
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|
|
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|
|
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|
|
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|
|
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|
|
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|
|
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|
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|
|
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|
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|