{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Vibrational structure" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Introduction \n", "\n", "The molecular Hamiltonian is \n", "\n", "$$\n", "\\mathcal{H} = - \\sum_I \\frac{\\nabla_{R_I}^2}{M_I} - \\sum_i \\frac{\\nabla_{r_i}^2}{m_e} - \\sum_I\\sum_i \\frac{Z_I e^2}{|R_I-r_i|} + \\sum_i \\sum_{j>i} \\frac{e^2}{|r_i-r_j|} + \\sum_I\\sum_{J>I} \\frac{Z_I Z_J e^2}{|R_I-R_J|}\n", "$$\n", "\n", "Because the nuclei are much heavier than the electrons they do not move on the same time scale and therefore, the behavior of nuclei and electrons can be decoupled. This is the Born-Oppenheimer approximation.\n", "\n", "Within the Born-Oppenheimer approximation, a molecular wave function is factorized as a product of an electronic part, which is the solution of the electronic Schroedinger equation, and a vibro-rotational one, which is the solution of the nuclear Schroedinger equation in the potential energy surface (PES) generated by sampling the eigenvalues of the electronic Schroedinger equation for different geometries.\n", "\n", "The nuclear Schroedinger equation is usually solved in two steps, in analogy with its electronic counterpart. \n", "A single-particle basis (the basis functions are called, in this case, modals) is obtained either by the harmonic approximation applied to the PES or from a vibrational self-consistent field (VSCF) calculation. \n", "Vibrational anharmonic correlations are added a-posteriori with perturbative or variational approaches.\n", "The latter include Vibrational Configuration Interaction (VCI) and Vibrational Coupled Cluster (VCC) for highly-accurate anharmonic energies. \n", "The main advantage of VCI and VCC over alternative approaches (such as perturbation theories) is that their accuracy can be systematically improved towards the complete basis set limit for a given PES. \n", "However, their applicability is limited to small molecules with up to about 10 atoms due to their unfavorable scaling with system size.\n", "\n", "To tackle the scaling problem we would like to use quantum algorithms.\n", "\n", "The nuclear Schroedinger equation is\n", "$$\n", "\\mathcal{H}_{\\text{vib}} |\\Psi_{n}\\rangle = E_{n} |\\Psi_{n}\\rangle\n", "$$\n", "\n", "The so-called Watson Hamiltonian (neglecting vibro-rotational coupling terms) is\n", "$$\n", " \\mathcal{H}_\\text{vib}(Q_1, \\ldots, Q_L) \n", " = - \\frac{1}{2} \\sum_{l=1}^{L} \\frac{\\partial^2}{\\partial Q_l^2} + V(Q_1, \\ldots, Q_L)\n", "$$\n", "where $Q_l$ are the harmonic mass-weighted normal coordinates.\n", "\n", "$\\mathcal{H}_\\text{vib}$ must be mapped to an operator that acts on the states of a given set of $N_q$ qubits in order to calculate its eigenfunctions on quantum hardware.\n", "In electronic structure calculations, the mapping is achieved by expressing the non-relativistic electronic Hamiltonian in second quantization, \\textit{i.e.} by projecting it onto the complete set of antisymmetrized occupation number vectors (ONV) generated by a given (finite) set of orbitals.\n", "To encode the vibrational Hamiltonian in an analogous second quantization operators, we expand the potential $V(Q_1, \\ldots, Q_L)$ with the $n$-body expansion as follows:\n", "\n", "$$\n", " V(Q_1, \\ldots, Q_L) = V_0 + \\sum_{l=1}^L V^{[l]}(Q_l) \n", " + \\sum_{lVersion Information
Qiskit SoftwareVersion
qiskit-terra0.24.0.dev0+2b3686f
qiskit-aer0.11.2
qiskit-ibmq-provider0.19.2
qiskit-nature0.6.0
System information
Python version3.9.16
Python compilerGCC 12.2.1 20221121 (Red Hat 12.2.1-4)
Python buildmain, Dec 7 2022 00:00:00
OSLinux
CPUs8
Memory (Gb)62.50002670288086
Thu Apr 06 08:54:06 2023 CEST
" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "

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" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import tutorial_magics\n", "\n", "%qiskit_version_table\n", "%qiskit_copyright" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.16" } }, "nbformat": 4, "nbformat_minor": 4 }