一维函数方法求解原子和分子薛定谔方程

Solutions of Atomic and Molecular Schrödinger Equations with One-dimensional Function Approach

对薛定谔方程的严格数值求解,尤其是发展标准方法之外的、包含新功能的解法,一直是物理学研究的基本关注点.本文介绍一种近些年发展的一维函数近似解方法,该方法通过对波函数的不同坐标分量进行处理来求解原子和分子体系的薛定谔方程.电子的试探波函数被离散化到实空间均匀格点上,因此可以通过残差矢量校正的方法对其进行改进.一维函数方法本身的特征决定其非常利于数值积分,避免了很多由常规的多电子、多中心势分子积分所带来的问题.计算中,最终能量是从严格的能量上限逐渐收敛所获得,计算出的两电子薛定谔波函数呈现出常规单电子近似方法所含有的电子关联效应.不同于密度泛函理论及Hartree-Fock的单电子解法,本方法对电子-电子排斥能的多体效应的处理更加精确.

Rigorous numerical techniques to solve the Schrödinger equation are both interesting and desirable,particularly with one that can include new features beyond the standard methods.In this article,we review one-dimensional function(1D function)approach developed recently by us to obtain the solutions of the Schrödinger equations of atomic and molecular systems where one-dimensional basis functions have been ap⁃plied to separate components.A uniform real-space grid representation of the electronic wavefunctions is em⁃ployed;hence,a refinement technique of residual vector correction can be implemented.The 1D function ap⁃proach facilitates such convenient numerical integrations that many problems related with the many-electron multi-center potential molecular integrals are circumvented.The converged energy is obtained from a strictly upper bound one,while the obtained two-electron Schrödinger wavefunction exhibits the electron correlation effect on one-electron distribution.Different from density functional theory or Hartree-Fock with the assumed particle-separability,the obtained solution treats more accurately many-body effect of electron correlation found in the electron-electron repulsion energy.