精确计算分子体系能级下界的平方库仑势多中心积分的方法
Methods for accurate calculations of multi-center integrals of squared Coulomb potentials for lower bounds to energy levels of molecular systems
摘要：
本文导出了平方库仑势和Slater型轨道多中心积分的计算方法。这些积分对于精确计算分子系统能级的下界是必要的。所有的多中心积分减少到基本积分使用的Gaunt系数和平移的STO。当势为通常的库仑势时，利用势的拉普拉斯展开式或诺依曼展开式，可以计算积分。然而，对于平方库仑势，这样的展开是未知的。对于带平方库仑势的单中心和双中心基本积分，用不需要这种展开的方法，导出了精确的解析表达式和解析函数的一维积分表达式。这些方法主要依赖于椭球坐标下的积分、傅立叶变换、霍布森定理和适合于计算的简单微分算子展开。给出了这些表达式的数值结果并进行了比较。
英文摘要：
In this paper methods for calculations of multi-center integrals of squared Coulomb potentials and Slater-type orbitals (STO) are derived. These integrals are necessary for accurate lower bounds to energy levels of molecular systems. All multi-center integrals are reduced to fundamental integrals using the Gaunt coefficients and translation of STO. When the potential is the usual Coulomb potential, using the Laplace expansion or the Neumann expansion of the potential the integrals can be calculated. However, for the squared Coulomb potentials such expansions are not known. For the fundamental one-center and two-center integrals with squared Coulomb potentials, by methods free from such expansions exact analytic expressions and expressions by one-dimensional integrals of analytic functions are derived. The methods mainly rely on the integration in ellipsoidal coordinates, the Fourier transform, Hobson's theorem and expansion of differential operators by simple ones suitable for the calculation. Numerical results by these expressions are given and compared.
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