Quantum Chemistry Research Institute, Kyoto Technoscience Center 16, 14 Yoshida Kawara-machi, Sakyo-Ku, Kyoto 606-8305, Japan.
J Chem Phys. 2023 Jul 14;159(2). doi: 10.1063/5.0155105.
We investigate, in this paper, the Gaussian (G) function with odd powers of r, rxaybzc exp(-αr2), called the r-Gaussian or simply the rG function. The reason we investigate this function here is that it is generated as the elements of the complement functions (cf's) when we apply the free complement (FC) theory for solving the Schrödinger equation to the initial functions composed of the Gaussian functions. This means that without the rG functions, the Gaussian set of functions cannot produce the exact solutions of the Schrödinger equation, showing the absolute importance of the rG functions in quantum chemistry. Actually, the rG functions drastically improve the wave function near the cusp region. This was shown by the applications of the present theory to the hydrogen and helium atoms. When we use the FC-sij theory, in which the inter-electron function rij is replaced with its square sij=rij2 that is integrable, we need only one- and two-electron integrals for the G and rG functions. The one-center one- and two-electron integrals of the rG functions are always available in a closed form. To calculate the integrals of the multi-centered rG functions, we proposed the rG-NG expansion method, in which an rG function is expanded by a linear combination of the G functions. The optimal exponents and coefficients of this expansion were given for N = 2, 3, 4, 5, 6, and 9. To show the accuracy and the usefulness of the rG-NG method, we applied the FC-sij theory to the hydrogen molecule.
本文研究了具有奇数幂 r 的高斯（G）函数 rxaybzc exp(-αr2)，称为 r-高斯函数或简称 rG 函数。我们研究这个函数的原因是，当我们将自由补（FC）理论应用于由高斯函数组成的初始函数来求解薛定谔方程时，它是互补函数（cf's）的元素生成的。这意味着，如果没有 rG 函数，高斯函数集就无法产生薛定谔方程的精确解，这表明 rG 函数在量子化学中的绝对重要性。实际上，rG 函数极大地改善了波函数在尖点区域附近的表现。这一点已经通过将本理论应用于氢原子和氦原子得到了证明。当我们使用 FC-sij 理论时，其中电子间函数 rij 被其可积的平方 sij=rij2 所替代，我们只需要 G 和 rG 函数的单电子和双电子积分。rG 函数的单中心单电子和双电子积分总是可以用封闭形式表示。为了计算多中心 rG 函数的积分，我们提出了 rG-NG 展开方法，其中 rG 函数由 G 函数的线性组合展开。对于 N=2、3、4、5、6 和 9，给出了这个展开的最佳指数和系数。为了展示 rG-NG 方法的准确性和实用性，我们将 FC-sij 理论应用于氢分子。
