由波函数的极化展开确定交换相互作用能
Determination of the exchange interaction energy from the polarization expansion of the wave function
摘要：
本文从波函数的极化展开出发，利用常规的表面积分公式和两个涉及体积积分的公式：自适应微扰理论（SAPT）公式和我们推荐的变分公式，计算了交换对氢原子与质子相互作用能量的贡献。在大的核间距$R$，所有三个公式产生正确的表达式$-(2/e)Re^{-R}$，但近似它与非常不同的收敛速度。在SAPT公式的情况下，收敛是几何的，误差为$3^{-K}$，其中$K$是所应用的偏振展开的阶数。曲面积分公式的误差随$a^K/(K+1)!$呈指数下降，其中$a=\ln2 -\tfrac{1}{2}$。变分公式的性能最好，其误差衰减为$K^{1/2} [a^{ K}/(K+1)!]^2$。这些收敛速度比通过多极展开近似波函数得到的收敛速度快得多。这表明了多极级数的部分重混受极化展开影响的效率。我们的结果也表明了将变分原理纳入分子相互作用的微扰理论的好处。
英文摘要：
The exchange contribution to the energy of the hydrogen atom interacting with a proton is calculated from the polarization expansion of the wave function using the conventional surface-integral formula and two formulas involving volume integrals: the formula of the symmetry-adapted perturbation theory (SAPT) and the variational formula recommended by us. At large internuclear distances $R$, all three formulas yield the correct expression $-(2/e)Re^{-R}$, but approximate it with very different convergence rates. In the case of the SAPT formula, the convergence is geometric with the error falling as $3^{-K}$, where $K$ is the order of the applied polarization expansion. The error of the surface-integral formula decreases exponentially as $a^K/(K+1)!$, where $a=\ln2 -\tfrac{1}{2}$. The variational formula performs best, its error decays as $K^{1/2} [a^{ K}/(K+1)!]^2$. These convergence rates are much faster than those resulting from approximating the wave function through the multipole expansion. This shows the efficiency of the partial resummation of the multipole series effected by the polarization expansion. Our results demonstrate also the benefits of incorporating the variational principle into the perturbation theory of molecular interactions.
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