Effective core potential (ECP) integrals are among the most difficult one-electron integrals to calculate due to the projection operators. The radial part of these operators may include r⁰, r^–1, and r^–2 terms. For the r⁰ terms, we exploit a simple analytic expression for the fundamental projected integral to derive new recurrence relations and upper bounds for ECP integrals. For the r^–1 and r^–2 terms, we present a reconstruction method that replaces these terms by a sum of r⁰ terms and show that the resulting errors are chemically insignificant for a range of molecular properties. The new algorithm is available in Q-Chem 5.0 and is significantly faster than the ECP implementations in Q-Chem 4.4, GAMESS (US) and Dalton 2016.

中文翻译：

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有效核心势能积分的有效计算方法

由于投影算子，有效核心电势（ECP）积分是最难计算的单电子积分之一。这些算子的径向部分可以包括r ⁰，r ^–1和r ^–2项。对于r ⁰项，我们利用基本投影积分的简单解析表达式来得出新的递归关系和ECP积分的上限。对于r ^–1和r ^–2项，我们提出了一种重构方法，该方法将这些项替换为r ^0之和并表明，对于一系列分子特性，所产生的误差在化学上无关紧要。新算法在Q-Chem 5.0中可用，并且比Q-Chem 4.4，GAMESS（US）和Dalton 2016中的ECP实现要快得多。