Valeev Edward F
Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, USA.
J Chem Phys. 2006 Dec 28;125(24):244106. doi: 10.1063/1.2403852.
Explicitly correlated R12 methods using a single short-range correlation factor (also known as F12 methods) have dramatically smaller basis set errors compared to the standard wave function counterparts, even when used with small basis sets. Correlations on several length scales, however, may not be described efficiently with one correlation factor. Here the authors explore a more general MP2-R12 method in which each electron pair uses a set of (contracted) Gaussian-type geminals (GTGs) with fixed exponents, whose coefficients are optimized linearly. The following features distinguish the current method from related explicitly correlated approaches published in the literature: (1) only two-electron integrals are needed, (2) the only approximations are the resolution of the identity and the generalized Brillouin condition, (3) only linear parameters are optimized, and (4) an arbitrary number of (non-)contracted GTGs can appear. The present method using only three GTGs and a double-zeta quality basis computed valence correlation energies for a set of 20 small molecules only 2.2% removed from the basis set limit. The average basis set error reduces to 1.2% using a near-complete set of seven GTGs with the double-zeta basis set. The conventional MP2 energies computed with much larger quadruple, quintuple, and sextuple basis sets all had larger average errors: 4.6%, 2.4%, and 1.5%, respectively. The new method compares well to the published MP2-R12 method using a single Slater-type geminal (STG) correlation factor. For example, the average basis set error in the absolute MP2-R12 energy obtained with the exp(-r12) correlation factor is 1.7%. Correlation contribution to atomization energies evaluated with the present method and with the STG-based method only required a double-zeta basis set to exceed the precision of the conventional sextuple-zeta result. The new method is shown to always be numerically stable if linear dependencies are removed from the two-particle basis and the zeroth-order Hamiltonian matrix is made positive definite.
与标准波函数方法相比，使用单个短程相关因子的显式相关R12方法（也称为F12方法）即使在使用小基组时，基组误差也显著更小。然而，仅用一个相关因子可能无法有效地描述多个长度尺度上的相关性。本文作者探索了一种更通用的MP2 - R12方法，其中每个电子对使用一组具有固定指数的（收缩）高斯型双电子基函数（GTG），其系数通过线性优化。以下特征将当前方法与文献中发表的相关显式相关方法区分开来：（1）仅需要双电子积分，（2）唯一的近似是单位分解和广义布里渊条件，（3）仅优化线性参数，（4）可以出现任意数量的（非）收缩GTG。仅使用三个GTG和双ζ质量基组的当前方法计算了一组20个小分子的价层相关能，与基组极限相比仅相差2.2%。使用七个GTG的近完备集和双ζ基组时，平均基组误差降至1.2%。用大得多的四重、五重和六重基组计算的传统MP2能量都有更大的平均误差，分别为4.6%、2.4%和1.5%。新方法与使用单个斯莱特型双电子基函数（STG）相关因子发表的MP2 - R12方法相比表现良好。例如，使用exp(-r12)相关因子获得的绝对MP2 - R12能量中的平均基组误差为1.7%。用本方法和基于STG的方法评估的原子化能的相关贡献仅需要双ζ基组就能超过传统六重ζ结果的精度。如果从双粒子基组中消除线性相关性并使零阶哈密顿矩阵正定，则新方法在数值上总是稳定的。
