Wang Yu, Kantorovich Lev
Department of Physics, King's College London, Strand, London WC2R 2LS, UK.
J Phys Condens Matter. 2009 Nov 25;21(47):474204. doi: 10.1088/0953-8984/21/47/474204. Epub 2009 Nov 5.
The group function theory by Tolpygo and McWeeny is a useful tool in treating quantum systems that can be represented as a set of localized electronic groups (e.g. atoms, molecules or bonds). It provides a general means of taking into account intra-correlation effects inside the groups without assuming that the interaction between the groups is weak. For non-orthogonal group functions the arrow diagram (AD) technique provides a convenient procedure for calculating matrix elements [Formula: see text] of arbitrary symmetrical operators [Formula: see text] which are needed, for example, for calculating the total energy of the system or its electron density. The total wavefunction of the system [Formula: see text] is represented as an antisymmetrized product of non-orthogonal electron group functions Φ(I) of each group I in the system. However, application of the AD theory to extended (e.g. infinite) systems (such as biological molecules or crystals) is not straightforward, since the calculation of the mean value of an operator requires that each term of the diagram expansion be divided by the normalization integral S = ⟨Ψ|Ψ⟩ which is given by an AD expansion as well. In our previous work, we cast the mean value [Formula: see text] of a symmetrical operator [Formula: see text] in the form of an AD expansion which is a linear combination of linked (connected) ADs multiplied by numerical pre-factors. To obtain the pre-factors, a method based on power series expansion with respect to overlap was developed and tested for a simple 1D Hartree-Fock (HF) ring model. In the present paper this method is first tested on a 2D HF model, and we find that the power series expansion for the pre-factors converges extremely slowly to the exact solution. Instead, we suggest another, more powerful, method based on a continued fraction expansion of the pre-factors that approaches the exact solution much faster. The method is illustrated on the calculation of the electron density for the 2D HF model. It provides a powerful technique for treating extended systems consisting of a large number of strongly localized electronic groups.
托尔皮戈和麦克韦尼提出的群函数理论是处理量子系统的一种有用工具，这类量子系统可表示为一组局域化电子群（如原子、分子或化学键）。它提供了一种通用方法，可在不假设群间相互作用微弱的情况下，考虑群内部的关联效应。对于非正交群函数，箭图（AD）技术为计算任意对称算符[公式：见正文]的矩阵元[公式：见正文]提供了一种便捷程序，例如计算系统总能量或其电子密度时就需要这些矩阵元。系统的总波函数[公式：见正文]表示为系统中每个群I的非正交电子群函数Φ(I)的反对称积。然而，将AD理论应用于扩展（如无限）系统（如生物分子或晶体）并非易事，因为算符平均值的计算要求图展开的每一项都除以归一化积分S = ⟨Ψ|Ψ⟩，而该积分同样由AD展开给出。在我们之前的工作中，我们将对称算符[公式：见正文]的平均值[公式：见正文]表示为AD展开形式，它是由带数值前置因子的相连（连通）AD的线性组合。为了得到前置因子，我们开发了一种基于重叠幂级数展开的方法，并在一个简单的一维哈特里 - 福克（HF）环模型上进行了测试。在本文中，该方法首先在二维HF模型上进行测试，我们发现前置因子的幂级数展开收敛到精确解的速度极慢。相反，我们提出了另一种更强大的方法，即基于前置因子的连分数展开，它收敛到精确解的速度要快得多。该方法通过二维HF模型电子密度的计算进行了说明。它为处理由大量强局域化电子群组成的扩展系统提供了一种强大的技术。
