ABSTRACT For molecules with constraints such as fixed lengths and angles, it is more efficient to consider the molecular movement in the space of generalised internal coordinate than in Cartesian coordinate. This paper presents a new framework in the simulation of molecular movement, especially for macro-molecules with massive length and angle constraints. The generalised forces are calculated to invert the dense mass matrix for integrating the constraints equation. The inverting of the mass matrix was made based on distance descending ordering method. The method does an reordering of the internal variables to make the Cholesky decomposition need no fill-in, which promises the time complexity in doing mass matrix inverting. The method was extended for application to loop structures. It is found that the mass matrix would become singular when the bond angles approach 0 or π; thus a rotation convention switch method was proposed to resolve the singularity. The time complexity has been demonstrated and the length and angle constraints can be arbitrarily applied. The long-time energy conservation in NVE ensemble was compared for results from symplectic and non-symplectic time integrators. The non-symplectic fourth-order Runge–Kutta method still has satisfactory long-time energy conservation if using small time step.

中文翻译：

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适用于具有任意约束和几何形状的分子的内坐标分子动力学框架

摘要对于具有固定长度和角度等约束条件的分子，在广义内坐标空间中考虑分子运动比在笛卡尔坐标空间中更有效。本文提出了一种模拟分子运动的新框架，特别是对于具有大量长度和角度限制的大分子。计算广义力以对密集质量矩阵求逆以对约束方程进行积分。质量矩阵的求逆是基于距离降序的方法进行的。该方法对内部变量进行重新排序，使 Cholesky 分解不需要填充，这保证了进行质量矩阵求逆的时间复杂度。该方法被扩展用于循环结构的应用。发现当键角接近0或π时，质量矩阵会变得奇异；因此提出了一种旋转约定转换方法来解决奇点问题。已经证明了时间复杂度，并且可以任意应用长度和角度约束。将 NVE 系综中的长时间能量守恒与辛时间积分器和非辛时间积分器的结果进行比较。非辛四阶 Runge-Kutta 方法在使用小时间步长的情况下仍然具有令人满意的长时间能量守恒。将 NVE 系综中的长时间能量守恒与辛时间积分器和非辛时间积分器的结果进行比较。非辛四阶 Runge-Kutta 方法在使用小时间步长的情况下仍然具有令人满意的长时间能量守恒。将 NVE 系综中的长时间能量守恒与辛时间积分器和非辛时间积分器的结果进行比较。非辛四阶 Runge-Kutta 方法在使用小时间步长的情况下仍然具有令人满意的长时间能量守恒。