Department of Chemistry, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Urbana, Illinois 61801, USA.
J Chem Phys. 2012 Jun 21;136(23):234109. doi: 10.1063/1.4729602.
In the size-extensive vibrational self-consistent field (XVSCF) method introduced earlier [M. Keçeli and S. Hirata, J. Chem. Phys. 135, 134108 (2011)], only a small subset of even-order force constants that can form connected diagrams were used to compute extensive total energies and intensive transition frequencies. The mean-field potentials of XVSCF formed with these force constants have been shown to be effectively harmonic, making basis functions, quadrature, or matrix diagonalization in the conventional VSCF method unnecessary. We introduce two size-consistent VSCF methods, XVSCF(n) and XVSCF[n], for vibrationally averaged geometries in addition to energies and frequencies including anharmonic effects caused by up to the nth-order force constants. The methods are based on our observations that a small number of odd-order force constants of certain types can form open, connected diagrams isomorphic to the diagram of the mean-field potential gradients and that these nonzero gradients shift the potential minima by intensive amounts, which are interpreted as anharmonic geometry corrections. XVSCF(n) evaluates these mean-field gradients and force constants at the equilibrium geometry and estimates this shift accurately, but approximately, neglecting the coupling between these two quantities. XVSCF[n] solves the coupled equations for geometry corrections and frequencies with an iterative algorithm, giving results that should be identical to those of VSCF when applied to an infinite system. We present the diagrammatic and algebraic definitions, algorithms, and initial implementations as well as numerical results of these two methods. The results show that XVSCF(n) and XVSCF[n] reproduce the vibrationally averaged geometries of VSCF for naphthalene and anthracene in their ground and excited vibrational states accurately at fractions of the computational cost.
在前面介绍的尺寸扩展振动自洽场 (XVSCF) 方法[M. Keçeli 和 S. Hirata，J. Chem. Phys. 135, 134108 (2011)]中，仅使用了一小部分可以形成连接图的偶数阶力常数来计算扩展总能量和密集跃迁频率。用这些力常数形成的 XVSCF 的平均场势已被证明是有效谐波的，使得传统 VSCF 方法中的基函数、求积或矩阵对角化变得不必要。除了包括由高达 n 阶力常数引起的非谐效应的能量和频率外，我们还引入了两种尺寸一致的 VSCF 方法 XVSCF(n)和 XVSCF[n]，用于振动平均几何形状。该方法基于我们的观察结果，即某些类型的少数奇数阶力常数可以形成与平均场势梯度图同构的开放、连接图，并且这些非零梯度会密集地移动势垒最小值，这被解释为非谐几何校正。XVSCF(n)在平衡几何形状处评估这些平均场梯度和力常数，并准确地但近似地忽略这两个量之间的耦合来估计这种移动。XVSCF[n]用迭代算法求解几何校正和频率的耦合方程，当应用于无限系统时，它给出的结果应该与 VSCF 的结果相同。我们给出了这两种方法的图论和代数定义、算法和初始实现以及数值结果。结果表明，XVSCF(n)和 XVSCF[n]在 Napthalene 和 Anthracene 的基态和激发振动状态下，以计算成本的几分之一准确地再现了 VSCF 的振动平均几何形状。
