非簡諧振動分析中振動座標的最佳化

Abstract

振動座標選擇能夠大幅影響非簡諧振動分析的收斂情形。理想情況下，我們會希望選擇的振動座標能夠對應到潛在的物理特徵，因而加速計算模擬的進行。在這個研究中，我們探討了在非簡諧振動分析中，幾種不同最佳化座標的方法。

在簡諧振動分析之中，簡正振動模式 (Normal mode) 是最常見且自然的選擇；然而若我們考慮非簡諧的位能面，各個簡正振動模式之間具有很強的耦合，因而使得對應的非簡諧振動計算的收斂十分緩慢。針對這類問題， Head 提出了「部分黑塞振動分析」(Partial Hessian Vibrational Analysis, PHVA) 對一部分原子、或是特定官能機團進行簡諧振動分析，取得對應的振動座標。這個方法在非簡諧振動分析中，能夠比傳統的簡正振動模式具有更好的收斂性。然而，這個方法需要人去手動將分子系統劃分區域，而這在許多系統中可能很難有效達成。參考軌域的局域化的方法，另一種更自動的座標選擇是「局部模式座標」(Local Mode Coordinates)。局部模式座標往往只涉及特定片段中的少數幾個原子的運動；這不僅符合人的直覺，同時使的我們能將對官能基的描述延伸到更大的系統中。然而，過度局域化會使結果嚴重偏離簡諧振動分析，因而使座標遠離其物理特徵。這意味著，最好的座標選擇，很可能介於簡正振動模式與完全局域化的局部模式之間。基於這個想法，1982年Thompson和Truhlar嘗試通過最小化基態能量來獲得最佳化的振動座標；除此之外，Yagi 提出了一種泛用且穩定的最佳化演算法。

在這份研究中，我們以有限基底表徵 (Finite Basis Representation, FBR) 描述振動波函數，基於雅可比掃描 (Jacobi Sweep) 與牛頓法進行最佳化的程序，透過變分原理來選擇使基態能量最小的座標。我們透過振動構型相互作用（VCI）和離散變量表示（DVR）的計算，對幾個氫鍵團塊進行了測試，以衡量該座標選擇的優勢。

The performance of reduced dimensional anharmonic vibrational calculations depends on the choice of vibrational coordinates. Ideally, the coordinates should be chosen to capture the underlying physics, interpret the vibrational features, and facilitate the computational simulations. In this study, we investigated different ideas on optimizing coordinates for anharmonic vibrational analysis. Normal mode coordinates are the most common choice for vibrational problems, however, for an anharmonic potential, the normal mode coordinates possess strong coupling constants among the modes and give slow convergence in n-mode potential representation and anharmonic calculations. One method of localizing the modes in specific functional groups is Partial Hessian vibrational analysis (PHVA), proposed by Head, which showed faster convergence compared with normal mode coordinates. However, this method is based on one’s chemical intuition, and it may be straightforward only for particular systems. Another more automatic approach which borrows ideas from orbital localization techniques is local mode coordinates. Localized modes tend to involve only a few atom movements in identifiable fragments, which not only follows our intuition but also means that the description of functional groups can be usefully transferable to understand bigger systems. Though, over-localization could deviate substantially from the harmonic picture and produce an unphysical representation. It implies that the optimal coordinates should be somewhere between fully localized and fully delocalized coordinates. The idea of optimizing coordinates by minimizing ground state energy dates back to early work from Thompson and Truhlar in 1982; furthermore, a robust and general optimization algorithm was proposed by Yagi. Our approach used the wavefunction as the product of one-dimensional solutions in the finite basis representation (FBR). The variational principle was applied to choose the coordinates that minimize the ground state energy. The procedure to optimize was based on a combination of the Jacobi sweep and Newton method. Several hydrogen-bonded clusters were tested to benchmark the advantages of this scheme in the vibrational configuration interaction (VCI) and discrete variable representation (DVR) calculations.