使用普通平面波Kohn-Sham方程計算的密度泛函理論（DFT）時，通過理論計算得到的帶隙往往會小於實際實驗得到的帶隙。一般而言，DFT+U可以通過調整U值、雜化密度泛函可以通過調整雜化比例，得到與實驗相符的帶隙，但是這些計算方法需要有經驗依據來設定參數。DFT+U在線性響應的計算方法可以得到符合Koopmans 定理的U值，並能得到合適的帶隙，但是這樣的方法不能用於閉殼系統。GW計算方法會消耗大量計算資源，根據其不同的收斂特徵可分為G0W0、GW0、GW等計算方法，而完全收斂的GW計算則往往得到比實驗值更大的帶隙。WKM（Wannier-Koopmans Method）是一種基於Koopmans定理、wannier函數以及常見的交換關聯泛函（如LDA、GGA等方法）的計算方法。它在半導體、離子晶體、有機共價晶體和二維材料等計算中都可得到與實驗一致的帶隙結果。因此，該計算方法是否有可能在不大量增加計算時間的情況下得到準確的電子結構？
來自美國的勞倫斯伯克利國家實驗室的汪林望博士與北京大學深圳研究生院的潘鋒教授合作，使用WKM方法針對過渡金屬氧化物作了研究。他們在計算中發現，通過去除交換關聯泛函中的d軌道wannier函數與內層電子的相互作用，即可在d0和d10閉殼系統中得到與實驗一致的帶隙大小。而對於開殼系統的過渡金屬氧化物，由於d軌道中佔據的wannier函數與非佔據的wannier函數間相互作用，WKM目前還無法計算得到與實驗一致的帶隙結果，但通過分析，本工作為後續改進WKM方法提供了新思路。
該文近期發表於npj Computational Materials 6: 33 (2020)，英文標題與摘要如下，點擊https://www.nature.com/articles/s41524-020-0302-0可以自由獲取論文PDF。
Wannier Koopmans method calculations for transition metal oxide band gaps
Mouyi Weng, Feng Pan, Lin-Wang Wang
The widely used density functional theory (DFT) has a major drawback of underestimating the band gaps of materials. Wannier–Koopmans method (WKM) was recently developed for band gap calculations with accuracy on a par with more complicated methods. WKM has been tested for main group covalent semiconductors, alkali halides, 2D materials, and organic crystals. Here we apply the WKM to another interesting type of material system: the transition metal (TM) oxides. TM oxides can be classified as either with d0 or d10 closed shell occupancy or partially occupied open shell configuration, and the latter is known to be strongly correlated Mott insulators. We found that, while WKM provides adequate band gaps for the d0 and d10 TM oxides, it fails to provide correct band gaps for the group with partially occupied d states. This issue is also found in other mean-field approaches like the GW calculations. We believe that the problem comes from a strong interaction between the occupied and unoccupied d-state Wannier functions in a partially occupied d-state system. We also found that, for pseudopotential calculations including deep core levels, it is necessary to remove the electron densities of these deep core levels in the Hartree and exchange–correlation energy functional when calculating the WKM correction parameters for the d-state Wannier functions.
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