# 瓦尼尔函数

## 定义

${\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{\mathbf {k} }(\mathbf {r} )}$

${\displaystyle \phi _{\mathbf {R} }(\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{\mathbf {k} }e^{-i\mathbf {k} \cdot \mathbf {R} }\psi _{\mathbf {k} }(\mathbf {r} )}$ ,
• R 表示任意格矢（即对于每一布拉菲格矢都有一与其对应的瓦尼尔函数）；
• ${\displaystyle N}$  为晶格中原胞的数量；
• k 的求和包含布里渊区（或倒易点阵中满足周期性边界条件的原胞）中的 ${\displaystyle N}$  个不同的 k，均匀地分布在整个布里渊区内。由于 ${\displaystyle N}$  的值通常较大，为了简化运算会使用如下关系来把此求和化为积分：
${\displaystyle \sum _{\mathbf {k} }\longleftarrow {\frac {N}{\Omega }}\int _{\text{BZ}}d^{3}\mathbf {k} }$

### 性质

• 对于任意格矢 R'
${\displaystyle \phi _{\mathbf {R} }(\mathbf {r} )=\phi _{\mathbf {R} +\mathbf {R} '}(\mathbf {r} +\mathbf {R} ')}$

${\displaystyle \phi (\mathbf {r} -\mathbf {R} ):=\phi _{\mathbf {R} }(\mathbf {r} )}$
• 借助瓦尼尔函数，布洛赫函数可被写作如下形式：
${\displaystyle \psi _{\mathbf {k} }(\mathbf {r} )={\frac {1}{\sqrt {N}}}\sum _{\mathbf {R} }e^{i\mathbf {k} \cdot \mathbf {R} }\phi _{\mathbf {R} }(\mathbf {r} )}$

• 波函数集 ${\displaystyle \phi _{\mathbf {R} }}$  是一组标准正交基
${\displaystyle \int _{\text{crystal}}\phi _{\mathbf {R} }(\mathbf {r} )^{*}\phi _{\mathbf {R'} }(\mathbf {r} )d^{3}\mathbf {r} ={\frac {1}{N}}\sum _{\mathbf {k,k'} }\int _{\text{crystal}}e^{i\mathbf {k} \cdot \mathbf {R} }\psi _{\mathbf {k} }(\mathbf {r} )^{*}e^{-i\mathbf {k'} \cdot \mathbf {R'} }\psi _{\mathbf {k'} }(\mathbf {r} )d^{3}\mathbf {r} ={\frac {1}{N}}\sum _{\mathbf {k,k'} }e^{i\mathbf {k} \cdot \mathbf {R} }e^{-i\mathbf {k'} \cdot \mathbf {R'} }\delta _{\mathbf {k,k'} }={\frac {1}{N}}\sum _{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {(R'-R)} }=\delta _{\mathbf {R,R'} }}$

## 现代的极化理论

${\displaystyle \mathbf {p_{c}} =-e\sum _{n}\int \ d^{3}r\,\,\mathbf {r} |W_{n}(\mathbf {r} )|^{2}}$

## 参考文献

1. "The structure of electronic excitation levels in insulating crystals," G. H. Wannier, Phys.
2. ^ "Dynamics of Band Electrons in Electric and Magnetic Fields", G. H. Wannier, Rev.
3. Marzari et al.: Exponential localization of Wannier functions in insulators
4. ^ Marzari et al.: An Introduction to Maximally-Localized Wannier Functions
5. A Bohm, A Mostafazadeh, H Koizumi, Q Niu and J Zqanziger. The Geometric Phase in Quantum Systems. Springer. 2003: §12.5, p. 292 ff. ISBN 3-540-00031-3.
6. ^ MP Geller and W Kohn Theory of generalized Wannier functions for nearly periodic potentials Physical Review B 48, 1993
7. ^ W. Kohn, Analytic Properties of Bloch Waves and Wannier Functions, Phys. Rev. 115, 809 (1959)
8. ^ E. Ö. Jónsson, S. Lehtola, M. Puska, and H. Jónsson: Generalized Pipek-Mezey orbital localization method for electronic structure calculations employing periodic boundary conditions
9. ^ Gerd Berghold et al. General and efficient algorithms for obtaining maximally localized Wannier functions
10. ^ SM Nakhmanson et al. Spontaneous polarization and piezoelectricity in boron nitride nanotubes, 2008
11. ^ D Vanderbilt Berry phases and Curvatures in Electronic Structure Theory.
12. ^ C. Pisani. Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society. Springer. 1994: 282. ISBN 3-540-61645-4.