赝势

范数守恒赝势和超软赝势

范数守恒赝势

${\displaystyle {\hat {V}}_{\textit {ps}}(r)=\sum _{l}\sum _{m}|Y_{lm}\rangle V_{lm}(r)\langle Y_{lm}|}$

1. 临界半径 ${\displaystyle r_{c}}$ 内，每一伪波函数的范数需与其所对应的全电子波函数相同，即[4]

${\displaystyle \int _{r

2. 全电子波函数和伪波函数在临界半径 ${\displaystyle r_{c}}$ 外需要完全一致。

超软赝势

${\displaystyle q_{\mathbf {R} ,ij}=\langle \phi _{\mathbf {R} ,i}|\phi _{\mathbf {R} ,j}\rangle -\langle {\tilde {\phi }}_{\mathbf {R} ,i}|{\tilde {\phi }}_{\mathbf {R} ,j}\rangle }$,

${\displaystyle {\hat {H}}|\Psi _{i}\rangle =\epsilon _{i}{\hat {S}}|\Psi _{i}\rangle }$,

${\displaystyle {\hat {S}}=1+\sum _{\mathbf {R} ,i,j}|p_{\mathbf {R} ,i}\rangle q_{\mathbf {R} ,ij}\langle p_{\mathbf {R} ,j}|}$,

${\displaystyle p_{\mathbf {R} ,i}}$ 是在截断频率内通过赝参照态（pseudo reference state）形成对偶空间的投影（projector），在截断频率外取的值为零：

${\displaystyle \langle p_{\mathbf {R} ,i}|{\tilde {\phi }}_{\mathbf {R} ,j}\rangle _{r.

（PAW）与此相关[6]

费米赝势

${\displaystyle V(r)={\frac {4\pi \hbar ^{2}}{m}}b\,\delta (r)}$,

参考文献

1. ^ Schwerdtfeger, P., The Pseudopotential Approximation in Electronic Structure Theory, ChemPhysChem, August 2011, doi:10.1002/cphc.201100387
2. ^ M. L. Cohen, J. R. Chelikowsky, "Electronic Structure and Optical Spectra of Semiconductors", (Springer Verlag, Berlin 1988)
3. ^ Hamann, D. R.; Schlüter, M.; Chiang, C. Norm-Conserving Pseudopotentials. Physical Review Letters. 1979-11-12, 43 (20): 1494–1497. doi:10.1103/PhysRevLett.43.1494.
4. ^ Bachelet, G. B.; Hamann, D. R.; Schlüter, M., Pseudopotentials that work: From H to Pu, Physical Review B (American Physical Society), October 1982, 26 (8): 4199–4228, Bibcode:1982PhRvB..26.4199B, doi:10.1103/PhysRevB.26.4199
5. ^ Vanderbilt, David, Physical Review B (American Physical Society), April 1990, 41 (11): 7892–7895, Bibcode:1990PhRvB..41.7892V, doi:10.1103/PhysRevB.41.7892 已忽略未知参数| title= (帮助); 缺少或|title=为空 (帮助)
6. ^ Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. 1999. Bibcode:1999PhRvB..59.1758K. doi:10.1103/PhysRevB.59.1758.
7. ^ E. Fermi, Motion of neutrons in hydrogenous substances, Ricerca Scientifica, July 1936, 7: 13–52
8. ^ Squires, Introduction to the Theory of Thermal Neutron Scattering, Dover Publications (1996) ISBN 0-486-69447-X