HOMO-LUMO gaps and excitation energies

sobereva

转载自http://chemical-quantum-images.blogspot.com/2013/06/why-is-homo-lumo-gap-not-good-guess-of.html
这是篇不错的博文，讨论了在HF/CIS框架下，HOMO-LUMO gap和激发能的关系（假设电子激发100%由HOMO->LUMO跃迁来描述）。可见，激发能除了gap外，还要考虑HOMO和LUMO间的库仑积分和交换积分（有时称为激子结合能）。由于这个原因，CIS算的激发能是要低于gap的。

HOMO-LUMO gaps and excitation energies

Why is the HOMO-LUMO gap not a good guess of the excitation energy? The HOMO energy is a good guess for electron detachment and (under certain idealized conditions) the LUMO for electron attachment. Excitation can formally be considered as an electron detachment with a subsequent attachment, which makes a consideration of the HOMO-LUMO gap sound reasonable. But the difference is that the LUMO energy is initially determined under the condition that the HOMO is occupied. To get the correct first order excitation energy one has to subtract the "electron-hole interaction" or the "exciton binding energy" (from last post).

The Hartree-Fock energy in terms of the occupied MOs (indexed i,j) of the neutral system is given by



If we remove an electron out of orbital k (for example the HOMO but this holds for any occupied orbital), the new total energy becomes



The detachment energy (or ionization potential) is simply defined as the difference



And considering the permutation symmetry of two-electron integrals this reduces to


which is nothing but the negative orbital energy of orbital k with respect to the original n-electron system. This is just Koopmans' theorem.

What happens if we take away an electron from the occupied orbital k and put it into the virtual orbital a? Then the energy of the resulting Slater determinant is given by




Subtracting the ground state Hartree-Fock energy from this (and considering permutation symmetry)




Part of this is of course just εk. But the other half is not exactly εa, which also includes the interaction with orbital k


However, we can still write the first order expression for the excitation energy in the following way


First, there is the orbital energy gap, as we expected. But second also Coulomb and exchange integrals between the occupied and virtual orbitals have to be considered, yielding a term, which can be identified with the exciton binding energy from last post. The first part can be identified with an attractive Coulomb interaction between the "electron" and the "hole", the second one with an exchange interaction, which is present for singlet excited states (or more generally if the ground and excited states are of the same multiplicity).

Since the Coulomb integral is larger than the exchange integral (which I believe it is always) the first order correction term is negative and the excitation energy is lower than the gap between the corresponding orbitals. The first order energy (i.e. CIS) is usually still too high because it neglects orbital relaxation in the excited state.

For DFT the story is different because of self-interaction and these rigorous equalities do not hold. On the one hand this is part of the reason why TDDFT is so unexpectedly good in many cases. On the other hand the failure to rigorously include the electron-hole interaction is the reason why TDDFT fails so badly for charge transfer states and does not retain the correct 1/r asymptotics [see e.g. this Ref.].

currynar

这篇讲解真的很有趣，非常感谢。我最近刚开始涉入这一领域，正好遇到一个组奇怪的数据，如下图：
根据我合作导师的讲法，excitonic和quasi gap分别对应的是HL gap和激子能。那么结合我看过这篇博文后的理解，图中这组数相等，就是说12个原子的碳链它的激子能对应与相同方法计算的无限长碳链的HL gap相等。那么请问sobereva大这之间是相差了库仑积分还是相差了激子结合能？这个真的是巧合吗？

sobereva

初初 发表于 2015-1-24 10:54
您好，sob：
(1)假设电子激发100%由HOMO->LUMO跃迁来描述，TD-DFT（垂直激发）算的激发能与E_LUMO - E_HOM ...

那个链接得翻墙
关于TDDFT的情况和更多的讨论，你看看这个文章吧
Chem. Rev. 2005, 105, 4009-4037

初初

您好，sob：
(1)假设电子激发100%由HOMO->LUMO跃迁来描述，TD-DFT（垂直激发）算的激发能与E_LUMO - E_HOMO的大小关系怎样呢？
(2)是否TD-DFT获得的激发能（不在上述的假设下）总是大于HOMO与LUMO间的能隙呢？
(3)关于激子结合能，您能不能多解释下（链接没能打开，可能是我电脑的问题），特别是在TD-DFT下如何处理？

谢谢

小范范1989

好帖子

trewqaz

的确不错

blueyangliu

谢谢，Sob