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'''库普曼斯定理'''（'''Koopmans' theorem'''）表明，在闭壳层的[[哈特里－福克方程|哈特里－福克]]近似下，一个体系的第一电离能等于其最高占据轨道（[[HOMO]]）的能量的负值，这个定理是1934年[[佳林·库普曼斯]]提出的。<ref>{{cite journal
  | last = Koopmans
  | first = Tjalling
  | title = Über die Zuordnung von Wellenfunktionen und Eigenwerten zu den Einzelnen Elektronen Eines Atoms
  | journal = Physica
  | year = 1934
  | volume = 1
  | issue = 1–6
  | pages = 104–113
  | publisher = Elsevier
  | doi = 10.1016/S0031-8914(34)90011-2
| bibcode=1934Phy.....1..104K}}
</ref>库普曼斯于1975年获得了[[诺贝尔奖]]，但不是在物理或化学领域，而是在[[经济学|诺贝尔经济学奖]]领域。

在哈特里－福克理论的框架之中，并引入电离前后轨道不发生改变这一假设后，库普曼斯定理精确成立。通过这一方法计算得到的电离能与实验结果基本一致，对于小分子来说，误差一般在 2 电子伏特以内。<ref name="politzer">{{cite journal|last1=Politzer|first1=Peter|first2=Fakher |last2=Abu-Awwad|year=1998|title=A comparative analysis of Hartree–Fock and Kohn–Sham orbital energies|journal=Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta)|volume=99|issue=2|pages=83–87|doi=10.1007/s002140050307}}</ref><ref name="hamel">{{cite journal|last1=Hamel|first1=Sebastien|first2=Patrick|last2= Duffyc|first3=Mark E. |last3=Casidad |first4= Dennis R. |last4=Salahub|year=2002|title=Kohn–Sham orbitals and orbital energies: fictitious constructs but good approximations all the same |journal=Journal of Electron Spectroscopy and Related Phenomena|volume=123|issue=2–3|pages=345–363|doi=10.1016/S0368-2048(02)00032-4}}</ref><ref>例如，可参见 {{Cite book|first1=A. |last1=Szabo |first2= N. S.|last2= Ostlund|title=Modern Quantum Chemistry|chapter=Chapter 3|isbn=0-02-949710-8}}</ref>因此，Koopmans定理的有效性与HF波函数的准确性密切相关。{{Citation needed|reason=Correlating the wave function with the HOMO seems to need more explanation than simply therefore|date=April 2009}}两个主要的误差来源为：


* '''轨道弛豫'''，指体系中的电子数目发生变化时，[[福克算符]]与哈特里-福克轨道的变化；

* '''[[强关联|电子相关作用]]'''，指的是在哈特里-福克近似中，用自洽场的福克算符的单电子本征函数（分子轨道）的[[斯莱特行列式]]来描述多体波函数时引入的误差

将实验结果与高级从头算方法的计算结果比较的经验表明，在大部分情况下，上述两部分误差相互抵消，但并不总是这样。{{Citation needed|date=April 2009}}

在[[密度泛函理论]]中有着将第一电离能与电子亲和能分别与HOMO/LUMO的单电子轨道能关联起来的类似结论，尽管其推导过程与确切表述都与原始的库普曼斯定理有差异。通常，从密度泛函理论轨道能计算得到的单电子轨道能误差较大，可远大于2电子伏特，并与交换相关势的形式相关。<ref name="politzer"/><ref name="hamel"/>在通常的近似下，DFT方法得到的LUMO轨道能与电子亲和能的相关性并不好。<ref>{{cite journal|last1=Zhang|first1=Gang|first2=Charles B. |last2=Musgrave|year=2007|title=Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations|journal=The Journal of Physical Chemistry A|volume=111|issue=8|pages=1554–1561|doi=10.1021/jp061633o|pmid=17279730}}</ref>在密度泛函理论的库普曼斯定理中，误差的主要来源是交换相关能量泛函的不准确性，这点与哈特里－福克近似不同，因此在改进计算结果上有较大的提高空间。

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==Koopmans定理的推广==
Koopmans定理的原始形式只适用于描述闭壳层HF波函数与第一电离能的关系，经推广后则成为从轨道能计算电子数目发生改变的过程中的能量变化的一种方法。
==Generalizations of Koopmans' theorem==
While Koopmans' theorem was originally stated for calculating ionization energies from restricted (closed-shell) Hartree-Fock wavefunctions, the term has since taken on a more generalized meaning as a way of using orbital energies to calculate energy changes due to changes in the number of electrons in a system.

===基态与激发态离子===
===Ground-state and excited-state ions===
Koopmans’ theorem applies to the removal of an electron from any occupied molecular orbital to form a positive ion. Removal of the electron from different occupied molecular orbitals leads to the ion in different electronic states. The lowest of these states is the ground state and this often, but not always, arises from removal of the electron from the HOMO. The other states are excited electronic states.

For example the electronic configuration of the H<sub>2</sub>O molecule is (1a<sub>1</sub>)<sup>2</sup> (2a<sub>1</sub>)<sup>2</sup> (1b<sub>2</sub>)<sup>2</sup> (3a<sub>1</sub>)<sup>2</sup> (1b<sub>1</sub>)<sup>2</sup>,<ref name=Levine>{{cite book|last=Levine|first= I. N.|title=Quantum Chemistry|url=https://archive.org/details/modeselectiveche0024jeru|edition=4th |publisher= Prentice-Hall|year= 1991|page=[https://archive.org/details/modeselectiveche0024jeru/page/475 475]|isbn=0-7923-1421-2}}</ref>  where the symbols a<sub>1</sub>, b<sub>2</sub> and b<sub>1</sub> are orbital labels based on [[molecular symmetry]]. From Koopmans’ theorem the energy of the 1b<sub>1</sub> HOMO corresponds to the ionization energy to form the H<sub>2</sub>O<sup>+</sup> ion in its ground state (1a<sub>1</sub>)<sup>2</sup> (2a<sub>1</sub>)<sup>2</sup> (1b<sub>2</sub>)<sup>2</sup> (3a<sub>1</sub>)<sup>2</sup> (1b<sub>1</sub>)<sup>1</sup>. The energy of the second-highest MO 3a<sub>1</sub> refers to the ion in the excited state (1a<sub>1</sub>)<sup>2</sup> (2a<sub>1</sub>)<sup>2</sup> (1b<sub>2</sub>)<sup>2</sup> (3a<sub>1</sub>)<sup>1</sup> (1b<sub>1</sub>)<sup>2</sup>, and so on. In this case the order of the ion electronic states corresponds to the order of the orbital energies. Excited-state ionization energies can be measured by [[Ultraviolet photoelectron spectroscopy|photoelectron spectroscopy]].

For H<sub>2</sub>O, minus the near-Hartree-Fock orbital energies of these orbitals in [[Electronvolt|eV]], are 1a<sub>1</sub> 559.5, 2a<sub>1</sub> 36.7 1b<sub>2</sub> 19.5, 3a<sub>1</sub> 15.9 and 1b<sub>1</sub> 13.8. The corresponding ionization energies are 539.7, 32.2, 18.5, 14.7 and 12.6 eV.<ref name=Levine/> As explained above, the deviations are due to the effects of orbital relaxation as well as differences in electron correlation energy between the molecular and the various ionized states.

===Koopmans' theorem for electron affinities===
It is sometimes claimed<ref>See, for example, {{Cite book|first1=A. |last1=Szabo |first2= N. S.|last2= Ostlund|title=Modern Quantum Chemistry|page=127|isbn=0-02-949710-8}}</ref> that Koopmans' theorem also allows the calculation of [[electron affinity|electron affinities]] as the energy of the lowest unoccupied molecular orbitals ([[LUMO]]) of the respective systems. However, Koopmans' original paper makes no claim with regard to the significance of eigenvalues of the [[Fock matrix|Fock operator]] other than that corresponding to the [[HOMO]]. Nevertheless, it is straightforward to generalize the original statement of Koopmans' to calculate the [[electron affinity]] in this sense.

Calculations of electron affinities using this statement of Koopmans' theorem have been criticized<ref>{{cite book
 | last1 = Jensen
 | first1= Frank
 | title= Introduction to Computational Chemistry
 | url = https://archive.org/details/introductiontoco00fran_327
 | year = 1990
 | publisher = Wiley
 | pages = [https://archive.org/details/introductiontoco00fran_327/page/n39 64]–65
 | isbn = 0-471-98425-6}}</ref> on the grounds that virtual (unoccupied) orbitals do not have well-founded physical interpretations, and that their orbital energies are very sensitive to the choice of basis set used in the calculation.  As the basis set becomes 
more complete; more and more "molecular" orbitals that are not really ''on'' the molecule of interest will appear, and care must be taken not to use 
these orbitals for estimating electron affinities.

Comparisons with experiment and higher-quality calculations show that electron affinities predicted in this manner are generally quite poor.

===Koopmans' theorem for open-shell systems===
Koopmans' theorem is also applicable to open-shell systems.  It was previously believed that this was only in the case for removing the unpaired electron,<ref>{{cite book
 | last1 = Fulde
 | first1 = Peter
 | title = Electron correlations in molecules and solids
 | url = https://archive.org/details/electroncorrelat00fuld_068
 | year = 1995
 | publisher = Springer
 | pages = [https://archive.org/details/electroncorrelat00fuld_068/page/n39 25]–26
 | isbn = 3-540-59364-0}}</ref> but
the validity of Koopmans’ theorem for ROHF in general has been proven  provided that the correct orbital energies are used.<ref>
{{cite journal | doi = 10.1063/1.2393223 | title = Koopmans' theorem in the ROHF method: Canonical form for the Hartree-Fock Hamiltonian | year = 2006 | last1 = Plakhutin | first1 = B. N. | last2 = Gorelik | first2 = E. V. | last3 = Breslavskaya | first3 = N. N. | journal = The Journal of Chemical Physics | volume = 125 | pages = 204110 | pmid = 17144693 | issue = 20 | bibcode=2006JChPh.125t4110P}}
</ref><ref>
{{cite journal | doi = 10.1063/1.3418615 | title = Koopmans's theorem in the restricted open-shell Hartree–Fock method. II. The second canonical set for orbitals and orbital energies | year = 2010 | last1 = Davidson | first1 = Ernest R. | last2 = Plakhutin | first2 = Boris N. | journal = The Journal of Chemical Physics | volume = 132 | issue = 18 | pages = 184110 | bibcode=2010JChPh.132r4110D}}
</ref><ref>
{{cite journal | doi = 10.1021/jp9002593 | title = Koopmans' Theorem in the Restricted Open-Shell Hartree−Fock Method. 1. A Variational Approach† | year = 2009 | last1 = Plakhutin | first1 = Boris N. | last2 = Davidson | first2 = Ernest R. | journal = The Journal of Physical Chemistry A | volume = 113 | pages = 12386–12395 | pmid = 19459641 | issue = 45 | bibcode=2009JPCA..11312386P}}
</ref><ref>
{{cite journal | last1 = Glaesemann | first1 = Kurt R. | last2 = Schmidt | first2 = Michael W. | title = On the Ordering of Orbital Energies in High-Spin ROHF† | journal = The Journal of Physical Chemistry A | volume = 114 | issue =33 | pages = 8772–8777 | year = 2010 | pmid = 20443582 | doi = 10.1021/jp101758y}}
</ref>
The spin up (alpha) and spin down (beta) orbital energies do not necessarily have to be the same.<ref>
{{cite journal|last1=Tsuchimochi|first1=Takashi|last2=Scuseria|first2=Gustavo E.|title=Communication: ROHF theory made simple|journal=The Journal of Chemical Physics|volume=133|issue=14|pages=141102|year=2010|pmid=20949979|doi=10.1063/1.3503173|bibcode = 2010JChPh.133n1102T }}
</ref>

==Counterpart in density functional theory==
In [[density functional theory]] (DFT) a similar theorem exists that relates the first ionization energy and electron affinity to the HOMO and LUMO energies. This is sometimes called the '''DFT-Koopmans' theorem'''. More generally, for a fixed geometry defined by the ''N''-electron system, the HOMO energy is equal to the ionization energy of the ''N''-electron system when the total number of electrons is in the range ''N''&nbsp;−&nbsp;δ''N'' for 0&nbsp;&lt;&nbsp;δ''N''&nbsp;&lt;&nbsp;1, and is equal to the electron affinity when the total number of electrons is in the range ''N''&nbsp;+&nbsp;δ''N'' for 0&nbsp;&lt;&nbsp;δ''N''&nbsp;&lt;&nbsp;1 (with the δ''N'' occupying what is the LUMO of the ''N''-electron state).<ref name="perdew82">{{cite journal
  |last1=Perdew
  |first1=John P.
  |first2=Robert G. |last2=Parr|first3=Mel |last3=Levy|first4=Jose L.|last4= Balduz, Jr.
  |year=1982
  |title=Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy
  |journal=Physical Review Letters
  |volume=49
  |issue=23
  |pages=1691–1694
  |doi=10.1103/PhysRevLett.49.1691 |bibcode=1982PhRvL..49.1691P}}</ref><ref>{{cite journal
  |last1=Perdew
  |first1=John P.
  |first2=Mel|last2= Levy
  |year=1997
  |title=Comment on "Significance of the highest occupied Kohn–Sham eigenvalue"
  |journal=Physical Review B
  |volume=56
  |issue=24
  |pages=16021–16028
  |doi=10.1103/PhysRevB.56.16021|bibcode = 1997PhRvB..5616021P }}</ref> While these are exact statements in the formalism of DFT, the use of approximate exchange-correlation potentials makes the calculated energies approximate. As in HF theory, the electron affinity calculated in this way is less accurate than the ionization energy.

A proof of the DFT counterpart of Koopmans' theorem usually employs [[Janak's theorem]]: that the derivative of the total DFT energy, ''E'', with respect to the occupation of a given orbital, ''n<sub>i</sub>'' is equal to the corresponding orbital energy, ''ε<sub>i</sub>'':<ref>{{cite journal
 | title = Proof that ∂''E'' / ∂''n<sub>i</sub>'' = ''ε<sub>i</sub>'' in density-functional theory
 | journal = Physical Review B
 | volume = 18
 | issue = 12
 | year = 1978
 | pages = 7165–7168
 | doi = 10.1103/PhysRevB.18.7165
 | first = J. F. |last=Janak 
|bibcode = 1978PhRvB..18.7165J }}</ref>
<ref>
{{cite book 
 | title = Advances in Quantum Chemistry: v. 6
 | first = J. C. |last=Slater
 | editor-first = Per-Olov |editor-last=Lowdin
 | year = 1972
 | isbn = 978-0-12-034806-0
 | publisher = Academic Press
 | pages = 1
}} </ref>
:<math>\frac{\partial E}{\partial n_i}=\epsilon_i.</math>


A tuning procedure of density functional theory approximation imposes the Koopmans' theorem on DFT approximations.
<ref>{{cite journal | title = Koopmans' springs to life | journal = The Journal of Chemical Physics | volume = 131 | issue = 23
 | year = 2009 | pages = 231101–4 | doi = 10.1063/1.3269030 | first1 = U. |last1 = Salzner  | first2 = R. |last2 = Baer }}{{cite journal | title = "Tuned" Range-separated hybrids in density functional theory | journal = Annual Review of Physical Chemistry | volume = 61 | year = 2010 | pages = 85–109 | doi = 10.1146/annurev.physchem.012809.103321 | first1 = R. |last1 = Baer | first2 = E. |last2 = Livshits | first3 = U. |last3 = Salzner | pmid = 20055678 }}{{cite journal | title = Fundamental gaps of finite systems from the eigenvalues of a generalized Kohn-Sham method | journal = Physical Review Letters | volume = 105 | year = 2010 | pages = 266802 | doi = 10.1103/PhysRevLett.105.266802 | first1 = T. |last1 = Stein  | first2 = H. |last2 = Eisenberg | first3 = L. |last3 = Kronik | first4 = R. |last4 = Baer | issue = 26 | pmid = 21231698 }}</ref>
-->

==参考文献==
{{reflist}}

==外部链接==
*{{cite web|url=http://www.chemistry.emory.edu/faculty/bowman/old_classes/chem531/lectures/koopman's_theorem.pdf |title=Emory大学上库普曼斯定理的介绍 |first=Joel |last=Bowman |deadurl=yes |archiveurl=https://web.archive.org/web/20050930115041/http://www.chemistry.emory.edu/faculty/bowman/old_classes/chem531/lectures/koopman%27s_theorem.pdf |archivedate=2005-09-30 }}
*{{cite web |url=http://nobelprize.org/nobel_prizes/economics/laureates/1975/koopmans-autobio.html |title=库普曼斯传 |publisher=诺贝尔基金会 |year=1975 |accessdate=2012-06-17 |archive-date=2012-04-08 |archive-url=https://web.archive.org/web/20120408174618/http://www.nobelprize.org/nobel_prizes/economics/laureates/1975/koopmans-autobio.html |dead-url=no }}



[[Category:量子化学]]
[[Category:计算化学]]
[[Category:理论化学]]