Janak’s theorem

Using the fact that the energy is linear with respect to the number of electrons and Janak s theorem [31], the orbital energies of the N—n and N+n electron system become equal to the exact ground state vertical ionization energy and electron affinity, respectively ... 

The first term in this expansion is the ground-state energy Eq = E no). For the second term we use Janak s theorem, Eqn (4) ... 

The first derivatives in a Taylor expansion, similar to Eqn (28), of the energy E with respect to the occupation numbers rii provide the KS-eigenvalues, as stated by Janak s theorem, and the second derivatives ... 

Since the KS-eigenvalues are defined through Janak s theorem as first derivatives of the total DF-energy, the ij-th element of the hardness matrix can be obtained as the first derivative of i with respect to rij ... 

Janak s theorem, valid for general OEL equations when occupation numbers are varied, holds for the UHF theory in the form... 

From Eq. (5.7), and Janak s theorem [185], the contribution of correlation energy to the mean energy of the continuum orbital within the //-matrix boundary is... 

The Janak s theorem (eq.17) and the hardness tensor definition (eq. 20) allows the calculations of Tiij as the first derivative of the Kohn-Sham orbital eigenvalues with respect to the orbital occupation numbers [17] ... 

The validity of Janak s theorem in OFT opens up the prospect of incorporating ab initio methods into Landau theory for applications to strongly correlated systems, especially when combined with the procedure of integrating over fractional occupation numbers to obtain physical energy differences. 

The quantities F(X) can be obtained straightforwardly if the calculations are carried out within the Density Fimctional Theory (DEI) formalism. Indeed, for ionization of an electron from molecular orbital (MO) Kohn-Sham type calculation in DFT, with an occupation 1 - A = 5 in level number k. In the Generalized Transition State (GTS) method, Williams et al. [16] proposed the use of ... 

There is something similar to the Koopmans theorem in the Kohn-Sham (KS) DPT [32], It is Janak s theorem [33], which states that the derivative of the total energy with respect to the occupation number of the KS orbital 4>i is exactly the KS orbital energy ... 

Janak s theorem (Janak 1978) is also one of the most significant theorems having to do with orbital energy. It is noteworthy that the Janak theorem is established not only for the Kohn-Sham and Hartree-Fock equations but also for overall one-electron SCF equations in the form of Eq. (4.7). Let us consider the Kohn-Sham equation in Eqs. (4.6) and (4.10). Introducing the occupation numbers of orbitals, ,, the Kohn-Sham equation is written as... 

By Janak s theorem [51], the highest partly-occupied Kohn-Sham eigenvalue ho equals dE/dN = fj, and so changes discontinuously [49,50] at an integer Z ... 

It can easily be shown that the uGTS approximation deviates from the above equation by only 4/9 and higher order terms. For the ionization of an electron from the th molecular orbital (MO) 4>k, represents the fraction of electron removed from this MO and 9 /9X is the negative of the orbital energy e (X), according to Janak s theorem. 

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