# A basic question about total energy expression of KS equation

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### Yike Huang

Aug 11, 2021, 5:19:12 PMAug 11
to cp2k
Dear cp2k developers and users,

Recently I plan to study more details of SCF and non-SCF calculation, however when I try to find differences between total energy expression of KS equation and that of Harris-Foulkes Functional, I find total energy of KS equation is not expressed directly as the sum of energies of KS orbitals, instead, there are two more terms substracted from it, i.e., Coulomb potential and the difference between XC energy density and XC chemical potential:

which is obtained from:

Could anybody tell me or share some reading material about why the total energy is not simply the sum of KS orbital energies (only the first term)?

Very best wishes,
Yike

### Fabian

Aug 18, 2021, 8:12:06 AMAug 18
to cp2k
Hello Yike,

Basically, the Harris-Foulkes functional can be seen as first order approximation to the Kohn-Sham functional:
Let's say you know the ground-state density up to a certain delta-rho ,  so that n^out = n^in + delta-rho.
Now you can replace n^out in the KS-functional, expand the expression, and truncate the resulting second order terms in delta-rho.
This results in the appearance of the Coulomb and XC "correction" terms which you are seeing,
and - more importantly - in an energy functional expression which is only dependent on the input density.

A good starting point is the book "Interatomic forces in condensed matter" by Mike Finnis.

Kind regards,
Fabian

### Thomas Kühne

Oct 13, 2021, 6:12:44 PMOct 13
to 'Dorothea Golze' via cp2k
Dear Like Huang,

a good introduction is the PhD thesis of Peter Haynes:
See Eq. 3.23 and 3.24 for the answer to your question. BTW, Eq. 3.25 & 3.26 you have also used
assumes the LDA!

Best,
Thomas Kühne

Am 11.08.2021 um 23:19 schrieb Yike Huang <ykhua...@gmail.com>:

Dear cp2k developers and users,

Recently I plan to study more details of SCF and non-SCF calculation, however when I try to find differences between total energy expression of KS equation and that of Harris-Foulkes Functional, I find total energy of KS equation is not expressed directly as the sum of energies of KS orbitals, instead, there are two more terms substracted from it, i.e., Coulomb potential and the difference between XC energy density and XC chemical potential:

<eq1.png>
which is obtained from:
<eq2.png>

Could anybody tell me or share some reading material about why the total energy is not simply the sum of KS orbital energies (only the first term)?

Very best wishes,
Yike

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<eq1.png><eq2.png>

==============================
Thomas D. Kühne
Dynamics of Condensed Matter
Chair of Theoretical Chemistry
Warburger Str. 100
Germany