[Pw_forum] fermi level(XQ Wang)

wangxinquan wangxinquan at tju.edu.cn
Sat Aug 16 09:57:06 CEST 2008

Dear Stefano, Eyvaz and Lazaro,

    I deeply appreciate your kindly help. I must be cautious in understanding 
the fundamental concepts of quantum mechanics indeed. I need spend
more time to master them as Stefano suggested. Although they are rather abstract
for me, I hope I can apprehend their physical essence. 

wrote by Stefano B:
>1) The eigenstates of the Hamiltonian of a system of non-interacting  
>electrons can ALWAYS be chosen as antisymmetrized products ("Slater  
>deteminants") of one-particle wavefunctions ("molecular orbitals" in  
>quantum chemistry, "Bloch states" in solid-state physics) which are  
>eigenfunctions of a one-electron Hamiltonian. Pay attention to the  
>conceptually simple, trivial, but often overlooked, difference between  
>the many-body Hamiltonian (with its-own eigenfunctions and  
>eigenstates) and the one-particle Hamiltonian. The many-body energy is  
>simply the sum of the one-electron energies of all the molecular  
>orbitals whose product is the many-body eigenstate.
>2) The antisymmetric nature of the many-body wavefunctions is such  
>that, if you construct  a product of a set of functions where two of  
>them are equal, the result will vanish. This is the PAULI PRINCIPLE.  
>No two electrons can occupy the same one-electron state.
>3) Because of (2), the lowest possible many-body energy (ground state)  
>is the sum of the lowest N one-particle energy eigenvalues (N being  
>the number of electrons). The highest occupied one-electron energy  
>level is the difference between the ground state energy of the system  
>with N electrons and that with N-1 electrons (ionization potential).  
>The lowest unoccupied energy is the difference between the ground  
>states with N+1 and N electrons (electron affinity).
The occupation of electrons depends on the sequence of their energies. 
>4) For any finite system (as well as for insulating infinite ones) the  
>electron affinity is different from the ionization potential. For  
>(infinite) metals, they coincide and the define the Fermi energy: by  
>definition, the energy necessary to add or to remove an electron from  
>the system (in classical thermodynamics this same quantity is called  
>the chemical potential). For insulators, it is not that the Fermi  
>energy "does not exist". Only, it is ill-defined it the zero- 
>temperature limit (it can be assumed to take any value between the  
>electron affinity and the ionization potential). At any finite  
>temperature, thermodynamic considerations remove this indeterminacy.
I need more time to understand these comments. 

>> The electron states(eigenvalue of the density matrix)

>what a mess, here! electron states, if ever, may be eigenSTATES of a  
>quantum operator, not eigenVALUES. it is true that for independent  
>electrons the "electron states" (i.e. the eigensSTATES of the one- 
>particle Hamiltonian) are also eigenstates of the one-particle density  
>matrix, but the viceversa is not true. Being an eigenstate of the  
>density matrix is not a sufficient condition for being a legitamate  
>"electron state" (in the sense of being an eigenstate of the  
>Hamiltonian). this is so because the density matrix is a projector,  
>whose eigenvalues (0 and 1) are highly degenerate ...
I'm sorry for my terrible opinion. eigenSTATE should be the 
eigenvector of matrix (quantum operator) and eigenVALUE should
be the mean value of quantum operator. I have read some quantum
mechanics textbooks as Eyvaz suggested (C.KITTEL), unfortunately 
I fail to make a connection between the book and application.  

>> My brain doesn't work.

>take it easy. you are probably one of the many victims of the modern  
>tendency to study advanced (at times, very advanced) topics without  
>having properly understood the fundamentals. as trivial as these  
>fundamentals may be, it takes time to master them. I am sure it is not  
>your fault.
In fact the chemical process is my interesting and the physical analysis 
are the most powerful and essential idea to get a deep insight into the 
mechanism of reactions. Honest to myself, the output information(
local relaxtion, electron state around fermi level and redox energy) are 
important to me. However the results largely depends on the input which 
can only be setted exactly by a systematic understanding of the quantum mechanic 
theory. (To what extent of the knowledge should I achieve in quantum mechanics 
field is another problem which has confused me for a long time...) Thanks again 
to Stefano for giving me confidence. I will try my best.

  wrote by Lazaro:
>... ... ...
>I think it would be better to keep the definition of Fermi energy
>separated from the definition of chemical potential. If we stick to the
>definition of the Fermi energy as the energy of the highest occupied
>electron state at T=0K then there is a Fermi energy in semiconductors,
>that is the top of the valence band, and a chemical potential somewhere
>in the gap depending on the temperature. If the Fermi energy is defined
>as a value of the energy that divides occupied and empty states (as in
>Ashcroft & Mermin I think) then any value in the gap could be taken as a
>Fermi energy.
I found that it is the clearest explanation of fermi energy. I deeply agree with you.
At last I deeply appreciate all your kindly helps once more.

Best regards,
XQ Wang
X.Q. Wang 
wangxinquan at tju.edu.cn
School of Chemical Engineering and Technology
Tianjin University
92 Weijin Road, Tianjin, P. R. China
tel:86-22-27890268, fax: 86-22-27892301
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