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<DIV><FONT face=Verdana size=2><FONT color=#000080>Dear Stefano, Eyvaz and
Lazaro,</FONT></FONT></DIV>
<DIV><FONT face=Verdana size=2><FONT color=#000080></FONT></FONT> </DIV>
<DIV><FONT face=Verdana size=2><FONT color=#000080> I
deeply appreciate your kindly help. I must be cautious in
understanding </FONT></FONT></DIV>
<DIV><FONT face=Verdana size=2><FONT color=#000080>the fundamental
concepts of quantum mechanics indeed. I need spend</FONT></FONT></DIV>
<DIV><FONT face=Verdana size=2><FONT color=#000080>more time to master them as
Stefano suggested. Although they are rather abstract</FONT></FONT></DIV>
<DIV><FONT color=#000080>for me, I hope I can apprehend their physical
essence. </FONT><FONT face=Verdana size=2></DIV>
<DIV>
<DIV> </DIV>
<DIV>wrote by Stefano B:</DIV>
<DIV>>1) The eigenstates of the Hamiltonian of a system of non-interacting </DIV>
<DIV>>electrons can ALWAYS be chosen as antisymmetrized products ("Slater </DIV>
<DIV>>deteminants") of one-particle wavefunctions ("molecular orbitals" in </DIV>
<DIV>>quantum chemistry, "Bloch states" in solid-state physics) which are </DIV>
<DIV>>eigenfunctions of a one-electron Hamiltonian. Pay attention to the </DIV>
<DIV>>conceptually simple, trivial, but often overlooked, difference between </DIV>
<DIV>>the many-body Hamiltonian (with its-own eigenfunctions and </DIV>
<DIV>>eigenstates) and the one-particle Hamiltonian. The many-body energy is </DIV>
<DIV>>simply the sum of the one-electron energies of all the molecular </DIV>
<DIV>>orbitals whose product is the many-body eigenstate.</DIV>
<DIV>
<DIV><FONT color=#000080>> </FONT></DIV></DIV>
<DIV>>2) The antisymmetric nature of the many-body wavefunctions is such </DIV>
<DIV>>that, if you construct a product of a set of functions where two of </DIV>
<DIV>>them are equal, the result will vanish. This is the PAULI PRINCIPLE. </DIV>
<DIV>>No two electrons can occupy the same one-electron state.</DIV>
<DIV>></DIV>
<DIV>>3) Because of (2), the lowest possible many-body energy (ground state) </DIV>
<DIV>>is the sum of the lowest N one-particle energy eigenvalues (N being </DIV>
<DIV>>the number of electrons). The highest occupied one-electron energy </DIV>
<DIV>>level is the difference between the ground state energy of the system </DIV>
<DIV>>with N electrons and that with N-1 electrons (ionization potential). </DIV>
<DIV>>The lowest unoccupied energy is the difference between the ground </DIV>
<DIV>>states with N+1 and N electrons (electron affinity).</DIV>
<DIV>The occupation of electrons depends on the sequence of their energies.
</DIV>
<DIV> </DIV>
<DIV>>4) For any finite system (as well as for insulating infinite ones) the </DIV>
<DIV>>electron affinity is different from the ionization potential. For </DIV>
<DIV>>(infinite) metals, they coincide and the define the Fermi energy: by </DIV>
<DIV>>definition, the energy necessary to add or to remove an electron from </DIV>
<DIV>>the system (in classical thermodynamics this same quantity is called </DIV>
<DIV>>the chemical potential). For insulators, it is not that the Fermi </DIV>
<DIV>>energy "does not exist". Only, it is ill-defined it the zero- </DIV>
<DIV>>temperature limit (it can be assumed to take any value between the </DIV>
<DIV>>electron affinity and the ionization potential). At any finite </DIV>
<DIV>>temperature, thermodynamic considerations remove this indeterminacy.</DIV>
<DIV>I need more time to understand these comments. </DIV>
<DIV> </DIV>
<DIV>>> The electron states(eigenvalue of the density matrix)</DIV>
<DIV> </DIV>
<DIV>>what a mess, here! electron states, if ever, may be eigenSTATES of a </DIV>
<DIV>>quantum operator, not eigenVALUES. it is true that for independent </DIV>
<DIV>>electrons the "electron states" (i.e. the eigensSTATES of the one- </DIV>
<DIV>>particle Hamiltonian) are also eigenstates of the one-particle density </DIV>
<DIV>>matrix, but the viceversa is not true. Being an eigenstate of the </DIV>
<DIV>>density matrix is not a sufficient condition for being a legitamate </DIV>
<DIV>>"electron state" (in the sense of being an eigenstate of the </DIV>
<DIV>>Hamiltonian). this is so because the density matrix is a projector, </DIV>
<DIV>>whose eigenvalues (0 and 1) are highly degenerate ...</DIV>
<DIV>I'm sorry for my terrible opinion. eigenSTATE should be the </DIV>
<DIV>eigenvector of matrix (quantum operator) and eigenVALUE should</DIV>
<DIV>be the mean value of quantum operator. I have read some quantum</DIV>
<DIV>mechanics textbooks as Eyvaz suggested (C.KITTEL), unfortunately </DIV>
<DIV>I fail to make a connection between the book
and application. </DIV>
<DIV> </DIV>
<DIV>>> My brain doesn't work.</DIV>
<DIV> </DIV>
<DIV>>take it easy. you are probably one of the many victims of the modern </DIV>
<DIV>>tendency to study advanced (at times, very advanced) topics without </DIV>
<DIV>>having properly understood the fundamentals. as trivial as these </DIV>
<DIV>>fundamentals may be, it takes time to master them. I am sure it is not </DIV>
<DIV>>your fault.</DIV>
<DIV><FONT color=#000080>In fact the chemical <FONT face=Verdana
size=2>process is my interesting</FONT></FONT><FONT face=Verdana
size=2><FONT color=#000080> and </FONT></FONT><FONT face=Verdana
size=2><FONT color=#000080>the physical analysis </FONT></FONT>
<DIV><FONT face=Verdana size=2><FONT color=#000080>are the most powerful
and essential idea to </FONT></FONT><FONT face=Verdana size=2><FONT
color=#000080>get a deep insight into the </FONT></FONT></DIV>
<DIV><FONT face=Verdana size=2><FONT color=#000080>mechanism of reactions.
</FONT></FONT><FONT face=Verdana size=2><FONT color=#000080>Honest to
myself, </FONT></FONT><FONT face=Verdana size=2><FONT
color=#000080>the output information(</FONT></FONT></DIV>
<DIV><FONT face=Verdana size=2><FONT color=#000080>local relaxtion, electron
state around fermi level and redox energy) are
<DIV><FONT face=Verdana size=2><FONT color=#000080>important to
me. However the results</FONT></FONT></FONT></FONT><FONT face=Verdana
size=2><FONT color=#000080> </FONT></FONT><FONT face=Verdana size=2><FONT
color=#000080>largely depends on the </FONT></FONT><FONT face=Verdana
size=2><FONT color=#000080>input</FONT></FONT><FONT face=Verdana size=2><FONT
color=#000080> which </FONT></FONT></DIV>
<DIV><FONT face=Verdana size=2><FONT color=#000080>can only be setted exactly
by a systematic understanding </FONT></FONT><FONT face=Verdana size=2><FONT
color=#000080>of the quantum </FONT></FONT><FONT color=#000080>mechanic
</FONT></DIV>
<DIV><FONT color=#000080>theory.</FONT><FONT
color=#000080> (To what </FONT><FONT color=#000080>extent of
the knowledge should I achieve in quantum mechanics </FONT></DIV>
<DIV><FONT color=#000080>field is another </FONT><FONT color=#000080>problem
which has </FONT><FONT color=#000080>confused me for a long time...)
</FONT><FONT color=#000080>Thanks again </FONT></DIV>
<DIV><FONT color=#000080>to Stefano for </FONT><FONT color=#000080>giving
me </FONT><FONT color=#000080>confidence. </FONT><FONT color=#000080>I will
try </FONT><FONT color=#000080>my best.</FONT></DIV></DIV>
<DIV><FONT color=#000080></FONT> </DIV>
<DIV><FONT color=#000080> wrote by Lazaro:</FONT></DIV>
<DIV>>... ... ...</DIV>
<DIV>>I think it would be better to keep the definition of Fermi
energy<BR>>separated from the definition of chemical potential. If we stick
to the<BR>>definition of the Fermi energy as the energy of the highest
occupied<BR>>electron state at T=0K then there is a Fermi energy in
semiconductors,<BR>>that is the top of the valence band, and a chemical
potential somewhere<BR>>in the gap depending on the temperature. If the Fermi
energy is defined<BR>>as a value of the energy that divides occupied and
empty states (as in<BR>>Ashcroft & Mermin I think) then any value in the
gap could be taken as a<BR>>Fermi energy.</DIV>
<DIV>I found that it is the clearest explanation of fermi energy. I deeply agree
with you.</DIV>
<DIV>At last I deeply appreciate all your kindly helps once more.<IMG
src="cid:__0@Foxmail.net"></DIV>
<DIV> </DIV>
<DIV><FONT color=#000080>Best regards,</FONT></DIV>
<DIV><FONT color=#000080>XQ Wang</FONT></DIV>
<DIV><FONT color=#000080>
<DIV>
<DIV><FONT face=Verdana size=2>
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