<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">OK. Let me try to throw my own 5 cents into the discussion. Let me start with some very general, very elementary statements, and I will comment later on some of the statements made by others. The concept of Fermi energy is a very simple one and is best explained thinking of a system of non-interacting electrons (of which Fermi liquids are generalizations I will not comment on).<div><br></div><div>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.</div><div><br></div><div>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.</div><div><br></div><div>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).</div><div><br></div><div>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.</div><div><br><div><div>On Aug 15, 2008, at 11:47 AM, wangxinquan wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div>Dear Eyvaz,<br><br><blockquote type="cite">1. Scf calculations do not give the Fermi level, because you use fixed occupations used by default for semiconductors (insulators). I.e. if nothing specified as occupations, >your system is considered as a semiconductor<br></blockquote><blockquote type="cite">(insulator). In this case the number of bands treated is exactly half of the number of valence electrons, i.e. 4 in your case.<br></blockquote>Do you mean that the occupations of electrons were constrained during iterations? To my limited experiences, <br>the occupations should be changed by setting the "starting_ns_eigenvalue" parameters (lda+u calculation) and<br>the occupation matrix will be tuned through the iterations process. </div></blockquote><div><br></div>the electron occupations are usually fixed following the "filling principle": states with lower energies are filled first. in an insulator, this is easily done because the number of filled states is independent of the wavevector in the Brilloion zone. in metals, this is more tricky because the number of occupied states will depend on the key point, and in order to sort out this dependence one has to know in advance the electronic structure (i.e. the sequence of energy levels) at ALL the k points in the Brillouin zone</div><div><br><blockquote type="cite"><div>I'm sorry for that I confused "electron density" with "occupation". In my opinion, both<br>concepts describe the probabilty of electrons locating at lattic site. </div></blockquote><div><br></div>I am sorry but your opinion is wrong. The "occupation" does not refer to the probability of finding an electron at any particular lattice site, but rather in any of the molecular orbitals (which, being generally delocalized, cannot be attached at any specific lattice site).</div><div><br><blockquote type="cite"><div>The "electron density" <br>which is derived from wavefunctions belongs to quantum mechanics field, while "occupation" <br>belongs to condensed matter physics field with respect to band structure.</div></blockquote><div><br></div>I do not quite see the difference between "belonging to quantum bechanics" as opposed to "belonging to condensed matter", but for the fact that you may have learnt one concept in one textbook and another in a different book. As a rule of thumb, what you learn tends to be rather useless up the time when you can forget where/when you learnt it and you can pretend that the concepts you are using are YOURS ...</div><div><br><blockquote type="cite"><div>The electron density <br>should be changed but the occupation of bands should be fixed in scf calculation. right?</div></blockquote><div><br></div>if so you like to think, this is not wrong ... (which indicates that you could have seen by yourself why your previous statement is wrong)</div><div><br><blockquote type="cite"><div>The electron states(eigenvalue of the density matrix) </div></blockquote><div><br></div>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 ...</div><div><br><blockquote type="cite"><div>which construct the band structures will not<br>be occupated entirely. For insulator, the valence band is full while the conduct band is empty. <br>In the output file of scf I have no idea what the "fermi energy" results are. Does it not mean the <br>"real" fermi energy? E.J.Yoffa and D.Adler (Phy. Rev. B, 12, 2260) have talked about the calculation <br>of fermi energy for Mott insulator. I'm afraid the definition of fermi energy were misunderstood.</div></blockquote><div><br></div>if you mean misunderstood by you, this is quite probable.</div><div><br><blockquote type="cite"><div>My brain doesn't work. </div></blockquote><div><br></div>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.</div><div><br><blockquote type="cite"><div>Any help to remove my understanding to fermi energy will be deeply appreciated.</div></blockquote><div><br></div>hope I did help, a little bit - stay tuned</div><div><br></div><div>Stefano B</div><div><br></div><div apple-content-edited="true"> <span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">---</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">Stefano Baroni - SISSA</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">&</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">DEMOCRITOS National Simulation Center - Trieste</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "><a href="http://www.sissa.it/~baroni">http://www.sissa.it/~baroni</a> / [+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; font-size: 12px; "><br style="font-size: 12px; "></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; font-size: 12px; "><span class="Apple-style-span" style="font-size: 14px; "><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">La morale est une logique de l'action comme la logique est une morale de la pensée - Jean Piaget</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "><br></span></font></div></span></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">Please, if possible, don't</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">send me MS Word or PowerPoint attachments</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">Why? See:</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" color="#0023E9" size="3"><span class="Apple-style-span" style="font-size: 12px; "><a href="http://www.gnu.org/philosophy/no-word-attachments.html">http://www.gnu.org/philosophy/no-word-attachments.html</a></span></font></div><div><font class="Apple-style-span" color="#0023E9" size="3"><span class="Apple-style-span" style="font-size: 12px; "><br class="webkit-block-placeholder"></span></font></div></div></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"> </div><br></div></body></html>