<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><br><div><div>On Aug 2, 2011, at 8:18 AM, mohaddeseh abbasnejad wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite">Dear Prof. Baroni<br> <br>Thanks again<br><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div style="word-wrap:break-word"><div><div class="im">
<blockquote type="cite"><div>Dear Prof. Baroni</div>
<div> </div>
<div>Thanks for your reply.</div>
<div>I wonder shouldn't the plot show the same frequency for a range of displacements?<span dir="ltr"> </span></div></blockquote><div><br></div></div>what range? why should it?</div><div><div class="im"><br></div></div>
</div></blockquote><div><br style="background-color: rgb(255, 255, 255);"><font style="background-color: rgb(255, 255, 255);">-- Because if we are in the
neighberhood of a minimum "where a quadratic could be fitted", any
displacement in that range should result the same frequency. By
displacement, I mean the same pattern but different in amplitude.</font><br></div></div></blockquote><div><br></div>who said this? around a minimum, the only thing you can say is that the Taylor expansion of the energy in powers of the atomic displacements lacks any linear terms. In general, it has both quadratic terms (which are the one you are interested in), as well as higher order terms ... No reason why the Taylor expansion should be *exactly* quadratic (which is what would give rise to a "same frequency"). Note however, that it is your definition of frequency that is probably flawed. I suspect that you define the frequency as the ratio between the energy variation and the square of the displacement, which obviously depends on the magnitude of the displacement, if the energy variation is not exactly quadratic. The correct definition, instead, is the *limit* of that ration when the displacement goes to zero. That limit is obviously well defined, independently of the magnitude of any particular distortion that you may consider.</div><div><br><blockquote type="cite"><div class="gmail_quote"><div> </div><blockquote class="gmail_quote" style="margin-top: 0pt; margin-right: 0pt; margin-bottom: 0pt; margin-left: 0.8ex; border-left-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex; position: static; z-index: auto; "><div style="word-wrap: break-word;">
<div><div class="im"><blockquote type="cite"><div><span dir="ltr">And the same freq. for the range to be the harmonic frequency? or the zero displacement limit, is the harmonic frequency?</span><br></div></blockquote><div>
<br></div></div>yes, by definition</div></div></blockquote></div></blockquote><div><br></div>See the above discussion</div><div><br><blockquote type="cite"><div class="gmail_quote"><blockquote class="gmail_quote" style="margin-top: 0pt; margin-right: 0pt; margin-bottom: 0pt; margin-left: 0.8ex; border-left-width: 1px; border-left-style: solid; border-left-color: rgb(204, 204, 204); padding-left: 1ex; position: static; z-index: auto; "><div style="word-wrap: break-word;"><div><div class="im"><br><blockquote type="cite"><div>
</div>
<div>And when I test the displacements with the same amplitude, but
different in sign (regarding the inward and outward atomic movements in
the basis) I do not get the same results. </div></blockquote><div><br></div></div>why should you?</div><div><div class="im"><br><blockquote type="cite"><div>Shouldn't the plot have
symmetry vs. the vertical axis?</div></blockquote><div><br></div></div>why?</div></div></blockquote><div><br><font style="background-color: rgb(255, 255, 255);">-- Consider for example a linear
chain of atoms, with 2 atoms per basis. The optical mode is the result
of the outward movement of the atoms (The one in +a/2 moves towards +x
and the one in -a/2 moves towards -x). I expect that if I displace the
atoms in an inward movement with the same amplitude as the outward, I
should see the same frequency (The one in +a/2 moves towards -x and the
one in -a/2 moves towards +x) because the 2 patterns present the same
mode, right? That is what I've done in the case of Diamond. Is my
expectation wrong?</font><br style="background-color: rgb(255, 255, 255);"></div></div></blockquote><div><br></div>Yes. If the Taylor expansion of the energy difference vs displacement has cubic terms (as it usually has, but in very special cases dictated by symmetry), there is no reason why the energy difference should be even with respect to energy displacement. Actually, it usually isn't ...</div><div><br></div><div>Hope this clarifies a bit the situation ...</div><div><br></div><div>Take care</div><div>SB</div><br><div>
<span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; 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: 0px; font-size: medium; "><div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font-size: 14px; "><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-size: 14px; "><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-size: 14px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "><a href="http://stefano.baroni.me/">http://stefano.baroni.me</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-size: 14px; "><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-size: 14px; "><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><div><font class="Apple-style-span" color="#0023E9" size="3"><br></font></div></span><br class="Apple-interchange-newline">
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