<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; ">Kostya: I beg to differ here. I do not want to open a thread "PW's vs. Gaussians" which would have the same chances to be settled as "coca vs. pepsi", but what you say, while perfectly (kind of) clear to me, may be misleading to the less experienced.<DIV><BR class="khtml-block-placeholder"></DIV><DIV>PW potential energy surfaces (PES) are as smooth as they can be (i.e. *perfectly smooth*, or of class C(\infty), if you prefer) *with fixed boundary conditions* (i.e. for a fixed shape/size of the unit cell which determines the periodic boundary conditions). This smoothness has strictly nothing to do with any cancellation of errors. With a fixed number of PW's, PES's are perfectly smooth, but the (variational) accuracy would depend on the size/shape of the unit cell. Discontinuities come in when varying the boundary conditions because PW basis sets depend on boundary conditions (conceptually) for exactly the same reasons why localized basis sets depend on nuclear positions. With Gaussians you have "Pulay forces" (which require a special treatment going beyond Helmann Feynman), with PW's you have "Pulay stresses" (which require an equally special ad hoc treatment). </DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>It is indeed the ease with which accuracy can be assessed and improved with PW's which make the problem appear. As said, if one was ready to use a fixed basis set for different geometries (as one does with localized basis sets without really caring about the dependence of the accuracy on the geometry), the potential energy surface would be perfectly smooth, though less accurate.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>As simple as that. No misterious powerful cancellation principles here.</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Stefano</DIV><DIV><BR class="khtml-block-placeholder"><DIV><DIV><DIV>On Sep 8, 2005, at 9:17 PM, Konstantin Kudin wrote:</DIV><BR class="Apple-interchange-newline"><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV> <BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Basically, as you enalarge your unit cell, your brillouin zone</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">shrinks,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">and at constant cutoff it means that more lattice vectors (i.e. plane</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">waves) enter into the fixed cutoff sphere. More plane waves means a<SPAN class="Apple-converted-space"> </SPAN></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">systematically larger basis set (this is one of the good things</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">of plane waves, it's easy to make the basis set more and more</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">complete -</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">a nightmare in Gaussian), and, variationally, a lower energy. You see</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">in<SPAN class="Apple-converted-space"> </SPAN></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">fact your energy drop going to the left.</DIV> <BR></BLOCKQUOTE><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><SPAN class="Apple-converted-space"> </SPAN>I'd like to chime in here. While indeed reaching the basis set limit</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">is sort of difficult with Gaussian basis sets, there is this powerful</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">principle called "cancellation of errors". This allows one to chase not</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">the full convergence, but convergence with the basis set that is good</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">enough. As it turns out, such convergence is actually rather easy to</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">achieve.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><SPAN class="Apple-converted-space"> </SPAN>Then one has to care that the energy is fully variational and smooth</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">(i.e. differentiable) with respect to changes in atomic coordinates and</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">lattice vectors. While indeed PWs do reach the basis set limits easier,</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">the smoothness of the potential energy surfaces on the other hand</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">leaves something to be desired. The smoothness issue is usually much</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">better taken care of with Gaussians. So there you go ... :-)</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><SPAN class="Apple-converted-space"> </SPAN>Kostya</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><P style="margin: 0.0px 0.0px 0.0px 0.0px; min-height: 14.0px"><SPAN class="Apple-converted-space"><SPAN class="Apple-converted-tab"> </SPAN></SPAN><BR class="khtml-block-placeholder"></P><P style="margin: 0.0px 0.0px 0.0px 0.0px; min-height: 14.0px"><SPAN class="Apple-converted-space"><SPAN class="Apple-converted-tab"> </SPAN></SPAN><BR class="khtml-block-placeholder"></P><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">______________________________________________________</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Click here to donate to the Hurricane Katrina relief effort.</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="http://store.yahoo.com/redcross-donate3/">http://store.yahoo.com/redcross-donate3/</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">_______________________________________________</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Pw_forum mailing list</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</A></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><A href="http://www.democritos.it/mailman/listinfo/pw_forum">http://www.democritos.it/mailman/listinfo/pw_forum</A></DIV> <BR class="Apple-interchange-newline"></BLOCKQUOTE></DIV><BR><DIV> <SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; 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; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; 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; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><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 class="Apple-style-span" style="font-size: 12px; ">---</SPAN></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;"><SPAN class="Apple-style-span" style="font-size: 12px; ">Stefano Baroni - SISSA</SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; ">&</SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; ">DEMOCRITOS National Simulation Center - Trieste</SPAN></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;"><SPAN class="Apple-style-span" style="font-size: 12px; ">[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)</SPAN></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 class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; ">Please, if possible, don't</SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; ">send me MS Word or PowerPoint attachments</SPAN></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;"><SPAN class="Apple-style-span" style="font-size: 12px; ">Why? See:</SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></FONT><A href="http://www.gnu.org/philosophy/no-word-attachments.html"><FONT class="Apple-style-span" color="#0023E9" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;; color: rgb(0, 35, 233); -khtml-text-decorations-in-effect: underline; "><SPAN class="Apple-style-span" style="color: rgb(0, 35, 233); font-size: 12px; -khtml-text-decorations-in-effect: underline; ">http://www.gnu.org/philosophy/no-word-attachments.html</SPAN></SPAN></FONT></A></DIV><DIV><BR class="khtml-block-placeholder"></DIV><BR class="Apple-interchange-newline"></SPAN></SPAN> </DIV><BR></DIV></DIV></BODY></HTML>