[ver] 4 [sty] [files] [charset] 82 ANSI (Windows, IBM CP 1252) [revisions] 0 [prn] Epson LQ-850 [port] LPT1.OS2 [lang] 2 [desc] Thesis Chapter IV, May 1995 Edition 802125496 19 800287967 8738 22 0 0 0 0 1 [fopts] 0 1 0 0 [lnopts] 2 Body Text 1 [docopts] 5 2 [GramStyle] Academic Writing [tag] Body Text 2 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Text 0 0 [tag] Body Single 3 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Single 0 0 [tag] Bullet 4 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*0> 360 1 1 0 0 0 0 [nfmt] 272 1 2 . , $ Bullet 0 0 [tag] Bullet 1 5 [fnt] Times New Roman 240 0 49152 [algn] 1 1 288 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*5> 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Bullet 1 0 0 [tag] Number List 6 [fnt] Times New Roman 240 0 49152 [algn] 1 1 360 360 360 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*:>. 360 1 1 0 16 0 0 [nfmt] 272 1 2 . , $ Number List 0 0 [tag] Subhead 7 [fnt] Times New Roman 240 0 49155 [algn] 1 1 0 0 0 [spc] 33 273 1 72 72 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 2 0 1 1 0 0 0 0 [nfmt] 272 1 2 . , $ Subhead 0 0 [tag] Title 8 [fnt] Arial 360 0 16385 [algn] 4 1 0 0 0 [spc] 33 446 1 144 72 1 100 [brk] 16 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 1 0 1 1 0 0 0 0 [nfmt] 272 1 2 . , $ Title 0 0 [tag] Header 9 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Header 0 0 [tag] Footer 11 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Footer 0 0 [frm] 1 537395392 1440 9631 3172 10216 0 1 3 0 0 0 0 0 0 0 0 16777215 6 0 0 1732 408 [frmname] Frame6 [frmlay] 10216 1732 1 0 0 1 9631 0 0 2 0 0 0 0 1 1440 2870 0 [isd] .X6 .tex .X6 1 1 0 0 1627 64946 100 0 0 .tex 0 65128 0 [frm] 1 537395392 1440 7727 3138 8353 0 1 3 0 0 0 0 0 0 0 0 16777215 5 1 0 1698 311 [frmname] Frame5 [frmlay] 8353 1698 1 0 0 1 7727 0 0 2 0 0 84 10798 1 1440 2836 0 [isd] .X5 .tex .X5 1 1 0 0 1614 64948 100 0 0 .tex 0 65225 0 [frm] 1 537395392 1440 5685 2714 5981 0 1 3 0 0 0 0 0 0 0 0 16777215 4 2 0 1274 230 [frmname] Frame4 [frmlay] 5981 1274 1 0 0 1 5685 0 0 2 0 0 0 0 1 1440 2412 0 [isd] .X4 .tex .X4 1 1 0 0 1178 65252 100 0 0 .tex 0 65306 0 [frm] 1 537395392 1440 3228 2918 3555 0 1 3 0 0 0 0 0 0 0 0 16777215 3 3 0 1478 261 [frmname] Frame3 [frmlay] 3555 1478 1 0 0 1 3228 0 0 2 0 0 0 0 1 1440 2616 0 [isd] .X3 .tex .X3 1 1 0 0 1393 65200 100 0 0 .tex 0 65275 0 [lay] Standard 513 [rght] 15840 12240 1 1440 1440 1 1440 1440 0 1 0 1 0 2 1 1440 10800 12 1 720 1 1440 1 2160 1 2880 1 3600 1 4320 1 5040 1 5760 1 6480 1 7200 1 7920 1 8640 [hrght] [lyfrm] 1 11200 0 0 12240 1440 0 1 3 1 0 0 0 0 0 0 0 0 1 [frmlay] 1440 12240 1 1440 72 1 792 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] <+B><:f180,,>IV. How Local Can One Make A Pseudopotential<:f> > [frght] [lyfrm] 1 13248 0 14400 12240 15840 0 1 3 1 0 0 0 0 0 0 0 0 2 [frmlay] 15840 12240 1 1440 792 1 14472 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] <+B><:f180,,> <+B><:f180,,><:P10,0,IV-><:f> > [elay] [l1] 0 [pg] 22 25 93 151 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 31 582 88 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 48 0 21 32 0 0 0 65534 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 59 433 48 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 74 403 182 0 0 0 0 65534 565 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 79 2456 98 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 91 571 93 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 98 1585 155 0 0 0 0 65534 1685 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 105 2136 148 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 114 427 100 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 116 3026 287 32 0 0 0 65534 3224 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 127 1082 121 0 0 0 0 65534 1176 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 129 993 100 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 135 369 144 0 0 0 0 65534 455 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 145 0 47 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 181 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 226 0 82 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 265 0 13 32 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 292 0 0 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 316 0 0 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 342 0 259 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 358 0 8 1025 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 [edoc] <+B><:s><:#566,9360><:f480,,> <+B><:s><:#566,9360><:f480,,><+!>IV<-!> <+B><:s><:#284,9360><+!><:f240,,> <+B><:s><:#566,9360><+!><:f480,,>Local and Partly Local Pseudopotentials <+B><:s><:#566,9360><+!><:f480,,>from <+B><:s><:#566,9360><+!><:f480,,>A Non-local Method<-!><:f><-!> <+B><:s><:#566,9360><+!><:f480,,> <+B><:s><:#379,9360><+!><:f320,,>( How Local Can One <-!><+!>Make <-!><+!>A Pseudopotential )<-!><:f> <+@><:s><:#284,9360> <+B><:s><:#284,9360><:f240,,> <+B><:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:S+-2><:#3408,9360>The purpose of this chapter is to demonstrate further how to apply the Projector Reduction method, introduced in Chapter III, in generating pseudopotentials of elements in various categories. This also allows us to carry out a systematic investigation of th e non-locality of pseudopotentials of the corresponding chemical elements, which helps us to understand why some elements can be reasonably represented by a local pseudopotential while for some others a non-local one is necessary. In the other words, we wan t to ask whether good local pseudopotentials for some particular elements probably exist but are yet to be found, or whether it is fundamentally impossible to treat them reliably by using local pseudopotentials. <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:#284,9360><:f240,,> <:#284,9360><:f240,,> <:#379,9360><:f320,2Times New Roman,><+!>IV.1. Introduction<-!><:f> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:S+-2><:#1746,9360>In Chapter III we showed how the systematic procedures of Chapter II can be used to generate pseudopotentials that are partially local i.e. have nearly the same <+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) for two or more different <+">l<-">, so that when a weighted average of them is chosen as <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>), those corresponding <:f240,2Symbol,0,0,0>d<+"><:f>V<+'>l <-'><-">(<+">r<-">) will be sufficiently small to be ignored, which makes the <+">l<-"> dependence of those <+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) disappear. <:s><:S+-2><:#426,9360> <:S+-2><:#2982,9360>In this chapter we apply this to different types of atom in different parts of the periodic table, to see how far one can push this in practice. The extreme situation is that, to a sufficient degree of approximation, one can use the same <+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) for all <+">l<-">, i.e. one can use a local pseudopotential. There are a few elements, particularly Al, Ge and As, for which local potentials have been used for several years with reasonable success. This raises the question whether there is something special about these elements, or whether there is a realistic possibility of generating equally good local pseudopotentials for more atoms. Computationally that would clearly be very desirable. <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#4686,9360><:f,2Times New Roman,>All the examples and tests discussed in this chapter are restricted to<:f><:f240,2Times New Roman,0,0,0> non-<+">f<-"> elements i.e. those not having any valence electron in the 4<+">f<-"> or 5<+">f<-"> shells in the atomic state,<:f> although there is no obvious reason why the basic idea presented in this chapter can not be applied to the elements with occupied <+">f<-"> or even higher <+">l<-">-states. Since the central issue of the <:f,2Times New Roman,>non-locality of a pseudopotential<:f> is the <+">l<-">-dependency of each <+">V<+'>l <-'><-">(<+">r<-">), it will be useful to know which and how many <+">V<+'>l<-'><-">(<+">r<-">) should be considered in general. <:f,2Times New Roman,>A non-local pseudopotential <:f><:f,2Times New Roman,>in practice <:f><:f,2Times New Roman,>contains <:f><:f,2Times New Roman,>only <:f><+">V<+'>l <-'><-">(<+">r<-">)<:f,2Times New Roman,> for a few lower <+">l<-">, <:f><:f,2Times New Roman,>which may seem inappropriate<:f><:f,2Times New Roman,> for <:f><:f240,2Times New Roman,0,0,0>Bloch functions which in general contain components of all angular momenta.<:f> <:f,2Times New Roman,>However,<:f><:f,2Times New Roman,> with a simple argument <:f><:f,2Times New Roman,>we can demonstrate<:f><:f,2Times New Roman,> the trend<:f><:f,2Times New Roman,> that higher the <+">l<-"> is, less important the <:f><+">V<+'>l <-'><-">(<+">r<-">)<:f,2Times New Roman,> <:f><:f,2Times New Roman,>will be, which justifies the use of only <:f><+">V<+'>l <-'><-">(<+">r<-">)<:f,2Times New Roman,> with lower <+">l<-"> in a non-local pseudopotential<:f><:f,2Times New Roman,> because<:f><:f,2Times New Roman,> they are the dominant components. Our estimate also<:f><:f,2Times New Roman,> suggests that <:f><:f,2Times New Roman,><+">l<-">=0,1,2 should be enough for non-<+">f<-"> elements. <:s><:S+-2><:#426,9360> <:S+-2><:#4344,9360><:f,2Times New Roman,>We make the argument <:f>by first assuming the delocalised (valence) electrons in a given system to be free-electron-like. The maximum linear momentum (i.e., the Fermi momentum) of the system can then be expressed in terms of the densi ty of electron gas. One can then estimate the typical value of angular momentum of the system in terms of the incoming momentum and the impact parameter, as in the case of a classical scattering problem. The magnitude of the angular momentum estimated this way depends purely on the density, which will give us some idea about the maximum angular momentum of the given electron gas system in terms of the number of delocalised electron per atom ratio. More precisely, <:f,2Times New Roman,>from the free electron approximation, the Fermi <:f,2Times New Roman,>momentum <+">k<+'>F<-'><-"> of free electron gas is determined by the charge density <:f240,2Symbol,0,0,0>r<:f><:f,2Times New Roman,>, in atomic units (<+">m<-"><+'>e<-'> = <+">h<-">/2<:f240,2Symbol,0,0,0>p<:f,2Times New Roman,> = <+">e<-"> = 1)<:f><:f,2Times New Roman,>, <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:s><:S+-2><:f,2Times New Roman,><:A3>. (1.1) <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#1278,9360><:f,2Times New Roman,>Let <+">r<-"><+'>0<-'> be the Wigner-Seitz radius of the atom. The classical scattering picture suggests that the largest angular momentum <+">l<-"><+'>max<-'> that can exist in the system<:f> is the largest possible linear momentum <+">k<-"><+'>F<-'> multiplied by the largest possible impact parameter <+">r<-"><+'>0<-'> : <:s><:S+-2><:#426,9360> <+@><:s><:S+-2><:f,2Times New Roman,><:A2> . (1.2) <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#894,9360><:f,2Times New Roman,>The mean charge density <:f240,2Symbol,0,0,0>r<:f,2Times New Roman,> in (1.1) can be written in terms of the total number of electrons <+">N<-"> in the <:f><:f,2Times New Roman,>Wigner-Seitz <:f><:f,2Times New Roman,>cell, <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:s><:S+-2><:f,2Times New Roman,><:A1> . (1.3) <:s><:S+-2><:#426,9360> <:S+-2><:#426,9360>If<:f,2Times New Roman,> we introduce (1.3) and (1.1) into the<:f><:f,2Times New Roman,><-'> <:f><:f,2Times New Roman,>(1.2),<:f><:f,2Times New Roman,> <:f><:f,2Times New Roman,>then the <+">l<-"><+'>max<-'> becomes<:f> <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:s><:S+-2><:f,2Times New Roman,><:A0>. (1.4) <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#2556,9360><:f,2Times New Roman,>Estimated this way, the <:f240,2Times New Roman,0,0,0><+">l<-"><+'>max<-'><:f,2Times New Roman,> is only a function of<:f><:f,2Times New Roman,> <+">N<-">, i.e. the number of valence (conduction) electrons <:f><:f,2Times New Roman,>per atom. We can now use (1.4) to find out<:f><:f,2Times New Roman,> the value of <+">l<-"><:f><+'><:f240,2Times New Roman,0,0,0>max<-'><:f><:f,2Times New Roman,> that corresponds to a given <+">N<-"><+">.<-"> For the state <:f240,2Times New Roman,0,0,0><+">l<-"><:f,2Times New Roman,> = 3 to start to be important, the value of <:f240,2Times New Roman,0,0,0><+">N<-"><:f,2Times New Roman,> has to reach about 10 (so that <+">l<-"><+'>max<-'> <;> 2), while for <+">l<-"> = 1 to be effective, the <:f240,2Times New Roman,0,0,0><+">N<-"><:f,2Times New Roman,> only needs to be larger than 1. It is obvious that in most cases in real applications, <:f240,2Times New Roman,0,0,0><+">N<-"><:f,2Times New Roman,> takes the value between these two numbers, which means that <:f240,2Times New Roman,0,0,0><+">l<-"><:f,2Times New Roman,> = 2 should be sufficient for a free-electron-like system. <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#2130,9360><:f,2Times New Roman,>When there are more localised electronic<:f><:f,2Times New Roman,> states exist in the system, such as in the case of a molecule or when there are <+">d<-"> or <:f240,2Times New Roman,0,0,0><+">f<-"><:f,2Times New Roman,> electrons <-"><:f,2Times New Roman,>in a solid, the situation will be different. However, in these systems these localised states should<:f><:f,2Times New Roman,> at least be less localised than those in an atom. Thus as long as a pseudopotential carries a component matching the highest <:f240,2Times New Roman,0,0,0><+">l<-"><:f,2Times New Roman,> of the occupied atomic ground state, it should be sufficient to describe the scattering in a solid. <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#2556,9360><:f,2Times New Roman,>From the above reasoning<:f><:f,2Times New Roman,> we believe<:f><:f,2Times New Roman,> that, for both a delocalised<:f><:f,2Times New Roman,> (free-electron-like) system and highly localised system such as a transition metal, our simp le picture explains why, for a non-<+">f<-"> element, a non-local potential<+"> <-">with <+">s<-">, <+">p<-">, and <+">d<-"> components together should be enough for a faithful representation<:f><:f,2Times New Roman,> of the major <+">l<-"> dependent non-local effect. <:f><:f,2Times New Roman,>In addition, we believe that a straight-forward extension of this argument can be made to cover the <+">f<-">-elemeents, in which case we think a non-local pseudopotential carrying <:f><+">V<-"><+'><+">l<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<:f,2Times New Roman,> of <:f240,2Times New Roman,0,0,0><+">s<-"><:f,2Times New Roman,>,<:f240,2Times New Roman,0,0,0><+"> p<-"><:f,2Times New Roman,>,<:f240,2Times New Roman,0,0,0><+"> d<-"><:f,2Times New Roman,> <:f240,2Times New Roman,0,0,0>and <+">f<-"><:f,2Times New Roman,> should be sufficient. <:s><:S+-2><:#426,9360> <:S+-2><:#852,9360>It worth emphasising that the above argument also justifies the conventional way of writing <+">V<-"><+'><+">l<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) as <:s><:S+-2><:#426,9360> <:S+-2><:#468,9360><+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) = <+">V <-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) + <:f240,2Symbol,0,0,0>d<+"><:f>V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) , (1.5) <:s><:S+-2><:#426,9360> <:S+-2><:#3024,9360>in which (for non-<+">f<-"> elements) those <:f240,2Symbol,0,0,0>d<+"><:f>V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) for <+"><:f240,2Times New Roman,0,0,0>l<-"><:f> <;> 2 are set to zero to save computing, <-">leaving only the <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) to operate on components of wave functions for all <+">l<-"> <;> 2 <-'><-">. It is widely known that the <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) can be fairly arbitrary chosen (to avoid ghost state in KB form) without affecting the results of calculation <[>Ref.K.2, G.1], which is co nsistent with our argument that the electron-ion interaction with <+">l<-"> <;> 2 is not important anyway and therefore the results of calculation are insensitive to the actual value of <+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) used (in this case, <+">V <-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)). As for <+">l<-"> = 2, we will see from the study of Al in Section 3 that there are cases when <+">V<-"><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) is important. <-"> <:s><:S+-2><:#426,9360> <:S+-2><:#1704,9360>In Section 2 we discuss making <+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) effectively the same for <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f> components in some elements, and for <:f240,2Times New Roman,0,0,0><+">p<-"><:f> and <:f240,2Times New Roman,0,0,0><+">d<-"><:f> in other elements. In Section 3 we consider Al, Si and Ge in detail in relation to finding fully local (approximate) pseudopotentials for these elements. A short dis cussion and conclusion is given in Section 4. <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:#379,9360><:f320,,><+!>IV.2. <:f320,2Times New Roman,>Reducing The Non-locality in Different Cases<-!><:f> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:S+-2><:#2982,9360><:f,2Times New Roman,>The feasibility<:f><:f,2Times New Roman,> of reducing the number of projectors in a pseudopotential<:f><:f,2Times New Roman,> depends mainly on the nature of the atomic orbitals of a given element. We find it convenient to discuss the procedure in two categories<:f><:f,2Times New Roman,>, namely those involving <+">pd<-"> reduction and those involving <+">sp<-"> reduction, with each category further classified by whether there exists a <+">cancellation<:f><:f,2Times New Roman,> effect<-"> <:f><:f,2Times New Roman,><[>Ref.H.2]<:f><:f,2Times New Roman,> in the pseudopotential<:f><:f,2Times New Roman,>. <:f><:f,2Times New Roman,>Atomic logarithmic derivative tests are always used to examine the success of the reduction<:f><:f,2Times New Roman,>. <:f><:f,2Times New Roman,>Detailed generating and tuning parameters as well as figures of <+">V<+'>l<-'><-">(<+">r<-">) and logarithmic derivatives are given for each pseudopotential<:f><:f,2Times New Roman,> mentioned.<:f> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360><+"><+!>pd<-"> reduction<-!> <:s><:S+-2><:#426,9360> <:S+-2><:#4260,9360><:f,2Times New Roman,>For elements<-"><:f,2Times New Roman,> such as C, N, O, <:f><:f,2Times New Roman,>both of their 2<+">p<-"> and 3<:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,> radial wavefunctions<:f><:f,2Times New Roman,> have no node and there is no cancellation effect<:f><:f,2Times New Roman,> from an inner she ll to contribute to pseudising<:f><:f,2Times New Roman,> the wavefunctions<:f><:f,2Times New Roman,>. We have found that for such elements it is relatively easy to redistribute<:f><:f,2Times New Roman,> (i.e. optimise) the pseudo wavefunctions for both the<:f> <:f,2Times New Roman,> 2<+">p<-"> and 3<:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,> <:f><:f,2Times New Roman,>states within the pseudo core <:f><:f,2Times New Roman,>without too much restriction, which makes a fairly flexible change of shape of <+">V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">d<-"><-'><:f,2Times New Roman,>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>) possible. <:f><:f,2Times New Roman,>An extremely high degree of similarity between <-"><:f><+"><:f,2Times New Roman,>V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">d<-"><-'><:f,2Times New Roman,>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>) <:f><:f,2Times New Roman,>of the pseudopotential can therefore<:f><:f,2Times New Roman,> be achieved. <:f><:f,2Times New Roman,>Two typical examples of such <+">pd<-"> "localisation<:f><:f,2Times New Roman,>" of the<:f><:f,2Times New Roman,> 2<:f240,2Times New Roman,0,0,0><+">p<-"><:f,2Times New Roman,> and 3<:f240,2Times New Roman,0,0,0><+">d<-"><:f><:f,2Times New Roman,> components<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>can be seen in the case of C <:f><:f240,2Times New Roman,0,0,0>and B, as shown in Fig.IV.2.1 and Fig.IV.2.2 respectively.<:f> (The detailed generating parameters of this C and B pseudopotentials can be found in Chapter V, which are numbered as C021 and B001a.) <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#5538,9360><:f,2Times New Roman,>When one or both of <+">p<-"> and <:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,> wavefunctions have nodes, <:f><:f,2Times New Roman,>the possibility of finding a common pseudopotential for both angular<:f><:f,2Times New Roman,> momentum<:f><:f,2Times New Roman,> channels<:f><:f,2Times New Roman,> is more case dependent<:f><:f,2Times New Roman,>.<:f><:f,2Times New Roman,> Redistributing charge density<:f><:f,2Times New Roman,> using <+">Q<-"><+'>c<-'>-tuning is usually not as easy as in the cases in which both <+">p<-"> and <:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,> wavefunctions are nodeless. <:f><:f,2Times New Roman,>(The worst case is presumably when <+">l<-">=1 has cancellation but <+">l<-">=2 does not, e.g. 3<+">d<-"> transition metals. <:f><:f,2Times New Roman,>But in fact even for 4<+">d<-"> and 5<+">d<-"> transition metals, the <:f><+"><:f,2Times New Roman,>V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">d<-"><-'><:f,2Times New Roman,>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>)<:f><:f,2Times New Roman,> are still quite different, which we believe due mainly to t he fact that the location of main charge density peak is very different for their <:f240,2Times New Roman,0,0,0><+">p<-"><:f,2Times New Roman,> and <+">d<-"> components.) <:f><:f,2Times New Roman,>Under this situation, the best we can do is to try to apply the Projector Reduction<:f><:f,2Times New Roman,> even though <:f><-"><+"><:f,2Times New Roman,>V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">d<-"><-'><:f,2Times New Roman,>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f,2Times New Roman,>) <:f><:f,2Times New Roman,>are not extremely similar. <:f><:f,2Times New Roman,>This works alright in some cases, especially those cases in which the <:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,>-state is unoccupied in the atomic ground state and should therefore be comparatively less important<:f><:f,2Times New Roman,> in solid<:f><:f,2Times New Roman,> than the occupied states<:f><:f,2Times New Roman,> . <:f><:f240,2Times New Roman,0,0,0>An example of the successful localisation <+">pd<-"> non-locality within this category is the Br pseudopotential<:f240,2Times New Roman,0,0,0>,<:f><:f240,2Times New Roman,0,0,0> shown in Fig.IV.2.3.<:f><-"><-'><-"><:f240,2Times New Roman,0,0,0> The generating parameter of this Br pseudopotential ca n be find in detail in the section III.2 of Chapter III. <:f> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:S+-2><:#426,9360><+"><+!>sp<-"> reduction<-!> <:s><:S+-2><:#426,9360> <:S+-2><:#9882,9360>Unlike in the case of <+">p<-"><:f240,2Times New Roman,0,0,0><+">d<-"><:f> localisation discussed above, we are less interested in the case which both <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f> that have no cancellation effect because only H and He satisfy such a condition. As for the cases which the valence states are 2<+">s<-"> (cancelled) and 2<:f240,2Times New Roman,0,0,0><+">p<-"><:f> (un-cancelled), such as C and O, their <+">V<-"><+'><+">s<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) <-"><-">components are very different even after applying <+">Q<+'>c<-"><-'>-tuning. We therefore think <-">that is essential to keep the individual <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) <-">in the pseudopotential for those elements. In the cases in which both <:f240,2Times New Roman,0,0,0><+">s<-"><:f> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f> wavefunctions have nodes, however, we found that it is possible to make <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<-"> similar. Among the cases explored, we have successfully reduced the <+">sp<-"> non-locality for the 3d transition metals that has more than four <+">d<-"> electrons, such as Fe, which is shown in Fig.IV.2.4. The full details of the parameter of projector reduced Fe pseudopotential is described in Chapter V as Fe002. From the Fig.IV.2.4(a) we can see that its <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) are very similar. The logarithmic derivative of the resulting <+">sp<-"> reduced Fe pseudopotential<:f240,2Times New Roman,0,0,0> is <:f>shown in Fig.IV.2.4(b). The similar quality of Projector Reduction has also been demonstrated in the cases of Co and Cu discussed in Chapter III. Experence has shown that the degree of agreement is very satisfactory, giving errors smaller than a typical LDA pseudopotential calculation. Unlike in the case of Fe, our best possible <+">sp<-"> localisation for Ge is only marginally acceptable, as shown by the logarithmic derivative tests in Fig.IV.2.5(a) and (b). The Ge pseudopotential was generated using atomic ground state configuration to calculate the <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f> wave functions, and the ionic state 4<+">s<+&><-&><-"><+&>1.00<-&> 4<:f240,2Times New Roman,0,0,0><+">p<-"><:f><+&> 0.75<-&> 4<:f240,2Times New Roman,0,0,0><+">d<+&> <-&><-"><:f><+&>0.25<-&><-"><-"><-"><-"><-"> to calculate the <:f240,2Times New Roman,0,0,0><+">d<-"><:f> wave function.<-"> The <:f240,2Times New Roman,0,0,0><+">r<-"><:f><+'>c<-'> took a typical value of 2.2 a.u. for Ge. To minimise <:f240,2Symbol,0,0,0>d<+"><:f>V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) the <+">Q<-"><+'>c<-'>/<+">q<-"><+'>3<-'>(<+">s<-">,<+"> p<-">,<+"> d<-">) was set to be (0.7, 0.9, 0.95). T<:f240,2Times New Roman,0,0,0>he weight<:f> 0.8 for the <+"><:f240,2Times New Roman,0,0,0>V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and 0.2 for the <+"><:f240,2Times New Roman,0,0,0>V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) were used to construct the <+"><:f240,2Times New Roman,0,0,0>V<:f240,2Times New Roman,0,0,0><-"><+&>L<-&><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>). A noticeable improvement of logarithmic derivatives can be seen with the <:f240,2Symbol,0,0,0>d<+"><:f>V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) of the pseudopotential kept, as shown in Fig.IV.2.5(c), which indicates that the non-locality of <+">s<-"> and <+">p<-"> may still be needed for a very accurate calculation.<-"><-"><-"><-"><-"><-"> <:S+-2><:#426,9360> <:S+-2><:#2556,9360><-"><-"><-"><-"><-"><-"><-"><-"><-"><-"><-"><-"><-">Although Ge is in the same long row of the periodic table as Fe, Co, Ni, Cu, Zn , we observed a clear difference in the degree of similarity that one can achieve between <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>), which in the case of Ge is more difficult than with the mentioned transition metals. The reason may be that for these me tals the 4<:f240,2Times New Roman,0,0,0><+">p<-"><:f> is an excited state, while for Ge it is occupied (in the atomic ground state). Thus we expect the 4<:f240,2Times New Roman,0,0,0><+">p<-"><:f> state of Ge to be more tightly bound and have its main charg e density peak at quite a different location from that of the unoccupied 4<+">p<-"> state of the transition metals.<-"><-"> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:#379,9360><:f320,2Times New Roman,><+!>IV.3. Case Studies of Al, Si and Ge<-!><:f> <:s><:#284,9360> <:s><:#284,9360> <:S+-2><:#1278,9360>Apart from the advantage of saving computing time and memory, the Projector Reduction provides an unambiguous procedure and test for discussing the degree of locality/non-locality of the pseudopotential of a given element, which may help to clarifies some l ong-term puzzles. <:S+-2><:#426,9360> <:S+-2><:#4728,9360>As mentioned in Section 1, reasonably good local pseduopotentials has been in use for Al <[>Ref.R.2] and Ge <[>Ref.N.2] but apperently none for Si. In view of the importance of calculations on Si, one may presume that efforts have been made to find a simila r local pseduopotential for Si but failed, and there is some folkelore to this effect <[>Ref.H.3]. Thus case studies of Si/Ge and Al/Si are devised with an attempt to clarify this issue by using our new technique, which generates local or partly local pse udopotentials from a non-local approach. More precisely, since in our method each local or partly local pseudopotential has a coreesponding fully non-local counterpart (i.e. the one with all <:f240,2Symbol,0,0,0>d<+"><:f>V<:f240,2Times New Roman,0,0,0><-"><+'><+">s<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) kept), the degree of the success in reducing the non-locality of Si, Al and Ge pseudopotemtials can be used as a measure of w hether a lequally good local Si pseduopotential is really more difficult to be found (with a reasonable <+">r<-"><+'>c<-'>, of course) than a Ge or an Al one. <:S+-2><:#426,9360> <:S+-2><:#4260,9360>For the cases stuided in this section, local pseudopotentials of Al, Si and Ge genertaed using Projector Reduction method was prepared. Atomic ground state was used for both Al and Si to calculate their <+">s<-"> and <+">p<-"> wave functions. The <+">r<-"><+'>c<-'> were chosen as the most typical and adequate values for corresponding elements based on our experience, with <+">Q<-"><+'>c<-'> tuned to make <+"><:f240,2Times New Roman,0,0,0>V<+'>s<-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p<-'><-"><:f>(<+">r<-">) similar. (We don't need to deal with <:f240,2Times New Roman,0,0,0><+">V<+'>d<-'><-"><:f>(<+">r<-">) for Al and Si here because the <+"><:f240,2Times New Roman,0,0,0>V<+'>d<-'><-"><:f>(<+">r<-">) is much less imporant than <+"><:f240,2Times New Roman,0,0,0>V<+'>s<-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p<-'><-"><:f>(<+">r<-">) due to its larger <+">l<-"> and the fact that the <+">d<-"> state is not occupied in the atomic ground state of these elements.) The resulting <+">r<-"><+'>c<-'> and <+"><:f240,2Times New Roman,0,0,0>Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f> for Al : <+">r<-"><+'>c <-'>= 2.4 a.u., <+"><:f240,2Times New Roman,0,0,0>Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>/<:f240,2Times New Roman,0,0,0><+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f>(<:f240,2Times New Roman,0,0,0><+">s<-"><:f>,<:f240,2Times New Roman,0,0,0> <+">p<-"><:f>) = (1.10, 1.00), and for Si : <:f240,2Times New Roman,0,0,0><+">r<:f240,2Times New Roman,0,0,0><-"><+'>c <-'><:f>= 2.0 a.u., <:f240,2Times New Roman,0,0,0><+">Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>/<:f240,2Times New Roman,0,0,0><+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f>(<:f240,2Times New Roman,0,0,0><+">s<-"><:f>,<+">p<-">) = (0.6, 0.9). Th e weight of mixing to construct <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) are 0.8<:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+"><+'>s<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and 0.2<:f240,2Times New Roman,0,0,0> <+">V<:f240,2Times New Roman,0,0,0><-"><+"><+'>p<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) for both cases. As for the local Ge pseudopotential, the exactly same one as described in Section 2 is used. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360><+!><:f,2Times New Roman,>Al and Si<-!><+!> Pseudopotentials<-!><:f> <:s><:S+-2><:#426,9360> <:S+-2><:#7668,9360>As already mentioned, it appears<:f,2Times New Roman,> that a local pseudopotential works better for Al than for Si<:f><:f,2Times New Roman,>. <:f><:f,2Times New Roman,>However, the valence orbitals occupied in the atomic ground states of Al and Si are both 3<+">s<-"> and 3<:f240,2Times New Roman,0,0,0><+">p<-"><:f>, so that one does not expect any difference to come from the inner level cancellat ion.<:f240,2Times New Roman,0,0,0> In fact,<:f> <:f,2Times New Roman,>by analysing the logarithmic derivative of both <:f><:f,2Times New Roman,>"localised" pseudopotentials of <:f><:f,2Times New Roman,>Al and Si described above (shown in Fig.3.1 and Fig.3.2 respectively<:f,2Times New Roman,>)<:f><:f,2Times New Roman,>,<:f><:f,2Times New Roman,> we have found that the degre e of non-locality in <:f><:f,2Times New Roman,>them is <:f240,2Times New Roman,0,0,0>indeed<:f,2Times New Roman,> fairly similar.<:f><:f,2Times New Roman,> <:f><:f,2Times New Roman,>This<:f><:f,2Times New Roman,> implies that <:f><:f,2Times New Roman,>the defferent performance<:f><:f,2Times New Roman,> of local pseudopotentials for these two elements, if it exists, <:f><:f,2Times New Roman,>is <:f><:f,2Times New Roman,>not due to their atomic nature such as cancellation<:f><:f,2Times New Roman,> from core states or the profile of the wavefunctions, but nust come mostly from their different situation<:f><:f,2Times New Roman,> in the solid state. <:f>An obvious difference between Al and Si in their solid form is that <:f,2Times New Roman,>Al is a close-packed metal and Si a semiconductor with a more open structure <:f><:f,2Times New Roman,>(mole volume of Al : 10 cm<+&>3<-&>/mole, Si : 12 cm<+&>3<-&>/mole)<:f>. <:f,2Times New Roman,>Since there are only 4 <:f><:f,2Times New Roman,>nearest neighbours for each Si atom <:f><:f,2Times New Roman,>in <:f><:f,2Times New Roman,>its diamond structure, in contrast to 12 for Al as an fcc metal, <:f><:f,2Times New Roman,>the chemical bonds between Si atoms are more directional than those between Al atoms.<:f><:f,2Times New Roman,> <:f><:f,2Times New Roman,>It may, therefore, <:f><:f,2Times New Roman,>be the covalency of the system which leads to the need for a more accurate non-local description from a Si pseudopotential. For example the directional bonds depend on <+">sp<-"> hybridisation and thus may be sensitive to the difference between the <+">s<-"> and <+">p<-"> pseudopotentials. In brief,<:f> we find that Al and Si have the same degree of non-locality, which suggests that the greater acceptability of an Al local potential is probably due to the higher tolerance of the system. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:S+-2><:#426,9360><+!><:f,2Times New Roman,>Si and Ge Pseudopotentials<-!><:f> <:S+-2><:#426,9360> <:S+-2><:#2982,9360>Although we have pointed out in the begining of this section that the<:f240,2Times New Roman,0,0,0><+"> V<:f240,2Times New Roman,0,0,0><+'>d<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) is less important than <:f240,2Times New Roman,0,0,0><+">V<+'>s<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<+'>p<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) for elements such as Al, Si and Ge. It can be understood from our free-electron analysis (1.4) that, since both Si and Ge have the number of valence electron <+">N <-">= 4 which results in the effective <+">l<-"><:f240,2Times New Roman,0,0,0><+'>max<-'><:f> for Si and Ge being 1.46 and larger than 1 (i.e. the <+">l<-"> for<+"> p<-">-wave), the electronic states with <+">d<-"> symmerty may not be completely ignored in the solid Si and Ge. Since a local potential acts on all <+">l<-"> state, it will be useful to see whether our local potentials of Ge and Si produce equally good or bad <+">d<-">-scattering, which might still have a modist importance for a solid. <:S+-2><:#426,9360> <:S+-2><:#6390,9360><:f240,2Times New Roman,0,0,0>We have tested the <+">d<-"> <:f><:f240,2Times New Roman,0,0,0>logarithmic derivatives<:f><:f240,2Times New Roman,0,0,0> (all-electron and pseudo) of the projector reduced<:f><:f240,2Times New Roman,0,0,0> local pseudopotential s<:f><:f240,2Times New Roman,0,0,0> for Si and Ge,<:f><:f240,2Times New Roman,0,0,0> as shown in Fig.IV.3.2(c) and Fig.3.3(b) respectively.<:f><-"><-"><:f240,2Times New Roman,0,0,0> Although the<:f><:f240,2Times New Roman,0,0,0> logarithmic derivatives of t he pseudo wave functions for both Ge and Si <:f><:f240,2Times New Roman,0,0,0>cases do not match the nearly perfect agreement with all-electron values as we had found with<:f><:f240,2Times New Roman,0,0,0> C and B with <:f><:f240,2Times New Roman,0,0,0>2<+">p<-">-3<+">d<-"> <:f><:f240,2Times New Roman,0,0,0>reduction (described in Section 2), they<:f><:f240,2Times New Roman,0,0,0> do clearly show that<:f><:f240,2Times New Roman,0,0,0> the local pseudopotential of Ge is no t as bad as that of Si in reproducing the<:f><:f240,2Times New Roman,0,0,0> <+">d<-">-scattering<:f><:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>This <:f><:f240,2Times New Roman,0,0,0>can be understood from the fact that the cancellation effect from inner shells exists for the 4<+">d<-"> orbital of Ge but not for the 3<+">d<-"> orbital of Si,<:f><:f240,2Times New Roman,0,0,0> which<:f><:f240,2Times New Roman,0,0,0> results in <:f240,2Times New Roman,0,0,0>the <:f><+"><:f240,2Times New Roman,0,0,0>V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-"><-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<-"><-"> and <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">d<-"><-"><-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<:f240,2Times New Roman,0,0,0> from the<:f><:f240,2Times New Roman,0,0,0> 4<+">p<-"> and 4<+">d<-"> <:f><:f240,2Times New Roman,0,0,0>states of Ge <:f><:f240,2Times New Roman,0,0,0>being more or less similar, whereas those from the 3<+">p<-"> and 3<+">d<-"> states of Si are very different. <:f><:f240,2Times New Roman,0,0,0>Therefore from the view point of the <+">d<-">-component, the Ge pseudopotential<:f><:f240,2Times New Roman,0,0,0> is more "local" than a Si one due to it atomic nature.<:f><:f><-"><-"><-"> As for <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f>, we have shown in Section 2 and Fig.IV.2.5(b) that the quality of the <+">sp<-"> partly-local Ge pseudopotential is only marginally acceptable, but it is still better, or at least not worse, than the similar case of Si, as shown in Fig.IV.3.2(b)<-"><-"><-"><-"><-'><-"><-"><-">. From the overall comparison of Ge and Si we find that there is a noticeable difference in locality, which supports the common experience that one can use a local potential for Ge but not for Si.<-"> <:s><:S+-2><:#426,9360> <:S+-2><:#5538,9360>The atomic nature of Ge not only allows its pseudopotential to be more local than that of Si, but it also <:f,2Times New Roman,>helps to reduce the pseudopotential error. <:f><:f,2Times New Roman,>This is because Ge has a larger principle<:f><:f,2Times New Roman,> quantum number <+">n<-"> <:f><:f,2Times New Roman,>for its valence <+">s<-">-state <:f><:f,2Times New Roman,>than Si has (<+">n <-">= 4 for Ge, <+">n<-"> = 3 for Si)<:f><:f,2Times New Roman,>. Due to this bigger <+">n<-"> <:f><:f,2Times New Roman,>as well as <:f><:f,2Times New Roman,>the larger core size<:f><:f,2Times New Roman,>, the<-"><:f,2Times New Roman,> overlapping of the <:f240,2Times New Roman,0,0,0><+">s<-"><:f,2Times New Roman,> orbitals of neighbouring atoms of Ge is less than in the case of Si, which leads to a narrower <:f240,2Times New Roman,0,0,0><+">s<-"><:f,2Times New Roman,> (<+">sp<-">) band. All those 4<+">s<-">-derived Bloch states of Ge are therefore not as far away from its atomic 4<:f240,2Times New Roman,0,0,0><+">s<-"><:f,2Times New Roman,> level compared with the 3<+">s<-">-derived Bloch state of Si from its Si 3<:f240,2Times New Roman,0,0,0><+">s<-"><:f,2Times New Roman,> level. <:f><:f,2Times New Roman,>Based on the fact that a pseudopotential<:f><:f,2Times New Roman,> only reproduces scattering<:f><:f,2Times New Roman,> within a finite range of energies<:f><:f,2Times New Roman,>, we can say that the smaller <+">s<-"> overlap in Ge makes the results of<:f><:f,2Times New Roman,> <:f><:f,2Times New Roman,>solid state calculation less sensitive<:f><:f,2Times New Roman,> to <:f><:f,2Times New Roman,>the error from a (local) pseudopotential<:f><:f,2Times New Roman,> of Ge compared with Si.<:f><:f,2Times New Roman,> <-"><:f><:f,2Times New Roman,>(Actually, this is the solid state analogue<:f><:f,2Times New Roman,> of an important chemical effect called the "inert <+">s<-">-pair effect". It explains why there is a <:f><:f,2Times New Roman,>tendency <:f><:f,2Times New Roman,>for the <+">s<-"> electrons of the heavier elements to have lower chemical reactivity than those from the lighter ones <:f><:f,2Times New Roman,>in the same column of the periodic table which<:f><:f,2Times New Roman,> have analogous valence electronic configuration<:f><:f,2Times New Roman,>.) <:f> <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#1278,9360><:f,2Times New Roman,>In conclusion, we have shown that the pseudopotential of Si is inherently somewhat more non-local, in agreement with general experience, for reasons which we have traced to differences in the atoms. <:S+-2><:#426,9360><-!> <:s><:S+-2><:#426,9360> <:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0><-">A critical re-appraisal of Al : a <+">p<-">-local<-!><+!> <+">s<-">-non-local Al<-!><:f><+!> Pseudopotential<-!> <:S+-2><:#426,9360> <:S+-2><:#6390,9360>Like in the case of Ge and Si discussed above, the free-electron estimatation also suggests that the <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><-"><+"><+'>d<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) can still play some role in a solid Al becasue it has 3 valence electrons, giveing an effective <+">l<-"><+'>max<-'> as 1.33, which is already larger than 1. In fact, there is a pure <+">d<-"> symmetry at the corner of the Brillouin zone less than 1 eV above the Fermi level so that significant <+">l <-">= 2 components whould be expected also below <+">E<-"><+'>F<-'>. However,<:f240,2Times New Roman,0,0,0> a local pseudopotential for Al works fine in reproducing some basic of solid-state bulk properties. This is probably because an accurate angula r dependency of the electronic structure is not so important in those calculations. <:f><:f240,2Times New Roman,0,0,0>The point is that angular effects depend critically on<:f><:f240,2Times New Roman,0,0,0> the degree of hybridisation between <+">s<-">, <:f240,2Times New Roman,0,0,0><+">p<-"><:f240,2Times New Roman,0,0,0> and <:f240,2Times New Roman,0,0,0><+">d<-"><:f240,2Times New Roman,0,0,0> orbitals in a chemical tight binding picture. In metals this translates into the right type and degree of mixing among the <:f240,2Times New Roman,0,0,0><+">l <-"><:f240,2Times New Roman,0,0,0>= 0, 1 an 2 components of Bloch functions. which depends sensitively on the differences between the <:f240,2Times New Roman,0,0,0><+">V<+'>l <-'><-"><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>) with <:f240,2Times New Roman,0,0,0><+">l <-"><:f240,2Times New Roman,0,0,0>= 0, 1, 2. The latter are swept aside in the approximation of a local pseudopotential, which can therefore be expected to fail in describing subtle structural properties, such as <:f><:f240,2Times New Roman,0,0,0>surface energy <[>Ref.G.1] and the energy <:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>stacking fault <:f><:f240,2Times New Roman,0,0,0>and defect<:f> <:f240,2Times New Roman,0,0,0><[><:f240,2Times New Roman,0,0,0>Ref.<:f240,2Times New Roman,255,0,255> <:f240,2Times New Roman,0,0,0>P.1<:f240,2Times New Roman,0,0,0>]<:f>. We must therefore have a pseudopotential which takes <+">V<+'>d<-'><-">(<+">r<-">) properly into account as well as <+">V<+'>s<-'><-">(<+">r<-">) and <+">V<+'>p<-'><-">(<+">r<-">). <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#8220,9360>On the other hand, due to the efficiency of using a pseudopotential with less non-local components, it will be useful to know whether one can use a Al pseudopotential without explicitly carrying <:f240,2Symbol,0,0,0>d<+"><:f>V<-"><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>). We have therefore generated a series of partly local pseudopotential for Al with only <:f240,2Symbol,0,0,0>d<+"><:f>V<-"><+'><+">s<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) non-zero, i.e. only requiring one non-local projector. We started by simply taking <+">V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) as local component <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and just ignoring <:f240,2Symbol,0,0,0>d<+"><:f>V<-"><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>). There pseudopotentials were tested in a systematic investigation of bulk properties <[>Ref.M.1]. <:f,2Times New Roman,>The results<:f><:f,2Times New Roman,> showed, <:f><:f,2Times New Roman,>interestingly,<:f><:f,2Times New Roman,> that<:f><:f,2Times New Roman,> <:f><:f,2Times New Roman,>although <:f><:f,2Times New Roman,>all such pseudopotentials reproduce very good scattering of <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f,2Times New Roman,> states in logarithmic derivative tests,<:f><:f,2Times New Roman,> pseudopotentials that have<:f><:f,2Times New Roman,> larger <+">r<-"><+'>c<-'> give better bulk properties (Table.IV.3.1). <-"><:f><-"><-"><-"><:f,2Times New Roman,>We regard this as an indication of the importance of the <:f240,2Times New Roman,0,0,0><+">d<-">-character<:f><:f,2Times New Roman,> in Al metal. <:f><:f,2Times New Roman,>Since no explicit non-local part for <+">d<-"> is used in these Al pseudopotentials, <:f><:f,2Times New Roman,>the <+">d<-">-scattering is <:f><:f,2Times New Roman,>represented<:f><:f,2Times New Roman,> by the local component <+">V<+&><-&><-"><+&>L<-&>(<+">r<-">), which is actually the <+">V<-"><+'><+">p<-"><-'>(<+">r<-">) from the 3<+">p<-"> state of Al. <:f><:f,2Times New Roman,>This <:f><+"><:f,2Times New Roman,>V<-"><+&>L<-&>(<+">r<-">)<:f><:f,2Times New Roman,> is <:f><:f,2Times New Roman,>repulsive around <+">r <-">= 0<:f><:f,2Times New Roman,> due to the cancellation effect from the inner 2<+">p<-"> shell, which leads to a low electron density at small <+">r<-"> and results in a repulsive <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f,2Times New Roman,>(<+">r<-">) there to exclude electrons. <:f><:f,2Times New Roman,>On the other hand, in the contrary, the 3<:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,> component of Al has no <:f><:f,2Times New Roman,>cancellation <:f><:f,2Times New Roman,>effect and <:f><+">V<-"><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<:f,2Times New Roman,> is therefore attractive near <+">r<-"> = 0, and this will give significant error in bulk properties involving the 3<+">d<-"> character<:f><:f,2Times New Roman,> of the electrons. <:f><:f,2Times New Roman,>We therefore<:f><:f,2Times New Roman,> expect the more repulsive <:f><+"><:f,2Times New Roman,>V<+&><-&><-"><+&>L<-&>(<+">r<-">)<:f><:f,2Times New Roman,> derived from <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<:f,2Times New Roman,> to give worse <:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,>-scattering,<:f240,2Times New Roman,0,0,0> as shown in Fig.IV.3.4. <:f><:f,2Times New Roman,><:f><:f,2Times New Roman,><:f><:f,2Times New Roman,><:f><:f,2Times New Roman,>Moreover, the smaller the <+">r<-"><+'>c<-'> is the more repulsive the 2<+">p<-"> component will be <:f><:f,2Times New Roman,>(Fig.3.5)<-"><-'><-"><-"><:f><:f,2Times New Roman,>, and hence the<:f><:f,2Times New Roman,> worse the result<:f><:f,2Times New Roman,>, which is opposite to the common impression that using a small <+">r<-"><+'>c<-'> usually keeps a pseudopotential accurate.<-"><-"><-"><-'> <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#4260,9360><:f,2Times New Roman,>To address this problem, <-'><:f,2Times New Roman,>I have generated a <:f><+"><:f,2Times New Roman,>V<-"><+'>p<-'>(<+">r<-">) for<:f><:f,2Times New Roman,> Al<-"><:f,2Times New Roman,> using <+">Q<-"><+'>c<-'>-tuning (Chapter II) to make it as attractive as possible <:f><:f,2Times New Roman,>so that it can mimic the shape of <-"><:f><+"><:f,2Times New Roman,>V<-"><+'><+">d<-"><-'>(<+">r<-">) <:f><:f,2Times New Roman,>, while keeping the scattering of the <:f240,2Times New Roman,0,0,0><+">p<-">-wave<:f><:f,2Times New Roman,> correct. <:f><:f,2Times New Roman,>This is shown in Fig.IV.3.6 in comparison with the original Al pseudopotential components. <:f><:f><:f,2Times New Roman,>The solid state tests <[>Ref.T.1,M.1] done by D. I. Thomson<:f><:f,2Times New Roman,> and N. Marzari showed that this "<:f240,2Times New Roman,0,0,0><+">d<-"><:f,2Times New Roman,>-aware" <+">sp<-">-pseudopotential of Al out-performed<:f><:f,2Times New Roman,> the best of the original<:f><:f,2Times New Roman,> pseudopotentials in giving much more improved structural-critical quantities such as elastic tensor constants C<+'>11<-'> C<+'>12<-'> and C<+'>44<-'>, as shown in Table.IV.3.2. <-'><:f><:f,2Times New Roman,>The result of this study reveals the role of the <+">d<-">-non-locality of a pseudopotential<:f><:f,2Times New Roman,> in the elastic properties of Al fcc metal, and more importantly, it demonstrates<:f><:f,2Times New Roman,> how can one generate a reliable pseudopotential<:f><:f,2Times New Roman,> with <:f><:f,2Times New Roman,>very few non-local components, which reinforces <:f>the usefulness of Projector Reduction.<-"><-'> <:s><:#284,9360> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:#379,9360><:f320,2Times New Roman,><+!>IV.4. Discussion <-!><+!>and Conclusion<-!><:f> <:s><:S+-2><:#426,9360><:f240,,> <:S+-2><:#426,9360><+!><:f240,2Times New Roman,>Discussion<-!><:f> <:S+-2><:#426,9360><:f240,,> <:S+-2><:#12354,9360><:f240,,>In Section 2 we have applied the Projector Reduction technique on a number of cases to investigate the non-locality of pseduopotentials, <:f><:f240,,>in which the<:f><:f240,,> cancellation effect is not only used as an insightful classification<:f><:f240,,> but also<:f><:f240,,> it helps to explain the success of a simple local pseudopotential for Ge. For Ge, all 4<+">s<-">, 4<+">p<-"> and 4 <+">d<-"> valence states are cancelled, yeilding very weak (and hence more similar) <:f><+"><:f240,,>V<+'>l<-'><-"><:f><:f240,,>(<+">r<-">)<:f><:f240,,> corresponding to <+">l<-">=0,1,2, this therefore makes the use of a local pseudopotential for Ge possible. <:f><:f240,,>However, <:f><:f240,,>one should be careful in using <:f><:f240,,>cancellation effect<:f><:f240,,> or <:f><:f240,,><+">cancellation theorem<-"> <[>Ref.H.2]<:f><:f240,,> in the discussion of the non-locality of a pseudopotential <:f><:f240,,>because <:f><:f240,,>the effect has <:f><:f240,,>more important influences in some types of pseduopotential than others, in other words it depend on the type of the pseudopotential method. <:f><:f240,,>We have found that although the cancellation argument can stilled be used in <:f><:f240,,>norm-conserving pseudopotential, <:f><:f240,,>its predictbility on whether a local norm-conserving pseudopotential can be used for a given element should not be overemphasised.<:f> <:f240,,>Take Si for example, <:f><:f240,,>although both 3<+">s<-"> and 3<+">p<-"> are cancelled, a satisfactory Projector Reduction on its <+">V<+'>s<-'><-">(<+">r<-">) and <+">V<+'>p<-'><-"><:f><:f240,,>(<+">r<-">) is still hard to achieve. <:f><:f240,,>It also worth to distiguish <:f><:f240,,>the <:f><:f240,,>great similarity between the <+">Q<-"><+'>c<-'>-tuned <:f><+"><:f240,,>V<-"><+'><+">p<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,> and <:f><+"><:f240,,>V<-"><+'><+">d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,> of<:f><:f240,,> B and C<:f><:f240,,> (Section 2) from<:f> <:f240,,>the exact identity<:f><:f240,,> of <:f><+"><:f240,,>V<-"><+'><+">p<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,> and <:f><+"><:f240,,>V<-"><+'><+">d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,> as a <:f><:f240,,>consequence<:f><:f240,,> of OPW (Orthogonal Plane Wave)<:f><:f240,,> pseudopotential method. <:f><:f240,,>The pseudisation<:f><:f240,,> of <:f><:f240,,>OPW type pseudopotential<:f><:f240,,> is done by adding back the feature<:f><:f240,,> of core states to a wavefunction which is orthogonal to these core states. Since there is no orthogonoal <:f><:f240,,>core states for those un-cancelled <+">l<-"> components to be added back, the resulting OPW pseudopotentials should all be identical to the true potential. <:f><:f240,,>In the other words, there is no pseudising<:f><:f240,,> effect at all on the potential of un-cancelled atomic states (if one invert the radial Schr<\v>dinger<:f><:f240,,> equation by using such atomic wave functions).<:f><:f240,,> As for the no rm-conserving<:f><:f240,,> pseudopotentials<:f><:f240,,>, however, valence states are mortified<:f><:f240,,> in most cases no matter whether there are underlying core states with the same <+">l<-">. <:f><:f240,,>In other words the cancellation theorem predicts the OPW <:f><+"><:f240,,>V<-"><+'><+">p<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,> and <:f><+"><:f240,,>V<-"><+'><+">d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,> for B and C should always be identical<:f><:f240,,>, but for norm-conserving ones the similar prediction does not apply. <:f><:f240,,>It therefore <:f><:f240,,>requires a genuine effort to<:f><:f240,,> reduce<:f><:f240,,> the projectors<:f><:f240,,> of a norm-conserving pseudopotential<:f><:f240,,> even its valence states have no low lying core states with the same <+">l<-">, <:f><:f240,,>which is exactly what happened in the cases presented in Section 2. <:f><:f240,,>W<:f><:f240,,>e are pleased to find that our method works for norm-conserving pseudopotentials<:f><:f240,,>. The general limitation<:f><:f240,,> of applying the method, as explained in Chapter III, is the <+">r<-"><+'>c<-'> of the pseudopotential<:f><:f240,,> in a practical<:f><:f240,,> application. <:s><:S+-2><:#426,9360> <:S+-2><:#8520,9360>The above reduction of non-locality between <+">l<-">=0 and 1 for B, C, N and O has so far been the most successful application of Projector Reduction. It has solved a previous dilemma in our research group, namely how best to generate pseudopotentials for these elemets in KB form. There are several reasons to for and against the inclusion of the <+">d<-"> component in such pseudopotentials, but neither was clearly more appropriate than the other. The reasons for not including <+"><:f240,,>V<-"><+"><+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f> are : (1) to avoid the extra computing cost (both in CPU and memory), (2) the energies of the <+">d<-">-states for these elements are too high to be important <[>Ref.W.1], and (3) using a somewhat arbitrary ionised <+">d<-"> reference state in the KB form is not always justifiable. On the other hand, the reasons to use <+"><:f240,,>V<-"><+"><+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f> are : (1) the contribution from the <+">d<-">-component of Bloch states may be important, (2) the recipe for using (bounded) ionised <+">d<-">-state published in the BHS paper <[>Ref.B.1] is well established and accepted as a useful approach, and (3) some other reputable group using only <+"><:f240,,>V<-"><+"><+'>s<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f> and <+"><:f240,,>V<-"><+"><+'>p<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f> seems to get less convincing results <[>Ref.N.1]. Fortunately, the success in reducing the <+">p<-"><:f240,2Times New Roman,0,0,0><+">d<-"><:f> non-locality of the pseudopotentials of these elements results in a much more satisfactory situation because these pseudopotentials are : (1) extremely fast with a low memory demand (only <+">s<-"> is treated non-local, making the potential up to 8 times more efficient), (2) the BHS <+">d<-">-state pseudopotential is used, and (3) no KB form is needed for either <+">d<-"> or <+">p<-"> components. Equipped with<:f240,,> the experiences acquired from these first row elements and the case study of (<+">s<-">-non-local <:f240,2Times New Roman,0,0,0><+">p<-"><:f240,,>-local) Al described in Section 3, we have a better understanding and a better handling of the non-local components of a pseudopotential : both are essential for<:f><:f240,,> a <:f><:f240,,>more accurate strategy of pseudopotential design. <:S+-2><:#426,9360><+!><:f240,,> <:S+-2><:#426,9360><+!><:f240,2Times New Roman,>Conclusion<-!><:f> <:S+-2><:#426,9360> <:S+-2><:#5112,9360>In conclusion, we have demonstrated how to apply the Projector Reduction method on elements in various categories, and have used this<:f240,2Times New Roman,> to<:f><:f240,2Times New Roman,> investigate<:f><:f240,2Times New Roman,> systematically<:f><:f240,2Times New Roman,> the non-locality of the pseudopotential <:f>according to the nature of the element. We have restricted our discussion to non-<+">f<-"> elements. A simple argument based on the Free Electron model and the classical scattering picture suggested that, for non -<+">f<-"> elements, <+">l<-">=0,1,2 should be sufficient to account for the dominant angular momentum components of the occupied Bloch states. <:f240,2Times New Roman,>The results of Projector Reduction<:f><:f240,2Times New Roman,> further show that <:f><:f,2Times New Roman,>not all three <:f><:f,2Times New Roman,>parts <:f><+"><:f240,,>V<-"><+"><+'>s<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f>,<-'><-"><-"> <+"><:f240,,>V<-"><+"><+'>p<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f> and <+"><:f240,,>V<-"><+"><+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f> of the pseudopotential have to be expressed separately always.<:f,2Times New Roman,> Therefore the non-locality<:f><:f,2Times New Roman,> of the pseudopotential in muc h cases can be reduced and the numerical efficiency improved<:f><:f,2Times New Roman,>. <:f><:f,2Times New Roman,>Since the quality of such projector<:f><:f,2Times New Roman,> reduced pseudopotentials can always be<:f><:f,2Times New Roman,> examined<:f><:f,2Times New Roman,> <:f><:f,2Times New Roman,>using the standard logarithmic derivative test <:f><:f,2Times New Roman,>on the corresponding <+">l<-">-dependent scattering<:f><:f,2Times New Roman,>, <:f><:f,2Times New Roman,>this strategy makes<:f><:f,2Times New Roman,> the use of local or partly-local pseudopotentials more rigorous and systematic.<:f> <:S+-2><:#426,9360><:f240,2Times New Roman,> <:S+-2><:#5964,9360><:f240,2Times New Roman,>The case studies on Al, Si and Ge have shown that it is not always appropriate<:f><:f240,2Times New Roman,> to use a local pseudopotential, <:f><:f240,2Times New Roman,>particularly for Si among these three. For example the difference between Si and Ge, and the greater appropriateness of a local pseudopotential for the latter can be understood in terms of the cancellation effect as already rema rked.<:f><:f240,2Times New Roman,> <:f><:f,2Times New Roman,>From the view point of reproducing the logarithmic derivatives for valence states, we found that a local pseudopotential for Al is not better than that for Si. Thus there is no fundamental reason (from the atomic view point) why <:f><:f,2Times New Roman,>it should be easier or better to make the Al pseduopotential fully local than the Si one.<:f><:f,2Times New Roman,> We therefore<:f><:f,2Times New Roman,> think that a local pseudopotential works be tter for Al than for Si simply because m<:f><:f,2Times New Roman,>any applications of Al are less sensitive to those pseudopotential errors caused by insufficient non-locality, in particular in the treatment of <:f><+"><:f240,,>V<-"><+"><+'>s<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f> and <+"><:f240,,>V<-"><+"><+'>p<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f>. We suggest this is related to the metallic close packed structure of Al which involves less directional effects. However the success of the local pseudopotential for Al breaks down when more directionally sensitive properties are calculated, e.g. the <+">C<-"><+'>44<-'> elastic constant and the stacking fault energy. This is consistent with a greater role <+"><:f240,,>V<-"><+"><+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>) in such properties and the need of a non-local potential to calculate them.<:f> <:S+-2><:#426,9360> <:S+-2><:#2172,9360><:f240,,>In this context we have successfully generated a new pseudopotential for Al which only needs a non-local projector for <:f240,2Symbol,0,0,0><-">d<+"><:f240,,>V<-"><+"><+'>s<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,>. This was achieved by using <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,,><+'>c<-'>-tuning to generate a <:f240,2Times New Roman,0,0,0><+">p<-"><:f240,,>-potential which is attractive in shape so that it mimics <:f><+"><:f240,,>V<-"><+"><+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>) and can be chosen as the local part <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>)<:f><:f240,,>.<:f> The significant improvement of this Al pseudopotential shows the usefulness of the Projector Reduction technique in reducing the non-locality while keeping a pseudopotent ial reliable. <:S+-2><:#426,9360> <:S+-2><:#3834,9360><:f,2Times New Roman,>The methodology developed in this chapter and Chapter III has allowed us to <:f>explore the fundamental problem of the non-locality of a pseudopotential. It also provides a robust procedure that <:f,2Times New Roman,>leads to a new generation of pseudopotentials whose non-locality has been designed in a specific way according the need of a given application. <:f><:f,2Times New Roman,>This well-controlled<:f><:f,2Times New Roman,> non-locality makes a pseudopotential as efficient as possible while keeping control over the<:f><:f,2Times New Roman,> important angular momentum components. The approach is equally us eful when the pseudopotential is expressed in<:f><:f,2Times New Roman,> Kleimann-Bylander form. <:f><:f,2Times New Roman,>The ability to <:f><:f,2Times New Roman,>adjust and test the non-locality of a pseudopotential<:f><:f,2Times New Roman,> <:f><:f,2Times New Roman,>systematically <:f>represents a further step toward the better understanding and the technical enhancement of generating pseudopotentials for large scale electronic structure calculations. <:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,> <:s><:S+-1><:#379,9360><:f320,2Times New Roman,><+!>Acknowledgements<-!><:f> <:s><:S+-2><:#426,9360><:f,2Times New Roman,> <:S+-2><:#2556,9360><:f,2Times New Roman,>I would like to thank Dr. J-S. Lin for helping with the logarithmic derivative test program, which I used extensively in the current work. Dr R. J. Needs, Dr M.C. Payne and Prof. M.W.<:f,2Times New Roman,> Finnis kindly informed<:f><:f,2Times New Roman,> me of useful source of information. <:f><:f240,2Times New Roman,0,0,0>Thanks are due to N. Marzari and D. I. Thomson for their rigorous<:f><:f240,2Times New Roman,0,0,0> tests on the Al pseudopotentials<:f><:f240,2Times New Roman,0,0,0> generated by the me thod described in this chapter. Without those tests <:f>being quickly available, the progress of this project would have been slower. <:s><:S+-1><:#284,9360> <:s><:#284,9360><:f240,,> <:#284,9360><:f240,,> <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:S+-1><:#379,9360><:f320,2Times New Roman,><+!>References<-!><:f> <:S+-1><:#284,9360><:f,2Times New Roman,> <:S+-1><:#284,9360><:f,2Times New Roman,><[>B.1] <:S+-1><:#289,9360><:f,2Times New Roman,>G. B. Bachelet, D. R. Hamman, and M. Schluter, Phys. Rev. B <+!>26<-!>, 4199 (1982)<:f> <:S+-1><:#284,9360><:f,2Times New Roman,> <:S+-1><:#284,9360><:f,2Times New Roman,><[>F.1] <:S+-1><:#284,9360><:f,2Times New Roman,>M. W. Finnis, private communication <:S+-1><:#284,9360><:f,2Times New Roman,> <:S+-1><:#284,9360><:f,2Times New Roman,><[>G.1] <:S+-1><:#289,9360><:f,2Times New Roman,>L. Goodwin, R. J. Needs and V. Hiene, Phys. Rev. Lett <+!>60<-!>, 2050 (1988) <:S+-1><:#284,9360><:f,2Times New Roman,> <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0><[>H.1] <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0>D. R. Hamann, M. Schluter and C. Chaing, Phys. Rev. Lett <+!>43<-!>, 1494 (1979) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0><[>H.2] <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0>V. Heine, Pseudopotential Concept, <+">Solid State Physics<-"> Vol.24 (1970)<:f> <:S+-1><:#284,9360><:f,2Times New Roman,> <:S+-1><:#284,9360><:f,2Times New Roman,><[>H.3] <:S+-1><:#284,9360><:f,2Times New Roman,>V. Heine, <:f><:f,2Times New Roman,>and R. J. Needs<:f><:f,2Times New Roman,>, private communication <:S+-1><:#284,9360><:f,2Times New Roman,> <:S+-1><:#284,9360><:f,2Times New Roman,><[>K.1] <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0>R. D. King-Smith, M. C. Payne and J-S. Lin, Phys. Rev. B <+!>44, 13063<-!> (1991)<:f> <:S+-1><:#284,9360><:f,2Times New Roman,> <:S+-1><:#284,9360><:f,2Times New Roman,><[>K.2] <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0>L. Kleinman and D.M. Bylander, Phys Rev. Lett. <+!>4<-!>, 1425 (1978)<:f> <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>K.3] <:S+-1><:#289,9360>G. P. Kerker, J. Phys. C <+!>13<-!>, L198 (1980) <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>L.1] <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0>J-S. Lin, A. Qteish, M. C. Payne and V. Heine<:f240,2Times New Roman,0,0,0>, Phys Rev B <+!>47,<-!> 4174 (1993)<:f> <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>M.1] <:S+-1><:#284,9360>N. Mazari, <+">TCM First Year Report <-">(The solid state tests was done mainly by N. Marzari) <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>N.1] <:S+-1><:#284,9360>R. J. Needs, private communication. <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>N.2] <:S+-1><:#289,9360>M. Needls, M. C. Payne and J. D. Joannopoulos, Phys. Rev. B <+!>38<-!>, 5543 (1998) <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>P.1] <:S+-1><:#284,9360>M. C. Payne, private communication. <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>R.1] <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0>A.M. Rappe, K.M. Rabe, K. Kaxiras and J.D. Joannopoulos<:f><:f240,2Times New Roman,0,0,0>, Phys. Rev. B <+!>41,<-!> 1227 (1990)<:f> <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>R.2] <:S+-1><:#289,9360>I. J. Robertson, M. C. Payne and V. Heine, Europhys. Lett., <+!>15<-!> (3), 301 (1991) <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>S.1] <:S+-1><:#289,9360>I. Stich, M.C. Payne, A De Vita, M. J. Gillan and L. J. Clarke, Chem. Phys. Lett. <+!>213<-!>, 422 (1993) <:S+-1><:#284,9360> <:S+-1><:#284,9360><[>T.1] <:S+-1><:#284,9360>D. I. Thomson, <+">TCM First Year Report <-"> <:#284,9360><:f240,,> <:#284,9360><:f240,,><[>T.2] <:#289,9360><:f240,,>N. Troullier and J. L. Martins, Phys. Rev. B<+!>43<-!>, 1993 (1991) <:#284,9360><:f240,,> <:#284,9360><:f240,,><[>W.1] <:#284,9360><:f240,,>R. Wentzcovich, private communication. <:#284,9360><:f240,,> <:#284,9360><:f240,,> <:#284,9360><:f240,,> <:#284,9360><:f240,,> <:#284,9360><:f240,,> <:S+-1><:#379,9360><:f320,,><+!>Table<-!><:f> <:S+-1><:#284,9360> <:S+-1><:#284,9360>TABLE.1 Lattice constants and bulk moduli predicted by Al pseudopotentials with different <+">r<-"><+'>c<-'>. <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:#202,9360><:f200,QCourier,>(E_cut=150 eV, brodening=3eV) <:#202,9360><:f200,QCourier,> <:S+-2><:#360,9360><:f200,2Times New Roman,>=======================================================<:f> <:S+-2><:#360,9360><:f200,2Times New Roman,> Potential <+">r<-"><+'>c<-'> (a.u.) <+">Q<:f200,2Times New Roman,0,0,0><-"><+'>c<-'><:f200,2Times New Roman,>/<:f200,2Times New Roman,0,0,0><+">q<:f200,2Times New Roman,0,0,0><-"><+'>3<-'><:f200,2Times New Roman,>(<:f200,2Times New Roman,0,0,0><+">s<-"><:f200,2Times New Roman,>) <:f200,2Times New Roman,0,0,0><+">Q<:f200,2Times New Roman,0,0,0><-"><+'>c<-'><:f200,2Times New Roman,>/<:f200,2Times New Roman,0,0,0><+">q<-"><:f200,2Times New Roman,><+'>3<-'>(<:f200,2Times New Roman,0,0,0><+">p<-"><:f200,2Times New Roman,>) <+">a<-"><+'>0<-'> (<\E>) <+">B <-">(GPa) <:S+-2><:#360,9360><:f200,2Times New Roman,>======================================================= <:S+-2><:#303,9360><:f200,QCourier,> Al009 1.6 0.65 0.75 4.15 68.8 <:S+-2><:#303,9360><:f200,QCourier,> Al007 2.2 0.65 0.90 4.12 69.4 <:S+-2><:#303,9360><:f200,QCourier,> Al006 2.2 0.50 0.90 4.12 70.0<:f> <:S+-2><:#303,9360><:f200,QCourier,> Al013 2.4 1.10 1.00 4.08 72.0 <:S+-2><:#303,9360><:f200,QCourier,>------------------------------------------------------- <:S+-2><:#303,9360><:f200,QCourier,> Expt. - - - 4.05 79.4 <:S+-2><:#303,9360><:f200,QCourier,>-------------------------------------------------------<:f> <:S+-2><:#303,9360><:f200,QCourier,> Al013a 2.4 1.10 1.10 4.02 81.5<:f> <:S+-2><:#360,9360><:f200,,>=======================================================<:f> <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:S+-1><:#284,9360>TABLE.1 The comparison of bulk properties of fcc metal Al using different Al pseudopotentials <:S+-1><:#284,9360> <:S+-1><:#240,9360><:f200,,> <:S+-2><:#360,9360><:f200,,>================================================================== <:S+-2><:#360,9360><:f200,,>Pseduopotential <:f><+"><:f200,2Times New Roman,0,0,0>a<-"><:f200,,><+'>0<-'> (<\E>) <:f200,2Times New Roman,0,0,0><+">B<-"><:f200,,><+'>0<-'> (GPa) <:f200,2Times New Roman,0,0,0><+">B<-"><:f200,,><+'>0<-'>' <+">C<-"><:f200,2Times New Roman,0,0,0><+'>11<-'><:f200,,> <:f200,2Times New Roman,0,0,0><+">C<-"><:f200,,><+'>12<-'> <:f200,2Times New Roman,0,0,0><+">C<-"><:f200,,><+'>44<-'> <:S+-2><:#360,9360><:f200,,>==================================================================<:f> <:S+-2><:#303,9360><:f200,QCourier,>al.4<+&>a<-&> <:f200,QCourier,>3.969 79.90 4.82 96.54 54.19 29.10 <:S+-2><:#303,9360><:f200,QCourier,>al2001<+&>b<-&> <:f200,QCourier,>3.960 83.47 4.51 103.6 51.76 29.60 <:S+-2><:#303,9360><:f200,QCourier,>Al013<+&>c<-&> <:f200,QCourier,>4.085 71.9 4.69 69.20 55.66 17.80 <:S+-2><:#303,9360><:f200,QCourier,>Al013a<+&>d<-&> <:f200,QCourier,> 4.022 76.7 5.56 103.8 60.8 28.5 <:S+-2><:#360,9360><:f200,2Times New Roman,>Experiment <:f200,QCourier,> <:f200,QCourier,> 4.05 79.4 5.15 114.3 61.92 31.62<:f> <:S+-2><:#360,9360><:f200,,>==================================================================<:f> <:S+-1><:#312,9360>a. Ref.T.2 (with <:f240,2Symbol,0,0,0>d<:f><+">V<+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)) <:S+-1><:#312,9360>b. Ref.K.3 (with <:f240,2Symbol,0,0,0>d<:f><+">V<+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)) <:S+-1><:#312,9360>c. The original <+">sp<-"> only pseudopotential with a less attractive <+">V <-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) (without <:f240,2Symbol,0,0,0>d<+"><:f>V<+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)) <:S+-1><:#312,9360>d. The specially <+">Q<-"><+'>c<-'>-tuned one with a more attractive <+">V <-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) (without <:f240,2Symbol,0,0,0>d<:f><+">V<+'>d<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)) <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:S+-1><:#284,9360> <:S+-1><:#379,9360><:f320,,><+!>Figures<-!><:f> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.1(a) <:S+-2><:#1278,9360><+">V<-"><+"><+'>l <-'><-">(<+">r<-">) components of the C pseudopotential, only <+">V<-"><+'>p<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'>d<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) are shown, the high degree of similarity between them ensures the logarithmic derivative of <+">d<-">-state to be faithfully reproduced by <+">V<-"><+'>p<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) solely, as can be seen from the comparison on Fig. 2.1(b) and 2.1(d). <:f200,,>(C021)<:f> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.1(b) <:S+-2><:#468,9360>The logarithmic derivative of <+">p<-"> and <+">d<-">-state by the C pseudopotential with its <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'>d<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) kept. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.1(c) <:S+-2><:#468,9360>The logarithmic derivative of <+">p<-"> and <+">d<-">-state by the C pseudopotential with its <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'>d<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) removed. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.2(a) <:S+-2><:#426,9360><+">V<-"><+"><+'>l <-'><-">(<+">r<-">) components of the B pseudopotential, only <+">V<-"><+'>p<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'>d<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) are shown. <:f200,,>(B001)<:f> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.2(b) <:S+-2><:#468,9360>The logarithmic derivative of <+">p<-"> and <+">d<-">-state by the B pseudopotential with its <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) kept. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.2(c) <:S+-2><:#468,9360>The logarithmic derivative of <+">p<-"> and <+">d<-">-state by the B pseudopotential with its <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) removed. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.3(a) <:S+-2><:#786,9360>The<-"> <:f240,2Times New Roman,0,0,0><+">p<-"><:f> and <+">d<-"> components <+">V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) of the Br pseudopotential, which are <+">Q<-"><+'>c<-'>-tuned similar.<-"> <:f200, Times New Roman,0,0,0>(Br000)<:f> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.3(b) <:S+-2><:#852,9360>The logarithmic derivative of<+"> p<-"> and <+">d<-">-states by the Br pseudopotential. (Pseudo : solid line, all-electron : dased line.) <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.4(a) <:S+-2><:#786,9360>The <+">V<-"><+'><+">s<-"><-'>(r) and <+">V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) <-"><-">components of the Fe pseudopotential, they are <+">Q<-"><+'>c<-'>-tuned <-"><-"><-">to be very similar. <:f200, Times New Roman,0,0,0>(Fe002)<:f> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.4(b) <:S+-2><:#894,9360>The logarithmic derivative of <+">s<-"> and <+">p<-">-state by the Fe pseudopotential with its <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'><+">s<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) removed<-">. (Pseudo : solid line, all-electron : dased line.) <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.5(a) <:S+-2><:#426,9360>The<-"><-"><-"> <+">V<-"><+'><+">s<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) components of the Ge pseudopotential which is to be made local. <:f200,,>(Ge010)<:f> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.5(b) <:S+-2><:#1320,9360>The logarithmic derivative of <+">s<-"> and <+">p<-">-states by the Ge pseudopotential with <[>0.8<+">V<-"><+'>s<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) + 0.2<+">V<-"><+'>p<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)] chosen as local component <+">V<+&> <-&><-"><+&>L<-&>(<+">r<-">), with all non-local parts <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'>s<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'>p<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><:f><+'>d<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) removed, thus this is actually a local Ge pseudopotential. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.2.5(c) <:S+-2><:#852,9360>The logarithmic derivative of <+">s<-"> and <+">p<-">-states by the same Ge pseudopotential but with the full projectors. A better accuracy of such potential then the local one is expected. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.1(a) <:S+-2><:#426,9360>The <+">V<-"><+'><+">s<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) used to construct the local pseudopotential for Al. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.1(b) <:S+-2><:#852,9360>The logarithmic derivative of <+">s<-"> and <+">p<-">-states by the local Al pseudopotential. (Pseudo : solid line, all-electron : dashed line.) <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.1(b) <:S+-2><:#852,9360>The logarithmic derivative <+">d<-">-state by the local Al pseudopotential. (Pseudo : solid line, all-electron : dashed line.) <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.2(a) <:S+-2><:#426,9360>The <+">V<-"><+'>s<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>), <+">V<-"><+'>p<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'>d<-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) which are used to construct the local Si pseudopotential. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.2(b) <:S+-2><:#852,9360>The logarithmic derivative of the<+"> s<-"> and <+">p<-">-states using the local Si pseudopotential. (Pseudo : solid line, all-electron : dashed line.) <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.2(c) <:S+-2><:#852,9360>The <+">d<-">-states logarithmic derivative by the local Ge pseudopotential. (Pseudo : solid line, all-electron : dashed line.) <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.3(a) <:S+-2><:#426,9360>The <+"> <-"><+">V<-"><+'><+">s<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<+">, V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) used to construct the local pseudopotential for Ge. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.3(b) <:S+-2><:#852,9360>The logarithmic derivative <+">d<-">-state by the local Ge pseudopotential. (Pseudo : solid line, all-electron : dashed line.) <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.4 <:S+-2><:#1278,9360>The logarithmic derivatives of d-state by the <+">V<-"><+&>L<-&>(<+">r<-">) of Al pseudopotentials with different degree of attractiveness. (All-electron : thick dots, attractive <+">V<-"><+&>L<-&>(<+">r<-">) : dashed line, flat <+">V<-"><+&>L<-&>(<+">r<-">) : solid line, repulsive <+">V<-"><+&>L<-&>(<+">r<-">) : dot-dash.) <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.5 <:S+-2><:#426,9360><+">V<+'>p<-'><-">(<+">r<-">) of Al pseudopotentials with different <+">r<-"><+'>c<-'>. <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360>FIG.IV.3.6 <:S+-2><:#1704,9360>The <+">V<:f240,2Times New Roman,0,0,0><-"><+"><+'>p<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) and <+">V<-"><+'><+">d<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) of a typical Al pseudopotential, here two possible <+">V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) are also shown, the best <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) generated from usual approach is shown in dashed line which is much less attractive then the typical <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">d<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) (dot-dash). The "<+">d<-">-aware" <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">p<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) in shown in full line, its more attractive feature and its effect to imitate <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">d<-"><-'><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) is obvious. <:#284,9360><:f240,,> > Times New Roman,18,12,0,0,0,0,0 $$l_{\text{max}}\ =\ \left( \frac 14\pi \,N\right) ^{\frac 13}$$SSh$h``@ÀùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿùùùÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿùÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùùÿÿÿÿùÿùÿÿÿùÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿùÿÿÿÿÿùÿùÿÿÿùÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿùùùùùùùùÿÿÿÿùÿÿùÿÿÿÿùÿÿùÿÿùÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿùÿÿÿùÿùÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿùÿÿÿùÿùÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿùÿÿÿÿÿùÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿÿùùÿÿùùùÿÿÿÿùÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿùùùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿTimes New Roman,18,12,0,0,0,0,0 $$\rho \ =\ \frac NV\ =\ \frac N{\frac 43\pi \,r_0^3}$$SSf&f``@ÀùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùùùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùùùùùùùùùùùùùùùùùùùùùùùùùùÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùùÿÿùùÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿùÿÿÿÿùÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùÿÿÿÿÿùÿÿùÿÿÿÿùùÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿùÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿùÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿùÿÿùÿÿÿÿùÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿÿÿùùÿÿÿùÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿTimes New Roman,18,12,0,0,0,0,0 $l_{\max }\ =\ r_0k_F$SSMN``@ÀùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿùùÿùùÿÿÿÿÿÿÿÿÿùÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿùùÿÿÿÿùÿùÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿùÿÿÿÿùÿÿùÿÿÿùùùÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿùÿÿÿøøÿøøÿøÿøÿÿøøøøøÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿùÿÿùÿÿÿùÿùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿùÿÿÿÿÿøøÿøøÿøÿøÿÿøÿøÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿùÿÿùÿÿùÿùÿÿÿÿÿÿùùùÿÿÿÿÿÿÿùùÿÿÿÿøÿÿøÿÿøøøÿÿÿøÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùÿÿùÿÿùÿÿÿùÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿøÿÿøÿÿøÿøÿÿøÿøÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿøøøøøøøøøøøøÿÿøøÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿTimes New Roman,18,12,0,0,0,0,0 $k_F\ =\ (3\pi ^2\rho )^{\frac 13}$SSYZ``@Àùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿÿÿùÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿÿùÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿùÿÿÿÿùùùùùùÿùÿùÿÿÿÿÿùùÿÿÿÿÿùÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿùùÿÿÿùÿùÿÿùÿùùùùÿÿÿÿùÿÿùÿÿÿÿùÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿùÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿùÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿùùùùùÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿùÿÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿùÿÿÿÿÿùùÿÿÿÿÿÿÿÿÿÿÿùÿùÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿùÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿùÿÿÿùÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿùÿùÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿùÿÿÿùÿÿùÿÿÿÿÿÿÿÿÿùÿÿùÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿùÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿùùùùÿÿÿùÿÿÿùùÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùùùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿùÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ [Embedded] 6 .tex 89167 97 89264 3762 5 .tex 93026 88 93114 3894 4 .tex 97008 55 97063 1422 3 .tex 98485 68 98553 1818 00100373