[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 III, June 1995 Edition 802056114 7 800149128 1968 15 0 0 0 0 1 [fopts] 0 1 0 0 [lnopts] 2 Body Text 1 [docopts] 5 2 [GramStyle] [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 537395328 5728 12096 8344 12452 0 1 3 0 0 0 0 0 0 0 0 16777215 6 0 4288 2616 222 [frmname] Frame6 [frmlay] 12452 2616 1 0 0 1 12096 0 0 2 0 18 1 0 1 5728 8344 0 [isd] .X6 .tex .X6 1 1 0 0 2614 65247 100 0 0 .tex 0 65314 0 [frm] 1 537395328 1440 12096 4421 12465 0 1 3 0 0 0 0 0 0 0 0 16777215 5 1 0 2981 222 [frmname] Frame5 [frmlay] 12465 2981 1 0 0 1 12096 0 0 2 0 0 0 0 1 1440 4421 0 [isd] .X5 .tex .X5 1 1 0 0 2992 65247 100 0 0 .tex 0 65314 0 [frm] 1 537395328 1440 10530 6395 10834 0 1 3 0 0 0 0 0 0 0 0 16777215 4 2 0 4955 222 [frmname] Frame4 [frmlay] 10834 4955 1 0 0 1 10530 0 0 2 0 0 0 0 1 1440 6395 0 [isd] .X4 .tex .X4 1 1 0 0 4962 65247 100 0 0 .tex 0 65314 0 [frm] 1 537395328 1440 1440 7057 2176 0 1 3 0 0 0 0 0 0 0 0 16777215 3 3 0 5617 442 [frmname] Frame3 [frmlay] 2176 5617 1 0 0 1 1440 0 0 2 0 0 0 8 1 1440 7057 0 [isd] .X3 .tex .X3 1 1 0 0 5700 64864 100 0 0 .tex 0 65094 0 [frm] 1 537395328 1825 9824 5815 10473 0 1 3 0 0 0 0 0 0 0 0 16777215 7 4 385 3990 366 [frmname] Frame7 [frmlay] 10473 3990 1 0 0 1 9824 0 0 2 0 8 0 1487 1 1825 5815 0 [isd] .X7 .tex .X7 1 1 0 0 3993 64987 100 0 0 .tex 0 65170 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,,>III. Non-local Projector Reduction in Optimised Pseudopotentials<: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,III-><:f> > [elay] [l1] 0 [pg] 15 20 286 99 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 29 0 29 0 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 38 121 130 0 0 0 0 65534 194 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 55 935 289 0 0 0 0 65534 1181 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 59 3234 275 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 64 0 0 96 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 73 0 29 0 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 80 0 29 0 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 90 0 0 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 0 100 1946 100 32 0 0 0 65534 1907 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 128 0 29 0 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 152 0 5 512 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 202 0 29 0 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 244 276 18 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 0 268 0 154 1025 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 [edoc] <+B><:#568,9360><:f480,,> <+B><:s><:#568,9360><:f480,,><+!>III<-!> <+B><:s><:#568,9360><+!><:f480,,> <+B><:s><:#568,9360><+!><:f480,,>Non-local P<-!><:f><+!><:f480,,>r<-!><:f><+!><:f480,,>o<-!><:f><+!><:f480,,>jector Reduction <+B><:s><:#568,9360><+!><:f480,,>in <+B><:s><:#568,9360><+!><:f480,,>Optimised Pseudopotentials <+B><:s><:#288,9360><+!><:f240,,> <+B><:s><:#288,9360><+!><:f240,,> <+B><:s><:#288,9360><+!><:f240,,> <+B><:s><:#288,9360><+!><:f240,,> <+B><:s><:#288,9360><+!><:f240,,> <+B><:s><:#288,9360><+!><:f240,,> <:S+-2><-!><:f240,2Times New Roman,0,0,0>In this chapter we describe<:f><:f240,2Times New Roman,0,0,0> a method to reduce the number of non-local components<:f><:f240,2Times New Roman,0,0,0> of a pseudopotential, i.e. the number of different<:f><:f240,2Times New Roman,0,0,0> radial pseudopotentials acting on different angular momenta<:f><:f240,2Times New Roman,0,0,0> in the spherical Schr<\v>dinger equation. The benefit<:f><:f240,2Times New Roman,0,0,0> of treating a pseudopotential in this way is significant for large scale <+">ab initio<-"><:f><:f240,2Times New Roman,0,0,0> calculations, especially when the non-local part of the pseudopotential is implemented<:f><:f240,2Times New Roman,0,0,0> in real-space in the manner of <:f><:f240,2Times New Roman,0,0,0>Ref.K.2<:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>Furthermore<:f><:f240,2Times New Roman,0,0,0>,<:f><:f240,2Times New Roman,0,0,0> when the non-local components of a pseudopotential are expressed<:f><:f240,2Times New Roman,0,0,0> in the form proposed by <:f><:f240,2Times New Roman,0,0,0>Kleinman and Bylander<:f><:f240,2Times New Roman,0,0,0> <[>Ref.K.1], <:f><:f240,2Times New Roman,0,0,0>eliminating some non-local components<:f><:f240,2Times New Roman,0,0,0> can sometimes help to improve the accuracy of the pseudopotential <:f><:f240,2Times New Roman,0,0,0>because<:f><:f240,2Times New Roman,0,0,0> the Kleinman-Bylander form is approximate and can be somewhat ill-behaved on some components<:f><:f240,2Times New Roman,0,0,0> in some cases. <:f><:f240,2Times New Roman,0,0,0>Some tests will be presented to show these features of the pseudopotentials <:f><:f240,2Times New Roman,0,0,0>generated using this Projector (Component) Reduction technique. We will also present a theoretical interpretation of the procedure<:f><:f240,2Times New Roman,0,0,0> and its limitations<:f><:f240,2Times New Roman,0,0,0>.<:f> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#376,9360><:f320,,><+!>III.1. Introduction<-!><:f> <:s><:S+-1><:#280,9360> <:S+-2><:#4260,9360>The introduction of <+">ab initio<-"> norm-conserving pseudopotentials<:f240,2Times New Roman,0,0,0> by<:f><:f240,2Times New Roman,0,0,0> Hamann, Schluter and Chiang<:f><:f240,2Times New Roman,0,0,0> <[>Ref.H.1]<:f> was a significant step in pseudopotential theory. The "norm-conserving" condition assures that the pseudopotentials reproduce the electrostatic interaction between atoms as well as the scattering of the electronic states by the ions. These features make the pseudopotentials extremely accurate and make them important for modern electronic structure calculations. To generate an <+">ab initio<-"> pseudopotential, one first finds the all-electron wave functions for an atom : <-">the pseudopotential is then generated by pseudising the <+">l<-">-dependent radial wave functions and then inverting the radial Schr<\v>dinger equation, as described in more detail in <:f240,2Times New Roman,0,0,0>Chapter I<:f>. The resulting pseudopotential is therefore non-local because of its <+">l<-">-dependence. More precisely, due to the construction from all the <+">l<-">-channels, the pseudopotential has the form <-"> <:s><:S+-2><:#420,9360> <:S+-2><:#444,9360><+"><:f240,2Times New Roman,0,0,0>V<:f><+&> <-">PS<-&><+"> = <:f240,2Symbol,0,0,0><-">S<+"><-'><:f><:f240,2Times New Roman,0,0,0><+'>lm<-'><:f><-'> |<:f240,2Times New Roman,0,0,0>Y<+'>lm <:f240,2Times New Roman,0,0,0><-'><-"><;><+"><:f240,2Times New Roman,0,0,0> <:f><-'><:f240,2Times New Roman,0,0,0>V<+'>l<-'><:f><:f240,2Times New Roman,0,0,0><+&> <-"><-&><:f>(<+">r<-">) <-'><:f><+&><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0><-&><<<+"><:f240,2Times New Roman,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'>lm <-'><-"><:f>|<-'><-'><-'><-'><-'><-'> , (1.1)<-'><-'> <:s><:S+-2><:#420,9360> <:S+-2><:#4308,9360>in which the <+"><:f240,2Times New Roman,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'>lm<-"><-'><:f> are the spherical harmonics. The pseudopotential in (1.1) is local in the radial coordinate but non-local in the angular coordinates, which is why it is also de scribed as "semi-local". In most cases such a pseudopotential is expressed in terms of a local part <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>), which is common for all <+">l<-">, and a non-local part <:f240,2Symbol,0,0,0>d<+"><: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>) = <+">V<+'>l<-"><-'>(<+">r<-">)<:f240,2Symbol,0,0,0>-<+"><:f>V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) for each angular momentum <:f240,2Times New Roman,0,0,0><+">l<-"><:f>, with only <:f240,2Symbol,0,0,0>d<+"><: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>) of a few lower <+">l<-"> treated explicitly in a calculation. In this way, the <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) is acting as the (pseudo) potential for the rest of the high <+">l<-"> states. Since the scattering of very high <+">l<-"> is not important (because it corresponds to either high kinetic energy or long interaction distance), as long as a sufficient number of <:f240,2Symbol,0,0,0>d<+"><: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>) have been treated explicitly, the choice of the local <+">V<-"><+&>L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) can be fairly arbitrary without losing the accuracy of the pseudopotential. With this transformation, (1.1) becomes <:s><:S+-2><:#420,9360> <:S+-2><:#444,9360><:f240,2Times New Roman,0,0,0><+">V<-"><:f><+&> PS <-&>= <:f240,2Symbol,0,0,0>S<-"><-'><:f><:f240,2Times New Roman,0,0,0><+"><+'>lm<-"><-'><:f><-'> |<:f240,2Times New Roman,0,0,0><+">Y<+'>lm<-'><-"><:f><;><:f240,2Times New Roman,0,0,0><+"> <-"><:f><-'><:f240,2Times New Roman,0,0,0><[><:f><+">V<-"><+&> L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) <:f240,2Symbol,0,0,0>+ d<+"><: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>)] <-'><-&><-'><-"><-"><+&><:f240,2Times New Roman,0,0,0> <-&><:f><<<:f240,2Times New Roman,0,0,0><+">Y<:f240,2Times New Roman,0,0,0><+'>lm<-'><-"><:f>| <-"> = <-"><-'><+">V<-"><+&> L<-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<-'><-&><-'><-"><-"><-'> <:f240,2Symbol,0,0,0>+<:f> <:f240,2Symbol,0,0,0>S<-"><-'><:f><:f240,2Times New Roman,0,0,0><+"><+'>lm<-"><-'><:f><-'> |<:f240,2Times New Roman,0,0,0><+">Y<+'>lm<-'><-"><:f><;><-"><-'><-"><:f240,2Symbol,0,0,0> d<+"><: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>) <-'><-&><-'><-"><-"><+&><:f240,2Times New Roman,0,0,0> <-&><:f><<<:f240,2Times New Roman,0,0,0><+">Y<:f240,2Times New Roman,0,0,0><+'>lm<-'><-"><:f>| . (1.2) <:s><:S+-2><:#420,9360> <:S+-2>Because of the need to perform 2<+">l<-">+1 projections with the <+">Y<+'>lm<-'><-"> for each <:f240,2Symbol,0,0,0>d<+"><: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 non-local (semi-local) pseudopotential is computationally much more expensive than a local one. Much effort has been expended to improve the efficiency, such as the use of the <:f240,2Times New Roman,0,0,0>Kleinman-Bylander (KB) <:f><:f240,2Times New Roman,0,0,0><[>Ref.K.1] <:f><:f240,2Times New Roman,0,0,0>form<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>which approximates<:f><:f240,2Times New Roman,0,0,0> the double-variable integral<:f><:f240,2Times New Roman,0,0,0> as a multiplication of two single-variable<:f><:f240,2Times New Roman,0,0,0> ones<:f><:f240,2Times New Roman,0,0,0> , <:s><:S+-2><:#420,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A3> , (1.3) <:s><:S+-2><:#420,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#2592,9360><:f240,2Times New Roman,0,0,0>in which <:f240,2Symbol,0,0,0>F<:f240,2Times New Roman,0,0,0><+&>0<-&><+'><+">lm<-"><-'>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)<+"> = <:f240,2Symbol,0,0,0>j<:f240,2Times New Roman,0,0,0><-"><+&>PS<-&><+'><+">l<-"><-'>(<+">r<-">)<+">Y<+'>lm<-'><-">(<:f240,2Symbol,0,0,0>q<:f240,2Times New Roman,0,0,0>,<:f240,2Symbol,0,0,0>f<:f240,2Times New Roman,0,0,0>), with <:f><+"><:f240,2Symbol,0,0,0>j<:f240,2Times New Roman,0,0,0><-"><-&><-&><-"><-'><+&>PS<-&><+'><+">l<-"><-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0> being the pseudo wavefunction used to create <+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>). <:f>A real-space formalism of KB form has been proposed by King-Smith <+">et al<-"> <:f240,2Times New Roman,0,0,0><[>Ref.K.2] <:f>to<-&> allow the KB form to be used reliably in real-space, which has the advantage of a much more favourable scaling with respect to the system size when performing non-local operations. Even with those significant improvements, the 2<+">l<-">+1 spherical harmonic projections for each <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><+'><+">l<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> <:f>are still a very heavy numerical burden in <+">ab initio<-"> calculations. <:s><:S+-2><:#420,9360> <:S+-2><:#3804,9360>On the other hand, the KB form can also be a source of error because it needs to use pseudo wave functions <+"><:f240,2Symbol,0,0,0>j<:f240,2Times New Roman,0,0,0><-"><-&><-&><-"><-'><+&>PS<-&><+'><+">l<-"><-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0> <:f>as reference states, as shown in (1.3). It is easy to verify that the KB form is most accurate (in fact it is exact) only when the wave functions of the system under study are identical to the reference wave functions within the region defined by the ps eudising radius. This explains why the accuracy of a KB pseudopotential depends on the choice of reference states, which also implies that using a reference state that is quite different from the typical wave function in the application can cause a large er ror in a KB pseudopotential. Apart from the problem of the reference states, the possibility of KB form being completely wrong, due to so called "ghost states", has also been reported and analysed by Gonze<:f240,2Times New Roman,0,0,0> <[>Ref.G.1]<:f>. <:s><:S+-2><:#420,9360> <:S+-2><:#2964,9360>Various ways have been used to reduce the number of spherical harmonic projection and to improve the accuracy of the KB form. Usually one simply ignores some higher <+">l<-"> non-local components that are not occupied in atomic ground states, hoping that the local part can account for them adequately. <:f240,2Times New Roman,0,0,0>To avoid the ghost states or to obtain a better accuracy, one can vary the pseudopotential<:f><:f240,2Times New Roman,0,0,0> generating parameters such as the electronic<:f><:f240,2Times New Roman,0,0,0> configuration, the <+">r<-"><+'>c<-'> and the choice of the local potential <:f><:f240,2Times New Roman,0,0,0><[>Ref.G.1]<:f><:f240,2Times New Roman,0,0,0>. One may also include more reference states in the KB form as suggested by <:f><:f240,2Times New Roman,0,0,0>Bl<\v>chl <:f><:f240,2Times New Roman,0,0,0><[>Ref.B.1]<:f><:f240,2Times New Roman,0,0,0>. <:s><:S+-2><:#420,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#3468,9360><:f240,2Times New Roman,0,0,0>The above approaches work in their own right, but lack the advantage of addressing both the question of efficiency and of accuracy together in a single method. <:f><:f240,2Times New Roman,0,0,0>In this chapter, we will show that some of the<:f> <:f240,2Symbol,0,0,0>d<+"><: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>) in (1.2)<:f240,2Times New Roman,0,0,0> can be eliminated without loss of the overall accuracy of th e <:f>pseudopotential<:f240,2Times New Roman,0,0,0>. This is achieved by making these <:f><:f240,2Symbol,0,0,0>d<+"><: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>) <:f240,2Times New Roman,0,0,0>very small so that they are negligible<:f><:f240,2Times New Roman,0,0,0>. <:f>It is clear that using the least possible number of non-local parts <:f240,2Symbol,0,0,0>d<+"><: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>) in a pseudopotential helps not only to avoid the need for the KB form, but also reduces the number of <+"><:f240,2Times New Roman,0,0,0>Y<+'>lm<-"><-'><:f> projections. The approach of safely eliminate <:f240,2Symbol,0,0,0>d<+"><: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>), which we shall call Projector Reduction, is the main issue to be discussed in this chapter. <:s><:S+-2><:#420,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#2100,9360><:f240,2Times New Roman,0,0,0>In the following sections, we will describe the <:f><:f240,2Times New Roman,0,0,0>Projector Reduction<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>method <:f><:f240,2Times New Roman,0,0,0>and give two examples in Section 2.<:f><:f240,2Times New Roman,0,0,0> <:f>In Section 3, solid state tests of <:f240,2Times New Roman,0,0,0>a projector reduced pseudopotential will be presented, and comparison with a commonly used <:f><:f240,2Times New Roman,0,0,0>pseudopotential<:f><:f240,2Times New Roman,0,0,0> will be given. The discussion of the simple theory and the limitations of this method forms the Section 4. Finally we will make a brief conclusion in Section 5.<:f> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#376,9360><:f320,,><+!>III.2. The Projector Reduction Method and Examples<-!><:f> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#1728,9360><:f240,2Times New Roman,0,0,0>The<:f><:f240,2Times New Roman,0,0,0><+"> Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>-tuning method described in Chapter II allow us to change the shape of <+"><:f240,2Times New Roman,0,0,0>V<-"><+'><+">l<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> individually<:f><:f240,2Times New Roman,0,0,0>. With this ability, <:f><+"><:f240,2Times New Roman,0,0,0>we can make two l-dependent<-"><:f><+"><:f240,2Times New Roman,0,0,0> potential components <-"><:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'><+">i<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f> <+">and<-"> <+"><:f240,2Times New Roman,0,0,0>V<-"><+'><+">j<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><-'><-'><-'><-'><-"><+"><:f240,2Times New Roman,0,0,0> with l = i, j as similar as possible<-">. <:f><:f240,2Times New Roman,0,0,0> Once this is<:f><:f240,2Times New Roman,0,0,0> achieved, and knowing that the local potential <:f><+"><:f240,2Times New Roman,0,0,0>V<+&> <-&><-"><+&>L <-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>can be arbitrarily<:f><:f240,2Times New Roman,0,0,0> chosen,<:f><:f240,2Times New Roman,0,0,0> we simply construct the <:f><:f240,2Times New Roman,0,0,0><+">V<+&> <-&><-"><+&>L <-&>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> from a weighted average of <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'><+">i<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f> and <:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">j<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0>, i.e.<:f><-"><-'><-"><-'><-"><-'> <:s><:S+-2><:#420,9360> <:s><:S+-2><:f240,2Symbol,0,0,0><:A2>.<:f> (2.1) <:s><:S+-2><:#420,9360> <:s><:S+-2><:#420,9360><:f240,2Times New Roman,0,0,0>This makes<:f><:f240,2Times New Roman,0,0,0> the non-local<:f><:f240,2Times New Roman,0,0,0> potential components <:f> <:s><:S+-2><:#420,9360> <:s><:S+-2><:A1> and <:A0> (2.2) <:s><:S+-2><:#420,9360> <:S+-2><:#2604,9360><:f240,2Times New Roman,0,0,0>extremely small because <:f><+">V<:f240,2Times New Roman,0,0,0><-"><+"><+'>i<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<:f240,2Symbol,0,0,0><+">-<-"><+"><:f240,2Times New Roman,0,0,0>V<+'>j<-'><-"><:f>(<:f240,2Times New Roman,0,0,0> <+">r<-"><:f>) is small, <:f240,2Times New Roman,0,0,0>and therefore both <:f><:f240,2Symbol,0,0,0>d<:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">i<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> and <:f><:f240,2Symbol,0,0,0>d<:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">j<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>can hopefully be thrown away<:f><:f240,2Times New Roman,0,0,0> completely. The value of <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0> is chosen so that the local potential <:f240,2Times New Roman,0,0,0><+">V<+&> <-&><-"><:f240,2Times New Roman,0,0,0><+&>L<-&>(r) alone will reproduce the best <:f><:f240,2Times New Roman,0,0,0>possible <:f><:f240,2Times New Roman,0,0,0>agreement <:f><:f240,2Times New Roman,0,0,0>between <:f><:f240,2Times New Roman,0,0,0>the logarithmic<:f><:f240,2Times New Roman,0,0,0> derivatives of the <:f><:f240,2Times New Roman,0,0,0>true and pseudo potentials for <+">both<-"> <:f240,2Times New Roman,0,0,0><+">l=i<-"><:f240,2Times New Roman,0,0,0> and <+">l=j<-"> scattering after <:f><:f240,2Symbol,0,0,0>d<:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">i<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> and <:f><:f240,2Symbol,0,0,0>d<:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">j<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> are discarded. <:f><:f240,2Times New Roman,0,0,0>This method obviously will only work when <:f><:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">i<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> and <:f><:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">j<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f> can be generated to be very similar but we will show that this can often be achieved in practice.<-"><-'><-"><-'> <:s><:S+-2><:#420,9360> <:S+-2><:#3876,9360>We now demonstrate the method by reducing the <:f240,2Symbol,0,0,0>d<:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">s<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> and <:f><:f240,2Symbol,0,0,0>d<:f240,2Times New Roman,0,0,0><+">V<-"><+'><+">p<-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> of a Co <:f>pseudopotential. (In the case of transition metals, we found in Chapter II that their <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) are very different from the corresponding <-"><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) or <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) in shape. Therefore we can only use the Projector Reduction technique to eliminate their non-local components <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">).) For comparison, two Co pseudopotentials were constructed, one with full projectors and the other with the Projector Reduction. For both pseudopotentials, a valence elec tronic configuration<:f240,2Times New Roman,0,0,0> 3<+">d<-"><+&> 7<-&> 4<+">s<-"><+&> 1.00<-&> 4<+">p<-"><+&> 0.75<-&><:f> was used with pseudising radii <:f240,2Times New Roman,0,0,0><+">r<-"><:f><+'>c <-'>= 2.0 a.u. for <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">),<-"> and <:f240,2Times New Roman,0,0,0><+">r<:f240,2Times New Roman,0,0,0><-"><+'>c <-'><:f>= 2.4 a.u. for <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">). Since we are mostly interested in using pseudopotentials in KB form for efficiency, all the tests of the logarithmic derivatives of the pseudopotentials were done in th eir KB form.<-!> <:s><:S+-2><:#420,9360> <:S+-2><:#9516,9360>To generate the Co pseudopotential with full projectors, a standard <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'>-tuned Optimisation procedure (Chapter II) was used to generate <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) independently, without any effort to make them similar. We just set <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>(<:f240,2Times New Roman,0,0,0><+">s<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>(<+">p<-">) equal to <:f240,2Times New Roman,0,0,0><+">q<-"><:f><:f240,2Times New Roman,0,0,0><+'>3<-'><:f>(<:f240,2Times New Roman,0,0,0><+">s<-"><:f>) and <:f240,2Times New Roman,0,0,0><+">q<-"><:f><+'>3<-'>(<:f240,2Times New Roman,0,0,0><+">p<-"><:f>) without further tuning. Such values of <+">Q<-"><+'>c<-'> for <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) <-"><-">already fall within the range of the reasonable <+">Q<-"><+'>c<-'> for these <+"><:f240,2Times New Roman,0,0,0>V<+'>l<-'><-"><:f>(<+">r<-">)<-'><-"><-"> to give accurate logarithmic derivatives compared with those of the true potential, and also there is no need to further tune these <+">Q<-"><+'>c<-'> parameter to improve the convergence of the pseudopotential because both <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) <-"><-"><-"><-"><-">are "soft". Unlike the case for <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f>, we have adjusted <+">Q<-"><+'>c<-'>(<+">d<-">)<-"> to 1.18<+"><:f240,2Times New Roman,0,0,0>q<-"><:f><+'>3<-'>(<+">d<-">) for <-'><+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) <-"><-">to give an accurate and soft <:f240,2Times New Roman,0,0,0><+">d<-"><:f>-component that satisfies the criterion of a <:f240,2Times New Roman,0,0,0><+">Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>-tuned Optimised Pseudopotential. Thus the <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'> parameters became <+"><:f240,2Times New Roman,0,0,0>Q<:f240,2Times New Roman,0,0,0><+'>c<-'>/q<+'>3<-'><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0><-">(<:f240,2Times New Roman,0,0,0><+">s<:f240,2Times New Roman,0,0,0>,<:f240,2Times New Roman,0,0,0>p<:f240,2Times New Roman,0,0,0>,<:f240,2Times New Roman,0,0,0>d<:f240,2Times New Roman,0,0,0><-">)<+"> =<:f><-"> (<:f240,2Times New Roman,0,0,0> 1.00, 1.00, 1.18)<:f> in terms of the ratio between <+"><:f240,2Times New Roman,0,0,0>Q<:f240,2Times New Roman,0,0,0><+'><-">c<-'> and<+"> q<-"><+'>3<-'><:f>. <-"><-'>The <-"><+">s<-">-potential <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) is chosen as local, which is usually the case for pseudopotentials for transition metals in KB form, because the KB form for <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) tends to give problems and is therefore best avoided in this way. Choosing <+"><:f240,2Times New Roman,0,0,0>V<+'>s<-'><-"><:f>(<+">r<-">) as the local potential for transition metals is also used by other authors <[>Ref.T.1, L.2]. The <-"><-'><+"><:f240,2Times New Roman,0,0,0>V<+'>s<-'><-"><:f>(<+">r<-">), <+"><:f240,2Times New Roman,0,0,0>V<+'>p<-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>d<-'><-"><:f>(<+">r<-">) <-"><-">of this Co pseudopotential are shown in Fig.1(a). We also show the logarithmic derivatives of the wave function for both pseudo and true potentials in Fig.1(b). If one drops the <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p<-'><-"><:f>(<+">r<-">) of this pseudopotential (the <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) is zero in this case because the <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) has been chosen as <:f240,2Times New Roman,0,0,0>local<:f240,2Times New Roman,0,0,0><-&>)<:f> then the logarithmic derivative becomes much worse as can be seen in Fig.1(c), which means that the local potential alone is not capable of reproducing the <+">p<-">-wave scattering. In fact the damaged <+">p<-">-scattering can not be recovered by just choosing an alternative local potential because the <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) are very different : if one makes <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) small, then the <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) will be large make s-scattering worse, so that there is not way to simply drop both <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) using the above <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">), <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">). <:s><:S+-2><:#420,9360> <:S+-2><:#5640,9360>For the projector reduced Co pseudopotential, the <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> parameters of <:f240,2Times New Roman,0,0,0><+">s<-"><:f>, <:f240,2Times New Roman,0,0,0><+">p<-"><:f> were varied to search for the greatest similarity between <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">). The procedure is shown in Fig.1(<:f240,2Times New Roman,0,0,0>f<:f>) and Fig.1(<:f240,2Times New Roman,0,0,0>g<:f>). The figures show how different <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'> give different shapes of <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">), and how we have made them approach gradually to becoming similar. In this case, reducing the <+">Q<-"><+'>c<-'>(<+">s<-">) makes the <+"><:f240,2Times New Roman,0,0,0>V<+'>s<-'><-"><:f>(<+">r<-">) less repulsive and reducing the <+">Q<-"><+'>c<-'>(<+">p<-">) makes the <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) less attractive. The optimal condition results in <+"><:f240,2Times New Roman,0,0,0>Q<:f240,2Times New Roman,0,0,0><+'>c<-'>/q<+'>3<-'><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0><-">(<:f240,2Times New Roman,0,0,0><+">s<:f240,2Times New Roman,0,0,0>,<:f240,2Times New Roman,0,0,0>p<:f240,2Times New Roman,0,0,0>,<:f240,2Times New Roman,0,0,0>d<:f240,2Times New Roman,0,0,0><-">)<:f240,2Times New Roman,0,0,0> = <:f>(<:f240,2Times New Roman,0,0,0>0.7, 0.965, 1.18) which makes <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> very similar<:f><-"> as shown in Fig.1(d). Based on the criterion that both <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f> scattering have to be reasonably reproduced by the local potential itself, which means a balance between eliminating <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) has to be consider, we found that the best situation occurred when the local potential was chosen from mixing the <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) with the weight <:f240,2Symbol,0,0,0>a<:f> for <+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">)<-'><-"> in (2.1) being 0.2 and weight 1<:f240,2Symbol,0,0,0>-a<:f> for <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">)<-"> being 0.8<-'><-'>. The logarithmic derivatives of both <+">s<-"> and <+">p<-">-wave of the pseudopotential without <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) are shown in Fig.1(e), which clearly show that the scattering of the true potential are nicely reproduced by the projector-reduced pseudopotential within a reasonable range of energy.<-"> <:s><:S+-2><:#420,9360> <:S+-2><:#2964,9360>We can now compare the efficiency of the above two pseudopotentials for Co. The first one (with <+"><:f240,2Times New Roman,0,0,0>V<+'>s<-'><-"><:f>(<+">r<-">) chosen as local potential) needs projectors for<:f240,2Times New Roman,0,0,0> 8 spherical<:f><:f240,2Times New Roman,0,0,0> harmonics<:f><:f240,2Times New Roman,0,0,0>, three from <+">p<-"> and five from <+">d<-">, whereas the projector reduced one has only 5 projectors from the <+">d<-"> potential in total, which means nearly 40% of the computer time spent on non-local operations can be saved if the later is used. <:f><-'><:f240,2Times New Roman,0,0,0>This projector reduced Co pseudopotential has in fact<:f><:f240,2Times New Roman,0,0,0> been used in various the applications to Co and CoSi<+'>2<-'><-'> <-'>and an extensive test of their ground state properties can be found in the published work. <[>Ref.M.1]<-'> <:s><:S+-2><:#420,9360> <:S+-2>We now turn to the example of Br to show that by reducing the number of projectors, some problematic KB potential components can be avoided in a most efficient way, compared with traditional strategies for dealing with such problems (such as changing <+">r<-"><+'>c<-'>, the configuration, the local potential etc.). Since atomic Br has a ground state configuration 4<+">s<-"><+&>2<-&> 4<+">p<-"><+&>5<-&>, a typical non-local pseudopotential of Br will carry <+">s<-">,<:f240,2Times New Roman,0,0,0><+"> p<-"><:f> and <:f240,2Times New Roman,0,0,0><+">d<-"><:f> components and with <+"><:f240,2Times New Roman,0,0,0>V<+'>p<-'><-"><:f>(<+">r<-">)<-"> taken as local in order to avoid using the less accurate KB potential to act on the heavily occupied <+">p<-"> states. <:f240,2Times New Roman,0,0,0>A standard Br pseudopotential was generated using electronic configuration <:f>4<+">s<-"><+&>2<-&> 4<+">p<-"><+&>5<-&><:f240,2Times New Roman,0,0,0> for <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s<-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> and <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>p<-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0>, and using <:f>4<+">s<-"><+&>1<-&> 4<+">p<-"><+&>3.75<-&><:f240,2Times New Roman,0,0,0> 4<+">d<-"><+&>0.25<-&><:f> for <+"><:f240,2Times New Roman,0,0,0>V<+'>d<-'><-"><:f>(<+">r<-">), with <+">r<-"><+'>c <-'>= 1.89 a.u. and <:f240,2Times New Roman,0,0,0><:f><+">Q<-"><+'>c<-'>/<+">q<-"><+'>3<-'>(<+">s<-">,<:f240,2Times New Roman,0,0,0><+">p<-"><:f>,<:f240,2Times New Roman,0,0,0><+">d<-"><:f>) = (1.0,1.0,1.0)<-"><-"><-"><-">. We see from the <+"><:f240,2Times New Roman,0,0,0>V<+'>l<-'><-"><:f>(<+">r<-">) in Fig.2(a) and their logarithmic derivatives in Fig.2(b) that the <+">d<-">-potential is ill-behaved for a straight-forward pseudopotential constructed in this way. In this particular case, the reason is that the denominator in (1.2) is very small which introduces significant errors at energies away from that of the refer ence state and which may even cause the calculation to be unstable. By using the Projector Reduction technique (with <+">Q<-"><+'>c<-'>/<+">q<-"><+'>3<-'>(<+"><-"><:f240,2Times New Roman,0,0,0><+"><-"><:f><:f240,2Times New Roman,0,0,0><+">d<-"><:f>) changed to 0.9<-"><-"><-"><-"> and <:f240,2Symbol,0,0,0>a<:f>(<+">p<-">)=0.7, <:f240,2Symbol,0,0,0>a<:f>(<+">d<-">)=0.3), we have successfully made <+"><:f240,2Times New Roman,0,0,0>V<+'>p<-'><-"><:f>(<+">r<-">)<-"> and <+"><:f240,2Times New Roman,0,0,0>V<+'>d<-'><-"><:f>(<+">r<-">) of Br sufficiently similar to share a common local potential, as shown in Fig.2(c). Both <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) can then be abandoned, leaving only one spherical harmonic projector for the <:f240,2Times New Roman,0,0,0><+">s<-"><:f> wave, while reproducing the <+">p<-"> and <:f240,2Times New Roman,0,0,0><+">d<-"><:f> scattering nicely, as shown in Fig.2(d). <:f240,2Times New Roman,0,0,0>It is obvious that the Projector Reduction technique provides us an elegant<:f><:f240,2Times New Roman,0,0,0> solution to avoid some ill-behaved<:f> KB components and also reduce the number of projections to be performed.<-"> <:s><:S+-1><:#280,9360> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#376,9360><:f320,,><+!>III.3. Atomic and Solid State Test<-!><:f> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#2532,9360>In this section we apply the Projector Reduction technique on the Cu pseudopotential that has was generated and tested in the previous chapter. <:f240,2Times New Roman,0,0,0>We took the original configuration of Cu used there <:f><:f240,2Times New Roman,0,0,0>and make necessary changes to the generating parameters to make <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>and <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> as similar as possible for the construction<:f><:f240,2Times New Roman,0,0,0> of a projector reduced pseudopotential<:f><:f240,2Times New Roman,0,0,0>. By comparing<:f><:f240,2Times New Roman,0,0,0> the calculated bulk properties using both pseudopotentials, we will know<:f><:f240,2Times New Roman,0,0,0> how well a projector-reduced pseudopotential<:f><:f240,2Times New Roman,0,0,0> reproduces the results of the original non-reduced pseudopotential.<:f> <:s><:S+-2><:#420,9360><:f240,2Times New Roman,0,0,0> <+@><:S+-2><:#6456,9360><:f240,2Times New Roman,0,0,0>The electronic configuration used to generate the original Cu pseudopotential was<:f><:f240,2Times New Roman,0,0,0> 3<+">d<-"><+&> 9<-&> 4<+">s<-"><+&> 0.75<-&> 4<+">p<-"><+&>0.25<-&><:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>The pseudising radii were <:f><+"><:f240,2Times New Roman,0,0,0>r<-"><:f><+'>c <-'>= 2.0 a.u. for <:f240,2Times New Roman,0,0,0>both<+"> V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">),<-"> and <:f240,2Times New Roman,0,0,0><+">r<:f240,2Times New Roman,0,0,0><-"><+'>c <-'><:f>= 2.5 a.u. for <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">). The values <+"><:f240,2Times New Roman,0,0,0>Q<:f240,2Times New Roman,0,0,0><+'>c<-'>/q<+'>3<-'><:f240,2Times New Roman,0,0,0><-">(<:f240,2Times New Roman,0,0,0><+">s<:f240,2Times New Roman,0,0,0>,<:f240,2Times New Roman,0,0,0>p<:f240,2Times New Roman,0,0,0>,<:f240,2Times New Roman,0,0,0> d<:f240,2Times New Roman,0,0,0><-">)<+"> =<:f><-"> (<:f240,2Times New Roman,0,0,0>0.80, 1.00, 1.20)<:f> were used and <:f240,2Times New Roman,0,0,0>the<+"> V<+'>s <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> was chosen as local. The <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0>, <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> and <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> of this original Cu pseudopotential are shown in Fig.3(a).<:f> <:f240,2Times New Roman,0,0,0>To apply the Projector Reduction on this Cu pseudopotential, we have set <:f><:f240,2Times New Roman,0,0,0><+">q<-"><:f240,2Times New Roman,0,0,0><+'>3<-'><:f240,2Times New Roman,0,0,0>(<+">s<-">) to the same value as <:f240,2Times New Roman,0,0,0><+">q<-"><:f240,2Times New Roman,0,0,0><+'>2<-'>(<:f240,2Times New Roman,0,0,0><+">s<-"><:f240,2Times New Roman,0,0,0>) so that effectively the <+">s<-"> pseudo wavefunction is expanded<:f><:f240,2Times New Roman,0,0,0> in terms of only two spherical Bessel functions<:f><:f240,2Times New Roman,0,0,0> that can change the shape of <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">).<:f240,2Times New Roman,0,0,0> (<:f><:f240,2Times New Roman,0,0,0>It is not always<:f><:f240,2Times New Roman,0,0,0> necessary to use such a procedure<:f><:f240,2Times New Roman,0,0,0> to change the shape of a soft pseudopotential<:f><:f240,2Times New Roman,0,0,0> component : we just demonstrate it here to show that it is possible to do so.)<:f><:f240,2Times New Roman,0,0,0> The shape of <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> is regulated by changing <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'> from <+">Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f240,2Times New Roman,0,0,0>/<:f240,2Times New Roman,0,0,0><+">q<:f240,2Times New Roman,0,0,0><-"><+'>3 <-'><:f240,2Times New Roman,0,0,0>= 1.00 to 0.95 to match the shape of <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">), which made <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) very similar<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>to <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>as shown in Fig.3(b). After some trials the weight <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0> = 0.3 for <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> and 1<:f240,2Symbol,0,0,0>-a<:f240,2Times New Roman,0,0,0> = 0.7 for <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) was used to construct the local potential and again both<:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> <:f>were discarded. The logarithmic derivative test of both pseudopotentials in Fig.3(c) shows that the projector reduced pseudopotential reproduces the phase shift of the original one reasonably well<:f240,2Times New Roman,0,0,0> . <:f>It is therefore not too surprising that this is also true for bulk properties, as we can see from the bulk tests listed in Table.1. <+@><:s><:S+-2><:#420,9360> <+@><:S+-2><:#4632,9360>To demonstrate that the actual electronic band structure of the solid is not compromised by using a Projector Reduced pseudopotential (as distinct from the bulk properties tested above which depend only on the total energy), we show in Fig.4 the band struct ure of Cu using the projector reduced Cu pseudopotential. This is in good agreement with the results from the original pseudopotential and other published works <[>Ref.P.1,M.3]. In Table.2 we list some of these calculated eigenvalues for a detailed comparis on. It worth mentioning that the more noticeable differences in the eigenvalues quoted in Table.2 are largely due to the fact that different lattice parameters were used in these calculations. This effect is demonstrated in Table.3 in which one can see that the eigenvalues in the band structure plots are volume sensitive quantities. <:f240,2Times New Roman,0,0,0>(Agreement of results in Table.2 <:f240,2Times New Roman,0,0,0>for <+">s<-">-band depends <:f240,2Times New Roman,0,0,0>mostly on the lattice constant.)<:f> We are therefore confident in claiming that the projector reduced Cu pseudopotential nicely reproduces the results of the original one. <+@><:s><:S+-1><:#280,9360> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#376,9360><:f320,,><+!>III.4. Discussion<-!><:f> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#4764,9360>The reason why we can do Projector Reduction, or more precisely, make two <+">V<-"><+"><+'>l <-'><-">(<+">r<-">) with different <+">l<-"> to be similar, is because of the non-uniqueness of the shape of a pseudopotential component <+">V<-"><+"><+'>l<-'><-">(<+">r<-">). However, the shape of <+">V<-"><+"><+'>l<-'><-">(<+">r<-">) is not as flexible as one might wish because one only has a limited degree of freedom in pseudising/modifying the wave function, especially for <+"><:f240,2Times New Roman,0,0,0>r<-"><:f> near <:f240,2Times New Roman,0,0,0><+">r<-"><:f><+'>c<-'> . This is because there is no pseudising effect (i.e. the wave function is not changed) in the region of <:f240,2Times New Roman,0,0,0><+">r<-"><:f> <:f240,2Symbol,0,0,0><;><:f> <+">r<-"><+'>c<-'> and that the wave function must be continuous across <+">r<-"><+'>c<-'> , which make the wave function near <:f240,2Times New Roman,0,0,0><+">r<-"><:f><+'>c<-'> (but less than <+">r<-"><+'>c<-'>) rather restricted. However in the more central part of the pseudising region, the pseudo wavefunction is easier to modify either because <+">r <-">is further away from the constraint at <+">r<-"><+'>c<-'> or/and because the limiting form of the wave function (~<+">r<-"><+"><+&>l<-&><-"><-">) at very <+">r<-"> which makes the effect of varying <+">V<-"><+"><+'>l <-'><-">(<+">r<-">) vanishingly small there and hence makes a larger variation of <+">V<-"><+"><+'>l <-'><-">(<+">r<-">) possible at small <+">r<-">. <:s><:S+-2><:#420,9360> <:S+-2><:#2124,9360>From the above arguments it is clear that a successful Projector Reduction depends on whether the corresponding <+">V<-"><+"><+'>l <-'><-">(<+">r<-">) can be made similar <-'>at the <:f240,2Times New Roman,0,0,0><+">r<-"><:f> near <+">r<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>. With such understanding, we realised that the fundamental limitation to applying the Projector Reduction technique to a pseudopotential is the nature<-"> of the pseudopotential near <+"><:f240,2Times New Roman,0,0,0>r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>. This can be seen by analysing the radial Schr<\v>dinger equation <:f240,2Times New Roman,0,0,0>(in atomic-Rydberg units)<:f> : <:s><:S+-2><:#420,9360> <:s><:S+-2> <:A4> . (4.1) <:s><:S+-2><:#420,9360> <:S+-2><:#2592,9360>The only term in equation (4.1) that distinguishes the different <+">l<-"> is the term <:f240,2Times New Roman,0,0,0><+">l<-"><:f>(<:f240,2Times New Roman,0,0,0><+">l<-"><:f>+1)/<+">r<-"><+&>2<-&>. With a larger <+">r<-"> this term is smaller and so is the <+">l<-">-dependence of the equation. It is therefore expected that at large <:f240,2Times New Roman,0,0,0><+">r<-"><:f> the solutions <+">P<-"><+"><+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) of different <+">l<-"> are similar, and again since <+"><:f240,2Times New Roman,0,0,0>l<-"><:f>(<:f240,2Times New Roman,0,0,0><+">l<-"><:f>+1)/<+">r<-"><+&>2<-&> is small at large <:f240,2Times New Roman,0,0,0><+">r<-"><:f>, the inverted<-"> <:f240,2Times New Roman,0,0,0><+">V<+'>l<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) at large <:f240,2Times New Roman,0,0,0><+">r<-"><:f> are automatically similar. In the other words, a large <+">r<-"><+'>c<-'> ensures the <+"><:f240,2Times New Roman,0,0,0>V<+'>l<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) of different <+">l<-"> are already similar around <+">r<-"><+'>c<-'>, making it possible for one to regulate further the shape of <+"><:f240,2Times New Roman,0,0,0>V<+'>l<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) within the pseudising region to reduce the projectors. <:s><:S+-2><:#420,9360> <:S+-2><:#3444,9360>We can have a closer look at these quantities by taking some typical values. For <+">r<-"><+'>c<-'> = 2.0 a.u., the <+">l<-">(<+">l<-">+1)/<:f240,2Times New Roman,0,0,0><+">r<-"><:f><+&>2<-&> is 0 and 0.5 for <+">l=<-"><-"><-"> 0 and 1, respectively. On the other hand, the potential has a typical value of <:f240,2Symbol,0,0,0>-<:f>5 to <:f240,2Symbol,0,0,0>-<:f>10 Ryd at <:f240,2Times New Roman,0,0,0><+">r<-"><:f> = 2.0 a.u. for the cases of Co, Cu, etc. (because of the electrostatic potential 2Z<+'>eff<-'>/<+">r <-">; see also the figures in Section 3). It is therefore clear that the "centrifugal" potential is nearly an order of magnitude smaller than the electrostatic one at <+">r <-">= 2.0 a.u. so that the effect of the <+">l<-">-dependence at such <+">r<-"> should be small. <:f240,2Times New Roman,0,0,0>With the tests of Cu shown in Section 3 and the published <:f><:f240,2Times New Roman,0,0,0>results on <:f><:f240,2Times New Roman,0,0,0>CoSi<+'>2<-'><:f><:f240,2Times New Roman,0,0,0>, we <:f>have demonstrated that a practical pseudising radius (2.0 a.u.) for common physical applications is indeed large enough for the <:f240,2Times New Roman,0,0,0>Projector<:f><:f240,2Times New Roman,0,0,0> Reduction<:f>.<-"><-"> <:s><:S+-2><:#420,9360> <:S+-2><:#2580,9360><:f240,2Times New Roman,0,0,0>On the other hand, we think that the feasibility of the Projector Reduction technique depends also on the pseudopotential generating scheme.<:f> <:f240,2Times New Roman,0,0,0>As long as<:f><:f240,2Times New Roman,0,0,0> a<:f><:f240,2Times New Roman,0,0,0> flexible<:f><:f240,2Times New Roman,0,0,0> control of the shape of individual <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>l<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>)<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>is possible, any <:f><:f240,2Times New Roman,0,0,0>pseudopotential <:f><:f240,2Times New Roman,0,0,0>generating<:f><:f240,2Times New Roman,0,0,0> method<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>can be used to generate <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>l<-'><-"><:f>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) <:f240,2Times New Roman,0,0,0>of different <+">l<-"> with similar shape<:f><:f240,2Times New Roman,0,0,0> for the further reduction of the corresponding <:f><:f240,2Symbol,0,0,0>d<+"><: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>)<:f240,2Times New Roman,0,0,0>. In this context <:f><:f240,2Times New Roman,0,0,0>the <:f240,2Times New Roman,0,0,0><+">Q<-"><+'>c<-'>-tuned<:f><:f240,2Times New Roman,0,0,0> Optimised Pseudopotentials are good candidates for Projector Reduction treatment both because of their well controllable shapes an d very<:f240,2Times New Roman,0,0,0> flexible<:f><:f240,2Times New Roman,0,0,0> choice of <+">r<-"><+'>c<-'>.<:f> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:#376,9360><:f320,,><+!>III.5. Conclusion<-!><:f> <:s><:#280,9360> <:S+-2><:#5136,9360><:f240,2Times New Roman,0,0,0>In conclusion, we have proposed a creative use of <+">Q<-"><+'>c<-'>-tuning in generating Optimised Pseudopotentials, which leads to a new way of treating the non-local projectors, and which we call Projector Reduction<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>The idea is to make some of the <:f><+"><: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>), the <+">l<-">-dependent components of <:f240,2Times New Roman,0,0,0>a pseudopotential, as similar as possible, and take a weighted average <:f>of them as the common local part <+"><: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>) so that the resulting <:f240,2Symbol,0,0,0>d<+"><: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>) are so small such that they can be ignored completely. <:f240,2Times New Roman,0,0,0>The elimination of some of the <:f><:f240,2Symbol,0,0,0>d<+"><: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>)<:f240,2Times New Roman,0,0,0> in a pseudopotential<:f><:f240,2Times New Roman,0,0,0> reduces the computational cost of performing 2<+">l<-">+1 spherical harmonic projections for each non-zero <:f><:f240,2Symbol,0,0,0>d<+"><: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>)<-"><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>in <:f><:f240,2Times New Roman,0,0,0>the pseudopotential. Sometimes this also improves the accuracy of the pseudopotential by avoiding the need for using the problematic <:f><:f240,2Times New Roman,0,0,0>Kleinman-Bylander<:f><:f240,2Times New Roman,0,0,0> form, such as in the case of Br.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>The logarithmic<:f><:f240,2Times New Roman,0,0,0> derivative tests over a range of energy have shown that these projector reduced pseudopotentials, such as for Co and Cu, reproduce <:f><:f240,2Times New Roman,0,0,0>very well the scattering behaviour<:f><:f240,2Times New Roman,0,0,0> of the unreduced ones. This suggests the reliability<:f><:f240,2Times New Roman,0,0,0> of the Projector Reduction technique which is also confirmed by our solid state tests. <:s><:S+-2><:#420,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0>A simple analysis of the generating procedure and the radial Schr<\v>dinger equation indicates that the fundamental limitation of this approach is the need for a large enough <+">r<-"><+'>c<-'> in a particular<:f><:f240,2Times New Roman,0,0,0> application. Our atomic and solid state tests have demonstrated that reasonable<:f><:f240,2Times New Roman,0,0,0> values of <:f><+"><:f240,2Times New Roman,0,0,0>r<-"><+'>c<-'><:f><:f240,2Times New Roman,0,0,0> can satisfy this requirement, and hence assure the usefulness of the method. The Projector <:f><:f240,2Times New Roman,0,0,0>Reduction is therefore an applicable and recommended strategy when using non-local (and Kleinman-Bylander) pseudopotentials.<:f> <:s><:#280,9360> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#376,9360><:f320,2Times New Roman,0,0,0><+!>Acknowledgement<-!><:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#1692,9360><:f240,2Times New Roman,0,0,0>I want to thank Dr. A. Qteish for his implementation of the method of arbitrary<:f><:f240,2Times New Roman,0,0,0> choice<:f><:f240,2Times New Roman,0,0,0> of local potential by mixing different <+">l<-"> components, together with other useful<:f><:f240,2Times New Roman,0,0,0> features he coded into the pseudopotential generation program. <:f><:f240,2Times New Roman,0,0,0>My deep appreciation<:f><:f240,2Times New Roman,0,0,0> also goes to Dr. V. Milman who worked together with me in this project and did the Cu and Co tests. <:s><:#280,9360> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0> <+C><:s><:#376,9360><:f320, Times New Roman,0,0,0><+!>References<-!><:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>B.1] <:#288,9360><:f240,2Times New Roman,0,0,0>P. E. Bl<\v>chl, Phys. Rev. B <+!>41<-!>, 5414 (1990) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>G.1] <:#288,9360><:f240,2Times New Roman,0,0,0>X. Gonze, P. Kackell and M. Scheffler, Phys. Rev. B <+!>41<-!>, 12264 (1990)<:f> <+C><:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>H.1] <:#288,9360><:f240,2Times New Roman,0,0,0>D.R. Hamann, M. Schluter and C. Chiang, Phys. Rev. Lett <+!>43<-!>, 1494 (1979)<:f> <+C><:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>K.1] <:#288,9360><:f240,2Times New Roman,0,0,0>L. Kleinman and D.M. Bylander, Phys Rev. Lett. <+!>48<-!>, 1425 (1982)<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>K.2] <:#288,9360><: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><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0><[>L.2]<:f> <:#288,9360><: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>, PRB <+!>47,<-!> 4174 (1993)<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><[>M.1] <:#288,9360>V. Milman, M-H. Lee and M.C. Payne, Phys. Rev. B <+!>49<-!>, 16300 (1994) <+C><:s><:#280,9360> <+C><:s><:#280,9360><[>M.2] <+C><:s><:#280,9360>V. Milman (CoSi<+'>2<-'>/Si interface , unpublished.) <+C><:s><:#280,9360> <+C><:s><:#280,9360><[><:f240,2Times New Roman,0,0,0>M.3<:f>] <+C><:s><:#288,9360><:f240,2Times New Roman,0,0,0>(Calculated at <:f><:f240,2Times New Roman,0,0,0> <+">a<-"><+'>0<-'> = <:f><:f240,2Times New Roman,0,0,0> 3.58 <\E>)<:f> <:#568,9360>V. L. Moruzzi, J. F. Janak, A. R. Williams, <+">Calculated electronic properties of metals, <-">Pergamon, (1978) <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <+C><:s><:#280,9360><[><:f240,2Times New Roman,0,0,0>P.1<:f>] <+C><:s><:#288,9360><:f240,2Times New Roman,0,0,0>(Calculated at <+">a<-"><+'>0<-'> = 3.61 <\E><:f>) <:#288,9360>D. A.Papaconstantopoulos,<+"> Handbook of the band structure of elemental solids<-">, (1986) <+C><:s><:#280,9360> <+C><:s><:#280,9360><[>T.1] <:#288,9360><:f240,2Times New Roman,0,0,0>N. Troullier and J. L. Martins, Phys. Rev. B <+!>43<-!>, 1993 (1991)<:f> <+C><:s><:#280,9360> <+C><:s><:p<* >><:f320,,><+!>Tables<-!><:f> <+C><:s><:#280,9360> <+C><:s><:#280,9360> <+C><:#1120,9360>TABLE.1. <:f240,2Times New Roman,0,0,0>Lattice parameters (in <\E>) and bulk moduli<:f> (in GPa) of Cu metal using the projector reduced pseudopotential (Cu-PR) and the original standard pseudopotential (Cu-OR). The experimental results are also listed whi ch suggest that the difference between Cu-OR and Cu-PR is smaller than a typical error from a pseudopotential calculation. <:s><:I720,0,0,0><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>=======================================<:f> <:I720,0,0,0><:#202,9360><:f200,QCourier,0,0,0> <+">a<-"><+'>0<-'>(<\E>) B(GPa) dB/dP <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>-------------------------------------------------------------------<:f> <:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0>Cu-PR 3.647 150 5.1 0.4<:f> <:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0>Cu-OR 3.658 145 4.8 0.2<:f> <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0>Expt. 3.61 142 5.28<:f> <:s><:I720,0,0,0><:#240,9360><:f200,,>=======================================<:f> <:s><:I720,0,0,0><:#280,9360> <:s><:#280,9360> <:s><:#280,9360> <:#1712,9360>TABLE.2. <:f240,2Times New Roman,0,0,0>Eigenvalues at special <+">k<-">-points <:f240,2Symbol,0,0,0>G<:f240,2Times New Roman,0,0,0>, X and L obtained from the band structure<:f><:f240,2Times New Roman,0,0,0> calculations<:f><:f240,2Times New Roman,0,0,0> of <:f><:f240,2Times New Roman,0,0,0>Cu <:f><:f240,2Times New Roman,0,0,0>metal <:f><:f240,2Times New Roman,0,0,0>using the original Cu pseudopotential (Cu-OR) at lattice parameter 3.658 <\E> (<+">E<-"><+'>F<-'> = -6.337 eV), the projector reduced Cu pseudopotential (Cu-PR)<:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>at lattice parameter 3.647 <\E> <:f240,2Times New Roman,0,0,0>(<+">E<-"><+'>F<-'> = -6.106 eV)<:f240,2Times New Roman,0,0,0>, and <:f><:f240,2Times New Roman,0,0,0>two published all-electron results <[>Ref.M.3] and <[>Ref.P.1]. <:f><:f240,2Times New Roman,0,0,0> The good quantitative<:f><:f240,2Times New Roman,0,0,0> agreement between<:f><:f240,2Times New Roman,0,0,0> Cu-PR and Cu-OR indicates the success of the Projector Reduction.<:f><:f> <:s><:#240,9360><:f200,2Times New Roman,0,0,0> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>==================================================<:f> <:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> <+">k<-"> band Cu-PR Cu-OR <[>1] <[>2] <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0><+"> <-"> n 3.647<\E> 3.658<\E> 3.61<\E> 3.58<\E><:f> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>==================================================<:f> <:s><:I720,0,0,0><:#192,9360><:f200,QCourier,0,0,0> <:f160,2Symbol,0,0,0>G<:f200,QCourier,0,0,0> 1 -8.616 -8.950 -9.44 -9.41 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 2 -3.209 -3.274 -3.06 -3.23 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 3 -2.447 -2.520 -2.33 -2.41 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 4 23.954 23.381 25.19 - <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 5 25.412 25.168<:f><:f200,QCourier,0,0,0> - -<:f> <:s><:I720,0,0,0><:#256,9360><:f200,BSystem,0,0,0>---------------------------------------------------------------------------------------<:f> <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> <:f><:f200,QCourier,0,0,0> <:f><:f200,QCourier,0,0,0>X<:f><:f200,QCourier,0,0,0> 1 -4.726 -4.801 -5.04 -5.26 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 2 -4.398 -4.458 -4.46 -4.68 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 3 -1.918 -2.000 -1.73 -1.82 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 4 -1.771 -1.856 -1.58 -1.62 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 5 1.428 1.646 1.79 1.63 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 6 8.165 7.926 7.12 - <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 7 13.093 13.072 12.29 - <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 8 21.424 21.394 - - <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 9 22.789 21.978<:f><:f200,QCourier,0,0,0> - -<:f> <:s><:I720,0,0,0><:#256,9360><:f200,BSystem,0,0,0>------------------------------------------------------------------------------<:f><:f200,BSystem,0,0,0>--------<:f> <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> <:f><:f200,QCourier,0,0,0> <:f><:f200,QCourier,0,0,0>L<:f><:f200,QCourier,0,0,0> 1 -4.840 -4.952 -4.97 -5.25 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 2 -3.228 -3.293 -3.16 -3.05 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 3 -1.907 -1.990 -1.77 -1.65 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 4 -1.040 -0.807 -1.07 -1.06 <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 5 4.432 4.167 3.73 - <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 6 22.050 21.621 - - <:s><:I720,0,0,0><:#280,9360><:f200,QCourier,0,0,0> 7 22.091 21.905<:f><:f240,2Times New Roman,0,0,0> <:f200,QCourier,0,0,0>- -<:f> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>==================================================<:f> <:s><:I720,0,0,0><:#280,9360><:f240,2Times New Roman,0,0,0> <:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0><[>1] Ref.P.1 <:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0><[>2] Ref.M.2<:f> <:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#576,9360><:f240,2Times New Roman,0,0,0>TABLE.3 <:f240,2Times New Roman,0,0,0>Bandwidth as a function of volume (calculated using the projector reduced Cu pseudopotential) illustrated <:f240,2Times New Roman,0,0,0>by eigenvalues at two <+">k<-">-points<:f><:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>G<:f240,2Times New Roman,0,0,0> an<:f><:f240,2Times New Roman,0,0,0>d L.<:f> <:s><:I720,0,0,0><:#160,9360><-"><+"><:f200,QCourier,0,0,0> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>======================================================<:f> <:I720,0,0,0><:#160,9360><+"><:f200,QCourier,0,0,0>k<-"> band<:f><:f200,QCourier,0,0,0> <+">E<-">(eV) <:f200,QCourier,0,0,0><+">E<-"><:f200,QCourier,0,0,0>(eV)<:f><:f200,QCourier,0,0,0> <:f200,QCourier,0,0,0><+">E<-"><:f200,QCourier,0,0,0>(eV)<:f><:f200,QCourier,0,0,0> band <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> n <:f><:f200,QCourier,0,0,0> 3.57<\E> 3.61<\E> 3.647<\E> <:f><:f200,QCourier,0,0,0> symmetry<:f> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>======================================================<-"><:f> <:s><:I720,0,0,0><:#192,9360><:f160,2Symbol,0,0,0>G<:f200,QCourier,0,0,0> 1 -9.02 -8.80 -8.56 (<:f200,QCourier,0,0,0><+">s<-"><:f200,QCourier,0,0,0>) <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 2 -3.43 -3.35 -3.22 (<:f200,QCourier,0,0,0><+">d<-"><:f200,QCourier,0,0,0>) <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 3 -3.41 -3.33 -3.20 (<:f200,QCourier,0,0,0><+">d<-"><:f200,QCourier,0,0,0>) <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 4 -2.58 -2.54 -2.45 (<:f200,QCourier,0,0,0><+">d<-"><:f200,QCourier,0,0,0>)<:f> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>--------------------------------------------------------------------------------------------<:f> <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0>L 1 -5.21 -5.05 -4.86 (<:f200,QCourier,0,0,0><+">s<-"><:f200,QCourier,0,0,0>) <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 2 -3.43 -3.35 -3.22 (<:f200,QCourier,0,0,0><+">d<-"><:f200,QCourier,0,0,0>) <:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 3 -2.00 -1.99 -1.93 (<:f200,QCourier,0,0,0><+">sd<-"><:f200,QCourier,0,0,0>) <:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> 4 -1.53 -1.45 -1.33 (<:f200,QCourier,0,0,0><+">sd<-"><:f200,QCourier,0,0,0>)<:f> <:s><:I720,0,0,0><:#240,9360><:f200,2Times New Roman,0,0,0>=====================================================<-"><:f> <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> <:s><:I720,0,0,0><:#160,9360><:f200,QCourier,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#376,9360><:f320,2Times New Roman,0,0,0><+!>Figure Captions<-!><:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#568,9360><:f240,2Times New Roman,0,0,0>FIG.1(a).<:f> The <-"><-'><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">), <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) <-"><-">of a standard Co pseudopotential which is not designed for Projector Reduction. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#560,9360><:f240,2Times New Roman,0,0,0>FIG.1(b). The logarithmic derivative for the <:f>standard Co pseudopotential. (For pseudo wave functions as solid lines, true wave functions as dashed lines.) <:s><:#280,9360> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#872,9360><:f240,2Times New Roman,0,0,0>FIG.1(c). <:f><:f240,2Times New Roman,0,0,0>The logarithmic derivative for the <:f>standard Co pseudopotential<:f240,2Times New Roman,0,0,0> if the <:f240,2Times New Roman,0,0,0><+">s<-"><:f240,2Times New Roman,0,0,0> and <+">p<-"> non-local parts <:f><:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">)<:f240,2Times New Roman,0,0,0> are dropped. (For pseudo wave function as solid lines, all-electron as dashed lines.) (The<:f><:f240,2Times New Roman,0,0,0> irrelevant<:f><:f240,2Times New Roman,0,0,0> <+">d<-"> scattering is not shown<:f><:f240,2Times New Roman,0,0,0>.) <:s><:#280,9360> <:s><:#280,9360> <:#576,9360><:f240,2Times New Roman,0,0,0>FIG.1(d).<:f><:f240,2Times New Roman,0,0,0> <:f>The <-"><-'><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">), <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) <-"><-">of the projector reduced Co pseudopotential<:f240,2Times New Roman,0,0,0>. The <:f><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) have been made as similar as possible. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#560,9360><:f240,2Times New Roman,0,0,0>FIG.1(e).<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>The logarithmic derivative of the<:f><:f240,2Times New Roman,0,0,0> <:f>projector reduced Co pseudopotential. (Solid line : pseudo, dashed line : all-electron) <:s><:#280,9360> <:s><:#280,9360> <:#568,9360>FIG.1(f). The effect of <+">Q<-"><+'>c<-'>-tuning on the <+">V<-"><+"><+'>s<-'><-">(<+">r<-">) of Co, with <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'>/<:f240,2Times New Roman,0,0,0><+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f>(<+">s<-">) = 1.00 (dot-dash), 0.80 (dash), and 0.70 (solid). <:s><:#280,9360> <:s><:#280,9360> <:#568,9360>FIG.1(g). The effect of <+">Q<-"><+'>c<-'>-tuning on the <+">V<-"><+"><+'>p<-'><-">(<+">r<-">) of Co, with <:f240,2Times New Roman,0,0,0><+"><-"><:f><+">Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>/<+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f>(<:f240,2Times New Roman,0,0,0><+">p<-"><:f>) = 1.00 (dot-dash), 0.98 (dash), and 0.965 (solid). <:s><:#280,9360><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,189,191,193> <:s><:#280,9360><:f240,2Times New Roman,189,191,193> <:f240,2Times New Roman,0,0,0>FIG.2(a).<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>T<:f><:f240,2Times New Roman,0,0,0>he <+">V<+'>s <-'><-"><:f>(<+">r<-">), <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) <-"><-">of the <-"><:f240,2Times New Roman,0,0,0>standard Br pseudopotential<:f><:f240,2Times New Roman,0,0,0>.<:f> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,189,191,193> <:f240,2Times New Roman,0,0,0>FIG.2(b). <:f>The <+">d<-">-wave logarithmic derivative of the standard. (Pseudo : solid line, diverged badly. All-electron : dashed line). <:s><:#280,9360><:f240,2Times New Roman,189,191,193> <:s><:#280,9360><:f240,2Times New Roman,189,191,193> <:f240,2Times New Roman,0,0,0>FIG.2(c). <:f><:f240,2Times New Roman,0,0,0>T<:f><:f240,2Times New Roman,0,0,0>he <+">V<+'>s <-'><-"><:f>(<+">r<-">), <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) <-"><-">of the <-"><:f240,2Times New Roman,0,0,0>projector-reduced<:f><:f240,2Times New Roman,0,0,0> Br pseudopotential<:f><:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0><:f><+"><-"> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>FIG.2(d). The <:f><+"><:f240,2Times New Roman,0,0,0>p<-"> and <+">d<-">-wave <:f><:f240,2Times New Roman,0,0,0>logarithmic derivatives of the projector-reduced Br. (Pseudo<:f><:f240,2Times New Roman,0,0,0> : solid line, all-electron : dashed line)<+"><-"><:f><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#568,9360><:f240,2Times New Roman,0,0,0>FIG.3(a). <:f>The <-"><-'><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">), <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) and <+"><:f240,2Times New Roman,0,0,0>V<+'>d <-'><-"><:f>(<+">r<-">) <-"><-">of the original Cu pseudopotential which is not projector-reduced. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#568,9360><:f240,2Times New Roman,0,0,0>FIG.3(b).<:f><:f240,2Times New Roman,0,0,0> <:f>The <-"><-'><+"><:f240,2Times New Roman,0,0,0>V<+'>s <-'><-"><:f>(<+">r<-">)<+"><:f240,2Times New Roman,0,0,0><+'><-'><-"><:f><+"><-"> and <+"><:f240,2Times New Roman,0,0,0>V<+'>p <-'><-"><:f>(<+">r<-">) <-"><-">of the projector-reduced Cu pseudopotential<:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0> They <:f><+"><:f240,2Times New Roman,0,0,0><+'><-'><-"><:f><+"><-"><+"><:f240,2Times New Roman,0,0,0><+'><-'><-"><:f><+"><-">have been made as similar as possible. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:#848,9360><:f240,2Times New Roman,0,0,0>FIG.3(c). <:f><:f240,2Times New Roman,0,0,0>The comparison of the logarithmic derivatives of wavefunctions from the original pseudopotential (dashed line) and the projector-reduced (solid line) one. Both <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f240,2Times New Roman,0,0,0> scattering of the original <:f><:f240,2Times New Roman,0,0,0>pseudopotential<:f><:f240,2Times New Roman,0,0,0> are nicely reproduced by the <:f><:f240,2Times New Roman,0,0,0>projector reduced <:f><:f240,2Times New Roman,0,0,0>pseudopotential<:f><:f240,2Times New Roman,0,0,0>. <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:s><:#280,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#280,9360><:f240,2Times New Roman,0,0,0>FIG.4. <:f><:f240,2Times New Roman,0,0,0>The band structure of Cu calculated using the Projector Reduced Cu pseudopotential.<:f> > Times New Roman,18,12,0,0,0,0,0 $\delta V_j(r)=-\alpha \cdot \left[ V_i(r)-V_j(r)\right] 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ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿTimes New Roman,18,12,0,0,0,0,0 $\delta V_i(r)=(1-\alpha )\cdot \left[ V_i(r)-V_j(r)\right] 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New Roman,18,12,0,0,0,0,0 $$V^{\text{ L}}(r)=\alpha \cdot V_i(r)+(1-\alpha )\cdot V_j(r)\text{, \quad with\quad }0\leq \alpha \leq 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New Roman,18,12,0,0,0,0,0 $\underset{lm}\to{\sum }\left| Y_{lm}\left\rangle \,\delta V_l(r)\,\right\langle Y_{lm}\right| \quad \approx \quad \underset{lm}\to{\sum }\dfrac{\left| \delta V_l(r)\Phi _{lm}^0(\text{\BF r})\rangle \,\langle \Phi _{lm}^0(\text{\BF r})\delta V_l(r)\right| }{\langle \Phi _{lm}^0(\text{\BF r})\left| \delta V_l(r)\right| \Phi _{lm}^0(\text{\BF r})\rangle 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New Roman,18,12,0,0,0,0,0 $$\frac{d^2P_l\,(r)}{dr^2}+\left[ E+V\,(r)-\,\frac{l\,(l+1)}{r^2}\right] 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[Embedded] 6 .tex 80918 91 81009 5674 5 .tex 86683 94 86777 6688 4 .tex 93465 141 93606 9138 3 .tex 102744 389 103133 24642 7 .tex 127775 122 127897 15318 00143217