[ver] 4 [sty] [files] [charset] 82 CHINESEBIG5 (Windows) [revisions] 0 [prn] QLII FaxDriver v1.0 [port] FX/MDM [lang] 2 [desc] PhD Thesis Chapter II, June edition 973367284 58 791582827 12512 15 4633 28916 149 149 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 [tag] Thesis Body 12 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 40 316 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 . , $ Thesis Body 0 0 [frm] 3 537395328 4140 4581 8953 5012 0 1 3 0 0 0 0 0 0 0 0 16777215 8 0 2700 4813 295 [frmname] Frame8 [frmlay] 5012 4813 1 0 0 1 4581 0 0 2 0 32 53 12855 1 4140 8953 0 [isd] .X8 .tex 1 1 0 0 4795 65087 100 0 0 .tex 0 65241 0 [frm] 1 537395328 2391 9950 9410 10565 0 1 3 0 0 0 0 0 0 0 0 16777215 7 1 951 7019 337 [frmname] Frame7 [frmlay] 10565 7019 1 0 0 1 9950 0 0 2 0 48 50 14368 1 2391 9410 0 [isd] .X7 .tex 1 1 0 0 6865 64952 100 0 0 .tex 0 65199 0 [frm] 1 537395328 2218 4164 6571 4581 0 1 3 0 0 0 0 0 0 0 0 16777215 3 2 778 4353 263 [frmname] Frame3 [frmlay] 4581 4353 1 0 0 1 4164 0 0 2 0 0 0 0 1 2218 6571 0 [isd] .X3 .tex 1 1 0 0 4340 65100 100 0 0 .tex 0 65287 0 [frm] 1 537395328 2762 12600 9064 13239 0 1 3 0 0 0 0 0 0 0 0 16777215 4 3 1322 6302 402 [frmname] Frame4 [frmlay] 13239 6302 1 0 0 1 12600 0 0 2 0 0 0 0 1 2762 9064 0 [isd] .X4 .tex 1 1 0 0 6185 64939 100 0 0 .tex 0 65134 0 [frm] 2 537395328 2760 13241 5246 13524 0 1 3 0 0 0 0 0 0 0 0 16777215 6 4 1320 2486 224 [frmname] Frame6 [frmlay] 13524 2486 1 0 0 1 13241 0 0 2 0 0 0 0 1 2760 5246 0 [isd] .X6 .tex 1 1 0 0 2327 65248 100 0 0 .tex 0 65312 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,,>II. Optimised Pseduopotentials with Kinetic Energy Filter Tuning > [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,II-> > [elay] [l1] 0 [pg] 15 21 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 26 1484 108 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 34 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 44 1616 133 0 0 0 0 65534 1721 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 48 100 348 0 0 0 0 65534 384 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 57 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 62 193 99 0 0 0 0 65534 172 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 67 0 8 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 75 1060 192 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 84 833 260 0 0 0 0 65534 1065 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 91 277 96 0 0 0 0 65534 182 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 100 506 44 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 0 137 0 29 32 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 173 0 30 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 0 186 152 355 1025 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 [edoc] <+B><-!><:f><:f480,,><+!><-!><:f><+!><:f480,,><-!><:f><+!><:f480,,> <+B><:#562,9360><+!><:f480,,>II<:f> <+B><:#284,9360><+!> <+B><:#284,9360><+!> <+B><:#562,9360><+!><:f320,,> <:f480,,>Optimised Pseudopotentials <+B><:#562,9360><+!><:f480,,>with <+B><:#562,9360><+!><:f480,,>Kinetic Energy Filter Tuning<-!><:f><-'><-'> <+@><:#284,9360> <+@><:#284,9360> <+@><:#284,9360> <+@><:#284,9360> <+@><:#284,9360> <:S+-2><:#426,9360><-!><-!> <:S+-2><:#426,9360> <+@><:S+-2><:#426,9360><-!> <+@><:S+-2><:#4344,9360><-!><-!><:f240, Times New Roman,0,0,0>We have developed an improved scheme for generating Optimised Pseudopotentials <[>1-3]<:f><:f240, Times New Roman,0,0,0> which is more systematic and leads to a better insight<:f><:f240, Times New Roman,0,0,0>.<:f> The control parameter <+">Q<-"><+'>c<-'> connected with the kinetic energy<-!> of the pseudo wavefunction <:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><-'><-'><-'><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">r<-">)<:f> is used in a new way to tune the pseudopotential. The scheme uses only three constraints and three spherical Bessel functions in the expansion of <:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><-'><-'><-'><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">r<-">)<:f> inside the pseudising radius <+">r<-"><+'>c<-'><-!><-'>, compared with four or more previously. The combined effect is that the fidelity of the pseudopotential as seen in the logarithmic derivative, can be improved in a simple and system atic way by tuning <+">Q<-"><+'>c<-'>.<-!><-'><-!><-'> The softness of the pseudopotential is also improved somewhat. <:f240, Times New Roman,0,0,0>The scheme opens the way to tailor-making pseudopotentials for specific requirements <:f><:f><:f240,2Times New Roman,0,0,0>which will be particularly useful for large scale <+">ab initio<-"> calculations.<:f> <+@><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <+@><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <+@><:S+-2><:#564,9360><:f320,,><+!>II.1. Introduction<-!><:f> <+@><:s><:S+-2><:#426,9360><+!><:f240,,> <+@><:S+-2><:#4686,9360>Soft and accurate pseudopotentials are essential for state-of-the-art large scale<+"> ab initio<-"> simulations for solids using plane-wave basis sets. This is in the context of periodic superlattice calculations based on Density Functional Theory with Loca l Density Approximation (LDA) for exchange and correlation with possibly a Generalised Gradient Approximation (GGA). Here, a "soft" pseudopotential means that a low energy cut-off, <+">E<-"><+'>cut<-'> , can be used in the planewave expansion of the wavefunction, and the "accuracy" of a pseudopotential can be measured by the agreement between the logarithmic derivatives of pseudo and true wavefunctions in a certain energy range, which is an equivalent characterisation of the scattering property of a pseudopotential in terms of the phase shifts of different incoming waves. Even with the accuracy and efficiency achieved by a Car-Parrinello type of algorithm <[>4,5], the computational cost still requires a soft pseudopotential to make the largest calculations affordable. <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#7668,9360>While the advantage of the softness in a pseudopotential can not be over-emphasised, what is equally important is to have a flexible scheme to generate quickly and systematically a new pseudopotential appropriate to a new physical situation, such as short i nter-atomic distances in some compound or at ultra high pressures. Another example would be some structural energy difference where errors cancel, so that one can compromise on convergence properties. A further situation might be one where the energy range over which the logarithmic derivative of the pseudo wavefunction has to be accurate is unusually narrow or wide.<:f240,2Times New Roman,0,128,255> <:f>Whereas one standard transferable pseudopotential for a given chemical element may be satisfactory for many purposes, this will not be so for some calculations. On the contrary, the flexibility of balancing the accuracy (logarithmic derivative) and the efficiency (<+">E<-"><+'>cut<-'>) of a pseudopotential in its construction will be useful, because then computing effort will not be wasted in achieving unnecessary precision. This utility can not be replaced by simply using an insufficient <+">E<-"><+'>cut<-'> in exchange for a less accurate result because the quality of computed physical quantities falls very rapidly when decreasing <+">E<-"><+'>cut<-'> below some point which is essentially determined by the way the pseudopotential is generated. A flexible scheme is therefore to be welcomed if it allows one to regulate the degree of approximation in different parts of a pseudopotential , such as the logarithmic derivative, <+">E<-"><+'>cut<-'> and a large pseudising radius ("cut-off" radius) <+">r<-"><+'>c<-'> to maximise the pseudopotential performance in a given problem without compromising the science. <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#3834,9360>In order to generate a new pseudopotential efficiently, it is a great help to understand how changes in the input parameters affect the pseudopotential that results. Such understanding also helps to one avoid an unrealistic choice of parameters. We present in this chapter a robust way of constructing pseudopotentials which addresses those essential points mentioned above, namely accuracy, softness, flexibility and understanding, through further improvements to the well established "Optimised Pseudopotentials" approach. <[>1-3]. We regard our method as an alternative to those very popular schemes proposed by <:f240,2Times New Roman,0,0,0>Troullier<:f> and Martins <[>9], by Vanderbilt <[>19], and <:f240,2Times New Roman,0,0,0>by Blöchl<:f> <[>20]. These schemes all aim to improve the softness and/or the accuracy of pseudopotentials. <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#5196,9360>Before we describe the technical detail of our current scheme, it is necessary to outline previous pseudopotential optimisation methods. <:f240,2Times New Roman,0,0,0>The "Optimised<:f><:f240,2Times New Roman,0,0,0> Pseudopotentials" proposed by Rappe, Rabe, Kaxiras and Joannopoulos<:f><:f240,2Times New Roman,0,0,0> (RRKJ) <[>1,2] are recognised<:f><:f240,2Times New Roman,0,0,0> as very soft pseudopotentials. <:f><:f240,2Times New Roman,0,0,0>Their scheme is very suitable<:f240,2Times New Roman,0,0,0> for transition<:f><:f240,2Times New Roman,0,0,0> metals and first-row elements which we are interested in and which usually need higher E<+'>cut <-'> then other elements due to their very localised 3<+">d<-"> (or 4<+">d<-">) or 2<+">p<-"> valence electrons. <:f><:f240,2Times New Roman,0,0,0>Based on the RRKJ idea, a modified strategy<:f240,2Times New Roman,0,0,0> of generating optimised<:f><:f240,2Times New Roman,0,0,0> pseudopotentials was suggested by Lin, Qteish, Payne and Heine (LQPH)<:f240,2Times New Roman,0,0,0> <[>3] in order to<:f><:f240,2Times New Roman,0,0,0> simplify the numerical procedure. <:f><:f240,2Times New Roman,0,0,0>Although using different options and procedures,<:f><:f240,2Times New Roman,0,0,0> the basic formulations <:f><:f240,2Times New Roman,0,0,0>in LQPH and the current work<:f><:f240,2Times New Roman,0,0,0> are the same as those in the original RRKJ. <:f><:f240,2Times New Roman,0,0,0>In all these schemes the pseudo wavefunction <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><-'><-'><-'><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">r<-">) <:f><:f240,2Times New Roman,0,0,0>of angular momentum<:f><:f240,2Times New Roman,0,0,0> <+">l<-"> is generated first, and then the pseudopotential <+">V<-"><+"><+'>l<-'><-"><-'><+'> <-'><-'><-'>(<+">r<-">) is derived from it by inverting the Schrödinger equation <:f><:f240,2Times New Roman,0,0,0><[>6]<:f><:f240,2Times New Roman,0,0,0>. The <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><-'><-'><-'><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0> is expressed in terms of some specially chosen spherical Bessel functions as follows :<:f><-'><-'> <+@><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <+B><:S+-2><:f240,2Times New Roman,0,0,0> <:A3> , (1.1) <:f> <:s><:S+-2> <:A4>, <:S+-2><:#426,9360> <:S+-2><:#2298,9360><:f240,2Times New Roman,0,0,0>in which the <+">j<:f240,2Times New Roman,0,0,0><-"><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">q<-"><+'>i <-'><+">r<-">) are spherical Bessel functions with (<+">i<:f240,2Symbol,0,0,0>-<-"><:f240,2Times New Roman,0,0,0>1<:f><:f240,2Times New Roman,0,0,0>) zeros for <+">r<-"> <:f240,2Symbol,0,0,0><<<:f240,2Times New Roman,0,0,0> <+">r<-"><+'>c, <-'>and <:f><-"><:f240,2Times New Roman,0,0,0> <+">j<-"><+'><+">l<-"><-'><+">'<-"><+'> <-'>(<+">q<-"><+'>i <-'><+">r<-">)<:f><:f240,2Times New Roman,0,0,0> their first derivative with respect to <+">r<-">. The <:f240,2Symbol,0,0,0>f<:f240,2Times New Roman,0,0,0><-'><+'><+">l<-"><:f> <-'>(<+">r<-">) is the proper all-electron atomic wavefunction and <:f240,2Symbol,0,0,0>f<:f><+'><+">l<-"><-'><+">'<-"><+'> <-'>(<+">r<-">) its first derivative. Since we start with <:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>f<:f240,2Times New Roman,0,0,0><-'><+'><+">l <-"><:f><-'>(<+">r<-"><+'>c<-'>) when generating a new pseudopotential, all the <+">q<-"><+'>i<-'> are fixed once the <+">r<-"><+'>c<-'> is chosen.<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>The portion of the kinetic energy of the pseudo wavefunction<:f><:f240,2Times New Roman,0,0,0> due to the <+">q<-"> <:f240,2Symbol,0,0,0><;><:f240,2Times New Roman,0,0,0> <+">Q<-"><+'>c<-'><:f><:f240,2Times New Roman,0,0,0> part of its Fourier<:f><:f240,2Times New Roman,0,0,0> components is denoted by <:f240,2Symbol,0,0,0>D<:f240,2Times New Roman,0,0,0><+">E<+'>k<-"><-'><:f><:f240,2Times New Roman,0,0,0> (in atomic-Rydberg unit) :<-"><-'><-"><-'><-"><-'> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><-'><-'><-'><-"><-'><-"><-'><-"><-'><-"><-'><:f240,2Times New Roman,0,0,0> <:A2> (1.2a) <:f><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0> <:A0> , (1.2b) <:S+-2><:#426,9360> <:S+-2><:#4086,9360><:f240,2Times New Roman,0,0,0>in which the <:f240,2Symbol,0,0,0>Y<+"><+'><:f240,2Times New Roman,0,0,0>l <-'><-"><:f240,2Times New Roman,0,0,0>(<+">q<-"><:f><:f240,2Times New Roman,0,0,0>) is the Fourier transform of the <:f><:f240,2Symbol,0,0,0>Y<+"><+'><:f240,2Times New Roman,0,0,0>l <-'><-"><:f240,2Times New Roman,0,0,0>(<+">r<-"><:f><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0>. The central idea of optimising a pseudopotential is that for a given <+">Q<-"><+'>c<-'>, the coefficients <:f240,2Symbol,0,0,0>a<+'><:f240,2Times New Roman,0,0,0>i<-'> of <:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"> <-'><:f240,2Symbol,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Symbol,0,0,0>)<:f240,2Times New Roman,0,0,0> in (1.1) can be obtained by minimising <:f240,2Symbol,0,0,0>D<:f240,2Times New Roman,0,0,0><+">E<+'>k<-"><-'><:f><:f240,2Times New Roman,0,0,0> in (1.2) with Lagrange multipliers constraining the normalisation<:f><:f240,2Times New Roman,0,0,0> and the<:f><:f240,2Times New Roman,0,0,0> continuity of the first and second derivatives of the pseudo wavefunction at <+">r<-"><+'>c<-'><:f><:f240,2Times New Roman,0,0,0>. Thus a smooth<:f><:f240,2Times New Roman,0,0,0> and norm-conserving<:f><:f240,2Times New Roman,0,0,0> pseudo <:f><:f240,2Times New Roman,0,0,0>wavefunction <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"> <-'><:f240,2Symbol,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Symbol,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> can be determined. Incidentally,<:f><:f240,2Times New Roman,0,0,0> the continuity of the pseudo wavefunction <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"> <-'><:f240,2Symbol,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Symbol,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>at <+">r<-"><+'>c<-'><-&> is not<:f><:f240,2Times New Roman,0,0,0> imposed explicitly<:f><:f240,2Times New Roman,0,0,0> because the optimisation procedure results in this condition<:f><:f240,2Times New Roman,0,0,0> being fu lfilled automatically.<:f><:f240,2Times New Roman,0,0,0> This can be understood from the definition of the<:f><:f240,2Times New Roman,0,0,0> <+">j<:f240,2Times New Roman,0,0,0><-"><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">q<-"><+'>i <-'><+">r<-">) <:f><:f240,2Times New Roman,0,0,0>in (1.1). We know that when the constrained<:f><:f240,2Times New Roman,0,0,0> minimisation<:f><:f240,2Times New Roman,0,0,0> is successful, one has <:f><:f240,2Symbol,0,0,0>Y<-&><:f240,2Times New Roman,0,0,0><+'><+">l<-'><-">'<-'><:f><-'><-&><-'><:f240,2Symbol,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<+'>c<-'><-"><:f240,2Symbol,0,0,0>) =<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>f<:f240,2Times New Roman,0,0,0><+'><+">l<-"><-'>'<-"><-"><:f><-">(<+">r<+'>c<-'><-">)<:f240,2Times New Roman,0,0,0> : therefore<:f><:f240,2Times New Roman,0,0,0><-&><-&><-&><-"><-&><-'><-"><-&><-'><-"><-&><-'> <:S+-2><:#426,9360> <:s><:S+-2><:f240,2Times New Roman,0,0,0> <:A1>, (1.3) <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#6642,9360><:f240,2Times New Roman,0,0,0>which gives the required continuity <:f><:f240,2Times New Roman,0,0,0>of <:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"> <-'><:f240,2Symbol,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Symbol,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> from the continuity of <:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>Y<-&><:f240,2Times New Roman,0,0,0><+'><+">l<-'><-">'<-'><:f><-'><-&><-'><:f240,2Symbol,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<+'>c<-'><-"><:f240,2Symbol,0,0,0>) <:f><:f240,2Times New Roman,0,0,0>due to the special choice of expansion functions in (1.1). Although the mathematical scheme is essentially the same, <:f><:f240,2Times New Roman,0,0,0>there are some differences in the way <:f><:f240,2Times New Roman,0,0,0>the procedure is used by the previous authors and in the present work. In <:f><:f240,2Times New Roman,0,0,0>the RRKJ method, typically ten or more spherical Bessel function terms <:f><:f240,2Times New Roman,0,0,0>in (1.1)<:f><:f240,2Times New Roman,0,0,0> are used, with <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'> being varied iteratively<:f><:f240,2Times New Roman,0,0,0> such that the <:f240,2Symbol,0,0,0>D<:f240,2Times New Roman,0,0,0><+">E<+'>k<-"><-'> is minimised<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>to a pre-chosen<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>tolerance, say 1 mRyd. It appears<:f><:f240,2Times New Roman,0,0,0> that<:f><:f240,2Times New Roman,0,0,0> using <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> in that way has the advantage of controlling<:f><:f240,2Times New Roman,0,0,0> the quality of the total energy convergence with respect to the energy cut-off used in the calculat ions.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>In the LQPH method,<:f><:f240,2Times New Roman,0,0,0> the number of spherical<:f><:f240,2Times New Roman,0,0,0> Bessel function terms is<:f><:f240,2Times New Roman,0,0,0> fixed to be four so that there are four <:f240,2Symbol,0,0,0>a<:f240,2Times New Roman,0,0,0><+'>i<-'> coefficients<:f><:f240,2Times New Roman,0,0,0> for the <:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"> <-'>(<+">r<-">) to be determined, making the number of free parameters<:f><:f240,2Times New Roman,0,0,0> equal to the number of constraints<:f><:f240,2Times New Roman,0,0,0>, namely norm-conservation, continuity of the first and second derivatives of <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"> <-'><:f240,2Symbol,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Symbol,0,0,0>)<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>at <+">r<-"><+'>c<-'>, and minimisation of <:f><:f240,2Symbol,0,0,0>D<:f240,2Times New Roman,0,0,0><+">E<+'>k<-"><-'><:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>Efficient numerical routines exist for such a problem, and together with the reasonably small number (four) of<:f><:f240,2Times New Roman,0,0,0> terms in <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">r<-">),<:f><:f240,2Times New Roman,0,0,0> it helps to stabilise<:f><:f240,2Times New Roman,0,0,0> the numerical<:f><:f240,2Times New Roman,0,0,0> procedure<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>In addition to using just four spherical Bessel function terms in <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0>, <:f><:f240,2Times New Roman,0,0,0>the <:f><-'><:f240,2Times New Roman,0,0,0>LQPH method<:f><:f240,2Times New Roman,0,0,0> always sets <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f><:f240,2Times New Roman,0,0,0> equal to the largest <+">q<-"><+'>n <-'>, i.e. <+">q<-"><+'>4 <-'>, <:f><:f240,2Times New Roman,0,0,0>which avoids the variation<:f><:f240,2Times New Roman,0,0,0> of <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> and therefore makes the numerical procedure significantly simpler.<:f><-&><-&><-&><-&><-&><-&><-&><-&><-&><-&> <:s><:S+-2><:#426,9360> <:S+-2><:#7382,9360>We recognise the success of the above mentioned schemes, but the consequences of some of their detailed assumptions, such as the value of <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>, the number of terms and the choice of constraints, were not fully clear to us. In particular, the <+"><+!>role<-!><-"> of <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> in optimising pseudopotentials attracted our attention. From the definition in (1.2), <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> can be regarded as a kinetic energy filter controlling the constrained minimisation of the kinetic energy of <:f240,2Symbol,0,0,0>Y<:f><+'><+">l<-"><-'> in the range <+">q <:f240,2Symbol,0,0,0><-"><;><:f> <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>. If the minimisation is effective, the resulting <:f240,2Times New Roman,0,0,0><+!>k<-!><:f>-space pseudo wavefunction <:f240,2Symbol,0,0,0>Y<:f><+'><+">l <-"><-'>(<:f240,2Times New Roman,0,0,0><+">q<-"><:f>) will be restricted as far as possible to the range 0 <:f240,2Symbol,0,0,0><< <:f240,2Times New Roman,0,0,0><+">q<-"><:f240,2Symbol,0,0,0> <<<:f> <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>, which will subsequently determine the analogous behaviour of the pseudopotential <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">l<-"><-'><+&>ps<-&>(<+">q<-">) in <+!>k<-!>-space in solid state applications. A unique correspondence is therefore likely to exist between <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> and the pseudopotential in <+">k<-">-space <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">l<-"><-'><+&>ps<-&>(<+">q<-">,<:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'>), which of course also applies to <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> and <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">l<-"><-'><+&>ps<-&>(<+">r<-">,<:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>) in <+!>r<-!>-space due to the duality of <+!>r<-!> and <:f240,2Times New Roman,0,0,0><+!>k<-!><:f> spaces. Most importantly, the scattering property of such a pseudopotential should also depends on <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> in some simple manner because it is all in the characteristics of <:f240,2Times New Roman,0,0,0><+">V<-"><:f><+'><+">l<-"><-'><+&>ps<-&>(<+">q<-">,<:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>). In the current scheme we therefore vary <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> to control the phase shift, i.e. logarithmic derivative, as will be demonstrated in Section 2. We shall call this <+">"Q<-"><+'>c<-'> tuning" and it will be the crux of the present work. Additionally, from the argument above we expect the<:f240,,> <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> to<:f><:f240,,> correspond roughly to <:f240,2Times New Roman,0,0,0><+">E<-"><:f240,,><+'>cut<-'><:f><:f240,,>. The <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> therefore controls both the scattering property and the energy convergence of our new optimised pseudopotential.<:f><:f240,,> <:f> <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#4344,9360>On investigating the choice of constraints, we realised that it is not necessary to impose strictly the continuity of <:f240,2Times New Roman,0,0,0> <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"><:f240,2Times New Roman,0,0,0><-'>''(<+">r<-">) <:f>at <+">r<-"><+'>c<-'> because minimisation of<:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>D<:f240,2Times New Roman,0,0,0><+">E<+'>k<-"><-'><:f><:f240,2Times New Roman,0,0,0> already more or less constrains the higher derivatives of <:f><:f240,2Symbol,0,0,0>Y<:f><+'><+">l <-"><-'>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f>) by reducing its high <+">q<-"> amplitude. Moreover, dropping unnecessary constraints means that a less restricted and more efficient minimisation can be performed. Thus not only can a softer pseudopotential be obtained but also the resulting pseudopotential will be more sensiti ve to the choice of <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>, which enhances the controllability of<-'> the pseudopotential by <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'>. <:f240,,>We have, in fact,<:f><:f240,,> tried applying the <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> tuning within the <:f><:f240,,>four-term/four-constraint <:f><:f240,,>framework, and found that both logarithmic derivative and the shape of the pseudopotential<:f><:f240,,> do not vary systematically with respect to <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,>, which is presumably<:f><:f240,,> due to the extra (unnecessary) constraint which somehow restricts the effect of <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> tuning. <+@><:s><:S+-2><:#426,9360><:f240,,> <+@><:S+-2><:#3834,9360><:f240,,>However with the use of three terms and three constraints, we found that the logarithmic derivative and pseudopotential varied smoothly with <+">Q<-"><+'>c<-'> so that one can use <+">Q<-"><+'>c<-'> tuning efficiently. This is the main reason for preferring<:f><:f240,,> the three-term/three-constraint framework. Incidentally, keeping the number of terms and unknown coefficients in (1.1) equal to the number of constraints is very helpful in maintaining<:f><:f240,,> a stable numerical procedure, as found by LQPH. We can also see why the pseudopotential becomes somewhat softer. To a first approximation, <:f240,2Times New Roman,0,0,0><+">E<-"><:f240,,><+'>cut<-'> = <+">Q<-"><+'>c<-'><+&>2<-&> Ryd. if <+">Q<-"><+'>c<-'> is in atomic units as will be assumed<:f><:f240,,> hereafter, and <:f><+"><:f240,2Times New Roman,0,0,0>Q<-"><:f240,,><+'>c<-'> <:f><:f240,,>is approximately<:f><:f240,,> the maximum <+">q<-"><+'>i<-'>. Thus omitting the term <+">j<:f240,2Times New Roman,0,0,0><-"><+"><+'>l<-'><-"><:f240,,>(<:f240,2Times New Roman,0,0,0><+">q<-"><:f240,,><+'>4<-'><:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>) reduces <+">E<-"><+'>cut<-'>, because <:f><:f240,,> <+">j<:f240,2Times New Roman,0,0,0><-"><+"><+'>l<-'><-"><:f240,,>(<:f240,2Times New Roman,0,0,0><+">q<-"><:f240,,><+'>4<-'><:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>) <:f><:f240,,>has the maximum number of nodes and hence the highest Fourier components of the <:f><+"><:f240,,>j<:f240,2Times New Roman,0,0,0><-"><+"><+'>l<-'><-"><:f240,,>(<:f240,2Times New Roman,0,0,0><+">q<-"><:f240,,><+'>i<-'><:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,>).<:f><:f240,,> <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#2556,9360>To summarise : using a three-term expansion in (1.1) and three constraints gives a stable numerical procedure allowing a good flexibility in the pseudo wavefunction. It combines well with tuning <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'> to optimise the accuracy of the pseudopotential, which is the main purpose of the present work. The effect of <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> tuning and relevant technical points are discussed in Section 2. In Section 3 some solid state tests of the pseudopotentials will be presented, which is followed by a discussion and conclusion as Section 4. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <+@><:S+-2><:#426,9360><:f240,,> <+@><:S+-2><:#426,9360><-!> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#564,9360><:f320,,><+!>II.2. The <+">Q<-"><+'>c<-'> Tuning Method<-!><:f> <:S+-2><:#426,9360> <:S+-2><:#10650,9360>To generate<:f240,,> a pseudopotential in<:f><:f240,,> the current scheme<:f><:f240,2Times New Roman,0,0,0>, as in all<:f><:f240,2Times New Roman,0,0,0> <+">ab initio<-"> pseudopotential generating procedures<:f><:f240,2Times New Roman,0,0,0>, <:f><:f240,2Times New Roman,0,0,0>an all-electron LDA or GGA atomic calculation is first performed to obtain all the atomic orbitals of a selected<:f><:f240,2Times New Roman,0,0,0> configuration : in the present work we just use the LDA. <:f><:f240,,>The procedure described in Section 1 is then implemented with three terms in (1.1) and the three constraints already discussed, while<:f><:f240,,> <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> remains as an adjustable input parameter.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,,>In <:f><:f240,,>Fig.1 we demonstrate<:f><:f240,,> the effect of varying <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> in the current scheme on the oxygen 2<+">p<-"> pseudopotential. Three different <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,,><+'>c<-'> were used to generate the corresponding pseudopotentials, and the logarithmic derivative was tested on these pseudopotentials. <:f><:f240,,>We can see that for a given atomic configuration and pseudising radius, there is a certain value of <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> which yields<:f><:f240,,> the best <:f><:f240,,>agreement with the logarithmic derivative<:f><:f240,,>, in this case that shown in Fig.1(b).<:f><:f240,,> <:f><:f240,,>We note that<:f><:f240,,> the logarithmic derivative curve of the pseudo wavefunction for a larger and a smaller <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> deviate<:f><:f240,,> from the curve with the best possible <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,,><+'>c<-'> in opposite directions, as shown in Fig.1(a) and Fig.1(c). <:f><:f240,,>In <:f><:f240,,>Fig.1 we <:f><:f240,,>also see that the shape of the pseudopotential changes with <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,>, which can be regarded as the reason why the scattering properties of the resulting pseudopotentials<:f><:f240,,> are <:f><:f240,,>different.<:f><:f240,,> <:f><:f240,,>The monotonic correspondence between the variation of (a) the <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,>, (b) the shape <:f><:f240,,>of the pseudopotential and (c) the logarithmic derivative <:f><:f240,,>of the pseudopotential<:f><:f240,,> is the most important feature<:f><:f240,,> in the current scheme. <:f><:f240,,>T<:f240,,>his feature<:f><:f240,,> enables us to establish a systematic procedure for <:f><:f240,,>updating <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> towards the best results judged by the following criterion.<:f><:f240,,> <:f><:f240,,>As mentioned in Section 1, <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> controls the softness of a pseudopotential<:f><:f240,,> as well as its accuracy because it affects both the <:f240,2Times New Roman,0,0,0><+">E<-"><:f240,,><+'>cut<-'> and<:f><:f240,,> the logarithmic derivative of the wavefunction<:f><:f240,,>. If transferability<:f><:f240,,> is a higher priority in a particular application, then <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> should be tuned<:f><:f240,,> to obtain the best match between the logarithmic<:f><:f240,,> derivatives<:f><:f240,,> of the<:f><:f240,,> pseudo and true wavefunctions. <:f><:f240,,>Depending on the application, one may require a good match over a wide range of energy for broad bands or only a narrower range in the case of narrow bands. <:f><:f240,,>If a satisfactory match<:f><:f240,,> can be obtained for<:f><:f240,,> a range of <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,>, then the smallest <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,,> should be used to achieve the lowest <:f240,2Times New Roman,0,0,0><+">E<-"><:f240,2Times New Roman,0,0,0><+'>cut<-'><:f240,,>.<:f> <+@><:s><:S+-2><:#426,9360><:f240,,> <+@><:S+-2><:#7326,9360><:f240,,>By removing <:f><:f240,,>the constraint <:f><:f240,,>on the second derivative <:f><:f240,,>of the wavefunction at <:f><:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,><+'>c<-'> <:f><:f240,,>, we allow our pseudopotential to have a discontinuity there because the kinetic energy, which is proportional to <:f240,2Symbol,0,0,0>Y<:f240,,><+&> ' '<-&>(<+">r<-">), is discontinuous<:f><:f240,,> across <+">r<-"><+'>c<-'>, and hence so is the potential. <:f><:f240,,>Although a large discontinuity in a pseudopotential can damage its scattering property, in the current scheme the best possible <+">Q<-"><+'>c<-'> is chosen to give the best fit of the logarithmic derivative, which thus guarantees<:f><:f240,,> that the discontinuity is harmless. This is also consistent with our observation that whenever the logarithmic derivative agreement is satisf actory, the discontinuity is always small. <:f><:f240,,>This can also be understood from the fact that in the current scheme<:f><:f240,,> <:f><:f240,,>the high <+">q<-"> components<:f><:f240,,> of the pseudo wavefunction<:f><:f240,,> are reduced as much as possible, <:f><:f240,,>both because they are expanded<:f><:f240,,> using the least<:f><:f240,,> possible number of spherical<:f><:f240,,> Bessel functions and also because of the minimising procedure imposed on <:f><:f240,,> <:f><:f240,2Symbol,0,0,0>D<:f240,,><+">E<+'>k<-"><-'> <:f><:f240,,>in (1.2)<:f><:f240,,>. <:f><:f240,,>A small<:f><:f240,,> discontinuity at <:f><:f240,2Times New Roman,0,0,0><+">r<-"><:f240,,><+'>c<-'> <:f>will give the pseudopotential some very high Fourier components of small weight spread over a wide range of <+">q<-">, which will not affect calculation significantly if they are cut off by <+">E<-"><+'>cut<-'>. Incidentally, <:f240,2Times New Roman,0,0,0>it is generally the case <:f>that<:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>the <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> yielding the best fit to the logarithmic derivative<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>of the true potential need not be <:f><:f240,2Times New Roman,0,0,0>the <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> that <:f><:f240,2Times New Roman,0,0,0>minimises the discontinuity of the pseudopotential<:f><:f240,2Times New Roman,0,0,0>, even though these two <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> are usually close.<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>We regard the quality of scattering being optimised by the <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> as being more significant than<:f><:f240,2Times New Roman,0,0,0> the existence<:f><:f240,2Times New Roman,0,0,0> of the discontinuity. <:f> <+@><:s><:S+-2><:#426,9360><:f240,,> <+@><:S+-2><:#4281,9360><:f240,,>The harmlessness<:f><:f240,,> of a small discontinuity<:f><:f240,,> is further confirmed by our experience that<:f><:f240,,> good agreement is obtained between the <:f><:f240,,>results of super-cell calculations<:f><:f240,,> <:f><:f240,,>using both <:f><:f240,,><+!>k<-!>-space and <:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,,>-space <:f><:f240,,>versions <[>7]<:f><:f240,,> of the same pseudopotential expressed in<:f><:f240,,> Kleinman-Bylander<:f><:f240,,> form <[>8]. <:f><:f240,,>To convert the <:f><:f240,,><+!>k<-!>-space pseudopotential to one in <:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,,>-space we use the method<:f><:f240,,> of King-Smith <+">et al<-">.<:f><:f240,,> <[>7]. It will <:f><-"><-'><:f240,,>modify the original pseudopotential in a way that depends on <:f><+"><:f240,,>E<-"><:f240,2Times New Roman,0,0,0><+'>cut<-'><:f><:f240,,> in minimising the<:f><:f240,,> aliasing error of the Fast Fourier Transform in planewave supercell calculations.<:f><:f240,,> T<:f><:f240,,>he discontinuity<:f><:f240,,> at <+">r<-"><+'>c<-'> in the original pseudopotential is smoothed out by<:f><:f240,,> the transformation. The fact that both the original and the transformed pseudopotentials gave <:f>almost identical results for the relaxation and energy of structures<:f240,,> shows that the high <+">q<-"> feature at <+">r<-"><+'>c<-'> is irrelevant<:f><:f240,,> to the super-cell results when a reasonable <+">E<-"><+'>cut<-'> is used.<:f> <+@><:s><:S+-2><:#426,9360><:f240,,> <+@><:s><:S+-2><:#426,9360><:f240,,> <+@><:S+-2><:#426,9360><:f240,,> <+@><:S+-2><:#564,9360><-!><+!><:f320,,> <+@><:S+-2><:#564,9360><:f320,,><+!>II.3. Generation and Test of Some Pseudopotentials<-!><:f> <:S+-2><:#426,9360> <:S+-2><:#2556,9360>Although the logarithmic derivative test gives a useful indication about the quality of a pseudopotential, there is no precise criterion of how good the agreement should be for a particular physical application. Also the test is evaluated at a given <+">r<-"> outside the pseudising radius <+">r<-"><+'>c<-'>, which does not give us the information whether the <+">r<-"><+'>c<-'> is small enough for the frozen core approximation to be valid for the given application. A solid state calculation is therefore always necessary for a serious test of a pseudopotential. <:s><:S+-2><:#426,9360> <:S+-2><:#5112,9360>To test the pseudopotential generated by our current scheme,<:f240,2Times New Roman,0,0,0> we have chosen some <:f><:f240,2Times New Roman,0,0,0>bulk<:f><:f240,2Times New Roman,0,0,0> properties of Cu metal <:f><:f240,2Times New Roman,0,0,0>because it is a popular case tested by other authors<:f><:f240,2Times New Roman,0,0,0> <[>1, 9]<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>We follow the RRKJ paper<:f><:f240,2Times New Roman,0,0,0> in using a slightly ionised Cu<:f><:f240,2Times New Roman,0,0,0> configuration 3<+">d<-"><+&> 9<-&> 4<+">s<-"><+&> 0.75<-&> 4<+">p<-"><+&> 0.25 <-&>from which to generate the pseudopotential. <:f><:f240,2Times New Roman,0,0,0>After generating the pseudopotential<:f><:f240,2Times New Roman,0,0,0> for each <+">l<-"> as described in Section 2, it was converted to <:f><:f240,2Times New Roman,0,0,0>Kleinman-Bylander<:f240,2Times New Roman,0,0,0> <[>8]<:f><:f240,2Times New Roman,0,128,255> <:f><:f240,2Times New Roman,0,0,0>form<:f><:f240,2Times New Roman,0,0,0> with the <+">s<-">-potential<:f><:f240,2Times New Roman,0,0,0> chosen as the local potential. Two Cu pseudopotentials<:f><:f240,2Times New Roman,0,0,0> were prepared (Fig.2<:f><:f240,2Times New Roman,0,0,0>), one with smaller <:f240,2Times New Roman,0,0,0><+">d-<-"><:f240,2Times New Roman,0,0,0>core and the other a larger <+">d<-">-core, with <+">r<-"><+'>c<-'>(<+">s,p,d<-">) = (2.0, 2.0, 2.0)<:f><:f240,2Times New Roman,0,0,0> a.u. and <:f><+"><:f240,2Times New Roman,0,0,0>r<-"><+'>c<-'><:f><:f240,2Times New Roman,0,0,0>(<+">s,p,d<-">) = (2.0, 2.0, 2,4) a.u. respectively.<:f> <:f240,2Times New Roman,0,0,0>The <+">Q<-"><+'>c<-'> for these two potentials are <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0>(<+">s,p,d<-"><:f><:f240,2Times New Roman,0,0,0>) = (3.17, 4.66, 6.47<:f><:f240,2Times New Roman,0,0,0>) and Q<+'>c<-'>(<+">s,p,d<-">) = (3.17, 4.66, 5.17<:f><:f240,2Times New Roman,0,0,0> ). <:f><:f240,,>In most cases, we found it useful to choose <:f240,2Times New Roman,0,0,0><+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f240,,> as the initial guess for <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f><:f> from which to start the tuning, so that it is convenient to express the final <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f> in terms of the ratio <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240, Times New Roman,0,0,0><+'>c<-'><:f>/<:f240,2Times New Roman,0,0,0><+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f>. For the Cu pseudopotentials in this section, this becomes <: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><+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f>(<:f240,2Times New Roman,0,0,0><+">s,p,d<-"><:f>) = (0.8, 1.0, 1.175) and <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'>/<+">q<:f240,2Times New Roman,0,0,0><-"><+'>3<-'><:f>(<:f240,2Times New Roman,0,0,0><+">s,p,d<-"><:f>) = (0.8, 1.0, 1.2). <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#2556,9360><:f240,2Times New Roman,0,0,0>The Cu pseudopotential with the smaller <+">d<-">-pseudocore (<+">r<-"><+'>c<-'> = 2.0 a.u.) allows our results to be <:f><:f240,2Times New Roman,0,0,0>compared<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>directly <:f><:f240,2Times New Roman,0,0,0>with those of other popular schemes<:f><:f240,2Times New Roman,0,0,0> in the literature<:f><:f240,2Times New Roman,0,0,0> <[>1,9]<:f><:f240,2Times New Roman,0,0,0>, while we shall use the one with a big <+">d<-">-pseudocore to demonstrate the flexibility<:f><:f240,2Times New Roman,0,0,0> of using <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'>-tuning to generate<:f><:f240,2Times New Roman,0,0,0> a pseudopotential with a larger <+">r<-"><+'>c<-'>. <:f><:f240,2Times New Roman,0,0,0>Although a pseudopotential with larger <+">r<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'> is always softer, it may not <:f><:f240,2Times New Roman,0,0,0>be accurate enough. In our current scheme we can tune the value of <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> so that we obtain a good <:f><:f240,2Times New Roman,0,0,0>logarithmic derivative even for such a large <:f><+"><:f240,2Times New Roman,0,0,0>r<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f><:f240,2Times New Roman,0,0,0>.<:f> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#6830,9360><:f240,2Times New Roman,0,0,0>For calculating the bulk properties of Cu metal, an 8<:f240,2Times New Roman,0,0,0> by 8 by 8 Monkhorst-Pack <+!>k<-!>-point grid <[>10] was used for a<:f><:f240,2Times New Roman,0,0,0> simple-cubic<:f><:f240,2Times New Roman,0,0,0> unit cell<:f><:f240,2Times New Roman,0,0,0> containing four atoms. <:f><:f240,2Times New Roman,0,0,0>With such a coarse grid of <+!>k<-!>-points, smearing<:f><:f240,2Times New Roman,0,0,0> of the occupation function at the Fermi level of 1eV was needed, and the energy was corrected appropriately<:f><:f240,2Times New Roman,0,0,0> <[>11]<:f><:f240,2Times New Roman,0,0,0>.<:f><:f240,2Times New Roman,0,0,0> <:f><-"><:f240,2Times New Roman,0,0,0>In the case of the small <+">d<-">-core pseudopotential, the <:f><:f240,2Times New Roman,0,0,0>convergence test was done and the sudden drop of the total energy in a super-cell calculation was found to occur<:f><:f240,2Times New Roman,0,0,0> at 650 eV<:f><:f240,2Times New Roman,0,0,0> where absolute convergence to ab out 0.1 eV<:f><:f240,2Times New Roman,0,0,0> per atom is reached (Fig.3).<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>To justify<:f><:f240,2Times New Roman,0,0,0> the results obtained at <:f><:f240,2Times New Roman,0,0,0><+">E<-"><:f><+'><:f240,2Times New Roman,0,0,0>cut<-'> <:f><:f240,2Times New Roman,0,0,0>= 650 eV<:f><:f240,2Times New Roman,0,0,0>, a similar calculation was also<:f><:f240,2Times New Roman,0,0,0> performed at 1000 eV where the total energy converged<:f><:f240,2Times New Roman,0,0,0> to within 0.01 eV per atom , and the<:f><:f240,2Times New Roman,0,0,0> results for the bulk properties, as shown in Table.1, were found to be essentially the same. <:f><:f240,2Times New Roman,0,0,0>This is consistent <:f>with our experience that the <:f240,2Times New Roman,0,0,0><+">E<-"><:f><+'>cut<-'> that gives the calculated total energy converged to around 0.1 eV per atom is usually high enough for reliable solid state bulk properties. <:f240,2Times New Roman,0,0,0>In the case of the<:f><:f240,2Times New Roman,0,0,0> pseudopotential with a <:f><:f240,2Times New Roman,0,0,0>large<:f><:f240,2Times New Roman,0,0,0> <+">d<-">-core<:f><:f240,2Times New Roman,0,0,0>, <:f><:f240,2Times New Roman,0,0,0>the convergence test was also done (Fig.3) and we chose<:f><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0><+">E<-"><:f240,2Times New Roman,0,0,0><+'>cut<-'> = 500 eV to run the simple bulk<:f><:f240,2Times New Roman,0,0,0> property tests which are shown as the third line in<:f><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0>Table.1<:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>As one can see <:f><:f240,2Times New Roman,0,0,0>from the <:f240,2Times New Roman,0,0,0>table,<:f><:f240,2Times New Roman,0,0,0> the overall result is satisfactory in comparison<:f><:f240,2Times New Roman,0,0,0> with experiment <:f><:f240,2Times New Roman,0,0,0>and other computational<:f><:f240,2Times New Roman,0,0,0> methods<:f><:f240,2Times New Roman,0,0,0>.<:f> We understand that the valid comparison is with the all-electron calculation because we are testing the pseudising, no t the accuracy of LDA. <:S+-2><:#426,9360> <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#426,9360><:f240,,> <+@><:s><:S+-2><:#426,9360><:f240,,> <+@><:S+-2><:#564,9360><:f320,,><+!>II.4. Discussion and Conclusion<-!><:f> <+@><:s><:S+-2><:#426,9360> <+@><:S+-2><:#6872,9360><:f240,,>In Section 2 we described how <+">Q<-"><+'>c<-'> may be varied to obtain the best fit to the logarithmic derivative of the original potential, and we turn now to look at how the variation of <:f><+"><:f240,,>Q<-"><+'>c<-'><:f><:f240,,> manifests itself in the resultant pseudopotential. <:f><:f240,,>Fig.1 shows that the main effect of <:f><:f240,2Times New Roman,0,0,0>varying the Kinetic Energy<:f><:f240,2Times New Roman,0,0,0> Filter parameter <+">Q<-"><+'>c<-'><:f><:f240,2Times New Roman,0,0,0> is to change the depth of the pseudopotential in<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0><+!>r<-!>-space. One can <:f><:f240,2Times New Roman,0,0,0>interpret<:f><:f240,2Times New Roman,0,0,0> qualitatively the<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>effect of the optimisation on the shape of a pseudopotential from an <+!>r<-!>-space view point, which is<:f><:f240,2Times New Roman,0,0,0> useful when using <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> to regulate the shape of the pseudopotential. <:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>If<:f><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'> is set to be relatively small, this pushes <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-"><:f240,2Times New Roman,0,0,0> <-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0> in the direction of having lower Fourier components, which means having lower kinetic energy inside <+">r<-"><+'>c<-'>. Since the energy eigenvalue is fixed and is equal to the kinetic energy plus potential energy, the low kinetic energy implies a rather shallow (weak) pseudopotential. On the other hand<:f><:f240,2Times New Roman,0,0,0> using a higher <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> results in a deeper pseudopotential as shown in Fig.1(c). <:f><:f240,2Times New Roman,0,0,0>If<:f><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> is reduced even further, <:f><:f240,2Times New Roman,0,0,0>the pseudopotential becomes even<:f><:f240,2Times New Roman,0,0,0> shallower (weaker) and<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>a barrier <:f><:f240,2Times New Roman,0,0,0>will be <:f><:f240,2Times New Roman,0,0,0>raised near <:f><:f240,2Times New Roman,0,0,0><+">r<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'> <:f><:f240,2Times New Roman,0,0,0>as a result of the norm-conserving constraint<:f><:f240,2Times New Roman,0,0,0> so<:f><:f240,2Times New Roman,0,0,0> that the pseudopotential preserves the correct amount of charge within the pseudo-core region. Such a barr ier may look strange but experience shows it does not affect <+">E<-"><+'>cut<-'> or the accuracy of the pseudopotential in solid state tests provided the logarithmic derivative fits well.<:f> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#4260,9360><:f240,2Times New Roman,0,0,0>The effect of<:f><:f240,2Times New Roman,0,0,0> <:f240,2Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0>-tuning <:f><:f240,2Times New Roman,0,0,0>on the shape of a<:f><:f240,2Times New Roman,0,0,0> pseudopotential<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>also depends on other factors<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>In the case of nodeless orbitals such as 2<+">p<-"> and 3<:f240,2Times New Roman,0,0,0><+">d<-"><:f240,2Times New Roman,0,0,0>, the<:f><:f240,2Times New Roman,0,0,0> pseudopotentials are highly attractive<:f><:f240,2Times New Roman,0,0,0> because there is no "Cancellation Effect"<:f><:f240,2Times New Roman,0,0,0> from<:f><:f240,2Times New Roman,0,0,0> inner shells <:f><:f240,2Times New Roman,0,0,0>in the sense of conventional pseudopotential theory <[>12]<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>Optimising<:f><:f240,2Times New Roman,0,0,0><-"><-"> these pseudopotentials therefore<:f><:f240,2Times New Roman,0,0,0> means shifting the electrons <+">outward<-"> from the centres of the atoms. <:f><:f240,2Times New Roman,0,0,0>On the other hand, <:f><:f240,2Times New Roman,0,0,0>in the case of (soft) pseudopotentials that do have a cancellation effect from inner shells, using a smaller<:f><:f240,2Times New Roman,0,0,0> <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> means spreading the charge distribution <+">inwards<-"> towards the centres of the atoms, which<:f><:f240,2Times New Roman,0,0,0> serves to lower the magnitude<:f><:f240,2Times New Roman,0,0,0> of the originally repulsive (or weakly attractive) pseudopotentials at <:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0> = 0, but has less<:f><:f240,2Times New Roman,0,0,0> effect on their shape near <+">r<-"><+'>c<-'><:f><:f240,2Times New Roman,0,0,0>. <:f240,2Times New Roman,0,0,0>Such<:f><:f240,2Times New Roman,0,0,0> a trend can be used to <:f>systematically regulate the shape of a pseudopotential by tuning <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><:f240,2Times New Roman,0,0,0><+'>c<-'><:f>.<-"><-"><-"> <:s><:S+-2><:#426,9360> <+@><:S+-2><:#1278,9360>The current scheme has been used to generate a significant number of pseudopotentials for a wide range of applications, some of which have already been published, namely those for Co <[>13]; Ge <[>14]; C, O and Pd <[>15]; Cu and Cl <[>16]. <+@><:s><:S+-2><:#301,9360><:f160,QCourier,> <+@><:s><:S+-2><:#301,9360><:f160,QCourier,> <:f> <:S+-2><:#5196,9360>In summary, therefore, <-!><-!><:f240, Times New Roman,0,0,0>we have introduced an improved scheme for generating Optimised Pseudopotentials. The <+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'><:f240, Times New Roman,0,0,0> parameter is used in a new way, and is tuned to give as accurate a pseudopotential as possible, meaning a good match to the all-electron logarithmic derivative of the wave functi on over a <:f><:f240,2Times New Roman,0,0,0>suitably wide range of energy. <-!><:f><:f240,2Times New Roman,0,0,0>The continuity constraint of <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-'>''<-"><:f240,2Times New Roman,0,0,0><-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>at <:f240, Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0><+'>c<-'> is dropped<:f><:f240,2Times New Roman,0,0,0> and t<:f><:f240,2Times New Roman,0,0,0>he number of terms in the expansion (1.1) is also reduced to three, to remain equal to the number of constraints in order to give a numerically stable algorithm. <:f><:f240,2Times New Roman,0,0,0>Dropping<:f><:f240,2Times New Roman,0,0,0> the constraint on continuity of the <:f><:f240,2Symbol,0,0,0>Y<:f240,2Times New Roman,0,0,0><+'><+">l<-'>''<-"><:f240,2Times New Roman,0,0,0><-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0> means the pseudopotential has a discontinuity at <+">r<-"><+'>c<-'>, but in practice the <:f240, Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'> is tuned<:f><:f240,2Times New Roman,0,0,0> in our scheme<:f><:f240,2Times New Roman,0,0,0> to match the logarithmic derivative<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>which always makes the discontinuity<:f><:f240,2Times New Roman,0,0,0> small, so that it does not <:f><:f240,2Times New Roman,0,0,0>adversely<:f><:f240,2Times New Roman,0,0,0> affect the accuracy or the softness <:f><:f240,2Times New Roman,0,0,0>of the pseudopotential<:f><:f240,2Times New Roman,0,0,0>. <:f><:f240,2Times New Roman,0,0,0>In some sense<:f><:f240,2Times New Roman,0,0,0> the dropping of one constraint <:f><:f240,2Times New Roman,0,0,0>allows<:f><:f240,2Times New Roman,0,0,0> the pseudo wavefunction (and hence<:f><:f240,2Times New Roman,0,0,0> pseudopotential) greater freedom for optimisation with regard to accuracy and convergence properties.<:f><-!><-'> <-!><-'> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#3834,9360><:f240,2Times New Roman,0,0,0>A most important point is that<:f><:f240,2Times New Roman,0,0,0> the generated pseudopotential and the corresponding logarithmic derivative vary with the chosen <:f240, Times New Roman,0,0,0><+">Q<-"><:f240,2Times New Roman,0,0,0><+'>c<-'> parameter in a systematic way. <:f><:f240,2Times New Roman,0,0,0>One <:f><:f240,2Times New Roman,0,0,0>therefore <:f><:f240,2Times New Roman,0,0,0>has a well controlled situation for generating and improving a pseudopotential for any given physical application, depending on the required balance<:f><:f240,2Times New Roman,0,0,0> between <+">E<-"><+'>cut<-'>, the accuracy of the pseudopotential and the width of the energy range over which it has to be accurate. This is important for many calculations. Moreover we have shown how one can <:f><:f240,2Times New Roman,0,0,0>physically <:f><:f240,2Times New Roman,0,0,0>understand <:f><:f240,2Times New Roman,0,0,0>the connection between <:f240, Times New Roman,0,0,0><+">Q<-"><:f240, Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> and the shape of the pseudopotential<:f><:f240,2Times New Roman,0,0,0>, which helps one to operate the scheme systematically and efficiently. <:f><:f240,2Times New Roman,0,0,0>The scheme represents a further<:f><:f240,2Times New Roman,0,0,0> significant step toward generating systematically good pseudopotentials for a wide variety of physical sys tems. <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#564,9360><:f320,2Times New Roman,0,0,0><+!>Acknowledgements<-!><:f> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#1704,9360><:f240,2Times New Roman,0,0,0>We are grateful to<:f><:f240,2Times New Roman,0,0,0> Dr. K. Rabe for the very useful discussion on the use <+">Q<-"><:f240, Times New Roman,0,0,0><+'>c<-'><:f240,2Times New Roman,0,0,0> at an early stage of this work. <:f><:f240,2Times New Roman,0,0,0>We also want to thank Dr. R. J. Needs for his interesting comments on<:f><:f240,2Times New Roman,0,0,0> basis set expansion for pseudo wavefunctions<:f><:f240,2Times New Roman,0,0,0>.<:f> I also want to thank Dr V. Milman a nd Dr S. Crampin for collaborating with me in doing the Cu tests. <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#496,9360><:f280,2Times New Roman,0,0,0><+!> <:S+-2><:#564,9360><:f320,2Times New Roman,0,0,0><+!>References<-!><:f> <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>1]<:f><:f240,2Times New Roman,0,0,0> <:f><: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><:f240,2Times New Roman,0,0,0> <:S+-1><:#568,9360><:f240,2Times New Roman,0,0,0><[>2] <:f><:f240,2Times New Roman,0,0,0>A. M. Pappe and J. D. Joannopoulos, in <+">Computer Simulation in Material Science<-">, edited by M. Mayer and V. Pontikis, pp. 409-422 (Kluwer, Dordrecht, 1991)<:f> <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><[>3] <: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><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>4]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>R. Car and M. Parrinello, Phys. Rev. Lett <+!>55,<-!> 2471 (1985)<:f> <:S+-1><:#284,9360> <:S+-1><:#573,9360><:f240,2Times New Roman,0,0,0><[>5]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>M. C. Payne, M. P. Teter, D. C. Allen, T. A. Arias and J. D. Joannopoulos, Rev. Mod. Phys. <+!>64<-!>, 1045 (1992)<:f> <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>6]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>D. R. Hamann, M. Schluter and C. Chiang, Phys. Rev. Lett <+!>43<-!>, 1494 (1979) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>7]<:f><:f240,2Times New Roman,0,0,0> <:f><: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><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>8]<:f><:f240,2Times New Roman,0,0,0> <:f><: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><:#289,9360><:f240,2Times New Roman,0,0,0><[>9]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>N. Troullier and J. L. Martins, Phys. Rev. B <+!>43<-!>, 1993 (1991)<:f> <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>10] <:f><:f240,2Times New Roman,0,0,0>H. J. Monkhorst and J. D. Pack, Phys. Rev. B <+!>13<-!>, 5188 (1976) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>11]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>M. J. Gillan, J. Phys : Condensed Matter <+!>1<-!>, 689 (1989); <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0>A. De Vita, Ph.D. Thesis, Keele University, 1992, and <:f><:f240,2Times New Roman,0,0,0>A. De Vita <+">et al<-">., to be published. <:S+-1><:#284,9360> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>12]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>V. Heine, Pseudopotential Concept, Solid State Physics <+!>Vol.24<-!> (1970) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>13]<:f><:f240,2Times New Roman,0,0,0> <:f>V. Milman, M. H. Lee, and M. C. Payne, Phys Rev B <+!>49<-!>, 16300 (1994) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#573,9360><:f240,2Times New Roman,0,0,0><[>14]<:f><:f240,2Times New Roman,0,0,0> <:f>V. Milman, D. E. Jesson, S. J. Pennycook, M. C. Payne, M. H. Lee, I. Stich, Phys Rev B <+!>50<-!>, 2663 (1994) <:S+-1><:#284,9360> <:S+-1><:#568,9360><:f240,2Times New Roman,0,0,0><[>15]<:f><:f240,2Times New Roman,0,0,0> <:f>P. Hu, D. A. King, S. Crampin, M. -H. Lee and M. C. Payne, Chem Phys Lett, 230, 501 (1994) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#568,9360><:f240,2Times New Roman,0,0,0><[>16]<:f> H. -C. Hsueh, J. R. Maclean, G. Y. Guo, M. -H. Lee, S. J. Clark, G. J. Ackland and J. Crain, To be published in Phys Rev B <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>17]<:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0>Z. W. Lu, S. -H. Wei, and A. Zunger, Phys. Rev. B <+!>41<-!>, 2699 (1990) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><[>18] P. van 't Kiooster, N. J. Trappeniers, and S. N. Biswas, Physica B <+!>97<-!>, 65 (1979) <:S+-1><:#284,9360> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>19] D. Vanderbilt, Phys. Rev. B <+!>41<-!>, 7892 (1990) <:S+-1><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-1><:#289,9360><:f240,2Times New Roman,0,0,0><[>20] <:f><:f240,2Times New Roman,0,0,0>P. E. Blöchl, Phys. Rev. B <+!>41<-!>, 5414 (1990)<:f> <:S+-1><:#284,9360><-!> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <+@><:S+-2><:#426,9360><-!> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#564,9360><:f320,,><+!>Tables<-!><:f> <+@><:S+-2><:#426,9360> <+@><:S+-2><:#2130,9360>TABLE.1. <:f240,2Times New Roman,0,0,0>The solid state bulk test of Cu pseudopotentials<:f>, comparing the results of the present calculations (first three lines) with those from another pseduopotential (TM <[>9]) and from an all-electron calculation (LAPW ), and with experiment : for lattice constant <+">a<-">, bulk modulus <:f240,2Times New Roman,0,0,0><+">B<-"><:f> and <:f240,2Times New Roman,0,0,0><+">B<:f240,2Times New Roman,0,0,0><-">'<:f> the pressure derivative of bulk modulus fitted from the equation of state. Percentage errors relative to experimental value are given in brackets. <+@><:S+-1><:#284,9360> <+@><:S+-1><:#211,9360><:f180,2Times New Roman,0,0,0>=================================================================================== <:S+-1><:#242,9360><:f180,QCourier,0,0,0> Type <+">E<-"><+'>cut<-'>(eV) <+">a<-">(Å) <+">B<-">(GPa) <+">B<-">'<:f> <:S+-1><:#221,9360><:f180,QCourier,0,0,0>____________________________________________________________________________ <:S+-1><:#221,9360><:f180,QCourier,0,0,0> <:S+-1><:#242,9360><:f180,QCourier,0,0,0> <+">r<-"><+'>c<-'>(<+">d<-">)=2.0 a.u. 1000 3.60 <<-0.3%<;> 166 <<17%<;> 5.0 <<-5.3%<;><:f> <:S+-1><:#242,9360><:f180,QCourier,0,0,0> <:f><+"><:f180,QCourier,0,0,0>r<-"><+'>c<-'>(<+">d<-">)=2.0 a.u.<:f><:f180,QCourier,0,0,0> 650 3.59 <<-0.6%<;> 163 <<15%<;> 5.4 << 2.3%<;> <:S+-1><:#242,9360><:f180,QCourier,0,0,0> <+">r<-"><+'>c<-'>(<+">d<-">)=2.4 a.u. 500 3.66 << 1.4%<;> 145 << 2%<;> 4.8 <<-9.1%<;> <:S+-1><:#221,9360><:f180,QCourier,0,0,0> <:S+-1><:#242,9360><:f180,QCourier,0,0,0><-&> <:f><+"><:f180,QCourier,0,0,0>r<-"><+'>c<-'>(<+">d<-">)=2.3<:f><:f180,QCourier,0,0,0> a.u.<+&>a<-&> 982 3.60 160 5.1<:f> <:S+-1><:#242,9360><:f180,QCourier,0,0,0> LAPW<+&>b<-&> 3.61 162 <:S+-1><:#242,9360><:f180,QCourier,0,0,0> Experiment<+&>c<-&> 3.61 142 5.28<:f> <:S+-1><:#211,9360><:f180,2Times New Roman,0,0,0>===================================================================================<:f> <+@><:S+-1><:#180,9360><:f180,QCourier,0,0,0> <+@><:S+-1><:#221,9360><:f180,QCourier,0,0,0> a<:f><:f180,QCourier,0,0,0> <:f180,QCourier,0,0,0>Ref.9<:f> <+@><:S+-1><:#221,9360><:f180,QCourier,0,0,0> b Ref.17 <+@><:S+-1><:#180,9360><:f180,QCourier,0,0,0> c Ref.18 <:S+-2><:#426,9360> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#426,9360> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#564,9360><:f320,2Times New Roman,0,0,0><+!>Figure Captions<-!><:f> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#1278,9360><:f240,2Times New Roman,0,0,0>FIG.1. <:f><:f240,2Times New Roman,0,0,0>Oxygen 2<+">p<-"> pseudopotential showing its variation with Q<+'>c<-'>. Upper panels, V<:f240,2Times New Roman,0,0,0><+"><+'>l<-">=1<-'><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0> <+">r<-"><:f240,2Times New Roman,0,0,0>) : lower panels, logarithmic derivatives of the true potential (dashed line) and pseudopotential (solid line). (a) 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>=0.98 (b) 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> =1.15 (c) 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>=1.20. <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#852,9360><:f240,2Times New Roman,0,0,0>FIG.2. <:f><:f240,2Times New Roman,0,0,0>The <: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> pseudopotentials for Cu. (a) Small <:f240,2Times New Roman,0,0,0><+">r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f240,2Times New Roman,0,0,0>=2.0 a.u. for <+">l<-">=0,1,2. (b) <:f240,2Times New Roman,0,0,0><+">r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f240,2Times New Roman,0,0,0>=2.0 a.u. for <:f240,2Times New Roman,0,0,0><+">l<-"><:f240,2Times New Roman,0,0,0>=0,1 but larger <:f240,2Times New Roman,0,0,0><+">r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f240,2Times New Roman,0,0,0>=2.4 a.u. for <:f240,2Times New Roman,0,0,0><+">l<-"><:f240,2Times New Roman,0,0,0>=2. (<+">s<-">: dashed line, <:f240,2Times New Roman,0,0,0><+">p<-"><:f240,2Times New Roman,0,0,0>: dot-dashed line, <:f240,2Times New Roman,0,0,0><+">d<-"><:f240,2Times New Roman,0,0,0>: solid line) <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#852,9360><:f240,2Times New Roman,0,0,0>FIG.3. <:f><:f240,2Times New Roman,0,0,0>Convergence of total energy per atom of copper metal with respective to the cut-off energy<:f><:f240,2Times New Roman,0,0,0> for two pseudopotentials<:f><:f240,2Times New Roman,0,0,0> wi th <:f><+"><:f240,2Times New Roman,0,0,0>r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f240,2Times New Roman,0,0,0>=2.5 a.u.<:f> (solid line) and <+"><:f240,2Times New Roman,0,0,0>r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f240,2Times New Roman,0,0,0>=2.0 a.u.<:f> (dashed line). > Times New Roman,18,12,0,0,0,0,0 $$=\ -\int_0^\infty d^3r\ \Psi _l^{*}(r)\ \nabla ^2\Psi _l(r)\ -\ \int_0^{Q_c}d^3q\ q^2|\Psi _l(q)|^2$$Times New Roman,18,12,0,0,0,0,0 $$\Psi _l(r_c)=\sum_{i=1}^n\alpha _i\ j_l(q_ir_c)=\sum_{i=1}^n\alpha _i\cdot \frac{\phi _l(r_c)}{\phi _l^{\prime }(r_c)}\cdot \ j_l^{\prime }(q_ir)=\Psi _l^{\prime }(r_c)\cdot \frac{\phi _l(r_c)}{\phi _l^{\prime }(r_c)}=\phi _l(r_c)$$Times New Roman,18,12,0,0,0,0,0 $$\,\Delta E_k(\alpha _1,\alpha _2,...,\alpha _n,Q_c)\ =\int_{Q_c}^\infty d^3q\ q^2|\Psi _l(q)|^2$$Times New Roman,18,12,0,0,0,0,0 $$\Psi _l(r)=\sum_{i=1}^n\alpha _i\ j_l(q_ir)\quad \text{for}\quad 0