[ver] 4 [sty] [files] [charset] 82 ANSI (Windows, IBM CP 1252) [revisions] 0 [prn] Digital Colormate PS [port] FILE: [lang] 2 [desc] Ph.D. Thesis Chapter VI 803503914 34 781849391 5284 16 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] 2 537526272 2225 10812 9937 13949 0 1 3 0 0 0 0 0 0 0 0 16777215 3 0 785 7712 2927 [frmname] Frame3 [frmlay] 13949 7712 1 0 0 1 10812 0 0 2 0 0 0 0 1 2225 9937 0 [isd] .X3 .sdw .X3 1 1 0 0 10886 62753 100 0 384 258 0 0 1 16 13895 24 13895 32 13895 40 13895 48 13895 56 13895 8 13895 15656 0 7989 11014 15694 14158 13 53 0 0 0 0 65535 255 0 0 0 0 20 0 0 21504 29549 21024 28269 0 0 0 0 0 0 0 0 0 0 0 0 0 224 0 0 0 0 0 0 1 360 180 360 720 1 0 0 16494 14414 14334 0 10415 1882 10415 115 14359 192 14359 418 14375 487 14375 288 14359 1 255 0 17151 30 33279 0 65535 0 65280 0 65280 178 65280 255 32768 255 0 255 16225 255 129 255 255 255 255 128 65535 255 0 0 1 49407 206 49407 182 49919 129 65535 194 65474 145 65473 213 65473 255 61378 255 61920 255 57568 255 49121 255 49663 253 49407 228 65535 255 36751 143 33535 160 33023 128 49151 24 65535 128 65408 128 65410 202 65408 255 58017 255 57792 255 49087 255 40930 255 32767 255 33535 194 63479 247 32896 128 16639 112 33279 65 33279 0 65535 0 65345 50 65346 199 65280 255 49215 255 40833 226 32896 255 33472 255 17151 249 16639 160 61423 239 29298 114 255 0 17151 30 25314 0 49087 0 65280 0 65280 178 57344 224 32768 255 33089 255 16705 255 17026 255 255 255 255 128 57825 225 24415 95 194 0 8417 0 16289 0 41377 0 49664 0 49408 150 49408 194 25088 225 0 255 65 255 129 255 161 159 192 127 53970 210 20303 79 129 0 160 0 17026 0 33153 0 32768 0 40704 130 33280 128 16384 128 0 128 34 161 65 194 128 128 130 64 49344 192 16448 64 45520 161 41440 117 45266 106 49856 124 49538 104 49281 151 49791 188 45681 207 40863 224 41408 224 40930 222 37359 235 40930 200 45746 178 12079 47 24703 79 25249 82 25216 16 33410 63 25151 31 24892 62 24631 94 16656 96 24674 161 16738 129 12640 129 8544 98 8546 82 41634 162 0 0 0 904 984 428 154 153 155 151 0 0 8 13887 24 13887 32 13887 40 13887 16 13887 48 13887 0 0 .sdw 13 65483 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,,>VI. Ideas for Futrue Research<: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] <:f180,,> <+B><:f180,,><:P10,0,VI-> <:f180,,> > [elay] [l1] 0 [pg] 16 25 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 38 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 46 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 55 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 67 1152 10 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 78 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 86 1510 95 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 96 394 92 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 0 109 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 0 116 775 94 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 0 147 0 33 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 0 191 0 27 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 0 236 0 54 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 0 286 0 79 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 0 333 0 0 0 0 0 0 65535 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 336 0 200 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><:#561,9360><:f480,,><+!> <+B><:s><:#561,9360><+!><:f480,,>VI <+B><:s><:#561,9360><+!><:f480,,> <+B><:s><:#561,9360><+!><:f480,,>Ideas for Further Research<-!><:f> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:#284,9360> <:#284,9360> <:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:S+-2><:#1704,9360>In the course of the work described in Chapters II to V, various problems have been encountered to which I have given some consideration without resolving them definitively. The present chapter will indicate some of my ideas and results as far as they have got, with thoughts for further research. The discussion is inevitably uneven and incomplete. <:s><:S+-2><:#426,9360> <:S+-2><:#1278,9360>The points to be discussed are collected under three headings. Section VI.1 is devoted to various issues connecting to overlap of the pseudo cores of radius <+">r<-"><+'>c<-'>. Section VI.2 discusses the effect of <+">Q<-"><+'>c<-'>-tuning optimisation on a pseudopotential which is already very soft initially. <:s><:S+-2><:#426,9360> <:S+-2><:#2130,9360>Elements in groups I-A and II-A of the periodic table has always presented problems for pseudopotential generation. These include the need for non-linear core correction, difficulty in reducing <+">r<-"><+'>c<-'> sufficiently, and others.<:f240,2Times New Roman,0,0,0> We need a practical method to generate accurate pseudopotentials in this family, and also a better understanding of the effect of highly excited reference states in the Kleiman-Bylan der non-local projector form.<:f> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:#376,9360><:f320,,><+!>VI.1. Pseudo Core Overlap<-!><:f> <:s><:#284,9360> <:s><:#284,9360> <:S+-2><:#4260,9360>The power of <+">ab initio<-"> total energy pseudopotential method in understanding the complex electronic structure problems is widely recognised. There are more and more challenging studies carried out involving systems far from the normal crystalline sol id state. Due to the wide variety of physical and chemical environments, the proper design and use of pseudopotential has never been so important. In some of these cases we are close to the limitation of pseudopotential approximation. However the overlappin g pseudo cores is a typical problem which frequently arises in molecular and high pressure studies. This motivates us to develop better strategies for using pseudopotentials when the pseudo-cores are inevitably overlapping. Note that we are discussing a mat hematical point concerning overlap between the spheres of radius <+">r<-"><+'>c<-'>, which have nothing to do with the size of the real atomic cores. <:s><:S+-2><:#426,9360> <:S+-2><:#2130,9360><:f240,2Times New Roman,0,0,0>There are two scenarios<:f><:f240,2Times New Roman,0,0,0> that will appear in our following discussion : If the error comes from the region near <+">r<-"><+'>c<-'> then the "Feed Back Loop" effect is important, and therefore the "lowest reliable <+">E<-"><+'>cut<-'>" approach should improve the situation. If the error is mainly due to the central region around the origin<:f><:f240,2Times New Roman,0,0,0>, the core penetration<:f><:f240,2Times New Roman,0,0,0> is significant and hence the "Non-line ar Core Correction" should help to correct it. Detailed discussions are presented in the following sections.<:f> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360><-!><+!>Pseudo Cores Overlapping and A Feed Back Loop<-!> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#2556,9360>It is extremely difficult to avoid pseudo cores overlapping in molecular system such as O<+'>2<-'> and CO. The bond length of O<+'>2<-'> is 1.207 <\E> and for CO it is 1.128 <\E>, whereas typical small-core optimised C and O pseudopotentials have pseudo core radii around 1.2 to 1.4 Bohr radius, which will obviously produce overlap. If an even smaller core is chosen, then there is not much left to be optimised and the pseudopotential will be extremely hard. It is therefore not recommended to use 2<+">p<-"> pseudopotentials having such a small core. <:s><:S+-2><:#426,9360> <:S+-2><:#1704,9360>It is natural to ask the question whether one can simply use overlapped core regions and still get reasonable results. Unfortunately, from the systems tested this seems not to be the case. In those tests we have seem the bond length shorten when <+">E<-"><+'>cut<-'> is increased. A heavily overlapped O<:f240,2Times New Roman,0,0,0><+'>2<-'><:f> usually leads to a very short bond length.<-'><-'> <:s><:S+-2><:#426,9360> <:S+-2><:#6390,9360>Based on this observation, we have proposed a possible mechanism to explain what might be happening during the testing process and how things start to go wrong. We think that between overlapped pseudopotentials, there is an unphysical feature which comes in when adding up two overlapped pseudopotentials. As shown in the following figures, adding up two Coulombic (convex) potentials leaves only a smooth maximum between two centres. However, adding up two flattened (because of pseudising) attractive concave-sha ped pseudopotentials will produce an artificially attractive well in the middle. This attractive well is fairly localised in space and therefore requires higher Fourier components to describe. There will certainly be some amount of charge accumulates in thi s region, but there may not be very much when the <+">E<-"><+'>cut<-'> is low as the attractive well will be numerically smoother (or flatter). When we increased <+">E<-"><+'>cut<-'> this part was further revealed, making it to attract more electron density which made the centre of the bond more negative and therefore brought two ions closer. This increased the overlap further and deepened the well, so the entire pr ocedure formed a feed back loop ! This explains why we have seen slow converging of bond length with respect to <+">E<-"><+'>cut<-'>. Of course the bond length will finally converge somewhere, but probably at a very very short on incorrect one. <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:A0> <:s><:#284,9360> <:s><:#284,9360> <:S+-2><:#2130,9360>If our suspicion is reasonable, then to correct this error we should not use the highest converged <+">E<-"><+'>cut<-'> obtained from a (core-overlapped) molecule test. To address this problem, we proposed a strategy to deal with the over estimated <+">E<-"><+'>cut<-'> which gave the bad quantities. The idea is to use the lowest possible <+">E<-"><+'>cut<-'> to minimise the possible error that will come from the overlapped region, but it should be high enough for calculating physical properties. <:s><:S+-2><:#426,9360> <:S+-2><:#2130,9360>Our strategy is to take the "Lowest Reliable <+">E<-"><+'>cut<-'>" from non-overlapping calculations from of that same pseudopotential, and then just stick with it for all the major calculations. By using this lowest-reliable <+">E<-"><+'>cut<-'> we de-emphasise the artificial well and therefore minimise the error from the overlapping region, while still providing an <+">E<-"><+'>cut<-'> that will satisfy the requirement of electronic structure studies. We know this <+">E<-"><+'>cut<-'> is usable because it works for the non-overlapping cases. <:s><:S+-2><:#426,9360> <:S+-2><:#3834,9360>Take the CO molecule as an example. In our case both C and O were tested before and we already know that O has the harder pseudopotential. We read from the non-overlapping MgO convergence test that 500eV is the <+">E<-"><+'>cut<-'> where the <+">E<-"><+'>tot<-'> reaches a sudden drop, which is an indication of the onset of the physical <+">E<-"><+'>cut<-'> region. We then use this 500eV for CO. Both ionisation energy and bond length have been looked at and the results turn out to be surprisingly satisfying for the study the chemisorption which this CO was aimed at. We<:f240,2Times New Roman,0,0,0> show tests on CO in Tables 1 and 2. <:f><:f240,2Times New Roman,0,0,0>The <:f>Table 2 is from <[>Ref.H.1]. Note that the calculation with more heavily overlapped cores has a worse convergence of the bond length, which might be suggesting the effect of the <+">E<-"><+'>cut<-'> feed back loop. <:s><:S+-2><:#426,9360> <:S+-2><:#2130,9360>Another possible evidence to link core-overlap with the bond length problem is from the tests on CH<+'>3<-'>OH and O<+'>2<-'> molecules. Using the same <+">E<-"><+'>cut<-'>, different O potentials that gave noticeable different bond lengths for O<+'>2<-'> turned out to be not affecting the CH<+'>3<-'>OH tests. This is consistent with the fact that in O<+'>2<-'> we have more serious core over-lapping than i n CH<:f240,2Times New Roman,0,0,0><+'>3<-'><:f>OH.<:f240,2Times New Roman,0,0,0> Some of the tests are shown in Tables 3 and 4, which were done by R. Shah.<:f> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#1278,9360>If the above analysis of the source of core-overlap error is true, then further correction can be done at the stage of super-cell calculations. We can try to reduce the double counting error by really picking up the double counted region in real space, and avoiding it. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:#376,9360><:f320,,><+!>VI.2. Softer Si Pseudopotentials<-!><:f> <:S+-2><:#426,9360><:f240,2Times New Roman,0,128,255> <:f><:f> <:S+-2><:#3408,9360>Although the Si pseudopotential is perhaps the softest one in the periodic table, there are occasions when calculations are needed on a system contains many Si atoms already. It will then be useful to reduce the <+">E<-"><+'>cut<-'> as much as possible. The Optimised Pseudopotential method was usually used to treat elements with hard potentials only. In this section we present a case study of using <+">Q<-"><+'>c<-'>-tuning to generate Si pseudopotentials with different degrees of optimisation. The Si potentials under study are Si001, Si014 and Si017, whose parameters can be found in Chapter V. A Kerker-type potential Si012 is also included in this s tudy as a comparison. Fig.VI.2.1(a), (b) and (c) compare these pseudopotentials in different aspects. <:s><:S+-2><:#426,9360> <:S+-2><:#4686,9360>It is obvious from the convergence plots Fig.VI.2.1(a) that the heavier optimisation, i.e. the smaller <+">Q<-"><+'>c<-'> , the more obviously the <+">E<-"><+'>tot<-'> converges in successive steps. The Fourier transform plots Fig.VI.2.1(b) reveal the mechanism of how <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'> works on the <:f240,2Times New Roman,0,0,0><+">E<:f240,2Times New Roman,0,0,0><-"><+'>tot<-'><:f>/<+">E<-"><+'>cut<-'> relation. We can see that in general terms, the main weight of the Fourier components is shifted to lower <+">q<-"> when a smaller <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'> is used, as expect from our understanding. It is understandable that the more oscillating the shape is, the more the convergence will proceed in steps as <+">E<-"><+'>cut<-'> hits each extremum in turn. Although the convergence of <+">E<-"><+'>tot<-'> with respect to <+">E<-"><+'>cut<-'> is not the same for all these Si pseudopotentials, the bulk tests done by Dr. R. Perze in the Tables 6, 7 and 8 show that these pseudopotentials all gave very good results when a very high cut-off of 400 eV was used, but the pseudopotentials optimised with smaller <+">Q<-"><+'>c<-'> do indeed converge to the correct value more quickly.<-!> <:s><:S+-2><:#426,9360> <:S+-2><:#4686,9360>As a test application, the 2x1 reconstruction on Si (111) surface was also calculated by Dr. Perez. <:f>An interesting aspect was found in the test : although the same Kerker-type Si pseudopotential (see Si012, Chapter V) was used, <+">E<-"><:f240,2Times New Roman,0,0,0><+'>cut<-'><:f> = 10 Ryd is necessary to start to simulate dimer bulking correctly, in contrast to the 7 Ryd <+">E<-"><+'>cut<-'> reported in the study of 7x7 reconstruction on Si (111) surface <[>Ref.S.2].<-'> The reason for this may be that the <+">sp<-"> hybridisation in these two cases is slightly different. A common Si pseudopotential, such as Si012, has its <+">s<-"> potential very much harder than their <+">p<-"> potential. It is possible that for a given <+">E<-"><+'>cut<-'> the <+">p<-"> character is fully converged but the <+">s<-"> hasn't. If a physical property is mainly due to p character and less sensitive to the <+">s<-">-feature than the <+">E<-"><+'>cut<-'> is sufficient to give a reasonable result. In the other word, if we imagine scanning <+">E<-"><:f240,2Times New Roman,0,0,0><+'>cut<-'><:f> from low to high then we will see the more <+">p<-">-related features converges before the more <+">s<-">-related. <:s><:S+-2><:#426,9360> <:S+-2><:#5112,9360>In most hard pseudopotentials the <+">Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f> is used mainly to fine-tune the scattering and control the shape. The use of <+">Q<-"><+'>c<-'> to improve convergence <-'>has been less emphasised because the value of <:f240,2Times New Roman,0,0,0><+">Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f> has been chosen based on an overall consideration of performance, namely accuracy, the possibility of reducing projectors, and core radius. There have not been very many syst ematic tests of <+">E<-"><:f240,2Times New Roman,0,0,0><+'>tot<-'><:f>/<+">E<-"><:f240,2Times New Roman,0,0,0><+'>cut<-'><:f> with respect to <:f240,2Times New Roman,0,0,0><+">Q<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>. From the case study of optimising Si pseudopotentials presented in this section, we know that even for a soft pseudopotential such as Si, we can still use the <+">Q<-"><+'>c<-'>-tuning method to make it softer, and using the <+">E<-"><+'>cut<-'> at the first convergence step of <+">E<-"><+'>tot<-'> already gives reliable results for bulk properties, which is consistent with the case of Cu reported in Chapter II, in which we found the convergence at the level of 0.1 eV is enough to reproduce bulk properties, instead of 0.01 eV. In the future, it will be useful to have a more systematic investigation on two questions, which are "to what extent can one push down the <+">Q<-"><+'>c<-'> for a soft pseudopotential ?" and "what <+">E<-"><+'>cut<-'> should be used for a reliable calculation ?". <:s><:S+-2><:#426,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:f320,,><+!>VI.3. The Special Cases of the I-A and II-A Elements<-!><:f> <:s> <:s> <:S+-2>The reason why we pay special attention to I-A and II-A Group elements is that their atomic ground states have only <+">s<-"> valence electrons, which has two negative effects on the pseudopotentials. Firstly, the heavier elements of these Groups have very large atomic radii because of their very loosely bound valence electrons, which makes the procedure of pseudising th e wavefunctions under a reasonable <+">r<-"><+'>c<-'> extremely difficult. Secondly, for those elements the non-<+">s<-"> states will be quite highly excited and therefore very different from solid state configurations. This is likely to cause a problem whe n Kleinman-Bylander projectors are used because they need to use these excited state as references states. We will discuss these two problems in the following two subsections. <:s><:S+-2> <:s><:S+-2><+!>Big Core Problem<-!> <:s><:S+-2> <:S+-2>The very large atomic radii of the heavy I-A and II-A elements, due to their very loosely bound valence electrons, make pseudopotential generation of them extremely difficult. This is because that the condition for successful pseudisation usually requires a very much larger <+">r<-"><+'>c<-'> that leads to a serious pseudo core overlap for most physical applications. One way to cure this problem is to "unfreeze" some of the core electrons and treat them as valence electrons in solid state calculations. By doi ng so we have much smaller core radii for those potentials, and this strategy also automatically solves another possible problem namely that of core polarisation, which is likely to occur for elements in the category of heavier I-A and II-A elements. <:f240,2Times New Roman,0,0,0>Freeing part of the core shell<:f240,2Times New Roman,0,128,255> <:f>inevitably makes a pseudopotential very hard, and this is the place where our powerful <+">Q<-"><+'>c<-'>-tuning Optimisation technique comes in. With that high degree of optimisation, we can make the solid state calculation with pseudopotentials still feasible. This strategy has been used in generating Ca, Sr, Ba <:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0>see Chapter V. Ca005, Sr002 and Ba015Rg<:f>) and tested for small molecules <[>Ref.M.1] as well as solids, which give very good results in CaO <[>Ref.M.2, R.1], and work reasonably in SrO (see Sr0 02 in Chapter V) and Ba<+'>2<-'>NH <[><:f240,2Times New Roman,0,0,0>Ref.W.1<:f>] respectively. It is worth emphasising that in these cases we have Kleinman-Bylander pseudopotential with no reference to any artificial excited state. (More details related to KB form will be discussed in next subsection.). <:s><:S+-2><:#426,9360> <:S+-2><:#2130,9360>It will be interesting to ask whether treating some of the core electrons as valence ones is the only way to improve the result of calculations using pseudopotential of I-A and II-A elements. Maybe one can just allow pseudo cores to overlap and will still o btain reasonable results. More tests and comparison on this aspect is need to find the best strategy for generating best pseudopotentials for I-A and II-A elements in the periodic table. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360><+!>Excited State Problem<-!> <:s><:#284,9360> <:S+-2><:#2130,9360>For I-A and II-A elements the valence electrons only occupy s-orbitals in the atomic ground states, as already mentioned. This means both the <+">p<-"> and <:f240,2Times New Roman,0,0,0><+">d<-"><:f> pseudopotentials need to be generated by using excited state configurations. Thus the number of pseudopotential components generated from excited states is larger for this group of elements than typi cally for other elements. In this section we take Mg as an example to discuss the excited state problem in I-A and II-A Groups. <:s><:S+-2><:#426,9360> <:S+-2><:#2982,9360>We have observed that a local Mg potential performs better than a non-local Mg potential generated from the excited states suggested in the BHS paper. This is seen in both MgO (as shown below, and also <[>Ref.R.1]) and MgCl<+'>2<-'> <[>Ref.Q.1] calculations . We think the reason for the non-local Mg pseudopotential to be unsatisfactory is due to its using those very artificial excited atomic states as references states in the Kleinman-Bylander non-local projectors. To identify the problem, we have performed a series of MgO tests using similar Mg and O pseudopotentials, both with or without <+">s<-">, <:f240,2Times New Roman,0,0,0><+">p<-"><:f>, <:f240,2Times New Roman,0,0,0><+">d<-"><:f> non-local components. <:S+-2><:#426,9360> <:S+-2><:#5112,9360>The Table 9 gave us the stresses in a MgO cell calculated at <+">E<-"><+'>cut<-'> = 450 eV when different Mg and O pseudopotentials are used, with the lattice parameter set equal to the experimental one in all cases. This gives a quicker measure of how well the pseudopotential reproduces the observed lattice constant than actually finding the equilibrium value with each pseudopotential. Clearly the smaller the absolute value of stresses the better, because it means the equilibrium lattice parameter will be cl oser to the experimental one. From this table we found that the results are worst (corresponds to 5% shorter lattice parameter than experimental result) whenever the Mg has a non-local <+">d<-">-potential, no matter whether the O pseudopotential carries only <+">s<-"> and <:f240,2Times New Roman,0,0,0><+">p<-"><:f> components or all <:f240,2Times New Roman,0,0,0><+">s<-"><:f>, <:f240,2Times New Roman,0,0,0><+">p<-"><:f>, <:f240,2Times New Roman,0,0,0><+">d<-"><:f> components. <-"><-">The non-local <+">d<-"> component in the O, which we have made<-'><-"><-"> similar to <+">p<-"> one<-'><-"><-">,<-"><-"> turn out to be not important. The table also shows that non-local p potential on the Mg is not nearly so damaging, although it also makes the lattice parameter somewhat worse, at least for the tests shown. <:s><:S+-2><:#426,9360> <:S+-2><:#3834,9360><:f240,2Times New Roman,0,0,0>Considering the fact that the scattering test of the Mg <+">d<-">-state itself is reasonably<:f><:f240,2Times New Roman,0,0,0> good, which means that the potential is correct at least for the configuration used to generate it, we believe that it is the highly excited-states nature in the state in the KB potentia l that makes the MgO results worse because it is too far away from the <+">d<-">-like states in the solid. Just to use a local potential in these elements seems to be a straight forward idea, at least it works in some systems<:f> (Mg in MgO, Na in Na/Si <[><:f240,2Times New Roman,0,0,0>Ref.M.1<:f>]). In the future we should te st the same properties of MgO with a Mg pseudopotential generated using the <+">d<-">-component as local so that there is no effect from the KB form of <+">d<-">-part. If the result is better than the one using Mg with the KB for <+">d<-">-component, then we know precisely it is the excited <+">d<-"> reference state cause the problem. <:s><:S+-2><:#426,9360><:f240,2Times New Roman,255,0,0> <:S+-2><:#1278,9360>In the case of MgO we are now convinced that the error is from the <+">d<-">-potential, but unfortunately <:f240,2Times New Roman,0,0,0><+">d<-"><:f>-states do not exist in any core or valence level of the neutral Mg atom, so that we can not take the advantage "un-freezing" a <+">d<-">-shell to avoid excited state. <:S+-2><:#426,9360>In the future, <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:S+-2><:#426,9360> <:s><:#284,9360> <:#376,9360><:f320,,><+!>VI.4. Conclusion (Suggested Future works)<-!><:f> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#1704,9360>In brief, we have discussed various issues of generating, using and testing pseudopotentials in this chapter, they are the problems of core-overlap, using <+">Q<-"><+'>c<-'>-tuning for soft pseudopotentials, and problems of big pseudo core and KB projectors related to I-A and II-A group elements in the periodic table. <:s><:S+-2><:#426,9360> <:S+-2><:#3408,9360>We have demonstrated that the pseudo-core overlapping is not an absolute disaster which need to be avoided without any compromise. A hand waving argument based on the feed back loop of the degree of overlap and its relation to <+">E<-"><+'>cut<-'> has been proposed, which leads to an alternative approach of using the "lowest but still reliable <+">E<-"><+'>cut<-'>" in a pseudo-core overlapped calculation. The test results of CO molecule seems to be very promising. The nest step will be to quantify the error caused by using overlapping pseudopotentials. It may be possible to introduce some sort of correction to reduce the error from the undesired situation without making <+">r<-"><+'>c<-'> of pseudopotentials very small to avoid overlap, which is obviously very expensive to compute. <:s><:S+-2><:#426,9360> <:S+-2><:#2556,9360>In the Section 2 we have applied the <+">Q<-"><+'>c<-'>-tuning method described in Chapter II to generate even softer Si pseudopotentials. Successive steps of <+">E<-"><+'>tot<-'> convergence with respect to <+">E<-"><+'>cut<-'> has been observed, which has a more pronounced effect when a smaller <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'> is set. What can be done next is to establish the optimum condition of using <+">E<-"><+'>cut<-'> with respect to <+">Q<-"><+'>c<-'> , so that a smaller <+">E<-"><+'>cut<-'> can be used. This can be done through systematic tests on the convergence of physical properties with respect to the <+">E<-"><+'>cut<-'> . <:s><:S+-2><:#426,9360> <:S+-2><:#6390,9360>Two of the major problems we have encountered in generating pseudopotentials are (1) the need to choose a large pseudo core which is larger than the larger than half of the inter-atomic distance, and (2) the difficulty in generate a reliable pseudopotential for atomic states which not occupied in its ground state configuration. Both of these two problems occur in pseudopotentials for I-A and II-A elements, making them more difficult to handle than pseudopotentials of other elements. By taking the advantage of <+">Q<-"><+'>c<-'>-tuning Optimisation method, we treated part of core electrons as valence one so that the a smaller <+">r<-"><+'>c<-'> can be used, and the resulting pseudopotentials are still soft enough to be handle. It worth to carry out some calculations to compare the performance between normal pseudopotentials with big pseudo-cores and those specially treated with extra valence electrons and smaller pseudo-cores one, thus a better strategy of dealing with a big core pseudopotential can be hopefully be found. As for the excited (reference) states and the (possible) KB form problem, a systematic investigation has been done to identify the source error for the case of Mg pseudopotential. A few more tests (in the future) for more elements to investigate whether or not one need to use certain components in a non-local pseudopotential, such as the one presented in Table. 9 , will be most useful. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360> <:S+-2><:#426,9360> <:#376,9360><:f320,,><+!>References<-!><:f> <:s><:#284,9360> <:s><:#284,9360><[>A.1] <:#568,9360>O.K. Andersen, A.V. Postnikov and S. Yu. Savrasov, Mat. Res. Soc. Symp. Proc. Vol. 253, 37 (1992) <:s><:#284,9360> <:s><:#284,9360><[>H.1] <:#284,9360>P. Hu, D. A. King, M. -H. Lee, S. Crampin and M. C. Payne, submitted to Phys. Rev. B <:s><:#284,9360> <:s><:S+-1><:#284,9360><[>K.1] <:S+-1><:#289,9360>G. P. Kerker, J. Phys. C <+!>13<-!>, L198 (1980) <:s><:#284,9360> <:s><:#284,9360><[>L.1] <:#289,9360>S.G. Louie, S. Froyen and M.L. Cohen, Phys. Rev. B <+!>26<-!>, 1738 (1982) <:s><:#284,9360> <:s><:#284,9360><[>M.1] <:#573,9360>V. Milman, M.C. Payne, V. Hiene, R.J. Needs, J.S. Lin and M-H. Lee, Phys. Rev. Lett. <+!>297<-!>, 3971 (1993) <:s><:#284,9360> <:s><:#284,9360><[>M.2] <:s><:#284,9360>V. Milman, private communication. <:s><:#284,9360> <:s><:#284,9360><[>Q.1] <:#284,9360>A. Qteish, private communication. <:s><:#284,9360> <:s><:#284,9360><[>R.1] <:#284,9360>K. Refson, private communication. <:s><:#284,9360> <:s><:#284,9360><[>S.1] <:#284,9360>R. Shah, TCM preprint & private communication. <:s><:#284,9360> <:s><:#284,9360><[>S.2] <:#573,9360>I. Stich, M. C. Payne, R. D. King-Smith, J.-S. Lin, and L. J. Clarke, Phys. Rev. Lett. <+!>68<-!>, 1351 (1992) <:s><:#284,9360> <:s><:#284,9360><[>W.1] <:#284,9360>B. Winkler, V. Milman, M-H. Lee and M.C. Payne, submitted <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#376,9360><:f320,,><+!>Tables<-!><:f> <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>TABLE.1. Test of the bond length of the CO molecule <:#284,9360>using LDA and pseudopotentials C009 (<+">r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>=1.2 a.u.), <:#284,9360>C019 (<:f240,2Times New Roman,0,0,0><+">r<:f240,2Times New Roman,0,0,0><-"><+'>c<-'><:f>=1.0 a.u.), O027 (<:f240,2Times New Roman,0,0,0><+">r<-"><:f><+'>c<-'>=1.4 a.u.). <:s><:#284,9360><:f,2Times New Roman,>=================================<:f> <:#284,9360><+"><:f,2Times New Roman,>E<-"><+'>cut<-'> C009/O027 C019/O027 <:f> <:s><:#284,9360><:f,2Times New Roman,>(eV) bond length error bond length error <:f> <:s><:#284,9360><:f,2Times New Roman,>=================================<:f> <:s><:#284,9360>400 +3.68% - <:s><:#284,9360>500 +0.01% +1.06% <:s><:#284,9360>600 -1.43% -1.05% <:s><:#284,9360>700 -1.76% -1.05% <:s><:#284,9360>800 -1.80% -0.98% <:s><:#284,9360>================================<:f,2Times New Roman,>= <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>TABLE.2. Convergence tests of CO using <:s><:#284,9360>LDA and two sets of pseudopotentials. <:s><:#284,9360><:f,2Times New Roman,>=========================== <:#284,9360><+">E<-"><+'>cut<-'> C009/O027 C019/O027 <:#284,9360>(eV) <+">E<-"><+'>tot<-'> (eV) <+">E<-"><+'>tot <-'>(eV) <:s><:#284,9360>=========================== <:s><:#284,9360>400 -577.9095286 - <:s><:#284,9360>500 -584.2689334 -583.4517404 <:s><:#284,9360>600 -587.1691199 -586.6544886 <:s><:#284,9360>700 -588.3228353 -588.0217869 <:s><:#284,9360>800 -588.7314478 -588.5846829 <:s><:#284,9360>=========================== <:s><:#284,9360><:f,2Times New Roman,> <:s><:#284,9360><:f,2Times New Roman,> <:s><:#284,9360><:f,2Times New Roman,> <:s><:#284,9360><:f,2Times New Roman,> <:s><:#284,9360><:f,2Times New Roman,> <:#284,9360><:f,2Times New Roman,>TABLE.3. CO properties (using pseudopotentials C009 and O027 at <+">E<-"><+'>cut<-'> = 500eV)<:f> <:s><:#284,9360>============================================================== <:#284,9360> bond length (<\E>) error vibrational frequency (cm<+&>-1 <-&>) error <:s><:#284,9360>============================================================== <:s><:#284,9360>LDA 1.1281 -0.02% 2167 1.1% <:#284,9360>GGA 1.1257 -0.23% 2206 2.9% <:s><:#284,9360>---------------------------------------------------------------------------------------------------- <:s><:#284,9360>Expt. 1.1283 - 2143 - <:s><:#284,9360>============================================================== <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>TABLE.4. Results of fully relaxed calculations of <:#284,9360>methanol molecule using different <+">E<-"><+'>cut<-'> , using the LDA <:s><:#284,9360>with pseudopotentials C009 and O027. <:s><:#284,9360><:f,2Times New Roman,>======================================== <:#284,9360><+"><:f,2Times New Roman,>E<-"><+'>cut<-'> <+"> E<-"><+'>tot<-'> C-H C-O O-H angle angle <:s><:#284,9360><:f,2Times New Roman,>(eV) (eV) (<\E>) (<\E>) (<\E>) C-O-H H-C-O <:s><:#284,9360><:f,2Times New Roman,>======================================== <:s><:#284,9360><:f,2Times New Roman,>450 -644.939 1.113 1.400 1.009 107.3 110.8 <:s><:#284,9360><:f,2Times New Roman,>500 -647.576 1.109 1.411 0.992 107.2 110.7 <:s><:#284,9360><:f,2Times New Roman,>550 -649.422 1.108 1.410 0.979 107.5 110.6 <:s><:#284,9360><:f,2Times New Roman,>600 -650.716 1.108 1.409 0.972 107.8 110.6 <:s><:#284,9360><:f,2Times New Roman,>650 -651.498 1.108 1.404 0.969 108.4 110.5 <:s><:#284,9360><:f,2Times New Roman,>700 -652.023 1.108 1.403 0.968 109.2 110.5 <:s><:#284,9360><:f,2Times New Roman,>------------------------------------------------------------------ <:s><:#284,9360><:f,2Times New Roman,>Expt. --- 1.09 1.42 0.96 107.2 108.0 <:s><:#284,9360><:f,2Times New Roman,>========================================<:f> <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>TABLE.5. Results of fully relaxed calculations of <:#284,9360>methanol molecule using different <+">E<-"><+'>cut<-'> and using GGA <:s><:#284,9360>with pseudopotentials C009 and O027. <:s><:#284,9360>======================================== <:#284,9360><+">E<-"><+'>cut<-'> <+"> E<-"><+'>tot<-'> C-H C-O O-H angle angle <:s><:#284,9360>(eV) (eV) (<\E>) (<\E>) (<\E>) C-O-H H-C-O <:s><:#284,9360>======================================== <:s><:#284,9360>450 -649.620 1.103 1.408 1.003 107.2 110.5 <:s><:#284,9360>500 -652.296 1.099 1.419 0.984 107.2 110.5 <:s><:#284,9360>550 -654.163 1.097 1.418 0.972 107.6 110.5 <:s><:#284,9360>600 -655.465 1.097 1.418 0.964 108.3 110.4 <:s><:#284,9360>650 -656.249 1.097 1.413 0.961 108.6 110.4 <:s><:#284,9360>700 -656.777 1.097 1.410 0.961 108.8 110.4 <:s><:#284,9360>---------------------------------------------------------------- <:s><:#284,9360>Expt. --- 1.09 1.42 0.96 107.2 108.0 <:s><:#284,9360>======================================== <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>TABLE.6. Lattice parameter (<\E>) test for Si (diamond structure) <:s><:#284,9360> <:#240,9360><:f200,2Courier,>==================================================================== <:#258,9360><+"><:f200,2Courier,>E<-"><+'>cut<-'>(eV) Si012 Si001 Si014 Si017 Experiment <:s><:#240,9360><:f200,2Courier,>-------------------------------------------------------------------- <:s><:#240,9360><:f200,2Courier,> 82 - 5.41386 5.42482 5.42723 5.429 <:s><:#240,9360><:f200,2Courier,> 96 - 5.43734 5.44538 5.44527<:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>100 5.38754 5.44349 - - <:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>125 - 5.45656 5.45701 5.45090<:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>150 5.41644 5.46373 5.46105 5.45243<:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>175 - 5.46613 5.46228 5.45258<:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>200 5.40306 5.46556 5.46190 5.45330<:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>300 5.39118 5.46478 5.45763 - <:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>400 5.38618 5.46520 5.46247 5.45824<:f><:f200,2Courier,> 5.429<:f> <:s><:#240,9360><:f200,2Courier,>====================================================================<:f> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>TABLE.7. Bulk modulus (GPa) tests for Si (diamond structure) <:s><:#284,9360> <:#240,9360><:f200,QCourier,>====================================================================<:f> <:#258,9360><+"><:f200,2Courier,>E<-"><+'>cut<-'>(eV) Si012 Si001 Si014 Si017 Experiment <:s><:#240,9360><:f200,2Courier,>--------------------------------------------------------------------<:f> <:s><:#240,9360><:f200,2Courier,> 82 - 0.95158 0.90941 0.90060 0.99 <:s><:#240,9360><:f200,2Courier,> 96 - 0.91392 0.89660 0.90426<:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>100 0.92465 0.84430 - - <:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>125 - 0.92255 0.91851 0.92335<:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>150 0.94260 0.89955 0.90196 0.91990<:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>175 - 0.90321 0.89641 0.91901<:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>200 0.99042 0.91040 0.90876 0.91863<:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>300 1.01318 0.91840 1.00568 - <:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>400 0.96332 0.91035 0.90969 0.91068<:f><:f200,2Courier,> 0.99<:f> <:s><:#240,9360><:f200,2Courier,>====================================================================<:f> <:s><:#240,9360><:f200,2Courier,> <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>TABLE.8. dB/dP tests for Si (diamond structure) <:s><:#284,9360> <:s><:#240,9360><:f200,2Courier,>==================================================================== <:#258,9360><+"><:f200,2Courier,>E<-"><+'>cut<-'>(eV) Si012 Si001 Si014 Si017 Experiment <:s><:#240,9360><:f200,2Courier,>-------------------------------------------------------------------- <:s><:#240,9360><:f200,2Courier,> 82 - 6.40779 6.15718 5.98730 4.23 <:s><:#240,9360><:f200,2Courier,> 96 - 5.39034 5.03432 4.79465<:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>100 7.06836 6.30272 - - <:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>125 - 4.46634 4.20827 4.26344<:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>150 4.80548 4.38808 4.24563 4.17850<:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>175 - 4.20176 4.30937 4.17551<:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>200 3.82422 4.07522 4.08639 4.10915<:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>300 2.33116 3.94766 2.71731 - <:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>400 4.16637 4.06751 4.08091 4.09504<:f><:f200,2Courier,> 4.23<:f> <:s><:#240,9360><:f200,2Courier,>====================================================================<:f> <:s><:#284,9360> <:s><:#240,9360><:f200,2Courier,> <:s><:#240,9360><:f200,2Courier,> <:#284,9360>TABLE.9. Tests of MgO crystal (at <+">E<-"><+'>cut<-'> = 450 eV) using different Mg and O <:s><:#284,9360>pseudopotentials. The non-local components carried by the pseudopotential <:#284,9360>is indicated in the ( ) next to the symbols of elements. The column fort.11 <:s><:#284,9360>indicates the actual pseudopotential data files used. <:S+-1><:#240,9360><:f200,QCourier,0,0,0> <:s><:S+-1><:#240,9360><:f200,QCourier,0,0,0>============================================================================ <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg O E_total <:f><:f200,QCourier,0,0,0>sig_xx sig_yy sig_zz fort.11<:f> <:s><:S+-1><:#240,9360><:f200,QCourier,0,0,0>---------------------------------------------------------------------------- <:S+-1><:#280,9360><:f200,QCourier,0,0,0>Mg(<:f240,QCourier,0,0,0>s<:f200,QCourier,0,0,0>) O(spd) -1818.4653461 <:f><:f200,QCourier,0,0,0>0.047905 0.043678 0.051855 Mg006 O020 <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(spd) O(spd) -1822.1352518 <:f><:f200,QCourier,0,0,0>0.169718 0.173849 0.167937 Mg007 O020 <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(sp) O(spd) -1819.0876292 <:f><:f200,QCourier,0,0,0>0.064164 0.059863 0.066161 Mg007sp O020 <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(sd) O(spd) -1821.8125848 <:f><:f200,QCourier,0,0,0>0.164425 0.161849 0.167122 Mg007sd O020 <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(s) O(spd) -1818.7248693 <:f><:f200,QCourier,0,0,0>0.056392 0.052678 0.059353 Mg007s O020a <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(s) O(spd) -1814.0010265 <:f><:f200,QCourier,0,0,0> 0.036187 0.034960 0.037841 Mg006 O020a <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(s) O(sp) -1813.4972199 <:f><:f200,QCourier,0,0,0> 0.018690 0.017259 0.020274 Mg006 O020asp <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(sp) O(sp) -1814.1255619 <:f><:f200,QCourier,0,0,0> 0.034687 0.033260 0.035264 Mg007sp O020asp <:S+-1><:#240,9360><:f200,QCourier,0,0,0>Mg(spd) O(sp) -1817.2512260 <:f><:f200,QCourier,0,0,0> 0.143403 0.142444 0.144053 Mg007 O020asp<:f> <:s><:S+-1><:#240,9360><:f200,QCourier,0,0,0>============================================================================ <:S+-2><:#426,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#376,9360><-!><+!><:f320,,>Figure Captions<-!><:f> <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>FIG.VI.2.1(a) <:#284,9360><+">E<-"><+'>tot<-'>/<:f240,2Times New Roman,0,0,0><+">E<-"><:f><+'>cut<-'> convergence tests of the Si pseudopotentials with different value of <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'>. <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>FIG.VI.2.1(b) <:#568,9360>Comparison of the Fourier transforms of the <+">V<-"><+"><+'>s<-'><-">(<+">r<-">) of the Si pseudopotentials generated with different <:f240,2Times New Roman,0,0,0><+">Q<-"><:f><+'>c<-'>. <:s><:#284,9360> <:s><:#284,9360> <:#284,9360>FIG.VI.2.1(c) <:#312,9360>The real space radial pseudo <+">s<-">-wave functions <+">R<-">(<+">r<-">) (i.e.<+"> r<-"><:f240,2Symbol,0,0,0>Y(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Symbol,0,0,0>) )<:f> for Si generated with different <+">Q<-"><+'>c<-'>. > SM 5|* 8 75R 5R 5Z5R 5 7sO rJO  rsO r dOsO 77.7% .777#. 7t t t t t | 75NNNN55NN 75S&D&&5S&5&S& 7(&|* (&. (&|* (&(&( |* 7 5  gr 5  5 3  7Pq UqqPqPq8 7c@] c@z @c]c]c4h@@ 73 33  33 b    Tms Rmn Overlapped 1/r potentials h   Tms RmnTms Rmn Overlapped concave pseudo coresSSF``#H^`^`X`YaZb[c[d]e^f_g_h`iajckdldmenfogphqirjsktlulvmwnxpyqzq{r|s}t~tv [Embedded] 3 .sdw 51355 829 52184 86734 00138920