[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 I 803115436 23 781849080 7043 20 2071 13204 715 716 1 [fopts] 0 1 0 0 [lnopts] 2 Body Text 1 [docopts] 5 2 [GramStyle] Academic Writing [tag] Body Text 2 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Text 0 0 [tag] Body Single 3 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 0 0 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 0 1 1 0 0 0 0 [nfmt] 280 1 2 . , $ Body Single 0 0 [tag] Bullet 4 [fnt] Times New Roman 240 0 49152 [algn] 1 1 0 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 10 10 1 [spec] 0 0 <*0> 360 1 1 0 0 0 0 [nfmt] 272 1 2 . , $ Bullet 0 0 [tag] Bullet 1 5 [fnt] Times New Roman 240 0 49152 [algn] 1 1 288 288 288 [spc] 33 273 1 0 0 1 100 [brk] 4 [line] 8 0 1 0 1 1 1 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An Introduction to Pseudopotentials and Total Energy Calculations<:f> > [frght] [lyfrm] 1 13248 0 14400 12240 15840 0 1 3 1 0 0 0 0 0 0 0 0 2 [frmlay] 15840 12240 1 1440 792 1 14472 1440 0 1 0 1 1 0 1 1440 10800 2 2 4680 3 9360 [txt] <+B><:f200,,> <+B><:f200,,><:P10,0,I-><:f> > [elay] [l1] 0 [pg] 20 20 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 27 90 121 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 40 281 22 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 50 812 97 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 71 0 111 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 92 453 97 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 0 109 264 116 32 0 0 0 65534 211 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 127 0 111 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 0 146 0 107 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 0 168 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 0 195 0 31 0 1 0 0 65534 4 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 221 503 132 0 0 0 0 65534 538 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 236 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 0 251 0 0 96 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 265 197 93 0 0 0 0 65534 65535 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 279 76 123 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 0 285 764 92 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 0 291 1193 97 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 0 329 128 11 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 0 345 0 8 1025 0 0 0 65535 2 Standard 65535 0 0 0 0 0 0 0 0 0 65535 0 0 65535 0 0 0 0 0 [edoc] <+B><:s><:#561,9360><:f480,,><-!><+!> <+B><:s><:#561,9360><+!><:f480,,>I <+B><:s><:#561,9360><+!><:f480,,> <+B><:s><:#561,9360><+!><:f480,,>An Introduction to Pseudopotentials <+B><:s><:#561,9360><+!><:f480,,>and <+B><:s><:#561,9360><+!><:f480,,>Total Energy Calculations<-!><:f> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:S+-2><:#564,9360><:f320,,><+!>I.1. Overview of Chapters<-!><:f> <:s><:S+-2><:#426,9360> <:S+-2><:#3408,9360>Almost the whole thesis (Chapter I to Chapter VI) is devoted to improving the generation of pseudopotentials for case in electronic structure calculations of solid (or systems that can be treated with supercell geometry). On the one hand I have improved the techniques of generating pseudopotentials. On the other hand these new ideas (which has been very largely my own) have also helped us to understand better the process of generating a good pseudopotential and thus to demystify the considerably what has been describe as its "Black Magic". These techniques and ideas have been applied to generate around 50 pseudopotentials for specific applications, almost all being carried out by other people. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#1278,9360>The remainder of the present chapter (Chapter I) will be devoted to background information, namely some relevant points from the history and theory of pseudopotentials and from the technique of total energy calculations. <:s><:S+-2><:#426,9360> <:S+-2><:#3408,9360>In Chapter II we will report some important insights into the popular Optimised Pseudopotential <[><:f240,2Times New Roman,0,0,0>Ref.R.1,L.1<:f>] methods. We will present a new way (with examples) to use the optimisation <+">q<-">-vector cut-off <+">Q<-"><+'>c<-'>, as well as other factors such as the number of constraints imposed and the number of special function terms used. The important message of this chapter is that the <+">Q<-"><+'>c<-'> can be varied to control the shape (and therefore plane wave energy cut-off) of the pseudopotential, and also to affect the logarithmic derivative of the wave function. Understanding the role of <+">Q<-"><+'>c<-'> in the optimisation procedure enable us to generate better pseudopotentials in a more systematic way based on established schemes for Optimised Pseudopotentials. <:s><:S+-2><:#426,9360> <:S+-2><:#4260,9360>In Chapter III we will propose an extension of the way we use our new <+">Q<-"><+'>c<-'>-tuning Optimisation scheme.<:f240,2Times New Roman,0,0,0> Optimisation has long been used solely<:f><:f240,2Times New Roman,0,0,0> to treat the harder components of a non-local pseudopotential (i.e. "hard" as the opposite of "soft", meani ng having significant high Fourier components : we shall always use "hard" in this sense) , and we will show the benefit of <:f><:f240,2Times New Roman,0,0,0>tuning<:f><:f240,2Times New Roman,0,0,0> <+">Q<-"><+'>c<-'> for other components<:f><:f240,2Times New Roman,0,0,0>. The idea is to make some of the components very similar so that they can be represented to sufficient accuracy by the same local potential, which will reduce the computational effort in subsequent total energy calculations. We called this technique <:f><:f240,2Times New Roman,0,0,0>Projector Reduction. Some atomic and solid state tests will be provided to show that the technique is reliable. <:f><:f240,2Times New Roman,0,0,0>The success of Projector Reduction emphasises the advantage of having a flexible control of the shape of a pseudopotential.<:f> <:s><:S+-2><:#426,9360> <:S+-2><:#2982,9360>Chapter IV is to ask the question : "How local can one make a pseudopotential ?". There has been an observation that some elements can be described by a local potential but others cannot. The success of the technique proposed in Chapter III motivates us to investigate this issue. We generate non-local pseudopotentials with the method introduced in Chapter II and III, and try to make these pseudopotentials as local as possible. In other words, we will present a new approach to construct local or partly-local p otentials from a non-local method, which allow us to investigate the degree of essential non-locality of a pseudopotential. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#1704,9360>A large proportion of our experience on pseudopotentials has come from collaborations and supplying a service. Chapter V <:f240,2Times New Roman,0,0,0>is a short one which <:f>comprises a brief description of the pseudopotentials generated both for general usage and for some special applications. Detailed generating parameters for these pseudopotentials will also be given. <:s><:S+-2><:#426,9360> <:S+-2>In Chapter VI we will mention some ideas for further research in the field. I have noticed several interesting aspect of pseudopotential generation and usage which might merit attention. <:f240,2Times New Roman,0,0,0>We will present some <:f><:f240,2Times New Roman,0,0,0>very preliminary<:f><:f240,2Times New Roman,0,0,0> results for them.<:f240,2Times New Roman,0,128,255> <:f>These relate to three aspects of pseudopotentials, namely pseudo core overlap, even softer potentials, and the some special cases among the I-A and II-A group elements of the periodic table. <:s><:S+-2><:#426,9360> <:S+-2><:#2598,9360><:f240,2Times New Roman,0,0,0>Finally, Chapter VII is an application of the <+">ab initio<-"> electro<:f>nic structure total energy method to study the origin of the disorder in a complex solid <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f>. In this study we have used a three-step approach which combines first principle calculations, a model Hamiltonian and then simulation. The methodology allows us to study the origin of disorder in this weakly ordered material with a modest computational cost. It is a preliminary survey for further studies in Al<:f240,2Times New Roman,0,0,0><+'>2<-'><:f>O<:f240,2Times New Roman,0,0,0><+'>3<-'><:f> systems that we have planned to do. <:s><:S+-2><:#426,9360> <:S+-2>Although the electronic structure total energy calculation is presented in this thesis as an independent chapter in the context of the study of the <:f240,2Symbol,0,0,0>g<:f>-Al<+'>2<-'>O<+'>3<-'> (Chapter VII), all the solid state tests for the pseudopotentials in the thesis were done by using the same <+">ab initio<-"> total energy method. In some cases these were done on semiconductors, in other cases insulators or metals, and even isolated molecules. These tests are not described separately, just mentioned briefly where relevant. <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360><-!> <:s><:#284,9360> <:s><:#376,9360><:f320,,><+!>I.2. Background Information: Essential<-!><+!> Theory of Pseudopotential<-!><:f> <:s><:#284,9360><-!> <:s><:S+-2><:#1704,9360>In the following sub-sections we will review some essential points from the theory of pseudopotential that are relevant to the following chapters of this thesis. The intention is not to develop a full comprehensive discussion but to provide the background i nformation that will be needed in this thesis. <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#289,9360><:f240,,><+!>I.2.0 What Is A Pseudopotential<-!><:f> <+!>and How Is It Used ?<-!> <:s><:#284,9360> <:S+-2><:#5112,9360>A pseudopotential is an effective potential for valence electrons in an atom, molecule or solid. It represents the effects of both the core-electrons and the nuclear Coulomb potential acting on the valence electrons. In other words, a pseudopotential is use d to reproduce the correct valence charge density outside the range of pseudopotential (i.e., outside the pseudising core region) without treating the core electrons and the atomic nucleus explicitly. Inside the pseudising core the distribution of the valen ce charge density is not correct, but this causes little disadvantage on most solid state problems because the physical properties and chemical bonding are dominated by the electronic structure of valence electrons between the ions. This means that the modi fication of the electronic structure inside the pseudising radius region has minimal effect on the calculated properties of solids and molecules. The decoupling of valence and core electrons is a very good approximation as can be seen throughout chemistry, which is based mainly on the effect of the valence electrons. <:s><:S+-2><:#426,9360> <:S+-2>The advantage of using pseudopotentials in electronic structure studies is most obvious when the wave functions are expressed in terms of a plane waves basis set. This is because the pseudopotential is much weaker than the true potential so that the Bloch w ave functions need only a fairly modest number of plane waves in the expansion. <:s><:S+-2><:#426,9360> <:S+-2>From the view point of the APW (Augmented Plane Wave) method <[>Ref.H.3] we know that the energy dependent logarithmic derivative <+">R<-"><+">'<-"><+"><+'>l<-'><-">(<+">r,E<-">)/<+">R<+'>l<-'><-">(<+">r,E<-">), in which the <+">R<+'>l<-'><-">(<+">r,E<-">) is the solution of the radial Schr<\v>dinger equation at energy <+">E<-">, plays the central role in representing the effect of the potential energy on the valence electrons in the APW formalism. Also from the wave scattering theory we know that the phase shift, which is closely related to the logarithmic derivative, of the scattered wave characterises the interaction between the wave function and the scattering potential. It is therefore not difficult to understand that the accuracy of a pseudopotential is determined by how closely it reproduces this logarithmic derivati ve. As an approximation, this logarithmic derivative is only correct in a certain range of the energy, namely that of occupied reference orbitals. This is generally true for all types of pseudopotential, and we will use the example of the OPW (Orthogonalise d Plane Wave) type pseudopotential later to demonstrate such energy dependence. <:s><:S+-2><:#426,9360> <:s><:S+-2><:#2556,9360>The logarithmic derivative of the wavefunctions is particularly important for a special family of pseudopotentials, namely Norm-conserving Pseudopotentials, as we will discuss later in this chapter. The condition of giving the correct phase shift also guara ntees the right electrostatic contribution from pseudopotentials, so that the phase shift (which is directly relate to the logarithmic derivative) is also seen from this point if view as a central criterion for pseudopotentials. <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:S+-2><:#426,9360><+!>I.2.1. The OPW (Orthogonalised Plane Wave) View Point<-!> <:s><:S+-2><:#426,9360> <:S+-2>The most significant effect of core electrons on the valence electrons is the orthogonality between their wavefunctions. Some basic ideas of pseudopotentials can be easily demonstrated by following the OPW type pseudopotential approach which we will discuss here. <:s><:S+-2><:#426,9360> <:S+-2><:#468,9360>Let <:f240,2Symbol,0,0,0>Y<:f> be a state vector and <:f240,2Symbol,0,0,0>e<:f> be its energy eigenvalue. Then the Schr<\v>dinger equation is <:s><:S+-2><:#426,9360> <:s><:S+-2><:A9> . (2.1.1) <:s><:S+-2><:#426,9360> <:S+-2><:#936,9360>We may write <:f240,2Symbol,0,0,0>Y<:f> in terms of a smooth function <:f240,2Symbol,0,0,0>F<:f> and an oscillatory part which is a linear combination of core eigen states <:f240,2Symbol,0,0,0>F<:f><+'>c<-'>, with coefficients <+">b<-"><+'>c<-'> <:s><:S+-2><:#426,9360> <:s><:S+-2><:A8> (2.1.2a) <:s><:S+-2><:#426,9360> <:S+-2>where the <:A3> can be determined by the orthogonality condition <-'> <:s><:S+-2><:#426,9360> <:s><:S+-2><:A7> (2.1.2b) <:s><:S+-2><:#426,9360> <:s><:S+-2>giving <:A2> . (2.1.2c) <:s><:S+-2><:#426,9360> <:s><:S+-2><:#468,9360>If we take this expression for <:f240,2Symbol,0,0,0>Y<:f> back into equation (2.1.1), we have <:s><:S+-2><:#426,9360> <:s><:S+-2><:A6> (2.1.3) <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360>which can be rewritten <:s><:S+-2><:#426,9360> <:s><:S+-2><:A5> (2.1.4a) <:s><:S+-2><:#426,9360> <:s><:S+-2>with <:A4> (2.1.4b) <:s><:S+-2><:#426,9360> <:S+-2><:#3066,9360>where <+">T<-"> is the kinetic energy. The message we receive from this OPW type approach is that the new pseudo Schr<\v>dinger equation (2.1.4) gives the <+">same energy eigenvalue<-"> as the original one. The true potential <+">V<-"> in original Hamiltonian <+">H<-"> has been replaced by an "<+">energy dependent<-">" effective "pseudopotential" <+">V<-"><+'>ps<-'> (2.1.4b) where major difference from true potential <+">lies in the core electron region<-">. At the same time the true wavefunction <:f240,2Symbol,0,0,0>Y<:f> is replaced by a pseudo-wavefunction <:f240,2Symbol,0,0,0>Y<:f><+'>ps<-'> which has <+">no effects from core-orthogonalisation<-"> and so should be much smoother than the original wavefunction. <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <-!><+!>I.2.2 <+">Ab initio<-"> Pseudopotentials from Pseudising and<-!><+!><:f240,2Times New Roman,0,0,0> Inverting <-!><:f><+!>Wave Functions<-!> <:s><:#284,9360> <:S+-2><:#3408,9360>Almost all modern pseudopotentials use <+">ab initio<-"> atomic wavefunction to obtain pseudopotentials. The valence wavefunction for each angular momentum <+">l<-"> obtained from solving radial Schr<\v>dinger equation is modified within a pre-chosen pseudising radius <+">r<-"><+'>c<-'> (called the pseudising or pseudo-core radius, or often just radius for short) to a create nodeless pseudo-wavefunction. The pseudo wavefunction is then introduced back into the radial Schr<\v>dinger equation to find the <+">l<-">-dependent pseudopotentials that will reproduce the energy eigenvalues when acting on the pseudo wave function. The formalism is briefly sketched as follows. <:s><:S+-2><:#426,9360><-!><-!><+!><:f240,2Times New Roman,0,0,0> <:S+-2><:#1320,9360><:f240,2Times New Roman,0,0,0>When we perform atomic an calculation, we are solving <:f><:f240,2Times New Roman,0,0,0>self-consistently<:f><:f240,2Times New Roman,0,0,0> the following set of equations with different <+">l<-"> <:f><:f240,2Times New Roman,0,0,0>for the radial wave functions defined by <:f240,2Symbol,0,0,0>Y<+"><+'><:f240,2Times New Roman,0,0,0>lm<-'><-">(<+!>r<-!>) = <:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0><+&>-1<:f240,2Symbol,0,0,0><-&>F<:f240,2Times New Roman,0,0,0><+"><+'>l<-'><-"><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>)<+">Y<-"> <:f240,2Times New Roman,0,0,0><+"><+'>lm<-'><-"><:f240,2Times New Roman,0,0,0>(<:f240,2Symbol,0,0,0>q<:f><:f240,2Times New Roman,0,0,0>,<:f240,2Symbol,0,0,0>f<:f240,2Times New Roman,0,0,0>) of the occupied orbitals with eigenvalues <+">E<+'>l<-'><-"> :<:f> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A29>. (2.2.1) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#852,9360><:f240,2Times New Roman,0,0,0>Keeping the same <+">E<+'>l<-'><-"> in the equation, we replace the real (screened) potential by a (screened) pseudopotential <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:A30>.<+!><:f240,2Times New Roman,0,0,0> <-!> (2.2.2) <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:S+-2><:#894,9360><:f240,2Times New Roman,0,0,0>How <:f240,2Symbol,0,0,0>F<:f><:f240,2Times New Roman,0,0,0><+'><+">l<-"><-'><+'>,<+!>ps<-!><-'> is generated from <:f240,2Symbol,0,0,0>F<:f240,2Times New Roman,0,0,0><+"><+'>l<-'><-"> varies from one method of generating pseudopotential to another. It is not unique and is the crux of each scheme. It will be discussed for our method in detail later.<:f> <:S+-2><:#1788,9360><:f240,2Times New Roman,0,0,0>The point common to all the methods is that <:f><:f240,2Symbol,0,0,0>F<:f240,2Times New Roman,0,0,0><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> has been modified such that <:f><:f240,2Symbol,0,0,0>F<:f><:f240,2Times New Roman,0,0,0><+'><+">l<-"><-'><+'>,<+!>ps<-!><-'><:f><:f240,2Times New Roman,0,0,0> is nodeless, but is equal to <:f><:f240,2Symbol,0,0,0>F<:f240,2Times New Roman,0,0,0><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> outside <+">r<-"><+'>c<-'>. Having first obtained a suitable <:f><:f240,2Symbol,0,0,0>F<:f><:f240,2Times New Roman,0,0,0><+'><+">l<-"><-'><+'>,<+!>ps<-!><-'><:f><:f240,2Times New Roman,0,0,0> we can then invert (2.2.2) to give<:f><:f240,2Times New Roman,0,0,0> the value of pseudopotential <+">V<-"><+'><+">l<-">,<-'><+'><+!>ps<-!><-'>(<+">r<-">) in terms of the inverse and the second derivative of the one-particle pseudo-wavefunction : <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:S+-2><+!><:f240,2Times New Roman,0,0,0><:A31> <-!> (2.2.3) <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A32> (2.2.4) <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0>This is the <+">l<-"> dependent (screened) pseudopotential<:f><:f240,2Times New Roman,0,0,0> that is seen by the <:f><:f240,2Times New Roman,0,0,0><+">l<-"><:f><:f240,2Times New Roman,0,0,0>-<:f><:f240,2Times New Roman,0,0,0>state pseudo-wavefunction <:A10>. Through this procedure, for each <+">l<-"> we have one <+">V<-"><+'><+">l<-">,<+!>ps<-!><-'>(<+">r<-">). These potentials act on the respective <+">l <-"><:f><:f240,2Times New Roman,0,0,0>state, and include the screening of the atomic core by the charge<:f><:f240,2Times New Roman,0,0,0> density of all <+">l<-"> states in valence shell. <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#852,9360><:f240,2Times New Roman,0,0,0>In short, the two stages of pseudising a wavefunction and inverting the Schr<\v>dinger equation from the pseudised wavefunction<:f><:f240,2Times New Roman,0,0,0> can be visualised with the following figures : <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:f240,2Times New Roman,0,0,0><:A28> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:#284,9360><+!><:f240,2Times New Roman,0,0,0>I.2.4 Unscreening Pseudopotentials<-!><:f> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#2982,9360><:f240,2Times New Roman,0,0,0>The way we want to use a pseudopotential in solid state calculations is to find the ground state of valence electrons in a self-consistent way, and therefore the information of valence electron charge density<:f><:f240,2Times New Roman,0,0,0> has to come from the solid state calculation. We want the pseudopotential not to carry the valence electron screening of the free atom so that we can add later the solid state one. This is the reason why we need to unscreen the atomic pseudopotential for e ach <+">l<-"> components. The (potential) energy contribution due to the valence charge density must be subtracted away to give an ionic pseudopotential : <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A33><+'><+!>free atom<-!><-'> (2.4.1) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0>where <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A34> , and (2.4.2) <:s><:S+-2><:#426,9360> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A35> . (2.4.3) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#3583,9360><:f240,2Times New Roman,0,0,0>Note that <+">V<-"><+'>XC <-'><[><:f240,2Symbol,0,0,0>r<:f240,2Times New Roman,0,0,0>(<+!>r<-!>)] = <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'>XC <-'><[><:f240,2Symbol,0,0,0>r<+'><+!><:f240,2Times New Roman,0,0,0>valence<-!><-'><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)<+"> <-"><:f240,2Times New Roman,0,0,0>+ <:f><:f240,2Symbol,0,0,0>r<+'><:f240,2Times New Roman,0,0,0><+!>core<-!><-'><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0>] is not e qual to <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'>XC <-'><[><:f240,2Symbol,0,0,0>r<+'><:f240,2Times New Roman,0,0,0><+!>core<-!><-'><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>) ]<:f><:f240,2Times New Roman,0,0,0> +<:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'>XC <-'><:f240,2Symbol,0,0,0> <:f240,2Times New Roman,0,0,0><[><:f240,2Symbol,0,0,0>r<+'><+!><:f240,2Times New Roman,0,0,0>valence<-!><-'><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)] because of the non-linearity of the function <:f200,2Symbol,0,0,0><+">r <-"><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>). <:f><:f><:f240,2Times New Roman,0,0,0>To subtract the valence charge density's contribution to exchange-correlation energy, we can follow the one of two option. The first is<:f><:f240,2Times New Roman,0,0,0> to freeze the<:f><:f240,2Times New Roman,0,0,0> c ore-valence exchange-correlation<:f><:f240,2Times New Roman,0,0,0> interaction<:f><:f240,2Times New Roman,0,0,0> so that the ionic pseudopotential carries along with itself this energy evaluated from the atomic state and uses it in all the solid state envir onment. Alternatively we can subtract the term <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'>XC <-'><[><:f240,2Symbol,0,0,0>r<:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)]<:f><:f240,2Times New Roman,0,0,0> entirely, just pass the core charge<:f><:f240,2Times New Roman,0,0,0> density<:f><:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>r<:f240,2Times New Roman,0,0,0><+!><+'>core<-'><-!>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>) to the solid state calculation and evaluate <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'>XC <-'><[><:f240,2Symbol,0,0,0>r<:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)]<:f><:f240,2Times New Roman,0,0,0> for the solid as <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+'>XC <-'><[><:f240,2Symbol,0,0,0>r<+'><+!><:f240,2Times New Roman,0,0,0>core<-!><-'><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>) + <:f><:f240,2Symbol,0,0,0>r<+'><:f240,2Times New Roman,0,0,0><+!>band<-!><-'><:f240,2Times New Roman,0,0,0>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)<:f><:f240,2Times New Roman,0,0,0>]<:f><:f240,2Times New Roman,0,0,0> later on. <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#894,9360><:f240,2Times New Roman,0,0,0>As for the term <+">V<-"><+'>Hatree<-'><[><:f240,2Symbol,0,0,0>r<:f240,2Times New Roman,0,0,0>(<+!>r<-!>)], since it is a linear function of <:f240,2Symbol,0,0,0>r<:f240,2Times New Roman,0,0,0>(<+!>r<-!>), so the subtraction of the energy contribution from the valence charge density is exact. <:s><:#284,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0>I.2.5 Norm-conserving<-!><:f><+!><:f240,2Times New Roman,0,0,0> Pseudopotentials<-!><:f> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0>Before the work of Hamann, Schluter and Chang (Ref.H.2, to be refer as HSC), pseudopotentials suffered from two problems. Firstly there was no guarantee<:f><:f240,2Times New Roman,0,0,0> that the amount of charge <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A37> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#1278,9360><:f240,2Times New Roman,0,0,0>inside the core-radius <+">r<-"><+'>c<-'> is conserved in going to the pseudo wave function. In order to obtain sensible electrostatic energy, for example in calculating the energy difference between two structures, we need to have equivalent electron densities, which is achieved nearly enough if <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A38> . (2.5.1) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0>The second problem concerned the logarithmic<:f><:f240,2Times New Roman,0,0,0> derivative. This was fixed to be the same for the <:f240,2Symbol,0,0,0>Y<+"><+'><:f240,2Times New Roman,0,0,0>l<-'><-"><:f240,2Times New Roman,0,0,0> and <:f240,2Symbol,0,0,0>Y<:f><:f240,2Times New Roman,0,0,0><+"><+'>l<-"><:f240,2Times New Roman,0,0,0>,ps<-'> at some energy eigenvalue <+">E<:f240,2Times New Roman,0,0,0><+'>l <-'><-"><:f240,2Times New Roman,0,0,0>, but for a good pseudopotential we would like it to be the same (or very nearly so) over the energy range of the occupied valence orbitals of the solid. This is to say that we w ould like also <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A39> . (2.5.2) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#1278,9360><:f240,2Times New Roman,0,0,0>HSC pointed out that if we make the pseudopotential satisfy the Norm-conserving condition (2.5.1), then (2.5.2) is also satisfied automatically. The reason is that the two quantities are linked by the identity <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A11>. (2.5.3) <:s><:S+-2><:#426,9360><-!><:f240,2Times New Roman,0,0,0> <:S+-2><:#852,9360><:f240,2Times New Roman,0,0,0>In the original paper of HSC only a hint is given to the proof. Here we will try to follow the hint and prove this condition. <: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><:f240,2Times New Roman,0,0,0>Let <:A25>. Then the <+">l<-">-state radial<:f><:f240,2Times New Roman,0,0,0> Schr<\v>dinger<:f><:f240,2Times New Roman,0,0,0> equation at different energies <:A27> and <:A26> becomes <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A12>, <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A13>. (2.5.4) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0>we multiply the first by <:f240,2Symbol,0,0,0>F<+"><+'><:f240,2Times New Roman,0,0,0>l<-'><-">' , the second by <:f><:f240,2Symbol,0,0,0>F<+"><+'><:f240,2Times New Roman,0,0,0>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> , subtract<:f><:f240,2Times New Roman,0,0,0> and eliminate common terms. We have <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A14> , <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0>Which integrated<:f><:f240,2Times New Roman,0,0,0> from <+">r<-"> = 0 up to <+">r<-"> = <+">R<-"> becomes <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A15> . (2.5.5) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0>The left hand side of the equation can be integrated<:f><:f240,2Times New Roman,0,0,0> out by parts to give <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0>LHS. of (2.5.5) = <:A16> , (2.5.6) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0>in which the terms in the square brackets <[> ] can be further simplified to <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A17> (2.5.7) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><-!><:f240,2Times New Roman,0,0,0><:A18> . <+!> <-!><:f>(2.5.8)<-!> <:s><:S+-2><:#426,9360> <:s><:S+-2><:#426,9360>Introducing the result back into equation (2.5.5), we have <:s><:S+-2><:#426,9360> <:s><:S+-2>(2.5.5) <:A19> . (2.5.9) <:s><:S+-2><:#426,9360> <:s><:S+-2>In the limit <+">E <-"><:f240,2Symbol,0,0,0>-<:f><+"> E<:f240,2Times New Roman,0,0,0><-">' <:A40>0<:f> case we obtain the differential form <:s><:S+-2><:#426,9360> <:s><:S+-2><+!><:f240,2Times New Roman,0,0,0><:A20>. <-!><:f><:f240,2Times New Roman,0,0,0>(2.5.10) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#852,9360><:f240,2Times New Roman,0,0,0>Note that the <:f><:f240,2Times New Roman,0,0,0>Norm-Conservation Condition only guarantees the correctness of the logarithmic derivative to first order in the energy variation.<:f> <: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><:f240,2Times New Roman,0,0,0>Since <:f><:f240,2Times New Roman,0,0,0><:A21>, we <:f240,2Times New Roman,0,0,0>have <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:S+-2><+!><:f240,2Times New Roman,0,0,0><:A22> <-!> <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:S+-2><+!><:f240,2Times New Roman,0,0,0><:A23> . <-!> (2.5.11) <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0><-!> <:s><:S+-2><:#852,9360><:f240,2Times New Roman,0,0,0>The factor of <+">r<-"><+&>2<-&> ensures that the limit <+">r <-">= 0 contributes zero in the last term in (2.5.11). This leaves only the upper limit at <:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0> = <+">R<-">, which is then substituted back into (2.5.10).<:f><+!><:f240,2Times New Roman,0,0,0><-!><:f><: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>We have therefore proved the Norm-Conserving Condition <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0> <:s><:S+-2><+!><:f240,2Times New Roman,0,0,0><:A24> <-!> <+!> <-!> (2.5.12), (2.5.3) <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><-!><+!> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0>I.2.6 The Kleinman-Bylander<-!><:f><+!><:f240,2Times New Roman,0,0,0> Form of A Pseudopotential<-!><:f> <:s><:S+-2><:#426,9360><+!><:f240,2Times New Roman,0,0,0><-!> <:S+-2><:#1278,9360><:f240,2Times New Roman,0,0,0>We now make two transformations on the pseudopotential. The first is rather trivial. Let <+">V<-"><+"><+'>l<-'><-"> be the <+">l<-">-component of the (unscreened) ionic pseudopotential in the sense of Section I.2.4. we can choose some arbitrary local potential <+">V<+&> <-&><-"><+&>L<-&><:f><:f240,2Times New Roman,0,0,0> , i.e. the same for all <+">l<-">, to and write <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#468,9360><:f240,2Times New Roman,0,0,0><:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> = <:f><+"><:f240,2Times New Roman,0,0,0> V <-"><+&>L<-&><:f><:f240,2Times New Roman,0,0,0> + <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> (2.6.1) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#468,9360><:f240,2Times New Roman,0,0,0>where <:f><:f><:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> = <:f><+"><:f240,2Times New Roman,0,0,0>V<-"><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0>-<+"><:f240,2Times New Roman,0,0,0> V <-"><+&>L<-&><:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0> . (2.6.2) <:f> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0>The whole ionic pseudopotential can then be conventionally expressed as <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#468,9360><:f240,2Times New Roman,0,0,0><+">V<-"><+'>ion<-'> = <:f240,2Symbol,0,0,0>S<+"><+'>l<:f240,2Times New Roman,0,0,0>m<-'><-"><:f240,2Times New Roman,0,0,0> |<:f240,2Times New Roman,0,0,0><+">Y<:f240,2Times New Roman,0,0,0><-"><+"><+'>lm<:f240,2Symbol,0,0,0> <-'><-"><;><:f240,2Times New Roman,0,0,0> <+">V<:f240,2Times New Roman,0,0,0><-"><+'><+">l<-'><-"><:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0><<<+"><:f240,2Times New Roman,0,0,0>Y<:f240,2Times New Roman,0,0,0><-"><+"><+'>lm<-'><-"><:f240,2Times New Roman,0,0,0><:f240,2Times New Roman,0,0,0>|<:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> = <+">V <-"><+&>L<-&> + <:f><:f240,2Times New Roman,0,0,0><:f240,2Symbol,0,0,0> S<+"><+'>l<:f240,2Times New Roman,0,0,0>m<-'><-"><:f240,2Times New Roman,0,0,0> |<:f240,2Times New Roman,0,0,0><+">Y<:f240,2Times New Roman,0,0,0><-"><+"><+'>lm<:f240,2Symbol,0,0,0> <-'><-"><;> <:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<:f240,2Times New Roman,0,0,0><-"><+'><+">l<-'><-"><:f240,2Times New Roman,0,0,0> <:f240,2Symbol,0,0,0><<<+"><:f240,2Times New Roman,0,0,0>Y<:f240,2Times New Roman,0,0,0><-"><+"><+'>lm<-'><-"><:f240,2Times New Roman,0,0,0><:f240,2Times New Roman,0,0,0>| , (2.6.3)<:f> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:f240,2Times New Roman,0,0,0>where <:f><:f240,2Times New Roman,0,0,0>|<:f240,2Times New Roman,0,0,0><+">Y<:f240,2Times New Roman,0,0,0><-"><+"><+'>lm<:f240,2Symbol,0,0,0><-'><-"><;><:f240,2Times New Roman,0,0,0> <+"><-"><:f><:f240,2Times New Roman,0,0,0> and <:f><:f240,2Symbol,0,0,0><<<+"><:f240,2Times New Roman,0,0,0>Y<:f240,2Times New Roman,0,0,0><-"><+"><+'>lm<-'><-"><:f240,2Times New Roman,0,0,0><:f240,2Times New Roman,0,0,0>|<:f><:f240,2Times New Roman,0,0,0> project out the <+">l<-">, <+">m<-"> component of the pseudo wave function. The Hartree field and exchange-correlation<:f><:f240,2Times New Roman,0,0,0> potential of valence electrons are also local functions with the use of Local Density Approximation for exchange and correlation<:f><:f240,2Times New Roman,0,0,0>) so that they can be added to <+">V <-"><+&>L<-&> without changing the <:f><:f240,2Symbol,0,0,0>d<+"><:f240,2Times New Roman,0,0,0>V<-"><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0> . The second transformation is to write the last term of (2.6.3) in the approximate form <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:f240,2Times New Roman,0,0,0><:A1> , (2.6.4) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#894,9360><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f240,2Times New Roman,0,0,0>in which <:f240,2Symbol,0,0,0>Y<:f><:f240,2Symbol,0,0,0><:f240,2Times New Roman,0,0,0><+&>0<-&><+'><+">lm<-"><-'>(<:f240,2Times New Roman,0,0,0><+!>r<-!><:f240,2Times New Roman,0,0,0>)<+"> = <:f240,2Symbol,0,0,0>j<:f240,2Times New Roman,0,0,0><-"><+&>PS<-&><+'><+">l<-"><-'>(<+">r<-">)<+">Y<+'>lm<-'><-">(<:f240,2Symbol,0,0,0>q<:f240,2Times New Roman,0,0,0>,<:f240,2Symbol,0,0,0>f<:f240,2Times New Roman,0,0,0>), with <:f><+"><:f240,2Symbol,0,0,0>j<:f240,2Times New Roman,0,0,0><-"><-&><-&><-"><-'><+&>PS<-&><+'><+">l<-"><-'>(<+">r<-">)<:f><:f240,2Times New Roman,0,0,0> being the pseudo wavefunction used to create <+">V<+'>l<-'><-">(<:f240,2Times New Roman,0,0,0><+">r<-"><:f240,2Times New Roman,0,0,0>). We therefore have<+"><-"><+'><-'><:f><:f240,2Times New Roman,0,0,0><-"><-&><-'> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:A0> , (2.6.5) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:S+-2><:#2682,9360><:f240,2Times New Roman,0,0,0>which is the Kleinman-Bylander form of the pseudopotential. To see its relation to (2.6.3) we operate with it on <:f240,2Symbol,0,0,0>Y<+&><:f240,2Times New Roman,0,0,0>0<-&><+"><+'>lm<-"><-'> to the left and right . We see that the two forms in (2.6.4) give identical results in this case because one factor of <:f240,2Symbol,0,0,0><<<:f240,2Symbol,0,0,0>Y<+&><:f240,2Times New Roman,0,0,0>0<-&><+"><+'>lm<-"><-'><:f><:f240,2Symbol,0,0,0>|d<+"><:f240,2Times New Roman,0,0,0>V<-"><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>|<:f> <:f240,2Symbol,0,0,0>Y<+&><:f240,2Times New Roman,0,0,0>0<-&><+"><+'>lm<-"><-'><:f240,2Symbol,0,0,0><;><:f> on the right side. We now make the argument that the <+">l<-">, <+">m<-"> angular momentum component of any pseudo wave function in the solid can be expected to be approximately like <:f240,2Symbol,0,0,0>Y<+&><:f240,2Times New Roman,0,0,0>0<-&><+"><+'>lm<-"><-'><:f> over region inside <+">r<-"><+'>c<-'>. With that assumption the two form in (2.6.4) become equivalent to a sufficient degree of approximation. <:s><:S+-2><:#426,9360> <:S+-2><:#2130,9360>What has been gained by this transformation is an enormous increase in computational efficiency. Suppose we use <+">N<-"> plane waves (<+">N<-"> ~ 10000) as basic functions. The <+">N<-"> x <+">N <-">representation of <+">V<-"><+'>ion<-'> requires in principle the calculation of <+">N<-"><+&>2<-&> numbers using the full non-local form (2.6.3). However the Kleinman-Bylander form (2.6.4) only requires evaluation (for each <+">l<-">, <+">m<-">) the <+">N<-"> quantities<:f240,2Symbol,0,0,0> <:s><:S+-2><:#426,9360> <:S+-2><:#468,9360><:f240,2Symbol,0,0,0><<<:f240,2Symbol,0,0,0>Y<+&><:f240,2Times New Roman,0,0,0>0<-&><+"><+'>lm<-"><-'><:f><:f240,2Symbol,0,0,0>|d<+"><:f240,2Times New Roman,0,0,0>V<-"><+"><+'>l<-"><-'><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0>| Pl ane-wave n<:f240,2Symbol,0,0,0><;><:f> (2.6.6) <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0>and taking products of them. <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0> <:s><:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0> <:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f><:f240,2Times New Roman,0,0,0><:f> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:S+-2><:#426,9360><:f240,2Times New Roman,0,0,0><+!>I.2.<-!><+!>7.<-!><+!> BHS table for excited states<-!><:f><+!><:f240,2Times New Roman,0,0,0>, and comments<-!><:f> <:s><:S+-2><:#426,9360> <:S+-2>A problem arises in generating pseudopotentails <+">V<-"><+"><+'>l<-'><-"> for the higher values of <+">l<-"> when there are no filled orbital in the free atom with that l, e.g. <+">V<-"><+'><+">l<-">=2<-'> for all elements in periodic table up to Ca. The problem is compounded when later transforming to Kleinman-Bylander form. Although there are no occupied orbitals with this <+">l<-"> in the free atom, in general any wave function in a solid will contain components of all <:f240,2Times New Roman,0,0,0><+">l<-"><:f> when expanded around an atomic site, so that the <:f240,2Times New Roman,0,0,0><+">V<:f240,2Times New Roman,0,0,0><+'>l<-'><-"><:f240,2Times New Roman,0,0,0><:f> for such <:f240,2Times New Roman,0,0,0><+">l<-"><:f> is needed in the solid state calculation. <:s><:S+-2><:#426,9360> <:S+-2><:#2598,9360>In generating the pseudopotential with such an l in the manner of Section I.2.3, we need the corresponding radial atomic orbital <:f240,2Symbol,0,0,0>F<+"><+'><:f240,2Times New Roman,0,0,0>lm<-'><-"><:f> and eigenvalue <:f240,2Times New Roman,0,0,0><+">E<:f240,2Times New Roman,0,0,0><-"><+"><+'>l<-'><-"><:f240,2Times New Roman,0,0,0><:f>. Such an unoccupied state in the atom may be totally unbound so that <:f240,2Times New Roman,0,0,0><+">E<:f240,2Times New Roman,0,0,0><+'>l<-'><-"><:f240,2Times New Roman,0,0,0><:f> may not exist : but even if it is bound, it is very likely loosely bound so that its <:f240,2Times New Roman,0,0,0><+">E<:f240,2Times New Roman,0,0,0><+'>l<-'><-"><:f240,2Times New Roman,0,0,0><:f> is not in the range of the occupied states of the solid. In that case matching the logarithmic derivative at <:f240,2Times New Roman,0,0,0><+">E<+'>l<-'><-"><:f> leads to significant errors in the energy range of interest in the solid. <:s><:S+-2><:#426,9360> <:S+-2>A further problem arises with the use of the (2.6.5). If the atomic <:f240,2Symbol,0,0,0>F<:f><:f240,2Times New Roman,0,0,0><+"><+'>l<-'><-"><:f240,2Times New Roman,0,0,0> i<:f>s very loosely bound, it will not have the form of the <:f240,2Times New Roman,0,0,0><+">l<-"><:f> components of the occupied states of the solid which are much more strongly bound. Hence the argument used to demonstrate the equivalence of the two forms in (2.6.4) is no longer valid, and significant errors are to be expected. <:s><:S+-2><:#426,9360> <:S+-2><:#1278,9360>Actually, it is convenient to test the logarithmic derivative of the pseudopotentail and to carry out the <:f240,2Times New Roman,0,0,0><+">Q<:f240,2Times New Roman,0,0,0><-"><+'><+">c<-'><-"><:f>-tuning in its ultimate Kleinman-Bylander form. This avoids, or rather cancels, any error inherent in the transformation. <:s><:S+-2><:#426,9360> <:S+-2>To overcome these problems, Bachelet, Hamann and Schluter (Ref.B.1, referred as BHS) published a table of atomic excited configuration in which an electron (or more often a fraction of one) is promoted into the orbital with the desired <:f240,2Times New Roman,0,0,0><+">l<-"><:f>. In this way more relevant atomic eigenvalue of <:f240,2Times New Roman,0,0,0><+">E<:f240,2Times New Roman,0,0,0><+'>l<-'><-"><:f240,2Times New Roman,0,0,0><:f> and the orbital <:f240,2Symbol,0,0,0>Y<:f><:f240,2Times New Roman,0,0,0><+"><+'>l<-'><-"><:f240,2Times New Roman,0,0,0><:f> are obtained. But the procedure is rather arbitrary and it may be necessary to check that <:f240,2Times New Roman,0,0,0><+">E<:f240,2Times New Roman,0,0,0><+'>l<-'><-"><:f240,2Times New Roman,0,0,0><:f> really lies in the range of occupied orbitals of the solid (taking into account the atomic unscreening and addition of the solid state Hartree p otential). <: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><:S+-2><:#426,9360> <:s><:#376,9360><:f320,,><+!>I.3. Total Energy Calculation<-!><:f> <:s><:#284,9360><-!> <:s><:S+-2><:#1278,9360><-!><-!><-!>Describing the bonding between atoms accurately is essential for computer experiments at the microscopic level. This is why a quantum mechanical treatment the electronic structure is necessary. <:s><:S+-2><:#426,9360> <:S+-2><:#3408,9360>In the study of a condensed matter system, a typical number of 10<+&>23<-&> electrons is involved, which makes it an extremely complicated quantum many-body problem. We therefore can only study solid state physics with a certain degree of approximation. On the other hand, we do not need to know detailed information for each electron in a solid. The Density Functional Theory (DFT) <[>Ref.D.1,P.4] is a useful representation of quantum mechanics for large number of interacting particles. The theory says that the re is a unique correspondence between the ground state total energy and the ground state charge density, i.e. the ground state total energy of a system is a functional of its charge density. <:s><:S+-2><:#426,9360> <:S+-2><:#2640,9360>If one treat ions as classical particles, then the ion-ion interaction is trivial ( and in a supercell geometry can be treated using Ewald summation.). Only the electronic part needs to be treated quantum mechanically. For a given ion arrangement, the total electronic energy in DFT can be expressed in terms of functionals of kinetic energy <+">T<-"><+'>m<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>], electron-electron repulsion <+"><:f240,2Times New Roman,0,0,0>E<-"><:f><+'>ee<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] and external (potential) energy <+"><:f240,2Times New Roman,0,0,0>E<-"><:f><+'>ext<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] (which contains effects such as electron-ion interaction<-!><-'>), as the following : <:s><:S+-2><:#426,9360> <:S+-2><:#468,9360><+">E<-"><+'>e<-'><[><:f240,2Symbol,0,0,0>r<:f><-'>] = <+">T<-"><+'>m<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] +<+"><:f240,2Times New Roman,0,0,0> E<-"><:f><+'>ee<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] + <:f240,2Times New Roman,0,0,0><+">E<-"><:f><+'>ext<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>], (3.1.1) <:s><:S+-2><:#426,9360> <:S+-2><:#2340,9360>where <:f240,2Symbol,0,0,0>r<:f>(<+!>r<-!>) is the electron density. However, the kinetic energy functional <+">T<-"><+'>m<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] for interacting particles is unknown, so that one can not evaluate <+">E<-"><+'>e<-'><[><:f240,2Symbol,0,0,0>r<:f><-'>] using (3.1.1), without using an approximate form for <+">T<-"><+'>m<-'><:f240,2Symbol,0,0,0><[>r<:f>] such as the Thomas Fermi model which is not very accurate. A practical scheme for evaluating <+">E<-"><+'>e<-'><[><:f240,2Symbol,0,0,0>r<:f><-'>] was proposed by Kohn and Sham <[>Ref.K.2] which expresses the <+">E<-"><+'>e<-'><[><:f240,2Symbol,0,0,0>r<:f><-'>] as<-!><-'><-!><-'> <:s><:S+-2><:#426,9360> <:S+-2><:#468,9360><+">E<-"><+'>e<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] = <+">T<-"><+'>s<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] + <+">E<-"><+'>xc<-'><[><:f240,2Symbol,0,0,0>r<:f>] + <+">E<:f240,2Times New Roman,0,0,0><-"><+'>H<-'><:f><[><:f240,2Symbol,0,0,0>r<:f><-'>] + <+">E<-"><+'>ext<-'><[><:f240,2Symbol,0,0,0>r<:f><-'>]. (3.1.2)<-!><-'><-!><-'> <:s><:S+-2><:#426,9360> <:S+-2>in which the <+">T<-"><+'>s<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>],<+"> <-"><+">E<-"><+'>xc<-'><[><:f240,2Symbol,0,0,0>r<:f>] and <+">E<-"><+'>H<-'><[><:f240,2Symbol,0,0,0>r<:f><-'>] are kinetic energy, exchange-correlation energy, and classical repulsion of electrons (Hartree energy), respectively. In the Kohn-Sham (KS) scheme <[>Ref.K.2], the interacting many-body problem (3.1.1) is mapped onto a non-interacting one (3.1.2) within the framework of Density Functional Theory, i.e. the energies are still functionals of the density. These non-interacting particles are described by wave functions called KS orbitals which will still give the same ground state density,<-!><-'><-!><-'> <:s><:S+-2><:#468,9360><:f240,2Symbol,0,0,0> <:S+-2><:#468,9360><:f240,2Symbol,0,0,0>r<:f240,2Times New Roman,0,0,0><+&>KS<-&><:f>(<+!>r<-!>) = <:f240,2Symbol,0,0,0>S<+'><:f>i<-'> <:f240,2Symbol,0,0,0>|Y<:f240,2Times New Roman,0,0,0><+&>KS<-&><+'><:f>i<-'>(<+!>r<-!>)<:f240,2Symbol,0,0,0>|<:f><+&>2<-&> <-&>= <:f240,2Symbol,0,0,0>r<-&><:f>(<+!>r<-!>) . (3.1.3) <:s><:S+-2><:#426,9360> <:S+-2><:#1320,9360>The advantage of KS scheme is that the exact form of the kinetic energy functional <+">T<-"><+'>s<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>] for the non-interacting particle is known, which is just the expectation value of the kinetic energy of the KS orbitals in the standard quantum mechanics sense. <:s><:S+-2><:#426,9360> <:S+-2><:#5406,9360>The non-trivial many-body effect in the Kohn-Sham scheme is included in the exchange-correlation term <+">E<-"><+'>xc<-'><[><:f240,2Symbol,0,0,0>r<:f>] in (3.1.2). This is essentially a sophisticated mean field approach, in which the non-locality of the exchange-correlation potential <-"><:f240,2Symbol,0,0,0><+">e<-"><+'><:f>xc<-'>(<+!>r<-!>), defined by <+"><:f240,2Symbol,0,0,0>e<-"><+'><:f>xc<-'>(<+!>r<-!>) = <:f240,2Symbol,0,0,0>d<+"><:f>E<-"><+'>xc<-'><[><:f240,2Symbol,0,0,0>r<:f>]/<:f240,2Symbol,0,0,0>d<:f240,2Symbol,0,0,0>r<:f>(<+!>r<-!>) , can be rationalised as the result of a dynamical mean field. However, the exact functional form of <+">E<-"><+'>xc<-'><[><:f240,2Symbol,0,0,0>r<:f>] is still unknown, so that some approximation has to be made to evaluate <+"><:f240,2Symbol,0,0,0>e<-"><+'><:f>xc<-'>(<+!>r<-!>). Fortunately, the Local Density Approximation (LDA) works surprisingly well. It approximates <+"><:f240,2Symbol,0,0,0>e<-"><+'><:f>xc<-'><[><:f240,2Symbol,0,0,0>r<:f>] as <+"><:f240,2Symbol,0,0,0>e<-"><+'><:f>xc<-'>(<:f240,2Symbol,0,0,0>r<:f>) which depends only on the charge density locally, and <+"><:f240,2Symbol,0,0,0>e<-"><+'><:f>xc<-'>(<:f240,2Symbol,0,0,0>r<:f>) can be parameterised using the accurate known for the ground state of the electron gas <[>Ref.C.1]. The LDA satisfies the sum-rule for the exchange and correlation hole and give very g ood structural properties for molecules and solids, but it overestimates the binding (cohesive) energy. The Generalised Gradient Approximation (GGA) includes the gradient terms in <+"><:f240,2Symbol,0,0,0>e<-"><+'><:f>xc<-'> , which improves the binding energy, but has been reported to give worse bulk moduli of solids <[>Ref.J.1]. <:s><:S+-2><:#426,9360> <:S+-2><:#1830,9360>To solve for the energy in (3.1.2), it is necessary to find the wave functions, charge density and the energies. For this one needs to minimise (3.1.2) to find the ground state. The <:f240,2Symbol,0,0,0>r<:f> in the ground state of the system satisfies the variational condition <:f240,2Symbol,0,0,0>d<:f><+">E<-"><+'>e<-'><[><:f240,2Symbol,0,0,0>r<:f><-!><-'>]/<:f240,2Symbol,0,0,0>d<:f240,2Symbol,0,0,0>r<:f>(<+!>r<-!>) = 0 which gives KS equations, a set of one-particle Schr<\v>dinger equations with an exchange-correlation potentia l <+"><:f240,2Symbol,0,0,0>e<-"><+'><:f>xc<-'>(<+!>r<-!>), <:s><:S+-2><:#426,9360> <:S+-2><[><+">H<-"><+&>KS<:f240,2Symbol,0,0,0><-&>-<:f><+">E<-"><+'>i<-'>] <:f240,2Symbol,0,0,0>Y<+'><:f>i<-'> = <:A36> . (3.1.4) <:s><:S+-2><:#426,9360> <:S+-2><:#5112,9360>The operator in (3.1.4) itself is the gradient operator of this energy minimisation problem, and operate with it on the trial wave functions yields the gradient vector pointing towards lower energy. When this vector eventually vanishes after iterative compu ting, (3.1.4) is satisfied and the solution is found. To actually perform such "direct" minimisation procedure, the steepest decent method seems to be a natural choice, but it is not suitable for successive searches for the minimum because it makes "right a ngle" turns everytime at each new step. The more efficient conjugate gradient minimisation <[>Ref.P.1, P.2, P.3] uses the information from the previous step to search for the minimum. In addition, since the gradient of the wavefunction has an explicit depen dence on the energy eigenvalue of the wave function <[>Ref.P.1, P.3], it is important to precondition these gradient vectors by re-scaling their (energy dependent) weight to achieve an even better overall smooth convergence. <:s><:S+-2><:#426,9360> <:S+-2><:#2577,9360>From the above description we know that carrying out the operation in (3.1.4) on a wave function is one of the main computing tasks (others including making the KS orbitals orthogonal). In the basis-set formalism, the kinetic energy operator of the Hamilton ian is diagonal in <+!>k<-!>-space but the potential energy one is only diagonal in <+!>r<-!>-space. A significant efficiency can be achieved by using Fast Fourier Transform (FFT) to convert wave functions back and forth between <+!>k<-!> and <+!>r<-!> spaces, and by performing these two types of operation in their diagonal from. <:s><:S+-2><:#426,9360> <:S+-2><:#2982,9360>After the electronic ground state is found, for a given ionic configuration, the force on each atom can be evaluated directly from wave function by the Hellmann-Feynman theorem <[>Ref.D.1,P.4], which means that there is no need to perform the derivative of the total energy with respect to ion positions numerically. The force on each atom can then be used to relax the ionic position in order to search for the overall minimum energy of the entire system and find the equilibrium atomic geometry. In a supercell c alculation, the stress on the cell can also be calculated from first principles <[>Ref.N.1]. <:s><:S+-2><:#426,9360> <:S+-2><:#6432,9360>Pseudopotentials are involved in all three terms <:f240,2Symbol,0,0,0><+">e<-"><:f><+'>xc<-'> , <:f240,2Symbol,0,0,0><+">e<-"><+'><:f>H<-'> and <:f240,2Symbol,0,0,0><+">e<-"><:f><+'>ext <-'>in (3.1.4) in several ways. These come from different aspects of the pseudopotential method and its approximations, namely the Non-linear Core-correction (Chapter I, Section I.2.4 and Chapter VI, Section VI.1), Core Over-lap (Chapter VI, Section VI.1) a nd Softness (Chapter II and Chapter VI, Section VI.2). We will discuss these points in more detail in the relevant Chapters. The reason that pseudopotentials play such a significant effect role in (3.1.2) and (3.1.4) is basically because pseudopotentials re present the combined effect of both core electrons and the nucleus on the valence electrons. It is therefore important to develop the best strategies for using pseudopotentials in total energy calculations. For large scale electronic structure calculations it is most favourable to use a low energy cut-off of the plane wave basis set, but many plane waves are needed to describe rather tightly bound orbitals such as those of 2<:f240,2Times New Roman,0,0,0><+">p<-"><:f> and 3<+">d<-"> valence states. How to make a pseudopotential softer therefore becomes important, which will be discussed in Chapter II. As we will see in the Chapter III and IV, non-locality in pseudopotentials is important for accurate calculation but it is expensive, and sometimes problematic. In this thesis we will provide a few new techniques to address these issue in relevant chapters. <:s><:S+-1><:#284,9360> <:s><:S+-1><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#284,9360> <:s><:#331,9360><:f280,,><+!>Reference of This Chapter<-!><:f> <:s><:#284,9360> <:s><:#284,9360><:f,2Times New Roman,><[>B.1] <:#289,9360><:f,2Times New Roman,>G.B. Bachelet, D.R. Hamann, and M. Schluter, Phys. Rev. B <+!>26<-!>, 4199 (1982)<:f> <:s><:#284,9360><:f,2Times New Roman,> <:s><:#284,9360><:f,2Times New Roman,><[>C.1] <:#289,9360><:f,2Times New Roman,>D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. <+!>45<-!>, 566 (1980) <:s><:#284,9360><:f240,,> <:s><:#284,9360><:f240,,><[>D.1] <:#568,9360><:f240,,>R. M. Dreizler and E. K. Gross, <+">Density Functional Theory - An Approach to the Quantum Many-Body Problem<-">, Springer-Verlag (1990) <:s><:#284,9360><:f,2Times New Roman,> <:s><:#284,9360><:f240,2Times New Roman,0,0,0><[>H.1] <:#289,9360><:f240,2Times New Roman,0,0,0>V. Heine, Pseudopotential Concept, <+">Solid State Physics<-"> <+!>Vol.24<-!> (1970) <:f> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0><[>H.2]<:f> <:#289,9360><:f240,2Times New Roman,0,0,0>D.R. Hamann, M. Schluter and C. Chiang, Phys. Rev. Lett <+!>43<-!>, 1494 (1979)<:f> <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0><[>J.1] <:#289,9360><:f240,2Times New Roman,0,0,0>Y-M. Juan and E. Kaxiras, Phys. Rev. B <+!>48<-!>, 14944 (1993) <:s><:#284,9360><:f240,2Times New Roman,0,0,0> <:s><:#284,9360><:f240,2Times New Roman,0,0,0><[>K.1] <:#289,9360><:f240,2Times New Roman,0,0,0>L. Kleinman and D.M. Bylander, Phys Rev. 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Joannopoulos<:f><:f240,2Times New Roman,0,0,0>, Phys. Rev. B <+!>41,<-!> 1227 (1990)<:f> <:#284,9360><:f240,,> <:#284,9360><:f240,,><[>P.4] <:#568,9360><:f240,,>R. G. Parr and W. Yang, <+">Density-Functional Theory of Atoms and Molecules<-">, Oxford University Press (1989)<:f> <:#236,9360><:f200,,> > Times New Roman,18,12,0,0,0,0,0 $$\ V_{\text{ion}\ }\approx \ V^{\text{L}}\,+\,\underset{lm}\to{\sum }\dfrac{\left| \Psi _{lm}^0\delta V_l\rangle \,\langle \Psi _{lm}^0\delta V_l\right| }{\langle \Psi _{lm}^0\left| \delta V_l\right| \Psi _{lm}^0\rangle }$$SS*``.Times New Roman,18,12,0,0,0,0,0 $$\underset{lm}\to{\sum }\left| Y_{lm}\left\rangle \,\delta V_l\,\right\langle Y_{lm}\right| \quad \approx \quad \underset{lm}\to{\sum }\dfrac{\left| \Psi _{lm}^0\delta V_l\rangle \,\langle \Psi _{lm}^0\delta V_l\right| }{\langle \Psi _{lm}^0\left| \delta V_l\right| \Psi _{lm}^0\rangle }$$SS*``.Times New Roman,18,12,0,0,0,0,0 $b_{\text{c}_{\text{i}}}=\left\langle \Psi _{\text{c}_{\text{i}}}\right| \Psi _{\text{ps}}\rangle $SSZZ``.Times New Roman,18,12,0,0,0,0,0 $b_{\text{c}_{\text{i}}}$SS``.Times New Roman,18,12,0,0,0,0,0 $V_{\text{ps}}=V\,\,+\underset{\text{c}}\to{\sum }\left( \epsilon -E_{\text{c}}\right) \left| \Psi _{\text{c}}\right\rangle \,\left\langle \Psi _{\text{c}}\right| $SS``.gt |Times New Roman,18,12,0,0,0,0,0 $\left( T+V_{\text{ps}}\right) \left| \Phi \right\rangle =\epsilon \left| \Phi \right\rangle $SSyz``.Times New Roman,18,12,0,0,0,0,0 $H\left| \Psi _{\text{ps}}\right\rangle =\epsilon \left| \Psi _{\text{ps}}\right\rangle -\underset{\text{c}}\to{\sum }b_{\text{c}}\left( \epsilon -E_{\text{c}}\right) \left| \Psi _{\text{c}}\right\rangle =\epsilon \left| \Psi _{\text{ps}}\right\rangle -\underset{\text{c}}\to{\sum }\left( \epsilon -E_{\text{c}}\right) \left| \Psi _{\text{c}}\right\rangle \left\langle \Psi _{\text{c}}\right| \Psi _{\text{ps}}\rangle $SS``.Times New Roman,18,12,0,0,0,0,0 $\left\langle \Psi \right| \Psi _{\text{c}_{\text{i}}}\rangle =0$SSFF``.Times New Roman,18,12,0,0,0,0,0 $\left| \Psi \right\rangle =\left| \Psi _{\text{ps}}\right\rangle -\underset{\text{c}}\to{\sum }b_{\text{c}}\left| \Psi _{\text{c}}\right\rangle $SS``._rTimes New Roman,18,12,0,0,0,0,0 $H\left| \Psi \right\rangle =\epsilon \left| \Psi \right\rangle \quad \quad \quad $SSLL``.Times New Roman,18,12,0,0,0,0,0 $\Phi _{l,\text{\BF p\BF s}}(r)$SS+,``.WVQLJyPxcrPyWiocrTimes New Roman,18,12,0,0,0,0,0 $$-2\pi \,\left[ \left( r\Psi \right) ^2\frac d{d\epsilon }\frac d{dr}\ln \Psi \right] _R\ =\ 4\pi \int_0^R\Psi ^2r^2dr$$SS``. 1ttVAT\D\Times New Roman,18,12,0,0,0,0,0 $$\frac{d^2\Phi _l}{dr^2}\ +\ 2\,\left[ E\,-\,U(r)\,-\,\frac{l(l+1)}{r^2}\right] \,\Phi _l\ =\ 0$$SS$``.Times New Roman,18,12,0,0,0,0,0 $$\frac{d^2\Phi _l^{\prime }}{dr^2}\ +\ 2\,\left[ E^{\prime }\,-\,U(r)\,-\,\frac{l(l+1)}{r^2}\right] \,\Phi _l^{\prime }\ =\ 0$$SS"``.Times New Roman,18,12,0,0,0,0,0 $$\Phi _l^{\prime }\frac{d^2\Phi _l}{dr^2}\,-\,\Phi _l\frac{d^2\Phi _l^{\prime }}{dr^2}\,+\,2\,\left( E\,-\,E^{\prime }\right) \,\Phi _l^{\prime }\Phi _l\ =\ 0$$SS"``.Times New Roman,18,12,0,0,0,0,0 $$\int_0^R\left( \Phi _l^{\prime }\frac{d^2\Phi _l}{dr^2}\,-\,\Phi _l\frac{d^2\Phi _l^{\prime }}{dr^2}\,\right) dr\ =\ -\,2\,\left( E\,-\,E^{\prime }\right) \,\int_0^R\Phi _l^{\prime }\Phi _ldr$$SSN$N``.Times New Roman,18,12,0,0,0,0,0 $$\int_0^R\frac d{dr}\,\left[ \Phi _l^{\prime }\frac{d\Phi _l}{dr}\,-\,\Phi _l\frac{d\Phi _l^{\prime }}{dr}\,\right] \,dr\ =\ \left[ \Phi _l^{\prime }\frac{d\Phi _l}{dr}\,-\,\Phi _l\frac{d\Phi _l^{\prime }}{dr}\right] _{\,r=0}^{\,r=R}$$SSG(H``. dt}pl ee u iSik24 epaTimes New Roman,18,12,0,0,0,0,0 $$\Phi _l^{\prime }\frac{d\Phi _l}{dr}\,-\,\Phi _l\frac{d\Phi _l^{\prime }}{dr}\,\ =\ \Phi _l^{\prime }\Phi _l\left( \frac 1{\Phi _l}\frac{d\Phi _l}{dr}\,-\,\frac 1{\Phi _l^{\prime }}\frac{d\Phi _l^{\prime }}{dr}\right) $$SS&``.Times New Roman,18,12,0,0,0,0,0 $$\quad =\ \Phi _l^{\prime }\Phi _l\left( \frac d{dr}\,\ln \Phi _l-\,\frac d{dr}\ln \Phi _l^{\prime }\right) \ =\ \Phi _l^{\prime }\Phi _l\,\frac d{dr}\left( \,\ln \Phi _l-\,\ln \Phi _l^{\prime }\right) $$SSef``.Times New Roman,18,12,0,0,0,0,0 $$\Rightarrow \ -\,\left[ \Phi _l^{\prime }\Phi _l\frac d{dr}\left( \frac{\ln \Phi _l-\,\ln \Phi _l^{\prime }}{E-E^{\prime }}\right) \right] _0^R\ =\ 2\int_0^R\Phi _l^{\prime }\Phi _l\,dr$$SS1)2``.t o Times New Roman,18,12,0,0,0,0,0 $$-\left[ \Phi _l^2\frac d{dr}\left( \frac d{dE}\,\ln \Phi _l\right) \right] _0^R\ =\ 2\int_0^R\Phi _l^2\,dr$$SS#``.Times New Roman,18,12,0,0,0,0,0 $\Phi _l=r\Psi _l$SS56``. MTimes New Roman,18,12,0,0,0,0,0 $$-\left[ \Phi _l^2\frac d{dr}\left( \frac d{dE}\,\ln \Phi _l\right) \right] _0^R=-\left[ (r\Psi _l)^2\frac d{dE}\frac d{dr}\,(\ln r+\ln \Psi _l)\right] _0^R$$SSH)H``.Times New Roman,18,12,0,0,0,0,0 $$=-\left[ r^2\Psi _l^2\frac d{dE}\,\,[\frac 1r(1)+\frac d{dr}\ln \Psi _l]\right] _0^R=-\left[ r^2\Psi _l^2\frac d{dE}\frac d{dr}\ln \Psi _l\right] _0^R$$SSO"P``.Times New Roman,18,12,0,0,0,0,0 $$-2\pi \,\left[ r^2\Psi _l^2\frac d{dE}\frac d{dr}\ln \Psi _l\right] _R=\ 4\pi \int_0^R\Psi ^2r^2dr$$SS ``.Times New Roman,18,12,0,0,0,0,0 $\Phi _l=r\Psi _l$SS66``.Times New Roman,18,12,0,0,0,0,0 $E^{\prime }$SS``.Times New Roman,18,12,0,0,0,0,0 $E$SS  ``.SM/.U6YN T!UT!Ucwd cwd  7$+ $D2+$ $ $#'+ 7'M)P&M)P'PM)M)M)(P'P 7(g *0g *(g * * *;%g(g 7lLl,lLllhL 7i$% i$e'i$%i$i$%% 7%' #5'%'''Z&% 7') '+G)'''"() 7(0 (q8(0((+&,0 7)"+ t'"+8+)"++*")" 7+J3  +: +J3 ++ / J3  7H*+ *, T+*H**=++ 7'*&%*&*'&**)&'&M)U6"M)Q*+/,q,,,l-y-- .f...*T/{//dA0N00w11)22253334dm4%556U6Q}Z).1~Z))v))-***\++O=,,,-b-.L..//00G.1T)O+/*~ &' &' Y Y 7(/ (b6/( ( (,/ Y  Y  B  B  7 8  ) y r yL Y r rE/ c6]W$f*/,sF>$fWng&9L H*57Y^11L1LY^1V, -V, -)O* )O*  7$ $ $ $ $' Q)E  Q|<VK1gsI{vFY,Z7E)Dk $E >17 Y !7 Yl f  ?  9 "  us-K 9 MH#! HCH;f.MI}!p$" }pCp ;zh@ !!M.%/-$%# .%$P&#'+((^)**V+/<,~,Q-6-YkY"9&$99XmbY k y!!!s "Y" M '% xTms RmnBLE!FSPRINTERFONTNAMEF ETV(r) OC(&C@Tms RmnBLE!FSPRINTERFONTNAMEF ETVps(r)D M)'xTms RmnBLE!FSPRINTERFONTNAMEF ETV(r) O ~I*( ~@Tms RmnBLE!FSPRINTERFONTNAMEF ETVps(r). M+)pTms RmnFSINVALIDATESCREENFONTCA EXR(r) MyEM,*yEpTms RmnFSINVALIDATESCREENFONTCA EXR(r)p Oy-+y8Tms RmnFSINVALIDATESCREENFONTCA EXRps(r) O7?.,78Tms Rmn Rps(r) p }/- } p(b:c.b:c(SS%&``.]-399zR*(4V!/KS!-Y X# 7&  %&B":L cTimes New Roman,18,12,0,0,0,0,0 $$\frac{d^2\Phi _l(r)}{dr^2}\,+2\,\left( E_l\,-\,V(r)\,-\,\frac{l(l+1)}{r^2}\right) \Phi _l(r)=0$$SS !``.Times New Roman,18,12,0,0,0,0,0 $$\frac{d^2\Phi _{l\BF ,\text{\BF p\BF s}}(r)}{dr^2}\,+2\,\left( E_l\,-\,V_{l,\text{\BF p\BF s}}(r)\,-\,\frac{l(l+1)}{r^2}\right) \,\Phi _{l,\text{\BF p\BF s}}(r)\ =\,0$$SS:":``.Times New Roman,18,12,0,0,0,0,0 $$V_{l,\text{\BF p\BF s}}(r)\,=\,\frac 1{2\Phi _{l,\text{\BF p\BF s}}(r)}\,\left\{ \frac{d^2\Phi _{l,\text{\BF p\BF s}}(r)}{dr^2}\,\,+\ 2\ \left[ E_l\ -\ \frac{l(l+1)}{r^2}\right] \ \Phi _{l,\text{\BF p\BF s}}(r)\right\} $$SSe.f``.6CQ G=3HcxTimes New Roman,18,12,0,0,0,0,0 $$V_{l,\text{\BF p\BF s}}(r)\,=\,\frac 1{2\Phi _{l,\text{\BF p\BF s}}(r)}\,\left[ \frac{d^2\Phi _{l,\text{\BF p\BF s}}(r)}{dr^2}\right] \ +\ \left[ E_l\ -\ \frac{l(l+1)}{r^2}\right] $$SS8$8``.Times New Roman,18,12,0,0,0,0,0 $V_{l,\text{\BF p\BF s}}^{\,\text{ion}}($$\text{\BF r}$$)\ =\ V_{l,\text{\BF p\BF s}}($$\text{\BF r})\ -\ \left\{ V_{\text{Hatree}}[\rho (\BF r)]\ +\ V_{\text{XC}}[\rho (\BF r)]\,\right\} $SS""``.Times New Roman,18,12,0,0,0,0,0 $$V_{\text{Hatree}}[\rho (r)]\ =\ \int \frac{\rho (r^{\prime })}{\left| r-r^{\prime }\right| }\,dr^{\prime }$$SS!``.Times New Roman,18,12,0,0,0,0,0 $$V_{\text{xc}}\left[ \rho (r)\right] \ =V_{\text{x}}\left[ \rho (r)\right] \ +V_{\text{c}}\left[ \rho (r)\right] \ =\ -\frac 34\left( \frac 3\pi \right) ^{\frac 13}\,\rho ^{\frac 13}(r)\ +\ V_{\text{c}}\left[ \rho (r)\right] $$SSv"v``.Times New Roman,18,12,0,0,0,0,0 $$\left[ -\frac 12\nabla ^2+\epsilon _{\text{xc}}+\int \frac{\rho (\text{\BF r}^{\prime })}{\left| \text{\BF r}-\text{\BF r}^{\prime }\right| }d\BF r^{\prime }+\epsilon _{\text{ext}}-E_i\right] \Psi _i=0$$SS*``.Times New Roman,18,12,0,0,0,0,0 $$4\pi \int_0^{r_c}\Psi ^2r^2dr$$SSRR``.Times New Roman,18,12,0,0,0,0,0 $$4\pi \int_0^{r_c}\Psi ^2r^2dr\ =\ 4\pi \int_0^{r_c}\Psi _{\text{\BF p\BF s}}^2\,r^2dr$$SS``.Times New Roman,18,12,0,0,0,0,0 $$\frac d{d\epsilon }\left[ \frac d{dr}\ln \Psi _l\right] _{r_c}=\frac d{d\epsilon }\left[ \frac d{dr}\ln \Psi _{l,\text{\BF p\BF s}}\right] _{r_c}$$SS ``.dTimes New Roman,18,12,0,0,0,0,0 $\rightarrow $SS``. [Embedded] 68 .tex 85900 257 86157 9174 67 .tex 95331 323 95654 11610 62 .tex 107264 132 107396 1638 61 .tex 109034 58 109092 342 45 .tex 109434 197 109631 3630 44 .tex 113261 127 113388 3312 42 .tex 116700 452 117152 8670 41 .tex 125822 98 125920 1278 40 .tex 127198 179 127377 2790 39 .tex 130167 116 130283 1310 11 .tex 131593 65 131658 810 20 .tex 132468 154 132622 7582 21 .tex 140204 131 140335 9090 22 .tex 149425 161 149586 8654 23 .tex 158240 194 158434 8994 24 .tex 167428 228 167656 12042 25 .tex 179698 269 179967 13138 26 .tex 193105 255 193360 10810 27 .tex 204170 238 204408 10758 28 .tex 215166 222 215388 12564 29 .tex 227952 143 228095 7788 30 .tex 235883 51 235934 936 31 .tex 236870 192 237062 13466 32 .tex 250528 187 250715 11442 34 .tex 262157 135 262292 7826 35 .tex 270118 51 270169 936 36 .tex 271105 46 271151 256 37 .tex 271407 36 271443 188 38 .sdw 271631 2894 274525 88568 47 .tex 363093 131 363224 8928 49 .tex 372152 203 372355 10694 50 .tex 383049 256 383305 16486 51 .tex 399791 217 400008 11250 52 .tex 411258 222 411480 5528 53 .tex 417008 143 417151 5694 54 .tex 422845 261 423106 12734 60 .tex 435840 238 436078 11442 63 .tex 447520 66 447586 2068 64 .tex 449654 122 449776 4818 65 .tex 454594 182 454776 6482 66 .tex 461258 47 461305 290 00461597