d, n mr' E,[n, V,,]=T[n]+fdr V,,(r)n(r) +E,[n] (r r I. INTRODUCTION New approach for solving the density-functional self-consistent-field problem

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PHYSCAL REVE% B VOLUME 26, NUMBER 6 5 SEPTEMBER 982 New approach for solving the density-functional self-consistent-field problem Paul Bendt and Alex Zunger Solar Energy Research nstitute, Golden, Colorado
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PHYSCAL REVE% B VOLUME 26, NUMBER 6 5 SEPTEMBER 982 New approach for solving the density-functional self-consistent-field problem Paul Bendt and Alex Zunger Solar Energy Research nstitute, Golden, Colorado 8040 and Department of Physics, Uniuersity of Colorado, Boulder, Colorado (Received 26 April 982) A new approach for obtaining the minimum of the density-functional total energy is developed by the application of the variational method to the effective potential rather than to wave functions. The resulting conditions on the effective potential are shown to reduce to a system of simultaneous nonlinear equations. This system can then be solved easily with the use of modern ideas from optimization theory. This also gives a unified description of most self-consistency convergence accelerators and enables us to design a superior procedure. The new approach has been implemented in a completely general band-structure method. A special construction of the potential and mixed basis set enables us to calculate efficiently the band structure of materials with both complex unit cells and interacting d states. The method is demonstrated on crystalline Si and ZnS and is used to obtain the first ab initio band structure for CunSe2 (8 atoms per unit and 292 electrons per unit cell).. NTRODUCTON The density-functional formalism of Hohenberg, Kohn, and Sham' (HKS) underlies much of the recent progress in understanding the electronic and structural properties of molecules and solids. HKS have shown that the ground-state charge density and total energy E,of electrons moving in any given external potential V,,(r) can be obtained from the expression E,[n, V,,]=T[n]+fdr V,,(r)n(r) d, n mr' (r r +E,[n] by finding the density n (r ) that minimizes E,. (Throughout the paper the asterisk denotes selfconsistent ground state. ) This form is motivated by the independent-particle model: The first three terms represent, respectively, the kinetic energy, applied potential energy, and classical mean-field interelectronic Coulomb energy of the independent particles. The universal functional E,[n] represents all corrections to the independent-particle model; i.e., the nonclassical many-body effects of exchange and correlation (xc). Whereas Eq. () is easily proven formally, the evaluation of n*[ V,,] can be difficult in practice. The major problems are the following: (i) The functional E,[n] is at present known only for a few simple systems and must in general be approximated, (ii) the only known prescription for calculating T[n] given only n(r) (no wave functions) is often insufficiently accurate on the scale of binding energies or conformational energies of polyatomic system, and (iii) current searching algorithms to find n (r) that minimizes E,are not effective for general polyatomic systems. The first problem is circumvented by applying the local-density approximation for E,[n]; it is determined by borrowing the known solution for the homogeneous electron gas. For the second difficulty, Kohn and Sham proposed bypassing the evaluation of T[n] by simultaneously constructing a density n (r ) and its kinetic energy T[n] from a set of wave functions of noninteracting particles. A recent theorem by Theophilou described below shows that these fictitious noninteracting particles form a rigorous foundation for the independent-particle model. The results of Theophilou have motivated us to find a new approach to solving the densityfunctional problem. n its final form our approach is quite similar to that of HKS, but it also provides an understanding for the common basis of several computational techniques currently in use. This understanding in turn suggests new techniques for obtaining the solution to the densityfunctional problem for polyatomic systems. These techniques have been implemented in this paper and result in a very effective procedure for finding the variational self-consistent-field (SCF) potential The American Physical Society QJ PJ(r)) NEW APPROACH FOR SOLVNG THE DENSTY-FUNCTONAL... Theophilou (see also Refs. 5 and 6) has shown that for any physical charge density n (r) there exists a potential U,,( r )(an ordinary function of r) in which the noninteracting particles will have the charge density n (r). The Schrodinger equation for the noninteracting particles is [, 7 +U,,(r))PJ(r)=ejgq(r). By construction, the charge density of the real system is identical to the charge density of the system of fictitious noninteracting particles given by n ( r) = +co J ( r ) where coj are occupation numbers. The groundstate kinetic energy in the independent-particle approximation is likewise constructed from the noninteracting particle orbitals as T[n, (r)]=gcoj(pj(r) J,V. (4) Theophilou proved a one-to-one-to-one correspondence among the generating potential U,,(r) (up to an additive constant), the ground-state charge density n (r), and the ground-state orbitals QJ(r) J for the noninteracting particles ( and a similar correspondence also for real electrons). Notice that U,,( r ) contains the information of the fixed external potential V,,(r) [Eq. (}]which, together with the number of electrons N =gjcoj, defines the physical system. Notice further that the only meaning for U,,(r ) is that it generates the charge density and kinetic energy via Eqs. (2) (4). n the density functional equation (},the density n (r) is formally the independent variable. However, given an n (r), it is difficult to find the corresponding T[n] (unless approximate gradient series expressions are used for T [n]). Theophilou's theorem, however, opens the way to treating either the orbitals QJ(r ) (as done by HKS and almost all subsequent applications of the density-functional formalism} or the generating potential U,,(r ) (as done here) as the independent variable for minimizing the total energy in Eq. (). While these two approaches are parallel, they still represent fundamentally different philosophical viewpoints, which in turn are translated into different computational schemes. As shown below, the scheme based on regarding U,,(r ) as the independent variable can be adapted to use modern and powerful minimization techniques, which greatly simplifies the solution of the densityfunctional problem. We will define the potential variation which is used here by first comparing it to other variational approaches. The choice between the two possible independent variables and the type of variation used leads to four fundamentally different approaches to minimizing the HKS total energy. n practice the minimization is always restricted to searching a finite-dimensional subspace from the total Hilbert space of wave functions or potentials. We will denote the independent variables as functions of both position and a set of variational parameters which specify a point in the given parameter subspace: P(r) =P,(r; a;, ), U,,(r)=U,,(r;[p J). Here [a, & and tp& may be either linear or nonlinear parameters. The four approaches to the energy-minimization problem are the following: (i) wave functio-n sampling, i.e., a discrete minimization of E«[n a,j ; V, ]by sampling points a,j J and choosing the lowest Eto (ii) the waue functiongradient method, i.e., the Kohn-Sham approach in which one solves the eigenvalue equations resulting from the condition V, E,=0, (iii) potential sampling, i.e., a discrete minimization of Eto, [n p&; V,,] by sampling p J, and (iv) the potential gradient method, i.e., solving the equations resulting from the condition V' -E«, 0. All but the last approach have been used previously, either in the context of the densityfunctional energy expression of Eq. () or in conjunction with other energy functionals. n the next section we discuss the extent to which these different approaches lend themselves to effective computational schemes. We will show that the potential-gradient method (iv) has important advantages. Section shows how these advantages can be used to simplify the solution of the noninteracting single-particle equation [Eq. (2)] for solids. Our completely general potential-variation mixed basis method introduced in this section can treat materials with both low symmetry and interacting d bands. n Sec. V we give an overview of Broyden's method, a powerful optimization method particularly well suited for solving the potential-variation energy-minimization problem. This method requires only quantities already calculated for other purposes but cleverly combines them with the iteration history to find the SCF potential quickly. The illustrative applications of Sec. V show our results for three very different semiconductors: covalently bonded Si, ionic ZnS 3il6 PAUL BENDT AND ALEX ZUNGER 26 with d electrons, and the structurally complex ternary semiconductor CunSe2.. FOUR VEWS ON THE TOTAL-ENERGY- MNMZATON PROBLEM A. The wave-function sampling method n this approach one must (i) select a trial parameter set a;] for Eq. (5) and construct the wave functions ) of Eq. (5); (ii) construct fz(r; a; the density n(r) and kinetic energy T[n] from Eqs. {3}and (4), respectively, and (iii) calculate E,[n, V,,] of Eq. () and repeat the above steps to obtain the minimum of E«,[nta&];V,,] as a function of a,j.. The primary advantage of this method is that it does not require any eigenvalue problem solving, such as Eq. (2), and does not require that any effective potential be constructed. Hence, it is possible to deal conveniently with nonlinear parameters and interelectronic correlation effects directly in the wave functions. Consequently, this method has been used extensively to calculate many-body interaction energies for bosons, nuclear matter, Fermi liquids, solids, ' and molecules with the use of nonlinear forms such as the Jastrow wave functions' Q( r iz, t a, z J ) or Feenberg wave functions Pj( r, 2, r, 2s,..., a,j ). t has also been used to define Wannier functions for solids. ' However, for our purpose here of treating systems with a large number of occupied single-particle orbitals tgj ], this method is very ineffective because it requires a good search method to converge at all and since the number of its variational parameters depends on the number of occupied orbitals. t is therefore limited, with the use of state-of-the-art nonstochastic search algorithms, to about 00 parameters a,j. The result is that this method is suitable for independent-particle problems only when the number of occupied states is small (as simultaneous minimization is required for all single-particle orbitals g ) or when the wave functions have a particularly simple form. Many problems in contemporary one-electron solid-state physics do not satisfy thee conditions. B. The wave-function gradient, or the HKS method Since the form of the energy-minimizing potential U,,(r ) of Eq. (2) is generally unknown, Kohn and Sham have provided a particular interpretation to it, based on a wave-function variational principle BE,/BQ=O. This leads to the condition for the gradient: BEtot Ba,j Ba,j 2 + VKS[«r)] to be satisfied by the final SCF wave functions g*(r). This is more commonly identified as the Kohn-Sham (KS) single-particle equation for the variational wave functions of the fictitious noninteracting particles: [ ', V'+ VKs[n(r)]]y, *(r)=e,yj(r). This is identical in form to the general Eq. (2) that generates orbitals QJ, except that U,,(r } of Eq. (2), which is an ordinary function of r, is replaced in Eq. (7b) by an integrodifferential operator, n (r ') B& [n] VKs[n ( r)] = V,,( r }+ dr + r r' Bn(r) f f = V,,(r)+ Vc, i[n,(r)]+ V,[n (r)], (7a) (7b) where Vc, [n(r)] and V,[n(r)] are the interelectron Coulomb and xc potentials, respectively. The explicit dependence of the Kohn-Sham potential on the charge density requires that the variational Eqs. (7) and (8) be solved iteratively. The general description of this iterative procedure is as follows: (i) Select a trial set a, z j and construct the wave functions [QJ(rJ., aj }]of Eq. (5); (ii} construct the charge density n (r) and kinetic energy T[n] using Eqs. (3) and (4); (iii) from the charge density calculate the Kohn-Sham potential VKs[n, (r)] of Eq. (8); {iv) solve the eigenvalue problem [Eq. (7)] to obtain new wave functions. Repeat all of the above steps until the input and output wave functions are equal (to within a prescribed tolerance). This method has an important advantage over the wave-function sampling method: At an arbitrary mth iteration it yields solutions for a set of wave functions QJ '(r)] which contain substantially more information than the single number E«, ' provided by the wave-function sampling method. This information then can be used as a guide for selecting the next trial wave functions. This procedure has been implemented in the past NEW APPROACH FOR SOLVNG THE DENSTY-FUNCTONAL... with various computational approximations, including (i) the size and form of the basis set describing QJ j, (ii) the calculation of Vs[n (r )] from n ( r ), (iii) the solution of the eigenvalue problem in Eq. (7), and (iv) the specification of a consistency between input and output wave functions. However, a number of annoying features remain. These include the high sensitivity of n (r') to computational fluctuations in VKs(r) (which often leads to instabilities in iterative schemes, e.g., Ref. 4) and the large size of the subspace a,j j needed to obtain a useful accuracy. ' These difficulties frequently lead to the introduction of (often severe) computational approximations to solve the allelectron Kohn-Sham problem. n part they also encourage a pseudopotential approach to the problem' in which Vxs(r) in Eq. (8) is replaced by an approximate functional identifying V,,(r) with an empirical pseudopotential and treating only the valence electrons. The method for selix:ting trial wave functions fz '( r ) for the mth iteration implied by the Kohn- Sham approach is simply QJ '(r }=ipj (r), where QJ (r) is the solution of Eq. (7) for the previous (m ) iteration. Although obvious and simple, this leads to an iteration sequence which is not guaranteed to converge and often does not, as wib be illustrated in Sec. V for ZnS. n attempts to overcome this problem, as well as those indicated above, various artificial devices for forcing convergence have been used. These include the potential-mixing method (e.g., Ref. 6), the Pratt scheme, ' improved mixing schemes, ' or potential-attenuation schemes. ' These methods all modify the potential of Eq. (8} by including contributions from previous iterations in various manners. Although these have yielded solutions for many interesting systems, the convergence is generally slow and the inherent instabilities in such methods remain formidable. Furthermore, in the context of the Kohn-sham variational scheme such methods are formally illegitimate: Given a charge density n' '(r) for some iteration m, the potential is formally determined by Eq. (8) alone. The potential-gradient method introduced in Sec. D below, in its treatment of U,,(r ) rather than [QJ(r) j as the independent variable, provides both an explanation for frequent failure of ad hoc convergence accelerators in the context of the Kohn- Sham approach and a formalism in which a rapidly convergent method is easily found. C. The potential-sampling method The potential-sampling nethod focuses on the original wave-function-generating potential of Eq. (2) rather than on the Kohn-Sham interpretation of it [Eq. (8)]. n this approach one (i) selects a trial set [pz j in Eq. (6) and constructs the generating potential U,,(r; [pz j) [Eq. (6)] from it; (ii) solves the independent-particle equation [Eq. (2)] to get the wave functions [PJ(r) j; (iii) constructs the density n (r) and kinetic energy T[n] from Eqs. (3) and (4), respectively; and (iv) calculates E,[n (p };V,,] and repeats the above steps to minimize the energy with respect to p }. n this approach U,,( r; [Juz } ) is only used to generate wave functions whereas the physical potential V,,( r) is used to evaluate the total energy. The primary advantages of this method are its conceptual simplicity and the ease of getting crude solutions to simple (e.g., one-dimensional) problems. Much like the wave-function sampling method, it does not require the calculation of the complicated Vzs[n(r)] but rather that of the far simpler generating potential U,,( r; [pz j }. However, as is true for any sampling method, the potential-sampling approach also has the serious drawback in that it provides at each iteration only the single number E,' '. ts low informational content makes it difficult to construct an effective trial potential for the subsequent iterations. Furthermore, because the convergence of the sampling becomes dramatically more difficult as the number of variables [pz } increases, in most applications only a small number of parameters were used (e.g., two in Ref. 9). With so little variational freedom in the potential, the results are not particularly accurate relative to those obtained by the Kohn- Sham wave-function-gradient method. This method was applied recently in the densityfunctional context to the problem of the jellium surface and electron-hole ' drops, and in the context of Thomas-Fermi and Hartree-Fock energy functionals to the calculation of the energies of isolated ions. ' No applications have been reported on the electronic structure of real solids. G. The potential-gradient method All of the advantages of the wave-function-gradient method and the potential-sampling method can be combined in the potential-gradient method. n this approach one solves the equations resulting from zeroing the analytic gradient of the total energy [Eq. ()] with respect to the potential variational parameters [pz j VKs Pq(r)} 38 PAUL BENDT AND ALEX ZUNGER [Eq. (6)]. n Appendix A we show that this condition results in BE, =0=2Re ggraj (fj(r) BU,, U,, QJ'(r) Pj'(r) (9) where Vs is given by Eq. (8) and U,, is an ordi nary local function of r, by Eq. (6). This is the fundamental equation in our approach, and is variationally equivalent to the Kohn-Sham condition in Eq. (7). To satisfy this condition, it is sufficient to have At the minimum of the total energy, Eq. (0) must be satisfied, implying that at the solution p*, the output potential VKs(r) can be written in the same analytic form as U,,(r ). We assume this is also true for some domain around p*', i.e., that there exist some v J such that V s(r) U,,(r; V, ] )=0, (0) V s[r;i, ]]=U,,[- which causes the first matrix element in every term in Eq. (9) to vanish. The steps involved in this approach are as follows: (i) Select a trial set pz ] and construct from it the potential U,,(r; pz j ) of Eq. (6); (ii) solve the independent-particle eigenvalue equation [Eq. (2)] to obtain ltj.(r) and ej ); (iii) construct the charge density n(r) from the orbitals g&.(r) [Eq. (3)]; (iv) from n(r) construct Vs[n(r)] of Eq. (8) and repeat the above steps uritil Vs[n(r;p&J)] = U..«r' V, ] ). The main advantage of this method over the potential-sampling technique is that each iteration yields the function VKs'(r) U, ','(r) rather than the single number E,' '. This provides much more information on which to base the next trial potential U,', + (r). Thus fewer wrong guesses are needed before finding the unique self-consistent potential. At the same time, one can include enough free parameters in the potential to obtain accurate results. The advantage of potential gradients over the Kohn-Sham approach is the ease of incorporating and understanding sophisticated strategies for selecting trial potentials. Although the major steps described are the same as for the Kohn-Sham method, there is no implication that the potential used for the eigenvalue problem [Eq. (2)] should be Vs(r) or any ad hoc manipulation of it. Thus the construction of the potential for the singleparticle problem is no longer viewed as an ad hoc procedure or as a convergence accelerator; instead it is viewed as a particular sear
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