PHYSICAL REVIEW B 66, 245104 ͑2002͒ Targeting specific eigenvectors and eigenvalues of a given Hamiltonian using arbitrary selection criteria Alan R. Tackett Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235 Massimiliano Di Ventra Department of Physics and Center for Self-Assembled Nanostructures and Devices, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435 ͑Received 9 May 2002; revised manuscript received 1 July 2002; published 5 December 2002͒ We present a method for calculating some select eigenvalues and corresponding eigenvectors of a given Hamiltonian. We show that it is possible to target the eigenvalues and eigenvectors of interest without diago- nalizing the full Hamiltonian, by using any arbitrary physical property of the eigenvectors. This allows us to target, for example, the eigenvectors based on their localization properties ͑e.g., states localized at a given surface or interface͒. We also show that the method scales linearly with system size. DOI: 10.1103/PhysRevB.66.245104 PACS number͑s͒: 71.15.Ϫm, 31.15.Ct, 31.15.Ew I. INTRODUCTION criteria other than the eigenenergies are more relevant to tar- get specific eigensolutions of a given Hamiltonian, e.g., the The calculation of the eigenvalues and corresponding localization properties of the wave functions at surfaces or eigenvectors of a given Hamiltonian H is of fundamental interfaces. importance in quantum mechanics. In many physical prob- We present in this paper an alternative approach to target lems, it is enough to determine the eigensolutions of H that select eigenvalues and corresponding eigenvectors by using correspond to the lowest-energy states of the spectrum. In arbitrary physical properties of the eigensolutions. The this case, several numerical methods are available that effi- method relies on the use of the Jacobi-Davidson technique10 ciently allow us to diagonalize H with respect to the lowest- that does not require ‘‘squaring’’ the Hamiltonian operator as energy states.1–5 However, there are many physics problems in the FS method, but solves the eigenvalue equation di- that require knowledge of only some select eigenvalues and rectly. This technique does not suffer from either of the FS corresponding eigenvectors of H, which are not the lowest- problems and can easily be extended to generalized eigen- energy states of the spectrum, and for which the diagonaliza- value problems resulting from the use of particular methods tion of the full Hamiltonian is computationally very expen- and technicalities for the solution of the Schro¨dinger equa- sive if not impossible.6–8 For these problems, the tion, such as the projector augmented wave ͑PAW͒ method11 determination of such eigensolutions presents a challenge. To or the use of ultrasoft pseudopotentials12 or a nonorthogonal this end, Wang and Zunger6 developed a method ͓the so- basis set. called ‘‘folded spectrum ͑FS͒’’ method͔ which scales linearly The paper is organized as follows. In Sec. II, we will with system size rather than the usual cubic scaling required briefly outline the problem we want to address and its current by traditional matrix diagonalization techniques.9 The solution within the FS method. In Secs. III and IV, we method consists of ‘‘folding’’ the eigenvalues of the spec- present the Jacobi-Davidson method that represents the core trum around a given reference energy and ‘‘squaring’’ the of the alternative approach we propose. In Sec. V its practi- resulting Hamiltonian operator.6 Using Hamiltonians con- cal implementation is outlined. In Sec. VI, we discuss its structed from semiempirical pseudopotentials, those authors convergence properties and scalability with system size. Fi- have successfully applied the technique to a number of inter- nally, in Sec. VII, we discuss few examples of application of esting problems including the calculation of the dielectric the present approach where different selection criteria are properties of quantum dots,7 variation of the band gap with used. quantum dot size, and the solution of the ‘‘inverse band- 8 structure problem.’’ In all of these cases only a relatively II. THE EIGENVALUE PROBLEM small fraction of the total number of eigensolutions of the Schro¨dinger equation was determined around a specific en- The main concern in electronic structure calculations is ergy. However, the ‘‘squaring’’ of the Hamiltonian operator solving the eigenvalue problem, in the FS method greatly increases the difficulty in solving ␺ ϭ␧ ␺ ͑ ͒ the original eigenvalue problem. Moreover, in electronic H i i i , 1 structure calculations, the solution of a generalized eigen- 11,12 ␺ value problem is sometimes required. The FS method where i’s are the electronic wave functions, H is the system ␧ could, in principle, be extended to such cases but at the cost Hamiltonian, and the i’s are the energy eigenvalues. Equa- of a significant increase in solving difficulty. In addition, the tion ͑1͒ can represent, for instance, a set of Kohn-Sham13 FS method can only handle reference energies as selection equations to determine the ground-state properties of a given criterion. However, in certain physical problems selection system. A generalized eigenvalue equation can occur if a 0163-1829/2002/66͑24͒/245104͑7͒/$20.0066 245104-1 ©2002 The American Physical Society A. R. TACKETT AND M. DI VENTRA PHYSICAL REVIEW B 66, 245104 ͑2002͒ nonorthogonal basis set is chosen or as a result of the dling generalized eigenvalue problems poses no difficulty. In pseudopotential formalism.11,12 The generalized eigenvalue the following, we refer to the latter case to illustrate the problem is defined as method. The method consists of solving the projected eigenprob- ␺ ϭ␧ ␺ ͑ ͒ H i iO i , 2 lem with different search and test subspaces. The search sub- ϭ͓ ͔ where the new operator O is called the overlap operator. space V v1 v2•••vn spans the space of the possible so- ϭ͓ ͔ There are several techniques for trying to find a few of the lutions, and the test subspace W w1 w2•••wn provides a smallest or largest eigenvalues, but few of them are effective space for testing the quality of the solutions. In most appli- in finding selected eigenvalues inside the spectrum.14 Fur- cations, the test and search subspaces are the same, leading thermore, all of these methods rely on the eigenvalues as the to the following equation for the projected generalized eigen- selection criterion, and cannot be generalized to using other value problem: selection criteria. † ϭ␧ † ͑ ͒ V HVui iV OVui . 5 ␺newϭ A. Folded spectrum method The new eigenvectors are then calculated as i Vui . However, when targeting selected interior eigenvalues There are few methods for solving interior eigenvalue ␧ problems.22 The most successful approach to date is the around some reference energy ref , it is more advantageous folded spectrum method.6 This technique is based upon fold- to make the test and search subspaces different. This leads to ␧ the following equation: ing the eigenvalue spectrum around a reference energy ref , thus shifting the lowest eigenstate of the resulting system to W†HVu ϭ␧ W†OVu . ͑6͒ that closest to the reference energy. The resulting eigenvalue i i i equation is If the test space is chosen as ͑HϪ␧ ͒2␺ ϭ͑␧ Ϫ␧ ͒2␺ , ͑3͒ ϭ Ϫ␧ ͒ ͑ ͒ ref i i ref i W ͑H refO V, 7 which is then solved with standard techniques to find the and W is made orthogonal, W†WϭI, the eigenvalue prob- 6 lowest-energy states. The main disadvantage of this tech- lem can be solved in an efficient way. First, let us shift the nique is that ‘‘squaring’’ the effective Hamiltonian operator ␧ spectrum with the reference energy ref , also ‘‘squares’’ the condition number, which is directly re- ␺ ϭ␧ ␺ lated to the difficulty in solving the FS equation. This in- H i iO i , crease in the condition number makes solving the FS method Ϫ␧ ͒␺ ϭ ␧ Ϫ␧ ͒ ␺ much more difficult. As stated by the authors of the FS ͑H refO i ͑ i ref O i , method,15 a typical case requires ϳ100 conjugate gradient ͑ ͒ ˜ ␺ ϭ˜␧ ␺ ͑ ͒ CG steps per energy band per iteration. Each CG step re- H i iO i , 8 quires two applications of H due to the squaring operation. ˜ ϭ Ϫ␧ ˜␧ ϭ␧ Ϫ␧ This corresponds to ϳ200 applications of H per band per where H H refO and i i ref . iteration. As the authors of FS method also point out, this is Now we can apply the oblique projection with WϭH˜ V ␺ ϭ about the ‘‘square’’ of the normal number of CG steps re- and i Vui to the eigenvalue equation giving quired for solving the original eigenvalue equation.15 † ˜ ϭ˜␧ † ͑ ͒ W HVui iW OVui . 9 B. Extending the FS method to generalized eigenvalue problems Because W is orthogonal, we obtain Even if never used in this context, the FS method can W†H˜ Vu ϭ˜␧ W†OVu , easily be extended to handle generalized eigenvalue prob- i i i lems. The resulting generalized folded spectrum equation is † ϭ˜␧ † W Wui iW OVui , Ϫ␧ ͒ Ϫ1 Ϫ␧ ͒␺ ϭ ␧ Ϫ␧ ͒2 ␺ ͑ ͒ ͑H refO O ͑H refO i ͑ i ref O i . 4 1 W†OVu ϭ u , However, from the above equation it is obvious that now i ˜␧ i the condition number and difficulty has increased ‘‘cubi- i cally’’ over the original eigenvalue problem, and we have 16 1 implicitly assumed that the O operator can be inverted. W†OH˜ Ϫ1Wu ϭ u , ͑10͒ i ˜␧ i i III. JACOBI-DAVIDSON METHOD since VϭH˜ Ϫ1W. The Jacobi-Davidson method,10 briefly described here, is Notice that this choice of the test subspace is mathemati- an oblique projection method that solves the eigenvalue cally equivalent to using a normal orthogonal projection equation directly.