On cluster embedding schemes based on orbital space partitioning

Transcription

1 On cluster embedding schemes based on orbital space partitioning Ulrich Gutdeutsch, Uwe Birkenheuer, Sven Krüger, and Notker Rösch a) Lehrstuhl ür Theoretische Chemie, Technische Universität München, D Garching, Germany Received 4 November 1996; accepted 3 January 1997 The embedding approach to the electronic structure o local perturbations in extended systems is based on the undamental assumption that beyond a certain region around the deect, the properties o the environment are not altered by the presence o the deect. In many computational schemes the resulting subdivision o the deect system into a central and an external region is deined in terms o orbital basis unctions. The undamental embedding assumption then translates into a partitioning o matrix representations, accompanied by ixing the external region contributions to their values in the unperturbed reerence system. With the help o density unctional cluster-in-cluster embedding calculations we have investigated the quality o this assumption without introducing any additional approximation as usually done to arrive at a computationally easible embedding scheme. The undamental embedding assumption is ound to cause spurious virtual orbital admixtures to the density matrix which lead to artiacts in the results o embedding calculations. To minimize these undesirable eects, a special class orthogonalization scheme has been employed. It allows a perect reproduction o the deect induced charge density changes as judged by cluster-in-cluster model calculations or a hydrogen substitutional deect in large Li n clusters with n up to 309. However, equilibrium geometries, total energies, and vibrational requencies calculated with this embedding scheme do not exhibit any improvement over results rom calculations employing the corresponding nonembedded model clusters. The reason or this ailure which prevents the expected convergence o the calculated results with increasing cluster size is analyzed. Thus, rom a pragmatic point o view, naked cluster models are preerable, at least or metal substrates, due to their relative computational simplicity. Possible techniques to either avoid the virtual orbital admixtures or to improve the quality o the total energies obtained rom the embedding calculations are discussed together with the drawbacks o these schemes American Institute o Physics. S X I. INTRODUCTION Ever since irst principles quantum mechanical calculations have been established as a powerul tool or studying the microscopic structure o solids and molecules, embedding schemes have been developed to extend the applicability o these new methods toward more complex systems. So ar, a bewildering variety o dierent embedding schemes have been suggested They all share the concept o partitioning the system under investigation into a central region which contains the structure o interest a deect, a unctional group, an adsorbate complex, etc. and a surrounding external region. The impact o this environmental region on the central region is considered as non-negligible, yet o less importance so that it may be represented only approximately to reduce the eort o the calculations or to even make the calculations easible at all. O course, the central region is described at as high a level o sophistication as deemed necessary or properly representing the physical situation at hand. This undamental partitioning into an embedded cluster and an indented environment is oten visualized as a direct subdivision in real space, as actually implemented by Greens unction matching techniques. 1 6 Alternatively, it may simply be given by the dierent levels o theory at which the two regions are treated, as in cases where a cluster is embedded in an array o point charges, 7,8 in a crystal ield, 9,10 in a homogeneous dielectric medium, or in a shell model representation o the environment More generally, the partitioning o the system is implemented as a division o the variational Hilbert space into appropriate unctional subspaces These subspaces are most commonly deined by means o localized basis unctions which are introduced to span not completely, but suiciently accurately the Hilbert space o either the central region only, or o the entire system o interest. Among these techniques one may urther distinguish between methods with a dynamic partitioning, such as the group unction schemes based on localized orbitals or the concept o Whitten, which takes into account the actual electronic structure at the region interace when constructing the two regions, and partitioning schemes which are based on a ixed choice o the unctional subspaces, like in the various embedding techniques proposed by Pisani To arrive at an embedding scheme o practical use, approximations concerning the electronic structure o the external region and its inluence on the embedded cluster have to be introduced. As long as there is a distinct chemical separation between the part considered as central region and its surrounding, as is obvious or molecules in solvents or or weakly bound adsorbates, a simple cluster-in-environment approach oten is appropriate. Thereby, the embedded cluster is treated as an almost isolated particle, except that contribua Author to whom all correspondence should be addressed J. Chem. Phys. 106 (14), 8 April /97/106(14)/6020/11/$ American Institute o Physics

2 Gutdeutsch et al.: Cluster embedding schemes 6021 tions o the external region to the Hamiltonian are represented by some approximative terms which are not necessarily simpler, but are at least easy enough to obtain. A similar approach may be applied to systems with strong ionic interactions between the two regions, as long as no signiicant covalent bonding and charge rearrangement occurs across the region boundary However, or less ionic, covalent, or even metallic environments with direct contact to the central region, like in deect and chemisorption systems, or when unctional groups within larger molecules are taken to make up the environment, proper boundary conditions or the wave unction o the embedded cluster have to be taken into account. 1 6,17 49 Such a physical situation requires a partitioning o the wave unctions or, alternatively, o the density matrix o the system. In these cases, one o the most commonly adopted assumptions, the so-called undamental embedding assumption, is to take the density matrix, or even its energy resolved contributions the projected density o states, in the external region to be essentially independent o the speciic structure and composition o the inner part o the central region. This allows one to evaluate the contribution to the Hamiltonian and to the total energy o the embedded cluster due to the indented environment once and orever in a separate and, in general, much easier calculation. The moderately large embedded cluster MLEC technique, which was originally designed as a Hartree Fock embedding scheme, 36,37 and which we have adapted to set up a density unctional embedding scheme DF-MLEC intended or applications to ininite metallic substrates, 45,46 is one o these methods which rest on this undamental embedding assumption. The EMBED program based on the perturbed cluster equations is another one The assumption o an essentially unperturbed external region density matrix is not beyond criticism, despite the successul application o both o these methods to a variety o deect and adsorbate systems with various substrates and host crystals ,50 52 Experience shows that the approximations unavoidably introduced by the undamental embedding assumption may aect binding and deect ormation energies up to the order o a ew tenth o an ev or some critical combinations o adsorbate, substrate, and basis sets employed, and thus may signiicantly exceed what is commonly accepted as chemical accuracy. I these observations pointed to a speciic methodological problem, embedded cluster calculations would not perorm better on energetics than common nonembedded model cluster calculations. 53 Such a conclusion would seriously question the motivation or using the more complicated and oten also signiicantly more demanding embedding approach, at least or situations where accurate total energies are important. It is the goal o the present work to assess this undamental embedding assumption, which is crucial or the applicability o many schemes. To this end, we will isolate the eects which occur when the density matrix in the external region o a complex system is replaced by a reerence density matrix rom the eects o additional approximations that are generally invoked to arrive at a computationally eicient embedding scheme. To have access to the exact electronic structure o the deect system, and to be able to restrict the approximation exclusively to introducing the reerence density matrix, we have adopted a cluster-in-cluster embedding approach in the present investigation. Although embedding schemes are generally designed to yield reasonable results already or relatively small central clusters, it is important to check cluster convergence. This may perhaps not be necessary in routine applications, but has to be done in benchmark studies like the present one in order to evaluate the reliability o a method. With this in mind, two rather simple, yet quite realistic test systems have been chosen or the present study: a vacancy and a hydrogen substitutional deect in icosahedral lithium clusters. By this choice, rather large host systems may be treated with a density unctional method. Beore turning to the actual cluster-in-cluster test calculations, various ways o establishing a partitioning in unctional space will be discussed. Subsequently, the results on the validation o the embedding assumption are presented in two steps. First, the necessity o a balanced orthogonalization scheme or the orbital space partitioning is demonstrated, and then the inluence o the approximated density matrix on the charge distribution and on the ormation energies o the deects is discussed. II. THE CONCEPT OF ORBITAL SPACE PARTITIONING In embedding schemes which employ localized, and preerentially atom-centered orbital basis unctions to represent the electronic one-particle wave unctions the partitioning o the density operator Pˆ is usually established by blocking its matrix representation P according to P P CC P DC P CD P DD. Here, C denotes the part o the matrix indices attributed to the embedded cluster which, in general, consists o an adsorbate or deect region A and a surrounding interace region B; D denotes the part assigned to the indented environment. The undamental embedding assumption then reads P P CC P CD P DC P DD or P P CC P CD P DC P DD 2 depending on whether only the cluster cluster subblock P CC is allowed to relax with respect to the reerence density matrix P as is the case in the MLEC ormalism 36 38,45,46 or whether the overlap contributions P CD and P DC are allowed to relax as well like in the EMBED methodology We envisage a modiied sel-consistent ield SCF procedure, Kohn Sham or Hartree Fock, where approximation 2 is applied in each cycle, though one might also conceive a strictly variational ormalism based on this undamental embedding assumption see Sec. VI or urther discussion. For two reasons the matrix blocking strategy o Eq. 2 does not lead to a unique partitioning o the density operator. First, the density matrix may either be taken as a twoold contravariant representation 1

3 6022 Gutdeutsch et al.: Cluster embedding schemes P P ij i,j 1,...,N with Pˆ ij i P ij j, 3 as is common in quantum chemistry, but it may also be regarded as a twoold covariant representation P P ij i,j 1,...,N with P ij i Pˆ j, 4 or as any o the possible mixed representations with co and contravariant indices. Unortunately, in the quantum chemistry literature the type o matrix representations actually employed is oten not clearly indicated and the tensor notation o Eqs. 3 and 4, ideally suited or this purpose, is hardly used. Second, the matrix representation P is only deined up to unitary transormations which may be applied to the complete basis set C D without actually changing the variational space o the total system. As soon as basis unctions assigned to region C and region D are mixed by such a unitary transormation, dierent partitioning schemes are described by Eq. 1. In principle, any o these partitioning schemes might be adopted, and the question on the most suitable scheme is still subject o discussion. There are, however, certain restrictions, in case an orbital space partitioning is not only applied to the density operator or to the Greens operator to which the ormer one is closely related, but also to quantities like the Hamilton or Fock operator. Because in practical implementations relations such as z Ĥ Ĝ z 1ˆ or  tr Pˆ  5 are interpreted as matrix equations, the representations adopted or the various operators are associated with each other, unless the matrix equations are extended appropriately by metric tensors g ij and g ij which are given by the overlap matrix g ij i j and its inverse, respectively. Here, we rerain rom giving any details or explicit expressions see, e.g., Re. 54, but will ocus on the partitioning o the density matrix as the central quantity to which the undamental embedding assumption is applied. In the course o optimizing the DF-MLEC ormalism 45,46 we have tested various partitioning schemes or the density matrix, among them the common twoold contravariant partitioning in the original atom-centered basis set. The use o the standard contravariant density matrix results in problems with the conservation o charge on that part o the system which is to be treated sel-consistently. As evident rom N tr SP tr S CC P CC 2tr S CD P DC tr S DD P DD 6 the number o electrons attributed to the relaxed part o the density matrix, N C tr S CC P CC, diers rom the sum o the nuclear charges Z C o the embedded cluster due to the overlap population 2 tr(s CD P DC ). For a metallic Li substrate an electron charge deect o up to 10% was encountered 45 even or unperturbed substrate clusters; conversely, the twoold covariant partitioning leads to electron numbers N C much larger than the sum o nuclear 7 charges o the embedded cluster. These nonvanishing purely ormal net charges o the unperturbed embedded substrate clusters do not cause any problems during the evaluation o the CC subblock o the Hamiltonian o the embedded system as long as the crystal ield correction H is evaluated exactly as in the Hartree Fock based EMBED program. 40 This crystal ield correction is deined as H H CC P H isolated P CC, 8 with H isolated being the Hamiltonian o the embedded cluster treated as an isolated species. It totally compensates those contributions to the Hartree potential o the isolated cluster Hamiltonian H isolated which result rom the arbitrary decomposition o the electron overlap population entering N C. However, the crystal ield correction is not easy to calculate as all non-negligible matrix elements between the central region and the, in principle, ininite external region have to be computed. A urther complication in case o a density unctional implementation is caused by the nonlinearity o the exchange-correlation unctional with respect to the density. On the other hand, with the crystal ield correction H treated approximately as H H H CC P H isolated P CC, 9 where H is the crystal ield correction rom a reerence calculation on a deect ree system, it turned out to be rather diicult to compensate the Coulomb potential o the artiicial charge contributions to Z C N C to suicient accuracy, especially in the vicinity o some adsorbates which was the kind o deect we were primarily interested in when developing the DF-MLEC scheme. 45,46 This situation arises because the variational space o the reerence system only partially covers the variational space o the adsorbate, and hence a kind o truncation in the approximated crystal ield correction H occurs near the adsorbate. In response to this problem, we have developed 55 an extrapolation scheme based on arrays o atom-centered point charges around the embedded cluster. The point charges are optimized in such a way that the matrix elements o H on substrate atoms in the direct vicinity o the adsorbate are reproduced, as well as possible, by the corresponding matrix elements o the electrostatic potential o the point charges. To arrive at a well-deined and numerically stable itting procedure, and also to guarantee a sensible extrapolation o the resulting electrostatic potential into the region o the adsorbate, the variational reedom o the point charges is reduced by constraints which only allow the major multipole contributions around the adsorbate to be corrected. This technique has successully been employed to extrapolate the missing Coulomb contribution to H into the adsorbate s variational space. 55 Yet other problems remain, the most important o them is related to the act that the elements o the MLEC matrices coupling the central region to the outer region or their deinition see Re. 37 were ound to become rather large when a twoold covariant density matrix partitioning scheme was employed; this entails serious numerical instabilities. Similar problems were encountered or twoold contravariant partitioning within the MLEC embedding scheme 45 in line with

4 Gutdeutsch et al.: Cluster embedding schemes 6023 previous indings o other research groups. 36,37,42,43 These diiculties are independent o how the cluster Hamiltonian H CC [P] is actually set up, and most likely they also occur in the EMBED ormalism since the EMBED coupling matrices are closely related to the MLEC matrices. 39 One way to avoid spurious charge contributions on embedded clusters as well as undesirable numerical instabilities is to partition the density matrix in an orthonormal basis set o the entire variational space C D Then the distinction between co and contravariant representation becomes superluous, but it is crucial choosing the orthogonalization in a balanced way. To illustrate this point consider an orthonormal basis set which is generated by simply orthogonalizing the basis unctions o the external region onto those o the central region beore any intraregion orthonormalization is carried out. This partitioning scheme would be equivalent to a twoold covariant partitioning as ar as the cluster cluster subblock o the density matrix is concerned, and thus the diiculties discussed above remain. Similarly, preorthogonalization o the central region basis unctions onto the basis unctions o the external region with subsequent intraregion orthogonalization leads to an embedding scheme which is equivalent to that where the twoold contravariant orm o the density matrix is partitioned directly. Symmetric orthogonalization appears to be the ideal procedure or generating an appropriate orthonormal basis set. This strategy will be employed or parts o the calculations o the present investigation. However, as will be discussed in Sec. IV, an even more balanced orthogonalization scheme had to be developed to inally arrive at an acceptable partitioning. III. METHOD AND MODEL SYSTEMS Two types o model deect systems will be studied in the present work: a lithium vacancy as well as a substitutional hydrogen impurity in metal-like lithium. The metallic host system is simulated by clusters o up to 309 Li atoms. For computational eiciency, these model systems are chosen to exhibit icosahedral symmetry, comprising up to our icosahedral shells. The intershell distance o the Li icosahedra is ixed at the experimental nearest-neighbor lithium distance o bcc bulk lithium, Å, 56 resulting in a slightly expanded intrashell lithium distance o Å. To create the deect the central Li atom is either removed or substituted by a hydrogen atom. The surrounding nearest-neighbor shell o 12 Li atoms is geometrically relaxed unless stated dierently. The geometry o all urther shells is chosen to coincide with that o the unperturbed reerence host system. Only the outermost shell will be regarded as the external region because size convergence o the embedded cluster is more important in the present context than convergence o the indented environment toward a realistic ininite bulklike host crystal. The deect systems constructed in this way will be denoted by either XLi n (12 m k ) or XLi n (12* m k ) depending on whether a geometrical relaxation o the irst icosahedral shell is taken into account 12* or models with an unchanged geometry o the nearest-neighbor shell around the deect are being considered 12. m and k denote the numbers o those lithium atoms in the remaining icosahedral shells which belong to the embedded cluster C and to the environment D, respectively. X designates the atom at the center o the icosahedron i any, and the upper index indicates the use o a reerence density P in the external region. For exact cluster calculations the corresponding notations are XLi n (12 m) and XLi n (12* m), where m is the total number o Li atoms beyond the irst icosahedral shell. The cluster calculations as well as the cluster-in-cluster embedding calculations were carried out with a suitably modiied version o the LCGTO-DF linear combination o Gaussian-type orbitals-density unctional code. 57 The orbital basis sets, 9,4 4,2 or Li and 6,1 3,1 or H, as well as the basis sets or representing the electron charge density and the exchange-correlation potential, (9s,2d r 2,5p) or Li and (6s) or H, are taken rom previous cluster-in-surace embedding studies. 45,46 The local density approximation to the exchange-correlation unctional as parametrized by Vosko et al. 58 is used throughout this study. In all embedding calculations presented in the ollowing, the orbital space partitioning according to Eq. 2 is realized by means o an orthonormal basis set which is generated via an intermediate orthonormalization step. To this end, the standard SCF procedure is modiied as ollows. Once the Kohn Sham orbitals o the complete deect system and the corresponding one-particle energies have been obtained, a trial Fermi level is chosen, and a ractional occupation numbers are determined by Gaussian broadening 0.3 ev o the cluster level spectrum. 59 Then, b the cluster density matrix is set up as usual and transormed into the representation o the intermediate orthonormal basis set. Next, c the external region subblocks o the density matrix are replaced by those o the reerence density matrix which has been obtained beore in a separate calculation o the corresponding unperturbed lithium cluster. Finally, d the total number o electrons is computed rom the new density matrix, and a new step a is carried out ater a suitable adjustment o the Fermi level. This procedure is iterated until the ormal, preassigned number o electrons o the complete deect system is reached. Except or the constraint imposed by Eq. 2 the charge distribution is allowed to relax reely over the whole deect system, and the total energy is evaluated as in a standard cluster calculation without resorting to any urther approximation. Since replacing density matrix subblocks, in general, violates charge conservation, the Fermi energies determined or the exact cluster and or the corresponding cluster-in-cluster embedding calculation will dier. However, or systems where the undamental embedding assumption is justiied, no signiicant dierences are expected. Both strategies o replacing the subblocks o the external region mentioned in Eq. 2, only the DD subblock and all but the CC subblock, will be considered in this study. The Fermi level adjustment described above closely ollows the concepts developed or the DF-MLEC orm-

5 6024 Gutdeutsch et al.: Cluster embedding schemes alism. 45,46 It is based on the idea that or overall neutral deects any charge concentration or abstraction at the deect site is completely screened within the embedded cluster. O course, this requires the embedded cluster to be suiciently moderately large. Alternatively, one could allow or some charge transer across the boundary o the embedded cluster even though the external region subblocks o the density matrix are kept ixed. 60 This is justiied because an ininite indented host system may act as a perect electron reservoir without any changes in its electronic structure. However, to account or the energy connected to this charge exchange, a chemical potential has to be introduced 60 which is ound to be quite distinct rom the Fermi level o the host crystal. Furthermore, this chemical potential may signiicantly depend on the shape and composition o the boundary o the embedded cluster. 60,61 To maniest such admixtures in the approximated density matrix P obtained rom a sel-consistent embedding calculation, the dierence P P P 14 with respect to the corresponding exact density matrix P resulting rom a standard SCF calculation on C D will be analyzed as ollows. To irst order the dierence in total energy between the embedding calculation and the exact calculation is given by 1 E tr PH n n P n n n n n, 15 with n being the eigenunctions rom the exact reerence calculation and n the associated eigenvalues. The dierence in total electron charge IV. OPTIMIZATION OF THE PARTITIONING Orbital space partitioning, though a quite natural scheme at irst glance, conceals an intrinsic problem as ar as the wave unctions are concerned. Let n be one o the occupied orbital wave unctions o the target system, i.e., the complete deect system, which is composed o contributions rom regions C and D n n C n D 10 according to the intermediate orthonormal basis set. Each o the two contributions may be reexpanded in the complete set o eigenunctions o the deect system, n C m m a C mn, D n m a D mn. 11 m In general, there will be admixtures o all kinds o orbitals, occupied as well as virtual ones, in both parts C n and D n, but as long as no approximations are introduced these contributions will cancel exactly, a C mn a D mn mn. 12 However, replacing the external region contribution D n by that o an unperturbed reerence system leads to n n C n D, m m a C mn a D, mn. 13 Since the cancellation according to Eq. 12 will no longer be perect, admixtures o high-energy virtual states may occur in the approximated occupied orbitals n which, in turn, signiicantly distort the electronic structure and the total energy o the approximated deect system. Replacing regional contributions to orbitals is not equivalent to replacing density matrix subblocks, nevertheless it provides some insight into the impact o a procedure where orbital space partitioning is combined with reezing contributions rom the external region. Most likely there will be some similar high-energy virtual states admixture in density matrix partitioning schemes as well. Q tr PS n n P n n n, 16 on the other hand, has to vanish because o the Fermi level adjustment discussed in Sec. III. However, the act that this sum over the orbital resolved density matrix dierence n vanishes does not guarantee that the energy weighted sum in Eq. 15 does so as well, or even stays small. To visualize the orbital resolved density matrix dierences, we introduce sign-sensitive, and locally integrated orbital resolved density matrix dierences Q, n n /2, 17 n where the prime indicates summation over orbitals energies n subject to 1 2 n 2. 1 We also deine the related energy contributions E, and E, to the irstorder energy dierence 1 E, e.g., E, n n F. 18 n 0 The quantities deined in Eqs. 17 and 18 are similar in spirit to density o states, and may easily be displayed as histograms. The reason or introducing the sign sensitivity is to document the random scattering signs o the individual contributions n which otherwise would cancel to a large extent during the local integration. For convenience, the Fermi level F has been chosen as zero energy in Eq. 18. Figure 1 shows the quantities Q, and E, or the deect system HLi The orbital partitioning in this example is based on a symmetrically orthogonalized basis set, and all but the CC subblocks o the density matrix have been replaced here. As expected, the main contributions are gathered around the Fermi level at 3.0 ev, but obviously, there also are signiicant high-energy virtual orbital admixtures with energies up to almost 200 ev. Even some admixtures o Li 1s core orbitals between 48.8 and 49.9 ev are discernible see the inset and it is evident rom Fig. 1 b that there is little hope or the energy weighted orbital resolved density matrix dierences n ( n F ) to sum up to zero. Actually 1 E 1.35 ev in

6 Gutdeutsch et al.: Cluster embedding schemes 6025 FIG. 1. Plain a and energy weighted b orbital resolved density matrix dierences n n P n o the HLi deect system due to the undamental embedding assumption. The orbital space partitioning scheme considered here is the one based on a symmetrically orthogonalized basis set with all but the CC subblocks o the density matrix being replaced see the text or details. Shown are the sign-sensitive and locally integrated quantities Q, and E, as deined in Eqs. 17 and 18 with 10 ev. The insert ocuses on contributions with small absolute values. this example. During symmetric orthogonalization localized basis unctions acquire admixtures rom diuse basis unctions o neighboring atoms, and vice versa, and thus a strong interlocking o the two unctional subspaces C and D takes place. As a consequence the region contribution coeicients a C mn and a D mn o each occupied orbital become much larger than one would expect rom the neglect o the deect induced changes in the electronic structure o the external region. Even core orbitals in the central region, i located near the region boundary, require admixtures o orthonormal basis unctions rom the external region, in order to compensate all o the diuse atom-centered basis unctions which have been added to the core orbital-like atomic basis unctions in the course o the symmetric orthogonalization. In systems with heavier elements this holds or both core as well as semicore states. All these admixtures are artiicial and only introduced by the intermediate basis set. The results displayed in Fig. 1 leave little hope or a successul ormalism that implements the undamental embedding assumption in a straightorward ashion. To arrive at a physically more meaningul orthogonalization, the original atom-centered basis unctions are divided into various classes o chemically separated orbitals, such as core orbitals, valence orbitals, and polarization unctions, and Schmidt orthogonalization in order o increasing energy is applied or the interclass preorthogonalization, while a symmetric orthonormalization is used or the intraclass orthogonalization. This scheme, which in the ollowing shall be denoted as class orthogonalization, takes advantage o the geometrical balance o the symmetric orthogonalization between orbitals o the same kind, and at the same time prevents that localized atom-centered basis unctions or the core and valence orbitals to become contaminated by polarization unctions and basis unctions or atomic orbitals o higher energy. A similar strategy has successully been applied previously 62 to avoid unbalanced core orbital admixtures to coniguration interaction spaces which are especially set up to study localized chemical bonds in extended systems. The positive eect o the class orthogonalization can be seen in Fig. 2 which shows data or the same system as in Fig. 1, yet is computed by means o a class orthogonalization. Three classes o basis unctions have been introduced or the Li atoms: the 1s core orbitals o all atoms, the 2s and 2p valence orbitals orthogonalized on the core class, and the 3s, 3p, and 4s polarization unctions orthogonalized on the core and the valence classes. Asexpected, the Li 1s core orbital contamination o the density matrix dierence vanishes completely see the insert o Figs. 1 and 2, and the overall amount o high-energy virtual orbital admixtures is signiicantly reduced c. Figs. 1 and 2. Furthermore, the energy weighted orbital resolved density matrix dierences E, now tend to zero with increasing one-particle energies, which was not the case or the embedding calculation based on orbitals that had been simply subjected to symmetric orthogonalization. In addition, the irst-order energy dierence 1 E, 0.50 ev, is signiicantly reduced in magnitude compared to the previous results. Clearly, class orthogonalization achieves a much more balanced orbital space partitioning o the density matrix than symmetric orthogonalization. Common contravariant partitioning, on the other hand, turned out to be much worse with 1 E 15.9 ev or HLi V. EVALUATION OF THE EMBEDDING SCHEME Orbital space partitioning based on class orthogonalization, as has been shown in Sec. IV, substantially reduces the negative impact o the undamental embedding assumption as compared to other orthogonalization/partitioning schemes. Yet, it remains to be veriied whether the approximation involved provides a reasonable description o the electronic structure o an embedded cluster. This will be done in two steps: irst, we will examine in detail the charge distribution o the deect system HLi , and then we will

7 6026 Gutdeutsch et al.: Cluster embedding schemes FIG. 2. Plain a and energy weighted b orbital resolved density matrix dierences n n P n o the HLi deect system due to the undamental embedding assumption as in Fig. 1, but with an orbital space partitioning based on a class orthogonalized basis set. FIG. 3. Charge density dierence maps o the deect system HLi displayed in one o the iteen equivalent mirror planes o the icosahedral cluster. Shown is the quantity HLi HLi or cases where a only the subblock P DD o the density matrix has been replaced or b all subblocks except P CC. Panel c displays the corresponding dierence map HLi HLi LiLi 54 LiLi 146 or a nonembedded cluster calculation. The values o the contour lines are , , , , , and a.u. with positive and negative dierences represented as solid and dashed lines, respectively. The positions o the atoms in the displayed plane are marked. compare typical deect ormation characteristics in a series o embedded cluster models comprising up to 308 Li atoms. Figure 3 a shows the dierence between the selconsistent density o an embedded cluster calculation or HLi , with only the external region subblock P DD being replaced the second variant o Eq. 2 and that o a normal SCF calculation o the whole deect system HLi The overall charge density dierence is remarkably small less than 10 4 a.u.. It is completely restricted to the external region which is the outer shell o the LiLi 146 icosahedron in the present case. The charge distribution in the central region is almost perectly reproduced by the embedding calculation the lowest value o the contour lines: a.u.. I the subblocks P CD and P DC are replaced as well the irst variant in Eq. 2, the undamental embedding assumption is expected to be ulilled to a lower degree. This is indeed the case Fig. 3 b, yet the charge density dierences remain essentially localized near the interace region between the second and the third icosahedral shells. Some minor impact on the deect center is also discernible. Obviously, the transer o electron charge density onto the more electronegative hydrogen atom is already suppressed to some extent when the overlap subblocks P CD and P DC o the density matrix are ixed to their values in the unperturbed reerence host system. Still, the dierences in the charge density are small compared to those typically obtained by nonembedded model cluster calculations, e.g., or the HLi model o the HLi 146 deect system shown in Fig. 3 c. Here, a proound dierence is discernible between the induced charge rearrangement around the substitutional hydrogen as predicted by the nonembedded cluster model and that o the complete deect system more than a.u. in the immediate vicinity o the hydrogen atom. In summary, proper cluster embedding certainly leads to a substantially improved description o a deect system as ar as the charge distribution is concerned, and consequently o all related quantities like the dipole moment and atomic charges.

8 Gutdeutsch et al.: Cluster embedding schemes 6027 TABLE I. Comparison o the deect ormation energies o a Li vacancy and a hydrogen substitutional deect in Li icosahedra obtained by various embedded and nonembedded model calculations. The reerence energies E d in the second column result rom ull SCF calculations on the complete deect Next we turn to a discussion o the energetics. Table I summarizes the deect ormation energies E d or a lithium vacancy and or a hydrogen substitutional deect in Li icosahedra. These quantities are deined according to the ormal reactions LiLi n E d Li n Li, H LiLi n E d HLi n Li. 19 The deect ormation energy o the reerence deect system is given in the second column o Table I while the deviations due to the two variants o ixing the external region subblocks o the density matrix are shown in the next two columns. For comparison, the deviations ound or nonembedded model clusters without an external region are also given the ith column. No geometrical relaxation o the nearestneighbor Li shell around the deect has been considered here. Two general trends are discernible. The deect ormation energy dierences become smaller with increasing cluster size, and the results or the less restrictive embedding scheme only P DD ixed are always worse and opposite in sign than those o the more restrictive embedding scheme all but P CC ixed. The ormer trend was to be expected since the undamental embedding assumption is satisied better, the larger the distance between the deect and the cluster boundary. On the other hand, the latter trend is rather surprising, and a much more detailed analysis o individual contributions to the total energy would be required to trace its origin beyond doubt. However, the most important observation in the present context is that the quality o the predicted deect ormation energies is generally not improved upon embedding. Only in some cases the deect ormation energy rom an embedding calculation with all subblocks o P but P CC ixed is closer to the target quantity than the corresponding value o the nonembedded cluster calculation, but the energy dierences are small 0.05 ev. Furthermore, the results o,p DC,P DD systems. The columns headed by P DD and P CD reer to the two variants o replacing the density matrix sub-blocks see Eq. 2. The nonembedded model clusters are constructed by removing the atoms in the external region denoted by k. Energies are given in ev. System E d P DD E d model E d P CD,P DC,P DD Nonemb. HLi HLi HLi HLi Li Li Li Li TABLE II. Comparison o various deect ormation characteristics o a hydrogen substitutional deect in Li icosahedra obtained rom calculations using embedded class orthogonalization and all but P CC replaced and nonembedded model clusters. The reerence values resulting rom ull SCF calculations on the corresponding complete deect systems are given in the second column. The quantities considered are: the optimized radius o the nearest-neighbor Li 12 shell r e, the corresponding deect ormation energy E d, and the vibrational requency e o the totally symmetric shell breathing mode. System Table I show that embedded clusters with just one neighboring shell around the deect treated sel-consistently do not provide appropriate models. So ar only total energies o geometrically unrelaxed deect systems have been discussed, and the question arises whether calculated equilibrium properties show a similar behavior. For this purpose, we decided to study models where the nearest-neighbor shell around the hydrogen substitutional deect is allowed to relax. We have considered only the DF- MLEC-like embedding scheme the second variant o Eq. 2. The computed optimized shell radius, the equilibrium deect ormation energy, and the vibrational requency o the Li 12 shell breathing mode o all our HLi n clusters investigated are summarized in Table II. No signiicant improvement, i any, is discernible upon embedding or any o these quantities. From these results we are lead to the conclusion that the approximations introduced by the undamental embedding assumption, though minimized by perorming the orbital space partitioning in a class orthogonalized intermediate basis set, are still too strong to yield improved results or the energetics and related quantities than those obtained rom the usual nonembedded cluster models. VI. DISCUSSION Reerence value Model induced dierence P CD,P DC,P DD Nonemb. Optimized deect energy E d ev HLi 12 12* 0.43 HLi 54 12* HLi * HLi * Equilibrium shell radius r e Å a HLi 12 12* 2.78 HLi 54 12* HLi * HLi * Vibrational requency e cm 1 HLi 12 12* 251 HLi 46 12* HLi * HLi * a Unrelaxed shell radius: Å. At this point it is instructive to recall that the reliability o the deect ormation energies computed with the EMBED program have improved substantially ater a variational charge balance correction has been introduced. 60 This energy correction takes into account the deect induced charge transer between the embedded cluster and the indented host crys-

9 6028 Gutdeutsch et al.: Cluster embedding schemes TABLE III. Correlation between the deviations E d o the deect ormation energy and o the Fermi level F due to the undamental embedding assumption or Li vacancies and hydrogen substitutional deects in Li icosahedra. P DD, and P CD,P DC,P DD reer to the irst and second variant o the approximations to the density matrix P see Eq. 2. Energies are given in ev. P DD P CD,P DC,P DD System E d F E d / F E d F E d / F HLi Li HLi Li tal even though the density matrix in the external region is assumed to remain unchanged. Empirically a remarkably linear correlation between the missing charge Q and the corresponding change E q in total energy was ound and a correction in the spirit o a chemical potential, E Q Q, has been suggested. 60,61 A similar correlation is also noticeable in our data see Table III. Recall the procedure described in Sec. III to ensure a constant electronic charge on the embedded cluster which leads to a change F o the Fermi level. Using the density o states F at the Fermi level, this change may directly be related to the missing cluster charge that would be ound i the Fermi level would have been kept ixed, Q ( F ) F. Furthermore, or adjustments F on the order o a ew hundredth o an ev as encountered in the present study Table III the linear relation E F clust ( F ) F provides a good estimate o the associated total energy change o the embedded cluster. Thus, together with the missing charge contribution E Q, the change E d o the deect ormation energy due to the application o the undamental embedding assumption should read E d clust F F. 20 For the EMBED type o density matrix ixing P DD only the correlation between E d and F is surprisingly good, the ratio E d / F varies only between 19 and 21. For the DF-MLEC type o ixing the density matrix, the range is much larger 30 to 60, nevertheless a qualitative correlation exists, too. Thereore some evidence exists that a charge balance correction amends the poor perormance o the undamental embedding assumption or the energetics, and, in turn, improves the geometry and geometry sensitive quantities. However, such a corrective device carries an expost lavor and may be viewed as interering with the irst principles character that one generally strives or when designing sophisticated embedding schemes like DF-MLEC or EMBED see also the remarks on the chemical potential at the end o Sec. III. From a pragmatic point o view it is also worth recalling that embedding schemes are, in general, computationally much more demanding than calculations employing nonembedded model clusters. According to our understanding, the main reason or the ultimate ailure o the orbital space partitioning schemes, at least or metal host systems tested in the present investigation, is related to the mismatch o the unctional space partitioning used to deine the embedded cluster, and that given by the occupied and virtual states o the deect system. This mismatch causes the nonvanishing virtual orbital admixtures vir C n,vir m m a C mn, D n,vir m vir D m a mn 21 to the region C and region D contributions o the occupied orbitals discussed in Sec. IV see Eq. 11 or the notation. A new method to avoid this mismatch was recently proposed by Head. 63 In the light o the present discussion and using the nomenclature introduced here, his idea may be phrased as ollows. Let U C be the unctional subspace spanned by the atom-centered basis unctions located in the region o the cluster that is to be embedded, and let Pˆ occ and Pˆ vir 1ˆ Pˆ occ be the projectors onto the occupied and virtual states o the complete, unperturbed reerence system which, or a closed shell system, are directly related to the density operator. Then Head s ormalism corresponds to deining the embedded cluster through its unctional subspace, U U 1 U 2 with U 1 Pˆ occ U C, U 2 Pˆ vir U C, 22 and the indented host crystal through the orthogonal complement V U. Speciically, the subspace U is constructed 63 by orbital rotations which are generated by augmenting the original basis unctions o U C by atomic orbitals rom the indented environment in two ways such that the rotated basis unctions, on the one hand, exclusively belong to the subspace o the occupied orbitals o the entire reerence system U 1, and on the other hand, exclusively belong to the subspace o the virtual orbitals o the complete unperturbed system U 2. By construction, U is an extension o the original subspace U C, which is almost twice as large as U C.As evident rom Eq. 22, U is invariant under Pˆ occ, which results in a block diagonal matrix representation, P occ P UU 0 0 P VV 23 o the projector Pˆ occ. Consequently, the occupied orbitals n o the complete system not only obey Pˆ occ n n, but also Pˆ occ n U n U and Pˆ occ n V n V, which immediately leads to U V 0, n,vir 0. n,vir 24

10 Gutdeutsch et al.: Cluster embedding schemes 6029 Hence, no virtual orbital admixtures occur in this orbital space partitioning scheme, and there may be a chance to substantially improve the quality o the undamental embedding assumption. However, there are two aspects to consider. First, it is crucial to recognize that there is no guarantee or the subspace deining the embedded cluster to stay localized in the common sense. Depending on the nature o the host crystal material, the basis unctions spanning U 1 and U 2 may become rather extended. Especially or metals one may run into serious problems because o the large number o matrix elements which have eventually to be evaluated and because o the orthogonalization steps necessary to set up the subspace U or details see Re. 63. The diiculties encountered 63 when this new orbital space partitioning scheme was applied to hydrogen adsorption on-top o a Li 001 monolayer may very well be related to a large implicit geometrical extension o the subspace U which is assigned to the embedded Li cluster, and urther investigations seem necessary to characterize the type o host systems which may be treated successully with the method o Head. The second aspect which has to be taken into account is that the projectors Pˆ occ and Pˆ vir depend on the electronic structure o the system considered. Thereore the subspaces U and V, which have been set up or the unperturbed reerence system, will in general no longer be perectly invariant under Pˆ occ once the deect is introduced. Without speciic tests it is hard to judge to which extent this will inluence the validity o Eq. 24. I the electronic structure o the deect system is treated within the constrained SCF approach as done in Re. 63, the block diagonal orm o the matrix P occ Eq. 23 is guaranteed and no problems arise with Eq. 24. On the other hand, i some coupling between orbitals o the embedded cluster and orbitals o the indented host crystal is allowed beore the undamental embedding assumption is applied, there will most likely be high-energy virtual orbital admixture in the density matrix again. To our understanding only a strictly variational approach like the constrained SCF scheme 64 is able to yield regional orbital contributions n U and n V or n C and n D which are ully consistent with the density matrix resulting rom the undamental embedding assumption, and which thereore rules out any spurious virtual orbital admixtures. However, a strictly variational scheme which in contrast to the constrained SCF approach also allows orbital coupling between the central region and its environment does not seem easy to design. VII. SUMMARY AND CONCLUSIONS In speciic implementations o embedding schemes it is generally diicult, i not impossible, to separate numerical and computational approximations rom those approximations which arise rom the undamental embedding assumptions o the underlying method. The ormer type o approximations are introduced to simpliy the algorithm or to save computer time; in principle, they can be eliminated as ar as necessary. On the other hand, the latter approximations have to be regarded as intrinsic to an embedding ormalism. They limit the overall accuracy o an embedding scheme, and thus it is important to check their quality and consequences. By restricting ourselves to pure cluster-in-cluster embedding we were able to exclusively ocus on the undamental embedding assumption which is generic to embedding schemes that are based on orbital space partitioning see Eq. 2 and to avoid any urther approximations with regard to the theoretical and computational treatment o the reerence deect system. Vacancy and hydrogen substitutional deects in large Li icosahedral clusters have been chosen as test cases. Matrix blocking and subsequent replacement o the subblocks which reer to the external region see Eq. 2 is an algorithmic scheme rather than a deinite procedure because its concrete meaning strongly depends on the underlying orbital basis set and on the matrix representation adopted. In order to arrive at an orbital space partitioning scheme which is well suited or embedding, it turned out to be extremely important, especially or total energy calculations on metal host systems, to reduce as much as possible the spurious admixtures o virtual orbitals. These contaminations o the density matrix are unavoidably introduced i one implements the undamental embedding assumption by means o orbital space partitioning. A special orthogonalization scheme, termed class orthogonalization, has been employed to minimize these undesirable consequences, and it was ound that perorming the orbital space partitioning in such a class orthogonalized basis set yields the best results with respect to the spurious virtual orbital admixtures. Deect induced charge rearrangement and related quantities were ound to be almost perectly reproduced by a cluster-in-cluster embedding calculation based on the class orthogonalization partitioning scheme. The undamental embedding assumption is obviously well-suited or that purpose. However, as ar as the energetics and the geometry o the deect systems investigated is concerned, we ound that the embedding calculations did not perorm any better than straightorward nonembedded cluster model calculations or the deect system. Although minimized by the class orthogonalization, an eect o the virtual orbital admixtures on the total energy, o the order o a ew tenths o an electron volt, seems to remain. Thus accuracy is not improved beyond that o nonembedded cluster models o comparable size which are generally computationally ar less demanding than the corresponding embedded cluster models. Well developed quantum chemistry computer codes or clusters and molecules are available that have been optimized or their task. Alternatively, one may resort to supercell calculations, employing one o the codes designed or periodic boundary conditions. Thus, or solving practical problems, one may question the virtue o embedding schemes based on orbital space partitioning. In this discussion, one has to keep in mind that the degree o approximation introduced by the undamental embedding assumption strongly depends on the amount o virtual orbital admixtures. Hence, host crystals with a large band gap will be much less aected than semiconductors or even metals, and metals with a low density o states near the Fermi level will be less sensitive to the orbital space partitioning than metals like aluminum. O course, the inluence