8538
J. Phys. Chem. B 2001, 105, 8538-8543
Ab Initio Embedded Cluster Study of F and M Centers in LiF Christoph Ko1 lmel† and Carl S. Ewig*,‡ Molecular Simulations Inc., 9685 Scranton Road, San Diego, California 92121 ReceiVed: June 5, 2001
Ab initio computations using an embedded cluster method have been employed to study the electronic structure and excitation energies of an M center (an electron pair associated with two neighboring vacancies) in LiF crystals. For comparison, the results for an F center (one electron and one vacancy) computed in the same way have also been studied. The ab initio approach consisted of the Hartree-Fock approximation applied to Li14F13e, Li15F20e27- and Li10F18e210- clusters of Li+ and F- ions about the crystal vacancies employing each of two differing basis sets. The contribution from the remainder of the infinite ionic lattice was computed using the Ewald transformation. No other constraints were applied to the wave function. For the M center, both 〈100〉 and 〈110〉 orientations were considered. The energy difference corresponding to the gain or loss of an electron by the F center, F + e- f F- and F f F+ + e-, and to loss of electrons from the M center, M f M+ + e- and M f M2+ + 2e-, were compared. The electron isodensity surfaces of the Σ-type states of the 〈110〉 M center and the computed energy differences confirm the “molecular” nature of the M center in the 〈110〉 orientation. The Π-type states, however, are delocalized. The computed energy of the vertical transition to the first Σ state, 2.41 and 1.47 eV for the two basis sets, respectively, are compared with the experimental absorption energy, 2.79 eV. These calculations provide a framework to interrelate the energies and electron densities of the differing species. The advantages and limitations of the present computational approach are discussed.
Introduction Despite their simple electronic structures and high symmetries, alkali halide crystals such as LiF can host a wide variety of metastable defects, such as the color centers, when irradiated with energetic photons or electrons.1-6 These defects correspond primarily to species in which an electron has been excited from an ion of the crystal, leading to separation of positive and negative charges and possibly migration of the resulting hole. For example, in KCl crystals an excited electron may associate with a vacancy in the crystal lattice normally occupied by a chloride anion, to form an F center, while chlorine atoms (holes) may combine with chloride ions to form H or Vk centers with stoichiometries Cl2-. These, in turn, may interact and combine either to return the crystal to its original state, or to lead to its degradation products (free metal and halogen).5,7-10 A particularly important and experimentally well-studied family of defects includes the color centers consisting of F centers and their combinations. On the basis of a variety of experimental approaches it has been established that F centers tend to combine to form agglomeration species, often denoted as F2, F3, etc. In the F center and each of the higher species, “excess” electrons are associated with vacancies surrounded by the cations.1-6,11-13 As the agglomeration continues, these combinations appear eventually to become unstable and finally the electrons and cations combine to form free metal. The F center consists of a single electron associated with a vacancy which is stabilized by the Madelung potential and which is generally pictured as having an electronic structure roughly analogous to a hydrogen atom. The F2 center, with two “excess” * Corresponding author. E-mail:
[email protected]. † Current address: HTE North America, 9685 Scranton Road, San Diego, CA 92121. ‡ Current address: IBM, 4660 La Jolla Village Drive, Suite 300, San Diego, CA 92122.
electrons, is considered analogous to the H2 molecule, the F3 center to an H3 molecule, etc.2,3 For clarity in this report we will employ a common alternative notation for these color centers and refer to the F2 species as the M center, F3 as the R center, and F4 as the N center. Despite their considerable importance in understanding the properties of ionic crystals, there have been remarkably few first-principles theoretical studies of these types of defects other than a small number of studies of F centers. In a previous paper14 the early theoretical work including the first principles calculations on F centers was reviewed, and ab initio molecular orbital methods were employed with a simple cluster model to study the electron distribution in the ground and first excited states of the F center in LiF crystals. These calculations indicated that in the ground electronic state the electron distribution that results from such calculations does, in fact, roughly resemble that of the 2S state of a hydrogen atom. In other words it approximately resembles a hydrogen atom but without a nucleus. That the ground state of the F center thus corresponds to a “nonnuclear attractor” in the electron density has been supported both experimentally and by other calculations.15 In the excited state, analogous to a hydrogen 2P state, we found that the F-center electron is primarily delocalized onto the surrounding ions. In this paper we extend the calculations to the simplest of the F-center agglomeration species in LiF crystals, the M center. This is an experimentally characterized color center, which gives LiF crystals a blue color. The objectives here are first to characterize the electronic structure, employing molecular orbital methods, to determine the extent to which the M center actually resembles an H2 molecule, i.e., whether there is a concentration of charge between the two vacancies analogous to a covalent bond in the H2 molecule. For this we have examined the energies and electron distributions in the ground and six excited electronic states. Second we wished to evaluate an improved computational
10.1021/jp012155e CCC: $20.00 © 2001 American Chemical Society Published on Web 08/17/2001
Embedded Cluster Study of F and M Centers in LiF technique, employing an all-electron Hartree-Fock calculation for a cluster of ions surrounding the defect plus Ewald summations16,17 to represent the remaining infinite lattice. In the following sections of this paper we first outline the computational method and the construction of appropriate cluster models for the F and M centers in LiF. We then present the results for the F center, including the energies required for the crystal to gain or lose an electron (i.e., the process F + e- f F- and F f F+ + e-) and the excitation energy to reach the first excited electronic state (F(2S) f F(2P) in atomic notation). The latter may be directly compared with the previous results.14 We then present the computed energies of the M center in each of two orientations (〈100〉 and 〈110〉 in the crystal) and the associated electron distributions. For comparison with F centers we report the energies of the M f M+ + e- and M f M2+ + 2e- transitions. Finally, we compare the computed energy differences with those of the electronic transitions observed experimentally. Models have previously been proposed for the structures and properties of M centers in semiconductors, such as InP18 and, more recently, ab initio calculations for the M center on AgBr surfaces (represented as a neutral Ag2 molecule) have been reported.19 These species have some similarities to those discussed here in that they correspond to defect pairs, but since they are applied to semiconducting or covalent rather than ionic materials, they are fundamentally different. Also, models have been suggested for the energies of M centers in alkali halide crystals.3,20,21 These have been based on forming the center from assumed electron distributions, such as from actual hydrogen wave functions20 or those of approximated F centers.21 However, to our knowledge this paper describes the first ab initio molecular orbital computation on any such agglomeration species in an ionic crystal. Methodology The computations reported herein were based on cluster models to represent a discrete region of the infinite crystal lattice surrounding a vacancy. The cluster must be sufficiently large to permit delocalization of the excess electron or electrons associated with the defect and polarization of the surrounding ions, but also for practical reasons it must not be so large that the computations become intractable. In the study of F centers in LiF14 it was found that the 26 ions forming the first three shells of ions surrounding the defect (the nearest 6 Li+ ions, the nearest 12 F-, and the next nearest 8 Li+) were sufficient to describe the electron delocalization. The electronic delocalization into the third shell was significantly less than the other two, even in the 2P state. This same cluster model for the F center is employed here. Throughout, the experimental lattice constant 2.014 Å is assumed.22 However, in this study we have extended this model by embedding the cluster in an infinite ionic lattice represented by Ewald sums.16,17 Thus the electrons were allowed to delocalize among the ions of the cluster, but their electrostatic interactions with all ions in the crystal beyond the cluster were also represented. Energies and wave functions were computed in the HartreeFock approximation with no additional constraints or approximations. Two basis sets were employed. For each ion the first, basis I, was the same as employed in the previous paper.14 It consisted of a minimal set (1s, 2s, and 2p) of Slater functions with exponents optimized in the crystal as described previously,14 plus a single diffuse sp primitive Gaussian function to describe each ion. The electron distribution of the defect was accounted for by four s and four p uncontracted primitive
J. Phys. Chem. B, Vol. 105, No. 36, 2001 8539
Figure 1. Cluster model of LiF crystals employed for the F-center calculations. The small open spheres represent Li+ ions, the larger filled spheres F- ions, and the large filled sphere is the center of the F-center electron distribution.
TABLE 1: Ab Initio Computational Models, Including Numbers of Centers and Basis Functions, for the F Center and Two Orientations of the M Center in LiF Crystals F center, Oh symmetry 1 defect site plus 26 ions (12 Li+ + 14 F-) 165 electrons Basis I: 4s4p primitive functions in the defect STO-3G + diffuse for each F- and Li+ center 250 total basis functions Basis II: 4s4p primitive functions in the defect 5111/311 plus diffuse for F-, 6311/311 for Li+ 366 total basis functions M center, 〈100〉 orientation, D4h symmetry 2 defect sites plus 35 ions (15 Li+ + 20 F-) 232 electrons Basis I: same as for F center 347 total basis functions Basis II: same as for F center 507 total basis functions M center, 〈110〉 orientation, D2h symmetry 2 defect sites plus 28 ions (10 Li+ + 18 F-) 202 electrons Basis I: same as for F center 284 total basis functions Basis II: same as for F center 414 total basis functions
functions optimized for the hydrogen atom. The second basis, basis II, is an appreciably larger and partially contracted set of Gaussian functions with five s and three p functions for F(5111/311 contraction) and four s and three p functions for the Li+ (6311/311). For the F-center model (shown in Figure 1) these two basis sets consist of 250 and 366 functions, respectively. The computational model for the F center is summarized in the top portion of Table 1. For the M center we considered two possible orientations: along the 〈110〉 and 〈100〉 axes. These represent the closest and next closest possible arrangements of two anionic vacancy sites, respectively. The local symmetry of the defect oriented along 〈110〉 is D2h and along 〈100〉 is D4h. Both defect orientations were studied with both basis sets I and II. However, due to the size of the computations, particularly for the lower-symmetry D2h species, we employed somewhat less ambitious cluster
8540 J. Phys. Chem. B, Vol. 105, No. 36, 2001
Ko¨lmel and Ewig
Figure 2. Cluster model of LiF crystals employed for the M-center calculations in which the center is oriented along a 〈100〉 axis. The small open spheres represent Li+ ions, the larger filled spheres F- ions, and the large filled spheres are the centers of the M-center electron distributions.
Figure 3. Cluster model of LiF crystals employed for the M-center calculations in which the center is oriented along a 〈110〉 axis. The small open spheres represent Li+ ions, the larger filled spheres F- ions, and the large filled spheres are centers of the M-center electron distributions.
models than for the F center. For the 〈100〉 defect the cluster contained 15 Li+ and 20 F- ions to represent the nearest neighbor ions about both vacancies. This is depicted in Figure 2. The basis sets consisted of the same functions as employed for the F center. Basis I contained 347 functions and II 507 functions. For the 〈110〉 orientation we similarly chose in total 10 Li+ and 18 F- ions about the two vacancies (Figure 3). In this case, basis sets I and II contained 284 and 414 functions, respectively. The computational details for both 〈100〉 and 〈110〉 orientations are summarized in the second and third sections of Table 1. All computations employed the Turbomole computer program.23 This program contains several features which have been introduced to make it especially appropriate for studies of highly symmetric cluster models for localized species in crystalline solids, particularly the explicit use of symmetry at each step of the computation. This permits (1) computation of excited electronic states within the Hartree-Fock approximation as the
lowest state of a given symmetry, and (2) straightforward convergence of the Hartree-Fock equations, which as noted previously14 are otherwise difficult or impossible to converge for the present types of systems. Also recent extensions to this program24 implement (3) Ewald sums16,17 for computation of electrostatic interactions with all ions of the infinite crystal external to the cluster. Results and Discussion 1. The F Center. Ab initio molecular orbital calculations14 have indicated that the excess electron density in the ground state of the F center in LiF is primarily localized in the vacancy (about 70%), while in the first excited state it is predominately delocalized into the structure bordering the defect. The present computations for the F center all confirmed this conclusion. Table 2 lists the energies of the F center in LiF computed using basis sets I and II. The energy is the sum of the energies
Embedded Cluster Study of F and M Centers in LiF
J. Phys. Chem. B, Vol. 105, No. 36, 2001 8541
TABLE 2: Relative Computed Energies of One- and Two-Vacancy Species in LiF Crystals (eV) Basis I
Basis II
F (2S)a f F (2P) F (2S) f F+ (1S) + eF (2S) + e- f F- (1S)
transition
4.2 5.9 1.5
3.2 4.9 1.8
M (1Σg+) 〈110〉b f M+ (2Σg+) 〈110〉 + eM (1Σg+) 〈110〉 f M2+ (1Σg+) 〈110〉 + 2eM (1Σg+) 〈100〉c f M2+ (1Σg+) 〈100〉 + 2e-
4.5
3.2
16.0
13.6
12.3
10.3
a Total energy: Basis I -1284.046900, Basis II -1300.252001 hartrees. b Total energy: Basis I -1851.204578, Basis II -1874.399934 hartrees. c Total energy: Basis I -2081.371736, Basis II -2107.525908 hartrees.
of the excess electron interacting with the electron distributions of the cluster ions, the cluster ions interacting with each other, and all of these interacting with the external field as represented by the Ewald expression. The energy of the vertical F(2S) f F(2P) transition computed in basis I, 4.2 eV, may be compared with the previous value computed in this same basis set, 4.1 eV.14 The only difference is the presence of the Ewald term in the present results, whereas the previous ones employed a set of four additional shells of point charges (totally 66 additional charges) with unit ( charges at the experimental crystalline lattice positions for the cations/anions. Therefore the computation of this electronic transition verifies that the Ewald approximation to the infinite set of point charges agrees to within 0.1 eV with the model in which these terms are given by the large but finite set of four additional shells of charges. In basis II the energy difference is computed to be 3.2 eV. Experimentally this transition has been measured as a broad (0.6 eV halfwidth) UV absorption peak with a maximum at 5.1 eV.25 Disagreement with experiment to this degree is to be expected since we have neglected both electron-correlation effects and also lattice relaxation from the experimental perfect crystal structure. The latter effect, differing ion-ion distances in the vicinity of the vacancy, is likely to be an especially important effect as seen, for example, from the Stokes shift between absorption and emission energies. To our knowledge the emission in LiF has not been measured, but it is known to be roughly one-half of the absorption energy in all the halides of Na, K, and Rb.2 For example, the maximum in the absorption spectrum of the F center in KCl is 2.313 eV while in emission it is 1.215 eV. This indicates that the excess electron is coupling appreciably to the geometry of the surrounding lattice, and implies an inherent limitation in the accuracy of both the Fand M-center energy computations reported here. For later reference we also report in Table 2 the binding energy required to remove an electron from the F-center cluster to form the F+ (or FR) vacancy, and also the electron affinity of the F center to add an electron forming the F- (or F′) defect. The latter is a localized (diamagnetic) electron pair analogous to a He atom.26 Since both energies are positive, these results agree with the experimentally well-established principle that the F center is stable with respect to both of the other two species. Both basis sets agree that the energy required to excite the trapped electron to the 2P state is about 1.7 eV less than the energy needed to remove it completely from the crystal. Both basis sets agree that the ground states of the F and F- centers are comparatively close in energy, the latter 1.5 and 1.8 eV higher than the F center in basis sets I and II, respectively. 2. The M Center. The vacancy-vacancy distance is appreciably longer in the model for an M center oriented along a
Figure 4. Electron isodensity surface corresponding to 0.012 e/Å3 for the 1Σg+ type state of the M center in LiF oriented along the 〈110〉 direction in the crystal, computed in basis II.
〈100〉 axis than along 〈110〉 . Also, comparing the cluster models depicted in Figures 2 and 3 it is clear that an M center, as opposed to two separated F centers, is unlikely to be formed in the 〈100〉 orientation as in Figure 2 since there is a Li+ cation separating the two vacancies. This is in accord with the experimental finding1-4 that the M centers in Li, Na, K, and Rb halides are oriented along the 〈110〉 direction. Therefore we have focused primarily on studying defects with the 〈110〉 orientation. M-Center Electron Density Distributions. Figure 4 shows the electron isodensity surface of the electron pair in the highest occupied orbital of the ground state (1Σg+ in molecular notation) of an M center oriented along the 〈110〉 axis as computed using basis II. This surface corresponds to the locus of points for which the electron density is 0.012 e/Å3. From this figure it is apparent that the ground 1Σg+ state does in fact exhibit an electron distribution resembling a covalent bond, with an accumulation of charge between the defects and remarkably similar to that in a free H2 molecule.27 Figure 5 shows the isodensity surface for the singly occupied orbital of first excited state (1Σu+) of the M center, formed by exciting an electron from the highest filled σg orbital to the lowest unfilled σu orbital. Since this orbital contains a single electron, for comparison with Figure 4 this surface corresponds to an electron density half as large as for the 1Σg+ state, 0.006 e/Å3. This excited state is again almost completely localized within the two defects and displays the covalent electron distribution resembling the first excited (1Σu+) state of free H2.
8542 J. Phys. Chem. B, Vol. 105, No. 36, 2001
Ko¨lmel and Ewig TABLE 4: Computed Total Electron Populationa of Each Defect Center of the Ground Electronic State of the F+ Center, Lowest 2S and 2P States of the F Center, the Ground 1S State of the F- Center, and of the Two Defect Sites of the 1Σ +, 1Σ +, and 3Σ + States of the M-Center in Its 〈110〉 g u u Orientation, in LiF species
site population
F+ (1S) F (2S) F (2P) F- (1S) M (1Σg+) M (1Σu+) M (3Σu+)
0.112 0.698 0.118 0.765 1.228 1.388 0.942
a From integration of the electron density from the defect center to two bohrs, the approximate first minimum.
Figure 5. Electron isodensity surface corresponding to 0.006 e/Å3 for the 1Σu+ type state of the M center in LiF oriented along the 〈110〉 direction in the crystal, computed in basis II.
TABLE 3: Computed Relative Electronic Energies of the M-Center, in the 〈110〉 Orientation, in LiF (eV) state
Basis I
Basis II
1Σ + g 1Σ + u 3Σ + u 1Π ux 3Π ux 1Π uy 3Π uy
0.00 2.41 -1.02 2.60 2.52 2.69 2.60
0.00 1.47 -1.03 1.47 1.43 1.49 1.46
No symmetry was imposed on the wave function in these calculations. However the computed results in all cases gave an electron distribution with each vacancy occupied equally, i.e., D2h and D4h symmetries for the 〈110〉 and 〈100〉 orientations, respectively. Starting with an asymmetric F+/F- pair, which places both the excess electrons in one vacancy, the computation again converged to the more symmetric distribution. This is also to be expected from the results shown in Table 2 that both the F (2S) f F+ (1S) + e- and F (2S) + e- f F- (1S) transitions are characterized by positive energy differences. M-Center Excitation Energies. Finally we list in Table 3 the vertical excitation energies to six excited electronic states of the M center in LiF, oriented along the 〈110〉 direction, computed in each of the two basis sets. The energies found with basis II are again consistently less than those from basis I. The excitation energies to the 1Σu+ state, whose electron distribution is shown
in Figure 5, are 2.41 and 1.47 eV, respectively, for the two basis sets. Surprisingly the corresponding triplet state, 3Σu+, is computed to be lower in energy than the presumed ground state, 1Σ +. This is likely due to neglect of electron correlation effects. g The larger correlation energy of the 1Σg+state, making the true value lower in energy that the triplet state, might be expected by analogy to the hydrogen molecule, in which the HartreeFock method to gives dissociation curves leading to excited states of the constituent atoms.28 This artifact of the singleconfiguration wave function raises the energy of the 1Σg+ state, while a similar effect does not occur for the 3Σu+ state (for which the Hartree-Fock method better describes dissociation to ground-state atoms). The possible existence of 3Σu+ in the experimental spectra is problematic since none of the triplet states of the M center have been conclusively identified experimentally and transitions to them from the 1Σg+ state are forbidden by symmetry. Note that, since the symmetry of the defect in the 〈110〉 orientation is D2h, the directions perpendicular to the axis of the two vacancies, here denoted as x and y, are no longer symmetrically equivalent. Therefore states with the unpaired π electron predominately in the x direction, 1Πux and 3Πux, may have differing energies from analogous states pointing in the y direction, 1Πuy and 3Πuy. Examination of the electron distributions for these four Π states shows that they are all almost entirely delocalized into the surrounding lattice, appearing in isodensity plots as only slight increases in the charges on neighboring ions. The two basis sets are in agreement on several key features of the excited electronic states: (1) the 1Σu+ and all four of the Π states are very similar in energy, (2) the splitting of the x and y components of the Π states is computed to be only a few hundredths of an eV, and (3) the differences between the excited electronic states and the ground 1Σg+ state are all much less than for the 2S f 2P transition in the F center. These results may all be usefully compared with the experimental M-center spectra. Alkali halide crystals typically have a series of overlapping bands associated with M centers, designated M1, M2, etc., often as high as M4.1-6 M1 has been identified with the 1Σg+ f 1Σu+ transition and M2 and M2′ with transitions to the Π states, 1Σg+ f 1Πux and 1Σg+ f 1Πuy. The maximum in the M1 band in LiF has been reported1,2 as 2.79 eV for absorption, which may be compared with our computed values of 2.41 and 1.47 eV. The Stokes shift is invariably less for M centers than for F centers, as would be expected since the excited M center electron is distributed between two vacancies and therefore should perturb the lattice to a lesser extent. The experimental energy for the emission, the 1Σg+ f 1Σ + transition, has been reported as 1.85 eV.2 u Apparently no experimental data are available concerning the Π states of the M center in LiF. However, KCl has been studied
Embedded Cluster Study of F and M Centers in LiF in particular detail and its electronic transitions provide a useful comparison with the present results. The 2S f 2P transition for the F center occurs at 2.31 eV, while the 1Σg+ f 1Σu+ transition for the M center in KCl occurs at 1.55 eV, 1Σg+ f 1Πux at 2.23 and 1Σg+ f 1Πuy transition at 2.30 eV.2 Thus two key and somewhat surprising results of our calculations on LiF also appear in the KCl spectra as well: the excited 1Σu+ and Π states have roughly comparable energies relative to the ground 1Σg+ state, even though the electron distributions of these excited states are much different, and also the splitting of the x and y components of the 1Πu state, only 0.09 and 0.02 in basis sets I and II, respectively, for LiF, are comparable to that observed experimentally in KCl, 0.07 eV. Comparison of Electron Distributions among Species. To show the extent to which the electron distributions of the differing species are delocalized out into the surrounding crystal lattice, we show in Table 4 the electron densities localized in the vacancy for the F+ center, two electronic states of the F center, the F- center, and the three localized states of the M center. These were computed by numerically integrating the total electron densities from the center of the vacancy or pair of vacancies out to a distance of two bohrs, the approximate position of the first minimum in the density. The F+ center and the excited F center have very little charge remaining in the defect. However, about 0.7e of the ground state of the F center is localized in the defect. The F- enter exhibits about the same electronic population in the defect as the F center, indicating that the additional electron is quite delocalized. The M center electrons are localized to about the same extent as in the F center, although in the highest electronic state of the M center, this population decreases to about 0.5e. RelatiVe Energies of Different Species. In the last three rows of Table 2 we show the analogous energies to remove either one electron or an electron pair from the M center oriented along the 〈110〉 axis, and also the energy to remove two electrons from an M center oriented along 〈100〉 . Both basis sets agree that roughly 1.5 eV less energy is required to remove an electron from the 〈110〉 M center than from an F center. However, the energy to remove the second electron from the 〈110〉 M center is much greater, 16.0 and 13.6 eV in basis sets I and II, respectively. This is a reflection of the covalent nature of this M-center electron distribution, since the highest energy electron of the M+ center is attracted by two anionic sites simultaneously. However the energy to remove two electrons from the 〈100〉 M center, M (1Σg+) 〈100〉 f M 2+ (1Σg+) 〈100〉 + 2e- is much less, 12.3 and 10.3 eV in basis sets I and II, respectively. These are both almost exactly twice the F f F+ + e- energy, as would be expected if the electron distribution for this orientation were characteristic of two noninteracting F centers rather than a single M center. (A precise comparison cannot be made since the F and M center computations employed differing cluster models for the surrounding crystal.) Thus these relative energies are in accord with our computed electron density distributions and also with the experimental observation that the spectroscopically observed M centers are oriented in the 〈110〉 direction. Conclusions The present computations of the electronic structure of the M center in LiF, with comparison to analogous results for F centers, substantiate the interpretation of experimental color center spectra that the M center may best be described as a pair of F centers oriented along a 〈110〉 axis but with an appreciable concentration of charge between the two vacancies. Thus the electron distribution in the ground and first excited states are very analogous to those in the H2 molecule (with no nuclei), supporting the “molecular” description of this F-center ag-
J. Phys. Chem. B, Vol. 105, No. 36, 2001 8543 glomeration species. To our knowledge these are the first ab initio quantum calculations to indicate the electron distributions or excitation energies of one of the molecular analogue species in ionic crystals. The electronic structures to be expected from still larger agglomeration species, such as the F3 or F4 (R and N) centers, are unclear since straightforward application of the molecular orbital picture, approximating these by H3 and H4 molecules, would predict unstable species. Our present approach, applying the Hartree-Fock approximation to a vacancy in an ionic cluster embedded in an infinite array of point charges as described by Ewald sums, is clearly limited by the lack of electron correlation and lattice relaxation effects. Nonetheless, by comparison of our computed results with the experimental spectra of M centers in LiF and KCl it appears that this computational approach gives a useful level accuracy, which may be applied to modeling the electronic energies of larger defect species. Acknowledgment. We are indebted to Dr. John Newsam for his encouragement and comments on this work. The MSI (now Accelrys Inc.) Catalysis and Sorption Project contributed to the development of the computer program used in this work. The Project is supported by a consortium of industrial, governmental, and academic institutions. We thank the membership for their input, advice, and suggestions. References and Notes (1) Schulman, J. H.; Compton, W. D. Color Centers in Solids; Pergamon: New York, 1962. (2) Fowler, W. B. Physics of Color Centers; Academic: New York, 1968. (3) Stoneham, A. M. Theory of Defects in Solids; Clarendon Press: Oxford, 1975. (4) Farge, Y.; Fontana, M. P. Electronic and Vibrational Properties of Point Defects in Ionic Crystals; North-Holland: Amsterdam, 1979. (5) Hayes, W.; Stoneham, A. M. Defects and Defect Processes in Nonmetallic Solids; Wiley: New York, 1985. (6) Perez, A.; Davenas, J.; Dupuy, C. H. S. Nucl. Instrum. Methods 1976, 132, 219. (7) Green, T. A.; Loubriel, G. M.; Richards, P. M.; Tolk, N. H.; Haglund, R. F., Jr. Phys. ReV. B 1987, 35, 781. (8) Itoh, N. AdV. Phys. 1982, 31, 491. (9) Tanimura, K.; Itoh, N. Nucl. Instrum.Methods B 1988, 33, 815. (10) Williams, R. T. Opt. Eng. 1989, 28, 1024. (11) Ha¨rtel, H.; Lu¨ty, F. Z. Phys. 1964, 177, 369. (12) Farge, Y.; Lambert, M.; Smoluchowski, R. Solid State Commun. 1966, 4, 333. (13) Bouchaala, N.; Kotomin, E. A.; Kuzovkov, V. N.; Reichling, M. Solid State Commun. 1998, 108, 629. (14) Ewig, C. S.; Tellinghuisen, J.; Mendenhall, M. H. Chem. Phys. Lett. 1992, 188, 501. For earlier theoretical work, see references therein. (15) Madsen, G. K. H.; Gatti, C.; Iversen, B. B.; Damjanovic, L.; Stucky, G. D.; Srdanov, V. I. Phys. ReV. B 1999, 59, 12359. (16) Ewald, P. P. Ann. Phys. 1921, 64, 253. (17) Deem, M. W.; Newsam, J. M.; Sinha, S. K. J. Phys. Chem. 1990, 94, 8356. (18) Wager, J. F.; Van Vechten, J. A. Phys. ReV. B 1985, 32, 5251. (19) Shelimov, K. B.; Safonov, A. A.; Bagatur’yants, A. A. Chem. Phys. Lett. 1993, 201, 84. (20) Evarestov, R. A. Opt. Spectrosk. 1964, 16, 198. (21) Meyer, A.; Wood, R. F. Phys. ReV. 1964, 133, A1436, and references therein. (22) Knox, R. S.; Teegarden, K. J. In ref 2, Appendix A. (23) TURBOMOLE User Guide; Molecular Simulations Inc.: San Diego, 2000. (24) Catalysis User Guide; Molecular Simulations Inc.: San Diego, 2000. (25) Rabin, H.; Reich, M. Phys. ReV. 1964, 135, A101. (26) Georgiev, M. F‘ Centers in Alkali Halides; Springer-Verlag: Berlin, 1988. (27) Van Wazer, J. R.; Absar, I. Electron Densities in Molecules and Molecular Orbitals; Academic Press: New York, 1975. (28) Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Introduction to AdVanced Electronic Structure Theory; McGraw-Hill: New York, 1989.
