Stability, local geometry and resonant vibrations of Cu^{+} impurity in alkali halides.

Victor Luaña, Miguel A. Blanco, M. Flórez, and L. Pueyo
Departamento de Quimica Fisica y Analitica
Universidad de Oviedo, 33006-Oviedo, Spain
email: victor@carbono.quimica.uniovi.es

(Presented to the First Electronic Computational Chemistry Conference, Celebrated in Internet on November 1994. Most of the papers are available on CD-ROM The Conference has been reviewed in the Chemical Abstract).
  1. Description of the problem
  2. Method of calculation
  3. Convergence of results when using impurity models of increasing size
  4. Cluster model used in production calculations
  5. Local geometry of Cu^{+} in the alkali halides
  6. Resonant vibrations of Cu:NaF
  7. Acknowledgements
  8. References

1. Description of the problem

Cu^{+} enters in the alkali halides, AX, substituting the alkaline cation and forming Cu_A centers. Considered to be rather simple closed-shell impurity systems they pose, however, several interesting and difficult problems:
  1. determine the detailed geometry of each center, not well established experimentally in most cases and for which quite significant differences exist among theoretical calculations,
  2. determine the stability of each center,
  3. explain the off-center position exhibited by Cu^{+} in some hosts,
  4. obtain the resonant vibrational modes, associated to the impurity and its near neighbors,
  5. characterize the UV (3d-4s) electronic spectra,
  6. analyze the thermal and pressure effects on the properties of the center.
Questions 1 to 4 will be addressed in the present communication.

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2. Method of calculation

We describe in this communication ab initio Perturbed Ion (aiPI) quantum mechanical calculations on the electronic ground state of the Cu:AX centers.

The aiPI method is a first-principles approach to the construction of the electronic structure of weakly overlapping pure and defective solids [1, 2]. Its foundation lies in the Theory of Electronic Separability for weakly overlapping groups and in the Adams-Gilbert formalism (See Francisco et al. [3] for a detailed discussion on this topic). Within aiPI the Hartree-Fock equation of the solid is solved in a localized Fock space by breaking the solid wavefunction into local nearly orthogonal group functions, each describing a single ion or atom. At the end of a self-consistent process we get the total energy of the system and a set of completely localized wave functions for every non-equivalent atom or ion. Once the electronic structure has been obtained, the correlation energy is estimated by means of the unrelaxed Coulomb-Hartree-Fock formalism, uCHF, where unrelaxed means here that the aiPI wave functions are not affected by the correlation correction [4]. The aiPI method is particularly well suited to ionic materials, and it has produced excellent results for the equilibrium properties and 0 K thermodynamics of halides, oxides, sulfides and others [5].

The basic idea of aiPI calculations on defects is rather simple. First of all, the defective crystal is divided into two regions: cluster (C) and lattice (L). The cluster region is assumed to contain all ions significantly affected by the defect. Thus, local wavefunctions for all ions in C are self-consistently obtained, while perfect crystal wavefunctions are kept rigid for those ions in L. Furthermore, we have found convenient to divide C into two other regions: C1, in which the geometry is allowed to relax, and C2, where the pure host geometry is held fixed. In this way, the C2 region attenuates the sudden boundary effects on the cluster wavefunctions and supplies an interface to accommodate the cluster electronic density to the fixed lattice density.

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3. Convergence of results when using impurity models of increasing size

Quantum mechanical calculations of impurity centers are usually restricted to very small clusters, most times limited to the impurity plus the first shell of neighbors. It is assumed that embedding this cluster within a frozen quantum representation of the rest of the crystal would suffice to give an accurate geometry for the impurity center.

We have explicitely checked this hypothesis in Cu:NaF and Cu:NaCl by doing aiPI calculations in a collection of clusters of increasing size [6]. We will denote as Cn-m a calculation in which the cluster is formed of the central ion and its n nearest shells of neighbors (forming the C region), the first m of them (defining the C1 set) being allowed to relax its position along independent breathing (totally symmetric) modes. Calculations on the equivalent cluster models for the pure hosts, Na:NaF and Na:NaCl, have also been carried to check the embedding self-consistency of the method.

Large Slater-type orbital (STO) basis sets are used for all ions. The lattice is fixed to the experimental geometry, and preliminar aiPI calculations on the pure crystals are used to get the wavefunctions of the lattice ions.

Fig. 1

The figure above shows the aiPI potential energy curves for the totally symmetric breathing of the NaF6 octahedron according to the C1-1, C2-1, C3-1, and C4-1 models. It is clear from the figure that the seven-ion cluster produces very different results from those obtained with larger clusters. Particularly, the C1-1 model predicts a rather significant inwards displacement of the fluorides, in contradiction with the presumed stability of the pure crystal. When the C2 region contains one or more shells, as in the C2-1 and larger clusters, the calculation predicts the crystallographic position to be the most stable one, as it should be.

Table I contains our results concerning the local relaxation around Cu^{+}. The relaxation is computed as the difference between the equilibrium distance found for Cu-X and that found for Na-X. In this way, our prediction is corrected for the systematic errors present in the reference Na:NaX calculation. It can be observed that the equilibrium distances in the C1-1 model largely differ from the results in the other models. The C3-1 and C4-1 models are, on the other hand, essentially equivalent.

The best calculation in Table I predicts an inwards relaxation: -0.089 Å for Cu:NaF and -0.085 Å for Cu:NaCl. This is a quite significant effect, that agrees with the relaxation of -0.10(+/-)0.02 Å measured on Cu:NaCl in x-ray absorption fine structure (EXAFS) experiments [7].


Table I Cu-X and Na-X nearest neighbors equilibrium distances (in Å) and totally symmetric vibration frequencies (*omega*, in wavenumbers). The error quoted for Rth(Na,X) refers to the experimental distances: 2.317 Å (NaF) and 2.820 Å (NaCl).
system   property      C1-1     C2-1     C3-1     C4-1
Cu:NaF   Rth(Cu,F)    2.138    2.184    2.203    2.196
         relax.      -0.112   -0.118   -0.089   -0.089
         *omega*        272      248      289      288
Na:NaF   Rth(Na,F)    2.250    2.302    2.291    2.285
         error       -0.067   -0.015   -0.026   -0.032
         *omega*        366      321      350      355
Cu:NaCl  Rth(Cu,Cl)   2.619    2.673    2.704    2.697
         relax.      -0.106   -0.114   -0.085   -0.085
         *omega*        190      170      193      195
Na:NaCl  Rth(Na,Cl)   2.725    2.787    2.789    2.782
         error       -0.095   -0.033   -0.031   -0.038
         *omega*        220      193      214      216

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4. Cluster model used in production calculations

To avoid the above undesirable effects related to the cluster-lattice interface, the impurity centers will be simulated in the rest of the work by a large cluster of 179 ions, CuX_92A_86^{5-}, formed of the central Cu^{+} plus 12 shells of neighbors embedded in the crystal. The first four shells will be allowed to relax along totally symmetric breathing modes. Accordingly, the calculations that we are going to discuss from now on correspond to the C12-4 cluster model. Figures 2 and 3 depict the C and C1 regions of the cluster, respectively.

Fig. 2 Fig. 3

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5. Local geometry of Cu^{+} in the alkali halides

In this section we report the results of a theoretical investigation on the electronic structure, equilibrium geometry, and stability of 9 Cu:AX impurity systems (A: Li, Na, K; X: F, Cl, Br) (Flórez et al.). The main objectives of this study are:

The calculations have been performed in two steps. In the first one, we have relaxed the geometry of the 4 first shells of neighbors around Cu^{+} by allowing independent breathing motions of each shell. The same treatment was applied to the equivalent cluster model of the pure hosts, in order to any systematic error. Then, we have verified whether the impurity ion is stable at the on-center position, and thus the impurity neighborhood maintains the nominal octahedral symmetry.

Consistency of the model

As discussed in section 3, it is important to check the embedding scheme for the so called self-embedding consistency, according to which the cluster-in-the-lattice calculations on the pure crystal should reproduce the quantum description attained in the calculation for the infinite crystal. It can be observed in Table II that both the 33-ion (C4-m) cluster and 179-ion (C12-m) cluster models give full self-embedding consistent descriptions of the 9 alkali halides. This means that the pure crystal calculations give negligible ionic relaxations (geometric self-embedding consistency) as well as negligible stabilization energies (energetic self-embedding consistency). We confirm with these 9 cases the significant role of the C1+C2 partition of the cluster in fulfilling this criterion of model consistency.

From Table II it is also interesting to note how the C4-1 calculations predict a slight inwards relaxation of the nearest neighbors. When two shells are allowed to relax, as in the C4-2 case, the first shell uniformly moves outwards, with respect to the C4-1 picture, by amounts ranging from 0.05 to 0.1 Å. The 179-ion cluster calculation, involving relaxation of four shells, tends to recover the C4-1 geometries. In addition, the C12-4 geometries are in excellent agreement with the observed data. It is satisfactory to see that the more complex model produces the better agreement with the experimental values.


Table II Optimal nearest-neighbors (nn) distances, Rth(A,X) (in Å), for the pure crystals, according to the aiPI calculations on 33-ion and 179-ion clusters.
 System    C4-1      C4-2      C12-4     Experiment
 LiF       1.991     2.076     2.021       2.014
 LiCl      2.533     2.580     2.555       2.570
 LiBr      2.724     2.772     2.753       2.751
 NaF       2.286     2.386     2.325       2.317
 NaCl      2.782     2.867     2.811       2.820
 NaBr      2.963     3.048     2.992       2.989
 KF        2.593     2.664     2.654       2.674
 KCl       3.071     3.158     3.135       3.147
 KBr       3.242     3.324     3.303       3.298

Local geometry of the Cu:AX systems

Table III contains our results for the local relaxation of the first shell around Cu^{+}. The relaxation is computed as the difference between the equilibrium distance found for Cu:AX, Rth(Cu,X), and that found for A:AX, Rth(A,X). The relaxations experienced by the second, third, and fourth shells around Cu^{+} are small and do not appreciably differ from those found on the A^{+}-centered clusters. Thus the best we can say is that no significant relaxation of those shells is predicted.


Table III Geometries for the Cu:AX and A:AX clusters. The relaxations are computed as the differences Rth(Cu,X)-Rth(A,X). Distances are given in Å.
System  C4-1    C4-2     C12-4    C12-4   C12-4   **a**  **b**
       relax.  relax.  Rth(Cu,X) Rth(A,X) relax.  relax. relax.
LiF     0.005  -0.026    2.021    2.021    0.000   0.177   0.14
LiCl   -0.015  -0.033    2.522    2.555   -0.033   0.174   0.10
LiBr   -0.004  -0.015    2.734    2.753   -0.020               
NaF    -0.089  -0.082    2.201    2.325   -0.124   0.040   0.00
NaCl   -0.084  -0.090    2.706    2.811   -0.105   0.011  -0.03
NaBr   -0.070  -0.072    2.906    2.992   -0.086               
KF     -0.267  -0.273    2.276    2.654   -0.378  -0.132  -0.19
KCl    -0.286  -0.261    2.769    3.135   -0.365  -0.120  -0.20
KBr    -0.250  -0.212    3.004    3.303   -0.299               
**a** ICECAP calculations by Zuo et al..
**b** rescaling of ionic radii by Bucher.

Our C12-4 results show qualitatively different nn relaxations depending on the cation being substituted. Negligible or very small inwards relaxations are obtained for Cu:LiX, inwards relaxations close to -0.1 Å are obtained for Cu:NaX, and values around -0.3 Å are encountered for Cu:KX. The effect of the anion is small, increasing slightly the inwards relaxations as the size of the anion decreases. Similar results are obtained in the C4-1 and C4-2 calculations.

We want to stress that our prediction for Cu:NaCl, -0.10 Å, agrees with the EXAFS measurement by Emura et al., -0.10(+/-)0.02 Å, that constitutes the only direct experimental evidence of the local geometry of a Cu_A center. Given the uniform behavior of the aiPI method shown above within these 9 crystals, this isolated experiment suggests that our method may give experimentally consistent descriptions of the lattice relaxations.

It is interesting to compare the aiPI results with other cluster-in-the-lattice theoretical calculations. Cu:NaF and Cu:NaCl are the most studied impurity centers. Winter et al., using an embedding method based on finite sets of point charges plus Effective Core Potentials representing the cationic shells, predict a negligible relaxation of the Cu^{+} neighborhood in Cu:NaF and Cu:NaCl. The same tendency is found by Meng et al. using the ICECAP technique in which a quantum cluster is embedded into a classical shell-model latice. In contrast, Barandiarán and Seijo, who proposed the use of free-ion ab initio Model Potentials to account for the quantum and classical lattice effects, found a -0.1 Å inwards relaxation in the case of Cu:NaCl, in close agreement with the Emura et al. experiment and the aiPI results. The X-*alpha* calculations of Till et al, on the other hand, appear to be extremely sensitive to the arbitrary parameters that define the Muffin-tin environment used to modelize the lattice, up to the point that the predicted Cu-F distance in Cu:NaF varies between 2.12 and 2.44 Å and the a1g breathing mode frequency varies between 428 and 1750 cm^{-1}.

The recent ICECAP calculations by Zuo et al. and the empirical arguments of Bucher based on a simple rescaling of ionic radii, both tend to predict large outwards relaxations (0.1 to 0.2 Å) for the Cu:LiX, negligible relaxations in the Cu:NaX, and large inwards relaxations (-0.1 to -0.2 Å) in the Cu:KX. Contrarily, Harrison and Lin suggest a small inwards relaxation for the Cu_Li center in Cu:LiCl and this was indeed found in the Local-Density-approximation calculation published by Jackson et al..

The differences between the several calculations evidence that the local geometry of the impurity is far more sensitive to the embedding scheme than to the quantum technique used to solve the cluster.

Of particular interest is to remark that the ligand relaxations upon impurity substitution, shown in Table III, largely contrasts with the picture that could be deduced by considering Pauling's octahedral ionic radii: r(Cu^{+})=0.96 Å, r(Li^{+})=0.60 Å, r(Na^{+})=0.95 Å, and r(K^{+})=1.33 Å. The aiPI results, on the other hand, follow the tendency that should be expected after Shannon's octahedral radii: r(Cu^{+})=0.91 Å, r(Li^{+})=0.90 Å, r(Na^{+})=1.16 Å, and r(K^{+})=1.52 Å.

Formation energy of the Cu_A centers

The formation of the Cu_A centers should be discussed in terms of the free energy difference for the exchange reaction:

A:AX (s) + Cu{+1} (g) ---> Cu:AX (s) + A{+1} (g)

The formation energy of the Cu_A centers is quite dependent on the size of the C1 and C2 regions. For instance, Cu:LiF is unstable by 1.10 eV in the C0-0 model, by 0.21 eV in C12-0, and it turns to be stable by -0.25 eV when the geometry is relaxed in the C12-4 model.

According to the C12-4 model, substituting Cu^{+} for the cation is energetically favored in the nine alkali halides. As Table IV shows, the formation energies can be broadly grouped by the cation, the Cu_Li centers being the less stable and the Cu_K centers being the most stable. The aiPI values can be compared with the results from atomistic HADES simulations, also presented in Table IV. The HADES and aiPI results are rather coincident for the three fluorides, but for the three chlorides the atomistic simulation predicts the Cu_A centers to be 1-2 eV more stable than the aiPI prognostic. The cause for this big difference remains yet unknown.


Table IV Formation energies (in eV) of the Cu_A centers.
         LiF    LiCl   LiBr   NaF    NaCl   NaBr
aiPI   -0.252 -0.688 -0.392 -1.022 -0.867 -0.684
HADES* -0.200 -2.704   ---  -1.598 -3.200   --- 

         KF     KCl    KBr
aiPI   -1.771 -1.646 -1.377
HADES* -1.978 -3.123   ---
* We have used Catlow's empirical set II potentials and shell parameters to describe the host ions, and ICECAP derived Meng's potentials for the Cu-X interactions.

Off-center displacements

As discussed in McClure and Weaver, several spectroscopic studies of Cu^{+} in alkali halides give information about the position of Cu^{+} in the lattice as well as on the potential function that describes its motions. In some alkali halides, Cu^{+} lies off the ion site in an appreciable extent. This has been shown to be the case for the ellectronic ground state of three of the systems analyzed here: Cu^{+} in NaBr has a shallow off-center position and Cu^{+} in KCl and KBr have deep off-center positions. For Cu:NaBr, Emura and Ishiguro have obtained the corresponding off-center parameters by analyzing the pressure effects on the absorption bands. The off-center stabilization energy is estimated to be 25(+/-)3 meV, and the off-center displacement 0.55 Å in the [110] or [111] directions. The small amount of this stabilization energy makes the prediction of the off-center geometry an extremely difficult task for any {\it ab initio} calculation.

To study these matters we have done calculations with the 179-ion cluster in the following way. Fixing the cluster ions at the positions optimized for the C12-4 model, we have moved the Cu^{+} ion along the [100] direction and computed the effective energy of the cluster for every new position of the impurity. In the same way, we hace obtained the [110] and [111] energy curves. As an example of the results, Fig. 4 represents the potential energy curves for the movement of Cu^{+} along the three axes [100], [110], and [111] in Cu:NaF.

The most relevant result of these calculations is that the Cu^{+} remains on-center in the 9 systems analyzed here, that is, the (0,0,0) site is the point of minimum energy along the [100], [110] and [111] axes. The three curves are rather flat and highly similar near the minimum, progressively separating each other for displacements of 0.3 Å or larger. The movement of the Cu^{+} ion is easiest along the [111] axis and most difficult along the [100] axis. In consequence, our model does not explain the off-center position observed for Cu^{+} in the three above mentioned compounds. This failure might probably be related to the absence of non-spherical deformations of the ionic wave functions in the present aiPI calculations.

Nagasaka has proposed that the stability of the off-center position of Cu^{+} ion in some alkali halides is due to the mixing between the closed-shell electronic ground state and a X^{-} --> Cu^{+} charge-transfer excitation state. This interpretation suggest that the polarization of the anions surrounding the Cu^{+} ion plays a key role in the stabilization of the off-center position, thus posing a particular difficulty for the present aiPI calculations to reproduce the phenomenon.

Fig. 4

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6. Resonant vibrations of Cu:NaF

To fully characterize, under the harmonic approach, the vibrational modes of Cu^{+} and its four first shells of neighbors (that is, the 33 innermost ions of the 179-ion model) we have to determine 138 independent force constants, as it is shown in Table V. The procedures used to determine the force constants using numerical derivative formulas applied to the aiPI energies are described in Luaña et al. That article also contains a complete report of the results found, from which only a brief excerpt will be presented in this communication.


Table V Oh local vibrational modes of Cu^{+} and its nearest four shells of neighbors.
shell         a1g  a2g   eg  t1g  t2g  a1u  a2u   eu  t1u  t2u

0       Cu     0    0    0    0    0    0    0    0    1    0
1     6 F      1    0    1    1    1    0    0    0    2    1
2    12 Na     1    1    2    2    2    0    1    1    3    2
3     8 F      1    0    1    1    2    0    1    1    2    1
4     6 Na     1    0    1    1    1    0    0    0    2    1

      total    4    1    5    5    6    0    2    2   10    5
force cons.   10    1   15   15   21    0    3    3   55   15   

It is interesting to analyze the convergence of the frequencies as more and more shells are coupled to produce the vibrational modes. We have found strong couplings among vibrational modes of adjacent shells. As an example, the lowest t1u vibration mode drops from 155 cm^{-1}, when the movement of Cu^{+} is considered alone, to 108 cm^{-1} when the concerted movement of the 33 innermost ions is calculated. Similar results are found for other modes (See figure 5).

Fig. 5. Convergence of the vibration frequencies of some normal modes when the number of coupled shells is sucessively augmented.

On the other hand, the C12-4 model of both the impurity center and the pure host crystal have been solved, and from the comparison between the normal modes of both we have determined that the only modes truly characteristic of the impurity center are the lowest a1g, eg and t1u vibrations (Compare Fig. 6. y Fig. 7.). Vibration frequencies characteristic of the doped system are found at 108 cm^{-1} (1t1u), 206 cm^{-1} (1a1g), and 173 cm^{-1} (1eg), being the corresponding modes mostly related to the motion of the CuF6 octahedron.

These frequencies fairly agree with the experimentally observed t1u frequency of 93 cm^{-1} (See McClure and Weaver), the value of 160(+/-)10 cm^{-1} empirically estimated for 1a1g upon the shape of the 1A1g<-T1g(3Eg) electronic band (McClure and Weaver), and the rather especulative 1eg frequency of 150 cm^{-1} (Payne et al.). It is particularly rewarding to observe that the coupled motion of several shells diminishes the frequencies of the lowest modes of each irreducible representation giving rise to a better agreement with the experimental values. This effect, on the other hand, turns nearly unuseful the frequencies calculated with small clusters like the C1-1 model of 7 ions.

Fig. 6. Coupled vibrational frequencies of C1 for Cu:NaF. Error bars are determined from the numerical precision of the force constants.

Fig. 7. Coupled vibrational frequencies of C1 for Na:NaF

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Acknowledgements

Finantial support was provided by the Spanish DGICyT of the Ministerio de Educación y Ciencia under Grant No. PB93/0327.


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