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).

- Description of the problem
- Method of calculation
- Convergence of results when using impurity models of increasing size
- Cluster model used in production calculations
- Local geometry of Cu^{+} in the alkali halides
- Resonant vibrations of Cu:NaF
- Acknowledgements
- References

- determine the detailed geometry of each center, not well established experimentally in most cases and for which quite significant differences exist among theoretical calculations,
- determine the stability of each center,
- explain the off-center position exhibited by Cu^{+} in some hosts,
- obtain the resonant vibrational modes, associated to the impurity and its near neighbors,
- characterize the UV (3d-4s) electronic spectra,
- analyze the thermal and pressure effects on the properties of the center.

The *ai*PI 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 *ai*PI 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 *ai*PI wave functions are not affected by the correlation
correction [4].
The *ai*PI 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 *ai*PI 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.

We have explicitely checked this hypothesis in Cu:NaF and Cu:NaCl by doing
*ai*PI 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
*ai*PI calculations on the pure crystals are used to get the
wavefunctions of the lattice ions.

Fig. 1

The figure above shows the *ai*PI 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:Na*X* 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].

system propertyC1-1C2-1C3-1C4-1Cu: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

- to determine the ionic displacements around the Cu^{+} ion in these 9 crystals;
- to compute the stabilization energies involved in the impurity substitution reaction.

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.

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.

SystemC4-1C4-2C12-4Experiment 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

SystemC4-1 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 Zuoet al.. **b** rescaling of ionic radii by Bucher.

Our

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 *ai*PI 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 *ai*PI 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
*ai*PI 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 *ai*PI 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.

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 *ai*PI values can
be compared with the results from atomistic HADES simulations, also
presented in Table IV. The HADES and *ai*PI
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 *ai*PI prognostic. The cause for
this big difference remains yet unknown.

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.

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 *ai*PI calculations to
reproduce the phenomenon.

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).

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

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