UHF MC INDO Calculations of the Interstitial Hydrogen in Alkali Halides

M. MATOS: UHF MC INDO Calculations of Hydrogen in Alkali Halides 399

phys. stat. sol. (b) 141, 399 (1987)

Subject classification: 61.70; 71.55; 59.11

Pontificia Uniuersidade Catolica do Rio de Janeirol)

UHF MC INDO Calculations of the Interstitial Hydrogen in Alkali Halide@)

BY MARIA MATOS

The applicability of a general scheme of parametrization of semiempirical methods for ionic crystalline systems is investigated in the study of pure LiF, NaF, LiCl, and NaCl crystals using different molecular clusters, and in the investigation of optical and magnetic hyperfine properties of the interstitial hydrogen (U,-center) in these crystals. MC-INDO (for the pure crystal) and UHF MC-INDO (for the defect) calculations are performed to investigate the band crystalline structure and to estimate the U2-band absorption peak energy and the electronic spin density a t the nucleus of the interstitial atom. It is shown that the procedure of scaling the defect Slater-type orbitals (STO) is not appropriate for the U,-center and that this fact is related to the covalent nature of the binding of the interstitial atom to the crystal.

Un scheme g6n6ral de parametrization des methodes semi-empiriques pour les cristaux ioniques est Btudie parmis des caIculs INDO dans LiF, NaF, LiCl et NaCl par rapport B des changes dans l’agglom6rat cristallin. Le centre U, est aussi BtudiB en utilisant la version Hartree-Fock unrestricte et l’energie de la bande U, aussitat que le densite Blectronique de spin dans l’atome d’hydrogene sont evalu6s. On montre que le scheme utilisB avec suochs dans le cristal ionique n’est pas indiquB pour le centre V, et que ce resultat ce doit au charactere covalent de la liason de l’atome interstitiel avec le cristal.

1. Introduction

I n three recent papers [l to 31 it has been proposed a general procedure to parameterize semi-empirical H F LCAO methods for ionic crystalline systems. Molecular cluster intermediate neglect of differential overlap (MC INDO) calculations have been performed to investigate the electronic structure of pure alkali halides [l, 21 and optical and magnetic hyperfine properties of substitutional defects (F, U, and U, centers) in these crystals [3]. For F and U, centers the unrestricted Hartree-Fock (UHF) scheme has been used to take into account their open shell structure. The procedure of parametrization was based on the original ideas of Pople and Beveridge [4] in their study of molecules. By adjusting atomic bonding parameters and Slater-type orbital (STO) exponents [ of Li, Na, F, and C1 to fi t experimental data on electronic and band structure of pure LiF and NaCl crystals, calculations have been successfully performed, without any additional fitting, in other two alkali halides, LiCl and NaF. In these two crystals the basis set exponents have been obtained from LiF and NaCl through a scaling transformation according to which an atomic STO is, in units of lattice constants, the same in all crystaIs with a given geometric structure.

1) Rua Marques de Sao Vicente, 225, Gavea, 22451 Rio de Janeiro, Brazil. 2) Work partially supported by Brazilian agencies FINEP and CNPq.

400 M. MATOS

The generality of the parametrization scheme has also been successfully tested in the study of optical and magnetic hyperfine properties of the defective crystals. By using the same crystal parameters as in the pure systems, the P- and U-center B and f parameters have been experimentally fitted to NaCl, being the 6’s obtained through scaling transformation in the other three crystals. The estimated P- and U-band peak transition energies for LiF, LiC1, and NaP and the Us-band peak transition energy, as well as the electronic spin density a t the nucleus of the hydrogen atom, in KaCl were in excellent agreement t o experiment. In all defect calculations the overall characteristics of the obtained crystalline band structures were preserved except for the presence of defect isolated levels. In all calculations the bonding parameters have been obtained through the formula B ~ l l = - i b x [l].

In the present paper I shall investigate the interstitial hydrogen (U, center) in alkali halides, a one-spin unpaired system, taking special consideration to the U,-band peak transition energy, AE, and the electronic spin density a t the hydrogen nucleus, a hyperfine magnetic property of the system. MC UHF calculations will be performed by using the INDOjl version of the method [5] and the basis set will be extended to include a 2p-hydrogen STO. To estimate AE limited configuration interactions (GI) will be included in calculations. This procedure takes into account electron-hole correlations, a non-negligible effect when localized MO’s are involved ; this is generally the case in self-consistent cluster calculations [l to 31. ps will be estimated through (5) of 131, which is reproduced for convenience,

@b = [mals(o)12 9 (1) qalg is the unpaired U,-center (located a t the origin) ground state MO.

Equation (1) is valid as long as spin polarization effects are neglected. As has been used with success in the study of the U, center [37, improvements in the calculation of this quantity will not be considered here.

I discuss in Section 2 some physical properties of the U, center concerned to the present work and describe in Section 3 the molecular cluster and boundary conditions used in calculations. Investigations on the defective systems will be preceded by the study of pure alkali halides, also reported in Section 3, which intend to analyse the appropriateness of the MC in describing the crystalline band structure. The parametrizations for the defect are described in Section 4 and results of several calculations are discussed in Section 5. In Section 6 I present the conclusions of this paper.

2. The UB Center

The U, center3), a neutral hydrogen atom trapped at interstitial sites, has been observed in several alkali halides. Since the work of Delbecq et al. [GI, who observed this center in KC1, much experimental [7 to 131 and theoretical [lo, 11, 14 to 171 work has been done which aimed a t the understanding of the interactions of the defect with the crystal. Optical [6, 7, 9, 121 and magnetic hyperfine [6 to 8, 10, 11, 131 properties have been investigated in several crystals. Fisher [9] measured the optical U, absorption band in the series of alkali halides formed froin the alkaline Na, K, and R b and halides C1, Br, and I ions. Spaeth and Sturm [ l l ] obtained paramagnetic resonance data in K F and in the series of Na, K, and R b chlorides and bromides. Hoentzsch and Spaeth [13] investigated hyperfine magnetic properties in NaP, KF, and RbF. In the last two papers the contact hyperfine parameter of the interstitial proton has been measured for all crystals under study. I n Pig. l AE has been plotted as a function of lattice

3, The author in indebted to the referee for calling her attention to theoretical and experimcrital work in this center.

UHF MC INDO Calculations of the Interstitial Hydrogen in Alkali Halides 40 1

1 Fig. 1. Experimental values for the U,-band absorption peak in several alkali halides plotted against the lattice constant d. Both (alkaline and halide) families of straight lines are adjusted to the experimental points

1 constant d and two distinct families of straight lines have been sketched, each curve corresponding to a given alkali or halide ion. The unpaired spin density a t the hydrogen nucleus, which come out from [ l l ] and [13], are shown in Table 1.

Theoretical studies have been mainly concerned with the interpretation of magnetic hyperfine data. Different one-electron wave functions for the unpaired spin density distribution have been used to estimate the proton isotropic parameter (1). Effects of

3 4 orthogonalization between orbitals of different . atoms (including both, defect and

crystal), which accounts for Yauli repulsion, and covalency in the hydrogen bonding have been discussed by using this simple model [lo, 13, 16, 171. As pointed out by Spaeth and Sturm [ll], covalency between the hydrogen and the crystal neighbors allows for spreading of the unpaired electronic distribution and could explain the observed decrease in the electronic spin density a t the hydrogen nucleus, as compared to the free atom value, in most alkali halides (see Table 1). Hagston [15] used similar arguments to stress the importance of covalent bonding in U, center. These effects have been investigated by Spaeth and Seidel [16] in the interstitial hydrogen center in several alkali halides and by Heder et al. [17] in the substitutional hydrogen center in KCI. Their results suggest the existence of a certain (non-negligible) amount of covalent bonding in these centers. In the present paper a distinction between suh- stitutional and interstitial defects, as regard to their binding nature, will be discussed.

Dynamic effects due to zero point vibrations of the interstitial hydrogen have been considered by Spaeth [14]. By introducing the variations of the hydrogen-ion core overlaps with the interatomic distances this author estimated crystal magnetic hyperfine constants in KC1. Differences up to 30% between static and dynamic calculations have been obtained.

I

d iIO-'nrnl -

Table 1

Experimental values of electron spin densities (in at. units) in the nucleus of the hydrogen atom for the U, center in several alkali halides. Free hydrogen value: 0.318

F C1 Br

Na 0.336 0.303 0.292 K 0.333 0.308 0.292 Rb - 0.310 0.302

402 M. MAWS

To appreciate theoretical results I will proceed in a similar way as was done for the U center [3]; in this case the empirical Ivey law has been used to obtain this quantity in crystals where experimental data were not available. I n the present case simple linear extrapolation of the curves shown in Fig. 1 will be done to approach numerical values for AE in LiCl and NaF. No extrapolation will be performed for lithium halide or alkali fluoride since corresponding curves are not available.

In calculating AE, only electronic spin up transitions will be considered. Although spin down transitions (“charge transfer” processes) can also occur in hydrogen im- purities [3, 151 I have disregarded this possibility here. Studies of U, centers in cesium halides 1121 partially support this procedure since, in that case, electronic transitions consisting of transference from a ground state unpaired - spin MO could be inferred from symmetry arguments.

3. The Molecular Cluster

The defect is represented by a cluster of 32 alkaline and halide ions alternating the vertices of one central and six adjacent cubes, centered on an interstitial site, where the hydrogen atom is located. This cluster is embedded in the electrostatic field of 968 external integer (for bulk sites) and fractional (1/2 for faces, 1/4 for edges, and 1/8 for vertices) positive and negative point charges, distributed among absent crystal ions sites, in such a way that the whole physical system (PS) - MC plus external charges - constitutes a neutral cube of 1000 crystalline sites, with the local symmetry (Td) of the defect. A two dimensional sketch is shown in Fig. 2.

The use of a finite set of external point charges is a convenient boundary condition for ionic crystals [l to 3, 81; its contribution is inserted into the Fock operator as a one-electron term and the Madelung potential can be correctly reproduced in all MC points, provided that the set in conveniently chosen. I n a non-self-consistent calculation i t has been computed, in all cluster sites, the electrostatic potential of a crystalline cube, constructed from the PS through substitution of normal cluster ions by integer plus and minus point charges. The results of this calculation are shown in Table 2, for NaC1. It is observed that the finite set reproduces correctly the Madelung energy inside the cluster.

Preliminary INDO calculations have been performed in the pure crystals - LiF, NaF, LiCl and NaCl - by using the 32-ion PS and the crystalline parametrization of [2], reproduced in Table 3, which was obtained through experimental adjusting in a 27-ion cluster (0, symmetry). The calculated electronic levels schemes are presented in Fig. 3 where are also shown those calculated through the 27-ion cluster. When 0, and T, calculated band structures are compared, no differences are observed between both MC results. This fact indicates that the crystalline parametrization of Table 3

f - f - + - + + - + - + - + -

f - 0-0 -t - + f - .-0--.--0 + -

- f 0- * -0 -0 - +

+ - f - + 0--. -

-

- 1 1

I l l 1 I 1

f - + - + Fig. 2. Two-dimensional representation of the physical + - -

+ - + - + + + - system (PS) plus external point charges

UHF MC INDO Calculations of the Interstitial Hydrogen in Alkali Halides 403

Tab le 2 Exact ( Vex(ri)) and approximate ( V p s ( r i ) ) Madelung potential a t some typical cluster sites. Vps has been computed from the physical system (PS) described in the text

(O,O, 0) 0.000001 0.0 (1/2, 1/23 1/2) 0.459880 0.46 (112, -1/2, l/2) -0.459880 -0.46 (3/2,1/2, 1/2) -0.459870 -0.46

Tab le 3 INDO parameters for LiF, NaF, LiCI, and NaCl crystals [2]. Bonding parameters in eV and STO exponents in at. units

LiF LiCl

i s independent of the molecular cluster used to represent the pure crystal, thus cor- roborating its generality. Further, the PS is suitable to describe the interstitial

L if

75 !=

hydrogen center, as departures NoCl from the pure crystal electronic

structure can now be unambig- 0, Td uously associated to the defect.

The crystal parameters of Table 3 wfil be used in the nest calculations.

Fig. 3. Calculated IXDO crystalline band structure for the four alkali halides by using two (Oh and Td) different MC’s. Oh: 27-ion cluster; Ta: 32-ion cluster

404 M. MATOS

-1. The Defect Parametrization

Five empirical parameters associated to the defect must be determined in the Fock equation : the hydrogen STO exponents tls and &, the corresponding one-center atomic integrals U , and the atomic bonding parameter &. The U integrals will be obtained, as for the other hydrogenic centers [3], through direct integration of the corresponding basis set orbitals, (pi U , /p) , being U , the one-electron atomic operator term, giving

4 1 s + t B s l 2 ; ,U = Is-STO , (2 a)

- t&/2 + &$,/2; ,U = 2p-STO . (2 b)

For the definition of the other three parameters two different schemes will be analyzed -- parametrizations I and I1 - in which direct as well as indirect experimental fittings are used.

4.1 Paramef+ixation Z

a) = 1.0 at. units, tZP = 0.5 at. units, the free H atom values in ail crystals. This choice is based upon observation of Table 1, in connection with (l), according to which i t is suggested that the electronic spatial distribution around the interstitial hydrogen is not too far from that of the free atom.

b) /IH is directly adjusted to obtain the experimental value for AE in NaC1. In this scheme it is not assumed a scaling transformation of the hydrogen STO’s. This transformation is defined through [2 , 31

ta = E x ’ , (3) where t(E’) corresponds to a crystal with lattice constant d(d’). We shall investigate the effects of scaling the hydrogen STO’s by using the next parametrization scheme.

4.2 Parametrizatiors I1

a) tls = 1.0 at. units, &, = 0.5 at. units in RbI, the alkali halide with the greatest lattice constant. For NaCi, LiCI, NaF, and LiF the exponents are obtained from RbI through (3). It is assumed that, if effects are to be observed due to volume contractions of the interstitial region, they are less important in RbI.

b) The same as Ib , by using exponents defined from I I a . The defect empirical parameters p and 6, for schemes I and 11, are presented in

Table 4.

T a b l e 4 INDO parameters of the U, center in the four crystals for two different schemes (defined in the text). Bonding parameters in eV and STO exponents in at. units

I I1 .- ~

NaCl LiCl NaF LIF XaCl LiCl XaF LiF ~ ~-

-BH“ 3.88 3.88 3.88 3.88 3.60 3.60 3.60 3.60 51s 1.0 1 .o 1.0 1 .0 1.30 1.42 1.58 1.82 52p 0.5 0.5 0.6 0.5 0.65 0.71 0.79 0.91


5. Results and Discussions

Calculations have been performed for the U, center in KaCl, LiC1, NaF, and LiF by using parametrizations I and 11, described in Section 4, and AE and es estimated. Results obtained for these quantities are shown in Table 5. It is observed that the general agreement between calculated and experimental quantities is better for parametrization I, where AE (calc.) is 6% (in LiC1) and 11% (in NaF) smaller than AE (exp.). This difference in parametrization I1 amounts to 4% in LiCl but to 29% in NaF. In this case AE (calc.) decreases strikingly for LiF exhibiting an opposite trend, with respect to changes in the halide ion, as that of Pig. 1. The calculated spin densities (es (calc.)) in NaCl and NaF as compared to the experimental values, present differences of 30% (NaC1) and 10% (NaF) in parametrization I and 117% (NaC1) and 250% (NaF) in parametrization 11. For the other two crystals a comparison between es (calc.) and the corresponding free atom value shows differences below 14% in parametrization I, while for parametrization I1 the differences are between 179 % (LiC1) and 388% (LiF). In crystals for which es was measured (Table 1) this difference is below 8%. In parametrization I the direct influence of scaled 1s-hydrogen orbitals is recognizable (see (1)); it clearly reflects contractions of this orbital when the lattice parameter decreases.

These results suggest that the scaling transformation procedure is not indicated for the parametrization of the hydrogen STO's in the study of interstitial defect properties, which are more appropriately described through the free atom orbitals. This conclusion is the reverse of that we have arrived a t in the study of the U center, a substitutional hydrogenic impurity [3]. It raises an apparent ambiguity in the proposed general scheme of parametrization according to which all basis set exponents must be scaled through (3). I n the next sequence I shall discuss this particular aspect, intending to establish a relationship between the scaling procedure and the nature of chemical bonding of the impurity in the crystalline environment.

5.1 Binding character of the defect

An important distinction concerning substitutional and interstitial defects in ionic crystals is related to the bonding with the crystalline surroundings. The long-range Coulomb potential provides a stabilizing well for substitutional defects (F, U, and U, centers) as i t can be inferred from the Madelung energy a t anionic sites, which for the alkali halides runs from -11.8 eV (Lip) to -6.5 eV (RbI). This term, on the other

Tab le 5 Results of UHF MC INDO calculations of the U, center in alkali halides, by using parametrization I and 11, defined in Table 4. Quantities are defined in the text. Energies in eV, other quantities in at. units

Parametrization 1/11

NaCl LiCl NaF LiF ~~

AE (calc.) 5.6315.62 5.6315.76 6.85/5.48 6.32/3.07 A E (exp.) 5.63*) 5.96 7.68 - es(0)calc. 0.209/0.662 0.285/0.887 0.304/1.179 0.275/1.551 es(0)exp. 0.305 - 0.336 -

*) Extrapolated from Fig. 1.

406 M. MATOS

A

i i i G

z g @ a Fig. 4. Electronic energy level scheme for the a-spin 3p-C1 valence band and - hydrogenic levels (ale) obtained through UHF MC INDO calculations for the U, center in NaCl, by using several bonding parameters (in eV)

- - - - - - - - -

7n. __- - - i c - I

- - - - - - - - -- --

Z Z 7 Z Z Z ~

b C d e f 1

75 20 30 4 0

- I 0 i a

-pHo = OeV 05

hand, vanishes a t interstitial sites so that an atom located there must be subject, to be stable, to covalent bondings with the neighboring crystal ions.

To investigate the binding of the defect several calculations have been performed for the U, center in NaCl by using different values for the bonding parameter &. Changes in PH produce changes in resonance integrals H C ~ H = ,6&3HCl appearing in the Fock operator; in general resonance terms are directly related to molecular covalent bondings [4], being responsible for shifts between bonding and antibonding MO’s. In Fig. 4 is shown the ct-spin electronic levels scheme for the crystalline p-valence band region obtained from these calculations. It is observed that covalent bonds are present in the U, center; this can be seen from the existence of two ale levels coming from mixtures of the 1s-hydrogen orbital (which is present in the calculation for PI< = 0) with 3p-C1 valence band MO’s; these levels are progressively split off when 16, increases, being this effect striking for values of this parameter near to those obtained in the experimental fittings I and I1 (see Table 4).

A similar investigation has been performed for the U, center in NaCl, a substitutional hydrogen atom center. Results are shown in Pig. 5, for two different values of PH. It is interesting to compare these results with those of the U, center; in U, center

- ‘ a 14

Z z i m i Fig. 5. Electronic UHF MC INDO levels scheme for the a-spin 3p-C1 valence band region of the U, center in NaCI, by using two different pa’s (in eV). A 27-ion cluster 1

b C has been used in this case


no relevant splittings are observed for sensitive changes in &, denoting the absence of important covalent bondings in this case.

A s a remark, it has been noted, from inspection of the CI matrix elements in the calculations of Fig. 4, that Hop = (Do/H/D,) tends to zero when #IH varies from zero to -4.0 eV (Do and D, are ground and singly excited state determinants and H the electronic Hamiltonian). The Brillouin theorem [19] states that these matrix elements are zero for actual Hartree-Fock MO’s. The fact that Hop increases when the absolute value of PH decreases, is probably related to difficulties in stabilizing the hydrogen atom in the crystal by means of small resonance terms in the corresponding Fock equations, confirming the assertion about the covalent character of the U, center.

From the analysis presented in this section, a relationship between the appropriateness of the scaling transformation of a given A 0 and the character of chemical bonding to which the corresponding atom is subject, becomes evident. It seems that this procedure is not indicated when covalency is present. The success obtained in describing important crystalline features of alkali halide crystals, both perfect and containing substitutional defects [2, 31, is closely related to the ionic characteristics of these systems. This can be understood from the following argument : let us consider a particular atomic orbital in a given alkali halide with lattice constant d. It is noted, from (3), that this orbital contracts in a new alkali halide with lattice constant d‘ < d. This effect prevents the increasing of overlap integrals (with a consequent increase in covalence) thus preserving the ionicity of the new crystal, with direct influence in calculated band gaps, band widths, charge distributions, and other electronic-dependent properties [2] .

In this study I have disregarded the influence of zero-point vibrations of the interstitial nucleus. Since covalence is directly related to overlap, dynamical effects would espectedly influence the results. Nevertheless the present conclusions would not change since covalency would increase in both substitutional and interstitial cases.

A systematic theoretical investigation of the U, center in alkali halides in which this defect has been experimentally observed is necessary to propose parametrization schemes for interstitial defects in ionic crystals. This preliminary study suggests that such an investigation will be useful for other covalent systems as it has been shown that the generality of a given parametrization scheme is strongly related to the chemical character of the bindings present in the crystal.

6. Conclusions

In this paper a proposed general scheme of parametrization of semi-empirical methods, which has been successfully used in the study of ionic crystalline systems, pure and containing substitutional defects, was tested in two new situations :

(i) the study of pure alkali halides by using different molecular clusters; (ii) the investigation of electronic properties of the interstitial hydrogen in these

crystals. It was shown that INDO empirical parameters for pure crystals determined through

direct experimental fitting in a 27-ion cluster were successful in describing the crystalline band structure by using a 32-ion cluster, different from the original “adjusting” MC in size and symmetry. It is concluded that the generality of the proposed parametrization scheme for ionic crystalline systems in confirmed by this test, being the empirical parameters independent of the molecular cluster.

I n the case of the interstitial defect i t was shown that the procedure of scaling transformation of atomic basis set orbitals is not appropriate to the defect STO’s and that this fact is related to the covalent binding of the interstitial atom to the

408 M. MATOS : UHF MC INDO Calculations of Hydrogen in Alkali Halides

crystal. The scaling of atomic orbitals reflects contraction of these orbitals due t o decreasing of the lattice constant, a truly ionic feature, being this indicated only when ionicity is an important characteristic of the system.

Acknowledgement

The author would like to acknowledge Dr. B. Maffeo for suggesting this investigation and for helpful and clarifying discussions.

References

[ l ] M. MATOS, B. MAFFEO, and M. C. ZERNER, J. Phys. C 15, 7351 (1952). [ 2 ] M. MATOS and B. MAFFEO, Solid State Commun. 48, 961 (1983). 131 M. MATOS and B. MAFPEO, phys. stat. sol. (b) 124, 757 (1984). [4] 5. A. POPLE and D. L. BEVERIDGE, Approximate Molecular Orbital Theory, JIcGraw-Hill

[51 J. RIDLEY and &I. C. ZERKER, Theor. Chim. Acta 32, 111 (1973). [6] C. J. DELBECQ, B. SMALLER, and P. H. YUSTER, Phys. Rev. 104,599 (1956). [7] F. KERXHOFF, W. MARTIENSSEN, and W. SANDER, 2. Phys. l i 3 , 184 (1963). [8] J. &I. SPAETH, Z. Phys. 1!)2, 107 (1966). [O] F. FISHER, Z. Phys. 204, 351 (1967).

Publ. Co., 1970.

[lo] G. LEHNERT and J. &I. SPAETH, phys. stat. sol. 31, 703 (1969). [I11 J. 31. SPAETH and RI. STURM, phys. stat. sol. 42, 739 (1970). [12] K. GUKTHER and F. FISHER, Z. Phys. 247, 304 (1971). [13] CH. HOENTZSCH and J. M. SPAETH, Solid State Commun. 29, 577 (1979). [14] J. M. SPAETH, phys. stat. sol. 34, 171 (1969). [15] W. E. HAGSTON, phys. stat. sol. 39, 551 (1970). [la] J. M. SPAETH and H. SEIDEL, phys. stat. sol. (b) 46, 323 (1971). [17] G. HEDER, 5. M. SPAETH, and A. H. HARKER, J. Phys. C 13,4965 (1980). [18] &I. R. HAYNS and L. DISSADO, Theor. Chim. Acta 37, 147 (1975). [19] See for example: F. L. PILAR, Elementary Quantum Chemistry, McGraw-Hill Publ. Co.,

(Received November 4,1986) 1968.

Documents

UHF MC INDO Calculations of the Interstitial Hydrogen in Alkali Halides