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Electronic band structure of ferroelectric HCI
Volume 45. number 1
ELECTRONIC
CHEMICAL PHYSICS LE-ITCRS
BAND STRUCTURE
OF FERROELECTRIC
1 lanuary L977
HCI
A. BLUMEN and C. MERKEL LehrstuhI ftir Theoretische Munich, Germany
Chemie. Technische
IJllbersitdt Mzinchen.
Received 24 September 1976
We report the ab-imtio electronic band structure of infinite HCI chains, evaluated using the BIUNICH-CHAIN programme and compare the results to calculations on HF chains obtained by us and by other authors.
The hydrogen halides HCl, HBr, and HI undergo in their solid state a series of phase transitions. The low temperature phase of the hydrogen bonded molecular solids HCl and HBr has been shown to be ferroelectric
[1,21. In the framework of studies on hydrogen bonded ferroelectrics [3--S], the hydrogen halides are, because of their relatively simple chemical nature, a valuable object of study. Although the structures of ferroelectric HCl and HF are very similar (see ref. [6] for comparison), in distinction to HF where a large amount of ab-initio and semi-empirical work on energy band structure has been published [7- 111, to our best knowledge for HCI such results are missing. Here we present an abinitio LCAO SCF band structure calculation for HCl. HCl in its ferroelectric phase HCl III consists of molecular chains; the crystal has Cif symmetry and the chains lie in zig-zag pattern in the a-b plane [12141. In the deuterated compound the DC1 length is 1.25 a at 77.4 K, the Cl-Cl distance being 3.69 A and the Cl-D-Cl angle 93.5” [12,13]. In this phase band structure calculations on only one chain are justified, since the hydrogen bonds are intrachain only and since the ferroelectric turn-over process maintains the identity of the chains [I 51. The interchain interactions are of electrostatic nature, the distance of one chain to the adjacent crystal plane being 3.70 A. Many approaches may be used in calculating the electronic properties of one-dimensional molecular sys-
terns. In our case, where hydrogen bonding and Iong intrachain interactions play a significant rGie [6], abinitio methods are best suited to account for such features. Moreover we prefer a programme where different sets of basis functions may be used freely. For these reasons in this work the calculations were performed with the MUNICH-CHAIN programme [16], which is an extension of the Diercksen-Kraemer MUNICH programme of Roothaan SCF MO LCAO type, designed for use with generalized gaussian type filnctions as a basis [ 171. MUNICH-CHAIN calculates crystal orbitals (CO’s) [ 181 in systems of infinite chains of moIecuIes, the CO being generated from a finite molecular complex by applying the operations of a cyclic space group. Here the starting density matrices were obtained through MO calculations and the interaction integrals were computed up to 4th molecular neighbours. In order to ensure fast convergence in the SCF iteration an efficient extrapolation scheme [19] was used. Convergence was assumed when the difference in the elements of the density matrices was less than 2 X 10e6 _ In all HCI calculations a 3/l s basis on H [ZO], a LO/3 s 6/2 p basis on Cl [21], and a 9 k-points grid for the Brillouin zone have been used. Enlarging the grid to 17 k-points led to energy changes o;less than 10e3 eV. At first the electronic band structure for an infinite HCl chain with the experimental geometry was evaluated. Its band complex is given in fig. 1, where the core levels are omitted. The elementary cell contains two HCl molecules_ The lowest band shown corresponds to the molecular 3s level of HCl and is very narrow. 47
Volume 45, number
1
CHEMICAL
PHYSICS LETTERS
Next is the broad o-like band of the intramolecular Cl-H bond, stemming from the Is H and the 3p Cl levels. Restricting the calculation only to second neighbours changes the width of this band by less than 10m2
-.6 r,
i _- 1.0 <
r Fig. 1. Electronic ab-initio band structure of the HCI III chain in the bent geometry. Energies in au (1 au = 27.2 eV).
48
1 January
1977
eV. The upper two valence bands originate from the 3p Cl states orthogonal to the HCl direction. From these one is a narrow n-band, due to the planar symmetry of the chain, while the wider one arises from the in-plane p Cl electrons. The energy gap to the conduction band is in the centre of the zone (k = 0) 20.1 eV, in agreement with the insulating properties of HCl. Comparing to molecular MUNICH MO calculations, the ionization potential of the chain is larger by 0.2 eV. Stretching the chain (i.e. setting the Cl-H-Cl angle to 180”, while keeping all other parameters futed) leads to the band structure of fig. 2. The upper two 3p Cl valence bands coalesce due to the higher symmetry, while the Cl-H bonding band gets broader (2.58 eV instead of 0.88 eV), due mainly to differences in the intermolecular hydrogen bonding. The band gap is 17.4 eV; thus the crystal ionization potential is reduced in the linear geometry by 2.5 eV, compared to the isolated molecule. A similar behaviour was found in HF [l l]. Compared to the stretched geometry the experimental HCl zig-zag configuration is energetically favoured. Calculations on HF lead to a similar band structure [22], where the 2s and 2p fluorine levels replace the 3s and 3p chlorine levels. In the bent geometry the band assignment of HF remains the same as found here for HCl; the upper occupied band in HF is a n-band [22 1, in agreement with ref. [ 111, differing from ref. [9]. Stretching the chain leads to a broadening of the u F-H bonding band by a factor of 1.7 (in HC12.9, where one should keep in mind while comparing, the difference in the angles of the bent geometries). Contrary to HCl, in HF at k = 0 the o-band lies slightly higher than the n-bands [8,11,22]. Keeping the chlorine atoms at their experimental positions while altering the position of the hydrogens along the Cl-Cl axis does not lead to spectacular changes in the band structure. For the proton in the middle of the Cl-Cl bond the upper two bands are energetically more separated, a behaviour which was found in HF too [9]. But in HCl the n-like band is only slightly wider in the symmetric than in the asymmetric geometry. The symmetrical HCI structure is by 0.25 eV per molecule disadvantaged as compared to the asymmetrical one. In conclusion, we assert that the electronic structure of HF and of ferroelectric HCI are very similar. In ref. [5] we have shown that the energy required to shift
Volume
45, number
1
CHEMICAL
PHYSICS LETTFRS
the polarization in HF should be only by about 2.74 larger than in HCI. The parallelity of the electronic structure of HF and HCI III reinforces our belief that
1 January
1977
the polarization in HF might be shifted by applying external eIectric field. This is of course true for HCL III, which is ferroelectric.
an
The authors are much indebted to Professors G.L. Hofackec and J. Ladik for kind support and to Professor G.H.F. Diercksen for his MUNICH programme. Grants from Stiftung Volkswagenwerk are gratefully acknowledged.
References
Fig. 2. Electronic ab-initio the linear geometry.
band structure
of the HCl chain in
II] S. Hoshino, K. Shimaoka and N. Ndmura. Phys. Rev. Letters 39 (1967) 1286. [2] S. Hoshino. K. Shimaoka, N. Niimura, H. Motegi and N. hiaruyama, J. Phys. Sot. Japan 28 Suppl. (1970) 189. [3] A. Blumen and K.D. God&, Sohd St&c Commun. I8 (1976) 1119. [4] K.D. Godzlk and A. Blumen, Phys. Stat. Sol. 666 (L974) 569. [ 5) A. Blumen and K.D. Godzlk. I-erroelectrics 12 (;976), to bc published. (61 C. hlerkcl and A. Blumen, Solid State Commun. I9 (I976:, to be pubhshed. [ 71 I:. Bassdni, L. Puxroncro dnd R. Rcsta, J. Phys C 6 (1973) 2133. [8 1 A. Karpfcn, J. Ladtk, P. Russegcr, P. Schuster and S. Suhai. Theoret. Chim. Acta 34 (1974) 115. [9] M. Kc&z, J. Koller and A. Azman, Chem. Phys- Letters 36 (1975) 576. [lo] L. Pietronero and N.O. Lipari, J. Chem. Phys. 62 (1975) 1796. [ll] A. Zunger, J.Chem. Phys.63 (1975) 1713. [12] L. Sindor and R.F.C. l-arrow, Nature 213 (L967) 171. _ [ 131 E. Sjndor and R.F.C. l-arrow, Discussions Faraday Sot. 48 (1969) 78. [14] E. Sindor and R.F.C. Farrow. Nature 215 (1967) 1265. [ 151 D.J. Genin, D.E. O’Rcilly, E.hf. Peterson and P. Tszna. J. Chcm. Phys. 48 (1968) 4525. [ 161 C. Merkel, h¶UNICHCHAlN, Crystal Orbital Program System Rcferencc h!anual, Lehrstuhl fir Thcoretischc Chemie, Tcchnische Universitat, htunich, Germany[ 171 C.H.F. Diercksen and W.P. Kracmer, MUNICH, MoIecxlar Program System Reference Manual, hfax-PIanck Irzstitut ftir Physik und Astrophysik, Munich. Germany[ 181 G. Del Re, J. Ladlk and C. Bicz6. Phys- Rev. I.55 (1967) 997. [ 19 1 P.-O. Ldwdin, J. Appl. Phys- Suppl. 33 (1962) 251. 1201 E. Clcmcnti. J. Chem. Phys. 46 (1967) 4737. [21] B. Roos and P. Siegbahn, Theoret. Chim. Acta 17 (1970; 209. 1221 C. Merkel and J. Ladik, to be published.
49
ELECTRONIC
CHEMICAL PHYSICS LE-ITCRS
BAND STRUCTURE
OF FERROELECTRIC
1 lanuary L977
HCI
A. BLUMEN and C. MERKEL LehrstuhI ftir Theoretische Munich, Germany
Chemie. Technische
IJllbersitdt Mzinchen.
Received 24 September 1976
We report the ab-imtio electronic band structure of infinite HCI chains, evaluated using the BIUNICH-CHAIN programme and compare the results to calculations on HF chains obtained by us and by other authors.
The hydrogen halides HCl, HBr, and HI undergo in their solid state a series of phase transitions. The low temperature phase of the hydrogen bonded molecular solids HCl and HBr has been shown to be ferroelectric
[1,21. In the framework of studies on hydrogen bonded ferroelectrics [3--S], the hydrogen halides are, because of their relatively simple chemical nature, a valuable object of study. Although the structures of ferroelectric HCl and HF are very similar (see ref. [6] for comparison), in distinction to HF where a large amount of ab-initio and semi-empirical work on energy band structure has been published [7- 111, to our best knowledge for HCI such results are missing. Here we present an abinitio LCAO SCF band structure calculation for HCl. HCl in its ferroelectric phase HCl III consists of molecular chains; the crystal has Cif symmetry and the chains lie in zig-zag pattern in the a-b plane [12141. In the deuterated compound the DC1 length is 1.25 a at 77.4 K, the Cl-Cl distance being 3.69 A and the Cl-D-Cl angle 93.5” [12,13]. In this phase band structure calculations on only one chain are justified, since the hydrogen bonds are intrachain only and since the ferroelectric turn-over process maintains the identity of the chains [I 51. The interchain interactions are of electrostatic nature, the distance of one chain to the adjacent crystal plane being 3.70 A. Many approaches may be used in calculating the electronic properties of one-dimensional molecular sys-
terns. In our case, where hydrogen bonding and Iong intrachain interactions play a significant rGie [6], abinitio methods are best suited to account for such features. Moreover we prefer a programme where different sets of basis functions may be used freely. For these reasons in this work the calculations were performed with the MUNICH-CHAIN programme [16], which is an extension of the Diercksen-Kraemer MUNICH programme of Roothaan SCF MO LCAO type, designed for use with generalized gaussian type filnctions as a basis [ 171. MUNICH-CHAIN calculates crystal orbitals (CO’s) [ 181 in systems of infinite chains of moIecuIes, the CO being generated from a finite molecular complex by applying the operations of a cyclic space group. Here the starting density matrices were obtained through MO calculations and the interaction integrals were computed up to 4th molecular neighbours. In order to ensure fast convergence in the SCF iteration an efficient extrapolation scheme [19] was used. Convergence was assumed when the difference in the elements of the density matrices was less than 2 X 10e6 _ In all HCI calculations a 3/l s basis on H [ZO], a LO/3 s 6/2 p basis on Cl [21], and a 9 k-points grid for the Brillouin zone have been used. Enlarging the grid to 17 k-points led to energy changes o;less than 10e3 eV. At first the electronic band structure for an infinite HCl chain with the experimental geometry was evaluated. Its band complex is given in fig. 1, where the core levels are omitted. The elementary cell contains two HCl molecules_ The lowest band shown corresponds to the molecular 3s level of HCl and is very narrow. 47
Volume 45, number
1
CHEMICAL
PHYSICS LETTERS
Next is the broad o-like band of the intramolecular Cl-H bond, stemming from the Is H and the 3p Cl levels. Restricting the calculation only to second neighbours changes the width of this band by less than 10m2
-.6 r,
i _- 1.0 <
r Fig. 1. Electronic ab-initio band structure of the HCI III chain in the bent geometry. Energies in au (1 au = 27.2 eV).
48
1 January
1977
eV. The upper two valence bands originate from the 3p Cl states orthogonal to the HCl direction. From these one is a narrow n-band, due to the planar symmetry of the chain, while the wider one arises from the in-plane p Cl electrons. The energy gap to the conduction band is in the centre of the zone (k = 0) 20.1 eV, in agreement with the insulating properties of HCl. Comparing to molecular MUNICH MO calculations, the ionization potential of the chain is larger by 0.2 eV. Stretching the chain (i.e. setting the Cl-H-Cl angle to 180”, while keeping all other parameters futed) leads to the band structure of fig. 2. The upper two 3p Cl valence bands coalesce due to the higher symmetry, while the Cl-H bonding band gets broader (2.58 eV instead of 0.88 eV), due mainly to differences in the intermolecular hydrogen bonding. The band gap is 17.4 eV; thus the crystal ionization potential is reduced in the linear geometry by 2.5 eV, compared to the isolated molecule. A similar behaviour was found in HF [l l]. Compared to the stretched geometry the experimental HCl zig-zag configuration is energetically favoured. Calculations on HF lead to a similar band structure [22], where the 2s and 2p fluorine levels replace the 3s and 3p chlorine levels. In the bent geometry the band assignment of HF remains the same as found here for HCl; the upper occupied band in HF is a n-band [22 1, in agreement with ref. [ 111, differing from ref. [9]. Stretching the chain leads to a broadening of the u F-H bonding band by a factor of 1.7 (in HC12.9, where one should keep in mind while comparing, the difference in the angles of the bent geometries). Contrary to HCl, in HF at k = 0 the o-band lies slightly higher than the n-bands [8,11,22]. Keeping the chlorine atoms at their experimental positions while altering the position of the hydrogens along the Cl-Cl axis does not lead to spectacular changes in the band structure. For the proton in the middle of the Cl-Cl bond the upper two bands are energetically more separated, a behaviour which was found in HF too [9]. But in HCl the n-like band is only slightly wider in the symmetric than in the asymmetric geometry. The symmetrical HCI structure is by 0.25 eV per molecule disadvantaged as compared to the asymmetrical one. In conclusion, we assert that the electronic structure of HF and of ferroelectric HCI are very similar. In ref. [5] we have shown that the energy required to shift
Volume
45, number
1
CHEMICAL
PHYSICS LETTFRS
the polarization in HF should be only by about 2.74 larger than in HCI. The parallelity of the electronic structure of HF and HCI III reinforces our belief that
1 January
1977
the polarization in HF might be shifted by applying external eIectric field. This is of course true for HCL III, which is ferroelectric.
an
The authors are much indebted to Professors G.L. Hofackec and J. Ladik for kind support and to Professor G.H.F. Diercksen for his MUNICH programme. Grants from Stiftung Volkswagenwerk are gratefully acknowledged.
References
Fig. 2. Electronic ab-initio the linear geometry.
band structure
of the HCl chain in
II] S. Hoshino, K. Shimaoka and N. Ndmura. Phys. Rev. Letters 39 (1967) 1286. [2] S. Hoshino. K. Shimaoka, N. Niimura, H. Motegi and N. hiaruyama, J. Phys. Sot. Japan 28 Suppl. (1970) 189. [3] A. Blumen and K.D. God&, Sohd St&c Commun. I8 (1976) 1119. [4] K.D. Godzlk and A. Blumen, Phys. Stat. Sol. 666 (L974) 569. [ 5) A. Blumen and K.D. Godzlk. I-erroelectrics 12 (;976), to bc published. (61 C. hlerkcl and A. Blumen, Solid State Commun. I9 (I976:, to be pubhshed. [ 71 I:. Bassdni, L. Puxroncro dnd R. Rcsta, J. Phys C 6 (1973) 2133. [8 1 A. Karpfcn, J. Ladtk, P. Russegcr, P. Schuster and S. Suhai. Theoret. Chim. Acta 34 (1974) 115. [9] M. Kc&z, J. Koller and A. Azman, Chem. Phys- Letters 36 (1975) 576. [lo] L. Pietronero and N.O. Lipari, J. Chem. Phys. 62 (1975) 1796. [ll] A. Zunger, J.Chem. Phys.63 (1975) 1713. [12] L. Sindor and R.F.C. l-arrow, Nature 213 (L967) 171. _ [ 131 E. Sjndor and R.F.C. l-arrow, Discussions Faraday Sot. 48 (1969) 78. [14] E. Sindor and R.F.C. Farrow. Nature 215 (1967) 1265. [ 151 D.J. Genin, D.E. O’Rcilly, E.hf. Peterson and P. Tszna. J. Chcm. Phys. 48 (1968) 4525. [ 161 C. Merkel, h¶UNICHCHAlN, Crystal Orbital Program System Rcferencc h!anual, Lehrstuhl fir Thcoretischc Chemie, Tcchnische Universitat, htunich, Germany[ 171 C.H.F. Diercksen and W.P. Kracmer, MUNICH, MoIecxlar Program System Reference Manual, hfax-PIanck Irzstitut ftir Physik und Astrophysik, Munich. Germany[ 181 G. Del Re, J. Ladlk and C. Bicz6. Phys- Rev. I.55 (1967) 997. [ 19 1 P.-O. Ldwdin, J. Appl. Phys- Suppl. 33 (1962) 251. 1201 E. Clcmcnti. J. Chem. Phys. 46 (1967) 4737. [21] B. Roos and P. Siegbahn, Theoret. Chim. Acta 17 (1970; 209. 1221 C. Merkel and J. Ladik, to be published.
49