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Ab initio relativistic lattice energy calculations for fluorides of the 7p series of superheavy elements
Volume 81, number 3
CHEMICALPHYSICSLETTERS
1 August 1981
AB INITIO RELATIVISTIC LATTICE ENERGY CALCULATIONS FOR FLUORIDES OF THE 7p SERIES OF SUPERHEAVY ELEMENTS C.P. WOOD and N.C. PYPER Department of Theoretical Chemistry, University Chemical Laboratory, Cambridge CB2 1EW, UK
Received 23 March 1981; in final form 8 May 1981
Lattice energies of the ionicfluorides of Ag, Pb, E113 and E116 are calculated by an ab initio relativistic method. Previous predictions of the exothermicity of the ionic fluorides of E113 and E116 are confirmed. For AgF and PbF2 small discrepancies between theory and experiment are explained by semi-empirical calculations of the dispersion energy.
1. Introduction There is continuing interest in trying to discover superheaw elements in terrestrial or extra-terrestrial matter and in attempting to synthesize them in nuclear reactions [ 1,2]. The location and identification of these, as yet unknown, elements would be considerably helped by a reliable prediction of their chemistry. Predictions based on simple extrapolation down groups of the periodic table are unreliable because relativistic effects upon even the valence orbitals increase dramatically with atomic number. Thus for example whilst the non-relativistic and relativistic 6s orbital eigenvahies in Hg are 7.10 and 8.93 eV the corresponding values for the 7s orbital in E112 (eka Hg) are 6.48 and 12.52 eV [3]. Similarly the nonrelativistic and relativistic 6s orbital radii in Hg are 3.328 and 2.842 au whilst the corresponding radii for the 7s orbital in E112 are 3.644 and 2.476 au. Thus much insight into the chemistry of such elements could be gained by the use of ab initio calculations both directly in the study of isolated molecules containing superheavy elements, and indirectly in the prediction of heats of formation of possible ionic compounds using the well known Born-Haber cycle. The latter requires knowledge of the ionization potentials and ionic radii of the superheaw elements and of a parameter characterizing tile short-range repulsion between a superheaw ion and its nearest neighbour in the crystal. The ionization potentials can be reliably predicted
by using the results of ab initio relativistic atomic Dirac-Fock calculations. However previous estimates of the ionic radii have relied on either simple extrapolation down groups of the periodic table or on an empirical relation [5], to be described elsewhere, between this radius and that of the outermost occupied DiracFock atomic orbital. The calculated lattice energies are less sensitive to the repulsion parameter for which it is not too hard to select a plausible value. It is the purpose of this paper to report ab inito relativistic calculations of lattice energies of halides of both known elements and the superheaw elements E113 and E116. These predictions do not require that either the ionic radii or the repulsion parameter be known so that these quantities are obtained as byproducts of the main calculation. It should be pointed out that these calculations seem to provide the only method of calculating the lattice energies of solids containing the E1162+ ion, which has been predicted [5] to play a role in the chemistry of E116, without recourse to the empirical relation between the ionic radius and the radius of the outermost Dirac-Fock atomic orbital. It is not even possible to predict the radius of this ion by extrapolation down groups of .the periodic table because there are no experimentally known ions with the same relativistic core ~2 configuration.
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2. Methods
2.1. The complete lattice energy The lattice energy U(R) of a given crystal structure composed of cations C and anions A for a value R of the cation-anion (AC) nearest-neighbour inter-ionic separation is given by
U(R) =M/R - nAC VAc(R ) -- ½nAA VAA (XAAR ) -- ½ncc Vcc(xccR ) - Udi~p(R),
in the crystal which determine Udisp(R ) differ significantly from those of the free ions [6,7]. However the dispersion term is far from being the major contributor to the lattice energy and can be calculated not unreliably by standard methods. Thus Udisp(R) can be written as the sum of dipole-dipole (UDD(R)) and dipole-quadrupole (UDQ(R)) terms Udisp(R ) = UDD(R ) + UDQ(R),
(3)
given by [6] (1)
if all the interactions excepting those between the nearest AC, AA and CC pairs are assumed to be composed only of point coulombic plus dispersion energy terms. Here M is the Madelung constant, n x y is the number of nearest Y ions around each X ion and VXy(R ) is the interaction potential of the ions X and Y after subtraction of both the dispersion energy and the point Coulomb expression qxqy/R where qx is the charge on the ion X. The quantities XAA and Xcc are purely geometrical factors which relate the AA and CC separations to the AC separation R. The quantity Udisp(R ) is the total energy arising from the dispersion interactions between the ions. The lattice energy (1) is dominated by the Madelung term and the nearest-neighbour cation-anion potential VAc(R). In this approximation eq. (1) can be expressed
U(R) = --nAC ~Ac(R) -- (nACqAqc -- M)/R,
(2a)
VOAc(R) = VAc(R ) + qAqc/R.
(2b)
The quantity VOAc(R),which is negative for separations R close to the equilibrium nearest-neighbour cation-anion distance in the crystal, is, neglecting the dispersion term, the difference between the energy of one cation-anion pair separated by a distance R and that of the infinitely separated ion pair. It should be pointed out that eq. (2b) does not imply that any decomposition of VOAc(R)into a purely coulombic qAqC/R and a short-range contribution VAc(R ) has any significance beyond providing one, possibly not very good, way of defining a short-range contribution to the interaction energy. The contribution Udisp(R ) to the lattice energy arising from dispersion cannot be calculated in a fully ab initio fashion because those properties of the ions
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UDD(R) = - R -6 [S6CAC + S6,(CAA + Ccc)/2],
(4a)
UDQ(R ) = - R - 8 [S8dAc + S8,(dA A + dcc)/2],
(45)
Here $6, $6, , S 8 and S 8, are, in the notation of ref. [8], constants depending on the crystal structure, whilst Cxy and d x y are coefficients describing respectively the dipole-dipole and dipole-quadrupole interactions of one ion X with all the Y ions in the crystal environment. These constants can be calculated from the expressions [6,9] 3 exey 2 ex + ey aX°~Y'
(5a)
9 ( a x e x aYeY 1 d x y = 4 C x y \ PXX + PY-]'
(5b)
Cxy
where a x , ex and PX are the polarizability, characteristic energy [6] and electron number [6] for the ion X. The polarizabilities of the known ions are derived from experimental data whilst values for E113 + and E116 2+ can be obtained from the approximate relation [101 a x = ~ (r2)/IPx,
(6)
by comparing the predictions of (6) with experiment for analogous known ions. Both the expectation value o f r 2 ((r2)) and the first ionization potential (IPx) of the ion X are taken from the results of atomic DiracFock calculations. For the cations the characteristic energy is taken to be the first ionization potential of the ion whilst values for the halide ions are derived [6] from experimental optical spectra of ionic crystals. The electron number was taken to be 8 and 10 for the F - and Ag+ ions respectively whilst for both Pb 2+ and E113 + this was taken to be the average of the value (12) obtained by considering the outermost s and d electrons and that derived [ 11] from the relation
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(Px = e2C~X)• For E116 2+ an average of the latter value and that (14) obtained by considering the outermost p, s and d electrons was used. Small uncertainties in the electron number are not important because this only enters the dipole-qnadrupole energy which is a very small fraction of the total lattice energy
U(R). The lattice energy calculation can be refined by retaining the short-range anion-anion interaction VAA(XAAR) in eq. (1). The potential VAA(R) is calculated as - q 2 / R plus the interaction energy, neglecting dispersion, between two anions separated by a distance R. The interaction between two fluoride or two chloride ions can be calculated by any standard non-relativistic ab initio method or by the electron gas density functional method [12,13] which has been shown [14] to describe reliably two systems having the same inert gas electronic structures. Furthermore since the dispersion energy Udisp(R ) can be calculated from (3)-(6) whilst the short-range anionanion interaction only constitutes a small correction to the predictions obtained from eq. (2a), it is clear that the principal difficulty lies in calculating the interaction energies I~Ac(R) between a halide ion and the ion of the superheavy element. Not only does such a system contain a large number of electrons but also relativistic effects for the superheavy ion are so large that non-relativistic calculations would be inadequate. The contribution to ~Ac(R) arising from electron correlation is only a small fraction of VOAc(R) so that an approximate calculation of this contribution using electron-gas theory is adequate. It has been shown [ 14,15 ] that the electron-gas overlap correlation energy should be regarded as different from the dispersion energy so that both these terms should be included. Thus the principal difficulty lies in calculating the much larger non-correlation contribution to
2.2. The non-correlated cation-anion potential
ab initio in that it does not use pseudo-potentials [ 16,17], exchange approximations or the discrete variational method [18] but calculates exactly the energy of relativistic molecular wavefunctions. The molecular wavefunction describing the interaction of two ions whose densities remain undistorted and which therefore describes the situation postulated by the simplest ionic model is Ig'(rl, ... rNT)) = S ~ '[I qba(r1 .... rna)) X [~b (rna+l , ... rNT)>],
(7)
where S is the normalization constant. Here [ ¢ba(rt , ... r n )} and I~b (rna+l, "'" rNT)}I (ATT= na +nb) are wave]unctions for the isolated ions containing n a and n b electrons respectively which are taken to be single determinants of Dirac-Fock atomic orbitals
I¢ia(ra,i)) ,
Iq~a(rl, ...rna)}=
[~5b(rna+1, ...rNT)} = d (/n=[~l I¢ib(rb,i+na))).
(8a)
(8b)
The quantity ru, i is the vector describing the position of electron i with respect to nucleus/~, ]~ip (r)) is the ith Dirac-Fock atomic orbital for atom/~, ~ is the anti-symmetrizer whilst s~' appearing in (7) is the partial anti-symmetrizer containing only the identity plus permutations which interchange coordinates belonging to the set r l , ... rna with those in the set rn +1, "- rNw. Direct evaluation of the total energy oi~the wavet~unction (7) as the expectation value over the relativistic hamiltonian (g;~T) NT 4 T = G {C0t(/)'/~i + c2 [fl(i) -- 1] + l)nuc(ri) } i=1
NT-1 NT
+ G The problem of calculating the major non-correlation contribution to the potential I~Ac(R ) was wercome by using a newly developed program for performing fully ab initio relativistic calculations for diatomic molecules. This program is fully relativistic in that it is based on the Dirac equation rather than low-order perturbation descriptions of relativity. It is also fully
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1,
(9)
i=1 /'=i+1
where =(i) and/~(i) are Dirac matrices for electron i, l)nuc(ri) is the operator for its interaction with the nuclei and c is the velocity of light, is complicated by the overlap of atomic orbitals belonging to one atom with those belonging to the other. However by noting that the wavefunction (7) is invariant with respect to 397
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a linear transformation of the orbitals I~b,,i(r)) among themselves, this function can be re-expressed as ]~(r 1 .... rNT)) = ~(iN_~lT
]t~i(ri)) ) ,
(10)
with the orbitals t ~i(r)), which are linear combinations of the I(~,,i(r)), chosen to be orthonormal. The energy of the function (10) then takes the standard form
lVT
i=1
(~ilcct'b + c2(/3 --
1) + 12nucl~bi)
+ .~.((~it~]lr~ll~i~i) - (t~i@lr~tl~]t~i)).
(11)
The interaction energy FOAc(R) is then calculated by subtracting the energies of the isolated ions from the sum of the internuclear repulsion and the molecular energy calculated from (11). It should be noted that this calculation excludes the dispersion and correlation energies which are calculated separately. Both the details of the choice of the orbitals I ~i(r)), their construction and the evaluation by purely numerical methods of the integrals appearing in the energy expression (11) will be described elsewhere. The Dirac-Fock ion wavefunctions (8) were computed using the Oxford Dirac-Fock program. It should be pointed out that the electron-gas density functional method cannot be relied upon to provide a good description of the potential VOAc(R),this method being unreliable when applied to the interaction of two ions one of which has occupied valence orbitals of a symmetry with respect to the molecular environment which is different from that of any occupied valence orbital of the other ion [14]. Both the superheavy-fluoride systems examined here fall into this category. However the correlation contribution to I~Ac(R), which is in any case a small fraction of the interaction energy computed exactly for the wavefunction (7), can be estimated using the electron-gas density functional method.
3. Results and discussion
For four fluorides the lattice energies and nearest398
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neighbour cation-anion separations in the crystal predicted from four slightly different calculations neglecthag dispersion are reported in table 1. These four methods differ only in whether the electron gas correlation contribution is included in the potentials IA)Ac(R) and VAA(R) and in whether the short-range F - F potential VAA(R) is considered. The results show not only that the predictions of all four methods are similar but also that the results of the simplest calculation based on eq. (2a) using the uncorrelated ab initio potential VAc(R ) are little changed from those obtained by including both the attractive electron-gas correlation energy and the repulsive short-range F - - F potential VAA(R). The catio radii can be predicted by subtracting the radius (1.33 A [19]) of F - from the computed (table 1) cation-anion separations. For both AgF and PbF 2 the computed lattice energies are slightly but appreciably smaller than the experirnental ones derived from the Born-Haber cycle. The predicted ionic radius of 1.21 A for Ag+ is slightly larger than the experimental value of 1.13 A whilst the Pb 2+ radius is more significantly overestimated (1.38 A compared with the experimental value of 1.21 A [19] ). These results by themselves, which have the advantage of being computed using an entirely ab initio method, show either that AgF and PbF 2 are not fully ionic or, as verified below, that the dispersion energy plays a small but important role in the cohesion of both these crystals. It should be pointed out that the semi-empirical calculation, neglecting the dispersion energy, which predicts the lattice energy ofPbF 2 to be 2433 kJ/mol (table 2.5 in ref. [19]) contains an element of circularity in that both the radius of Pb 2+ and the parameter used to describe the short-range potential VAc(R ) are obtained from experimental data assuming that various lead compounds are fully ionic. Consequently the effects of the neglected dispersion terms is absorbed by these parameters. Although the results for AgF and PbF 2 show that the cation radii may be overestimated, particularly if the ion is polarizable, the relative smallness of the discrepancies between the computed and the experimental lattice energies shows that the results for E113F and E116F 2 can be used as a basis for the prediction of the chemistry of these elements. Furthermore the AgF and PbF 2 results show that for E113F and E116F 2 the predicted lattice energies of 721 and 2097 kJ/mol and cation radii of 1.65 and 1.54 A are
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1 August 1981
Table 1 Lattice energies and interionic distances predicted by ab initio calculations without the dispersion energy a) Lattice energy (kJ/mol) no correlation
AgF PbF2 E113F EI16F 2
with correlation
expt. [13]
eq. (2a)
(2a) + VAA
eq. (2a)
(2a) + VAA
848 2239 699 2072
839 2219 698 2061
879 2278 72t 2104
874 2262 721 2097
953 2460
Equilibrium nearest-neighbour cation- anion distance (au) no correlation
AgF PbF2 E113F Ell6F 2
with correlation
expt. [24]
eq. (2a)
(2a) + VA A
eq. (2a)
(2a) + VAA
4.85 5.13 5.73 5.46
4.91 5.20 5.75 5.50
4.77 5.07 5.63 5.40
4.81 5.12 5.63 5.43
4.65 4.84
a) AgF and E113F results calculated using the NaC1 crystal structure whilst those for PbF2 and E116F2 used the CaF2 structure. almost certainly lower and upper bounds respectively to the true lattice energies and radii. Thus the differences between the ab initio lattice energies and those (748 and 2341 kJ/mol) previously calculated using the Kapustinskii equation and taking the radii of E l l 3 + and E l l 6 2 + to be 1.49 and 1.35 A are sufficiently small that the predictions that E113F will be exothermic and that E116F 2 will be strongly exothermic compounds remain unchanged. The predicted heats of formation of E113F and E116F 2 are reduced f r o m - 1 2 5 [20] a n d - 8 3 7 kJ/mol [5] t o - 9 8
and - 5 9 5 kJ/mol. The confirmation of these predictions by an ab initio method, which will under- rather than over-estimate the lattice energies, provides evidence that they are correct. The lattice energies, equilibrium internuclear distances and cation radii predicted by including both the dispersion energy, the overlap correlation energy and the short-range F - - F - interaction in (1), are reported in table 2. The polarizability and characteristic energy of the fluoride ion were taken to be 5.978 au [21] and 11.86 eV [6,7] whilst the experimental Ag +
Table 2 Lattice energies, interionic distances and cation radii predicted by including correlation, dispersion and the short-range F - - F interaction a)
AgF PbF2 Ell3F Ell6F 2
Lattice energy (kJ/mol)
A-C distance (au)
Cation radius (A)
calc.
expt. [23]
calc.
expt. [24]
calc.
expt. [19]
939 2403 813 2279
953 2460
4.61 4.85 5.21 5.08
4.65 4.84
1.11 1.24 1.43 1.36
1.13 1.21 b)
a) See note to table 1. b) pb~+ radius derived from experimental Pb-F separation in PbF2 is 1.23 A. 399
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Table 3 Ionic polarizabilities predicted from (6)
Ag÷ T1÷ Pb2÷ El13 ÷ Ell62÷
(r2) (au)
IP (eV)
a (au)
C~expt(au)
34.39 52.42 46.04 68.64 65.24
21.49 20.43 31.94 23.14 29.86
29.03 46.55 26.15 53.81 39.63
11.62[7] 23.64[7] 24.32[25]
and Pb2 + polarizabilities reported in table 3 were used, thus avoiding recourse to eq. (6). The results (table 2), which agree well with experiment, show that the discrepancies between experiment and the fully' ab initio predictions reported in table 1 can be attributed t o t h e neglect of the dispersion energy. The polarizabilities of E113 + and E1162+ had to be based upon eq. (6). The cation polarizabilities predicted from this equation using the values (table 3) of (r 2) obtained from Dirac-Fock calculations and the experimental ionization potentials [22] are compared with experiment in table 3 for Ag+ and Pb 2+. The predictions for E113 + and E1162+ used the most reliable estimates of the ionization potentials obtained by adding 0.8 eV which takes some account of electron correlation, to the ab initio single manifold Dirac-Fock results (table 1 in ref. [3] ). Since the Pb 2+ polarizability predicted by (6) agrees moderately well with experiment, the value of 39.63 obtained directly from (6) was used for E1162+ . However the results of table 3 show that eq. (6) seriously overestimates the polarizabilities of the singly charged ions so that the value of 53.81 predicted for E113 + is almost certainly too large. Consequently the E113F results reported in table 2 were calculated using the value 27.36 for the E113 + polarizability derived by reducing the value obtained from (6) by the factor needed to convert the T1+ prediction of (6) to the experimental value. The predicted ionic radii are less insensitive to uncertainties in the dispersion term than that lattice energies for which these uncertainties seem to be sufficiently small as to be quite unimportant. The predicted E113 + and E1162+ radii (table 2) of 1.43 and 1.36 A are close to the values of 1.49 and 1.36 A obtained previously [5,20] by using the empirical relation between the ionic radius and that of the outermost occupied Dirac-Fock atomic orbital. Since 400
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these predictions require the inclusion of the dispersion energy which cannot be introduced in a fully ab initio fashion, there is not sufficient evidence to distinguish between the present 1.43 A prediction for the radius of E113 +, that of 1.49 [20] and 1.48 A obtained [4] by extrapolating down group IIIB. Furthermore the predicted (table 2) lattice energy of E116F 2 of 2279 kJ/mol is little changed from that obtained previously [5] by more empirical methods, although the present 813 kJ/mol prediction for E113F is somewhat greater than the previous estimate [20]. The close agreement between theory and experiment for AgF and PbF 2 shows that the results for both the ionic radii of E113 + and E 1162+ and the lattice energies of the fluorides are probably the most reliable predictions of these quantities currently available, even though the magnitude of the dispersion contribution may be a little uncertain.
4. Conclusions It has been shown by purely ab initio relativistic computations that the lattice energies of AgF and PbF~ are underestimated and that the ionic radius of pb 224"is overestimated by the ionic model unless the dispersion interactions between the ions are considered. It has been shown by the fully ab initio relativistic approach, neglecting the dispersion energy and thus~ underestimating the lattice energy, that ionic E 113F is predicted to be exothermic whilst ionic E116F 2 is predicted to be highly exothermic. These results confirm and greatly strengthen confidence in predictions [5,20] based on a much less rigorous approach. It has been shown that dispersion interactions involving the large and polarizable E113 + and E1162+ ions not only yield a minor but significant part of the lattice energies but also that they play an important role in determining the radii of these ions. The application of further calculations of the type reported here to the prediction of the chemistry of the 7p series of superheavy elements will be described elsewhere.
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Acknowledgement One of us (CPW) acknowledges the Science Research Council for the award of a Research Studentship and the Cambridge Philosophical Society for financial support.
References [1] B. Fricke and J.T. Waber, Actinides Rev. 1 (1971) 433. [2] M.A.K. Lodhi, Superheavy elements (Pergamon Press, Oxford, 1978). [3] N.C. Pyper and I.P. Grant, Proc. Roy. Soc. (1981), to be published. [4] O.L. Keller, J.L Burnett, T.A. Carlson and C.W. Nestor, J. Phys. Chem. 74 (1970) 1127. [5] I.P. Grant and N.C. Pyper, Nature 265 (1977) 715. [6] J.E. Mayer, J. Chem. Phys. 1 (1933) 270. [7] J.E. Mayer, J. Chem. Phys. 1 (1933) 327. [8] T.C. Waddington, Advan. Inorg. Chem. Radiochem. 1 (1959) 157. [9] H. Margenau, Phys. Rev. 38 (1931) 747. [10] H. Eyring, J. Walter and G.E. Kimball, Quantum chemistry (Wiley, New York, 1944) ch. 18.
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[11] H.D.B. Jenkins and K.F. Pratt, Proc. Roy. Soc. A356 (1977) 115. [12] R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 (1972) 3122. [13] M.J. Clugston, Advan. Phys. 27 (1978) 893. [14] C.P. Wood and N.C. Pyper, Mol. Phys. (1981), to be published. [15] M.J. Clugston and N.C. Pyper, Chem. Phys. Letters 63 (1979) 549. [16] Y.S. Lee, W.C. Ermler and K.S. Pitzer, J. Chem. Phys. 67 (1977) 5861. [17] N.C. Pyper, Mol. Phys. 39 (1980) 1327. [18] A. Rosen and D.E. Ellis, Chem. Phys. Letters 27 (1974) 595. [19] D.A. Johnson, Some thermodynamic aspects of inorganic chemistry (Cambridge Univ. Press, London, 1968). [20] N.C. Pyper, unpublished work. [21] A.J. Michael, J. Chem. Phys. 51 (1969) 5730. [22] C.E. Moore, Ionization Potentials and Ionization Limits Derived from the Analysis of Optical Spectra, NSRDSNBS 34 (US Govt. Printing Office, Washington, 1971). [23] R.C. Weast, ed., Handbook of chemistry and physics (CRC Press, Cleveland, 1979). [24] A.F. Wells, Structural inorganic chemistry, 4th Ed. (Clarendon Press, Oxford, 1975). [25] C.S.G. Phillips and R.J.P. Williams, Inorganic chemistry (Clarendon Press, Oxford, 1965).
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AB INITIO RELATIVISTIC LATTICE ENERGY CALCULATIONS FOR FLUORIDES OF THE 7p SERIES OF SUPERHEAVY ELEMENTS C.P. WOOD and N.C. PYPER Department of Theoretical Chemistry, University Chemical Laboratory, Cambridge CB2 1EW, UK
Received 23 March 1981; in final form 8 May 1981
Lattice energies of the ionicfluorides of Ag, Pb, E113 and E116 are calculated by an ab initio relativistic method. Previous predictions of the exothermicity of the ionic fluorides of E113 and E116 are confirmed. For AgF and PbF2 small discrepancies between theory and experiment are explained by semi-empirical calculations of the dispersion energy.
1. Introduction There is continuing interest in trying to discover superheaw elements in terrestrial or extra-terrestrial matter and in attempting to synthesize them in nuclear reactions [ 1,2]. The location and identification of these, as yet unknown, elements would be considerably helped by a reliable prediction of their chemistry. Predictions based on simple extrapolation down groups of the periodic table are unreliable because relativistic effects upon even the valence orbitals increase dramatically with atomic number. Thus for example whilst the non-relativistic and relativistic 6s orbital eigenvahies in Hg are 7.10 and 8.93 eV the corresponding values for the 7s orbital in E112 (eka Hg) are 6.48 and 12.52 eV [3]. Similarly the nonrelativistic and relativistic 6s orbital radii in Hg are 3.328 and 2.842 au whilst the corresponding radii for the 7s orbital in E112 are 3.644 and 2.476 au. Thus much insight into the chemistry of such elements could be gained by the use of ab initio calculations both directly in the study of isolated molecules containing superheavy elements, and indirectly in the prediction of heats of formation of possible ionic compounds using the well known Born-Haber cycle. The latter requires knowledge of the ionization potentials and ionic radii of the superheaw elements and of a parameter characterizing tile short-range repulsion between a superheaw ion and its nearest neighbour in the crystal. The ionization potentials can be reliably predicted
by using the results of ab initio relativistic atomic Dirac-Fock calculations. However previous estimates of the ionic radii have relied on either simple extrapolation down groups of the periodic table or on an empirical relation [5], to be described elsewhere, between this radius and that of the outermost occupied DiracFock atomic orbital. The calculated lattice energies are less sensitive to the repulsion parameter for which it is not too hard to select a plausible value. It is the purpose of this paper to report ab inito relativistic calculations of lattice energies of halides of both known elements and the superheaw elements E113 and E116. These predictions do not require that either the ionic radii or the repulsion parameter be known so that these quantities are obtained as byproducts of the main calculation. It should be pointed out that these calculations seem to provide the only method of calculating the lattice energies of solids containing the E1162+ ion, which has been predicted [5] to play a role in the chemistry of E116, without recourse to the empirical relation between the ionic radius and the radius of the outermost Dirac-Fock atomic orbital. It is not even possible to predict the radius of this ion by extrapolation down groups of .the periodic table because there are no experimentally known ions with the same relativistic core ~2 configuration.
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2. Methods
2.1. The complete lattice energy The lattice energy U(R) of a given crystal structure composed of cations C and anions A for a value R of the cation-anion (AC) nearest-neighbour inter-ionic separation is given by
U(R) =M/R - nAC VAc(R ) -- ½nAA VAA (XAAR ) -- ½ncc Vcc(xccR ) - Udi~p(R),
in the crystal which determine Udisp(R ) differ significantly from those of the free ions [6,7]. However the dispersion term is far from being the major contributor to the lattice energy and can be calculated not unreliably by standard methods. Thus Udisp(R) can be written as the sum of dipole-dipole (UDD(R)) and dipole-quadrupole (UDQ(R)) terms Udisp(R ) = UDD(R ) + UDQ(R),
(3)
given by [6] (1)
if all the interactions excepting those between the nearest AC, AA and CC pairs are assumed to be composed only of point coulombic plus dispersion energy terms. Here M is the Madelung constant, n x y is the number of nearest Y ions around each X ion and VXy(R ) is the interaction potential of the ions X and Y after subtraction of both the dispersion energy and the point Coulomb expression qxqy/R where qx is the charge on the ion X. The quantities XAA and Xcc are purely geometrical factors which relate the AA and CC separations to the AC separation R. The quantity Udisp(R ) is the total energy arising from the dispersion interactions between the ions. The lattice energy (1) is dominated by the Madelung term and the nearest-neighbour cation-anion potential VAc(R). In this approximation eq. (1) can be expressed
U(R) = --nAC ~Ac(R) -- (nACqAqc -- M)/R,
(2a)
VOAc(R) = VAc(R ) + qAqc/R.
(2b)
The quantity VOAc(R),which is negative for separations R close to the equilibrium nearest-neighbour cation-anion distance in the crystal, is, neglecting the dispersion term, the difference between the energy of one cation-anion pair separated by a distance R and that of the infinitely separated ion pair. It should be pointed out that eq. (2b) does not imply that any decomposition of VOAc(R)into a purely coulombic qAqC/R and a short-range contribution VAc(R ) has any significance beyond providing one, possibly not very good, way of defining a short-range contribution to the interaction energy. The contribution Udisp(R ) to the lattice energy arising from dispersion cannot be calculated in a fully ab initio fashion because those properties of the ions
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UDD(R) = - R -6 [S6CAC + S6,(CAA + Ccc)/2],
(4a)
UDQ(R ) = - R - 8 [S8dAc + S8,(dA A + dcc)/2],
(45)
Here $6, $6, , S 8 and S 8, are, in the notation of ref. [8], constants depending on the crystal structure, whilst Cxy and d x y are coefficients describing respectively the dipole-dipole and dipole-quadrupole interactions of one ion X with all the Y ions in the crystal environment. These constants can be calculated from the expressions [6,9] 3 exey 2 ex + ey aX°~Y'
(5a)
9 ( a x e x aYeY 1 d x y = 4 C x y \ PXX + PY-]'
(5b)
Cxy
where a x , ex and PX are the polarizability, characteristic energy [6] and electron number [6] for the ion X. The polarizabilities of the known ions are derived from experimental data whilst values for E113 + and E116 2+ can be obtained from the approximate relation [101 a x = ~ (r2)/IPx,
(6)
by comparing the predictions of (6) with experiment for analogous known ions. Both the expectation value o f r 2 ((r2)) and the first ionization potential (IPx) of the ion X are taken from the results of atomic DiracFock calculations. For the cations the characteristic energy is taken to be the first ionization potential of the ion whilst values for the halide ions are derived [6] from experimental optical spectra of ionic crystals. The electron number was taken to be 8 and 10 for the F - and Ag+ ions respectively whilst for both Pb 2+ and E113 + this was taken to be the average of the value (12) obtained by considering the outermost s and d electrons and that derived [ 11] from the relation
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(Px = e2C~X)• For E116 2+ an average of the latter value and that (14) obtained by considering the outermost p, s and d electrons was used. Small uncertainties in the electron number are not important because this only enters the dipole-qnadrupole energy which is a very small fraction of the total lattice energy
U(R). The lattice energy calculation can be refined by retaining the short-range anion-anion interaction VAA(XAAR) in eq. (1). The potential VAA(R) is calculated as - q 2 / R plus the interaction energy, neglecting dispersion, between two anions separated by a distance R. The interaction between two fluoride or two chloride ions can be calculated by any standard non-relativistic ab initio method or by the electron gas density functional method [12,13] which has been shown [14] to describe reliably two systems having the same inert gas electronic structures. Furthermore since the dispersion energy Udisp(R ) can be calculated from (3)-(6) whilst the short-range anionanion interaction only constitutes a small correction to the predictions obtained from eq. (2a), it is clear that the principal difficulty lies in calculating the interaction energies I~Ac(R) between a halide ion and the ion of the superheavy element. Not only does such a system contain a large number of electrons but also relativistic effects for the superheavy ion are so large that non-relativistic calculations would be inadequate. The contribution to ~Ac(R) arising from electron correlation is only a small fraction of VOAc(R) so that an approximate calculation of this contribution using electron-gas theory is adequate. It has been shown [ 14,15 ] that the electron-gas overlap correlation energy should be regarded as different from the dispersion energy so that both these terms should be included. Thus the principal difficulty lies in calculating the much larger non-correlation contribution to
2.2. The non-correlated cation-anion potential
ab initio in that it does not use pseudo-potentials [ 16,17], exchange approximations or the discrete variational method [18] but calculates exactly the energy of relativistic molecular wavefunctions. The molecular wavefunction describing the interaction of two ions whose densities remain undistorted and which therefore describes the situation postulated by the simplest ionic model is Ig'(rl, ... rNT)) = S ~ '[I qba(r1 .... rna)) X [~b (rna+l , ... rNT)>],
(7)
where S is the normalization constant. Here [ ¢ba(rt , ... r n )} and I~b (rna+l, "'" rNT)}I (ATT= na +nb) are wave]unctions for the isolated ions containing n a and n b electrons respectively which are taken to be single determinants of Dirac-Fock atomic orbitals
I¢ia(ra,i)) ,
Iq~a(rl, ...rna)}=
[~5b(rna+1, ...rNT)} = d (/n=[~l I¢ib(rb,i+na))).
(8a)
(8b)
The quantity ru, i is the vector describing the position of electron i with respect to nucleus/~, ]~ip (r)) is the ith Dirac-Fock atomic orbital for atom/~, ~ is the anti-symmetrizer whilst s~' appearing in (7) is the partial anti-symmetrizer containing only the identity plus permutations which interchange coordinates belonging to the set r l , ... rna with those in the set rn +1, "- rNw. Direct evaluation of the total energy oi~the wavet~unction (7) as the expectation value over the relativistic hamiltonian (g;~T) NT 4 T = G {C0t(/)'/~i + c2 [fl(i) -- 1] + l)nuc(ri) } i=1
NT-1 NT
+ G The problem of calculating the major non-correlation contribution to the potential I~Ac(R ) was wercome by using a newly developed program for performing fully ab initio relativistic calculations for diatomic molecules. This program is fully relativistic in that it is based on the Dirac equation rather than low-order perturbation descriptions of relativity. It is also fully
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1,
(9)
i=1 /'=i+1
where =(i) and/~(i) are Dirac matrices for electron i, l)nuc(ri) is the operator for its interaction with the nuclei and c is the velocity of light, is complicated by the overlap of atomic orbitals belonging to one atom with those belonging to the other. However by noting that the wavefunction (7) is invariant with respect to 397
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a linear transformation of the orbitals I~b,,i(r)) among themselves, this function can be re-expressed as ]~(r 1 .... rNT)) = ~(iN_~lT
]t~i(ri)) ) ,
(10)
with the orbitals t ~i(r)), which are linear combinations of the I(~,,i(r)), chosen to be orthonormal. The energy of the function (10) then takes the standard form
lVT
i=1
(~ilcct'b + c2(/3 --
1) + 12nucl~bi)
+ .~.((~it~]lr~ll~i~i) - (t~i@lr~tl~]t~i)).
(11)
The interaction energy FOAc(R) is then calculated by subtracting the energies of the isolated ions from the sum of the internuclear repulsion and the molecular energy calculated from (11). It should be noted that this calculation excludes the dispersion and correlation energies which are calculated separately. Both the details of the choice of the orbitals I ~i(r)), their construction and the evaluation by purely numerical methods of the integrals appearing in the energy expression (11) will be described elsewhere. The Dirac-Fock ion wavefunctions (8) were computed using the Oxford Dirac-Fock program. It should be pointed out that the electron-gas density functional method cannot be relied upon to provide a good description of the potential VOAc(R),this method being unreliable when applied to the interaction of two ions one of which has occupied valence orbitals of a symmetry with respect to the molecular environment which is different from that of any occupied valence orbital of the other ion [14]. Both the superheavy-fluoride systems examined here fall into this category. However the correlation contribution to I~Ac(R), which is in any case a small fraction of the interaction energy computed exactly for the wavefunction (7), can be estimated using the electron-gas density functional method.
3. Results and discussion
For four fluorides the lattice energies and nearest398
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neighbour cation-anion separations in the crystal predicted from four slightly different calculations neglecthag dispersion are reported in table 1. These four methods differ only in whether the electron gas correlation contribution is included in the potentials IA)Ac(R) and VAA(R) and in whether the short-range F - F potential VAA(R) is considered. The results show not only that the predictions of all four methods are similar but also that the results of the simplest calculation based on eq. (2a) using the uncorrelated ab initio potential VAc(R ) are little changed from those obtained by including both the attractive electron-gas correlation energy and the repulsive short-range F - - F potential VAA(R). The catio radii can be predicted by subtracting the radius (1.33 A [19]) of F - from the computed (table 1) cation-anion separations. For both AgF and PbF 2 the computed lattice energies are slightly but appreciably smaller than the experirnental ones derived from the Born-Haber cycle. The predicted ionic radius of 1.21 A for Ag+ is slightly larger than the experimental value of 1.13 A whilst the Pb 2+ radius is more significantly overestimated (1.38 A compared with the experimental value of 1.21 A [19] ). These results by themselves, which have the advantage of being computed using an entirely ab initio method, show either that AgF and PbF 2 are not fully ionic or, as verified below, that the dispersion energy plays a small but important role in the cohesion of both these crystals. It should be pointed out that the semi-empirical calculation, neglecting the dispersion energy, which predicts the lattice energy ofPbF 2 to be 2433 kJ/mol (table 2.5 in ref. [19]) contains an element of circularity in that both the radius of Pb 2+ and the parameter used to describe the short-range potential VAc(R ) are obtained from experimental data assuming that various lead compounds are fully ionic. Consequently the effects of the neglected dispersion terms is absorbed by these parameters. Although the results for AgF and PbF 2 show that the cation radii may be overestimated, particularly if the ion is polarizable, the relative smallness of the discrepancies between the computed and the experimental lattice energies shows that the results for E113F and E116F 2 can be used as a basis for the prediction of the chemistry of these elements. Furthermore the AgF and PbF 2 results show that for E113F and E116F 2 the predicted lattice energies of 721 and 2097 kJ/mol and cation radii of 1.65 and 1.54 A are
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Table 1 Lattice energies and interionic distances predicted by ab initio calculations without the dispersion energy a) Lattice energy (kJ/mol) no correlation
AgF PbF2 E113F EI16F 2
with correlation
expt. [13]
eq. (2a)
(2a) + VAA
eq. (2a)
(2a) + VAA
848 2239 699 2072
839 2219 698 2061
879 2278 72t 2104
874 2262 721 2097
953 2460
Equilibrium nearest-neighbour cation- anion distance (au) no correlation
AgF PbF2 E113F Ell6F 2
with correlation
expt. [24]
eq. (2a)
(2a) + VA A
eq. (2a)
(2a) + VAA
4.85 5.13 5.73 5.46
4.91 5.20 5.75 5.50
4.77 5.07 5.63 5.40
4.81 5.12 5.63 5.43
4.65 4.84
a) AgF and E113F results calculated using the NaC1 crystal structure whilst those for PbF2 and E116F2 used the CaF2 structure. almost certainly lower and upper bounds respectively to the true lattice energies and radii. Thus the differences between the ab initio lattice energies and those (748 and 2341 kJ/mol) previously calculated using the Kapustinskii equation and taking the radii of E l l 3 + and E l l 6 2 + to be 1.49 and 1.35 A are sufficiently small that the predictions that E113F will be exothermic and that E116F 2 will be strongly exothermic compounds remain unchanged. The predicted heats of formation of E113F and E116F 2 are reduced f r o m - 1 2 5 [20] a n d - 8 3 7 kJ/mol [5] t o - 9 8
and - 5 9 5 kJ/mol. The confirmation of these predictions by an ab initio method, which will under- rather than over-estimate the lattice energies, provides evidence that they are correct. The lattice energies, equilibrium internuclear distances and cation radii predicted by including both the dispersion energy, the overlap correlation energy and the short-range F - - F - interaction in (1), are reported in table 2. The polarizability and characteristic energy of the fluoride ion were taken to be 5.978 au [21] and 11.86 eV [6,7] whilst the experimental Ag +
Table 2 Lattice energies, interionic distances and cation radii predicted by including correlation, dispersion and the short-range F - - F interaction a)
AgF PbF2 Ell3F Ell6F 2
Lattice energy (kJ/mol)
A-C distance (au)
Cation radius (A)
calc.
expt. [23]
calc.
expt. [24]
calc.
expt. [19]
939 2403 813 2279
953 2460
4.61 4.85 5.21 5.08
4.65 4.84
1.11 1.24 1.43 1.36
1.13 1.21 b)
a) See note to table 1. b) pb~+ radius derived from experimental Pb-F separation in PbF2 is 1.23 A. 399
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Table 3 Ionic polarizabilities predicted from (6)
Ag÷ T1÷ Pb2÷ El13 ÷ Ell62÷
(r2) (au)
IP (eV)
a (au)
C~expt(au)
34.39 52.42 46.04 68.64 65.24
21.49 20.43 31.94 23.14 29.86
29.03 46.55 26.15 53.81 39.63
11.62[7] 23.64[7] 24.32[25]
and Pb2 + polarizabilities reported in table 3 were used, thus avoiding recourse to eq. (6). The results (table 2), which agree well with experiment, show that the discrepancies between experiment and the fully' ab initio predictions reported in table 1 can be attributed t o t h e neglect of the dispersion energy. The polarizabilities of E113 + and E1162+ had to be based upon eq. (6). The cation polarizabilities predicted from this equation using the values (table 3) of (r 2) obtained from Dirac-Fock calculations and the experimental ionization potentials [22] are compared with experiment in table 3 for Ag+ and Pb 2+. The predictions for E113 + and E1162+ used the most reliable estimates of the ionization potentials obtained by adding 0.8 eV which takes some account of electron correlation, to the ab initio single manifold Dirac-Fock results (table 1 in ref. [3] ). Since the Pb 2+ polarizability predicted by (6) agrees moderately well with experiment, the value of 39.63 obtained directly from (6) was used for E1162+ . However the results of table 3 show that eq. (6) seriously overestimates the polarizabilities of the singly charged ions so that the value of 53.81 predicted for E113 + is almost certainly too large. Consequently the E113F results reported in table 2 were calculated using the value 27.36 for the E113 + polarizability derived by reducing the value obtained from (6) by the factor needed to convert the T1+ prediction of (6) to the experimental value. The predicted ionic radii are less insensitive to uncertainties in the dispersion term than that lattice energies for which these uncertainties seem to be sufficiently small as to be quite unimportant. The predicted E113 + and E1162+ radii (table 2) of 1.43 and 1.36 A are close to the values of 1.49 and 1.36 A obtained previously [5,20] by using the empirical relation between the ionic radius and that of the outermost occupied Dirac-Fock atomic orbital. Since 400
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these predictions require the inclusion of the dispersion energy which cannot be introduced in a fully ab initio fashion, there is not sufficient evidence to distinguish between the present 1.43 A prediction for the radius of E113 +, that of 1.49 [20] and 1.48 A obtained [4] by extrapolating down group IIIB. Furthermore the predicted (table 2) lattice energy of E116F 2 of 2279 kJ/mol is little changed from that obtained previously [5] by more empirical methods, although the present 813 kJ/mol prediction for E113F is somewhat greater than the previous estimate [20]. The close agreement between theory and experiment for AgF and PbF 2 shows that the results for both the ionic radii of E113 + and E 1162+ and the lattice energies of the fluorides are probably the most reliable predictions of these quantities currently available, even though the magnitude of the dispersion contribution may be a little uncertain.
4. Conclusions It has been shown by purely ab initio relativistic computations that the lattice energies of AgF and PbF~ are underestimated and that the ionic radius of pb 224"is overestimated by the ionic model unless the dispersion interactions between the ions are considered. It has been shown by the fully ab initio relativistic approach, neglecting the dispersion energy and thus~ underestimating the lattice energy, that ionic E 113F is predicted to be exothermic whilst ionic E116F 2 is predicted to be highly exothermic. These results confirm and greatly strengthen confidence in predictions [5,20] based on a much less rigorous approach. It has been shown that dispersion interactions involving the large and polarizable E113 + and E1162+ ions not only yield a minor but significant part of the lattice energies but also that they play an important role in determining the radii of these ions. The application of further calculations of the type reported here to the prediction of the chemistry of the 7p series of superheavy elements will be described elsewhere.
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Acknowledgement One of us (CPW) acknowledges the Science Research Council for the award of a Research Studentship and the Cambridge Philosophical Society for financial support.
References [1] B. Fricke and J.T. Waber, Actinides Rev. 1 (1971) 433. [2] M.A.K. Lodhi, Superheavy elements (Pergamon Press, Oxford, 1978). [3] N.C. Pyper and I.P. Grant, Proc. Roy. Soc. (1981), to be published. [4] O.L. Keller, J.L Burnett, T.A. Carlson and C.W. Nestor, J. Phys. Chem. 74 (1970) 1127. [5] I.P. Grant and N.C. Pyper, Nature 265 (1977) 715. [6] J.E. Mayer, J. Chem. Phys. 1 (1933) 270. [7] J.E. Mayer, J. Chem. Phys. 1 (1933) 327. [8] T.C. Waddington, Advan. Inorg. Chem. Radiochem. 1 (1959) 157. [9] H. Margenau, Phys. Rev. 38 (1931) 747. [10] H. Eyring, J. Walter and G.E. Kimball, Quantum chemistry (Wiley, New York, 1944) ch. 18.
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[11] H.D.B. Jenkins and K.F. Pratt, Proc. Roy. Soc. A356 (1977) 115. [12] R.G. Gordon and Y.S. Kim, J. Chem. Phys. 56 (1972) 3122. [13] M.J. Clugston, Advan. Phys. 27 (1978) 893. [14] C.P. Wood and N.C. Pyper, Mol. Phys. (1981), to be published. [15] M.J. Clugston and N.C. Pyper, Chem. Phys. Letters 63 (1979) 549. [16] Y.S. Lee, W.C. Ermler and K.S. Pitzer, J. Chem. Phys. 67 (1977) 5861. [17] N.C. Pyper, Mol. Phys. 39 (1980) 1327. [18] A. Rosen and D.E. Ellis, Chem. Phys. Letters 27 (1974) 595. [19] D.A. Johnson, Some thermodynamic aspects of inorganic chemistry (Cambridge Univ. Press, London, 1968). [20] N.C. Pyper, unpublished work. [21] A.J. Michael, J. Chem. Phys. 51 (1969) 5730. [22] C.E. Moore, Ionization Potentials and Ionization Limits Derived from the Analysis of Optical Spectra, NSRDSNBS 34 (US Govt. Printing Office, Washington, 1971). [23] R.C. Weast, ed., Handbook of chemistry and physics (CRC Press, Cleveland, 1979). [24] A.F. Wells, Structural inorganic chemistry, 4th Ed. (Clarendon Press, Oxford, 1975). [25] C.S.G. Phillips and R.J.P. Williams, Inorganic chemistry (Clarendon Press, Oxford, 1965).
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