Calculation of Chemical Shifts in the Mossbauer Spectra of some Tin(1V) Compounds BY N. N. GREENWOOD P. G. PERKINS AND D. H. WALL Dept. of Chemistry The University Newcastle upon Tyne. Received 15th September 1967 The Pople-Segal-Santry SCMO method is used to calculate the electronic structures of a selected series of compounds of Sn(IV) with basis sets including 5s 5p and 5d orbitals on tin. The occupa- tion number of the 5s orbital of tin in each compound is correlated with their known experimental chemical shifts and an approximately linear relationship is found. The 5s electron density at the tin nucleus calculated from the Fenni-Segrk equation using a modified form of Burns’ screening rules also exhibits a linear relation with the observed shifts. The inclusion of 5d orbitals does not appear to exert a marked effect on the calculated s-electron densities.From the results the value of AR/R for the l19Sn nucleus on excitation to the 23.8 keV level is 3 . 5 ~ The differential contribution due to 5s screening of 4s electron density to the total at the nucleus is calculated for the series. Finally the occupation numbers of the three individual 5p orbitals lead to a qualitative understanding of experimental quadrupole splitting data. 1. INTRODUCTION Much discussion has centred on the magnitude and sign of AR/R for the tin nucleus on excitation to the 23.8 keV level and some qualitative and semi-quantita- tive attempts have been made to reconcile Mossbauer data with a particular sign and magnitude of this quantity. Furthermore the chemical shift in the Mossbauer spectrum for a particular compound depends on the s-electron density at the nucleus of the Sn atom.This follows from the eq~ation,~ 2pAR 2 2 2 p 1 3-2p R 0 a r ( 2 p ) 3 + 2 p 6 = n e - R - - 2 -($abs.(0)2 - $em.(o)2). Z The s-electron densities $(0)2 on the right hand side may be written as The 5s electron density does not necessarily correspond to integral numbers of 5s electrons (as would be supposed if we write “ divalent ” tin as Sn2+ and “ quadrivalent ” tin as Sn4+). This is because in tin compounds the bonding between atoms almost always has some covalent character and hence the bond electron density is distributed between the tin atom and the ligand moiety. Moreover the fraction which remains on the tin atom will vary depending on the nature of the ligands e.g. their electro- negativities.This means that if a quantitative relationship between 6 and s-electron density is required and if we are to calculate chemical shifts directly then the problem of calculating the valence electronic structures of tin compounds must be first approached. 51 52 This paper attempts to throw light on these problems by use of a method not previously tried in this context. This is the SCMO method introduced by Pople Santry and Segal which takes account of all valence electrons. This technique has been successfully applied to molecules containing first-row elements of the periodic table in order to obtain information on ground-state properties e.g. electron densities on atoms bond orders between them and electronic energies. In the present treatment it has proved necessary to adapt the method for use in connection with a fourth row element.The salient point is that if the distribution of the s and p electrons of tin bonded in a series of compounds can be calculated then we have a starting point from which the s-electron density at the tin nucleus may be obtained via the Fermi-Segrk expres~ion.~ This should then lead to a value of ARIR. CALCULATION OF Sn(IV) MOSSBAUER SPECTRA 2. THE SELF-CONSISTENT MOLECULAR ORBITAL METHOD The Pople-Segal-Santry SCF method assumes that molecular orbitals for both electrons may be written as linear combinations of orbitals centred on 0 and different atomic sites ; thus The coefficients xLP are calculated by diagonalizing a Fock interaction matrix with elements The quantities HPP HPv are the diagonal and off-diagonal elements of the core Hamiltonian matrix.They are molecular integrals representing the energy of an electron in the field of one or two atoms respectively. The terms YAA YAB have a corresponding function in the electron repulsion matrix whilst the set of Ppv form the spinless density matrix. The set of calculated coefficients xip may then be used to form a new density matrix and the whole process repeated until self-consistency is achieved. The elements of the self-consistent density matrix then yield directly the s and p orbital occupation numbers. ADAPTATION OF THE TECHNIQUE The modifications suggested are similar to those made to the n SCF method by Pariser and Parr and concern the values assigned to the above molecular integrals. (i) The one-centre electron repulsion integrals are approximated according to the formula YAA = IA-AA in which IA and A A are the s orbital valence state ionization potential and electron affinity respectively.The two-centre type are then calculated from these by a series expan~ion.~ (ii) The core Hamiltonian matrix elements are obtained directly or indirectly from the valence state ionization potentials listed by Hinze and Jaffk * rather than by Pople’s technique. Where an orbital initially supplies two electrons I +YAA i s used in place of I:. The off-diagonal elements of the core matrix may N. N. GREENWOOD P. G. PERKINS AND D . H. WALL 53 then be computed from the diagonals via the Mulliken-Wolfsberg-Helmholtz expres- ion,^ i.e. H p Y = q l v ( ~ p + ~ v ) * In selecting compounds to be studied by this method two considerations were of primary importance (a) the compound should in the solid state consist of discrete species of known geometry.This condition makes a study of Sn(I1) systems difficult. (b) The number of different ligands attached to the tin atom was to be restricted in order to minimize the number of parameters in the calculations. The final list of compounds chosen for investigation comprised SnX4 (X = F Cl Br I H Me) SnMe,X (X = F C1 Br I H) SnX,2- (X = F Cl Br I) SnMeH, SnMe,C12 and SnMe,H,. PARAMETERS REQUIRED FOR THE CALCULATIONS The diagonal elements of the core Hamiltonian matrix and of the electron repul- Orbital overlaps required for computing Hpv were sion matrix are given in table 1. TABLE 1 .-VALUES OF MOLECULAR INTEGRALS (eV) H,pW HPc((P) Hwc( cpx) HppW Y A A Sn - 16.16 - 8.32 - - 1.50 8-43 F - 53.26 - 20.86 - 31.98 - 13-87 c1 - 34.78 - 15.03 -22.18 - 9.55 Br -31.31 - 13.10 - 19.15 - 7.56 I - 2546 - 12.67 - 15.45 - 4-43 H - 13.60 - - - 12.84 C - 21 -01 - 11-27 - - 12-1 1 obtained directly from the appropriate “ master formulae ” derived for the purpose.The effective nuclear charges for each atom were estimated by Burns’1o method and the Sn-X bond length in SnX was assumed for all compounds based on tetrahedral symmetry. The Sn-F bond length in SnF,2- is unknown but may be estimated from the radii of the atoms. The two extreme cases i.e. the neutral pair Sn-F and Sn-F- were both examined in the calculations. PARTICIPATION OF ORBITALS OTHER THAN 5s AND 5p A minimum basis set comprising the 5s and 5p valence shells on tin was first considered.The inner filled 4s and 4p levels were not included and were assumed to remain unperturbed except for shielding effects. This approximation is reasonable because these latter atomic levels are considerably more stable than those of the valence shells and so are not expected to enter into bonding to any extent. In order to render the treatment more comprehensive comparable calculations were carried out in which the empty 5d orbitals on tin were also included The inclusion of these orbitals is important because certain of the d-orbitals may mix with the 5s orbital in a way which is governed by the symmetry of the ligand field around the central atom. Furthermore it is important to assess the contribution of d orbitals to the 7~ bonding with the ligands and hence their effect on the electric field gradients.The effective ionization potential of an electron in a 5d-orbital has not previously 54 CALCULATION OF Sn(1V) MOSSBAUER SPECTRA been obtained but it may reasonably be calculated from the known 5p ionization potential and the energy of the process 5s 5p2 +5s 5p5d. Interaction of the tin 6s level with the ligands was tested for stannane. 3. RESULTS ORBITAL OCCUPATION NUMBERS To a first approximation the number of s electrons in the 5s orbital will affect directly the density at the nucleus and so we would expect an approximately linear relationship between the chemical shift and the occupation number of the tin 5s orbital. Table 2 and fig. 1 show the results for the set of compounds calculated with and without inclusion of the 5d orbitals. The reference point for the experi- mental isomer shifts is a-tin.In general a linear relation emerges and as the s-electron orbital density increases so the shift increases. A comparison of the two sets of points of fig. 1 shows that the contribution of d-orbitals to the bonding affects the occupation of the 5s-orbital slightly as also does the inclusion of the 6s orbital in the case of SnH,. The calculated s electron densities for SnI are badly off the line. This is probably due to our neglect of the 5d orbitals on iodine. Inclusion of these excited states is however not practicable in the present calculations. A further important point emerges from this graph. It is often assumed that because of tetrahedral symmetry and ‘‘ sp3 hybridization ” the occupation number of the 5s orbital in a-tin must be exactly one electron.This assumption is theoretically unjustified and the graph indicates a 5s electron population near 1.2 for a-tin. That each Sn atom in a-tin must have exactly four electrons associated with it means that the occupation of each 5p orbital is 3(4-s) and not that the 5s and 5p orbitals each contain exactly one electron. Stannic fluoride has not been considered explicitly because it is not simply tetrahedral in the solid but made up of octahedral units linked by fluorine bridges. It may however be seen that SnF, (6 = -2.5 mm sec-l) would have a 5s occupation of -0.7 electrons. Moreover if it is taken into account that there must also be some 5p occupation in this compound (probably -0.3 electrons per 5p orbital) it is clear that the electronic environment of the tin atom in SnF does not correspond to Sn4+.The latter situation constituted a basic assumption of a previous attempt to calculate ARIR. The present results better satisfy chemical intuition. 4. ESTIMATION OF THE EFFECTIVE ELECTRON DENSITY AT THE TIN NUCLEUS Earlier the chemical shift 6 was correlated directly with the 5s orbital occupation number in the series of compounds. The correct quantity to be used however is the effective s electron density at the nucleus a term explicit in eqn. (1.1). Our knowledge of the occupation numbers of both the 5s and 5p valence orbitals makes direct calculation possible of the effective nuclear charge experienced by any particular s-electron in the outer orbital regions. The Fermi-Segrk equation then allows the relevant s-electron density at the nucleus to be computed ; thus, N.N. GREENWOOD P. G. PERKINS AND D. H. WALL 55 compound SnC14 SnBr4 SnI4 SnH4 SnMe4 SnFMe3 SnClMe3 SnBrMe3 SnIMe3 SnC12Me2 SnHMe3 SnHzMep SnH3Me SnF - (2.2A) SnFi- (2.3 A) SnCla- SnBra- SnTi- TABLE 2.-cHEMICAL SHIFTS AND ELECTRON DENSITIES Electron occupation numbers and densities at the nucleus experimental d mm sec-1 - 1.30~ - 1.00a -0.30~ - 0.83~ - 0.56~ - 0.85u - 0.70~ - 0.65~ - 0.63~ - 0.57e (-0.SSC) - 0.86~ - 0.87~ - 0.86~ - 2 . W (- 2.60b) - 1.60d - 1.23d - 0.85d 5s 0.928 1 0,9874 1.108 (1 * 103)* 1.263 1.146 1.112 1-144 1.015 1.174 1.137 1.120 0.6856 0.7494 0.9105 0.9634 0.9367 - - 5d orbitals included 5d orbitals not included 5p yuSS(O)2 a.u.-3 y4,(0)2 a.u.4 5s 5p ySS(O)2 a u.-3 1.930 11.442 1.941 12-129 2.643 12-872 (2+43) 2.697 14.520 2.264 13.664 2.374 13-174 2.395 13.514 2-145 12.273 2.7 18 13.528 2.692 13.145 2.646 13.003 1-308 8-92 1 1 -406 9.655 1.995 11.177 1.965 11.827 2.152 11.359 - - - - 256.158 256.1 52 256.140 256.125 256.137 256.140 256.1 37 256.150 256.1 34 256.138 256-139 256.183 256.176 256.160 256.155 256.1 58 - - 0.940 1 1.021 1-006 1.108 1.266 1.149 1.115 1-149 1.101 1.176 1.138 1.120 0.6857 0.7600 0.9343 0.9975 0,9882 - 1.996 11.525 2.015 12.458 2-323 12.015 2.659 12.859 2.753 14.493 2.311 13.652 2 - 4 0 13.144 2.460 13.505 2.474 12.955 2.761 13.505 2.727 13.123 2.672 12.979 1-325 8.902 1.419 9.774 2.064 11.401 2.058 12.147 2.332 11.803 - I *6s orbital also included.a M. Cordey-Hayes Applications of the Massbauer Effect in Chemistry and Solid-state Physics Technical Reports b V . I.Gol'danskii E. F. Makarov P. A. Stukan T. N. Sumarokova V. A. Trukhtanov and V. V. Khrapov Dokl. c R. H. Herber and G. I. Parisi Inorg. Chem. 1966 5,769. d N. N Greenwood and J. N. R. Ruddick J. Chem. Soc. A 1967,1679. e R. H. Herber H. A. Stockler and W. T. Reichle J. Chem. Physics 1965 42,2447. Series No. SO,(I.A.E.A. Vienna 1966) p. 156. Akad. Nauk. S.S.S.R. 1964,156,400. 5s orbital occupation number FIG. 1.-Chemical shift plotted against 5s orbital electron occupation. A d-orbitals not included ; 0 d-orbitals included. 56 CALCULATION OF s n ( m MOSSBAUER SPECTRA In this expression 2 is the atomic number Zo is the nuclear charge experienced by any particular s-electron and CT is the quantum defect at the nth level. The factor P is the occupancy of the s orbital under consideration.It is straightforward to apply the equation to evaluate t,hns(0)2 for 12 = 1 2 3 but it may reasonably be assumed that the sum of these densities will remain constant from absorber to emitter and hence will cancel out (see eqn. (1.2)). The electron density in the 4s level will vary to some extent in different cases because of shielding by the 5s electrons. For the present it is also taken as constant in eqn. (1.2). Now the outer valence shells of the tin atom in compounds are incomplete and the problem of screening by a fractional number of electrons arises. In order to deal with this situation we propose a modification of Burns’s screening rules ; thus 2 = Zb(inner shells) - (42) x 0-4 - m x 0.35 (4.2) in which n is the number of 5s electrons and rn the number of 5p electrons.If the derived Zo for each case is now substituted into the Fermi-Segr6 equation the value of t,k5s(0)2 for the tin atom can be calculated. This quantity to first order deter- mines the Mossbauer chemical shift in tin compounds. Table 2 lists the computed densities and fig. 2 illustrates the relation between $5s(0)2 and 6 for the series. The zero of electron density is the constant sum over the inner electron shells and as 0 \ 7 8 9 10 II 12 13 14 15 5s electron density ( a . ~ . ) - ~ FIG. 2.4hemical shift plotted against 5s electron density at the nucleus. (d orbitals included) before the reference point for the chemical shifts is grey tin. The plot is strikingly similar to that of fig. 1 and confirms that there is a linear relation between the s-electron density at the nucleus and the isomer shift.It is perhaps unexpected that fig. 1 and 2 should be so similar in view of the fact that for the first p orbital occupancy is ignored whereas it enters explicitly into the screening factor Zo which yields the second relation. It does however suggest that qualitative predictions of Mossbauer N. N. GREENWOOD P. G. PERKINS AND D . H. WALL 57 chemical shifts in tin(1V) compounds based on intuitive estimates of s orbital occupancy should be reliable. It is now possible to evaluate AR/R from fig. 2 because its slope is equal to A6/A$5s(0)2. Hence AR/R = A6/At,b5s(0)2 x 1.55 x 10-29 eV cm3 = 3.5 x This figure may be compared with the value 1.1 x lo- derived by Boyle Bunbury and Edwards l1 on the supposition that SnF contains Sn4+ and SnCl contains Sn2+ and with the value 1.9 x lo-, calculated by Gol’danskii et a2.l’ The latter resulted from a more realistic appraisal of the electronic structures of tin compounds but was only obtained by recourse to empirical (NQR) data.A value of 3.3 x has also recently been obtained.13 In view of our disagreement with two of these values recalculation of the electron densities at the nucleus using Hartree-Fock wave functions for Sn rather than Burns’ radial functions is in progress. 5. EFFECTS OF SHIELDING OF INNER S-ELECTRONS BY THE 5s ELECTRONS Heretofore it has been assumed that the shells 18-4s are totally unperturbed by bonding and do not make any contribution to the isomer shift. The outer 5s electrons may in fact shield the 1s-4s shells from the nucleus because they penetrate the core up to the nucleus.In covalent compounds the extent of this shielding will depend on the occupation number of the 5s orbital. We can calculate roughly the size of this effect by a modification of the approach of Crawford and Schawlow :14 for a fraction p of the total time a 5s electron is interposed between the ns electron and the nucleus and so reduces the effective nuclear charge Zo on the ns electron by one unit. Hence the fractional change in ns electron density at the nucleus is and so for the fraction of time p t,bns(0)2 is reduced by the fraction 2p/Z0 i.e. it becomes $119(0)2(1 - 2p/Z0). Finally p may be evaluated from the integral (which must be scaled to the occupation number of the 5s orbital) 03 Ti The R-functions are the radial functions for $ns and 1c/5s and so using Burns’ orbitals the integrals can be evaluated in closed form.The correction would be expected to be most important for 4s electrons and indeed it turns out to be small for 4s and negligible for all shells inside this. Accordingly this correction was applied to allow for the screening of the 4s by the 5s electrons and table 2 shows the variation of 4s electron density for the series of compounds. This correction was used when calculating the value of AR/R. The average correction to $4s(0)2 was 0.04 %. 6. QUADRUPOLE SPLITTING AND ELECTRIC FIELD GRADIENTS The hyperfine splitting of a single Mossbauer line results from a non-zero electric field gradient at the nucleus. The diagonal components of the E.F.G. tensor may be related directly to the individual occupation of the three 5p orbitals of the tin atom because these may produce a spherically assymmetric charge distribution around the nucleus.The present treatment is particularly well adapted to throw 58 CALCULATION OF Sn(IV) MOSSBAUER SPECTRA light on this situation because it yields explicitly the occupation numbers of the individual p orbitals. A z z cc M P J - $%(P,) -fn,(py>>(1+ v2/3>+7 (6.1) where q is an asymmetry parameter and Table 3 lists the calculated values of p-orbital imbalance together with the observed quadrupole splittings for the compounds under investigation. The results are not yet particularly quantitative but some points of interest emerge (i) In all cases a molecule with largep-orbital imbalance exhibits a large quadrupole splitting. (ii) Two types of molecule have zero quadrupole splitting ; those with symmetry in which the three p orbitals remain triply degenerate and so are equally populated.This is as expected. Secondly there are those which lack such symmetry but which possess similar occupations of the p orbitals. The latter type of compound repre- sented by SnHMe, SnH,Me should in principle show quadrupole splitting but it seems to be too small to be detected. From the present results the value of p-orbital TABLE 3 .-QUADRUPOLE SPLITTING DATA AND ORBITAL OCCUPATION p-orbital imbalance total 5d average pn-& A nim sec-1 (eqn. (6.1)) occupation bond order Sn-X Sn-Me SnCl SnBr SnI SnH4 SnMe4 SnFMe3 SnC1Me3 SnBrMe SnIMe3 SnC12Me2 SnHMe3 SnH2Me2 SnH3Me SnFg- SnClg- SnBr g- SnIg- 0 0 0 0 0 4.03 3.55 3 a40 3.19 3.41 0 0 0 0 0 0 0 0 0 0 0 0 0.286 0.146 0.148 0.068 0.09 1 0.022 0.007 0.020 0 0 0 0 0.295 0.3 15 0-120 0.07 1 8 0.073 0.130 0.133 0.214 0.050 0-038 0.025 0-125 0-576 0.61 3 1.181 - - 0.0950 0.1060 0-0327 0.0888 0-0989 0.0879 - 0.08 15 0.2005 0.21 95 0,1785 0-0261 0.0300 0,0333 0.0326 0.0265 0.0308 0.0302 0.0292 - imbalance below which no quadrupole splitting will be observed is N 0-03.It is worth- while to examine the electronic structures of certain of these compounds in more detail in order to try to gain some insight into the large differences in the quadrupole effects. Table 3 lists the total occupation of the 5d orbitals and also the total n bond orders between the ligands with filled p or pseudo p orbitals (e.g. F C1 Br Me) and the central atom d orbitals. It is evident that a high quadrupole splitting in a compound can be associated with a high pn-d bond order between ligands and the central atom.This point was made by Gibb and Greenwood l6 and the present calculations appear to confirm their suggestions. The only compound which violates the generalization is SnFMe which exhibits a large quadrupole splitting yet both its 5d occupation number and F-Snp,-d, N. N. GREENWOOD P . G. PERKINS A N D D . H . WALL 59 bond order are small. However whereas for the calculations pyramidal symmetry was assumed for this system it is known that it possesses trigonal bipyramidal units in the solid having the three methyls coplanar with the tin atom and the two fluorines bridging to neighb0urs.l' We thank the General Electric Company for financial support. V. I. Gol'danskii The Mossbauer Effect and its Applications in Chemistry (Consultants Bureau New York 1964) chap.6. I. B. Bersuker V. I. Gol'danskii and E. F. Makarov J. Expt. Theor. 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