The polarizabilities of perchlorate, bisulphate and dihydrogen phosphate anions

26 May 1995

CHEMICAL PHYSICS LETTERS Chemical Physics Letters 238 (1995) 180-186

The polarizabilities of perchlorate, bisulphate and dihydrogen phosphate anions S.T. Howard, G.A. Attard, H.F. Lieberman Department of Chemistry, University of Wales. Card@ Card@ CFI 3TB, UK Received 23 November

1994; in final form 20 March 1995

Abstract The structures and dipole polarizabilities of the isoelectronic perchlorate, bisulphate and dihydrogen phosphate anions have been determined at the Hartree-Fock self-consistent field level. The predicted structures are compared with mean experimental solid-state anion geometries. Three quantities derived from the electronic structure are considered as possible indicators of anion adsorption strength at electrode surfaces: the polarizability, the HOMO-LUMO gap, and electron density lone pair properties. The effects of electron correlation and solvation on geometry and polarizability are also considered, using hypochlorite and chloride ions, and the Cl-...HzO complex as model systems.

1. Introduction Few polarizability calculations on molecular anions have been reported in the literature. Maroulis and Bishop [ 1] and Pluta et al. [ 21 have reported results at the Hartree-Fock self-consistent field (HF SCF) and many-body-perturbation theory (MBPT) levels for OH-; Fowler and Diercksen for (CN) - [ 31; and Inoue and Iwata reported HF configuration interaction (CI) calculations for SiO- [4]. Much of the pioneering work on accurate electric properties of atoms, small molecules and anions has been carried out by Sadlej and co-workers [ 2,5-81. Sadlej has recently remarked: “Since not too much is known about the electric properties of molecules involving second-row atoms, even the SCF level of approximation is fairly attractive.” [ 81. The calculations presented here are very much in this spirit, although we have a particular applied interest in the polarizabilities of these oxy-anions. This stems from numerous electrochemical studies, partic0009-2614/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(95)00335-5

ularly halides, adsorbed at electrode surfaces whereby the relative strengths of adsorption at zero charge potential (so-called specific adsorption) are found to follow the sequence

F- < Cl- < Br- < I-.

That is, iodide anions display a greater propensity to adsorb than other halides [9]. Correlations between the adsorption bond strength and anion size [ IO], the extent of ion hydration [ 91, the partial covalent character of the metal-halide bonds [ 1 l] and the donoracceptor properties of the metal-anion complex [ 121 have all been cited as fundamental to understanding the nature of specific adsorption. Unfortunately, many of these correlations fail when oxy-anions are considered. That is, arguments based on anion size/ hydration cannot explain the experimental observation that anion adsorption bond strength is described well by the sequence

ST. Howard et al. /Chemical

F- < (ClO,)-

O

< (HS04)-

< (HzPOd)-

< Cl-

< Br-

< I-.

In an effort to elucidate the molecular basis of specific adsorption for all anions, it has been proposed that two (related) electronic factors may be important for a given electrode at constant pH and constant charge [ 131: (i) the anion polarizability, and (ii) the HOMO-LUMO energy gap. This first point is difficult to establish due to the paucity of quantitative data concerning the polarizability of oxy-anions. Furthermore, since the Sadlej calculations pertain to the naked anions and a de-solvation step is thought necessary in specific adsorption [9], such calculations are thought likely to yield better agreement between anion adsorption strength and polarizability, should such a correlation exist. The second point is more straightforward, in that copious data concerning far UV absorption is available [ 141. As an approximation, we may associate the peak wavelength for UV absorption A,, with the anion HOMO-LUMO gap. Since these energies will be obtained en route to calculating the polarizability, this presents an opportunity to verify or otherwise the postulated trend between specific adsorptivity and the HOMO-LUMO gap.

2. Computational

181

Physics Letters 238 (1995) 180-186

details

Geometry optimizations were performed with a DEC Alpha RISC workstation running GAMESS [ 151, using direct SCF methods and cutoffs. Dipole polarizabilities were computed with the largest basis set using finite-field (FF) techniques. The molecules were initially optimized using the 321 -t-G basis set [ 161, followed by the 6-3 1 l++G(d, p) basis [ 171. Td symmetry was assumed throughout for the perchlorate ion, and C, symmetry for the bisulphate and dihydrogen phosphate ions These ions are illustrated together with their atom labelling schemes in Fig. 1. Polarizabilities at this 6-31 l+G(d, p) level were computed with the coupled-Hartree-Fock (CPHF) technique [ 181, as implemented in GAUSSIAN 92 [ 191. The latter program was also used for geometry optimizations and CPHF polarizability calculations with and without electron correlation, on the X ‘I: ground state of the diatomic anion ClO-.

’ I-

o/y\O 0

[

Ioi IH/“\sP1

‘---=o 2’

L

_I

Fig. 1.

It is well-known that anion polarizability calculations typically need very large basis sets to give reliable results. However, an alternative has been provided by Sadlej [ 71. He has shown that basis sets of medium size may come near to the results of very large basis sets for selected electronic properties, most notably the static dipole moment and dipole polarizability, if the contraction coefficients and exponents are optimized in the presence of finite electric fields. These basis sets were not developed speci$caZly for anions, but later we show that (for example) for Clthe Sadlej basis gives a reasonably near Hat-tree-Fock value for the Cl- polarizability. Notwithstanding this, it should be borne in mind that the results presented here are obtained with basis sets which have not been re-optimized for anions. Consequently, we employed Sadlej’s electric-field optimized (EFO) bases [ 7,8] in further geometry optimizations and FF polarizability calculations. These correspond to a (lOs6p4d) [%3p2d] basis on oxygen, (6~4~) [ 3s2p] on hydrogen, and ( 14slOp4d) [ 7s5p2d] on the second-row atoms phosphorous, sulphur and chlorine. The latter were used with precisely the same contraction recommended by Sadlej [ 81. In the case of oxygen, the second d function consisting of two contracted Gaussian primitives was left uncontracted, giving a slightly more flexi-

182

S.T. Howard

et al. /Chemical

ble ( lOs6p4d) [ 5s3p3d J basis set. The final basis set sizes for (ClO4)-, (HS04)and (H2PO4)were therefore: 162, 171 and 180 functions, respectively. We shall refer to these basis sets as ‘Sadlej’ bases, although the oxygen basis has been slightly modified in the way described above. SCF calculations with these largest basis sets encountered some problems with convergence, and geometry optimizations were also slowly convergent. In FF polarizability calculations it is desirable to use tight SCF convergence criteria: (C104) - was sufficiently well behaved that the SCF density matrix could be converged to 8 decimal places, but only fifth decimal place convergence could be achieved with the other two ions, when finite-fields were applied. Test calculations on (C104) - and ClO- showed that this represents an accuracy (in au) of 7 decimal places for the total energy; the third decimal place of in dipole moment; and one decimal place in the FF polarizability tensor components. In (HS04) -, it was also necessary to use an applied electric field of 0.0005 au rather than the conventional 0.001 au, in order to obtain this level of SCF convergence. Significant computing power was needed to carry out these calculations with the largest basis sets: optimizations on the polyatomic ions required up to 96 h of processor time (on a DEC RISC Alpha 400) ; and the largest FF polarizability calculations (H2PO4) needed some 27 h per tensor component (the applied field generally negates the possibility of using even C, symmetry constraints). So in practise, calculations including electron correlation for these systems are currently out of range. However, we may estimate the effect of correlation with calculations on smaller, related systems. The Cambridge Crystallographic Data Base [20] running the QUEST software has been employed to determine the mean geometries of these three anions in the solid state.

Physics Letters 238 (1995) 180-186 Table 1 Results for (Clod) -

r(Cl-0) total energy quadrupole

(au)

Sadlej (Cl, 0) 162 functions

1.445 -758.676877

1.457 ( 1.400) -758.68955071

-42.5 29.2 -0.2811 0.2458

-42.0 35.0 -0.2769 0.1826

115640

100840 ( >56000)

ItIOment

W’Aja

a (au) HOMO energy (au) LUMO energy (au) HOMO-LUMO gap (cm-‘) aOrigin

6-3ll+G(d) 118 functions

at centre-of-mass;

(z2). b A,,,, for UV absorption

expectation of aqueous

b

value of (x2) = (y*) = ClO;

[ 141.

Table 2 Results for (HS04) 6-31 l++G(d, 125 functions r(S-01) r(S-02)= r(S-02’) r(S-04) r(04-H) S-04-H 02-S-02’ 01-s-02 total energy (au) dipole moment (D) a axx ‘YYY azz cr

HOMO energy (au) LUMO energy (au) HOMO-LUMO gap (cm-‘)

p)

Sadlej (S, 0, H) 17 1 functions

1.447 1.435 1.628 0.941 106.7 114.6 102.1 -697.643933

1.456 ( 1.442) 1.446 (1.442) 1.631 (1.525) 0.945 105.3 114.5 102.0 -697.641399

3.07 32.03 30.07 29.02 30.37 -0.2459 0.1677

2.80 37.4 35.8 35.1 36.1 -0.245 0.1433

90770

85240 (60000)

a Origin at centre-of-mass. b A,,,= for UV absorption of aqueous (HS04)-

I

b

[ 141.

3. Results and discussion

stantially from those with the larger basis sets, especially the Cl-O, S-O and P-O bond lengths (up to

Tables l-3 report the optimized structures, energies, first non-zero multipole moment, and polarizabilities for the three anions. The atom labelling scheme is depicted in Fig. 1. Geometry optimizations with the 3-21+G basis are stable, but the results differ sub-

0.15 8, discrepancy with the Sadlej basis results). The 6-31 l++G(d, p) basis sets give very similar structures to the larger Sadlej EFO bases, but significantly smaller polarizabilities. In Tables l-3, immediately after the EFO basis set optimized bond lengths, the mean bond lengths as de-

ST. Howard et al. /Chemical

Physics Letters 238 (1995) 180-186

Table 3 Results for (H#O4)6-311++G(p, d) Sadlej (P, 0, H) ( 137 functions) ( 180 functions) r(P-01) (A) r(P-02) (A) r(P-03)= r(P-04) (A) r(03-Hl)=r(04-H2) (A) 01-P-02 (deg) 03-P-04 (deg) total energy (au) dipole moment a (D) axx (au)

ayy (au) oy;: (au) (Y HOMO energy (au) LUMO energy (au) HOMO-LUMO gap (cm-‘)

1.463 I.482 1.633 0.940 101.5 124.9 -641.607606 4.36 35.7 30.9 30.6 32.4 -0.2370 0.1619

1.470 (1.505) 1.489 (1.505) 1.637 ( 1.562) 0.943 101.4 125.9 -641.617520 4.08 40.1 37.6 36.6 38.1 -0.2404 0.1403

87540

83540 (59000)

a Origin at the centm-of-mass. b All,x for absorption of aqueous H#‘OT

b

[ 141.

termined by single-crystal X-ray diffraction are given. (We have also considered whether EXAFS measurements of bond lengths could have been made in the amorphous state on these anions, but a literature search revealed nothing [ 2 1] .) The Cambridge Crystallographic Database contains many thousands of structures with (Clod) - counter-ions, so we have chosen to average over only the most ‘accurate’ structures, which were ‘error-free’ and reported no disorder. A cut-off R-value of 4% was employed, giving a subset of 31 perchlorate ions. The database contains relatively fewer structures with (HS04) - or (HzP04) counter-ions, so all available structures were used in these cases, giving 57 ions for (H2PO4) - and 27 for (HS04) -. The sample standard deviation of these mean bond lengths is x 0.005 A. The formally double bonds in (HS04)and (H2PO4) - have solid-state mean geometries in good agreement with the ab initio optimized values. However, the mean solid-state bond length in (Clod) is almost 0.06 8, longer than the free-ion computed value, and the discrepancy between computed and measured bond lengths is even larger for the single S-O and P-O bonds, being 0.11 A in the former case. If solid-state and predicted gas-phase geometries of anions differ systematically, then there are

183

many good reasons to suppose that the former should be shorter. Firstly, anion wavefunctions in crystals are compacted by Pauli repulsion with the nearest neighbours. Secondly, the Madelung potential at the anion site is effectively a potential well, acting to further compact the anion wavefunction. The fact that it is the single bonds which give the large discrepancies also lends credence to the idea that this is a solid-state effect, since these would be most easily deformed by intermolecular interactions. Lastly, in the solid state there is necessarily a finite amount of charge transfer between ions: the transfer of negative charge from anion to cation may be expected to cause contraction of the anion bonds. However, we should also be concerned with the effect of omitting electron correlation, as this might be expected to have a significant effect on anion structure. We have investigated the effect of electron correlation on the Cl-O bond by carrying out HF and MP2 optimizations on the X ‘Z ground state of the hypochlorite anion ClO- using the same Sadlej basis sets. The optimized bond lengths at the HF and MP2 levels of theory are 1.7 17 and 1.707 A, respectively. The MP3 correction would be likely to slightly increase the bond length again, so this calculation does not suggest that missing electron correlation explains the experiment/theory discrepancy seen in the larger anions. Further support comes from the calculations of Inoue and Iwata on SiO- [ 41, who found a 0.04 A increase in the Si-0 bond length on improving the calculation with CI. The FF polarizabilities (computed with the Sadlej bases) have been computed from the induced dipole moments, rather than total energy changes. The tensors were computed with respect to symmetry axes, which gave one non-zero off-diagonal component which was small (less than 2 au) in both of the anions with C, symmetry. Consqeuently, although the tensors reported in Tables 2 and 3 are in diagonalized form, the new rotated axes are nearly coincident with the symmetry axes. The key result is that the mean polarizabilities Y vary only slightly between the three compounds: from 35.0 au in ClO; to 38.1 au in (H2P04)

-.

We now turn to the reliability of the Sadlej bases used here for computing various properties. Using the very large basis of Diercksen and Sadlej [5] for Cl-, we compare the performance of the medium-sized

184 Table 4 Properties

XT. Howard et al. /Chemical

of Cl6-31 l++G(d,

total energy (au) a (au) HOMO energy (au) LUMO energy (au) gap (cm-‘) HOMO-LUMO

aAnax for

Physics Letters 238 (1995) 180-186

UV absorption

of aqueous Cl-

]171 ( 3 1 functions) -459.568127 13.07 -0.1501 0.3078 100490

p)

Sadlej basis

Diercksen-Sadlej

[81

151

(34 functions)

(77 functions)

-459.541472 29.14 -0.1502 0.1882 74270

-459.570007 31.56 -0.1503 0.0566 45410 (54500)

basis

a

[ 141,

Sadlej basis in computing the key quantities of interest, namely LYand the HOMO-LUMO gap. The results in Table 4 show that the smaller EFO Sadlej basis is remarkably effective for computing LY(the deficit with respect to the Diercksen-Sadlej basis, which is more than twice as large, is only 8%). The 6-31 l++G(d, p) basis actually gives a lower energy than the Sadlej EFO basis, but despite being smaller by only three functions it gives less than half the LYvalue. The representation of the HOMO-LUMO gap with the Sadlej EFO bases is much less satisfactory: a value some 20000 higher cm-’ than the experimental (aqueous) basis Amax value is obtained. The Diercksen-Sadlej gives a result somewhat closer (9000 cm-t lower than experiment). We may estimate the likely effect of electron correlation on the polarizability by turning to various smaller model systems, including ClO-. Our CPHF calculations using the Sadlej basis sets give the following results (in atomic units) : HF level, aXX (perpendicular to the C, axis) = 25.89, aZZ (parallel) = 4l.l7;MP2level, LY,, = 30.84,~~~~ = 44.28.ThusZof the hypochlorite anion is increased by 14% on adding correlation at the MP2 level. We may also consider the accurate published results for the anions (up to MP4 SDTQ level) F- [22], Cl- [5], and OH- 121. In F-, electron correlation accounts for some 50% of the total dipole polarizability. In Cl- and OH-, the corresponding enhancements due to correlation are 20% and (on average) 47%, respectively. Consequently the SCF results reported here probably underestimate the absolute polarizabilities by similar factors, of some 20%-50%. We now turn to the question of solvation effects how are electronic properties such as the polarizabil-

Table 5 Properties

of H20...CI-

a Sadlej EFO bases (78 functions)

r(CI...O)

(A)

r(O-H) (A) H-O-H (deg ) total energy (au) axx (au) qy (au) LYE:(au) cr HOMO energy (au) LUMO energy (au) HOMO-LUMO gap (cm-t) a For Hz0 in the same basis: a,,=7.79;

3.318 0.949 99.3 -535.613818 34.55 35.36 39.74 36.6 -0.1712 0.1646 73700 cryY=8.96

; a:; =8.34.

ity modified in the presence of solvent molecules? The work of Fowler et al. [23,24 J mostly on atomic anions has shown that effective polarizabilities in crystals and melts are usually smaller than gas-phase values. Calculations on clusters of molecules may give some idea of the importance of these interactions in solution. Here we consider simply the interaction of one water molecule with Cl- (Table 5), using the same Sadlej basis sets. These values refer to geometryoptimized H20...Classumed to be co-planar with Czv symmetry. The essential result is that Z is not greatly depressed in the complex (36.6 au) compared to the summed Z in the free ion and molecule (37.5 au). Nor is the HOMO-LUMO gap greatly affected, falling by less than 1% compared to the gap in isolated Cl-. However, it must be pointed out that the compression effect of surrounding molecules on anion polarizabilities is likely to be much larger when the coordination ‘sphere’ around the anion is more com-

XT. Howard et al. /Chemical Table 6 Oxygen (3, -3)

‘lone pair’ CP properties

CD(HS04)01 (double) 02=03 (double) 04 (single) (HzP04)Ol(double) 02 (double) 03=04 (single)

d (bohr)

Te bohrw3)

V2P (e bohc5)

0.650 0.658 0.658 0.640 0.668 0.667 0.647

0.903 0.865 0.866 0.950 0.821 0.823 0.918

-4.03 -3.48 -3.49 -4.46 -2.91 -2.99 -4.04

plete, e.g. in (HzO)4Cl-. Such a study is certainly worthy of attention. Since the gross anion polarizabilities show so little variation, we finally consider a property which looks directly at the electronic differences in the lone pair (LP) regions. The Laplacian distribution of the electron density V2p has a rich topology, and local maxima in this distribution effectively locate the centroid of LPs [ 251. In Table 6 we report the positions (&distance of LP centroid from the the oxygen nucleus) and values of p and V2p at these stationary points. More negative values of V2p correspond to higher concentrations of p. The comparison is complicated by the fact that (HS04) - and (H2PO4) - anions contain both formally single-bonded and doublebonded oxygens. But if we confine ourselves to the latter, both the values of p and V2p fall in the sequence (C104)-

< (HS04)-

< (H2PO4)-

which, like the HOMO-LUMO gap, mimics the experimental sequence of specific adsorption.

4. Conclusions Anion polarizability is not the overriding factor which determines the strength of specific adsorption, since i? for (C104)-, (HS04)and (H2PO4)is always in excess of the SCF value for Cl-, a more strongly adsorbing anion. Although the absolute values of the transition energies (as estimated from the HOMO-LUMO gap) are not well-reproduced in these calculations, the relative values reported in Tables l-4 do follow the sequence (C104)-

< (HS04)-

< (HzPO4)-


Physics Letters 238 (1995) 180-186

185

If a plot is made of ion adsorbtion strength versus h,, [ 131, the correct sequence of specific adsorption is produced. However, given the tenuous connection between the HOMO-LUMO gap and the polarizability, it is probably unwise to attach much significance to this successful explanation of a trend (after all, there are only four ions in our ‘sample’). The structures of these anions are fairly well predicted by medium-sized basis sets such 6-3 11 ++G( d, p), but small bases such as 3-21+G give poor results. Even with the Sadlej basis sets, some of the HFoptimized bond lengths are significantly shorter than the mean structures obtained from solid-state X-ray diffraction. These differences are consistent with the types of effects expected to influence anion structure in the solid state, and are probably not indicative of absent electron correlation. Hence the structures presented here should be reasonable representations of the gas phase geometries in these anions. The polarizabilities obtained with the Sadlej basis sets are expected to be within 10% of the respective HF-limiting values. Since the mean molecular polarizabilities differ very little between the three anions, it appears that this is too crude a quantity to explain experimentally observed differences in adsorption. The LP properties, measured in terms of electron density and electron density concentration at the LP centroids, do show a pronounced trend which correlates with the known sequence of relative specific adsorption. A decomposition of the polarizability tensors into atomic contributions along the lines of the schemes of Bader et al. [ 261 or Lesueur and Stone [ 271 might also show that the terminal oxygen ‘atoms’ have markedly different properties in these three anions. Howard has recently reported results for the former type of decomposition for a number of small anions [28]. Unfortunately, the anions studied here proved just beyond available computing resources for a comparable treatment. The LUMO energies are so sensitive to the choice of basis set, it is clearly unwise to ascribe significance to the absolute values of the HOMO-LUMO gaps. However, the trends in HOMO-LUMO gap and Z are consistent with simple models of polarizability, i.e. less polarizable species have bigger energy gaps. A calculation on Cl- interacting with one water molecule indicates only a very small effect on either the HOMOLUMO gap or the anion polarizability.

186

ST. Howard et al. /Chemical

Acknowledgement STH would like to thank the UK Engineering and Physical Sciences Research Council for an Advanced Fellowship; Dr. Paul Mallinson (University of Glasgow) for providing access to an extra DEC alpha computer; and the University of London Computer Centre, for a generous allocation of computing time on the Convex C3800. Thanks to David Hibbs for preparing Fig. 1. GAA acknowledges stimulating discussions with A. Wieckowski concerning the nature of specific adsorption.

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