Electronic structure and related properties of ferrocyanide ion calculated by the SCF Xα-scattered wave method

I. Phyr C&m. So&

1977. vol. 38, pp. 35-39,

Pqaaon

Pmss.

Printed

inGreatBritain

ELECTRONIC STRUCTURE AND RELATED PROPERTIES OF FERROCYANIDE ION CALCULATED BY THE SCF Xa-SCA’ITERED WAVE METHOD D. ~UEN~~GER Centro Brasileiro de Pesquisas Ffsicas, Avenida Wenceslau Braz 71, Rio de Janeiro, Brasil

B. MAFFEO Pontiffcia Universidade Catdlica do Rio de Janeiro, Departamento de Fisica, Rua Marquis de S. Vicente 225,200OO Rio de Janeiro, Brasil and

M. L. DE SIQUEIRA Departamento de Fisica ICEX, Universidade Federal de Minas Gerais, Caixa Postal 1621,30000Belo Horizonte, Brasil (Received 1I December 1975;accepted

22 Aptif 1976)

SCF-XlrSW method is used to calculate the electronic structure of the fe~~yanide ion. Optical transitions and X-ray phot~~ectron emission are obtained from the energy level scheme and compared with experimental results. The charge density in the Fe nucIeus is also computed and the result is correlated with isomer shift measurements made on this and other Fe complexes for which theoretical calculations have been performed.

AbsWet-The

Fe nucleus) and tangent to the spheres of the outer N atoms, and region II is the interstitial space between atomic spheres. In regions I and III the molecular potential is spherically averaged, in region II it is volume averaged. The limiting sphere of region III (“outer sphere”) also serves as a “Watson sphere” where a charge is dis~ibut~ to simulate the stabilizing effect of the lattice on the ion. In the present calculation, we have made the spheres centered on neighbou~ng atoms tangent to each other. The parameters used for octahedral [Fe(CN),lm4 are: inter-atomic distances, Fe-C = 1.92A, C-N = 1.1578([7]; values of the atomic exchange parameter (Ywere taken from Schwarz(8al: a(Fe) = 0.71, a(C) = 0.76 and a(N) = 0.75; the value used in regions II and III was a = 0.75. The following reasoning was followed to provide the muffintin radii: the radii of the C and N spheres were chosen to be proportional to the covalent radii of these atoms in a triple bond[9]. Once these were determined, the Fe sphere radius was automatic~y fixed. In this manner, we obtained the following values for the radii: Fe= 2.4818a.u., C = 1.1464a.u., N = 1.0399a.u., and “outer sphere” = 6.855 a.u. All occupied atomic orbitals were included in the calculation, where we have a total of 108 electrons. Self-consistency in the energies was carried out up to the fourth decimal in all cases. Convergence of the ground state was obtained for two values of the Watson sphere charge, namely, t4 and +5. Although the former is enough to neutralize the ion, it places the lowest virtual orbitals uncomfortably near the continuum. However, it was verified that the use of Watson sphere charge +5 merely dislocates the energies by an almost constant amount, as would be expected.

1.lNlRODUcIlON

During the last years, transition metal complexes have been the subject of a considerable amount of experimental and theoretical study[l]. Calculations of the electronic structure of these ions have been mainly of semiempirical nature, due to computational di%culties. More recently, the elf-consistent-meld X&-Scattered Wave method (XaSW), proposed by Johnson and Slaterf21, has proved to be a useful tool in underst~di~ the properties of these complexes[3]. In spite of the X&SW method being based on first principles, the computer time spent on a calculation is considerably smaller than for ab initio methods. In view of the encouraging results obtained for similar systems, we decided to study by the XaSW method the low-spin covalent [Fe(CN)61-4 ion. In this paper the calculation of its electronic structure is made and comparisons with experimental results concerning optical transitions 141and X-ray phot~lec~on emission [S] are presented. The charge density in the Fe nucleus is also computed in order to obtain the quantity (1:= AlfAp, using other theoretical calculations of Fe complexes[6] for which isomer shift measurements were done.

2. DETALLS OF TIDE-CALCULATIONS The theoretical development of the Xa!SW method has been described in detail in several publications [2]. The ion is studied as a cluster of atoms, which is divided in three regions, where the muffin-tin approximation to the potential is used. Region I consists of spheres centered on each atom, region III is defined as the outer region limited by a sphere with origin at the center of the molecule (the 35

36

D. 3. RESULTS

3.1 Electronic

AND

GUENZBURGER~~LI~.

number of electrons inside a muffin-tin sphere to a “charge”, the final charge of Fe in [Fe(CN)J’ is greater than that associated to atomic Fe as for other metal halogenides[6], though the difference obtained (-2.38) seems exageratedly large. In contrast, the number of electrons in the C and N spheres is well below the atomic numbers of these elements, indicating that their electrons have “spread out” into the interstitial region. This is due partly to the muffin-tin approximation, and partly to overlap integrals between ligands being different from zero. We must mention that the “Fe charge” is sensitive to the value chosen for the Fe sphere radius, as shown in a test calculation: taking a value 0.087 a.u. smaller the “Fe charge” changes to approximately 26. I.

DISCUSSION

structure

The results obtained for the ground state electronic structure of ferrocyanide ion are presented in Table I. The electronic scheme obtained predicts a closed shell structure up to the 2t,, level, as expected. The ordering highest filled t2y(metal) below the lowest empty e, (metal) is also obtained, which agrees with ligand field theory. In Table I is also given the final distribution of the electrons in the three regions. It is seen that a large amount of the electronic charge is distributed in region II, which is partly a consequence of the latter having a large volume compared to the total volume of the cluster. This could indicate that the muffin-tin approximation is rather poor for this particular case. The adoption of overlapping spheres [ IO] would perhaps improve this situation. Regarding the final charges in the different atomic spheres, one notices that the total number of electrons in the Fe sphere is -0.8 greater than 26, the atomic number of Fe. This indicates a large “invasion” of electrons from the cyanide ligands. We have compared (see Table 1) the number of electrons in a sphere of equal volume for atomic Fe and for the ion Fe”. These last were calculated with an atomic Xo-Hermann-Skillman Hartree-FockSlater program for the same value of (Y (0.71) as in the calculation of the complex. If one is to correlate the

3.2 Optical transitions The optical spectrum of IFe(CN)J4 shows a relatively simple structure. Since the highest filled tzg level is completely occupied, only a restricted number of transitions are possible. The only allowed transitions are the charge transfer ‘A ,g - ‘T,.. In addition, bands of low intensity, corresponding to the Laporte forbidden d - d transitions, are expected to occur. The optical spectrum of [Fe(CN)J4 was obtained and interpreted by Gray and Beach[4]. As is known, the one-electron energy difference li - 4, corresponding to

Table 1. Energy levels and charge distribution in [Fe(CN)J? (a) Watson sphere charge +5. The five lowest empty orbitals are included; (b) All energies in Rydbergs; (c)Charge distribution in % of one electron;(d) II and III stand for interstitial region and “outer sphere” region respectively; (e) Orbitals marked (*) are empty Energy

levels

Charge

(b)(e) Fe sphere

Distribution

C spheres

-509.355

100

lls)

-

59.363

100

(2.5)

- 51.351

100

(2p)

(c)

N spheres

II(d)

LOO (1s)

- 28.801 - 20.912

100

(1s)

III(d)

_ _

-

_

_ 0

-

6.841

(lyg)

99.96

0

0

0.04

-

4.506

(ltlU)

99.80

0

0

0.20

-

1.730

(2a1g)

0.53

19.77

32.27

46.63

0.79

-

1.721

(2tlU)

0.11

19.67

33.75

45.70

0.69

-

1.715

(leg)

0.08

19.96

34.21

44.53

1.22

-

1.259

(3alg)

26.19

21.51

5.84

46.33

0.13

-

1.083

(3tl")

-

1.081

(2eg)

-

0.986

-

0

8.25

18.02

17.37

55.87

0.49

38.36

22.01

7.67

30.84

1.11

(4.yg)

2.74

10.29

38.15

45.66

3.15

0.959

(lt2g)

22.16

14.14

11.89

50.16

1.64

-

0.958

Ueg)

12.43

10.69

35.54

35.53

5.80

-

0.956

(4tlJ

6.04

16.75

26.96

47.06

3.18

-

0.658

(lt2u)

0.31

14.51

27.77

55.04

2.36

-

0.847

(5tl"l

3.97

18.26

24.10

52.99

0.68

-

0.825

Utlg)

0.15

14.27

32.06

50.39

3.12

-

0.760

(2t2g)

62.59

0.68

13.91

21.41

1.40

-

0.516

(5alg)*

3.28

2.06

1.71

64.37

28.50

-

0.437

c3t2g1*

8.93

3.42

22.64

59.05

5.96

-

0.435

(leg)*

54.33

16.90

3.65

22.90

2.22

-

0.431

16tlJ*

1.38

17.08

16.66

49.65

15.23

-

0.341

(2t2Jf

1.30

23.24

24.23

51.02

0.21

Overall Fe II.

electron

sphere, 29.94;

distribution,

26.83; III,

C sphere,

to a total

3.68

each;

of

108 electrons:

N sphere,

4.65

each;

1.22. -

Total [Fe(CN)rl-' 26.83

electronic

charge Free

within

atom

24.45

the

Fe sphere Free

ion 23.82

(3d')

37

Electronic structure and related properties of ferrocyanide ion Table 2. Optical transitions of [Fe(CN)J’ in units of IO’cm-’ xam Eqxximental transition state concept

ci-cj

(2t

*4eg1

35.65

35.85

bands

and

(from reference

assignment (4))

23.70(cmx=4.73)

1Alg+3Tlg

(A=33.8) I 31.00(cmax=302)

lAlv*lT1v

29

(2t2v+6tlu)

37.04(~~~~%1000)

1Alv*1T2g

36.03

38.48

45.87(~~~~=24200)

lA1v+lT1u

(2t2v-2t2" ) 45.95

48.33

50.00(e,,,=23700)

'A

(2t2v+3t2v)

35.39

37.28

(2t*g+5a1g)

26.74

29.66

the one-electron levels i and j involved in an optical transition, obtained from a ground state calculation using the XaSW method does not in general compare well the experimental result. To achieve a better agreement the transition state concept [8b] must be used, the calculation being done for a state where half an electronic charge is transferred to the excited one-electron level involved in the optical transition. In fact, it is seen in Table 2 that, whereas the calculated value of the 2rZg+4e, transition energy is almost the same whether using the transition state concept or not, the charge transfer transitions are shifted to higher energies, giving a better agreement with the experimental results. According to our calculations, two other transitions from the 2tzg would occur in this region (see Table 2). The parity forbidden 2fIg -+ 3t, is predicted by the transition state calculation to occur at -37,000 cm-‘. The parity and symmetry forbidden 2tZg+5a,, would be predicted to occur at -3O,OOOcm-‘. However, the Se,, level is not reliable, since this orbital is almost entirely localized in the interstitial region and “outer sphere”. A more reliable value would require the inclusion of the next neighbors of [Fe(CN)J4 in the crystal lattice. The rather poor agreement between the 2t2g-+ 6t,. transition energy found and the experimental value may also be a consequence of the latter level having 15% of its charge in the “outer sphere” region. Although the crystal field parameter A = IODq and the transition energy between the one electron levels 2tzg + 4e, of the Xa SW calculation are not equivalent, since the former is a purely one-electron quantity and the latter involves many-electron effects, their values can be expected to be similar. The 2tzg -+4e, transition energy found, as seen in Table 2, is quite near the value of A. The transition energies of the XaSW method are rather heavily dependent on the choice of muffin-tin radii. An almost linear relationship between the 1, + e, transition energy and muffin-tin radii was found for other octahedral clusters [ 1I]. In the case of [Fe(CN)J4, a test calculation, in which the Fe sphere radius was made 0.087 a.u. smaller than the value previously used, shifted the 2t, +4e, transition energy to -3l,OOOcm-‘. Nevertheless, in the case of [Fe(CN)J4, the choice of radii is limited to a large extent by the small C-N distance. A relatively small deviation from the Fe radius used results in a relatively large difference between the C and N radii, which would be an unphysical model.

1 1g* Tl"

3.3 X-ray photoelectron spectrum Recently, the valence band X-ray photoelectron spectrum of [Fe(CN)J4 has been measured by Prins and Biloen [5]. We have compared the binding energies of the valence electrons obtained theoretically with the experimental peaks. Since the former were calculated to be at considerably lower energies, even for Watson sphere charge t4, we have shifted the XaSW energies by an equal amount, in order to fit the binding energy of the highest occupied (2t4) orbital with the peak at -2eV, following the assignment made by Prins and Biloen. This discrepancy of the absolute values occurs also for [Pt(CN)J* [ 121. We performed transition state calculations to determine the ionization energies from the different levels. Figure 1 shows the XaSW energies superposed to the experimental peaks. As far as the 9 and 6eV peaks are concerned the distribution of the theoretically obtained binding energies supports the interpretation given by Prins and Biloen, in that the 9eV peak corresponds to ionization from a c level (which, in our calculation, has a large Fe(4s) mixture) and the 6 eV peak to ionization from levels of both (T and ?r nature. Some of the energies of this last group are calculated to be quite near to that associated to the highest filled tze level. There is a large discrepancy between calculated values concerning the first group of valence levels (- 15eV) and the experimental peak at 23.5 eV, assigned in [S] to the c, orbital of CN-. This difference is indeed very large, and leads us to suggest the possibility of assigning the 18.5eV peak to the u, orbital of CN-. Although this hypothesis

Blndingenergy, eV Fig. 1. X-ray photoelectron spectrum of Li.[Fe(CN)J[S] and theoretical ionization energies (see text).

3x

D. GUENZBE~GER et al.

values of p(O) obtained against the experimental isomer

isomer shiff in mm seCi~5)

Values of p(O)(Is and 2s orbitals excluded) against isomer _ shifts of “Fe: (a) Isomer shift values measured against sodium ferr~~an~de at 3BOX (Tmutwein A.. Remmrd J. R.. Harris F. E, Fie

2.

and Gaeda Y., Whys.&TV.IV,947 @?f$. fb) FeFi-‘-value of p(U)from Ref. [61, Fe-F distance = 2.18A (estimated) and Fe muffin-tin radius = 1.76a.u.; (cl) FeFs-‘-value of p(0) from Ref, IS]. Fe-F distance= 2.06h and Fe radius = t.76a.u.: (c2f Fe&-*-vague of p(0) from f6J, same distance and Fe radius = 2.04a&.; (dll FeF,-‘--value of p(O)from 161interpolated from the curve for Fe radius = I.76 a.u. Fe-F distance = 1.85ii (see Ref. f@f; (d2) FeF,-3-approximate value of p(O),estimated from %I+ for Fe-F distance = 1.85A and Fe radius 2.04a.u., supposing the cnrve for thisradiusparallel to the curve for Fe radius = I .76a.u.

does not agree with tke assignment of &ins and Bilaen, it is nevertheless true that each of their spectra, including the one of N&N, shows a peak in this region (1g.SeV for the transition metal cyanides and - 15 eV far NaCN) whose origin is not well understood. Moreover, it is recognized by these authors that the peak at 23.5 eV has non negligible contributjons due to oxygen co~t~nin~ impurities.

3.4 M&sbauer isomer shift Iron is a Miissbauer element; the MGssbauer isomer shift is defined asIl3j:

shifts, ~~o~nnate~y, the values depend on the choice of men-TV radii, as was dicta out by tke authors in Ref. &I. However, as explained before, in our case the choice of radii is more limited. It was also noticed that the vtiue of p(0) seems to be muck less sensitive to the radii chosen than optical transitions: for the test calculation in which the Fe spkere radius was made &OR?a,~. smalfer, p(Q) of the valence orbit~s diminished by only -0.3%. While the charge at the nucleus becomes larger for some of the orbitafs involved, it becomes smaffer for the others. Tkis compensating effect results in the small change observed, Figure 2 shows the p(0) values obtained for the three other hexa&~rd~n~ed iuns FeFhm5,Fe&+ and FeFse3 (from Ref. 161)and OUTvalue of p(0) for ~F~(~N)~]-~. Though a definite value for a! cannot be given, due to the amb~ity brought in by the muon-in a~~rox~at~on~ it is gratifying to notice that the trend obtained is quite reasonable. In particular, taking the values for the Fe radius in the three fluorine complexes equal to 1.76, an almost hear re~atjonskjp is obtained Lading to a value of a = -0.29, which is very similar to the vaiue by= -0.31, obtained by ab itlitio cafeulations on iron cyanides and fluoridesli51. Trantwein and Harrisfl6J obtained values of p(0) for FeF,-I, FeFse4, [Fe(CN)J’, and other covalent complexes using a semi-empirical method, but could not cm&e them successfully to isomer shift values due TV parametrization di~culties. Calculations with the XaSW method are in progress for the ion ~~e(~~)~~-3~ for wkick the isomer skift value has been also measured. ~~knuwl~~~~m~n~s~~e of the authors (D. G.) wishes to thank Prof. J, &non for interesting discussions regarding M&sbauer parameters. Thanks are also due to L. E. de Oliveira and E. L. de Albuquerque for helpful discussions on computational problems. RElwtYmcEs

I. See, for example: &or&nation Chemistry (Edited by A. E, Marteli), Vol. i, p, 168.A&S Monograf, R&b&d, New York (f97lf: Davies I). R. and WebbG. A., Coo&. Ckem. I&B.& 95

imj:

where R, and R. are the radii of a nucleus with charge 2

in tke excited and ground states respective&. S(Z) is a correction factor due to rerat~viat~~effects, which must be used if Jm and & are non relativistic wave functions. For a given nucleus, ali factors are constant except (&+“(O) ~~(~~)=i\p(O), which is due to the difference in the electronic environment between the absorber (A) and the source (S). The quantity (Y= U&(O) is then a constant for “Fe in different com~o~nds, and many su~s~ons have been made for its value, ranging from -0.62 to -0.11[14]. The electron density at the nucleus was obtained in this calculation of [Fe(CN)J-4. The value of p(O) = @(Ofwas found to be 1187S.88a0-‘, of which 146.84aG’ is the

~~ntr~bot~ondue ?o the valence orbitals (Inrs-4urpf. The same result was obtained for Watson charge +4 and +5. Since the XffSW method has been used to study isomer ski&s of other Fe ciusters[6], we have plotted (Fig. 2) the

2. Slater J. C. and Johnson K. H., Pkys Rev. SB, 844 (1972); Slater J. C., ~~~~~~~s itl Q~~n~um C&&fry @J&XI by P.-O. Liiwdinf, Vol. 6, p. 1. Academic Press, New York (1972); Johnson K H., Advances in Quantum Chemistry (Edited by P.-O. Lawdin), Vol. 7, p, 143. Academic Press, New York (@73). 3. See, for example: Johnson K. H. and Weaken U., fn& L Quanf. Ckem. 6,243 (1972); Johnson K. H. and Smith Jr. F. C., Phys. &co. SB, 831 {lY72); Messmer R. P., Wahlgren U. and Johnson K. H., C&em Pkys. Lefters 18, 7 (1973j. 4. Gray H. 8. and Beach N. A., J. Am. Ckem. SK 85.2922 (I%%);Alexander J. J. and Gray H. B., J. Am. Ckem. Sot. 90, 4260 W58~: Alexander J. 1. and Grav H. 3.. Coot-d. Ckem. &?a i, 29 $6?). 5. Prins R. and Biloen P., C&m. P&s. Letters 30, 340 (19’75). 6. Larsson S., Viinikka E.-K,, de Siiueira M. L. and Connolly J.

7. Wyeoff R. W. G., Crysfai Structures, 2nd Edn, Vol. 3,6. @3?, Interscience, New York (1965). 8. (a) Slater J. C., Q&u~~~mTkeary ofMoleah and LWids, Vol. 4, p. 27.McGraw-Hi& New York ff974); (b) Same references, p. 51. 9. Pau& L., The Nature of the C~rnfe~ Bond. p. 230. Cornell University Press, Ithaca (1960).

Electronic structure and related properties of ferrocyanide 10. Riisch N., Klemperer

W. G. and Johnson K. H., C/rem. Phys.

Lefterr 23, 149 (1973). I I. Albuquerque E. L., Maffeo B., Brandi H. S. and de Siqueira M. L., Solid State Comm. 18, 1381 (1976). 12. Iterrante L. V. and Messmer R. P., Chem. fhys. Letters 26, 225 (1974). 13. Herber R. H. and Goldanskii V. I., Chemical Applicafions of

ion

39

Miissbouer Spectroscopy. Academic Press, New York (1968). 14. Duff K. J., Phys. Rev. 98. 66 (1974). IS. Sawatzky G. A. and van der Woude F., .I. Phys. 12C6, 47 (1974). 16. Trautwein A. and Harris F. E., Theoret. Chim. Ada 30, 45 (1973).