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Electronic structure of ferrocyanide ion calculated by the SCF Xα-scattered wave generalized partitioning method
J. Fbys, C/m+. So/ids Vol. 43, No. Printed in Great Britain.
1I.pp.lOSflW1, 1982
ELECTRONIC STRUCTURE OF FERROCYANIDE ION CALCULATEDBYTHE SCF Xc-x-SCATTERED WAVE GENERALIZED PARTITIONING METHOD? KAZUNORI WATAIUand Josh R. LEITE Institute de Fisica, Universidade de 90 Paulo, Caixa Postal 20516,.%o Paulo, Bra&l
and MANOELL. DE SIQUEIRA Departamento de Ffsica ICEx, Universidade Federal de Minas Gerais, Caixa Postal 1621,3OOOO Belo Horizonte, Brasil (Received 24 February 1982;accepted
26 March 1982)
Abstract-A generalized partitioning (GP) version of the Xo scattered-wave (SW) method is used to carry out self-consistent calculations of the electronic states for the ferrocyanide ion. The GP scheme is based on a generali~tion of the standard muffus-tin petitioning of the molecular space in which the neighboring atomic regions are main~ned into local clusters. The calculated energy spectrum leads to inte~re~tion for the X-ray pho~iectron emission and optical transitions for the ion in fairly good a~eement with the ex~rimen~ resufts. It is concluded that the self-consistent GP scheme offers very good possibilities to improve the physical realism of the SW theory for a wide range of open structures. 1. INTRODUClYON
The Xn-scattered wave method (SW), developed by Johnson and collaborators at the end of sixtiesIll, has been applied successfully to a wide range of molecules and molecular clusters. The method has been extensively used in its muffin-tin (MT) form[2-51. However, the unsatisfactory features of the MT approximation applied to open structures, such as diatomic@] and large planar molecuIes[7], are welt known. The rigorous extension of the SW method to non-MT poten~als, that has been tried until now, turned out to be difhcuit and costly to implement [8,9J. A practical and useful way to introduce non-MT corrections in the SW method is the straightforward use of overlapping atomic spheres (OS). Most of the recent applications of the SW method has been made within the framework of the OS model [ 10-121. In the SW method the molecular system is partitioned into three different regions: atomic, interatomic and extramolecular. The atomic and extramolecular regions are limited by spherical surfaces. The one-electron Slater Xo equation is solved inside each region, using the MT approximation: the potential is spheric~ly symmetric in the atomic and extramolecuiar regions and constant in the interatomic region[l]. For some molecular species the MT approximation is poor because of the large volume where the potential is constant. The OS method circumvent this limitation, by reducing the bad effects of a constant potential in a much extended region. In the present work we implement another alternative way to increase the physical realism of the MT model in the SW method. The proposed model is mathematically rigorous and retains most of the computational efficiency of the original scheme. It is based on a generalization of the SW theory in which the standard MT p~tition~g of tWork supportedby FAPESP and FINEP, Brasil.
the molecular space is extended by maintaining neighboring atomic regions into local clusters. The formal extension of the SW theory to this generalized partitioning scheme (GP) was already made by Kjellander[l3]. The GP theory was then applied to study the coupling between units in partionable systems. This was performed by treating each local cluster separately, subject to boundary conditions’ which describe the coupling of the cluster to the rest of the molec~e[l3,14]. Our work is d~eren~y motivated. We sdve the GP secular equation seIf-consistently as a whole. Our procedure is exact, therefore no approximations are assumed to handle the couphng between units of the system. The self-consistent field scattered wave GP method has the essential features to improve on the standard MT model. As the first test case for the application of the scheme devised in this paper, ferrocyanide ion was chosen. The SW method within the framework of the standard MT approximation was recently applied to calculate the electronic states of this system[Z], It was found that a large amount of the electronic charge is distributed in the interatomic region. Due to the Iarge volume of this region, the MT approximation is rather poor for the [Fe(CN),J-. The limitation of the MT model in this case is consequence of the small C-N distance. On the other hand, within the context of the GP formalism, the definition of six local CN clusters for the ferrocyanide ion avoids most of the undesirable charge transfer to the interatomic region, increasing considerably the physical realism of the SW method.
The theoretics formulation of the SW method has been described in detail in several papers& 21. In order 1053
IlMOqS
OS[e S!
salaqds aq) JO%u!Jaqutnuau ‘smotex!s bq pasodmozalmalow JOJ slalsn[:, p2sof OMI ql!~ Bu!uog!$mdpaz!fvlauag ‘1 ‘SF6
se “p 3 b ~alua:, E 30 Al&q+ aql le uo!suedxa la$uas-a@s e u! ual$!jM aq uw (z) uba ‘[~]sura~oaql uo!suadxa aaem-pJ!$red p~pue$s aql Su!sn 1(8 *lalsnl3 1~01 e Su!sopua s alaqds aql 30 uo!lala~&alu! Ienp aql 01 anp luellodw! s! xplew rrqn3as aql 30 6$!3y!w3aq ayJ, ‘dYv sluapylao3 aql u! papnl:, -u! aJ= ahoqe pauopaw s1013e3aql asneDaq uql!wlaq lou s! [f~] ‘Jag u! paq!IDsap xgew nqn3as aqL .w103 uugwIaq aql u! d9 30 xym .u2lmas aql 01 spaal (z) uba u! 61lgdxa uall!lm “a,+,([-) pue “2 s~ope3 aqL
‘0 zn’ 3! ‘“Sflnd~ 0 = TIJ! ‘alaqds lalno ~OsflO” 11eIgun ‘s aJaqds palaqwnu Jarno lxau aql 01 sampaDold !slua!sgaoD uo!suedxa alz ,,Yv !(rl uo!Sar c)!wollzJa$u! ql3y puu qlrno3 am leadal ‘qlx!s tqvno3 u! s bq paugap Ialsn13 1~301nqn3!ln?d aq$ 01 Su!SuoIaq salaqds ~!wole sasolsua l!)Os 01Su!SuoIaq s aJaqds ~eln~!lIed t! 30 aX3ms ‘ql3y !(AI uo!SaI wol3 alaqds ,,lalno,, se paMa!A Ial rauu! aq$ slaqwnu “s fsalaqds 3!wole SEuaas alo3alaql -snp [e~o[) Sugaqwnu 1saMoIql!m s aIaqds laIn3!$1ed 30 ‘(0 = ti) 11uo!Sal u10.13 uaas (s alaqds) slalsnls 1~30130las e s! 0s ! d uo;rSal 3!wolwalu! 01 Su!SuoIaq salaqds 3!woltz aX3’ns Jauu! ‘qvno3 :(II uo!Sal woq salaqds ,,~!wole,, se pama!A slalsrqa [WOI) s salaqds 30 saaepns Jalno 30 las I! s! “m tpu!~ luy aql30 uog3un3 IayuvH p+aqds ‘pnqi f(sialsnl3 1e90101%u!Suolaq$0~) 11uo@aI aql utq$!M 1 luoqDun3 utzwnaN p+Iaqds heup~o paytpow e s! (,,Y . salaqds ~!mole ‘puoDas :(I palaqwnu s!) aIaqds lalno ue s!(x)luIdIaA!l3adsa~‘suo!l%n3 Iassaa IcD!ladspay!pow pue h2wp10 ale (x)1! pue (x)y !(ur ‘1)= y !9!uowJaq ‘iuy :saraqds 30 Su!Iaqwnu aq$ 01 Su!laplo p?guanbas aql $dopa aM ‘I ‘Sg aql u! Moqs aM sy p+aqds ItraI B SF(di)Yd ! d raluaa sauyap dx ‘da - J = do SU!MO~IOJ .samnlDnrls@ulalxa aql alou%! aM aDup aJaqds ,,lalno,, U’I!se l! lea31 kern aM ‘(~1 uoySar) ap!su! aql ~0.13 s alaqds 30 acejms aql q3eoIdde aM uaw .sp$ap
“n.> 23
‘%
3
“d>El “fj
d 10 “~11% 3 d “‘s = d JO 1 = d
“‘m 3 d JO 0~~0” 3 d “4 > B
‘“s = d
10
1= d
‘(“.oI\r)‘C”a
I
I
-
a3aqm Y
(Z)
(dd)y~(dol)d’ldyv x
+3*
x =(J)Y’
uo!suedxa ahEM Iegled IaluaD!$Inwe u! ual$gM aq Aew (I) uba 30 uo!ln[os aql ‘uo!Sal 3!woleJalu! qciea uI ‘0 +d uaqM AI uo!Sal ~!wol~~alu! 01 pw 0 = rl 3111uo!SaJ 3!wolelalu! 01 sla3al rl xapu! aql fhaqpdx u! uaA!S d%laua aql s! g !uo!Sar ?woleJalu! q2ea u! Iegualod luelsuo~ s! *A aJaqm
(I)
. . . ‘z ‘0 =
‘I
‘0
d
=
(J)‘+d’(‘+d - 3 t ,A)
uogenba laStnpaolq3s aql d3sges lsnw suo!Sal 3!wolelalu! aql u! suogsun3 aAvM aqL .uo!leluawaldw! Iauogelndwos aql pus X~$L?W Iepvas aqi 30 uogela~cLra$u!aq$ Qydw!s sa1n.I Suilaqwnu asaqJ .paJappuo2 uaaq aAeq sJa$snI:, 18301
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1055
Electronicstructure of ferrocyanideion where I h(K,r,), g,‘YK,r,) =
h(K,r,),
q= 1 or qEo,JJS,, q=l
k,“‘(K,r,), i i0Lr,),
q= s,, or
qEti,,
or q=s,,
q E ooUSo
or
034
E> V&
(6b)
E>V*
64
E
q E o,,
E < P&;
6-3 (7)
with ‘I’+=-‘nL(K,R4J,
q,pEwoUSo
1
q=l q=S,,
1 ‘J’+L-’j,(K,R,,), #,‘YK,R,,)
= . (-l)“+‘k,“‘(K,R,),
and
or
q=l q=S,,
or p=l or p=S,
p E &,
p# u.
The wave functions inside atomic spheres and outside the outer sphere are written as $“(r,) = c C,‘u,P(r,, E)Y,VJ,
q,pEo,,
@a)
E%
or p=l when q,pE& E>V* or p = S, when q,p E 6,, ’
q,p EocUSo
~-~)~‘+‘j~~K~R~), 0 if q E (j,
or
q,p E +,
(gc)
E
when q,pE& when
0%)
E
(gd) (Se)
The same conditions for the wave functions at the boundaries of region IV (p+ 0 in eqn 5) and atomic region q E op, p f 0, lead to the equations
(9)
where $‘(r,, Ef are numerical solutions of the radial Schroedinger equation 4 f %-
V&,)-E
I
u:(r,,E)=O
(10)
for the spherically averaged potential Vp(rP) and a given energy E. The coefficients C” are related to A” by means of the equation
(13)
Note that eqns (12) and (13) are exactly those of the standard SW method the only difference being the sum over the scattering centers that belong to interatomic regions of different hinds. The diagonal matrix elements, t;/, are the inverse of the relative amplitudes of scattering of the Ith partial wave of energy E, given by
AiP = (I- 2~*,)b~*W[g~(K~b~), HtP(bp,E)lChP
(11)
when p = 1 or p E o,_ p = 0,1,2, . . . [ 11,The parameter b, is the radius of sphere p and W[f(x),gfx)] refers to the Wronskian of two functions f(x) and g(x). The requirement of the continuity of the wave functions (and their normal derivatives at the boundaries) of region II (CL= 0 in eqn (5)) with the atomic region q E too, or the extramolecular region, leads to the set of equations
q=I IPCS Vol. 43, No. I I-C
or
qEo*
(12)
4
65@,7 )L=o, 1,2 , * . .*
(14)
The off-diagonal elements GE. are caIled structure factors, and we may prove that they are symmetric (hermitian). We must also require the continuity of the wave functions and their derivatives on the surfaces of the spheres S. At the vicinity of the sphere S, numbered 4, enclosing the region IL,we have the h component of the wave functions
K. WATARIet al.
1056
in the outer side and
(16) in the inner side of the same sphere. Imposing the continuity of expressions (15) and (16) and their normal derivatives on the surface of sphere S(r, = bp), and eliminating the term
we have
where
Wg,SrW,b,), fiPW&J
ta’=W,"(K,b,),
g?(KobJ'
(18)
and 1 8%
h
1
=
b,2Wk,S’(&h_A
d’Woba)l’
(19)
3. APPLICATIONTO FERROCYANIDEION
3.1 GP partitioning for [Fe(CN)JIn this section we report on the results for the electronic states of [Fe(CN),J- as calculated by the selfconsistent Xo scattered wave GP method. The GP partitioning for this system is schematically shown in Fig. 2. The standard MT division of the molecular space is modified by introducing six spheres S which define six new regions of constant potential (regions IV). The S spheres are tangent to each group CN. The potential V,, is obtained by the usual procedure of volume averaging the molecular potential in
The same statement with elimination of the term
P’P
leads to the equations Ih&PAhB + t-’$1 A*% +p~Up~G:RLP=O
Now, we have a complete set of homogeneous linear equations composing a secular equation for GP theory. These equations are given by (12), (13), (17) and (20). From them we may infer the general structure of the secular matrix of GP. There are blocks along the diagonal with, in general, non zero elements. Each of these blocks corresponds to the secular equation of the standard SW method for each local cluster. There are also blocks where all the elements are null. These blocks lie in the crossing region of the indexes corresponding to the different local clusters in the secular equation. Thus, there is no direct coupling between the internal regions of different local clusters. Finally, there exist blocks with only coupling factors as non-zero elements. These blocks give the coupling between inner side and outside of the local clusters. Another fact apparent in GP is that the inclusion of one sphere S enclosing a local cluster is equivalent to adding two spheres in SW. This increases the dimension of the secular equation, but it does not increase significantly the difficulty in the computations since the secular matrix of GP is filled with a considerable amount of zeros. We have said above that the structure factors are symmetric (hermitean). Therefore, due to eqn (22), the secular matrix of GP is hermitean and hence there are only real eigenvalues corresponding to zeros of the associated determinant.
(20)
where ti;, =
WLf?“W,b,), dWobs)l Wk,SCW,b,h dWobp)l'
(21)
Equation (17) and (20) are the equations of SW method except for the coupling factors lhps* and I,‘@. Equation (17) shows that the S spheres enclosing the local clusters may be seen as atomic spheres if we ignore the.internal structures, as we have discussed earlier. In the same way, eqn (20) allows us to see the same sphere as an outer sphere to the local cluster if we ignore the external structure. The terms t ;i and t 2, play the same role as t ;,’ at re = b,+ The terms lhes* and I*‘*@couple the interatomic regions II and IV and we shall call them coupling factors.
Fig. 2. Generalized partitioning for [Fe(CN)$.
Electronic structure of ferrocyanide
Interatomic
Radius
molecular orbitals (MO’s) which we classify according to the irreducible representations of Oh symmetry group.
Distances Fe-C:
3.6282
C
2.1863
-N:
near the
2.4818
Rc
=
1.1464
RN
=
1.0399
=
2.1863
=
6.1355
RS R
Exchange
(in
=
RFe
3.2 Electronic structure The GP calculation was carried out by starting with the SW self-consistent potential. Therefore, in the first iteration V,” = VI,. It is interesting to compare the transference of charge to the local clusters for different iterations during the convergence process. This is done in Fig. 3 where we show the behaviour of the one electron energy spectrum of the ion as a function of the number of iterations. The values of the constant potentials V,, and V,, are also shown. Due to the charge transfer to the local clusters the potential V,r becomes more shallow and the potential V,, deeper than the
(in
of the spheres
The calculations were performed with a charge of t5 at the Watson’s sphere. The use of 4 positive charges
0.0 I_ E(Ry)
1057
Table 1. Parameters used in GP scheme for [Fe(CN)&
the region IV. In order to compare our calculations with the previous one, carried out within the framework of the SW method, we are adopting here the same geometrical and exchange parameters used by Guenzburger et al. [15]. The values are indicated in Table 1. The outer sphere is also used as a Watson’s sphere to simulate the stabilizing effect of the lattice on the ion. The total of 108 electrons in [Fe(CN)$- ion is filling
places the lowest virtual orbitals irrealistically continuum as in the case of SW calculation.
ion
out
a
a.".
)
a.".
)
parameters
cc(Fe)
=
0.71
a(C)
=
0.76
a(N)
=
0.75
a(111
=
(r(III)
=
cr(IV)
_/-
,
-0.5
,
.
-1.0 I-
- 1.5,is
L._._,* -_ .___._
. . . ..__.
VIV _.-
% I
l
2
f-Mm--
5
IO
20
25
30
Iteration Fig. 3. Convergence process in GP scheme. The potential for the first iteration was that obtained from the self-consistent SW calculation with a Watson’s sphere charge t 4. The empty levels were shown only for the first iteration. The constant potentials VI, and VIVare also shown.
=
0.75
1058
K. WATARIet al.
self-consistent interatomic potential obtained from the SW calculation. The self-consistent one electron energy spectra for the ground state of the ion obtained from the SW and GP calculations are shown in Fig. 4. The levels that lie above 2fZg in the spectra are the first empty levels. One important difference between the two spectra is that in GP calculation the highest occupied level is 2tz, and the lowest empty level is 4e,, while in SW calculation they are 2tzn and 5a,g, respectively. The ordering of these levels in GP is according to that obtained by Gray et al. by using the Linear Combination of Atomic Orbitals (LCAO) method[l6]. This result seems to indicate an improvement of the molecular potential in the GP method. The charge distribution for the ground state of the ferrocyanide ion for the different molecular regions, obtained according to GP calculation, are shown in Table 2. The comparison between these results and those derived from the SW calculation (see Table 1 of Ref. [ 151)has to be made with some caution, because there is only one interatomic region in SW while there are seven in GP. Furthermore, the regions II in SW and GP have different volumes. In order to make a fair comparison between the results we decided to divide the charge of region II in SW in two parts. One occupying a volume corresponding to the region IV of GP and other the volume equivalent to the region II of GP. With this procedure one obtains the results shown in Table 3.
One observes that the net effect of the generalized partitioning scheme is to transfer charge from the MT standard region II to the region IV of the local clusters. According to Table 3, the charge within the region II in SW calculation is reduced by a factor of two and the charge within the region IV in GP is four times greater than the charge found in the equivalent region in SW. By adding the chdrges in the C and N atomic spheres to the charge of region IV one obtains -9.1 electrons according to SW results. The total charge within the local cluster in the GP calculation is - 11.2 electrons, which is in better agreement with the expected value for the cyanide group, say -13 electrons. These results are strong evidence of improvement on the MT potential, indicating that the generalized partitioning scheme leads to more realistic charge distributions. The definition of the local clusters avoids the transference of charge to region II, gathering it in the correct places. 3.3 X-Ray photoelectron spectrum The valence band X-ray photoelectron spectrum of the ferrocyanide ion was obtained by Prins and Biloen[lll]. In order to compare our results with this experiment, the transition state concept was used to obtain the one electron ionization energies [ 181. The binding energy of the highest occupied orbital (2t,,) was used as a reference to adjust the experimental peak at -2 eV [15]. Figure 5 shows the GP energies superposed to the experimental peaks. The 6eV peak corresponds essen-
o.o-
,_-’ .2’2”
,a’
.-6t,. _,.* $I._. - - 4 l p
-0.5--
,,,. ‘,.59”
“..3126 “‘..60
-..
-’ 16
,,’ 2’26 ,’ *“lq ,’ ,3t,u .’ ..’
-’
-:.y,
cr” QJ
..
.3*
,/ .*
-‘*___--“2”
. .
I’
,’ ,’ , ’
D
.401q I
I’
,‘,’ : I, <._
4’1” .I110
,’
.2cq
-:_::.--..,t2q
_.-41,”
-I.O--
-,
-*.
,’
-~_:__-.I~2” ,_:::L.._,eq --:---.._
,‘_. ,:_’
..__ .3’,” “...._5a,q
-..
“nzq
_ _ “*40,q ‘..:‘.2e . ..
a 3’1”
-. -----...%3
-1.5-.,lC
_4ff:;....2’l”
,-
q
“‘.20,q
-2.0 --
0 0
Fig. 4. One electron energy spectra according to SW and GP calculations. Watson’s sphere charge t 5.
1059
Electronic structure of ferrocyanide ion Table 2. Energy levels and charge distribution for [Fe(CN)$ according to GP calculation. The value t5 was assumed for the charge at the Watson’s sphere. The six lowest empty orbitals are included: (a) all energies in Rydberg; (b) charge distribution in % of one electron; (c) orbitals marked (*) are empty lEnergy (a)
levels
Charge Fe sphere
(c)
stributior
C sphere
N sphere
(b) II
III
-509.029
lOO(lS)
-
-
-
-58.988
lOO(2S)
-
-
-
-
-
lOO(2Pf -
-
-
-
-
-
-
-
-
-
-
-
-50.981 -28.205
-
-20.275 -6.520(1a,g) -4.196(ltlu)
-
16.67(1s)
-
too
16.67(1s) -
-
-
-
too
--
-
-
-
-
-l.725(2a1g)
0.50
2.90
4.32
9.54
7.66
0.74
-1.717(2t,")
0.17
2.89
4.48
8.70
7.71
0.65
-1,714(1eg)
0.20
2.90
4.52
7.73
7.74
1.17
-0.993(3a,g)
18.23
2.17
2.48
33.85
3.12
1.25
-0.907(3tl")
3.14
2.04
3.53
29.18
5.89
0.01
-0.904(lt2g)
2.03
1.92
3.30
24.21
6.81
1.60
-0.864(lt2U)
0.11
2.10
3.77
19.29
7.32
1.44
15.56
1.80
4.72
19.29
3.41
5.56
-0.843(ltlg)
0.05
2.17
3.98
16.18
7.51
1.74
-0.827(4tfu)
2.34
1.89
5.47
21.41
4.57
4.66
-0.855(2eg)
3.21
3.36
6.53
26.90
0.30
8.73
29.84
3.70
4.31
16.95
0.0
5.18
5.29
5.29
4.16
32.21
0.35
3.68
)
90.87
0.06
0.20
6.20
0.20
0.15
-0.273(4eg 4*
54.10
3.12
0.62
la.41
0.59
-0.218(5a,g)'
3.25
0.32
1.35
12.02
0.0
-o.l98(3t*g):
0.07
0.05
0.63
19.62
0.28
74.63
-0.173(6t1,)
0.35
0.49
1.27
0.0
0.0
89.07
-O.l12(5eg)*
0.19
0.24
0.48
0.0
0.0
95.50
-o.l10(2t*,)'
0.0
0.01
0.34
10.12
II.10
87.19
26.50
3.61
4.53
12.61
3.09
-0.684(4a1g) -0.595(3eg) -0.577(5t,") -0.493c2t2
Electron distribution. Total of 108 electrons
1.54 74.71
1.53
Table 3. Total charge distribution for the different regions of the molecular space. The interatomic region in the SW scheme was divided artificially in regions II and IV, equivalent to those defined in the GP theory
tially to ionization of u and B MO’s according to our calculations. This is in agreement with the interpretation given by Prins and Biloen. The theoretically obtained binding energy of 8 eV corresponds to a (I MO. This value differs slightly from the 9eV peak assigned by Prins and Biloen to the ionization of a u MO. By comparing with the previous SW results, we observe a decreasing in the iron 4s contribution for this u statell51. Although the inte~re~tion of the low energy peaks is practically the same provided by Guenzb~ger et al., the GP model leads to an interpretation apparently more
realistic to the 18.5eV experimental peak. According to
our calculation, it is originated from the first group of cr valence MO’s with ionization energies of -18eV. According to the SW results this peak is located at -15 eV. 3.4 Optical transitions In this section the GP results are used to interpret the optical spectrum of (Fe(CN)&” obtained by Gray et a[.[ 161.The catculated ground state energies, associated to the measured optical transitions, are dispIayed in Table 4. The SW results as well as the expe~men~ values are shown. It is observed that the energy
1060
K. WATARI
et al.
1
I
20
IO
0
energy(ev) Fig. 5. X-Ray photoelectronspectrum of L&[Fe(CN)G][17] and GP theoreticalionizationenergies. Table 4. Interpretation of the optical spectrumof the ferrocyanideion accordingto the SW and GP calculations. Energiesin units of 10’cm-’ GP
Experlmental
2t2g
g
35.65
36.03
24.08
38.48
(A.33.8)
34.98
42.13
48.33 37.28 26.74
a: Reference b: Transition c: Reference
and
assignmentC
‘i-‘j
+ 4e
bands
23.70
1Atg +3Tlg
31.00
1Alg+
37.04
1Alg+. 'T 2g
45.87
1Alg +'T,u
50.00
1Alg+
'T ,g
'TlU
32.26
29.66
36.98
15 state
concept
- Reference
18
16
differences, ei - q, corresponding to one-electron levels i and j involved in the optical transitions (Koopmans’ method), do not compare well with the experimental results. Hence, the transition state concept[ll], where half of an electronic charge is transferred to the excited one-electron state involved in the optical transition, was used in order to achieve better agreement with the experimental results. According to the values shown in Table 4, the electron relaxation effects affect more the GP results than the SW one. This is expected since the GP method leads to a rather diierent charge distribution for the ion when compared with SW result. Table 4 shows that the GP method leads to an interpretation for the optical spectrum of [Fe(CN)Jin fairly good
agreement with that inferred from the experimental results by Gray et al. [ 161. 4. CONCLUSIONS
The SW method, in its standard MT form, has only one constant potential for the interatomic region. When this method is applied to the study of a variety of materials that have radicals or molecules surrounding a metallic ion, the charges of the ligands are spread out. Thus the method leads to a non-realistic electronic structure for the molecules or the clusters. The GP theory allows for the systematic study of this kind of materials. The simple application to the [Fe(CN)J- shows that the main difiiculty with the SW
Electronicstructure of ferrocyanideion theory is overcame. On the other hand, one expects that the GP formalism will be a useful tool to deal with open structures in general. Before concluding this work a final comment should be made: the GP scheme retains most of the flexibility of the standard SW method. By considering the previous application of the SW scheme to the ferrocyanide ion we conclude that only a small extra computational effort was necessary to implement the calculations reported in this paper. Acknowledgements-The authors wish to thank Profs. JosC R. P. Neto and A. Fazzio for many helpful discussions. Thanks are also due to Dr. Marflia J. Caldas for the critical reading of the manuscript. REFERENCES
1. Johnson K. H., Adoances in Quantum Chemistry (Edited by 2. 3. 4. 5. 6.
P. 0. Liiwdin), Vol. 7, p. 143. Academic Press, New York (1973). Johnson K. H., Ann. Rev. Phys. Chem. 26,39 (1975). Braga M., Larsson S. and Leite J. R., J. Am. Chem. Sot. 101, 3867(1979). Fazzio A. and Leite J. R., Phys. Reo. B 21,471O(1980). De Leo G. G., Watkins G. D. and Fowler W. B., Phys. Rev. B 23, 1851(1981). Weinberger P. and Konowalow D. D., Int. J. Quant. C/tern. s7, 353 (1973).
1061
7. Herman F. and Batra I. P., Phys. Reo. Lett. 33, 94 (1974); Batra I. P., Bennett B. I. and Herman F.. Phvs. _ Reo. B 11. 4927(1975). 8. DaneseJ. B. and ConnollyJ. W. D., J. Chem.Phys. 61,3063 (1974):Danese J. B.. J. Chem.Phvs. 61. 3071 (1974): Chem. Phys.’iett. 45, 150(i977). ’ . ” 9. Ferreira L. G., Agostinho A. and Lida D., Phys. Rev. B 14, 354 (1976). 10. Rosch N., Klemperer W. G. and Johnson K. H., Chem. Phys. Lett. 23, 149(1973). 11. Howard I. A., Pratt G. W., Johnson K. H. and Dresselhaus G., J. Chem. Phys. 74,3415 (1981). 12. Braga M., Pavao A. C. and Leite J. R., Phys. Reu. B 23,4328 . _ (1981). I’. Kjellander R., Chem. Phys. Lett. 29,270 (1974);Chem. Phys. 12,469 (1976);Chem. Phys. 20, 153(1977). 14. Johnson K. H., Herman F. and Kjellander R., Electronic Structure of Polymers and Molecular Clusters (Edited by J. AndrC, J. Ladik and J. Delhalle), Vol. IV, p. 601. Plenum Press, New York (1975). 15. Guenzburger D., MatTeoB. and De Siqueira M. L., J. Phys. Chem. Solids 38, 35 (1977). 16. Gray H. B. and Beach N. A., J. Am. Chem. Sot. 85, 2922 (1%3); Alexander J. J. and Grav H. B.. J. Am. Chem. Sot. 90. 4260 (1968); Alexander J. J. and Gray H. B., Coord. Chem: Rev. 2, 29 (1%7). 17. Prins R. and Biloen P., Chem. Phys. Lett. 30,340 (1975). 18. Leite J. R. and Ferreira L. G., Phys. Rev. A 3, 1224(1971);J. C. Slater, Quantum Theory of Molecules and Solids, Vol. 4. McGraw-Hill, New York (1974).
1I.pp.lOSflW1, 1982
ELECTRONIC STRUCTURE OF FERROCYANIDE ION CALCULATEDBYTHE SCF Xc-x-SCATTERED WAVE GENERALIZED PARTITIONING METHOD? KAZUNORI WATAIUand Josh R. LEITE Institute de Fisica, Universidade de 90 Paulo, Caixa Postal 20516,.%o Paulo, Bra&l
and MANOELL. DE SIQUEIRA Departamento de Ffsica ICEx, Universidade Federal de Minas Gerais, Caixa Postal 1621,3OOOO Belo Horizonte, Brasil (Received 24 February 1982;accepted
26 March 1982)
Abstract-A generalized partitioning (GP) version of the Xo scattered-wave (SW) method is used to carry out self-consistent calculations of the electronic states for the ferrocyanide ion. The GP scheme is based on a generali~tion of the standard muffus-tin petitioning of the molecular space in which the neighboring atomic regions are main~ned into local clusters. The calculated energy spectrum leads to inte~re~tion for the X-ray pho~iectron emission and optical transitions for the ion in fairly good a~eement with the ex~rimen~ resufts. It is concluded that the self-consistent GP scheme offers very good possibilities to improve the physical realism of the SW theory for a wide range of open structures. 1. INTRODUClYON
The Xn-scattered wave method (SW), developed by Johnson and collaborators at the end of sixtiesIll, has been applied successfully to a wide range of molecules and molecular clusters. The method has been extensively used in its muffin-tin (MT) form[2-51. However, the unsatisfactory features of the MT approximation applied to open structures, such as diatomic@] and large planar molecuIes[7], are welt known. The rigorous extension of the SW method to non-MT poten~als, that has been tried until now, turned out to be difhcuit and costly to implement [8,9J. A practical and useful way to introduce non-MT corrections in the SW method is the straightforward use of overlapping atomic spheres (OS). Most of the recent applications of the SW method has been made within the framework of the OS model [ 10-121. In the SW method the molecular system is partitioned into three different regions: atomic, interatomic and extramolecular. The atomic and extramolecular regions are limited by spherical surfaces. The one-electron Slater Xo equation is solved inside each region, using the MT approximation: the potential is spheric~ly symmetric in the atomic and extramolecuiar regions and constant in the interatomic region[l]. For some molecular species the MT approximation is poor because of the large volume where the potential is constant. The OS method circumvent this limitation, by reducing the bad effects of a constant potential in a much extended region. In the present work we implement another alternative way to increase the physical realism of the MT model in the SW method. The proposed model is mathematically rigorous and retains most of the computational efficiency of the original scheme. It is based on a generalization of the SW theory in which the standard MT p~tition~g of tWork supportedby FAPESP and FINEP, Brasil.
the molecular space is extended by maintaining neighboring atomic regions into local clusters. The formal extension of the SW theory to this generalized partitioning scheme (GP) was already made by Kjellander[l3]. The GP theory was then applied to study the coupling between units in partionable systems. This was performed by treating each local cluster separately, subject to boundary conditions’ which describe the coupling of the cluster to the rest of the molec~e[l3,14]. Our work is d~eren~y motivated. We sdve the GP secular equation seIf-consistently as a whole. Our procedure is exact, therefore no approximations are assumed to handle the couphng between units of the system. The self-consistent field scattered wave GP method has the essential features to improve on the standard MT model. As the first test case for the application of the scheme devised in this paper, ferrocyanide ion was chosen. The SW method within the framework of the standard MT approximation was recently applied to calculate the electronic states of this system[Z], It was found that a large amount of the electronic charge is distributed in the interatomic region. Due to the Iarge volume of this region, the MT approximation is rather poor for the [Fe(CN),J-. The limitation of the MT model in this case is consequence of the small C-N distance. On the other hand, within the context of the GP formalism, the definition of six local CN clusters for the ferrocyanide ion avoids most of the undesirable charge transfer to the interatomic region, increasing considerably the physical realism of the SW method.
The theoretics formulation of the SW method has been described in detail in several papers& 21. In order 1053
IlMOqS
OS[e S!
salaqds aq) JO%u!Jaqutnuau ‘smotex!s bq pasodmozalmalow JOJ slalsn[:, p2sof OMI ql!~ Bu!uog!$mdpaz!fvlauag ‘1 ‘SF6
se “p 3 b ~alua:, E 30 Al&q+ aql le uo!suedxa la$uas-a@s e u! ual$!jM aq uw (z) uba ‘[~]sura~oaql uo!suadxa aaem-pJ!$red p~pue$s aql Su!sn 1(8 *lalsnl3 1~01 e Su!sopua s alaqds aql 30 uo!lala~&alu! Ienp aql 01 anp luellodw! s! xplew rrqn3as aql 30 6$!3y!w3aq ayJ, ‘dYv sluapylao3 aql u! papnl:, -u! aJ= ahoqe pauopaw s1013e3aql asneDaq uql!wlaq lou s! [f~] ‘Jag u! paq!IDsap xgew nqn3as aqL .w103 uugwIaq aql u! d9 30 xym .u2lmas aql 01 spaal (z) uba u! 61lgdxa uall!lm “a,+,([-) pue “2 s~ope3 aqL
‘0 zn’ 3! ‘“Sflnd~ 0 = TIJ! ‘alaqds lalno ~OsflO” 11eIgun ‘s aJaqds palaqwnu Jarno lxau aql 01 sampaDold !slua!sgaoD uo!suedxa alz ,,Yv !(rl uo!Sar c)!wollzJa$u! ql3y puu qlrno3 am leadal ‘qlx!s tqvno3 u! s bq paugap Ialsn13 1~301nqn3!ln?d aq$ 01 Su!SuoIaq salaqds ~!wole sasolsua l!)Os 01Su!SuoIaq s aJaqds ~eln~!lIed t! 30 aX3ms ‘ql3y !(AI uo!SaI wol3 alaqds ,,lalno,, se paMa!A Ial rauu! aq$ slaqwnu “s fsalaqds 3!wole SEuaas alo3alaql -snp [e~o[) Sugaqwnu 1saMoIql!m s aIaqds laIn3!$1ed 30 ‘(0 = ti) 11uo!Sal u10.13 uaas (s alaqds) slalsnls 1~30130las e s! 0s ! d uo;rSal 3!wolwalu! 01 Su!SuoIaq salaqds 3!woltz aX3’ns Jauu! ‘qvno3 :(II uo!Sal woq salaqds ,,~!wole,, se pama!A slalsrqa [WOI) s salaqds 30 saaepns Jalno 30 las I! s! “m tpu!~ luy aql30 uog3un3 IayuvH p+aqds ‘pnqi f(sialsnl3 1e90101%u!Suolaq$0~) 11uo@aI aql utq$!M 1 luoqDun3 utzwnaN p+Iaqds heup~o paytpow e s! (,,Y . salaqds ~!mole ‘puoDas :(I palaqwnu s!) aIaqds lalno ue s!(x)luIdIaA!l3adsa~‘suo!l%n3 Iassaa IcD!ladspay!pow pue h2wp10 ale (x)1! pue (x)y !(ur ‘1)= y !9!uowJaq ‘iuy :saraqds 30 Su!Iaqwnu aq$ 01 Su!laplo p?guanbas aql $dopa aM ‘I ‘Sg aql u! Moqs aM sy p+aqds ItraI B SF(di)Yd ! d raluaa sauyap dx ‘da - J = do SU!MO~IOJ .samnlDnrls@ulalxa aql alou%! aM aDup aJaqds ,,lalno,, U’I!se l! lea31 kern aM ‘(~1 uoySar) ap!su! aql ~0.13 s alaqds 30 acejms aql q3eoIdde aM uaw .sp$ap
“n.> 23
‘%
3
“d>El “fj
d 10 “~11% 3 d “‘s = d JO 1 = d
“‘m 3 d JO 0~~0” 3 d “4 > B
‘“s = d
10
1= d
‘(“.oI\r)‘C”a
I
I
-
a3aqm Y
(Z)
(dd)y~(dol)d’ldyv x
+3*
x =(J)Y’
uo!suedxa ahEM Iegled IaluaD!$Inwe u! ual$gM aq Aew (I) uba 30 uo!ln[os aql ‘uo!Sal 3!woleJalu! qciea uI ‘0 +d uaqM AI uo!Sal ~!wol~~alu! 01 pw 0 = rl 3111uo!SaJ 3!wolelalu! 01 sla3al rl xapu! aql fhaqpdx u! uaA!S d%laua aql s! g !uo!Sar ?woleJalu! q2ea u! Iegualod luelsuo~ s! *A aJaqm
(I)
. . . ‘z ‘0 =
‘I
‘0
d
=
(J)‘+d’(‘+d - 3 t ,A)
uogenba laStnpaolq3s aql d3sges lsnw suo!Sal 3!wolelalu! aql u! suogsun3 aAvM aqL .uo!leluawaldw! Iauogelndwos aql pus X~$L?W Iepvas aqi 30 uogela~cLra$u!aq$ Qydw!s sa1n.I Suilaqwnu asaqJ .paJappuo2 uaaq aAeq sJa$snI:, 18301
Iuulalu! aq$ arou%! aM a+ alaqds ,,~!wole,, ue se l! luag Law aM ‘(11uo!Sal ~0.13)ap!s$no aql ~0~3 alaqds q%?olddu aM UaqM X&M aAgrruJalp20~1 u! s alaqds q;3ea$aJdIalu! 6em a& ~luelsuo:, aq 01 pawnsse s! uo!Sal s!ql 103 Iwlualod aqL ‘~1 suo!Sai 11~sIIeqs aM q3!qm ‘s alaqds q%a ap!su! Jeadde suo!Sal s!uolaJalu! Mau pue (s) salaqds [euo!qppe dq pasolDua a.w slalsnp 1e3o1 aqJ ‘1 9~ u! umoqs li~p2Xluwaq3ss! df) aIq!ssod v .r(Iluaqsuo9 3las uoysnba Jelnaas aql aA[os 01 lapJo u! ape:, [auog~lndwo3 MS prepuels aql u! apsw aq 01 peq ltql suo~le3ylpow hessa3au aql Supap!suo9 Lq paqsgqE$sa SEM ylom -awsq ~e3gewaq$aw aqJ, *asodmd UMOmo 01 paldepe ‘lapuella~g Lq paA!‘ap leql Al@3!sEqs! alaq paluasald uogeInwlo3 aqL .suo!SaJ J,K lua3aEp aql 30 sain2punoq aql le snonwluo:, aq I@qs saAyeA!lap ~ew~ou +aql pw suo!pun3 aAeMaql leql Su!.qnbal Kq pauF$qo uaql s! uoyenba nqnDas aqL .lapow J,K prepuels aql Swpualxa ‘ua$snIs 1~901altl!rdoldda aql auyap aM ‘sproa -laqlo UI .aDeds &qn3alow aql 30 Su!uojved pazyeJaua% aql Suiuyap lsry Aq l~e$s aM wsyew103 df) aql luawaldw! 01
1055
Electronicstructure of ferrocyanideion where I h(K,r,), g,‘YK,r,) =
h(K,r,),
q= 1 or qEo,JJS,, q=l
k,“‘(K,r,), i i0Lr,),
q= s,, or
qEti,,
or q=s,,
q E ooUSo
or
034
E> V&
(6b)
E>V*
64
E
q E o,,
E < P&;
6-3 (7)
with ‘I’+=-‘nL(K,R4J,
q,pEwoUSo
1
q=l q=S,,
1 ‘J’+L-’j,(K,R,,), #,‘YK,R,,)
= . (-l)“+‘k,“‘(K,R,),
and
or
q=l q=S,,
or p=l or p=S,
p E &,
p# u.
The wave functions inside atomic spheres and outside the outer sphere are written as $“(r,) = c C,‘u,P(r,, E)Y,VJ,
q,pEo,,
@a)
E%
or p=l when q,pE& E>V* or p = S, when q,p E 6,, ’
q,p EocUSo
~-~)~‘+‘j~~K~R~), 0 if q E (j,
or
q,p E +,
(gc)
E
when q,pE& when
0%)
E
(gd) (Se)
The same conditions for the wave functions at the boundaries of region IV (p+ 0 in eqn 5) and atomic region q E op, p f 0, lead to the equations
(9)
where $‘(r,, Ef are numerical solutions of the radial Schroedinger equation 4 f %-
V&,)-E
I
u:(r,,E)=O
(10)
for the spherically averaged potential Vp(rP) and a given energy E. The coefficients C” are related to A” by means of the equation
(13)
Note that eqns (12) and (13) are exactly those of the standard SW method the only difference being the sum over the scattering centers that belong to interatomic regions of different hinds. The diagonal matrix elements, t;/, are the inverse of the relative amplitudes of scattering of the Ith partial wave of energy E, given by
AiP = (I- 2~*,)b~*W[g~(K~b~), HtP(bp,E)lChP
(11)
when p = 1 or p E o,_ p = 0,1,2, . . . [ 11,The parameter b, is the radius of sphere p and W[f(x),gfx)] refers to the Wronskian of two functions f(x) and g(x). The requirement of the continuity of the wave functions (and their normal derivatives at the boundaries) of region II (CL= 0 in eqn (5)) with the atomic region q E too, or the extramolecular region, leads to the set of equations
q=I IPCS Vol. 43, No. I I-C
or
qEo*
(12)
4
65@,7 )L=o, 1,2 , * . .*
(14)
The off-diagonal elements GE. are caIled structure factors, and we may prove that they are symmetric (hermitian). We must also require the continuity of the wave functions and their derivatives on the surfaces of the spheres S. At the vicinity of the sphere S, numbered 4, enclosing the region IL,we have the h component of the wave functions
K. WATARIet al.
1056
in the outer side and
(16) in the inner side of the same sphere. Imposing the continuity of expressions (15) and (16) and their normal derivatives on the surface of sphere S(r, = bp), and eliminating the term
we have
where
Wg,SrW,b,), fiPW&J
ta’=W,"(K,b,),
g?(KobJ'
(18)
and 1 8%
h
1
=
b,2Wk,S’(&h_A
d’Woba)l’
(19)
3. APPLICATIONTO FERROCYANIDEION
3.1 GP partitioning for [Fe(CN)JIn this section we report on the results for the electronic states of [Fe(CN),J- as calculated by the selfconsistent Xo scattered wave GP method. The GP partitioning for this system is schematically shown in Fig. 2. The standard MT division of the molecular space is modified by introducing six spheres S which define six new regions of constant potential (regions IV). The S spheres are tangent to each group CN. The potential V,, is obtained by the usual procedure of volume averaging the molecular potential in
The same statement with elimination of the term
P’P
leads to the equations Ih&PAhB + t-’$1 A*% +p~Up~G:RLP=O
Now, we have a complete set of homogeneous linear equations composing a secular equation for GP theory. These equations are given by (12), (13), (17) and (20). From them we may infer the general structure of the secular matrix of GP. There are blocks along the diagonal with, in general, non zero elements. Each of these blocks corresponds to the secular equation of the standard SW method for each local cluster. There are also blocks where all the elements are null. These blocks lie in the crossing region of the indexes corresponding to the different local clusters in the secular equation. Thus, there is no direct coupling between the internal regions of different local clusters. Finally, there exist blocks with only coupling factors as non-zero elements. These blocks give the coupling between inner side and outside of the local clusters. Another fact apparent in GP is that the inclusion of one sphere S enclosing a local cluster is equivalent to adding two spheres in SW. This increases the dimension of the secular equation, but it does not increase significantly the difficulty in the computations since the secular matrix of GP is filled with a considerable amount of zeros. We have said above that the structure factors are symmetric (hermitean). Therefore, due to eqn (22), the secular matrix of GP is hermitean and hence there are only real eigenvalues corresponding to zeros of the associated determinant.
(20)
where ti;, =
WLf?“W,b,), dWobs)l Wk,SCW,b,h dWobp)l'
(21)
Equation (17) and (20) are the equations of SW method except for the coupling factors lhps* and I,‘@. Equation (17) shows that the S spheres enclosing the local clusters may be seen as atomic spheres if we ignore the.internal structures, as we have discussed earlier. In the same way, eqn (20) allows us to see the same sphere as an outer sphere to the local cluster if we ignore the external structure. The terms t ;i and t 2, play the same role as t ;,’ at re = b,+ The terms lhes* and I*‘*@couple the interatomic regions II and IV and we shall call them coupling factors.
Fig. 2. Generalized partitioning for [Fe(CN)$.
Electronic structure of ferrocyanide
Interatomic
Radius
molecular orbitals (MO’s) which we classify according to the irreducible representations of Oh symmetry group.
Distances Fe-C:
3.6282
C
2.1863
-N:
near the
2.4818
Rc
=
1.1464
RN
=
1.0399
=
2.1863
=
6.1355
RS R
Exchange
(in
=
RFe
3.2 Electronic structure The GP calculation was carried out by starting with the SW self-consistent potential. Therefore, in the first iteration V,” = VI,. It is interesting to compare the transference of charge to the local clusters for different iterations during the convergence process. This is done in Fig. 3 where we show the behaviour of the one electron energy spectrum of the ion as a function of the number of iterations. The values of the constant potentials V,, and V,, are also shown. Due to the charge transfer to the local clusters the potential V,r becomes more shallow and the potential V,, deeper than the
(in
of the spheres
The calculations were performed with a charge of t5 at the Watson’s sphere. The use of 4 positive charges
0.0 I_ E(Ry)
1057
Table 1. Parameters used in GP scheme for [Fe(CN)&
the region IV. In order to compare our calculations with the previous one, carried out within the framework of the SW method, we are adopting here the same geometrical and exchange parameters used by Guenzburger et al. [15]. The values are indicated in Table 1. The outer sphere is also used as a Watson’s sphere to simulate the stabilizing effect of the lattice on the ion. The total of 108 electrons in [Fe(CN)$- ion is filling
places the lowest virtual orbitals irrealistically continuum as in the case of SW calculation.
ion
out
a
a.".
)
a.".
)
parameters
cc(Fe)
=
0.71
a(C)
=
0.76
a(N)
=
0.75
a(111
=
(r(III)
=
cr(IV)
_/-
,
-0.5
,
.
-1.0 I-
- 1.5,is
L._._,* -_ .___._
. . . ..__.
VIV _.-
% I
l
2
f-Mm--
5
IO
20
25
30
Iteration Fig. 3. Convergence process in GP scheme. The potential for the first iteration was that obtained from the self-consistent SW calculation with a Watson’s sphere charge t 4. The empty levels were shown only for the first iteration. The constant potentials VI, and VIVare also shown.
=
0.75
1058
K. WATARIet al.
self-consistent interatomic potential obtained from the SW calculation. The self-consistent one electron energy spectra for the ground state of the ion obtained from the SW and GP calculations are shown in Fig. 4. The levels that lie above 2fZg in the spectra are the first empty levels. One important difference between the two spectra is that in GP calculation the highest occupied level is 2tz, and the lowest empty level is 4e,, while in SW calculation they are 2tzn and 5a,g, respectively. The ordering of these levels in GP is according to that obtained by Gray et al. by using the Linear Combination of Atomic Orbitals (LCAO) method[l6]. This result seems to indicate an improvement of the molecular potential in the GP method. The charge distribution for the ground state of the ferrocyanide ion for the different molecular regions, obtained according to GP calculation, are shown in Table 2. The comparison between these results and those derived from the SW calculation (see Table 1 of Ref. [ 151)has to be made with some caution, because there is only one interatomic region in SW while there are seven in GP. Furthermore, the regions II in SW and GP have different volumes. In order to make a fair comparison between the results we decided to divide the charge of region II in SW in two parts. One occupying a volume corresponding to the region IV of GP and other the volume equivalent to the region II of GP. With this procedure one obtains the results shown in Table 3.
One observes that the net effect of the generalized partitioning scheme is to transfer charge from the MT standard region II to the region IV of the local clusters. According to Table 3, the charge within the region II in SW calculation is reduced by a factor of two and the charge within the region IV in GP is four times greater than the charge found in the equivalent region in SW. By adding the chdrges in the C and N atomic spheres to the charge of region IV one obtains -9.1 electrons according to SW results. The total charge within the local cluster in the GP calculation is - 11.2 electrons, which is in better agreement with the expected value for the cyanide group, say -13 electrons. These results are strong evidence of improvement on the MT potential, indicating that the generalized partitioning scheme leads to more realistic charge distributions. The definition of the local clusters avoids the transference of charge to region II, gathering it in the correct places. 3.3 X-Ray photoelectron spectrum The valence band X-ray photoelectron spectrum of the ferrocyanide ion was obtained by Prins and Biloen[lll]. In order to compare our results with this experiment, the transition state concept was used to obtain the one electron ionization energies [ 181. The binding energy of the highest occupied orbital (2t,,) was used as a reference to adjust the experimental peak at -2 eV [15]. Figure 5 shows the GP energies superposed to the experimental peaks. The 6eV peak corresponds essen-
o.o-
,_-’ .2’2”
,a’
.-6t,. _,.* $I._. - - 4 l p
-0.5--
,,,. ‘,.59”
“..3126 “‘..60
-..
-’ 16
,,’ 2’26 ,’ *“lq ,’ ,3t,u .’ ..’
-’
-:.y,
cr” QJ
..
.3*
,/ .*
-‘*___--“2”
. .
I’
,’ ,’ , ’
D
.401q I
I’
,‘,’ : I, <._
4’1” .I110
,’
.2cq
-:_::.--..,t2q
_.-41,”
-I.O--
-,
-*.
,’
-~_:__-.I~2” ,_:::L.._,eq --:---.._
,‘_. ,:_’
..__ .3’,” “...._5a,q
-..
“nzq
_ _ “*40,q ‘..:‘.2e . ..
a 3’1”
-. -----...%3
-1.5-.,lC
_4ff:;....2’l”
,-
q
“‘.20,q
-2.0 --
0 0
Fig. 4. One electron energy spectra according to SW and GP calculations. Watson’s sphere charge t 5.
1059
Electronic structure of ferrocyanide ion Table 2. Energy levels and charge distribution for [Fe(CN)$ according to GP calculation. The value t5 was assumed for the charge at the Watson’s sphere. The six lowest empty orbitals are included: (a) all energies in Rydberg; (b) charge distribution in % of one electron; (c) orbitals marked (*) are empty lEnergy (a)
levels
Charge Fe sphere
(c)
stributior
C sphere
N sphere
(b) II
III
-509.029
lOO(lS)
-
-
-
-58.988
lOO(2S)
-
-
-
-
-
lOO(2Pf -
-
-
-
-
-
-
-
-
-
-
-
-50.981 -28.205
-
-20.275 -6.520(1a,g) -4.196(ltlu)
-
16.67(1s)
-
too
16.67(1s) -
-
-
-
too
--
-
-
-
-
-l.725(2a1g)
0.50
2.90
4.32
9.54
7.66
0.74
-1.717(2t,")
0.17
2.89
4.48
8.70
7.71
0.65
-1,714(1eg)
0.20
2.90
4.52
7.73
7.74
1.17
-0.993(3a,g)
18.23
2.17
2.48
33.85
3.12
1.25
-0.907(3tl")
3.14
2.04
3.53
29.18
5.89
0.01
-0.904(lt2g)
2.03
1.92
3.30
24.21
6.81
1.60
-0.864(lt2U)
0.11
2.10
3.77
19.29
7.32
1.44
15.56
1.80
4.72
19.29
3.41
5.56
-0.843(ltlg)
0.05
2.17
3.98
16.18
7.51
1.74
-0.827(4tfu)
2.34
1.89
5.47
21.41
4.57
4.66
-0.855(2eg)
3.21
3.36
6.53
26.90
0.30
8.73
29.84
3.70
4.31
16.95
0.0
5.18
5.29
5.29
4.16
32.21
0.35
3.68
)
90.87
0.06
0.20
6.20
0.20
0.15
-0.273(4eg 4*
54.10
3.12
0.62
la.41
0.59
-0.218(5a,g)'
3.25
0.32
1.35
12.02
0.0
-o.l98(3t*g):
0.07
0.05
0.63
19.62
0.28
74.63
-0.173(6t1,)
0.35
0.49
1.27
0.0
0.0
89.07
-O.l12(5eg)*
0.19
0.24
0.48
0.0
0.0
95.50
-o.l10(2t*,)'
0.0
0.01
0.34
10.12
II.10
87.19
26.50
3.61
4.53
12.61
3.09
-0.684(4a1g) -0.595(3eg) -0.577(5t,") -0.493c2t2
Electron distribution. Total of 108 electrons
1.54 74.71
1.53
Table 3. Total charge distribution for the different regions of the molecular space. The interatomic region in the SW scheme was divided artificially in regions II and IV, equivalent to those defined in the GP theory
tially to ionization of u and B MO’s according to our calculations. This is in agreement with the interpretation given by Prins and Biloen. The theoretically obtained binding energy of 8 eV corresponds to a (I MO. This value differs slightly from the 9eV peak assigned by Prins and Biloen to the ionization of a u MO. By comparing with the previous SW results, we observe a decreasing in the iron 4s contribution for this u statell51. Although the inte~re~tion of the low energy peaks is practically the same provided by Guenzb~ger et al., the GP model leads to an interpretation apparently more
realistic to the 18.5eV experimental peak. According to
our calculation, it is originated from the first group of cr valence MO’s with ionization energies of -18eV. According to the SW results this peak is located at -15 eV. 3.4 Optical transitions In this section the GP results are used to interpret the optical spectrum of (Fe(CN)&” obtained by Gray et a[.[ 161.The catculated ground state energies, associated to the measured optical transitions, are dispIayed in Table 4. The SW results as well as the expe~men~ values are shown. It is observed that the energy
1060
K. WATARI
et al.
1
I
20
IO
0
energy(ev) Fig. 5. X-Ray photoelectronspectrum of L&[Fe(CN)G][17] and GP theoreticalionizationenergies. Table 4. Interpretation of the optical spectrumof the ferrocyanideion accordingto the SW and GP calculations. Energiesin units of 10’cm-’ GP
Experlmental
2t2g
g
35.65
36.03
24.08
38.48
(A.33.8)
34.98
42.13
48.33 37.28 26.74
a: Reference b: Transition c: Reference
and
assignmentC
‘i-‘j
+ 4e
bands
23.70
1Atg +3Tlg
31.00
1Alg+
37.04
1Alg+. 'T 2g
45.87
1Alg +'T,u
50.00
1Alg+
'T ,g
'TlU
32.26
29.66
36.98
15 state
concept
- Reference
18
16
differences, ei - q, corresponding to one-electron levels i and j involved in the optical transitions (Koopmans’ method), do not compare well with the experimental results. Hence, the transition state concept[ll], where half of an electronic charge is transferred to the excited one-electron state involved in the optical transition, was used in order to achieve better agreement with the experimental results. According to the values shown in Table 4, the electron relaxation effects affect more the GP results than the SW one. This is expected since the GP method leads to a rather diierent charge distribution for the ion when compared with SW result. Table 4 shows that the GP method leads to an interpretation for the optical spectrum of [Fe(CN)Jin fairly good
agreement with that inferred from the experimental results by Gray et al. [ 161. 4. CONCLUSIONS
The SW method, in its standard MT form, has only one constant potential for the interatomic region. When this method is applied to the study of a variety of materials that have radicals or molecules surrounding a metallic ion, the charges of the ligands are spread out. Thus the method leads to a non-realistic electronic structure for the molecules or the clusters. The GP theory allows for the systematic study of this kind of materials. The simple application to the [Fe(CN)J- shows that the main difiiculty with the SW
Electronicstructure of ferrocyanideion theory is overcame. On the other hand, one expects that the GP formalism will be a useful tool to deal with open structures in general. Before concluding this work a final comment should be made: the GP scheme retains most of the flexibility of the standard SW method. By considering the previous application of the SW scheme to the ferrocyanide ion we conclude that only a small extra computational effort was necessary to implement the calculations reported in this paper. Acknowledgements-The authors wish to thank Profs. JosC R. P. Neto and A. Fazzio for many helpful discussions. Thanks are also due to Dr. Marflia J. Caldas for the critical reading of the manuscript. REFERENCES
1. Johnson K. H., Adoances in Quantum Chemistry (Edited by 2. 3. 4. 5. 6.
P. 0. Liiwdin), Vol. 7, p. 143. Academic Press, New York (1973). Johnson K. H., Ann. Rev. Phys. Chem. 26,39 (1975). Braga M., Larsson S. and Leite J. R., J. Am. Chem. Sot. 101, 3867(1979). Fazzio A. and Leite J. R., Phys. Reo. B 21,471O(1980). De Leo G. G., Watkins G. D. and Fowler W. B., Phys. Rev. B 23, 1851(1981). Weinberger P. and Konowalow D. D., Int. J. Quant. C/tern. s7, 353 (1973).
1061
7. Herman F. and Batra I. P., Phys. Reo. Lett. 33, 94 (1974); Batra I. P., Bennett B. I. and Herman F.. Phvs. _ Reo. B 11. 4927(1975). 8. DaneseJ. B. and ConnollyJ. W. D., J. Chem.Phys. 61,3063 (1974):Danese J. B.. J. Chem.Phvs. 61. 3071 (1974): Chem. Phys.’iett. 45, 150(i977). ’ . ” 9. Ferreira L. G., Agostinho A. and Lida D., Phys. Rev. B 14, 354 (1976). 10. Rosch N., Klemperer W. G. and Johnson K. H., Chem. Phys. Lett. 23, 149(1973). 11. Howard I. A., Pratt G. W., Johnson K. H. and Dresselhaus G., J. Chem. Phys. 74,3415 (1981). 12. Braga M., Pavao A. C. and Leite J. R., Phys. Reu. B 23,4328 . _ (1981). I’. Kjellander R., Chem. Phys. Lett. 29,270 (1974);Chem. Phys. 12,469 (1976);Chem. Phys. 20, 153(1977). 14. Johnson K. H., Herman F. and Kjellander R., Electronic Structure of Polymers and Molecular Clusters (Edited by J. AndrC, J. Ladik and J. Delhalle), Vol. IV, p. 601. Plenum Press, New York (1975). 15. Guenzburger D., MatTeoB. and De Siqueira M. L., J. Phys. Chem. Solids 38, 35 (1977). 16. Gray H. B. and Beach N. A., J. Am. Chem. Sot. 85, 2922 (1%3); Alexander J. J. and Grav H. B.. J. Am. Chem. Sot. 90. 4260 (1968); Alexander J. J. and Gray H. B., Coord. Chem: Rev. 2, 29 (1%7). 17. Prins R. and Biloen P., Chem. Phys. Lett. 30,340 (1975). 18. Leite J. R. and Ferreira L. G., Phys. Rev. A 3, 1224(1971);J. C. Slater, Quantum Theory of Molecules and Solids, Vol. 4. McGraw-Hill, New York (1974).