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Normal mode analysis of the octacyanomolybdate(IV) ion in a crystalline environment
Journal Elsevier
of Molecular Structure, 77 (1981) 165-172 Scientific Publishing Company, Amsterdam
-
Printed
in The Netherlands
NORMAL, MODE ANALYSIS OF THE OCTACYANOMOLYBDATE(IV) ION IN A CRYSTALLINE ENVIRONMENT
E. HAHN, M. ACKERMANN, H. BGHLIG and J. FRUWERT Sek tion Chemre
der Karl-Marx-Universitiit,
Leipzig
(D.D.R.)
(Received 6 May 1981)
ABSTRACT A normal coordinate calculation has been carried out applying a modiPied valence force field and a dodecahedron model. The frequencies of the 45 normal modes of the octacyanomolybdate(IV) ion have been calculated with an average error of 1.3% or 4.4 cm-‘. All frequencies are completely assigned_ The investigation of IR and Raman spectra of the solid K,[Mo(CN).] - 2H,O enables calculation of ten force constants by use of a least-squares refinement .
INTRODUCTION
The IR and Raman spectra of Kq [Mo(CN),] - 2H10 have been reported several times [l-6] without satisfactory assignment of the frequencies. Therefore a normal mode analysis has been carried out applying a dodecahedral configuration and a modified valence force field (MVFF). This normal coordinate calculation was done not only to obtain an assignment of the 45 normal modes but also to obtain a set of force constants. The calculations were performed by use of the program FPERT [7] which is based on the GF-matrix method of Wilson. FPERT includes an iterative least-squares refinement. A former normal mode calculation of the octacyanomolybdate(IV) ion using the GF-matrix method and a strong simplified potential function only provided the frequencies of the normal vibrations [ 83 . Our work is based on the previous IR and Raman spectra [l-S] investigations of solid K,[Mo(CN),] 2H20. l
MOLECULAR
MODEL
AND INTERNAL
COORDINATES
There are two favoured configurations for eight-coordinated ions: the Archimedian antiprism (Dad) and the dodecahedron (D2d). The calculation of the matrix G of an Archimedian antiprism has already been done [9]. However, an exact Archimedian antiprism demands a distance between its two squares of a [ (12 sin* q - 3)/(12 sin’ q - l)] where (a3 = volume of the initial cube [9] and q = 37r/8). The dodecahedron model used is shown in OOZZ-2860/81/0000-0000/$02.75
0 1981
Elsevier Scientific Publishing Company
Fig. 1. Dodecahedron u, I uz.
model of [Mo(CN),]‘-,
C = 1 _ _ -8, MO = 9, N = 10 . , . 17,
Fig. 1, in which the eight ligands are divided into two non-equivalent sets of four (four A and four B positions). Each set forms a special tetrahedron. It is possible to describe the dodecahedron by means of only two characteristic angles q and 6 with q(L396 or L297) = 145.6” and 6 (L198 or L495) = 69.1’ [lo]. For the calculations LMoCN = 180”, r(Mo-C) = 2.163 A, r(C=N) = 1.156 A [ll], m(12C) = 12.0, m(14N) = 14.003074 and m(=Mo) = 97.9055, were used. TABLE
1
Internal coordinates Internal coordinates=
tipe
(sign)
Stretching MO-C (v, ) Stretching CkN (v: 1 Bending MoCN (6,)
Position to cr, or uzb
Ligand position
i
B A B A A B B
A i i 0
i 0
Bending CMoC (6,)
aAll internal coordinates were labelled only with the numbers of the carbon atoms (see Fig. 1). bi = in plane, o = out of plane.
167
The internal coordinates selected (Table 1) agree with the demands of group theory because they guarantee a dodecahedral distribution of the 45 normal modes among the symme+ types A,, Al, B1, I& and E. In other words, the eight internal coordinate: of type Aa must be included to calculate one of the three A, modes: only two A, modes will result if the eight Aa co-ordinates are not taken into account. The angles a, /3 and 7 are obtained from the following equations cos CX= cos 77/ 2 cos s/2
(1)
OL= 75.89” cos p = -cos2
6/2
(2)
fl = 132.72” Y =7T -
l/2(7, + 6) = 72.67”
(3)
The difference between 48 internal coordinates and 45 normal modes includes three redundancies (Table 2). The conslzuction of internal symmetry coordinates (Table 3) was carried out to distribute the calculated vibrational frequencies among the symmetry types of the point group Dzd . CHOICE
OF AN
INITIAL
FORCE
FIELD
A set of significant interaction force constants is necessary for the physical description of the real vibrational behaviour of the anion. The Jacobi matrix of an eigenvalue calculation with a “maximal force field” (i.e. with all interaction force constants of interactions being between connected internal displacements) provides some knowledge about this set. This “maximal force field” contains the values of all diagonal force constants and of some interaction force constants which have been transferred from octahedral cyan0 complexes [12] . The unknown values of the other interaction force constants were set to zero. It was then possible to compute the eigenvalues and to use the computed Jacobi matrix for a selection of the most important TABLE
2
Distribution Symmetry
of the normal
modes
the symmetry
Cartesian coordinates
Activity
type A
among
R(P)
A,
B,
Wdp)
&
Wdp),
IR
E
R(G),
m
types Internal coordinates
8
9
3
3
4 8
4 8 12
11
Redundancies 1
2
168 TABLE3 Internal symmetry
coordinates
A,
S, = l/2 (a, + a4 + 0, + a,) S,,=1/2(bz+ b,+ b,+ b,) SC = 112 (e, + c4 + c, + c,) ~3~ = 112 (d, + d, + if, + d,) s, = l/2 (w, + w4 + a5 + w,) srp = l/2 to, + cp3+ Qs + @,) s, = l/J% (cyII + Q,, + Q,, + Q, + %, + Q, + a,, + &.B) sp = l/2 (Pl, + PI, + P, + &,‘I s, = l/2 (71, + Yw + Ys7 + Y&.9)
4
s,
l/2 (7, - 7, - Ts + T8) =1i2(e1--e,-e6+ 6,) Se = 1/,,%(a,,- ~,,-&~--~y =
se
B,
B3
s, Se S, Sp
1/2 (7, + 7-,+ r5 + 7.) 112 (ff %+ B, + 8, + 0,) 1tJs (CY,,- QI, - QZB+ Qy l/2 (Pl, - P,5- is.3 + P,,)
S, = If2 (a, -a, -f.7, + 4,) Sb = l/2 (b, - b, - b, + b,) SC=1/2(c,-c,-c5 1 c,) S~=1/2(d,--d,-dd,+ d,) s,=1/2(w,--w,--w,c WB) s@=~/2(#,-9,-~Qs+ 05) S& = l/G (a:, + =a, + =ZB- &,., s,
E
= = = =
f
S,,
=
112
(Y,s
= W/3
-
(a,
Yz*
-
Y57
+
p33
+
p4b-a~6
Q,* -a*
&35
-
+
%R)
+ QSC, -c %a)
Qde
-
%b
+
%8)
Ycss,)
-ao,)
S az = WE@, --aa,) s bl = l/fi (b, - be) Sbz = lk/?! s,, = mn
(b,
-
(c,
-cc,)
s,,
= l/Jz
(c,
-c,)
sdz
= I/v’3
(d,
-de)
so1 = I/4 sp, = llJ3;
(6,
-
@,a -
b,)
0,) Pm1
S
= 1/Jz t-r,, -yea) s,‘:= 112 (cx,z+ aI7 - a- --P,) S c$ = 112 (%.I - (135+ %6 - 4 S a!3 = 1/2(cn,l-a,, + a,,-or,) S p.v= l/2 I%3 + Q.35 - %s - Q,,)
The resulting eigenvalues have to agree closely with the experimental frequencies. A slight modification of the force constant values (by hand) was necessary to obtain such an agreement. Different sets of interaction force constants were improved by means of eigenvalue cdculations and refinements. All these refinements and zero-order ~~~~ations were made to obtain a significant initial force field. interaction
force constants.
169
Despite the availability of twenty-four experimental frequencies of IR and Raman spectra, it was not possible to refine more than ten force constants. Two of the twelve included force constants (Table 4) were fixed during the refinement. These two constants are transferable and they have a remarkable influence only on the CN-stretching frequencies. Further refinements were necessary to obtain a convergence with this force field, but this was also a problem of the number of experimental frequencies and their assignment. EXPERIMENTAL
VIBRATIONAL
FREQUENCIES
Most experimental frequencies used for the refinements were taken from previously published IR and Raman spectra [l-6] of solid K,[Mo(CN),] . 2Hz0. Our IR and Raman spectra of this substance should have been an additional source of experimental data but they were in relatively good agreement with earlier work. It was not possible to measure the depolarization ratios of the Raman bands because the material was polycrystalhne, and it has therefore been more difficult to assign the frequencies. Detailed assignments for the 2100 cm-l region were offered in refs. 2, 4 and 5. Consideration of crystal effects of crystalline substances is always necessary. However, the internal vibrations of different ions in the unit cell are not measurably coupled and the site symmetry of the ion (C,) causes only a very slight effect [ 41. RESULTS
AND DISCUSSION
The refinements were carried out with a damping factor [ 71 of only 0.002 and all results are given in Tables 4 and 5. The values of the (refined) force constants fi, f2, far f5, f6, f, and fs do not differ much from the expected values [12-151. An estimate of the calculated force constants fs, fro, fil and fiz is difficult because there are no comparable values in the literature. The relatively high value of f3 probably results from slight steric hindering of bond angle bendings in eight-coordinated ions.
TABLE
4
Calculated force constantsa of the octacyanomolybdate(IV) f,
(MoC)
=
2.14 (0.04)b (0.05) 1.71 (0.10) 0.31(0.02) 0.43 (0.02) 0.03
f, (CN)
= 16.79
f, (MoCN) f, (MoC/MoC)
=
f, (CMoC) f, WWCW
=
=
=
ion
f, (MoWCaN 1 f, (MoCbICaN) f, (MoC/CMoC) f,,(CMoC/CMoC) f,,(MoCN/CMoC) f,,(MoCN/CMoC)
0.31 = 0.11 (0.02) = 0.17 (0.07) = 0.33 (0.05) = 4.12 (0.03) = -0.06 (0.02)
=
af,, f,, f, . - . f, in mdyn A-’ , f, in mdyn and f,, f,, f,, - m - f,, in mdyn B.. bErrors (relative uncertainties) of the refined force constants in parentheses.
170 TABLE
5
Assignment Symmetry
of the 45 calculated normal modes (in cm-‘)= No.
hbs.
i'c2Jc.
2138.0 2116.0 568.0 476.0 387 .O 325.0 0.0 0.0 0.0
2137.9 2116.2 568.3 477.7 386.7 331.2 151.9 119.8 0.0
Assignmentb
type A
E
1 2 3 4 5 6 7 8 9 10 11 12
0.0 0.0 0.0
419 .o 324.3 115.6
13 14 15 16
0.0 0.0 0.0 00
486.1 337.1 132.9 52.4
17 18 19 20 21 22 23 24
2125.0 2107 .o 511.0 436.0 377.0 332.0 155.0 112.0
2116.5 2116.2 518.7 436.9 380.5 328.2 156.1 100.6
25,26 27.28 29,30 31, 32 33,34 35,36 37,38 39.40 41,42 43,44 45,46 47.48
2130.0 2104.0 622.0 466.0 404.0 363.0 332.0 0.0 178.0 131.0 87.0 0.0
2116.4 2116.2 615.7 463.5 403.4 361.8 328 .O 320.6 173.0 136.0 86.0 0.0
aValues of the potential energy distribution matrix below 10% were neglected. b 6 z = 6 (CMoC). WI = v(MoC), vz = u(CN), 6, = 6 (MoCN),
171
All twelve force constants are the basis of a complicated potential function (eqn. 4) which describes the vibrational behaviour better than the potential function used by Salvetti [S] . 2V=
f
f,i (ri)2
(4)
i-1
+f51~aiaah~22ajbh+~bib +
2f7
+
2fgIZajaj,
+
fl0
+
2fll
C IL aj
C C ajx Z
Wj
CJ +
x
by
kl +fe[ZCiCtz dj3
+
2fs
[ 1
+ IL ajPjx +E Yjy Pjx
+ E +
Pjx
Pjy
2f12CZWjajx
+
aj
ajij, 2
+
+C
x
IX
aj
biajx
Pjy +
Eajx
+
Ck
+ 2 Ecjdk dk
+
x
+ E
2 1 “jx
bi
Ck
+
+
Eddid,]
z
bj
dk
1
bjYjx1
Yjy +
2 IX Pjx Yjy
1
@jajxl
(wherej,k,x,yvaryfromlto8andx#y,jfk) Equation (4) only gives the structure of the potential function. The assignment of the normal modes (Table 5) is based on the potential energy distribution matrix. Table 5 shows that the vibrational behaviour of the ion is very complex. There are only two regions of frequencies: that between 630 cm-l and 50 cm-l with very strongly coupled stretching and bending modes, and the CN-stretching region at 2100 cm-l. Only the v-ibrations near 330, 568 and 2100 cm-l may be described as non-mixed. It was possible to reproduce the 45 normal modes of the octacyanomolybdate(IV) ion with an average error of 1.3%. But five CN-stretching frequencies were calculated with a value of about 2116 cm-l. The reason for this is the absence of Adme interaction force constants in our refinements. Only ten force constants could be refined but the computational uncertainties of their final values were relatively small. The results of this work are the basis of a detailed assignment of additional vibrational frequencies of the solid K4[Mo(CN),] .2H20 in the region 2090 cm-’ to 2020 cm-1 1161. All c al cu 1at ions were based on the dodecahedron model. This model is suitable for a description of the vibrational behaviour of the octacyanomolybdate(IV) ion. ACKNOWLEDGEMENTS
The authors thank Prof. Dr. G. Geiseler for the many discussions which stimulated this study. We also thank Dr. P. Reich, AdW der DDR, for his Raman measurements. REFERENCES 1 H. Stammreich and 0. Sala, Z. Elektrochem., 64 (1960) 741: 65 (1961) 149. 2 E. Kiinig, Theor. Chim. Acta, 1 (1962) 23. 3 S. F. A. Kettle and R. V. Parish, Spectrochim. Acta, Part A, 21 (1965) 1087.
172 4 K. 0. Hartman and F. A. Miller, Spectrochim. Acta, Part A, 24 (1968) 669. 5 R. V. Parish, P. G. Simms, M. A. Wells and L. A. Woodward, J. Chem. Sot. A, (1968) 2882. 6 T. V. Long and G. A. Vernon, J. Am. Chem. Sot., 93 (1971) 1919. 7 R. N. Jones, Computer Programs For Infrared Spectrophotometry, N.R.C.C. Bulletin Ottawa, 1976. 8 0. Salvetti, Ann. Chim. (Rome), 48 (1958) 1293. 9 H. L. Schltifer and H. F. Wasgestian, Theor. Chim. Acta, 1 (1963) 369.
10 G. Racah, J. Chem. Phys., 11 (1943) 214. 11 12 13 14 15 16
J. L Hoard, T. A. Hamor and M. D. Glick, J. Am. Chem. Sot., 90 (1968) 3177. L. H. Jones and B. I. Swanson, Act. Chem. Res., 9 (1976) 128. I. Nakagawa and T. Shimanouchi, Spectrochim. Acta, 18 (1962) 101. K. H. Schmidt, Spectrochim. Acta, Part A, 33 (1977) 369. J. J. Rafalko, I3 I Swanson and L. H. Jones, J. Chem. Phys., 67 (1977) 1. E. Hahn, H. Bijhlig, M. Ackermann and J. Fruwert, Spectrochim. Acta, in press.
15,
of Molecular Structure, 77 (1981) 165-172 Scientific Publishing Company, Amsterdam
-
Printed
in The Netherlands
NORMAL, MODE ANALYSIS OF THE OCTACYANOMOLYBDATE(IV) ION IN A CRYSTALLINE ENVIRONMENT
E. HAHN, M. ACKERMANN, H. BGHLIG and J. FRUWERT Sek tion Chemre
der Karl-Marx-Universitiit,
Leipzig
(D.D.R.)
(Received 6 May 1981)
ABSTRACT A normal coordinate calculation has been carried out applying a modiPied valence force field and a dodecahedron model. The frequencies of the 45 normal modes of the octacyanomolybdate(IV) ion have been calculated with an average error of 1.3% or 4.4 cm-‘. All frequencies are completely assigned_ The investigation of IR and Raman spectra of the solid K,[Mo(CN).] - 2H,O enables calculation of ten force constants by use of a least-squares refinement .
INTRODUCTION
The IR and Raman spectra of Kq [Mo(CN),] - 2H10 have been reported several times [l-6] without satisfactory assignment of the frequencies. Therefore a normal mode analysis has been carried out applying a dodecahedral configuration and a modified valence force field (MVFF). This normal coordinate calculation was done not only to obtain an assignment of the 45 normal modes but also to obtain a set of force constants. The calculations were performed by use of the program FPERT [7] which is based on the GF-matrix method of Wilson. FPERT includes an iterative least-squares refinement. A former normal mode calculation of the octacyanomolybdate(IV) ion using the GF-matrix method and a strong simplified potential function only provided the frequencies of the normal vibrations [ 83 . Our work is based on the previous IR and Raman spectra [l-S] investigations of solid K,[Mo(CN),] 2H20. l
MOLECULAR
MODEL
AND INTERNAL
COORDINATES
There are two favoured configurations for eight-coordinated ions: the Archimedian antiprism (Dad) and the dodecahedron (D2d). The calculation of the matrix G of an Archimedian antiprism has already been done [9]. However, an exact Archimedian antiprism demands a distance between its two squares of a [ (12 sin* q - 3)/(12 sin’ q - l)] where (a3 = volume of the initial cube [9] and q = 37r/8). The dodecahedron model used is shown in OOZZ-2860/81/0000-0000/$02.75
0 1981
Elsevier Scientific Publishing Company
Fig. 1. Dodecahedron u, I uz.
model of [Mo(CN),]‘-,
C = 1 _ _ -8, MO = 9, N = 10 . , . 17,
Fig. 1, in which the eight ligands are divided into two non-equivalent sets of four (four A and four B positions). Each set forms a special tetrahedron. It is possible to describe the dodecahedron by means of only two characteristic angles q and 6 with q(L396 or L297) = 145.6” and 6 (L198 or L495) = 69.1’ [lo]. For the calculations LMoCN = 180”, r(Mo-C) = 2.163 A, r(C=N) = 1.156 A [ll], m(12C) = 12.0, m(14N) = 14.003074 and m(=Mo) = 97.9055, were used. TABLE
1
Internal coordinates Internal coordinates=
tipe
(sign)
Stretching MO-C (v, ) Stretching CkN (v: 1 Bending MoCN (6,)
Position to cr, or uzb
Ligand position
i
B A B A A B B
A i i 0
i 0
Bending CMoC (6,)
aAll internal coordinates were labelled only with the numbers of the carbon atoms (see Fig. 1). bi = in plane, o = out of plane.
167
The internal coordinates selected (Table 1) agree with the demands of group theory because they guarantee a dodecahedral distribution of the 45 normal modes among the symme+ types A,, Al, B1, I& and E. In other words, the eight internal coordinate: of type Aa must be included to calculate one of the three A, modes: only two A, modes will result if the eight Aa co-ordinates are not taken into account. The angles a, /3 and 7 are obtained from the following equations cos CX= cos 77/ 2 cos s/2
(1)
OL= 75.89” cos p = -cos2
6/2
(2)
fl = 132.72” Y =7T -
l/2(7, + 6) = 72.67”
(3)
The difference between 48 internal coordinates and 45 normal modes includes three redundancies (Table 2). The conslzuction of internal symmetry coordinates (Table 3) was carried out to distribute the calculated vibrational frequencies among the symmetry types of the point group Dzd . CHOICE
OF AN
INITIAL
FORCE
FIELD
A set of significant interaction force constants is necessary for the physical description of the real vibrational behaviour of the anion. The Jacobi matrix of an eigenvalue calculation with a “maximal force field” (i.e. with all interaction force constants of interactions being between connected internal displacements) provides some knowledge about this set. This “maximal force field” contains the values of all diagonal force constants and of some interaction force constants which have been transferred from octahedral cyan0 complexes [12] . The unknown values of the other interaction force constants were set to zero. It was then possible to compute the eigenvalues and to use the computed Jacobi matrix for a selection of the most important TABLE
2
Distribution Symmetry
of the normal
modes
the symmetry
Cartesian coordinates
Activity
type A
among
R(P)
A,
B,
Wdp)
&
Wdp),
IR
E
R(G),
m
types Internal coordinates
8
9
3
3
4 8
4 8 12
11
Redundancies 1
2
168 TABLE3 Internal symmetry
coordinates
A,
S, = l/2 (a, + a4 + 0, + a,) S,,=1/2(bz+ b,+ b,+ b,) SC = 112 (e, + c4 + c, + c,) ~3~ = 112 (d, + d, + if, + d,) s, = l/2 (w, + w4 + a5 + w,) srp = l/2 to, + cp3+ Qs + @,) s, = l/J% (cyII + Q,, + Q,, + Q, + %, + Q, + a,, + &.B) sp = l/2 (Pl, + PI, + P, + &,‘I s, = l/2 (71, + Yw + Ys7 + Y&.9)
4
s,
l/2 (7, - 7, - Ts + T8) =1i2(e1--e,-e6+ 6,) Se = 1/,,%(a,,- ~,,-&~--~y =
se
B,
B3
s, Se S, Sp
1/2 (7, + 7-,+ r5 + 7.) 112 (ff %+ B, + 8, + 0,) 1tJs (CY,,- QI, - QZB+ Qy l/2 (Pl, - P,5- is.3 + P,,)
S, = If2 (a, -a, -f.7, + 4,) Sb = l/2 (b, - b, - b, + b,) SC=1/2(c,-c,-c5 1 c,) S~=1/2(d,--d,-dd,+ d,) s,=1/2(w,--w,--w,c WB) s@=~/2(#,-9,-~Qs+ 05) S& = l/G (a:, + =a, + =ZB- &,., s,
E
= = = =
f
S,,
=
112
(Y,s
= W/3
-
(a,
Yz*
-
Y57
+
p33
+
p4b-a~6
Q,* -a*
&35
-
+
%R)
+ QSC, -c %a)
Qde
-
%b
+
%8)
Ycss,)
-ao,)
S az = WE@, --aa,) s bl = l/fi (b, - be) Sbz = lk/?! s,, = mn
(b,
-
(c,
-cc,)
s,,
= l/Jz
(c,
-c,)
sdz
= I/v’3
(d,
-de)
so1 = I/4 sp, = llJ3;
(6,
-
@,a -
b,)
0,) Pm1
S
= 1/Jz t-r,, -yea) s,‘:= 112 (cx,z+ aI7 - a- --P,) S c$ = 112 (%.I - (135+ %6 - 4 S a!3 = 1/2(cn,l-a,, + a,,-or,) S p.v= l/2 I%3 + Q.35 - %s - Q,,)
The resulting eigenvalues have to agree closely with the experimental frequencies. A slight modification of the force constant values (by hand) was necessary to obtain such an agreement. Different sets of interaction force constants were improved by means of eigenvalue cdculations and refinements. All these refinements and zero-order ~~~~ations were made to obtain a significant initial force field. interaction
force constants.
169
Despite the availability of twenty-four experimental frequencies of IR and Raman spectra, it was not possible to refine more than ten force constants. Two of the twelve included force constants (Table 4) were fixed during the refinement. These two constants are transferable and they have a remarkable influence only on the CN-stretching frequencies. Further refinements were necessary to obtain a convergence with this force field, but this was also a problem of the number of experimental frequencies and their assignment. EXPERIMENTAL
VIBRATIONAL
FREQUENCIES
Most experimental frequencies used for the refinements were taken from previously published IR and Raman spectra [l-6] of solid K,[Mo(CN),] . 2Hz0. Our IR and Raman spectra of this substance should have been an additional source of experimental data but they were in relatively good agreement with earlier work. It was not possible to measure the depolarization ratios of the Raman bands because the material was polycrystalhne, and it has therefore been more difficult to assign the frequencies. Detailed assignments for the 2100 cm-l region were offered in refs. 2, 4 and 5. Consideration of crystal effects of crystalline substances is always necessary. However, the internal vibrations of different ions in the unit cell are not measurably coupled and the site symmetry of the ion (C,) causes only a very slight effect [ 41. RESULTS
AND DISCUSSION
The refinements were carried out with a damping factor [ 71 of only 0.002 and all results are given in Tables 4 and 5. The values of the (refined) force constants fi, f2, far f5, f6, f, and fs do not differ much from the expected values [12-151. An estimate of the calculated force constants fs, fro, fil and fiz is difficult because there are no comparable values in the literature. The relatively high value of f3 probably results from slight steric hindering of bond angle bendings in eight-coordinated ions.
TABLE
4
Calculated force constantsa of the octacyanomolybdate(IV) f,
(MoC)
=
2.14 (0.04)b (0.05) 1.71 (0.10) 0.31(0.02) 0.43 (0.02) 0.03
f, (CN)
= 16.79
f, (MoCN) f, (MoC/MoC)
=
f, (CMoC) f, WWCW
=
=
=
ion
f, (MoWCaN 1 f, (MoCbICaN) f, (MoC/CMoC) f,,(CMoC/CMoC) f,,(MoCN/CMoC) f,,(MoCN/CMoC)
0.31 = 0.11 (0.02) = 0.17 (0.07) = 0.33 (0.05) = 4.12 (0.03) = -0.06 (0.02)
=
af,, f,, f, . - . f, in mdyn A-’ , f, in mdyn and f,, f,, f,, - m - f,, in mdyn B.. bErrors (relative uncertainties) of the refined force constants in parentheses.
170 TABLE
5
Assignment Symmetry
of the 45 calculated normal modes (in cm-‘)= No.
hbs.
i'c2Jc.
2138.0 2116.0 568.0 476.0 387 .O 325.0 0.0 0.0 0.0
2137.9 2116.2 568.3 477.7 386.7 331.2 151.9 119.8 0.0
Assignmentb
type A
E
1 2 3 4 5 6 7 8 9 10 11 12
0.0 0.0 0.0
419 .o 324.3 115.6
13 14 15 16
0.0 0.0 0.0 00
486.1 337.1 132.9 52.4
17 18 19 20 21 22 23 24
2125.0 2107 .o 511.0 436.0 377.0 332.0 155.0 112.0
2116.5 2116.2 518.7 436.9 380.5 328.2 156.1 100.6
25,26 27.28 29,30 31, 32 33,34 35,36 37,38 39.40 41,42 43,44 45,46 47.48
2130.0 2104.0 622.0 466.0 404.0 363.0 332.0 0.0 178.0 131.0 87.0 0.0
2116.4 2116.2 615.7 463.5 403.4 361.8 328 .O 320.6 173.0 136.0 86.0 0.0
aValues of the potential energy distribution matrix below 10% were neglected. b 6 z = 6 (CMoC). WI = v(MoC), vz = u(CN), 6, = 6 (MoCN),
171
All twelve force constants are the basis of a complicated potential function (eqn. 4) which describes the vibrational behaviour better than the potential function used by Salvetti [S] . 2V=
f
f,i (ri)2
(4)
i-1
+f51~aiaah~22ajbh+~bib +
2f7
+
2fgIZajaj,
+
fl0
+
2fll
C IL aj
C C ajx Z
Wj
CJ +
x
by
kl +fe[ZCiCtz dj3
+
2fs
[ 1
+ IL ajPjx +E Yjy Pjx
+ E +
Pjx
Pjy
2f12CZWjajx
+
aj
ajij, 2
+
+C
x
IX
aj
biajx
Pjy +
Eajx
+
Ck
+ 2 Ecjdk dk
+
x
+ E
2 1 “jx
bi
Ck
+
+
Eddid,]
z
bj
dk
1
bjYjx1
Yjy +
2 IX Pjx Yjy
1
@jajxl
(wherej,k,x,yvaryfromlto8andx#y,jfk) Equation (4) only gives the structure of the potential function. The assignment of the normal modes (Table 5) is based on the potential energy distribution matrix. Table 5 shows that the vibrational behaviour of the ion is very complex. There are only two regions of frequencies: that between 630 cm-l and 50 cm-l with very strongly coupled stretching and bending modes, and the CN-stretching region at 2100 cm-l. Only the v-ibrations near 330, 568 and 2100 cm-l may be described as non-mixed. It was possible to reproduce the 45 normal modes of the octacyanomolybdate(IV) ion with an average error of 1.3%. But five CN-stretching frequencies were calculated with a value of about 2116 cm-l. The reason for this is the absence of Adme interaction force constants in our refinements. Only ten force constants could be refined but the computational uncertainties of their final values were relatively small. The results of this work are the basis of a detailed assignment of additional vibrational frequencies of the solid K4[Mo(CN),] .2H20 in the region 2090 cm-’ to 2020 cm-1 1161. All c al cu 1at ions were based on the dodecahedron model. This model is suitable for a description of the vibrational behaviour of the octacyanomolybdate(IV) ion. ACKNOWLEDGEMENTS
The authors thank Prof. Dr. G. Geiseler for the many discussions which stimulated this study. We also thank Dr. P. Reich, AdW der DDR, for his Raman measurements. REFERENCES 1 H. Stammreich and 0. Sala, Z. Elektrochem., 64 (1960) 741: 65 (1961) 149. 2 E. Kiinig, Theor. Chim. Acta, 1 (1962) 23. 3 S. F. A. Kettle and R. V. Parish, Spectrochim. Acta, Part A, 21 (1965) 1087.
172 4 K. 0. Hartman and F. A. Miller, Spectrochim. Acta, Part A, 24 (1968) 669. 5 R. V. Parish, P. G. Simms, M. A. Wells and L. A. Woodward, J. Chem. Sot. A, (1968) 2882. 6 T. V. Long and G. A. Vernon, J. Am. Chem. Sot., 93 (1971) 1919. 7 R. N. Jones, Computer Programs For Infrared Spectrophotometry, N.R.C.C. Bulletin Ottawa, 1976. 8 0. Salvetti, Ann. Chim. (Rome), 48 (1958) 1293. 9 H. L. Schltifer and H. F. Wasgestian, Theor. Chim. Acta, 1 (1963) 369.
10 G. Racah, J. Chem. Phys., 11 (1943) 214. 11 12 13 14 15 16
J. L Hoard, T. A. Hamor and M. D. Glick, J. Am. Chem. Sot., 90 (1968) 3177. L. H. Jones and B. I. Swanson, Act. Chem. Res., 9 (1976) 128. I. Nakagawa and T. Shimanouchi, Spectrochim. Acta, 18 (1962) 101. K. H. Schmidt, Spectrochim. Acta, Part A, 33 (1977) 369. J. J. Rafalko, I3 I Swanson and L. H. Jones, J. Chem. Phys., 67 (1977) 1. E. Hahn, H. Bijhlig, M. Ackermann and J. Fruwert, Spectrochim. Acta, in press.
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