2. N. Winograd and T. Kuwana, Electroanal. Chem. 7, 1 (1974). 3. J. F. Tyson and T. S. West, Talanta 27, 335 (1980). 4. R. Pruiksma and R. L. McCreery, Anal. Chem. 53, 202 (1981). 5. J. D. Brewster and J. L. Anderson, Anal. Chem. 54, 2560 (1982). 6. J. Zak, M. D. Porter, and T. Kuwana, Anal. Chem. 55, 2219 (1983). 7. P. Rossi and R. L. McCreery, J. Electroanal. Chem. 151, 47 (1983). 8. M. D. Porter and T. Kuwana, Anal. Chem. 56, 529 (1984). 9. L. E. Fosdick and J. L. Anderson, Anal. Chem. 60, 156 (1988).

10. L. E. Fosdick and J. L. Anderson, Anal. Chem. 60, 163 (1988).

11. M. C. Yappert, S. M. Kimbrell, and E. S. Yeung, Appl. Opt. 26, 3536 (1987).

12. R. N. Jones, J. Am. Chem. Soc. 74, 2681 (1952). 13. J. L. Koenig, Anal. Chem. 36, 1045 (1964). 14. T. Hirschfeld, Anal. Chem. 51, 495 (1979). 15. A. Bewick, J. M. Mellor, and S. Pons, Electrochim. Acta 25, 93

(1980). 16. J. D. Brewster, Ph.D. Dissertation, University of Georgia, Athens

(1987).

Optically Active Vibrations of Lithium and a-Sodium Uranium Hexafluorides (LiUF6 and a-NaUFe)

K E N O H W A D A Division of Chemistry, Japan Atomic Energy Research Institute, 319-11, Tokai-mura, Naka-gun, Ibaraki-ken, Japan

Normal coordinate analyses of the optically active vibrations of LiUF6 and a-NaUFe have been made, with the assumption of an idealized structural model. On the basis of the results of this treatment, all of the observed bands so far reported have been successfully assigned to the intramolecular vibrations (UFe-) as well as the optically active lattice vibrations [M+(Li, Na) . . . UFe-]. The force constants concerning U-F and M-F bonds have been obtained within the framework of the modified valence force field. It has been found that the present analysis, based on a whole crystal, is powerful not only for assigning the optically active lattice vibrations but also for elucidating the vibronic structure of the electronic absorption spectra obtained at iow-temperature measure- ments.

Index Headings: Infrared; Molecular structure; Raman spectroscopy; UV-visible spectroscopy.

I N T R O D U C T I O N

There is' no doub t t ha t the studies of vibronic struc- tures of pen tava len t act inide halide complexes provide significant informat ion on the typical electronic t ransi- t ions and energies of 5 /e lec t rons . Therefore , it is useful to analyze in detai l the electronic absorpt ion spec t ra of various pen tava len t act inide halide complexes p repa red so far. 1-1~ Recently, H e c h t et al. 1~ have measured elec- t ronic absorp t ion spec t ra of complex pen tava len t ura- n ium halides such as LiUF6, a-NaUF6, and CsUF6 at low tempera tu res , and have sys temat ica l ly in te rpre ted their spec t ra as consist ing of sequences of vibronic t ransi t ions coupled to well-defined electronic excited states. In ad- dition, crystal-field calculat ions have been pe r fo rmed successfully. However , it seems tha t there are a few un- sa t i s fac tory spectra l ass ignments of observed bands. For instance, the mode labeled vc(340 cm-1), * only found in the excited s ta te of L iUF 6, is not assigned clearly; fur- the rmore , the ass ignment of the low-frequency mode a t 53 cm -1 is obscure. In order to remove such ambiguit ies , it would be be t te r to pe r fo rm a normal coordinate anal- ysis of optical ly active latt ice vibrat ions as a whole crys- tal.

Received 2 December 1988. * Hecht et al.lg have designated a vibronic band observed at ~ 340 cm-'

in LiUF6 as vc.

In the p resen t s tudy, we take up two examples of LiUF6 and a -NaUF6 t h a t have the same crystal s t ruc ture and assume, to a first approx imat ion , an idealized s t ruc ture for bo th crystals, to which the GF mat r ix me thod 16 de- veloped for analyzing the optical ly active latt ice vibra- t ions of a crystal is applied. I t is shown tha t the presen t analysis is useful for elucidating unassignable bands found in the electronic excited s ta tes of LiUF6 and a -NaUFs .

N O R M A L C O O R D I N A T E A N A L Y S I S

The crystal s t ruc ture of LiUF6 and a -NaUFs , which has been de te rmined by P e n n e m a n and co-workers ) ,5 is rhombohedra l , with space group R3. Each u ran ium is compr ised of six oc tahedra l ly bonded fluorine a toms (six- fold coordinat ion) a t a dis tance U-F - 1.98 A. t On the other hand, M ÷ (Li, Na) is su r rounded with six UF6- units in MUF6 (dis tor ted NaCl - type s t ructure) . In the p resen t s tudy, we assume an idealized s t ruc ture (face- centered cubic type) in which each UF6- is a regular oc tahedron (Fig. 1, Oh symmet ry ) and M ÷ is also coor- dinated, as a regular octahedron, by six UF6- uni ts (Fig. 2). Here , i t should be no ted t ha t this mode l s is somewha t different f rom the actual crystal s t ructure . However , such model ing m a y be allowed, as a first s tep in exact analyses of vibronic s t ructures , when one takes into account (1) simplif icat ion of group theoret ical and normal coordinate analyses, and (2) reasonable ass ignments of observed bands.

A group theoret ical considerat ion shows t h a t for the model s t ruc ture descr ibed above there should be eight opt ical ly active fundamen ta l vibrat ions, i.e., one A16 (Ra- man-act ive) , one EG (Raman-act ive) , one Flc (Raman- active), three FlU ( infrared-act ive) , one F2c (Raman-ac- tive), and one F2u (both- inact ive) vibrat ions. I t is found at first sight t ha t two vibra t ional modes are added anew to six fundamen ta l modes of the UF6- uni t alone. One is the Raman-ac t ive FIG vibra t ion expressing the rota-

t This value is assumed to be nearly the same as the U-F bond in CsUF6 (see Ref. 4), and therefore is somewhat arbitrary. ~: The present model is close to the crystal structure of B-NaUF~ (see Ref. 3).

"/'14 Volume 43, Number 4, 1989 0003-702s/89/4304-071452,00/0 APPLIED SPECTROSCOPY © 1989 Society for Applied Spectroscopy

( %*,%%

Z

t )5

x

FIG. 1.

Uronium

Q Fluorine

• Lithium (Sodium) Specification of internal coordinates for LiUF6 and a-NaUF6.

tional lattice vibration, and the other is the infrared- active F w vibration corresponding to the translational lattice vibration. These new modes will be considered in detail later.

In performing the normal coordinate analysis by the GF matrix method, 1G one takes the optically active Carte- sian coordinates Xop and the optically active internal coordinates Roe as follows:

Xop (24): Ax., Aye, Az~, Axe, Ay. Az~, Ax2, Ay2, Az2, Ax3, Ay3, Az3, Ax4, Ay4, Az4, Axs, AYs, Az~, Ax6, Ay6, AZs, AXM, AyM, AZM.

Rop (36): Ar,, Ar2, Ar3, Ar4, Ars, Ar6, Aql, Aq2, Aq3, Aq4, Aqs, Aq6, A¢12, A¢23, A¢34, A¢14, A¢51, 5¢52, A(~53, A(~54, A(~6I, A¢62, A(~63, 5¢64, A(~ly, n¢lz, A(~2x, A¢2z, A(~3y, A~/~3z, A~/~4x, A¢4 z, n(~5x, A¢5~, A¢6~, A¢6~.

Here, r's and q's are the internuclear distances of the U-F and M-F bonds, and ¢'s are the bond angles around the uranium and fluorine atoms, as shown in Fig. I. The inverse kinetic energy matrix Gop c for the optically active vibrations is given with respect to the optically active Cartesian coordinates as

AxvA yuAzu Ax~ A y~ Azi . . . AXMAYMAZM ~u

#g #u 0

#F GoeC = ~tF (1)

#F

0 #M #M

#M

Z

I y

I

1 - -

FIG. 2.

(~) UF6

0 Lithium (Sodium) Idealized coordination of UF6- to lithium (or sodium) ion.

where tt,, ~F, and ]-£M are the reciprocal masses of ura- nium, fluorine, and lithium (or sodium) atoms, respec- tively.

In order to obtain the corresponding potential energy matrix Fop c with respect to the optically active Cartesian coordinates, we need a transformation matrix Bop, which is defined by

Rop = BopXop. (2)

The Bop matrix in Eq. 2 can easily be obtained by simple geometric considerations as a 24 x 36 order matrix• By using this Bop matrix, we obtain the FoS matrix in ques- tion by the relation

Fop c= (Bop)tFopiBov (3)

where Foe i is the potential energy matrix expressed in terms of the optically active internal coordinates Rop, and the superscript t indicates the transposed matrix. On the assumption of a modified valence force field, the potential energy V of the MUF6 lattice is given by

2V = ZKI(Ar.) 2 + EK2(Aq.) 2 + ~-,HI(Adpnm) 2 + EH2(ACm,) 2 + 2Zk(Ar,) (Arm) + 2Z/(Arm) (At,) + 2Eh(A¢,m) (ACres) + 2Zg(A¢,m) (ACsi) (4)

where K1 and K2 are the U-F stretching and M-F inter- action force constants, H, and/ /2 are the bending force constants around the uranium and fluorine atoms, k and f are the bond-bond interaction force constants, and h and g are the angle-angle interaction force constants. The specification of the latter four interaction constants (k, f, h, and g) is schematically shown in Fig. 3. According to this model, the Foe i matrix is constructed as a 36th- order matrix with eight kinds of the elements of KI, K2, H1, H2, k, f, h, and g.

Since it is assumed in the present study that MUF6 has the face-centered cubic lattice, the Gop c and Fop c matrices may be further factored by a proper choice of

APPLIED SPECTROSCOPY "/15

( (

(k) (f)

%

()

FIG. 3. field.

(h) (g) Interaction force constants introduced in a simple valence force

opt ical ly active Car tes ian s y m m e t r y coordinates Sop, which are given by

Sop = UXop . (5)

Here , U is, of course, the t r ans fo rmat ion ma t r ix which t ransforms the optically active Cartesian coordinates into the opt ical ly active Car tes ian s y m m e t r y coordinates. One set of the s y m m e t r y coordinates is given by the following expressions (6a-6h):

S 1 = ( 1 / ~ / f 6 ) ( A X l -~- A y 2 - - AX3

- Ay4 + Az5 - Az6), (6a) S~ = ( 1 / V T ~ ) ( ~ x , + ay~ - ax~

- Ay4 -- 2Az5 + 2Az6), (6b) $3 = (1/2)(Az, - Az3

-- Ax~ + Ax6), (6c) $4 = ( 1 / k / 5 6 ) ( 7 A Z v - Az l -- Az2 -- Az3

- - Az4 -- Az~ -- Az6 -- AZM), (6d) $5 = ( 1 / V ~ 2 ) ( 6 A Z M -- AZl -- AZ2 -- AZ3

- - AZ4 -- AZ~ -- AZ6), (6e) $6- - (1 /VT2) ( -Az~ - Az2 - Az~

- Az~ + 2Az~ + 2Az~), (6f) Sv = (1/2)(Az~ - Az3 + Az~ - Az6), (6g) Ss = (1/2)(Az, - Az2

+ Az~ -- Az4). (6h)

The s y m m e t r y coordinates S,, S~, and S~ express the Alc, E~, and FI~ types of vibrat ions, respectively. Those of $4, S~, and S~ are the three F , u types of vibrat ions, and $7 and $8 are the F2~ and F2u types of vibrat ions. Here , i t should be no ted t h a t two s y m m e t r y coordinates, S~ (Eq. 6c) and S~ (Eq. 6e), are added anew to those of the U F ~ - unit. These two s y m m e t r y coordinates are sche- mat ica l ly shown in Fig. 4.

T h e inverse kinetic energy mat r ix Gop cs and the po- tent ia l energy mat r ix Fop cs expressed in t e rms of the Sop coordinates are calculated by the re la t ionships

Gop cs = UGopCU t, (7)

and

Fop cs = UFopCU '. (8)

The Gop cs and Fop cs matrices obtained are shown in Eqs. 9 and I0:

$2 $3 $4 $5 $6 $7 Ss S1 A

GopCS=

A A

B

S y m m e t r i c

0

D 0 C 0

A A

A

(9)

where

and

A = ],.tF, B = (49#u + 6~F + #M)/56, C = (6#M + ~F)/7, D = (V~/14)(#F -- #M),

FovCS=

$1 $2 $3 St A

B C

D

where

Symmet r i c

A = K 1 -F g 2 -F 4k + f , B = K l + K 2 - 2 k + f , C = (1/r , + 1/r2)2H2,

$5 $6

E F

G H

I

$7 Ss

0

(10)

J

K

D = (16/7)[K1 - f + 4(H, + 2h - g ) / r l 2 + 2H2/ri2], E = (4V~/21)[Ka - f + 4(H~ + 2h - g ) / r l 2

+ 2H2/r l 2 + 14HJr l r2] , F = (4V~2)[K1 - [ - 2(/-/1 + 2h - g)/r~ 2 - H2/r~ 2

- H2/r,r2], G = (1/21)[K~ - f + 49K2 + 4(//1 + 2h - g)/ra 2

+ 2HJr~ 2 + 2SH2/r~r2 + 98H2/r22], H = (V~4/42)[K1 - f + 7K2 - 2(/-/1 + 2h - g ) / r l 2

- H J r ~ 2 - 7H2/r22 - 8H2/rlr2], I = (1/6)[g~ - f + K2 + 10(Ht + 2h - g)/r~ 2

+ 5( l / r , + 1/r2)2H2], J = 4 ( / / 1 + g ) / r ? + (1/rl + 1/r=)=H~, g = 2(H~ - 2h - g ) / r l 2 + (1/r~ + 1/r2)2H2.

In Eq. 10, rl and r2 denote the in ternuclear dis tances of the U-F and M - F bonds.

The secular equat ion for calculat ing optical ly active normal v ibra t ion frequencies (~) is finally ob ta ined as

I GopCSFoe cs - EAI = 0 (11)

716 Volume 43, Number 4, 1989

%%% l

I

Y:s (F,Ig) g5 (Flu) Fro. 4. Optically active lattice vibrational modes.

where E is the unit matrix with the same order as that of the Gop cs and Fop cs matrices, and h = 47r2C2v 2.

DISCUSSION ON THE VIBRATIONAL ASSIGNMENTS OF OBSERVED FREQUENCIES AND FORCE CONSTANTS

The infrared and Raman spectra of LiUF6, a-NaUF6, and CsUF6 were measured by Hecht et al. ~5 They also measured the electronic absorption spectra at low tem- peratures. The results have been assigned to the normal modes of an isolated UF6- group through the traditional way. 4,t7 In so far as we are concerned with only the iso- lated UF6- group, their assignments seem to be exact. However, as has been pointed out previously, the mode labeled Vc(340 c m - 1 ) , a s well as the low-frequency mode (53 cm -~) found in the excited-state spectrum of LiUF6, is not assigned to any normal modes of the isolated UF6 group. This argument applies in the cases of a-NaUF6 and CsUF6, in which no band corresponding to v~ for LiUF6 is observed, but the low-frequency mode is ob- served. Therefore, it is apparent that, in due course, new and reasonable assignments must be made for all the observed bands of LiUF6, a-NaUF6, and CsUF6.

In the present study, we have taken up two examples of LiUF6 and a-NaUF6 that have the same crystal struc- ture and have tried to give new assignments for such compounds on the basis of the idealized structure. Our new results are listed in Tables I (LiUF6) and II (a- NaUF6). In these tables, the highest Raman-active bands observed at 618-626 cm -1 for both compounds are as- signed to the A~c vibrational mode (vl), expressing the

TABLE I. Observed ~ and calculated vibrational frequencies (in cm-') of LiUF6.

Observed Mode

(Symm. species) Fund. b Excit. ° Calculated

I"/1 (A,c) 622 626 622 :'2 (Ec;) 439 461 439 :'3 (F,c) - - 53 53 :'4 (F~) 515 510, 582 515 P5 (F1.) - - 340 340 ~,~ (F,.) 152 152, 183 152 :'7 (F2a) 222, 232 240 222 :'8 (F~.) - - 107 107

Data taken from Ref. 15. b Observed infrared- and Raman-active vibrational frequencies of fun- damental modes. Deduced from measurements in excited-state spectra.

TABLE II. Observed ~ and calculated vibrational frequencies (in cm -1) of a-NaUF6.

Observed Mode

(Symm. species) Fund. b Excit. ° Calculated

v, (A~c) 621 618 621 :'2 (Ea) 449 - - 449 :'3 (Flo) - - 48 48 :'4 ( F ~ , ) 520 512 520 :'~ (F,.) - - - - 190 V 6 (Flu) - - 122, 132 122 :'7 (F2c) 206, 209 - - 206 :'s (F2, , ) - - 72 72

Data taken from Ref. 15. " Observed infrared- and Raman-active vibrational frequencies of fun-

damental modes. c Deduced from measurements in excited-state spectra.

totally symmetric U-F stretching vibration in the UF6- group. The Raman-active bands at 439-461 cm -1 are also assigned to the Ec vibrational mode (~2), expressing the degenerate symmetric U-F stretching vibration in the U F c group. The lowest bands observed at 48-53 cm -1 in the excited electronic spectra of both compounds are newly attributed to the F~c vibrational mode (~3), which just corresponds to the triply degenerate rotational lat- tice vibration, as shown in Fig. 4. The relatively high infrared-active bands observed at 510-520 cm -1 are as- signable to one (v4) mode of the three F1, vibrational modes, which mainly expresses the triply degenerate asymmetric U-F stretching vibration in the UF~- group. The model labeled ~c(340 cm -1) by Hecht et al., which was only found in the excited electronic spectra of LiUF6 and not in a-NaUF6, where it would be obscured by the UF6- vibrations, may possibly be assigned to one (v~) mode of the three F1, vibrational modes. This triply de- generate mode arises from the translational lattice vi- bration between M + and UF6% as shown in Fig. 4. The infrared-active bands near 122-183 cm -1 are assigned to the last (~6) of the three F~, vibrational modes, which is mainly interpreted as being the triply degenerate U-F bending vibration in the UF6- group. The Raman-active bands observed at 190-240 cm -1 are certainly assigned to the triply degenerate U-F bending vibration, in the UF6- group, corresponding to the F2G vibrational mode (VT). The bands observed at 72-107 cm -1 in the excited electronic spectra are attributed to both the infrared- and the Raman-inactive mode of the triply degenerate U-F bending vibration.

The vibrational assignments, as interpreted, may be supported by normal coordinate analyses of the optically active lattice vibrations as already described above. Eight

TABLE III . Modified valence force constants (in md/A,) used in the frequency calculation.

Force constant LiUF6 a-NaUF~

K1 3.3262 3.4952 K2 0.2313 0.2245 HI ~ 0.4007 0.3030 H2 ° 0.0316 0.0259 k 0.3622 0.3433 f -0.6759 -0.7764 h ~ 0.0510 0.0512 ga 0.1091 0.1373

a In md-/~/rad 2 units.

APPLIED SPECTROSCOPY 717

kinds of the force c o n s t a n t s - - K , K2, H1,//2, k; f, h, and g in Eq. 10--are adjusted in a least-squares way, so tha t the calculated frequencies give the best agreement with observed ones. The force constants obtained are listed in Table III. Here, it is helpful to compare the force constants in Table I I I with those of hexavalent uranium fluoride, UF6 .ls,19 As might be expected, the value of the U-F stretching force constant K~ is somewhat smaller than tha t (3.76 md//~) of UF~. The other force constants, such as H~, k, f, h, and g, seem reasonable in comparison with those of UF6. The magni tude of the force constants K2 and/-/2 are also reasonable when it is taken into con- sideration tha t they govern the lattice interaction be- tween M ÷ and UF6-.

The calculated frequencies for LiUF6 and a-NaUF6 are listed in Tables I and II together with the observed ones. ~5 I t is seen from these tables tha t the agreement between the calculated and observed frequencies is very good for both compounds. This fact supports the supposit ion tha t the present vibrational assignments made on the as- sumpt ion of the idealized structure are adequate. There- fore, we may conclude tha t the normal coordinate anal- ysis of the optically active lattice vibrations is useful for the theoretical interpretat ion of the vibronic structures of crystals such as LiUF6 and a-NaUF6.

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3, 126 (1964). 4. M. J. Reisfeld and G. A. Crosby, Inorg. Chem. 4, 65 (1965). 5. G. D. Sturgeon, R. A. Penneman, F. H. Kruse, and L. B. Asprey,

Inorg. Chem. 4, 748 (1965). 6. L. B. Asprey, T. K. Kennan, R. A. Penneman, and G. D. Sturgeon,

Inorg. Nucl. Chem. Lett. 2, 19 (1966). 7. R. A. Penneman, G. D. Sturgeon, L. B. Asprey, and F. H. Kruse,

J. Am. Chem. Soc. 87, 5803 (1965). 8. J. Selbin and J. D. Ortego, Chem. Rev. 69, 657 (1965). 9. J. L. Ryan, J. Inorg. Nucl. Chem. 33, 153 (1971).

10. A. F. Leung, J. Phys. Chem. Solids 38, 529 (1977). 11. E. Soulie, J. Phys. Chem. Solids 39, 695 (1978). 12. Y. M. Poon and D. J. Newman, J. Chem. Phys. 77, 1077 (1982). 13. H. G. Hecht, L. P. Varga, W. B. Lewis, and A. M. Boring, J. Chem.

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Symp. Ser. 216, 199 (1983). 15. H. G. Hecht, J. G. Malm~ J. Foropoulos, and W. T. Carnall, J.

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1597 (1961). 17. W. A. Yeranos, Z. Physik. Chem. Neue Folge 45, 72 (1965). 18. H. H. Claassen, J. Chem. Phys. 30, 968 (1959). 19. R. S. McDowell, L. B. Asprey, and R. Paine, J. Chem. Phys. 61,

3571 (1974).

718 Volume 43, Number 4, 1989