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Electron spin resonance study on phase transitions and structure of frozen ionic solutions
Volume 44, number
CHEMICAL
3
15 Deccmbcr
PHYSICS LETTERS
1976
ELECTRON SPLNRESONANCE STUDY ON PHASE TRANSITIONS AND STRUCFURE OF FROZEN IONIC SOLUTIONS
W.W. SCHMIDT and K.G. BREITSCHWERDT Institut frrr Angcwandte Rcccivcd
Physik, Universitat fieidelbcrg,
ffcidelhcrg,
Germany
16 July 1976
Rapidly frozen dllutc aqueous solutions tra mdsatc. that at 77 K the Cu2+ Ions arc These rcglons crystallize at 212 K and melt between the c.rystaIlmc and liquid phase of the propagation vcloclty of the solidlticatlon
of Cu(N03)2 have been investigated by means of ESR spectroscopy. The spccscgrcgatcd into regions of high ion concentration with amorphous structure. at 247 K. A quantitatlvc study of the solute transport through the interface the solvent shows how the ion distrrbutlon in tlx frozen solution dcpcnds on front.
1. Introduction Rapidly frozen ionic solutions have so far mainly been studied by means of Mbssbaucr spectroscopy and differential thermal analysis [l-8]. Different mutually exclusive suppositions about the low-temperature structure of dilute solutions have been reported, e.g., cooling an aqueous FcCl, solution with a rate of 10 K/s has been assumed to result in a homogeneous glass [6], whereas precipitation of the crystalline hydrated salt in aqueous SnC12 solution even at a cooling rate of lo4 K/s has been suggested [4] and, finally, a mixture of crystalline and glassy phases has been assumed to exist in a quenched Fc(ClO4)2 solution [ 1,2]. With increasing temperature, the quenched solutions show several structural transitions which have been ascribed to phase transitions between a number of crystalline and amorphous phases [l-3,
and 273 K. The accuracy of the temperatures given for the observed phase transitions is estimated to be -3 K.
2. Results and discussion In the temperature range bchveen 77 K and 212 K the spectrum of an aqueous solution of0.05 M Cu(NO3)2 is a simple symmetrical line (fig. la). The
WI.
In this letter ESR measurements arc reported which yield information concerning the structure of quickly frozen solutions. The results are used to test the physical significance of the proposed structure models. The CL? ion (electronic spin l/2, nuclear spin 3/2) is especially useful in the ESR investigation because of its simple spectrum [9]. The solutions were quenched to 77 K within about 5 s. Then, 10 to 15 spectra were recorded at 9.3 GHz between 77 K
b
2800
3oOa
Fig. 1. ESR spectra of an aqueous at dlffercnt temperatures.
32ao HiG1 8.05 hf Cu(NOs)z
soIution
Volume 44, number 3
CHEMICALPHYSICS
absence of hyperfine structure and of anisotropic magnetic properties may be explained by spin-spin interaction between neighboring copper ions. This means that the copper ions are segregated in high-concentration regions with amorphous or crystalline structure as a result of diffusion processes during the freezing procedure. Redistribution of a solute caused by the advancing interface of the crystalline phase of the soIvent has been discussed in the literature [IO] assuming partitioning of the solute concentration on both sides of the interface according to the equilibrlurn distributton coefficient. However, high crystahization velocities may invaliditate this assumption. Thercfore, the mass transport through the interface is analyzed in terms of nonequilibrium thermodynamics. Neglecting t.hermaI diffusion and a possible center of mass motion the diffusion velocity u of the solute in a binary solution is [ 1 l] F-L11
gradyiT/p(l
-X)T,
(1)
where X = p/p0 is the mass fraction, p and p” arc the mass densities of the solute and of the solution, rcspectively, L lt is a phenomenological coefficient, and ECis the chemical potential of the solute. The index Tin eq. (1) indicates that the gradient has to be taken at constant temperature. The mass transport is studied in a one-dimensional, inlinitely extended system where the solid-liquid interface propagates with constant veIocity vo. With respect to an axis x, which is perpendicular to and moving with the interface, the solid phase is at x < 0, the liquid phase at x > d. At 0
=D4p:A,(l
(21
-&&Y/R.,
(3)
where a! = s, !? refers to the solid and liquid phase, respectively, c(z is independent of x, M is the molar mass of the solute, R the gas constant, and D, the diffusion coefficient. At thermodynamic equilibrium the distribution coefficient of the solute, g, = ~Jps, is ge = exp [-elf
- &M/RTl-
(4)
The continuity equation for the diffusion process can easily be satisfied if one. considers a stationary diffu558
15
LETTERS
Dcccmbcr 1976
sion. Then, P(X) =o(O) [u(O) -
v01/tw - vol.
(5)
In general, D, in the solid phase, x G 0, is very small, so L(O) can be neglected in eq. (5). Definingg = p(O)/~(d) as the dynamical distnbution coefficient that may depend on ‘lo, one obtams from eqs. (I), (3) and (5) g = (MD,/RTuO)
gradp(d
+ 1.
(6)
gradB(d)(., is not known it is approximated here by the mean gradient in the interface As
run (4 - @)I
IT/d = -(R T/Md) In (g/g,) .
(7)
This expression, inserted into cq. (6) yields uo = (Q&0
ln (&Ml
-g)
.
(8)
According to eq. (8), forg < 1 and a crystaI.lization velocity u0 = Dp/d a dynamical distribution coefficient of 2.7 gc results. With D, = 10s5 cm2/s and an interface thickness of d = 10 A this velocity is u. = 100 cm/s. Very much higher velocities are necessary in order to bringg close to 1. Then, a homogeneous solid solution should grow with the same concentration as in the liquid phase. A value ofg, 4 1 is to be expected for solutions of an ionic species in polar soIvents like water. Experimentally a growth velocity of about 10 cm/s has been observed in pure water crystallizing at -lO°C [ 121. Velocities in this range arc too smaII to bring the dynamical distribution coefficient close to 1. Thercfore, most of the solute is rejected at the advancing crystalline phase of the solvent. In a limited sample volume the solute is increasingly concentrated in the remaining liquid regions until the temperature is low enough for solidification. As quenched concentrated aqueous solutions solidify in the glassy state [ 131 the high-concentration regions may have an amorphous structure. The irreversible phase transition observed a‘t 212 K further supports the assumption of an amorphous phase. This transition leads to a new line shape up to 247 K (fig. I b) which may be ascribed to a crystalline phase with copper ions on magnetically equivalent lattice sites corresponding to the following structure model: N copper ions with electronic spins RSi (i = 13. .. ) N) and nucIear spins hli (i = 1, .. . ,N) occupy magneticaIIy equivaIent lattice sites in a crystaI.line
phase. The g-tensors, G, and hyperfine tensors, A, have axiaI symmetries with the principa1 tensor clements gl, gll and A,, n ,,, respcct~vely. There exists an exchange interaction Ji, between the ions i and j. The spin hamiltonian of the system is
N
N
JC=Fo~r-G.CSi+
N
C.I,Si.SJ i,J
1
(9)
+CSi*A*I,, I
where H is the magnetic field and PO the B&r magncton. Sy extending to
a dlagonalization
include the anisotropic
transitions
are
proccclurc
interactions
[ 141
of cq. (9), ESII
under the conditmn l&,HGl% IAl and I.II S IAI at magnetic II according to the expression found
11~ = &-,IIG + Am/N
15 December L976
CtfEhIlCAL I’IIYSICS LET-I ERS
Volurnc 44. number 3
that field strengths
,
where v is the microwave
wo
frequency,
nz = Xnz, the
total nuclear magnetic number, tlZi that of an individual nuclear spin, (; and /I arc the following functions of the angle 0 between the magnetic field and the symmetry axes of the g and hyperfinc tensors: (i2 =gzcos”O +gzsin?-0 ,
(11)
(12) For large N, cq. (10) predicts that the cnvclope of all hyperfine lines is obscrvcd experimentally, which is very peaked at the center [14]. If there arc many clusters with a statistically isotropic dlstrlbution of the angle 0 a powder spectrum results from eqs. (lo), (I I), and (12) that has the features of the axialIy symmetrical g-tensor and no hyperfine structure. Under these conditions the spin hamiltonian, eq. (9), can be fitted to the observed spectrum of fig. lb, givinggl = 2.11 and gll = 2.32. In order to further identify the structure, polycrystalline samples of Cu(N03)~~xH~0, x = 3 and 6, were mvestigated. The spectrum of the hexahydrate turns out to be identical with that of fig. lb. At 247 K a reversible structural transition leads to a simple symmetrical line (fig. lc) up to 273 K. Spcctra like these have been reported for concentrated aqueous CU(NO~)~ solutions in the liquid state [ 151. Therefore, the reversible transition at 247 K is ascnbed to the melting process of the concentrated phase. This conclusion is supported by the fact that the Cu(N03)2-water system has a eutectic composi-
tion of about 5 mot% CU(NO~)~ at 247 K [13]. Above 273 K (fig. Id) the spectra show the poorIy resolved hyperfine structure of the magneticaIIy dilute hquid
solution
[ 16,17]_
Solute segregation in frozen solutions may be rcduced by the addition of a glass-forming agent [ 181. The influence of such a substance on the structure of the quickly frozen solution was studied using an aqueous 0.05 M CU(NO~)~ solution with 1 M nitric acid. Rctwecn 77 and 170 K the spectra (fig. 23) suggest that the Cu2+ cot~~ploxes nre in the map,neticdiy dilute state with magnetic
parameters
(gL = 2.09, gn = 2-40,
and IA u/hci = 0.0128 cm-l) nearly identical with those reported for the Cu(H20)r complex in other media [ 16,171. In the temperature range 170-200 K the frozen solution undergoes a continuous irrevcrsibIc structural transformation (fig. 2b). Between 200 and 232 K the spectra (fig. 2c) art? identical with the [ine shape of fig. lb for an aqueous CU(NO~)~ soIution_ Because of this identity one may conclude that C$* is contained m the crystalline hydrate Cu(NO&GH,O. Above 232 K a new line shape (fig. 2d) indicates the begmmng of a reversible transition which leads to nearly unresolved hyperfine iines above 240 K (fig. 2e). This transition is assigned to the
bl172’=K
cl 2lO’K -__
dl 23YK
2700-
2900
3100
IISR spectra of m aqueous 0.05 hf Cu(NOz)+ wrth 1 hi nitric acid at different temperatures. FIN. 2.
c
HEI
soIution
559
Volume 44, number
3
CHEMICAL
PIIYSICS
melting process of the highconcentration regions since the hyperfine structure between 240 and 273 K suggests that the copper ions are in a liquid solution.
References [ 1] J-A. Cameron, L. Kcszthclyi, G. Nagy and L. Kaccoh. J. Chem. Phys. 58 (1973) 46 IO. (21 K. Frohlich and L. Keszthclyi, J. Chem. Phys. 58 (1973) 4614. [3] A.S. Plachmda and E.F. hfakarov. Chem. Phys. Lcttcrs 25 (1974) 364. [4] R.L. Cohen and K.W. West, Chcm. Phys. Lcttcrs 13 (1972) 482. [S] P.M. Thomas, M. Sanders, 1. Deszi and P.J. Ouscph. Chcm. Phys. Lcttcrs 11 (1971) 42. (61 B. Burnot, U. Iiauser and W. Ncuwuth. 2. Physrk 249 (1971) 134.
560
LETTERS
I5 Dcccmbcr
1976
I71 AS. Plachinda and E.F. Makarov, Chem. Phys. Letters 15 (1972) 627. 181A. Du Fresnc, J.hf. Knudsen. J-E. Morcrra and KS. Ncto, Chcm. Phys. Letters 20 (1973) 108. PI W. Schmidt and K. Breitschwcrdt, Bcr. Bunscngcs. Physrk. Chcm. 78 (1974) 1268. [IOl W.A. TilIer and R.F. Sckerka, J. Appl. Phys. 35 (1964) 2726. thenno1111S.R. dc Groot and P. Mazur, Nonequrhbrium dynamics (North-lIolland, Amsterdam, 1962). 1121C.S. Lindenmeyer, G.T. Orrok, K.A. Jackson and B. Chalmers, J. Chcm. Phys. 27 (1957) 822. CA Angcll and E.J. Sax, J. Chcm. Phys. 52 (1970) 1058. Y. Zshikawa, J. Phys. Sot. Japan 21 (1966) 1473. S. Fujiwara and 11. Ii~yashi, J. Chcm. Phys. 43 (1965) 23. R. Poupko and Z. Luz, J. Chcm. Phys. 57 (1972) 3311. W 8. LCWIS,hf. AIci and L.O. Morgan, J. Chcm. Phyg. 44 (1966) 2409. [I81 R.T. Ross, J. Chcm. Phys. 42 (1965) 3919.
CHEMICAL
3
15 Deccmbcr
PHYSICS LETTERS
1976
ELECTRON SPLNRESONANCE STUDY ON PHASE TRANSITIONS AND STRUCFURE OF FROZEN IONIC SOLUTIONS
W.W. SCHMIDT and K.G. BREITSCHWERDT Institut frrr Angcwandte Rcccivcd
Physik, Universitat fieidelbcrg,
ffcidelhcrg,
Germany
16 July 1976
Rapidly frozen dllutc aqueous solutions tra mdsatc. that at 77 K the Cu2+ Ions arc These rcglons crystallize at 212 K and melt between the c.rystaIlmc and liquid phase of the propagation vcloclty of the solidlticatlon
of Cu(N03)2 have been investigated by means of ESR spectroscopy. The spccscgrcgatcd into regions of high ion concentration with amorphous structure. at 247 K. A quantitatlvc study of the solute transport through the interface the solvent shows how the ion distrrbutlon in tlx frozen solution dcpcnds on front.
1. Introduction Rapidly frozen ionic solutions have so far mainly been studied by means of Mbssbaucr spectroscopy and differential thermal analysis [l-8]. Different mutually exclusive suppositions about the low-temperature structure of dilute solutions have been reported, e.g., cooling an aqueous FcCl, solution with a rate of 10 K/s has been assumed to result in a homogeneous glass [6], whereas precipitation of the crystalline hydrated salt in aqueous SnC12 solution even at a cooling rate of lo4 K/s has been suggested [4] and, finally, a mixture of crystalline and glassy phases has been assumed to exist in a quenched Fc(ClO4)2 solution [ 1,2]. With increasing temperature, the quenched solutions show several structural transitions which have been ascribed to phase transitions between a number of crystalline and amorphous phases [l-3,
and 273 K. The accuracy of the temperatures given for the observed phase transitions is estimated to be -3 K.
2. Results and discussion In the temperature range bchveen 77 K and 212 K the spectrum of an aqueous solution of0.05 M Cu(NO3)2 is a simple symmetrical line (fig. la). The
WI.
In this letter ESR measurements arc reported which yield information concerning the structure of quickly frozen solutions. The results are used to test the physical significance of the proposed structure models. The CL? ion (electronic spin l/2, nuclear spin 3/2) is especially useful in the ESR investigation because of its simple spectrum [9]. The solutions were quenched to 77 K within about 5 s. Then, 10 to 15 spectra were recorded at 9.3 GHz between 77 K
b
2800
3oOa
Fig. 1. ESR spectra of an aqueous at dlffercnt temperatures.
32ao HiG1 8.05 hf Cu(NOs)z
soIution
Volume 44, number 3
CHEMICALPHYSICS
absence of hyperfine structure and of anisotropic magnetic properties may be explained by spin-spin interaction between neighboring copper ions. This means that the copper ions are segregated in high-concentration regions with amorphous or crystalline structure as a result of diffusion processes during the freezing procedure. Redistribution of a solute caused by the advancing interface of the crystalline phase of the soIvent has been discussed in the literature [IO] assuming partitioning of the solute concentration on both sides of the interface according to the equilibrlurn distributton coefficient. However, high crystahization velocities may invaliditate this assumption. Thercfore, the mass transport through the interface is analyzed in terms of nonequilibrium thermodynamics. Neglecting t.hermaI diffusion and a possible center of mass motion the diffusion velocity u of the solute in a binary solution is [ 1 l] F-L11
gradyiT/p(l
-X)T,
(1)
where X = p/p0 is the mass fraction, p and p” arc the mass densities of the solute and of the solution, rcspectively, L lt is a phenomenological coefficient, and ECis the chemical potential of the solute. The index Tin eq. (1) indicates that the gradient has to be taken at constant temperature. The mass transport is studied in a one-dimensional, inlinitely extended system where the solid-liquid interface propagates with constant veIocity vo. With respect to an axis x, which is perpendicular to and moving with the interface, the solid phase is at x < 0, the liquid phase at x > d. At 0
=D4p:A,(l
(21
-&&Y/R.,
(3)
where a! = s, !? refers to the solid and liquid phase, respectively, c(z is independent of x, M is the molar mass of the solute, R the gas constant, and D, the diffusion coefficient. At thermodynamic equilibrium the distribution coefficient of the solute, g, = ~Jps, is ge = exp [-elf
- &M/RTl-
(4)
The continuity equation for the diffusion process can easily be satisfied if one. considers a stationary diffu558
15
LETTERS
Dcccmbcr 1976
sion. Then, P(X) =o(O) [u(O) -
v01/tw - vol.
(5)
In general, D, in the solid phase, x G 0, is very small, so L(O) can be neglected in eq. (5). Definingg = p(O)/~(d) as the dynamical distnbution coefficient that may depend on ‘lo, one obtams from eqs. (I), (3) and (5) g = (MD,/RTuO)
gradp(d
+ 1.
(6)
gradB(d)(., is not known it is approximated here by the mean gradient in the interface As
run (4 - @)I
IT/d = -(R T/Md) In (g/g,) .
(7)
This expression, inserted into cq. (6) yields uo = (Q&0
ln (&Ml
-g)
.
(8)
According to eq. (8), forg < 1 and a crystaI.lization velocity u0 = Dp/d a dynamical distribution coefficient of 2.7 gc results. With D, = 10s5 cm2/s and an interface thickness of d = 10 A this velocity is u. = 100 cm/s. Very much higher velocities are necessary in order to bringg close to 1. Then, a homogeneous solid solution should grow with the same concentration as in the liquid phase. A value ofg, 4 1 is to be expected for solutions of an ionic species in polar soIvents like water. Experimentally a growth velocity of about 10 cm/s has been observed in pure water crystallizing at -lO°C [ 121. Velocities in this range arc too smaII to bring the dynamical distribution coefficient close to 1. Thercfore, most of the solute is rejected at the advancing crystalline phase of the solvent. In a limited sample volume the solute is increasingly concentrated in the remaining liquid regions until the temperature is low enough for solidification. As quenched concentrated aqueous solutions solidify in the glassy state [ 131 the high-concentration regions may have an amorphous structure. The irreversible phase transition observed a‘t 212 K further supports the assumption of an amorphous phase. This transition leads to a new line shape up to 247 K (fig. I b) which may be ascribed to a crystalline phase with copper ions on magnetically equivalent lattice sites corresponding to the following structure model: N copper ions with electronic spins RSi (i = 13. .. ) N) and nucIear spins hli (i = 1, .. . ,N) occupy magneticaIIy equivaIent lattice sites in a crystaI.line
phase. The g-tensors, G, and hyperfine tensors, A, have axiaI symmetries with the principa1 tensor clements gl, gll and A,, n ,,, respcct~vely. There exists an exchange interaction Ji, between the ions i and j. The spin hamiltonian of the system is
N
N
JC=Fo~r-G.CSi+
N
C.I,Si.SJ i,J
1
(9)
+CSi*A*I,, I
where H is the magnetic field and PO the B&r magncton. Sy extending to
a dlagonalization
include the anisotropic
transitions
are
proccclurc
interactions
[ 141
of cq. (9), ESII
under the conditmn l&,HGl% IAl and I.II S IAI at magnetic II according to the expression found
11~ = &-,IIG + Am/N
15 December L976
CtfEhIlCAL I’IIYSICS LET-I ERS
Volurnc 44. number 3
that field strengths
,
where v is the microwave
wo
frequency,
nz = Xnz, the
total nuclear magnetic number, tlZi that of an individual nuclear spin, (; and /I arc the following functions of the angle 0 between the magnetic field and the symmetry axes of the g and hyperfinc tensors: (i2 =gzcos”O +gzsin?-0 ,
(11)
(12) For large N, cq. (10) predicts that the cnvclope of all hyperfine lines is obscrvcd experimentally, which is very peaked at the center [14]. If there arc many clusters with a statistically isotropic dlstrlbution of the angle 0 a powder spectrum results from eqs. (lo), (I I), and (12) that has the features of the axialIy symmetrical g-tensor and no hyperfine structure. Under these conditions the spin hamiltonian, eq. (9), can be fitted to the observed spectrum of fig. lb, givinggl = 2.11 and gll = 2.32. In order to further identify the structure, polycrystalline samples of Cu(N03)~~xH~0, x = 3 and 6, were mvestigated. The spectrum of the hexahydrate turns out to be identical with that of fig. lb. At 247 K a reversible structural transition leads to a simple symmetrical line (fig. lc) up to 273 K. Spcctra like these have been reported for concentrated aqueous CU(NO~)~ solutions in the liquid state [ 151. Therefore, the reversible transition at 247 K is ascnbed to the melting process of the concentrated phase. This conclusion is supported by the fact that the Cu(N03)2-water system has a eutectic composi-
tion of about 5 mot% CU(NO~)~ at 247 K [13]. Above 273 K (fig. Id) the spectra show the poorIy resolved hyperfine structure of the magneticaIIy dilute hquid
solution
[ 16,17]_
Solute segregation in frozen solutions may be rcduced by the addition of a glass-forming agent [ 181. The influence of such a substance on the structure of the quickly frozen solution was studied using an aqueous 0.05 M CU(NO~)~ solution with 1 M nitric acid. Rctwecn 77 and 170 K the spectra (fig. 23) suggest that the Cu2+ cot~~ploxes nre in the map,neticdiy dilute state with magnetic
parameters
(gL = 2.09, gn = 2-40,
and IA u/hci = 0.0128 cm-l) nearly identical with those reported for the Cu(H20)r complex in other media [ 16,171. In the temperature range 170-200 K the frozen solution undergoes a continuous irrevcrsibIc structural transformation (fig. 2b). Between 200 and 232 K the spectra (fig. 2c) art? identical with the [ine shape of fig. lb for an aqueous CU(NO~)~ soIution_ Because of this identity one may conclude that C$* is contained m the crystalline hydrate Cu(NO&GH,O. Above 232 K a new line shape (fig. 2d) indicates the begmmng of a reversible transition which leads to nearly unresolved hyperfine iines above 240 K (fig. 2e). This transition is assigned to the
bl172’=K
cl 2lO’K -__
dl 23YK
2700-
2900
3100
IISR spectra of m aqueous 0.05 hf Cu(NOz)+ wrth 1 hi nitric acid at different temperatures. FIN. 2.
c
HEI
soIution
559
Volume 44, number
3
CHEMICAL
PIIYSICS
melting process of the highconcentration regions since the hyperfine structure between 240 and 273 K suggests that the copper ions are in a liquid solution.
References [ 1] J-A. Cameron, L. Kcszthclyi, G. Nagy and L. Kaccoh. J. Chem. Phys. 58 (1973) 46 IO. (21 K. Frohlich and L. Keszthclyi, J. Chem. Phys. 58 (1973) 4614. [3] A.S. Plachmda and E.F. hfakarov. Chem. Phys. Lcttcrs 25 (1974) 364. [4] R.L. Cohen and K.W. West, Chcm. Phys. Lcttcrs 13 (1972) 482. [S] P.M. Thomas, M. Sanders, 1. Deszi and P.J. Ouscph. Chcm. Phys. Lcttcrs 11 (1971) 42. (61 B. Burnot, U. Iiauser and W. Ncuwuth. 2. Physrk 249 (1971) 134.
560
LETTERS
I5 Dcccmbcr
1976
I71 AS. Plachinda and E.F. Makarov, Chem. Phys. Letters 15 (1972) 627. 181A. Du Fresnc, J.hf. Knudsen. J-E. Morcrra and KS. Ncto, Chcm. Phys. Letters 20 (1973) 108. PI W. Schmidt and K. Breitschwcrdt, Bcr. Bunscngcs. Physrk. Chcm. 78 (1974) 1268. [IOl W.A. TilIer and R.F. Sckerka, J. Appl. Phys. 35 (1964) 2726. thenno1111S.R. dc Groot and P. Mazur, Nonequrhbrium dynamics (North-lIolland, Amsterdam, 1962). 1121C.S. Lindenmeyer, G.T. Orrok, K.A. Jackson and B. Chalmers, J. Chcm. Phys. 27 (1957) 822. CA Angcll and E.J. Sax, J. Chcm. Phys. 52 (1970) 1058. Y. Zshikawa, J. Phys. Sot. Japan 21 (1966) 1473. S. Fujiwara and 11. Ii~yashi, J. Chcm. Phys. 43 (1965) 23. R. Poupko and Z. Luz, J. Chcm. Phys. 57 (1972) 3311. W 8. LCWIS,hf. AIci and L.O. Morgan, J. Chcm. Phyg. 44 (1966) 2409. [I81 R.T. Ross, J. Chcm. Phys. 42 (1965) 3919.