E. G. McRm
61i2
mony elcctrodo in cell V was replaced by a plaitinuin elect'rode, the e.m.f. was found to be about the snine as for cell V. This result indicates that the antimony electrode in cell V is not reversible at 300" hiit shows some kind of passivity which prevents ndjustrnent of the sulfur activity in AgSbSz-SbzSa Inixt,urrs to the two-phase equilibrium value of the system RbSbzSa. This interpretation is in accord wit,Ii the fact that antimony reacts very slowly with sulfur. An antimony pellet having a surface area .~ a weight increase of only 4.6 of 0.5 c ~ n showed mg. after heating in sulfur vapor at 400" during 04 hours. Thermodynamics of the System AgzS-Sb2S8.In the first place, A F e q for the phases AgsSbSa and AgSbSz may be calculated from eq. 11 by using reduct ion equilibrium data obtained by Schenck and his associates2 and data for A g S b alloys reported above. Upon reduction of AgaSbSi(4 = 0.5)with hydrogen, the H2S/H2 ratio was found to be constant over most of the ranges of q . Thus the integrals in cq. 11 may be evaluated as products. When AgSbS& = 0.75) was reduced, two plateaus of the plot q vs. q were found. In addition, a further decrease of q at low values of T was observed. This tail in the Q vs. q plot has been ascribed to the formation of a phase designated as Z with the approximate formula Sb& 2AgzS 31Ag, but the existence of this phase has not yet been proved by other methods, e.g., X-ray investigations. In the following evaluation, the tail of the q vs. q plot has been disregarded because detailed data are not available and the contribution of the tail is only minor.
+
+
Vol. 01
'Substituting numerical values in eq. 11, we obtain for Ag,SbSa AFeq
(E
3
0.50, 9
and for AgSbSg
p
-
1; 400')
-675 - 176 -761 cal./equivalent (26) 5
AFsp ( E = 0.76, 7 = 1; 400") = -288 - 260 = -638 cal./equivalent (26)
In each of these equations, the first term represents the value of AFeq([, 11 = 0) for the alloy, and t h e second term is the sum of the integrals involving values of q. In addition, values of AFeqfor AgaSbSaand AgSbS2 have been calculated from eq. 15 with partial equivalent free energies of AglS derived from e.m.f. measurements listed in Table I. Equation 15 yields for Ag3SbS3 APeq( E 0 . 5 , ~ 1; 276') = -710cal./equivalent (27) and for AgSbSa
- -
AFeq ( E
0 . 7 5 , ~ 1; 275") -665 cal./equivalent (28)
In spite of different temperatures, 400 and 275" the values of AF, in eq. 25 and 27 for Ag3SbS8and in eq. 26 and 28 lor AgSbSl differ only to a minor extent in accord with the rule that the free energy change of reactions involving solid phases of virtually invariable composition varies only slightly with temperature. Acknowledgment.-This work wa.s supported by Office of Ordnance Research, U. 8. Army, under Contract DA-19-020-ORD-3061.A. G. Verduch acknowledges gratefully the receipt of a Fulbright Fellowship.
v
THEORY OF SOLVENT EFFECTS ON MOLECULAR ELECTRONIC SPECTRA. FREQUENCY SHIFTS1 BY E. G. M c R A ~ Department of Chemistry, Florida Slate Universitu, Tallahassee, Florida Received OcLober 16, I966
The effects of electric dipole interactions on electronic band frequencies in solution spectra are analyzed theoretically. A eneral expression for the frequency shift is derived by perturbation theory. The frequency shift is the sum of contributions from dis ersive and static dipole interactions. The dispersive contribution represents the eneral red shift, which is present in all s o h i o n spectra; it depends in part on the weighted mean wave lengthcharacteristic o!thesolvent. In theelectrostatic contribution the role of the uadratic Stark effect is emphasized. The introduction of a simple electrostatic model permits the derivation of formulas rdating frequency shifts in both emission and absorption spectra to the refractiveindex and static dielectric constant of the solvent. Illustrative numerical applications are described.
I. Introduction In several recent discussions of solvent effects on electronic spectra, solvent-induced frequency shifts have been interpreted in terms of electric dipole interactions. The well-known red shift2 .in the apectra of non-polar solutes has been related by ( I ) This work was carried out under a contract between the U. 8. Air Force. OWce of Scientific Research, ARDC, and the Florida State University. (2) Shifts to lower frequencies relative to the vapor frequency are called red shifts, and shifts to higher freqiienay, bluc shifts. In dgehrrtic' cxprnaniona;.rrd shifts will bo doaiRnater1 by a negative sign, blue nhifts by H yoaitive sign.
Baylissa to the solvation energy of the transition dipole, while the work of Ooshika4 indicates that the red shift is caused by dispersive interactions. Several authors'-6 have discussed frequenc shifts in the spectra of polar solutes in terms of t e relative solvation energies of the permanent dipoles appropriate either to the combining states of the
g
(3) N. 8. Bayliss, J . C l e m Phya., 18, 282 (1850). (4) Y. Ooshika, J. P h y s . Soe. Japan, 0 , 684 (1854). (6) L. G. 8. Brooker, Erperienlia Supplementurn I I , (XIVCh InLernalional Congress of Pure and Applied Chemistry), 220 (1855). (6) N. 6. Baylisa and E. G . McRae, THIBJOURNAL, 58, 1002 (1954).
*
THEORY OF SOI~VENT EFFECT# ON MOLECULAR ICIJCCTI~ONIC SIWCT~~A
May, 1067
503
sohitme,or to the resomice structures contributing turbed component molecules; e.g. , the energy t o those states. As has been stressed by Bayliss corresponding to is given by Wzj = ( N and McRae,6 it, is necessary in general to consider 1)w: wi wy,where w denotes the electronic the frcqueucy shifts arising from interactions in- state energy of an unperturbed molecule. Energy volving the solute permanent dipoles as super- differences will be expressed in cm.-l, and denoted posed on a general rcd sliijtt,?which is present, in all by Y ; thus, for example, the energy difference besolutioii spectra. tween the ith and j t h states of the solute will be In this paper we present a further discussion of expressed by frequency shifts caused hy dipole interactions. The most important contribution of this study u;, = (tu; - w:')/hc = -u:; is the derivation, by perturbation theory, of a general expression for the frequency shift. It is where h and c have the usual meanings. We represent the energy of interaction of the hoped that this forniulation will serve to correlate previous interpretations, and provide the basis molecules in solution by the classical energy of for a more precise and complete qualitative inter- dipole interaction, which is given in the pointpretation. A second contribution of the present dipole approximation by work is the derivation, from the general expresN sion, of formulas linking the frequency shift to the x' = - (1/2) x solvent refractive index and static dielectric conP-1 stant. The formulas presented here are more N N ev(P)v(e)mv(P)nv(d (1) widely applicable than those previously put forp=lq=t ward. where 3K denotes the instantaneous magnitude of the 11. Theoretical dipole moment of a molecule and 8 is a geometrical The method of treatment consists of the ap- factor dependent on the mutual orientation and plication of second-order perturbation theory to separation of two dipoles. the calculation of the electronic state energies of a The energy of the state @i, corresponding in the solution containing N identical solvent molecules zeroth order of approximation to +!, is governed and one solute molecule. We suppose in the be- principally by the matrix elements Hi,, = W0& ginning that the molecules have fixed positions and J@X'qi!dT ( S j i is the Kronecker delta), H a ( p ) i , i = orientations. I n view of the comparative rapidity of electronic transitions, the same positions and J @'oocp)j,X'Q#r and Ha(P)b(q)i,i= .f@oa(p)b(q)i orientations are appropriate to the ground and ex- x'@!dT. Here and in what follows, the state functions are for simplicity taken to be real. In view cited states of the solute. I n the zeroth order of approximation, we con- of (l),the matrix elements may be written sider the molecules not to interact. The zerothN order electronic state functions of the solution are H = WP&i eK$) M4Ml;o then made up of products of state functions for the P-=l N N unperturbed component molecules. Neglecting (1/2) O$&'Q1(M&,)*S, the non-orthbgonality of the latter, we shall emp - 1 q=1 ploy simple product functions, for which we introduce notations such as (2) N * : ( p ] b ( Q ) , = +;(I) . .. . . . I#Jp . . . & ( N ) +I B$%"'"'K&m Ha(p),,i - BK$MZMblo - (1/2) *:[D)f = &,('I . . . @v(rl . . . +;(wl . . . +;(N)+; and q=1, *P @:(u)o = #p . . . +p. . . I$p. . . +p & and where for example @&p)t,(q)j denotes the zerothN order function representing the state of the solution H . ( ~ ~ = ~ ( ~ ~ ~ ( , ~ 2 ) e:%(wlM:om (3) in which the solute molecule is in its j t h excited ,l'P electronic state #, and the pth and qth solvent *P molecules are in their ath and bth excited states Here M denotes a matrix element of the dipole +i(P),&(q), respectively.8 Throughout this paper moment (e.g., M $ = f +~3nU#dr),and 8 denotes a the notations u and u(p) will refer to the solute and geometrical factor of proportionality between the the pth solvent molecules, and the subscript zero product of the moments of two rigid point dipoles will indicate the ground electronic state of a single and their energy of interaction (e.g., the energy of molecule. interaction of two rigid point dipoles whose moThe zeroth-order electronic state energies are sums of the electronic state energies of the unper- ments are, respectively, MYi and M:,,, and which belong, respectively, to the solute and pth solvent (7) Bayliss and MeRae8 used the term polarization red shill. molecules, is given by -9x:$MyiiM&). ( 8 ) (a) It should be noticed that the subscripts for unexcited solConfining our attention to non-degenerate states vent rnoleeirlcs are omitted, but the aubseript for the unexcited aolute of the solute, the energy of the state 9i is given molecule is rotained. (b) The above notations embrace all of the required zeroti]-order functions, for in tho approximations of the present according to second-order perturbation theory9 by
+
+
~ U V ( P I ~ ~ U ~ V ( P )
s94
+
,(
+Up)
E
0
treatment, matrix elements involving zeroth-order functions eorresponding to the simultaneous excitation of more than two solvent molecules are zero.
(9) H. Eyring, .T. Walter and C. E. Kimball, "Quantum Chemistry," John Wiley and Sons. Inc., New York, N. Y., 1944,p. 96.
E. G. MCRAE
564
VOl. 61
ev&I&vId&Mv 00
A similar expression for Wois obtained by replacing i by 0. The frequency shift, Av, is given by kAv (Wr - Wa)- (W," - W t ) (6) where the bar indicates a time-average value. General Formula.-In order to obtain a simple cxpression for the frequency shift, it will be assumed that all the (point) dipoles associated with any one molecule (Le., transition dipoles as well as permanent dipoles) may be considered to lie at the same point in the molecule. Equation 4 can then he reduced to a form in which the energy is related explicitly t o environmental contributions to the electric field a t the solute dipoles. The timeaverage of this expression yields, through ( 6 ) , a general formula for the frequency shift. In the explicit reduction of (4),it will be assumed that the molecules can be considered as optically isotropic. However, as will be shown later, the formulas given below require only minor modification, if either the solute or the solvent molecules or bot,h can be considered rod-like, in the sense of having all dipole moments parallel. The term by tcrm ~*cduction of (4) can be accomplished without furthor gross approximations. Let E" denote the ficltl at the solute dipoles, due to the permanent dipoles of thc surrounding solvent molecules. In view of (2), the first term in (4) reduces to * N
WP
- M,"r.E"- ( l / 2 ) c
e$&$.'"(M&,)z
J*i .
&
*p
The second sum represents the energy of interaction of permanent and induced solvent dipoles, and makes no contribution to the frequency shift. The first sum can be written
- -I cN c
4
(EI(D)u,Mvb))
f a )
E:$.'P'U +
p-1 a t 0
rlr)rk'
Q"L
. M V "
SP
VL
where E:(PIU denotes the field at the dipoles belonging to the pth solvent molecule, due to the permanent dipole of the solute molecule in its ith excited state, and Ev(p)v(q)denotes the field a t the same point, due to the permanent dipole of the qth solvent molecule. The second factor in parentheses in the above expression represents the field at the pth solvent molecule, due to the permanent dipoles of the surrounding molecules. We shall neglect the contribution of solvent dipoles to this field, on the ground that the percentage error thereby incurred in the above sum will be large only when the solvent is highly polar, and in that case terms of the type we are discussing at present make a relatively small contribution to the frequency shift. Making use of the propertiecl of isotropic molecules, we obtain
r,
(MuE")* 4 --J'..-
P-1
where a: denotes the static isotropic polarizability of a solvent molecule in its ground state. Let mi(p) denote the moment of the dipole induced in a The above exsolvent molecule by the field pression then becomes
- (1/2)
N E(pl~.m;(p)
F-1
Ii'or an isotropic molecule, the value of (Mjui.EU/ which can be written alternatively - ( 1/2)M;e: (8) Mi\EU)2averaged over all states j # i is 1/3. Consequently the second tcrm in (4) becomes approxi- where e: denotes the field at the solute dipoles, mately due to solvent induced dipoles such as that with moment mT(p). - -( E 9 2 (M:)a 3hc J f i 4 Utilizing (2), the fourth term in (4) can bo written Making use of the usual expression for the isotropic polarizability of a molecule,l0 this becomes
41/2 )( @ ) Z d
(7)
where a: denotes the static isotropic polarizability of the solute molecule in its ith excited electronic state. With the aid of (2), the third term in (4) can be written (10) Rofcrence 9, p. 121.
*
N
(6)
The terms under summation in (6) represent the energy of interaction of solvent permanent dipoles, and makc 110 contribution to the frequency shift. The second term in ( 2 ) can be written IlC
b
N
p=iq=l *P
- -1 E
&$130~q41*
m)
From the above expression, we obtain
which for isotropic molecules rnay be written approximately
*
THEORY OF SOLVENT EFFECTS ON MOI~TCCULAR ELECTRONIC SPECTRA
Mny, 1957
565
zation of the solvent by an oscillating field whose frequency is v;; therefore it should be related to the square of the solvent refractive index at this freI t is conveiiicnt to introduce a weighted m a n wave quency. A slightly different interpretation should Zcnglh, defiiicd by be placed on the reaction fields ~1 and TO, which arc to be identified, respectively, with and. : e These reaction fields arise from the contribution of induced dipoles to the static polarization of the Wc thcn ol)tain, for the fourth term in (4) solvent, and should therefore be related to the N square of the solvent refractive index, extrapolated - (1/2) (1 - Ylr,,,)(~(.)")*(YX(vB) to zero frequency. The third term in (10) may be p - I ,*i (l/hc)(Mk - Myi).R, where R denotes the written Here cr,"(v,"i) dcnotes the isotropic polarizability of a reaction field corresponding to Eu. (Obviously, R solvcn1 Inolecule, appropriate to an oscillating field should not be identified with E;). R arises from the of frcqiiciicy v,". Lct m$p) denote the amplitude of the solvent permanent dipoles to contribution of thc dipolr moment induced in a solvent molecule the static polarization of the solvent, and should by an oscillntiiig field of amplitude E3V1(P)U and fre- therefore be related to that part of the static diquency vyi. Wc then obtain electric constant attributable to orientation polN arization. A difficulty arises in the treatment of the fourth - (1/2) C (1 - v;L{,)Ei;'"'"m;l" ?>=l>+i term in (lo), since (EU) cannot be related to macrowhich can t x written alternatively scopic solvent properties through identifications such as those indicated above. However we may be sure that when R is comparatively large, ( E U ) 2 as R2. Here el;, for cxninple, denot,es the field a t the solute will be of the same order of magnitude __ dipolen, diio to a solvent induced dipole with mo- Somewhat arbitrarily we adopt ( E U ) 2= 3R2 ( R large). For a rod-like solute molecule, we choose ment m;(p). As is made clcnr by (3), the fifth term in (4) (M:o.Eu/M,U,)2= R2 ( R large). The advantage of represents the encrgy of an interaction between these assumptions is that the last term in (10) and solvent molecules only. Consequently, this term the corresponding expression for a rod-like solute molecule become the same, viz., (3/2hc)(ru: makes no contribution t'o the frequency shift. From (G), (7), (8) and (I)) we find an expression a:> R2. for the energy Wi, and a similar expression for Wo We shall utilize the well known expression for the results on replacing i by 0. The frequency shift reaction field, derived on the basis of a point dipole in the transition between the ground state and the a t the center of a spherical cavity in a homogeneous ith excited state is given, according to ( 5 ) , by dielectric." For an absorption transition starting from the ground state, we have for an isotropic Avio (1/2hc) (1 - ~~oLjo)MyoG solute molecule [ j $0
2
(1
- v;Lir)Wde31
j *i -.
+ (1/2hc)(M&b: - /lf:$j) + (I/hcj(M,",--M;;).E" + (t/2ac)(a: - a r ) ( ~ u ) "
(10)
where the bars indicate time-avcrage values. If either the solute or solvent molecules or both can be considered rod-like instead of isotropic, (4) can be reduced in a manner similar to that described above. If the solute molecule is rodlike, the final forinula differs from (10) only in the fourth term, which becomes (3/2hc)(ag
- w~)(W~.E"/WW)~
where a: and CY: again denote isotropic polarizabilities. If the solute molecule is not rod-like, the final formula is the same as (10). Relationship to Macroscopic Properties.-Equation 10 may be written in a form involving the reaction fields of the solute dipoles. These may in turn be related to macroscopic properties of the solvent. We may identify%, for example, with the reaction field ~ of i ithe dipole whose moment is J!l+ The reaction ficld ~ j ariscs i formally from the polnri-
where D' denotes the contribution of the permanent dipoles to the static dielectric constant of the solvent, and a denotes the cavity radius. The expression in parentheses represents the dipole moment of the solute molecule in solution. For most molecules, the ground-state isotropic polarizability is about one-half the cube of the most suitable cavity radiusli2 so that we derive approximately
+
where we have substituted for (D' - l)/(D' 2) by the factor in brackets, involving D,the static dielectric constant of the solvent, and no,the solvent refractive index extrapolated to zero frequency. The value of R appropriate to emission bands depends on the degree 'of dipole reorientation occurring in the interval between population of the excited state, and emission. We consider two limiting cases: if there is negligible dipole reorientation, (11) C.$1. F. Bottcher, "Theory of Electric Tolarization," Rlscvier Publishing Co., Amsterdam, 1052, p, 64. (12) Ileferrnce 11, p. 200.
E. G . MCRAE
566
the value of R is the same as that for the corresponding absorption band. On the other hand, if there is complete dipole reorientation, we have for a t,ransition starting from the ith excited state
+
a?R) D' - 1 R = 2(M$ ____-_____ a3
2D'+ 1
Neglecting terms involving powers of (D'- l)/(D'+ 2) higher than the second, we obtain after a short c*alculatjionsimilar to the derivation of (11)
2 Mg.(SMI:
G
Vol. 61
- 2M&)(&- (YY) aa
D -1 -, [D+2
4-1 -4 + 2
1 4
(140
where we have neglected terms involving powers of [(D- l)/(D 2) (nf- l)/(nf 2)] higher than the second. From (14) and (14'), the frequency difference between the 0-0 bands in the absorption and emission transitions between the ground and the excited states of an isotropic solute molecule is predicted to be
+ -
Au(abaorption)
+
- Av(emi8sion) = qhc( M & -a* M i ) *
Icor the remaining reaction fields, we have in both absorption and emission bands
where nji, for example, denotes the solvent refractivc index for light of frequency v;. In (13) it would have been more accurate to write aYR in place of M:, and a similar expression in place of M:. However this refinement would not greatly improve the accuracy of the final expression for the frequency shift, because whenever R is large, the term involving the reaction fields r; and r0 makes a relatively small contribution to the shift. In view of the above discussion, including (ll), (12) and (13)) we obtain for the frequency shift in an absorption transition from the ground state to the ith excited state of an isotropic molecule
III. Discussion
The result of the treatment given in Sec. I1 is embodied in eq. 10, 14 and 14.14 Equation 10 is limited to transitions between non-degenerate states of the solute, but is otherwise generally applicable. It is fundamentally correct even though it is approximate, and therefore provides the basis for a sound qualitative interpretation of frequency shifts caused by dipole interactions. Because of the simplifications invoked in its derivation, especially the point-dipoIe approximation and the approximations inherent in second-order perturbation theory, the environmental fields appearing in (10) Bhould not always be taken literally. For example, if the solute permanent charge distribution consists of positive and negative charges separated by several A., it does not make sense to speak literally of the field at the solute dipole. Instead, the environmental fields should be considered simply as parameters whose qualitative variation from solvent to solvent can often be inferred, either from the macroscopic properties of the solvent or through consideration of hydrogen bonding. Again, if either the solute or where in the first term we have introduced the solvent molecule is non-polar, but contains highly oscillator strength, which, for a transition between polar groups whose moments cancel, it is not realisthe ith and j t h states of t,he solute, for example, is tic to put the time-average of the field EUequa1 to zero, even though this is implied formally in the given byla derivation of eq. 10. = -j; ( M ; in e.s.u.) j ; = 4.704 X 10eg V; Equation 14 represents the frequency shift in Equation 14 is applicnhle to emission transitions absorption bands. Unless the solvcnt and solute only in the cases in which there is no appreciable di- are both polar, frequency shifts in corresponding pole reorientation in tthe excited state. If there is absorption and emission bands are predicted to be complete dipole reorientation, (1 I!), (12) and (18) equal. When the solvent and solute are both polar, are applicable, and give for the frequency shift iii frequency shifts in emission bands are represented the emission transition from t,he ith excited state by (14) if there is negligible dipole reorientation in to the ground state of an isotropic solute moleculo the excit,ed stat,e, or by (14') if there is complete di-
+
(13) (1041).
R. 9. Miillikon
niid
C. A. Ripkc, Rap. I'vog. P h y s . , 8 ,
231
(14) Those eqiiatione roprrscnt the Rolvcnt efleck on the frequency of tho 0-0 band. A s tlic 0-0 band cannot always be identified in solution spectra, the cqoations mrist bo compared with observed freqiiency ahifts in band msxirna, relative to the corresponding band msxinia in t,lie vapor sp~ctruin. The latter should he determincd by drawing tlia onvdope of the vihmt.iona1 structure. Unless the frequency shifts arc comparable with or larger than the band width, the nbovc! procedure for applying the equat.iona cited is valid only insofar an there is no significnnt solvent effrct on band shape. Similar remarks apply t o eq. 15.
. h
May, 1057
THEORY OF SOLVENT EFFECTS ON MOLECULAR ELICCTRONIC SPECTRA
pole rcorientation. For example, frequency shifts induced in fluorescence spectra by rigid-glass solvent's are represented by (14), since in such solvents the dielectric relaxation times are some orders of magnitude greater than the lifetime of the excited state (-10-8 second). On the other hand the relaxation times of most liquid solvents at room tomperature are about 1O-Io second,16so that shifts induced by them in fluorescence spectra are represented by (14'). I n the application of (14) t o frequency shifts induced by solid polar solvents, the adopted dielectric constant should be the value extrapolated from that of the liquid solvent. Equations 14 and 14' are of course subject to the same limitations as (lo), and an additional limitat,ion is imposed by the use of the simple model for the solut,e in solution. I n particular, (14)and (14') do not apply to frequency shifts caused in part by hydrogen bonding. Because of the assumption of identical solvent molecules in the derivation of (lo), this formula atid the formulas derived from it apply to frequency shifts induced by pure solvents only. The generalization to mixed solvents can be achieved in cases in which the solvent composition near the solute molecule can be assumed equal to the bulk composition. I n that case, the weighted mean wave length of a mixed solvent is a linear combination of weighted mean wave lengths of the solvent components, the coefficients being the appropriate mole fractions. Dispersive Interactions.-The first term in (10) represents a contribution from dispersive interactions. I t will ordinarily be negative, because > uyi for j>i (the states being numbered in order of increasing energy). In practice we observe frequency shifts in only the first few electronic bands of the solute, and these lie a t lower frequencies than that of the onset of solvent absorption. The first term represents the general red shift. The expression for the general red shift consists of a sun1 of terms, each of which corresponds to a virtual transition starting from one of the combining states of the solute. Each term contains the reaction field of a dipole whose moment oscillates with a frequency equal to the appropriate transition frequency and with amp1it)ude equal t o the appropriate transition dipole moment. According to the present formulation, the general red shift depends in part on the weighted mean wave length, which is a function of the frequency, characteristic of the solvent. If none of the transitions which make an important contribution to the general red shift lie close in frequency t o a solvent absorption band, the weighted mean wave length may be considered approximately as a const,ant. Whenever this is the case, we shall for definiteness adopt the value a t zero frequency, denoting it by Lo. The weighted mean wave length may he taken as a constant in all applications to strong transil), because any frequctiicg shifts are tions ( f governed mainly by the term corrcspoiitling to the transition in question. In applications to weak transitions, tha frcclunncy-d~~ei~dence of thc N
(15) 1). 11. Whiffcn, Qtrnrt. Ren. Clteni. S o r . , 4 , 131 (I95o).
567
weightcd mean wave lcngth may be important if the solvent begins to absorb a t a frequency close to that of the transition in question. Limits are set on LO,the weighted mean wave length a t zero frequency, by the inequalitieR
where the primes indicate summation over terms correR onding to known trmsitions in the absorption spectrum of tRe solvent in uestion. As long as $e frequency and intensity of a t least one strong transition is known, the limits are sufficiently close that their arithmetic mean is a good estimate of LO. For example, we have for the solvent benzene, utilizing the known polarizability, the formula relating the oscillator strength to the transition dipole moment and the information in Table I: 1850 > LO> 650 (4.)yielding LO= 1250 b. This value is quite insensitive to refinements such as taking into account some Rydberg transitions. It is appro riate to many benzene derivatives as well as to benzene itself), Unfortunately the ultraviolet absorption spectra of most other common solvents are not sufficiently well known for the above method to be useful. However we discuss a few typical solvents as follows.
TABLEI FREQUENCY SHIFTS IN THE BENZENE SPECTRUM Transitiona
AG-+B& AG-8; Al,-+E+ I3$ --c Ezr B1,-+E
V > IX.a4 Almost certainly,, this is the order of decreasing ground state permanent dipole moment. The outstanding feature of the froquency shifts, as revealed by Fig. 2a, is that, tho hand of the most highly polar dye IV shifts t,o higher frequencies as the water content of the solvcnt is increased, while the band of the least polar dye IX shifts initially to lower and then to higher frcquencitts. Intermediate behavior is exhibited hy the dye of intermediate dipole moment. The phenomena have been interpreted by Brookcr5 in terms of the relative solvent stabilizat,ioii of polar and non-polar resonance structures, mid this approach has been elaborated by Simpson3band by Platt.3s As a particular application of thr prcscnt theory, we advance an alternative (but not fiindament8nllydifferent) explanation in terms of thc qi1:idratic Stark effect. 7’hc merocyanine dyes have just the right combinnt,ion of properties to favor the quadratic Stark effect. First,, they are highly polar, implying that in inotlcratoly or highly polar solvents R (and hence R2) is pnrticiilarly large; second, the first singlet 1). The second property transif.ion is strong (f makns it rcaaonable to assume, RS is necessary in the following int,crpretation, that the polarizability of any givrn tlyc in its lowest singlet excited state is consitlcrtibly less than in its ground state. This can he iintlerstood upon inspection of the usual formnla for the polarizability.1° The lowest singlet t)ransition, bcing strong, makes a sizable contribut’iun to t’hc ground state polarizability, and a negative oontrihiit8ion of the same magnitude to thc lowest excited singlet state polarizability. ‘4 part)icularly simple discussion is made possible by writing (10) in a form appropriate to the limiting case of large R, zriz.
-
AV
-
AVmf
= (l/hc)(M& -
+ (3/2hc)(a; - a:)
M:?,).R
11‘2
(18)
where Avref represents the frequency shift induced (32) C. P. Snipth. “Diclrct,ric n o h a l k r and Striioture,” McGrawHill Rook C o . , Inn., New York. N. Y . , 19.55, p. 348. (33) 1,. G. S. Hrookrr, C. 11. Krye-8, I t . IT. Spragur, R. H. Van Dykn, 16. V a n I,aro, Cr. Van Zniidt., V. 1,. White, € W. I.J . Crsssman and 8. G . Dnnt,. J . Am. Chsm. Roc., 73, 5332 (10.51). (34) 11. G . S. Rrookor, G. 11. Iiryrs nnd D. W. Hrnoltine, ibid., 73, 5350 (1051). Drookcr and ro-workers hayo stiidied solvent effects on maximum extinction ooefFicionts as wrll a s bnnd freqnencirs. (35) W. T. Simpson, J . Am. Chem. Noc., 73, 53.59 (1051). (36) J. R . Platt, J . Chem. Phya., 26, 80 (1!)56).
Observed relative frequency shifts (cm.-l), 0.2(
+
0,lS
h
?
v
p
0.10
2 X
a: O.O! 0
AY
- Avref. (om.-’).
Fig. 2.-(a, top) variation of merocyanine band frequency with percentage of water in aqueous pyridine solvent. The abscissa ia the frequency shift relative to the shift induced by pure pyridine. Curves are drawn for three merocyanine dyes, whose ground state dipole momenta are considered to decrease in the order: IV > V > I X (see text). The optical data and dye notations are those of Brooker, Keyes and Heseltine.*‘ (b, bottom variation of band frequency with the reaction field, accor ing to e 18 (duo- aUi 0.67 X lo-” cm.8, M% - M ~ i i 1.5,%, - 1.5 Debyes). These curves are meant to illustrate the general behavior redicted by the theory. The numbers are considered to {e of the rorrect order of magnitude (see text), but are otherwise The ciirves do not refer specificdly to the dyes
d
p:
-
p.@&
hy a non-polar reference solvent, and the frequency shift as again expressed in terms of R instead of E“. We assume that in order to preserve the orthogonality of ground and excited electronic state functions in a series of merocyanines of decreasing ground state dipole moment, the excited state dipole moments either stay nearly constant or tend to increase in the same order as the ground state dipole moments decrease. I n Fig. 2b, the frequency shift Au - Avt-ef is plot,ted against R in the three possible cases: M:o - M:i greater than, less than and equal to zero. (In the discussion of visible merocyanine spectra, tho ground and excited state permanent dipole moments may be considercd parallel.) The comparison of Pigs. 2a and 2b shows that (18) is capable of reproducing the gross features of the phenomena, for R may be assumed to increase
E. G. McRm
572
monotonically with wat8cr content in an aqueous pyridine solvent. The comparison must remain purely qualitative, because there is in this case no reliable way of relating R to the solvent composition. Indeed, the reaction field has only formal significance in the present application. The reason for this has been explained, with explicit referenre tto EU,a t the beginning of this section. Equation 18 itsclf is far from realistic, because the first two terms in (10) may well suffer significant changes on passing from solvent to solvent. However it is unlikely that the consideration of the general rcd shift, for example, would alter the qrirtlitativc conclusions drawn here. In spite of the drastic simplificationR which have been introdriccd, the resent discussion is readily shown to involve qi~nntit~ies of t l e correct order of magnitude. For this purpoae the ordinate in Fig. 2b was chosen to run to 2 X 10’ e.s.u., which is the order of magnitude of the reaction field indiratcd by t8hesimple model, for a solvent of high dielectric constant and 111& = IO debyes. The differenre between the ground and excited state permanent dipole momenta is ulmoat certniiily of the order of a few debyes, and 1.5 debyes waH chosen for the sake of illustration. In order to reprodwe the gross features of the phenomena within the choscii range of R , the difference between the ground and excit8edstat,(! isotropic polarizabilities must he 0.5-1 .O X cm.*, z.e., nbout one-third of the contribution of a strong visible trarisition ta the ground state polarizctbility.
In his discussion of the phenomena, Brooker6 has focused attention on the plot of the peak extinction coefficient e, against the corresponding wave length A,, and he has associated the maxima in such curves with the isoenergetic point, a t which the principal polar and non-polar resonance structures have equal energies. At the isoenergetic point the ground and excited state dipole moments of the solute in solution should be equal. From (18) we find that the condition for a minimum value of AV - Avref (M: > M&) is ill& 3 4 R = M; 3 4 R . For a rod-like solute molecule, 301; and 3a;’ are polarizabilities appropriate to a field directed parallel to the dipole moments, so that M$ 3aPR and M i 3aYR rcpresent dipole moments of a rod-like solute molecule in solution. Since the merocyariinea may be considered nearly rod-like, the present theory suggests that in the approximation of neglecting the first two terms in (lo), the minimum in Av - Avref should be associated with the isoenergetic point. The minimum in Av Avpf and the maximum in the plot of e, us. A, ordinarily occur a t nearly the same solvent cornposition.37 The present discussion thus tends to support the views previously expressed by Brooker,6 and in fact indicates that Drooker’s conclusions
+
+
+
+
(37) This ia not the mse in the curves for Brooker, Key- and Heseltine’s dyes I X and X.84 IIowcver, since these dye8 are relat,ively weakly polar, the first two terms in (10) may make an important contribution.
Vol. G1
remain valid even if there are more than two rcsonance structures making important contributions to tho combining states. Other Interactions.-Finally we point out that not all frequency shifts are caused entirely by dipole interactions, but that other types of interaction may make important or even dominant contribu: tions in certain cases. For example, chargetransfer complex formation is generally accomanied by shifts in band frequencies; this topic s been reviewed recently.88 Again, a large frequency shift may arise from intermolecular repulsive forces, in the event that the solute moleciile suffers a marked change of equilibrium sizeae or shapeqDv41 upon excitation. Since the completion of the present work, some important contributions to the theory of solvent effects have come to the author’s attention. From Ooshika’s f o r m ~ l a Mataga, ,~ Kaifu and Kdzumi4* have deduced an expression for the solvent effect on the frequency separation of the 0 4 bands in corresponding absorption and fluorescence transitions. This expression differs only slightly from the first term of that derived in the present work (eq. 16). Mataga, Kaifu and Koizumi have applied their formula in the evaluation of excited-state dipole moments.4a Another method for evaluating excited-state dipole moments has been proposed by Lippert. 44 The writer has been privileged to read, in rnanuscript form, a new treatment by Longuet-Higgins and P 0 p 1 e ~of ~ solvent-induced frequency shifts arising from dispersive interactions. The conclusions drawn by Longuet-Higgins and Paple are similar to those reached in the present work, but are expressed in a simpler form. Acknowledgments.-The writer gladly acknowledges the award of a Hackett Studentshi by the University of Western Australia, and the ind interest of Dr. N. S. Bayliss, Dr. H. C. LonguetHiggins, Dr. M. Kasha and Mr. M. F. O’Dwyer.
L
. 4
z
(38) (8) L. J. Andrewa, Cham. Rmr., 64, 713 11984); (b) I ,. E. Oreel. Quart. Rm. Cham. SOL. 8, 422 (1954). (39) (a) N. 5. Bayliss and A. L. 0 . Rees, J . Cksm. Phus., 8, 377 (1940): (b) N. 9. Bayliss, A. R. H. Cole and B. 0 . Oreen, Australian 3. Sei. Rea., Ai, 472 (1948); (c) A. L. a. R e a , J . Cham. Phua., 8,429 (1940). (40) K. W. Hausser, R. Kuhn and E. Kuhn, 2. phyaik. Chsm., 8 9 9 , 417 (1935). Thia work has been summarized snd discussed by F6ra-
ter.“ (41) Reference 25, p. 136. (42) N. Mataga. Y . Kaifu end M . Eloizurni, Bull. Cham. Soc. Japan S9, 115 (1950). (43) N. Mataga. Y. Kaifu and M . Koizumi, ibid., 99, 466 (1958). (44) E. Lippert. Z. phusik. Chsm. (N.F.), 6, 125 (1950); Z. Naturforrsh.. lo&,641 (1955). (45) H. C. Longuet-Hippins and J. A. Pople, to be published in the Journal of Chemical Phusics.
.b
