J. Phys. Chem. 1994, 98, 11226- 11229
11226
COMMENTS Comment on “Time-Dependent Theoretical Treatment of Intervalence Absorption Spectra. Exact Calculations in a One-Dimensional Model”
TABLE 1: Equivalent Parameters in the SRTZ’ and PKS2y3 Models with hv = 450 cm-’ for Mode q and M = 17 a m d PKS parameters*
SRTZ parameters
Paul N. Schatz* and Susan B. Piepho
e = era = 2800 cm-I Q = Qsm AQ, = AQ = 0.150 8,
Chemistry Department, University of Virginia, Charlottesville, Virginia 22901, and Chemistry Department, Sweet Briar College, Sweet Briar, Virginia 24595
Values given are those used in Table 1 and Figures 2 and 3 of SRTZ. Note that Wong and Schatz3and Piepho6 use normal (massweighted) coordinates so that Qws = Qpb &.
Received: February 22, 1994 We comment on a recent paper in this Journal by Simoni, Reber, Talaga, and Zink’ which we hereafter refer to as SRTZ. We have two purposes in mind. First, we wish to discuss the relation of the SRTZ treatment to previous theoretical models of mixed-valence systems. Second, we show that there are systematic errors in the SRTZ calculations which invalidate their results. Specifically, we conjecture below that the SRTZ method, though using a different formalism, is precisely equivalent to the 1978 treatment by Piepho, Krausz, and Schatz,2 often referred to as the PKS model, and we believe that the SRTZ systematic errors are a consequence of parity (strictly, interchange symmetry) errors in determining vibronic selection rules. We note first that SRTZ are proposing to treat the same problem treated by PKS, namely, the quantitative calculation of the transition energies and intensities of the intervalence band (IVB) and its associated charge-transfer-induced infrared (ctiir) transitions. As is the case with PKS, SRTZ intend, right from the start, to include vibronic coupling between the two potential surfaces so that the vibronic eigenfunctions of the system are represented by a sum of products of an electronic and vibrational function from each of the two surfaces. That is, it is explicitly recognized by both PKS and SRTZ that the Born-Oppenheimer approximation is inadequate, and so nonadiabatic vibronic functions are employed. Both treatments use the same single coordinate (q or Q) to describe the nuclear motion. Mathematically, the vibronic Schrodinger equation is being solved, viz.
where H e l includes the nuclearArepulsionterms as well as all terms involving electrons and Tn is the nuclear kinetic energy term. PKS solve eq 1 using standard methods employed in JahnTeller and pseudo-Jahn-Teller problems by transforming to a nonadiabatic basis in which the electronic wave functions are not expicit functions ofAnuclear coordinates (thus introducing off-diagonal terms in He& The result is a pair of coupled equations in the vibrational functions associated with the two surfaces. These equations are solved by expanding the unknown vibrational functions in a single complete set of harmonic oscillator functions in coordinate q. The result is an infinite tridiagonal matrix which can be swiftly diagonalized to arbitrary accuracy by truncating the basis. The most detailed exposition of the PKS model may be found in the review article by Wong and S ~ h a t zthe ; ~ treatment just outlined is contained in eqs 4756 of that article. We note that precisely this method was used
* Author to whom correspondence should be addressed at the University of Virginia. 0022-365419412098- 11226$04.5010
cpks= -cs,Jhv = -2800 cm-’/450 cm-I = -6.222 q = 2~(Mv/h)”*Q-; Q- = (1/&) 1 = h Q ( 4 n * M ~ / h )= ~ ”2.259
(QA- QB)= Qrm
TABLE 2: Comparison of Calculated Eigenvalues (in cm-l) of SRTZ’ and PKS2J for the Table 1 Parameters PKS PKS SRTZ SRTZ eigenvalues“ eignevaluesb eigenvalues” eigenvaluesb 0 238 506 79 1 1089 1396 1711 2032 2359 2691 3027 3367 3711 4057 4406 4758 5112 5469
1. 0.0 (+) 2.237.9 (-) 3. 505.8 (+) 4. 790.7 (-) 5.1088 (+) 6.1395 (-) 7. 1709 (+) 8.2030 (-) 9.2357 (+) 10.2689 (-) 11.3024 (+) 12.3364 (-) 13.3707 (+) 14.4053 (-) 15.4402 (+) 16.4753 (-) 17.5107 (+) 18.5463 (-)
5793 C C
6392 6550 C
6984 7280 7568 C
8015 8148 8385 8722 C
9128 929 1
19.5792 (-) 20.5821 (+) 21.6181 (-) 22.6391 (+) 23.6543 (+) 24.6907 (-) 25.6982 (-) 26.7272 (+) 27.7566 (+) 28.7638 (-) 29.8007 (+) 30.8145 (-) 31.8376 (-) 32.8718 (+) 33.8747 (+) 34.9119 (-) 35.9287 (-)
“Exact eigenvalues” from left column of Table 2, SRTZ.’ Eigenusing the parameters of Table 1; (-) and values by the PKS (+) designate odd and even parity (interchange symmetry),respectively. No value corresponding to the PKS value-see text. many years ago by Fulton and Gouterman to calculate the vibronic manifold in the excited states of symmetrical dimem4z5 In the mixed-valence case, the transitions all occur within this manifold. SRTZ solve eq 1 using a wave packet formalism that is described in their article. Thus, it would seem evident that if the same physical model is used in both treatments, the same results must be obtained. As far as we can tell, precisely the same model is indeed used, since both treatments assume the same single nuclear coordinate (q), a single vibronic coupling parameter (A), a single electronic coupling parameter (c), and hannonic zero-order potentials with a single force constant. In Table 1, we give the correspondence between SRTZ’ and PKS2 parameters, and in Table 2 we compare the eigenvalues obtained by the two methods. The first column in Table 2 gives the “exact eigenvalues” listed in Table 1 of SRTZ, and the second column gives the corresponding eigenvalues (numbered for ease of reference) obtained by the PKS method using equivalent parameters. One notices that there is excellent agreement between the two methods except that PKS lists five additional eigenvalues in the range 5800-8700 cm-’ for which there are no corresponding SRTZ values. (We note that a sufficiently large basis was used in the PKS calculation that the eigenvalues listed in the second column of Table 2 do not change if the basis size is increased.) Were it not for the five discrepant eigenvalues, we would conclude that the two methods give the same eigenvalues, as we believe they should. 0 1994 American Chemical Society
Comments
J. Phys. Chem., Vol. 98, No. 43, 1994 11227
TABLE 3: Vibronic Basis in the PKS Model” lower surface eigenfunction eigenvalue (cm-’)
w +xo w+x1 w +x2
6000 5000
0 450 900 1350
*+x3
upper surface eigenfunction
eigenvalue (cm-l)
w-xo
5600 6050 6500 6950
w-x1
w-x2 w-x3
A
.
IVB
4000. 3000.
T = any value
2000
c = -6.222
1000
I
’
X=O
A 2000 40oo 6Ooo so00 loo00 Frequency/cm-’
The basis is diagonal in the delocalized limit (1= 0; 6 f 0) and has the indicated eigenvalues for 1 = 0 and the Table 1 values: cPb =
20
-6.222, hv = 450 cm-’.
-
--
6000 -
5000 - Is
Absorption Spectrum in the PKS Model We turn now to the calculation of transition intensities where there are clearly demonstrable errors in SRTZ. Perhaps the easiest way to see the problem is to begin with the PKS model in the delocalized limit (1= 0, E 0) and consider the allowed transitions between vibronic states. In Table 3 we list representative vibronic basis functions (see ref 3, pp 386-388) together with their energies for 1 = 0 and the Table 1 values of eph and hv. The PKS vibronic basis is diagonal in the delocalized limit, so the entries in Table 3 are also the actual eigenvalues and eigenfunctions of the system in this case. The electronic basis consists of the states
A
4000-10-
3000-
*
ctiir
5 -
-
2000 -
T = 4.2K
IVB
c=-6.222
50
2000
X = 2.259
4000
1
10
8000
10000
Frequency/cm-’
5000 -
4Ooo-
A In molecular orbital the upper surface is an antibonding MO, (+a - v b ) , and the lower surface is a bonding MO, (wa f v b ) , where v a and v b are appropriate atomic orbitals. ly, have no explicit q dependence, and either do not change Sign ( v a f v b ) or do change sign (va- VI,), when the identical metal centers (A, B) comprising the mixed-valence dimer are interchanged. For simplicity of language, we hereafter use the term “parity” in place of “interchange symmetry.” The vibrational basis is the orthonormal set of harmonic oscillator functions in coordinate q (QA - QB):
-
(3) where H,(q) are the Hermite polynomials. Since q (Table 1) goes into -q upon interchange of the centers (Le., q has odd parity), it follows from the mathematical form of the Hn(q)that Xn(q)goes into (-l)nxn(q) under the same operation; thus, when n is even, xn(q)has even parity, and when n is odd, Xn(q)has odd parity. Let us now consider the selection rules for electric-dipoleallowed transitions between vibronic states. As discussed in detail in ref 3, pp 405-409, in our v h electronic basis, the Franck-Condon approximation is valid, and the vibrational functions may be factored out of the transition moment integral. For the remaining electronic part, simple group theoretical q+)are electricselection rules show that v+ v- (and lydipole-allowed (and z-polarized) while I$+ v+ and vare parity-forbidden. In the delocalized limit, vibronic coupling is zero (1 = 0), and since equal force constants are assumed in the two metal-center oxidation states, it follows that the I)+ and the v- potential surfaces in q-space consist of identical parabolas directly over one another, separated in energy
-
+-
--
-
ctiir
3000 -
I
2000 1000 -
X = 2.259
II
A 4O0O 2000
6000
SO00
10000
Frequency/cmFigure 1. (a, top) Calculated intervalence transitions by PKS model in the delocalized limit (2, = 0) using the Table 1 value of the electronic coupling ( E ) and hv = 450 cm-’. Stick heights are directly proportional to absorbance and numerically are equal to the product of dipole strength and transition energy (cm-l) with .AT= (yj+lmz@.) set to unity. For simplicity of display, the sticks are not convoluted with a bandshape function. (b, middle) Calculated intervalence transitions by PKS model using the Table 1 parameters at low temperature. The charge-transferinduced infrared (ctiir) transition is shown in the left inset. Stick heights are scaled as in (a). (c, bottom) Calculated intervalence transitions by PKS model using the Table 1 parameters at room temperature. The charge-transfer-inducedinfrared (ctiir) transitions are shown in the left inset. Stick heights are scaled as in (a). by 21€pksl hv. The only electric-dipole-allowed transition moments in our vibronic basis are of the type
w+xo
-
w-209
++XI-
W-Xl,
+-
-
w+xz w-xz,
... (4)
(or the above with v+ and interchanged) because of the orthogonality of the vibrational basis functions. In the delocalized limit, eq 4 corresponds to the actual allowed vibronic transitions, all of which occur at 2 I ~ ~ k ~= l h5600 v cm-’ (Table 3). Thus, as illustrated in Figure la, a single line is predicted (at all temperatures) in the delocalized limit.
11228 J. Phys. Chem., Vol. 98, No. 43, 1994 TABLE 4: Selected Vibronic Eigenfunctions and Eigenvalues in the PKS Model for the Table 1 Parameters: Cnkr = -6.222,A = 2.259, and hv = 450 Cm-‘ “lower surface” eigenfunction eigenvalue (cm-’)
--
Comments
contribution. (The single line in Figure l a is pure ++XO +-XO at low temperatures.) Additional lines arise, for ai5(6982 cm-’) transition, while example, from the (6391 cm-’) transition is, of course, paritythe forbidden. +-,yo
- ai2
-
Comparison of SRTZ Results with Those of the PKS Model
-
“upper surface” eigenfunction
eigenvalue (cm-’)
a.19= v+[o.l15X1 - 0.025313 + ...I
+ ?)-[0.987~0- 0.105312 + ...] = ~ + [ 0 . 1 1 1+ ~ 00.143312 + ...] + ?)-[0.967~1- 0.172313 + ...I ?)+[0.159x1+ 0.152113 + ...I + ~ - [ 0 . 0 9 0 ~+00.940312 + ...I
5792
@2;
6391 6982
When vibronic coupling is turned on (A f 0), pseudo-JahnTeller coupling in q mixes our Table 3 basis states: vibronic basis states from the lower surface mix with those from the upper surface which differ by f l in the vibrational quantum number n (eqs 3 and 4). The general form of the resultingnonadiabatic vibronic eigenfunctions is ca
m
‘l=v+
‘k,dn+Wn=1,3,5, ...
c
c ca
ca
a;
= q+
“k,dn
n=0,2,4, ...
Ck‘,dn
n= 1,3,5,,..
+ v-
n=0,2,4,
...
C’k‘,Jn
(5)
and we have now taken explicit account of the previously discussed parity properties, with sub- or superscript plus designating even parity and sub- or superscript minus designating odd parity. Using the parameters of Table 1, we obtain the PKS eigenvalues (energies) shown in the second column of Table 2, and a few of the corresponding eigenfunctions are shown in Table 4. The resulting spectrum (Figure lb,c) now has two sets of lines. The lower energy group of transitions are usually called charge-transfer-inducedinfrared (ctiir) transitions (though we originally referred to them as tunneling transitions8). They are a consequence of the vibronic mixing (1f 0) described above. Inspection of the vibronic eigenfunctions (Table 4) for (at 0 cm-’) and (at 237.9 cm-’), together with the selection rules for our Table 3 vibronic basis given in eq 4, shows why the @: transition at 237.9 cm-’ is allowed: nonzero contributions include those from the ++XO +-XO and ?+-XI $J+x~ components. In contrast, the @: (505.8 cm-’) transition is forbidden since all of the allowed electronic parts (q+ $J- or I)- ++) have zero Franck-Condon overlap due to the orthogonality of their associated vibrational basis functions. (The above are just two explicit examples of the general selection rule that transitions are allowed only between vibronic states (a’s) of opposite parity.) The higher energy group of lines in the spectrum (Figure lb,c) constitute what is conventionally termed the intervalence band. The additional lines in this region (as compared to the delocalized limit of Figure la) arise from the same type of mixing which gives rise to the ctiir bands. The strongest line : (5792 cm-’) in the spectrum is due to the a transition; it derives most of its intensity from the large W+XO
-
-
-
--
-
-
In contrast with the above results, SRTZ calculate the most CP.: (505.8 cm-’) intense ctirr transition to be the (a: transition and the most intense intervalence transition to be the CJ&(6391 cm-’) transition. As is clear from the above discussion, both of these transitions are parity-forbidden and thus have zero intensity! In fact, all the transitions that SRTZ designate as allowed are forbidden, and vice versa. These errors lead SRTZ to a clearly unphysical result for the ctiir transitions. We can see this by examining the bottom panel of their Figure 7, which shows their calculated spectra obtained holding A fixed at 2.259 and varying E (in our notation) from -280/450 to -2800/450 cm-’, i.e., from a localized to a delocalized case. Their figure shows that the calculated ctiir transition intensity increases relative to the intervalence band as the system becomes more delocalized, but in fact exactly the opposite must occur. This is evident on simple physical grounds. As the vibronic coupling (A) goes to zero, intensity borrowing from the intervalence band goes to zero, and conventional infrared selection rules become applicable. But the q mode is infrared (dipole)-forbidden despite the fact that it has odd parity. This is so because it is simply a linear combination of totally symmetric modes (QA and QB) on the two monomer centers;2expansion of the transition moment leads to a sum of contributions from QA and QB which are all zero since QA and QB are infrared-inactive. (This is discussed in detail on p 408, ref 3.) In addition, the SRTZ calculated intensity for the ctiir transitions (SRTZ, Figure 3) becomes orders of magnitude too large. For the Table 1 parameters, the SRTZ integrated ctiir intensity is (by eye) roughly 30% that of the intervalence band whereas it should be (our Figure 1, b or c) roughly 0.5% that of the intervalence band! The disappearance of the ctiir transitions in the delocalized limit can also be established analytically. We have previously used perturbation theory to treat the ctiir transitions in both limit^,^ and the result to first order in the delocalized regime is
-
D = 8(n + 1 ) ~ ~ ~ ~ ~ -1 1)2 ~ & / ( (7) 4 ~ ~ ~ ~ where TE is the transition energy between successive low-lying vibronic states, D is the corresponding transition dipole strength, (V+lm,l+-) = (Qalmzl$Jb) is the electronic transition dipole, and n is the vibronic quantum number. (Note that an erroneous comparison between theory and experiment was made in the original PKS treatmentlo) Equation 6 states that, to first order in 1,successive low-lying vibronic levels will be equally spaced. As a rough check, if we use the Table 1 parameters cph = -2800/450,;1 = 2.259, and hv = 450 cm-‘, we get TE = 264 cm-‘, which is in rough accord with Table 2 and confirms that the Epks and parameters correspond to a moderately delocalized system, the crossover between localized and delocalized being at (lEpksl/j1*) x L213 Thus, although eqs 6 and 7 are not quantitatively applicable to the specific parameters in Table 1, which correspond to a case beyond the first-order limit,
J. Phys. Chem., Vol. 98, No. 43, 1994 11229
Comments
/
/
0.3c 0.lt
\
\
Note Added in Proof. Since Talaga, Reber, and Zinc (TRZ) concede in their Reply that both the PKS and SRTZ treatments give exactly the same eigenvalues, it must be the case that the PKS and SRTZ treatments also give exactly the same eigenfunctions. But we have explicitly shown above that the transition claimed to be most intense by SRTZ is in fact between states of the same parity. Such a transition must be exactly forbidden by simple group theoretic principles (viz.,the parity selection rule) independent of any of the detailed properties of the transition dipole operator discussed in the TRZ Reply. The error in the TRZ Reply is in “the assumption that pi*, 0, pi=, = 0”; no such assumption is necessary, since pu is readily calculated (eq 3 1 in ref 2 or eq 113 in ref 3) with the result pi*, - 0, pi=j 0.
*
*
References and Notes Figure 2. Vibrational wave functions associated with ly+ (upper curve) and V- (lower curve) for the lowest vibronic eigenstate (@: of Table 4); see eq 5 .
these equations clearly show the required limiting behavior as the ratio of vibronic to electronic coupling ( A 2 k p ~ )goes to zero. The nonadiabatic nature of our eigenfunctions is illustrated in Figure 2 for the ground vibronic state (a: of Table 4) using the Table 1 parameters. The upper and lower curves show the respective coefficients of v+ and v- in the fiist of eq 5 . Note the opposite parity of these functions and the small amplitude of the vibrational function from the upper surface basis (Table 3). The latter feature is expected and simply means that the vibronic coupling in this fairly delocalized system mixes only a relatively small amount of upper surface vibronic function into the lowest vibronic state. Finally, we note that more recent work by 0 n d r e ~ h e n l l - l ~ and Piepho63l4shows that the q mode used in the original PKS treatment (and by SRTZ’) plays only a minor role in determining the bandshape of delocalized systems such as, for example, the Creutz-Taube ion. In summary, we believe that if SRTZ correct their parity errors, their results will agree with the PKS calculation. Indeed, it is hard to see how a different formalism for solving the same Schrodinger equation, and using exactly the same physical assumptions, can give a different result.
Acknowledgment. This work was supported by the National Science Foundation under Grant CHE9207886.
(1) Simoni, C.; Reber, C.; Talaga, D.; Zink, J. I . J . Phys. Chem. 1993, 97, 12678-12684. (2) Piepho, S. B.; Krausz, E. R.; Schatz, P. N. J . Am. Chem. SOC. 1978, 100, 2996-3005. (3) Wong, K. Y.;Schatz, P. N. Prog. lnorg. Chem. 1981, 28, 369449. (4) Fulton, R. L.; Gouterman, M. J . Chem. Phys. 1961, 35, 1059. ( 5 ) Fulton, R. L.; Gouterman, M. J . Chem. Phys. 1964, 41, 2280. (6) Piepho, S. B. J . Am. Chem. SOC. 1988, 110, 6319-6326. (7) Schatz, P. N. Vibronic Coupling Models of Mixed Valency: Relation of the PKS and MO Models for One- and Two-Electron Systems; Prassides, K., Ed.; Mixed Valency Systems: Applications in Chemistry, Physics and Biology; Kluwer: Dordrecht, 1991; pp 7-28. (8) Schatz, P. N.; Piepho, S. B.; Krausz, E. R. Chem. Phys. Lett. 1978, 55, 539. (9) Prassides, K.; Schatz, P. N. J . Phys. Chem. 1989, 93, 83-89. (10) In Figure 3 of the original PKS paper? and in Figure 14 of the Wong and Schatz review calculated dipole strengths were used in the theoretical fit of the experimental intervalence band of the CreutzTaube ion. The correct procedure requires the product of dipole strength and transition energy, as was done in later work.15 This emor makes the calculated ctiir bands in the cited figures far too intense relative to the intervalence band. The error would have a relatively small effect on the fit of the intervalence band. (11) KO,I.; Zhang, L.-T.; Ondrechen, M. J. J . Am. Chem. SOC. 1986, 108, 1712-1713. (12) Zhang, L.-T.; KO,J.; Ondrechen, M. J. J . Am. Chem. SOC.1987, 109, 1666-1671 and references therein. (13) Ondrechen, M. J.; KO, J.; Zhang, L.-T. J . Am. Chem. SOC. 1987, 109, 1672-1676 and reference therein. (14) Piepho, S. B. J . Am. Chem. SOC. 1990, 112, 4197-4206. (15) Neuenschwander, K.; Piepho, S. B.; Schatz, P. N. J . Am. Chem. SOC.1985, 107, 7862-7869.
