J . Phys. Chem. 1990, 94, 8542-8547
8542
TABLE 111: Rate Constants for the Various Processes of 2PBI in the Excited Singlet State in AQNS Solution reaction deactivation radiative deactivation rate constants rate constants, s+ rate constants, s-' k'NT = 3.1 X IO8 s'I kN = 9.7 X IO9 krN = 3.2 X IO8 kcT = 6.7 x lo8 S-l kc = 7.3 X IO8 krc = 1.3 X IO8 kcD = I , 3 X IO" 5-I M-' k T = 6.0 X 10' krT = 1.0 X IO8 k D T l k D = 0.5
reasonable as regards their order of magnitude and agree well with the values estimated from the Forster cycle pK*,: the quenching constant kco is of the order of the diffusion-controlled limit and agrees well with the value calculated from the quenching experiments in lifetime measurements; the large tautomerization constant kCT confirms the efficiency of this process; and k;rlT is as expected large enough to compete with the rate of deactivation of N*.
The values obtained for the above parameters estimated from fluorescence steady-state data, together with the lifetimes of the fluorescent species, the quantum yields of N* and C*, and the ratios hN and 4Tc(Table II), jointly allow calculation of the rate constants of the various processes undergone by excited 2PBI in neutral and acid media, and also of the radiative constants of the three fluorescent species. The results'7 (Table 111) are quite
Acknowledgment. Our special thanks go to Dr. Mapnita and the Structural Chemistry Centre, Lisbon, for their invaluable assistance in obtaining time-resolved fluorescence measurements with their time-correlated single-photon-counting apparatus. This work was supported directly by the Spanish Interministerial Science and Technology Commission (CICYT) and the University of Santiago and indirectly by a CICYT postgraduate training grant awarded to M.N.
(17) It is worth noticing that those rate constants derived from lifetime data obtained at the limit of the temporal resolution capacity of the apparatus, notably those referring to N*, probably have a considerable percentage error.
Registry No. D, 129364-68-7; C, 129364-69-8;N, 1137-68-4; A,
129364-70-I .
Boxing Procedure for Estimating Shape Resonance Energies from Stabilization Graphs with the MSXa Method Maurizio Cuerra Istituto dei Composti del Carbonio Contenenti Eteroatomi e lor0 Applicazioni, CNR, Via della Chimica 8, 40064 Ozzano Emilia, Bologna, Italy (Received: February 5, 1990; In Final Form: May 16, 1990)
Shape resonance energies have been estimated with the M S X a method by stabilizing transition-state energies with a positively charged sphere (Watson sphere). Stabilization graphs have been investigated by using as a variable parameter either the Watson sphere radius rw or its charge qw. The importance of determining resonance energies from stabilization graphs as a function of rw by fixing the ratio qw/rw to a large value (boxing procedure) is clearly demonstrated. Resonance energies are better determined from the stability of the transition-state energies of valence anion states rather than by localizing their avoiding crossings with single-centersolutions, owing to a sharp variation of the electronic relaxation energy in the neighborhood of avoided crossing points in the transition-state procedure.
Introduction Over the past few years temporary anion states have been investigated experimentally by using the sharp variations occurring at specific energies in the electron-scattering cross section (shape resonance) in electron transmission (ET) spectra.' A shape resonance arises as a result of the temporary capture of the incoming electron into a normally vacant molecular orbital (MO). The resonance energies are the negative of the gas-phase electron affinities (EAs) and can be estimated with bound-state calculations by using stabilization methods2 The multiple scattering X a (MSXa) method3 has proven to be a powerful tool for characterizing shape resonances4 In fact,
resonance energies have been accurately reproduced for calcophenesI0 and transition-metal comple~es'*~J with little Computational effort. Tossell and c o - ~ o r k e r s 'first ~ attempted to estimate shape resonance energies with the M S X a method, using the transition-state procedure,I4 by stabilizing the positive energy of the unbound electron with an attractive potential generated by an uniformly positive charged (qw = 2e) sphere (Watson sphere). The Watson sphere radius rw was taken as large as twice the outer-sphere radius (To) so as to ensure that all negative eigenvalues were stabilized by qw/rw. The positive energy was then computed by adding this amount to the corresponding transition-state ei-
Schulz, G . J. Reu. Mod. Phys. 1973, 45, 378. Hazi, A . U.; Taylor, H. S.Phys. Rev. A 1970, I , 1109. Johnson, K. H. Adu. Quantum Chem. 1973, 7, 143. A more rigorous theoretical approach for assigning resonances is to compute the total electron-scattering cross section by means of the continuum MSXa method.' However the agreement with experiment is so far only a qualitative one,&* owing to the difficulty of describing accurately the interaction between the target molecule and the incoming e l e ~ t r o n . ~ ( 5 ) Dehmer, J . L.; Dill, D. I n Electron-Molecule and Photon-Molecule Collisions;Rescigno. T.,Mc Koy, V.,Schneider, B., Us.; Plenum Press: New York, 1979. (6) Tossell, J. A.; Davenport, J. W . J . Chem. Phys. 1984, 80, 813. (7) Guerra, M.; Jones, D.; Distefano, G.; Foffani, A,; Modelli, A. J . Am. Chem. SOC.1988, 110. 315
(8) Modelli, A.; Foffani, A.; Scagnolari, F.; Torroni, S.; Guerra, M.; Jones, D. J . Am. Chem. SOC.1989, I l l , 6040. (9) Krylstedt, P.; Elander, N.; Brandas, E. In?. J . Quantum Chem. 1987, 31, 755. (IO) Modelli, A.; Guerra, M.; Jones, D.; Distefano, G.; Irgolic, K. J.; French, K.; Pappalardo, G. C . Chem. P h p . 1984,88, 455. ( I I ) Modelli, A.; Foffani, A.; Guerra, M.; Jones, D.; Distefano, G. Chem. Phys. Lett. 1983, 99, 5 8 . (12) Modelli, A,; Distefano, G.; Guerra, M.; Jones, D. J . Am. Chem. Soc. 1987, 109, 4440. ( I 3) Giordan, J. C.; Moore, J. H.; Tossell, J. A. J . Am. Chem. SOC.1981, 103, 6632. (14) Slater, J. C . In Computational Methods in Band Theory; Marcus, P. M . , Janak, J. F., Williams, A. R., Eds.; Plenum Press: New York, 1971.
(1) (2) (3) (4)
0022-3654/90/2094-8542$02.50/0
0 1990 American Chemical Society
Estimating Shape Resonance Energies
The Journal of Physical Chemistry, Vol. 94, No. 23# I990 8543
genvalue. The features appearing in the ET spectra were assigned to the computed low-lying unstable anion states also when the charge distribution in the normally unoccupied MO indicated that the state was single center (continuum-Iikels) rather than valence in character. On the contrary, we have taken into account only valence states for assigning shape reonance with this procedure.16 To avoid a plethora of continuum-like states, the Watson sphere charge has also been increased. Recently, Chao and Jordan1' have outlined a procedure to estimate shape resonance energies from stabilization graphs within the MSXa framework. Investigation of stabilization graphs can lead to a more accurate estimate of shape resonance energies but at a loss of simplicity and demanding a greater computational effort. Shape resonance energies are usually estimated by localizing avoided crossings'* between the eigenvalues of a CI matrix by varying either the size of the box where the system is confined (boxing procedure) or a real scaling parameter in the Hamiltonian of the system (scaling p r o c e d ~ r e ) . ' ~The procedure suggested by Chao and Jordan consists of localizing avoided crossings between transition-state energies by plotting them against the Watson sphere charge qw keeping its radius rw fixed to a value just a little larger than that of the outersphere radius, r,. Thus, this procedure corresponds to neither the boxing nor scaling procedure. Only when avoided crossings were not found in this way, rw was varied keeping qw fixed. However, the effect of the value of the fixed parameter on the estimate of resonance energies was not investigated at all. In addition, it could be inappropriate to associate resonance energies with avoiding crossings in conjunction with the transition-state procedure. In fact, the transition-state energies of the valence and continuum-like states are not solutions of the same Hamiltonian. Therefore, we have carried out a detailed study on the factors that influence the estimate of shape resonance energies determined from stabilization graphs by using the transition-state procedure with the MSXa method.
Method and Computational Details The details of the MSXa method can be found in Johnson's review3, and only a brief description of the method relevant to the discussion will be given here. I n the X a theory,20the molecular orbitals fii are solutions of a set of one-electron differential equations (in atomic units): [H(1) + V C ( ~+) Vxc(l)Ifii(l) = tifii(1)
(1)
where H is a sum of the kinetic and nuclear attraction potentials
ff( 1 ) = -1 / 2 v 2 - zAZA/IrI - RAl
(2)
Vc is the electronic Coulomb potential
and V,, is an approximate local exchange-correlation potential related to the electronic charge density, p , and a scaling factor a
Vxc(l) = -6a[3p(r1)/8r]1/3
(4)
The multiple scattering procedure allows an effective solution of eq 1. The coordinate space of the molecule is partitioned into three regions: the atomic regions (A), which are inside the spheres centered on the atoms, the outer-sphere region (OUT), which is the region outside an outer sphere surrounding the cluster of the atomic spheres, and the intersphere region (INT), which is the (IS) A continuum-like orbital is defined as the one having an electron charge outside the atomic spheres greater than 80%. (1 6) A detailed discussion on the reassignment of resonances previously attributed to continuum-like states is reported in ref 7. (17) Chao, J. S.-Y.; Jordan, K. D. J . Phys. Chem. 1987, 91, 5578. (18) Simons, J. J . Chem. Phys. 1981, 75, 2465. (19) Garcia-Sucre, M . ; Lefebvre, R. Chem. Phys. Lett. 1986, 130,240. and references therein. (20) Slater, J. C. Adu. Quantum Chem. 1972, 6, 1.
region between the atomic spheres and the outer sphere. In the atomic and outer-sphere regions the potential is assumed to be spherically symmetric and the electronic wave function is expanded in terms of real spherical harmonics. In the intersphere region the potential is taken to be a constant, determined by the volume average of the potential, and a multicenter expansion of the wave function is used. The radial part of eq 1 is numerically solved for trial values of the orbital energy t i . The expansion coefficients are determined by solving a set of homogeneous linear equations under the condition that the wave function and its first derivative are continuous at the sphere boundaries. According to the geometrical decomposition of the coordinate space, the molecular orbitals may be classified as either valence (multicenter) or continuum-likeI5 (single-center) depending on the relative charge computed inside the atomic spheres and outside the outer sphere. The X a theory can be view as an approximation of the local density Hohenberg-Kohn theory.z1 The orbital eigenvalues t i are related not to the ionization potentials or electron affinities through the Koopmans theoremz2as in the Hartree-Fock (HF) theory but to the X a statistical total energy: Ex, = Cni(+ilH(l)l+i) + f / 2 j d r l p(rl)[VdI) i
+ Vd1)l
(5)
as the first derivatives of the total energy with respect to the orbital occupation number ni: fi
= aEx,/ani
(6)
The EA of the ith orbital can be then approximate to the negative of the orbital eigenvalue ci calculated in the state where one-half electron is added to the vacant orbital (transition-state proced~re):~~ EA, = Ex,(ni=O) - Ex,(ni=l) = -ti(ni=y2) - f/24a3EX,(ni=f/2)/a3ni ... (7)
+
EA,
= -ei(ni=y2)
(8)
This procedure takes into account the electron relaxation occurring during the electron capture as in the ASCF procedure in the H F approach. MSXa calculations were performed on carbon tetrachloride and fluorobenzene at the geometry reported in refs 6 and 17, respectively. To circumvent the limitations deriving from the assumption of a constant potential in the intersphere region, the atomic spheres were allowed to partially overlap23and the double-counting correction in normalizing the molecular orbitals was employed.24 The radii ratios for the atomic spheres were determined by using the nonempirical procedure of Norman.2s The sphere radii were chosen as 5.8691uo, 1.8138ao, 2.5317ao, for the outer, C, and C1 spheres, respectively, in CC14 and 6.6761ao, 1.665543, 1.7462U0, 1.7502~0,1.7502U0, 1.2908U0, 1.292OU0, 1.2900uo, 1 . 7 7 8 0 ~for~ the outer, C I , C2, C3, C4, Hz, Hj, H4, and F spheres, respectively, in C6HsF. These sphere radii satisfy the virial ratio condition in the neutral molecules. Core electrons were not frozen during the SCF procedure,and transition-state energies converged to better than 0.001 eV. Wave functions were expanded in partial waves up to I = 0 (s) on H, I = 1 (sp) on C and F, I = 2 (spd) on CI, and 1 = 3 (spdf) on the outer sphere. A positively charged (qw) sphere of radius rwwas employed to stabilize orbital eigenvalues. Transition-state energies, obtained by adding the amount of qw/rw to the corresponding eigenvalue, were computed at either 1.25e or 0 . 1 2 5 ~intervals. ~ The step size was reduced near the avoided crossing points. The calculations were performed on the FPS array processor attached to the VAX 1 1 /780 computer of the Theoretical Chemistry Group in Bologna. (21) Hohenberg, P.;Kohn, W. Phys. Reo. B 1964, 136, 864. (22) Koopmans, T. A. Physica 1933, I , 104. (23) Herman, F.;Williams, A. R.; Johnson, K. H. J . Chem. Phys. 1974, 61, 3508. (24) Herman, F. In Electrons in Finite and Infinite Structures; Phariseau, F., Scheire, L., Eds.; Plenum Press: New York, 1977. ( 2 5 ) Norman Jr., J. G. J . Chem. Phys. 1974. 61, 4630.
8544
The Journal of Physical Chemistry, Vol. 94, No. 23, 1990
Guerra
(Bil
I
nt -l.O
i 7
6
8
9
10
11
Watson sphere radius ( ao) Figure 2. Stabilization graphs for carbon tetrachloride. Transition-state energies as a function of the Watson sphere radius rw keeping its charge qw fixed to 4e (solid) and 12e (dashed).
Figure 1. (a) Attractive potential generated by a uniformly charged sphere. (b) MSXa potential inside the atomic spheres (A), in the intersphere region (INT), and outside the outer sphere (OUT).(c) MSXa potential stabilized with a Watson sphere having small qw and rw, (d) small rw and large qw, (e) large rw and the q w / r w ratio as in (c), (f) large rw and the qw/rw ratio as in (d). Dashed lines represent potentials obtained by varying 9w in (d) and rw in (0. Results and Discussion
The attractive potential ( Vw) generated by an uniformly positive charged sphere is displayed in Figure 1. Knowledge of its shape is fundamental for understanding its influence on the value of negative EAs computed with the transition-state method. Vw is a constant inside the Watson sphere and decreases as the inverse of r outside the Watson sphere:
r < rw (9) r > rw (10) Figure 1 also illustrates, schematically, the variation in the molecular potential produced by the attractive potential Vw for selected values of the Watson sphere parameters. For small rw and qw (Figure IC) the potential is stabilized weakly inside the Watson sphere, and hence only resonance energies lying just above threshold can be estimated. Besides, the potential rises slowly outside the Watson sphere, and therefore the computed valence anion states are expected to have appreciable electronic charge density outside it. For small rw and large qw (Figure Id) the stabilization inside the Watson sphere is strong. The potential rises rapidly outside the Watson sphere, and the resulting potential well allows one to estimate the energy of highly unstable valence anion states. Figure le,f show the effect of increasing r, on the molecular potential, keeping the ratio qw/rw fixed. The potential does not change inside the Watson sphere but rises more rapidly outside the Watson sphere, and its attractive effect extends at large distances away from the atomic spheres. First, we have examined the effect of rw on the estimate of resonance energies. Information can be obtained by studying molecules that have bound and unbound anion states. In fact, bound anion states have sizable electronic charge density outside the outersphere, although less than that expected for unbound anion states, and the corresponding positive EA can also be computed without the stabilizing potential. Both bound (*A,) and unbound (2T2)anion states were found for carbon tetrachloride by performing MSXa calculations;6 therefore the transition-state energies of these states have been computed as a function of rw for selected values of qw. The vw = qw/rw Vw = -qw/r
TABLE I: Resonance Energies (eV) of the ’B, States in Fluorobenzene Estimated from Both the Energy Stability and Avoided Crossing Point by Varying the Watson Sphere Charge 9w (Value of pw(e) at the Avoided Crossing Point Reported in Parentheses) rW=
energy stability
7.0 11.0 14.0
avoided crossing
42B, 1.80 1.48 1.47
1.28 (2.8125) b 1.40 (10.125) 1.37(3.75)
52BI 7 .O
5.20
11.0
4.70
14.0
4.68
‘Watson sphere radius in
a,,.
5.00(23.75) 4.60 (2.75) 4.55 (2.5625) 4.30(7.5) 4.24 (5.625) 4.41 (4.5625) 4.37 (15.0) 4.44 (10.625) 4.18 (6.25) 4.07 (5.0)
bNot found.
stabilization graphs are reported in Figure 2. For values of rwjust a little larger than the outer-sphere radius r, (To < rw < r, + 2 . 0 ~the ~ ) transition state energy of the bound 2A, state is sizeably higher than that computed without the Watson sphere, the deviation being proportional to qw. Otherwise, for large rw the transition-state energy remains stationary, its value being equal to that computed without the Watson sphere. The transition-state energy of the unbound 2T2state shows the same behavior, its destabilization for small rw being slightly stronger than that obtained for the ZA,state. The reason for this is that the extra electron experiences a stronger electron repulsion when rw decreases, because the electronic cloud is compressed by the potential well, which becomes narrower. The destabilization is proportional to the strength of Vw, since the capability of constraining the electron charge inside the Watson sphere increases (compare Figure Ic,d). For qw = 12e and small rw the transition-state energy of the 2A, state even becomes positive, that is, the anion state is unstable (negative EA). Incidentally, the transition-state energy (0.94 eV) of the unbound 2T2state estimated from the energy stability, ;.e., by using large rw, closely reproduces the experimental resonance energy (1 .O eV).26 (26)Burrow, P.D.;Modelli, A.; Chiu, N. S.; Jordan, K. D.J . Chem. Phys. 1982, 77. 2699.
Estimating Shape Resonance Energies
The Journal of Physical Chemistry, Vol. 94, No. 23, 1990 8545
TABLE II: Transition-State Energies t (eV) for the Valence zB1 States and for the Continuum-like State Nearest the SZBIState in Fluorobenzene, Together with the Electron Charge Distribution (a) rw = 70, rw = l l a o 9w f A INT OUT W S c A INT OUT W S 4'Bl 2.5 1.33 31 51 18 12 1.46 54 38 8 1 5.0 1.41 43 47 IO 6 1.46 55 38 7 1 10.0 1.63 57 40 3 1 1.47 55 39 6 0 15.0 1.66 59 39 2 1 1.48 55 39 6 0 20.0 1.70 60 39 1 0 1.48 55 39 6 0 25.0 1.72 60 39 I 0 1.48 55 39 6 0 1 0 1.48 55 39 30.0 1.74 60 39 6 0 40.0 1.77 61 38 0 1.48 55 39 1 6 0 2.5 5.0 10.0 15.0 20.0 25.0 30.0 40.0
4.51 4.90 5.10 5.17 5.16 4.83 4.94 5.10
32 49 58 54 45 37 44 52
40 39 37 40 46 53 49 44
28 12 5 6 9
IO 7 4
52BI 21 9 3 3 4 4 3 2
C6H5F
rw = 7ao
...... ......................
5.0
-
a 4.14 4.43 4.53 4.46 4.49 4.61 4.68
33 38 41 33 36 43 45
16 28 29 28 30 31 31
51 34 30 39 34 26 24
0.0
33 7 4 6 5 2
3.01 3.52 4.09 4.43 4.66 5.22 5.27 5.38
5 8 12 20 29 35 25 16
45 52 61 61 57 54 62 70
50 40 27 19 14 11
13 14
36 25 14 9 6 4 5 4
2.54 2 2.99 2 3.54 4 3.91 IO 4.21 23 4.27 7 4.38 5 4.54 4
30
20
10
40
Watson sphere charge ( e ) C6H5F
I
1
I
I
I
rw = I l a ,
........................ 5.0
2BI(Continuum-like) 2.5 5.0 10.0 15.0 20.0 25.0 30.0 40.0
4.0
4 13 20 24 27 24 2s 26
94 85 76 66 50 69 70 70
49 39 26 18
IO
2 ->. P 5
12
'3
IO 8
OThe stabilizing potential is too low for estimating this energy.
These calculations suggest that the resonance energies evaluated by Chao and Jordan from stabilization graphs for fluorobenzene and pyridine (rw r r, 0 . 5 ~ should ~) have been overestimated. We have then carried out M S X a calculations on fluorobenzene to establish the effect of rw on the computed resonance energies when qw is varied. The stabilization graphs as a function of qw for 2Bi anion states are shown in Figure 3 for selected values of rw.27 Low-lying (42Bi)and high-lying (52B1)unbound valence anion states have been computed for this symmetry. In the 42Bl state the extra electron occupies the MO deriving from the interaction of the symmetric component of the vacant degenerate eZuMO of benzene with the 2p lone pair on the F atom, whereas the electronic charge distribution in the 52Bl state does not appreciably differ from that found in the vacant b MO of benzene. The resonance energies, determined both from %e stability of the transition state energies of valence states and from their avoided crossings with the transition-state energies of continuum-like states, are reported in Table I. The electronic charge density in the atomic spheres (A), in the intersphere region (INT), and outside both the outer (OUT) and Watson (WS) spheres are reported in Table I1 as a function of the variable parameter qw for selected values of the fixed parameter rw. The electronic charge distribution is used to classify a state as either valence or continuumlike.ls Knowledge of the amount of electron charge located outside the Watson sphere is fundamental for establishing whether the transition-state energies have been computed correctly. Indeed, they are overestimated if a nonnegligible electron charge is located outside the Watson sphere as it results from comparing eqs 9 and
4.0
.....................
3.0
0
1
s
._ 2.0 .-v)
;"
1 .o
...............................................................
0.0
+
I
I
I
I
10
20
30
40
Watson sphere charge ( e )
..
i
I
r.
s
.......................
:::::::::::::i::.i--.-
.......
. + V I
- d ,::............................................................... +::A
.- 2.0 ._ Lo
E
I-
.
1.0
1:.
-
..................................................
5:.,._
1
................................................................... 0.0
I
I
I
I
10
20
30
40
Watson sphere charge ( e ) Figure 3. Stabilization graphs for fluorobenzene. Transition-state energies of valence (solid) and continuum-like (dashed) states as a function of the Watson sphere charge qw keeping its radius rw fixed to ( a ) 7ao, (b) 1 lao, (c) 1400. The horizontal dash-dotted lines indicate the experimental resonance energies.
10.
Table I shows that the resonance energies estimated from the energy stability are also, in the present case, sizeably higher when computed fixing rw to a small rather than to a large value. The (27) In the neighborhood of avoided crossings the valence and continuum-like states can mix to a large extent; the state localized at the smaller extent in the atomic spheres has been assigned to the continuum-like curve in stabilization graphs even though the criterion of classifying it as a continuum-like state is no longer satisfied.
difference between the resonance energies estimated with rw = 7.0ao, the value used by Chao and Jordan:* and with rw = 1 lao is as large as 0.32 and 0.5 eV for the 42Bl and 52Bl states, re~ p e c t i v e l y . ~By~ contrast, there is no more variation of the res(28) Unfortunately, the atomic sphere radii were not reported in ref 17. The present results for fluorobenzeneobtained by fixing rw to 7.Mare slightly different from those previously reported.
8546 The Journal of Physicnl Chemistry, Vol. 94, No. 23, 1990
Guerra
TABLE 111: Resonance Energies (eV) of the 'B, States in Fluorobenzene Estimated by the Boxing Procedure from Both the Energy Stability and Avoided Crossing Point (Value of rw (so) at the Avoided Crossing Point Rewrted in Parentheses) VWU energy stability avoided crossing
4*BI 0.8 8.0 40.0
I .47 1.48 1.48
1.27(7.0) I .27 (8.75) 1.33 (9.0)
8.0
40.0
4.54
4.74
4.75
-
4.0
r
P 5
3.0
m
I c
5'Bl 0.8
5.0
4.69(7.0625) 4.52(7.875) 4.58(8.25) 4.40(9.75) 4.21 (10.375) 4.38 (I 1.625) 5.16 (7.0) 4.63 (9.125) 4.65 (10.1875) 4.55 (11.375) 5.05 (7.75) 4.62 (9.625) 4.65 (10.625) 4.60(1 I .75)
6 2.0 E c
...... . _ L
I .o
.....
'
0.0
I
I
I
I
I
7
8
9
10
ii
J
Watson sphere radius ( ao)
'Stabilizing potential in rydbergs.
onance energies with increasing rw from 1 lao to 14no. The estimate of resonance energies, especially of high-lying anion states, is less accurate if avoided crossings are considered. The stabilization graphs show that several avoided crossings occur between the high-lying 52B,state and continuum-like states in a narrow range for 9w smaller than 9e. The mixing of the valence state with more than one continuum-like state complicates the localization of the avoided crossing point. In addition, the resonance energy is estimated to be lower by about 0.5 eV than that computed from the energy stability owing to the large mixing of the valence state with continuum-like states even though the transition-state energies should be somewhat overestimated because of the large amount of electron charge located outside the Watson sphere (see Table 11). As mentioned earlier, the transition-state energies are, in this case, stabilized by an amount less than qw/rw. It is much easier to localize avoided crossing points for low-lying valence anion states since few continuum-like states are expected to occur at low energies. Sometimes, for this reason, it is not possible to find avoided crossings as shown in Figure 3b for the 42BIanion state. On the other hand, the probability of finding avoided crossings is low in the region where the energy can be evaluated with accuracy, viz., 9w > 10e. In fact, a variation of 9w does not significantly modify the potential well outside the outer sphere (see Figure Id); as a consequence the transition-state energies slowly change, the variation being slightly larger for continuum-like than for valence states. For example, Figure 3a and Table I1 show that the 52Blstate is mixed to a large extent with a continuum-like state for values of 9w going from 15e to 40e. Figure 3b and Table I1 show that the 52BIstate mixes with a continuum-like state near 9w = 20e even if an avoided crossing does not occur. Finally, Figure 3c shows that the curve of the S2B, state and that of a continuum-like state are very close one another, the energy difference being less than 0.05 eV, for a qW range as large as 20e. Again, resonance, energies estimated from avoided crossing points are higher fixing rw to a low rather than to a large value as found above from the energy stability (see Table 1).
It is evident from all the prior discussion that the Watson sphere charge cannot be utilized as a variable parameter keeping rw fixed to construct stabilization graphs. We have then varied the Watson sphere radius keeping either the value of 9w or the ratio qw/rw (29) For the 4'8, state the energy difference should be wen higher; in fact, Figure 3 and Table 11 show that the energy becomes stationary only by fixing rw to a large value, whereas an energy drift of small entity occurs fixing r, to 7 . 0 ~For~ the 5*B1state an energy drift of comparable amount occurs for all of the selected values of rw, and hence it should negligibly affect the energy difference.
0.0
'
I
I
I
I
I
7
8
9
10
ii
I
Watson sphere radius ( a , ) ) C6H5F I
5.0
4.0
%,
Y
0.0
I
I
I
I
I
7
8
9
io
. .'..,.
,
I
11
Watson sphere radius (a(,) Figure 4. Stabilization graphs for fluorobcnzene. Transition-state energies of valence (solid) and continuum-like (dashed) states as a function of the Watson sphere radius keeping the attractive potential fixed to (a) 0.8 Ry, (b) 8 Ry, (c) 40 Ry. The horizontal dash-dotted lines indicate the experimental resonance energies.
fixed. Exploratory stabilization graphs constructed by fixing qw at 4e (0.65 < Vw < 1.I Ry) and Vw at 0.8 Ry showed a similar behavior, since the transition-state energies are more sensitive to variations in rw than in qw, hence we have constructed stabilization graphs by keeping the attractive potential Vw constant inside the Watson sphere to better simulate a boxing procedure. The corresponding stabilization graphs are displayed in Figure 4, the computed resonance energies and the charge distribution are
Estimating Shape Resonance Energies
The Journal of Physical Chemistry, Vol. 94, No. 23, 1990 8541
TABLE IV: Transition-State Energies t (eV) for the Valence *B1 States and for the Lowest 'B, Continuum-like State in Fluorobenzene, Together with the Electron Charge Distribution (9%)
Vw
0.8 Ry INT OUT WS 4'81 51 18 I1 6 2 40 6 1 39 39 6 1 38 7 0 38 7 0
rw
z
A
7.0 8.0 9.0 10.0 11.0 12.0
1.23 1.48 1.47 1.47 1.46 1.45
31 54 55 55 55 55
7.0 8.0 9.0 10.0 11.0 12.0
4.58 4.59 4.57 4.31 4.49 4.26
32 29 43 35 36 28
40 26 28 26 20 26
1.34 22 0.89 16 0.70 I I 0.60 8 11.0 0.52 7 12.0 0.47 7
55 53 51 49 47 45
28 45 29 39 44 46
S2BI 21 25 17 19 33 32
Vw = 40 Ry INT OUT WS
z
A
1.88 1.65 1.30 1.48 1.48 1.47
62 58 37 52 54 55
38 40 50 41 40 39
5.35 5.14 4.86 4.76 4.75 4.62
63 55 50 41 47 42
36 39 40 35 29 30
24 24 28
0 0 0 0 0 0
86 76 62 55 53 50
6 16 23 27 35 41
0 0 0 0 0
0
2 13 7 6 6 1
6
IO
0 0 0 0 0 0
2B,(Continuum-like) 7.0 8.0 9.0 10.0
23 31 38 43 46 48
15 8 7 4 3 2
3.15 8 1.95 8 1.38 15 0.98 18 0.76 12 0.64 9
1
reported in Tables 111 and IV, respectively, for selected values of the attractive potential Vw. The stabilization graphs show that the transition-state energies of valence states sizeably decrease as rw increases and then remain stationary for rw larger than r, by about 3.0ao as previously found for CC14. On the other hand, the transition-state energies of continuum-like states rapidly decrease with increasing rw since, at variance from varying qw, the potential well becomes wider (compare Figure Id,f). Consequently, avoided crossing points can be easily localized. However, for weakly stabilizing potentials (for example, Vw = 0.8 Ry; see Figure 4a) the resonance energy of the high-lying S2B, state cannot be determined with accuracy, since Table IV shows that large amount of the electron charge is located outside the Watson sphere. On the other hand, Table IV shows that for strongly stabilizing potentials the electron charge outside the Watson sphere is negligible even for continuum-like states so as to ensure that all energies have been computed correctly. Besides, Figure 4b,c shows that the curves of valence states are not sensitive to the variation of the attractive potential from 8 to 40 Ry, whereas those of continuum-like states are pushed up to higher energies. Thus, this approach is equivalent to the boxing procedure. The space between two avoided crossings is so wide as to permit an accurate estimate of resonance energies from the energy stability. The accuracy is estimated to be better than 0.02 eV. The question now is whether or not resonance energies can be estimated by localizing avoided crossing points. Figure 4 shows that the transition-energy of both the 42B1and 52Bl states sizeably decreases near the avoided crossing points owing to a variation of the relaxation energy. Indeed, the extra electron experiences a lower electron repulsion, being more delocalized because of the large mixing of the valence state with continuum-like states. The energy lowering depends on the extent of mixing. Notwithstanding this, the resonance energy is estimated at a higher energy than that determined from the energy stability when the avoided crossing point occurs in the region where anion states are desta-
bilized, that is, at rw lower than 8.5ao,as shown in Table I11 and Figure 4b,c for the 52Bl state. As expected, the resonance energy is higher when the avoided crossing occurs at lower rw. For these reasons, the resonance energies estimated by localizing avoided crossing points strongly depends on the value of rw at which the avoided crossing point occurs as shown in Tables I and 111. It is then clear that resonance energies are better determined from the stability of transition-state energies rather than by localizing avoided crossing points. As previously found for CC14 the resonance energies estimated from the energy stability at 1.48 and 4.75 eV for the 42BI and S2BIstates, respectively, closely reproduce the experimental shape resonance energies (1.41 and 4.77 eV).17 In our previous studies, concerning the characterization of temporary anion states with the M S X a method, the attractive potential employed to stabilize orbital eigenvalues was strong enough to allow a reliable estimate of the resonance energies except in the case of disubstituted benzene derivative^.^^.^' In particular, the computed resonance energies of high-lying MOs were underestimated with respect to experiment. The appreciable deviation was ascribed to the underestimation of the constant potential in the intersphere region due to the large size of this region in these compounds. In view of the present study, it should be attributed to the large mixing of the valence states with continuum-like states, since the attractive potential utilized in these cases was slightly less than 0.8 Ry. A more accurate estimate of the resonance energies of disubstituted benzene derivatives is now in progress.32 However, the resonances in these molecules should have been properly characterized. Conclusions
Stabilization graphs constructed with the boxing procedure are more suitable for an accurate estimate of resonance energies with the M S X a method. In fact, transition-state energies of continuum-like states are much more sensitive to variations in the value of rw than in that of qw. Besides, resonance energies obtained by varying qw are sizeably higher when computed by fixing rw to a small rather than to a large value owing to the compression of the electronic cloud inside the Watson sphere. Resonance energies are better determined from the stability of the transition-state energies rather than by localizing their avoiding crossings with continuum solutions owing to a sharp variation of the electronic relaxation energy in the neighborhood of avoided crossing points in the transition-state procedure. A computational strategy can be then suggested. The transition-state energies should be computed with rw larger than ro by 3-4ao by using a strongly stabilizing potential ( Vw > 8 Ry). The comparison with the energies estimated with increasing rw by 0.5ao,keeping Vw fixed, provides information about the stability of the transition-state energies of valence states and permits to identify rw ranges where the probability of finding an avoided crossing is low. Few calculations could then be sufficient to estimate resonance energies with accuracy from the energy stability in these regions. For broad-shape resonances, the energy stability criterion could fail. To overcome this problem, procedures that permit the extraction of both the resonance energies and widths from stabilization graphsI9 need to be developed within the M S X a framework. Further study in this direction is now being made. (30) Guerra, M.; Distefano, G.;Jones, D.; Colonna, F. P.; Mcdelli, A. Chem. Phys. 1984, 9 / , 383. (31) Modelli, A.; Distefano, G.;Guerra, M.; Jones, D.; Rossini, S.Chem. Phys. 1988, 125, 389. (32) Guerra, M., unpublished results,
