10756
J. Phys. Chem. 1992, 96, 10756-10768
(13) (a) Krusic, P. J.; Wasserman, E.; Parkinson, B. A.; Malone, B.; Holler, E. R., Jr.; Keizcr, P. N.; Morton, J. R.; Preston, K. F. J. Am. Chem. Soc. 1991, 113, 6274. (b) Krusic, P. J.; Wasserman, E.;Keizer, P. N.; Morton, J. R.; Preston, K. F. Science 1991, 254, 1183. (14) (a) Scuseria, G. E. Chem. Phys. Lerr. 1991, 176, 423. (b) Gu,0.; Scuseria, G. E. Chem. Phys. Lett. 1992,191,427. This latter work appeared after completion of our work. (15) Ciaslowski, J. Chem. Phys. Lett. 1991, 181,68. (16) Dunlap, B. I.; Brenner, D. W.; Mintmire, J. W.; Mowrey, R. C.; White, C. T. J . Phys. Chem. 1991, 95, 5763. (17) Saunders, M. Science 1991, 253, 330. (18) (a) Cioslowski, J. J. Am. Chem. Soc. 1991,113.4139. (b) Ciaslowski, J.; Fleischmann, E. D. J . Chem. Phys. 1991, 94, 3730. (19) Bakowies, D.; Thiel, W. J . Am. Chem. Soc. 1991, 113, 3704. (20) Matsuzawa, N.; Dixon, D. A.; Fukunaga, F. J . Phys. Chem. 1992, 96, 1594. (21) Dixon, D. A.; Matsuzawa, N.; Fukunaga, T.; Tebbe, F. N. J. Phys. Chem. 1992, 96, 6107. (22) Stewart, J. J. P. J . Comput.-Aided Mol. Des. 1990, 4. 1. (23) Stewart, J. J. P. QCPE Program 455, 1983; version 5.00 (1987). (24) Dewar, M. J. S.;Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J . Am. Chem. SOC.1985, 107, 3902. (25) Stewart, J. J. P. J. Compur. Chem. 1989, 10, 209, 221. (26) This criterion corresponds to the PRECISE option in MOPAC. (27) Dixon, D. A.; Matsuzawa, N. J . Phys. Chem., in press. (28) The A H f O ' s of the eclipsed ethanes calculated at the AM1 (PM-3) level are-16.17 (-16.71), -111.67 (-100.19), -29.79 (-21.24), -5.10 (-2.85), and +17.40 (+30.93) kcal/mol for X = H, F, Cl, Br, and I, respectively. The AHHp's at the AM1 level for the trans conformers are -17.41, -1 14.29, -33.81, -7.94, and 15.75 kcal/mol in the order given above. (29) Pauling, L. The Nature of rhe Chemical Bond; Cornell University Press: Ithaca, NY, 1960. (30) For example, the minimum of the Lennard-Jones potential for the fluorine molecule has a well depth of -0.2 kcal/mol (Hirschfelder, J. 0.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley and Sons: New York, 1954). (31) Matsuzawa, N.; Dixon, D. A. J . Phys. Chem. 1992, 96, 6241. (32) For example, the experimental C-C bond length of ethane is 1.536 A, as tabulated in ref 22.
(33) These values are experimental C-X bond lengths for ethane, fluoromethane, chloromethane, bromomethane, and iodomethane, as tabulated in ref 22. (34) The van der Waals radii are taken from ref 29. (35) For 1,2-added C & ,with n = 10 and 18, a stable structure could not be optimized as four and eight iodine atoms separated from the CWframework, respectively. (36) The AM1 calculated AHfo'swhich were used to obtain the W ' s are -5.2, -22.5, -14.2, -5.3, and +19.8 kcal/mol for H2, F2. C12, Br2, and I*. respectively. These can be compared to respective experimental values ( t a b ulated in ref 22) at 298 K of 0.0, 0.0, 0.0, +7.4, and +14.9 kcal/mol. (37) The values of AHo for reaction 3 can be calculated from the data in refs 28 and 36 and the AM1 AHf' of ethylene of 16.4 kcal/mol. (38) (a) Pedley, J. B.; Naylor, R. D.; Kirby, S . P. Thermochemical Data of Organic Compounds; Chapman and Hall: London, England, 1986. (b) Chase, M. W., Jr.; Davies, C. A.; Downey, J. R., Jr.; Frurip, D. J.; MacDonald, R. A.; Syverud, A. N. J . Phys. Chem. Ref. Data 1985, 14 (Suppl. 1). (39) Recently, intercalation of solid CWwith I2 was reported (Zhu, Q.; Cox. D. E.: Fischer. J. E.: Kniaz. K.: McGhie. A. R.: Zhou. 0. Nature 1992. 355,'712).' No reaction between the iodine molecule and Cm was observed; consistent with our LDF results (ref 21). (40) Thiel, W. MNW90 program, as integrated into the Cray UniChem program system, Cray Research; Eagan, MN. (41) AMI calculated values of group additivity are obtained as follows; AHfo(propane)- AHfo(ethane) = -6.9 kcal/mol for C(H),(C),, AHf0(2,2difluoropropane) - AHfo(ethane) = -101.9 kcal/mol for C(F)2(C)2, AHf"(benzene)/6 = 3.7 kcal/mol for CB-(H), AHfo(toluene) 5(CB-(H)) AHfo(ethane)/2 = 4.8 kcal/mol for CB-(C), and AHf0(1,3,5-tri!luorobenzme) - 3(CB-(H)))/3 = -40.9 kcal/mol for C,-(F). The AM1 calculated AHfo's used in the group additivity calculations are AHfo(ethane) = -17.4 kcal/mol, AHfo(propane) = -24.3 kcal/mol, AHf0(2,2-difluoropropane) = -1 11.7 kcal/mol, Mfo(benzene) = +22.0 kcal/mol, AHfo(toluene) = +14.4 kcal/ mol, and AHf0(1,3,5-trifluorobenzcne)= -1 19.3 kcal/mol (obtained from ref 22 or calculated by us). The group additivity values obtained here are in good agreement with the experimental group additivity values in ref 42. (42) Benson, S. W. Thermochemical Kinetics; John Wiley and Sons: New York, 1968. The values used are -4.95, -97.0, +3.30, +5.51, and -42.8 kcal/mol for C(H),(C),, C(F),(C),, C,-(H), Ca-(C), and CB-(F), respectively.
-
Electronlcally Excited States of Ethylene Kenneth B. Wiberg,* Christopher M. Hadad, James B. Foresman, and William A. Oiupka Department of Chemistry, Yale University, New Haven, Connecticut 0651 1 (Received: May 18, 1992)
The transition energies for ethylene have been calculated via configuration interaction including all singly excited configurations (CIS) using a variety of basis sets. The minimum requirement for a satisfactory basis set is 6-31 1(2+)G* having two sets of diffuse functions on the carbon atoms. The excited states were examined via charge density difference plots. The CIS and MP2-c~rrectedCIS (CIS-MP2) methods provided good agreement with experiment for both vertical and adiabatic energies. The equilibrium structures and vibrational frequencies of the excited states and the radical cations were explored, and the CIS, UHF, and UMP2 levels agreed very well with experiment. The nature of the excited states was clearly differentiated by the charge density difference plots and allowed for a reliable and unambiguous assignment of the excited states into both Rydberg and valence classes. Rydberg-valence mixing and the role of these conjugates are also discussed.
1. Introduction We have recently examined the electronically excited states of bicyclo[ 1.1.O]butane via a combination of experimental and computational studies.' The use of the 6-31 1(2+)G* basis set with configuration interaction including all singly excited confgurations (CIS)2gave a calculated spectrum in good accord with the observed spectrum. Charge density difference plots between the excited states and the ground state showed that all of the states corresponded to Rydberg states having a radical cation-like core and the excited electron in a diffuse orbital. We are now studying a series of small organic molecules in the same fashion. Among polyatomic molecules, ethylene has probably received more extensive study than any other compound? The spectrum of ethylene is shown in Figure !l It consists of a weak transition at about 56 OOO cm-I, followed by a strong transition at 58 OOO cm-I, and a series of bands that are believed to be Rydberg 0022-3654/92/2096-10756$03.00/0
transitions. There have been many experimental studies of ethylene in the vapor and crystallinephases, and much of this work has been summarized by Robin.3 Theoretical studies similarly abound for ethylene. Ethylene has been studied by many computational methods including semiempiricalsand nonempirid techniques. Within the ab initio realm, there have been studies with the HartreeFock p r d u r e , 6 single and double configuration interaction (SDCI),' multireference CI (MRD-CI)? multiconfigurational self-consistent field (MCSCF)? quasidegenerate variational perturbation theory (QDVPT),'O random phase approximation (RPA)," symmetryadapted cluster CI (SAC-CI),IZ and, most recently, the configuration interaction with single excitations (CIS) method.* The MRD-CI and SDCI methods have been used most extensively. There is now a generally good agreement between the experimental data and theoretical calculations. 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10757
Electronically Excited States of Ethylene 70
00
,
I
I/
1 J, 55'00
70 00 CHI-I!
61'00
60'0 0
wavenumber
75'00 x
80 00
05 00
10-91
Figure 1. Ultraviolet spectrum of ethylene.
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-
The ground state of ethylene has been termed the N state, while the x x* singlet valence state is the V state." The 'K 3s Rydberg state is commonly called the R state, and the triplet T x* state is the T state. It is known that the vertical R state is very close in energy to the V state, with the vertical excitation energy of the V state assigned at 7.60 eV (-62 000 cm-I) and an oscillator strength of about 0.35. On going to its equilibrium structure, the V state undergoes twisting around the C = C bond and is predicted to have a DM geometry (dihedral angle of 90') with an elongated C-C bond length of 1.4 A.14The R state is also known to be twisted, although only by 37", and to have D2 symmetry. The vertical T state has a significantly lower excitation energy (4.36 eV) than the V state, and the T state is also known to be twisted with a torsion angle of 90°; however, the C-C bond length is elongated to 1.54 A in the T state.I5 The SD.CI and MRD-CI calculations have been used to examine the Rydberg-valence mixing of the llBlu (* **)and 2IB,, ( x dxz)states.I6 On the basis of their (x2) values, Le., the outsf-plane expectation values, and configuration coefficients, Buenker and Peyerimhoff have stated that the V state has about 50% Rydberg character due to mixing and that the inclusion of diffuse functions in the basis set is very important for the proper description of the excited states.l& McMurchie and Davidson,'d from their SDCI studies, stated that u u* configurations are also important, but Buenker and Peyerimhoff have shownsathat this was due to the use of an inflexible basis set. Cave has discussed many of these studies and cited size inconsistency as a potential problem for accurate determination of excitation energies of the different states." The T state has been calculated to be less diffuse than the V state, and its compactness has been attributed to less mixing with any Rydberg state.* There has also been much discussion of the concept of Rydberg-valence mixing in ethylene. The discussion has focused on whether the r 3d, Rydberg state of ethylene has an independent existence from the x x* valence state. Since both have the same nodal structure, Mulliken16b.chas stated emphatically that both the valence r x* and the Rydberg z 3d, states cannot coexist for ethylene but, instead, that the V state is conveniently described as a linear combination of wave functions:
-
-
-
-
-
-
V = A(?r
-
-
+ E(*
**)
+
3d,)
The relative contribution of each wave function, Le., the coefficients A and E, depends on the molecular geometry. In the united atom limit, the wave function for the V state becomes that of an atomic Rydberg orbital; that is, A will go to zero. For Mullikefi, the 2B1, state would then be a x 4 ,state with three nodes (two angular and one radial) as compared to two nodes (two angular) for the V state. Buenker and Peyerimhoff,16' as well as Robin,3 have stated that both the x x* and the x 3dX2states should exist as independent states (the lBI, and the 2B1, states, respectively), and mixing between the two states would occur. Thus, their individual characters would be spread between both states, such that the V state would have Rydberg character and vice versa. The recent implementation of the CI-singles (CIS) method in the Gaussian suite of programsls has provided a size-consistent method for the study of ethylene. Foresman et a1.2 have recently
-
-
demonstrated that the CIS method provides a reasonable approach to the study of the excited states of ethylene and, moreover, yields excitation energies (both vertical and adiabatic) that are close to experiment. We have therefore used the CIS method to study the character of the different excited states of ethylene. The two main objects of this investigation were to provide an additional test for the CIS method and to make use of the method to study the nature of the excited states. Despite the many experimental and theoretical studies, there is still a controversy about the Rydberg vs valence character of the excited states. It appeared likely that charge density difference plots could be useful in studying ethylene, and this will be the subject of this report. 2. Singlet States In order to establish what basis set would be appropriate for the calculation of the transition energies, the effect of basis set on the CIS energies was examined using valence triple-zeta basis sets with polarization functions and zero, one, or two sets of diffuse functions. The basis sets that were considered are the following: 1, 6-311G*; 2, 6-311+G*; 3, 6-311(2+)G*; and 4, 6-311(2+,2+)G**. (Basis set 3 signifies two different diffuse sp functions on carbon with exponents of 0.0438 and 0.013 1928, while basis set 4 uses two diffuse sp shells on carbon, as above, and two diffuse s shells on hydrogen with exponents of 0.0360 and 0.010 843 4.) The results using the MP2/6-31G* geometry which is very close to the experimental geometry are summarized in Figure 2. Other calculations with methane and trans-1,3butadiene19 have shown that additional polarization functions (a further set of compact d functions or even f functions) do not affect the transition energies, but as in the case of bicyclo[1.1.O]butane,' two sets of diffuse functions were found to be important. For ethylene, two sets of diffuse functions on both carbons and hydrogens provided the best transition energies. The calculated transition energies, with both the CIS and the MP2-corrected CIS (CIS-MP2) methods, appear to have converged using the two largest basis sets, and the results with the larger set are summarized in Table 1. The energies and forms of the occupied orbitals (Figure 3) were found to be little affected by changes in basis set size. However, the virtual orbitals were strongly affected, and the first four are shown in Figure 3. It can be seen that they are much more diffuse than the occupied orbitals. The virtual orbitals calculated using an adequate basis set are properly those for the radical anion, and they are appropriate for calculations of transition energies. In order to obtain virtual orbitals corresponding to the ground state, it is necessary to carry out a calculation for the radical cation at the ground-state geometry of ethylene. The first four virtual orbitals of the cation are shown in Figure 4. These virtual orbitals are much less diffuse than those presented in Figure 3. The calculations can also provide the wave function for each excited state. Of these, there are two choices: the one-particle density matrix (1PDM) and the generalized density matrix (GDM).2 Since incomplete CI expansions do not satisfy the Hellmann-Feynman theorem, properties derived as an expectation value and those derived as a derivative are not equal. The lPDM constitutes a density that is an expectation value, while the GDM is obtained via derivative methods and includes a relaxation of the charge density following the rearrangement due to excitation. (From a practical point of view, the GDM allows cross terms in the density matrix between the occupied and virtual orbitals, whereas the lPDM has these cross terms as being zero. These cross terms allow for a relaxation of the density matrix due to the excitation. For further details, ref 2 is recommended.) The GDM has been shown to be the better representation for correlated densities, and it produces the correct magnitude and sign of the dipole moment and other properties.2*20We have therefore obtained the GDM for each state and transformed the corresponding density into natural orbitals. The latter makes it possible to obtain the charge density as a function of coordinate. Cubes of charge density (80 X 80 X 80 points with 40 au on each side) were calculated, and the differences with respect to the ground state were obtained. The 1 X lo4 e/bohr3contours are shown in Figure
10758 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Wiberg et al.
Vertical CIS Transition Energy
A
RC V i r t u a l Y O 4 a g ( I )
RC V i r t u a l MO 3 b l u (2)
RC V i r t u a l Y O 2 b z u ( 3 )
RC V i r t u a l YO 2 b 3 g ( 4 )
.
5
v
9z
Lrl
E? U
Figure 4. Four lowest virtual orbitals of the 2B3uu ethylene radical cation at the 0.01 e/bohr3 contour. 6 0
I
1
I
1
1
2
3
4
5
Basis Set
Vertical CIS-MP2Transition Energy
-
I
I
14 J
12
-
10
-
8-
V
From the charge density difference plots (Figure 5 ) , one can see the nature of the excited states, and they allow for a simple assignment of each state. The atomic orbitals are readily assigned under D2,, symmetry, and they are listed in Table 11. The orientation of ethylene in our calculations has the molecule in the y z plane with the z axis extending along the C=C bond. The x axis is therefore the out-of-plane axis. All of the states that terminate in a Rydberg-type atomic orbital are very diffuse. The valence states (3 and 13) are less diffuse than the Rydberg states. Comparison of states 3 (a a*)and 13 (aCH2 a*), both of which are valence excitations to the same final state, shows that the a a* state is the more diffuse one. It is also interesting to note state 10, the a 4d, Rydberg state, which is also of B1, symmetry. It is significantlymore diffuse than the a a* state. We shall return to the assignments of these states and to our choice of 4d, rather than 3d,, in a moment. To further clarify the difference between valence and Rydberg states, we have plotted only the charge density depletion regions (the dashed contours) for the vertical excitations. This is shown for the first eight excited states in Figure 6. The charge depletion regions are generally quite similar, except for the valence states (for example, the first B1,state). The depletion region for most of the states is quite similar to that for going from the '4ethylene ground state to the 2B3u (a)radical cation (Figure 6). Thus, they correspond to Rydberg states with a radical cation core and the excited electron in a diffuse orbital. The first B1, state corresponds to the a a* transition, is calculated to be intense, and is quite different in character from the other transitions. The depletion zone for the lB,, state shows significantly more structure than any of the other states (except for state 13), and it certainly is different than the radical cation of ethylene. Thus, there is significant charge reorganization of the u system in the a a* states. The structure in the depletion zone, therefore, seems to be a noticeable characteristicof the valence states and serves to separate them from the Rydberg excited states. We should note that previous worker^'^*^ have analyzed plots of $ for the primary natural orbital of the excited electron in order to assign the states, but this approach can be difficult when more than one excited configuration is involved in the final state. The analysis of the excited natural orbitals also does not help in distinguishing the characteristics of the valence and Rydberg states, as does the charge density depletion regions. From Table I, comparison of the CIS and MP2-corrected CIS (CIS-MP2) energies for the excited states of ethylene shows that the CIS-MP2 excitation energies are systematically higher than those at the CIS level. This effect was also observed by Foresman et a1.2 This behavior is opposite to that found for bicyclo[ 1.1.O]butane' and formaldehyde but similar to that observed for tr~ns-1,3-butadiene.~~ We believe that effect is due to the ability of Hartee-Fock (HF) theory to adequately describe the alkene. The ground states of compounds with strained rings or strongly polarized bonds, such as ketones, are poorly treated at the H F level, and electron correlation is very important. This effect is most evident in the calculated geometries; for example, C-O bonds are underestimated at the H F level, but geometries at the MP2
I
I
I
1
I
I
0
1
2
3
4
5
Basis Set
Figure 2. Effect of basis set on the calculated (CIS and CIS-MP2) vertical transition energies for ethylene. The basis sets are as follows: 1, 6-311G*; 2, 6-31 1+G*; 3, 6-311(2+)G*; 4, 6-311(2+,2+)G**.
O c c u p i e d Y O Ib3! ( 7 )
Occupied Y O Ib3,,
(8)
" i r t u a ! !IO l a g ( I )
V i r t u a l Y O 3 b l u (2)
V i r t u a l Y O 2bZu (3)
V i r t u a l YO Ibgg ( 4 )
Figure 3. Two highest occupied and four lowest virtual orbitals of ethylene. The ratio of the scales for the occupied and virtual orbitals was 3: 1. The contour values for the occupied and virtual orbitals are 0.10 and 0.01 e/bohr3, respectiveiy.
5 , where the solid and dashed contours correspond to charge accumulation and depletion, respectively, in the excited state with respect to the ground-state charge distribution at the same geometry.
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The Journal of Physical Chemistry, Vol. 96,No. 26, I992 10759
Electronically Excited States of Ethylene
TABLE I: Calculrted Vertical Transition Energies for Ethylene (CIS/6-311(2+,2+)C**//MP2/6-3lC*)" CIS CIS-MP2 assignment cm-l eV osc str cm-l eV state 7.13 0.0913 60.65 7.52 x 3s 57.47 1 1 B3u 65.67 8.14 7.71 0.0000 62.19 = 3PY 2 1B1, 62.39 7.74 0.5125 67.63 8.39 3 1BIu I T* 1.86 o.ooO0 65.51 8.12 = 3p2 63.37 4 B2g 8.09 0.0000 67.94 8.42 u 3Px 65.27 5 2 4 8.63 0.0061 u 3dzz 69.61 7 1.98 8.92 6 2B3u 7 1 AU x 4dyz 70.77 8.77 0.0000 72.62 9.00 8 3Bh u 3d,z + 4s 72.05 8.93 0.0108 73.71 9.14 9 7 4Py 73.29 9.09 0.0000 75.12 9.31 9.09 0.0543 75.64 9.38 x 4 x 2 73.35 10 2BI" 11 4B3u T 3dXy 73.50 9.11 0.0998 76.51 9.49 9.19 0.0000 15.43 9.35 = 4p2 74.08 12 2B2g $CHI ** 14.86 9.28 0.0000 75.11 9.31 13 3BI, 9.56 0.0000 78.66 9.75 x 4f2(xl-y1) 77.08 14 3 B2, u 5dy, 77.84 9.65 0.0000 79.30 9.83 15 2AU 81.13 10.06 0.0000 84.08 10.42 16 ~BI, u 4fyzz4-$3, 82.75 10.26 0.0000 86.61 10.74 17 4B2, T 4fz(x2-y2)
--
m,
-------- --- - -
QCHz, uCC
exptb eV 7.11 7.80 7.60 8.01 8.29 8.62 9.34 9.33 8.90 9.2
T*
0.0000 86.33 10.70 4Px 83.02 10.29 T 4d2z 84.34 10.46 0.0618 87.63 10.86 0.0000 86.91 10.78 x 4fz(xiy2) 84.47 10.47 Q C H ~QCC 4dxz "The ground-state energies (6-31 1(2+,2+)G**) are as follows: RHF, -78.055 635 hartrees; MPZ(full), -78.388 229 hartrees. bAll excitations are advert (as defined in ref 3), except for the T u* state which is vertical. Experimental energies are from: (i) Reference 3. (ii) Gedanken, A.; Kuebler, N. A,; Robin, M. B. J . Chem. Phys. 1982, 76, 46. (iii) State 13 as cited in ref l l b . 18 19 20
3 4 5B3u
x
-.
TABLE II: Symmetry of Atomic Orbitals under D2,, Symmetry sYm orbital sYm orbital A, s, d,z, dxz-yz Au fxyz BI, dxy BIU Pn fz39 fz(x2-y2) B2g dxz B2u Py9 f 22, fy(3x2-9) B3g dYZ B3u Px7 fYA fx(xZ-3y2)
level provide C-O bond lengths which are in good agreement with experimentqZ1Thus, when the H F level is adequate for the treatment of the ground state, we expect that the CIS method will also provide good agreement with the experimental information. This is the case for ethylene and the longer polyenes, as electron correlation does not have a large effect on the calculated properties of the ground states of these olefin^.'^**^ In cases where electron correlation is needed for the ground state, for example in formaldehyde, transition energies will be overestimated at the CIS level, and CIS-MP2 corrections will be needed to provide better agreement with e ~ p e r i m e n t . ~This J ~ behavior is also observed for bicycle[ 1.l.OIbutane.l Therefore, the good agreement with experiment at the CIS level seems to be due to the ability of H F theory to describe the ground state adequately. One can see remarkably good agreement with the CIS energies for all of the excited-state transition energies for which experimental data are available, including both valence and Rydberg states. Agreement is typically better than 0.2 eV at the CIS level for the different excited states.z2 The CIS-MP2 energies are consistently about 3000 cm-' (-0.4 eV) too high as compared to experiment, similar to our previous results with bicyclo[l.l.O]butane.' The vertical r r* state, in particular, is calculated as 7.74 (CIS) and 8.39 eV (CIS-MPZ), as compared to 7.60 eV experimentally.' Our excited-state assignment seems quite reasonable, and unlike previous calculations and even experiment, our charge density analysis clearly reveals the type of state. Of particular note is the r r* valence state and what we denote, for m n s to be discussed, the T 4d, Rydberg state. These states are both of BI, symmetry, and the question of Rydberg-valence mixing can be addressed. For some time there has existed a controversy, or at least an apparent inconsistency, regarding what Robin' calls Rydbergvalence conjugate states, i.e., a pair of states consisting of a valence state and the lowest Rydberg state of the sume symmetry. In our calculations on ethylene, the relevant states are states 3 and 10. However, Robin denotes these two states as r r* and r 3d, states, emphasizing that they would be significantly mixed. This
-
- -
-
-
mixing would decrease the oscillator strength of the first state and increase that of the latter, while adding diffuse character to the first and a concomitant compactness to the latter. Calculations by Buenker and Peyerimhoff (BP)16ahave been used by Robin to support this viewpoint. However, Mulliken16b-c has disputed this viewpoint, saying of the r* MO, "the lbz, MO is not quite a semi-Rydberg MO. However, it must be the precursor of a series nbzr of Rydberg M O s 4dr,, 5drx, etc. just as luu in H2 is the precursor of an npa series. BP's formulation in which the MOs T * and 3drXor the corresponding lBIustates, are regarded as independent valenceshell and Rydberg entities, is not acceptable." (A precursor of a series, according to Mulliken,16b*C is a valence orbital which has the same angular nodal structure as the orbitals in its series; see section 3 for further details.) Thus, Mulliken would insist that the d, Rydberg orbital, which in Robin's terminology is the 3d,, orbital conjugate to the r* orbital, should have one more radial node than the r* orbital and thus more properly be termed the 4d,, orbital. To explore this controversy, we wished to determine the number of radial and angular nodes for each state under discussion. Our charge density difference plots do not have the resolution in the interior region near to the nuclei because they are only a coarse grid of only 0.5 au between points. Instead, we have looked at the actual wave function for each state. Within each wave function, there is generally one primary natural orbital which has the excited electron (usually 0.9-0.95 occupancy). We have, therefore, chosen to look at the plots of # for this primary natural orbital which would best describe the excited electron for the different states of ethylene. Similar plots were shown by McMurchie and Davidson in their analysis of the V state of eth~lene.'~ In Figure 7 we have shown the contour plots of J. for the primary excited natural orbital of the configurations r 3py, u 4py, r r * ,and T 4d,, (states 2,9, 3, and 10, respectively). One can easily see that the contour plots for the 3py and 4py configurations both have one angular node but possess one and two radial nodes, respectively, as expected in a semiunited atom f o r m u l a t i ~ n .These ~ ~ orbitals have the occupied 2py ( 1b2J orbital with one angular and zero radial nodes as a precursor. Orthogonality of states 2 and 9 to their precursor requires that there be one additional radial node, and this is clearly seen in Figure 7. The contour plots for the r* and 4d,, configurations each have two angular nodes, and they also have zero and one radial node, respectively, in accord with Mulliken's expectation.
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10760 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
There is no precursor to the a * state, and the 4d,, state must orthogonalize itself with the a * state by possessing one more additional radial node. Mulliken describes the u* orbital by the linear combination'6b a(3da,)
+ b(2pa,
- 2pa,)
with b > a at the equilibrium geometry (one 2pax orbital resides on each carbon atom) and denies the existence of 'independent valence-shell and Rydberg entities". As is discussed later, this apparent controversy is, in this case at least, perhaps more a matter of notation and convention than of physical significance. To further test this idea of Rydberg-valence mixing, we have calculated many of the excited states of a "squeezed" ethylene having a central C-C bond length of 1.0349 A with all other parameters held fixed at the MP2/6-31GS optimized geometry and using the 6-311(2+,2+)G** basis set. As the molecule approaches the semiunited atom, Mulliken would predict that, in the linear combination of his description of the a * orbital, the coefficient a would increase relative to b, and the orbital would become more diffuse and Rydberg-like, specifically 3d,,-like. The energies of both this 3dx,-like state and of the 4d,, state should increase, but their difference should decrease and their quantum defects should become small and nearly equal since the now 3d,, orbital has no "precursor in the core" (Le., becomes less penetrating). The 4d,, orbital should increase in size as well, but the fractional increase should not be as great as that of the 3d,, orbital. Our charge density difference plots can help to approach this question of mixing. In our calculations, there was one complication as the two MO highest occupied M O s switched their order, and the "ICH2 (occupied MO 7 in Figure 3) became the HOMO, and there was a small energy gap between the U'C+ and a orbitals. Thus, many of the states that were generated were excitations from the dCH2 or the r MO (or from both); however, one could still assign each of the states due to its Rydberg and valence character via their charge density difference plots. In Figure 8, we have plotted the u*/d,, conjugates for some of the states of ethylene at the MP2/6-3 1G* equilibrium ground-state geometry and at the squeezed geometry. The most prominent change in the r r* state on going from the equilibrium to the "squeezed" configuration is the dramatic increase in size. The a 4d, state on the other hand shows little change in size, probably due to an insufficiency of diffuse functions in our basis set. However, it does begin to show the inner radial node and the resemblance of the inner region, including the depletion zone, to that of its precursors, the a a * state. All of this is in agreement with Mulliken's viewpoint. The expected approach toward equality (and decreasing values) of the quantum defects of both states on approaching the semiunited atom is not clearly demonstrable due to the great sensitivity of quantum defects to the value of the ionization potential used. However all reasonable methods of calculation show such a trend. For instance, if one assumes a quantum defect (6) of zero for the 4d,, state in both the equilibrium and squeezed configurations and calculates the ionization potential from the calculated energies of the 4d,, state, one can then obtain for the r* state values of n* = 2.49 (6 = 0.51) and n* = 2.66 (6 = 0.34) for the equilibrium and squeezed configurations, respectively. Figure 8 also depicts the a*/d,, conjugates that are due to MO. The squeezed d C H 2 a * state excitations from the "ICH2 appears to be very different from the equilibrium state, but the squeezed state is composed of both "ICH2 a * and a 3py excitations, both of which are of the same overall BIBsymmetry. The a 3pYexcitation is only in tpe y z plane, and therefore, the out-of-plane radial distribution is due solely to the dCH2a * excitation. Thus, with respect to the radial extension in the out-of-plane direction, the squeezed *ICH2 a * state is very similar to the equilibrium a a * state, and they each have two angular nodes. The squeezed "IcH~ 4, is likewise very similar to the equilibrium r 4d, state, and there are three nodes (two angular and one radial). There is only a slight distortion of the squeezed dcH24d, depletion zone (dashed contours) away from
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Wiberg et al.
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the radical cation's featurelessappearance, and the nodal character of the squeezed dcH2 4d,, state becomes slightly apparent. Since the energy separation for the squeezed utcH2 to its a*/dx, states (1.44 eV) is almost the same as the equilibrium r to its a*/d, states (1.45 eV), we would expect similar appearances but much different behavior for the squeezed r to its r*/d,, states since the energy separation is only 1.08 eV. This behavior is observed. Thus, the squeezed V state becomes predominantly a a 3d,, state as the valence contribution is decreased, but the radial character of the squeezed u 4dx, state is unchanged. Thus, the V state of ethylene is a linear combination of the a a * and the a 3dx, configurations, and each does not have a separate existence. The 2B,, state is truly a d Rydberg state, but due to the presence of a precursor (i.e., the r* state), it is best described as a 4dx, Rydberg state. Finally, although they are not shown, the other excited states of ethylene were similarly analyzed in order to determine the principal quantum number, and the assignments are listed in Table I.
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3. Rydberg Precursors and Rydberg-Valence Conjugates According to Mulliken,'6bsca valence orbital (excited or not) is called the precursor of a specific Rydberg orbital if they both have the same angular nodal structure. Thus, for diatomic molecules, to which these concepts have most often been applied, the orbitals must have the same value of X and the same (approximate) angular momentum 1. In the case of ethylene, the requirement of identical angular nodal structure is exemplified by the a* and 4d,, orbitals (according to Mulliken's formulation). Robin,' on the other hand, defines a Rydberg-valenceconjugate pair of orbitals as a valence orbital and the lowest lying Rydberg orbital of the same symmetry. In the case of ethylene, he has designated the a* orbital and what he denotes as the 3d,, orbital as constituting such a pair. Critical to Robin's viewpoint is the notion that since these orbitals (and states) have the same symmetry, they can be mixed. This apparent discordance of viewpoint is reminiscent of an exactly analogous situation in the case of the nd Rydberg states of NO. The latter situation is analyzed at length in a very insightful paper by J ~ n g e n . In ~ ~the case of NO, Huber and MiescherZ5have assigned a Rydberg complex as due to the u, a , and 6 states of the 3d Rydberg states on an empirical basis, as all of these levels lie within a -700 cm-' interval. However, from their assignment the 3d6 level lies nearly 700 cm-I below the 3do and 3dr although it is less penetrating (it has no precursors in the core) and therefore would be expected to lie well above the other two. In fact, the u and a components actually have small positive quantum defects, completely unexpected for penetrating Rydberg orbitals. As stated by J ~ n g e n ?this ~ apparent anomaly is due to the fact that the empirical assignment of Huber and Miescher "differs, however, from the systematic united-atom interpretation given by Mulliken, according to which the principal quantum numbers n and quantum defects 6 of the pu, du, and d a components have to be increased by one unit." The pu, du, and d r orbitals all have precursors in the core and, therefore, must have an additional radial node in order to satisfy the orthogonality requirement, In the case of ethylene, if the lowest d,, Rydberg orbital is described in first approximation by a hydrogenic 3d,, orbital, it must be made orthogonal to the a* orbital. Since the two orbitals have identical angular nodal structures, orthogonality must be ensured by the radial part of the Rydberg orbital. Following J~ngen,2~ we denote the overlap integral between the wave function &(ir*)and the hydrogenic function 4(3d,,) by S and write the orthogonalized Rydberg orbital function as @ = (1
- S2)-'/2[4(3d,,) - S+(r*)]
We note that this function now has one radial node, and if n is to have any physical meaning rather than being simply an arbitrary number used to order Rydberg states, the function CP should be denoted as 4d,, as it is in the Mulliken picture. Following nd,, orthogonalized Rydberg orbitals will have successively additional radial nodes corresponding to 5d,,, etc. Thus, the simple re-
Electronically Excited States of Ethylene
The Journal of Physical Chemistry, Vol. 96, No. 26, 1992 10761
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lBgU
A +
33 ( 1 )
lAu
A +
4dlz ( 7 )
2Blu
II +
4dxZ (10)
3Blg A'CH2
+ A'
(13)
lBln
2Au
3Ag Figure 5. Charge density differences between the ethylene singlet excited states and the ground state (1
quirement of orthogonality, without any calculation or consideration of interaction, leads to an orbital structure in accord with the Mulliken picture.
X
A +
A +
A +
II' (3)
5dlZ (15)
4 ~ 1 (18)
lo4 e/bohr3 contour).
In Robin's a p p r ~ a c hthe , ~ zero-order ?r* and 3d, orbitals are considered to interact, giving in-phase and out-of-phase linear combinations of the zero-order orbitals which have zero and one
10762 The Journal of Physical Chemistry, Vol. 96, No. 26, 1992
Wiberg et al.
