Theoretical Studies of High-Spin Organic Molecules. 1. Enhanced

33405 Talence, France, and Department of Chemistry, City UniVersity of New YorksHunter ... would have all the unpaired electrons in the same spin stat...
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J. Phys. Chem. 1996, 100, 9631-9637

9631

Theoretical Studies of High-Spin Organic Molecules. 1. Enhanced Coupling between Multiple Unpaired Electrons J. J. Dannenberg,*,† Daniel Liotard,‡ Philippe Halvick,‡ and Jean Claude Rayez‡ Laboratoire de Physicochimie The´ orique, UniVersite´ Bordeaux I, 351 Cours de la Libe´ ration, 33405 Talence, France, and Department of Chemistry, City UniVersity of New YorksHunter College and The Graduate School, 695 Park AVenue, New York, New York 10021 ReceiVed: NoVember 30, 1995; In Final Form: February 23, 1996X

Molecular orbital calculations using CASPT2 ab initio and AM1/CI methods show that polymeric hydrocarbons with multiple nonbonding π-orbitals can attain higher spin states than their smaller analogs due to (a) the quadratic increase of the quantity of exchange (Kij) terms as one unpairs the spins of additional electrons and (b) the removal of the disjoint relationship between nonbonding orbitals (NBO’s) upon low-energy excitations from the highest doubly occupied molecular orbital (HDOMO) to a NBO or LUMO. We illustrate the effect with the example of the dimethylenepolycyclobutadienes containing up to nine cyclobutadiene rings. AM1/ CI calculations (which are in good agreement with CASPT2 calculations for systems where the complete π-active space could be accommodated) predict that all dimethylenepolycyclobutadienes with three or more rings will have spin states S g (n/2) + 1, where n is the number of rings, in contrast to earlier reports that suggested that S ) 0 when n is even and S ) 1 when n is odd.

High-spin organic systems have received much recent attention, as they provide the potential components of possible organic ferromagnetic materials. Several groups have approached the prediction of the spin state of a molecule with multiple nonbonding molecular orbitals (NBO) using somewhat different approaches.1 In 1950, Longuet-Higgins suggested that alternant hydrocarbons2 (AH’s) should have at least X - 2T (where X is the number of carbons in the π-system and T is the maximum number of π-bonds in any resonance structure) nonbonding π-MO’s.3 In these MO’s the amplitude would be found exclusively on either the starred or unstarred carbons. Besides the X - 2T NBO’s, there could be additional NBO’s, which he called “supernumerary”, that did not necessarily have this characteristic. He concluded that neutral molecules of this type would have one electron in each NBO. Applying Hund’s rule, he further suggested that the ground state of such systems would have all the unpaired electrons in the same spin state. Somewhat later, Ovchinnikov4 and Borden and Davidson5 provided additional insight into understanding which molecules with degenerate NBO’s will have high or low spin. Ovchinnikov suggested that the spin state should be (starred - unstarred)/ 2. This is equivalent to suggesting that the electrons in the supernumerary NBO’s be paired. Borden and Davidson specifically considered the cases with two electrons in two singly occupied NBO’s. They suggested that the pairs of NBO’s could be classified as either “disjoint” (where the amplitudes of two NBO’s do not involve any of the same atomic orbitals), or “nondisjoint” (where they do). They suggested that the exchange integral, Kij, between the two NBO’s (i and j) would be zero for a disjoint pair. This would predict a first-order degeneracy between the singlet and triplet states. However, the singlet would be lowered relative to the triplet by interaction with other states. Thus, they would predict the same ground states as Ovchinnikov for systems with two electrons in two NBO’s. †

City University of New York. Universite´ Bordeaux I. X Abstract published in AdVance ACS Abstracts, April 15, 1996. ‡

S0022-3654(95)03552-0 CCC: $12.00

Dougherty6 has discussed the spin states of molecules of the general structure, I. He extended the reasoning of Borden and

Davidson by applying it to systems with more than two electrons in two NBO’s Pranata later has published a more general discussion of this reasoning to larger systems.7 Except for Longuet-Higgins, all parity models agree that the spin states of I with n odd should be triplets, while those with n even should be singlets. Previous calculations agree with these predictions for the cases of n ) 1 and n ) 2.6c Much of the reasoning behind the prediction of the spin states derives from the study of the individual exchange term, Kij, between two unpaired electrons in degenerate orbitals. To the best of our knowledge, the study of the global effect of the Kij’s in a system containing many unpaired electrons in degenerate (or nearly so) orbitals has not been considered for organic π-systems. In this paper we illustrate that (1) the quantity of Kij terms increases greatly with the number of unpaired electrons; (2) excitations from highest doubly occupied molecular orbitals (HDOMO’s) to NBO’s or LUMO’s can destroy the disjoint relationships between singly occupied molecular orbitals (SOMO’s); and (3) as one increases the number of unpaired electrons in SOMO’s, the effect of the increased quantity of Kij’s might eventually overcome the arguments made for systems with two electrons in NBO’s. We illustrate this with MCSCF ab initio methods for n ) 1 to 4; and AM1/CI calculations on structures of the type I, with n ) 1-9. occ

a+b a+b

Etot ) ∑ ei + ∑ i

i

a

a

b

b

i

j

i

j

∑Jij - ∑∑KijR - ∑∑Kijβ j

© 1996 American Chemical Society

(1)

9632 J. Phys. Chem., Vol. 100, No. 23, 1996 Following the general outline of the unrestricted HartreeFock development8 leads to eq 1, where Etot is the total electronic energy, ei the energy of orbital i, Jij and Kij the Coulomb and exchange integrals between molecular orbitals i and j; a and b are the numbers of R and β electrons, respectively. While the Coulomb term, Jij, is operative between orbitals with electrons of both spins, the exchange term, Kij, can only be nonzero between electrons with the same spin. The relative stabilities of the different spin states of a molecule can be related to the stabilizing exchange interactions. Thus, Borden and Davidson5 have discussed the relative energies of singlet and triplet states of organic molecules with delocalized π-systems in terms of the magnitude of the exchange term, Kij, in the triplet.5 They have argued that the exchange term should be greater between those orbitals where the electron density can be localized on the same atoms (nondisjoint) than for orbitals where the electron density can be separated so that there is little or none on the corresponding atoms for MO’s i and j. For systems that may have more than two unpaired electrons, there will be several more Kij terms between the R-electrons than the β (where we assume that the number of R > β electrons). To analyze the structural effect(s) upon the relative energies of various spin states, we will need to consider both the magnitudes and the quantities of the various terms that contribute to the energies of these species. The quantity of exchange terms between electrons in singly occupied molecular orbitals (SOMO’s)9 will increase as more spins become unpaired. One can easily show that the number of Kij terms increases by m2/4 (when m is even) or (m + 1)(m - 1)/4 (when m is odd) upon unpairing m electrons. Thus, going from a singlet to a triplet (unpairing two electrons) increases the number of Kij’s by one, but unpairing 10 electrons to go from a singlet to a state of S ) 5 increases the number by 25. The additional 25 Kij’s represent the 45 Kij’s between 10 electrons of the same spin for the S ) 5 state, less twice 10 Kij’s for the five electrons of each spin in the singlet. As the number of exchange terms increases quadratically, one might expect the higher spin states to become relatively more stable as the number of SOMO’s increases. While we shall illustrate that this is indeed the case, the demonstration is not that simple, as one must also consider the values of the Kij terms. The MO’s of any molecule can be transformed by unitary transformations between sets of MO’s with the same orbital occupation (0, 1, or 2) without affecting the molecule’s total wave function or its properties. As the result of such a transformation, the one-electron orbitals change; therefore, the Jij’s and Kij’s must also change. The total energy is invariant, as are the individual aggregate energies of the doubly occupied and singly occupied orbitals. Thus, the appropriate sums over Jij - Kij’s between the singly and doubly occupied orbitals are invariant. However, the individual terms are not invariant between orbitals with the same occupation, as these summations compensate for the changes in the orbital energies keeping the total energies constant with respect to any unitary transformation. Cyclobutadiene provides a simple example. One can define cyclobutadiene in the two different orientations in Cartesian space give in Figure 1. If one performs an AM1

Dannenberg et al. calculation on the same fixed geometry (square, C-C and C-H distances 1.50 and 1.10 Å, respectively) one obtains values of Jij of 7.0167 and 7.1727 eV and values of Kij of 0.0060 and 0.1620 eV for the orientations defined by A and B (Figure 1) respectively. Thus, Jij - Kij remains constant at 7.0107 eV while the individual components change. The heats of formation are the same to 10-5 kcal/mol. It is evident from this example that one cannot simply consider the sum of the K’s, but must rather consider the sum of J - K’s. However, let us consider the case of a molecule of (a) fixed geometry and (b) fixed in a defined coordinate system. In this case, there will be no changes in the individual K and J terms as one changes the individual spins of the electrons in the SOMO’s. Thus, the changes in the energies will result from the changes in the indices of the summation over Jij - Kij, as all the individual terms will remain constant. From consideration of this case, together with the observation that the quantity of K terms increases quadratically with the number of unpaired electrons, one sees that the higher spin states could become relatively more stable as n increases. Consideration of the arguments made by Ovchinnikov,4 Borden and Davidson,5 and Dougherty6 (who also did PariserParr-Pople (PPP)) calculations6c all lead to the conclusion that I should be a singlet for even and triplet for odd n. We show here that spins as high as and higher than n + 1/2 can become the ground states when n is odd, and n/2 or higher when n is even. Calculational Methods Semiempirical Calculations. The present calculations use the AM110 approximation to MO theory with extensive configuration interaction (CI). The AMPAC 4.511 program was modified to allow for more extensive CI. The modifications are described later in this section. The calculations use the “OPEN” option of the AMPAC program, which solves the SCF equations using the half-electron method,12 for p electrons in p orbitals (in all the cases studied here, there are an equal quantity of SOMO’s and electrons occupying them). A natural approach to CI could be performed using the full configuration active space (full CI) for these m electrons in p orbitals. Such a calculation artificially favors the low-spin states. For example, for p ) 6, there are 175 singlet, 189 triplet, 35 quintet, but only one septet state (S ) 3) possible. Thus, the singlet and triplet are stabilized by many interactions, the quintet by several, but the septet cannot be stabilized by any. In order to relieve this artifactual stabilization of the lower spin states, we included additional configurations in the CI. As the number of possible configurations rapidly becomes intractable, we must choose a criterion for selecting the configurations that are to be retained for calculation. The AMPAC program first calculates the Epstein-Nesbet (EN) energies,13 〈∆iH∆i〉, and retains several (usually 7) of the lowest determinants, ∆i. If there are other determinants which are degenerate with any of this set, it also retains them. The resulting target group of determinants will then be g7. Next, the (EN) perturbation interactions, between the target group and all other possible determinants, is calculated. A criterion described by

〈∆iH∆k〉

∑ | | i)1 〈∆ H∆ 〉 - 〈∆ H∆ 〉 n

i

Figure 1. Two possible orientations of square cyclobutadiene in Cartesian space.

i

k

k

where the subscript i denotes the target group, is then used to choose the other determinants, ∆k’s that interact most strongly

High-Spin Organic Molecules with the target group. Determinants are chosen taking care to either include or eliminate all determinants that have MøllerPlesset (MP) energies,14 〈∆iF∆i〉, that are degenerate (within 10-4 eV). If there is not enough room among the 676 determinants retained for consideration for all of them, the complete set of degenerate determinants is deleted from the calculation. Elimination of some determinants required for a spin state can lead to incorrect calculation of the state. This process accounts indifferently for spin or space degeneracies but does not yield a well-balanced basis set of configurations. In order to obtain a balanced comparison of the energies of the various states within limitations discussed above, we calculated each molecule using configurations constructed from linear combinations of determinants of each spin projection from Sz ) 0 to the highest value of Sz appropriate to that spin state. For example, the triplet, Rβ + βR, rather than RR or ββ, would be constructed from determinants with Sz ) 0. States of higher spin than the value of Sz used for the individual determinants can be constructed from them, but not states with lower spin. Thus, the S ) 5 state can be constructed from Sz ) 1 states, but the S ) 0 state cannot be. The result is that as one uses determinants with increasing Sz to construct the spin states of interest, one obtains more determinants that interact with the higher spin states for two reasons: (a) one eliminates the lower spin states from consideration, and (b) the higher spin states are constructed from linear combinations of fewer individual determinants, thus allowing more configurations to be constructed from the 676 determinants retained in the calculation. Eventually, the highest spin state is calculated only from determinants of its own maximum Sz. While this procedure relieves some bias toward stabilization of the low-spin states, it may introduce a different bias in the other direction. For example, many of the determinants necessary for a good description of the singlet state may be eliminated from consideration if higher spin states are more stable. In this situation, the target group of determinants may belong to these higher spin states. The full complement of determinants would be constructed on the basis of interactions with this group. The singlet states constructed from these determinants would be linear combinations of only those retained on the basis of interaction with the higher spin states. Thus, determinants that lower the singlet states might be unavailable. The AMPAC 4.5 program was modified in several ways: (a) the number of orbitals that could be included in the CI was increased from 10 to 26; (b) the number of determinants considered in the selection procedure was increased from ∼2000 to ∼35 000; (c) if the number of determinants in the full CI would be greater than 35 000, then the generation is confined to single, double, triple, and quadruple excitations. These would be progressively reduced to single, double, and triple, then single and double, etc., as space requirements dictate; (d) the number of determinants retained in the final calculation was increased from 100 to 676. In addition, a test was introduced to verify that the spin states were not contaminated. This was accomplished by verifying that S2 be correct to within 0.001. With nearly degenerate NBO’s, spin contamination could be introduced by lack of precision in calculating the MP energies, artifactually placing one or more beyond the 10-4 eV criterion used for retaining all determinants of each configuration. Other new capabilities, which are not essential for the calculations presented here, include the ability to include only single or single and double excitations; the ability to choose which orbitals will be used in the CI; and the ability to conserve the same configurations for different calculations.

J. Phys. Chem., Vol. 100, No. 23, 1996 9633 The last two modifications can be important when comparing different geometries or optimizing a molecule, since the components of the CI calculations may change as the geometry varies. The calculations reported include the lowest energy of each spin state as calculated from individual linear combinations of determinants for each possible Sz up to the maximum for the state in question. Usually, this energy was equivalent to that of the state constructed from determinants each having the maximum Sz for the state in question. The geometries of the molecules were kept idealized with all C-C and C-H distances were kept equivalent, at 1.4254 and 1.0824 Å, and angles of 135°, 120°, and 90°. This allowed us to consider the purely electronic effects independently of the geometry optimization for each of the individual states of each molecule I (n ) 1-9). Ab Initio Calculations. Ab initio CASSCF and CASPT2 calculations15 were performed using the MOLCAS program16 with the D95 and 6-31G basis sets. In these calculations multiconfigurational SCF calculations were performed using excitations to the full π-active space (FP) where possible, or restricted to part of the π-active space (RP) for some of the larger systems,17 followed by second-order perturbation treatment of the multireference wave function. A similar procedure was used by Borden to evaluate the spin states of 1,2,4,5tetramethylenebenzene.18 For I (n ) 1-2), full π-active space (FP) calculations using 6 or 10 electrons in 6 or 10 orbitals was used. For I (n ) 3 or 4), the restricted π-active space (RP) calculations were performed using 12 electrons in 12 orbitals (the practical limit of our computer resource). The CASPT2 many-body perturbation implementation was written for systems where the CASSCF is assumed to have eliminated near degeneracies. While this is not always the case for the present calculations, the report by Borden suggests that calculations of this type might be useful. Nevertheless, we performed the same kind of calculations for I (n ) 4) with 4 electrons in 4 MO’s, followed by the second-order perturbation treatment. The CASSCF calculations with 4 electrons in 4 MO’s gave higher energies than the corresponding calculations for 12 electrons in 12 MO’s as expected. However, the total energies after the second-order perturbation were lower for 4 electrons in 4 MO’s than for the corresponding calculation with 12 electrons in 12 MO’s. The reason for this discrepancy is not entirely clear but may be due to the near degeneracy of the states. Since these energies are lower than the single configuration energies, the perturbational stabilizations for each configuration become smaller as the active space increases. This result suggests that the perturbation results might be unreliable for near degenerate spin states of the same system, as reported for CH2.15a They are nevertheless included for completeness. Results and Discussion Tables 1-4 provide details of the calculations, while Table 5 summarizes the ab initio results for I (n ) 1-4). The values cited in the text are for the D95 basis set after CASPT2, unless otherwise noted. For n ) 1, the 3B2u triplet is about 16 kcal/ mol more stable than the 1B2u singlet, similar to the 18.2 kcal difference obtained by Borden.19 In contrast, for n ) 2, the 1A singlet is about 2 kcal/mol more stable than the 3B triplet. g 3u In both cases the quintet states are much more (100 and 38 kcal/mol) energetic. When n ) 3, the 3B2u triplet is again the ground state, but the 5B2u quintet is only about 4 kcal/mol higher. The 1B2u singlet is about 23 kcal/mol higher than the triplet. For n ) 4, our best calculation predicts that the 1Ag singlet to be slightly lower than the 3B3u triplet by about 2 kcal/mol, but

9634 J. Phys. Chem., Vol. 100, No. 23, 1996

Dannenberg et al.

TABLE 1: D95/CASSCF Results for I (n ) 1)a con- detersym figurn minants 1A

g 1B 2u 3A g 3 B2u 5A g 5B 2u

56 39 42 51 6 14

104 104 48 65 6 14

CASSCF

Erel

CASPT2

TABLE 4: D95/CASSCF Results for I (n ) 4)a Erel

-230.497 820 31.99 -231.004 216 28.10 -230.510 551 24.00 -231.023 130 16.24 -230.347 940 126.04 -230.851 082 124.20 -230.548 802 0.00 -231.049 004 0.00 -230.219 121 206.87 -230.699 172 219.52 -230.396 310 95.69 -230.889 947 99.81

a Complete π-active space of six electrons in six orbitals was used with the D95 bsais set. Configurations and determinants refer to the quantity of each used in each symmetry. CASSCF and CASPT2 energies in hartrees, Erel in kcal/mol.

TABLE 2: D95/CASSCF Results for I (n ) 2)a condetersym figurn minants 1A g 1B 3u 3A g 3 B3u 5A g 5B 3u

5048 4948 7408 7508 3080 3055

16072 16072 11024 11124 3616 3616

CASSCF

Erel

CASPT2

Erel

-383.099 273 -382.977 468 -383.014 044 -383.094 073 -383.038 184 -382.980 141

0.00 76.43 53.48 3.26 38.33 74.76

-383.926 781 -383.807 099 -383.847 281 -383.922 998 -383.866 935 -383.814 967

0.00 75.10 49.89 2.37 37.55 70.16

a Complete π-active space of 10 electrons in 10 orbitals was used with the D95 basis set. Configurations and determinants refer to the quantity of each used in each symmetry. CASSCF and CASPT2 energies in hartrees, Erel in kcal/mol.

TABLE 3: D95/CASSCF Results for I (n ) 3)a condetersym figurn minants 1A g 1B 2u 3A g 3B 2u 5 Ag 5 B2u

4960 4812 7352 7468 3100 3115

15912 15912 10952 11100 3600 3632

CASSCF

Erel

CASPT2

Erel

-535.486 775 -535.500 324 -535.480 522 -535.549 189 -535.428 421 -535.534 284

39.16 30.66 43.09 0.00 75.78 9.35

-536.713 484 -536.728 407 -536.708 572 -536.765 464 -536.677 315 -536.759 542

32.62 23.25 35.70 0.00 55.31 3.72

a Restricted π-active space of 10 electrons in 10 orbitals was used with the D95 bsais set. Configurations and determinants refer to the quantity of each used in each symmetry. CASSCF and CASPT2 energies in hartrees, Erel in kcal/mol.

the 5Ag quintet is only 9 kcal/mol above the singlet. In fact, the 7B3u septet is only 16 kcal above the singlet (40 kcal/mol, before CASPT2). For the case of n ) 4, where the limitations of the MOLCAS program effectively prevent us from using the full π-active space, we performed the CASSCF calculations in three different ways to explore the effect of the different sized active spaces. For four electrons in four orbitals (the NBO’s) we obtained roughly degenerate spin states with the singlet < triplet < quintet. Upon increasing the active space to 10 electrons in 10 orbitals, the energy differences between the spin states increase. However, upon further increasing the active space to 12 electrons in 12 orbitals, these energy differences decrease. If we consider the first calculation (4 electrons in 4 orbitals), we can construct 12 1Ag and 8 1B3u singlets, and 7 3Ag and 8 3B 5 3u triplets, but only one Ag quintet. The numerical results of Table 4 suggest that upon increasing the active space to 10 electrons in 10 orbitals, there are more stabilizing interactions with the lower spin states but that this tendency might be reversed by further increasing the active space to 12 (or more) electrons in 12 (or more) orbitals. It is significant that the CASPT2 calculations with 4 electrons in 4 orbitals gave lower total energies than the CASPT2 calculations with larger active spaces, as noted above. As noted by Dougherty,6c one can transform the NBO’s of I (n even) into orbitals localized on each cyclobutadienyl ring.

con- detersym figurn minants 1A

12 8 7 8 1

g

1B 3u 3 Ag 3B 3u 5A g

1

4936 4836 7360 7460 3130 3095 470 495

1

57008 56608 95391 95616 49305 49080 11028 11064

Ag 1B 3u 3 Ag 3B 3u 5A g 5 B3u 7A g 7B 3u Ag 1 B3u 3 Ag 3B 3u 5A g 5B 3u 7 Ag 7B 3u

CASSCF

Erel

4 Electrons in 4 Orbitals -687.884 732 0.00 -687.699 878 116.00 -687.884 199 0.33 -687.884 514 0.14 -687.883 838 0.56

20 16 8 8 1

CASPT2

Erel

-689.652 475 0.00 -689.519 589 83.4 -689.647 583 3.07 -689.647 503 3.12 -689.642 611 6.19

10 Electrons on 10 Orbitals 15912 -688.010 444 0.00 -689.643 348 15912 -687.926 439 52.71 -689.565 965 10976 -687.997 224 8.30 -689.632 095 11076 -688.004 078 3.99 -689.637 929 3616 -687.994 356 10.10 -689.629 582 3616 -687.958 104 32.84 -689.603 000 496 -687.894 378 72.83 -689.583 972 521 -687.951 196 37.18 -689.602 218

0.00 48.56 7.06 3.40 8.64 25.32 37.26 25.81

12 Electrons in 12 Orbitals -688.047 281 0.00 -689.629 836 -687.941 373 66.46 -689.573 557 -688.037 544 6.11 -689.614 865 -688.043 746 2.22 -689.627 247 -688.033 740 8.50 -689.614 771 -687.970 527 48.16 -689.610 120 -687.939 258 67.78 -689.568 135 -687.983 910 39.77 -689.604 544

0.00 35.32 9.39 1.62 9.45 12.37 38.72 15.87

213840 213440 156831 156832 61441 61216 12136 12136

a Reduced π-active space was used with the D95 bsais set. Active spaces of 4, 10, and 12 electrons in the same number of orbitals were used. Configurations and determinants refer to the quantity of each used in each symmetry. CASSCF and CASPT2 energies in hartrees, Erel in kcal/mol.

TABLE 5: Relative Energies (kcal/mol) of the Lowest Energies of Various Spin States of I Using CASSCF and CASSCF Calculations with and without Second-Order Perturbation Interactions (CASPT2) CASSCF spin n

CI

0

1

2

1 2 3 4

6 10 10 12

24.0 0 30.7 0

0 3.3 0 2.2

95.7 38.3 9.35 8.5

CASPT2 spin 3

0

1

2

3

39.8

16.2 0 23.3 0

0 2.4 0 1.6

99.8 37.6 3.7 9.5

15.9

These orbitals are degenerate in π-MO theory and nearly so using more sophisticated MO methods. As these orbitals are completely disjoint, all the possible K terms between electrons in these orbitals will be zero (in π-MO) theory or nearly so (using more sophisticated methods). Thus, one would expect the possible spin states (singlet, triplet, and quintet) to be degenerate in a first approximation. This is what we obtain in the 4 electron in 4 MO CASSCF calculation. Upon increasing the active space, the 1Ag singlet benefits most from simple configuration interactions to become the favored state. Let us continue with the example of I (n ) 4). If one excites an electron from the highest doubly occupied orbital (HDOMO) to a NBO that is initially a singly occupied orbital (SOMO), one now has a new set of 4 SOMO’s which cannot be transformed into a set where the K’s are zero (or nearly so) as the HDOMO has density at each carbon (see Figure 3). Thus, if the excitation required is sufficiently low, the stabilization provided by the new set of 4 (nonzero) exchange interactions created upon unpairing the four electrons can cause the excited quintet to be significantly lowered. Following similar logic, exciting an electron from the HDOMO to the LUMO would provide 6 SOMO’s that are no longer disjoint. The 9 new (nonzero) exchange interactions would have to counteract an excitation that is somewhat higher than that of the first example

High-Spin Organic Molecules

J. Phys. Chem., Vol. 100, No. 23, 1996 9635

TABLE 6: Relative AM1 Enthalpies (kcal/mol) for Different Spin States of I for n ) 1, 9a spin n

s

CI

0

1

1 2 3 4 5 6 7

2 2 4 6 8 8 8

12 12 12 14 14 14 14

19.9 0 5.7 15.2b 23.0c 27.1d 19.0

0 4.2 5.7 7.3 17.3 16.8 19.0

2

0 4.3 12.2 15.5 17.4

3

0 0 10.8 9.9

4

12.1 0 0

5

25.7 11.4

6

7

∆Hf 106.4 213.5 316.1 421.7 532.4 656.5 753.6

a The lowest energy obtained from all calculations for each spin state is entered (see text for discussion). Except as noted, s represents the number of singly occupied orbitals used in the SCF calculation; CI represents the number of orbitals in the CI active space, ∆Hf refers to the lowest energy spin state. The numbers of doubly occupied and virtual orbitals are always equal. b 4 SOMO’s. c 6 SOMO’s. d 10 SOMO’s.

(roughly double if one invokes the pairing rule for alternant hydrocarbons). Thus, one might expect the septet to be similarly lowered. These “excited” states will only be stabilized by interactions with configurations that can be conceived as excitations of themselves; thus a large active space is required to properly describe them. The observations that for I (n ) 4) the 5B3u quintet becomes almost as stable as the 1Ag, and that the 7B3u septet becomes relatively stable, upon increasing CI illustrate this point. Obvious questions that arise are (1) would one of the highspin states (quintet and septet) become the ground state if one could do a more exact calculations (better basis and full CI)?; and (2) if the high-spin states do not prevail for I (n ) 4), would they eventually do so if one examined larger molecules in the series (n > 4), where more exchange terms will be created and the necessary excitations will be lower? The combined limitations of the MOLCAS program and our computational facilities prevent us from addressing both questions using this methodology. Moving to the AM1 semiempirical method allows us to address the second question. The AM1 energies of the various spin states for molecules of structure I (where n varies from 1 to 9) are presented in Table 6. For I n ) 1 and 2, the results are in very good accord with the CASSCF calculations which used a complete π-active space, as well as with previous work.6c For the n ) 3 and 4, AM1 predicts increasing stabilization of the higher spin states, paralleling the trends of the ab initio CASSCF calculations discussed above. However, AM1 predicts the high-spin states to be relatively more stabilized than the CASSCF calculations, where the active spaces were truncated due to size. Thus, AM1 predicts the quintet and the septet to be the ground states for n ) 3 and 4, respectively. There is a natural tendency to suspect that the ab initio calculations are more accurate; however, many examples exist in the literature where AM1 calculations are clearly preferable to ab initio calculations with such small basis sets.20 The important feature of the comparison is that both predict the same trends: relative lowering of the energies of the higher spin states. One should remember that neither the AM1 nor MOLCAS calculations use the full π-active space for I (n ) 3 or 4). MOLCAS uses all the possible configurations for a given spin state in a given symmetry that can be constructed from the designated CI. AMPAC truncates the number of determinants that can be constructed from a given Sz. Thus, when the active space is truncated, MOLCAS will generally favor a low-spin state since there will be relatively fewer important interactions with the high (rather than low) spin states. On the other hand, AMPAC can include the full π-active space for some high-spin states when Sz is at its maximum value for

Figure 2. HDOMO, NBO’s, and LUMO for I (n ) 3) viewed from the top of the π-orbitals. The relative sizes of the lobes are approximate.

that spin. However, AMPAC cannot do so for low-spin states as the higher spin states are not excluded when a low Sz is used. Thus, it is possible that AMPAC might favor the higher spin states when one uses many MO’s in the CI. To minimize this problem, we generally used several different levels of CI and all possible values of Sz for the AM1 calculations. We chose the lowest energy result obtained for each spin state. Thus the observations that AM1 in AMPAC predicts greater stability for the higher spin states for I (n ) 3 and 4) than does MOLCAS, while the two give very similar results for I (n ) 1 and 2) are consistent with the likely direction of the errors induced in each method due to the truncation of the π-active space. The molecules I (n ) 5-9) could only be studied using AMPAC/AM1. The results are summarized in Table 6. All of these molecules with n odd are predicted to have ground states with S g (n + 1)/2. When n ) 5 and 7, all have S ) (n + 1)/2. However, when n ) 9, the ground state is predicted to have S ) 6, which is (n + 1)/2 + 1. This requires that two electrons in what previously had been the HDOMO become unpaired, presumably due to the added stability of the extra 10 Kij terms that stabilize the system when one of these electrons is excited to the LUMO. From these results, one might anticipate that spin states even higher than (n + 1)/2 + 1 might become the ground state for larger molecules of type I. In all the cases for n odd, the energy decreases as the spin is increased from 0 to (n + 1)/2. For the cases where n is even, this progression is not so smooth. Nevertheless, except for the case where n ) 2 (which is a singlet), all cases for n even have ground states with S ) (n/2) + 1. Let us consider the nature of the NBO’s involved in these molecules. In the tradition of π-MO theory, all of those MO’s which have a node through the longitudinal axis of the molecule will be π-NBO’s. In addition, the n ) odd molecules will have an additional NBO that can be imagined by placing nodes perpendicular to the longitudinal axis through each starred atom as defined for AH’s. Using Dougherty’s enhancement of the terminology of Borden and Davidson, one sees that each of the NBO’s with the longitudinal node can be localized in a single ring (illustrated in Figure 2 for n ) 3). Thus, they (NBO’s 2-4) are disjoint with respect to each other. The NBO which lacks the longitudinal node (NBO 1) will necessarily be nondisjoint with the orbitals localized on every other ring starting with that nearest an end (NBO’s 2 and 4), and disjoint with the others (NBO 3). In the case of I (n ) 3), this leads to

9636 J. Phys. Chem., Vol. 100, No. 23, 1996

Figure 3. HDOMO, NBO’s, and LUMO for I (n ) 4) viewed from the top of the π-orbitals. The relative sizes of the lobes are approximate.

three NBO’s that always have nonzero K’s between them (NBO’s 1, 2, and 4 of Figure 2), and one NBO (NBO 3) whose K’s can be made to be zero with the others. Thus, the electrons in NBO’s 1, 2, and 4 will have the same spin, while that in NBO 3 can have either spin, leading to a degenerate triplet and quintet (at the first approximation). Additional interactions would be expected to favor the lower spin (triplet) state. However, an excitation from the HDOMO to a NBO or the LUMO will render all the resulting SOMO’s nondisjoint, favoring the high-spin states, as mentioned above. In such a case, the quantity of K’s does not increase, as the number of unpaired electrons does not change. The increased stabilization comes from the increase in the magnitude of the K’s that occurs when there are no longer any SOMO’s that are disjoint with all the others. When n ) 9, AM1 predicts the ground state to have spin of 6. In this case, the excitation obtained from promoting an electron from the HDOMO to the LUMO unpairs two additional electrons to form 12 instead of 10 SOMO’s is overcome by the additional 11 Kij’s as well as the disappearance of the disjoint relationship between some SOMO’s and all the others. The situation for molecules with n even is slightly different (see Figure 3). In principle, all the NBO’s could be written as localized on the two radical sites of one of the rings. These orbitals would have no exchange interaction, leaving all possible spin states containing one electron per NBO degenerate before CI, as in cyclobutadiene. The HDOMO is bonding at every second double bond, with nodes between the radical sites of the rings and every bonding double bond position. The LUMO (the pair of the HDOMO in the AH approximation) is similar except its bonding and antibonding positions are reversed. As previously indicated by Dougherty,6c upon increasing n, the difference in energy between the HDOMO and the LUMO progressively decreases. The energy that corresponds to promoting an electron form the HDOMO to the LUMO corresponds to the energy gap between the orbitals less any stabilization that might be obtained from unpairing the spins of the electrons originally in the HDOMO. Since the HDOMO and LUMO are paired in the AH approximation, the magnitude of the coefficients at each position will be the same (although their signs will not be), assuring a significant Kij between them. As n increases, the stabilization gained from unpairing the electrons can overcome the “excitation” energy of promoting an electron from the HDOMO to the LUMO. The AMPAC data of Table 6 suggest that this happens when n(even) g 4. The CASSCF data of Table 4 suggest that this may happen

Dannenberg et al. when n > 4. As the HDOMO and LUMO will have small but significant Kij’s with the other NBO’s, they provide a mechanism for coupling of the other NBO’s, creating a high-spin system. Since the orbital coefficients have small values at the radical sites of the rings, the exchange coupling remains weaker than for the cases of n odd. Let us consider the specific case of I (n ) 4). An openshell calculation with four electrons in the four NBO’s without any further CI (full CI within the active space of the four NBO’s is automatically included), the states of S ) 0 to S ) 2 are degenerate to less than 0.6 and 0.1 kcal/mol by AM1 and CASSCF, respectively (see Tables 4 and 6). These results are consistent with the Borden/Davidson analysis. If one modifies the AM1 calculation to use six electrons in six orbitals (the SCF state would now correspond to that obtained by exciting one electron from the HDOMO to the LUMO), one obtains approximately degenerate (∆∆H e 0.8 kcal/mol by AM1) states of S ) 0 to S ) 3, all of which are lower in energy than the lowest obtained from the previous calculation (with four unpaired electrons). In particular, the septet (S ) 3) is more stable than the singlet constructed from the calculation with four electrons in four orbitals. Thus, the energy required to excite an electron from the HDOMO to the LUMO is more than compensated for by the other factors discussed above. The HDOMO and LUMO are illustrated in Figure 3. Following the pairing rule for AH’s, they differ in that the HDOMO is symmetric and the LUMO antisymmetric with respect to a plane perpendicular to that of the molecule that bisects the central double bond. Inspection of the orbitals indicates that the bonding and antibonding interactions all approximately cancel except at the central bond. Thus, the energy difference between the two orbitals arises from the bonding vs antibonding nature of the interaction between the two central carbons. In this molecule the coefficients of the central bond are approximately (0.3, leading to an energy gap of about 0.2 π-bonds per excitation. If one extrapolates to large n even structures of I, the coefficients on the central carbons will become increasingly small until the gap becomes negligible. Eventually, these orbitals will become energetically indistinguishable from the NBO’s. We shall call them proto-nonbonding (PNBO). By the Borden/Davidson approach, the set of NBO’s and PNBO’s orbitals cannot be converted by unitary transformation into a set which are completely disjoint, opening the possibility of lifting the first-order degeneracy of the states. Examination of the CASSCF data for I (n ) 4) leads to similar conclusions (Table 4). The 4 electron in 4 orbital calculation has the singlet, triplet, and quintet degenerate within 0.6 kcal/mol, as previously mentioned. Increasing the active space to 10 electrons in 10 orbitals favors the singlet, triplet, and quintet states in that order, as seen by the number of configurations used for each. This results in the triplet and quintet being raised to 4.0 and 10.1 kcal/mol above the singlet, respectively. Further increasing the active space to 12 electrons in 12 orbitals increases the number of configurations used in the triplet and quintet states relative to the singlet. As a result, the triplet and quintet descend to 2.2 and 8.5 kcal/mol above the singlet. We must consider why our treatment leads to different predictions from those of Ochinnikov and Dougherty. The proof offered by Ochinnikov actually was that S g (starred unstarred).4 He assumed that the “>” did not correspond to a physical reality, so he simplified his expression to S ) (starred - unstarred). Thus, there is no conflict with Ochinnikov’s proof, only with his simplification of it.

High-Spin Organic Molecules Dougherty’s conclusions6c are based largely upon his PPP/ CI calculations. As he noted, these calculations only used configurations that are based upon single and double excitations of SCF state that led to no more than four unpaired electrons. This restriction on the choice of CI has the artifactual effect of lowering the relative energies of the lower spin states. In conclusion, the calculations presented here suggest that it may be possible to obtain high-spin molecules that were previously thought to be inaccessible. This is due to (1) the relatively large number of K’s created upon unpairing many electrons; and (2) the eventual excitation from HDOMO’s to NBO’s and LUMO’s which can destroy the disjoint relationship between NBO’s. The coupling between chemical motifs containing unpaired spins that are weakly ferromagnetically coupled may become more robust as their number (therefore the quantity of K’s between them) increases. This may overcome the necessity to use only “robust” coupling interactions in large multispin systems. Current strategies for the synthesis of ferromagnetic organic materials might bear reconsideration in the light of these results. Acknowledgment. This work was supported, in part, by a grant from the research foundation of CUNY. J.J.D. thanks the Ministe`re de la Recherche et de la Technologie for support during a sabbatical leave at Bordeaux. References and Notes (1) For reviews see: (a) Dougherty, D. A. Acc. Chem. Res. 1991, 24, 88. (b) Kollmar, C.; Kahn, O. Acc. Chem. Res. 1993, 26, 259. (c) Iwamura, H.; Koga, N. Acc. Chem. Res. 1993, 26, 346. (d) Borden, W. T.; Iwamura, H.; Berson, J. A. Acc. Chem. Res. 1994, 27, 109. (2) For a discussion of the properties of alternant hydrocarbons, see: Salem, L. Molecular Orbital Theory of Conjugated Systems; Benjamin: New York, 1966; p 36.

J. Phys. Chem., Vol. 100, No. 23, 1996 9637 (3) Longuet-Higgins, H. C. J. Chem. Phys. 1950, 18, 265. (4) Ovchinnikov, A. A. Theor. Chim. Acta 1978, 47, 297. (5) Borden, W. T.; Davidson, E. R. J. Am. Chem. Soc. 1977, 99, 4587. (6) (a) Dougherty, D. A. Acc. Chem. Res. 1991, 24, 88. (b) Pranata, J.; Marudarajan, V. S.; Dougherty, D. A. J. Am. Chem. Soc. 1989, 111, 2026. (c) Pranata, J.; Dougherty, D. A. J. Am. Chem. Soc. 1987, 109, 1621. (7) Pranata, J. J. Am. Chem. Soc. 1992, 114, 10537. (8) Pople, J. A.; Nesbet, R. K. J. Chem. Phys. 1954, 22, 571. Berthier, G. Molecular Orbitals in Chemistry, Physics and Biology; Lowdin, P. O., Pullman, B., Eds.; Academic Press: New York, 1964; p 57. (9) We make the distinction between NBO’s and SOMO’s for two reasons: (a) the NBO’s may be singly, doubly or unoccupied; and (b) some of the normally bonding and antibonding orbitals may be singly occupied due to increased exchange stabilization. (10) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902. (11) SemiChem, Inc., Shawnee, KS. (12) Dewar, M. J. S.; Hashmall, J. A.; Venier, C. G. J. Am. Chem. Soc. 1968, 90, 1953. (13) (a) Epstein, P. S. Phys. ReV. 1926, 28, 695. (b) Nesbet, R. K. Proc. R. Soc. (London) 1955, A230, 312. (c) Proc. R. Soc. (London) 1955, A230, 322. (d) Diner, S.; Malrieu, J. P.; Claverie, P. Theor. Chim. Acta 1969, 13, 1. (14) Møller, C.; Plesset, M. S. Phys. ReV. 1934, 618. (15) (a) Andersson, K.; Malmqvist, P.-A.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. J. Phys. Chem. 1990, 94, 5483. (b) Andersson, K.; Malmqvist, P. A.; Roos, B. O. J. Chem. Phys. 1992, 96, 1218. Andersson, K.; Roos, B. O. Int. J. Quantum Chem. 1993, 45, 591. (16) MOLCAS, version 2. Andersson, K.; Fu¨lscher, M. P.; Lindh, R.; Malmqvist, P-A° .; Olsen, J.; Roos, B. O.; Sadlej, A. J.; Widmark, P-O., Department of Theoretical Chemistry, University of Lund, Sweden. (17) We make the distinction between the full π-active space (FP) when the entire π-active space is used, and restricted π-active space (RP) when the π-active space is truncated due to its size. (18) Hrovat, D. A.; Borden, W. T. J. Am. Chem. Soc. 1994, 116, 6327. (19) Du, P.; Hrovat, D. A.; Borden, W. T. J. Am. Chem. Soc. 1989, 111, 3773. (20) For example, see: Turi, L.; Dannenberg, J. J. J. Phys. Chem. 1992, 96, 5819.

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