Electron Propagator Calculations on the Ground and Excited States of

Article pubs.acs.org/JPCA

Electron Propagator Calculations on the Ground and Excited States of C−60 V. G. Zakrzewski, O. Dolgounitcheva, and J. V. Ortiz* Department of Chemistry and Biochemistry Auburn University Auburn Alabama 36849−5312, United States ABSTRACT: Electron propagator calculations in two approximationsthe third-order algebraic, diagrammatic construction and the outer valence Green’s function (OVGF)have been performed on the vertical electron affinities of C60 and the vertical electron detachment energies of several states of C−60 with a variety of basis sets. These calculations predict bound 2T1u and 2T1g anions, but fail to produce 2T2u or 2Hg anionic states that are more stable than ground-state C60. The electron affinity for the 2Ag state is close to zero, but no definitive result on its sign has been obtained. This state may be a resonance or marginally bound anion. The OVGF prediction for the vertical electron detachment energy of 2T1u C−60, 2.63 eV, is in excellent agreement with recent anion photoelectron spectra.

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INTRODUCTION Buckminsterfullerene, the icosahedral C60 molecule, may be capable of binding up to four electrons in the gas phase1 and up to six electrons in solution.2 The vertical electron detachment energy (VEDE) of gaseous C−60 was estimated to be in the range of 2.6−2.8 eV as early as 1987.3 Precise experimental values near 2.7 eV were obtained from high-resolution, anion 6,7 and of photoelectron spectra.4,5 Reports of C2− 60 species multiply charged anions, the robust, ground state of C−60, and the abundance of low-lying, unoccupied orbitals in carbon cages have inspired interest in bound (that is, more stable than C60) excited states of the Buckminsterfullerene anion in the gas phase. Experimental observation of such states5,8,9 and tantalizing evidence of electron−molecule complexes formed near vanishing collision energies10−20 provide additional impetus to the search for bound states of gas-phase C−60. Recently, two papers dedicated to the ground and excited anionic states of C60 have appeared.21,22 In these reports, two equation-of-motion, coupled-cluster methods, EOM-EAMP223,24 and EOM-EA-CCSD,25 were used to obtain vertical electron affinities (VEAs) of C60. In the EOM-EA-CCSD method, the single-replacement and double-replacement cluster amplitudes of a CCSD (coupled-cluster singles and doubles) wave function for C60 are employed. To avoid this computationally intensive, preliminary step, the EOM-EA-MP2 method, which ignores the single-replacement cluster amplitudes and substitutes first−order, double−replacement amplitudes for their CCSD counterparts, has been used. The former work was concerned with the prediction and analysis of a weakly bound, 2Ag state of C−60.21 EOM-EA-MP2 calculations employed atomic natural orbital (ANO) basis functions with various sets of diffuse s, p, and d functions located in the center of mass of the cluster. An EOM-EACCSD calculation was performed with the smallest ANO basis and a set of six s and six p center-of-mass functions. In all cases, the 2Ag state was bound. By assuming the additivity of various basis-set and correlation effects, the authors estimated that an EOM-EA-CCSD calculation with a 4s3p2d ANO basis © 2014 American Chemical Society

augmented by six s, six p, and four d center-of-mass functions would yield an electron affinity of 0.120 eV for the 2Ag anion. In addition to the above calculations, He−C60, Ne−C60, and H2O−C60 clusters were examined with the EOM-EA-MP2 method. The smallest, modified ANO basis21 and the aug-ccpVDZ basis26,27 for He, Ne, and H2O were used. The resulting 2 Ag states of the He and Ne clusters and their lower-symmetry counterpart for the H2O cluster were still bound, but their binding energies were smaller than for the C60 case. The 2T1u, 2T1g, 2Ag, 2T2u, and 2Hg states of C−60 were studied with the EOM-EA-CCSD method and various basis sets.22 Results of Outer Valence Green’s Function (OVGF) and nondiagonal, second-order electron propagator calculations also were reported in this paper. The cc-pVDZ26 basis for carbon augmented with the diffuse s function from the aug-ccpVDZ basis27 was used in these calculations. The total number of basis functions in this main basis was 900. Electron attachment to the t1u, t1g, and t2u virtual orbitals produced bound anionic states at the EOM-EA-CCSD level. When d functions were dropped from the initial basis, the 2T1u and the 2 T1g states remained bound. Addition of six s, six p, and six d center-of-mass functions to the previous basis without d functions resulted in a bound 2Ag state with an electron binding energy of 0.128 eV. When the initial cc-pVDZ + s basis was augmented with the same set of 54 diffuse functions, four anionic states were predicted to be bound. Further expansion of the center-of-mass basis functions with six f and six g sets had no crucial effect on the results. Thus, four out of five possible low-lying anionic states were found to be bound at the EOMEA-CCSD level. (Only the 2Hg anion was not found to be more stable than C60.) No attempts to improve the carbon basis beyond double-ζ plus diffuse s were reported. Special Issue: Kenneth D. Jordan Festschrift Received: December 31, 2013 Revised: May 7, 2014 Published: May 9, 2014 7424

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that pertain to these methods have been described and reviewed.30,38−44 Computational Details. All calculations were performed with the developers’ version of the Gaussian suite of codes.45 The equilibrium geometry of ground−state C60 was obtained with the B3LYP functional46,47 and the 6-311G* basis48,49 (carbon−carbon bond lengths of 1.454 and 1.396 Å result). This geometry was used in subsequent calculations of C60 electron affinities. To obtain accurate values of the adiabatic electron affinity (AEA), geometries of both neutral C60 and its ground state anion were optimized with the MP2 method and the 6-311+G* basis.50 Electron propagator calculations of VEAs were performed in the ADC(3) and OVGF approximations. For these calculations, the 6-311+G* basis was augmented with extra diffuse s-type functions placed on each carbon center. The exponent of this function (α) was optimized to maximize the ADC(3) electron affinity of C60 for the 2Ag final state; the optimium value was 0.003. The total number of MOs in the electron propagator calculations was 1380. The Gaussian code purges linearly dependent combinations of basis functions from Hartree−Fock and correlated calculations. Attempts also were made to introduce a set of extra diffuse functions placed at the center of mass of the molecule as described in previous reports.21,22 In such a procedure, molecular symmetry cannot be taken into account by Gaussian routines, and thus electron propagator calculations of good quality become infeasible. To bypass this obstacle, calculations were performed for a system in which a He atom was placed at the center of mass of C60. Thus, the high symmetry of the cluster was maintained. To minimize the He atom’s influence on the results, a 1s STO-3G basis function was used for the He atom, and sets of diffuse functions used previously21 were added to the same center. The following notations for different basis sets will be used henceforth: • 6-311G* + extra s on each carbon = B1 • 6-311+G* + extra s on each carbon = B2 • cc-pVDZ + s (from aug-cc-pVDZ) + 4s,4p in the center (or He atom) = B3 • B3 + STO-3G on He = B3a • 6-311G* + 4s,4p = B4 • 6-311G + extra s on each carbon +4s,4p = B5 More detailed descriptions of these sets are given in the next section.

Electron binding energies obtained with EOM-EA-CCSD and the cc-pVDZ(+s)+6s6p6d basis were compared with those from electron propagator calculations in the second-order, the third-order, and the OVGF approximations. All five anionic states were bound at the second-order level. Only the 2T1u and the 2T1g states were bound in the third-order and the OVGF approximations. The authors concluded that finite-order, perturbative methods are not capable of describing the correlation effects that produce a bound 2Ag anion. Recent electron propagator calculations have employed the GW0 approximation with a 6-311+G(3df) basis in a search for super-atom molecular orbitals.28 In a demonstration of the Cholesky decomposition method for reducing the rank of the electron repulsion matrix, second-order electron propagator calculations with double-ζ and triple-ζ, correlation-consistent basis sets were performed on the electron affinity of C60.29 However, the second-order self-energy is demonstrably unreliable for predictions of electron affinities.22 In this report, electron propagator methods in the third-order Algebraic Diagrammatic Construction, or ADC(3), and the OVGF approximations are applied to VEAs of the C60 fullerene. VEDEs from anionic states also are examined with the OVGF and second-order, perturbation-theory (ΔMP2) methods.

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THEORY AND COMPUTATIONAL DETAILS Theoretical Background. The electron propagator methods used in this work have been widely applied and repeatedly reviewed and therefore require presently only a brief description. Two main types of electron propagator methods are in general use. Quasiparticle approximations, such as the OVGF, are applicable when the one-electron picture of electron detachment or attachment of Hartree−Fock theory remains qualitatively valid. In this class of methods, nondiagonal matrix elements of the self-energy are neglected. Approximations with nondiagonal, renormalized self-energies, such as ADC(3), are capable of describing stronger orbital relaxation and electron correlation effects. Additional background can be found in various reviews.30−36 In both cases, OVGF and ADC(3), the self-energy is complete through third order in the fluctuation potential, However, the nondiagonal character of the ADC(3) self-energy allows the pth Dyson orbital for electron attachment to an N-electron molecule, ϕpDyson(xN + 1) =

N+1

∫ Ψ*neutral(x1, x2 , x3 , ..., xN )

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RESULTS AND DISCUSSION MP2 Adiabatic Electron Affinities. The AEA of C60 was obtained as the difference of the total energies of Ih C60 and its Ci anion, which is descended from the 2T1u anion of the former point group. The equilibrium structures were optimized with the MP2/6-311+G* method. The AEA values obtained presently are 2.95 eV with ΔUMP2 and 3.03 eV with the latter method’s spin-projected counterpart, ΔPUMP2.51 Spin contamination in the unrestricted Hartree−Fock (UHF) calculation on the anion has only a minor effect on the ΔUMP2 result. ADC(3) Electron Affinities. Electron affinities of C60 were calculated with the ADC(3) and OVGF methods. Because the system is large and electron propagator calculations required considerable memory, disk space, and time, some restrictions to the active orbital space were needed. The total number of MOs for the C60 system in the B2 basis described above is 1380, 1200 of which are virtual. In accord with the usual strategies,

Ψanion,p(x1, x 2 , x3 , ..., xN , xN + 1) dx1 dx 2 dx3...dxN

to be expressed as a linear combination of all molecular orbitals (MOs), whereas the OVGF approximation requires that the Dyson orbital be proportional to a canonical, Hartree−Fock orbital. The renormalized self-energy of the ADC(3) method includes all self-energy terms through third order and certain kinds of terms in all orders. (A method is said to renormalize when it includes terms in all orders.) The OVGF method relies on evaluation of diagonal, third-order self-energy terms and estimates higher-order terms with simple extrapolation techniques. The ADC(3) approximation usually employs a procedure to calculate renormalized, constant-diagram contributions,37 but in this report, a finite expansion through third order was used unless otherwise noted. The ADC(3) selfenergy therefore supersedes its OVGF counterpart. However, the OVGF method’s arithmetic demands are lower and therefore facilitate calculations with larger basis sets. Algorithms 7425

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extra diffuse function on each carbon atom was used, the 2Ag binding energy depended on the exponent of the extra s function. In such cases, a bound state is produced by the second-order propagator. The third-order approximation and the OVGF both fail to produce a bound 2Ag state. Note that the s-exponent of 0.003, which gave the best binding energy with ADC(3) and certain restrictions of the virtual orbital space, leads to OVGF and second-order results that are less bound then those obtained with the smaller exponent. OVGF results of Table 2 and ADC(3) results of Table 1 for the largest VEA of C60 are in close agreement at 2.14 eV. MP2 Vertical Electron Affinities. For the sake of comparison, the VEAs corresponding to the lowest t1u and ag virtual orbitals were calculated with the MP2 method and the B2 basis. Only core MOs were omitted in these correlated calculations. The corresponding VEAs calculated as ΔPUMP2 total energies are 2.30 eV for the ground anionic state and 0.043 eV for the 2Ag state. These results are somewhat more positive than their OVGF and ADC(3) counterparts. UHF calculations with initial 2T1g and 2T2u configurations converged to 2Ag and 2T1u solutions. Vertical Electron Detachment Energies. VEDEs were calculated for the 2T1u, 2T1g, and 2Ag anions. The B2 basis was used for the 2T1u and 2Ag anions. The 2T1g state had to be treated with the 6-311+G* basis due to convergence problems with the B2 basis. The ground-state anion geometry was optimized with the UMP2/6-311+G* method. The C60 geometry was used for the 2T1g and 2Ag states. Results obtained with electron propagator methods and ΔMP2 are presented in Table 3.

carbon core MOs were dropped from the active space. The number of retained, virtual orbitals was varied. Table 1 presents Table 1. C60 ADC(3)/B2 Vertical Electron Affinities (eV) 2

2

2.14

0.73

T1u

2

T1g

Ag

−0.001

2

Hg

−0.232

2

T2u

−0.334

the results of ADC(3) calculations for the 2T1u, 2T1g, 2Ag, 2Hg, and 2T2u states of C−60. At the Koopmans level, the order of these states is 2T1u, 2Ag, 2Hg, 2T2u and 2T1g; only the first state is bound. At the ADC(3) level, the 2Hg and 2T2u anionic states are not bound, independent of the size of the active virtual space. However, the 2T1u and 2T1g states are bound. The 2Ag anionic state is an intermediate case in which the sign of the electron binding energy is not as clear. The anionic state is bound when the active space is small. The maximum value of 0.089 eV is obtained with 385 active, virtual MOs. This VEA decreases as the number of virtual MOs grows. The maximum value of virtual MOs is 1100, since 100 orbitals have been discarded because of linear dependencies in the atomic basis. The 2Ag state becomes unbound when all but 20 virtual MOs (i.e., 1080) are included. It is likely that the sign of this VEA will be negative in a calculation with the full set of virtual MOs. Calculations with 585 or 785 virtual MOs indicate that use of a renormalized procedure for constant self-energy terms37 decreases the 2Ag electron affinity by approximately 0.03 eV. Quasiparticle Electron Affinities. C60 VEAs also were calculated with the OVGF. Second-order and third-order quasiparticle results were generated in the course of the OVGF calculations. Results obtained with various basis sets are compiled in Table 2. (In all cases, the C version of OVGF30,33

Table 3. C−60 Vertical Electron Detachment Energies (eV) state

Table 2. C60 Vertical Electron Affinities (eV) Basis 6-311G* α = 0.00005

State

second

third

OVGF

2

T1u Ag

2.86 −0.002

1.52 −0.002

2.00 −0.002

T1u Ag

2.91 −0.002

1.58 −0.004

2.05 −0.003

T1u T1g 2 Ag

3.00 1.84 0.011

1.65 0.09 −0.039

2.14 0.90 −0.022

2

3.00 1.83 0.057

1.65 0.09 −0.113

2.14 0.74 −0.053

2

6-311G* α = 0.01, α = 0.001, α = 0.0001 6-311+G* α = 0.001

2

2

2

2

6-311 + G* α = 0.003 i.e., B2

T1u T1g 2 Ag

2

ΔPUMP2 second order third order OVGF exp.4 exp.5

2

T1u

3.07 2.45 2.80 2.63 2.683 ± 0.008 2.689 ± 0.008

2

T1g

1.17 0.76 1.24 0.84

2

Ag

0.043 0.039 −0.099 −0.060

2 T1u. The lowest unoccupied MO of C60 is triply degenerate, and Jahn−Teller distortion of the ground−state anion therefore is expected. The geometry optimization converged to a Ci structure with a 2 A u electronic configuration. OVGF calculations were performed for the singly occupied au MO. The VEDE thus obtained, 2.63 eV, is in excellent agreement with the experimental values of 2.683 ± 0.008 eV4 and 2.689 ± 0.008 eV.5 VEDEs obtained with the second-order and the third-order electron propagator are 2.45 and 2.80 eV, respectively. The ΔPUMP2 results are notably more positive. 2 T1g. The OVGF VEDE for this anion is 0.84 eV. The second-order electron propagator gives a slightly smaller detachment energy of 0.76 eV, whereas third order and the ΔMP2 produced greater values. 2 Ag. VEDE values for the 2Ag anion obtained with OVGF and third-order electron propagator methods and with retention of all valence occupied and virtual MOs are negative. The secondorder approximation produces a bound state and so does the ΔMP2 method. Close agreement between the two latter methods suggests that orbital relaxation in the anion is unimportant and that the Dyson orbital is diffuse.

produces the recommended value.) The full set of valence occupied and virtual MOs was used, i.e., only C core MOs were omitted. For the 2T1u state, consistent results were obtained by all approximations. In each basis, the second-order values were much larger than the ones produced by the third-order selfenergy, whereas the OVGF gave values in between. For the 2Ag state, no binding was established if the basis lacked regular + diffuse s functions (as defined in the 6-311+G* basis) even when three extra diffuse s-type functions were placed on each carbon atom. When the 6-311+G* basis augmented with one 7426

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Effects of Center-of-Mass Functions. To facilitate comparisons with EOM-EA-CCSD results,21,22 ADC(3) calculations were performed with a basis, used in a recent report,22 which comprised the cc-pVDZ + s set augmented with s and p diffuse functions deployed at the C60 cage’s center of mass. Four diffuse s-functions (α = {0.005, 0.002806, 0.001575, 0.000884}) and four sets of p-functions (α = {0.01, 0.005518, 0.003045, 0.00168}) were used. The total number of basis functions was 916 (basis B3). With such an approach, molecular symmetry cannot be recognized by the Gaussian routines and highly inefficient computations would follow. Instead, calculations were done on a system with a He atom in the center of the C60 cluster (basis B3a). The following order of the lowest virtual MOs was obtained with the basis described above and a 1s STO-3G basis function on He: t1u, ag, t1u, ag, t1u, ag, t1u, t1g, ag, t2u. The same order of orbitals was obtained with Hartree−Fock calculations in which the diffuse s and p functions were placed at the center of the cluster. ADC(3) VEAs obtained for the 2T1g and 2Ag states with the B3 and B3a bases are presented in Table 4. The 2T1g state is

CONCLUSIONS The OVGF method, a quasiparticle approximation that is computationally efficient but approximate in its treatment of renormalization effects, produces a vertical electron detachment energy for the 2T1u anion of 2.63 eV that is in excellent agreement with the experimental value of 2.69 eV5 (see Table 3). The same method yields a vertical electron detachment energy of 0.84 eV for the 2T1g state of C−60. OVGF also predicts a vertical electron detachment energy for the 2Ag state of C−60 that is close to zero, but negative. ADC(3) calculations with the B2 basis set, which includes the 6-311+G* set and an extra, diffuse s function on each carbon atom, do not yield bound 2Ag, 2T2u or 2Hg anions and confirm the positive electron affinities that pertain to the 2T1u and 2T1g states. This renormalized method makes similar predictions when diffuse, center-of-mass basis functions are employed. The discrepancy between these results and those obtained with the EOM-EA-MP2 and EOM-EA-CCSD methods and smaller basis sets21,22 indicates that definitive calculations will require renormalized methods and triple-ζ basis sets with multiple polarization and diffuse functions. The best ADC(3)/B2 result of Table 1 and the data of Table 4 on the 2Ag case indicate that the corresponding electron affinity is close to zero. However, the sign of this electron affinity cannot be inferred definitively from these results. Inclusion of the constant-diagram renormalizations that usually are included in ADC(3) calculations,37 use of more efficient, renormalized electron propagator methods52 with larger basis sets, or application of renormalized electron propagator methods that employ Brueckner orbitals and coupled−cluster, double− substitution amplitudes33,34,53 may enable the resolution of the present contradictions.

Table 4. C60 ADC(3) Vertical Electron Affinities (eV)

a

molecule

basis

C60 He−C60 He−C60

a

2

B3 B3aa B3ab

2

T1g

Ag

0.65 0.65 0.74

0.009 −0.006 −0.008

418 active virtual orbitals. bComplete active orbital space.

bound by 0.65 eV in the C60 calculation with center-of-mass functions and in the He−C60 model. This value is close to the 0.73 eV result of Table 1. With the number of virtual orbitals set to 418, the 2Ag state is marginally bound with the B3 basis for C60 and is marginally unbound with the B3a basis for He− C60. Restoration of the full set of virtual orbitals in the last line of Table 4 has a negligible effect on the He−C60 results. A calculation with the full set of virtual orbitals from the B3 basis could yield a positive VEA for the 2Ag case. Calculations were performed on the He−C60 cluster with the 6-311G* basis augmented with the aforementioned diffuse s and p functions centered on He and with 6-311G*, plus an extra s-function (α = 0.005) on each carbon atom and the same s,p diffuse set on He. The total numbers of basis functions were 1097 (basis B4) and 1157 (basis B5). The order of virtual MOs obtained with the B4 set coincided with the one produced by the B3 and B3a sets. The results are given in Table 5. ADC(3)

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a

NVMOa

B4 B5

977 1010

2

2

1.99 2.01

0.59 0.60

T1u

T1g

2

Ag

−0.018 −0.013

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation through a grant to Auburn University. REFERENCES

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Table 5. He−C60 ADC(3) Vertical Electron Affinities (eV) basis

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