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Coupled Cluster Studies of Ionization Potentials and Electron Affinities of Single Walled Carbon Nanotubes Bo Peng, Niranjan Govind, Edoardo Aprà, Michael Klemm, Jeff R. Hammond, and Karol Kowalski J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b10874 • Publication Date (Web): 19 Jan 2017 Downloaded from http://pubs.acs.org on January 30, 2017
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Coupled Cluster Studies of Ionization Potentials and Electron Affinities of Single Walled Carbon Nanotubes Bo Peng,† Niranjan Govind,† Edoardo Apr´a,† Michael Klemm,‡ Jeff R. Hammond,¶ and Karol Kowalski∗,† †William R. Wiley Environmental Molecular Sciences Laboratory, Battelle, Pacific Northwest National Laboratory, K8-91, P.O. Box 999, Richland, WA 99352, USA ‡Intel Deutschland GmbH, Feldkirchen/Munich, Germany ¶Intel Corporation, Portland, Oregon, USA E-mail:
[email protected] Phone: (509) 371-6464
Abstract In this paper we apply the equation-of-motion coupled cluster (EOM-CC) methods in the studies of the vertical ionization potentials (IPs) and electron affinities (EAs) for a series of single walled carbon nanotubes (SWCNT). The EOM-CC formulations for IPs and EAs employing excitation manifolds spanned by single and double excitations (IP/EA-EOM-CCSD) are used to study the IPs and EAs of the SWCNTs as a function of the nanotube length. Several armchair nanotubes corresponding to C20n H20 models with n = 2 − 6 have been used in benchmark calculations. In agreement with previous studies, we demonstrate that the electronegativity of C20n H20 systems remains, to a large extent, independent of the nanotube length. We also compare
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IP/EA-EOM-CCSD results with those obtained with the coupled cluster models with single and double excitations corrected by perturbative triples, CCSD(T), and density functional theory (DFT) using global and range-separated hybrid exchange-correlation functionals.
Introduction The sp2 hybridization/bonding-state of the carbon atom leads to various molecular structures with extraordinary properties. Since the discovery of C60 molecule by Smalley, Kroto and co-workers, 1 many other structures have been synthesized and studied (for review see Ref. 2,3 ) including tubular carbon nanostructures. 4,5 The impressive properties of various types of nanotubes are directly tied to their structural properties corresponding to quasione-dimensional structure and the graphite-like arrangements of carbon atoms. 2 For example, structural parameters can be used to tune the conducting properties of nanotubes from semiconducting to metallic, which makes them ideal building blocks for molecular electronic devices such as field-effect transistors, single-electron transistors and rectifying diodes. 6 Carbon nanotubes have also attracted considerable attentions for their optical properties and possibility of using covalently modified and/or functionalized form of the nanotubes in combination with suitable electron donors. Recent progress in the separation of semiconducting single walled carbon nanotubes (SWCNTs) and derivatization techniques have opened new avenues to engineer charge separation at donor/acceptor interfaces. 3 In designing organic photovoltaic (OPV) materials, one has to take into account several parameters which impact the short circuit current density, the open circuit voltage, and the fill factor. The open circuit voltage is dictated by the relative ionization potential (IP) of the donor and electron affinity (EA) of the acceptor. Typically, it is approximated by the energy gap between the highest occupied molecular orbital (HOMO) of donor and the lowest unoccupied molecular orbital (LUMO) of acceptor. 7,8 Therefore, it is clear that the appropriate material design and screening of the D/A materials combination is fundamental 2
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for the optimization of the open circuit voltage. In many cases, however, crude approximations of the IPs and EAs made by the HOMO and LUMO orbital energies cannot provide the desired level of accuracy. For example, discrepancies between Koopmans’ theorem and experimental IPs are of the order of several electron volt (eV). 9 In these cases proper inclusion of correlation effects plays a critical role in obtaining accurate results. Among several methods which allow for the accurate description of the IPs and EAs Green’s function based approaches 10–38 and IP/EA equation-of-motion coupled cluster (IP/EA-EOM-CC) formulations 39,40 have been shown to provide accurate IPs and EAs for medium-size molecular systems. In this paper, we apply the IP/EA-EOM-CC formulation with singles and doubles (IP/EAEOM-CCSD) 39–42 and CCSD(T) 43–47 approaches to calculate the IPs and EAs of a series of SWCNT models. Due to the high numerical cost, we perform the IP/EA-EOM-CCSD calculations with a larger basis set (aug-cc-pVDZ) 48 only for the C40 H20 system. For the general trend with the larger systems (C20n H20 systems (n = 2, 3, 4, 5, 6)), we calculate the IPs and EAs using the smaller 6-31G basis set. 49 CCSD(T) calculations for the same systems have been performed with a novel implementation of the CCSD(T) approach which avoids local memory bottlenecks encountered in our previous implementations and at the same R R Many Integrated Core Architecture). 50 We MIC (Intel time takes advantage of the Intel
also compare our results with those obtained with density functional theory (DFT) using global and range-separated hybrid density functionals. 51–56 To evaluate basis set effect we also discuss DFT results for larger cc-pVDZ, aug-cc-pVDZ, and cc-pVTZ basis sets. 48
Theory The IP/EA-EOM-CC formulations have been extensively discussed in the literature. 39,40 Here, we give only the basic tenets of this approach. In close analogy to the equationof-motion coupled cluster (EOM-CC) formulations for calculating the excitation energies
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of N -electron system, the IP/EA-EOM-CC formulations use the EOM-CC-type of wave function parametrization for K-th state of (N − 1)-/(N + 1)-electron system, (N −1)
IP T i = RK e |Φi ,
(1)
(N +1)
EA T i = RK e |Φi ,
(2)
|ΨK
|ΨK
IP EA where the RK and RK are the IP/EA excitation operators, T is the cluster operator for
N -electron system, and |Φi represents the Hartree-Fock (HF) Slater determinant for the N -electron system. (N −1)
The energies of the (N − 1)-/(N + 1)-electron system (EK
(N +1)
/EK
) can be obtained
by diagonalizing the corresponding electronic Hamiltonians (we will denote them generally by H without specifying the number of electrons), i.e., (N −1)
IP T e |Φi . RK
(3)
(N +1)
EA T RK e |Φi ,
(4)
IP T HRK e |Φi = EK
EA T HRK e |Φi = EK
By premultiplying the above equations from the left by e−T one can cast these equations in the form of eigen-problems, ¯ IP |Φi = E (N −1) RIP |Φi , HR K K K
(5)
(N +1) EA EA ¯ K RK |Φi , |Φi = EK HR
(6)
with RK ’s and EK ’s representing the eigenvectors and eigenvalues of the similarity trans¯ formed Hamiltonian H, ¯ = e−T HeT , H
(7)
in the (N − 1)- and (N + 1)-electron Hilbert spaces. Additionally, if one uses the normal
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¯ N of the similarity transformed Hamiltonian ordered form H ¯N = H ¯ − hΦ|H|Φi ¯ ¯ − E0(N ) , H =H (N )
where the ground-state energy of the N -electron system E0
(8) ¯ is equal to hΦ|H|Φi, then
Eqs.(5) and (6) can be rewritten as IP IP IP ¯ N RK H |Φi = ωK RK |Φi .
¯ N REA |Φi = ω EA REA |Φi , H K K K
(9) (10)
IP EA IP where the eigenvalues ωK and ωK are the IPs and EAs, respectively. The eigenvalues ωK EA and ωK are identical to the poles of the CC Green’s function formulations. 57–59
In our calculations of vertical IPs and EAs we used IP/EA-EOM-CC with singles and doubles (IP/EA-EOM-CCSD) approximations 41,42 where
T =
X i,a
IP RK =
X
X
tia a†a ai +
=
X
(11)
raij (K) a†a aj ai ,
(12)
i
