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J. Phys. Chem. C 2010, 114, 20870–20876
Nonlinear Optical Effects Induced by Nanoparticles in Symmetric Molecules† Tim Hansen, Thorsten Hansen, Vaida Arcisauskaite, and Kurt V. Mikkelsen* Department of Chemistry, H. C. Ørsted Institute, UniVersity of Copenhagen, DK-2100 Copenhagen Ø, Denmark
Jacob Kongsted Department of Physics and Chemistry, UniVersity of Southern Denmark, DK-5230 Odense M, Denmark
Vladimiro Mujica Department of Chemistry, Arizona State UniVersity, Tempe, Arizona 85287-1604, United States ReceiVed: August 12, 2010; ReVised Manuscript ReceiVed: October 8, 2010
We present theoretical methods and computations of the effects of nanoparticles on nonlinear optical properties of symmetric molecules. We utilize quantum mechanical/molecular mechanics (QM/MM) response methods for calculating electromagnetic properties of molecules interacting with nanoparticles, and we report calculations of the frequency-dependent first hyperpolarizability. The frequency-dependent first hyperpolarizability of (p)benzenedithiol in the presence of zero, one, or two gold nanoparticles of varying sizes and distances is calculated by DFT/MM. The hyperpolarizability of the molecule depends strongly on the distance between the nanoparticles and the molecule, whereas the size of the nanoparticle is of little importance. We clearly show that metal nanoparticles are able to induce a first hyperpolarizability in symmetric molecules. I. Introduction The field of molecular electronics has been the subject of extensive study over the past decade, and many key aspects concerning molecular wires are now largely understood, and a semiquantitative agreement between theory and experiment is emerging.1-5 Meanwhile, several intriguing experiments have demonstrated a rich diversity of phenomena in molecular transport junctions. (i) The Coulomb blockade has been observed, thereby realizing the molecular transistor.6-9 (ii) The Kondo resonance, a highly correlated magnetic phenomena, has been seen.7,8 (iii) Hysteresis and switching behavior have demonstrated the importance of multiple molecular states for certain systems.10-12 These phenomena illustrate the complexity and challenges of molecular electronics. Another outstanding challenge is the understanding of electro-optical properties of molecular systems at the nanoscale, and nanooptics is an actively investigated research field in its own right.13-18 The interplay between molecular transport junctions with light and plasmons poses an intriguing scientific problem with immense perspectives for applications. Specifically, we mention electro-optical devices at the molecular level, such as molecular light-emitting diodes or molecular optical switches. A prerequisite for such progress is a thorough understanding of the optical properties of molecular transport junctions, and in this presentation, we will present a powerful and flexible strategy for calculating optical properties of molecules near or between nanoparticles. Note that the geometries studied in this work are not representative of a fully formed molecular transport junction. In such case, the hydrogen-sulfur bonds will dissociate, and the sulfur atoms will bind directly to gold. This requires a careful treatment of the bonding region and a method that †
Part of the “Mark A. Ratner Festschrift”. * To whom correspondence should be addressed.
can handle the radical nature of the molecule without hydrogen. We are currently addressing these challenges. Previously, we have utilized a heterogeneous dielectric continuum model for investigating the electrostatic properties of a molecular junction.19,20 In the present investigation, we improve the flexibility in the choice of molecular and environmental structures and demonstrate the use of a quantum mechanics/molecular mechanics (QM/MM) model that allows for an atomistic description of the metal electrodes. Previously, we have successfully applied this method to the calculation of both linear and nonlinear optical properties of molecules in solution, both at the Hartree-Fock (HF), coupled cluster (CC), and density functional theory (DFT) levels of electronic structure theory.21-24,26-29 The molecule is treated fully quantum mechanically with the desired electronic structure method, and the electrodes are treated at a classical level as a structured atomistic environment where each atom is assigned an isotropic atomic polarizability. The electromagnetic response of the electrodes is included as a dynamic electric field in the electronic structure calculation, and the entire problem is solved self-consistently. The electronic structure calculations are performed using density functional theory (DFT)30-32,32-44 using the B3LYP,30,31 and CAM-B3LYP32 functionals. We perform QM/MM quadratic response calculations in order to determine the frequencydependent first hyperpolarizability of the molecular system interacting with one or two gold nanoparticles. II. Theory We utilize our QM/MM method for calculating the hyperpolarizability of a molecular system interacting with nanoparticles, and we divide the system of interest into several parts or subsystems, which are treated at different levels of theory using quantum mechanics and molecular mechanics.21-24,26-29,58 This gives the total energy of the system as E ) EQM +
10.1021/jp107633z 2010 American Chemical Society Published on Web 11/15/2010
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EQM/MM + EMM, where EQM is the quantum mechanical energy, EQM/MM gives the interaction energy between the quantum and classical subsystems, and the final term, EMM, is the energy of the classical subsystem treated by molecular mechanics. For the quantum mechanical part of the total system, we use second quantization, and the molecular electronic Hamiltonian is given as
ˆ ) H
∑ hpqEˆpq + 21 ∑ gpqrseˆpqrs + hnuc pq
at the MM atoms having the partial charge qs and positioned according to the position vector b R s. The interaction energy related to the induced dipole moments (µaind) where the index a refers to sites in the classical system is given as A
E[µind a ] ) -
s b n b ˆ ∑ µind a (E (Ra) + 〈Rra〉 + E (Ra)) -
a)1
(1)
A
∑
1 µindT µind (9) 2 a,a'(a*a') a aa' a'
pqrs
where
hpq )
∫
gpqrs )
(
)
M
where Taa′ is the dipole interaction tensor
Zm 1 r - ∇2 φ*( φq(b) r db r p b) 2 r -b Rm | m)1 | b
∫∫
∑
r 1)φ*( r 2)φq(b φ*( r 1)φs(b r 2) p b r b r2 db r 1 db |b r1 - b r 2|
(2)
(3)
where the fermion creation and annihilation operatores are † denoted aˆpσ and aˆpσ, respectively, and they act on the electron b)〉} represents in the pth orbital with spin σ. We let the set {|φp(r the molecular orbitals, and hnuc is the nuclear repulsion energy. The one-electron excitation operator is defined as
Taa' )
[
]
ba - b ba - b 3(R Ra')(R Ra')T 1 -1 ba - b ba - b |R Ra' | 3 |R Ra' | 2
ba) denotes the electric field due to the QM nuclei at and En(R the MM site a. The electric field due to the partial charges ba), and the situated at the other MM molecules is given by Es(R QM electronic electric field is written as
ˆ ra ) R
b r -b R
∑ 〈φp| |br - bR a|3 |φq〉Eˆpq pq
Eˆpq )
∑ aˆpσ† aˆqσ
(4)
σ
and the two-electron excitation operator as
eˆpqrs )
∑ aˆpσ† aˆrτ† aˆsτaˆqσ
A
ˆ vdw ) H
The interactions between the quantum mechanical and classical subsystems are given by
ˆ QM/MM ) H ˆ el + H ˆ vdw + H ˆ pol H
(6)
ˆ el represents the electrostatic interactions where H S
ˆ el ) H
el,nuc ∑ Nˆs + ES,N
(7)
s)1
(11)
a
where the expectation value of this operator over the electronic wave function determines the electric field at position b Ra. For the van der Waals interaction, we have selected the 6-12 Lennard-Jones potential and modeled the van der Waals contributions as
(5)
στ
(10)
∑ ∑
a)1 m:center
[
Ama bm - b |R Ra |
12
-
Bma bm - b |R Ra | 6
]
(12)
where a and m denote the MM and QM sites, respectively. The DFT-based hybrid QM/MM approach is used to investigate the nonlinear interactions between light and a centrosymmetric molecule adsorbed on metal nanoparticles. We start out by considering the expectation value of a time-independent operator Aˆ. For a time-dependent perturbation given by the operator Vˆ(t), we observe that the expectation value of Aˆ becomes time-dependent, and therefore, we expand the expression for the time-dependent expectation value in orders of the perturbation.
and where the electronic contribution is
Nˆs )
∑ 〈φp| |Rb pq
s
qs - b| r
〈t|Aˆ |t〉 ) 〈t|Aˆ |t〉(0) + 〈t|Aˆ |t〉(1) + 〈t|Aˆ |t〉(2) + ... |φq〉Eˆpq
(13)
(8)
We describe the electrostatic interactions between the partial charges in the classical subsystem and the electrons and the nuclei in the QM system by this term, and we let the term Eel,nuc S,N denote the interactions between the MM partial charges and the QM nuclei, and the index s runs over all of the sites in the MM system. Typically, the sites in the MM system are located
and furthermore, we denote Vˆω as the Fourier transformation of Vˆ(t)
Vˆ(t) )
∫ Vˆω exp(-iωt) dω
(14)
We obtain the Fourier representations of the time-dependent terms in eq 13 as
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〈t|Aˆ |t〉(1) ) 1 〈t|Aˆ |t〉(2) ) 2
Hansen et al.
∫ 〈〈Aˆ;Vˆω〉〉ω exp(-iωt) dω
∫∫ 〈〈Aˆ;Vˆω1, Vˆω2〉〉ω ,ω 1
2
(15)
exp(-i(ω1 + ω2)t) dω1 dω2 (16)
We find that the expressions in eqs 15 and 16 define the linear and quadratic response functions implicitly. We perform model calculations on molecule nanoparticle systems where the molecule is treated by quantum mechanics and the nanoparticles are described by molecular mechanics. The gold atoms constituting the nanoparticles are described using molecular mechanics, and we assign an atomic dipoledipole polarizability, R, to each of these gold atoms, which gives induced dipole moments on each site due to the electric fields from the QM and MM subsystems. The sites do not contain partial charges. We have the following expression for the induced dipole moments µ (eq 17) -1 µ __)-1E _ QM _ ) (R 1__ - T
(17)
and having N polarizable sites, we get µ and EQM as 3Ndimension vectors containing the induced dipole moments and the electric field due to the QM subsystem, respectively. The relay matrix (R-11_ - T _) is a 3N × 3N matrix, and it contains the inverse of the isotropic atomic polarizability tensor R in 3 × 3 diagonal blocks that are coupled by the dipole field tensors T, and we have
(R-1 __ 1 1
-T __) )
(
R-1 1
0
0
0
0
R-1 ··· 2
···
l
···
0
0
·
·. 0
···
(
R-1 n
)
[
T _12µ to ≈ µ _2 and, therefore, we damp the electric field by _1 T modifying the dipole field tensors T _ij using the exponential damping scheme that we have utilized previously.57-59 The damping of the QM electric field at the polarizable MM sites is crucial because the MM sites do not contain electrons repelling QM site electrons, and therefore, significant QM electric fields might appear at the polarizable MM sites, resulting in overpolarization. In order to obtain a physically correct view of the QM and MM interactions, we damp these electric fields according to the damping function in eq 20
f(d) ) (1 - e-AD)3
(20)
where A is a damping coefficient and D is the distance between the center-of-mass of the QM system and the polarizable MM sites. For A ) 0, the QM electric field is fully damped at the polarizable MM sites, and by increasing the value of A, we enlarge the coupling between QM and MM sites. III. Computational Methods
-
0 T12 · · · T1N T21 0 · · · · · · l l · ·. · · · TN1 · · · · · · 0
)
(18)
The dipole field tensors T are defined as
Taa' )
Figure 1. (a) Structure of the staggered conformation of (p)benzenedithiol. (b) Eclipsed conformation of the same.
]
ba - b ba - b 3(R Ra')(R Ra')T 1 -1 3 2 ba - b ba - b |R Ra' | |R Ra' |
(19)
and the vector b Ra is a position vector to an induced dipole moment in the MM part of the system, and the index a refers to the center of each MM atom. The induced dipole moments at the atomic sites in the respective nanoparticles lead to a polarization field interacting with the molecular system, and therefore, the quantum mechanical operators have to be modified.25,27 In our approach, we utilize two kinds of damping mechanisms, (i) damping between the MM-induced dipole moments and (ii) damping of the QM electric field at the polarizable MM sites. We need to include a procedure for avoiding infinite polarization57-59 due to cooperative interaction between two induced dipole moments in the direction of the line connecting these two. This occurs because the polarization interaction between two induced dipole moments is proportional
We have considered two conformations of (p)-benzenedithiol, a staggered and an eclipsed, with the hydrogen atoms on the sulfur atoms on the same or opposite sides of the plane, defined as containing both sulfur atoms and being orthogonal to the molecular plane, respectively (see Figure 1). The x-axis connects the left gold nanoparticle through the two two sulfur atoms to the right gold nanoparticle. The geometries of these conformations were optimized in Gaussian0354 using the B3LYP functional together with a 6-311G** basis set.45-49 The QM/MM response methods were implemented in the DALTON quantum chemical code.52 Test calculations were carried out for different density functional theories (B3LYP, VWN5, CAM-B3LYP)30-32,34-44 and basis sets (6-31G**, ccpVDZ, cc-pVTZ, aug-cc-pVDZ and aug-cc-pVTZ),45-51 and we found that the best strategy for calculating frequencydependent first hyperpolarizabilities for (p)-benzenedithiol was CAM-B3LYP/aug-cc-pVTZ.33,58,62 The frequency-dependent first hyperpolarizabilities were then calculated in the presence of zero, one, or two gold clusters of varying size (treated by MM) and with different sulfur-to-gold distances. The binding site on the gold clusters was chosen to be a fcc site on a (111) surface as this is the most likely.53-55 The gold clusters were designed to be half-spheres by choosing a point and including all gold atoms of an infinite gold structure within a given radius with the flat (111) surface toward the molecule.55 The opposite clusters were designed similarly but first inverted and then translated to the other side of the molecule to the desired distance. The sites (atoms) in the gold clusters were attributed with no permanent electrical effects but were assigned a polarizability of 4.6 Å3, which was extracted from a DFT calculation of the polarizability of gold clusters, and a damping coefficient of 0.415 Å-1 established by calibration of
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Figure 2. (a) Gold-to-sulfur distance dependence of β| for the staggered conformation of (p)-benzenedithiol against one gold cluster. (b) Same for the eclipsed conformation. In both cases, the gold cluster size is 85 atoms. The point ∞ corresponds to the case with no gold cluster present. The frequency-dependent first hyperpolarizabilities are calculated for four frequencies, 0.0, 0.03, 0.42823, and 0.65625 au.
Figure 3. (a) Gold cluster size dependence of β| for the staggered conformation of (p)-benzenedithiol against one gold cluster. (b) Same for the eclipsed conformation. In both cases, the gold-to-sulfur distance is 1.58 Å. The frequency-dependent first hyperpolarizabilities are calculated for four frequencies, 0.0, 0.03, 0.42823, and 0.65625 au.
excitation energies.56-59 As discussed in the previous section, this approach is used to damp the DFT electric field as there are no MM potentials to repel the electrons. The results from the different basis sets only differed numerically but showed the same general behavior with variation of the different parameters, such as the metal-molecule distance, the size of the metal nanoparticle, and the number of metal nanoparticles. Therefore, we have chosen to present the
Figure 4. Symmetric gold-to-sulfur distance dependence of β| for the eclipsed conformation of (p)-benzenedithiol against two gold clusters. The gold cluster size is 85 atoms. The point ∞ corresponds to the case with no gold cluster present. The frequency-dependent first hyperpolarizabilities are calculated for four frequencies: 0.0, 0.03, 0.42823, and 0.65625 au.
results for the aug-cc-pVTZ basis set. Similarly, there was little qualitative difference between the results from the calculations with the different functionals, but the results from the CAM-B3LYP functional were found to be more trustworthy as this functional previously has shown to give a better performance over B3LYP.33 The density functional method CAM-B3LYP is an extension of the B3LYP functional where the 1/r operator in the exact exchange term is separated into short- and long-range components. The
Figure 5. Symmetric gold cluster size dependence of β| for the eclipsed conformation of (p)-benzenedithiol against two gold clusters. The goldto-sulfur distance is 1.58 Å. The frequency-dependent first hyperpolarizabilities are calculated for four frequencies, 0.0, 0.03, 0.42823, and 0.65625 au.
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Hansen et al.
Figure 6. (a) Asymmetric gold-to-sulfur distance dependence of β| for the staggered conformation of (p)-benzenedithiol against two gold clusters, where one cluster is kept at 1.58 Å and the distance to the other is varied. (b) Same for the eclipsed conformation. In both cases, the gold cluster size is 85 atoms. The point ∞ corresponds to the case with one gold cluster present with a gold-to-sulfur distance of 1.58 Å.The frequencydependent first hyperpolarizabilities are calculated for four frequencies, 0.0, 0.03, 0.42823, and 0.65625 au.
Figure 7. (a) Asymmetric gold cluster size dependence of β| for the staggered conformation of (p)-benzenedithiol against two gold clusters, where one cluster is kept at 85 atoms and the size of the other is varied. (b) Same for the eclipsed conformation. In both cases, the gold-to-sulfur distance is 1.58 Å. The frequency-dependent first hyperpolarizabilities are calculated for four frequencies, 0.0, 0.03, 0.42823, and 0.65625 au.
parametrization of the CAM-B3LYP functional as proposed in the original work32 was employed. We performed calculations of the frequency-dependent first hyperpolarizabities in relation to second harmonic generation experiments, and we utilized the following four frequencies (ω) of the external electromagnetic field: 0.0, 0.03, 0.042823, and 0.065625 au. We present here the results of the frequency-dependent mean first hyperpolarizability in terms of β||(- 2ω;ω,ω), which is given by the following formula
β|(-2ω;ω, ω) )
∑
1 (β (-2ω;ω, ω) + 5 i)x,y,z xii βixi(-2ω;ω, ω) + βiix(-2ω;ω, ω)) (21)
where βijk(-2ω;ω,ω) is the ijk tensor component of the second harmonic generation first hyperpolarizability.60,61 IV. Results and Discussion The variations of the hyperpolarizability for the two conformations of (p)-benzenedithiol (staggered or eclipsed) with
changing sulfur-to-gold distance and changing gold cluster size for one gold cluster are depicted in Figures 2 and 3. For both conformations of (p)-benzenedithiol, we observe that the hyperpolarizability decreases with increasing sulfur-to-gold distance, whereas there is little effect in varying the size of the gold nanoparticle. The latter indicates that we have utilized gold nanoparticles of a sufficient size that the effects of enlarging the nanoparticle are rather small, but we expect that much smaller nanoparticles will give a stronger dependence on the size of the nanoparticle. The presence of one gold nanoparticle breaks the molecular symmetry, and the nanoparticle induces a hyperpolarizability in a centrosymmetric molecular system. In this case, with only one gold cluster, there is no qualitative difference to be found for the two conformations. In Figures 4 and 5, we present the hyperpolarizabilities for the eclipsed conformation of (p)-benzenedithiol placed between the two gold nanoparticles for the symmetric variations of the parameters, that is, the same cluster sizes and sulfur-to-gold distances. Here, the effect of the conformations is visible. For the staggered conformation, a center of inversion is maintained with the addition of the gold clusters, and no effect is observed;
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Figure 8. (a) Asymmetric gold-to-sulfur distance dependence of βxzz for the staggered conformation of (p)-benzenedithiol against two gold clusters, where one cluster is kept at 1.58 Å and the distance to the other is varied. (b) Same for the eclipsed conformation. In both cases, the gold cluster size is 85 atoms. The point ∞ corresponds to the case with one gold cluster present with a gold-to-sulfur distance at 1.58 Å. The frequencydependent first hyperpolarizabilities are calculated for four frequencies, 0.0, 0.03, 0.42823, and 0.65625 au.
therefore, for the staggered conformation, the two nanoparticles do not induce a molecular hyperpolarizability. However, the eclipsed conformation of (p)-benzenedithiol has no such center of inversion, and we observe that the two nanoparticles are able to induce a molecular hyperpolarizability. This effect falls off dramatically with distance, with a convergence many times faster than that for the case with one gold cluster, and the effect increases with the size of the nanoparticle. For the asymmetric variation, that is, one cluster is kept at 1.58 Å for the variation in sulfur-to-gold distances and one nanoparticle is kept at 85 gold atoms for the variation in cluster size, we obtain the results presented in Figures 6 and 7. Here, the center of inversion is broken because of an asymmetrical variation, and thus, a response is seen for both conformations. In both cases, the hyperpolarizability converges toward that found with only one gold cluster. Again, only little effect is seen in varying the cluster size, but it is sufficient to provoke a response in the staggered conformation, although of minute proportions. If the value of interest is chosen to be βxzz instead of β||, an even larger effect of the asymmetric variation of distance is observed, as seen in Figure 8. Here, the effect of breaking symmetry is seen for both conformations, which justifies the assumption that the π-system is being perturbed as the hydrogen atoms take no or little part in this, and thus, a center of inversion of the π-system can be found in both systems, which is broken by an asymmetric variation. It can also be seen for the eclipsed conformation in the symmetric variation, where the value of βxzz is relatively low. V. Conclusion We have presented theoretical methods and computations of the effects of nanoparticles on frequency-dependent first hyperpolarizabilities of symmetric molecules. We have utilized quantum mechanical/molecular mechanics (QM/MM) response methods for calculating electromagnetic properties of molecules interacting with nanoparticles and frequency-dependent electromagnetic fields. The frequency-dependent first hyperpolar-
izability of (p)-benzenedithiol in the presence of zero, one, or two gold nanoparticles of varying sizes and distances is calculated by DFT/MM. The hyperpolarizability of the molecule depends strongly on the distance between the nanoparticles and the molecule, whereas the size of the nanoparticle is of little importance. We clearly show that metal nanoparticles are able to induce a first hyperpolarizability in symmetric molecules. The plasmon response (the oscillations of the free electron density against the fixed positive ions in a metal) of the gold cluster is not included; nevertheless, we observe an induced hyperpolarizability in centrosymmetric molecules. The enhanced hyperpolarizability arises because of the space symmetry breaking induced by the presence of a nanoparticle or an asymmetric arrangement of two gold nanoparticles. Acknowledgment. The authors thanks the Danish Center for Scientific Computing for computational resources. J.K. and K.V.M. thank the Danish Natural Science Research Council/ The Danish Councils for Independent Research and the Villum Kann Rasmussen Foundation for financial support. Th.H. thanks the Carlsberg Foundation for financial support. References and Notes (1) Aviram, A.; Ratner, M. A. Chem. Phys. Lett. 1974, 29, 277. (2) Mujica, V.; Kemp, M.; Roitberg, A.; Ratner, M. A. J. Chem. Phys. 1996, 104, 7296. (3) Mujica, V.; Kemp, M.; Ratner, M. A. J. Chem. Phys. 1994, 101, 6849. (4) Nitzan, A.; Ratner, M. A. Science 2003, 300, 1384. (5) Lindsay, S. M.; Ratner, M. A. AdV. Mater. 2007, 19, 23. (6) Park, H.; Park, J.; Lim, A. K. L.; Anderson, E. H.; Alivisatos, A. P.; McEuen, P. L. Nature 2000, 407, 57. (7) Park, J.; Pasupathy, A. N.; Goldsmith, J. I.; Chang, C.; Yaish, Y.; Petta, J. R.; Rinkoski, M.; Sethne, J. P.; Abrun˜a, H. D.; McEuen, P. L.; Ralph, D. C. Nature 2002, 417, 722. (8) Liang, W.; Shores, M. P.; Bockrath, M.; Long, J. R.; Park, H. Nature 2002, 417, 725. (9) Kubatkin, S.; Danilov, A.; Hjort, M.; Cornil, J.; Bre`das, J. -L.; StuhrHansen, N.; Hedegård, P.; Bjørnholm, T. Nature 2003, 425, 698. (10) Chen, J.; Reed, M. A.; Rawlett, A. M.; Tour, J. M. Science 1999, 286, 1550.
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J. Phys. Chem. C, Vol. 114, No. 48, 2010
(11) Blum, A. S.; Kushmerick, J. G.; Long, D. P.; Patterson, C. H.; Yang, J. C.; Henderson, J. C.; Yao, Y. X.; Tour, J. M.; Shashidhar, R.; Ratna, B. R. Nat. Mater. 2005, 4, 167. (12) van der Molen, S. J.; Liljeroth, P. J. Phys.: Condens. Matter 2010, 22, 133001. (13) Galperin, M.; Nitzan, A. Phys. ReV. Lett. 2005, 95, 206802. (14) Galperin, M.; Nitzan, A. J. Chem. Phys. 2006, 124, 234709. (15) Orrit, M. Nature 2009, 460, 42. (16) Hwang, J.; Pototschnig, M.; Lettow, R.; Zumofen, G.; Renn, A.; Goetzinger, S.; Sandoghdar, V. Nature 2009, 460, 76. (17) Ward, D. R.; Halas, J. N.; Ciszek, J. W.; Tour, J. M.; Wu, Y.; Nordlander, P.; Natelson, D. Nano Lett. 2008, 8, 919. (18) Sukharev, M.; Galperin, M. Phys. ReV. B 2010, 81, 165307. (19) Hansen, T.; Mikkelsen, K. V. Theor. Chem. Acc. 2004, 111, 122. (20) Hansen, T.; Pedersen, T. B.; Mikkelsen, K. V. Chem. Phys. Lett. 2005, 405, 118. (21) Kongsted, J.; Osted, A.; Mikkelsen, K. V.; Christiansen, O. J. Phys. Chem. A 2003, 107, 2578. (22) Kongsted, J.; Osted, A.; Mikkelsen, K. V.; Christiansen, O. J. Chem. Phys. 2003, 118, 1620. (23) Kongsted, J.; Osted, A.; Mikkelsen, K. V.; Christiansen, O. J. Chem. Phys. 2003, 119, 10519. (24) Kongsted, J.; Osted, A.; Mikkelsen, K. V.; Christiansen, O. J. Chem. Phys. 2004, 120, 3787. (25) Kongsted, J.; Osted, A.; Mikkelsen, K. V.; Christiansen, O. Mol. Phys. 2002, 100, 1813. (26) Mikkelsen, K. V. Annu. ReV. Phys. Chem. 2006, 57, 365. (27) Nielsen, C. B.; Christiansen, O.; Mikkelsen, K. V.; Kongsted, J. J. Chem. Phys. 2007, 126, 154112. (28) Hansen T. A Molecule Wired. Ph.D. Thesis, University of Copenhagen, 2005. (29) (a) Kongsted, J.; Osted, A.; Mikkelsen, K. V.; Christiansen, O. J. Phys. Chem. A. 2003, 107, 2578. (b) Poulsen, T. D.; Ogilby, P. R.; Mikkelsen, K. V. J. Chem. Phys. 2001, 115, 7843. (c) Kongsted, J.; Osted, A.; Mikkelsen, K. V.; Astrand, P. O.; Christiansen, O. J. Chem. Phys. 2004, 121, 8435. (d) Poulsen, T. D.; Kongsted, J.; Osted, A.; Ogilby, P. R.; Mikkelsen, K. V. J. Chem. Phys. 2001, 115, 2393. (30) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (31) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. (32) Yanai, T.; Tew, D. P.; Handy, N. C. Chem. Phys. Lett. 2004, 393, 51. (33) Peach, M. J. G.; Helgaker, T.; Sałek, P.; Keal, T. W.; Lutnæs, O. B.; Tozer, D. J.; Handy, N. C. Phys. Chem. Chem. Phys. 2006, 8, 558. (34) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864. (35) Jensen, F. Introduction to Computational Chemistry, 2nd ed.; Wiley: New York, 2007. (36) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133. (37) Gritsenko, O. V.; Schipper, P. R. T.; Baerends, E. J. J. Chem. Phys. 1997, 107, 5007. (38) Thomas, L. H. Proc. Cambridge Philos. Soc. 1927, 23, 542. (39) Fermi, E. Rend. Accad. Naz. Lincei 1927, 6, 602. (40) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press Inc.: New York, 1989.
Hansen et al. (41) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (42) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (43) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (44) Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K. J. Chem. Phys. 2001, 115, 3540. (45) Hehre, W. J.; Ditchfield, R. J.; Pople, J. A. J. Chem. Phys. 1972, 56, 2257. (46) Binkley, J. S.; Pople, J. A. J. Am. Chem. Soc. 1980, 102, 939. (47) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J. Chem. Phys. 1980, 72, 650. (48) Frisch, M. J.; Pople, J. A.; Binkley, J. S. J. Chem. Phys. 1984, 80, 3265. (49) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; DeFrees, D. J.; Pople, J. A. J. Chem. Phys. 1982, 77, 3654. (50) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007. (51) Kendell, R. A.; Dunning, T. H.; Harrison, R. J. J. Chem. Phys. 1992, 96, 6796. (52) Dalton Quantum Chemistry Program, release 2.0. http://www. kjemi.uio.no/software/dalton/dalton.html 2005. (53) Ceperley, D. M.; Alder, B. J. Phys. ReV. Lett. 1980, 45, 566. (54) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, revision B.05; Gaussian, Inc.: Wallingford, CT, 2004. (55) Bilic, A.; Reimers, J. R.; Hush, N. S. J. Chem. Phys. 2005, 122, 094708. (56) Wang, J.; Yang, M.; Jellinek, J.; Wang, G. Phys. ReV. A 2006, 74, 023202. (57) van Duijnen, P. T.; Swart, M. J. Phys. Chem. A. 1998, 102, 2399. (58) Arcisauskaite, V.; Kongsted, J.; Hansen, T.; Mikkelsen, K. V. Chem. Phys. Lett. 2009, 470, 285. (59) Jensen, L.; Schmidt, O. H.; Mikkelsen, K. V.; Astrand, P. O. J. Phys. Chem. B 2000, 104, 10462. (60) Boyd, R. W. Nonlinear Optics; Academic Press: London, 1992. (61) Luo, Y.; Agren, H.; Mikkelsen, K. V.; Jorgensen, P. AdV. Quantum Chem. 1995, 26, 168. (62) Jacquemin, D.; Perpe`te, E. A.; Scalmani, G.; Frisch, M. J.; Kobayashi, R.; Adamo, C. J. Chem. Phys. 2007, 126, 114105.
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