Calculations of One- and Two-Photon Absorption Spectra for

Article pubs.acs.org/JPCA

Calculations of One- and Two-Photon Absorption Spectra for Molecular Metal Chalcogenide Clusters with Electron-Acceptor Ligands Published as part of The Journal of Physical Chemistry virtual special issue “Mark S. Gordon Festschrift”. Kiet A. Nguyen,*,† Ruth Pachter,*,† and Paul N. Day†,‡ †

Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, United States UES, Inc. Dayton, Ohio 45432, United States

‡

ABSTRACT: We present calculated one- and two-photon absorption (OPA, TPA) spectra for molecular neutral, cation, and anion cadmium chalcogenide nonstoichiometric clusters [CdnE′m′(ER)m, E = S and Se, R = hydrogen, methyl, phenyl, para-nitrophenyl, para-cyanophenyl], ranging from less than 1 nm to more than 2 nm in size with well-defined structures. A systematic treatment of the clusters is carried out to assess the effects of size and ligand on their linear and nonlinear optical properties. Ligands and cluster size were found to have a large influence on the color and intensity of the electronic absorption spectra. TPA cross sections were found to increase linearly with cluster size. Electron-accepting ligands were also found to induce linear enhancement in TPA cross sections. Blue shifts of TPA maxima were observed for the first band with reduced molecular size. The effects of phenyl, para-nitrophenyl, and para-cyanophenyl substitutions, as well as changes in the chalcogenide atom, have been analyzed in detail. one-photon absorption (OPA).11 Although not fully characterized, experiments with a larger cluster were also reported to have strong CT linear absorption.11 Thus, similar to the effects of substituents on the enhancement of optical properties of organic molecules, nonstoichiometric chalcogenide clusters are promising candidates for tuning linear and nonlinear optical properties using electron-acceptor/donor ligands. In general, cadmium chalcogenide nonstoichiometric clusters belong to the supertetrahedral (Tn), penta-supertetrahedral (Pn), and capped supertetrahedral (Cn) (n is the number of metal layers in each cluster; see Figure 1) topological classes,23 covering a size range of less than 1 nm [Cd4(SeR)6L42−, R = hydrogen, phenyl (Ph); L is a coordinating ligand such as a halogen atom, water, or phosphine) to more than 2 nm [M54Se32(SeR)48L4]. Tn clusters are regular tetrahedrally shaped fragments of the cubic ZnS-type lattice, while each Pn is an assembly of four Tn clusters tetrahedrally distributed onto four faces of one Tn cluster with interconverted cationic and anionic sites. The Cn cluster cores of these molecular clusters form a tetrahedrally bound II−IV atoms network consisting of primarily fused adamantane cages. For IInVIm Tn clusters, the largest experimentally observed cluster is T3. However, larger CdSe tetrahedral clusters with interconverted cationic and anionic sites (Tin) have been synthesized with carboxylate

1. INTRODUCTION The elements of Groups II and VI that form nonstoichiometric chalcogenide (designated as IInVIm) clusters with well-defined size and structure serve as model material systems with sizedependent properties.1,2 A wide range of such ultrasmall semiconductor quantum dots (QDs) have been successfully produced using IInVIm molecular clusters with unique structures and sizes that are similar to those of small stoichiometric (n = m) QDs.3−9 Because these clusters serve as fundamental building blocks for new materials that overcome relatively low photostability and two-photon absorption (TPA) cross sections, extensive experimental3−9 and theoretical studies9−17 have been carried out to understand the role of size variance, solvent, and surface ligands on linear10−15 and nonlinear18 optical properties. Depending on the constituent material, large enhancement of TPA cross sections has been reported with both a decrease and an increase in QD particle size due to resonance enhancement and the increase in the density of states, respectively.19 Using design strategies involving electron-donor and electron-acceptor (EA) substituents in organic chromophores that have been applied to enhance nonlinear optical properties,20−22 hybrid systems based on organic cation dyes and a small CdS anion cluster have also been shown to increase TPA-based upconversion emission.18 A small CdS cluster with (covalently attached) nitrobenzyl substituents has also been synthesized and spectroscopically characterized.11 The cluster with EA substitution, a derivative of Cd4(SC6H5)102−, was reported to have strong and broad donor−acceptor charge-transfer (CT) linear © 2017 American Chemical Society

Received: October 31, 2016 Revised: January 6, 2017 Published: January 25, 2017 1748

DOI: 10.1021/acs.jpca.6b10955 J. Phys. Chem. A 2017, 121, 1748−1759

Article

The Journal of Physical Chemistry A

Figure 1. Structures of molecular chalcogenide clusters belonging to supertetrahedral (Tn), penta-supertetrahedral (Pn), and capped supertetrahedral (Cn) topologies considered, Cd4(ER)6L42− (E = S, Se; R = H, Ph, PhCN, PhNO2; L = Cl, SPh, SPhNO2), Cd8E(ER)6Cl42− (E = S, Se, Te; R = H, Ph, PhCN, PhNO2), Cd10Se4(SeR)12L4 (R = H, Ph, PhCN, PhNO2; L = Cl, PnPr3), Cd17Se4(SeH)282−, Cd32Se14(SeH)36PH3, Cd54Se32(SeH)48(H2O)4, and Cd54Se32(SH)48(H2O)4.

and amine ligands.24 The Tn, Pn, and Cn clusters have been synthesized with various bridging and coordinating ligands. Their crystal structures and OPA spectra have also have reported, allowing spectral interpretation and detailed comparisons with computed results reported in our previous work.16 In this work, we performed density functional theory (DFT) and linear and quadratic response time-dependent DFT (TDDFT) calculations on a series of cadmium chalcogenide nonstoichiometric clusters (Figure 1). OPA spectral assignments are also presented for clusters with EA [para-cyanophenyl (PhCN) and para-nitrophenyl (PhNO2)] ligands that have not been previously considered. The effects of Ph, PhCN, and PhNO2 substitutions, as well as changes in the chalcogen atoms, are discussed for the P1 Cd8E(ER)6L42− (E = S, Se, and Te; R = Ph, PhCN, and PhNO2; L = Cl) cluster. Our quantum chemical calculations also provide insights into the origin of ligand-induced changes in optical properties by examining the density of excited states and the electronic nature of participating orbitals.

0.11 eV and are, therefore, employed to study structures and spectra in the present work. The errors predicted by the PBE0 functional for CdS and CdSe clusters with phenyl and alkyl ligands may not however persist for other clusters and/or ligands, especially for ligands that induce strong CT transitions. Thus, RS hybrid functionals were also examined for their applicability. Kohn−Sham (KS)27 DFT and TDDFT electronic structure calculations were done using the Stuttgart/Dresden (SD) valence basis set, and effective core potentials were used for Cd, (7s7p5d)/[5s5p2d],28,29 and Se, (4s5p2d)/[2s2p2d],30 including two additional sets of d functions for Se (ζ = 0.475412, 0.207776)31 atoms. The addition of polarization functions for atoms with high coordination in the clusters is crucial for obtaining accurate structures and energetics.32,33 Other atoms were treated with the 6-31G(d) basis set.34,35 The effects of basis set going from SD-6-31G(d) to the larger Def2-TZVP36 basis set were found to be small; a mean absolute deviation of 0.04 eV in excitation energies was obtained between the two basis sets for the first 10 states of the Cd4(SePh)6Cl42− cluster.16 For the T3 cluster with p-nitrophenyl ligands and the large Cn clusters, calculations for OPA and TPA were also done using the smaller LANL2DZ basis set augmented with an additional set of d-functions (ζ = 0.363)37 for Se [LANL2DZ(d)]. Using the PBE0/SD-6-31G(d) structures, the PBE0/LANL2DZ(d) excitation energies have mean deviations within 0.2 and 0.4 eV for the T3 (with p-nitrophenyl ligands) and C3 clusters, respectively. TDDFT singlet excitation energies and oscillator strengths using the dipole length representation were calculated using the Gaussian 0938 and GAMESS39 programs. For some clusters, there are a number of conformational isomers due the orientations at each of the pyramidal ER groups.40 Thus, Boltzmann averaging of excited-state properties was done with low-energy isomers (Qi) of these clusters. The OPA extinction coefficients

2. COMPUTATIONAL METHODS Structures for clusters without cyano and nitro groups were determined in our previous16 work using the PBE025 functional. The computed bond distances are about 0.03 Å longer for both CdS and CdSe clusters compared to the X-ray structures.16 The low-energy structures for clusters with less than 54 Cd atoms have been verified to be minima on the potential energy surface by previous harmonic frequency calculations. In benchmark calculations compared to a number of hybrid and rangeseparated (RS) hybrid functionals, the PBE0 functional was previously found to provide a good description for both the ground and excited states for ZnS and CdSe stoichiometric clusters.26 For CdS and CdSe nonstoichiometric clusters,16 PBE0 excitation energies have a mean absolute error (MAE) of 1749


Article


Table 1. Summary of Computeda OPA and TPA Peak Maximab (in eV) and Intensities for Clusters with Phenyl, p-Nitrophenyl, and p-Cyanophenyl Ligands Using the PBE0 Functionalc OPA

TPA

symmetryd

Emax

εmax

Cd4(SPhNO2)102− Cd4(SePh)6Cl42−

C3 C3

3.55 4.34

1.05 × 105 1.01 × 105

Cd4(SePhNO2)6Cl42− Cd4(SePhCN)6Cl42− Cd8S(SPh)12Cl42−

C3 C3 T

Cd8S(SPhCN)12Cl42−

T

Cd8Se(SPhNO2)12Cl42− Cd8Se(SePh)12Cl42−

T T

Cd8Se(SePhCN)12Cl42−

T

Cd8Se(SePhNO2)12Cl42− Cd8Te(TePh)12Cl42−

T T T

Cd8Te(TePhCN)12Cl42−

T

3.47 4.06 4.34 4.94 4.15 4.71 4.03 4.02 4.61 3.86 4.49 3.54 3.77 3.94 4.18 3.72 4.24

1.80 × 105 2.11 × 105 1.08 × 105 1.01 × 105 1.23 × 105 3.93 × 105 3.10 × 105 1.10 × 105 6.69 × 104 1.44 × 105 2.47 × 105 1.67 × 105 2.22 × 105 1.12 × 105 5.06 × 104 1.75 × 105 2.09 × 105

Cd8Te(TePhNO2)12Cl42−

T

Cd10Se4(SePh)12(PnPr3)4

T

Cd10Se4(SePh)12Cl44−

T

3.32 3.60 4.13 4.65 4.04

1.93 × 105 1.77 × 105 1.27 × 105 6.36 × 104 1.38 × 105

Cd10Se4(SePhNO2)12Cl44− Cd10Se4(SePhCN)12Cl44−

T T

2.92 3.80 4.03

1.75 × 105 2.91 × 105 1.78 × 105

cluster

Emax

δmax

4.12 4.76 3.61 4.43 4.37 5.22 4.16 4.74 4.11 4.48 4.87 4.21 4.56 3.83

4 22 330 299 15 185 14 287 475 52 52 146 529 567

3.93 4.46 3.77 4.15 4.32 3.45

8 43 67 340 337 675

4.63

50

3.86 4.01 4.50 2.96 4.06

2 2 55 445 317

a

Computed values for the Cd10Se4(SePhNO2)12Cl44− cluster are obtained with the LANL2DZ(d) basis set. OPA and TPA maxima obtained with LANL2DZ(d) for Cd10Se4(SePhCN)12Cl44− are 3.84 (1.82 × 105) and 4.04 (307), respectively. bComputed maxima were obtained using a Gaussian line shape with a fwhm of 0.2 eV; the experimental maximum obtained in methanol was 4.76 eV for Cd4(SPh)102−; experimental values obtained in Nujol are 4.24 [Cd4(SePh)6Cl42−], 4.084,41 [Cd8Se(SePh)12Cl42−], 3.88 (5.7 × 105),4 3.92, and 4.1758 [Cd10Se4(SePh)12(PnPr3)4]. cOPA extinction coefficients are in M−1 cm−1, while TPA cross sections are in GM. dHighest symmetry of the cluster.

Table 2. Summary of Computeda OPA and TPA Peak Maximab (in eV) and Intensities for Clusters with Hydride Ligands Using the PBE0 Functionalc OPA cluster

symmetry

Cd4(SeH)6Cl42−

C3

Cd8(SeH)12Cl42−

T

Cd8(SeH)162−

T

Cd10Se4(SeH)12(PH3)4

T

Cd17Se4(SeH)282−

D2

Cd17Se4(SeH)24(PH3)42+

T

Cd32Se14(SeH)36(PH3)4

T

d

Emax 4.96 5.48 4.30 4.86 4.35 5.02 3.92 4.41 4.98 3.06 3.66 3.98 4.45 3.91 4.37 4.66 3.38 1750

TPA εmax 1.08 6.93 9.66 2.57 1.09 7.21 6.31 6.64 9.02 3.10 1.16 5.49 9.05 1.48 9.05 5.06 1.03

× × × × × × × × × × × × × × × × ×

105 103 104 104 105 104 104 104 104 103 105 104 104 105 104 104 105

Emax

δmax

4.90 5.34 4.60 5.10 4.84 5.22 4.58 4.92

2 3 5 136 32 55 69 38

3.05 3.93 4.47

5 161 157

4.02 4.42

9 34

3.69

19


Article

The Journal of Physical Chemistry A Table 2. continued OPA cluster

symmetry

Cd54Se32(SeH)48(H2O)44−

D2

Cd54Se32(SH)48(H2O)44−

D2

d

Emax 3.67 4.31 2.98 3.23 3.53 3.05 3.33 3.62

TPA εmax 6.54 1.02 1.06 4.59 7.21 1.05 3.35 7.54

× × × × × × × ×

104 105 105 104 104 105 104 104

Emax

δmax

4.16

356

3.55

505

3.68

463

Computed values obtained with the LANL2DZ(d) basis set are 3.93 eV (1.07 × 105 M−1 cm−1), 4.16 eV (19 GM), and 4.65 eV (263 GM) for Cd32Se14(SeH)36(PH3)4; 3.49 eV (1.17 × 105 M−1 cm−1), 3.65 eV (13 GM), and 3.99 eV (388 GM) for Cd54Se32(SeH)48(H2O)44−; and 3.52 eV (1.12 × 105 M−1 cm−1), 4.04 eV (8.98 × 104 M−1 cm−1), and 3.68 eV (12 GM), 4.07 eV (365 GM) for Cd54Se32(SH)48(H2O)44−. bComputed maxima were obtained using a Gaussian line shape with fwhm of 0.2 eV. Experimental values are 3.57 eV9 [in Nujol for Cd17Se4(SePh)282−], 3.32 eV4 [in Nujol for Cd32Se14(SePh)36(PPh)4], and 3.18 eV6 [in DMF for Cd54Se32(SPh)48(H2O)44−]. cOPA extinction coefficients are in M−1 cm−1, while TPA cross sections are in GM. dHighest symmetry of the cluster. a

are computed with the normalized Gaussian line shape function as22 ε(ν)̃ =

2 ln 2 4.32 × 10−9 π

∑ g i (Q i ) ∑ i

f

⎡ ⎤ −4 ln 2 × exp⎢ fwhm 2 (ν ̃ − ν0̃ f (Q i))2 ⎥ ⎢⎣ (ν f̃ ⎥⎦ )

f0f (Q i) ν f̃fwhm

(1)

where g(Qi), f 0f(Qi), and ν̃0f(Qi) are the Boltzmann factor, oscillator strength, and transition frequency for a given Qi isomer, respectively. The OPA broadening factor, ν̃fwhm , corresponds f to the full width at half-maximum (fwhm) of the Gaussian line shape. The effects of counterions on the computed16 and experimental41 electronic spectra were found to be quite small and were not included in this work. For cadmium thiophenolate and selenophenolate clusters examined in our previous work, the TDDFT excitation energies and extinction coefficients using the PBE0 functional have MAEs of 0.11 eV and 1200 M−1 cm−1, respectively. By relating the absorption rate42,43 to the TPA transition probability using the normalized Gaussian line shape function for linearly polarized photons with parallel polarization, the rotationally averaged44,45 degenerate cross section is obtained as22 16π 4 ⎛⎜ ln 2 ⎞⎟ c 2h ⎝ π ⎠

1/2

δ(2Eλ) =

Eλ 2 ∑ f

|Sf 0|2 E fwhm f

⎡ ⎤ −4 ln 2 × exp⎢ fwhm 2 (2Eλ − Ef )2 ⎥ ⎢⎣ (E f ) ⎥⎦

Figure 2. Experimental and computed absorption spectra in methanol for the Cd4(SPhNO2)102− cluster. (2)

3. RESULTS AND DISCUSSION Supertetrahedral (T2, T3) and Penta-supertetrahedral (P1) Clusters. The computed OPA and TPA absorption maxima and intensities are given in Tables 1 and 2 along with the available OPA experimental values (given in footnotes of the tables) for comparison. In the present work, we mainly focus on the CdSe clusters with 4−54 metallic sites that have been characterized in the chemically inert Nujol oil for the selenophenolate derivatives.3−5 However, no experimental data exist for CdSe clusters with EA ligands. Thus, we begin with the small adamantane-like CdS cluster with PhNO2 ligands that has been

where c is the speed of light, h is Planck’s constant, Eλ is the photon energy, Efwhm is the fwhm of the Gaussian line shape, Sf 0 is f the two-photon matrix element for a two-photon transition between the ground- (0) and exited-state ( f). The two-photon matrix elements are computed with the DFT-based singleresidue quadratic response theory46−48 using the locally modified version of the Dalton program49,50 that includes restartable linear and nonlinear response calculations for the PBE0 functional. TPA cross sections are reported in Goppert-Mayer51 (GM) units (1 GM = 1 × 10−50 cm4 photon−1 molecule−1). 1751


Article

The Journal of Physical Chemistry A Table 3. Computeda Ionization Potential (IP) and Axial Ligand Binding Energy (LBE, in kcal/mol) cluster

IP

LBEb

Cd10Se4(SePh)12(PnPr3)4 Cd10Se4(SePh)12Cl44− Cd10Se4(SePhNO2)12Cl44− Cd10Se4(SePhCN)12Cl44−

135.3 −16.9 23.8 19.9

16.3 5.5 37.3 35.6

0.6−3.2 kcal/mol in solvent) for Cd4(SPh)102− isomers, larger (≥2.0 kcal/mol) energy differences for the lowest-energy isomer of Cd4(SPhNO2)102− were found relative to other isomers. Thus, the most intense (3.12 × 105 M−1 cm−1) transition predicted by the PBE0 functional in methanol with the PCM solvation model occurs at 3.42 eV (3.55 eV and 1.05 × 105 M−1 cm−1 in the gas phase) for the lowest-energy isomer, which is about the same as the Boltzmann-averaged value (see Table 1 and Figure 2). The observed broad band comprises several excited states with large oscillator strengths. Good agreement between the experimental and computed PBE0 spectra provides the basis for our selection of the functional and basis sets in the present work. Note that RS functionals53 such as the CAM-B3LYP,54 CAM-QPT(00),55 and CAM-QPT(01)56 that are designed to treat Rydberg and CT transitions show less satisfactory results for Cd4(SPhNO2)102−. These RS functionals have been optimized with a percentage of exact Hartree−Fock (HF) exchange and long-range HF and a range-separation parameter to produce various ground- and excited-state properties in different data sets. The QPT family also requires that the KS eigenvalues of the occupied orbital approximately reproduce the vertical ionization energies. A reasonable range of errors in excitation energies for systems with varying degrees of CT character remains a challenge.53 Because the range-separation parameter is not a

a

PBE0/SD-6-31G(d)//PBE0/SD-6-31G(d). bThe energy per ligand included zero-point energy corrections.

synthesized and characterized by Yoon et al.11 in order to establish the accuracy of the computational methods. The OPA spectrum for the Cd4(SPhNO2)102− cluster in methanol was reported to have an intense (13600 M−1 cm−1) transition that was assigned to a cluster-to-nitrobenzene CT transition with a maximum at 3.30 eV (376 nm), demonstrating a large red shift from the absorption peak of 4.76 eV (260 nm) of the phenyl derivative in the same solvent. Isomerization can occur through the axial−equatorial inversion of the nitrophenyl groups at the bridging sulfur atom, leading to different Cd4(SPhNO2)102− conformers, in analogy to those observed in the Cd4(SPh)102− cluster.16,52 While the variation in the groundstate energies is small (0.3−0.5 kcal/mol in the gas phase and

Figure 3. Lowest (L) virtual, higher virtual, highest (H) occupied, and lower occupied orbitals with symmetry labels at a cutoff value of 0.01 au for CdSe clusters. Orbital energies (in eV) are in parentheses. 1752


Article


Figure 4. Lowest (L) virtual, higher virtual, highest (H) occupied, and lower occupied orbitals with symmetry labels at a cutoff value of 0.01 au for CdSe clusters. Orbital energies (in eV) are in parentheses.

more than double the corresponding peak intensity. For the phenolate P1 and T3 clusters, the cyanophenyl ligands induce red shifts of about 0.2 eV and increase the intensities by 36 and 111%, respectively. The PhNO2 ligands appear to have much stronger effects on the absorption spectra of these clusters, shifting the maxima by 1.1 and 0.9 eV for the T3 Cd10Se4(SePh)12Cl44− and T2 Cd4(SePh)6Cl42− anion clusters, respectively. Their OPA peak intensities modestly increase by 52 (T3) and 72% (T2). In contrast, the PhNO2 groups exert smaller red shifts on the first absorption maximum of the P1 Cd8Se(SePh)12Cl42− anion (0.5 eV) clusters. For the T3 clusters, the Cd10Se4(SePh)12Cl44− cluster has not been synthesized, although their CdS halide counterparts have been prepared and characterized. 59 The Cd 10 Se 4(SePh)12 Cl 44− cluster was found to be less stable toward ionization (Cd10Se4(SePh)12Cl43−) and has a lower ligand binding energy compared with that of the neutral Cd10Se4(SePh)12(PnPr3)4 cluster (see Table 3). However, EA ligands were found to increase the ionization potential and binding energy of the Cd10Se4(SePh)12Cl44− cluster.

universal constant, optimizing it to satisfy the ionization potential theorem for a specific system has been shown to provide better results for RS hybrid functionals.57 Turning to the linear absorption of CdSe phenolate clusters that have been reported in chemically inert Nujol oil,3−5 we first consider a small Cd4(SePh)6Cl42− cluster and then examine the effects EA ligands (PhNO2 and PhCN) as the cluster grows larger in size to Cd8Se(SePh)12Cl42−, Cd10Se4(SePh)12Cl44−, and Cd10Se4(SePh)12(PPr3)4. The effects of changing both chalcogen atoms (E = S, Se, Te) and ligands (R = H, Ph, PhCN, PhNO2) are examined in the P1 Cd8E(ER)12Cl42−cluster. The first computed maximum for the T1 Cd4(SePh)6Cl42− was located at 4.34 eV (1.01 × 105 M−1 cm−1) and red shifted to slightly higher than 4 eV for Cd8Se(SePh)12Cl42−, Cd10Se4(SePh)12Cl44−, and Cd10Se4(SePh)12(PnPr3)4. The predicted values for P1 and neutral T3 clusters were found to be in good agreement with the lowest-energy observed maxima at around 4 eV4,5,58 in Nujol.17 These computed values provide references for spectral shifts induced by EA ligands. The PhCN ligands shift the T1 Cd4(SePh)6Cl42− maximum by about 0.3 eV to the red but 1753


Article


Figure 5. Lowest (L) virtual, higher virtual, highest (H) occupied, and lower occupied orbitals with symmetry labels at a cutoff value of 0.01 au for CdSe clusters. Orbital energies (in eV) are in parentheses.

HOMOs are linear combinations of nonbonding π orbitals involving major contributions from the four Se atoms, while the LUMO has dominant contributions from 5s orbitals of the inner Cd atoms. More intense bands (OPA: 3.66 eV and 1.16 × 105 M−1 cm−1; TPA: 3.93 eV and 161 GM) for the C1 clusters are obtained from excitations from the lower-energy occupied MOs with π orbitals of the outer and terminal Se atoms to the LUMO. Because the effects of the phenyl on the excitation energies are modest, they are not included in the C1 and larger clusters for TPA calculations. For the larger C2 clusters, the first (3.38 eV) and second (3.67 eV) OPA bands obtained for Cd32Se14(SeH)36·4PH3 were found to be strong (1.03 × 105 M−1 cm−1) and moderate (6.54 × 104 M−1 cm−1) in intensity. The predicted maxima were in good agreement with the experimental maximum of 3.3 and 3.6 eV taken in Nujol.16 Interestingly, the corresponding TPA bands for the cluster were blue shifted from those of Cd17Se4(SeH)282−, located at the second OPA peak (3.69 eV and 19 GM) and at higher energy (4.16 eV and 356 GM). For C3 clusters [Cd54Se32(SeH)48·4H2O4− and Cd54Se32(SH)48·4H2O4−], the first strong transitions were located at 3.5 eV (1.1−1.2 × 105 M−1 cm−1) and ∼3 eV (1.1 × 105 M−1 cm−1)

For Cd4(SePh)6Cl42−, the first transitions were found to arise from the HOMO−LUMO excitations.16 The HOMO and lower occupied MOs have major contributions from the π orbitals involving SePh (Figure 3). The π character of these MOs also extends to SePhCN and SePhNO2 ligands for other EA clusters. The LUMOs and the higher virtual MOs are characterized by the π orbitals from the ligands (Cl, SePh) and the 5s orbitals of the Cd atoms, which are reduced in the higher virtual MOs. In clusters with EA ligands (SePhCN and SePhNO2), metallic characters of virtual MOs are significantly reduced. Thus, weak CT characters can be invoked in the spectral interpretation of these clusters. The combination of metallic and π-ligand bonding patterns is also observed for the larger clusters (Figures 4 and 5). The MOs, however, are more delocalized, leading to red shifts in excitation energies as the cluster grows larger in size. Capped Supertetrahedral (C1, C3, C3) Clusters. For the C1 anionic clusters, the first four nearly degenerate transitions (1A, 1B1, 1B2,and 1B3) were predicted to underlie the first weak OPA (3.10 × 103 M−1 cm−1) and TPA (5 GM) bands at about 3 eV for Cd17Se4(Seh)282− (Table 2), involving excitations from HOMO and lower-energy MOs to the LUMO.16 The first four 1754


Article


Figure 6. Computed OPA spectra for Cd8E(ER)12Cl42− using a Gaussian line shape of 0.2 eV for the fwhm.

(0.5−0.6 eV) red shifts are calculated by both basis sets upon going from the C2 to the C3 Cd54Se32(SH)48·4H2O4− cluster. TPA spectra for the C3 clusters are found to be weak at low energy. Strong TPA bands are located at about the same energy as the second intense OPA bands, 3.55 eV (505 GM) and 3.68 eV (463 GM) for the Cd54Se32(SeH)48·4H2O4− and Cd54Se32(SH)48· 4H2O4− clusters, respectively, as calculated by the SD-6-31G(d)

in the gas phase with the SD-6-31G(d) and LANL2DZ(d) basis sets, respectively. Note that the size dependence shifts are not qualitatively changed when using the same basis set. For Cd54Se32(SH)48·4H2O4−, predicted SD-6-31G(d) and LANL2DZ(d) first absorption maximum are slightly lower and higher, respectively, than the experimental value of 3.2 eV taken in DMF. However, about the same OPA (0.3−0.4 eV) and TPA 1755


Article


Figure 7. Computed TPA spectra for Cd8E(ER)12Cl42− using a Gaussian line shape of 0.2 eV for the fwhm.

Size-Dependent TPA. An empirical correlation between CdSe (2.1−4.8 nm) and CdTe (4.4−5.4 nm) QDs’ size and TPA cross sections was found to have a power dependence on the diameter of the dots.60 Measurements of TPA cross sections in hexane for 2.0−3.9 nm diameter CdSe dots using 775 nm 160 fs laser pulses have been found to grow with the dot size.60 Models based on the effective mass approximation for general shapes (sphere, rod, cube) have been developed to account for the observed TPA cross sections.61−63 However, the performance of these simple models was reported to deteriorate for CdSe QDs with diameters less than 3 nm due to the effects of confinement and surface ligands.63 Indeed, even CdSe clusters with diameters larger than 2 nm have been shown to have well-defined structures.6,24 For the nonstoichiometric CdSe molecular clusters, computed TPA cross sections increase linearly with molecular size, increasing from 2 to 3 GM for Cd4(SeH)6Cl42− to about 500 GM for Cd54Se32(SeH)48(H2O)44− (Table 2 and Figure 8). A similar increase was also obtained for the [Cd4(SePh)6Cl42−−Cd10Se4(SePh)12(PnPr3)4] clusters with phenyl

basis set. The corresponding LANL2DZ(d) energies and cross sections are higher (∼0.4 eV) and lower (∼100 GM), respectively. Chalcogen P1 Clusters. The OPA and TPA spectra for Cd8E(ER)12Cl42− (E = Te, Se, S; R = Ph, PhCN, PhNO2) are shown in Figures 6 and 7, respectively. The OPA and TPA bands red shift for heavier chalcogen atoms and stronger EA groups. The first absorption peaks follow the order S > Se > Te, Ph > PhCN > PhNO2 from high to low energy. Substituting the chalcogen atoms in Cd8S(SPh)12Cl42− with more polarizable ones red shifts the first OPA maximum (4.34 eV) by 0.32 (Se) and 0.40 eV (Te). Magnitudes of chalcogen-induced shifts are larger than those of the corresponding PhCN shifts (∼0.2 eV). Within a given cluster, PhNO2 ligands produce a spectral red shift of 0.32, 0.48, and 0.62 eV for the CdS, CdSe, and CdTe clusters, respectively. With regard to OPA and TPA absorption intensities, stronger EA ligands and heavier chalcogen atoms in a cluster yield more intense absorption in the first bands, with the largest values (1.93 × 105 M−1 cm−1 and 675 GM) obtained for the Cd8Te(TePhNO2)12Cl42− cluster. 1756


Article


also found to induce linear enhancement in TPA cross sections and OPA extinction coefficients. Blue shifts are observed for the OPA and TPA transitions with reduced molecular size. Because both the system size and ligands can extend the length of conjugation, the general effects for size dependence and ligand are analogous to the particle in a box model. As the number of ligands and the cluster size increase, leading to the increase in density of overlapping excited states and transition dipole moments due to delocalization, the resulting optical band intensity becomes amplified. Although the present study focuses on electron-acceptor ligands, some electron-donor groups are also found to enhance linear and nonlinear absorption in our preliminary calculations, to be further addressed in future work. Thus, in addition to the ligand surface density, different ligand types can be selected to enhance optical properties by extending the length of conjugation for participating orbitals of the excited states/energy levels of interest.

■

Figure 8. Summary of TPA maximum cross sections for CdSe clusters. Energies are in eV.

AUTHOR INFORMATION

ORCID

Kiet A. Nguyen: 0000-0003-0363-313X Paul N. Day: 0000-0002-6333-6359

substituents. A modest increase in molecular size by EA (PhNO2 and PhCN) ligand substitutions results in much larger TPA cross sections. An increase of over 2 orders of magnitude for the first TPA maximum is observed for the nitro (330 GM) and cyano (299 GM) substituents for the smallest Cd4(SePh)6Cl42− cluster (2 GM). TPA cross sections of clusters with EA ligands also grow linearly with the size of the systems, nearly doubling for the Cd8Se(SePhNO2)12Cl42− (567 GM) and Cd8Se(SePhCN)12Cl42− (529 GM) clusters. The effects of ligands appear to reduce with increasing cluster size. The maximum cross sections slightly decrease upon going to the larger Cd10Se4(SePhNO2)12Cl44− (445 GM) and Cd10Se4(SePhCN)12Cl44− (317 GM) clusters compared to the corresponding P1 clusters. The decrease in cross sections might be attributed to the computational limit in the number of TPA states that can be obtained in our calculations. As the cluster grows larger in size, the density of states, many of which become degenerate, increase. The overlap of the degenerate and near-degenerate transitions produces an additive effect on the resultant band intensity, which is amplified by EA ligands and polarizable atomic substitutions.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We gratefully acknowledge support from the Air Force Office of Scientific Research and computational resources and helpful assistance provided by the AFRL DSRC.

■

REFERENCES

(1) Corrigan, J. F.; Fuhr, O.; Fenske, D. Metal Chalcogenide Clusters on the Border betweenMoleculesand Materials. Adv. Mater. 2009, 21, 1867−1871. (2) Malik, M. A.; Afzaal, M.; O’Brien, P. Precursor Chemistry for Main Group Elements in Semiconducting Materials. Chem. Rev. 2010, 110, 4417−4446. (3) Soloviev, V. N.; Eichhöfer, A.; Fenske, D.; Banin, U. Molecular Limit of a Bulk Semiconductor: Size Dependence of the “Band Gap” in CdSe Cluster Molecules. J. Am. Chem. Soc. 2000, 122, 2673−2674. (4) Soloviev, V. N.; Eichhöfer, A.; Fenske, D.; Banin, U. Molecular Limit of a Bulk Semiconductor: Size Dependent Optical Spectroscopy Study of CdSe Cluster Molecules. Phys. Status Solidi B 2001, 224, 285− 289. (5) Soloviev, V. N.; Eichhöfer, A.; Fenske, D.; Banin, U. Size Dependent Optical Spectroscopy Study of a Homologous Series of CdSe Cluster Molecules. J. Am. Chem. Soc. 2001, 123, 2354−2364. (6) Zheng, N.; Bu, X.; Lu, H.; Zhang, Q.; Feng, P. Crystalline Superlattices from Single-Sized Quantum Dots. J. Am. Chem. Soc. 2005, 127, 11963−11965. (7) Bendova, M.; Puchberger, M.; Pabisch, S.; Peterlik, H.; Schubert, U. Studies on the Formation of CdS Nanoparticles from Solutions of (NMe4)4[Cd10S4(SPh)16]. Eur. J. Inorg. Chem. 2010, 2010, 2266−2275. (8) Bendova, M.; Puchberger, M.; Schubert, U. Characterization of “Cd 10 S 4 (SPh) 12 ”, the Thermal Decomposition Product of (NMe4)4[Cd10S4(SPh)16]: Synthesis of a Neutral Cd54 Sulfide Cluster and of a Polymeric Chain of Thiolate-Bridged Cd17 Sulfide Clusters. Eur. J. Inorg. Chem. 2010, 2010, 3299−3306. (9) Matheis, K.; Eichhöfer, A.; Weigend, F.; Ehrler, O. T.; Hampe, O.; Kappes, M. M. Probing the Influence of Size and Composition on the Photoelectron Spectra of Cadmium Chalcogenide Cluster Dianions. J. Phys. Chem. C 2012, 116, 13800−13809. (10) Kuz’mitskii, V. A.; Gael, V. I.; Filatov, I. V. Quantum-chemical Calculation of the Electronic Structure and Excited States of ZnS and CdS Clusters.J. J. Appl. Spectrosc. 1996, 63, 594−602.

4. CONCLUSIONS OPA and TPA calculations were carried out to explore optical tunability for molecular cadmium chalcogenide clusters [CdnE′m′(ER) m, E, E′ = S and Se, R = hydrogen, phenyl, paranitrophenyl, para-cyanophenyl], ranging from less than 1 nm to more than 2 nm in size with well-defined structures. Different RS and hybrid functionals were examined to quantify their accuracy in comparison with experiment. Good agreement between the experimental and computed spectra was observed for the small Cd4(SPhNO2)102− cluster using the PBE0 functional. The same functional was, therefore, applied to study other clusters. Our systematic study for clusters of different sizes and ligands carried out with nonlinear response theory to obtain the effects of the clusters’ size and ligand on their nonlinear optical properties has shown some similarities with the corresponding linear results. The size and ligand were found to have a large influence on the color and intensity of the OPA and TPA spectra of the clusters. Due to the increase in the density of states that are nearly degenerate, TPA cross sections were found to increase linearly with cluster size. Electron-withdrawing ligands, delocalized over participating orbitals involved in the electronic transitions, are 1757


Article


Core Potentials: The Atoms Ga - Kr and In - Xe. J. Chem. Phys. 2001, 114, 3408−3420. (32) Nguyen, K. A.; Day, P. N.; Pachter, P. Understanding Structural and Optical Properties of Nanoscale Cdse Magic-Size Quantum Dots: Insight from Computational Prediction. J. Phys. Chem. C 2010, 114, 16197−16209. (33) Matxain, J. M.; Mercero, J.; Fowler, J. E.; Ugalde, J. M. Clusters of II-VI Materials: CdX (X = S,Se, Te). J. Phys. Chem. A 2004, 108, 10502− 10508. (34) Hehre, W. J.; Ditchfield, R.; Pople, J. A. SelfConsistent Molecular Orbital Methods. XII. Further Extensions of Gaussian Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules. J. Chem. Phys. 1972, 56, 2257−2261. (35) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; DeFrees, D. J.; Pople, J. A.; Gordon, M. S. Self-Consistent Molecular Orbital Methods. 23. A Polarization-Type Basis Set for 2nd-Row Elements. J. Chem. Phys. 1982, 77, 3654−3665. (36) Weigend, F.; Ahlrichs, R. Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297−3305. (37) Check, C. E.; Faust, T. O.; Bailey, J. M.; Wright, B. J.; Gilbert, T. M.; Sunderlin, L. S. Addition of Polarization and Diffuse Functions to the LANL2DZ Basis Set for P-Block Elements. J. Phys. Chem. A 2001, 105, 8111−8116. (38) 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.; et al. Gaussian 09, revision A.11.4; Gaussian, Inc.: Wallingford, CT, 2009. (39) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; et al. General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347−1363. (40) Dance, I. G.; Choy, A.; Scudder, M. L. Syntheses, Properties, and Molecular and Crystal Structures of (Me4N)4[E4M10(SPh)16] (E = S, Se; M = Zn, Cd): Molecular Supertetrahedral Fragments of the Cubic Metal Chalcogenide Lattice. J. Am. Chem. Soc. 1984, 106, 6285−6295. (41) Eichhöfer, A.; Hampe, O.; Blom, M. Synthesis, Structures, and Properties of a Series of [Cd8Se(SePh)12Cl4]2− Cluster Compounds with Different Counterions, Vol, 2003, (2003). Eur. J. Inorg. Chem. 2003, 2003, 1307−1314. (42) Sutherland, R. L. Handbook of Nonlinear Optics; Marcel Dekker, Inc.: New York, 1996. (43) Day, P. N.; Nguyen, K. A.; Pachter, R. A TDDFT Study of Oneand Two-Photon Absorption Properties: Donor-π-Acceptor Chromophores. J. Phys. Chem. B 2005, 109, 1803−1814. (44) Monson, P. R.; McClain, W. M. Polarization Dependence of the Two-Photon Absorption of Tumbling Molecules with Application to Liquid 1-Chloronapthaline and Benzene. J. Chem. Phys. 1970, 53, 29− 37. (45) McClain, W. M.; Harris, R. A. In Excited States; Lim, E. C., Ed.; Academic: New York, 1977; Vol. 3, p 1−56. (46) Olsen, J.; Jørgensen, P. Linear and Nonlinear Response Functions for an Exact State and for and MCSCF State. J. Chem. Phys. 1985, 82, 3235−3264. (47) Salek, P.; Vahtras, O.; Helgaker, T.; Agren, H. Density-Functional Theory of Linear and Nonlinear Time-Dependent Molecular Properties. J. Chem. Phys. 2002, 117, 9630. (48) Salek, P.; Vahtras, O.; Guo, J.; Luo, Y.; Helgaker, T.; Agren, H. Calculations of Two-Photon Absorption Cross Sections by Means of Density-Functional Theory. Chem. Phys. Lett. 2003, 374, 446−452. (49) Aidas, K.; Angeli, C.; Bak, K. L.; Bakken, V.; Bast, R.; Boman, L.; Christiansen, O.; Cimiraglia, R.; Coriani, S.; Dahle, P.; et al. The Dalton quantum chemistry program system. Wiley Interdisciplinary Reviews: Computational Molecular Science 2014, 4, 269−284. (50) Dalton, release 2015. http://daltonprogram.org (2015). (51) Goppert-Mayer, M. Elementary Acts with Two Quantum Jumps. Ann. Phys. 1931, 401, 273−294.

(11) Yoon, D. I.; Selmarten, D. C.; Lu, H.; Liu, H.-J.; Mottley, C.; Ratner, M. A.; Hupp, J. T. Spectroscopic and Photophysical Studies of Apparent Cluster-to-Organic-Acceptor Charge Transfer in a Molecular Cadmium Sulfide Assembly. Chem. Phys. Lett. 1996, 251, 84−89. (12) Eichkorn, K.; Ahlrichs, R. Cadmium Selenide Semiconductor Nanocrystals: A Theoretical Study. Chem. Phys. Lett. 1998, 288, 235− 242. (13) Lopez del Puerto, M.; Tiago, M. L.; Chelikowsky, J. R. Excitonic Effects and Optical Properties of Passivated CdSe Clusters. Phys. Rev. Lett. 2006, 97, 096401/1−096401/4. (14) Lopez del Puerto, M.; Tiago, M. L.; Chelikowsky, J. R. Ab Initio Methods for the Optical Properties of Cdse Clusters. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 77, 045404/1−045404/10. (15) Frenzel, J.; Joswig, J.-O.; Seifert, G. Optical Excitations in Cadmium Sulfide Nanoparticles. J. Phys. Chem. C 2007, 111, 10761− 10770. (16) Nguyen, K. A.; Pachter, R.; Day, P. N.; Su, H. Theoretical Analysis of Structures and Electronic Spectra in Molecular Cadmium Chalcogenide Clusters. J. Chem. Phys. 2015, 142, 234305/1−234305/ 11. (17) Zhu, X.; Chass, G. A.; Kwek, L.-C.; Rogach, A. L.; Su, H. Excitonic Character in Optical Properties of Tetrahedral CdX (X = S, Se, Te) Clusters. J. Phys. Chem. C 2015, 119, 29171−29177. (18) Zhang, X.; Tian, Y.; Jin, F.; Wu, J.; Xie, Y.; Tao, X.; Jiang, M. SelfAssembly of an Organic Chromophore with Cd−S Nanoclusters: Supramolecular Structures and Enhanced Emissions. Cryst. Growth Des. 2005, 5, 565−570. (19) Padilha, L. A.; Nootz, G.; Olszak, P. D.; Webster, S.; Hagan, D. J.; Van Stryland, E. W.; Levina, L.; Sukhovatkin, V.; Brzozowski, L.; Sargent, E. H. Optimization of Band Structure and Quantum-Size-Effect Tuning for Two-Photon Absorption Enhancement in Quantum Dots. Nano Lett. 2011, 11, 1227−1231. (20) Day, P. N.; Nguyen, K. A.; Pachter, R. Calculation of Two-Photon Absorption Spectra of Donor-π-Acceptor Compounds in Solution using Quadratic Response Time-Dependent Density Functional Theory. J. Chem. Phys. 2006, 125, 094103/1−094103/13. (21) Nguyen, K. A.; Day, P. N.; Pachter, R. Effects Conjugation in Lenght and Dimension on Two-Photon Properties of Fluorene-Based Chromophores. Theor. Chem. Acc. 2008, 120, 167−175. (22) Nguyen, K. A.; Day, P. N.; Pachter, R. One- and Two-photon Spectra of Platinum Acetylide Chromophores: A TDDFT Study. J. Phys. Chem. A 2009, 113, 13943. (23) Feng, P.; Bu, X.; Zheng, N. The Interface Chemistry between Chalcogenide Clusters and Open Framework Chalcogenides. Acc. Chem. Res. 2005, 38, 293−303. (24) Beecher, A. N.; Yang, X.; Palmer, J. H.; LaGrassa, A. L.; Juhas, P.; Billinge, S. J. L.; Owen, J. S. Atomic Structures and Gram Scale Synthesis of Three Tetrahedral Quantum Dots. J. Am. Chem. Soc. 2014, 136, 10645−10653. (25) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0Model. J. Chem. Phys. 1999, 110, 6158−6170. (26) Nguyen, K. A.; Pachter, R.; Day, P. N. Computational Prediction of Structures and Optical Excitations for Nanoscale Ultrasmall ZnS and CdSe Clusters. J. Chem. Theory Comput. 2013, 9, 3581−3596. (27) Kohn, W.; Sham, L. J. Self-Consistent Field Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133−A1138. (28) Dolg, M.; Wedig, U.; Stoll, H.; Preuss, H. Energy Adjusted Pseudopotentials for the First Row Transition-Elements. J. Chem. Phys. 1987, 86, 866−872. (29) Andrae, D.; Haeussermann, U.; Dolg, M.; Stoll, H.; Preuss, H. Energy-adjusted Ab Initio Pseudopotentials For the 2nd and 3rd Row Transition-Elements. Theor. Chem. Acc. 1990, 77, 123−141. (30) Bergner, A.; Dolg, M.; Kuechle, W.; Stoll, H.; Preuss, H. Ab-initio Energy-Adjusted Pseudopotentials for Elements of Groups 13−17. Mol. Phys. 1993, 80, 1431−1441. (31) Martin, J. M. L.; Sundermann, A. Correlation Consistent Valence Basis Sets for Use with the Stuttgart-Dresden-Bonn Relativistic Effective 1758


Article

The Journal of Physical Chemistry A (52) Dean, P. A. W.; Vittal, J. J.; Payne, N. C. Syntheses of (Me4N)2[(μ-EPh)6(MX)4](M = Cd, Zn; E = S, Se; X = C1, Br, I). Crystal and Molecular Structures of (Me4N)2[(p-EPh)6(CdBr)4E4 ]= S, Se) and Characterization of [(μ-EPh)6(CdX)4]2− in Solution by 113 Cd and 77Se NMR. Inorg. Chem. 1987, 26, 1683−1689. (53) Nguyen, K. A.; Day, P. N.; Pachter, R. The performance and relationship among range-separated schemes for density functional theory. J. Chem. Phys. 2011, 135, 074109/1−074109/10. (54) Yanai, T.; Tew, D. P.; Handy, N. C. A New Hydrid ExchangeCorrelation Functional using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51−57. (55) Verma, P.; Bartlett, R. J. Increasing the Applicability of Density Functional Theory. IV. Consequences of Ionization-Potential Improved Exchange-Correlation Potentials. J. Chem. Phys. 2014, 140, 18A534/1. (56) Jin, Y.; Bartlett, R. J. The QTP Family of Consistent Functionals and Potentials in Kohn-Sham Density Functional Theory. J. Chem. Phys. 2016, 145, 034107/1−034107/10. (57) Kronik, L.; Stein, T.; Refaely-Abramson, S.; Baer, R. Excitation Gaps of Finite-Sized Systems from Optimally Tuned Range-Separated Hybrid Functionals. J. Chem. Theory Comput. 2012, 8, 1515−1531. (58) Eichhöfer, A.; Hampe, O. Investigating the Thermolysis Products of [Cd10Se4(SePh)12(PnPr3)4] − The New Cluster Ion [Cd17Se4(SePh)28]2−). J. Cluster Sci. 2007, 18, 494−504. (59) Adams, R. D.; Zhang, B.; Murphy, C. J.; Yeung, L. K. Halide enhancement of the luminescence of Cd10S4 thiolate clusters. Chem. Commun. 1999, 383−384. (60) Pu, S.-C.; Yang, M.-J.; Hsu, C.-C.; Lai, C.-W.; Hsieh, C.-C.; Lin, S. H.; Cheng, Y.-M.; Chou, P.-T. The Empirical Correlation between Size and Two-Photon Absorption Cross Sections of CdSe and CdTe Quantum Dots. Small 2006, 2, 1308−1313. (61) Fedorov, A. V.; Baranov, A. V.; Inoue, K. Two-Photon Transitions in Systems with Semiconductor Quantum Dots. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 8627−8632. (62) Feng, X.; Ji, W. Shape-Dependent Two-Photon Absorption in Semiconductor Nanocrystals. Opt. Express 2009, 17, 13140−13150. (63) Makarov, N. S.; Lau, P. C.; Olson, C.; Velizhanin, K. A.; Solntsev, K. M.; Kieu, K.; Kilina, S.; Tretiak, S.; Norwood, R. A.; Peyghambarian, N.; Perry, J. W. Two-Photon Absorption in CdSe Colloidal Quantum Dots Compared to Organic Molecules. ACS Nano 2014, 8, 12572− 12586.

1759


Calculations of One- and Two-Photon Absorption Spectra for

Recommend Documents