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Article Cite This: ACS Omega 2019, 4, 920−931
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Electronic Structure with Dipole Moment and Rovibrational Calculation of Cadmium Chalcogenide Molecules CdX (X = Se, Te) Khalil Badreddine and Mahmoud Korek*
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Faculty of Science, Beirut Arab University, P.O. Box 11-5020 Riad El Solh, Beirut 1107 2809, Lebanon ABSTRACT: Ab initio calculations of 51 electronic states in the representation 2s+1Λ(±) of CdX (X = Se, Te) molecules have been carried out by using the complete active space self-consistent field and multireference configuration interaction (single and double excitations with the Davidson correction). The potential energy along with the static and transition dipole moment curves for the investigated electronic states of the CdX molecules has been mapped. Consequently, the spectroscopic constants Re, ωe, Be, and Te have been computed for the bound states. The spectroscopic dissociation energy De, the zero-point energy, and the ionicity are also calculated for the bound electronic states X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+. Rovibrational calculation is performed for the X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+ states of CdSe together with the X3Π, (1)1Σ+, and (1)1Π states of a CdTe molecule. The Einstein coefficients of spontaneous and induced emissions, A21 and B21, are computed for the transition between the electronic states (1)3Σ+ and X3Π. In the present work, the values are well-consistent with those available in the literature. that the ground state is the 1Σ+ state, and the spectroscopic constants were calculated with and without the inclusion of spin−orbit coupling for the lowest two electronic states 1Σ+ and 3Π of the CdX molecules. The lack of theoretical studies on the halides and chalcogenides of transition metals motivated us to investigate the transition-metal halides15−19 in previous works. In the present research, we examine the CdSe and CdTe molecules. Through ab initio calculations (CASSCF/MRCI), we investigate the potential energy and dipole moment curves (DMCs) for the lowest electronic states in the representation 2s+1Λ(±) of the singlet, triplet, and quintet electronic states of the CdX molecules. For the bound states, the spectroscopic constants are determined. The ionicity20−22 for X3Π and the three lowlying excited states 1Σ+, 3Σ+, and 1Π are evaluated using the computed constants. The emission coefficients are calculated by examining the transition between (1)3Σ+ and X3Π. With regard to the theoretical calculations of the CdX molecules, it is obvious that there are discrepancies in identifying the ground state of the titled molecules. Because of the absence of the experimental data, our investigated values should provide a basis for future experimental studies on these molecules.
1. INTRODUCTION Among the II−VI semiconductors, cadmium chalcogenides have allured special recognition.1,2 On the basis of the quantum confinement effect, the optical and electric properties of their nanoparticles may be modified, reflecting a great potential in many applications as photovoltaic and optoelectronic devices.3 In particular, CdSe quantum dots are used in a wide range of applications including solar cells,4 white light phosphors and light-emitting diode fabrication,5−8 optical chemical sensors,9 and in biomedical imaging.10 Cadmium telluride is important in the fabrication of solar cells, detectors of infrared and gamma radiations, and field-effect transistors.11 The lack of theoretical investigation with regard to the excited electronic states of CdX (X = Se, Te) molecules and the relationship between the energy of solids and molecules12 provoked us to carry out high-level ab initio computations in the present work. Concerning our calculation on the CdX molecules, two very close and low-lying electronic states, 3Π and 1Σ+, exist with dissimilar electric properties and barely different equilibrium geometries. The ground state of CdX alters suddenly with a little adjustment of the geometry. In our study, the X3Π state, associated with the first covalent dissociation asymptote originating from a combination of the Cd(1S) and X(3P) ground states, is identified as a ground state. In 2007, Feng et al.13 reported the equilibrium positions, the spectroscopic constants, and the vibrational levels of the ground-state X3Π and the three low-lying excited states 1Σ+, 3 + Σ , and 1Π of the CdSe diatomic via the complete active space self-consistent field (CASSCF) method followed by multireference configuration interaction (MRCI) calculation. On the other hand, the near-equilibrium potential energy functions of the 1Σ+ and 3Π states have been calculated by MRCI and coupled cluster calculations by Peterson et al.14 They found © 2019 American Chemical Society
2. COMPUTATIONAL DETAILS In the present study, we report the electronic properties and the spectroscopy of CdSe and CdTe molecules in the neutral state. We examine the low-lying singlet, triplet, and quintet electronic states via SA-CASSCF23 followed by MRCI Received: June 11, 2018 Accepted: October 31, 2018 Published: January 10, 2019 920
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Table 1. Lowest Dissociation Asymptotes of the CdX Molecules and the Resulting Low-Lying Electronic States CdSe molecule dissociation limit of atomic levels Cd + Se
total dissociation energy limit of Cd + Se atoms (cm−1)
Cd (1S) + Se (3P) Cd (1S) + Se (1D) Cd (3P0) + Se (3P)
0 9576.149 30 113.99
Cd (3P0) + Se (1D)
39 690.13
molecular states of CdSe (X)3Π, (1)3Σ− (1)1Σ+, (1)1Π, (1)1Δ (1)1Σ+, (2)1Π, (1)1Δ, (2)1Σ− (1)3Σ+, (2)3Π, (1)3Δ, (2)3Σ− (1)5Σ+, (2)5Π, (1)5Δ, (2)5Σ− (1)3Σ+, (1)3Δ
dissociation energy limit of CdSe levels (cm−1) 0 11 007.53 26 342
37 349
CdTe molecule dissociation limit of atomic levels Cd + Te Cd Cd Cd Cd
(1S) + Te (3P) (1S) + Te (1D) (1S) + Te (1S) (3P0) + Te (3P)
Cd (3P0) + Te (1D)
total dissociation energy limit of Cd + Te atoms (cm−1)
molecular states of CdTe (X)3Π, (1)3Σ− (1)1Σ+, (1)1Π, (1)1Σ+ (1)1Σ+, (2)1Π, (2)1Σ− (1)3Σ+, (2)3Π, (2)3Σ− (1)5Σ+, (2)5Π, (2)5Σ− (1)3Σ+, (1)3Δ
0 10 557.87 23 198.39 30 113.99
40 671.86
calculations along with the Davidson correction that estimates the contributions to higher excitation terms.24,25 The ab initio calculations have been achieved via the computational chemistry program Molpro,26 together with the Gabedit interface.27 The quasi-relativistic effective core potentials ECP28MWB and ECP46MWB28 are employed for the Cd/ Se and Te atoms, respectively. For each molecule, 26 electrons are regarded as active ones, and their wave functions are determined by the CASSCF computations. For each molecule, 19 active orbitals are established (Cd: 4s, 4p, 4d, 5s, 5p, 6s; Se: 4s, 4p; 5s; and Te: 5s, 5p, 6s). Among these orbitals, the 4s, 4p, and 4d of cadmium and the 4s of selenium together with the 5s of Te are frozen, so that six valence electrons are clearly considered for each molecule. In the C2v symmetry group, the effective active space includes 5σ (Cd: 5s, 5p0, 6s; Se: 4p0, 5s) and 2π (Cd: 5p±1; Se: 4p±1) for the CdSe molecule along with 5σ (Cd: 5s, 5p0, 6s; Te: 5p0, 6s) and 2π (Cd: 5p±1; Te: 5p±1) for CdTe that are distributed into irreducible representations: 5a1, 2b1, 2b2, 0a2, well-known by [5, 2, 2, 0]. For both molecules, and in the representation 2s+1Λ±, the potential energy curves (PECs) together with the permanent DMCs of singlet, triplet, and quintet electronic states have been investigated as a function of the internuclear distance R in the range 1.7 Å ≤ R ≤ 8.27 Å with a step of 0.03 Å. In addition, the transition DMCs (TDMCs) for the lowest electronic transitions (1)3Σ+−X3Π and (1)1Π−(1)1Σ+ are explored.
(1)1Δ (1)1Δ,
dissociation energy limit of CdTe levels (cm−1) 0 9822.76 24 100.42 26 465.65
(1)3Δ, (1)5Δ, 36 222.83
Figure 1. Potential energy curves of the lowest singlet states of the molecule CdSe.
3. RESULTS AND DISCUSSION The CdX molecules, exhibiting 3Π and 1Σ+ as the lowest lying bound molecular states, are supposed to be mostly ionic around the equilibrium position. Consequently, the chalcogen anion possesses either a pσ or pπ hole resulting in Σ+ or Π electronic states (singlets and triplets), respectively. The 3Π molecular state, correlating with the ground-state neutral atoms Cd(1S) and X(3P), is the ground state of the CdX molecules if Cd and X remain uncoupled in their ground
atomic states. The 1Σ+ state, dissociating to the 1D excited state of X and Cd(1S) in its covalent limit, is due to the interaction between the 5s2 orbital of Cd and the empty 4pσ orbital of X. Consequently, if the Cd atom loses its two valence electrons to the chalcogenide atom that possesses four electrons in its p orbital, Cd and X create a closed shell, and the CdX molecules exist in the 1Σ+ molecular state. The transfer of electrons causes the dissociation energy of 1Σ+ to be greater than that of 3Π.13 921
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Figure 4. Potential energy curves of the lowest singlet states of the molecule CdTe.
Figure 2. Potential energy curves of the lowest triplet states of the molecule CdSe.
Figure 5. Potential energy curves of the lowest triplet states of the molecule CdTe.
Figure 3. Potential energy curves of the lowest quintet states of the molecule CdSe.
X3Π and (1)3Σ− molecular states. Because of the Cd(1S) ground state and the X(1D) excited state, (1)1Σ+, (1)1Π, and (1)1Δ match up with the second covalent asymptote. In the CdTe molecule, the third asymptote arises from the [Cd (1S) + Te (1S)] combination leading to the (1)1Σ+ molecular state. The third dissociation asymptote of CdSe and the fourth asymptote corresponding to CdTe, involving the excited Cd(3P0) state and the X(3P) ground state, harmonize with the (1)1Σ+, (2)1Π, (1)1Δ, (2)1Σ−, (1)3Σ+, (2)3Π, (1)3Δ, (2)3Σ−, (1)5Σ+, (2)5Π, (1)5Δ, and (2)5Σ− electronic states.
To pinpoint the low-lying electronic states of the CdSe and CdTe molecules, the energies of the lowest covalent dissociation asymptotes of the investigated states have been calculated. The computed energies are compared with the NIST experimental data in the literature29 and the results are listed in Table 1. The association of the Cd(1S) and X(3P) ground states identifies the first covalent dissociation asymptote of the CdX molecules that correlates with the 922
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Figure 7. Permanent dipole moment curves of the lowest singlet states of the molecule CdSe.
Figure 6. Potential energy curves of the lowest quintet states of the molecule CdTe.
Table 2. Spectroscopic Constants, Zero-Point Energy, and Dipole Moments of the Lowest Electronic Bound States of the CdX Molecules molecules
states
Te (cm−1) 0.0a 0.0b 852.27a 1132.49b 2765.59a
CdSe
(1)3Π Calc. (1)1Σ+ Calc. (1)1Π Calc. (1)3Σ+
15 356.13a
(2)3Π (2)1Π (2)1Σ+ (1)3Δ (1)1Σ− (2)3Δ (2)1Δ (1)5Π (2)3Σ− (1)3Π (1)1Σ+ (1)1Π (1)3Σ+ (2)3Π (2)1Π (2)1Σ+ (1)3Δ (2)3Δ (1)1Σ− (3)1Σ+ (2)1Δ (1)5Π (2)3Σ−
18 277.34a 20 988.89a 22 921.18a 25 790.07a 25 848.29a 25 981.28a 26 122.22a 26 878.55a 26 994.91a 0.0a 822.32a 2774.88a 16 375.16a 18 518.16a 20 641.37a 21 865.64a 24 619.45a 24 800.00a 24 814.78a 24 821.69a 25 103.86a 25 573.51a 25 900.29a
CdTe
ωe (cm−1)
Be (cm−1)
Re (Å)
|μe| (D)
De (cm−1)
G(0) (cm−1)
157.59 162.32b 236.31 240.31 200.17 209.74b 173.11 209.40b 135.23 109.27 97.63 185.14 168.82 164.20 119.81 81.32 149.31 125.68 195.89 163.97 141.39 113.01 97.28 87.57 158.51 138.34 141.50 124.15 105.39 80.65 131.70
0.0501 0.05 0.061 0.06 0.053 0.05 0.051 0.05 0.034 0.027 0.034 0.052 0.048 0.051 0.0511 0.036 0.049 0.0342 0.0418 0.0363 0.0353 0.0242 0.0201 0.0242 0.036 0.0356 0.0341 0.0339 0.0341 0.0268 0.0346
2.69 2.64 2.44 2.43 2.61 2.59 2.64 2.58 3.22 3.61 3.26 2.62 2.73 2.66 2.69 3.15 2.70 2.87 2.59 2.78 2.82 3.40 3.74 3.40 2.77 2.81 2.87 2.88 2.87 3.23 2.85
2.15
2705.28 3148.23 12 861.87 13 032.89 10 947.59 11 351.53 13 690.67 25 541.07
78.79
5.70 2.51 4.01 2.92 2.72 1.12 1.12 0.83 1.74 0.81 2.80 0.92 1.66 5.15 1.95 3.50 2.62 2.54 0.62 1.21 0.54 0.38 0.38 0.40 3.71 0.40
2351.40 11 374.38 9443.53 12 314.56
118.15 100.08 86.55
62.84 97.94 81.98 70.69
a
For the present work with MRCI calculation. bReference 13. 923
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Figure 8. Permanent dipole moment curves of the lowest triplet states of the molecule CdSe.
Figure 10. Permanent dipole moment curves of the lowest singlet states of the molecule CdTe.
Figure 11. Permanent dipole moment curves of the lowest triplet states of the molecule CdTe.
Figure 9. Permanent dipole moment curves of the lowest quintet states of the molecule CdSe.
spectroscopic constants, Re, ωe, Be, and Te, have been computed by putting the calculated energy data of these electronic states, near Re, into a polynomial with reference to the bond length R. The spectroscopic dissociation energy De and the zero-point energy G(0) are calculated for the electronic states X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+. The spectroscopic results and the dipole moments of these states are given in Table 2. In the present work, 25 electronic states are investigated for the CdSe molecule. The ground state is found to be X3Π, similar to the result obtained by Feng et al.,13 whereas in the study by Peterson et al.,14 the ground state is (1)1Σ+. For X3Π,
The (2)3Σ+ and (2)3Δ electronic states are in agreement with the fourth covalent dissociation asymptote of CdSe and the fifth asymptote of CdTe arising from the Cd(3P0) and X(1D) excited states. Our calculated energy values with regard to the dissociative asymptotes are in acceptable agreement with the experimental data,29 as illustrated in Table 1. The spin-free PECs of CdX molecules, correlating with the covalent asymptotes for the investigated singlet, triplet, and quintet electronic states, are computed at the MRCI + Q level and mapped in Figures 1−6. For the bound states, the 924
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Table 4. Calculation of Ionicity for the X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+ States of CdX Molecules X3Π (1)1Σ+ (1)1Π (1)3Σ+ X3Π (1)1Σ+ (1)1Π (1)3Σ+
CdSe
CdTe
|μ| (a.u.)
Re (Å)
μ/eRe
0.84696 2.24403 0.98941 1.58109 0.65584 2.02655 0.76797 1.38015
2.69 2.44 2.61 2.64 2.87 2.59 2.78 2.82
0.16 0.48 0.2 0.31 0.12 0.41 0.14 0.25
the value of ωe is 157.59 cm−1, revealing a qualitative agreement in comparison with the reported theoretical value13 of 162.32 cm−1, with an error of 2.9%. The value of Re is computed to be 2.69 Å, which differs from the latest theoretical value of 2.64 Å13 by 0.05 Å. According to our calculations, the value of Te of (1)1Σ+ is only 852.27 cm−1 above X3Π at Re = 2.44 Å, with ωe = 236.31 cm−1. Comparing our data with the theoretical calculations found in the literature, Feng et al.13 found the values of Te, Re, and ωe as 1132.49 cm−1, 2.43 Å, and 240.31 cm−1 respectively. In the study by Peterson et al.,14 the ground state is 1Σ+, and the first excited state is 3Π which lies 2039.1 cm−1 above the ground state. The electronic states of identical quantum number Λ and symmetry can avoid a cross, whereas the states of dissimilar symmetries can cross, leading to a modification in the stability of the molecules. If these crossings are ignored, an inaccurate chemical picture will appear.30 Τhe (1)1Σ+ and (1)1Π states of different symmetries and close potential energies interact, causing crossing points between their PECs at R = 2.72 Å which are near the minimum of the later state. The calculated values of Re for the (1)1Σ+ and (1)1Π states are 2.44 and 2.61 Å, respectively; moreover, the vibrational harmonic frequency ωe and the dissociation energy of (1)1Σ+ are larger than that of (1)1Π by 36 cm−1 and 1914.28 cm−1, respectively. Τhe discrepancies in the spectroscopic parameters between these two states can aid to identify them experimentally. The (1)1Π state shows a minimum at Re = 2.61 Å, which corresponds to a transition energy of 2765.59 cm−1, with regard to the minimum of the X3Π state. The value of Re = 2.61 Å agrees with the earlier reported values theoretically,13 with a relative difference ΔRe/Re = 0.76%. Our calculated value of ωe = 200.17 cm−1 reveals a good correspondence with the theoretical value13 of 209.74 cm−1, with a relative difference Δωe/ωe = 4.5%. For (1)3Σ+, the spectroscopic parameters ωe = 173.11 cm−1 and Re = 2.64 Å show reasonable consistency when compared to the theoretical results,13 where the relative differences are 17.3 and 2.2%, respectively. The electronic energy of (1)3Σ+ is equivalent to Te = 15356.13 cm−1 above the ground state. According to Figure 2, the potential energies of (1)3Σ+, (1)3Σ−, and (2)3Π are very close to each other;
Figure 12. Permanent dipole moment curves of the lowest quintet states of the molecule CdTe.
Figure 13. TDM between (1)3Σ+ and X3Π of the molecules CdSe and CdTe.
Table 3. TDM Values of the Upper State at Its Equilibrium Position |μ21|, Wavelength λ21, the Emission Angular Frequency ω21, the Einstein Coefficients of Spontaneous and Induced Emissions (A21 and Bω21), the Spontaneous Radiative Lifetime τSpon, and the Emission Oscillator Strength f 21 of the (1)3Σ+−(X)3Π Transition of CdX Molecules |μ21| (a.u.)
|μ21| × 1030 (C m)
ω21 × 10−15 (rad s−1)
(1)3Σ+−(X)3Π
0.0849
0.72
2.89
(1)3Σ+−(X)3Π
0.1265
1.07
3.08
transition
A21 × 10−5 (s−1)
CdSe molecule 0.527 CdTe molecule 1.41 925
τspon (μs)
Bω21 × 10−18 (m3 rad J−1 s−2)
λ21 (nm)
|f 21|
18.9
5.5
651.6
0.0003
12.16
611.1
0.00079
7.09
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Table 5. Rovibrational Calculation of the X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+ States of the Molecule CdSe states
V 0 1 2 3 4 5 6 7
X3Π
8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4
(1)1Σ+
5 6 7 8 9 10 11 12 13 14 15 16 17
Ev (cm−1)
Bv × 102 (cm−1)
Rmin (Å)
Rmax (Å)
a
4.99
2.09
2.628
2.76495
4.96
2.15
2.58527
2.82407
4.92
2.20
2.55785
2.8685
4.88
2.31
2.5367
2.90726
4.84
2.35
2.51925
2.94293
4.80
2.47
2.50427
2.97682
4.75
2.64
2.49113
3.00979
4.71
2.77
2.47945
3.0415
4.65
3.1053
2.46896
3.07719
4.60
3.1052
2.45951
3.10913
4.55
2.71
2.45087
3.14111
4.51 4.44 4.39 4.32 4.25 4.17 4.08 3.99 3.89 3.77 6.08
3.44 3.90 3.41 4.45 4.42 6.08 5.54 7.23 8.10 9.71 1.615
2.44283 2.43544 2.42866 2.42234 2.41653 2.41116 2.4063 2.40183 2.39779 2.39418 2.38759
3.17414 3.20811 3.24256 3.27744 3.31553 3.35674 3.40049 3.44834 3.50104 3.56038 2.49846
6.06
1.613
2.35159
2.5441
6.03
1.618
2.32808
2.57726
6.00
1.621
2.30969
2.60534
5.98
1.620
2.29429
2.63046
5.95
1.642
2.2809
2.65363
5.93
1.632
2.26899
2.67537
5.90
1.659
2.2582
2.69603
5.87
1.652
2.24833
2.71584
5.85
1.663
2.23921
2.73497
5.82
1.678
2.23073
2.75353
5.79 5.77 5.74 5.72 5.69 5.66 5.63
1.679 1.698 1.704 1.710 1.727 1.734 1.741
2.22278 2.2153 2.20824 2.20155 2.19518 2.18911 2.18332
2.77163 2.78933 2.80669 2.82376 2.84059 2.85721 2.87364
0 0b 153.55a 159.32b 304.39a 315.42b 452.49a 468.24b 597.44a 617.69b 739.50a 763.70b 878.33a 906.18b 1013.57a 1045.05b 1144.92a 1180.20b 1271.66a 1311.54b 1394.63a 1438.93b 1515.37a 1631.93a 1743.78a 1852.36a 1956.01a 2054.93a 2147.36a 2234.51a 2315.14a 2389.00a 0a 0b 235.69a 239.00b 469.98a 476.38b 702.72a 712.12b 933.87a 946.20b 1163.50a 1178.63b 1391.41a 1409.39b 1617.79a 1638.47b 1842.45a 1865.87b 2065.49a 2091.56b 2286.87a 2315.53b 2506.55a 2724.59a 2940.92a 3155.57a 3368.53a 3579.77a 3789.32a
926
Dv × 108 (cm−1)
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Table 5. continued states
V 18 19 20 0 1 2 3 4 5 6 7 8
(1)1Π
9 10 11 12 13 14 15 16 17 18 19 20 0
(1)3Σ+
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Ev (cm−1) a
3997.15 4203.26a 4407.63a 0a 0b 198.51a 208.84b 395.80a 416.59b 592.00a 623.25b 787.02a 828.79b 980.89a 1033.21b 1173.65a 1236.49b 1365.22a 1438.62b 1555.67a 1639.58b 1744.95a 1839.36b 1933.08a 2037.95b 2120.07a 2305.88a 2490.54a 2674.03a 2856.35a 3037.50a 3217.47a 3396.25a 3573.84a 3750.24a 0a 0b 172.29a 194.03b 343.68a 387.09b 514.15a 579.20b 683.71a 770.35b 852.36a 960.56b 1020.11a 1149.81b 1186.97a 1338.13b 1352.94a 1525.52b 1518.03a 1711.97b 1682.23a 1897.50b 1845.53a 2007.86a 2169.03a 2328.82a
Bv × 102 (cm−1)
Dv × 108 (cm−1)
Rmin (Å)
Rmax (Å)
5.61 5.58 5.55 5.29
1.760 1.763 1.775 1.502
2.17776 2.17244 2.16732 2.55856
2.88993 2.90608 2.92212 2.6794
5.27
1.508
2.51875
2.72856
5.25
1.507
2.4926
2.7641
5.23
1.511
2.47206
2.7941
5.21
1.5154
2.45482
2.82086
5.19
1.5156
2.43979
2.84546
5.17
1.528
2.42639
2.86849
5.15
1.520
2.41425
2.89032
5.13
1.532
2.40312
2.91121
5.11
1.529
2.39283
2.93132
5.09
1.534
2.38323
2.9508
5.07 5.05 5.03 5.01 4.99 4.97 4.95 4.93 4.91 4.89 5.16
1.544 1.542 1.546 1.5568 1.5567 1.566 1.567 1.569 1.585 1.589 1.847
2.37424 2.36578 2.35778 2.35018 2.34296 2.33606 2.32946 2.32314 2.31708 2.31124 2.58763
2.96974 2.98822 3.00631 3.02407 3.04153 3.05873 3.07571 3.09249 3.1091 3.12557 2.71735
5.14
1.843
2.54514
2.7703
5.12
1.839
2.51729
2.80862
5.10
1.837
2.49548
2.84095
5.08
1.833
2.47718
2.86978
5.06
1.834
2.46126
2.89627
5.03
1.826
2.44708
2.92104
5.01
1.829
2.43425
2.9445
4.99
1.823
2.42249
2.96689
4.97
1.821
2.41162
2.98843
4.95
1.820
2.40149
3.00931
4.93 4.91 4.89 4.86
1.833 1.872 1.926 1.904
2.39201 2.38308 2.37466 2.36669
3.02932 3.04908 3.06956 3.08904
927
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Table 5. continued states
V 15 16 17 18 19 20
Ev (cm−1)
Bv × 102 (cm−1)
Dv × 108 (cm−1)
Rmin (Å)
Rmax (Å)
4.84 4.83 4.81 4.79 4.77 4.75
1.722 1.599 1.770 1.905 1.762 1.669
2.35913 2.35189 2.34493 2.33825 2.33184 2.32567
3.10716 3.12509 3.14295 3.16051 3.17775 3.1948
a
2487.40 2645.53a 2803.60a 2960.93a 3116.98a 3272.32a
a
For the present work with MRCI calculation. bReference 13.
consequently, (1)3Σ+ crosses with (1)3Σ− and (2)3Π at 2.51 and 3.11 Å, respectively, which leads to trouble in the experimental observation. The comparison between our computed values of Be and De of the ground and the first three excited states shows a good accordance with those given theoretically in the literature,13 except for De of (1)3Σ+. However, the comparison between the values of [De − G(0)] (Table 2) in the present work for (1)1Σ+, (1)1Π, and (1)3Σ+ and the value of D031 reveals an appreciable consistency. The bound states (2)3Π, (2)1Σ+, (2)1Π, (1)1Σ−, (2)1Δ, (1−2)3Δ, (2)3Σ−, and (1)5Π are present here for the first time. All other singlet, triplet, and quintet states are found to be repulsive, as shown in Figures 1−3. A total of 26 electronic states for cadmium telluride are explored and mapped in Figures 4−6. The ground state is X3Π and the first excited state is 1Σ+ with Te = 822.32 cm−1. The results of the spectroscopic parameters along with the dipole moments of 14 bound states are listed in Table 2. The spectroscopic dissociation energy De and the zero-point energy are also given in Table 2 for the bound electronic states X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+. Peterson et al.14 reported that the ground state of CdTe is 1Σ+ and the first excited state is 3Π lying 1741.8 cm−1 above 1Σ+. The dissimilarities in the identification of the ground and the first excited states with regard to the CdTe molecule may provoke further theoretical and experimental studies to confirm the results. Beyond these two states, the others are investigated here for the first time. By comparing the results obtained in the present work with regard to the CdSe and CdTe molecules, the following conclusions are drawn: (i) in the same range of experimental dissociation energy limit of [Cd + X] atoms, the number of molecular states in the CdTe molecule is more than that of the CdSe molecule by one, as the Te atom exhibits an excited (1S) state that is not present for the Se atom, leading to an additional 1Σ+ state; (ii) the PECs of the singlet, triplet, and quintet electronic states of the CdSe and CdTe molecules follow the same trend, where the crossings and the avoided crossings between these states reveal an identical behavior; (iii) for identical bound states, the equilibrium bond distance Re increases along with the less electronegative chalcogen; (iv) being inversely proportional to the reduced mass of the molecule, the values of ωe and Be for the same molecular bound states of the CdX molecules are higher for the selenides; (v) the spectroscopic dissociation energies of the first three excited states 1Σ+, 3Σ+, and 1Π are greater for the CdSe molecule, revealing a satisfactory consistency with the values of D0.31 For the spin-free electronic states of CdX molecules, the permanent dipole moments (PDMs) at the MRCI + Q level were investigated and mapped in Figures 7−12. Table 2 shows the PDMs at the equilibrium bond length of the bound states. The PDMs have been computed by considering the more
electronegative atom which is chalcogen (X) at the origin, and the cadmium (Cd) atom moves along the positive z-axis. Upon losing its ionic character, an adiabatic state becomes neutral and the corresponding DMC approaches zero. Some states exhibit a positive dipole moment as in (1)5Σ+, (1)5Π, (1)5Δ, and (2)5Σ−, indicating an ionic structure of Cdδ+Xδ− type. Other states, as in (2)3Δ and (1)1Σ+, reveal an ionic structure of Cdδ−Xδ+ along with the negative value of their dipole moment. The other investigated electronic states have a change in the sign of the PDM values relative to R. As two electronic states avoid cross, an important modification in their dipole moment occurs where the polarity of the atoms is inversed and the electronic character is exchanged in this region. The avoided crossings at different internuclear distances are mirrored as crossings in the DMCs, as mapped in the avoided crossings among (1)1Δ/(2)1Δ, (2)1Π/(3)1Π, and (1)3Δ/(2)3Δ. A large value of the magnitude of the PDM is useful to align molecules in an optical lattice and to study long-range dipole−dipole forces.32,33 The TDM functions of (1)3Σ+−X3Π and (1)1Π−(1)1Σ+ transitions in the CdX molecules are investigated and represented in Figure 13. Near the equilibrium bond distance of the X3Π state in the CdSe molecule, the TDM of the (1)1Π−(1)1Σ+ transition is much greater than that of the other transition, but it decreases rapidly with the increase of the bond length. The sudden variations and jumps that appear in the TDMCs at different internuclear distances can be interpreted by (i) the presence of the crossings and avoided crossings among the PECs at these values of R, leading to a change in the sign of the electronic wave functions and (ii) the strong interaction between the ion-pair covalent states. It is well-observed that the TDMCs vanish for large values of R, which is attributed to the spin-forbidden transitions between two atomic orbitals at asymptotic limits. For instance, the (1)3Σ+−X3Π transition tends to be zero at large values of R because of the spin-forbidden transition 3Pu−1Sg for the Cd atom. The significance of the PDMs or TDMs rises owing to the implementation of polar molecules in the area of ultracold molecular gases.34 The PDM and TDM curves are essential, as they are useful in designing the photoassociation experiments for a molecule35 and in predicting transitions. Using the value μ21 of the TDM in the transition between (1)3Σ+ and X3Π at the equilibrium bond length Re of the higher state, we calculated the wavelength λ21, the emission angular frequency ω21, and the Einstein coefficient of spontaneous emission A21 of the CdX molecules. As a result, the spontaneous radiative lifetime τspon, the Einstein coefficient of induced emission Bω21, and the emission oscillator strength f 21 are calculated by applying Hilborn’s equations.36 The calculated constants are shown in Table 3. Strong electronic transitions are identified by A21 values in the zone 108 to 109 s−1 and radiative lifetimes in the range 1−10 ns. The radiative 928
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Table 6. Rovibrational Calculation of the X3Π, (1)1Σ+, and (1)1Π States of the Molecule CdTe States
X3Π
(1)1Σ+
(1)1Π
V 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Ev (cm−1) a
62.66 185.76a 306.94a 426.02a 542.83a 657.59a 770.28a 881.23a 990.39a 1095.93a 1197.38a 1296.56a 1391.71a 1483.08a 1572.26a 1658.32a 1740.18a 1818.65a 1892.81a 1962.87a 2028.12a 97.98a 292.54a 486.22a 678.74a 870.28a 1060.55a 1249.77a 1437.72a 1624.50a 1810.06a 1994.37a 2177.46a 2359.28a 2539.87a 2719.20a 2897.27a 3074.10a 3249.66a 3423.95a 3596.92a 3768.54a 81.85a 244.89a 407.10a 568.49a 729.07a 888.85a 1047.86a 1206.03a 1363.32a 1519.69a 1675.17a 1829.77a 1983.46a 2136.26a 2288.15a 2439.14a 2589.23a 2738.40a 2886.65a
Bv × 102 (cm−1)
Dv × 108 (cm−1)
Rmin (Å)
Rmax (Å)
4.99 3.38 3.36 3.34 3.31 3.29 3.26 3.24 3.21 3.18 3.14 3.11 3.07 3.03 3.00 2.96 2.91 2.87 2.81 2.75 2.69 4.17 4.15 4.14 4.12 4.10 4.09 4.07 4.06 4.04 4.03 4.01 4.00 3.98 3.96 3.95 3.93 3.92 3.90 3.88 3.87 3.85 3.63 3.62 3.60 3.59 3.58 3.57 3.56 3.55 3.53 3.52 3.51 3.50 3.48 3.47 3.46 3.45 3.43 3.42 3.41
1.037 1.061 1.093 1.137 1.150 1.192 1.187 1.199 1.480 1.671 1.395 1.849 1.935 1.621 1.945 2.420 2.274 2.845 2.875 3.654 3.786 0.7635 0.7607 0.7642 0.7609 0.7681 0.7659 0.7771 0.7750 0.7823 0.7851 0.7867 0.7968 0.7971 0.8031 0.8081 0.8092 0.8147 0.8199 0.8304 0.8435 0.8504 0.7186 0.7200 0.7212 0.7227 0.7221 0.7237 0.7262 0.7316 0.7361 0.7373 0.7390 0.7421 0.7449 0.7479 0.7491 0.7532 0.7576 0.7600 0.7622
2.80761 2.7654 2.73822 2.71722 2.69986 2.68491 2.67175 2.65992 2.64918 2.63949 2.63073 2.62262 2.61521 2.60841 2.60202 2.59609 2.59063 2.58556 2.5809 2.57661 2.5727 2.57523 2.57399 2.57271 2.57135 2.53622 2.51365 2.49921 2.47477 2.45942 2.44665 2.4348 2.42392 2.41402 2.40494 2.39652 2.38864 2.38121 2.37418 2.36752 2.36118 2.35514 2.78391 2.78365 2.7834 2.78316 2.78293 2.78269 2.78247 2.78225 2.78202 2.78181 2.78159 2.78137 2.78116 2.78094 2.78072 2.7805 2.78028 2.78006 2.77983
2.94242 3.00026 3.04361 3.08126 3.11574 3.14877 3.17909 3.20709 3.239 3.26999 3.30056 3.33384 3.36787 3.3998 3.43372 3.46907 3.50592 3.54511 3.58682 3.63196 3.68114 2.59794 2.5985 2.59904 2.59958 2.60014 2.60073 2.60137 2.60206 2.60281 2.60365 2.60462 2.60578 2.60731 2.61041 2.88794 2.90474 2.92113 2.93716 2.95288 2.96832 2.98351 2.80192 2.80238 2.80284 2.80329 2.80374 2.80418 2.80462 2.80506 2.80551 2.80595 2.8064 2.80686 2.80732 2.80779 2.80827 2.80876 2.80927 2.80981 2.81037
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Table 6. continued States
V 19 20
Ev (cm−1) a
3033.98 3180.38a
Bv × 102 (cm−1)
Dv × 108 (cm−1)
Rmin (Å)
Rmax (Å)
3.40 3.38
0.7577 0.7715
2.77959 2.77935
2.81097 2.81161
a
For the present work with MRCI calculation.
lifetimes of (1)3Σ+−X3Π transitions corresponding to the CdSe and CdTe molecules are determined to be 18.9 and 7.09 μs, respectively. The Cd atom is responsible for the (1)3Σ+− X3Π transition, as given in Table 1. In the NIST database,29 a transition between Cd (1S) and Cd (3P0) in the range 0− 30656.087 cm−1 at a wavelength λ = 326.1 nm gives Aki = 4.06 × 105 s−1, f ik = 0.00194, and τspon = 2.46 μs for a CdSe molecule as an illustration. The following comparison reveals the reliability of our work. The calculation of the ionicity of a molecule by qionicity = μ/ eRe37 aids to recognize the nature of a molecular bond where μ and Re are taken from PECs and DMCs at the equilibrium positions. The ionicity value indicates the amount of the ionic character relative to a molecular bond,38,39 where 0 ≤ qionicity ≤ 1 with qionicity = 1 corresponds to the extreme ionic character.40 We compute the ionicity of the CdSe and CdTe molecules for the X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+ states at their equilibrium bond distance Re. The results are given in Table 4 where the calculated values reveal the ionic and covalent nature of the bonding. At large values of R, the dipole moment of the titled molecules tends to zero because the molecule separates into its natural fragments. Comparing our results with regard to the ionicity of identical electronic states of CdX molecules, we conclude that the ionicity of these states in the case of a CdSe molecule is higher, together with smaller values of Re, recalling that Se is more electronegative than Te. Rovibrational calculation has been performed for the X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+ states of a CdSe molecule along with the X3Π, (1)1Σ+, and (1)1Π states of a CdTe molecule. Consequently, the eigenvalue Ev, the rotational constant Bv, the distortion constant Dv, and the abscissas of the turning points Rmin and Rmax have been computed up to the vibrational level v = 20 via the canonical functions approach41−45 along with the cubic spline interpolation among each two successive points of the PECs. These values are given in Table 5 for the CdSe molecule and in Table 6 for the CdTe molecule. The values presented in Table 5 are in good agreement with the data given in the literature.13
very weakly bound. The ionicity has been reported for the X3Π, (1)1Σ+, (1)1Π, and (1)3Σ+ electronic states of the CdX molecules, and it is found that the bond of these states is of mixed nature. The (1)3Σ+−X3Π transition is studied and some of the transition parameters are computed. The radiative lifetimes are of the order of microseconds. The agreement between our results and the available data in the literature is satisfactory.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected].
[email protected]. Phone: +961 7 985 858 ext. 3304. ORCID
Mahmoud Korek: 0000-0001-9684-8433 Notes
The authors declare no competing financial interest.
■
REFERENCES
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4. CONCLUSIONS Internally contracted MRCI + Q calculations have been performed on 25 and 26 molecular states of the CdSe and CdTe molecules, respectively. The PECs and DMCs of the singlet, triplet, and quintet electronic states of the CdX molecules are fitted along with the calculation of the spectroscopic constants Re, ωe, Be, and Te for the bound states. According to our study, the 3Π state is classified as the ground state, and it dissociates with the ground-state neutral atoms Cd(1S) and X(3P). Τhe PECs of the (1)1Σ+ and (1)1Π states cross, as they have close energy levels. The experimental inspection of the ground state and the first two excited states is difficult by energy difference; however, the differences in the spectroscopic parameters Re, ωe, and De between these states can aid to identify them experimentally. The dissociation energies of the ground-state 3Π of he CdSe and CdTe molecules are 2705.28 and 2351.40 cm−1, respectively, and it is 930
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