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The low-lying singlet electronic excited states of TiO2: A symmetry adapted cluster-configuration interaction (SAC-CI) study
Accepted Manuscript The low-lying singlet electronic excited states of TiO2: A symmetry adapted cluster–configuration interaction (SAC-CI) study Yung-Ching Chou PII: DOI: Reference:
S2210-271X(15)00083-3 http://dx.doi.org/10.1016/j.comptc.2015.02.012 COMPTC 1747
To appear in:
Computational & Theoretical Chemistry
Received Date: Revised Date: Accepted Date:
10 February 2015 16 February 2015 16 February 2015
Please cite this article as: Y-C. Chou, The low-lying singlet electronic excited states of TiO2: A symmetry adapted cluster–configuration interaction (SAC-CI) study, Computational & Theoretical Chemistry (2015), doi: http:// dx.doi.org/10.1016/j.comptc.2015.02.012
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The low-lying singlet electronic excited states of TiO2: A symmetry adapted cluster–configuration interaction (SAC-CI) study
Yung-Ching Chou
Department of Applied Physics and Chemistry, University of Taipei , No. 1, Ai-Guo West Road, Taipei 10048, Taiwan
Corresponding author: Yung-Ching Chou (E-mail Address: [email protected]) Department of Applied Physics and Chemistry University of Taipei No. 1, Ai-Guo West Road, Taipei 10048, Taiwan
Abstract The ground electronic state (X1A1) and nine low-lying singlet electronic excited states (21A1, 31A1, 11A2, 21A2, 11B1, 21B1, 31B1, 11B2, and 21B2) of TiO2 were studied using SAC-CI theory. The geometries of the nine electronic excited states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels. The adiabatic excitation energies of the nine electronic excited states were obtained from single-point calculations at the SAC-CI/def2-TZVPD//SAC-CI/def2-SVPD and SAC-CI/def2-TZVPPD//SAC-CI/def2-SVPD levels. Stable Cs structures for the 11B1 (A") and 31B1 (A") states were determined in this study. For comparison, the X1A1, 11A2, 11B1, and 11B2 states were also studied using the unrestricted DFT theory (UBPW91) with the def2-SVP, def2-SVPD, def2-TZVPD, def2-TZVPPD, and aug-cc-pVTZ basis sets.
1. Introduction TiO2 has versatile applications owing to its various advantageous properties, especially its photocatalytic activity [1]. Therefore, the study of its electronic excited states is important for understanding the photochemistry of the molecule. Several experimental [2–10] and theoretical [11–22] studies of the ground and low-lying electronic states of TiO2 and its clusters (TiO2)n have been presented in the literature. However, theoretical studies for the low-lying electronic excited states of the TiO2 monomer are comparatively few [19–22]. Grein [19] studied the lowest singlet, triplet, and quintet states (A1, A2, B1, and B2 states in C2v symmetry) by using density functional theory (DFT) at the BPW91/6-311+G(3df) level, and obtained the optimized geometries, vibrational frequencies, and adiabatic excitation energies for the electronic states. The vertical excitation energies calculated using multi-reference configuration interaction (MRCI) theory for the singlet and triplet states with excitation energies up to approximately 4.6 eV were also given in his study. Grein showed that the optimized C2v structure of the 11B1 state has an imaginary frequency for the b2 vibration (anti-symmetric stretching), and he suggested that the 11B1 state could have a lower energy in a Cs structure (A" state). Taylor and Paterson [20] used several coupled cluster response theories [CC2, CCSD, CCSDR(3), and CC3], and time-dependent DFT (TD-DFT) theory to study the low-lying excited states. Lin et al [21] used various levels of theory, including TD-DFT, complete active space self consistent field (CASSCF), CASPT2, CCSD(T), and equation-of-motion CCSD (EOM-CCSD), to study the low-lying electronic excited states of TiO2. In their study, the 11B2 (S1) state was optimized to two possible stable structures: one corresponding to a bent structure (C2v) and the other to a symmetric linear structure (D∞h). The 11A2 (S2) state was optimized to a symmetric linear structure in which the 11A2 and 11B2 states are degenerate and correspond to the 11Δu state in D∞h symmetry. They suggested that the 21B2 (S3) state has only one minimum with C2v symmetry by performing a two-dimensional potential energy surface scan at the TD-B3LYP/aug-cc-pVTZ level. In this study, we used the symmetry adapted cluster–configuration interaction (SAC-CI) method [23–27] to study the ground electronic state [X1A1(S0)] and the nine low-lying singlet excited states [21A1(S4), 31A1(S8), 11A2(S2), 21A2(S7), 11B1(S5), 21B1(S6), 31B1(S10), 11B2(S1), and 21B2(S3) states] of TiO2. The vertical excitation energies, optimized geometries, and adiabatic excitation energies of the considered excited states were calculated and compared with the experimental values. For more information of the SAC-CI theory, please refer to SAC-CI’s website [28]. For comparison, the unrestricted DFT theory (UBPW91) was also applied to study the X1A1, 11A2, 11B1, and 11B2 states. Stable Cs structures for the 11B1 (A") and 31B1 (A")
states, which were not found in the literature, were optimized in the SAC-CI calculations; the stable Cs structure for the 11B1 (A") state was also optimized in the UBPW91 calculations. 2. Computational details In the DFT calculations, the UBPW91 method with def2-SVP, def2-SVPD, def2-TZVPD, def2-TZVPPD [29–31], and aug-cc-pVTZ [32, 33] basis sets were applied to study the X1A1, 11A2, 11B1, and 11B2 states. In the SAC-CI calculations, only the singles and doubles linked excitation operators were included (default); the key word “NoLinkedSelection” was applied to include all singles and doubles linked operators and “NonVariational” was applied to solve the SAC-CI equations. The geometries of the electronic states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels; the vibrational frequencies were calculated at the SAC-CI/def2-SVP level to characterize the stationary points. Single-point calculations at the SAC-CI/def2-TZVPD//SAC-CI/def2-SVPD and SAC-CI/def2-TZVPPD//SAC-CI/def2-SVPD levels were performed to obtain the adiabatic excitation energies. All the calculations were performed with Gaussian 09 program packages [34]. 3. Results and Discussion 3.1. Vertical excitations Table 1 shows the SAC-CI-calculated vertical excitation energies, electronic transitions, and oscillator strengths of the first ten singlet excited states and the previous results calculated using MRCI [19], CCSD [20], and CASPT2 [21] theories. Figure 1 shows the selected molecular orbitals for the TiO2 molecule. The vertical excitation energies were calculated at the experimental equilibrium geometry of the ground state of TiO2 with a bond length (rTi-O) of 1.651Å and a bond angle (θO-Ti-O) of 111.57° [5]. The vertical excitation energies calculated using SAC-CI are similar to those calculated by the MRCI [19], CCSD [20], and CASPT2 [21] theories. As shown in Table 1, the SAC-CI-calculated vertical excitation energies are consistent with each other when using the def2-SVPD, def2-TZVPD, def2-TZVPPD, and aug-cc-pVTZ basis sets. This suggests that the def2-SVPD basis set is suitable for describing the excitation energies of the low-lying excited states of TiO2 in the SAC-CI calculations. The calculated vertical excitation energies of the 11B2 and 11B1 states were approximately 2.44–2.50 eV and 3.64–3.68 eV, respectively, which are reasonably higher than the experimental adiabatic excitation energies of 2.18 – 2.40 eV [3, 8–10] for the 11B2 state and 3.37 eV [9] for the 11B1 state. The SAC-CI-calculated vertical excitation energy for the 11B2 state is in good agreement with the experimental value
of 2.46 eV [3]. More information concerning the calculated vertical excitation properties of the excited states with excitation energies up to approximately 6 eV is shown in the supplementary materials. 3.2. DFT calculations Table 2 shows the calculated adiabatic excitation energies, geometry parameters, vibrational frequencies, and dipole moments of the X1A1, 11B2, 11A2, and 11B1 states using UBPW91 theory. The current DFT-calculated results are similar to those shown in previous DFT calculations [19, 21] except for the Cs conformation of the 11B1(A") state which is not found in the literature. For the X1A1 state, the calculated geometry parameters, vibrational frequencies, and dipole moments agree with the experimental values when using basis sets larger than the def2-SVP basis set. For the 11B2 state, the calculated adiabatic energies reasonably agree with the experimental values; the calculated bond lengths are in good agreement with the experimental values when using basis sets larger than the def2-SVP basis set, but the calculated bond angles are somewhat smaller than the experimental angles. The calculated a1 vibrational frequencies (ω1 referring to symmetric stretching and ω2 to bending) reasonably agree with the experimental values, but the calculated b2 frequencies (ω3) are approximately 1.5 times larger than the experimental values. For the 11B1 state, the geometry was optimized to two stationary points: one has a C2v symmetry and the other has a Cs symmetry. The C2v conformation of the 11B1 state is similar to those presented in previous calculations [19] and has an imaginary vibrational frequency for the b2 vibration. For the Cs conformation, the optimized structure has two bond lengths (Ti–O) of approximately 1.92 and 1.63 Å and a bond angle (O–Ti–O) of approximately 116°; the vibrational frequencies are all positive for the Cs conformation. For confirming the relation between the C2v and Cs conformations, we conducted an intrinsic reaction coordinate (IRC) calculation at the UBPW91/def2-SVPD level and found that the C2v conformation is the transition state on the reaction path between the two equivalent C s conformations. The calculated adiabatic excitation energies were approximately 2.93–3.02 eV for the C2v conformation and 2.81–2.91 eV for the Cs conformation, which are smaller than the experimental value of 3.37 eV. As sown in Table 2, the UBPW91-calculated results are consistent with each other when using basis sets larger than the def2-SVP basis set; this suggests that the def2-SVPD basis set is suitable for describing the low-lying electronic states of TiO2 in the UBPW91 calculations. 3.3. SAC-CI calculations
The X1A1, 11B2, 21B2, 11B1, 21B1, 31B1, 11A2, 21A2, 21A1, and 31A1 states were studied using the SAC-CI theory and are discussed as follows. Table 3 lists the optimized geometrical parameters and vibrational frequencies of the electronic states. Table 4 lists the calculated adiabatic excitation energies and dipole moments of the electronic excited states. 3.3.1. The X1A1 state TiO2 in its ground electronic state (X1A1) has a C2v point group symmetry. The equilibrium geometry of the ground state was optimized at the SAC/def2-SVP, SAC/def2-SVPD, and SAC/def2-TZVPD levels, and the vibrational frequencies were calculated at the SAC/def2-SVP and SAC/def2-SVPD levels. As shown in Table 3, the optimized geometrical parameters at the SAC/def2-SVPD level were rTi-O = 1.649 Å and θO-Ti-O = 112.40°, which agree with the experimentally determined values of rTi-O = 1.651 Å and θO-Ti-O = 111.57° [5]. The optimized geometry parameters at the SAC/def2-SVPD level are consistent with those at the SAC/def2-TZVPD level (rTi-O = 1.647 Å, θO-Ti-O = 112.83°), suggesting that the def2-SVPD basis set is suitable for describing the geometry of TiO2 in the SAC calculations. The calculated vibrational frequencies at the SAC/def2-SVP and SAC/def2-SVPD levels are on average approximately 11% and 5% higher than the experimental values, respectively. The calculated dipole moments were 6.89, 7.08, and 7.13 D at the SAC/def2-SVP, SAC/def2-SVPD, and SAC/def2-TZVPD level, respectively, which agree with the experimental values of 6.33 and 6.79 D [6, 8]. 3.3.2. The 11B2 and 21B2 electronic states The 11B2 and 21B2 states, in the Franck–Condon region, are characterized by the electronic transitions 6b2 → 10a1 ( + 6b2 →11a1) and 6b2 → 11a1 ( – 6b2 →10a1), respectively where the transition shown in parentheses indicates the minor one. The 6b2 , 10a1 , and 11a1 orbitals mainly involve O(2pz) – O(2pz), Ti(4s), and Ti(3dz2– x2) orbitals, respectively (see Fig. 1). Lin et al. [21] used several levels of theories to study the 11B2 state and showed that this state has two possible stable structures: one corresponding to a bent structure (C2v), and the other to a symmetric linear one (D∞h). In this study, the optimized geometrical parameters for the C2v conformation of the 11B2 state were (rTi-O = 1.672 Å and θO-Ti-O = 98.93°) and (rTi-O = 1.690 Å and θO-Ti-O = 98.81°) at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels, respectively. These calculated values agree with the experimental bond lengths of 1.704–1.709 Å and bond angles of 98.92–101.66° (at different vibational levels) [6, 8]. Compared with the geometrical parameters optimized using the ECM-CCSD and TD-B3LYP theories [21], the bond
length calculated using the SAC-CI/def2-SVPD method is slightly better than those calculated using the EOM-CCSD and TD-B3LYP methods. The SAC-CI-calculated bond angle is close to the experimental value of 98.92°, while the values calculated using the EOM-CCSD and TD-B3LYP methods are close to the experimental value of 100.1° (see Table 3). The vibrational frequencies for the C2v conformation were calculated at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels. The deviations between the calculated and experimental vibrational frequencies are on average approximately 17% and 9% at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels, respectively. The calculation using the def2-SVPD basis set significantly improves the quality of the vibrational frequency calculation compared with that using the def2-SVP basis set. However, for saving computational cost, the SAC-CI/def2-SVP calculation was used to characterize the optimized stationary points in this study. The calculated adiabatic excitation energies were 2.29–2.36 eV in the SAC-CI calculations, which agree with the experimental values of 2.18–2.40 eV [3, 8–10] (see Table 4). The SAC-CI-calculated dipole moments of 3.6–4.1 D, which are similar to those calculated by the EOM-CCSD (4.31 D) and TD-B3LYP (3.57 D) methods [21], are approximately 40%–60% larger than the experimental value of 2.55 D [6]. The linear equilibrium conformation of the 11B2 state, which correlates to the 11Δu state in D∞h symmetry, is characterized by the 5σu → 1δg electronic transition; the 5σu and 1δg orbitals mainly correspond to the O(2py) + O(2py) and Ti(3dz2 – x2) orbitals, respectively. The optimized bond lengths were 1.694 Å at the SAC-CI/def2-SVP level and 1.713 Å at the SAC-CI/def2-SVPD level, which are similar to those calculated using the TD-B3LYP and EOM-CCSD theories (see Table 3). The calculated adiabatic excitation energies were 2.47–2.55 eV in the SAC-CI calculations, which are similar to those calculated using the TD-B3LYP (2.609 eV) and EOM-CCSD (2.387 eV) theories [21]. As shown in Table 3, the vibrational frequencies calculated at the SAC-CI/def2-SVP level are similar to those calculated by the TD-B3LYP and EOM-CCSD theories except for the πu bending vibration. The calculated πu frequency at the SAC-CI/def2-SVP level was 80.0 cm−1, which is much smaller than those calculated by the TD-B3LYP (253.8 cm−1) and EOM-CCSD (297.3 cm−1) theories. This small πu frequency is related to a larger linearization energy [E(11∆u) – E(11B2)] in the SAC-CI calculations. The linearization energies were 0.12 – 0.20 eV in the SAC-CI calculations, which are larger than those in the TD-B3LYP (0.030 eV) and EOM-CCSD (0.063 eV) calculations [21]. Compared with those in the TD-B3LYP and EOM-CCSD calculations, the larger linearization energy suggests that the potential energy surface near the optimized geometry of the 11∆u state could be more flattened along the bond-angle coordinate in the SAC-CI calculations. Therefore, the πu frequency calculated by the SAC-CI method is
smaller than those calculated by the TD-B3LYP and EOM-CCSD methods. Lin et al. [21] conducted a two dimensional scan on the potential energy surface of the 21B2 state; they suggested that this state has only one minimum (at rTi-O = 1.680 Å and θO-Ti-O = 130.6°) on the potential energy surface and has an adiabatic excitation energy of 3.066 eV. In this study, the geometry of the 21B2 state was optimized to a minimum (in C2v symmetry) on the potential energy surface at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels; the calculated vibrational frequencies are all positive in the SAC-CI/def2-SVP calculation. The optimized equilibrium structure has a bond length of 1.695 Å and a bond angle of 131.97º at the SAC-CI/def2-SVPD level, which are similar to the values reported by Lin et al. [21]. The calculated adiabatic excitation energies were 2.84–2.90 eV in the SAC-CI calculations, which are slightly smaller than the calculated value of 3.066 eV. At this optimized geometry, the 21B2 state is mainly characterized by the electronic transition 6b2 → 10a1 (– 6b2 →11a1), where the major transition is the same as that in the 11B2 state in the Franck-Condon region; this phenomenon suggests that there could be some interaction between the two 1B2 states. To better understand the nature of the selected excited states, we conducted a potential energy surface scan along the bond-angle coordinate for these states and constructed a Walsh diagram. Figures 2 and 3 show the energy profile for the energy scan and Walsh diagram, respectively. As shown in Fig. 2, the minimum on the potential energy surface of the 21B2 state is formed due to the “avoid crossing” between the two 1B2 energy surfaces. At the SAC-CI/def2-SVP optimized geometry of the 21B2 state, the calculated vibrational frequencies of the 11B2 state were ω1(a1) = 920 cm-1, ω2(a1) = 316i cm-1, ω3(b2) = 465 cm-1 at the SAC-CI/def2-SVP level. The imaginary frequency on the bending mode (ω2) suggests that this geometry is a saddle point on the potential energy surface between the equilibrium geometries of the 11B2 and 11Δu states. At the SAC-CI/def2-SVPD optimized geometry of the 21B2 state, the calculated energy gaps between the two 1B2 states were 0.329, 0.291, and 0.292 eV at the SAC-CI/def2-SVPD, SAC-CI/def2-TZVPD, and SAC-CI/def2-TZVPPD levels, respectively. The calculated energies of the 11B2 state at the saddle point are higher than those at the equilibrium geometry by 0.217, 0.224, and 0.226 eV at the SAC-CI/def2-SVPD, SAC-CI/def2-TZVPD, and SAC-CI/def2-TZVPPD levels, respectively. As shown in Fig. 3, the energy of the 11a1 orbital becomes lower than that of the 10a1 orbital when the bond angle is larger than approximately 130°. The 21B2 and 11B1 states are thereby characterized by the electronic transitions 6b2 → 10a1 and 6b2 → 11a1, respectively, and correlate to the 1Σu+ and 1Δu states when the bond angle becomes 180° (see Fig. 2).
3.3.3. The 11B1, 21B1, and 31B1 electronic states The 11B1, 21B1, and 31B1 states in the Franck-Condon region are characterized by the electronic transitions 3b1 → 10a1 + 3b1 →11a1, 9a1 → 4b1, and 3b1 → 11a1 – 3b1 →10a1, respectively. The 3b1, 9a1, and 4b1 mainly involve O(2px) + O(2px), O(2py) – O(2py), and Ti(3dxz) orbitals, respectively. Grein [19] optimized the C2v equilibrium structure of the 11B1 state at the BPW91/6-311+G(3df) level and obtained an adiabatic excitation energy of 3.03 eV. Wei et al. [22] used the TD-B3LYP/cc-pVxZ (x = D, T, and Q) level of theories to optimize the C2v equilibrium structure and obtained the adiabatic excitation energies of 3.42–3.57 eV in the TD-B3LYP calculations and 3.14–3.33 eV in MRCI single-point calculations. In this study, the structures of the first three 1B1 states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels in both C2v and Cs symmetry. As shown in Table 3, the optimized C2v structure of the 11B1 state at the SAC-CI/def2-SVP level has a bond length of 1.752 Å and a bond angle of 138.77°. Similar to previous studies [19, 22], the C2v conformation has an imaginary frequency for the b2 vibration and could not be a stable one. From the SAC-CI/def2-SVPD calculation, the optimized C2v structure has a bond length of 1.772 Å and a bond angle of 142.35°. At this C2v geometry, the 11B1 state is characterized by the 3b1 → 11a1 transition mixed with some 9a1 → 4b1 characteristics. The calculated adiabatic excitation energies for the C2v conformation were 3.15–3.30 eV in the SAC-CI calculations, which are similar to the MRCI calculated values reported by Wei et al. [22]. For the Cs conformation, the optimized structure at the SAC-CI/def2-SVP level has two bond lengths of 1.907 and 1.608 Å and a bond angle of 120.69°; the vibrational frequencies are all positive for the Cs conformation. From the SAC-CI/def2-SVPD calculation, the Cs conformation has two bond lengths of 1.936 and 1.630 Å and a bond angle of 127.76°. The 11B1 (A") state at the Cs geometry is characterized by the 3b1 (a") → 11a1 (a') transition, where the 3b1 (a") orbital mainly involves the Oa(2px) and Ti(3dxz) orbitals (the Oa atom is on the longer Ti–O bond) and the 11a1 (a') orbital involves the Ti(3dz2–x2) orbital. The calculated adiabatic excitation energies were 2.78–2.87 eV for the Cs conformations in the SAC-CI calculations, which are smaller than those for the C2v conformation. As compared with the experimental adiabatic excitation energy of 3.37 eV [9], the calculated energies for the C2v conformation (3.15–3.25 eV) are in better agreement than those for the Cs conformation (2.78–2.87 eV). The stable Cs conformation of the 11B1 state could result from the pseudo-Jahn–Teller effect by which the 11B1 state couples with the 21A2 state via the b2 vibration. Figure 4 shows the pseudo-Jahn–Teller effect between the 11B1 and 21A2
states. It must be noted that the pseudo-Jahn–Teller effect should be checked by the CASSCF-calculated vibrational frequency [17, 35]; however, the CASSCF calculation is beyond the scope of this study. For the 21B1 state, the optimized C2v structure at the SAC-CI/def2-SVP level has a bond length of 1.736 Å and a bond angle of 135.76°; the C2v conformation has an imaginary frequency for the b2 vibration and could not be a stable conformation. From the SAC-CI/def2-SVPD calculation, the bond length is 1.766 Å and the bond angle is 143.24°. The C2v conformation of the 21B1 state, characterized by the 9a1 → 4b1 transition mixed with some 3b1 → 11a1 characteristics, has adiabatic excitation energies of 3.11–3.20 eV according to the SAC-CI calculations. No reliable Cs conformation was optimized in the SAC-CI calculation. The energies of the 21B1 and 11B1 states are very close to each other. As shown in Fig. 2, the energy differences between the two 1B1 states are within 0.12 eV, which are close to or within the accuracy of the current calculations. This phenomenon can be realized by looking at the Walsh diagram. As shown in Fig. 3, as the bond angle increases, the energies of the 9a1 and 4b1 orbitals approach to those of the 3b1 and 11a1 orbitals, respectively; therefore, the two b1 transitions, 3b1 → 11a1 and 9a1 → 4b1, have similar excitation energies and are somewhat mixed with each other. For the 31B1 state, the optimized C2v structure at the SAC-CI/def2-SVP level has a bond length of 1.726 Å and a bond angle of 131.62°; the C2v conformation has an imaginary frequency for the b2 vibration and could not be a stable conformation. From the SAC-CI/def2-SVPD calculation, the optimized C2v conformation, which is characterized by the 3b1 → 10a1 transition, has a bond length of 1.742 Å and a bond angle of 132.73°. The calculated adiabatic excitation energies of the C2v conformation were 3.61–3.70 eV in the SAC-CI calculations. For the Cs conformation, the optimized structure at the SAC-CI/def2-SVP level has two bond lengths of 1.880 and 1.603 Å and a bond angle of 126.29°; the vibrational frequencies are all positive for the Cs conformation. From the SAC-CI/def2-SVPD calculation, the optimized Cs conformation has two bond lengths of 1.897 and 1.615 Å and a bond angle of 127.42°. The optimized Cs conformation is characterized by the 3b1 (a") → 10a1 (a') transition, where the 3b1 (a") orbital mainly involves the Oa (2px) and Ti (3dxz) orbitals (the Oa atom is on the longer Ti–O bond) and the 10a1 (a') orbital involves the Ti (4s) orbital. The calculated adiabatic excitation energies for the Cs conformations were 3.29–3.39 eV in the SAC-CI calculations, which were smaller than those for the C2v conformation. The C2v conformations of the 11B1 and 31B1 states are mainly characterized by the transitions 3b1 → 11a1 and 3b1 → 10a1 at their respective optimized geometries, respectively, but by the combinations of 3b1 → 10a1 and 3b1 → 11a1 transitions in the
Frank–Condon region. This phenomenon is similar to that shown in the 11B2 and 21B2 state which are mainly characterized by 6b2 → 10a1 and 6b2 → 11a1 and can be realized by exploring the Walsh diagram (Fig. 3). Similar to the 11B1 state, the stable Cs conformation of the 31B1 state could result from the pseudo-Jahn–Teller effect (see Fig. 4). At the C2v structure (rTi-O = 1.752Å, θO-Ti-O = 123.29°) shown in Fig.4, the energy differences between the 31B1 and 21A2 states are 0.05 eV at the SAC-CI/def2-SVPD level and 0.07 eV at the SAC-CI/def2-TZVPD level, which are within the accuracy of the current calculations. 3.3.4. The 21A1 and 31A1 electronic states The 21A1 and 31 A1 states are mainly characterized by the combination of the 9a 1 → 10a1 and 9a1 → 11a1 transitions in the Franck–Condon region. For the 21A1 state, the C2v conformation optimized by the SAC-CI/def2-SVP calculation has an imaginary frequency for the b2 vibation, and the stable conformation could have a Cs symmetry. However, no reliable C s conformation was found for this state. From the SAC-CI/def2-SVPD calculations, the C2v structures have optimized geometrical parameters rTi-O = 1.767 Å and 1.731 Å and θO-Ti-O = 142.14° and 131.18° for the 21A1 and 31A1 states, respectively. The 21A1 state is characterized by the combination of the 9a1 → 11a1 and 3b1 → 4b1 transitions and the 31A1 state by the 9a1 → 10a1 transition mixed with some 3b1 → 4b1 characteristics at their respective optimized C2v structures. This phenomenon can be realized by checking the Walsh diagram, and it is similar to that for the 11B1 and 31B1 states, which are characterized by the 3b1 → 11a1 and 3b1 → 10a1 transitions mixed with some 9a1 → 4b1 characteristics. The calculated adiabatic excitation energies for the C2v conformations were 3.11–3.22 eV for the 21A1 states and 3.50–3.55 eV for the 31A1 states in the SAC-CI calculations. 3.3.5 The 11A2 and 21A2 electronic states The 11A2 state is characterized by the electronic transition 6b2 → 4b1 in the Franck–Condon region. Lin et al. [21] showed that the stable conformation of the 11A2 state has a symmetric linear form from their TD-DFT and EOM-CCSD calculations. At this linear geometry, the 11A2 and 11B2 states are degenerate and correlate to the 11Δu state in D∞h symmetry group. From the SAC-CI calculations, similar to those reported by Lin et al. [21], the 11A2 state has an optimized symmetric linear structure and is correlated to the 11Δu state. The 21A2 state is characterized by the combination of the transitions 1a2 → 10a1 and 1a2 → 11a1, where the 1a2 orbital involves the O(2px) – O(2px) orbital. From the SAC-CI/def2-SVPD calculation, the optimized C2v conformation has a bond length of 1.747 Å and a bond angle of 105.50°. The calculated adiabatic excitation energies
were 3.57–3.65 eV in the SAC-CI calculations. No Cs structure was optimized for this state. 4. Summary Nine low-lying electronic excited states, 21A1, 31A1, 11A2, 21A2, 11B1, 21B1, 31B1, 11B2, and 21B2, were studied using SAC-CI theory with various basis sets. The C2v conformations of the nine states and the Cs conformations of the 11B1 and 31B1 states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels. For the 11B1 and 31B1 states, the Cs conformations are stable ones in which the vibrational frequencies are all positive and the C2v conformation are saddle points that have an imaginary frequencies for the b2 vibrations in the SAC-CI/def2-SVP calculations. The Cs conformations of the 11B1 and 31B1 states could result from the pseudo-Jahn–Teller effect by which the two 1B1 states couple with the 21A2 state via the b2 vibrations. The calculated geometrical parameters and vibrational frequencies of the X1A1 and 11B2 states and the calculated adiabatic excitation energy of the 11B2 state agree with the experimental values. For the 11B1 state, the calculated adiabatic excitation energy of the C2v conformation is in better agreement with the experimental value of 3.37 eV than that of the Cs conformation. More experimental and theoretical studies are needed for understanding the nature of the low-lying electronic states of TiO2. Acknowledgements The author is grateful to the National Center for High-performance Computing for computer time and facilities.
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Figure captions Figure 1 The selected molecular orbitals obtained from the RHF/def2-SVPD calculation. The RHF-calculated orbitals were used as the reference orbitals in the SAC-CI calculations. Figure 2 The energy profile for the energy scan along the bond-angle coordinate. The energy scan was conducted along the bond angle θ O-Ti-O from 95° to 180° with a step size of 5° where the relative energies were calculated by single-point SAC-CI/def2-SVPD calculations at the geometries of the X1A1 state optimized at the BPW91/def2-SVPD level. Figure 3 The calculated orbital energies as a function of the bond angle. The orbital energies were calculated at the optimized bond length with a fixed bond angle at the BPW91/aug-cc-pVTZ level. Figure 4 The energy profile for the pseudo-Jahn–Teller effect between the 1B1 and 21A2 states. The relative energies were calculated by single-point SAC-CI/def2-SVPD calculations at the geometries obtained from the IRC calculation (UBPW91/def2-SVPD) for the C2v and Cs conformations of the 11B1 state.
Table 1 The calculated vertical excitation energies ∆Eve (eV), oscillator strengths f (a. u.), and electronic transitions of the low-lying singlet excited states of TiO2. SAC-CI calculations States
def2SVPa
def2SVPDa
def2TZVPD
Previous calculations
def2TZVPPD
aug-ccpVTZ
fb
Electronic transitionsc
MRCId CCSDe
CASPT2f
∆Eve 11B2
2.356 (2.423)
2.485 (2.501)
2.445
2.443
2.460
0.010
-0.73 (6b2 → 10a1) -0.53 (6b2 → 11a1)
2.43
2.386
2.386
11A2
2.999 (3.127)
3.061 (3.063)
3.053
3.049
3.062
fb
0.71 (6b2 → 4b1) -0.61 (6b2 → b1)
3.09
3.045
3.150
21B2
3.193 (3.323)
3.252 (3.250)
3.233
3.233
3.248
0.011
0.64 (6b2 → 11a1) -0.50 (6b2 → 10a1) -0.44 (6b2 → a1)
3.21
3.213
3.365
21A1
3.297 (3.388)
3.444 (3.448)
3.391
3.390
3.402
0.000
-0.68 (9a1 → 10a1) -0.53 (9a1 → 11a1)
3.13
3.315
3.480
11B1
3.602 (3.751)
3.679 (3.683)
3.643
3.644
3.657
0.033
-0.68 (3b1 → 10a1) -0.58 (3b1 → 11a1)
3.57
3.599
3.601
21B1
3.613 (3.765)
3.731 (3.726)
3.728
3.730
3.739
0.002
0.68 (9a1 → 4b1) -0.64 (9a1 → b1)
3.74
3.711
3.979
21A2
3.970 (4.123)
4.039 (4.072)
3.973
3.971
3.982
fb
-0.64 (1a2 → 10a1) -0.60 (1a2 → 11a1)
4.07
3.969
3.948
31A1
3.978 (4.145)
4.074 (4.063)
4.054
4.055
4.066
0.004
3.83
4.027
4.586
31B2
4.367 (4.527)
4.427 (4.445)
4.373
4.373
4.380
0.002
31B1
4.356 (4.574)
4.393 (4.381)
4.374
4.378
4.390
0.002
0.51 (9a1 → 10a1) -0.41(9a1 → 11a1) -0.40 (3b1 → 4b1) 0.53 (5b2 → 11a1) 0.45 (5b2 → 10a1) 0.40 (1a2 → 4b1) -0.57 (3b1 → 11a1) 0.56 (3b1 → 10a1) 0.46 (3b1 → a1)
a. Values in parentheses indicate the calculated vertical excitation energies at the SAC-optimized geometry of the ground electronic state of TiO2. b. SAC-CI/aug-cc-pVTZ calculated results; fb denotes "forbidden". c. SAC-CI/def2-SVPD calculated results; only the coefficients larger than 0.4 are listed. d. Ref. [19]. e. Ref. [20]. f. CASPT2(16,14) calculated results in Ref. [21].
4.19
4.39
4.250
Table 2 The calculated adiabatic excitation energies ΔEae (eV), optimized geometrical parameters [rTi-O (Å), θO-Ti-O (°)], vibrational frequencies ω (cm-1), and dipole moments μ (Debye) of the singlet X1A1, 11B2, 11B1, and 11A2 states of TiO2. Methods UBPW91/def2-SVP UBPW91/def2-SVPD UBPW91/def2-TZVPD UBPW91/def2-TZVPPD UBPW91/aug-cc-pVTZ BPW91/6-311+G(3df)a B3LYP/aug-cc-pVTZb
ΔEae
rTi-O
θO-Ti-O
ω1(a1)
ω2(a1)
ω3(b2)
μ
ΔEae
rTi-O
0 0 0 0 0 0 0
1.638 1.649 1.652 1.651 1.652 1.651 1.642 1.651c
109.36 110.56 110.65 110.61 110.60 110.7 111.7
1033.3 999.0 990.2 987.6 985.0 986 1023.0
111.57c
962d
UBPW91/def2-TZVPD UBPW91/def2-TZVPPD UBPW91/aug-cc-pVTZ BPW91/6-311+G(3df)a
3.042 3.020 2.934 2.938 2.929 3.03
1.730 1.752 1.754 1.753 1.755 1.753
118.54 123.29 123.21 123.13 123.12 123.20
834.3 792.5 788.5 786.8 785.2 786
357.6 343.0 340.4 340.7 340.2 339 341.9 329d
1000.9 958.8 953.0 947.8 946.5 946 975.2 935d
6.32 6.55 6.48 6.49 6.48 6.73 6.73 6.33e
2.538
1.710
144.26
802.6
ω3(b2)
μ 3.92 4.33 4.32 4.34 4.36 5.07
2.053 2.169 2.086 2.087 2.084 2.14
1.687 1.700 1.704 1.703 1.704 1.703
95.38 96.07 96.60 96.53 96.52 96.3
925.7 885.6 876.9 874.6 871.7 875
223.8 206.4 198.1 197.8 197.2 196
530.7 493.0 485.8 479.0 477.9 480
2.1813 2.37 2.34 2.4f
1.704, 1.708 1.709, 1.704g
100.1 101.66 98.92 100.1g
876h 825i
184h 194i
316h
227.2 195.4 196.3 197.6 198.6
1046.8 989.0 984.5 983.6 980.2
2.55e
11B1(A", Cs) 238.9 199.4 199.8 200.9 201.3 200
728.9i 640.4i 654.5i 661.4i 662.4i 659i
4.31 4.84 4.74 4.72 4.73
133.7
629.3
2.88
11A2(S2; C2v) UBPW91/def2-SVP
ω2(a1)
1 B2(C2v)
11B1(C2v) UBPW91/def2-SVPD
ω1(a1) 1
X A1(C2v)
Exp.
UBPW91/def2-SVP
θO-Ti-O
1
2.895 2.907 2.817 2.814 2.806
1.897, 1.613 1.918, 1.628 1.920, 1.631 1.921, 1.629 1.922, 1.630
112.09 116.15 116.03 115.82 115.82
615.5 583.2 583.3 582.2 583.8
4.43 4.92 4.86 4.84 4.85
UBPW91/def2-SVPD
2.549
1.724
143.00
771.0
123.4
562.6
3.10
UBPW91/def2-TZVPD
2.441 2.438
1.724 1.724
143.94 143.56
770.1 768.4
126.6 128.5
584.3 575.0
2.86 2.86
UBPW91/def2-TZVPPD
2.430 1.725 143.07 765.9 126.4 569.1 2.88 BPW91/6-311+G(3df) 2.5 1.724 143.9 767 127 575 a. Ref. [19]. b. Ref. [21]. c. Ref. [5]. d. Ref. [7]. e. Ref. [6] f. 2.1813 eV in Ref. [8], 2.37 eV in Ref. [9], 2.34 eV in Ref. [3], and 2.4 eV in Ref. [10]. g. (rTi-O, θO-Ti-O) = (1.704 Å, 100.1°) in Ref. [6], and (1.708, 101.66°), (1.709, 98.92°) and (1.704 Å, 100.1°) in Ref. [8]. h. Ref. [8]. i. Ref. [9]. UBPW91/aug-cc-pVTZ a
Table 3 The optimized geometrical parameters [rTi-O (Å), θO-Ti-O (°)] and vibrational frequencies ω (cm-1) of the low-lying singlet electronic excited states of TiO2. States
X1A1
11B2
Methods SAC/def2-SVP SAC/def2-SVPD SAC/def2-TZVPD CCSD(T)a Exp SAC-CI/def2-SVP SAC-CI/def2-SVPD TD-B3LYPa EOM-CCSDa Exp
11Δug
21B2
11B1
21B1 31B1
SAC-CI/def2-SVP SAC-CI/def2-SVPD TD-B3LYPa EOM-CCSDa SAC-CI/def2-SVP SAC-CI/def2-SVPD SAC-CI/def2-SVP SAC-CI/def2-SVPD SAC-CI/def2-SVP (Cs) SAC-CI/def2-SVPD (Cs) SAC-CI/def2-SVP SAC-CI/def2-SVPD SAC-CI/def2-SVP
rTi-O
θO-Ti-O
ω1(a1)
ω2(a1)
ω3(b2)
1.633 1.649 1.647 1.652 1.651b 1.672 1.690 1.673 1.662 1.704, 1.708 1.709, 1.704d 1.694 1.713 1.700 1.684 1.680 1.695 1.752 1.772 1.907, 1.608 1.936, 1.630 1.736 1.766 1.726
110.78 112.40 112.83 111.2 111.57b 98.93 98.81 100.2 100.4 100.1, 101.66 98.92, 100.1d 180(fixed) 180(fixed) 180.0 180.0 133.590 131.974 138.77 142.35 120.69 127.76 135.76 143.24 131.62
1071.1 1028.4
355.8 333.2
1047.7 986.1
998.5 962c 1011.4 947.3 945.6 1009.1 825f 876f 881.7
333.7 329c 222.2 205.9 212.5 198.3 194e 184f 80.0
955.2 935c 366.3 296.0 374.7 499.0
837.2 887.9 921.9
253.8 297.3 371.6
658.8 879.4 462.4
858.9
172.1
1243.4i
662.1
147.7
1102.1
900.9
311.7
1290.2i
316f 776.0
SAC-CI/def2-SVPD 1.742 132.73 SAC-CI/def2-SVP (Cs) 1.880, 1.603 126.29 711.1 285.3 1095.7 SAC-CI/def2-SVPD (Cs) 1.897, 1.615 127.42 SAC-CI/def2-SVP 1.712 122.69 857.7 104.5 977.7i 21A1 SAC-CI/def2-SVPD 1.767 142.14 SAC-CI/def2-SVP 1.711 128.74 31A1 SAC-CI/def2-SVPD 1.731 131.18 SAC-CI/def2-SVP 1.724 103.11 21A2 SAC-CI/def2-SVPD 1.747 105.50 a. Ref. [21]. b. Ref. [5]. c. Ref. [7]. d. (rTi-O, θO-Ti-O) = (1.704 Å, 100.1°) in Ref. [6], and (1.708, 101.66°), (1.709, 98.92°) and (1.704 Å, 100.1°) in Ref. [8]. e. Ref. [9]. f. Ref. [8]. g. In D∞h symmetry. ω1, ω2, and ω3 correspond to σg+, πu, and σg– vibrations, respectively
Table 4 The SAC-CI-calculated adiabatic excitation energies ΔEae and dipole moments μ for the low-lying singlet electronic excited states of TiO2. Basis sets
11B2
11∆ua
21B2
11B1 (A")b
21B1
31B1 (A")b
21A1
31A1
21A2
ΔEae (eV) def2-SVP
2.286
2.468
2.838
3.304 (2.872)
3.201
3.700 (3.393)
3.222
3.549
3.648
def2-SVPD
2.355
2.474
2.902
3.146 (2.782)
3.108
3.633 (3.342)
3.109
3.541
3.593
def2-TZVPD
2.335
2.525
2.849
3.218 (2.833)
3.167
3.583 (3.294)
3.167
3.496
3.570
def2-TZVPPD
2.344
2.548
2.862
3.254 (2.849)
3.200
3.607 (3.306)
3.199
3.516
3.590
μ (Debye) def2-SVP
3.61
0
0.73
4.15 (5.40)
3.80
1.08 (2.00)
4.47
1.70
5.22
def2-SVPD
4.13
0
0.56
3.85 (5.45)
3.52
0.97 (1.88)
3.78
2.01
5.86
def2-TZVPD
4.06
0
0.78
3.74 (5.36)
3.46
0.94 (2.06)
3.71
1.88
5.67
def2-TZVPPD
4.06
0
0.76
3.72 (5.34)
3.43
0.94 (2.05)
3.69
1.87
5.65
1
c
For 1 B2 state : ΔEae = 2.1813, 2.37, 2.34, and 2.4 eV. μ = 2.55 D. For 11B1 stated: 3.37 eV. a. In D∞h symmetry. b. Values in parentheses indicate the calculated values for the Cs conformation. c: ΔEae = 2.1813 eV in Ref. [8], 2.37 eV in Ref. [9], 2.34 eV in Ref. [3], and 2.4 eV in Ref. [10]. μ = 2.55 D in Ref. [6]. d: Ref. [9]. Exp
1
Intensity
1 B1
1
2 B2
1
1 B2
1
3 A1
1
1
1 A2 2.0
2.5
3.0
1
2 A1
2 B1
3.5
1
1
2 A2
3 B1
4.0
4.5
Vertical Excitation Energy (eV)
The nine low-lying singlet electronic excited states of TiO2 were studied using SAC-CI theory. The nature of the low-lying electronic states was discussed in this study.
Highlights 1. Nine low-lying singlet excited states of TiO2 were studied using SAC-CI theory. 2. The adiabatic excitation energies of the excited states were calculated. 3. Stable Cs structures for the 11B1 (A") and 31B1 (A") states were determined.
S2210-271X(15)00083-3 http://dx.doi.org/10.1016/j.comptc.2015.02.012 COMPTC 1747
To appear in:
Computational & Theoretical Chemistry
Received Date: Revised Date: Accepted Date:
10 February 2015 16 February 2015 16 February 2015
Please cite this article as: Y-C. Chou, The low-lying singlet electronic excited states of TiO2: A symmetry adapted cluster–configuration interaction (SAC-CI) study, Computational & Theoretical Chemistry (2015), doi: http:// dx.doi.org/10.1016/j.comptc.2015.02.012
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The low-lying singlet electronic excited states of TiO2: A symmetry adapted cluster–configuration interaction (SAC-CI) study
Yung-Ching Chou
Department of Applied Physics and Chemistry, University of Taipei , No. 1, Ai-Guo West Road, Taipei 10048, Taiwan
Corresponding author: Yung-Ching Chou (E-mail Address: [email protected]) Department of Applied Physics and Chemistry University of Taipei No. 1, Ai-Guo West Road, Taipei 10048, Taiwan
Abstract The ground electronic state (X1A1) and nine low-lying singlet electronic excited states (21A1, 31A1, 11A2, 21A2, 11B1, 21B1, 31B1, 11B2, and 21B2) of TiO2 were studied using SAC-CI theory. The geometries of the nine electronic excited states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels. The adiabatic excitation energies of the nine electronic excited states were obtained from single-point calculations at the SAC-CI/def2-TZVPD//SAC-CI/def2-SVPD and SAC-CI/def2-TZVPPD//SAC-CI/def2-SVPD levels. Stable Cs structures for the 11B1 (A") and 31B1 (A") states were determined in this study. For comparison, the X1A1, 11A2, 11B1, and 11B2 states were also studied using the unrestricted DFT theory (UBPW91) with the def2-SVP, def2-SVPD, def2-TZVPD, def2-TZVPPD, and aug-cc-pVTZ basis sets.
1. Introduction TiO2 has versatile applications owing to its various advantageous properties, especially its photocatalytic activity [1]. Therefore, the study of its electronic excited states is important for understanding the photochemistry of the molecule. Several experimental [2–10] and theoretical [11–22] studies of the ground and low-lying electronic states of TiO2 and its clusters (TiO2)n have been presented in the literature. However, theoretical studies for the low-lying electronic excited states of the TiO2 monomer are comparatively few [19–22]. Grein [19] studied the lowest singlet, triplet, and quintet states (A1, A2, B1, and B2 states in C2v symmetry) by using density functional theory (DFT) at the BPW91/6-311+G(3df) level, and obtained the optimized geometries, vibrational frequencies, and adiabatic excitation energies for the electronic states. The vertical excitation energies calculated using multi-reference configuration interaction (MRCI) theory for the singlet and triplet states with excitation energies up to approximately 4.6 eV were also given in his study. Grein showed that the optimized C2v structure of the 11B1 state has an imaginary frequency for the b2 vibration (anti-symmetric stretching), and he suggested that the 11B1 state could have a lower energy in a Cs structure (A" state). Taylor and Paterson [20] used several coupled cluster response theories [CC2, CCSD, CCSDR(3), and CC3], and time-dependent DFT (TD-DFT) theory to study the low-lying excited states. Lin et al [21] used various levels of theory, including TD-DFT, complete active space self consistent field (CASSCF), CASPT2, CCSD(T), and equation-of-motion CCSD (EOM-CCSD), to study the low-lying electronic excited states of TiO2. In their study, the 11B2 (S1) state was optimized to two possible stable structures: one corresponding to a bent structure (C2v) and the other to a symmetric linear structure (D∞h). The 11A2 (S2) state was optimized to a symmetric linear structure in which the 11A2 and 11B2 states are degenerate and correspond to the 11Δu state in D∞h symmetry. They suggested that the 21B2 (S3) state has only one minimum with C2v symmetry by performing a two-dimensional potential energy surface scan at the TD-B3LYP/aug-cc-pVTZ level. In this study, we used the symmetry adapted cluster–configuration interaction (SAC-CI) method [23–27] to study the ground electronic state [X1A1(S0)] and the nine low-lying singlet excited states [21A1(S4), 31A1(S8), 11A2(S2), 21A2(S7), 11B1(S5), 21B1(S6), 31B1(S10), 11B2(S1), and 21B2(S3) states] of TiO2. The vertical excitation energies, optimized geometries, and adiabatic excitation energies of the considered excited states were calculated and compared with the experimental values. For more information of the SAC-CI theory, please refer to SAC-CI’s website [28]. For comparison, the unrestricted DFT theory (UBPW91) was also applied to study the X1A1, 11A2, 11B1, and 11B2 states. Stable Cs structures for the 11B1 (A") and 31B1 (A")
states, which were not found in the literature, were optimized in the SAC-CI calculations; the stable Cs structure for the 11B1 (A") state was also optimized in the UBPW91 calculations. 2. Computational details In the DFT calculations, the UBPW91 method with def2-SVP, def2-SVPD, def2-TZVPD, def2-TZVPPD [29–31], and aug-cc-pVTZ [32, 33] basis sets were applied to study the X1A1, 11A2, 11B1, and 11B2 states. In the SAC-CI calculations, only the singles and doubles linked excitation operators were included (default); the key word “NoLinkedSelection” was applied to include all singles and doubles linked operators and “NonVariational” was applied to solve the SAC-CI equations. The geometries of the electronic states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels; the vibrational frequencies were calculated at the SAC-CI/def2-SVP level to characterize the stationary points. Single-point calculations at the SAC-CI/def2-TZVPD//SAC-CI/def2-SVPD and SAC-CI/def2-TZVPPD//SAC-CI/def2-SVPD levels were performed to obtain the adiabatic excitation energies. All the calculations were performed with Gaussian 09 program packages [34]. 3. Results and Discussion 3.1. Vertical excitations Table 1 shows the SAC-CI-calculated vertical excitation energies, electronic transitions, and oscillator strengths of the first ten singlet excited states and the previous results calculated using MRCI [19], CCSD [20], and CASPT2 [21] theories. Figure 1 shows the selected molecular orbitals for the TiO2 molecule. The vertical excitation energies were calculated at the experimental equilibrium geometry of the ground state of TiO2 with a bond length (rTi-O) of 1.651Å and a bond angle (θO-Ti-O) of 111.57° [5]. The vertical excitation energies calculated using SAC-CI are similar to those calculated by the MRCI [19], CCSD [20], and CASPT2 [21] theories. As shown in Table 1, the SAC-CI-calculated vertical excitation energies are consistent with each other when using the def2-SVPD, def2-TZVPD, def2-TZVPPD, and aug-cc-pVTZ basis sets. This suggests that the def2-SVPD basis set is suitable for describing the excitation energies of the low-lying excited states of TiO2 in the SAC-CI calculations. The calculated vertical excitation energies of the 11B2 and 11B1 states were approximately 2.44–2.50 eV and 3.64–3.68 eV, respectively, which are reasonably higher than the experimental adiabatic excitation energies of 2.18 – 2.40 eV [3, 8–10] for the 11B2 state and 3.37 eV [9] for the 11B1 state. The SAC-CI-calculated vertical excitation energy for the 11B2 state is in good agreement with the experimental value
of 2.46 eV [3]. More information concerning the calculated vertical excitation properties of the excited states with excitation energies up to approximately 6 eV is shown in the supplementary materials. 3.2. DFT calculations Table 2 shows the calculated adiabatic excitation energies, geometry parameters, vibrational frequencies, and dipole moments of the X1A1, 11B2, 11A2, and 11B1 states using UBPW91 theory. The current DFT-calculated results are similar to those shown in previous DFT calculations [19, 21] except for the Cs conformation of the 11B1(A") state which is not found in the literature. For the X1A1 state, the calculated geometry parameters, vibrational frequencies, and dipole moments agree with the experimental values when using basis sets larger than the def2-SVP basis set. For the 11B2 state, the calculated adiabatic energies reasonably agree with the experimental values; the calculated bond lengths are in good agreement with the experimental values when using basis sets larger than the def2-SVP basis set, but the calculated bond angles are somewhat smaller than the experimental angles. The calculated a1 vibrational frequencies (ω1 referring to symmetric stretching and ω2 to bending) reasonably agree with the experimental values, but the calculated b2 frequencies (ω3) are approximately 1.5 times larger than the experimental values. For the 11B1 state, the geometry was optimized to two stationary points: one has a C2v symmetry and the other has a Cs symmetry. The C2v conformation of the 11B1 state is similar to those presented in previous calculations [19] and has an imaginary vibrational frequency for the b2 vibration. For the Cs conformation, the optimized structure has two bond lengths (Ti–O) of approximately 1.92 and 1.63 Å and a bond angle (O–Ti–O) of approximately 116°; the vibrational frequencies are all positive for the Cs conformation. For confirming the relation between the C2v and Cs conformations, we conducted an intrinsic reaction coordinate (IRC) calculation at the UBPW91/def2-SVPD level and found that the C2v conformation is the transition state on the reaction path between the two equivalent C s conformations. The calculated adiabatic excitation energies were approximately 2.93–3.02 eV for the C2v conformation and 2.81–2.91 eV for the Cs conformation, which are smaller than the experimental value of 3.37 eV. As sown in Table 2, the UBPW91-calculated results are consistent with each other when using basis sets larger than the def2-SVP basis set; this suggests that the def2-SVPD basis set is suitable for describing the low-lying electronic states of TiO2 in the UBPW91 calculations. 3.3. SAC-CI calculations
The X1A1, 11B2, 21B2, 11B1, 21B1, 31B1, 11A2, 21A2, 21A1, and 31A1 states were studied using the SAC-CI theory and are discussed as follows. Table 3 lists the optimized geometrical parameters and vibrational frequencies of the electronic states. Table 4 lists the calculated adiabatic excitation energies and dipole moments of the electronic excited states. 3.3.1. The X1A1 state TiO2 in its ground electronic state (X1A1) has a C2v point group symmetry. The equilibrium geometry of the ground state was optimized at the SAC/def2-SVP, SAC/def2-SVPD, and SAC/def2-TZVPD levels, and the vibrational frequencies were calculated at the SAC/def2-SVP and SAC/def2-SVPD levels. As shown in Table 3, the optimized geometrical parameters at the SAC/def2-SVPD level were rTi-O = 1.649 Å and θO-Ti-O = 112.40°, which agree with the experimentally determined values of rTi-O = 1.651 Å and θO-Ti-O = 111.57° [5]. The optimized geometry parameters at the SAC/def2-SVPD level are consistent with those at the SAC/def2-TZVPD level (rTi-O = 1.647 Å, θO-Ti-O = 112.83°), suggesting that the def2-SVPD basis set is suitable for describing the geometry of TiO2 in the SAC calculations. The calculated vibrational frequencies at the SAC/def2-SVP and SAC/def2-SVPD levels are on average approximately 11% and 5% higher than the experimental values, respectively. The calculated dipole moments were 6.89, 7.08, and 7.13 D at the SAC/def2-SVP, SAC/def2-SVPD, and SAC/def2-TZVPD level, respectively, which agree with the experimental values of 6.33 and 6.79 D [6, 8]. 3.3.2. The 11B2 and 21B2 electronic states The 11B2 and 21B2 states, in the Franck–Condon region, are characterized by the electronic transitions 6b2 → 10a1 ( + 6b2 →11a1) and 6b2 → 11a1 ( – 6b2 →10a1), respectively where the transition shown in parentheses indicates the minor one. The 6b2 , 10a1 , and 11a1 orbitals mainly involve O(2pz) – O(2pz), Ti(4s), and Ti(3dz2– x2) orbitals, respectively (see Fig. 1). Lin et al. [21] used several levels of theories to study the 11B2 state and showed that this state has two possible stable structures: one corresponding to a bent structure (C2v), and the other to a symmetric linear one (D∞h). In this study, the optimized geometrical parameters for the C2v conformation of the 11B2 state were (rTi-O = 1.672 Å and θO-Ti-O = 98.93°) and (rTi-O = 1.690 Å and θO-Ti-O = 98.81°) at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels, respectively. These calculated values agree with the experimental bond lengths of 1.704–1.709 Å and bond angles of 98.92–101.66° (at different vibational levels) [6, 8]. Compared with the geometrical parameters optimized using the ECM-CCSD and TD-B3LYP theories [21], the bond
length calculated using the SAC-CI/def2-SVPD method is slightly better than those calculated using the EOM-CCSD and TD-B3LYP methods. The SAC-CI-calculated bond angle is close to the experimental value of 98.92°, while the values calculated using the EOM-CCSD and TD-B3LYP methods are close to the experimental value of 100.1° (see Table 3). The vibrational frequencies for the C2v conformation were calculated at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels. The deviations between the calculated and experimental vibrational frequencies are on average approximately 17% and 9% at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels, respectively. The calculation using the def2-SVPD basis set significantly improves the quality of the vibrational frequency calculation compared with that using the def2-SVP basis set. However, for saving computational cost, the SAC-CI/def2-SVP calculation was used to characterize the optimized stationary points in this study. The calculated adiabatic excitation energies were 2.29–2.36 eV in the SAC-CI calculations, which agree with the experimental values of 2.18–2.40 eV [3, 8–10] (see Table 4). The SAC-CI-calculated dipole moments of 3.6–4.1 D, which are similar to those calculated by the EOM-CCSD (4.31 D) and TD-B3LYP (3.57 D) methods [21], are approximately 40%–60% larger than the experimental value of 2.55 D [6]. The linear equilibrium conformation of the 11B2 state, which correlates to the 11Δu state in D∞h symmetry, is characterized by the 5σu → 1δg electronic transition; the 5σu and 1δg orbitals mainly correspond to the O(2py) + O(2py) and Ti(3dz2 – x2) orbitals, respectively. The optimized bond lengths were 1.694 Å at the SAC-CI/def2-SVP level and 1.713 Å at the SAC-CI/def2-SVPD level, which are similar to those calculated using the TD-B3LYP and EOM-CCSD theories (see Table 3). The calculated adiabatic excitation energies were 2.47–2.55 eV in the SAC-CI calculations, which are similar to those calculated using the TD-B3LYP (2.609 eV) and EOM-CCSD (2.387 eV) theories [21]. As shown in Table 3, the vibrational frequencies calculated at the SAC-CI/def2-SVP level are similar to those calculated by the TD-B3LYP and EOM-CCSD theories except for the πu bending vibration. The calculated πu frequency at the SAC-CI/def2-SVP level was 80.0 cm−1, which is much smaller than those calculated by the TD-B3LYP (253.8 cm−1) and EOM-CCSD (297.3 cm−1) theories. This small πu frequency is related to a larger linearization energy [E(11∆u) – E(11B2)] in the SAC-CI calculations. The linearization energies were 0.12 – 0.20 eV in the SAC-CI calculations, which are larger than those in the TD-B3LYP (0.030 eV) and EOM-CCSD (0.063 eV) calculations [21]. Compared with those in the TD-B3LYP and EOM-CCSD calculations, the larger linearization energy suggests that the potential energy surface near the optimized geometry of the 11∆u state could be more flattened along the bond-angle coordinate in the SAC-CI calculations. Therefore, the πu frequency calculated by the SAC-CI method is
smaller than those calculated by the TD-B3LYP and EOM-CCSD methods. Lin et al. [21] conducted a two dimensional scan on the potential energy surface of the 21B2 state; they suggested that this state has only one minimum (at rTi-O = 1.680 Å and θO-Ti-O = 130.6°) on the potential energy surface and has an adiabatic excitation energy of 3.066 eV. In this study, the geometry of the 21B2 state was optimized to a minimum (in C2v symmetry) on the potential energy surface at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels; the calculated vibrational frequencies are all positive in the SAC-CI/def2-SVP calculation. The optimized equilibrium structure has a bond length of 1.695 Å and a bond angle of 131.97º at the SAC-CI/def2-SVPD level, which are similar to the values reported by Lin et al. [21]. The calculated adiabatic excitation energies were 2.84–2.90 eV in the SAC-CI calculations, which are slightly smaller than the calculated value of 3.066 eV. At this optimized geometry, the 21B2 state is mainly characterized by the electronic transition 6b2 → 10a1 (– 6b2 →11a1), where the major transition is the same as that in the 11B2 state in the Franck-Condon region; this phenomenon suggests that there could be some interaction between the two 1B2 states. To better understand the nature of the selected excited states, we conducted a potential energy surface scan along the bond-angle coordinate for these states and constructed a Walsh diagram. Figures 2 and 3 show the energy profile for the energy scan and Walsh diagram, respectively. As shown in Fig. 2, the minimum on the potential energy surface of the 21B2 state is formed due to the “avoid crossing” between the two 1B2 energy surfaces. At the SAC-CI/def2-SVP optimized geometry of the 21B2 state, the calculated vibrational frequencies of the 11B2 state were ω1(a1) = 920 cm-1, ω2(a1) = 316i cm-1, ω3(b2) = 465 cm-1 at the SAC-CI/def2-SVP level. The imaginary frequency on the bending mode (ω2) suggests that this geometry is a saddle point on the potential energy surface between the equilibrium geometries of the 11B2 and 11Δu states. At the SAC-CI/def2-SVPD optimized geometry of the 21B2 state, the calculated energy gaps between the two 1B2 states were 0.329, 0.291, and 0.292 eV at the SAC-CI/def2-SVPD, SAC-CI/def2-TZVPD, and SAC-CI/def2-TZVPPD levels, respectively. The calculated energies of the 11B2 state at the saddle point are higher than those at the equilibrium geometry by 0.217, 0.224, and 0.226 eV at the SAC-CI/def2-SVPD, SAC-CI/def2-TZVPD, and SAC-CI/def2-TZVPPD levels, respectively. As shown in Fig. 3, the energy of the 11a1 orbital becomes lower than that of the 10a1 orbital when the bond angle is larger than approximately 130°. The 21B2 and 11B1 states are thereby characterized by the electronic transitions 6b2 → 10a1 and 6b2 → 11a1, respectively, and correlate to the 1Σu+ and 1Δu states when the bond angle becomes 180° (see Fig. 2).
3.3.3. The 11B1, 21B1, and 31B1 electronic states The 11B1, 21B1, and 31B1 states in the Franck-Condon region are characterized by the electronic transitions 3b1 → 10a1 + 3b1 →11a1, 9a1 → 4b1, and 3b1 → 11a1 – 3b1 →10a1, respectively. The 3b1, 9a1, and 4b1 mainly involve O(2px) + O(2px), O(2py) – O(2py), and Ti(3dxz) orbitals, respectively. Grein [19] optimized the C2v equilibrium structure of the 11B1 state at the BPW91/6-311+G(3df) level and obtained an adiabatic excitation energy of 3.03 eV. Wei et al. [22] used the TD-B3LYP/cc-pVxZ (x = D, T, and Q) level of theories to optimize the C2v equilibrium structure and obtained the adiabatic excitation energies of 3.42–3.57 eV in the TD-B3LYP calculations and 3.14–3.33 eV in MRCI single-point calculations. In this study, the structures of the first three 1B1 states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels in both C2v and Cs symmetry. As shown in Table 3, the optimized C2v structure of the 11B1 state at the SAC-CI/def2-SVP level has a bond length of 1.752 Å and a bond angle of 138.77°. Similar to previous studies [19, 22], the C2v conformation has an imaginary frequency for the b2 vibration and could not be a stable one. From the SAC-CI/def2-SVPD calculation, the optimized C2v structure has a bond length of 1.772 Å and a bond angle of 142.35°. At this C2v geometry, the 11B1 state is characterized by the 3b1 → 11a1 transition mixed with some 9a1 → 4b1 characteristics. The calculated adiabatic excitation energies for the C2v conformation were 3.15–3.30 eV in the SAC-CI calculations, which are similar to the MRCI calculated values reported by Wei et al. [22]. For the Cs conformation, the optimized structure at the SAC-CI/def2-SVP level has two bond lengths of 1.907 and 1.608 Å and a bond angle of 120.69°; the vibrational frequencies are all positive for the Cs conformation. From the SAC-CI/def2-SVPD calculation, the Cs conformation has two bond lengths of 1.936 and 1.630 Å and a bond angle of 127.76°. The 11B1 (A") state at the Cs geometry is characterized by the 3b1 (a") → 11a1 (a') transition, where the 3b1 (a") orbital mainly involves the Oa(2px) and Ti(3dxz) orbitals (the Oa atom is on the longer Ti–O bond) and the 11a1 (a') orbital involves the Ti(3dz2–x2) orbital. The calculated adiabatic excitation energies were 2.78–2.87 eV for the Cs conformations in the SAC-CI calculations, which are smaller than those for the C2v conformation. As compared with the experimental adiabatic excitation energy of 3.37 eV [9], the calculated energies for the C2v conformation (3.15–3.25 eV) are in better agreement than those for the Cs conformation (2.78–2.87 eV). The stable Cs conformation of the 11B1 state could result from the pseudo-Jahn–Teller effect by which the 11B1 state couples with the 21A2 state via the b2 vibration. Figure 4 shows the pseudo-Jahn–Teller effect between the 11B1 and 21A2
states. It must be noted that the pseudo-Jahn–Teller effect should be checked by the CASSCF-calculated vibrational frequency [17, 35]; however, the CASSCF calculation is beyond the scope of this study. For the 21B1 state, the optimized C2v structure at the SAC-CI/def2-SVP level has a bond length of 1.736 Å and a bond angle of 135.76°; the C2v conformation has an imaginary frequency for the b2 vibration and could not be a stable conformation. From the SAC-CI/def2-SVPD calculation, the bond length is 1.766 Å and the bond angle is 143.24°. The C2v conformation of the 21B1 state, characterized by the 9a1 → 4b1 transition mixed with some 3b1 → 11a1 characteristics, has adiabatic excitation energies of 3.11–3.20 eV according to the SAC-CI calculations. No reliable Cs conformation was optimized in the SAC-CI calculation. The energies of the 21B1 and 11B1 states are very close to each other. As shown in Fig. 2, the energy differences between the two 1B1 states are within 0.12 eV, which are close to or within the accuracy of the current calculations. This phenomenon can be realized by looking at the Walsh diagram. As shown in Fig. 3, as the bond angle increases, the energies of the 9a1 and 4b1 orbitals approach to those of the 3b1 and 11a1 orbitals, respectively; therefore, the two b1 transitions, 3b1 → 11a1 and 9a1 → 4b1, have similar excitation energies and are somewhat mixed with each other. For the 31B1 state, the optimized C2v structure at the SAC-CI/def2-SVP level has a bond length of 1.726 Å and a bond angle of 131.62°; the C2v conformation has an imaginary frequency for the b2 vibration and could not be a stable conformation. From the SAC-CI/def2-SVPD calculation, the optimized C2v conformation, which is characterized by the 3b1 → 10a1 transition, has a bond length of 1.742 Å and a bond angle of 132.73°. The calculated adiabatic excitation energies of the C2v conformation were 3.61–3.70 eV in the SAC-CI calculations. For the Cs conformation, the optimized structure at the SAC-CI/def2-SVP level has two bond lengths of 1.880 and 1.603 Å and a bond angle of 126.29°; the vibrational frequencies are all positive for the Cs conformation. From the SAC-CI/def2-SVPD calculation, the optimized Cs conformation has two bond lengths of 1.897 and 1.615 Å and a bond angle of 127.42°. The optimized Cs conformation is characterized by the 3b1 (a") → 10a1 (a') transition, where the 3b1 (a") orbital mainly involves the Oa (2px) and Ti (3dxz) orbitals (the Oa atom is on the longer Ti–O bond) and the 10a1 (a') orbital involves the Ti (4s) orbital. The calculated adiabatic excitation energies for the Cs conformations were 3.29–3.39 eV in the SAC-CI calculations, which were smaller than those for the C2v conformation. The C2v conformations of the 11B1 and 31B1 states are mainly characterized by the transitions 3b1 → 11a1 and 3b1 → 10a1 at their respective optimized geometries, respectively, but by the combinations of 3b1 → 10a1 and 3b1 → 11a1 transitions in the
Frank–Condon region. This phenomenon is similar to that shown in the 11B2 and 21B2 state which are mainly characterized by 6b2 → 10a1 and 6b2 → 11a1 and can be realized by exploring the Walsh diagram (Fig. 3). Similar to the 11B1 state, the stable Cs conformation of the 31B1 state could result from the pseudo-Jahn–Teller effect (see Fig. 4). At the C2v structure (rTi-O = 1.752Å, θO-Ti-O = 123.29°) shown in Fig.4, the energy differences between the 31B1 and 21A2 states are 0.05 eV at the SAC-CI/def2-SVPD level and 0.07 eV at the SAC-CI/def2-TZVPD level, which are within the accuracy of the current calculations. 3.3.4. The 21A1 and 31A1 electronic states The 21A1 and 31 A1 states are mainly characterized by the combination of the 9a 1 → 10a1 and 9a1 → 11a1 transitions in the Franck–Condon region. For the 21A1 state, the C2v conformation optimized by the SAC-CI/def2-SVP calculation has an imaginary frequency for the b2 vibation, and the stable conformation could have a Cs symmetry. However, no reliable C s conformation was found for this state. From the SAC-CI/def2-SVPD calculations, the C2v structures have optimized geometrical parameters rTi-O = 1.767 Å and 1.731 Å and θO-Ti-O = 142.14° and 131.18° for the 21A1 and 31A1 states, respectively. The 21A1 state is characterized by the combination of the 9a1 → 11a1 and 3b1 → 4b1 transitions and the 31A1 state by the 9a1 → 10a1 transition mixed with some 3b1 → 4b1 characteristics at their respective optimized C2v structures. This phenomenon can be realized by checking the Walsh diagram, and it is similar to that for the 11B1 and 31B1 states, which are characterized by the 3b1 → 11a1 and 3b1 → 10a1 transitions mixed with some 9a1 → 4b1 characteristics. The calculated adiabatic excitation energies for the C2v conformations were 3.11–3.22 eV for the 21A1 states and 3.50–3.55 eV for the 31A1 states in the SAC-CI calculations. 3.3.5 The 11A2 and 21A2 electronic states The 11A2 state is characterized by the electronic transition 6b2 → 4b1 in the Franck–Condon region. Lin et al. [21] showed that the stable conformation of the 11A2 state has a symmetric linear form from their TD-DFT and EOM-CCSD calculations. At this linear geometry, the 11A2 and 11B2 states are degenerate and correlate to the 11Δu state in D∞h symmetry group. From the SAC-CI calculations, similar to those reported by Lin et al. [21], the 11A2 state has an optimized symmetric linear structure and is correlated to the 11Δu state. The 21A2 state is characterized by the combination of the transitions 1a2 → 10a1 and 1a2 → 11a1, where the 1a2 orbital involves the O(2px) – O(2px) orbital. From the SAC-CI/def2-SVPD calculation, the optimized C2v conformation has a bond length of 1.747 Å and a bond angle of 105.50°. The calculated adiabatic excitation energies
were 3.57–3.65 eV in the SAC-CI calculations. No Cs structure was optimized for this state. 4. Summary Nine low-lying electronic excited states, 21A1, 31A1, 11A2, 21A2, 11B1, 21B1, 31B1, 11B2, and 21B2, were studied using SAC-CI theory with various basis sets. The C2v conformations of the nine states and the Cs conformations of the 11B1 and 31B1 states were optimized at the SAC-CI/def2-SVP and SAC-CI/def2-SVPD levels. For the 11B1 and 31B1 states, the Cs conformations are stable ones in which the vibrational frequencies are all positive and the C2v conformation are saddle points that have an imaginary frequencies for the b2 vibrations in the SAC-CI/def2-SVP calculations. The Cs conformations of the 11B1 and 31B1 states could result from the pseudo-Jahn–Teller effect by which the two 1B1 states couple with the 21A2 state via the b2 vibrations. The calculated geometrical parameters and vibrational frequencies of the X1A1 and 11B2 states and the calculated adiabatic excitation energy of the 11B2 state agree with the experimental values. For the 11B1 state, the calculated adiabatic excitation energy of the C2v conformation is in better agreement with the experimental value of 3.37 eV than that of the Cs conformation. More experimental and theoretical studies are needed for understanding the nature of the low-lying electronic states of TiO2. Acknowledgements The author is grateful to the National Center for High-performance Computing for computer time and facilities.
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Figure captions Figure 1 The selected molecular orbitals obtained from the RHF/def2-SVPD calculation. The RHF-calculated orbitals were used as the reference orbitals in the SAC-CI calculations. Figure 2 The energy profile for the energy scan along the bond-angle coordinate. The energy scan was conducted along the bond angle θ O-Ti-O from 95° to 180° with a step size of 5° where the relative energies were calculated by single-point SAC-CI/def2-SVPD calculations at the geometries of the X1A1 state optimized at the BPW91/def2-SVPD level. Figure 3 The calculated orbital energies as a function of the bond angle. The orbital energies were calculated at the optimized bond length with a fixed bond angle at the BPW91/aug-cc-pVTZ level. Figure 4 The energy profile for the pseudo-Jahn–Teller effect between the 1B1 and 21A2 states. The relative energies were calculated by single-point SAC-CI/def2-SVPD calculations at the geometries obtained from the IRC calculation (UBPW91/def2-SVPD) for the C2v and Cs conformations of the 11B1 state.
Table 1 The calculated vertical excitation energies ∆Eve (eV), oscillator strengths f (a. u.), and electronic transitions of the low-lying singlet excited states of TiO2. SAC-CI calculations States
def2SVPa
def2SVPDa
def2TZVPD
Previous calculations
def2TZVPPD
aug-ccpVTZ
fb
Electronic transitionsc
MRCId CCSDe
CASPT2f
∆Eve 11B2
2.356 (2.423)
2.485 (2.501)
2.445
2.443
2.460
0.010
-0.73 (6b2 → 10a1) -0.53 (6b2 → 11a1)
2.43
2.386
2.386
11A2
2.999 (3.127)
3.061 (3.063)
3.053
3.049
3.062
fb
0.71 (6b2 → 4b1) -0.61 (6b2 → b1)
3.09
3.045
3.150
21B2
3.193 (3.323)
3.252 (3.250)
3.233
3.233
3.248
0.011
0.64 (6b2 → 11a1) -0.50 (6b2 → 10a1) -0.44 (6b2 → a1)
3.21
3.213
3.365
21A1
3.297 (3.388)
3.444 (3.448)
3.391
3.390
3.402
0.000
-0.68 (9a1 → 10a1) -0.53 (9a1 → 11a1)
3.13
3.315
3.480
11B1
3.602 (3.751)
3.679 (3.683)
3.643
3.644
3.657
0.033
-0.68 (3b1 → 10a1) -0.58 (3b1 → 11a1)
3.57
3.599
3.601
21B1
3.613 (3.765)
3.731 (3.726)
3.728
3.730
3.739
0.002
0.68 (9a1 → 4b1) -0.64 (9a1 → b1)
3.74
3.711
3.979
21A2
3.970 (4.123)
4.039 (4.072)
3.973
3.971
3.982
fb
-0.64 (1a2 → 10a1) -0.60 (1a2 → 11a1)
4.07
3.969
3.948
31A1
3.978 (4.145)
4.074 (4.063)
4.054
4.055
4.066
0.004
3.83
4.027
4.586
31B2
4.367 (4.527)
4.427 (4.445)
4.373
4.373
4.380
0.002
31B1
4.356 (4.574)
4.393 (4.381)
4.374
4.378
4.390
0.002
0.51 (9a1 → 10a1) -0.41(9a1 → 11a1) -0.40 (3b1 → 4b1) 0.53 (5b2 → 11a1) 0.45 (5b2 → 10a1) 0.40 (1a2 → 4b1) -0.57 (3b1 → 11a1) 0.56 (3b1 → 10a1) 0.46 (3b1 → a1)
a. Values in parentheses indicate the calculated vertical excitation energies at the SAC-optimized geometry of the ground electronic state of TiO2. b. SAC-CI/aug-cc-pVTZ calculated results; fb denotes "forbidden". c. SAC-CI/def2-SVPD calculated results; only the coefficients larger than 0.4 are listed. d. Ref. [19]. e. Ref. [20]. f. CASPT2(16,14) calculated results in Ref. [21].
4.19
4.39
4.250
Table 2 The calculated adiabatic excitation energies ΔEae (eV), optimized geometrical parameters [rTi-O (Å), θO-Ti-O (°)], vibrational frequencies ω (cm-1), and dipole moments μ (Debye) of the singlet X1A1, 11B2, 11B1, and 11A2 states of TiO2. Methods UBPW91/def2-SVP UBPW91/def2-SVPD UBPW91/def2-TZVPD UBPW91/def2-TZVPPD UBPW91/aug-cc-pVTZ BPW91/6-311+G(3df)a B3LYP/aug-cc-pVTZb
ΔEae
rTi-O
θO-Ti-O
ω1(a1)
ω2(a1)
ω3(b2)
μ
ΔEae
rTi-O
0 0 0 0 0 0 0
1.638 1.649 1.652 1.651 1.652 1.651 1.642 1.651c
109.36 110.56 110.65 110.61 110.60 110.7 111.7
1033.3 999.0 990.2 987.6 985.0 986 1023.0
111.57c
962d
UBPW91/def2-TZVPD UBPW91/def2-TZVPPD UBPW91/aug-cc-pVTZ BPW91/6-311+G(3df)a
3.042 3.020 2.934 2.938 2.929 3.03
1.730 1.752 1.754 1.753 1.755 1.753
118.54 123.29 123.21 123.13 123.12 123.20
834.3 792.5 788.5 786.8 785.2 786
357.6 343.0 340.4 340.7 340.2 339 341.9 329d
1000.9 958.8 953.0 947.8 946.5 946 975.2 935d
6.32 6.55 6.48 6.49 6.48 6.73 6.73 6.33e
2.538
1.710
144.26
802.6
ω3(b2)
μ 3.92 4.33 4.32 4.34 4.36 5.07
2.053 2.169 2.086 2.087 2.084 2.14
1.687 1.700 1.704 1.703 1.704 1.703
95.38 96.07 96.60 96.53 96.52 96.3
925.7 885.6 876.9 874.6 871.7 875
223.8 206.4 198.1 197.8 197.2 196
530.7 493.0 485.8 479.0 477.9 480
2.1813 2.37 2.34 2.4f
1.704, 1.708 1.709, 1.704g
100.1 101.66 98.92 100.1g
876h 825i
184h 194i
316h
227.2 195.4 196.3 197.6 198.6
1046.8 989.0 984.5 983.6 980.2
2.55e
11B1(A", Cs) 238.9 199.4 199.8 200.9 201.3 200
728.9i 640.4i 654.5i 661.4i 662.4i 659i
4.31 4.84 4.74 4.72 4.73
133.7
629.3
2.88
11A2(S2; C2v) UBPW91/def2-SVP
ω2(a1)
1 B2(C2v)
11B1(C2v) UBPW91/def2-SVPD
ω1(a1) 1
X A1(C2v)
Exp.
UBPW91/def2-SVP
θO-Ti-O
1
2.895 2.907 2.817 2.814 2.806
1.897, 1.613 1.918, 1.628 1.920, 1.631 1.921, 1.629 1.922, 1.630
112.09 116.15 116.03 115.82 115.82
615.5 583.2 583.3 582.2 583.8
4.43 4.92 4.86 4.84 4.85
UBPW91/def2-SVPD
2.549
1.724
143.00
771.0
123.4
562.6
3.10
UBPW91/def2-TZVPD
2.441 2.438
1.724 1.724
143.94 143.56
770.1 768.4
126.6 128.5
584.3 575.0
2.86 2.86
UBPW91/def2-TZVPPD
2.430 1.725 143.07 765.9 126.4 569.1 2.88 BPW91/6-311+G(3df) 2.5 1.724 143.9 767 127 575 a. Ref. [19]. b. Ref. [21]. c. Ref. [5]. d. Ref. [7]. e. Ref. [6] f. 2.1813 eV in Ref. [8], 2.37 eV in Ref. [9], 2.34 eV in Ref. [3], and 2.4 eV in Ref. [10]. g. (rTi-O, θO-Ti-O) = (1.704 Å, 100.1°) in Ref. [6], and (1.708, 101.66°), (1.709, 98.92°) and (1.704 Å, 100.1°) in Ref. [8]. h. Ref. [8]. i. Ref. [9]. UBPW91/aug-cc-pVTZ a
Table 3 The optimized geometrical parameters [rTi-O (Å), θO-Ti-O (°)] and vibrational frequencies ω (cm-1) of the low-lying singlet electronic excited states of TiO2. States
X1A1
11B2
Methods SAC/def2-SVP SAC/def2-SVPD SAC/def2-TZVPD CCSD(T)a Exp SAC-CI/def2-SVP SAC-CI/def2-SVPD TD-B3LYPa EOM-CCSDa Exp
11Δug
21B2
11B1
21B1 31B1
SAC-CI/def2-SVP SAC-CI/def2-SVPD TD-B3LYPa EOM-CCSDa SAC-CI/def2-SVP SAC-CI/def2-SVPD SAC-CI/def2-SVP SAC-CI/def2-SVPD SAC-CI/def2-SVP (Cs) SAC-CI/def2-SVPD (Cs) SAC-CI/def2-SVP SAC-CI/def2-SVPD SAC-CI/def2-SVP
rTi-O
θO-Ti-O
ω1(a1)
ω2(a1)
ω3(b2)
1.633 1.649 1.647 1.652 1.651b 1.672 1.690 1.673 1.662 1.704, 1.708 1.709, 1.704d 1.694 1.713 1.700 1.684 1.680 1.695 1.752 1.772 1.907, 1.608 1.936, 1.630 1.736 1.766 1.726
110.78 112.40 112.83 111.2 111.57b 98.93 98.81 100.2 100.4 100.1, 101.66 98.92, 100.1d 180(fixed) 180(fixed) 180.0 180.0 133.590 131.974 138.77 142.35 120.69 127.76 135.76 143.24 131.62
1071.1 1028.4
355.8 333.2
1047.7 986.1
998.5 962c 1011.4 947.3 945.6 1009.1 825f 876f 881.7
333.7 329c 222.2 205.9 212.5 198.3 194e 184f 80.0
955.2 935c 366.3 296.0 374.7 499.0
837.2 887.9 921.9
253.8 297.3 371.6
658.8 879.4 462.4
858.9
172.1
1243.4i
662.1
147.7
1102.1
900.9
311.7
1290.2i
316f 776.0
SAC-CI/def2-SVPD 1.742 132.73 SAC-CI/def2-SVP (Cs) 1.880, 1.603 126.29 711.1 285.3 1095.7 SAC-CI/def2-SVPD (Cs) 1.897, 1.615 127.42 SAC-CI/def2-SVP 1.712 122.69 857.7 104.5 977.7i 21A1 SAC-CI/def2-SVPD 1.767 142.14 SAC-CI/def2-SVP 1.711 128.74 31A1 SAC-CI/def2-SVPD 1.731 131.18 SAC-CI/def2-SVP 1.724 103.11 21A2 SAC-CI/def2-SVPD 1.747 105.50 a. Ref. [21]. b. Ref. [5]. c. Ref. [7]. d. (rTi-O, θO-Ti-O) = (1.704 Å, 100.1°) in Ref. [6], and (1.708, 101.66°), (1.709, 98.92°) and (1.704 Å, 100.1°) in Ref. [8]. e. Ref. [9]. f. Ref. [8]. g. In D∞h symmetry. ω1, ω2, and ω3 correspond to σg+, πu, and σg– vibrations, respectively
Table 4 The SAC-CI-calculated adiabatic excitation energies ΔEae and dipole moments μ for the low-lying singlet electronic excited states of TiO2. Basis sets
11B2
11∆ua
21B2
11B1 (A")b
21B1
31B1 (A")b
21A1
31A1
21A2
ΔEae (eV) def2-SVP
2.286
2.468
2.838
3.304 (2.872)
3.201
3.700 (3.393)
3.222
3.549
3.648
def2-SVPD
2.355
2.474
2.902
3.146 (2.782)
3.108
3.633 (3.342)
3.109
3.541
3.593
def2-TZVPD
2.335
2.525
2.849
3.218 (2.833)
3.167
3.583 (3.294)
3.167
3.496
3.570
def2-TZVPPD
2.344
2.548
2.862
3.254 (2.849)
3.200
3.607 (3.306)
3.199
3.516
3.590
μ (Debye) def2-SVP
3.61
0
0.73
4.15 (5.40)
3.80
1.08 (2.00)
4.47
1.70
5.22
def2-SVPD
4.13
0
0.56
3.85 (5.45)
3.52
0.97 (1.88)
3.78
2.01
5.86
def2-TZVPD
4.06
0
0.78
3.74 (5.36)
3.46
0.94 (2.06)
3.71
1.88
5.67
def2-TZVPPD
4.06
0
0.76
3.72 (5.34)
3.43
0.94 (2.05)
3.69
1.87
5.65
1
c
For 1 B2 state : ΔEae = 2.1813, 2.37, 2.34, and 2.4 eV. μ = 2.55 D. For 11B1 stated: 3.37 eV. a. In D∞h symmetry. b. Values in parentheses indicate the calculated values for the Cs conformation. c: ΔEae = 2.1813 eV in Ref. [8], 2.37 eV in Ref. [9], 2.34 eV in Ref. [3], and 2.4 eV in Ref. [10]. μ = 2.55 D in Ref. [6]. d: Ref. [9]. Exp
1
Intensity
1 B1
1
2 B2
1
1 B2
1
3 A1
1
1
1 A2 2.0
2.5
3.0
1
2 A1
2 B1
3.5
1
1
2 A2
3 B1
4.0
4.5
Vertical Excitation Energy (eV)
The nine low-lying singlet electronic excited states of TiO2 were studied using SAC-CI theory. The nature of the low-lying electronic states was discussed in this study.
Highlights 1. Nine low-lying singlet excited states of TiO2 were studied using SAC-CI theory. 2. The adiabatic excitation energies of the excited states were calculated. 3. Stable Cs structures for the 11B1 (A") and 31B1 (A") states were determined.