The low-lying triplet electronic excited states of TiO2 and ZrO2: A symmetry adapted cluster–configuration interaction (SAC-CI) study

Computational and Theoretical Chemistry 1143 (2018) 20–28

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The low-lying triplet electronic excited states of TiO2 and ZrO2: A symmetry adapted cluster–configuration interaction (SAC-CI) study

T

Yung-Ching Chou Department of Applied Physics and Chemistry, University of Taipei, No. 1, Ai-Guo West Road, Taipei 10048, Taiwan

A R T I C LE I N FO

A B S T R A C T

Keywords: TiO2 ZrO2 Electronic excited states SAC-CI Electron detachment energy

The vertical excitation energies and electron detachment energies to the low-lying triplet excited states of TiO2 and ZrO2 were calculated at the SAC-CI/aug-cc-pVTZ levels. The vertical electron detachment energies to the triplet excited states of TiO2 were compared with previous photoelectron spectra. Possible candidate states for the unassigned bands in the previous photoelectron spectrum were found. Several triplet excited states of TiO2 and ZrO2 were optimized at the SAC-CI/aug-cc-pVDZ and SAC-CI/def2-SVPD levels, respectively. The adiabatic excitation energies and electron detachment energies to the selected triplet states were obtained by single-point calculations at the SAC-CI/def2-TZVPPD level. Density functional theory (BPW91/aug-cc-pVTZ) was also applied to study some of the triplet states for comparison. The calculated electron detachment energies and optimized geometries of the triplet excited states should be useful for studying the photoelectron spectra of TiO2− and ZrO2−.

1. Introduction TiO2 and ZrO2 belonging to the Group IVB metal oxides are of significant industrial and scientific interest due to their versatile applications. TiO2 has significant photocatalytic activity in water photolysis [1,2]. ZrO2 is an important material in the fields of ceramics, catalysts, and gas sensors [3–5]. TiO2 and ZrO2 both have wide band gaps and can work as n-type semiconductors [6]. Studies of MO2 (M = Ti and Zr) monomers and their clusters (MO2)n are meaningful for understanding the chemistry of the MO2-based materials. Molecular clusters provide connections between a single molecule and solid state materials and have been considered as simplified models for investigating the behavior of bulk materials. Investigations of the MO2 monomers can help in understanding the chemistry of (MO2)n clusters. In addition, the TiO2 molecule which was observed at radio wavelengths around the red supergiant VY CMa [7] is an interstellar molecule and is of interstellar interest. The MO2 molecule and their clusters (MO2)n have been extensively studied, both experimentally [8–21] and theoretically [22–37]. The rotational spectrum of TiO2 was recorded using Fourier transform microwave spectroscopy, and the effective bond length of 1.651 Å and bond angle of 111.57° were obtained [11]. The excitation energy and vibrational frequencies of the A1B2 state of TiO2 were determined from the electronic spectrum [13,14]. The photoelectron spectra of TiO2− anion and its cluster anions (TiO2)n− were recorded [15–17], and the

electron affinities of the neutral clusters were determined. The investigation of the photoelectron spectra showed that the band gaps of (TiO2)n clusters increased with the size of the clusters and approached the bulk limit when n = 7–10 [16]. The X and A bands in the photoelectron spectra were assigned to the Ti(3d)-derived and O(2p)-derived bands, respectively. The bond length and bond angle of ZrO2 were determined to be 1.7710(7) Å and 108.11(8)°, respectively, by Fourier transform microwave spectroscopy [18]. The excitation energy and vibrational frequencies of the A1B2 state of ZrO2 were obtained from the electronic spectrum [19]. The photoelectron spectra of ZrO2− [17,20] and Zr2O4− [21] anions were recorded, and the electron affinities were obtained. The electron affinities of TiO2 and ZrO2 were determined to be 1.5892(5) and 1.6397(5) eV, respectively [17]. As far as the author knows, no spectral data concerning the triplet excited states of ZrO2 or its clusters (ZrO2)n were presented in the literature. The vertical excitation energies of the low-lying singlet and triplet excited states of TiO2 have been studied using the multi-reference configuration interaction (MRCI) theory [30], time-dependent density functional theory (TD-DFT) [31,32], coupled cluster (CC) theory [31–33], complete active space self-consistent field (CASSCF and CASPT2) [32], and equation-of-motion CCSD (EOM-CCSD) theory [32,34]. The equilibrium geometries and energies of the lowest singlet, triplet, and quintet states in each of the irreducible representations in the C2v point group symmetry were computed at the BPW91/6-

E-mail address: [email protected]. https://doi.org/10.1016/j.comptc.2018.09.009 Received 29 July 2018; Received in revised form 14 September 2018; Accepted 21 September 2018 Available online 22 September 2018 2210-271X/ © 2018 Elsevier B.V. All rights reserved.

Computational and Theoretical Chemistry 1143 (2018) 20–28

Y.-C. Chou

Table 1 The calculated vertical excitation energies VEEs (eV) and vertical electron detachment energies VDEs (eV) to the low-lying triplet excited states of TiO2. VEEsa

States (Types) Electronic transitions

VDEsa

SAC-CI d-S

X1A1 (S0) 13B2 (A) 13A2 (B) 23B2 (C) 13A1 (A) 13B1 (A) 23B1 (B) 23A1 (B) 23A2 (A) 33A1 (C) 33B2 (A) 43B2 (B) 33B1 (C) 33A2 (B) 43B1 43A1 (A) 43A2 (C) 53B2 (C) 53A1 (C) 53B1 (B) a b c

4b2 → 6, 7a1 4b2 → 3, 4b1 4b2 → 7, 9, 6a1 5a1 → 7, 6a1 2b1 → 7, 6a1 5a1 → 3, 4b1 5a1 → 6a1/2b1 → 3, 4b1 1a2 → 7, 6a1 2b1 → 3, 4b1/5a1 → 6a1 3b2 → 7, 6a1 1a2 → 4, 3b1 2b1 → 6, 7, 9a1 3b2 → 4, 3b1 4b2 → 2a2 4a1 → 7, 6a1 1a2 → 6, 7, 9a1 3b2 → 6, 9a1 4a1 → 6, 9a1 4a1 → 3, 4b1

2.456 3.066 3.206 3.210 3.476 3.562 3.800 3.862 3.950 4.239 4.295 4.309 4.398 4.500 4.557 4.767 4.823 5.229 5.317

Previous calculations d-T

(2.268) (2.981) (3.096) (3.051) (3.313) (3.479) (3.643) (3.701) (3.869) (4.056) (4.222) (4.186) (4.310) (4.445) (4.403) (4.619) (4.605) (5.051) (5.210)

2.411 3.050 3.188 3.187 3.439 3.561 3.784 3.809 3.947 4.199 4.284 4.292 4.390 4.508 4.512 4.715 4.755 5.195 5.278

A-T

(2.196) (2.795) (2.948) (2.957) (3.225) (3.322) (3.545) (3.600) (3.712) (3.950) (4.046) (4.069) (4.155) (4.244) (4.312) (4.502) (4.505) (4.963) (5.032)

2.429 3.062 3.202 3.197 3.452 3.570 3.798 3.820 3.957 4.204 4.291 4.305 4.395 4.514 4.524 4.726 4.764 5.208 5.287

(2.212) (2.793) (2.953) (2.960) (3.233) (3.315) (3.550) (3.607) (3.710) (3.950) (4.038) (4.071) (4.144) (4.231) (4.319) (4.507) (4.512) (4.970) (5.026)

SAC-CI

EOM-CCSDb

CASPT2b

MRCIc

2.324 2.996 3.114 3.072 3.332 3.474 3.657 3.696 3.843 4.044

2.495 2.991 3.253 3.425 3.612 3.616 3.886 3.932 4.449 4.122

2.40 3.07 3.20 3.12 3.43 3.59 3.85 3.81 4.00 4.02 4.43 4.34

d-S 1.573 3.933 4.428 4.569 4.610 4.785 4.899 5.068 5.185 5.265 5.568 5.496 5.550 5.651 5.771 5.860 6.039 6.140 6.498 6.549

d-T (1.737) (3.914) (4.507) (4.617) (4.642) (4.798) (4.988) (5.125) (5.204) (5.290) (5.559) (5.614) (5.582) (5.732) (5.879) (5.884) (6.046) (6.073) (6.480) (6.605)

1.679 3.992 4.522 4.656 4.707 4.859 5.014 5.186 5.245 5.339 5.649 5.597 5.633 5.755 5.895 5.930 6.084 6.164 6.564 6.618

A-T (1.826) (3.915) (4.400) (4.552) (4.622) (4.785) (4.916) (5.082) (5.180) (5.251) (5.544) (5.506) (5.553) (5.665) (5.771) (5.868) (6.013) (6.060) (6.473) (6.507)

1.704 4.033 4.559 4.695 4.742 4.896 5.047 5.223 5.281 5.376 5.678 5.628 5.671 5.784 5.926 5.965 6.119 6.198 6.601 6.650

(1.846) (3.948) (4.418) (4.578) (4.643) (4.812) (4.929) (5.101) (5.205) (5.277) (5.559) (5.520) (5.578) (5.674) (5.777) (5.892) (6.039) (6.087) (6.500) (6.521)

The values shown in parentheses were calculated using Space 2. Ref. [32]. Ref. [30].

needed for understanding the photoelectron spectra of MO2− anions.

311+G(3df) level [30]. The energies relative to the X1A1 state were 2.31 and 2.29 eV for the 11B2, and 13B2 states, respectively, at the anion geometry at the MRCI level. These energy values agreed with the experimental term values of 2.4 (2) eV [15] and 2.22 (10) eV (the A band) [16]. The 11B2(S1), 11A2(S2), 13B2(T1), 13A2(T2) states of TiO2 were optimized at various levels of theory, and the potential energy surfaces of the S0-S5 states were investigated at the TD-B3LYP/aug-cc-pVTZ level [32]. Several low-lying singlet excited states of TiO2 and ZrO2 were investigated using the symmetry adapted cluster-configuration interaction (SAC-CI) method [35,36]. Some singlet excited states of these two molecules were optimized, and the adiabatic excitation energies were obtained. The spectrum for the photodetachment process ZrO2− (X2A1) → ZrO2 (X1A1) + e was simulated by combining high level ab-inito calculations and Franck-Condon factor calculations including allowance of Duschinsky rotation and anharmonicity [37]. The simulated spectrum was in excellent agreement with the experimental spectrum [20]. The (MO2)n (n = 1–4) clusters and their anions (MO2)n− were studied using CCSD(T) and DFT methods [31,34]. Possible conformations for the ground electronic states of these clusters and anions were optimized, and their relative energies were obtained. The vertical and adiabatic electron detachment energies to the ground states and first excited states of the (MO2)n clusters were computed. These calculated electron detachment energies were compared with the energies of the X and A bands in the experimental spectra [16]. The theoretical results shown in Refs. [31,34] can be used as benchmarks for the computational studies of (MO2)n and (MO2)n−. In the photoelectron spectra of (TiO2)n− clusters with n = 1–4 [16], the excited states related to the A bands have been theoretically studied in Ref. [31]. However, the spectral structures with higher binding energies are still unassigned. These unassigned structures could be related to the triplet excited states of the neutral clusters. Knowledge of the triplet excited states of (MO2)n is important for studying the photoelectron spectra of (MO2)n− anions. To the best of the author’s knowledge, for the triplet excited states of MO2, only the 13A1 (TiO2), 13A2 (TiO2), 13B1 (TiO2), and 13B2 (MO2) states have been optimized [30–32,34]. More investigations on the triplet excited states of MO2 are

2. Computational details The vertical excitation energies (VEEs) and vertical electron detachment energies (VDEs) were calculated using the SAC-CI method [38–43] at the experimental geometries [11,17,18] of the ground electronic states of MO2 and MO2−, respectively. The def2-SVPD, def2TZVPPD [44–46], and aug-cc-pVTZ [44,47,48] (aug-cc-pVTZ-pp for Zr [44,49]) basis sets were applied to calculate the vertical energies (VDEs and VEEs). For simplification, the notions d-S, d-T, and A-X were used to indicate the def2-SVPD, def2-TZVPPD, and aug-cc-pVXZ (aug-ccpVXZ-pp for Zr) basis sets, respectively. Two active spaces (Spaces 1 and 2) were used to study the triplet states of MO2. Space 1 included 16 valence electrons for MO2 [M(ns2(n – 1)d2)] and O(2s22p4)], where n = 4 for M = Ti and n = 5 for M = Zr. Space 2 included 8 semi-core electrons [M((n – 1)s2(n – 1)p6)] and the 16 valence electrons. All the unoccupied molecular orbitals were included in the two active spaces. The singles and doubles linked excitation operators were used in the SAC-CI calculations. Non-variational procedures [40] and direct algorithm [43] were adopted in the calculations. Restricted Hartree-Fock (RHF) orbitals for the X1A1 states of MO2 were used as the reference orbitals. All the calculations were performed using Gaussian 09 program packages [50]. The geometries of the triplet states of MO2 were optimized under C2v symmetry constraint at the SAC-CI levels. The adiabatic excitation energies (AEEs) and adiabatic electron detachment energies (ADEs) were obtained by single-point calculations at the SAC-CI/d-T level at the SAC-CI-optimized geometries. For simplification, the relative energies described in the context refer to those computed at the SAC-CI/dT (Space 1) level. 3. Results and discussion The calculated vertical energies and adiabatic energies (ADEs and AEEs) to the low-lying triplet excited states of TiO2 are listed in Tables 1 and 2, respectively. The vertical and adiabatic energies to the low-lying 21

Computational and Theoretical Chemistry 1143 (2018) 20–28

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Table 2 (continued)

Table 2 The adiabatic excitation energies AEEs (eV), adiabatic electron detachment energies ADEs (eV), and geometrical parameters [rTi-O (Å), θO-Ti-O (°)] of the selected triplet excited states of TiO2. States (Types)

Methodsa

AEEsb

ADEsb

rTi-Ob

θO-Ti-Ob

13B2-a (A)

SAC-CI/d-S

2.354 (2.208) 2.280 2.336 (2.150) 2.349

3.880 (3.874)

1.6872 (1.6766) 1.6883

99.56 (100.08) 99.57

3.85c

EXP.

2.24c, 2.30d, 2.25e, 2.135f 2.22(10)g

13B2-b

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

2.514 2.481 2.573 (2.322)

13A2 (B)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp) Previous calculations

23B2 (C)

SAC-CI/A-D SAC-CI/dT(sp) SAC-CI/AT(sp) Previous calculations

3.855 3.952 (3.885)

States (Types)

43A1 (A)

101.9f

4.039 4.056 4.189 (4.057)

1.7116 1.7139

168.07 168.00

2.514 2.481 2.573 (2.322) 2.402f

4.039 4.056 4.189 (4.057)

1.7116 1.7138

168.07 168.00

1.684f

180.0f

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

2.827 2.756 2.789 (2.595)

4.353 4.331 4.405 (4.330)

1.6944 1.6967

134.89 136.09

13A1 (A)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

2.959 2.935 3.048 (2.872)

4.484 4.510 4.664 (4.607)

1.7457 1.7496

128.79 130.22

13B1 (A)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

3.027 2.972 3.075 (2.917)

4.552 4.547 4.691 (4.652)

1.7536 1.7535

129.71 129.53

23B1 (B)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

3.074 3.035 3.161 (2.974)

4.599 4.611 4.777 (4.709)

1.7556 1.7599

136.67 138.86

23A1 (B)

SAC-CI/ d-S SAC-CI/A-D SAC-CI/dT(sp)

3.079 3.033 3.164 (2.983)

4.604 4.608 4.780 (4.718)

1.7649 1.7661

136.89 138.22

23A2 (A)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

3.442 3.378 3.454 (3.321)

4.967 4.953 5.070 (5.056)

1.7476 1.7476

107.93 107.48

33A1 (C)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

3.517 3.439 3.488 (3.338)

5.043 5.014 5.104 (5.073)

1.7230 1.7286

130.43 132.19

43B2 (B)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

3.609 3.575 3.691 (3.551)

5.134 5.150 5.307 (5.286)

1.7793 1.7807

119.30 119.26

33B1 (C)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

3.586 3.495 3.569 (3.438)

5.112 5.071 5.185 (5.173)

1.7445 1.7487

134.50 136.17

33B2 (A)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

3.793 3.765 3.860 (3.723)

5.318 5.340 5.476 (5.458)

1.7678 1.7701

126.30 126.86

33A2 (B)

SAC-CI/d-S SAC-CI/A-D

3.799 3.771

5.325 5.347

1.7719 1.7745

126.85 127.70

AEEsb

ADEsb

SAC-CI/dT(sp)

3.874 (3.742)

5.490 (5.477)

SAC-CI/d-S SAC-CI/A-D SAC-CI/dT(sp)

4.142 4.097 4.172 (4.040)

5.668 5.672 5.788 (5.774)

rTi-Ob

θO-Ti-Ob

1.7515 1.7503

112.92 111.76

a “sp” indicates single-point calculations at the SAC-CI/A-D (Space 1)-optimized geometries. b The values shown in parentheses indicate the results calculated using Space 2. c CCSD(T)/aD//B3LYP/aD calculated values in Ref. [31]. d EOM-CCSD-calculated results in Ref. [34]. e MRCI-calculated results in Ref. [30]. f EOM-CCSD-calculated results in Ref. [32]. g In Ref. [16]. The estimated experimental uncertainties are shown in parentheses.

3.993 1.660f

Methodsa

3.81(10)g

triplet excited states of ZrO2 are listed in Tables 3 and 4, respectively. The notations for the electronic transitions shown in Tables 1 and 3 are similar to those used in Ref. [30]. The optimized geometrical parameters of the selected triplet states of MO2 are also shown in Tables 2 and 4. The potential energy scanning along the bond angle coordinate for the triplet excited states of MO2 are shown in Fig. 1 for M = Ti and in Fig. 2 for M = Zr. The details of the vertical excitation properties of the triplet states of MO2 are listed in Tables S1–S6 (Supplementary materials). The valence electronic configurations of the X1A1 states were assigned to [Ti(core), 2O(1s2)] … (1a2)2(5a1)2(2b1)2(4b2)2 for TiO2 and [Zr(core), 2O(1s2)] … (1a2)2(2b1)2(5a1)2 (4b2)2 for ZrO2, based on the RHF reference orbitals [51]. The 4b2 orbital is the highest occupied molecular orbital (HOMO), and the 6a1 orbital is the lowest unoccupied molecular orbital (LUMO). The (ma1, ma2, mb1, mb2) orbitals described here correspond to the [(m + 4)a1, ma2, (m + 1)b1, (m + 2)b2] orbitals in Refs. [30,35] for TiO2 and [(m + 8)a1, (m + 1)a2, (m + 3)b1, (m + 4) b2] orbitals in Ref. [36] for ZrO2. The selected molecular orbitals visualized by GaussView program [52] are shown in Fig. S1. Most of the triplet states considered here mainly arise from the electronic transitions from the highest six occupied molecular orbitals (OMOs) to the 6a1/7a1 and 3b1/4b1 orbitals. The expression “6a1/7a1” indicates a mixture of the 6a1 and 7a1 orbitals. The 4b2, 5a1, 2b1, 1a2, 3b2, and 4a1 orbitals, which are the highest six OMOs, mainly involve the 2p orbitals of the two oxygen atoms. The 6a1, 7a1, and (3b1, 4b1) orbitals mainly involve the M(ns), M[(n – 1)dz2-x2], and M[npx(n – 1) dxz] characters, respectively. The low-lying triplet excited states of MO2 can be related to the X2A1, 12B1 and 22A1 states of MO2− by one-electron processes (see Fig. S2). The 12B1 and 22A1 states of MO2− could be bound states or have some bound characters [53,54]. To simplify the discussion, the triplet neutral states related to removing an electron from the doubly occupied molecular orbitals (DOMOs) of the X2A1, 12B1 and 22A1 anion states are denoted as Type A, B, and C states, respectively. The photodetachment processes from the X2A1 anion state to the Type A triplet neutral states reasonably agree with the one-electron approximation. The one-electron approximation describes that removing an electron from the anion state does not affect the molecular orbitals and the occupation of the other electrons in the photodetachment processes. Therefore, the Type A triplet states are conventionally expected to have larger intensities in the photoelectron spectra starting from the X2A1 anion state. As shown in Tables 1 and 3, the VEEs of MO2 are not significantly affected by the size of the selected basis sets in the SAC-CI calculations. The VEEs of MO2 calculated using Space 2 are approximately 0.2–0.3 eV lower than those calculated using Space 1. The VEEs of TiO2 22

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Table 3 The calculated vertical excitation energies VEEs (eV) and vertical electron detachment energies VDEs (eV) to the low-lying triplet excited states of ZrO2. States (Types)

Electronic transitions

VEEsa SAC-CI d-S

X1A1 13B2 (A) 13A1 (A) 13B1 (A) 13A2 (B) 23B2 (C) 23A2 (A) 23B1 (B) 33B2 (A) 23A1 (C) 33A1 (A) 43A1 (B) 33B1 (C) 43B2 (B) 33A2 (B) 43B1 43A2 (C) 53B2 (C) 53A1 53B1 (B) 63A1 (C) a b

4b2 → 6, 7a1 5a1 → 6, 7a1 2b1 → 6, 7a1 4b2 → 3, 4b1 4b2 → 7, 6a1 1a2 → 6, 7a1 5a1 → 3, 4b1 3b2 → 6, 7a1 5a1 → 7, 6a1 4a1 → 6, 7a1 2b1 → 3, 4b1 2b1 → 7, 6a1 1a2 → 3, 4b1 3b2 → 3, 4b1 4b2 → 2a2 1a2 → 7, 6a1 3b2 → 7, 6a1 4b2 → 5, 6b2 4a1 → 3, 4b1 4a1 → 7, 6a1

2.186 2.969 3.342 3.430 3.601 3.792 4.028 4.110 4.127 4.391 4.532 4.775 4.976 5.116 5.229 5.264 5.266 5.307 5.749 5.893

d-T

(1.932) (2.658) (3.079) (3.291) (3.463) (3.521) (3.857) (3.760) (3.946) (4.138) (4.383) (4.630) (4.825) (4.915) (5.114) (5.100) (5.045) (5.123) (5.586) (5.732)

2.229 2.985 3.352 3.450 3.618 3.793 4.024 4.120 4.128 4.398 4.527 4.767 4.956 5.100 5.189 5.247 5.253 5.312 5.736 5.885

A-T

(1.988) (2.693) (3.099) (3.272) (3.443) (3.529) (3.814) (3.784) (3.914) (4.154) (4.330) (4.586) (4.753) (4.853) (5.002) (5.047) (4.998) (5.130) (5.532) (5.689)

2.267 3.008 3.388 3.466 3.631 3.823 4.025 4.139 4.129 4.421 4.543 4.776 4.967 5.104 5.186 5.250 5.251 5.311 5.732 5.883

(2.027) (2.720) (3.135) (3.276) (3.446) (3.559) (3.802) (3.807) (3.906) (4.178) (4.332) (4.584) (4.748) (4.843) (4.977) (5.041) (4.988) (5.130) (5.518) (5.678)

TD-DFTb

VDEsa SAC-CI

B3LYP, BP86

d-S

2.12, 2.81, 3.14, 3.25, 3.32,

1.88 2.71 3.05 3.02 3.08

d-T

1.644 3.726 4.518 4.749 4.887 5.030 5.222 5.488 5.570 5.555 5.777 5.877 6.075 6.312 6.476 6.621 6.592 6.616 6.784 7.080 7.195

(1.808) (3.632) (4.379) (4.651) (4.910) (5.053) (5.117) (5.491) (5.392) (5.545) (5.705) (5.874) (6.094) (6.327) (6.450) (6.665) (6.591) (6.568) (6.761) (7.087) (7.201)

1.677 3.801 4.569 4.793 4.938 5.079 5.257 5.520 5.614 5.593 5.815 5.910 6.102 6.325 6.494 6.620 6.609 6.639 6.820 7.099 7.223

A-T (1.847) (3.725) (4.454) (4.710) (4.922) (5.067) (5.164) (5.482) (5.455) (5.550) (5.754) (5.861) (6.086) (6.289) (6.423) (6.596) (6.575) (6.557) (6.796) (7.066) (7.200)

1.701 3.862 4.617 4.852 4.978 5.116 5.310 5.546 5.657 5.621 5.857 5.954 6.135 6.360 6.522 6.647 6.636 6.662 6.842 7.118 7.244

(1.870) (3.785) (4.503) (4.767) (4.948) (5.091) (5.215) (5.494) (5.500) (5.566) (5.793) (5.889) (6.106) (6.305) (6.435) (6.598) (6.590) (6.569) (6.815) (7.071) (7.211)

The values shown in parentheses were calculated using Space 2. TD-DFT calculated results (B3LYP/aD, BP86/AD) in Table S11 in Ref. [34].

Table 4 The adiabatic excitation energies AEEs (eV), adiabatic electron detachment energies ADEs (eV), and geometrical parameters [rZr-O (Å), θO-Zr-O (°)] of the selected triplet excited states of ZrO2. States (Types)

Methodsa

AEEsb

ADEsb

rZr-Ob

θO-Zr-Ob

13B2 (A)

SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/A-D Previous calculations SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/ d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp) SAC-CI/d-S SAC-CI/d-T(sp)

2.024 (1.817) 2.098 (1.919) 1.983 2.05c, 2.14d, 1.82e 2.836 (2.594) 2.889 (2.665) 2.861 (2.719) 2.928 (2.812) 2.897 (2.896) 2.951 (2.891) 2.905 (2.912) 2.961 (2.906) 3.276 (3.068) 3.340 (3.148) 3.561 (3.560) 3.627 (3.574) 3.508 (3.523) 3.589 (3.551) 3.772 (3.527) 3.837 (3.617) 3.617 (3.647) 3.688 (3.672) 3.940 (3.827) 4.031 (3.919) 3.701 (3.742) 3.778 (3.777) 4.183 (4.191) 4.244 (4.210) 4.396 (4.392) 4.464 (4.418)

3.696 (3.606) 3.780 (3.709) 3.695

1.8365 (1.8211)

100.64 (99.46)

1.8458

102.88

1.8387 (1.8139)

114.11 (110.86)

1.8795 (1.8616)

119.40 (117.56)

1.8604 (1.8445)

150.55 (142.74)

1.8611 (1.8457)

154.51 (146.84)

1.8787 (1.8597)

103.77 (101.72)

1.8932 (1.8656)

129.71 (124.53)

1.9068 (1.8805)

130.63 (126.70)

1.8675 (1.8458)

111.13 (108.79)

1.9098 (1.8922)

134.26 (130.39)

1.8879 (1.8759)

110.01 (113.44)

1.9197 (1.9026)

138.50 (134.94)

1.9215 (1.9025)

113.15 (110.98)

1.9190 (1.8964)

120.58 (117.40)

13A1 (A) 13B1 (A) 3

1 A2 (B) 3

2 B2 (C) 23A2 (A) 23B1 (B) 3

2 A1 (C) 33B2 (A) 33A1 (A) 3

4 A1 (B) 3

3 B1 (C) 43B2 (B) 33A2 (B)

a b c d e

4.508 4.571 4.533 4.610 4.569 4.633 4.577 4.642 4.948 5.022 5.232 5.309 5.180 5.270 5.444 5.519 5.288 5.370 5.612 5.713 5.373 5.459 5.854 5.926 6.068 6.146

“sp” indicates single-point calculations at the SAC-CI/d-S (Space 2)-optimized geometries. The values shown in parentheses were calculated using Space 2. CCSD(T)/aD//B3LYP/aD-calculated value in Ref. [34] EOM-CCSD-calculated value in Ref. [34]. TD-DFT(B3LYP)-calculated value in Ref. [34].

23

(4.384) (4.455) (4.508) (4.602) (4.685) (4.682) (4.702) (4.696) (4.858) (4.939) (5.349) (5.364) (5.313) (5.341) (5.316) (5.407) (5.437) (5.462) (5.617) (5.709) (5.532) (5.568) (5.981) (6.000) (6.182) (6.208)

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Fig. 1. The potential energy scans along the bond angle coordinate for the triplet excited states of TiO2. The relative energies were calculated by single-point SAC-CI/ d-S calculations at the BPW91/d-S-optimized geometries of the X1A1 state.

3.1.1. Type A states (the 13B2, 13A1, 13B1, 23A2, 33B2, and 43A1 states) The 13B2, 13A1, 13B1, 23A2, 33B2, and 43A1 states of TiO2 are mainly characterized by the electronic transitions from the 4b2, 5a1, 2b1, 1a2, 3b2, and 4a1 orbitals to the 6a1/7a1 orbitals, respectively (see Tables 1 and S1). The effective electronic configurations (EECs) of the Ti atom for the Type A states can be assigned to Ti+[4s13d2] according to the gross orbital populations (GOPs) on the Ti atom (see Table S7). The VDE to the 13B2 state is 3.992 eV, and it agrees with the peak position of the A band (∼3.9 eV) in the photoelectron spectrum of TiO2− (see Fig. 1 in Ref. [16]) and the CCSD(T)-calculated value of 4.01 eV [31]. The 13B2 state was optimized to two C2v structures: one has a bond angle of 99.57° (13B2-a), and the other has a bond angle of 168.00° (13B2-b) in the SAC-CI/A-D calculations (see Table 2). The adiabatic energies to the 13B2-a state are 3.952 eV for ADE and 2.336 eV for AEE. These adiabatic energy values agree with the experimental values of 3.81(10) eV for ADE and 2.22(10) eV for AEE [16]. The zero-point energy (ZPE) corrections to the ADE and AEE for the 13B2 state are −0.037 and −0.046 eV, respectively, at the BPW91/A-T

computed at the SAC-CI levels reasonably agree with those computed at the EOM-CCSD [32], CASPT2 [32], and MRCI [30] levels. The VEEs of the lowest five triplet excited states of ZrO2 obtained at the SAC-CI levels are similar to those obtained at the TD-DFT(B3LYP) level [34]. The VDEs of MO2 computed using the two active spaces are consistent with each other except for the X1A1 states. The electron affinities of MO2 were somewhat overestimated in the SAC-CI (Space 2) calculations [53,54].

3.1. TiO2 In the previous calculations [53], the geometrical parameters optimized using Space 1 were in better agreement with the experimental values of the ground electronic states of TiO2 and TiO2− than those optimized using Space 2. Therefore, the AEEs and ADEs to the triplet excited states of TiO2 were calculated at the SAC-CI/d-T//SAC-CI/A-D (Space 1) level.

24

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Fig. 2. The potential energy scans along the bond angle coordinate for the triplet excited states of ZrO2. The relative energies were calculated by single-point SAC-CI/ d-S calculations at the BPW91/d-T-optimized geometries of the X1A1 state.

level (see Table S8). The VDEs/ADEs to the 13A1 and 13B1 states are 4.707/4.664 and 4.859/4.691 eV, respectively. These two VDEs agree with the peak position (4.7–4.8 eV) of the third band ranging from approximately 4.2 eV to 5 eV in the photoelectron spectrum [16]. The 23A2, 33B2, and 43A1 states have VDEs/ADEs of 5.245/5.070, 5.649/5.476, and 5.930/ 5.788 eV, respectively. These three states may have major contributions to the forth band ranging from approximately 5 to 6.2 eV. As shown in Fig. 1, the 13B2, (13A1, 13B1), and (23A2, 33B2) adiabatically correlate to the 13Δu, 13Πu, and 13Πg states in the symmetric linear form, respectively. The upper singly occupied molecular orbitals (SOMOs) of the 13Δu, 13Πu, and 13Πg states (see Table S5) are the 1δg orbitals mainly involving the Ti(3d) character. This phenomenon can be understood by examining the Walsh diagram of TiO2 (see Fig. 3 in Ref. [35]). As shown in the Walsh diagram of TiO2 [35], a “crossing” occurs between the energy surfaces of the 6a1 and 7a1 orbitals, and the 6a1 and 7a1 orbitals adiabatically correlate to the 1δg and 4σg orbitals, respectively. The (mσg, mπg, mπu, mσu, mδg) orbitals mentioned here for TiO2

correspond to the [(m + 3)σg, mπg, (m + 1)πu, (m + 2)σu, mδg] orbitals in Ref. [35]. The 43A1 state may interact with the upper A1 state characterized by the electronic configuration of (4b2)1(6b2)1 and adiabatically correlate to the 43Πu state characterized by the (3σu)1(2πg)1 configuration. 3.1.2. Type B states (the 13A2, 23B1, 23A1, 43B2, 33A2, and 53B1 states) The 13A2, 23B1, 43B2, 33A2, and 53B1 states arise from the electronic transitions from the 4b2, 5a1, 1a2, 3b2, and 4a1 orbitals to the 3b1/4b1 orbitals, respectively. The 23A1 and 33A1 states are characterized by the mixtures of the electronic transitions 5a1 → 6a1 and 2b1 → 3b1/4b1 at the geometry of the X1A1 state. However, the 23A1 and 33A1 states are mainly characterized by the electronic configurations of 2b1 → 3b1/4b1 and 5a1 → 6a1, respectively, at the anion geometry and their respective optimized geometries. Hence, the 23A1 and 33A1 states are tentatively classified into Type B and C states, respectively. The EEC(Ti) for the type B triplet states can be assigned to Ti+[4s03d3]. The Type B states may have larger intensities in the photoelectron 25

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spectrum starting from the 12B1 anion state. The VDEs from the 12B1 anion state to the Type B states are listed in Table S9. The ADEs from the 12B1 anion state to the 13A2, 23B1, 23A1, 43B2, and 33A2 states are 2.873, 3.428, 3.425, 3.967, and 4.164 eV at the SAC-CI/A-D level, respectively. These energy values can be obtained from the ADEs shown in Table 2 and the AEE (1.183 eV) of the 12B1 anion state [53]. The ADEs from the 12B1 anion state suggest that the energies required to remove an electron from the O(2p) orbitals in the 12B1 anion state are smaller than those in the X2A1 anion state by approximately 1–1.2 eV. The adiabatic energies and optimized geometries of the 13A2 state are similar to those of the 13B2-b state. These two states are nearly degenerate and are the Renner-Teller components of the 13Δu state (see Fig. 1). As shown in Fig. 1, the potential energy surface of the 13A2 state becomes flattened when the bond angle is larger than approximately 150°. This observation resembles that suggested in Ref. [30]. The (23B1, 23A1), (43B2, 33A2) and 53B1 states correlate to the 23Πu, 23Πg, and 13Δg states, respectively.

3.2.2. Type B states (the 13A2, 23B1, 43A1, 43B2, 33A2, and 53B1 states) The 13A2, 23B1, 43A1, 43B2, 33A2, and 53B1 states mainly arise from the electronic transitions from the highest six OMOs to the 3b1 orbital (LUMO + 1). The VEEs and VDEs to the Type B triplet states in ZrO2 are apparently larger than those in TiO2. This phenomenon could be due to that the energy gap between LUMO and LUMO + 1 in ZrO2 is significantly larger than that in TiO2 [35,36]. The EEC(Zr) for these triplet states can be assigned to Zr+[5s04d3]. The VDEs from the 12B1 anion state to the Type B states are listed in Table S9. The ADEs from the 12B1 anion state to the 13A2, 23B1, 43A1, 43B2, and 33A2 neutral states are 3.104, 3.768, 3.856, 4.400, and 4.601 eV, respectively, at the SAC-CI/d-S (Space 2) level. These ADEs suggest that the energies required to remove an electron from the O(2p) orbitals in the 12B1 anion state are approximately 0.4–0.7 eV smaller than those in the ground anion state. As shown in Fig. 2, the 13A2, 23B1, and (43B2, 33A2) states correlate to the 13Δu, 23Πu, and 23Πg states, respectively. A crossing or avoided crossing may occur between the potential energy surfaces of the 43B2 and 53B2 (Type C) states when the bond angle is approximately 150°155°. The 53B1 state may interact with the 43B1 state (4b2 → 2a2) and is not shown in Fig. 2.

3.1.3. Type C states (the 23B2, 33A1, 33B1, 43A2, 53B2, and 53A1 states) Similar to those in the Type A states, the 23B2, 33B1, 43A2, 53B2, and 3 5 A1 states also arise from the electronic transitions from the OMOs to the 6a1/7a1 orbitals. The 43A2, 53B2, and 53A1 states may have some contributions to the spectral structure with VDEs higher than 6 eV [16] because of the mixing of the 6a1 and 7a1 orbitals. The GOPs on the Ti(p) orbitals in these five triplet states are larger than those in Type A and B states, suggesting that the type C states may have more Ti(4pz) character. The 23B2, (33A1, 33B1), (43A2, 53B2), and 53A1 states correlate to the 13Σu+, 33Πu, 33Πg, and 13Δg states, respectively.

3.2.3. Type C states (the 23B2, 23A1, 33B1, 43A2, 53B2, and 63A1 states) The 23B2, 23A1, 33B1, 43A2, 53B2, and 63A1 states of ZrO2 mainly arise from the electronic transitions from the highest six OMOs to the 7a1 (LUMO + 2) orbital. Similar to those in Type B triplet states, the vertical energies to the Type C triplet states in ZrO2 are apparently larger than those to their corresponding states in TiO2. This observation may be because the energy of the 7a1 orbital is close to that of the 3b1 orbital. The 23B2, 23A1, 33B1, (43A2, 53B2) states correlate to the 13Δu, 23Πu, 33Πu, and 33Πg states, respectively.

3.2. ZrO2 In the previous calculations [54], the geometrical parameters optimized using the A-T basis set and Space 2 were in better agreement with the experimental values of the ground electronic states of ZrO2 and ZrO2− than those optimized using other basis sets and active space. However, the geometrical parameters optimized at the SAC-CI/d-S (Space 2) level also reasonably agreed with the experimental values. For saving computational cost, the adiabatic energies of the triplet excited states of ZrO2 were calculated at the SAC-CI/d-T//SAC-CI/d-S (Space 2) level.

3.3. BPW91 calculations The triplet states of MO2 arising from the electronic transitions from the 1a2, 5a1, 2b1, and 4b2 orbitals to the 6a1 and 3b1 orbitals were calculated at the BPW91/A-T level for comparison (see Table S8). The calculated results for the 13A1, 13A2, 13B1, and 13B2 states of TiO2 at the BPW91/A-T level are similar to those at the BPW91/6-311+G(3df) level [30]. The bond lengths of the triplet states of MO2 optimized at the BPW91/A-T levels are similar to those optimized at the SAC-CI levels. The bond angles for most of the triplet states optimized at the BPW91/A-T level are smaller than those optimized at the SAC-CI levels. The bond angle for the 13A2 (TiO2) state optimized at the BPW91/A-T level is approximately 25° smaller than that optimized at the SAC-CI levels. The bond angles of the 13A2 (MO2) states are difficult to determine because the potential energy surfaces of the 13A2 states become flattened when the bond angles increase (see Figs. 1 and 2). The AEEs and ADEs to the triplet states of MO2 computed at the BPW91 levels without ZPE corrections are within approximately 0.31 and 0.39 eV lower than those computed at the SAC-CI/d-T (Space 1) levels, respectively.

3.2.1. Type A states (the 13B2, 13A1, 13B1, 23A2, 33B2, and 33A1 states) The 13B2, 13A1, 13B1, 23A2, 33B2, and 33A1 states of ZrO2 mainly arise from the electronic transitions from the highest six OMOs to the 6a1 orbital (see Tables 3 and S3). The VEEs and VDEs to the Type A triplet states in ZrO2 are close to or slightly lower than those to their corresponding states in TiO2. This phenomenon resembles that found in their corresponding singlet excited states [36] and may be because the HOMO-LUMO energy gap of ZrO2 is smaller than that of TiO2. The EEC(Zr) for the Type A triplet states can be assigned to Zr+[5s14d2] (see Table S7). The electron detachment energies to the 13B2 state are 3.801 eV for VDE and 3.780 eV for ADE. The VDE and ADE computed at the SAC-CI level are similar to those computed at the CCSD(T) level (3.82 eV for VDE and 3.72 eV for ADE) [34]. The 13B2, (13A1, 13B1), and (23A2, 33B2) states correlate to the 3 1 Σu+, 13Πu, and 13Πg states, respectively (see Fig. 2). The 13Σu+, 13Πu, and 13Πg states have the same upper SOMO (4σg orbital); the 4σg and 6a1 orbitals mainly involve the Zr(5s) character. As shown in Fig. 2, an avoided crossing occurs between the potential energy surfaces of the 33A1 and 43A1 (Type B) states. The 33A1 and 43A1 states are mainly characterized by the electronic configurations of 2b1 → 3b1 and 4a1 → 6a1, respectively, at their respective optimized geometries. Therefore, the 33A1 and 43A1 states adiabatically correlate to the 33Πu and 13Σg+ states, respectively.

3.4. Similarities between the triplet and singlet excited states of MO2 The low-lying electronic excited states of MO2 mainly arise from the electronic transitions from the O(2p) orbitals to the M[ns(n − 1)d] orbitals. The triplet states and their corresponding singlet states have similar spatial orbital occupations. The lower and upper SOMOs of theses singlet and triplet pair states are localized on different atoms. This phenomenon leads to poor overlaps of the lower and upper SOMOs and small exchange energies. Therefore, the relative energies and geometries of the triplet states of MO2 are similar to those of their corresponding singlet states. This similarity in TiO2 has been reported and explained in Ref. [30]. The VDEs to the low-lying singlet excited states of MO2 are listed in 26

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Tables S10 and S11. The vertical energies to the considered triplet states of MO2 are lower than those to their corresponding singlet states except for the 13A2 (TiO2) state. The vertical energies to the 13A2 (TiO2) state are approximately the same as those to the 11A2 state. The vertical energies to most of the triplet excited states are within approximately 0.25 eV lower than those to their corresponding singlet states at the SAC-CI/A-T (Space 1) levels. The AEEs and optimized geometries of the selected triplet and singlet states at the SAC-CI/d-S (Space 1) levels are summarized in Table S12 for comparison. As shown in Table S12, the AEEs and optimized geometries of the triplet states of ZrO2 are very similar to those of their corresponding singlet states. The absolute values of the parameters ΔEST, ΔrST, and ΔθST for most of the singlet-triplet pairs of ZrO2 are smaller than 0.1 eV, 0.01 Å, and 5°, respectively. The parameters ΔEST, ΔrST, and ΔθST indicate the energy difference, bond-length difference, and bond-angle difference between the singlet and triplet states, respectively. The ΔEST, ΔrST, and ΔθST for the (13B1, 11B1) and (13A1, 21A1) states of TiO2 are approximately 0.15 eV, 0.02 Å, and 13°, respectively. These values are apparently larger than those for the other singlet-triplet pairs, and may be due to the “crossing” between the energy surfaces of the 6a1 and 7a1 orbitals (see Fig. 3 in Ref. [35]).

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4. Summary The vertical and adiabatic energies to the low-lying triplet excited states of MO2 were obtained in the SAC-CI calculations. Most of the lowlying excited states of MO2 mainly arise from an electron promotion from the O(2p) orbitals to the M[ns(n − 1)d] orbitals. The calculated energetic parameters and geometrical parameters for most of the triplet states are similar to those for their corresponding singlet states. The VDEs to the Type A triplet states of TiO2 reasonably agree with the peak positions in the photoelectron spectrum of TiO2− [16]. Possible candidate states for the unassigned bands were found. The VDEs to the Type A triplet states in ZrO2 are similar to those to their corresponding states in TiO2. However, the VDEs to the Type B and C triplet states in ZrO2 are significantly larger than those to their corresponding states in TiO2. There seem no photoelectron spectra concerning the triplet excited states of ZrO2. The calculated VDEs, ADEs, and geometries for the triplet excited states of ZrO2 will be useful for investigating the photoelectron spectra of ZrO2−. Acknowledgements The author is grateful to the National Center for High-performance Computing for computer time and facilities. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.comptc.2018.09.009. References [1] A. Fujishima, K. Honda, Electrochemical photolysis of water at a semiconductor electrode, Nature 238 (1972) 37–38. [2] K. Hashimoto, H. Irie, A. Fujishima, TiO2 photocatalysis: a historical overview and future prospects, Jpn. J. Appl. Phys. 44 (2005) 8269–8285. [3] T. Ishikawa, H. Yamaoka, Y. Harada, T. Fujii, T. Nagasawa, A general process for in situ formation of functional surface layers on ceramics, Nature 416 (2002) 64–67. [4] T. Yamaguchi, Application of ZrO2 as a catalyst and a catalyst support, Catal. Today 20 (1994) 199–218. [5] J. Riegel, H. Neumann, H.-M. Wiedenmann, Exhaust gas sensors for automotive emission control, Solid State Ionics 152–153 (2002) 783–800. [6] J.-M. Herrmann, J. Disdier, P. Pichat, Oxygen species ionosorbed on powder photocatalyst oxides from room-temperature photoconductivity as a function of oxygen pressure, J. Chem. Soc. Faraday Trans. I (77) (1981) 2815–2826. [7] T. Kamiński, C.A. Gottlieb, K.M. Menten, N.A. Patel, K.H. Young, S. Brünken, H.S.P. Müller, M.C. McCarthy, J.M. Winters, L. Decin, Pure rotational spectra of TiO and TiO2 in VY Canis Majoris, A&A 551 (2013) A113-1-A113-13.

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