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Electronic structure and chemical bonding in W2 molecule
Chemical Physics Letters 490 (2010) 24–28
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Electronic structure and chemical bonding in W2 molecule Antonio Carlos Borin a,*, João Paulo Gobbo a, Björn O. Roos b a b
Instituto de Química, Universidade de São Paulo, Av. Prof. Lineu Prestes 748, 05508-900 São Paulo, SP, Brazil Department of Theoretical Chemistry, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden
a r t i c l e
i n f o
Article history: Received 14 December 2009 In final form 8 March 2010 Available online 11 March 2010
a b s t r a c t The electronic structure of the lowest-lying electronic states of W2 were investigated at the CASPT2 level. 1 3 3 þ The ground state is a X1 Rþ g state, followed by the a Du , b Ru and A Du electronic states. Seven low-lying þ X-states were computed: ð1Þ0g , ð2Þ3u , ð3Þ2u , ð4Þ1u , ð5Þ0u , ð6Þ1u , and ð7Þ2u , with the ground state corre1 þ sponding to the ð1Þ0þ g ðX Rg Þ state. Comparison with the other VIB transition metal group dimers indicates a common pattern of electronic structure and spectroscopic properties. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The dimers of the VIB transition metal group are especially interesting, due to the possibility of forming chemical bonds of order six, involving the five nd and the ðn þ 1Þ s orbitals. We have discussed general aspects of the bonding in these three molecules [1], and reviewed important aspects of the electronic structure and spectroscopic properties of seven of the lowest-lying singlet and triplet electronic states of the Mo2 [2]. Based on an effective bond order (EBO) analysis [1], the metal–metal chemical bond in the Mo2 X1 Rþ g state was described by us [2] as a fully developed sextuple bond. In addition, the so-called 3 K state was unequivocally as3 signed to the b3 Rþ u state, the existence of a low-lying c Cu state 1 (T e ¼ 15 671 cm ) was confirmed, and the experimentally observed A1 Rþ u state described in detail, reinforcing the analysis of the A–X band system. Among the VIB transition metal dimers, W2 is the least known. From the experimental data reported by Hu et al. [3], the ground 1 state symmetry was determined to be 1 Rþ g , with xe ¼ 336:8 cm and xe xe < 1 cm1 . They also reported an absorption band at about X1 Rþ 434–436 nm, analogous to the A1 Rþ u g transition observed in Cr2 (460–469 nm) and Mo2 ð524 nmÞ. Kraus et al. [4] have also obtained tungsten clusters in rare gas matrices, which were investigated by laser induced fluorescence and cavity ringdown absorption techniques, but no spectra unequivocally attributed to W2 could be observed. The experimental binding energy is estimated to be 5 1 eV [5]. Theoretical investigations, focused on the ground electronic state, have been reported. Kraus et al. (Re ¼ 2:048 Å and xe ¼ 401:2 cm1 ) [4] and Wu and Ma (Re ¼ 2:039 Å and xe ¼ 410:6 cm1 ) [6] used density functional theory, Bourdreaux
* Corresponding author. Fax: +55 11 3815 5579. E-mail address: [email protected] (A.C. Borin). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.03.022
and Baxter extended Hückel [7], and Angeli et al. [8] second and third order multireference perturbation theory. At the highest level of calculation, Angeli et al. reported Re ¼ 2:0561 Å, xe ¼ 326:69 cm1 , xe xe ¼ 0:81 cm1 , and De ¼ 4:5110 eV, in agreement with the experimental values reported by Hu et al. [3] and within the error bar for the dissociation energy. In the following sections, we shall discuss the chemical bonding and spectroscopic properties of the W2 molecule. The results were obtained by using the multiconfigurational second-order perturbation theory (CASPT2) [9–11] and basis sets of the atomic natural orbital (ANO-RCC) type [12–14], following closely the procedure used for Mo2 [2]. 2. Methodology The lowest-lying singlet and triplet electronic states of W2 were investigated by the CASPT2//CASSCF protocol [9–11], including scalar relativistic effects via the Douglas–Kroll–Hess (DKH) approximation [15,16]. Spin–orbit effects (SOC) were included via an effective one-electron spin–orbit Hamiltonian based on an atomic mean field approximation of the two-electron part [17]. The CASSCF zeroth-order wavefunction was obtained including the W 5d and 6s orbitals in the active space (12 electrons in 12 active orbitals). The 5s and 5p electrons were kept in the inactive space at this level. Dynamic correlation effects were included by using the complete-active-space second-order perturbation theory (CASPT2), with the zeroth order Hamiltonian suggested by Ghigo et al. [18]. The 5s and 5p electrons were correlated at this level of theory and the core electrons were kept frozen. Intruder state problems were treated by using an imaginary shift [19] of 0.1 Hartree. A quadruple-f atomic ANO-RCC basis set [14] was used, derived from a primitive 24s21p15d11f 4g2h set contracted to 9s8p6d4f 3g2h. The ANO–RCC basis set was generated from a calculation that includes the average density matrix of the ground
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A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28 4
5
(5d 6s2 ) and lowest excited states (5d 6s1 ) of the W atom, its cat4 5 ion (5d 6s1 ) and anion (5d 6s2 ), and the atom in an electric field of strength 0.01 au. The semicore electrons (5s; 5p) were also correlated and scalar relativistic effects were included using the Douglas–Kroll–Hess Hamiltonian [14]. The computations were carried out using C2h point group symmetry. State average CASSCF calculations were performed for 10 states in 1 Ag , nine states in 1 Au and 10 states in 3 Au . Other states were also studied (P and U symmetries), but will not be discussed here because they are located in a higher energetic region. Orbital rotations were restricted in order to avoid mixing between states of different angular momenta. Potential energy functions and spectroscopic constants were obtained with the program VIBROT [20]. Dissociation energies were computed as the difference between the total energies at the equilibrium internuclear distance and at the dissociation limit (100.0 au). Calculations were performed with the MOLCAS-6 software [20].
3. Results and discussion 3.1. The X1 Rþ g electronic state: bonding and spectroscopic characteristics The ground state of the tungsten atom has the electronic config4 uration 5d 6s2 [21], 5 D0 . From the same electronic configuration, four other J-couplings are possible: 5 D1 ð0:21 eVÞ, 5 D2 ð0:41 eVÞ, 5 D3 ð0:60 eVÞ, and 5 D4 ð0:77 eVÞ, with excitation energies relative to the 5 D0 ground state given in parentheses (Table 1). The excita5 tion 6s ! 5d gives rise to the 7 S3 ð5d ð6 SÞ6s1 Þ excited state, lying 0.37 eV higher than the atomic ground state (5 D0 ). Therefore, the W2 diatom molecular ground state, 1 Rþ g , can correlate with both atoms in either ground or excited states. If both tungsten atoms 4 are in the ground atomic state (5 D0 ð5d 6s2 Þ), the maximum bonding order would be four; in this case, the 6s shell is not suitable for bonding and no significant sr bond can occur. But, if both atoms 5 are in the first excited state (7 S3 ð5d ð6 SÞ6s1 Þ), a sextuple bond could, in principle, be formed, as is the case in Mo2 and, to a lesser extent, in Cr2 [2]. The W2 ground state was computed to be of 1 Rþ g symmetry (Fig. 1 and Table 2), in agreement with previous experimental and theoretical findings. The computed spectroscopic constants (Re ¼ 2:010 Å, xe ¼ 354 cm1 ) and dissociation energy (De ¼ 5:37 eVÞ are also in accord with experimental findings. Around the equilibrium geometry, the ground state wavefunction is dominated by the valence electronic configuration 0:82j1r2g 1d4g 2r2g 1p4u i (Table 3), with the relevant atomic parentage contribution (in parentheses) to the valence molecular orbitals described as: 1rg ð5drÞ, 1dg ð5ddÞ, 2rg ð6sÞ, and 1pu ð5dpÞ (Fig. 2). The corresponding natural occupation numbers are given in Table 4 and Mulliken populations in Table 3. The effective bond order (EBO) [1] for the ground state is computed to be 5.2. Therefore, a fully developed Table 1 Low-lying electronic states of W2, their dissociation channels and energy separation at the dissociation limit. Atomic limit
a b
4
4
5
D0 ð5d 6s2 Þ þ 5 D0 ð5d 6s2 Þ
5
D0 ð5d 6s2 Þ þ 7 S3 ð5d 6s1 Þ
7
S3 ð5d 6s1 Þ þ 7 S3 ð5d 6s1 Þ
4
5
5
5
Molecular statesa
DE experimental (eV)b
3;7 þ 3;7 Rþ Ru ð3Þ; 1;5;9 R Rg ð2Þ g ð3Þ; u ð2Þ; 1;3;5;7;9 Pg;u ð2Þ; 1;5;9 Dg ð2Þ; 1;5;9 Du ; 3;7 Dg 3;7 Du ð2Þ; 1;3;5;7;9 Ug;u ð2Þ; 1;5;9 Cg ; 3;7 Cu 3;5;7;9;11 þ 3;5;7;9;11 Rg;u ; Pg;u ; 3;5;7;9;11 Dg;u 1;5;9 þ 3;7;11 þ Rg ; Ru
0.00
1;5;9
Number of states in parentheses. Relative to the lowest-lying J value. See Ref. [21].
0.37 0.73
Fig. 1. Potential energy curves for the low-lying singlet and triplet electronic states of W2.
Table 2 Spectroscopic constants for the low-lying electronic states of W2. State
Re (Å)
Te (cm1)
xe (cm1)
X1 Rþ g
2.010
a3 Du b3 Rþ u
2.089 2.111
7397 8547
329 300
A1 Du c3 Cu B1 Rþ g
2.079 2.078 2.173
11 208 16 873 18 083
350 342 306
C1 Cg
2.174
18 904
302
D1 R u
2.075
19 268
370
E1 Dg
2.161
19 768
301
F1 Rþ u
2.111
20 534
285
G1 Rþ u
2.021
25 249
480
354
sextuple bond is formed in the ground state of W2, with a small population of electrons in the antibonding orbitals. Besides, from the analysis of the natural occupation numbers and Mulliken populations, we conclude that the W2 ground state is derived by cou5 pling both atoms in their first excited state (7 S3 ð5d ð6 SÞ6s1 Þ). As for the Cr2 and Mo2 diatoms [2], in W2 the contributions of the rg ðsÞ and pu ðdpÞ molecular orbitals (occupation numbers: 1.93 and 3.82, respectively) to the chemical bond are also larger than that from the dg ðddÞ (occupation number: 3.56) orbitals. The contributions from the rg ðsÞ and dg ðddÞ orbitals to the dimer chemical bonding increase as one goes down in the VIB transition metals (rg ðsÞ : 1; 90; 1:91; 1:93; dg ðddÞ : 3:16; 3:52; 3:56, in Cr2, Mo2, and W2, respectively), as a result of the contraction of the ðn þ 1Þs shell and increasing diffuseness of the nd shells, resulting in orbitals of more equal size. Thus, contribution from the d-electron becomes more effective, and the chemical bonding stronger [2] (EBO ¼ 3:5; 5:2; and 5.2, for Cr2, Mo2, and W2, respectively). The X1 Rþ g dissociation energy was computed to be 5.37 eV (Table 2), being the highest among the VIB transition metal dimers (Cr2 1:5 eV, Mo2 4:41 eV) [2].
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A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28
Table 3 Dominant configurations, leading excitations, and Mulliken population analysis for the low-lying electronic states of W2 at the CASSCF level. State X
1
Rþ g
c
ð67Þj1r ð71Þj1r
b3 Rþ u
ð50Þj1r
A1 Du
ð74Þj1r
c Cu
b
4 2 g 1dg 2 3 2 g 1dg 2 3 2 1d g2 g 3 2 1d g2 g 3 2 g 1dg 2 2 2 g 1dg 2 2 2 1d g2 g 3 2 1d g2 g 2 2 g 1dg 2 4 1 g 1dg 2 4 2 1d g2 g
a3 Du
3
a
Configurationa
ð75Þj1r
B1 Rþ g
ð48Þj1r
C1 Cg
ð48Þj1r
D1 R u
ð77Þj1r
E1 Dg
ð41Þj1r
F1 Rþ u
ð61Þj1r
G1 Rþ u
ð46Þj1r
Leading excitationb 2 g1
r p r2g 1p4u 2r1u i r2g 1p4u 1d1u i r2g 1p4u 2r1u i r2g 1p4u 1d1u i r2g 1p4u 2r2u i r2g 1p4u 2r2u i r2g 1p4u 1d1u i r2g 1p4u 1r1u 1d1u i r2g 1p4u 2r1u i r1g 1p4u 2r1u i
EBOc
Mulliken population
5.2
5d
5:00
1dg ! 2ru
4.4
5d
4:47
6s1:30
1dg ! 1du
4.1
5d
4:90
6s0:98
1dg ! 2ru
4.4
5d
4:46
6s1:28
1dg ! 1du
4.4
5d
4:89
6s1:02
1d2g 1d2g
2 u
! 2r
3.5
5d
4:29
6s1:37
! 2r2u
3.4
5d
4:14
6s1:50
1dg ! 1du
4.5
5d
4:87
6s1:02
1d2g ! 1r1u 1d1u
3.6
5d
4:43
6s1:30
1rg ! 2ru
4.2
5d
4:52
6s1:22
2rg ! 2ru
4.2
5d
4:52
6s1:22
4 ui
6s1:00
Total weight (%) in parentheses. With respect to the ground state configuration. Effective bond order.
Fig. 2. Valence natural molecular orbitals (VMO) of W2. The numbers correspond to the natural occupation numbers in the W2 X1 Rþ g electronic state.
3.2. The excited electronic states The a3 Du electronic state is the lowest triplet excited state, placed 7397 cm1 above the X1 Rþ g state with Re ¼ 2:089 Å and xe ¼ 329 cm1 (Table 2). The dominant configuration is 0:84j1r2g 1d3g 2r2g 1p4u 2r1u i (Table 3), derived from the leading
configuration of the ground state by the single excitation 1dg ! 2ru , which transfers one electron from the bonding 1dg molecular orbital to the 2ru antibonding orbital. The natural occupation numbers for the active orbitals (Table 4) give an effective bond order of 4.4. In comparison to the other VIB transition metal group diatoms, a low-lying 3 Du for Cr2 was computed by Andersson [22] at the CASPT2 level at 17 341 cm1 ðRe ¼ 2:13 Å , DG1=2 ¼ 230 cm1 Þ above the ground state, being derived from the ground state by a dg ! ru single excitation. To the best or our knowledge, there is no mention to a low-lying 3 Du state for Mo2 [2]. The electronic X1 Rþ transition 3 Du g is unlikely to be observed because it is both electric dipole and spin forbidden. 1 above the The b3 Rþ u electronic state is located 8547 cm ground state, with an equilibrium internuclear distance of 2.111 Å and xe ¼ 300 cm1 (Table 2). Its multiconfigurational wavefunction (Table 3) is dominated by the configuration, 0:71j1r2g 1d3g 2r2g 1p4u 1d1u i. The b3 Rþ u is derived from the ground state by a single excitation (1dg ! 1du ). The EBO is 4.1. A low-lying 3 Rþ u is also found in Cr2 and Mo2 diatoms. Experimentally, Pellin and Gruen located a 3 Rþ u in Cr2 [23] at about 8000 cm1 . Andersson [22], at the CASPT2 level, places a 3 Rþ u state at 4517 cm1 above the ground state. As to the Mo2 molecule, the experimental study by Kraus et al. [4] placed a low-lying 3 Rþ u state 7800 cm1 above the ground state. Balasubramanian and Zhu [24], at the first-order configuration interaction level, found the corresponding state at 5499 cm1 , while at the CASPT2 level 1 above the ground we found a low-lying 3 Rþ u [2] at 8912 cm state. Another common aspect of the low-lying 3 Rþ u state of the Cr2 [22], Mo2 [2], and W2 (Table 3) diatoms is that their wavefunctions have a high contribution from the single excitation (dg ! du ), around the equilibrium internuclear geometry. The low-lying 3 Rþ u electronic state of the Cr2 and Mo2 diatoms corre5 lates with the two corresponding ground state 7 Sðnd ðn þ 1Þs1 Þ atoms, the same separated atom limit of the molecular ground 3 þ state (X1 Rþ g ). For W2, the low-lying Ru and the ground electronic 5 states correlate with the 7 S3 ð5d ð6 SÞ 6s1 Þ atomic limit. Therefore, X1 Rþ for the VIB transition metal group diatoms the 3 Rþ u g electronic transition corresponds to a redistribution of electrons within the same compact atomic d-shell. The A1 Du is located 11 208 cm1 above the X1 Rþ g state. With Re ¼ 2:079 Å and xe ¼ 350 cm1 (Table 2), its multiconfigurational wavefunction is dominated by the electronic configuration 0:86j1r2g 1d3g 2r2g 1p4u 2r1u i, originating from the ground state by the excitation 1dg ! 2ru (Table 3). The EBO is 4.4. According to the computation carried out by the CASSCF//CASPT2 approach [22],
27
A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28 Table 4 Natural orbital occupation numbers for the low-lying electronic states of W2 at the CASSCF level. State
Natural orbitals occupation numbers 1rg ðdrÞ
1pu ðdpÞ
1dg ðddÞ
2rg ðsÞ
1ru ðdrÞ
1pg ðdpÞ
1du ðddÞ
X1 Rþ g
1.88
3.82
3.56
1.93
0.13
0.18
0.42
0.08
a3 Du b3 Rþ u
1.88 1.65
3.92 3.76
2.71 2.84
1.87 1.88
0.10 0.12
0.18 0.24
0.34 1.16
1.02 0.34
A1 Du c3 Cu B1 Rþ g
1.89
3.82
2.71
1.97
0.10
0.18
0.30
1.02
1.89 1.68
3.83 3.88
2.78 2.04
1.92 1.92
0.09 0.10
0.17 0.22
1.20 0.60
0.12 1.65
C1 Cg
1.84
3.78
1.80
1.96
0.10
0.20
0.44
1.85
D1 R u E1 Dg
1.89 1.81
3.80 3.74
2.83 2.02
1.93 1.93
0.09 0.13
0.16 0.24
1.12 1.00
0.12 1.10
F1 Rþ u
0.99
3.79
3.17
1.96
0.07
0.21
0.53
0.99
G1 Rþ u
0.99
3.79
3.47
1.96
0.07
0.21
0.53
0.99
Cr2 also exhibits a low-lying 1 Du ðTe ¼ 19 357 cm1 , Re ¼ 2:06 Å, DG1=2 ¼ 180 cm1 ), but we did not find it in Mo2 [2]. At ðT e Þ16 873 cm1 above the ground state, we computed the c3 Cu ðRe ¼ 2:078 Å, xe ¼ 342 cm1 ) electronic state (Table 2), with a wavefunction dominated by the 0:86j1r2g 1d3g 2r2g 1p4u 1d1u i electronic configuration, derived from the ground state by a single excitation (1dg ! 1du ) (Table 2), and active orbital occupation numbers given in Table 4. The corresponding EBO is 4.4. Again, the W2 c3 Cu excited state has its counterpart in the Cr2 and Mo2 diatoms. For Cr2, Anderson [22] computed a 3 Cu state ðT e Þ19 438 cm1 above the ground state, with Re ¼ 1:78 Å, DG1=2 ¼ 420 cm1 , and wavefunction dominated by an electronic configuration best represented by the 3ddg ! 3ddu single excitation from the ground state. As to the Mo2 diatom, in our previous work [2] we placed the c3 Cu ðRe ¼ 2:010 Å, xe ¼ 447 cm1 Þ T e ¼ 15 671 cm1 above the X1 Rþ g ground state, in agreement with the results obtained by Balasubramanian and Zhu [24] ðT e ¼ 12 947 cm1 , Re ¼ 2:100 Å, xe ¼ 477 cm1 Þ at the MRSDCI (multireference single and doubles configuration interaction) level; the wavefunction is best represented by the electronic configuration j9r2g 2d3g 10r2g 5p4u 2d1u i [2,24]. 1 above the The B1 Rþ g excited state (Table 2) is placed 18 083 cm ground state, with Re ¼ 2:173 Å and xe ¼ 306 cm1 , its wavefunction is dominated by the 0:69j1r2g 1d2g 2r2g 1p4u 2r2u i electronic configuration, differing from the ground state by a double excitation 1d2g ! 2r2u , which transfers two electrons from a bonding to an antibonding molecular orbital. Therefore, in comparison to the ground state, the equilibrium internuclear distance increases by 0.16 Å, reflecting the counterbalance of the 2rg and 2ru populations. From the active orbital occupation numbers (Table 4) one gets an EBO of 3.5, almost two units less than that for the ground state. For Cr2 and Mo2, there is no mention of the existence of a low-lying 1 Rþ g state in the energy region considered by Anderson [22] and by us [2], respectively. Nonetheless, Balasubramanian 1 ðRe ¼ and Zhu [24] reported a 1 Rþ g state for Mo2 22 019 cm
2ru ðsÞ
2:146 Å, xe ¼ 381 cm1 ) higher than the ground state, computed at the first-order configuration interaction (FOCI) level. The C1 Cg ðRe ¼ 2:174 Å and xe ¼ 302 cm1 Þ state computed by us at 18 904 cm1 (Table 2) above the X1 Rþ g ground state is isoconstate described above. As expected, its figurational with the B1 Rþ g wavefunction is very similar to that for the B1 Rþ g state, with active orbital occupation numbers (Table 4) resulting in an EBO of 3.4. In this energy region, there is no evidence for a 1 Cg state in Cr2 [22] and Mo2 [2]. The next two excited states (Table 2) are the D1 R u ðRe ¼ 2:075 Å and xe ¼ 370 cm1 Þ and E1 Dg ðRe ¼ 2:161 Å and xe ¼ 301 cm1 Þ states located, respectively, 19 268 cm1 and 19 768 cm1 above the ground state. Around the equilibrium internuclear distance, the D1 R u state is best described by a wavefunction dominated by the 0:88j1r2g 1d3g 2r2g 1p4u 1d1u i electronic configuration, which differs from the ground state by the 1dg ! 1du single excitation. From the corresponding active orbital occupation numbers (Table 4) the EBO is 4.5. As for the E1 Dg state, the wavefunction is dominated by the 0:64j1r2g 1d2g 2r2g 1r1u 1p4u 1d1u i electronic configuration, derived from the ground state wavefunction by a double excitation: 1d2g ! 1d1u 1r1u . The corresponding active orbital occupation numbers (Table 4) result in an EBO of 3.6. In comparison to the other dimers of the VIB transition metal group, we found a counterpart for the D1 R u state only in Mo2, computed by Balasubramanian and Zhu [24] at the FOCI level to be 17 924 cm1 above the ground state, with Re ¼ 2:093 Å and xe ¼ 485 cm1 . 1 þ The last two states computed by us are the F1 Rþ u and G Ru elec1 state is placed 20 534 cm above tronic states (Table 2). The F1 Rþ u the ground state, with an equilibrium internuclear distance of 2:111 Å and xe ¼ 285 cm1 , a wavefunction dominated by the 0:78j1r1g 1d4g 2r2g 1p4u 2r1u i electronic configuration. The EBO is 4.2. The G1 Rþ u state is the last excited state included in this work, computed to be 25 249 cm1 above the ground state, with Re ¼ 2:021 Å and xe ¼ 480 cm1 . Around the equilibrium internuclear distance, its wavefunction is dominated by the 0:68j1r2g 1d4g 2r1g 1p4u 2r1u i elec-
Table 5 Electronic configurations, in terms of spin-free electronic states (coefficients in parenthesis), and spectroscopic constants for the lowest-lying electronic states of W2 with spin– orbit coupling.
X
Composition
xe
Re (Å)
(cm1)
Te (cm1)
DG0 (cm1)
DG1=2 (cm1)
ð1Þ0þ g
ð1:00ÞX1 Rþ g
2.009
381
189
373
ð2Þ3u ð3Þ2u
ð1:00Þa3 Du
2.088 2.086
359 362
6844 7677
177 178
353 355
ð4Þ1u ð5Þ0 u ð6Þ1u ð7Þ2u
ð0:80Þa3 Du þ ð0:20Þ A1 Du ð0:86Þb3 Rþ u
2.106
333
8615
166
326
ð1:00Þb3 Rþ u
2.111
331
9875
166
323
ð1:00Þa3 Du
2.091 2.082
367 369
10 175 12 053
180 182
358 361
ð0:20Þa3 Du þ ð0:80Þ A1 Du
28
A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28
(DKH) approximation, has been employed to investigate the electronic structure and chemical bonding of the lowest-lying singlet and triplet electronic states of W2. The ground state of W2 is state (Re ¼ 2:010 Å, xe ¼ 354 cm1 , computed to be a X1 Rþ g De ¼ 5:37 eV), with three low-lying electronic states (a3 Du : T e ¼ 1 7397 cm1 , Re ¼ 2:089 Å, xe ¼ 329 cm1 ; b3 Rþ u : T e ¼ 8547 cm , 1 1 1 Re ¼ 2:111 Å, xe ¼ 300 cm , and A Du : T e ¼ 11 208 cm , Re ¼ 2:079 Å, xe ¼ 350 cm1 ). Comparison with the other VIB transition metal group dimers (Cr2 and Mo2) indicates a common pattern of electronic structure and spectroscopic properties. Seven low-lying X-states were computed: ð1Þ0þg , ð2Þ3u , ð3Þ2u , ð4Þ1u , ð5Þ0u , ð6Þ1u , and ð7Þ2u , with the ground state corresponding to the 1 þ ð1Þ0þ g ðX Rg Þ state. Acknowledgements A.C.B. acknowledges academic support by the CNPq (Conselho Nacional de Desenvolvimento Cientı´fico and Tecnológico) and FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo). J.P.G. is grateful to FAPESP for a graduate fellowship. B.O.R. acknowledges support from the vice chancellor of Lund University. The authors thank the Laboratório de Computação Cientı´fica Avançada (LCCA) of the Universidade de São Paulo for the services and computer time.
References
Fig. 3. Potential energy curves, with spin–orbit coupling, for the lowest-lying singlet and triplet electronic states of W2.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
tronic configuration. The EBO is 4.2 (Tables 2 and 3). It is interest1 þ ing to note that the F1 Rþ u and G Ru electronic states are placed in the 20 000 cm1 energy region, the A–X band system observed is known to exist for Mo2 [2] and Cr2.
[10] [11]
3.3. Spin–orbit coupling
[12]
Spin–orbit effects (SOC) were included following a procedure that has been used in other applications [1,17,25–28]. All CASSCF wavefunction described here were used as basis functions for the spin–orbit Hamiltonian, with the diagonal elements replaced by the corresponding CASPT2 energies. We have focused our attention on the seven lowest X-states (Table 5, Fig. 3): ð1Þ0þ g , ð2Þ3u , ð3Þ2u , 3 ð4Þ1u , ð5Þ0 u , ð6Þ1u , and ð7Þ2u . The a Du splits into the ð2Þ3u (the lowest in energy), ð3Þ2u , and ð6Þ1u X-states, while the b3 Rþ u splits into the ð4Þ1u and ð5Þ0 u X-states. The ð7Þ2u X-state is dominated by the A1 Du wavefunction. The ground state in the relativistic 1 þ scheme has X ¼ 0þ g ðX Rg Þ. Although SOC splits the excited states, it has little influence on the computed spectroscopic constants (Table 5). 4. Conclusions The CASPT2//CASSCF protocol, with extended atomic basis sets and inclusion of relativistic effects via the Douglas–Kroll–Hess
[13] [14] [15] [16] [17] [18] [19] [20] [21]
[22] [23] [24] [25] [26] [27] [28]
B.O. Roos, A.C. Borin, L. Gagliardi, Angew. Chem., Int. Ed. 46 (2007) 1469. A.C. Borin, J.P. Gobbo, B.O. Roos, Chem. Phys. 343 (2008) 210. Z. Hu, J.G. Dong, J.R. Lombardi, D.M. Lindsay, J. Chem. Phys. 97 (1992) 8811. D. Kraus, M. Lorenz, V.E. Bondybey, Phys. Chem. Commun. 4 (2001) 44. M.D. Morse, Chem. Rev. 86 (1986) 1049. Z.J. Wu, X.F. Ma, Chem. Phys. Lett. 371 (2003) 35. E.A. Boudreaux, E. Baxter, Int. J. Quantum Chem. 100 (2004) 1170. C. Angeli, A. Cavallini, R. Cimiraglia, J. Chem. Phys. 127 (2007) 074306. K. Andersson, P.-Å. Malmqvist, B.O. Roos, A.J. Sadlej, K. Wolinski, J. Phys. Chem. 94 (1990) 5483. K. Andersson, P.-Å. Malmqvist, B.O. Roos, J. Chem. Phys. 96 (1992) 1218. J. Finley, P.-Å. Malmqvist, B.O. Roos, L. Serrano-Andrés, Chem. Phys. Lett. 288 (1998) 299. B.O. Roos, R. Lindh, P.-Å. Malmqvist, V. Veryasov, P.-O. Widmark, J. Phys. Chem. A 108 (2004) 2851. B.O. Roos, V. Veryazov, P.-O. Widmark, Theor. Chim. Acta 111 (2004) 345. B.O. Roos, R. Lindh, P.-Å. Malmqvist, V. Veryazov, P.-O. Widmark, J. Phys. Chem. A 109 (2005) 6575. M. Douglas, N.M. Kroll, Ann. Phys. 82 (1974) 89. B.A. Hess, Phys. Rev. 33 (1986) 3742. B.A. Hess, C.M. Marian, U. Wahlgren, O. Gropen, Chem. Phys. Lett. 251 (1996) 365. G. Ghigo, B.O. Roos, P.-Å. Malmqvist, Chem. Phys. Lett. 396 (2004) 142. N. Forsberg, P.-Å. Malmqvist, Chem. Phys. Lett. 274 (1997) 196. G. Karlström et al., Comput. Mater. Sci. 28 (2003) 222. NIST, National Institute of Standards and Technology (NIST) Reference on Constants, Units, and Uncertainty. Fundamental Constants Data Center of the NIST Physics Laboratory. The contents of this site have been prepared by Barry N. Taylor of the Data Center in close collaboration with Peter J. Mohr of the Physics Laboratory’s Atomic Physics Division, 2005. Available from:
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Electronic structure and chemical bonding in W2 molecule Antonio Carlos Borin a,*, João Paulo Gobbo a, Björn O. Roos b a b
Instituto de Química, Universidade de São Paulo, Av. Prof. Lineu Prestes 748, 05508-900 São Paulo, SP, Brazil Department of Theoretical Chemistry, Chemical Center, P.O. Box 124, S-221 00 Lund, Sweden
a r t i c l e
i n f o
Article history: Received 14 December 2009 In final form 8 March 2010 Available online 11 March 2010
a b s t r a c t The electronic structure of the lowest-lying electronic states of W2 were investigated at the CASPT2 level. 1 3 3 þ The ground state is a X1 Rþ g state, followed by the a Du , b Ru and A Du electronic states. Seven low-lying þ X-states were computed: ð1Þ0g , ð2Þ3u , ð3Þ2u , ð4Þ1u , ð5Þ0u , ð6Þ1u , and ð7Þ2u , with the ground state corre1 þ sponding to the ð1Þ0þ g ðX Rg Þ state. Comparison with the other VIB transition metal group dimers indicates a common pattern of electronic structure and spectroscopic properties. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The dimers of the VIB transition metal group are especially interesting, due to the possibility of forming chemical bonds of order six, involving the five nd and the ðn þ 1Þ s orbitals. We have discussed general aspects of the bonding in these three molecules [1], and reviewed important aspects of the electronic structure and spectroscopic properties of seven of the lowest-lying singlet and triplet electronic states of the Mo2 [2]. Based on an effective bond order (EBO) analysis [1], the metal–metal chemical bond in the Mo2 X1 Rþ g state was described by us [2] as a fully developed sextuple bond. In addition, the so-called 3 K state was unequivocally as3 signed to the b3 Rþ u state, the existence of a low-lying c Cu state 1 (T e ¼ 15 671 cm ) was confirmed, and the experimentally observed A1 Rþ u state described in detail, reinforcing the analysis of the A–X band system. Among the VIB transition metal dimers, W2 is the least known. From the experimental data reported by Hu et al. [3], the ground 1 state symmetry was determined to be 1 Rþ g , with xe ¼ 336:8 cm and xe xe < 1 cm1 . They also reported an absorption band at about X1 Rþ 434–436 nm, analogous to the A1 Rþ u g transition observed in Cr2 (460–469 nm) and Mo2 ð524 nmÞ. Kraus et al. [4] have also obtained tungsten clusters in rare gas matrices, which were investigated by laser induced fluorescence and cavity ringdown absorption techniques, but no spectra unequivocally attributed to W2 could be observed. The experimental binding energy is estimated to be 5 1 eV [5]. Theoretical investigations, focused on the ground electronic state, have been reported. Kraus et al. (Re ¼ 2:048 Å and xe ¼ 401:2 cm1 ) [4] and Wu and Ma (Re ¼ 2:039 Å and xe ¼ 410:6 cm1 ) [6] used density functional theory, Bourdreaux
* Corresponding author. Fax: +55 11 3815 5579. E-mail address: [email protected] (A.C. Borin). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.03.022
and Baxter extended Hückel [7], and Angeli et al. [8] second and third order multireference perturbation theory. At the highest level of calculation, Angeli et al. reported Re ¼ 2:0561 Å, xe ¼ 326:69 cm1 , xe xe ¼ 0:81 cm1 , and De ¼ 4:5110 eV, in agreement with the experimental values reported by Hu et al. [3] and within the error bar for the dissociation energy. In the following sections, we shall discuss the chemical bonding and spectroscopic properties of the W2 molecule. The results were obtained by using the multiconfigurational second-order perturbation theory (CASPT2) [9–11] and basis sets of the atomic natural orbital (ANO-RCC) type [12–14], following closely the procedure used for Mo2 [2]. 2. Methodology The lowest-lying singlet and triplet electronic states of W2 were investigated by the CASPT2//CASSCF protocol [9–11], including scalar relativistic effects via the Douglas–Kroll–Hess (DKH) approximation [15,16]. Spin–orbit effects (SOC) were included via an effective one-electron spin–orbit Hamiltonian based on an atomic mean field approximation of the two-electron part [17]. The CASSCF zeroth-order wavefunction was obtained including the W 5d and 6s orbitals in the active space (12 electrons in 12 active orbitals). The 5s and 5p electrons were kept in the inactive space at this level. Dynamic correlation effects were included by using the complete-active-space second-order perturbation theory (CASPT2), with the zeroth order Hamiltonian suggested by Ghigo et al. [18]. The 5s and 5p electrons were correlated at this level of theory and the core electrons were kept frozen. Intruder state problems were treated by using an imaginary shift [19] of 0.1 Hartree. A quadruple-f atomic ANO-RCC basis set [14] was used, derived from a primitive 24s21p15d11f 4g2h set contracted to 9s8p6d4f 3g2h. The ANO–RCC basis set was generated from a calculation that includes the average density matrix of the ground
25
A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28 4
5
(5d 6s2 ) and lowest excited states (5d 6s1 ) of the W atom, its cat4 5 ion (5d 6s1 ) and anion (5d 6s2 ), and the atom in an electric field of strength 0.01 au. The semicore electrons (5s; 5p) were also correlated and scalar relativistic effects were included using the Douglas–Kroll–Hess Hamiltonian [14]. The computations were carried out using C2h point group symmetry. State average CASSCF calculations were performed for 10 states in 1 Ag , nine states in 1 Au and 10 states in 3 Au . Other states were also studied (P and U symmetries), but will not be discussed here because they are located in a higher energetic region. Orbital rotations were restricted in order to avoid mixing between states of different angular momenta. Potential energy functions and spectroscopic constants were obtained with the program VIBROT [20]. Dissociation energies were computed as the difference between the total energies at the equilibrium internuclear distance and at the dissociation limit (100.0 au). Calculations were performed with the MOLCAS-6 software [20].
3. Results and discussion 3.1. The X1 Rþ g electronic state: bonding and spectroscopic characteristics The ground state of the tungsten atom has the electronic config4 uration 5d 6s2 [21], 5 D0 . From the same electronic configuration, four other J-couplings are possible: 5 D1 ð0:21 eVÞ, 5 D2 ð0:41 eVÞ, 5 D3 ð0:60 eVÞ, and 5 D4 ð0:77 eVÞ, with excitation energies relative to the 5 D0 ground state given in parentheses (Table 1). The excita5 tion 6s ! 5d gives rise to the 7 S3 ð5d ð6 SÞ6s1 Þ excited state, lying 0.37 eV higher than the atomic ground state (5 D0 ). Therefore, the W2 diatom molecular ground state, 1 Rþ g , can correlate with both atoms in either ground or excited states. If both tungsten atoms 4 are in the ground atomic state (5 D0 ð5d 6s2 Þ), the maximum bonding order would be four; in this case, the 6s shell is not suitable for bonding and no significant sr bond can occur. But, if both atoms 5 are in the first excited state (7 S3 ð5d ð6 SÞ6s1 Þ), a sextuple bond could, in principle, be formed, as is the case in Mo2 and, to a lesser extent, in Cr2 [2]. The W2 ground state was computed to be of 1 Rþ g symmetry (Fig. 1 and Table 2), in agreement with previous experimental and theoretical findings. The computed spectroscopic constants (Re ¼ 2:010 Å, xe ¼ 354 cm1 ) and dissociation energy (De ¼ 5:37 eVÞ are also in accord with experimental findings. Around the equilibrium geometry, the ground state wavefunction is dominated by the valence electronic configuration 0:82j1r2g 1d4g 2r2g 1p4u i (Table 3), with the relevant atomic parentage contribution (in parentheses) to the valence molecular orbitals described as: 1rg ð5drÞ, 1dg ð5ddÞ, 2rg ð6sÞ, and 1pu ð5dpÞ (Fig. 2). The corresponding natural occupation numbers are given in Table 4 and Mulliken populations in Table 3. The effective bond order (EBO) [1] for the ground state is computed to be 5.2. Therefore, a fully developed Table 1 Low-lying electronic states of W2, their dissociation channels and energy separation at the dissociation limit. Atomic limit
a b
4
4
5
D0 ð5d 6s2 Þ þ 5 D0 ð5d 6s2 Þ
5
D0 ð5d 6s2 Þ þ 7 S3 ð5d 6s1 Þ
7
S3 ð5d 6s1 Þ þ 7 S3 ð5d 6s1 Þ
4
5
5
5
Molecular statesa
DE experimental (eV)b
3;7 þ 3;7 Rþ Ru ð3Þ; 1;5;9 R Rg ð2Þ g ð3Þ; u ð2Þ; 1;3;5;7;9 Pg;u ð2Þ; 1;5;9 Dg ð2Þ; 1;5;9 Du ; 3;7 Dg 3;7 Du ð2Þ; 1;3;5;7;9 Ug;u ð2Þ; 1;5;9 Cg ; 3;7 Cu 3;5;7;9;11 þ 3;5;7;9;11 Rg;u ; Pg;u ; 3;5;7;9;11 Dg;u 1;5;9 þ 3;7;11 þ Rg ; Ru
0.00
1;5;9
Number of states in parentheses. Relative to the lowest-lying J value. See Ref. [21].
0.37 0.73
Fig. 1. Potential energy curves for the low-lying singlet and triplet electronic states of W2.
Table 2 Spectroscopic constants for the low-lying electronic states of W2. State
Re (Å)
Te (cm1)
xe (cm1)
X1 Rþ g
2.010
a3 Du b3 Rþ u
2.089 2.111
7397 8547
329 300
A1 Du c3 Cu B1 Rþ g
2.079 2.078 2.173
11 208 16 873 18 083
350 342 306
C1 Cg
2.174
18 904
302
D1 R u
2.075
19 268
370
E1 Dg
2.161
19 768
301
F1 Rþ u
2.111
20 534
285
G1 Rþ u
2.021
25 249
480
354
sextuple bond is formed in the ground state of W2, with a small population of electrons in the antibonding orbitals. Besides, from the analysis of the natural occupation numbers and Mulliken populations, we conclude that the W2 ground state is derived by cou5 pling both atoms in their first excited state (7 S3 ð5d ð6 SÞ6s1 Þ). As for the Cr2 and Mo2 diatoms [2], in W2 the contributions of the rg ðsÞ and pu ðdpÞ molecular orbitals (occupation numbers: 1.93 and 3.82, respectively) to the chemical bond are also larger than that from the dg ðddÞ (occupation number: 3.56) orbitals. The contributions from the rg ðsÞ and dg ðddÞ orbitals to the dimer chemical bonding increase as one goes down in the VIB transition metals (rg ðsÞ : 1; 90; 1:91; 1:93; dg ðddÞ : 3:16; 3:52; 3:56, in Cr2, Mo2, and W2, respectively), as a result of the contraction of the ðn þ 1Þs shell and increasing diffuseness of the nd shells, resulting in orbitals of more equal size. Thus, contribution from the d-electron becomes more effective, and the chemical bonding stronger [2] (EBO ¼ 3:5; 5:2; and 5.2, for Cr2, Mo2, and W2, respectively). The X1 Rþ g dissociation energy was computed to be 5.37 eV (Table 2), being the highest among the VIB transition metal dimers (Cr2 1:5 eV, Mo2 4:41 eV) [2].
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A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28
Table 3 Dominant configurations, leading excitations, and Mulliken population analysis for the low-lying electronic states of W2 at the CASSCF level. State X
1
Rþ g
c
ð67Þj1r ð71Þj1r
b3 Rþ u
ð50Þj1r
A1 Du
ð74Þj1r
c Cu
b
4 2 g 1dg 2 3 2 g 1dg 2 3 2 1d g2 g 3 2 1d g2 g 3 2 g 1dg 2 2 2 g 1dg 2 2 2 1d g2 g 3 2 1d g2 g 2 2 g 1dg 2 4 1 g 1dg 2 4 2 1d g2 g
a3 Du
3
a
Configurationa
ð75Þj1r
B1 Rþ g
ð48Þj1r
C1 Cg
ð48Þj1r
D1 R u
ð77Þj1r
E1 Dg
ð41Þj1r
F1 Rþ u
ð61Þj1r
G1 Rþ u
ð46Þj1r
Leading excitationb 2 g1
r p r2g 1p4u 2r1u i r2g 1p4u 1d1u i r2g 1p4u 2r1u i r2g 1p4u 1d1u i r2g 1p4u 2r2u i r2g 1p4u 2r2u i r2g 1p4u 1d1u i r2g 1p4u 1r1u 1d1u i r2g 1p4u 2r1u i r1g 1p4u 2r1u i
EBOc
Mulliken population
5.2
5d
5:00
1dg ! 2ru
4.4
5d
4:47
6s1:30
1dg ! 1du
4.1
5d
4:90
6s0:98
1dg ! 2ru
4.4
5d
4:46
6s1:28
1dg ! 1du
4.4
5d
4:89
6s1:02
1d2g 1d2g
2 u
! 2r
3.5
5d
4:29
6s1:37
! 2r2u
3.4
5d
4:14
6s1:50
1dg ! 1du
4.5
5d
4:87
6s1:02
1d2g ! 1r1u 1d1u
3.6
5d
4:43
6s1:30
1rg ! 2ru
4.2
5d
4:52
6s1:22
2rg ! 2ru
4.2
5d
4:52
6s1:22
4 ui
6s1:00
Total weight (%) in parentheses. With respect to the ground state configuration. Effective bond order.
Fig. 2. Valence natural molecular orbitals (VMO) of W2. The numbers correspond to the natural occupation numbers in the W2 X1 Rþ g electronic state.
3.2. The excited electronic states The a3 Du electronic state is the lowest triplet excited state, placed 7397 cm1 above the X1 Rþ g state with Re ¼ 2:089 Å and xe ¼ 329 cm1 (Table 2). The dominant configuration is 0:84j1r2g 1d3g 2r2g 1p4u 2r1u i (Table 3), derived from the leading
configuration of the ground state by the single excitation 1dg ! 2ru , which transfers one electron from the bonding 1dg molecular orbital to the 2ru antibonding orbital. The natural occupation numbers for the active orbitals (Table 4) give an effective bond order of 4.4. In comparison to the other VIB transition metal group diatoms, a low-lying 3 Du for Cr2 was computed by Andersson [22] at the CASPT2 level at 17 341 cm1 ðRe ¼ 2:13 Å , DG1=2 ¼ 230 cm1 Þ above the ground state, being derived from the ground state by a dg ! ru single excitation. To the best or our knowledge, there is no mention to a low-lying 3 Du state for Mo2 [2]. The electronic X1 Rþ transition 3 Du g is unlikely to be observed because it is both electric dipole and spin forbidden. 1 above the The b3 Rþ u electronic state is located 8547 cm ground state, with an equilibrium internuclear distance of 2.111 Å and xe ¼ 300 cm1 (Table 2). Its multiconfigurational wavefunction (Table 3) is dominated by the configuration, 0:71j1r2g 1d3g 2r2g 1p4u 1d1u i. The b3 Rþ u is derived from the ground state by a single excitation (1dg ! 1du ). The EBO is 4.1. A low-lying 3 Rþ u is also found in Cr2 and Mo2 diatoms. Experimentally, Pellin and Gruen located a 3 Rþ u in Cr2 [23] at about 8000 cm1 . Andersson [22], at the CASPT2 level, places a 3 Rþ u state at 4517 cm1 above the ground state. As to the Mo2 molecule, the experimental study by Kraus et al. [4] placed a low-lying 3 Rþ u state 7800 cm1 above the ground state. Balasubramanian and Zhu [24], at the first-order configuration interaction level, found the corresponding state at 5499 cm1 , while at the CASPT2 level 1 above the ground we found a low-lying 3 Rþ u [2] at 8912 cm state. Another common aspect of the low-lying 3 Rþ u state of the Cr2 [22], Mo2 [2], and W2 (Table 3) diatoms is that their wavefunctions have a high contribution from the single excitation (dg ! du ), around the equilibrium internuclear geometry. The low-lying 3 Rþ u electronic state of the Cr2 and Mo2 diatoms corre5 lates with the two corresponding ground state 7 Sðnd ðn þ 1Þs1 Þ atoms, the same separated atom limit of the molecular ground 3 þ state (X1 Rþ g ). For W2, the low-lying Ru and the ground electronic 5 states correlate with the 7 S3 ð5d ð6 SÞ 6s1 Þ atomic limit. Therefore, X1 Rþ for the VIB transition metal group diatoms the 3 Rþ u g electronic transition corresponds to a redistribution of electrons within the same compact atomic d-shell. The A1 Du is located 11 208 cm1 above the X1 Rþ g state. With Re ¼ 2:079 Å and xe ¼ 350 cm1 (Table 2), its multiconfigurational wavefunction is dominated by the electronic configuration 0:86j1r2g 1d3g 2r2g 1p4u 2r1u i, originating from the ground state by the excitation 1dg ! 2ru (Table 3). The EBO is 4.4. According to the computation carried out by the CASSCF//CASPT2 approach [22],
27
A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28 Table 4 Natural orbital occupation numbers for the low-lying electronic states of W2 at the CASSCF level. State
Natural orbitals occupation numbers 1rg ðdrÞ
1pu ðdpÞ
1dg ðddÞ
2rg ðsÞ
1ru ðdrÞ
1pg ðdpÞ
1du ðddÞ
X1 Rþ g
1.88
3.82
3.56
1.93
0.13
0.18
0.42
0.08
a3 Du b3 Rþ u
1.88 1.65
3.92 3.76
2.71 2.84
1.87 1.88
0.10 0.12
0.18 0.24
0.34 1.16
1.02 0.34
A1 Du c3 Cu B1 Rþ g
1.89
3.82
2.71
1.97
0.10
0.18
0.30
1.02
1.89 1.68
3.83 3.88
2.78 2.04
1.92 1.92
0.09 0.10
0.17 0.22
1.20 0.60
0.12 1.65
C1 Cg
1.84
3.78
1.80
1.96
0.10
0.20
0.44
1.85
D1 R u E1 Dg
1.89 1.81
3.80 3.74
2.83 2.02
1.93 1.93
0.09 0.13
0.16 0.24
1.12 1.00
0.12 1.10
F1 Rþ u
0.99
3.79
3.17
1.96
0.07
0.21
0.53
0.99
G1 Rþ u
0.99
3.79
3.47
1.96
0.07
0.21
0.53
0.99
Cr2 also exhibits a low-lying 1 Du ðTe ¼ 19 357 cm1 , Re ¼ 2:06 Å, DG1=2 ¼ 180 cm1 ), but we did not find it in Mo2 [2]. At ðT e Þ16 873 cm1 above the ground state, we computed the c3 Cu ðRe ¼ 2:078 Å, xe ¼ 342 cm1 ) electronic state (Table 2), with a wavefunction dominated by the 0:86j1r2g 1d3g 2r2g 1p4u 1d1u i electronic configuration, derived from the ground state by a single excitation (1dg ! 1du ) (Table 2), and active orbital occupation numbers given in Table 4. The corresponding EBO is 4.4. Again, the W2 c3 Cu excited state has its counterpart in the Cr2 and Mo2 diatoms. For Cr2, Anderson [22] computed a 3 Cu state ðT e Þ19 438 cm1 above the ground state, with Re ¼ 1:78 Å, DG1=2 ¼ 420 cm1 , and wavefunction dominated by an electronic configuration best represented by the 3ddg ! 3ddu single excitation from the ground state. As to the Mo2 diatom, in our previous work [2] we placed the c3 Cu ðRe ¼ 2:010 Å, xe ¼ 447 cm1 Þ T e ¼ 15 671 cm1 above the X1 Rþ g ground state, in agreement with the results obtained by Balasubramanian and Zhu [24] ðT e ¼ 12 947 cm1 , Re ¼ 2:100 Å, xe ¼ 477 cm1 Þ at the MRSDCI (multireference single and doubles configuration interaction) level; the wavefunction is best represented by the electronic configuration j9r2g 2d3g 10r2g 5p4u 2d1u i [2,24]. 1 above the The B1 Rþ g excited state (Table 2) is placed 18 083 cm ground state, with Re ¼ 2:173 Å and xe ¼ 306 cm1 , its wavefunction is dominated by the 0:69j1r2g 1d2g 2r2g 1p4u 2r2u i electronic configuration, differing from the ground state by a double excitation 1d2g ! 2r2u , which transfers two electrons from a bonding to an antibonding molecular orbital. Therefore, in comparison to the ground state, the equilibrium internuclear distance increases by 0.16 Å, reflecting the counterbalance of the 2rg and 2ru populations. From the active orbital occupation numbers (Table 4) one gets an EBO of 3.5, almost two units less than that for the ground state. For Cr2 and Mo2, there is no mention of the existence of a low-lying 1 Rþ g state in the energy region considered by Anderson [22] and by us [2], respectively. Nonetheless, Balasubramanian 1 ðRe ¼ and Zhu [24] reported a 1 Rþ g state for Mo2 22 019 cm
2ru ðsÞ
2:146 Å, xe ¼ 381 cm1 ) higher than the ground state, computed at the first-order configuration interaction (FOCI) level. The C1 Cg ðRe ¼ 2:174 Å and xe ¼ 302 cm1 Þ state computed by us at 18 904 cm1 (Table 2) above the X1 Rþ g ground state is isoconstate described above. As expected, its figurational with the B1 Rþ g wavefunction is very similar to that for the B1 Rþ g state, with active orbital occupation numbers (Table 4) resulting in an EBO of 3.4. In this energy region, there is no evidence for a 1 Cg state in Cr2 [22] and Mo2 [2]. The next two excited states (Table 2) are the D1 R u ðRe ¼ 2:075 Å and xe ¼ 370 cm1 Þ and E1 Dg ðRe ¼ 2:161 Å and xe ¼ 301 cm1 Þ states located, respectively, 19 268 cm1 and 19 768 cm1 above the ground state. Around the equilibrium internuclear distance, the D1 R u state is best described by a wavefunction dominated by the 0:88j1r2g 1d3g 2r2g 1p4u 1d1u i electronic configuration, which differs from the ground state by the 1dg ! 1du single excitation. From the corresponding active orbital occupation numbers (Table 4) the EBO is 4.5. As for the E1 Dg state, the wavefunction is dominated by the 0:64j1r2g 1d2g 2r2g 1r1u 1p4u 1d1u i electronic configuration, derived from the ground state wavefunction by a double excitation: 1d2g ! 1d1u 1r1u . The corresponding active orbital occupation numbers (Table 4) result in an EBO of 3.6. In comparison to the other dimers of the VIB transition metal group, we found a counterpart for the D1 R u state only in Mo2, computed by Balasubramanian and Zhu [24] at the FOCI level to be 17 924 cm1 above the ground state, with Re ¼ 2:093 Å and xe ¼ 485 cm1 . 1 þ The last two states computed by us are the F1 Rþ u and G Ru elec1 state is placed 20 534 cm above tronic states (Table 2). The F1 Rþ u the ground state, with an equilibrium internuclear distance of 2:111 Å and xe ¼ 285 cm1 , a wavefunction dominated by the 0:78j1r1g 1d4g 2r2g 1p4u 2r1u i electronic configuration. The EBO is 4.2. The G1 Rþ u state is the last excited state included in this work, computed to be 25 249 cm1 above the ground state, with Re ¼ 2:021 Å and xe ¼ 480 cm1 . Around the equilibrium internuclear distance, its wavefunction is dominated by the 0:68j1r2g 1d4g 2r1g 1p4u 2r1u i elec-
Table 5 Electronic configurations, in terms of spin-free electronic states (coefficients in parenthesis), and spectroscopic constants for the lowest-lying electronic states of W2 with spin– orbit coupling.
X
Composition
xe
Re (Å)
(cm1)
Te (cm1)
DG0 (cm1)
DG1=2 (cm1)
ð1Þ0þ g
ð1:00ÞX1 Rþ g
2.009
381
189
373
ð2Þ3u ð3Þ2u
ð1:00Þa3 Du
2.088 2.086
359 362
6844 7677
177 178
353 355
ð4Þ1u ð5Þ0 u ð6Þ1u ð7Þ2u
ð0:80Þa3 Du þ ð0:20Þ A1 Du ð0:86Þb3 Rþ u
2.106
333
8615
166
326
ð1:00Þb3 Rþ u
2.111
331
9875
166
323
ð1:00Þa3 Du
2.091 2.082
367 369
10 175 12 053
180 182
358 361
ð0:20Þa3 Du þ ð0:80Þ A1 Du
28
A.C. Borin et al. / Chemical Physics Letters 490 (2010) 24–28
(DKH) approximation, has been employed to investigate the electronic structure and chemical bonding of the lowest-lying singlet and triplet electronic states of W2. The ground state of W2 is state (Re ¼ 2:010 Å, xe ¼ 354 cm1 , computed to be a X1 Rþ g De ¼ 5:37 eV), with three low-lying electronic states (a3 Du : T e ¼ 1 7397 cm1 , Re ¼ 2:089 Å, xe ¼ 329 cm1 ; b3 Rþ u : T e ¼ 8547 cm , 1 1 1 Re ¼ 2:111 Å, xe ¼ 300 cm , and A Du : T e ¼ 11 208 cm , Re ¼ 2:079 Å, xe ¼ 350 cm1 ). Comparison with the other VIB transition metal group dimers (Cr2 and Mo2) indicates a common pattern of electronic structure and spectroscopic properties. Seven low-lying X-states were computed: ð1Þ0þg , ð2Þ3u , ð3Þ2u , ð4Þ1u , ð5Þ0u , ð6Þ1u , and ð7Þ2u , with the ground state corresponding to the 1 þ ð1Þ0þ g ðX Rg Þ state. Acknowledgements A.C.B. acknowledges academic support by the CNPq (Conselho Nacional de Desenvolvimento Cientı´fico and Tecnológico) and FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo). J.P.G. is grateful to FAPESP for a graduate fellowship. B.O.R. acknowledges support from the vice chancellor of Lund University. The authors thank the Laboratório de Computação Cientı´fica Avançada (LCCA) of the Universidade de São Paulo for the services and computer time.
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Fig. 3. Potential energy curves, with spin–orbit coupling, for the lowest-lying singlet and triplet electronic states of W2.
[1] [2] [3] [4] [5] [6] [7] [8] [9]
tronic configuration. The EBO is 4.2 (Tables 2 and 3). It is interest1 þ ing to note that the F1 Rþ u and G Ru electronic states are placed in the 20 000 cm1 energy region, the A–X band system observed is known to exist for Mo2 [2] and Cr2.
[10] [11]
3.3. Spin–orbit coupling
[12]
Spin–orbit effects (SOC) were included following a procedure that has been used in other applications [1,17,25–28]. All CASSCF wavefunction described here were used as basis functions for the spin–orbit Hamiltonian, with the diagonal elements replaced by the corresponding CASPT2 energies. We have focused our attention on the seven lowest X-states (Table 5, Fig. 3): ð1Þ0þ g , ð2Þ3u , ð3Þ2u , 3 ð4Þ1u , ð5Þ0 u , ð6Þ1u , and ð7Þ2u . The a Du splits into the ð2Þ3u (the lowest in energy), ð3Þ2u , and ð6Þ1u X-states, while the b3 Rþ u splits into the ð4Þ1u and ð5Þ0 u X-states. The ð7Þ2u X-state is dominated by the A1 Du wavefunction. The ground state in the relativistic 1 þ scheme has X ¼ 0þ g ðX Rg Þ. Although SOC splits the excited states, it has little influence on the computed spectroscopic constants (Table 5). 4. Conclusions The CASPT2//CASSCF protocol, with extended atomic basis sets and inclusion of relativistic effects via the Douglas–Kroll–Hess
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