Configuration Interaction Study on the AlBr Molecule Including Spin

Article pubs.acs.org/JPCA

Configuration Interaction Study on the AlBr Molecule Including Spin−Orbit Coupling Xiaoting Liu, Dandan Shi, Shimin Shan, Peiyuan Yan, Haifeng Xu,* and Bing Yan* Jilin Provincial Key Laboratory of Applied Atomic and Molecular Spectroscopy, Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China ABSTRACT: High-level ab initio calculations on the ground and the excited states of aluminum monobromide (AlBr) have been carried out by utilizing the internally contracted multireference configuration interaction method plus Davidson correction (icMRCI+Q) method. The core−valence correlation (CV) correction and spin−orbit coupling (SOC) effect have been investigated in the calculations. The potential energy curves (PECs) of the 13 Λ−S states, as well as those of the 24 Ω states generated from the Λ−S states under the SOC effect, have been obtained. The spectroscopic constants of the bound states have been determined, which are in accordance with the available experiment results. The SOC induced predissociation mechanisms of the a3Π and A1Π states have been analyzed with the aid of the spin−orbit matrix element. The transition properties of 0+(2)-X0+, 1(1)-X0+ and 1(2)X0+ transitions are predicted, including the transition dipole moments (TDMs), Franck−Condon factors (FCFs), and the radiative lifetimes. Finally, the possibility of AlBr to be used for molecular laser cooling has been discussed based on our calculations.

1. INTRODUCTION Aluminum monohalides (AlX, X = F, Cl, Br, I) have been detected in astrophysical environments,1 and are also readily produced in the vapor phase by high temperature pyrolysis and from chemiluminescent reactions.2 In order to understand the physical and chemical processes involving these molecules, it is essential to investigate the properties of their electronic states. Recently, aluminum monohalides have attracted increasing research interest because of their potential application in laser cooling of molecules. The significant development in laser cooling reported by Shuman et al.,3 in which the SrF molecule has been directly cooled by Doppler and Sisyphus cooling for the first time, has stimulated the search for other molecules to be candidates for molecular laser cooling. Di Rosa4 conducted a brief survey of potential laser cooling candidates and the list of candidates included aluminum compounds. Wells and Lane5 performed ab initio studies on electronic states of AlH and AlF, and predicted that the A1Π-X1Σ+ transition of both radicals can be a strong candidate for laser cooling. It is well-known that there are two criteria for a molecular transition to be used for laser cooling, i.e., highly diagonal FCFs to suppress decays to unwanted sublevels and short lifetime of the excited states, both of which require accurate ab initio calculations on the electronic states and transition properties of the possible molecular candidates. Here, we report a comprehensive high level ab initio study on the electronic excited of aluminum monobromide, AlBr. To date, the majority of the studies of AlBr reported in the literature were focused on the ground state X1Σ+ and the lowest two excited states a3Π and A1Π. The information about higherenergy electronic states is quite limited. In addition, the core− © XXXX American Chemical Society

valence correlation (CV) correction and spin−orbit coupling (SOC) effect, which play vital roles in electronic states of molecules containing heavy halogen atoms (such as bromine), has never been considered in previous studies on AlBr. Early experiments6−10 observed spectrum of the A1Π-X1Σ+ transition of AlBr. It was indicated that the A1Π state is a predissociative state, since only 0−3 vibrational levels of the A1Π state were observed in the spectra. However, the mechanism of the predissociation of the A1Π state is still unclear. William and Bredohl6,11 observed spin-forbidden transition a3Π-X1Σ+, and fitted spectroscopic constants of 3Π0 and 3Π1, the spin-mixed components of a3Π states. By utilizing averaged complete active space self-consistent field (CASSCF) method, Langhoff et al.12 reported the spectroscopic constants of X1Σ+ and A1Π states, and also provided an approximate radiative lifetime for the A1Π state of AlBr. Hamade et al.13 performed the CASSCF and multireference configuration interaction (MRCI) calculations on 12 spin-free electronic states for the AlBr molecule. Very recently, Gao et al.14 reported a MRCI calculation on the X1Σ+, A1Π, and a3Π states of AlBr, and predicted it as a possible candidate for direct laser cooling. However, the accurate and sufficient information on the fine-structures of the electronic states could not be obtained without considering the CV and SOC effects in AlBr. In our study, we performed an internally contracted MRCI method plus Davidson correction (icMRCI+Q) to compute the PECs of the 13 Λ−S states for the AlBr molecule. The CV Received: June 27, 2016 Revised: October 18, 2016 Published: October 21, 2016 A

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A corrections were taken into account in present paper, and the SOC between the interacting Λ−S states were evaluated with the help of the spin−orbit matrix elements. The SOC effect leads to splitting of the 13 Λ−S states into 24 Ω states. The transition dipole moments (TDMs), Franck−Condon factors (FCFs), and the radiative lifetimes of the several lowest transitions were also predicted. The possibility of AlBr for laser cooling is discussed based on the calculated results.

The SOC matrix elements are calculated with Breit−Pauli spin−orbit operator employing the state-interacting technique.33−35 The off-diagonal SO matrix elements are calculated with the MRCI wave functions, while the diagonal SO matrix elements are substituted by the energies of the MRCI+Q calculation. The energies including the SOC effect are obtained by the diagonalization of the SO matrix. The SOC effect makes the 13 Λ−S electronic states of AlBr split into 24 Ω states. The PECs of the Ω states are plotted with the help of avoided crossing rule. On the basis of the Λ−S PECs, the spectroscopic constants, including equilibrium internuclear distance Re, excitation energy Te, vibrational constants ωe and ωexe, rotation constant Be, and vibrational−rotational coupling constant αe, were determined by numerical solution of the one-dimensional nuclear Schrödinger equation for 27Al79Br with the aid of the LEVEL program.36 The dissociation energy De and D0 were obtained by subtracting the molecular energy at Re from the energy at a large separation. The spin−orbit matrix elements of a number of electronic states were calculated at the CASSCF level, while the TDMs and the permanent dipole moments (PDMs) were evaluated at the icMRCI level. The FCFs were calculated with the aid of the LEVEL program.36 The radiative lifetimes for several of the lowest energy transitions were predicted based on the calculated TDMs and FCFs.

2. METHODS All of electronic structure calculations of the AlBr molecule were performed with the MOLPRO 2012 quantum chemistry package.15,16 The symmetry point group of AlBr is the C∞ν point group. Because of the limitation of the MOLPRO program, the C2v point group symmetry was chosen in the present calculation. The C2v point group holds A1/B1/B2/A2 irreducible representations, and the corresponding relationships between the C2v and C∞ν point group are Σ+ = A1, Π = B1 + B2, Δ = A1 + A2, and Σ− = A2. For the AlBr, 4a1, 2b1, and 2b2 symmetry molecular orbitals (MOs) were determined as the active space, corresponding to the atomic orbitals 3s3p for Al and 4s4p for Br. The outermost 3s23p1 electrons of the Al atom and 4s24p5 electron of the Br atom were placed in the active space. Additional electrons were put into closed orbitals and correlated via single and double electron excitation in the subsequent icMRCI procedure. Generally, the 10 electrons in the 3d shell of Br were placed in the closed shell. While in the test calculations, we found the PEC of ground state became unsmooth in the short internuclear distance when only 3d shell was put into closed orbital. After further examination of molecular orbitals, we found the 3d orbital of Br atom and 2p orbital of Al atom partially reorder in the short internuclear distance. Thus, to overcome the problem of MOs reordering and to balance accuracy and computational cost, the 3d orbital of Br and 2p orbital of Al were put into closed orbital and correlated in electronic calculations. That is, there are a total of 26 electrons in AlBr included in the calculation of electronic correlation energy. The adiabatic PECs of the ground state and excited states of AlBr were constructed from 38 single point energies corresponding to internuclear distances from 1.8 to6.0 Å. In these calculations, the Gaussian-type contracted basis set augcc-pwCVQZ-DK17,18 was selected for the Al atom and the Br atom. The calculations were performed via the following three steps: the single-configuration wave function of the ground state for the AlBr was first calculated with the restricted Hartree−Fock (RHF) method; then, the state-averaged complete active space self-consistent field (SA-CASSCF) method19,20 was used to construct the multiconfiguration wave function that could consider the degeneracy or near degeneracy of the states; finally, on the basis of the optimized reference wave function determined from the SA-CASSCF calculation, the internally contracted icMRCI approach21−23 is employed to calculate the correlation energies of the electronic states. At the same time, the calculation is extended to include the Scalar relativistic effect with the aid of the one electron integral third-order Douglas-Kroll integrals.24−28 Adding the Davidson correction29 (+Q) balances the size-consistency error of icMRCI method. The PECs of these 13 Λ−S electronic states were drawn with the help of the avoided crossing rule of the states that have the same symmetry. The SOC effect is introduced as perturbation via a two-step procedure which is used in previous theoretical works.30−32

3. RESULTS AND DISCUSSION 3.A. PECs, Spectroscopic Constants, and PDMs of the Λ−S states. The 13 Λ−S states of AlBr, including 7 singlet states and 6 triplet states, were studied by using the icMRCI+Q method. These states are associated with two dissociation limits i n c l ud i n g n e u t r a l A l ( 2 P u ) + B r ( 2 P u ) a n d i o n - p a i r Al+(1Sg)+Br−(1Sg), as listed in Table 1. In contrast to the Table 1. Dissociation Relationships of the Λ−S States of AlBr Λ−S states

atomic state 2

2

Al( Pu) + Br( Pu) Al+(1Sg) + Br−(1Sg)

1 −

X Σ , Σ (2), Σ , A Π, 1Π(2), 1Δ, 3Σ+(1), 3Σ+(2), 3 − 3 Σ , a Π, 3Π(2), 3Δ 1 + Σ (3) 1 + 1 +

1

neutral atomic dissociation limits, the ion-pair dissociation limit Al+(1Sg)+Br−(1Sg) corresponds to the one-electron transfer from Al to Br. And only the 1Σ+(3) state is associated with the ion-pair limit. The adiabatic PECs of the 13 Λ−S states were constructed from 38 single point energies corresponding to internuclear distances from 1.8 to 6.0 Å and were depicted in Figure 1. From these calculated PECs, it is found that three singlet states X1Σ+, A1Π, 1Σ+(3) and one triplet state a3Π are bound or quasi-bound states; and the 1Π(2), 1Σ−, 1Δ, 3Π(2), 3Σ−, 3Δ states are typically repulsive states. The ground state X1Σ+ of the AlBr molecule is mainly characterized by the closed-shell electronic configuration 10σ211σ212σ213σ05π46π0 (84.06%). The 10σ and 11σ molecular orbitals mainly correspond to the 3s (Al) and 4s (Br) atomic orbitals, while the 12σ and 13σ orbitals to the 3pz and 4pz orbitals. The molecular 5π and 6π orbitals correspond to the atomic 3px + 3py (Al) and 4px + 4py (Br) orbitals. The calculated PEC of the first excited state A1Π state confirms the presence of a potential barrier against dissociation into Al(2Pu) + Br(2Pu). The wave function of A1Π is mainly described by the electronic configuration B

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

a very important step in the CASSCF and icMRCI+Q computations. In our study, we first calculated the spectroscopic constants of the ground state X1Σ+ using different basis sets of aug-cc-pVTZ-DK, aug-cc-pVQZ-DK, and aug-ccpwCVQZ-DK which includes the CV correction. The results show the convergence tendency of the basis sets, and indicate that the CV correction has a non-negligible effect on the accurate calculation of the spectroscopic constants of AlBr. As shown in Table 2, our calculated spectroscopic constants of the X1Σ+ state, as well as those of the a3Π and A1Π states, well reproduce the results of recent available experimental measurements.37 Neither theoretical nor experimental studies have been reported in the literature regarding the 1Σ+(3) state, which is the only state of the 13 Λ−S states that correlates to the ionpair limit Al+(1Sg)+Br−(1Sg). The 1Σ+(3) state has very large equilibrium internuclear distance of 4.9990 Å and the excitation energy Te is calculated to be 39762.0 cm−1. The wave function of the 1Σ+(3) state exhibits the obvious multiconfiguration character, consisting of the two important electronic configurations 10σ211σ212σ213σ05π46π0 (55.6%) and 10σ211σ212σα13σβ5π46π0 (30.3%). The presented results calculated at the icMRCI+Q/aug-cc-pwCVQZ-DK level provide an accurate prediction of the spectroscopic constants, which will lead to future experimental studies on the excited electronic states of the AlBr molecule. The PDMs of the 13 Λ−S states are calculated in the internuclear distance range from 2.0 to 6.0 Å and the curves are plotted in Figure 2. As shown in Figure 2, except for that of the 1 + Σ (3) state, all the calculated PDMs lead to consistent asymptotic limit close to 0 au This is because of the center of positive charges with the position vector |λ| = 0, which is corresponding to the dissociation limit of Al + Br. The PDM of the 1Σ+(3) state, is negative and shows linear dependence on R at large bond distance because of the center of negative charges with the position vector |λ| = R, demonstrating that the dissociation limit of the 1Σ+(3) state is Al+ + Br− instead.

Figure 1. PECs of the Λ−S states.

10σ211σ212σα13σ05π46πα. The electronic configuration indicates that the A1Π state corresponds to the one-electron 13σ → 6π excitation as in the X1Σ+ state. By solving the nuclear Schrödinger equation, the spectroscopic constants of the bound states are determined and listed in Table 2. For comparison, the available experimental and other theoretical results are also listed in the table. The selection of the basis set and number of correlated electrons are

Table 2. Computed Spectroscopic Constants of the Λ−S States of AlBr Λ−S state XΣ

1 +

a3Π

A1Π

Σ (3)

1 +

Te (cm−1)

Re (Å)

ωe (cm−1)

ωexe (cm−1)

Be (cm−1)

102αe (cm−1)

De (cm−1)

D0 (cm−1)

reference

0 0 0 0 0 0 0 0 24836.8 23025 23308 23801 37183.3 36396 36418 36450 35879.5 39762.0

2.3205 2.3115 2.289 2.31 2.306 2.32 2.30 2.294 2.2485 2.27 2.263 2.25 2.358 2.32 2.325 2.34 2.322 4.9990

370.488 374.771 382.693 381 378.72 377 382 378 405.710 413 409.46 403 271.492 333 314.59 316 297.2 156.324

1.2810 1.3163 1.0657 1.5 1.20

0.1552 0.1565 0.1595

0.075805 0.091982 0.077344

36210.9 37147.1 36877.2

36026.0 36960.0 36686.1

0.086044 0.105551

12444.3

12241.9

0.310005

909.9

777.4

0.216 0.016 75

1195.3

1118.6

this worka this workb this workc theory13 theory14 theory12 theory38 expt37 this workc theory13 theory14 expt37 this workc theory13 theory14 theory38 expt37 this workc

0.1577

1.28 1.9149 2.0 1.72 2.1 12.8125 4.8 7.61

0.1592 0.1653

6.40 5.6699

0.1555 0.0336

0.1637 0.1505 0.1551

a c

MRCI+Q/aug-cc-pVTZ-DK values with valence electrons correlated. bMRCI+Q/aug-cc-pVQZ-DK values with valence electrons correlated. MRCI+Q/aug-cc-pwCVQZ-DK values with valence, 3d of Br, and 2p of Al electrons correlated. C

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

(ν′ = 0−3) can be held in the well of the PEC, in accord with the experimental results.6,8,10 We computed the SO matrix elements, which could be used for a qualitative analysis on the predissociation of the a3Π and A1Π states induced by the spin−orbit coupling between singlet and triplet states. The calculated SO matrix elements as a function of the internuclear distance are plotted in Figure 4.

Figure 2. Permanent dipole moments of Λ−S states for the AlBr molecule.

B. Spin−Orbit Coupling of the Λ−S States. There are two curve-crossing regions for the PECs of the excited Λ−S states at energy of 36500−38000 cm−1. The first one is between the a3Π state and the 3Σ+(1), 1Δ, 3Δ, 3Σ−, 1Σ−, and 1Σ+(2) states in the R range of 3.5−4.5 Å; the other one is between the A1Π state and the 3Σ+(1), 1Δ, 3Δ, 3Σ−, 1Σ−, and 1Σ+(2) states in the R range of 2.75−3.5 Å. An expanded view of the crossing regions together with the vibrational levels of the a3Π and A1Π states are given in Figure 3. As shown in Figure 3, the curve crossings occur above the ν′ = 38 vibrational level of the a3Π state, indicating that the vibrational levels with ν′ < 38 could be long-lived because of being free from any interaction with other states. For the A1Π state, however, only four vibrational levels

Figure 4. Spin−orbit matrix elements of several Λ−S states. See eq 1 for the definition of the SOi symbols.

The SOi operator is represented in the basis of the real spinelectronic functions labeled as |term>, as in the case of CaO studied by Khalil et al.39 BP 3 + BP 3 SO1 = ⟨A1Π |HSO | Σ (1)⟩y SO2 = −i⟨A1Π |HSO | Δ⟩x

BP 3 − BP 3 SO3 = −i⟨A1Π |HSO | Σ ⟩x SO4 = −i⟨A1Π|HSO |a Π⟩z BP 3 BP 1 − SO5 = −i⟨a 3Π |HSO | Δ⟩x SO6 = −i⟨a 3Π |HSO | Σ ⟩x

BP 1 BP 3 − SO7 = −i⟨a 3Π |HSO | Δ⟩x SO8 = −i⟨a 3Π |HSO | Σ ⟩x BP 3 + BP 1 + SO9 = ⟨a 3Π |HSO | Σ (1)⟩y SO10 = ⟨a 3Π |HSO | Σ (2)⟩y

(1)

The SO matrix element values of A1Π-3Σ+(1) (SO1), A1Π-3Δ (SO2), and A1Π-3Σ− (SO3) are evaluated to be about 780, −600, and 1200 cm−1 at their respective crossing points, which are sufficient to induce predissociation of the A1Π state via coupling with the triplet states. It is noted that the A1Π-3Δ and A1Π-3Σ− curve-crossings lie at the repulsive section of the PEC of the A1Π state (Figure 3). Thus, the predissociation of the A1Π state is mainly through the channel A1Π → 3Σ+(1) at energy just above the ν′ = 3 level, the predissociation of the A1Π state is mainly through the channel A1Π → 3Σ+(1) at energy just above the ν′ = 3 level (at R ≅ 3.00 Å). This is consistent with the experimental results in which no vibrational

Figure 3. Expamded view of the crossing regions together with the vibrational levels of the a3Π and A1Π states. D

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A Table 3. Dissociation Relationships of the Ω States of AlBr atomic state

Ω states

Al(2P1/2) + Br(2P3/2) Al(2P3/2) + Br(2P3/2) Al(2P1/2) + Br(2P1/2) Al(2P3/2) + Br(2P1/2) Al+(1S0) + Br−(1S0)

2(1), 1(1), 1(2), X0+, 0−(1) 3(1), 2(2), 2(3), 1(3), 1(4), 1(5), 0+(2), 0+(3), 0−(2), 0−(3) 1(6), 0+(4), 0−(4) 2(4), 1(7), 1(8), 0+(5), 0−(5) 0+(6)

Figure 5. PECs of the Ω states.

Table 4. Computed Spectroscopic Constants of Ω States of AlBr Ω state

Te (cm−1)

Re (Å)

ωe (cm−1)

ωexe (cm−1)

Be (cm−1)

102αe (cm−1)

De (cm−1)

D0 (cm−1)

reference

+

0 24641.6 24654.7 23647 24823.9 23779.3 25008.2 36988.7

2.2890 2.2485 2.2485 2.26 2.2500 2.26 2.2515 2.3805

383.147 406.329 406.223 411.2 404.641 410.32 403.536 224.390

1.1315 2.0067 1.9696 1.75 1.9953 1.75 2.0291 22.8617

0.1596 0.1653 0.1645

0.092463 0.096069 −0.075390

35797.6 11338.2 11401.9

35606.3 11135.5 11199.3

0.1651 0.1645 0.1649 0.1491

0.091217 0.1 0.091325 0.832133

11142.7

10940.9

10958.4 97.3

10757.1 ―

this worka this worka this worka expt37 this worka expt37 this worka this worka

X0 0−(1) 0+(2) 1(1) 2(1) 1(2) a

icMRCI + Q + CV + SOC value.

Br(2Pu) and Al+(1Sg)+Br−(1Sg) split into five asymptotes, and the detailed relationships between the dissociation limits and Ω states are listed in Table 3. According to the avoided crossing rule between the states of the same symmetry, the PECs of these 24 Ω states are depicted in Figure 5. The spectroscopic constants of the bound Ω states are determined and tabulated into Table 4. And Figure 6 shows the R-dependent Λ−S components in the wave functions of the bound Ω states. It can be seen from Figure 6 that the ground Ω state X0+ is completely composed of the X 1 Σ + state. Hence, the spectroscopic constants of X0+ are almost the same as those of the Λ−S ground state X1Σ+ (see Table 4). The De and D0 values of the ground state X0+ are 413.3 and 419.7 cm−1 lower than that of the X1Σ+ state after considering the SOC, due to the SOC splitting of atomic ground states of Al and Br. The PECs for the four Ω components of the a3Π state, 0−(1), 0+(2), 1(1), and 2(1), have well-depths of 11338.2, 11401.9, 11142.8,

levels higher than ν′ = 3 were observed in the spectra.6,8,10 The absolute values of SO matrix element involving the a3Π state are about 600−1200 cm−1 at the corresponding curve-crossing points. This suggests predissociation of the a3Π state via SOC at energy above ν′ = 38. The corresponding channels would be more complicated than that of the A1Π state, which are induced by the coupling of the a3Π state with various electronic states including 3Σ+(1) (at ν′ = 38), 3Δ and 1Δ (at ν′ = 41) and 3Σ−, 1 − Σ and 1Σ+(2) (at ν′ = 43). Our study indicates that the SOC effect is of significant importance in understanding the behavior of the electronic excited states of AlBr. C. PECs and Spectroscopic Constants of the Ω States. Taking the SOC effect into account, a Λ−S state may split into different Ω states. In the case of AlBr, the 13 Λ−S states studied in this study split into 24 Ω states due to the SOC effect, including six Ω = 0+, five Ω = 0−, eight Ω = 1, four Ω = 2 and one Ω = 3. The original dissociation limits Al(2Pu) + E

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

Figure 7. Transition dipole moments of AlBr.

X0+ are from the spin forbidden transition a3Π-X1Σ+. The spin forbidden transitions are evaluated according to the formulas given by Minaev et al.40−42 However, the absolute TDM values of 1(2)-X0+ reduce quickly as R is increased. As shown in Figure 6, the Λ−S component of the 1(2) state changes from the singlet A1Π state to the triplet state 3Σ+(1) at ∼2.7 Å, leading to the decrease of the transition probability from the singlet ground state. The FCFs of the 0+(2)-X0+, 1(1)-X0+, and 1(2)-X0+ transitions are calculated with the aid of the LEVEL program36 and listed in Table 5. The radiative lifetime τ of the vibrational level ν′ for a given state is defined as the inverse of the total transition probability

Figure 6. Λ−S compositions for the X0+, 0+(2), 1(1), 1(2), 0−(1), and 2(1) states.

and 10958.4 cm−1 respectively. Near the equilibrium bond lengths, the energy increases in the order of 0−(1), 0+(2), 1(1), and 2(1), and the largest splitting among the components of the a3Π state is computed to be only 366.6 cm−1. Our calculated spectroscopic constants of the 1(1) state are in good accordance with the existing experimental results,23 for example, the errors of Re, Be, and 102αe are within 0.4%, 0.36%, and 8.7%, respectively. As shown in Figure 6, the Λ−S components for the 0−(1), 0+(2), 1(1), and 2(1) states switch between the a3Π, 3Σ+(1), 1Δ, 3Δ, 3Σ−, 1Σ−, and 1Σ+(2) states around the internuclear distance 3.5−4.5 Å, strongly indicating the coupling between those Λ−S states. With consideration of the influence of the SOC effect, the Ω states that have higher excitation energies exhibit mixtures and interactions with each other more strongly. Their PECs have a very complicated structure due to a lot of avoided crossing points between various Ω states (see Figure 5). It should be pointed out that only one state of Ω = 3 is generated in the calculation, which purely arises from splitting of the 3Δ state. Therefore, the shape of PEC for the Ω = 3 is similar to that of the corresponding Λ−S state. For the rest Ω states at high energy, the shapes of PECs are very complicated and some of them (e.g., 0+(6), 0−(3), and 1(5)) have two or more shallow potential wells due to the SOC effect. On the other hand, the depths of these wells are rather small, which can support less than two vibrational levels. Thus, it is expected that the spectra of the AlBr in this energy range will be very diffuse and thus is hard to be observed experimentally. D. TDMs, FCFs and radiative lifetimes of AlBr. The TDMs of the transitions 0+(2)-X0+, 1(1)-X0+, and 1(2)-X0+ are calculated as a function of the internuclear distance R from 1.8 to 6.0 Å. The corresponding TDM curves have been depicted in Figure 7. In the Franck−Condon region from 2.25 to 3.0 Å, the absolute values of TDMs of 1(2)-X0+ are much larger than those of the transitions 0+(2)-X0+ and 1(1)-X0+. This is because the 1(2)-X0+ transition is mainly from the spin-allowed transition A1Π-X1Σ+, while the transitions 0+(2)-X0+ and 1(1)-

τ = (∑ A ν ′ ν ″)−1 (2)

ν″

The Einstein coefficient Aν′ν″ between vibrational levels ν′and ν″ is determined by A ν ′ ν ″ = 2.026 × 10−6ν 3̃ (TDM )2 qν ′ ν ″

(3)

−1

where ν̃ (in unit of cm ) is the energy difference between vibrational levels ν′ and ν″, TDM (in atomic unit) is the average electronic transition dipole moment in the region of classical turning point, qν′ν″ is the FCF of the two vibrational levels ν′ and ν″, and the radiative lifetime τ is in unit of second. The radiative lifetimes of the aforementioned transitions are listed in the Table 6. It can be seen from Table 6 that our calculated lifetimes for the ν = 0 and 1 levels of 1(2)-X0+ transitions are 5.74 and 6.29 ns, which are much smaller than the results of 0+(2)-X0+ and 1(1)-X0+ transitions. This is because the transition 1(2)-X0+ has larger TDMs in the Franck−Condon region. Since the A1Π state predissociates and the 1(2)-X0+ transition is mainly involves the spin-allowed transition A1Π-X1Σ+, we computed the line width of vibrational levels for quasibound 1(2) state to estimate the predissociation lifetimes, which are 0.232 ns for ν = 0 and 0.424 ps for ν = 1. With inclusion of spontaneous emission and predissociation, the total decay lifetimes of 1(2) state are 0.223 ns (ν = 0) and 0.424 ps (ν = 1). With the aid of the present icMRCI+Q calculations including the SOC effect, we now discuss the possibility of AlBr to be used for molecular laser cooling. The optically bright transition A1Π-X1Σ+ is corresponding to the 1(2)-X0+ transition in the Franck−Condon region after taking the SOC effect into account. The lifetimes of the vibrational levels are only several F

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A Table 5. Franck-Condon Factors of the Selected Transitions for the AlBr ν″ = 0

ν″ = 1

ν″ = 2

ν″ = 3

ν′ ν′ ν′ ν′

= = = =

0 1 2 3

0.827095 0.161926 0.010729 0.000249

0.151948 0.548202 0.270530 0.028408

0.018846 0.236571 0.351512 0.341053

ν′ ν′ ν′ ν′

= = = =

0 1 2 3

0.839662 0.151236 0.008933 0.000169

0.142036 0.577258 0.256454 0.023669

0.016551 0.224654 0.388399 0.327550

ν′ = 0 ν′ = 1

0.364720 0.249962

0.301379 0.009755

0.174205 0.033646

ν″ = 4

0+(2)-X0+ 0.001922 0.000172 0.045975 0.006471 0.276088 0.075341 0.215548 0.284770 1(1)-X0+ 0.001601 0.000138 0.040794 0.005388 0.265695 0.067342 0.254272 0.278939 1(2)-X0+ 0.086512 0.040088 0.102347 0.128050

ν″ = 5

ν″ = 6

ν″ = 7

ν″ = 8

0.000015 0.000766 0.013580 0.102960

0.00002 0.000081 0.001958 0.022648

0.000000 0.000008 0.000237 0.003788

0.000000 0.000000 0.000024 0.000515

0.000011 0.000602 0.011422 0.092733

0.000001 0.000062 0.001554 0.019194

0.000000 0.000006 0.000181 0.003036

0.000000 0.000000 0.000018 0.000391

0.018065 0.118532

0.008091 0.095885

0.003659 0.072797

0.000197 0.053787

Table 6. Radiative Lifetimes of the Transitions of the AlBr Moleculea radiative lifetime

a

transition

unit

ν′ = 0

ν′ = 1

ν′ = 2

ν′ = 3

0+(2)-X0+ 1(1)-X0+ 1(2)-X0+

μs μs ns

152 1.08 5.74(0.22)

152 1.07 6.29(0.42 × 10−3)

152 1.07

151 1.07

The values in parentheses are total decay lifetimes including predissociation lifetimes.

account, the PECs of the 24 Ω states generated from the 13 Λ− S states have been calculated. The transition properties of 0+(2)-X0+, 1(1)-X0+, and 1(2)-X0+ have been investigated, including the TDMs, FCFs, and radiative lifetimes, which are predicted for the first time. It is indicated that the AlBr is not a good candidate for laser cooling based on the present high-level calculations. Our study strongly suggest that, for molecules containing heavy atoms, the SOC effect should be considered in theoretical searching for laser cooling candidates, as well as in achieving deep insights on spectroscopy and dynamics of the electronic excited states.

nanoseconds (Table 6), indicating high spontaneous-emission rates that are desirable for rapid laser cooling. However, the Re value of the 1(2) state is as large as 0.09 Å longer than that of the ground X0+ state (Table 4), which means large difference in PECs of the states (see also Figure 5). Accordingly, the FCFs will not be highly diagonal. Indeed, as we have shown in Table 6, the calculate FCFs are highly off-diagonal for the 1(2)-X0+ transition, q00 = 0.364720 and q11 = 0.009755. Thus, it strongly indicates that, different to those of the other AlX molecules with a lighter X atoms (X = H, F) which have been predicted as candidates for laser cooling,5 the A1Π-X1Σ+ transition of AlBr is not suitable to be used for laser cooling. This result is opposite to that of the recent ab initio calculations,14 in which the calculated FCFs were diagonal without considering CV and SOC. Our study indicates the CV and SOC effects are essential to achieve accurate information on the molecular electronic states, especially for a molecule containing high-Z atoms; thus, they cannot be neglected in theoretical searching for laser cooling candidates.

■

AUTHOR INFORMATION

Corresponding Authors

*Telephone: 86-431-85168817. Fax: 86-431-85168816. E-mail: [email protected] (H.X.). *E-mail: [email protected] (B.Y.). Notes

The authors declare no competing financial interest.

■

4. CONCLUSION High-level icMRCI+Q calculations have been performed to obtain the PECs of the 13 Λ−S states of AlBr, which are associated with neutral atomic dissociation limit Al(2Pu) + Br(2Pu) and ion-pair dissociation limit Al+(1Sg) + Br−(1Sg). From the PECs, the corresponding spectroscopic constants of the bound states have been determined by solving the radial Schrödinger equation. The results are in good agreement with previous available experimental measurements. The PDMs of the 13 Λ−S states have been calculated at the region of R = 1.8−6.0 Å, which exhibit two asymptotic limits because of the different dissociation products of Al + Br and Al+ + Br− at large distance. Our calculations show that the A1Π and a3Π states cross with some other excited states, and the SOC coupling has the significant effect on the PECs′ shapes. The predissociation mechanisms of the A1Π and a3Π states are analyzed with aid of computed SO matrix elements. Taking the SOC effect into

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant Nos. 11574114, U1532138 and 11274140) and Natural Science Foundation of Jilin Province (Grant No. 20150101003JC).

■

REFERENCES

(1) Cernicharo, J.; Guélin, M. Metals in IRC+10216: detection of Nacl, AlCl and KCl, and tentative detection of AlF. Astron. Astrophys. 1987, 183, L10−L12. (2) Rosenwaks, S. Chemiluminescent reactions of Al atoms and halogens. J. Chem. Phys. 1976, 65, 3668−3673. (3) Shuman, E. S.; Barry, J. F.; Demille, D. Laser cooling of a diatomic molecule. Nature 2010, 467, 820−823. (4) Di Rosa, M. D. Laser-cooling molecules: Concept, candidates, and supporting hyperfine-resolved measurements of rotational lines in the A-X(0,0) band of CaH. Eur. Phys. J. D 2004, 31, 395−402. G

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A (5) Wells, N.; Lane, I. C. Electronic states and spin-forbidden cooling transitions of AlH and AlF. Phys. Chem. Chem. Phys. 2011, 13, 19018− 19025. (6) Griffith, W. B.; Mathews, W. Rotational analyses of the A1Π-X1Σ and the a3Π-X1Σ systems of AlBr. J. Mol. Spectrosc. 1984, 104, 347− 352. (7) Wolf, U.; Tiemann, E. Spectroscopy and potential inversion for the predissociated C1Π state of AlBr. Electronic structure of group IIIa halides. Chem. Phys. 1988, 119, 407−418. (8) Bredohl, H.; Dubois, I.; Mahieu, E.; Melen, F. New rotational analysis of the A1Π-X1Σ+ transition of AlBr. J. Mol. Spectrosc. 1991, 145, 12−17. (9) Uehara, H.; Horiai, K.; Ozaki, Y.; Konno, T. Infrared emission spectrum of AlBr. Chem. Phys. Lett. 1993, 214, 527−530. (10) Fleming, P. E.; Mathews, C. W. A Reanalysis of the A1Π-X1Σ+ Transition of AlBr. J. Mol. Spectrosc. 1996, 175, 31−36. (11) Bredohl, H.; Danguy, P.; Dubois, I.; Mahieu, E.; Saouli, A. The triplet states of AlBr. J. Mol. Spectrosc. 1992, 151, 178−183. (12) Langhoff, S. R.; Bauschlicher, C. W.; Taylor, P. R. Theoretical studies of AlF, AlCl, and AlBr. J. Chem. Phys. 1988, 88, 5715−5725. (13) Hamade, Y.; Taher, F.; Monteil, Y. Theoretical electronic studies of aluminum monobromide and aluminum monoiodide. Int. J. Quantum Chem. 2009, 110, 1030−1040. (14) Yang, R.; Tang, B.; Gao, T. Ab initio study on the electronic states and laser cooling of AlCl and AlBr. Chin. Phys. B 2016, 25, 043101. (15) Werner, H. J.; Knowles, P.; Knizia, G.; Manby, F. R.; Schütz, M.; Celani, P.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G. Molpro, version 2010.1, a package of ab initio programs; 2010. (16) Werner, H. J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. Molpro: a general-purpose quantum chemistry program package. WIREs Comput. Mol. Sci. 2012, 2, 242−253. (17) Peterson, K. A.; Dunning, T. H. Accurate correlation consistent basis sets for molecular core-valence correlation effects: The second row atoms Al-Ar, and the first row atoms B-Ne revisited. J. Chem. Phys. 2002, 117, 10548−10560. (18) DeYonker, N. J.; Peterson, K. A.; Wilson, A. K. Systematically Convergent Correlation Consistent Basis Sets for Molecular CoreValence Correlation Effects: The Third-Row Atoms Gallium through Krypton. J. Phys. Chem. A 2007, 111, 11383−11393. (19) Werner, H. J.; Knowles, P. J. A second order multiconfiguration SCF procedure with optimum convergence. J. Chem. Phys. 1985, 82, 5053−5063. (20) Werner, H. J.; Meyer, W. A quadratically convergent multiconfiguration-self-consistent field method with simultaneous optimization of orbitals and CI coefficients. J. Chem. Phys. 1980, 73, 2342−2356. (21) Knowles, P. J.; Werner, H. J. An efficient method for the evaluation of coupling coefficients in configuration interaction calculations. Chem. Phys. Lett. 1988, 145, 514−522. (22) Shiozaki, T.; Knizia, G.; Werner, H. J. Explicitly correlated multireference configuration interaction: MRCI-F12. J. Chem. Phys. 2011, 134, 034113−1−12. (23) Werner, H. J.; Knowles, P. J. An efficient internally contracted multiconfiguration−reference configuration interaction method. J. Chem. Phys. 1988, 89, 5803−5814. (24) Douglas, M.; Kroll, N. M. Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 1974, 82, 89− 155. (25) Peng, D.; Hirao, K. An arbitrary order Douglas-Kroll method with polynomial cost. J. Chem. Phys. 2009, 130, 044102. (26) Reiher, M.; Wolf, A. Exact decoupling of the Dirac Hamiltonian. I. General theory. J. Chem. Phys. 2004, 121, 2037−47. (27) Reiher, M.; Wolf, A. Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas-Kroll-Hess transformation up to arbitrary order. J. Chem. Phys. 2004, 121, 10945−10956. (28) Wolf, A.; Reiher, M.; Hess, B. A. The generalized Douglas-Kroll transformation. J. Chem. Phys. 2002, 117, 9215.

(29) Langhoff, S. R.; Davidson, E. R. Configuration interaction calculations on the nitrogen molecule. Int. J. Quantum Chem. 1974, 8, 61−72. (30) Alekseyev, A.; Liebermann, H. P.; Buenker, R. Spin-orbit multireference configuration interaction method and applications to systems containing heavy atoms. In: Hirao, K., Ishikawa, Y., editors. Recent advances in relativistic molecular theory; World Scientific: Singapore. 2004; pp 65−105. (31) Balasubramanian, K. Spectroscopic Properties and Potential Energy Curves for Heavy p-Block Diatomic Hydrides, Halides, and Chalconides. Chem. Rev. 1989, 89, 1801−1840. (32) Balasubramanian, K. Spectroscopic Constants and Potential Energy Curves of Heavy p-Block Dimers and Trimers. Chem. Rev. 1990, 90, 93−167. (33) Berning, A.; Schweizer, M.; Werner, H.-J.; Knowles, P. J.; Palmieri, P. Spin-orbit matrix elements for internally contracted multireference configuration interaction wavefunctions. Mol. Phys. 2000, 98, 1823−1833. (34) Pitzer, R. M.; Winter, N. W. Electronic-Structure Methods for Heavy-Atom Molecules. J. Phys. Chem. 1988, 92, 3061−3063. (35) Tilson, J. L.; Ermler, W. C. Massively parallel spin-orbit configuration interaction. Theor. Chem. Acc. 2014, 133, 1564. (36) Le Roy, R. J. LEVEL 8.0: A computer program for solving the radial Schrödinger equation for bound and quasibound levels; Research Report CP-663; University of Waterloo Chemical Physics: 2007. (37) Huber, K. P.; Herzberg, G. Molecular structure and molecular spectra IV: constants of diatomic molecules; Van Rostrand-Reinhold: New York. 1979. (38) Hargittai, M.; Varga, Z. Molecular Constants of Aluminum Monohalides: Caveats for Computations of Simple Inorganic Molecules. J. Phys. Chem. A 2007, 111, 6−8. (39) Khalil, H.; Le Quéré, F.; Brites, V.; Léonard, C. Theoretical Study of the Rovibronic States of CaO. J. Mol. Spectrosc. 2012, 271, 1− 9. (40) Minaev, B.; Norman, P.; Jonsson, D.; Ågren, H. Response theory calculations of singlet-triplet transitions in molecular nitrogen. Chem. Phys. 1995, 190, 11−29. (41) Minaev, B.; Plachkevytch, O.; Ågren, H. Multiconfiguration Response Calculations on the Cameron Bands of the CO Molecule. J. Chem. Soc., Faraday Trans. 1995, 91, 1729−1733. (42) Olsen, J.; Minaev, B.; Vahtras, O.; Ågren, H.; Jørgensen, P.; Jensen, H. J. A.; Helgaker, T. The Vegard-Kaplan band and the phosphorescent decay of N2. Chem. Phys. Lett. 1994, 231, 387−394.

H

DOI: 10.1021/acs.jpca.6b06471 J. Phys. Chem. A XXXX, XXX, XXX−XXX