Relativistic Four-Component Potential Energy Curves for the Lowest

Article pubs.acs.org/JPCA

Relativistic Four-Component Potential Energy Curves for the Lowest 23 Covalent States of Molecular Bromine (Br2) José da Silva Gomes,† Ricardo Gargano,‡ Joaõ B. L. Martins,§ and Luiz Guilherme M. de Macedo*,† †

Faculdade de Biotecnologia, Instituto de Ciências Biológicas, Universidade Federal do Pará (UFPA), Belém, Pará 66075−110, Brazil Instituto de Física, Universidade de Brasília (UnB), P.O. Box 04455, Brasília, Federal District 70919−970, Brazil § Instituto de Química, Universidade de Brasília (UnB), P.O. Box 04478, Brasília, Federal District 70919−970, Brazil ‡

S Supporting Information *

ABSTRACT: The covalent excited states and ground state of the Br2 molecule has been investigated by using four-component relativistic COSCI and MRCISD methods. These methods were performed for all covalent states in the representation Ω(±). Calculated potential energy curves (PECs) were obtained at the four-component COSCI level, and spectroscopic constants (Re, De, D0, ωe, ωexe, ωeye, Be, αe, γe, Te, Dv) for bounded states are reported. The vertical excitations for all covalent states are reported at COSCI, MRCISD, and MRCISD+Q levels. We also present spectroscopic constants for two weakly bounded states (A′: (1)2u and B′:(1)0−u) not yet reported in the literature, as well as accurate analytical curves for all five relativistic molecular bounded sates [the ground state X:0g+ and the excited states A:(1)1u, B:(1)0u+, C: (2)1u, and B′:(1)0u−] found in this work.

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INTRODUCTION Molecular bromine has attracted a lot of attention due to its role in ozone depletion1 and because it is widely used in technically challenging dynamics studies of ultrafast neutral molecules,2,3 imaging4 and femtochemistry.5 Its spectrum has been a topic of study since 19266 and a good review from the literature up to 1977 can be found in the book by Huber and Herzberg.7 It should be remarked that the visible spectrum is very complex because natural bromine contains two isotopic species, 79 Br and 81Br, of nearly equal abundance, and as a consequence the rotational analysis of the visible spectrum of natural bromine has three closely overlapping isotopic bands.8 Near ultraviolet and visible spectra are mainly due to transitions from the ground state to a few covalent states − A:(1)1u, B:(1)0u+ and C:(2)1u. Each band is Gaussian in shape, and the spectrum is an unsolved overlap of these bands.9 Nevertheless, although these four states are well-known experimentally [the ground state X:0g+ and the excited states A:(1)1u, B:(1)0u+ and C:(2)1u ], very little is known about the other 19 covalent states from both experiment or theoretical calculations. From the theoretical point of view, it is a tour de force to study the electronic structure of molecular bromine, because of the influence of relativistic effects and crossing of several repulsive excited states. This means that both electron correlation and relativistic effects should be treated with highlevel correlated relativistic calculations in order to obtain reliable results. As a result, there are relatively few theoretical calculations10−13 on excited states of Br2 and, to the best of our © 2014 American Chemical Society

knowledge, only one work in the literature for the excited states of Br22+ in a four-component relativistic framework.14 The purpose of this work is to complement the available data through the report of the potential energy curves (PEC) for the covalent 23 lowest states of molecular bromine, their spectroscopic constants when available (nonrepulsive states), their vertical excitation energies and analytical forms for the bounded states. To this aim, the spin−orbit coupling is introduced from the onset due to the four-component relativistic framework, and the electron correlation is treated by complete open shell configuration interaction (COSCI), as well multireference configuration interaction with all single and double excitations (MRCISD) and MRCISD+Q (including Davidson size-extensivity correction) levels.

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COMPUTATIONAL APPROACH The methodology is similar to the previous articles for molecular iodine15 and chlorine.16 All calculations were performed by means of the fully relativistic MOLFDIR17 program using the Dirac−Coulomb−Gaunt Hamiltonian and aug-cc-PVTZ basis set18 taken from Visscher and Dyall recontracted at a four-component relativistic framework. The speed of light used was 137.0359895 au, and the bromine Special Issue: Energetics and Dynamics of Molecules, Solids, and Surfaces - QUITEL 2012 Received: November 20, 2013 Revised: April 17, 2014 Published: April 29, 2014 5818

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15890, 19580, and 23265 cm−1, respectively. The notation used to label the relativistic states follows Hund’s case (c). The results regarding the bounded covalent states are summarized in Table 1, including experimental data7,25−27 and other theoretical10−13,28−31 works. There are three wellcharacterized molecular states (X:(1)0g+, A:(1)1u, and B: (1)0u+), and the results from this work are in agreement with experimental values. The A′:(1)2u has only experimental values for harmonic vibration ωe (165 cm−1), ωexe (2.5 cm−1), and ωeye (5.689 × 10−2cm−1), and the respective theoretical values of 157 cm−1, 1.92 cm−1, and 6.021 × 10−2cm−1 are in very good agreement. There are no theoretical or experimental values for spectroscopic constants for the B′:(1)0u− state. The results from this work show that this state should be weakly bounded with De of 0.088 eV, and it should have values for Re, ωe, ωexe, and Te as 2.82 Å, 101 cm−1, 3.95 cm−1, and 1.95 eV, respectively. Another remark that should be made concerns the centrifugal distortion constant, Dv. We could not find experimental values for the states studied in this work, but the value obtained for X:(1)0g+ is consistent with the Dv value for v = 0 obtained calculated in a self-consistent manner32 from a corrected RKR (Rydberg−Klein−Rees) potential energy curve, 2.097 × 10−8 cm−1. The calculated values range from 2.214 × 10−8 cm−1 for the X:(1)0g+ state to 2.214 × 10−8 cm−1 for the B′:(1)0u− state. The calculated vertical transitions at ground state bond length are presented in Table 2. The electronic states are derived following the promotion of electrons into active space: there are configurations σkg πulπgmσun, where k+l+m+n = 10 electrons into 12 spin orbitals. Therefore, the X: (1)0g+ ground state is mainly represented by 2440 configuration, that means σ2g π4uπ4g σ0u. First, the adiabatic correlation linking the atomic Br(2PJ) states with the molecular states in Hund’s case (c) and (a) and the dominant configuration of each molecular orbitals can be inferred from Table 2. For example, the ground state X:(1)0g+ is originated from nonrelativistic 1Σg+, dissociates at the Br+Br channel, and has 2440 dominant configurations. The same reasoning can be extended for all molecular states. The theoretical vertical excitation from this work is in good agreement with the experimental28 available data. The differences for B′:(1)0u− and B″:(2)1u states are 0.08 and 0.15 eV, respectively. The previous theoretical data33 from the literature does not include spin−orbit effects, so comparison with the results from this work is meaningless. The choice of the degree of the curve fitting was made considering the balance between accuracy and the smallest number of variables in order to minimize the root-mean-square (RMS) deviations defined as34

nucleus was represented by a Gaussian charge distribution19 with an exponential value of 243199191.61. PECs for all states were calculated at 47 different bond distances using the average of Dirac−Hartree−Fock configurations followed by the COSCI approach, where full CI is performed in the open shell space of the 4p spin orbitals. Due to computational limitations, instead of performing MRCISD calculations at all points of the PEC, the curves were pragmatically improved by a simple empirical correction20,15,16 matching the ground-state PEC to the experimental Rydberg− Klein−Rees ground state8 and shifting all the excited states in a similar way. In order to estimate the effects of dynamical correlation and relaxation on the vertical excitation energies, MRCISD calculations were also performed, as well as an estimate of high-order corrections using the Davidson correction, labeled21 MRCISD+Q. The COSCI orbitals were used as the reference wave function for multireference calculations. The multireference active space has the 4s spinors in RAS1, the 4p spinors in RAS2, and 104 virtual spinors in RAS3. Finally, spectroscopic constants were obtained with the aim of Vibrot program implemented in MOLCAS 7.4,22 and analytical expressions for bounded states were obtained through the use of an extended Rydberg23 (ERyd) function of the type n = 10

V (R ) = −De[1 +

∑

Ck(R − R e)k ]e−C1(R − R e)

k=1

(1)

where De stands for dissociation energy, Re is equilibrium distance (both of which were fixed prior to the fittings), and Ck stands for the coefficients to be determined (in this work, they were obtained by Powell’s method24).

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RESULTS AND DISCUSSION First, it is necessary to point to some remarks about the notation employed. The Br2 dissociation in this work correlates

δ= Figure 1. PECs for the 23 relativistic covalent states of Br2.

1 N

N

∑ [Ee(R i) − V (R i)]2 i=1

(2)

where Ee(Ri) represents the ab initio electronic energies obtained in this work, V(Ri) is the fitted expression calculated at ith point, and N is the number of fitted electronic energies, with respect to each molecular relativistic state. In order to fit, the equilibrium distances and dissociation energies were fixed at atomic units from the values obtained theoretically in this work. All these values are listed in Table 3, and none of the RMS exceeds 1.53 × 10−4 a.u., suggesting a good description of the PECs.

into two neutral Br(2P) atoms as the product channel. Due to spin−orbit coupling, these atoms can exist in two different states, with J = 1/2 and J = 3/2. Following the notation used in our previous paper of molecular chlorine, the 2P1/2 state is denoted by Br and the 2P3/2 state as Br*. Therefore, the asymptotic PEC region of Br2 (nonrelativistic) is then split into three different channelsBr+Br, Br+Br*, and Br*+Br*as one can observe in Figure 1. These three channels occur at 5819

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Table 1. Spectroscopic Properties for Ground and Bounded Excited States of Molecular Bromine (Based on COSCI Results) Experimental state: Re (Å) De (eV) D0 (eV) ωe (cm−1) ωexe (cm−1) ωeye (cm−1) 10−2 Be (cm−1) 10−2 αe (cm−1) 10−4 γe (cm−1) 10−6 Te (eV) Re (Å) De (eV) D0 (eV) ωe (cm−1) ωexe (cm−1) ωeye (cm−1) 10−2 Be (cm−1) 10−2 αe (cm−1) 10−4 γe (cm−1) 10−6 Te (eV) Dv (cm−1)10−8

Other Work (Theoretical)

1 X: (1) 0g+

2 A′: (1) 2u

3 A: (1) 1u

4 B′: (1) 0u‑

5 B: (1) 0u+

2.281a 1.97b 325a 1.08a −22.98a

165c 2.50c 5.689c

2.691a 0.257d 153a 2.70a -

-

2.678a 0.466d 167a 1.64a

8.2107a 3.19a −1.045a -

5.9a 8.0a 1.64c 1.72a This Work

-

5.96a 4.89a 1.97a

2 A′: (1)2u

3 A: (1) 1u

2.820 0.088 0.082 101 −3.95 14.067 5.375 13.11 19.248 1.85 4.26 4 B′: (1) 0u‑

2.304e 2.282f 2.298g 2.298 2.33

2.758e 2.730f 5.153g s

2.712 0.438 0.427 163 −1.64 −4.933 5.810 5.11 −4.719 1.96 2.96

ωeye =

5 B: (1) 0u+

2.320

CASSCFe CASSCF+ECPf CASSCF+RECPg DK+CASPT2h SOCIi

1.954e 1.969f 1.81g 1.80 1.77

KRMP2+RECPj

1.69

0.2339e 0.219f 0.2077g

0.45095e 0.430f 0.4131g

144.8e 143.6f 142.3g

163.1e 161.36f 160.2g

2.85e 2.74f 2.76g

1.91e 1.79f 1.88g

e

CASSCF CASSCF+ECPf CASSCF+RECPg DK+CASPT2h SOCIi

320.7 320.07f 319.9g 317 321

KRMP2+RECPj

318

CASSCFe CASSCF+ECPf CASSCF+RECPg

1.24e 1.12f 1.10g

ωexe (cm−1)

a

Reference 7. bReference 25. cReference 26. dReference 27. eAbdelHafiez and co-workers’ Complete Active Space Self-Consistent Field (CASSCF) level, from ref 10. fZhang and Zhang’s CASSCF and Effective Core Potential (ECP), from ref 11. gAsano and Yabushitás Spin−Orbit Configuration Interaction (SOCI) method with Relativistic Effective Core Potential (RECP), from ref 12. hRoos et al.’s second order perturbation theory (CASSCF/CASPT2) and DouglassKroll (DK) Hamiltonian, taken from ref 29. iBalasubramaniańs SOCI, from ref 30. jLee and Lee’s Kramers Restricted Møller−Plesset perturbation theory calculations (KRMP2) with RECP, taken from ref 31.

5 B: (1) 0u+ 2.760e 2.679f 5.048g

smallest error compared to the experimental values. The error is larger for the last fit stated compared to the experimental values.

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FINAL REMARKS

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ASSOCIATED CONTENT

We obtained PECs for the 23 lowest molecular states of Br2 through the all-electron relativistic four-components method. The potential energy curves were generated at the COSCI level, and the vertical transitions were also obtained at the MRCI and MRCI+Q levels. The main feature is that we present spectroscopic constants for two weakly bounded states not yet reported in the literature. The A′:(1)2u state has Re, De, ωe, and MRCI+Q vertical energy values of 2.718 Å, 0,329 eV, 157 cm−1, and 2.33 eV, respectively, whereas the B′:(1)0−u has values of 2.820 Å, 0,088 eV, 101 cm−1, and 2.64 eV.

1 [14(ε1,0 − ε0,0) − 93(ε2,0 − ε0,0) 24 + 23(ε3,0 − ε1,0)]

ωexe =

4 B′: (1) 0u‑

ωe (cm−1)

Once the analytical form for the potential was obtained considering the data from Table 3, we were able to obtain novel analytical expressions for the several considered systems to be used in the solution of the Schrödinger nuclear equation through the application of the DVR methodology35 and, in this fashion, to obtain vibrational energies ε(υ) (Br2 reduced mass was fixed as 72824.33565 atomic unit), where υ denotes the vibrational quantum numbers. From these energies, one can obtain vibrational spectroscopic constants using the following equations:35 ωe =

3 A: (1) 1u

De (eV)

Re (Å) CASSCFe CASSCF+ECPf CASSCF+RECPg DK+CASPT2h SOCIi

2 A′: (1)2u Re (Å)

KRMP2+RECPj

e

2.279 2.718 2.746 1.964 0.329 0.238 1.944 0.320 0.229 317 157 143 −1.05 −1.92 −2.31 6.870 −6.021 −2.266 8.224 5.783 5.665 3.53 5.70 6.72 3.4528 −8.1969 −8.2093 1.61 1.7 2.21 3.13 3.52 Other Work (Theoretical) 1 X: (1) 0g+

1 X: (1) 0g+

1 [13(ε1,0 − ε0,0) − 11(ε2,0 − ε0,0) 4 + 3(ε3,0 − ε1,0)] 1 [3(ε1,0 − ε0,0) − 3(ε2,0 − ε0,0) + (ε3,0 − ε1,0)] 6

S Supporting Information *

(3)

Full details of the present 23 PECs can be downloaded as Supporting Information in an EXCEL file. This material is available free of charge via the Internet at http://pubs.acs.org.

The results for ωe, ωexe, and ωeye obtained from this approach are listed in Table 4. The lowest states have the 5820

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Table 2. Calculated Relativistic Vertical Excitation Energies (eV) at the Experimental Ground State Equilibrium Bond Length, Compared with Experimental and Other Theoretical Data (Non-Relativistic, NR)a nonrelativistic states Λ−Σ

relativistic states Ω

dissociation products

Σg+ 3 Πu

X: (1)0g+ A′: (1)2u A: (1)1u B′: (1)0u− B: (1)0u+ C: (2)1u (1)2g a: (1)1g a′: (2)0g+ (1)0g− (2)1g (3)0g+ (3)1g (2)0u− (3)1u (2)2g (4)0g+ (3)0u− (1)3u b′: (2)2u (4)1u (5)1u (4)0u−

P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2 P3/2

1

Πu Πg

1 3

Πg Σg‑

1 3

Σu+

3

Δg Σg+ 1 ‑ Σu 3 Δu 1 1

Σu+

1

a

+ + + + + + + + + + + + + + + + + + + + + + +

P3/2 P3/2 P3/2 P3/2 P1/2 P3/2 P3/2 P3/2 P3/2 P1/2 P1/2 P1/2 P1/2 P3/2 P3/2 P1/2 P1/2 P1/2 P1/2 P1/2 P1/2 P1/2 P1/2

COSCI

MRCISD

MR-CISD +Q

dominant configuration (klmn)

0.00 2.39 2.52 2.71 2.74 3.28 4.74 4.83 5.04 5.06 5.31 5.60 5.76 5.48 5.49 6.07 6.49 6.59 6.59 6.90 7.00 7.21 7.23

0.00 2.35 2.47 2.67 2.69 3.19 4.73 4.83 5.02 5.03 5.35 5.57 5.71 5.75 5.76 6.10 6.52 6.82 6.83 7.14 7.22 7.44 7.45

0.00 2.33 2.45 2.64 2.66 3.16 4.70 4.80 4.99 4.99 5.33 5.54 5.67 5.79 5.80 6.09 6.52 6.86 6.88 7,18 7.26 7.48 7.49

2440 2431 2431 2431 2431 2431 2341 2341 2341 2341 2341 2422 2422 1441 1441 2422 2422 2332 2332 2332 2332 2332 2332

theor. (other work, NR)

exp.

2.05b 2.58c 2.89b

3.01c

5.02b 5.02b

5.02b

The notation klmn used for the main configuration is based on σgk πul πgm σun. bReference 28. cReference 29.

Table 3. Adjusted Coefficients for Each Electronic Level (With Their RMS Deviation) for the Bounded States of Br2 Molecule Ck

1 X:(1)0g+

2 A′:(1)2u

3 A:(1)1u

4 B′:(1)0u−

5 B:(1)0u+

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 Re (a.u.) De (a.u.) RMS (10−4)

3.01975283 3.47191666 2.25658412 0.93170290 0.14052157 -0.43342059 -0.14911232 0.38906167 -0.16997387 0.02100420 4.30687 0.07213889 0.430328589

2.70809360 1.88993756 0.47187434 1.4376364 1.1132938 -1.21822383 -0.28684910 0.59446130 -0.20005288 0.02218818 5.06068 0.01713909 0.823630084

2.39823268 1.38759536 0.76388665 0.94585760 -0.13412554 -0.62442006 0.15788163 0.15057231 -0.06984493 0.00859972 5.10226 0.01390827 0.761118375

2.94519424 2.86167940 0.38884475 1.50476118 -3.44553116 1.35946773 0.19416801 0.00076016 -0.12448990 0.02774975 5.10226 0.00982766 1.5313783

2.58679820 1.62987120 2.256994741 1.135378319 -2.21796817 0.34861152 0.42050278 -0.20932801 0.01668938 0.00416956 5.29123 0.00461612 0.321332345

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Table 4. Some Spectroscopic Constants Obtained from a Fitted Analytical Curve and the Respective Experimental Values from the Literature Ck

1 X:(1)0g+

analytical curve (this work) ωe (cm−1) 322 ωexe (cm−1) 1.23 ωeye (cm−1) 10−2 1.03 experimental ωe (cm−1) 325a −1 ωexe (cm ) 1.08a ωeye (cm−1) 10−2 -22.98a a

2 A′:(1)2u

3 A:(1)1u

4 B′:(1)0u−

5 B:(1)0u+

166 3.80 8.72

134 5.30 5.80

101 5.77 52.40

201 4.01 1.92

b

a

165 2.50b 5.689b

153 2.70a ---

-------

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. Phone: +55(91)-3201-7918. Notes

The authors declare no competing financial interest.

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a

167 1.64a ---

ACKNOWLEDGMENTS

This work was supported by Conselho Nacional de ́ Desenvolvimento Cientifico e Tecnológico (CNPq) under

Reference 7. bReference 25.

Grant 47556/2009-7 and funding from Universidade Federal do Pará (UFPA) under the PADRC program. 5821

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