Electronic Structure Calculations with the Spin Orbit Effect of the Low-Lying Electronic States of the YbBr Molecule.

This work presents an electronic structure study employing multireference configuration interaction MRCI calculations with Davidson correction (+Q) of the ytterbium monobromide YbBr molecule. Adiabatic potential energy curves (PECs), dipole moment curves, and spectroscopic constants (such as R e, ωe, B e, D e, T e, and μe) of the low-lying bound electronic states are determined. The ionic character of the YbBr molecule at the equilibrium position is also discussed. With spin-orbit effects, 30 low-lying states in Ω = 1/2, 3/2, 5/2, 7/2 representation are probed. The electronic transition dipole moment is calculated between the investigated states and then used to determine transition coefficients, for example, the Einstein coefficient of spontaneous emission A ij and emission oscillator strength f ij . Vibrational parameters such as E ν, B ν, D ν, R min, and R max of the low vibrational levels of different bound states in both Λ and Ω representations are also calculated. Upon calculating the Franck-Condon factors, they are found to be perfectly diagonal between three couples of low-lying excited states. Vibrational Einstein coefficients and radiative lifetimes are computed as well for the lowest vibrational transitions. Most of the data reported in this work are presented here for the first time in the literature. Very good accordance is obtained in comparison with the previously reported constants by means of experimental methods.

Introduction

Since early 1970s, considerable interest has emerged in studying the electronic properties of lanthanide-containing diatomic molecules. The molecular structure, spectra, chemical properties, and bonding mechanisms of ytterbium-containing molecules have received special attention. Kramer[1] was the first to conduct vibrational analysis and spectroscopic characterization of the A2Π1/2,3/2–X2Σ+ transition of YbX (X = Cl, Br, and I)to determine the bandhead positions and transition. Kaledin et al.[2] performed ligand field theory calculations to study the electronic structure of lanthanide monohalides including YbBr. Dickinson et al.[3] used Fourier transform microwave spectrometry to analyze the low-lying vibrational bands (ν = 0, 1) of the B2Σ+–X2Σ+ system of YbBr. They also conducted a rotational analysis of the 0–0, 1–0, 0–1, and 1–1 bands of the same system[4] and the 0–0 and 1–0 bands of the A2Π–X2Σ+ system, as an extension to their work.[5] Su et al.[6] calculated the ground-state potential energy curves (PECs) and the ground-state spectroscopic parameters of the YbX (X: F, Cl, Br, I, and At) molecules using the unrestricted coupled cluster UCCSD(T) method with the ECP28MDF basis set for the Yb atom. A thermal excitation technique was used to study the emission spectrum and vibrational transitions of YbBr[7] from which the band sequence (Δν = ±3, ±4) of the A2Π–X2Σ+ system and those of the B2Σ+–X2Σ+ system (up to ν = 7 of the higher state) is obtained. As far as it is reported in literature, the investigation of the PECs and spectroscopic properties of YbBr has been limited to the ground state only. However, none of the previous studies has reported any information on the dipole moment of any of its molecular states, including the ground state. Such a lack of information has motivated us to thoroughly investigate the electronic structure of the low-lying states of the ytterbium monobromide molecule, using ab initio quantum chemistry calculations. This study also comes as a continuation to our previous publication,[8] dealing with the spectroscopic properties of YbX molecules with X: F, Cl, Br, and I.

In the present work, multireference configuration interaction MRCI level of theory with Davidson correction (+Q) is employed to investigate the PECs of eight doublet 2Λ(+/−) and six quartet 4Λ(+/−) states with 30 spin–orbit Ω low-lying electronic states of YbBr. The spectroscopic constants as equilibrium internuclear distance Re, harmonic frequency ωe, rotational constant Be, transition energy with respect to ground state minimum Te, and permanent dipole moment μe are calculated for the investigated states. The static dipole moment curves (DMCs) are computed for 14Λ–S electronic states at the spin-free level and for 30 lowest states with spin–orbit effects in the representation of Ω = 1/2, 3/2, 5/2, 7/2. The transition dipole moment μ between three couples of 2Λ states is also investigated and used to determine Einstein spontaneous emission coefficients A, spontaneous radiative lifetime τ, and emission oscillator strength f among them. For low vibrational levels (with J = 0), we computed the vibrational parameters (Eν, Bν, Dν, Rmin, and Rmax) in both Λ and Ω representations, the Franck–Condon factors (FCFs), Einstein coefficients, and radiation lifetimes for transitions between them. Good accordance is obtained upon comparison with the constants’ values reported in the literature, and new data are reported for the first time.

Computational Method

The present study of the YbBr electronic structure is conducted by using the high-level ab initio calculations as applied in the MOLPRO 2012 computing package[9] and the graphical interface GABEDIT.[10] Regarding coupling of spin and orbital angular momenta, they are uncoupled for the Λ–S states and coupled for the Ω states. The state-average complete active-space self-consistent field (CASSCF) approach is employed to generate multiconfiguration wave functions, which are used to perform internally contracted multireference single and double configuration interaction calculations, including Davidson correction (MRCI +Q).[11] After testing several basis sets to describe the ytterbium atom, we selected small-core quasi-relativistic energy-consistent pseudopotential ECP28MWB[12] with the segmented (14s13p10d8f6g/10s8p5d4f3g) basis set[13] which is augmented by the s2pdfg diffuse functions.[14] The bromine atom is described by the correlation consistent basis set including spdfg functions in conjunction with small-core relativistic pseudopotential ECP10MDF_AV5Z.[15] The active space in CASSCF calculations contains 3σ (Yb: 6s,6p; Br: 4p) and 2π (Yb: 6p; Br: 4p) molecular orbitals to be occupied by seven valence electrons. In the C2 point group symmetry, this active space is distributed into 3a1, 2b1, 2b2, 0a2, and noted by [3,2,2,0]. The molecular orbitals involved in the active space of YbBr are 1σ1π 2σ 2π3σ. The first two σ MOs arise from the active atomic orbitals of (Yb: 6s) and (Br: 4p). The third σ MO comes mainly from (Yb: 6p), whereas the remaining MOs, 1π and 2π, come from (Yb: 6p6p) and (Br: 4p4p) atomic orbitals. For the sake of time traceability, the orbitals of the 4f shell of ytterbium were kept frozen by being strictly doubly occupied, and thus the states that are associated with the excitations of the corresponding electrons are not treated in this work. However, considering the dynamical electronic correlation of the electrons occupying 4f14 of Yb and 4s2 of Br in the MRCI +Q calculations has an augmenting effect on the molecular states correlating with the lowest dissociation limit at Yb(1S) + Br(3P) fragments. Therefore, 23 electrons in all of the YbBr molecules are correlated in the MRCI +Q step of calculations. For the spin–orbit coupling (SOC) effect to be considered in the MRCI +Q calculations,[16] the effective core potential (ECP) method[17−19] is used and spin–orbit effects are introduced for both atoms through spin–orbit pseudopotentials. The energies of the molecular states, with Ω = 1/2, 3/2, 5/2, 7/2 are computed by diagonalizing the total Hamiltonian (Ht = He + WSO), that is, the Born–Oppenheimer approximation and the spin–orbit perturbation pseudopotential. The internuclear distance R at which calculations were conducted ranges between 2.24 and 10 Å at an increment of 0.02 Å around the equilibrium position Re and 0.1 Å elsewhere.

Results and Discussion

PECs and Dipole Moments of the Low-Lying Λ–S and Ω States of YbBr

The spin-free PECs of 14 low-lying doublet and quartet Λ–S electronic states of YbBr are presented in Figure . The lowest molecular dissociation asymptote Yb(1S) + Br(2P) associates with the ground state X2Σ+ and the first excited state, 12Π. The second asymptote Yb(3P) + Br(2P) is associated with the following states: two 2,4Σ+, two 2,4Π, one 2,4Σ–, and one 2,4Δ. An undulation in the PECs of the 32Π and 12Σ– states is noticed after 5 Å. This may be related to the electronic wave functions of these states leading to the breakdown of the Born–Oppenheimer approximation. Among the investigated states; 32Π, 14Π, 24Π, and 24Σ+ are found unbound, and the PEC of the 22Σ+ state shows two minima. Figure shows several crossings and avoided crossings between energy curves. Their corresponding positions, RC and RAC, and the energy separation at the avoided crossing region ΔEAC are given in Table S1 as the Supporting Information. A region of PECs interaction is noticed within the range 3.2 ≤ R ≤ 3.5 Å in which the energy minimum of each of 22Π and 32Σ+ is located. Such interaction affects the stability of the molecule in these states. Concerning the valence configuration of the ground-state molecular orbitals of YbBr, the orbitals of the closed 4f shell of the Yb atom are not involved in bonding.[6] Therefore, the covalent configuration (4f14) 1σ21π4 2σ1 of the X2Σ+ state is dominant at long internuclear distances, whereas the ionic configuration (4f14) 1σ11π4 2σ2 is dominant at short distances indicating a charge transfer from Yb to Br near the equilibrium position.[7] The molecular configurations of the 22Σ+ and 12Π electronic states close to the equilibrium distance are (4f14) 1π4 2σ23σ1 and 1π4 2σ2 2π1, respectively. The spectroscopic constants (Te, Re, ωe, and Be) of the investigated Λ–S bound states are calculated by fitting the energy values around the equilibrium position into a polynomial in terms of the internuclear distance R while considering the weighted average over the different isotopes. The ionic character fionic = μe/eRe is calculated as an indicator of the molecular bond nature at the state’s equilibrium position. The constants of the 22Π and 32Σ+ states are not calculated as an avoided crossing interaction takes place at their equilibrium positions, neither those of the 12Δ and 12Σ– as their shallowness prevents generating correct polynomial coefficients. Table lists the calculated parameters and the corresponding values available in the literature from both experimental and theoretical methods. In close comparison, this table shows that our calculated spectroscopic constants of the ground state X2Σ+ are in excellent agreement with those determined experimentally. The relative differences of equilibrium internuclear distance Re and rotational constant Be are δRe/Re = 0.72% (ref (4)), 0.64% (ref (3)), and δBe/Be = 2.03% (refs (4) and (5)). For harmonic frequency ωe, the relative difference ranges between 0.49% ≤ δωe/ωe ≤ 1.34% (ref (4)). These constants are in better accordance with the experimental values than those calculated by the UCCSD(T) method for the 174Yb79Br isotopologue.[6] In terms of the dissociation energy De, our calculated value is 295.2 kJ/mol and those calculated at the UCCSD(T) level of theory are 505.8 and 451.7 kJ/mol (ref (6)), whereas the experimental value is 316 kJ/mol (ref (20)). From above, it is noticed that our MRCI calculation significantly improves the theoretical De value. However, there are many factors which may influence the theoretically calculated dissociation energy such as the selected basis sets, the chosen active space, the number of valence electrons, the correlation relation among them, the interaction of the ground state with the close-lying excited states at long internuclear distances, the theoretical method of calculation, and so forth.

Figure 1

Potential energy curves for the 2,4Σ(+/−), 2,4Π, and 2,4Δ states of the YbBr molecule using spin-free MRCI +Q calculation.

Table 1

Spectroscopic Constants, Permanent Dipole Moment μe, and Ionic Character of the Lowest Electronic States of YbBr at the Spin-Free MRCI +Q Level

state	Te (cm–1)	Re (Å)	ωe (cm–1)	Be (cm–1)	De (kJ/mol)	μe (Debye)	fionic	refs
X2Σ+	0.00	2.664	193.89	0.0434	295.2	4.81	0.376	a
		2.619	214.54	0.0453	505.8			b
		2.838	178.81	0.0386	451.7			c
		2.645	196.52	0.0443	316 ± 9			d
		2.647	194.84	0.0443				e
								f
								g
								h
								i
12Π	18476.0	2.611	208.69	0.0452	74.6	5.77	0.461	a
		2.595		0.0461				d
22Σ+	19933.83	2.600	208.92	0.0456	235.7	5.66	0.453	a
		2.583	215.23	0.0465				d
			213.39					e
								f
14Σ+	38697.19	3.486	44.56	0.0254	12.2	2.31	0.138	a
14Δ	39141.36	3.604	34.46	0.0237	7.0	2.02	0.117	a
14Σ–	39464.19	3.810	23.68	0.0212	3.2	1.64	0.089	a

Present work using the CASSCF/MRCI method, ECP28MWB_segmented for Yb, and the ECP10MDF_AV5Z basis set for Br.

Reference (6): UCCSD(T) method, ECP28MDF basis set for Yb.

Reference (6): UCCSD(T) method, WTBS basis set for Yb.

Reference (4): exp.

Reference (4): experimental study for Yb(174)Br(79).

Reference (4): experimental study for Yb(174)Br(81).

Reference (5): exp.

Reference (20): exp.

Reference (3): exp.

For the first excited state, 12Π, the relative differences with respect to the experimentally reported values of Re and Be are δRe/Re = 0.61% (ref (5)) and δBe/Be = 1.99% (ref (5)) and those for the 22Σ+ state constants are δRe/Re = 0.65% (ref (4)), δBe/Be = 1.97% (ref (4)), and 2.14% ≤ δωe/ωe ≤ 3.02% (ref (4)). The adaptation of 23 correlated electrons as in the present work has shown significant improvement of the constants calculated. When the electrons occupying 4f14 of Yb and 4s2 of Br are included in the core such that the valence electrons are only seven, the calculated constants show large dispersion from the experimental values. The percentage of ionic character fionic in the molecular bond is calculated for the lowest doublet states and is provided in Table . It is 37.6% for X2Σ+, 46.1% for 12Π, and 45.3% for 22Σ+. These percentages indicate the strong ionic character of ytterbium halides.

For further investigation of the YbBr low-lying electronic states, 30 electronic states in the Ω representation are generated from the 14Λ–S states by considering the SOC effect. The PECs of symmetries Ω = 1/2, 3/2, 5/2, 7/2 are plotted as a function of the internuclear distance in the region between 2.24 and 10.0 Å as shown in Figures –4. When the SOC effects are taken into consideration, the spin-free dissociation asymptote Yb(1S) + Br(2P) splits into two branches: Yb(1S0) + Br(2P3/2) and Yb(1S0) + Br(2P1/2), while the dissociation limit Yb(3P) + Br(2P) splits into six branches: Yb(3P0) + Br(2P3/2), Yb(3P1) + Br(2P3/2), Yb(3P2) + Br(2P3/2), Yb(3P0) + Br(2P1/2), Yb(3P1) + Br(2P1/2), and Yb(3P2) + Br(2P1/2). The first asymptote correlates with the (1)1/2 and (1)3/2 states; meanwhile, the second one correlates with the (2)1/2 state lying at approximately 3500 cm–1 above the first asymptote. The 4.6% relative difference with the NIST experimental energy value of 3685.24 cm–1 (ref (21)) indicates a very good agreement. The third asymptote correlates with (3)1/2 and (2)3/2 states lying approximately at 15 116 cm–1, where the NIST value is 17 288.439 cm–1. These calculations are obtained considering diffuse functions in describing the ytterbium atom. Contrary to this consideration, the dissociation asymptote energies are shifted higher causing larger relative difference at the dissociation limit. Again, the large number of correlated electrons, 23 electrons as in this work, promotes the dispersion forces effectively on the dissociation limits as using diffuse functions constrains their effect. There are fifteen states of Ω = 1/2, ten of Ω = 3/2, four of Ω = 5/2, and one of Ω = 7/2. The (3)1/2 state shows several avoided crossings with lower and higher Ω = 1/2 states, which renders two potential wells for the corresponding state. The interaction with the Ω components of the lower excited 2Σ+ and 2Π states causes many avoided crossings to take place between R = 3.0 and 3.6 Å. Several avoided crossings have been recorded for the PECs of the Ω = 1/2, 3/2, and their positions are given in Table . Such behaviors in the PECs of the low excited states lead to perturbations in the vibrational levels of the Ω components, for example, (4)1/2, (2)3/2, and (5)1/2 of 22Π and 32Σ+ states, hence causing difficulty for an accurate description of these levels. The spectroscopic constants such as Re, Te, ωe, Be, and μe along with the De values of different Ω bound states and the dominant SΛ compositions at Re are determined and given in Table . The dominant SΛ component of (1)1/2 is 99.87% X2Σ+; hence, these two states, (1)1/2 and X2Σ+, have nearly the same spectroscopic constants. The main parent SΛ of the (2)1/2 and (1)3/2 states is 12Π with percentage compositions of 94.80 and 99.94%, respectively. The spectroscopic parameters of the (2)1/2 and (1)3/2 states are very close to those of 12Π. Also, the main SΛ component of the (3)1/2 is 22Σ+ (94.90%) with congruent parameters. The comparison between our calculated transition energy Te of the (2)1/2 and (1)3/2 states and those experimental values available in the literature[1] shows excellent agreement, where the relative differences are 0.26 and 0.82%, respectively.

Figure 2

PECs of YbBr: 15Ω = 1/2 states.

Figure 4

PECs of YbBr: 4Ω = 5/2 and one Ω = 7/2 states.

Table 2

Positions of the Avoided Crossings RAC with the Corresponding Crossings and Avoided Crossings of Λ States for Ω States of the YbBr Molecule

Ω	(n + 1)Ω/nΩ	RAC (Å)	avoided crossings of Λ states	crossings of Λ states
1/2	3/1	4.72	X2Σ+/22Σ+
	4/2	3.30	12Π/22Π
	4/3	3.56		22Σ+/22Π
	5/3	3.36	22Σ+/32Σ+
	8/4	4.80		22Π/14Σ–
	8/5	4.00		32Σ+/14Σ–
	11/10	3.00	14Π/24Π
	12/11	3.50	14Π/24Π
	14/12	2.96	14Π/24Π
	12/9	5.02		12Δ/14Π
3/2	2/1	3.12	12Π/22Π
	5/2	4.80		22Π/12Δ
	6/2	5.06		22Π/14Σ–
	7/6	4.16		14Σ–/14Π
	8/5	5.00		12Δ/24Π
	9/5	5.04		12Δ/32Π
	8/7	4.26	14Π/24Π
		4.36
	9/8	5.20		24Π/32Π

Table 3

Spectroscopic Parameters and Permanent Dipole Moment μe of the Lowest Bound Ω States of YbBr at Spin–Orbit Configuration Interaction Level of Theory

Ω state	Te (cm–1)	Re (Å)	ωe (cm–1)	Be (cm–1)	De (kJ/mol)	μe (Debye)	% (SΛ-parent) at respective Re	refs
(1) 1/2	0.00	2.665	193.84	0.0434	281.74	4.74	99.87% X2Σ+	a
(2) 1/2	17743.42	2.612	208.43	0.0452	69.82	6.60	94.80% 12Π,5.04% 22Σ+	a
	17789.9							b
(1) 3/2	19155.14	2.609	208.72	0.0453	94.45	6.39	99.94% 12Π	a
	19313.0							b
(3) 1/2	20072.18	2.601	208.80	0.0456	222.45	6.58	94.90% 22Σ+, 5.04% 12Π	a
(3) 3/2	37156.11	3.590	35.20	0.0239	4.79	2.12	59.82% 14Σ+, 38.76% 14Σ–	a
(1) 7/2	38691.73	3.602	35.22	0.0238	7.05	2.03	100% 14Δ	a
(8) 1/2	39627.06	3.734	24.96	0.0220		1.85	41.85% 14Δ, 31.27% 14Σ–, 22.12% 14Σ+	a

Present work.

Reference (1): exp.

PECs of YbBr: 10Ω = 3/2 states.

At the MRCI +Q level, the dipole moments for the investigated spin-free Λ–S states are also computed and given in Figure . However, the dipole moments of the high doublet states, that is, 32Π and 12Σ– are not presented at long distances due to curve discontinuity. With the ytterbium atom at the origin, the negative value corresponds to the Ybδ+Brδ− polarity. Figure shows that the DMCs of the investigated states smoothly approaches zero at large distances, which is the natural behavior for a molecule dissociating into its neutral fragments. This figure also shows drastic changes in the DMC of different states. By identification with respect to the PECs, it can be noticed that these changes occur at the avoided crossing positions RAC that are given in Table S1.·In the region near R = 3.3 Å, the polarity of some states flips indicating an ionic character exchange between two states, for example, between X2Σ+ and 22Σ+ at R = 5 Å. The difference between the electronegativity of the two atoms EN(Br) > EN(Yb) is reflected by the negative values that the dipole moment of the ground state possesses at different distances. Its largest amplitude of |μ| = 4.368 a.u. is calculated at R = 4.4 Å. States with large dipole moment amplitudes are known to be useful for molecular alignment in an optical lattice, and they are helpful while studying long-range dipole–dipole forces. For the bound states, the DM values μe at the equilibrium distances are listed in Table . We can notice that the ionic character of the doublet states around the equilibrium position is greater than that of the quartet states. The DMs of the investigated Ω states are also plotted as function of internuclear distance R and are given in Figures –8. However, the dipole moments of the (9)3/2 and (3)5/2 states are not presented at long distances due to curve discontinuity. One can notice the consistency between the positions of the avoided crossings of the PECs (Figures –4 and Table ) and the drastic crossings of DMCs of these Ω states. In addition, the DMCs of (1)1/2 and (3)1/2 show maximum values |μ| = 3.444 a.u. near R = 3.9 Å and |μ| = 5.031 a.u. around R = 5.6 Å, respectively. Table lists the μe values of the lowest Ω bound states.

Figure 5

Permanent dipole moments for the 2,4Σ(+/−), 2,4Π, and 4Δ states of YbBr using spin-free MRCI +Q calculation.

Figure 6

DMCs of YbBr: 15Ω = 1/2 states.

Figure 8

DMCs of YbBr: 4Ω = 5/2 and one Ω = 7/2 states.

DMCs of YbBr: 10Ω = 3/2 states.

The transition DMCs between the obtained states are computed at the spin-free MRCI +Q level and are shown in Figure . These curves approach zero as R ≥ 7.5 Å due to spin-forbidden transitions from Yb(1S) to Yb(3P) atomic orbitals at asymptotic limits. Transition constants as the Einstein coefficient of spontaneous emission A, classical radiative decay rate of the single-electron oscillator at the emission frequency γcl, and emission oscillator strength f, for the X2Σ+–12Π, 12Π–22Σ+, and X2Σ+–22Σ+ electronic transitions are determined and listed in Table S2. These constants are calculated using the upper state μ TDM value at the equilibrium position as follows[22]where ν is the transition frequency between the two states, h is Plank’s constant, ε0 is the vacuum permittivity, me is the mass of an electron, e is the electronic charge, and c is the speed of light. The spontaneous radiative lifetimes (τ = 1/∑A where j runs for the underlying states of the i state) are 14.97 and 17.55 ns for the 12Π and 22Σ+ states, respectively. The X2Σ+–12Π transition has the largest oscillator strength. No comparison is conducted for the calculated dipole moments (DM and TDM) and the emission coefficients as they are provided here for the first time.

Figure 9

TDMs for the lowest transitions of YbBr.

FCFs and Einstein Coefficients between the Lowest Vibrational Levels of YbBr

Vibrational energy Eν, rotational constant Bν, centrifugal distortion constant Dν, and the abscissas of turning points Rmin and Rmax are determined for the lowest Λ–S and Ω bound states by using the canonical function approach.[23−25] The reduced mass of the standard atomic weights of Yb and Br is used by ignoring isotopic effects. The rovibrational parameters of the X2Σ+, 12Π, and 22Σ+ states are presented in Table S3 in the Supporting Information file. The calculated vibrational transition energies of 22Σ+(ν′)–X2Σ+(ν) and 12Π(ν′)–X2Σ+(ν) systems, that are given in Table S4 as the Supporting Information, are in good accordance with the values reported in refs.[4,5]Table lists the parameters for the lowest vibrational levels of the (1)1/2, (2)1/2, (1)3/2, and (3)1/2 states. Table S5 lists the energy separation values between the lowest vibrational levels of the (2)1/2, (1)3/2, (3)1/2, and the ground state (1)1/2 showing significant agreement upon comparison with the experimentally determined values,[1,7] where the relative difference is lower than 2%.

Table 4

Values of Eigenvalue Eν, Rotational Constants Bν and Dν, and the Abscissas of the Turning Points Rmin and Rmax for the Lowest Vibrational Levels of the (1)1/2, (2)1/2, (1)3/2, and (3)1/2 Low-Lying Bound States of YbBr

state	ν	Eν (cm–1)	Bν × 102 (cm–1)	Dν × 108 (cm–1)	Rmin (Å)	Rmax (Å)
(1)1/2	0	96.39	4.3357	8.8081	2.610	2.723
	1	288.41	4.3229	8.8090	2.572	2.769
	2	479.65	4.3102	8.8085	2.547	2.801
	3	670.11	4.2975	8.8106	2.528	2.829
	4	859.78	4.2848	8.8091	2.511	2.853
	5	1048.67	4.2721	8.8121	2.497	2.875
	6	1236.78	4.2595	8.8141	2.484	2.896
	7	1424.11	4.2468	8.8147	2.472	2.916
	8	1796.43	4.2215	8.8195	2.451	2.952
(2)1/2	0	103.79	4.5133	8.5672	2.559	2.668
	1	310.61	4.5010	8.5602	2.523	2.712
	2	516.69	4.4886	8.5517	2.498	2.743
	3	722.05	4.4764	8.5431	2.479	2.769
	4	926.68	4.4642	8.5328	2.463	2.792
	5	1130.59	4.4521	8.5235	2.449	2.814
	6	1333.80	4.4400	8.5173	2.436	2.833
	7	1536.30	4.4282	8.5208	2.425	2.852
(1)3/2	0	103.80	4.5224	8.6202	2.556	2.665
	1	310.59	4.5093	8.6314	2.520	2.709
	2	516.47	4.4962	8.6426	2.496	2.741
	3	721.46	4.4831	8.6601	2.477	2.767
	4	925.52	4.4698	8.6798	2.461	2.791
	5	1128.64	4.4565	8.7092	2.447	2.812
	6	1330.80	4.4432	8.7522	2.434	2.832
(3)1/2	0	103.98	4.5506	8.7511	2.548	2.657
	1	311.14	4.5377	8.7462	2.512	2.701
	2	517.52	4.5248	8.7397	2.488	2.732
	3	723.12	4.5119	8.7381	2.469	2.759
	4	927.93	4.4990	8.7341	2.453	2.782
	5	1131.95	4.4862	8.7351	2.439	2.803
	6	1335.17	4.4733	8.7396	2.426	2.823

By using Le Roy’s LEVEL8.2 program,[26] FCFs fν′ν for the X2Σ+–12Π, X2Σ+–22Σ+ and 12Π–22Σ+ transitions are calculated and tabulated in Table S6. For more accurate description of the lowest vibrational wave functions overlap of YbBr, the FCF values are also calculated and given in Table for the (2)1/2–(1)1/2, (1)3/2–(1)1/2, (3)1/2–(1)1/2, (3)1/2–(2)1/2, and (3)1/2–(1)3/2 transitions. With close-lying equilibrium distances Re and turning points (Rmin and Rmax) as shown in Table for the lowest vibrational levels of the (3)1/2, (2)1/2, and (1)3/2 states, the (3)1/2–(2)1/2 and (3)1/2–(1)3/2 transitions have a high diagonal FCF array with f00 = 0.98086 and f00 = 0.98902, respectively.

Table 5

FCF Values for the Transitions between the Lowest Ω States of YbBr

	ν′ = 0	1	2	3	4	5	6
(2) 1/2 (ν′)–(1) 1/2 (ν)
ν = 0	0.63418	0.29588	0.06164	0.00763	0.00063	0.00003	0.00000
1	0.28090	0.18696	0.36470	0.13915	0.02529	0.00278	0.00020
2	0.06973	0.32379	0.02223	0.31815	0.20603	0.05204	0.00731
3	0.01289	0.14489	0.25946	0.00413	0.22751	0.24945	0.08500
4	0.00199	0.03898	0.19575	0.16518	0.04740	0.13490	0.26576
5	0.00027	0.00792	0.07276	0.21377	0.08179	0.10249	0.06214
6	0.00003	0.00134	0.01877	0.10716	0.20229	0.02650	0.14418
(1) 3/2 (ν′)–(1) 1/2 (ν)
ν = 0	0.60571	0.31266	0.07131	0.00945	0.00080	0.00004	0.00000
1	0.29425	0.15064	0.36536	0.15565	0.03042	0.00342	0.00024
2	0.08043	0.31686	0.00914	0.30015	0.22312	0.06082	0.00869
3	0.01635	0.15931	0.23349	0.01325	0.19985	0.26206	0.09674
4	0.00276	0.04754	0.20447	0.13264	0.06672	0.10795	0.27168
5	0.00041	0.01063	0.08516	0.21115	0.05457	0.12000	0.04292
6	0.00005	0.00197	0.02430	0.12017	0.18777	0.01133	0.15243
(3) 1/2 (ν′)–(1) 1/2 (ν)
ν = 0	0.51760	0.35375	0.10717	0.01905	0.00222	0.00018	0.00001
1	0.32737	0.06027	0.33644	0.20891	0.05700	0.00900	0.00092
2	0.11612	0.27743	0.00532	0.20791	0.26312	0.10534	0.21771
3	0.03061	0.19818	0.14355	0.07254	0.08813	0.26562	0.15378
4	0.00671	0.07956	0.21323	0.04266	0.13823	0.01845	0.22944
5	0.00130	0.02353	0.12606	0.17700	0.01859	0.16312	0.00010
6	0.00023	0.00574	0.04879	0.15502	0.11805	0.00915	0.14834
(3) 1/2 (ν′)–(2) 1/2 (ν)
ν = 0	0.98086	0.01911	0.00002	0.00000	0.00000	0.00000	0.00000
1	0.01866	0.94394	0.03732	0.00007	0.00000	0.00000	0.00000
2	0.00046	0.03556	0.90917	0.05466	0.00013	0.00000	0.00000
3	0.00001	0.00132	0.05084	0.87648	0.07113	0.00020	0.00000
4	0.00000	0.00004	0.00253	0.06455	0.84583	0.08676	0.00027
5	0.00000	0.00000	0.00010	0.00401	0.07680	0.81719	0.10151
6	0.00000	0.00000	0.00000	0.00020	0.00574	0.08763	0.79061
(3) 1/2 (ν′)–(1) 3/2 (ν)
ν = 0	0.98902	0.01097	0.00000	0.00000	0.00000	0.00000	0.00000
1	0.01078	0.96725	0.02196	0.00000	0.00000	0.00000	0.00000
2	0.00019	0.02119	0.94556	0.03302	0.00002	0.00000	0.00000
3	0.00000	0.00056	0.03133	0.92387	0.04419	0.00004	0.00000
4	0.00000	0.00002	0.00109	0.04123	0.90206	0.05552	0.00007
5	0.00000	0.00000	0.00004	0.00176	0.05100	0.88001	0.06705
6	0.00000	0.00000	0.00000	0.00009	0.00256	0.06068	0.85763

Using the same program,[26] the average radiative lifetimes and Einstein coefficients of the vibrational transitions between (X2Σ+ and 22Σ+), (X2Σ+ and 12Π), and (12Π and 22Σ+) states are determined. Einstein coefficients for a transition (from a vibrational state ν′ to a vibrational state ν) are calculated by using[27]where σνν′ is the wavenumber of the transition between upper vibrational level ν′ and lower vibrational level ν (in cm–1); Λ′ and Λ are the projections of electronic orbital angular momentum on the internuclear axis for the upper and lower electronic levels. Reνν′ is the electronic-vibrational transition moment expectation value, which can be obtained from the vibrational wave functions (ν and ν′) and electronic transition dipole moment (in atomic units). Table shows that the radiative lifetime between vibrational transitions of states X2Σ+ and 12Π varies between 15 and 30 ns, and that between vibrational transitions of states X2Σ+ and 22Σ+ varies between 17 and 30 ns. On the other hand, the radiative lifetime between the vibrational transitions of states 12Π and 22Σ+ are thousand times longer with values that vary between 0.37 and 0.40 ms.

Table 6

Radiative Lifetimes of the Vibrational Transitions between the Electronic States X2Σ+–12Π, X2Σ+–22Σ+, and 12Π–22Σ+

	ν′ (12Π) = 0	1	2	3	4	5	6
ν (X2Σ+) = 0	4.12 × 107	2.02 × 107	4.40 × 106	5.63 × 105	4.70 × 104	2.71 × 103	1.10 × 102
1	1.87 × 107	1.14 × 107	2.44 × 107	9.81 × 106	1.84 × 106	2.04 × 105	1.46 × 104
2	4.75 × 106	2.10 × 107	1.08 × 106	2.08 × 107	1.43 × 107	3.74 × 106	5.27 × 105
3	8.94 × 105	9.70 × 106	1.63 × 107	4.75 × 105	1.45 × 107	1.71 × 107	6.04 × 106
4	1.40 × 105	2.67 × 106	1.29 × 107	1.00 × 107	3.64 × 106	8.29 × 106	1.80 × 107
5	1.93 × 104	5.51 × 105	4.91 × 106	1.38 × 107	4.65 × 106	7.27 × 106	3.61 × 106
6	2.44 × 103	9.46 × 104	1.29 × 106	7.13 × 106	1.28 × 107	1.31 × 106	9.82 × 106
∑Aν′ν (s–1)	6.57 × 107	6.56 × 107	6.53 × 107	6.26 × 107	5.17 × 107	3.79 × 107	3.81 × 107
τ = 1/Aν′ν (s)	1.52 × 10–8	1.52 × 10–8	1.53 × 10–8	1.60 × 10–8	1.93 × 10–8	2.64 × 10–8	2.63 × 10–8
τ (ns)	15.2	15.2	15.3	16.0	19.3	26.4	26.3

Conclusions

At the MRCI +Q level of theory and with 23 correlated electrons, the electronic structure of 14 spin-free low-lying electronic states of the ytterbium monobromide molecule is investigated in addition to that of 30 states upon considering the SOC effect. The PECs and DMCs of the investigated states are plotted then used to determine the ionic character and spectroscopic constants (Te, Re, ωe, Be, and De) for the bound states. These curves have shown regions of considerable interaction in terms of crossings and avoided crossings especially between 3.0 and 3.6 Å. Perturbed states as (4)1/2, (2)3/2, and (5)1/2 are identified as interactions take place around their potential energy minimum. The TDM between low-lying doublet states is calculated at the spin-free level from which emission coefficients and spontaneous radiative lifetimes are determined. A rovibrational study is conducted with J = 0 to compute the vibrational parameters (Eν, Bν, Dν, Rmin, and Rmax) of the low vibrational levels. All of the molecular constants presented in this work are in very good agreement with the few data available in the literature as obtained from both experimental and theoretical methods. Except for the ground state, this work represents, for the first time, a detailed theoretical study concerning the electronic structure of the YbBr low-lying states.