High level ab initio calculations are carried out by using the Molpro 2010 package [22]. We use the internally contracted MRCI method. Since the lithium dimer is a diatomic molecule with symmetry group D_∞h and there is only Abelian point group in the Molpro package, the largest subgroup D_2h of D_∞h is selected in our calculations. The D_2h point group includes A_g/B_3u/B_2u/B_1g/B_1u/B_2g/B_3g/A_u irreducible representations. The augmented correlation consistent basis sets (aug-cc-pVXZ) (X = T, Q, 5) are employed to describe the lithium dimer. In the complete active space self-consistent field (CASSCF) calculations, ten molecular orbitals including three orbitals of A_g symmetry, one orbital of B_3u symmetry, one orbital of B_2u symmetry, three orbitals of B_1u symmetry, one orbital of B_2g symmetry, and one orbital of B_3g symmetry are chosen as the active space for the six electrons. Based on the CASSCF wave functions, we perform MRCI calculations at a series of given internuclear distances from 1.2 to 20 Å for the PECs of the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state and the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state of the lithium dimer with different basis sets.

To construct an analytical potential energy function (APEF), many analytical function formulas have been proposed, such as Murrel–Sorbie (MS) [23], Tietz [24], and Wei [25] potential energy functions, etc. Among them, the MS potential energy function is widely and successfully applied in the construction of APEFs for many diatomic molecules [26–28]. The MS function can be described as [23]where ρ = R – R_e, D_e is the potential well depth, R_e is the equilibrium bond length and R is the internuclear distance. The parameters D_e, R_e, and a_i can be determined by the nonlinear least square fitting method. In general, the accuracy of the MS function increases with the term number, n, and for light molecules, satisfactory results can usually be obtained when n is equal to 3 or 4 [26, 27]. However, because the values of the potential well depths of the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ and ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ states of the lithium dimer are similar to some heavy diatomic molecules which have deep potential well [28, 29], we choose n = 11 to get satisfactory results after a series of attempts.

The spectroscopic constants could be obtained based on the MS function. The quadratic, cubic, and quartic force constants can be expressed as and the spectroscopic constants are expressed as where μ is the reduced mass of Li₂, and c is the speed of light in vacuum. The spectroscopic parameters, B_e and α_e are the rotational constants at the equilibrium; ω_e and ω_eχ_e are the harmonic vibrational frequency and the second term of vibrational constant, respectively.

For each basis sets (AVTZ, AVQZ, AV5Z), we calculate 194 and 198 energy points for the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ and ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ states with R ranging from 1.2 to 20 Å, respectively. Generally, there are two approaches to increase the fitting accuracy. One is to optimize the distribution of the ab initio energy points by increasing the density of the energy points in the small R region where the interaction of the two atoms is strong. The other is to increase the fitting terms n of equation (1) to increase the accuracy with a high-order fitting function. Here, we adopt both approaches. The former is easy to achieve, and the latter is also computationally affordable for the diatomic molecular potential which only contains one dimension. To describe the important interaction region of the two atoms, we consider a dense grid for R < 10 Å with the grid gap of roughly 0.05 Å, while for the large internuclear distance of R ≥ 10 Å, the sparse grid with the gap of roughly 0.5 Å is used. Taking the calculation of MRCI/AV5Z for example, the ab initio energy points for the two electronic states are presented in figure 1. The quantum chemical calculations for these energy points converge well, and the potential energy varies smoothly with the increase of R.

We obtain the APEFs of the ground and first excited states of Li₂ by fitting the ab initio energy points to equation (1) with the fitting term number of n = 11. The fitting parameters for the PECs based on the ab initio energy points of AVTZ, AVQZ, and AV5Z are presented in table 1, respectively. R_e for the three sets of ab initio data are close to each other for the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state. However, D_e presents significant difference between the AVTZ and the AVQZ data (∼80 cm⁻¹), while D_e for AVQZ and AV5Z are much closer (∼14 cm⁻¹). Thus, out of the three bases, AV5Z is considered to be the most suitable for the lithium dimer. The corresponding fitting results for the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state with AV5Z basis set are listed in the last column of table 1. As shown in figure 2, the fitting curves for both the ground and excited states can well pass through all the ab initio energy points and the fitting curves are smooth with the variation of R. The adiabatic excitation energy (T_e) is 13 986.2 cm⁻¹ which is close to the previous experimental report (14 068 cm⁻¹) [30]. The atomic excitation energy (E_a) is 14 849.2 cm⁻¹ compared to the experiment (14 903 cm⁻¹) [31].

To further illustrate the validity of the MS fitting functions, we present the fitting error for each energy point in figure 3. Here, we compared two kinds of fittings: one is to set the term number n in the fitting function of equation (1) to be 9, and the other is to set n = 11. Obviously, with the increase of the term number, the fitting error decreases dramatically. The root means square (rms) errors can be calculated as ${\rm{RMS}}=\sqrt{\tfrac{1}{N}\displaystyle {\sum }_{i=1}^{N}{\left({V}_{{\rm{APEF}}}-{V}_{ab\,initio}\right)}^{2}},$ where N is the number of the data. As shown in figures 3(a) and (b), for n = 9, the rms = 3.0284 cm⁻¹ for the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state is smaller than that (rms) for the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state. The distribution of the fitting error with the variation of R is also different between the two electronic states. The fitting error for the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state is large for the short R region (R < 2 Å) with the largest error of roughly 14 cm⁻¹, while the fitting for the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state presents large errors (∼40 cm⁻¹) in both the short R region (R < 2 Å) and the asymptote region (R > 10 Å). This is because that the potential energy varies drastically in the short R region for both electronic states, and that potential energy of the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state presents a relatively long-range interaction to approaching the asymptote limit than the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state does. Nevertheless, the fitting error can be declined by increasing the fitting terms n. As shown in figures 3(c) and (d), for n = 11, the fitting errors for both states in all R region are appreciably smaller than those for n = 9. The fitting error for the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state for the short R region (R < 2 Å) has been decreased to within roughly 4 cm⁻¹. The fitting errors for the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state in the short R region (R < 2 Å) and the asymptote region (R > 10 Å) are now decreased to within 5 cm⁻¹ and 10 cm⁻¹, respectively. These errors are actually considerable small compared to the deep well depths of the two states (D_e = 8428 cm⁻¹ for ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ and D_e = 9291 cm⁻¹ for ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$). In addition, the fitting errors for the two states at the equilibrium R_e are both extremely small (1.176 cm⁻¹ for ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ and 0.7788 cm⁻¹ for ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$). The rms errors for n = 11 and for different basis sets are also listed in table 1. The rms is 0.8166 cm⁻¹ for the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state in our fitting to MRCI/AV5Z. Even for the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state, the rms is only 2.788 cm⁻¹ which is much smaller than the permitted chemical accuracy (1.0 kcal mol⁻¹ or 349.755 cm⁻¹) and proves the high quality of the fitting process. The small fitting error also indicates that the MS function is suitable for the description of the ground and first excited states of Li₂.

Hereinbefore, we demonstrate the fitting process and determine the APEFs with n = 11 based on the energy points calculated by MRCI/AV5Z. Now, we further compare the highly accurate fitting functions with previous reported spectroscopic constants in theory and experiment. The comparison for the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state is shown in table 2. Obviously, the values calculated from our APEFs, including the spectroscopic constants, equilibrium bond lengths R_e (Å), and potential well depth D_e (cm⁻¹) are all in good agreement with the experiments [21, 30], compared with previous theoretical reports [14, 18, 19, 33, 34, 38, 39]. The spectroscopic constants, equilibrium bond lengths R_e, and potential well depth D_e of the first excited state ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ are shown in table 3. It can be seen that the fitting APEFs for the excited state can also present accurate spectroscopic constants in good agreement with experimental and theoretical reports [14, 18, 32–34, 36, 40]. Thus, these APEFs based on MRCI/AV5Z for the ground and first excited states of the Li₂ dimer are reliable for future studies in spectroscopy and molecular dynamics.

Based on this newly constructed APEFs, we further calculated the vibrational energy levels by solving the following time-independent Schrödinger equation of nuclear motion using the Fourier grid Hamilton method [37].where μ is reduced mass, j is rotational quantum number, v is vibrational quantum number, and V⁽ⁱ⁾(R) is the APEF of the ith electronic state, i = ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ and ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}.$

Within the R range from 1.2 to 20 Å, we use 1024 grid points in the computation of the vibrational levels of the ground and first excited states of the lithium dimer. The density of the grid points has been checked to be converged by using even denser grids. The obtained vibrational levels (with j = 0) of the ground state of ⁷Li₂, ⁶Li₂, and ⁶Li⁷Li are listed in table 4, where δ is the relative difference between the present calculation and the previous experiment measurement [21]. The ground vibrational level of the ${\rm{X}}{}^{{\rm{1}}}{\rm{\Sigma }}_{g}^{+}$ state is close to the experimental result with the rather small difference of 1 cm⁻¹. The relative difference δ for every vibrational level is less than 1.5%. In general, the APEF of the ground state of Li₂ from equation (1) is reliable. Due to the decrease of the reduced mass, the number of vibrational levels for ⁶Li₂ (⁶Li⁷Li) is smaller than that for ⁷Li₂, and for given vibrational level, the eigenenergy for the ⁶Li₂ (⁶Li⁷Li) dimer is higher than that for the ⁷Li₂ dimer.

The vibrational levels (with j = 0) of the first excited state ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ of Li₂ are listed in table 5. The differences between our calculations and the previous experimental reports for the vibrational levels of the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state are within 1.5%. Thus, the fitting APEFs for the two electronic states can also well describe the corresponding vibrational levels.

Based on the obtained vibrational level v′ = 0 of the ground state and all the bound vibrational levels of the ${\rm{A}}{}^{{\rm{1}}}{\rm{\Sigma }}_{u}^{+}$ state of Li₂, we calculate the corresponding FCFs, $\left\langle {{\rm{\Psi }}}_{v^{\prime} ={\rm{0}}}^{X}\left|{{\rm{\Psi }}}_{v^{\prime\prime} =0}^{A}\right.\right\rangle ,$ which are illustrated in figure 4 and the specific values are listed in table 6. As seen in figure 4 and table 6, the maximum FCF corresponds to the overlap between v′ = 0 and v″ = 0, and the value of FCF decreases dramatically with the increase of the vibrational quantum number v″ of the excited state. For v″ greater than 7, one cannot figure out its contribution from figure 4. It is because that the equilibriums for the PECs of the two electronic states are similar, roughly 2.7 and 3.1 Å and that the width and depth of the two potential wells are also comparable. Thus, the ground vibrational wavefunctions for two electronic states are similar in shape and position, which can present the largest FCF.