•An improved analytical spectral formula with Hυ term for P-branch is suggested.•A physical requirement is added to minimize the possible error in prediction.•Accurate P-branch rotational lines of NaF and CuCl are predicted by the new formula.•The accuracy of the new rotational lines is much better.•Comparisons between physical predictions and mathematical extrapolations are given.The analytical formula derived by Sun et al. in 2011 and used to predict the rotational lines for rovibrational diatomic systems is improved in this study. The new formula is obtained by adding a higher order spectral term Hυ that is neglected in our previous expression. A physical requirement is also added to the converging process to minimize the possible error of the predicted rotational line. All these are applied to study some rovibrational transition systems of 63Cu35Cl and NaF molecules. The results indicate that the accuracy of the P-branch rotational lines predicted by this new formula is about one order of magnitude better than the results obtained using the previous formula, and that both the small Hυ contribution and the improved converging requirement may play a vital role in predicting the high-lying rovibrational energies and the rotational lines. Comparisons between physical predictions and mathematical extrapolations on the rotational lines are also given.Fig. 1 shows the spectral differences between P-branch experimental and theoretical rotational lines calculated using Eq.  (1) and Eq. (2) for the (0–0) band of the A′3Σ+ − X1Σ+ transition of the 63Cu35Cl molecules. It can be seen that the magnitude of the largest spectral differences (○) between experimental and theoretical rotational lines with the Hυ term are much smaller than those (+) between experimental and theoretical rotational lines without the Hυ term, so the precision of the current P-branch rotational lines is higher than the results obtained using the previous formula in Eq.  (2).Fig. 1 Comparisons of the spectral differences between P-branch experimental and theoretical rotational lines calculated using Eq.  (1) (○) and Eq. (2) (+) for the (0–0) band of the A′3Σ+ − X1Σ+ transition for the 63Cu35Cl molecules.

•Improved analytical spectral formulae with Hυ for R and Q branches are proposed.•A new converging standard is added to minimize the possible error in prediction.•Accurate rotational lines of TiF and 193IrN are predicted using new formulae.•The new rotational lines have much better accuracy.The difference converging method (DCM) used to predict the R-branch and the Q-branch high-lying rotational lines for diatomic systems is improved in this study. The key analytical formulae of the DCM method are modified by adding a higher order spectral term Hυ, and adding a physical converging criterion to improve the accuracy of predictions. Applications of the improved DCM method to the R-branch of the TiF molecule and the Q-branch of the 193IrN molecule show that the accuracy of the R-branch and the Q-branch rotational lines is about one order of magnitude better than the results obtained using the previous formulae, which demonstrate the necessity of the added small term Hυ and the physical converging criterion. The DCM results are also shown to be better than the extrapolated rotational lines using the least-squares method.Fig. 1 shows the spectral differences between Ree branch experimental νexpt and theoretical rotational lines calculated using Eq. (1)νh and Eq. (3)νd for the (0–1) band of the F4Δ7/2 − X4Φ9/2 transition of the TiF molecule. It is known that the magnitudes of the spectral differences (○) between experimental and theoretical spectral lines with the Hυ term are much smaller than those (+) between experimental and theoretical spectral lines without the Hυ term, therefore the accuracy of present Ree branch rotational lines is about one order of magnitude better than the results obtained using the previous formula in Eq. (3). Similar improvement and observations are also obtained for Q branch rotational lines.

•We modify our algebraic method (AM) to be a variational AM (VAM).•The VAM can adapt to individual physical nature of different diatomic systems.•The VAM can offset the possible errors that may exist in experimental data.•The VAM can predict the full vibrational spectra for some specific diatomic systems.The algebraic method (AM) proposed by Sun et al. is improved to be a variational AM (VAM) to offset the possible experimental errors and to adapt to the individual energy expansion nature of different molecular systems. The VAM is used to study the full vibrational spectra {Eυ} and the dissociation energies De of 4HeH+–X1Σ+, 7Li2–13Δg, Na2–C1Πu, NaK–71Π, Cs2–B1Πu and 79Br2–β1g(3P2) diatomic electronic states. The results not only precisely reproduce all known experimental vibrational energies, but also predict correct dissociation energies and all unknown high-lying levels that may not be given by the original AM or other numerical methods or experimental methods. The analyses and the skill suggested here might be useful for other numerical simulations and theoretical fittings using known data that may carry inevitable errors.The results of the Na2–C1Πu   state have shown that the new VAM (variational algebraic method) energies EυVAM(5) beautifully reproduce the experimental data and converge to the correct dissociation limit, while the original AM energies EυAM(8) converge to a wrong dissociation limit that is 2159.066 cm−1 lower than the experimental dissociation energy! Their RKR curves are shown in Fig 1.

•An algebraic method for rotational energies (AMr) is proposed.•The accurate rotational energies {ɛJ  } of 7Li27Li2 and NaF are obtained at first time.•The accurate rovibrational interaction energies ευJint have been studied at first time.•The error of the well known rigid rotor rotational energies has been studied.An algebraic method for rotational energies (AMr) is proposed to unearth the rotational spectrum {ɛJ  } and the rovibrational interaction energies ευJint that are hidden in the rovibrational energies EυJ  . The applications to the excited electronic state a3Σu+ of 7Li27Li2 and the ground state X1Σ+ of NaF molecules show that: (1) the rotational energies ɛJ   of the lighter 7Li27Li2 molecule have better accuracies than the widely used rigid rotor rotational energies εJrr particularly for the lowest two rotational states, while the rigid rotor model produces satisfied rotational energies for the heavier NaF   molecule and (2) the attractive rovibrational interaction energies ευJint stabilize a molecular rovibrational system.The figure shows the rovibrational interaction energies ευJint for the state a3∑u+ of 7Li27Li2 molecule, and indicates that the absolute values of rovibrational interactions are getting greater as either vibrational state υ or rotational state J   increases. The fact that all interaction energies ευJint are negative confirms that the rovibrational interaction energies are attractive, and that it is the rovibrational coupling makes the rovibrational energy EυJEυJ smaller than the sum Eυ+JEυ+J of its corresponding vibrational energy ευευ and the rotational energy εJεJ and makes the rovibrational system stable

VO and other vanadium oxides are important in catalysts, semiconductors and optical devices. Studying its interior microstructure is necessary for fully understanding its intrinsic nature and better applications. The P and R-branch emission spectra of the (0, 0) band in the 2Φ5/2–12Δ3/2 system of VO molecule are studied using the analytical formulae derived by Sun group in their previous work. The calculated result reproduced all known experimental spectral lines accurately, and the correct values of the unknown spectral lines up to J = 80.5 were predicted in this work that were not given experimentally.Graphical abstractExperimental (“—”) and theoretical (“- - - -”) emission spectra of P-branch of (0, 0) band in the 2Φ5/2–12Δ3/2 system of VO.Highlights► High-lying values of spectral lines of rovibrational transition are important. ► Those values are difficult to obtain experimentally or quantum mechanically. ► Analytical formula which are proposed to predict the values in our previous work. ► This study generates correct data of VO. ► Supplies reliable spectral lines with high-lying rotational states of VO.

•VN is an important metallurgical additive and a good electrode material molecule.•It is necessary to understand its intrinsic microstructure for better applications.•Analytical formula which are proposed to predict the values in our previous work.•This study generates four groups’ correct data of VN.•Supplies a reliable and economical physical method theoretically.Vanadium nitride is an important metallurgical additive and a good electrode material molecule. It is necessary to understand its intrinsic microstructure for better applications. Four groups of known experimental transition data of low-lying rotational quantum states and the analytical formula developed recently by Sun group are used to study the Ree, Pee, Rff and Pff  -branch emission spectra of the (0, 0) band of the e1Π-a1Δe1Π-a1Δ system of VN molecule respectively. The results not only reproduce all known experimental spectral lines accurately, but also generate valid data of the unknown spectral lines up to J = 80 that may not be available experimentally.Graphical abstract

•High-lying values of spectral lines of rovibrational transition are important.•Those values are difficult to obtain experimentally or quantum mechanically.•Analytical formula which are proposed to predict the values in our previous work.•This study generates correct data of NbN.•Supplies a reliable and economical physical method theoretically.The P and R-branch spectral lines of high-lying rotational quantum states of (0, 0), (1, 1) bands in d1Σ+-A3Σ0- electronic transition of NbN are obtained in this work by using the analytical formula proposed recently by Sun et al. The formula have generated unknown P and R-branch emission spectral lines up to J = 80 those may not be available experimentally for these bands of NbN molecule.Graphical abstractFig. 1. Experimental (“∗”) and theoretical (“○”) transition line of P-branch emission spectra of (0, 0) band of d1Σ+–A3Σ0- system of NbN.

An analytical formula is suggested to predict the accurate Q-branch spectral lines of rovibrational transitions for diatomic systems by taking multiple spectral differences, and is applied to study the high-lying Q  -branch emission spectra of the (0, 0) band of the Γ5/24−Φ3/24 and Γ7/24−Φ5/24 systems of TiF molecule using fifteen known accurate experimental transition data. The results show that not only the known experimental transition lines are accurately reproduced, but also the correct values of the unknown spectral lines are predicted.Graphical abstractHighlights► We propose a formula to predict Q-branch rovibrational transition spectral lines. ► The formula can reproduce all known experimental rovibrational transition lines. ► The formula can predicts the transition spectral lines missed in experiments.

The P-branch spectral lines of high-lying rotational quantum states of (2, 1), (1, 1) bands in [12.8] 2Φ7/2-a2Φ7/2 electronic transition of TiCl are obtained in this work by using an analytical formula proposed recently by Sun et al. The formula has generated 31 unknown P-branch emission spectral lines up to J = 80.5 those may not be available experimentally for these bands of TiCl molecule.Graphical abstractExperimental (“○”) and theoretical (“+”) transition line differences (ΔJ = νJ−1 − νJ) of P-branch emission spectra of (1, 1) band of [12.8] 2Φ7/2-a2Φ7/2 system of TiCl.Highlights► High-lying values of spectral lines of rovibrational transition are important. ► Those values are difficult to obtain experimentally or quantum mechanically. ► An analytical formula which is proposed to predict the values in our previous work. ► This study generates correct data of TiCl. ► Supplies a reliable and economical physical method theoretically.

An analytical formula based on the Herzberg's conventional rovibrational energy levels for diatomic system is proposed by taking multiple differences of spectral lines to predict the R-branch high-lying rovibrational emission spectroscopy, where only 15 accurate known transition lines and rotational constants Dυ′Dυ′, Dυ″Dυ″ are needed. Using the formula, the R11ee and R22ff branches of (0, 2) and (0, 3) transition bands in the B2Σ+–X2Σ+ system of 12C17O+ are studied. The results show that not only the relatively lower order rovibrational transition lines given by experiments are reproduced but also the higher and the absent spectral lines are correctly predicted for each band.Graphical abstractHighlights► An analytical formula used to predict R-branch rovibrational transition spectral lines is proposed without using any mathematical approximation and physical model. It uses only 15 accurate known transition lines and rotational constants Dυ′Dυ′ and Dυ″Dυ″. ► The formula proposed can reproduce all known experimental rovibrational transition lines and the spectral line differences. ► The formula proposed can accurately predict the absent or the high-lying transition spectral lines that may not be available experimentally.