Accurate vibrational energy spectra and dissociation energies of some diatomic electronic states

Abstract

An algebraic method (AM) used to study the full vibrational spectra of diatomic systems, and an analytical formula used to calculate accurate molecular dissociation energies are applied to study the full vibrational spectra and molecular dissociation energies of some electronic states of homonuclear and heteronuclear diatomic molecules and diatomic ions. Studies show that the AM method and the analytical expression are reliable and economical physical methods for studying full vibrational spectra and molecular dissociation energies of diatomic electronic systems theoretically. They are particularly useful for those diatomic systems whose high-lying vibrational energies may not be available.

Graphical abstract

Keywords

diatomic molecules / molecular ions / vibrational spectrum / dissociation energy / algebraic method

1 Introduction

Molecular vibrational energies, especially accurate high-lying vibrational energies, have very important significance for investigating molecular long-rang interactions and reaction dynamics in dissociation regions. Studies show that the process of chemical reactions not only depends on the system energies, but also on the energy forms. Thus, studying molecular excited states may help to selectively provide effective energies for the reaction systems, such that molecular electronic-vibrational-rotational states are in favor of chemical reactions towards the anticipated routes [1, 2]. Molecular high-lying vibrational (rotational) excited data are also important for studying molecular scattering [3]. In recent decades, many theoretical and experimental studies on vibrational energies of many diatomic electronic states have been made. However, due to the limitations of experimental conditions and techniques, studies on high-lying vibrational energies, especially those near molecular dissociation limits, are rather limited. Therefore, it is still very important to study full vibrational energy spectra of diatomic molecules [1].

The dissociation energy of stable diatomic electronic states plays a significant role in atomic and molecular physics, chemical physics, thermodynamics, molecular spectroscopy and molecular collision physics [4]. Precise knowledge of dissociation energies is particularly important to the spectroscopy of long-range molecules, the light coupling ionization, the ultra-fine pre-dissociation process and the collision physics between ultra-cold atoms. For example, the binding energy (dissociation energy) of transition from the highest vibrational energy to rotationless ground state level determines the s-wave scattering length which gives the low-energy elastic-scattering cross section that is relevant to Bose-Einstein condensation of cold atoms. Accurate molecular dissociation energies must be known in investigations of molecular potentials either using quantum mechanics or analytical potential method. In chemical reactions, correct dissociation energy of each molecular state may help one to understand chemical reaction route, reaction mechanism and reaction products.

However, it is usually difficult to directly measure dissociation energy. Only few works of direct measurements on dissociation energy have been reported. For example, in 1996, Jones et al. [4] obtained accurate dissociation energy of the ground state of Na₂ using three kinds of laser-induced fluorescence spectroscopies. There have been no reports about direct measurements on dissociation energies of molecular electronic excited states, and usually these dissociation energies are indirectly obtained [5]. Theoretically, due to limitations of ab initio method itself, it is difficult to obtain accurate dissociation energies for most of molecular electronic states. The computational process of accurate ab initio studies is usually very complicated, even if for the ground state of simplest molecule H₂, the molecular wave function for calculating accurate dissociation energy contains many constituents and many correction terms [6]. Therefore, it is also necessary to study new physical method for obtaining molecular dissociation energy.

This study uses a new tactic which combines the advantages of accurate experimental measurements and solid theoretical tool to study the full vibrational spectrum {E_υ}, and uses an analytical parameter-free formula to evaluate molecular dissociation energy for a group of stable diatomic electronic states. Section 2 describes the method. Section 3 presents the results and discussions. Section 4 concludes this study.

2 Methods

2.1 Algebraic method (AM) for full vibrational spectra of diatomic molecules

For a stable diatomic molecular electronic state, the non-relativistic analytical expression of vibrational energy of nuclear motion is obtained using second perturbation theory as [7]:

\begin{array}{}\text{(1)}& \begin{array}{ll}{E}_v=\hfill & {\omega }_0+({\omega }_e+{\omega }_{e0})\left(v+\frac{1}{2}\right)-{\omega }_e{x}_e{\left(v+\frac{1}{2}\right)}^2\hfill \\ \hfill & +{\omega }_e{y}_e{\left(\upsilon +\frac{1}{2}\right)}^3+{\omega }_e{z}_e{\left(\upsilon +\frac{1}{2}\right)}^4\hfill \\ \hfill & +{\omega }_e{t}_e{\left(\upsilon +\frac{1}{2}\right)}^5+{\omega }_e{s}_e{\left(\upsilon +\frac{1}{2}\right)}^6\hfill \\ \hfill & +{\omega }_e{r}_e{\left(\upsilon +\frac{1}{2}\right)}^7+\cdots \hfill \end{array}\end{array}

Comparing with Herzberg’s energy expression [8]:

\begin{array}{}\text{(2)}& \begin{array}{ll}{E}_v=\hfill & {\omega }_e\left(v+\frac{1}{2}\right)-{\omega }_e{x}_e{\left(v+\frac{1}{2}\right)}^2\hfill \\ \hfill & +{\omega }_e{y}_e{\left(v+\frac{1}{2}\right)}^3+{\omega }_e{z}_e{\left(v+\frac{1}{2}\right)}^4\hfill \\ \hfill & +{\omega }_e{t}_e{\left(v+\frac{1}{2}\right)}^5+{\omega }_e{s}_e{\left(v+\frac{1}{2}\right)}^6\hfill \\ \hfill & +{\omega }_e{r}_e{\left(v+\frac{1}{2}\right)}^7+\cdots \hfill \end{array}\end{array}

ω₀ and ω_e0 are new small quantities, but they are not neglectable in calculating high-lying rovibrational energies. ω₀ is the vibrational constants of the zero order of (υ + 1/2), ω_e0 is the correction term to harmonic vibrational constant ω_e of the first order of (υ + 1/2). Comparing Eq. (1) with Dunham’s energy expression [9]:

\begin{array}{}\text{(3)}& \begin{array}{ll}{E}_{\upsilon J}\hfill & ={\displaystyle \sum _l}{\displaystyle \sum _m}{Y}_{lm}{\left(v+\frac{1}{2}\right)}^l{[J(J+1)]}^m\hfill \end{array}\end{array}

One has

\begin{array}{}\text{(4)}& \{\begin{array}{ll}{Y}_{00}={\omega }_0;Y_{10}=({\omega }_e+{\omega }_{e0})\hfill & \hfill \\ Y_{20}=-{\omega }_e{x}_e;Y_{30}={\omega }_e{y}_e\hfill & \hfill \\ Y_{40}={\omega }_e{z}_e;Y_{50}={\omega }_e{t}_e\hfill & \hfill \\ Y_{60}={\omega }_e{s}_e;Y_{70}={\omega }_e{r}_e\hfill & \hfill \end{array}\end{array}

The algebraic method (AM) proposed by Sun et al. [7] is started by rewriting the vibrational energy expressions in Eq. (1) into a matrix form:

\begin{array}{}\text{(5)}& AX=E\end{array}

where the vector matrix X of vibrational spectroscopic constants and the energy matrix E are

\begin{array}{}\text{(6)}& X=\left(\begin{array}{c}{\omega }_0\\ {\omega }_e^{\prime }\\ -{\omega }_e{x}_e\\ {\omega }_e{y}_e\\ ⋮\\ {\omega }_e{r}_e\end{array}\right)\mathrm{,}E=\left(\begin{array}{c}{E}_{\upsilon }\\ E_{\upsilon +i}\\ E_{\upsilon +j}\\ ⋮\\ E_{\upsilon +s}\end{array}\right)\mathrm{,}v=\mathrm{0,1,2,}\cdots \end{array}

and the matrix element of the 8 × 8 coefficient matrix A is A_υk = (υ + 1/2)^k, k = 0, 1, 2, ···, 7. In the vector matrix X, {\omega }_e^{\prime }={\omega }_e+{\omega }_{e0}.

Although modern experimental techniques and accurate quantum methods have difficulty obtaining high-lying vibrational energies, especially those near the dissociation limit regions, they can usually obtain an accurate energy subset [E_υ] for most stable diatomic electronic systems. For a given electronic state, provided that m accurate energies in energy subset [E_υ] are known, one may choose 8 of the m known energies to form the vibrational energy matrix E. There are N=C_m^8 groups of choices. Solving Eq. (5) without using any mathematical approximation and physical model, and substituting the obtained N sets of vibrational spectroscopic constants {ω₀, ω_e + ω_e0,−ω_ex_e, ω_ey_e, ω_ez_e, ω_et_e, ω_es_e, ω_er_e} into Eq. (1) respectively, N sets of vibrational spectrum {E_υ} will be generated. One can always find a spectrum {E_υ} which best satisfies the following physical requirements:

\begin{array}{}\text{(7)}& {\frac{\text{d}{E}_v}{\text{d}v}|}_{v=v_{\text{max}}}=0\end{array}

\begin{array}{}\text{(8)}& \overline{\text{Δ}E(e\mathrm{,}c)}=\sqrt{\frac{1}{m}{\displaystyle \sum _{v=0}^{m-1}{\left|E_{v\mathrm{,}\exp }-E_{v\mathrm{,}cal}\right|}^2}}\to 0\end{array}

\begin{array}{}\text{(9)}& \text{Δ}{E}_{v_{\max }\mathrm{,}{v}_{\max }-1}=E_{v_{\max }}-E_{v_{\max }-1}\to \text{Small enough}\end{array}

\begin{array}{}\text{(10)}& D_e^{\exp t}-E_{v_{\max }}\to \text{Small}\text{enough}\end{array}

\begin{array}{}\text{(12)}& D_e^{cal}\approx E_{{\upsilon }_{\max }}+\frac{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-1}^2}{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-2}-\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-1}}\end{array}

The so determined vibrational spectroscopic constants should be the best physical representation of the true vibrational information for a specified molecular system, and the resulting vibrational energies form a full vibrational spectrum {E_υ} containing all correct vibrational energies. This AM spectrum is obtained by combining a limited number of accurate experimental energies with the advantages of standard algebraic computation subjecting to a set of physical converging conditions.

2.2 Formula for dissociation energies of diatomic molecules

A parameter-free analytical formula for dissociation energies of stable diatomic electronic states has been proposed recently by Sun et al. [10, 11]. A brief outline is given below. LeRoy and Bernstein studied the relationship among asymptotic potential V (R), vibrational energy E_υ and dissociation energy D_e of a stable diatomic system [12, 13]. When near molecular dissociation limit, they obtained [12]

\begin{array}{}\text{(14)}& D_e=E_v-\frac{n+2}{2n}\cdot \frac{E_v^{\prime }{}^2}{E_v^{″}}\end{array}

which is the analytical expression for dissociation energy D_e. They further approximated the first order derivative E_{\upsilon }^{\prime } of vibrational energy by [13]

\begin{array}{}\text{(15)}& E_v^{\prime }=\frac{\text{d}{E}_v}{\text{d}\upsilon }\approx \frac{1}{2}\left(E_{v+1}-E_{v-1}\right)+\cdots \end{array}

and shown [12]

\begin{array}{}\text{(16)}& \frac{E_{\upsilon }^{\prime }}{E_{\upsilon }^{″}}=-{\left(\text{π}\hslash \right)}^{-1}{\left(\frac{1}{2}\mu C_2\right)}^{\frac{1}{2}}\end{array}

One may substitute Eq. (15) and Eq. (16) into Eq. (14), and get

\begin{array}{}\text{(17)}& \begin{array}{ll}{D}_e=\hfill & E_v+\frac{1}{2}\cdot \frac{n+2}{2n}(E_{v+1}-E_{\upsilon -1}){(\text{π}\hslash )}^{-1}{\left(\frac{1}{2}\mu C_2\right)}^{\frac{1}{2}}\hfill \end{array}\end{array}

Eq. (17) is an alternative expression of Eq. (14). In LeRoy and Bernstein’s studies [12, 13], a nonlinear least squares fitting was applied to solve Eq. (14) or Eq. (17) to calculate the molecular dissociation energy by using a set of experimental vibrational energies E_υ and a group of initial trial values of dispersion coefficient C₂, asymptotic exponent n and D_e. However, it is not trivial for one to choose a reasonable initial values of C₂, n and D_e. The molecular dissociation energies fitted using these initial values sometimes may have notable errors. Therefore, the LB formula in Eq. (14) has not been widely used.

Based on the LB formula, a new analytical formula for dissociation energy D_e which is independent of dispersive coefficient C_n and asymptotic exponent n can be obtained. Apparently, the vibrational states in Eq. (17) should be those near the molecular dissociation limit, otherwise, the resulting dissociation energies from Eq. (17) will have big errors. Therefore, one may take υ + 1 = υ_max and Eq. (17) becomes

\begin{array}{}\text{(18)}& \begin{array}{ll}{D}_e^{cal}=\hfill & E_{v_{\max }-1}+\frac{1}{2}\cdot \frac{n+2}{2n}\left(E_{v_{\max }}-E_{v_{\max }-2}\right)\hfill \\ \hfill & \cdot {\left(\text{π}\hslash \right)}^{-1}{\left(\frac{1}{2}\mu C_2\right)}^{\frac{1}{2}}\hfill \end{array}\end{array}

Eq. (18) is used to calculate the molecular dissociation energies using the three highest vibrational energies by Ren et al. for some electronic states of N₂ molecule [14]. Their study only uses the known values of n and C₂ without using numerical fitting. However, the n and C₂ values of many electronic states are unknown, so Eq. (18) is also hardly applicable.

In order to obtain an analytical expression of dissociation energies which is free of parameters C₂ and n, one may take υ = υ_max and then Eq. (17) becomes

\begin{array}{}\text{(19)}& \begin{array}{ll}{D}_e^{cal}=\hfill & E_{{\upsilon }_{\text{max}}}+\frac{1}{2}\cdot \frac{n+2}{2n}(E_{{\upsilon }_{\text{max}}+1}-E_{{\upsilon }_{\text{max}}-1})\hfill \\ \hfill & \cdot {\left(\text{π}\hslash \right)}^{-1}{\left(\frac{1}{2}\mu C_2\right)}^{\frac{1}{2}}\hfill \end{array}\end{array}

Subtracting Eq. (18) from Eq. (19) gets

\begin{array}{}\text{(20)}& \begin{array}{ll}{C}_2^{1/2}=\hfill & 2\text{π}\hslash \frac{2n}{n+2}{\left(\frac{2}{\mu }\right)}^{1/2}\hfill \\ \hfill & \cdot \frac{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-1}}{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-2}-\text{Δ}{E}_{{\upsilon }_{\max }+\mathrm{1,}{\upsilon }_{\max }-1}}\hfill \end{array}\end{array}

Substituting Eq. (20) into Eq. (19), one has

\begin{array}{}\text{(21)}& \begin{array}{ll}{D}_e^{cal}\approx \hfill & E_{{\upsilon }_{\max }}+(E_{{\upsilon }_{\max }+1}-E_{{\upsilon }_{\max }-1})\hfill \\ \hfill & \cdot \frac{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-1}}{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-2}-\text{Δ}{E}_{{\upsilon }_{\max }+\mathrm{1,}{\upsilon }_{\max }-1}}\hfill \end{array}\end{array}

Since E_{{\upsilon }_{\text{max}}} is the maximum vibrational energy of a diatomic electronic state, E_{v_{\text{max}}+1} is a “virtual” energy which is not physical and is numerically slightly smaller than E_{v_{\text{max}}}. When E_{v_{\text{max}}+1}-E_{v_{\text{max}}-1} is negative, it may cause an unphysical result . In order to satisfy D_e^{cal}>E_{v_{\text{max}}}, the second term of Eq. (21) must be greater than zero. So one may replace E_{v_{\text{max}}+1} with E_{{\upsilon }_{\text{max}}}, and Eq. (21) becomes

\begin{array}{}\text{(22)}& D_e^{cal}\approx E_{{\upsilon }_{\max }}+\frac{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-1}^2}{\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-2}-\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-1}}\end{array}

where

\begin{array}{}\text{(22)}& \begin{array}{l}\text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-1}=E_{{\upsilon }_{\max }}-E_{{\upsilon }_{\max }-1}\\ \text{Δ}{E}_{{\upsilon }_{\max }\mathrm{,}{\upsilon }_{\max }-2}=E_{{\upsilon }_{\max }}-E_{{\upsilon }_{\max }-2}\end{array}\end{array}

Eq. (22) is the new analytical formula for molecular dissociation energy suggested recently [10, 11]. It indicates that one may obtain the correct molecular dissociation energy D_e^{cal} of a stable diatomic electronic state using its three highest vibrational energies E_{{\upsilon }_{\text{max}}}, E_{v_{\text{max}}-1} and E_{{\upsilon }_{\text{max}}-2}. Since the right hand side of Eq. (15) is always smaller than the left hand side by a very small quantity, the generated D_e^{cal} in Eq. (22) will be a little smaller than the correct experimental dissociation energy D_e^{\exp t}, namely

If Eq. (23) is violated, the three highest vibrational energies used to calculate D_e^{cal} have errors. Since Eq. (22) should use the three accurate highest vibrational energies, it may also be used as a physical criterion judging whether the full vibrational spectrum of a diatomic electronic state is well converged and it is so used in Eq.(12).

3 Results and discussion

3.1 Applications to homonuclear diatomic molecules

The improved computational program of the AM method is used to study the vibrational spectroscopic constants and the full vibrational spectra, respectively, for 10 diatomic molecular electronic states of ⁶Li₂, Na₂, K₂ and Br₂ molecules: the 1¹Π_g state of ⁶Li₂; the X^1{\text{Σ}}_g^{+}, 2^3{\text{Σ}}_g^{+}, 4^3{\text{Σ}}_g^{+}, 2¹Π_g and 2³Π_g states of Na₂; the X^1{\text{Σ}}_g^{+}, 2¹Π_g and 3³Π_g states of K₂; and the X^1{\text{Σ}}_g^{+} state of Br₂. The dissociation energies of the above electronic states are calculated using Eq. (22) and their three highest AM vibrational energies, respectively.

The vibrational spectroscopic constants from references and the AM method of these electronic states are listed in Table 1. It can be seen that the harmonic vibrational constant ω_e from the AM method has an excellent agreement with those from the references, the inharmonic vibrational constant ω_ex_e from the AM method agrees with those from references for most of the molecular electronic states, and some high-lying vibrational spectroscopic constants of several electronic states are not available in literatures. If one uses the literature low-lying vibrational spectroscopic constants to calculate all vibrational energies of these electronic states, the obtained high-lying vibrational energies would not be converged to their dissociation energies correctly and have big errors or are even incorrect.

For the above molecular electronic states, the known experimental vibrational energies, the full AM vibrational energies calculated using the AM vibrational spectroscopic constants and the theoretical vibrational energies calculated using the literature vibrational spectroscopic constants are listed in Table 2, respectively. For each electronic state, the 8 experimental energies chosen in the AM method are marked in bold face. Since the vibrational information and the measurement error contained in different experimental data are different, it is necessary to choose 8 energies which contain the most useful vibrational information and the smallest errors from a set of known experimental vibrational energies by applying a group of physical criteria in Eqs. (7)−(13) that judge the convergence of the calculation in the AM studies. Thus, the so obtained AM vibrational spectra not only reproduce known experimental energies, but also generate all unknown high-lying vibrational energies. Using the three highest AM vibrational energies and the analytical expression in Eq. (22), the correct molecular dissociation energies can be calculated.

It can be seen from Table 2 that for all electronic states of these homonuclear diatomic molecules, the obtained AM dissociation energies D_e^{cal} agree very well with the experimental values D_e^{\exp t}. The percent error of D_e^{cal} (AM) from D_e^{\text{exp}t} is less than 1 %. For example, for the K₂-2¹Π_g electronic state, Zhao et al. [22] gave 23 experimental energies in 1997, the maximum vibrational energy E_{v_{\text{max}}=22}^{\exp t} is 1161.4226 cm ⁻¹ which is 1156.5674 cm⁻¹ smaller than the experimental dissociation energy D_e^{\exp t}(\text{=2318}{\text{.0900 cm}}^{-\text{1}}). However, the full AM vibrational spectrum of this electronic state obtained using the AM method contains 62 vibrational energies with the maximum vibrational energy E_{{\upsilon }_{\text{max}}=61}^{\text{AM}} (=2317.9793 cm⁻¹) which is only 0.1107 cm⁻¹ less than experimental dissociation energy D_e^{\exp t}. Substituting the three highest AM vibrational energies into Eq. (22), the obtained AM dissociation energy D_e^{cal} is 2318.0426 cm⁻¹. The percent error between D_e^{cal} and D_e^{\text{exp}t} is only 0.002 045%. The accuracy of the AM energy spectrum uniquely depends on that of the experimental energies. If the experimental energies are accurate enough, the obtained vibrational spectroscopic constants are accurate. Consequently, the resulting AM vibrational spectrum will reproduce the known experimental energies, generate all high-lying vibrational energies including those in the dissociation region and produce correct dissociation energies.

It is also seen from Table 2 that the theoretical vibrational spectrum \left\{E_{\upsilon }^{\text{Theo}}\right\} obtained using the literature vibrational spectroscopic constants usually reproduce the known experimental energies, but the generated high-lying vibrational energies may have errors, or may even be wrong. For example, for the 3³Π_g electronic state of K₂, Ref. [23] gives 26 experimental energies. However, the number of vibrational energies obtained by using the literature spectroscopic constants from Ref. [23] is 151, and the maximum vibrational energy E_{v_{\text{max}}=150}^{\text{Theo}}=5491.8951{\text{cm}}^{-1}, which severely violates Eq. (11). In order to make a direct comparison, Fig. 1 plots the known (incomplete) experimental energies [E_{\upsilon }^{\exp t}], the AM vibrational spectrum \left\{E_v^{\text{AM}}\right\}, the theoretical vibrational spectrum \left\{E_v^{\text{Theo}}\right\} evaluated using vibrational constants from reference and the RKR potential curves for the state. Similar figures for K₂-2¹Π_g and {\text{Na}}_2-4^3{\text{Σ}}_g^{+} states are also plotted in Fig. 2 and Fig. 3, respectively. However, for some electronic states, the maximum vibrational energies E_{{\upsilon }_{\text{max}}}^{\text{Theo}} obtained using the literature spectroscopic constants are smaller or much smaller than their dissociation energies D_e^{\exp t} (like the {\text{Na}}_2-2^3{\text{Σ}}_g^{+} and the {\text{K}}_2-X^1{\text{Σ}}_g^{+} electronic states), or are not converged (such as the HgI-X²Σ⁺ electronic state in Table 4). This is because many experimental spectroscopic constants are obtained by fitting Dunham’s or Herzberg’s energy expression on known experimental energies. The fitting process not only obtains vibrational information of each energy, but also takes different errors contained in each experimental energy. The resulting spectroscopic constants are contaminated someway by these errors. The high-lying vibrational energies evaluated using the fitted vibrational constants might be poor or very poor, especially those near the dissociation limit. However, although the AM method also uses known experimental energies, it doesn’t use any mathematic fitting and physical approximation and instead selects the best physical representation of the known energies by using a group of physical criteria. This is the reason why AM spectroscopic constants and AM energies are different from those obtained by numerical fitting in literatures.

is an useful empirical criterion judging the correctness of the calculated theoretical dissociations D_e^{cal}. It is seen from Table 2 that all E_rror (AM) except for {\text{K}}_2-X^1{\text{Σ}}_g^{+} and ⁶Li₂- 1¹Π_g states satisfy this criterion. For {\text{K}}_2-X^1{\text{Σ}}_g^{+} and ⁶Li₂-1¹Π_g states, E_rror (AM) are slightly greater than 1. This is because the high-lying vibrational energies near dissociation energies are so dense that the last energy difference \text{Δ}{E}_{v_{\text{max}}\mathrm{,}{v}_{\text{max}}-1} is too small. Furthermore, all AM energy spectra of these states satisfy other criteria of Eqs. (7)−(13). However, for the theoretical vibrational spectra \left\{E_{\upsilon }^{\text{Theo}}\right\} obtained using the corresponding literature spectroscopic constants, all E_rror do not satisfy this criterion.

3.2 Applications to heteronuclear diatomic molecules

Compared with homonuclear diatomic molecules, the atoms in heteronuclear diatomic molecules have different electronegativity, so these molecules usually behave with a certain polarity. The molecules formed from the atoms with large electronegative difference usually have ionic bonds. In this situation, certain potential barriers may exist in the long-range asymptotic region, which will result in a maximum vibrational energy that would be greater than the molecular dissociation energy. This energy does not exceed the maximum value of the potential behavior. For those molecules having a potential barrier, the maximum vibrational energy criterion in Eq. (11) will become , and the potential barrier E_bar is the difference between the maximum value of the potential barrier and the dissociation limit. For many stable heteronuclear systems which have no potential barriers, the AM method can be directly used to study full vibrational spectra and then the analytical expression in Eq. (22) can be used to calculate their dissociation energies.

In this section, the vibrational spectroscopic constants and full vibrational spectra are studied for 12 heteronuclear diatomic electronic states without having noticeable potential barriers: ⁶LiH-C¹Σ⁺, ⁶LiD-C¹Σ⁺, NaK-X¹Σ⁺, NaRb-a³Σ⁺, NaLi-B¹Π, KRb-2¹Π, CsH-X¹Σ⁺, RbH-X¹Σ⁺, ICl-A³Π(1), ClF-A(³Π₁), CO-X¹Σ⁺, and HgI-X²Σ⁺ states. Their dissociation energies D_e^{cal} are calculated using the analytical expression in Eq. (22). The AM vibrational spectroscopic constants and vibrational spectra of these electronic states are listed in Table 3 and Table 4, respectively.

It can be seen from Table 3 and Table 4 that, for the above heteronuclear diatomic electronic states, the qualities of the AM vibrational spectroscopic constants and vibrational spectra are the same as those of homonuclear molecules. The full AM vibrational spectra not only reproduce known experimental energies, but also correctly generate all unknown high-lying vibrational energies. The percent errors between D_e^{cal} and D_e^{\exp t} are all less than 1 %. The greatest is 0.1906 % (CsH-X¹Σ⁺ state), and the smallest is only 0.000 081 72 % (CO-X¹Σ⁺ state).

Table 4 also lists the theoretical vibrational spectrum \left\{E_{\upsilon }^{\text{Theo}}\right\} calculated using vibrational spectroscopic constants from references. Most of these spectra do not correctly represent the vibrational information of high-lying vibrational states near molecular dissociation limit for corresponding electronic systems. In order to make a direct comparison, the AM vibrational spectra \left\{E_{\upsilon }^{\text{AM}}\right\}, theoretical vibrational spectra \left\{E_{\upsilon }^{\text{Theo}}\right\} obtained using the literature spectroscopic constants of ⁶LiH-C¹Σ⁺ and CO-X¹Σ⁺ states and their RKR potential curves are plotted in Fig. 4 and Fig. 5, respectively.

3.3 Application to diatomic molecular ions

The vibrational spectroscopic constants and full vibrational spectra of BeH⁺-X¹Σ⁺, CO⁺-X²Σ⁺, {\text{O}}_2^{+}-A^2{\text{Π}}_u, {\text{F}}_2^{+}-X^2{\text{Π}}_g and {\text{Li}}_2^{+}-X^2{\text{Σ}}_g^{+} states are studied using the AM method. The results are listed in Table 5 and Table 6, respectively.

It is seen from Table 6 that, the AM vibrational spectra \left\{E_{\upsilon }^{\text{AM}}\right\} of the above electronic states of some diatomic molecular ions reproduce the known experimental energies or quantum theoretical vibrational energies ({\text{Li}}_2^{+}-X^2{\text{Σ}}_g^{+} state) and correctly generate all high-lying vibrational energies which may be difficult to obtain experimentally or theoretically. All the AM vibrational spectra satisfy the physical criteria in Eqs. (7)−(13). Therefore the AM vibrational spectrum \left\{E_v^{\text{AM}}\right\} is the correct full vibrational spectra of a given diatomic ionic system. For example, for the {\text{O}}_2^{+}-A^2{\text{Π}}_u electronic state, Coxon et al. [48] observed the second negative band system of the A²Π_u → X²Π_g transition of {\text{O}}_2^{+} ion and 16 vibrational energies of the {\text{O}}_2^{+}-A^2{\text{Π}}_u state are fitted based on the positions of spectroscopic lines in each spectrum band. Using this set of vibrational energies, 29 AM vibrational energies are obtained using the AM method. The maximum AM vibrational energy E_{{\upsilon }_{\text{max}}=28}^{\text{AM}} is 14 168.285 0 cm⁻¹ and the percent error of E_{{\upsilon }_{\text{max}}=28}^{\text{AM}} from experimental dissociation energy D_e^{\exp t}(=14195.3520{\text{cm}}^{-1}) is only 0.1907 %. Therefore, the AM spectroscopic constants and corresponding vibrational spectrum \left\{E_v^{\text{AM}}\right\} of this state can correctly represent the true vibrational information of this system. The relative percent error of the AM dissociation energy D_e^{cal}(=14181.4705{\text{cm}}^{-1}) from experimental dissociation energy D_e^{\exp t} is 0.0978 %which is within the error between different experimental measurements.

{\text{Li}}_2^{+} is one of the simplest diatomic molecular ions except for {\text{H}}_2^{+}. Although it has been the object of many experimental and theoretical studies, there has been no experimental report on the accurate vibrational energies of the {\text{Li}}_2^{+}-X^2{\text{Σ}}_g^{+} state. Schmidt-Mink et al. [46] used self-consistent field configuration interaction (SCF-CI) with core polarization potentials (CPP), including closed-shell correlation effect, to obtain a theoretical subset [E_{\upsilon }^{\text{Theo}}] of the vibrational spectrum of this state. Since the kinetic energy of nuclear motion is large and the coupling interaction between electrons and nuclei is not negligible when {\text{Li}}_2^{+} is in high-lying vibrational motion, there will be a notable error due to the breakdown of the Born-Oppenheimer approximation if the SCF-CPP is used to calculate high-lying vibrational energies. Therefore, Schmidt-Mink only obtained some energies of its vibrational spectrum {E_υ}. The highest vibrational energy E_{{\upsilon }_{\text{max}}=57}^{\text{SCF}-\text{CPP}}(=9929{\text{cm}}^{-1}) is 535 cm⁻¹ smaller than its experimental dissociation energy D_e^{\exp t}(=10464{\text{cm}}^{-1}) [44]. The 58 SCF-CPP theoretical energies of this state are used for the AM study, and the obtained AM vibrational spectrum reproduces those SCF-CPP vibrational energies and generates all high-lying vibrational energies which are not given by the SCF-CPP method. The maximum vibrational energy E_{{\upsilon }_{\text{max}}=72}^{\text{AM}}(=10463.6901{\text{cm}}^{-1}) is only 0.3099 cm⁻¹ smaller than the experimental dissociation energy D_e^{\exp t}.

For the BeH⁺-X¹Σ⁺ and CO⁺-X²Σ⁺ states, the relative errors E_rror are 1.3516 and 1.0622, respectively. This is because their (D_e^{\exp t}-D_e^{cal}) and \text{Δ}{E}_{v_{\max }\mathrm{,}{v}_{\max }-1} have comparable values.

The vibrational spectra \left\{E_{\upsilon }^{\text{Theo}}\right\} obtained using the literature spectroscopic constants are also listed in Table 6. These spectra may not represent the correct high-lying vibrational information of a given ionic system. For example, for the {\text{F}}_2^{+}-X^2{\text{Π}}_g state, the vibrational spectrum \left\{E_{\upsilon }^{\text{Theo}}\right\} using spectroscopic constants from Ref. [43] gives 61 energies and has many wrong energies since υ=36. The obtained theoretical energy D_e^{cal}(=33088.5471{\text{cm}}^{-1}) is much greater than its experimental dissociation energy D_e^{\exp t}(=27472.8409{\text{cm}}^{-1}), namely, Eq. (11) is not satisfied. This indicates that this set of spectroscopic constants have notable physical deficiencies. However, the AM dissociation energy D_e^{\text{AM}}(=27412.6285{\text{cm}}^{-1}) is only 60 cm⁻¹ smaller than the D_e^{\exp t}. The AM spectrum \left\{E_v^{\text{AM}}\right\} and the \left\{E_{\upsilon }^{\text{Theo}}\right\} are plotted in Fig. 6 together with their RKR potential curves. It should be noted that there are only 16 experimental vibrational energies available for this state. If two or three more accurate energies are given experimentally, the accuracy of the full AM vibrational spectrum \left\{E_{\upsilon }^{\text{AM}}\right\} and the dissociation energy D_e^{cal}(\text{AM}) would be better. A similar figure for CO-X²Σ⁺ state is shown in Fig. 7.

3.4 Influence of experimental energy subsets [E_υ] on the AM vibrational energy spectra

From the AM process of obtaining molecular full vibrational spectra, it can be seen that when constructing algebraic equation A·X = E using the known experimental energies, different input energies E have different influences on the obtained vibrational spectrum. That is to say, the correctness and accuracy of the AM vibrational spectrum are determined by those of the chosen experimental energies and the vibrational information thereof. On one hand, if there are many accurate experimental energies which contain enough vibrational information, the extracted vibrational spectroscopic constants using the AM method will be accurate, and so is the AM spectrum \left\{E_v^{\text{AM}}\right\}. On the other hand, low-lying vibrational energies usually contain more low-lying vibrational information and high-lying vibrational energies contain more high-lying vibrational information. If the excited vibrational states are not high enough, the high-lying vibrational information embedded in the input energies is limited. Therefore, the vibrational spectroscopic constants obtained using these energies may not well represent the true high-lying physical pictures and although the obtained full vibrational spectrum can assure the accuracy of low-lying vibrational energies, it may not give accurate high-lying vibrational energies. The more the number of correct high-lying experimental energies are used, the more accurate the high-lying AM vibrational energies will be. Therefore, we should choose 8 energies which contain more high-lying ones and best satisfy Eqs. (7)−(13) from all known energies.

As an example, the influence of different input energy combinations on AM vibrational spectroscopic constants and vibrational spectra of NaRb-a³Σ⁺ state is given in Table 7 and Table 8, respectively. M represents the number of experimental energies listed in the 2nd column used in the energy subset [E_υ]_M in Table 8. One chooses 8 experimental energies which best satisfy the physical requirements in Eq. (7)−Eq. (13) from each energy subset (each M) and obtains the best AM vibrational spectrum for the corresponding subset using the AM method. For instance, the first 15 experimental energies in the 2nd column are included in the subset of M =15. In the 6th column of M =15, all AM vibrational energies are listed and the 8 energies used in the AM study are printed in bold face. One can see from Table 8 that except for M =16, as M is increasing (as more experimental energies are included in a subset), the values of E_{{\upsilon }_{\text{max}}}(M) and D_e^{cal}(M) are increasing and the values of the percent errors ΔD_e %(M) and the relative errors E_error (M) are decreasing. Therefore, the accuracy of the AM spectrum \left\{E_{\upsilon }^{\text{AM}}\right\} gets better as M increasing. For M =18, the last AM energy difference \text{Δ}{E}_{{\upsilon }_{\text{max}}\mathrm{,}{\upsilon }_{\text{max}}-1} is 0.2489 cm⁻¹ and the difference between the maximum AM vibrational energy E_{v_{\text{max}}=19}(=181.9440{\text{cm}}^{-1}) and D_e^{\exp t}(=182.00{\text{cm}}^{-1}) is only 0.056 cm⁻¹. Thus, the corresponding AM spectrum \left\{E_{\upsilon }^{\text{AM}}\right\} obtained from M =18 can be taken as the correct full vibrational spectrum for this state.

It should be noted that since the accuracy of the AM vibrational spectrum is uniquely determined by the accuracy of experimental energies, if the accuracies of the known experimental energies are not enough or if the given energies do not contain enough correct vibrational information, then the obtained AM vibrational spectrum would be less accurate or may even be wrong. For example, for the NaK-3¹Π state in Table 9, Laub et al.[51] obtained 23 vibrational energies using optical-optical double resonance spectroscopy (OODR). The highest vibrational energy E_{v_{\text{max}}=22}^{\exp t}(=894.3010{\text{cm}}^{-1}) is 396.599 cm⁻¹ smaller than the experimental dissociation energy D_e^{expt}(\text{=}1290.90{\text{ cm}}^{-1}). When this set of experimental energies is used on the AM study, one obtains 39 vibrational energies of which the maximum vibrational energy E_{{\upsilon }_{\text{max}}=38}(=1130.8147{\text{cm}}^{-1}) is still 160 cm⁻¹ smaller than the D_e^{\exp t}. Therefore, this spectrum is not the best representation of the true vibrational spectrum of this state. If the experimental energies are more accurate or there are more high-lying experimental vibrational energies available, the quality of the obtained full AM vibrational spectrum would be much better. The AM vibrational spectra of the NaK-5¹Σ⁺ and Rb₂-1¹Π_g states are also listed in Table 9, which have similar results as those of NaK-3¹Π state. The vibrational spectroscopic constants of these electronic states are listed in Table 10.

4 Summary

An algebraic method (AM) used to study the full vibrational energy spectra of diatomic systems proposed by Weiguo Sun et al. and an analytical formula used to calculate accurate molecular dissociation energies are described. At present, accurate vibrational energies near molecular dissociation limits for many diatomic electronic states may not be available experimentally or theoretically. The AM method and the new formula for dissociation energy are applied to study the vibrational spectroscopic constants, the full AM vibrational spectra and the molecular dissociation energies of some electronic states of some homonuclear and heteronuclear diatomic molecules and diatomic ions. Studies show that:

(1) For many diatomic molecular or ionic electronic states, although there are many results on their vibrational energies, the data of high-lying vibrational energies near dissociation limits are still sorely lacking. The AM method generates accurate vibrational spectroscopic constants and full vibrational spectrum for a diatomic electronic state by using some accurate experimental energies and a set of physical requirements satisfied by correct vibrational energies. The AM vibrational spectrum not only reproduces the known experimental energies, but also obtains all high-lying vibrational energies near dissociation limits which may be difficult to obtain experimentally.

(2) For some diatomic molecular or ionic electronic states, it may be difficult to obtain their accurate molecular (or ionic) dissociation energies using modern quantum methods and it is costly to obtain them experimentally. However, the parameter-free analytical expression we proposed recently is only the function of the three highest vibrational energies near the dissociation limit. Therefore, if we use the three accurate highest vibrational energies, the analytical formula will generate accurate molecular dissociation energies. The AM dissociation energies D_e^{\text{AM}} are in excellent agreement with the experimental dissociation energies D_e^{\exp t} and the percent error of D_e^{\text{AM}} from D_e^{\exp t} is only several thousandths or even several ten millionths.

In summary, the AM method, together with the analytical formula for dissociation energy, supplies a reliable and economical physical method to study the full vibrational spectrum and the molecular dissociation energy of a stable diatomic molecular and ionic system theoretically. This method is particularly useful for those diatomic systems whose high-lying vibrational energies may not be available experimentally. The parameter-free analytical formula also provides an effective physical tool to check the quality of any (experimental or theoretical) full vibrational spectrum.

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