Studies on the full vibrational energy spectra for some electronic states of diatomic molecular ions XY⁺

Abstract

The first accurate studies on the vibrational spectroscopic constants and the corresponding full vibrational energy spectra of some electronic states of diatomic molecular ions XY ⁺ were performed using algebraic method(AM). The AM is applied on the {\mathrm{X}}^1{\mathrm{\Sigma }}^{+} state of \mathrm{B}\mathrm{e}{\mathrm{H}}^{+}, the \mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}^{+} state of \mathrm{C}{\mathrm{O}}^{+}, the \mathrm{X}{}_{}{}^2\mathrm{\Pi }{}_{\mathrm{g}} state of {\mathrm{F}}_2^{+}, the \mathrm{A}{}_{}{}^2\mathrm{\Pi }{}_u state of {\mathrm{O}}_2^{+} and the \mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}_{\mathrm{g}}^{+} state of \mathrm{L}{\mathrm{i}}_2^{+}. The results show that AM can generate accurate vibrational spectroscopic constants as well as accurate full vibrational energy spectra by using some accurate experimental vibrational energies, and that the AM vibrational energies are better than other theoretical data.

Graphical abstract

Keywords

diatomic molecular ion / vibrational energy / algebraic method / electronic state

1 Introduction

The electronic states, spectral properties, vibrational energy spectra and potential energy curves of diatomic molecular ions XY⁺ have been studied by spectroscopists, theorists and chemists [1–6]. However, data on the full vibrational energy spectra including the maximum vibrational energy are wanting experimentally and theoretically. The accurate spectroscopic constants and the full vibrational energy spectra of \mathrm{B}\mathrm{e}{\mathrm{H}}^{+}, \mathrm{C}{\mathrm{O}}^{+}, {\mathrm{F}}_2^{+}, {\mathrm{O}}_2^{+} and \mathrm{L}{\mathrm{i}}_2^{+} are particularly needed.

The accurate vibrational energies (\upsilon =0–10) of the \mathrm{B}\mathrm{e}{\mathrm{H}}^{+}-\mathrm{X}{}_{}{}^1\mathrm{\Sigma }{}^{+} state were determined by Coxon and Colin [7] through the latest experimental spectral information. Nevertheless, they admitted that some other subtle spectra between E_{\upsilon } (\upsilon =10) and the correct dissociation energy D_e^{\exp } [8] could not be given precisely, due to the limits of resolution and computational accuracy. So, it is now necessary to further search for the higher vibrational energies to make a complete understanding of the spectra and vibrational structure of this state. The experimental vibrational energies of the \mathrm{C}{\mathrm{O}}^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}^{+} state fitted by Coxon and Foster [9] appeared continuous in high-lying vibrational transition because of the resolution of the device. Therefore, it is important to accurately figure out the high-lying vibrational spectra nearly reaching the dissociation region using an appropriate theoretical method based on reliable experimental spectral data on this state. {\mathrm{F}}_2^{+} plays a central role in the rare gas-halogen laser system. The limited vibrational energies of the {\mathrm{F}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Pi }{}_{\mathrm{g}} state were evaluated theoretically by Cartwright and Hay [4], taking advantage of the generalized valence bond-configuration interaction (GVB-CI) approach. However, this set of GVB-CI energies did not coincide with the accurate experimental data of \left[E_{\upsilon }^{\exp }\right][4]. Furthermore, the accuracy of existing experimental or theoretical vibrational spectroscopic constants of this state were somewhat suspect, and deserved to be focused on continuously. The experimental energies of the {\mathrm{O}}_2^{+}-\mathrm{A}{}_{}{}^2\prod {}_{\mathrm{u}} state and the theoretical results of the \mathrm{L}{\mathrm{i}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}_{\mathrm{g}}^{+} state were both incomplete, so the accurate vibrational spectroscopic constants and the full vibrational energy spectra of these two electronic states need to be pursued as well. The algebraic method (AM) is briefly described in Section 2 of this paper. In Section 3, the vibrational spectroscopic constants and the full vibrational energy spectra of the \mathrm{B}\mathrm{e}{\mathrm{H}}^{+}-\mathrm{X}{}_{}{}^1\mathrm{\Sigma }{}^{+}, \mathrm{C}{\mathrm{O}}^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}^{+}, {\mathrm{F}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Pi }{}_{\mathrm{g}}, {\mathrm{O}}_2^{+}-\mathrm{A}{}_{}{}^2\mathrm{\Pi }{}_{\mathrm{u}} and \mathrm{L}{\mathrm{i}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}_{\mathrm{g}}^{+} states were studied using AM. Our research works are summarized in Section 4.

2 Algebraic method(AM)

The analytical non-relativistic vibrational energy formula for a stable diatomic molecular system was derived by Sun et al. [14,15] using second-order perturbation theory, and was expressed as

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

Both {\omega }_0 and {\omega }_{e0} are small quantities, but they may be important in calculating high-lying rovibrational energies.

The analytical non-relativistic vibrational energy formula for diatomic molecular ion XY⁺ may also be expressed as Eq.(1), whose generation was not relevant to the expression of potential function V\left(R\right), which is quite unlike for diatomic molecular ion XY⁺ and its neutral molecule [6]. Then, the algebraic method (AM) [10] proposed to study the vibrational spectroscopic constants and energies may be stated by rewriting Eq.(1) in a matrix form

\begin{array}{}\text{(2)}& AX=E\end{array}

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

\begin{array}{}\text{(3)}& X=\left(\begin{array}{l}{\omega }_0\\ {\omega }_e^{\text{'}}\\ -{\omega }_e{x}_e\\ {\omega }_e{y}_e\\ ⋮\\ {\omega }_e{r}_e\end{array}\right),\text{and}E=\left(\begin{array}{l}{E}_{\upsilon }\\ E_{\upsilon +i}\\ E_{\upsilon +j}\\ ⋮\\ E_{\upsilon +s}\end{array}\right),\upsilon =0,1,2,\cdots \end{array}

and the matrix element of the n\times 8 coefficient matrix A is A_{\upsilon k}={\left(\upsilon +\frac{1}{2}\right)}^k, k=0,1,2,3,\cdots ,7. In Eq.(3) {\omega }_e^{\prime }={\omega }_e+{\omega }_{e0}.

For a given diatomic molecular electronic system, it is better to choose 8 energies of the known accurate vibrational energy subset of \left[E_{\upsilon }\right] to form the energy matrix E to generate the vibrational spectroscopic constants matrix X through solving Eq. (2) using the standard algebraic method. Subsequently, a set of full vibrational energy spectra \left\{E_{\upsilon }\right\} of system can be obtained by putting the vibrational spectroscopic constants \{{\omega }_0,{\omega }_e+{\omega }_{e0},-{\omega }_e{x}_e,{\omega }_e{y}_e,{\omega }_e{z}_e,{\omega }_e{t}_e,{\omega }_e{s}_e,{\omega }_e{r}_e\}, namely matrix X solved above into Eq.(1). The best solution for the vibrational spectroscopic constants which truly reflects the full vibrational information for a given electronic state should best satisfy the requirements

\begin{array}{}\text{(4)}& E_{{\upsilon }_{\max }}\le D_e\end{array}

\begin{array}{}\text{(5)}& D_e-E_{{\upsilon }_{\max }}\to \text{small enough}\end{array}

\begin{array}{}\text{(6)}& \Delta E\left({\upsilon }_{\max },{\upsilon }_{\max }-1\right)=E_{{\upsilon }_{\max }}-E_{{\upsilon }_{\max }-1}\to \text{small enough}\end{array}

\begin{array}{}\text{(7)}& {\frac{\mathrm{d}{E}_{\upsilon }}{\mathrm{d}\upsilon }|}_{\upsilon ={\upsilon }_{\max }}=0\end{array}

\begin{array}{}\text{(8)}& \overline{\mathrm{\Delta }E\left(c,e\right)}={\left(\frac{1}{m+1}{\displaystyle \sum _{\upsilon =0}^m\left[E_{\upsilon ,\exp }-E_{\upsilon ,\mathrm{c}\mathrm{a}\mathrm{l}}\right]}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\to 0\end{array}

The five relations are the principal requirements and criteria of choosing the correct full vibrational energy spectra \left\{E_{\upsilon }\right\} of a system, where Eq.(4) shows the convergency limit of the maximum vibrational energy, and D_e corresponds to the equilibrium dissociation energy of the molecular system when inter-nuclear distance incline to infinity for a long-range molecular potential without potential rampart. In this way, the best subset satisfying the requirements i.e., Eqs.(4)–(8) can be found out of the known reliable vibrational spectra \left[E_{\upsilon }\right], thus obtaining the vibrational spectroscopic constants \{{\omega }_0,{\omega }_e+{\omega }_{e0},-{\omega }_e{x}_e,{\omega }_e{y}_e,{\omega }_e{z}_e,{\omega }_e{t}_e,{\omega }_e{s}_e,{\omega }_e{r}_e\} which is the true representation of a given electronic state, thereby generating the full vibrational energy spectra \left\{E_{\upsilon }^{}\right\}. This is the algebraic method (AM) used to obtain the vibrational spectroscopic constants and the full vibrational energy spectra of diatomic molecules and its ions.

3 Application of AM to some diatomic molecular ions XY⁺

The AM vibrational energy spectra of the \mathrm{B}\mathrm{e}{\mathrm{H}}^{+}-\mathrm{X}{}_{}{}^1\mathrm{\Sigma }{}^{+} state listed in Table 1 is calculated using experimental energies \left[E_{\upsilon }^{\exp }\right] [7] given by Coxon and Colin in 1997. It is shown that the absolute errors between the AM values and the input data is no more than 0.037 cm^–1, and the percentage error between the maximum energy of AM and the dissociation energy D_e{}^{\exp }=26 426.637 cm^–1 [8] is 3.909 5 \times10^–2 %. All the properties satisfy the requirements Eqs.(4)–(8). Our research shows that the AM spectra of the \mathrm{B}\mathrm{e}{\mathrm{H}}^{+}-\mathrm{X}{}_{}{}^1\mathrm{\Sigma }{}^{+} state, which compensates for the lack of report on high-lying vibrational energies in the results of Coxon and Colin, is the correct full vibrational energy spectra of this system.

The AM vibrational spectroscopic constants and the full vibrational energy spectra \left\{E_{\upsilon }^{\mathrm{A}\mathrm{M}}\right\} of the \mathrm{C}{\mathrm{O}}^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}^{+} state are given in Table 1. Obviously, the AM full vibrational energy spectra \left\{E_{\upsilon }^{\mathrm{A}\mathrm{M}}\right\} generated by Coxon and Foster’s experimental data \left[E_{\upsilon }^{\exp }\right][9] not only can reproduce the input energies very well, but the percentage error between the maximum energy E_{{\upsilon }_{\max }}^{\mathrm{A}\mathrm{M}}=68350.287cm^–1 ({\upsilon }_{\max }=49) and the dissociation energy D_e{}^{\exp }[8] of this state is just \text{5}\text{.1702}\times {\text{10}}^{-\text{3}}\%. Moreover, the RKR potential of this state given by Coxon and Foster [9] based on experimental energy subset \left[E_{\upsilon }^{\exp }\right] failed to describe the physical performance close to dissociation region. As a result, the complete RKR potential generated by the AM full vibrational energy spectra in our work is presented here, and is shown in Fig.1.

The spectroscopic properties and vibrational structure of {\mathrm{F}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Pi }{}_{\mathrm{g}} state has been studied for a long time. The GVB-CI energies [4] evaluated by Cartwright et al is given in Table 1. As we can see, the GVB-CI values of which the absolute error with the accurate experimental energies \left[E_{\upsilon }^{\text{exp}}\right][4] was about 80 cm^–1 at zero point energy, did not accurately represent the vibrational information of this state. The AM full vibrational energy spectra is listed in Table 1. It is shown that the AM energies can reproduce the experimental energies excellently, and that the percentage error between the maximum vibrational energy of AM and the experimental dissociation energy D_e{}^{\text{exp}}[8] of this state is just 0.452 6 %. So, the AM vibrational spectroscopic constants and the full vibrational energy spectra satisfying the requirements Eqs.(4)–(8) are better than GVB-CI results.

The AM vibrational spectroscopic constants and the full vibrational energy spectra of the {\mathrm{O}}_2^{+}-A{}_{}{}^2\mathrm{\Pi }{}_u state are also given in Table 1. The absolute error between AM spectra \left\{E_{\upsilon }^{\mathrm{A}\mathrm{M}}\right\} and the experimental energies \left[E_{\upsilon }^{\text{exp}}\right] is no more than 0.34 cm^–1. The percentage error between the maximum vibrational energy of AM and the dissociation energy D_e{}^{\text{exp}} [3] is just 1.207 4×10^–2%.

The theoretical vibrational energies \left[E_{\upsilon }^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}}\right] of the \mathrm{L}{\mathrm{i}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}_{\mathrm{g}}^{+} state was evaluated by Schmidt-Mink et al. using self consistent field-core polarization potentials method (SCF-CPP) [5] based on the Born-Oppenheimer approximation. At the same time, the difference between the maximum vibrational energy of SCF-CPP E_{{\upsilon }_{57}}^{\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}}=9929cm^–1 and the experimental dissociation energy D_e^{\text{exp}}=10 464 cm^–1 [12] is huge, largely because the Born-Oppenheimer approximation breakdowns at the region of high-lying vibrational and rovibrational energies nearly reaching to dissociation energy, where the nuclear kinetic energy is getting larger, and all kinds of coupling interactions are non-negligible, thus unavoidably producing the notable errors in SCF-CPP.

4 Conclusions

The accurate vibrational spectroscopic constants and the corresponding full vibrational energy spectra of the \mathrm{B}\mathrm{e}{\mathrm{H}}^{+}-\mathrm{X}{}_{}{}^1\mathrm{\Sigma }{}^{+}, \mathrm{C}{\mathrm{O}}^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}^{+}, {\mathrm{F}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Pi }{}_{\mathrm{g}}, {\mathrm{O}}_2^{+}-\mathrm{A}{}_{}{}^2\mathrm{\Pi }{}_{\mathrm{u}} and \mathrm{L}{\mathrm{i}}_2^{+}-\mathrm{X}{}_{}{}^2\mathrm{\Sigma }{}_{\mathrm{g}}^{+} states are first obtained using the algebraic method (AM) . The applied studies of the electronic states of these five diatomic molecular ions XY⁺, covering homo-nuclear ions and hetero-nuclear ions in ground states and /or in excited states, show that the AM technique can give the correct full vibrational energy spectra which, in addition to accurately reproducing all the known energies of experiment or quantum theory, include those energies of more excited vibrational states involving the maximum vibrational energy E_{{\upsilon }_{\text{max}}}^{\mathrm{A}\mathrm{M}} immediate to the dissociation energy of the system concerned, which may not be easily determined experimentally or theoretically. In conclusion, our research may provide reliable data for studies requiring accurate vibrational spectroscopic constants and high-lying excited vibrational states of the electronic states for diatomic molecular ions.

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