1 Introduction

The Schrödinger equation for diatomic oscillators can be exactly solved only for a limited class of Born-Oppenheimer potential energy functions [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], in which the Kratzer-Fues (KF) potential [1, 2]

$$\begin{aligned} V(q)=D_0\left( \frac{q-1}{q} \right) ^2 -D_0\qquad q=\frac{r}{r_0} \end{aligned}$$
(1)

(\(D_0, r_0\) stand for potential depth and bond length) deserves special attention due to its numerous applications in mathematics[16], physics [17,18,19], chemistry [20,21,22,23,24,25], and especially in femto- [26] and high-resolution spectroscopy. In the latter case the IR and MW spectra have been analysed using the polynomial, multiparametric Simons-Parr-Finlan (SPF) expansion [27,28,29]

$$\begin{aligned} V(q)=D_0\left( \frac{q-1}{q}\right) ^{2}\left[ 1+\sum _{i=1}^N D_i\left( \frac{q-1}{q}\right) ^{i}\right] , \end{aligned}$$
(2)

which enables the reproduction of highly resolved rovibrational spectra of diatomic molecules. Unfortunately, applying the higher potential expansion terms makes it impossible to obtain analytical solutions of the Schrödinger equation, which forces researches to use approximate methods or numerical integration [30, 31] generating values of the rovibrational energies indispensable to quantitative analysis of the spectral data. To avoid this disadvantage, a simple modification of the original SPF approach was proposed by Molski and Konarski [32] by introducing the KF potential in the form dependent on the rotational quantum number. As a result, the rovibrational energies and associated wave functions have retained their analytical forms, and, additionally, they have been enriched with supplementary parameters enabling the satisfactory reproduction of the rotational transitions in the given vibrational state [32]. The J-dependent KF model [32] was generalized by Pliva [25] to a form dependent not only on the rotational J, but also on the vibrational quantum number v. While the dependence of the KF potential on J was properly justified by the effect of the centrifugal force generated by rotation [32], the v-dependence of the KF potential parameters proposed by Pliva [25] has been an ad hoc procedure aimed at obtaining a satisfactory reproduction of the IR and MW spectra by a closed KF energy formula (\(J=0\))

$$\begin{aligned} E^{P}_v= & {} D\left[ 1-\frac{D/B_0}{\left( v+\frac{1}{2}+\sqrt{D/B_0+\frac{1}{4}}\right) ^2}\right] ,\qquad B_0=\frac{\hbar ^2}{2mr_0^2} \end{aligned}$$
(3)
$$\begin{aligned} D= & {} D_0\left\{ 1+\left[ \alpha _5(v+1/2)+\alpha _6(v+1/2)^2\right] /\left[ 1+\alpha _7(v+1/2)^2\right] \right\} , \end{aligned}$$
(4)

which in its original form contains too few parameters to fit effectively the spectral data. For this purpose, a set of additional parameters was introduced to the model [25], which enabled the correct reproduction of the rovibrational transitions for the range of vibrational states depending on the rigidity (high value of the force constant and small bond length) of the molecules under consideration. Although the bond length v-dependence was included in the Pliva model [25] in the form

$$\begin{aligned} B=\frac{\hbar ^2}{2mr_v^2}=B_0\left[ 1+\beta _4(v+1/2)+\beta _5(v+1/2)^2\right] , \end{aligned}$$
(5)

it appears only in the J-dependent rovibrational energy \(E^P_{vJ}\), while it does not occur in the pure vibrational \(E^P_v\) one. This fact can be considered as an inconsistency of this model.

In light of the above facts, the main goal of this work is to improve the KF model to a form capable of reproducing vibrational transitions and energy levels of highly vibrationally excited diatomic molecules, with accuracy greater than the original KF formula offers. The usefulness of the model proposed in reproduction of highly excited vibrational states of the selected diatomic molecules will be demonstrated. We shall also be concerned with relating the parameters defining the model with the spectroscopic parameters determined from high-resolution MW and IR spectra, which makes it possible to compare the accuracy of the reproduction of the spectral data by the proposed and standard approaches.

2 Method

The starting point for deriving the improved energy formula of the KF oscillator (IKF) is a supersymmetric-algebraic approach [15, 33, 34], in which a wide spectrum of analytical solutions of the Schrödinger equation is obtained from the generating function

$$\begin{aligned} f(x)=c_1(x\pm c_0/c_1)+c_2(x\pm c_0/c_1)^2\pm ... \end{aligned}$$
(6)

expanded into a power series of \(|x|<1\) variable in the vicinity of \(x=\pm c_0/c_1\). Application of the zero-, first-, second- and higher-order terms as well as the relationship

$$\begin{aligned} \frac{dx(q)}{d q}=-f(x) \end{aligned}$$
(7)

allow us to calculate x(q) as a function of vibrational coordinate q by solving the equation

$$\begin{aligned} \int {\frac{dx}{f(x)}}=-q. \end{aligned}$$
(8)

In the supersymmetric approach \(-x(q)=W(q)\) plays a role superpotential, while in the algebraic treatment x(q) is a generalized vibrational variable, which form depends on the oscillator type. On the basis of x(q) one may construct annihilation and creation operators

$$\begin{aligned} \hat{A}= \frac{d}{d q}- x(q),\qquad \hat{A}^{\dag }= -\frac{d}{d q}-x(q), \end{aligned}$$
(9)

satysfying

$$\begin{aligned} \left[ \hat{A},\hat{A}^{\dag }\right] =-\frac{dx(q)}{dq}=f(x), \end{aligned}$$
(10)

which can be employed to derive the Schrödinger equation (\(\hbar ^2/2m=1\))

$$\begin{aligned} \left[ \hat{p}^2+V(q)-E_0\right] \psi _v=\Delta E_{v0}\psi _v,\qquad \hat{p}=-i\frac{d}{dq} \end{aligned}$$
(11)

in the factorized form

$$\begin{aligned} \hat{A}^{\dag }\hat{A}\psi _v=\Delta E_{v0}\psi _v,\qquad \Delta E_{v0}=E_v-E_0 \end{aligned}$$
(12)

in which

$$\begin{aligned} V(q)-E_0=x^2(q)+\frac{dx(q)}{dq} \end{aligned}$$
(13)

is a well-known Riccati equation. In view of this, the potential energy function V(q) and eigenvalue \(E_0\) are obtained from (13), whereas the associated eigenfunction \(\psi _0\) can be derived by integration of the annihilation equation

$$\begin{aligned} {\hat{A}\psi _0=\left[ \frac{d}{d q} - x(q)\right] }\psi _0=0, \end{aligned}$$
(14)

yielding

$$\begin{aligned} \psi _0=\exp \left[ \int x(q)dq\right] . \end{aligned}$$
(15)

Having determined the ground state eigenfunction \(\psi _0\) one may generate the excited solutions \(\psi _v\) using the SUSYQM formalism with \(W(q)=-x(q)\) [15] providing the original and partener potentials in the forms

$$\begin{aligned} V(q)-E_0=W(q)^2-\frac{dW(q)}{dq}, \qquad V(q)^{\dagger }-E_0^{\dagger }=W(q)^2+\frac{dW(q)}{dq}. \end{aligned}$$
(16)

In the algebraic approach applied in this work, the eigenfunctions of the excited states are calculated using \(\psi _v\) in the form of the product of the ground state solution\(\psi _0\) and the polynomial function \(F(z)_v\)

$$\begin{aligned} \psi _v=\psi _0F(z)_v \end{aligned}$$
(17)

expressed in the variable z(q) [8] adequate to the problem considered. Here, \(F(z)_v\) represents the generalized hypergeometric function including Bessel, Jacobi (Legendre, Gegenbauer, Chebychev), Laguerre or Romanovski functions as the special cases. Introducing a trial function (17) into (11) one gets the general equation [35, 36]

$$\begin{aligned} \left( aq^2+bq+c\right) F(z(q))_v^{''}+\left( dq+e\right) F(z(q))_v^{'}-\lambda _nF(z(q))_v=0,\qquad v=0,1,2... \end{aligned}$$
(18)

generating a polynomial term depending on a, b, c, d, e parameters. Application of the proper boundary conditions to solutions of (18) permits calculation of the wavefunctions \(\psi _v\) and associated eigenvalues \(E_v\) of (11) for \(v>0\).

3 Results

The mixed scheme described in the previous section can be employed to obtain the solutions for different forms of generating function f(x) in the form (6) applying the zero-, first-, second- and higher-order terms of the expansion. In this work, we use only the second-order term and the parameter relationships

$$\begin{aligned} f(x)=c_2\left( x+c_0/c_1\right) ^2\qquad c_2=c_1 \end{aligned}$$
(19)

Inserting the function (19) into (8) and then integrating it with respect to x-coordinate produces the equation

$$\begin{aligned} -\left[ c_1(x+c_0/c_1)\right] ^{-1}=-q, \end{aligned}$$
(20)

which can be reverted, generating x(q) in the q-dependent form

$$\begin{aligned} x(q)=\frac{1}{c_1q}-\frac{c_0}{c_1}. \end{aligned}$$
(21)

In the next step, we pass to calculate V(q), \(E_0\) and \(\psi _0\) by taking advantage of Eqs. (13) and (15)

$$\begin{aligned} V(q)-E_0= & {} \left[ \frac{1}{c_1q}-\frac{c_0}{c_1} \right] ^2-\frac{1}{c_1q^2}=\frac{c_0^2}{c_1^2(1-c_1)}\left[ \frac{q-\frac{(1-c_1)}{c_0}}{q}\right] ^2-\frac{c_0^2}{c_1^2(1-c_1)}+\frac{c_0^2}{c_1^2} \end{aligned}$$
(22)
$$\begin{aligned} \psi _0= & {} N_0q^{1/c_1}\exp \left( -\frac{c_0}{c_1}q\right) . \end{aligned}$$
(23)

Here, \(N_0\) is a normalisation constant. Now, one may construct the Schrödinger equation (11) for the ground state solution (13)

$$\begin{aligned} \left[ -\frac{d^2}{dq^2}+\frac{c_0^2}{c_1^2(1-c_1)}\left[ \frac{q-\frac{1-c_1}{c_0}}{q}\right] ^2-\frac{c_0^2}{c_1^2(1-c_1)}+\frac{c_0^2}{c_1^2}\right] \frac{1}{c_1q}\exp \left( -\frac{c_0}{c_1}q\right) =0 \end{aligned}$$
(24)

which can be converted to the form including original KF constants \(\lambda \), \(\gamma \) and \(\beta _0\) (\(2mr_0^2/\hbar ^2=1\))

$$\begin{aligned} \left[ -\frac{d^2}{dq^2}+\frac{\gamma ^2}{k}\left( \frac{q-k}{q} \right) ^2 \right] q^{\lambda }\exp \left( -\beta _0 q\right) =0 \end{aligned}$$
(25)

satisfying the following parameter relationships

$$\begin{aligned} \lambda= & {} \frac{1}{c_1},\qquad \beta _0=\sqrt{D_0/k-E_0}=\frac{c_0}{c_1},\qquad \gamma ^2=D_0=\frac{c_0}{c_1^2}\qquad k=\frac{1-c_1}{c_0}, \end{aligned}$$
(26)
$$\begin{aligned} \lambda ^2-\lambda= & {} k\gamma ^2 \qquad \gamma ^2=\beta _0\lambda . \end{aligned}$$
(27)

Taking advantage of the above specified relations, one may derive the ground state energy formula and the associated wave function of the improved KF model (IKF)

$$\begin{aligned} E_0=D_0\left( \frac{1}{k}-\frac{\gamma ^2}{\lambda ^2}\right) ,\qquad \psi _0=N_0q^{\lambda }\exp \left( -\beta _0 q\right) ,\qquad \lambda =\frac{1}{2}+\sqrt{k\gamma ^2+\frac{1}{4}}. \end{aligned}$$
(28)

A detailed analysis of the expressions derived reveals that the approach proposed reproduces the KF model enriched by an additional k-parameter, consequently both potential V(q) as well as the solutions of the Schrödinger equation depend on parameters \(D_0\), \(r_0\) and additionally k. Hence, one may expect that the IKF formula will better reproduce the spectral data than its original 2-parametric form. It should be pointed out that the results reported in this section are similar to, but not identical with those obtained by [37] in the algebraic-supersymmetric scheme using the vibrational variable \(q=r-r_0\) for \(q\in (-\infty , \infty )\) instead of the KF variable \(q=r/r_0\) for \(q\in (0, \infty )\). Additionally, calculations have been carried out with respect to the boundary condition \(x(q=0)=(1-c_0)/c_1\) which does not apply to the present formulation of the problem. Consequently, the parameter relationships (26) and (27) as well as the ground state energy formula (28) differ from those obtained previously.

Having derived the ground state solutions \(\psi _0\) and \(E_0\), one may pass to calculate \(\psi _v\) and \(E_v\) for the excited vibrational states characterized by the quantum number \(v=1,2...\). Application of the algebraic approach produces the general solutions in the form (17) [22]

$$\begin{aligned} \psi (q)_v=N_vq^{\lambda }\exp (-\beta _vq)_1F_1(\lambda -\gamma ^2/\beta _v,2\lambda ;2\beta _vq), \end{aligned}$$
(29)

in which

$$\begin{aligned} _1F_1(c,b;z)=\sum _{v=0}^{\infty }\frac{\Gamma (c+v)\Gamma (b)z^v}{\Gamma (b+v)\Gamma (c)v!} \end{aligned}$$
(30)

is a confluent hypergeometric function defined by the \(\Gamma (u)\) function. The eigenvalues characterizing the ground and excited vibrational states are calculated from the relation

$$\begin{aligned} \lambda -\frac{\gamma ^2}{\beta _v}=-v,\qquad v=0,1,2...\qquad \beta _v^2=\frac{D_0}{k}-E_v \end{aligned}$$
(31)

yielding

$$\begin{aligned} E_v=D_0\left[ \frac{1}{k}-\frac{\gamma ^2}{(\lambda +v)^2}\right] ,\qquad \lambda =\frac{1}{2}+\sqrt{k\gamma ^2+\frac{1}{4}}. \end{aligned}$$
(32)

The latter expression in (32) represents the positive solution of Eq. (27).

In order to test the ability of the IKF energy formula to reproduce the highly excited vibrational states of molecular systems, one may generalize the SPF potential expansion to include the k-parameter

$$\begin{aligned} V(q)=\frac{D_0}{k}\left( \frac{q-k}{q}\right) ^{2}\left[ 1+\sum _{i=1}^N D_i\left( \frac{q-k}{q}\right) ^{i}\right] , \end{aligned}$$
(33)

and then to apply it [37] for reduction of the spectral IR and MW data to the set of parameters \((D_i, k)\) using the approximate (semi-classical, variational, perturbational) methods or numerical integration procedure. In this way, parameters defining potential (33) are determinable with an accuracy dependent on the number of the expansion terms used in calculations. In this work, we apply an alternative approach: the k-parameter is directly fitted by the IKF formula (32) to the vibrational transitions in the rotationless state \(J=0\), producing the coherent anti-Stokes Raman spectra by the nitrogen molecule \(^{14}\)N\(_2\) in the ground electronic state \(X^1\Sigma _g^+\). In this way, one may investigate only the pure vibrational effects undisturbed by rotation, responsible for the rovibrational couplings of different orders. In the calculations, we have used the parameters recommended by NIST (https://www.nist.gov): bond length \(r_0=1.09768\) [Å], reduced mass \(m=7.001537005\) [amu] for \(^{14}\)N\(_2\) and \(r_0=1.208\) [Å], \(m=7.997457504\) [amu] for \(^{16}\)O\(_2\). Because in the KF energy formula the potential depth \(D_0\) instead of the dissociation energy \(D_e\) is used, we recalculated the values of the experimentally evaluated \(D_e= 9.756\) [eV] for \(^{14}\)N\(_2\) [38] and \(D_e=5.1156\) [eV] for \(^{16}\)O\(_2\) [39] (they are consistent with the results reported by [40] as well as [41]) onto \(D_0\) using the relationship \(D_0=ZPE+D_e\), in which \(ZPE=E_0\) stands for the zero-point energy of the KF oscillator. In this way, one may calculate \(D_0\) using the relationship (in original units)

$$\begin{aligned} D_e=\frac{D_0\gamma ^2}{\left( \frac{1}{2}+\sqrt{\gamma ^2+\frac{1}{4}}\right) ^2}\Longrightarrow D_0=D_e+\sqrt{D_eB_0},\qquad \gamma ^2=\frac{2mr_0^2D_0}{\hbar ^2}=\frac{D_0}{B_0}\qquad \end{aligned}$$
(34)

yielding \(D_0=9.808171\) [eV] and \(D_0=5.145868\) [eV] for nitrogen and oxygen molecules, respectively. In the preliminary calculations, we used 5 vibrational transitions \(0\rightarrow 1\), \(1\rightarrow 2\), \(2\rightarrow 3\), \(3\rightarrow 4\), \(4\rightarrow 5\), observed [42] in the coherent anti-Stokes Raman spectra of the nitrogen molecule \(^{14}\)N\(_2\) in the ground electronic state \(X^1\Sigma _g^+\) and the energy levels \(v<42\) of dioxygen \(^{16}\)O\(_2\) in the ground electronic state \(X^3\Sigma _g^-\) reported by [43]. Because the standard errors of measurements of the vibration transitions in dinitrogen have been determined by [42], they enable the calculation of a normalized standard deviation (NSD) that enriches a wide range of goodness of fit indicators used: coefficient of determination \(R^2\), standard deviation (SD), Akaike information criterion (AIC) [44], Bayesian information criterion (BIC) [45].

4 Discussion

Table 1 The values of k-parameter calculated for the experimental \(E^{Exp}\) vibrational transitions in the nitrogen molecule \(^{14}\)N\(_2\) in the ground electronic state \(X^1\Sigma _g^+\) reported by Orlov et al. [42]
Full size table

In the calculations performed for \(^{14}\)N\(_2\), we determined the k-parameter for each vibrational transition to check its dependence on the vibrational quantum number v. The results presented in Table 1 indicate that this relation actually exists, therefore in the next stage of calculations, we tested k in the v-dependent form

$$\begin{aligned} k(v)=k_0+k_1v+k_2v^2+.... \end{aligned}$$
(35)

which results in the modification of the energy formula (32)

$$\begin{aligned} E^{IKF}_v=D_0\left[ \frac{1}{k(v)}-\frac{\gamma ^2}{(\lambda (v)+v)^2}\right] ,\qquad \lambda (v)=\frac{1}{2}+\sqrt{k(v)\gamma ^2+\frac{1}{4}} \end{aligned}$$
(36)

the associated wave function (29)

$$\begin{aligned} \psi (q)_v=N_vq^{\lambda (v)}\exp (-\beta _vq)_1F_1[\lambda (v)-\gamma ^2/\beta _v,2\lambda (v);2\beta _vq], \end{aligned}$$
(37)

and the potential energy

$$\begin{aligned} V(q,v)=\frac{D_0}{k(v)}\left[ \frac{q-k(v)}{q}\right] ^{2} \end{aligned}$$
(38)

having a minimum for \(q=k(v)\) or \(r=r_0k(v)\). The normalization constant in (37) takes the form [22]

$$\begin{aligned} N_v=\frac{1}{\sqrt{\lambda (v)+v}}\frac{2\beta _v^{\lambda (v)+1/2}}{\Gamma [2\lambda (v)]}\sqrt{\frac{\Gamma [2\lambda (v)+v]}{2v!}}. \end{aligned}$$
(39)

Returning to the original displacement variable \(r=qr_0\), the potential (38) can be rewritten in the form

$$\begin{aligned} V(r,v)=\frac{D_0}{k(v)}\left[ \frac{r-k(v)r_0}{r}\right] ^{2}, \end{aligned}$$
(40)

which reveals the vibrational v-dependence of the equilibrium bond length \(r_0\) and potential depth \(D_0\) (see Figures 1 and 2). With a high degree of probability, nonadiabatic vibrational effects are responsible for this type of effects, which have a significant impact on the energy levels in highly excited vibrational states [46]. Because such effects are not explicitly included in the KF model, the original KF energy formula does not reproduce experimental vibrational transitions and energy levels accurately (see Tables 2 and 4). Since the nonadiabatic effects affect the equilibrium position of the atoms and modify the parameters of the Born-Oppenheimer potentials [47], the model presented includes such effects by introducing v-dependent potential depth \(D_0\rightarrow D_v\) and interatomic equilibrium separation \(r_0\rightarrow r_v\). For this reason, the IKF model can be used for the description of highly excited vibrational states of diatomic molecules in which vibrational nonadiabatic effects play a significant role.

Fig. 1
figure 1

Plots of IKF potential (38) for \(^{16}\)O\(_2\) in the vicinity of the modified equlibrium position \(q=k(v)\) or \(r=r_0(k_0+k_1v+k_2v^2)\) for the selected vibrational quantum numbers v and \(k_0\), \(k_1\) and \(k_2\) from Table 5

Full size image
Fig. 2
figure 2

Plots of IKF potential (38) for \(^{16}\)O\(_2\) in the dissociation limit \(D_0/(k_0+k_1v+k_2v^2)\) for the selected vibrational quantum numbers v and \(k_0\), \(k_1\) and \(k_2\) from Table 5

Full size image
Table 2 The vibrational transitions (cm\(^{-1}\)) in the nitrogen molecule \(^{14}\)N\(_2\) \(X^1\Sigma _g^+\) reproduced by the original \(E^{KF}\) and improved \(E^{IKF}(k_0,k_1,k_2)\) KF energy formulae
Full size table
Table 3 The parameters employed in reproduction of the spectral data of \(^{14}\)N\(_2\) \(X^1\Sigma _g^+\) presented in Table 2
Full size table

In order to verify the correctness of the model proposed and chemical meaning of the parameters determined, the IKF energy formula (36) can be expanded into a series of the vibrational variable \(v+1/2\) (using MAPLE vs 2019 processor for symbolic calculations)

$$\begin{aligned} E^{IKF}_v= & {} Y_0+\omega _e(v+1/2)-\omega _ex_e(v+1/2)^2+\omega _ey_e(v+1/2)^3 \end{aligned}$$
(41)
$$\begin{aligned} Y_0= & {} {\frac{4\,{ D_0}}{ \left( 4\,{ k_0}-2\,{ k_1}+{ k_2} \right) ^{2}{\gamma }^{2}+4\,{ k_0}-2\,{ k_1}+{ k_2}}} \end{aligned}$$
(42)
$$\begin{aligned} \omega _e= & {} { D_0}\, \left[ {\frac{4({ k_1}-{ k_2})}{ \left( 4\,{ k_0}-2\,{ k_1}+{ k_2} \right) \left( { k_0}-{ k_1}/2+{ k_2}/4 \right) }}\right. \nonumber \\{} & {} \left. -16\,{\frac{{\gamma }^{2} \left( { \gamma }^{2}{} { k_1}-{\gamma }^{2}{} { k_2}+\sqrt{4\,{\gamma }^{2} { k_0}-2\,{\gamma }^{2}{} { k_1}+{\gamma }^{2}{} { k_2}+1} \right) }{ \left( 4\,{\gamma }^{2}{} { k_0}-2\,{\gamma }^{2}{} { k_1}+{\gamma } ^{2}{} { k_2}+1 \right) ^{2}}} \right] \end{aligned}$$
(43)
$$\begin{aligned} \omega _ex_e= & {} -16 { D_0}\left\{ \left[ \frac{ \left( 4\,{ k_2}\,{ k_0}-4\,{{ k_1}}^{2 }+6\,{ k_2}\,{ k_1}-3\,{{ k_2}}^{2}\right) }{\left( 4\,{ k_0}-2\,{ k_1}+{ k_2} \right) ^{3}}\right] \right. \nonumber \\{} & {} + {\gamma }^{2}\left[ \frac{ \left( -4\,{ \gamma }^{4}{} { k_0}\,{ k_2}+4\,{\gamma }^{4}{{ k_1}}^{2}-6\,{ \gamma }^{4}{} { k_1}\,{ k_2}+3\,{\gamma }^{4}{{ k_2}}^{2}\right) }{\left( 4\,{\gamma }^{2}{} { k_0}-2\,{\gamma }^{2}{} { k_1} +{\gamma }^{2}{} { k_2}+1 \right) ^{3}}\right] \nonumber \\{} & {} - 6 {\gamma }^{4}\left[ \frac{{ k_1}\sqrt{1+ \left( 4\,{ k_0}-2\,{ k_1}+{ k_2} \right) {\gamma }^{2}}-{ k_2}\sqrt{1+ \left( 4\,{ k_0}-2 \,{ k_1}+{ k_2} \right) {\gamma }^{2}}}{ \left( 4\,{\gamma }^{2}{} { k_0}-2 \,{\gamma }^{2}{} { k_1}+{\gamma }^{2}{} { k_2}+1 \right) ^{3}}\right] \nonumber \\{} & {} - \left. {\gamma }^{2}\left( \frac{ 12\,{\gamma }^{2}{ k_0}-6\,{\gamma }^{2}{} { k_1}+2\,{\gamma }^{2}{} { k_2}+3}{ \left( 4\,{\gamma }^{2}{} { k_0}-2 \,{\gamma }^{2}{} { k_1}+{\gamma }^{2}{ k_2}+1 \right) ^{3}}\right) \right\} \end{aligned}$$
(44)
$$\begin{aligned} \omega _ey_e= & {} {{\frac{393216 D_0}{ \left( 4{ k_0}-2{ k_1}+{ k_2} \right) ^{4} \left( 4{\gamma }^{2}{ k_0}-2{\gamma }^{2}{ k_1}+{\gamma }^{2}{ k_2}+1 \right) ^{4}}}} \nonumber \\{} & {} \left\{ -1/4\, \left( { k _0}-{ k _1}/2+{ k _2}/4 \right) ^{4}{\gamma }^{2}\sqrt{1+ \left( 4 { k _0}-2 { k _1}+{ k _2} \right) {\gamma }^{2}}\right. \nonumber \\{} & {} \left. \left[ -1/3+ \left( -{{ k _2}}^{2}+ \left( { k _0}+2 { k _1}\right) { k _2}-5/4 {{ k _1}}^{2} \right) {\gamma }^{4} \right. \right. \nonumber \\ {}{} & {} + \left( -4/3 { k _0}+2/3 { k _1}-{ k _2}/12 \right) {\gamma }^{2}\nonumber \\{} & {} \left. \left. + \left( -1/24{{ k_2}}^{2}+ \left( 3/4{ k_0}-{ k_1}/24 \right) { k_2}+{{ k_0}}^{2}-{ k_0}{ k_1}+1/12{{k_1}}^{2} \right) \right. \right. \nonumber \\{} & {} \left. \left( { k_0}{-}{ k_1}/2{+}{ k_2}/4 \right) ^{3}{\gamma }^{6}\right] \left( { k_1}{-}{ k_2} \right) ] \nonumber \\{} & {} + 1/4 \left[ -1/16{{ k_2}}^{2}{+}{ k_2}{ k_0}{+}{ k_0 } \left( { k_0}-{ k_1} \right) \right] \left( { k_0}{-}{ k_1}/2{+}{ k_2}/4 \right) ^{2}{\gamma }^{4}] \left( { k_1}-{k_2} \right) \nonumber \\{} & {} + \left[ 1/48 \left( -1/4{{ k_2}}^{2}+ \left( { k_0}+{ k_1}/2 \right) { k_2}-1/2{{ k_1}}^{2} \right) \left( { k_0}-{ k_1}/2+{ k_2}/4 \right) {\gamma }^{2}\right] \nonumber \\{} & {} \left( { k_1}-{ k_2} \right) + \left. \left[ -{\frac{{{ k_2}}^{2}}{3072}}+ \left( {\frac{{ k_0}}{768}}+{ \frac{{ k_1}}{1536}} \right) { k_2}-{\frac{{{ k_1}}^{2} }{1536}}\right] \left( { k_1}-{ k_2}\right) \right\} \end{aligned}$$
(45)

whose coefficients represent spectroscopic parameters with known values cataloged by NIST (https://www.nist.gov). Substituting the parameters from Tables 3 and 5 into the equations (43), (44) and (45) one obtains the values of the spectroscopic constants presented in Table 6. On the other hand, using the inverse spectroscopic method [48], it is possible to determine the potential parameters from the equations specified above, which in general include not only \(k_i\) but also the bond length \(r_0\) and dissociation energy \(D_e\) defining \(\gamma ^2\), which represent important in chemistry molecular characterisics. The analysis of the results obtained reveals that the reproduction of the N\(_2\) spectroscopic parameters is excellent or acceptable (\(\omega _ey_e\)), unlike the O\(_2\) ones. From a theoretical point of view, the reason for the discrepancies may be inaccurate values of the energy levels reported in [43] or greater softness (anharmonicity) of O\(_2\) (force constant 1142 \(Nm^{-1}\)) compared to N\(_2\) (force constant 2243 \(Nm^{-1}\)). To resolve this dilemma, the NIST spectral parameters from Table 6 and the energy formula (41) were used to generate N\(_2\) and O\(_2\) energy levels from \(v=0\) up to the dissociation limits \(v=v_D\), which were then reproduced using the IKF (36) and the Pliva (3) fomulas. The results of the calculations presented in Table 7 prove that: (i) the discrepancies observed for O\(_2\) are caused by inaccurate values of the energy levels reported by [43], that differ significantly from those determined from the NIST spectral parameters; (ii) the IKF formula correctly reproduces the energy levels of N\(_2\) and O\(_2\) up to the dissociation limit (see Tables 8 and 9), (iii) the accuracy of the data reproduction by IKF fomula is greater than that offered by the Pliva model [25].

Table 4 The vibrational energy levels \( \Delta E_{v}=E_v-E_0\) [eV] of the oxygen molecule \(^{16}\)O\(_2\) in the ground electronic state \(X^3\Sigma _g^-\) reproduced by the original \(E_v^{KF}\) and improved \(E_v^{IKF}\) KF energy formulae
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Table 5 The parameters employed in reproduction of the energy levels of the oxygen molecule \(^{16}\)O\(_2\) in the ground electronic state \(X^3\Sigma _g^-\) presented in Table 4
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Table 6 Spectral coefficients (cm\(^{-1}\)) of \(^{14}\)N\(_2\) \(X^1\Sigma _g^+\) and \(^{16}\)O\(_2\) \(X^3\Sigma _g^-\) calculated from parameters in Tables 3 and 5
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Table 7 The potential parameters defining IKF and Pliva models for \(^{14}\)N\(_2\) \(X^1\Sigma _g^+\) and \(^{16}\)O\(_2\) \(X^3\Sigma _g^-\) molecules fitted to the N energy levels generated from the spectral NIST (https://www.nist.gov) data
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Table 8 The vibrational energy levels (eV) of \(^{14}\)N\(_2\) \(X^1\Sigma _g^+\) generated from the spectral NIST parameters presented in Table 6, and reproduced by the IKF energy formula \(E^{IKF}(k_0,k_1,k_2)\) with parameters from Table 7
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Table 9 The vibrational energy levels (eV) of \(^{16}\)O\(_2\) \(X^3\Sigma _g^-\) generated from the spectral NIST parameters in Table 6, and reproduced by the IKF energy formula \(E^{IKF}(k_0,k_1,k_2)\) including parameters from Table 7
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It is a well-known fact [1, 2] that the KF oscillator (\(k(v)=1\)) is characterized by an infinite number of quantum states, i.e. the relation

$$\begin{aligned} \lim _{v\rightarrow \infty }E^{KF}_v=D_0 \end{aligned}$$
(46)

is satisfied. In the case of the IKF oscillator (\(k(v)\ne 1\)), the situation changes radically - it has a finite number of quantum states to be determined from Eq. (41) including experimental spectral coefficients and the relationship

$$\begin{aligned} E^{IKF}_{v_D}-E^{IKF}_0=D_e \end{aligned}$$
(47)

in which \(v_D\) is a quantum number describing the dissociation state. To this aim, we employed the MAPLE processor for symbolic calculations producing the values \(v_D=47.40328127\approx 47\) and \(v_D=33.83170312\approx 33\) indispensable to prepare the data set to be fitted. Having determined \(k_i\) parameters one may calculate \(v_D\) from Eq. (47) given in an alternative form

$$\begin{aligned} E^{IKF}_{v_D}=D_0=D_0\left[ \frac{1}{k(v_D)}-\frac{\gamma ^2}{(v_D+\frac{1}{2}+\sqrt{k(v_D)\gamma ^2+\frac{1}{4}})^2}\right] , \end{aligned}$$
(48)

in which

$$\begin{aligned} k(v_D)=k_0+k_1v_D+k_2v_D^2. \end{aligned}$$
(49)

The system of the above equations can be solved with respect to \(v_D\) employing the MAPLE processor or manually using the solution of Eq. (48) for \(k(v_D)=const\)

$$\begin{aligned} v_D=\gamma \sqrt{\frac{k(v_D)}{1-k(v_D)}}-\sqrt{\gamma ^2k(v_D)+\frac{1}{4}}-\frac{1}{2} \end{aligned}$$
(50)

and the following iterative procedure: for the approximate value of \(k(v_D)=k_0\), one calculates rough \(v_D\) from (50) and substitute into (49), which allows us to calculate more accurate values of \(k(v_D)\), and consequently \(v_D\). The procedure is fast-converging and requires 4 -5 iterations to calculate \(v_D=33\) for O\(_2\) and \(v_D=47\) for N\(_2\). So, we have \(N=34\) and \(N=48\) quantum states including the ground state \(v=0\).The results obtained and the Eq. (50) cleary indicate that IKF oscillator for \(k(v)\ne 1\) is endowed with a finite number \(N=v_D+1\) of quantum states and satisfies

$$\begin{aligned} \lim _{v\rightarrow v_D}E^{IKF}_v=D_0. \end{aligned}$$
(51)

5 Conclusion

The analysis of the results presented in Tables 2-5 shows that the inclusion of only one additional \(k_0\)-parameter in the original KF model significantly increases its ability to reproduce the vibrational transitions in the nitrogen molecule and the energy levels of dioxygen. Taking into account additional parameters \(k_1\) and \(k_2\) results in a further increase in the accuracy of spectral reproduction usually achievable with the use of advanced computational methods and sophisticated theoretical models, including adiabatic as well as vibrational and rotational nonadiabatic efects. One of the most important advantages of the present approach is exact analytic energy formula and the associated wave function to be easily obtained by the substitutions \(D_0\rightarrow D_v\) and \(r_0\rightarrow r_v\) in the original KF formula. Consequently, the matrix elements of quantum-mechanical operators, the Franck-Condon factors and intensities of the vibrational transitions can be directly calculated [49, 50]. The proposed description of the vibrational states of diatomic molecules can be generalized by taking into account the rotational degrees of freedom according to the scheme proposed by Molski and Konarski [32]. However, this requires an in-depth study of the effects of rotation on vibration and vice versa, resulting in rovibrational couplings that play a key role in the correct reproduction of spectral data. Work on this issue is in progress and the results will be presented soon.