Bound-state energy spectrum and thermochemical functions of the deformed Schiöberg oscillator

Abstract

In this study, a diatomic molecule interacting potential such as the deformed Schiöberg oscillator (DSO) have been applied to diatomic systems. By solving the Schrödinger equation with the DSO, analytical equations for energy eigenvalues, molar entropy, molar enthalpy, molar Gibbs free energy and constant pressure molar heat capacity are obtained. The obtained equations were used to analyze the physical properties of diatomic molecules. With the aid of the DSO, the percentage average absolute deviation (PAAD) of computed data from the experimental data of the ⁷Li₂ (2 ³Π_g), NaBr (X ¹Σ⁺), KBr (X ¹Σ⁺) and KRb (B ¹Π) molecules are 1.3319%, 0.2108%, 0.2359% and 0.8841%, respectively. The PAAD values obtained by employing the equations of molar entropy, scaled molar enthalpy, scaled molar Gibbs free energy and isobaric molar heat capacity are 1.2919%, 1.5639%, 1.5957% and 2.4041%, respectively, from the experimental data of the KBr (X ¹Σ⁺) molecule. The results for the potential energies, bound-state energy spectra, and thermodynamic functions are in good agreement with the literature on diatomic molecules.

Introduction

The relevance of wave functions in quantum mechanics is the adequate information they provide about the quantum mechanical system being studied, hence the growing need for accurate numerical and approximate analytical bound state solutions of the non-relativistic and relativistic wave equations for a given potential energy function. The potential energy function is the medium through which a quantum mechanical system interacts with particles and molecules in its neighborhood. Examples of interaction potentials include the Tietz potential, Schiöberg potential, Hua potential, Rosen-Morse potential and Woods-Saxon potential^1,2,3,4,5.

A diatomic molecule oscillator is a potential energy function used to describe the rotational-vibrational states of a molecular system. To qualify as a diatomic molecule oscillator, a potential energy function must satisfy some prescribed conditions also known as the Varshni conditions⁶. Oscillators are modeled using molecular spectroscopic parameters such as the vibrational–rotational coupling coefficient (α_e), equilibrium harmonic vibrational frequency (ω_e), equilibrium bond length (r_e), and equilibrium dissociation energy (D_e). These parameters are usually determined experimentally or by ab initio calculations.

The solutions of the Schrödinger equation (SE) have been obtained analytically with different potential energy models. Expressions for the bound-state energies have been successfully used to investigate the thermodynamic, optical, magnetic and other physical properties of substances^{7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}. Some of the techniques used to obtain the bound state solutions of the wave equations are the asymptotic iteration method²², quantization rules^23,24, the supersymmetric quantum mechanics approach²⁵, the Nikiforov-Uvarov method and its parametric versions^26,27, and the Laplace transformation method²⁸.

Thermodynamic functions have immense applications in science and technology. For instance, enthalpy and entropy measurements have been used to determine the melting points of organic molecules, and to detect diseases in plants^29,30. The thermoplastic property, transition and melting points of nanostructures have also been investigated through measurements of heat capacity^31,32. In a very recent advancement, the Gibbs free energy equation developed from the well-known Fu-Wang-Jia (FWJ) oscillator has been used to establish the equilibrium constant for the water gas shift reaction³³.

The solution canonical partition function is a prelude to obtaining statistical-mechanical models (or analytical equations) for the calculation of thermo-chemical properties of gaseous molecules. The partition function takes into account; the vibrational, rotational and translational effects of the diatomic system. Analytical equations for the prediction of the molar entropy (S), enthalpy (H), Gibbs free energy (H), and isobaric heat capacity (C_p) exist in the literature, some examples can be found in Refs.^{34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58}. Different potential functions have been employed in the literature to construct analytical model equations^{6,59,60,61,62,63,64,65,66}.

The present study is centered on the Schiöberg potential energy function. Previously, the bound-state solutions of the Schrödinger and Dirac equations have been obtained with the Schiöberg potential^67,68. Using the Varshni conditions⁶, Wang and coworkers demonstrated the equivalence of the Manning-Rosen, Deng-Fan and Schiöberg potentials⁶⁴. The Schiöberg oscillator incorporates three independent input parameters viz D_e, r_e and ω_e.

In the quest to model an efficient version of the Schiöberg oscillator, the authors in Ref.⁶⁹ employed the transformation r → r−r₀ and the Varshni conditions⁶ to construct the reparameterized Schiöberg potential. The reparameterized Schiöberg oscillator is expected to encapsulate four independent input parameters: D_e, r_e, ω_e and α_e, nevertheless, the explicit form of the parameter r₀ was not deduced. Many diatomic molecule oscillators have been used by physicists and chemists to predict the thermochemical properties of gaseous molecules^{6,55,59,61,62,63,64,65}. However, for this purpose, the deformed Schiöberg potential has not been considered in the literature. It must be emphasized that q-deforming a potential energy function and subsequently subjecting it to the Varshni conditions for a diatomic molecule potential yields an equivalent model to the reparameterized version³⁵. For this reason, the present study primary objectives are to obtain the energy spectra and thermochemical functions of the deformed Schiöberg oscillator. The remaining parts of the paper is organized as follows: In section “Construction of the DSO”, the deformed Schiöberg oscillator is constructed. In section “Equation for the energy spectra of the DSO”, explicit equation for the energy spectra is derived. Thermochemical functions are obtained in section “Thermochemical functions of the DSO”. The results of numerical calculations are presented in section “Results and discussion”. A brief conclusion of the work is given in section “Conclusions”.

Construction of the DSO

In this section, the deformed Schiöberg oscillator (DSO) is constructed by employing the Varshni conditions⁶. The suggested model potential is given by

$$ {\text{U }}\left( r \right) \, = {\text{ U}}_{0} \left\{ {{1 }{-}\sigma {\text{coth}}_{q} \left( {\alpha r} \right)} \right\}^{{2}} , $$

(1)

where, coth_q (αr) = cosh_q (αr)/sinh_q (αr), cosh_q (αr) = ½ (e^αr + e^−αr), sinh_q (αr) = ½ (e^αr−e^−αr), r is the interparticle separation, U₀ is the depth of the potential well, q, α and σ are potential parameters. Evidently, (1) is the q-deformed version given in Ref.⁶⁴. The main difference between Eq. (1) and expression (1) of Ref.⁷⁰ lies in the functional forms of the two models.

Equation (1) is a diatomic molecule oscillator if it satisfies the following conditions⁶

$$ {\text{U}}\prime \, \left( r \right) \, = \, 0, {\text{U }}\left( {r \to \, \infty } \right) \, {-}{\text{ U }}\left( {r \to r_{{\text{e}}} } \right) \, = D_{{\text{e}}} ,\;\;{\text{U}}\prime \prime \, \left( {r_{{\text{e}}} } \right) \, = M_{0} \left( {{2}\pi c\omega_{{\text{e}}} } \right)^{{2}} , $$

(2)

where the prime in (2) denotes the derivative with respect to r, the speed of light is designated c, and μ is the reduced mass of a molecule. Inserting Eq. (1) into each of the expressions in (2) gives

$$ q = \, \left( {{1 }{-}\alpha \varepsilon } \right){\text{ e}}^{{{2}\gamma }} , $$

(3)

where σ = tanh_q γ, U₀ = D_e (1−σ)⁻², $\varepsilon = \tfrac{1}{{{\uppi }c\omega_{\text{e}} }}\left( {\tfrac{{2D_{\text{e}} }}{{M_{0} }}} \right)^{{\tfrac{1}{2}}}$, and γ = αr_e. The next step is to determine the potential screening parameter, α. The α_e-ω_e relationship given in publication⁶⁴ can be used, viz

$$ \alpha_{\text{e}} = - \frac{{6{\text{B}}_{\text{e}}^{2} }}{{\omega_{\text{e}} }}\left\{ {1 + \frac{{r_{\text{e}} }}{3}\frac{{{{{\rm U}^{\prime\prime\prime}}}\left( {r_{\text{e}} } \right)}}{{{{{\rm U}^{\prime\prime}}}\left( {r_{\text{e}} } \right)}}} \right\}, $$

(4)

where B_e = ħ/4πcM₀r_e², ħ = h/2π, h being the Planck constant. U″(r_e) and U‴(r_e) are obtained from Eq. (1) as

$$ {\text{U}}^{\prime \prime} \, \left( {r_{{\text{e}}} } \right) \, = { 2}\alpha^{{2}} {\text{U}}_{0} \left( {{1}/\sigma {-}\sigma } \right)^{{2}} , \;\;\; {{{\rm U}^{\prime\prime\prime} }}\left( {r_{{\text{e}}} } \right) \, = \, {-}{ 12}\alpha^{{3}} {\text{U}}_{0} \left( {{1 }{-}{ 1}/\sigma } \right)^{{2}} /\sigma . $$

(5)

Putting Eq. (5) into (4) and simplifying leads to

$$ \alpha = - \frac{1}{{2r_{\text{e}} }} + \frac{2}{\varepsilon } - \frac{{\alpha_{\text{e}} \omega_{\text{e}} }}{{12{\text{B}}_{\text{e}}^{2} r_{\text{e}} }}. $$

(6)

Equation for the energy spectra of the DSO

In this section, an analytical equation for the energy spectra is derived by solving the radial SE confined by the DSO. Different analytical methods for solving the SE exist in the literature^{22,23,24,25,26,27,28}. However, owing to the simplicity of the parametric Nikiforov-Uvarov (PNU) technique²⁷, it is considered in this work.

A brief outline of the PNU method

The PNU method gives that with the aid of a suitable coordinate transformation, a second-order differential equation of the hypergeometric-type can be expressed as²⁷

$$ {{{\rm u}^{\prime\prime}}}_{n\ell } \left( z \right) + \frac{{\alpha_{1} - \alpha_{2} z}}{{z\left( {1 - \alpha_{3} z} \right)}}{{{\rm u}^{\prime}}}_{n\ell } \left( z \right) + \frac{{ - \tau_{1} z^{2} + \tau_{2} z - \tau_{3} }}{{z^{2} \left( {1 - \alpha_{3} z} \right)^{2} }}{\text{u}}_{n\ell } \left( z \right) = 0, $$

(7)

where α_j (j = 0, 1, 2) are constant coefficients, n = 0, 1, 2, … is the vibrational (or principal) quantum number and ℓ = 0, 1, 2, … is the rotational (or orbital momentum) quantum number. The quantization condition leading to energy spectra is written as²⁷

$$ \left( {\alpha_{2} - \alpha_{3} } \right)n + \alpha_{3} n^{2} - \left( {2n + 1} \right)\alpha_{5} + \left( {2n + 1} \right)\left( {\sqrt {\alpha_{9} } + \alpha_{3} \sqrt {\alpha_{8} } } \right) + \alpha_{7} + 2\alpha_{3} \alpha_{8} + 2\sqrt {\alpha_{8} \alpha_{9} } = 0, $$

(8)

where

$$ \begin{aligned} \alpha_{4} & = \tfrac{1}{2}\left( {1 - \alpha_{1} } \right),\quad \alpha_{5} = \tfrac{1}{2}\left( {\alpha_{2} - 2\alpha_{3} } \right),\quad \alpha_{6} = \alpha_{5}^{2} + \tau_{1} \\ \alpha_{7} & = 2\alpha_{4} \alpha_{5} - \tau_{2} ,\;\alpha_{8} = \alpha_{4}^{2} + \tau_{3} ,\;\alpha_{9} = \alpha_{3} \alpha_{7} + \alpha_{3}^{2} \alpha_{8} + \alpha_{6} . \\ \end{aligned} $$

(9)

Analytical equation for the energy levels of the DSO by the PNU method

The radial SE for a particle of mass M₀ moving in a radial potential field, U (r) is given by

$$ \frac{{{\text{d}}^{2} {\text{u}}_{n\ell } \left( r \right)}}{{{\text{d}}r^{2} }} + \frac{{2M_{0} }}{{\hbar^{2} }}\left\{ {E_{n\ell } - {\text{U}}\left( r \right) - \frac{{J\hbar^{2} }}{{2M_{0} r^{2} }}} \right\}{\text{u}}_{n\ell } \left( r \right) = 0, $$

(10)

where J = ℓ (ℓ + 1) is the angular momentum of the system, u_nℓ (r) is the radial wave function and E_nℓ is the bound-state energy eigenvalue. Owing to the presence of the factor r^-2 in the centrifugal term, expression (10) has no exact solution with the potential (1), except for the special case where ℓ = 0 (the pure vibrational state). Nevertheless, an approximate analytical solution can be obtained with the help of approximation models. For small values of r, a Pekeris-type approximation scheme can be written for r^-2 as follows

$$ r^{ - 2} \approx d_{1} + d_{2} \coth_{q} \left( {\alpha r} \right) + d_{3} \coth_{q}^{2} \left( {\alpha r} \right), $$

(11)

where the constant coefficients d_j (j = 1, 2, 3) are deduced by the procedures outlined in Ref.⁷¹ as

$$ \begin{aligned} d_{1} & = \frac{1}{{r_{\text{e}}^{2} }}\left( {1 - \frac{{3\sinh_{{q^{2} }} 2\gamma }}{2\gamma q} + \frac{{3\sinh_{{q^{2} }}^{2} 2\gamma }}{{4\gamma^{2} q^{2} }} - \frac{{\sinh_{{q^{4} }} 4\gamma }}{{4\gamma q^{2} }}} \right) \\ d_{2} & = \frac{{\sinh_{q}^{2} \gamma }}{{r_{\text{e}}^{2} }}\left( {\frac{4}{\gamma q} - \frac{{3\sinh_{{q^{2} }} 2\gamma }}{{\gamma^{2} }} + \frac{{2\cosh_{{q^{2} }} 2\gamma }}{{\gamma q^{2} }}} \right) \\ d_{3} & = \frac{{\sinh_{q}^{2} \gamma }}{{r_{\text{e}}^{2} }}\left( { - \frac{3}{{2\gamma^{2} q}} + \frac{{3\cosh_{{q^{2} }} 2\gamma }}{{2\gamma^{2} q^{2} }} - \frac{{\sinh_{{q^{2} }} 2\gamma }}{{\gamma q^{2} }}} \right) \\ \end{aligned} $$

(12)

Inserting Eqs. (1) and (11) into (10) gives

$$ \frac{{{\text{d}}^{2} {\text{u}}_{n\ell } \left( r \right)}}{{{\text{d}}r^{2} }} + \left\{ {\frac{{2M_{0} \left( {E_{n\ell } - {\text{U}}_{0} } \right)}}{{\hbar^{2} }} - Jd_{1} + \left( {\frac{{4M_{0} {\text{U}}_{0} \sigma }}{{\hbar^{2} }} - Jd_{2} } \right)\coth_{q} \left( {\alpha r} \right) - \left( {\frac{{2M_{0} {\text{U}}_{0} \sigma^{2} }}{{\hbar^{2} }} + Jd_{3} } \right)\coth_{q}^{2} \left( {\alpha r} \right)} \right\}{\text{u}}_{n\ell } \left( r \right) = 0. $$

(13)

Using the substitution z^-1 = 1−q^-1 e^2αr, Eq. (13) is transformed to

$$ {{{\rm u}^{\prime\prime}}}_{n\ell } \left( r \right) + \frac{1 - 2z}{{z\left( {1 - z} \right)}}{{{\rm u}^{\prime}}}_{n\ell } \left( r \right) + \frac{{ - \tau_{1} z^{2} + \tau_{2} z - \tau_{3} }}{{z^{2} \left( {1 - z} \right)^{2} }}{\text{u}}_{n\ell } \left( r \right) = 0, $$

(14)

where

$$ \tau_{1} = \frac{{2M_{0} {\text{U}}_{0} \sigma^{2} }}{{\alpha^{2} \hbar^{2} }} + \frac{{Jd_{3} }}{{\alpha^{2} }},\;\tau_{2} = \frac{{2M_{0} {\text{U}}_{0} \sigma \left( {\sigma - 1} \right)}}{{\alpha^{2} \hbar^{2} }} + \frac{{J\left( {2d_{3} + d_{2} } \right)}}{{2\alpha^{2} }},\;\tau_{3} = \frac{{M_{0} \left( {D_{\text{e}} - E_{n\ell } } \right)}}{{2\alpha^{2} \hbar^{2} }} + \frac{{J\left( {d_{3} + d_{2} + d_{1} } \right)}}{{4\alpha^{2} }}. $$

(15)

By comparing expressions (14) and (7), one obtains α₁ = 1, α₂ = 2, α₃ = 1. Using these results in Eq. (9) gives α₄ = α₅ = 0, α₆ = τ₁, α₇ = – τ₂, α₈ = τ₃ and α₉ = τ₁−τ₂ + τ₃. Inserting the values of α₂, α₃, α₄, α₅, α₆, α₇, α₈ and α₉ into (8) yields

$$ \tau_{3} = \frac{1}{4}\left\{ {n + \frac{1}{2} \pm \sqrt {\tau_{1} + \frac{1}{4}} - \frac{{\tau_{1} - \tau_{2} }}{{n + \frac{1}{2} \pm \sqrt {\tau_{1} + \frac{1}{4}} }}} \right\}^{2} . $$

(16)

Using the expressions in (15) to eliminate τ₁, τ₂ and τ₃ in (16), the expressions for bound-state energy

$$ E_{n\ell } = D_{\text{e}} + \frac{{J\left( {d_{3} + d_{2} + d_{1} } \right)\hbar^{2} }}{{2M_{0} }} - \frac{{\alpha^{2} \hbar^{2} }}{{2M_{0} }}\left\{ {n + \frac{1}{2} \pm \sqrt {\frac{{2M_{0} {\text{U}}_{0} \sigma^{2} }}{{\alpha^{2} \hbar^{2} }} + \frac{{Jd_{2} }}{{\alpha^{2} }} + \frac{1}{4}} - \frac{{\frac{{2M_{0} {\text{U}}_{0} \sigma }}{{\alpha^{2} \hbar^{2} }} - \frac{{Jd_{2} }}{{2\alpha^{2} }}}}{{n + \frac{1}{2} \pm \sqrt {\frac{{2M_{0} {\text{U}}_{0} \sigma^{2} }}{{\alpha^{2} \hbar^{2} }} + \frac{{Jd_{2} }}{{\alpha^{2} }} + \frac{1}{4}} }}} \right\}^{2} . $$

(17)

The pure vibrational state energy E_n0 → E_n is obtained by letting ℓ = 0 in Eq. (17) to obtain

$$ E_{n} = D_{\text{e}} - \frac{{\alpha^{2} \hbar^{2} }}{{2M_{0} }}\left\{ {n + \frac{1}{2} \pm \sqrt {\frac{{2M_{0} {\text{U}}_{0} \sigma^{2} }}{{\alpha^{2} \hbar^{2} }} + \frac{1}{4}} - \frac{{\frac{{2M_{0} {\text{U}}_{0} \sigma }}{{\alpha^{2} \hbar^{2} }}}}{{n + \frac{1}{2} \pm \sqrt {\frac{{2M_{0} {\text{U}}_{0} \sigma^{2} }}{{\alpha^{2} \hbar^{2} }} + \frac{1}{4}} }}} \right\}^{2} . $$

(18)

The maximum vibrational quantum number is deduced from the expression E′_n (n_max) = 0, substituting (18) into this expression gives

$$ n_{\max } = \pm \left( {\frac{{2M_{0} {\text{U}}_{0} \sigma }}{{\alpha^{2} \hbar^{2} }}} \right)^{{\tfrac{1}{2}}} - \left\{ {\frac{1}{2} \pm \left( {\frac{{2M_{0} {\text{U}}_{0} \sigma^{2} }}{{\alpha^{2} \hbar^{2} }} + \frac{1}{4}} \right)^{{\tfrac{1}{2}}} } \right\}. $$

(19)

n_max is essentially a positive integer, which is the value of n at which the energy of the system is a maximum.

Thermochemical functions of the DSO

Having obtained the equation for vibrational state energies, in this section, some important analytical models for the prediction of thermochemical properties of substances are developed for the DSO. The canonical partition function from which the thermodynamic expressions are deduced is first derived. The canonical partition function is written as Z (T) = Z_vibZ_rotZ_tra, where T is the temperature of the system, Z_vib, Z_rot, and Z_tra are the vibrational, rotational and translational partition functions, respectively^44,55. The vibrational partition function depends on the oscillator used to model the diatomic system, it is given as³⁴

$$ {\text{Z}}_{{{\text{vib}}}} = \sum\limits_{n\,\, = \,\,0}^{{n_{\max } }} {{\text{exp}}\left( { - \frac{{E_{n} }}{{{\text{k}}_{{\text{B}}} T}}} \right)} , $$

(20)

where β = 1/(k_BT), k_B is the Boltzmann constant. Putting Eq. (18) into (20) gives

$$ {\text{Z}}_{{{\text{vib}}}} = \exp \left( { - \omega } \right)\sum\limits_{n\,\, = \,\,0}^{{n_{\max } }} {\varphi \left( n \right)} , $$

(21)

where

$$ \omega = \frac{{D_{\text{e}} }}{{{\text{k}}_{B} T}},\quad \varphi \left( n \right) = \exp \left\{ {\varsigma \left( {n + \delta - \frac{\kappa }{n + \delta }} \right)} \right\}^{2} ,\quad \varsigma = \frac{\alpha \hbar }{{\sqrt {2M_{0} {\text{k}}_{{\text{B}}} T} }}. $$

(22)

The series in (21) can be evaluated with the help of the modified Poisson summation formula⁷². The modified Poisson summation approach is used here because it is simple to implement and has yielded very accurate results with many oscillator models such as those in Refs.^35,38,73,74. Other methods for evaluating the vibrational partition function including the phase space sampling method and the Euler-Maclaurin summation approach are given in Refs.^75,76. Based on the modified Poisson summation formula, one can write⁷²

$$ \sum\limits_{n\,\, = \,\,0}^{{n_{\max } }} {\varphi \left( n \right)} = \frac{1}{2}\left\{ {\varphi \left( 0 \right) - \varphi \left( {n_{\max } + 1} \right)} \right\} + \sum\limits_{k = \, - \infty }^{k = \,\,\infty } {\int\limits_{0}^{{n_{\max } + 1}} {\varphi \left( y \right){\text{exp}}\left( { - {\text{i}}2{\uppi }ky} \right){\text{d}}y} } . $$

(23)

Substituting the second expression in (22) into the right-hand side of (23) and expanding out the summation gives

$$ \begin{aligned} \sum\limits_{n\,\, = \,\,0}^{{n_{\max } }} {\varphi \left( n \right)} & = \frac{1}{2}\left\{ {\exp \left( {\lambda_{0}^{2} } \right) - \exp \left( {\lambda_{1}^{2} } \right)} \right\} + \int\limits_{0}^{{n_{\max } + 1}} {\exp \left\{ {\varsigma \left( {y + \delta - \frac{\kappa }{y + \delta }} \right)} \right\}^{2} {\text{d}}y} \\ & \;\;\;\, + \sum\limits_{k = - \infty }^{ - 1} {\int\limits_{0}^{{n_{\max } + 1}} {{\text{exp}}\left\{ {\varsigma^{2} \left( {y + \delta - \frac{\kappa }{y + \delta }} \right)^{2} - {\text{i}}2{\uppi }ky} \right\}{\text{d}}y} } + \sum\limits_{k = 1}^{\infty } {\int\limits_{0}^{{n_{\max } + 1}} {{\text{exp}}\left\{ {\varsigma^{2} \left( {y + \delta - \frac{\kappa }{y + \delta }} \right)^{2} - {\text{i}}2{\uppi }ky} \right\}{\text{d}}y} } . \\ \end{aligned} $$

(24)

where $\lambda_{0} = \varsigma \left( {\delta - \tfrac{\kappa }{\delta }} \right)$, $\lambda_{1} = \varsigma \left( {n_{\max } + 1 + \delta - \tfrac{\kappa }{{n_{\max } + 1 + \delta }}} \right)$. The last-two terms in (24) include quantum correction terms. For the moderate to high temperature range of diatomic systems, the quantum correction terms are small and can be ignored. Therefore, expression (24) is recast as

$$ \sum\limits_{n\,\, = \,\,0}^{{n_{\max } }} {\varphi \left( n \right)} = \frac{1}{2}\left\{ {\exp \left( {\lambda_{0}^{2} } \right) - \exp \left( {\lambda_{1}^{2} } \right)} \right\} + \int\limits_{0}^{{n_{\max } + 1}} {\exp \left\{ {\varsigma \left( {y + \delta - \frac{\kappa }{y + \delta }} \right)} \right\}^{2} {\text{d}}y} . $$

(25)

Using the substitution z = ς {y + δ−κ/(y + δ)}, followed by the mapping x = (z² + 2ςκ²)^½ to evaluate the integral, the summation in (25) is obtained as

$$ \sum\limits_{n\,\, = \,\,0}^{{n_{\max } }} {\varphi \left( n \right)} = \frac{1}{2}\left\{ {\exp \left( {\lambda_{0}^{2} } \right) - \exp \left( {\lambda_{1}^{2} } \right)} \right\} - \frac{{\sqrt {\uppi } }}{4\varsigma }\left\{ {{\text{Erfi}}\lambda_{0} - {\text{Erfi}}\lambda_{1} + \exp \left( { - 4\kappa \varsigma^{2} } \right)\left( {{\text{Erfi}}\eta_{0} - {\text{Erfi}}\eta_{1} } \right)} \right\}, $$

(26)

Thus, inserting (26) into (21), the vibrational partition function is obtained in compact form as

$$ {\text{Z}}_{{{\text{vib}}}} = {\text{ A}}_{0} {-}{\text{ A}}_{{1}} {-}{\text{ A}}_{{2}} {-}{\text{ A}}_{{3}} , $$

(27)

where

$$ \begin{aligned} {\text{A}}_{0} & = \frac{1}{2}\exp \left( {\lambda_{0}^{2} - \omega } \right),\quad {\text{A}}_{1} = \frac{1}{2}\exp \left( {\lambda_{1}^{2} - \omega } \right),\quad {\text{A}}_{2} = \frac{{\sqrt {\uppi } }}{4\varsigma }\exp \left( { - \omega } \right)\left( {{\text{Erfi}}\lambda_{0} - {\text{Erfi}}\lambda_{1} } \right){,} \\ {\text{A}}_{3} & = \frac{{\sqrt {\uppi } }}{4\varsigma }\exp \left( { - 4\kappa \varsigma^{2} - \omega } \right)\left( {{\text{Erfi}}\eta_{0} - {\text{Erfi}}\eta_{1} } \right), \\ \end{aligned} $$

(28)

Based on the formalism of the rigid-rotor approximation for diatomic molecules, the rotational and translational components of the partition function are expressed as^36,40,46,50

$$ {\text{Z}}_{{{\text{rot}}}} \left( T \right) = \frac{1}{\upsilon }\left\{ {\frac{1}{3} + \frac{T}{{\Theta_{{{\text{rot}}}} }} + \frac{1}{15}\frac{{\Theta_{{{\text{rot}}}} }}{T} + \frac{4}{315}\left( {\frac{{\Theta_{{{\text{rot}}}} }}{T}} \right)^{2} } \right\}, $$

(29)

$$ {\text{Z}}_{{{\text{tra}}}} \left( T \right) = \left( {\frac{{m{\text{k}}_{{\text{B}}} T}}{{2{\uppi }\hbar^{2} }}} \right)^{{\tfrac{3}{2}}} {\text{V,}} $$

(30)

where V is satisfied by pV = RT, m is the mass of gas molecules enclosed in volume V, the gas pressure is denoted by p, R is the molar gas constant, $\Theta_{{{\text{rot}}}} = \hbar^{{2}} /{2}\pi \mu r_{{\text{e}}}^{{2}} {\text{k}}_{{\text{B}}}$ is the characteristic temperature of the gas. The parameter υ takes the value 2 if the gas is homonuclear, and 1 for heteronuclear gas molecules. Using the expression for the partition function, explicit equations for molar entropy, enthalpy, Gibbs free energy and constant pressure heat capacity are developed for the DSO as follows.

Molar entropy equation for the deformed Schiöberg oscillator

The molar entropy (J mol⁻¹ K⁻¹) of the system can be evaluated from the relation⁵³

$$ {\text{S}}\left( T \right) = {\text{R}}\ln {\text{Z}} + {\text{R}}T\left( {\frac{\partial }{\partial T}\ln {\text{Z}}} \right)_{{\text{V}}} . $$

(31)

Substituting the expression Z (T) = Z_vibZ_rotZ_tra into (31) and using Eqs. (27), (29) and (30) in the result, one obtains

$$ {\text{S}}\left( T \right) = \frac{5}{{2}}{\text{R}} + {\text{R}}\left( {\ln {\text{Z}} + \frac{{T{{{\rm Z}^{\prime}}}_{{{\text{vib}}}} }}{{{\text{Z}}_{{{\text{vib}}}} }}} \right) + \frac{{\text{R}}}{\upsilon }\left\{ {\frac{T}{{\Theta_{{{\text{rot}}}} }} - \frac{1}{15}\frac{{\Theta_{{{\text{rot}}}} }}{T} - \frac{8}{315}\left( {\frac{{\Theta_{{{\text{rot}}}} }}{T}} \right)^{2} } \right\} $$

(32)

where for compactness, the following abbreviation is used

$$ \begin{aligned} T{{{\rm Z}^{\prime}}}_{{{\text{vib}}}} & = \left( {\omega - \lambda_{0}^{2} + \frac{{\lambda_{0} }}{2\varsigma }} \right){\text{A}}_{0} - \left( {\omega - \lambda_{1}^{2} + \frac{{\lambda_{1} }}{2\varsigma }} \right){\text{A}}_{1} - \left( {\omega + \frac{1}{2}} \right){\text{A}}_{2} \\ & \;\;\;\, - \left( {\omega + 4\kappa \varsigma^{2} + \frac{1}{2}} \right){\text{A}}_{3} + \frac{1}{4\varsigma }\left( {\eta_{0} {\text{e}}^{{\eta_{0}^{2} }} - \eta_{1} {\text{e}}^{{\eta_{1}^{2} }} } \right){\text{e}}^{{ - 2\omega - 4\kappa \varsigma^{2} }} . \\ \end{aligned} $$

(33)

Molar enthalpy model of the DSO

The molar enthalpy (J mol⁻¹) of the DSO can be deduced from the expression⁵⁴

$$ {\text{H}}\left( T \right) = {\text{R}}T^{2} \left( {\frac{\partial }{\partial T}\ln {\text{Z}}} \right)_{{\text{V}}} + {\text{R}}T{\text{V}}\left( {\frac{\partial }{{\partial {\text{V}}}}\ln {\text{Z}}} \right)_{T} . $$

(34)

The substitution Z (T) = Z_vibZ_rotZ_tra and Eqs. (27), (29) and (30) and (34) yields

$$ {\text{H}}\left( T \right) = \frac{5}{{2}}{\text{R}}T + {\text{R}}T\left( {\frac{{T{{{\rm Z}^{\prime}}}_{{{\text{vib}}}} }}{{{\text{Z}}_{{{\text{vib}}}} }}} \right) + \frac{{{\text{R}}T}}{\upsilon }\left\{ {\frac{T}{{\Theta_{{{\text{rot}}}} }} - \frac{1}{15}\frac{{\Theta_{{{\text{rot}}}} }}{T} - \frac{8}{315}\left( {\frac{{\Theta_{{{\text{rot}}}} }}{T}} \right)^{2} } \right\}. $$

(35)

Equation (35) can be used to compute molar enthalpy data for diatomic substances. However, to enable the results obtained in this study to be compared with available literature, scaled values of (35) are needed. The scaled molar enthalpy is written as^44,45

$$ {\text{H}}_{{{\text{scaled}}}} = {\text{ H }}{-}{\text{ h}}_{{{298}.{15}}} , $$

(36)

where h_298.15 is given by (35), it denotes the molar enthalpy of the molecules calculated at temperature of 298.15 K and pressure of 0.1 MPa.

Molar Gibbs free energy of the DSO

Here, the analytical equation for molar Gibbs free energy is derived for the DSO. The Gibbs free energy is given by

$$ {\text{G }} = {\text{ H }}{-}T{\text{S}}. $$

(37)

Replacing (34) and (31) into (37) gives

$$ {\text{G }} = \, {-}{\text{ lnZ}}_{{{\text{vib}}}} {-}{\text{ lnZ}}_{{{\text{rot}}}} {-}{\text{ lnZ}}_{{{\text{tra}}}} . $$

(38)

For the purpose of relating to observed data, the scaled Gibbs free energy is defined as^44,45

$$ {\text{G}}_{{{\text{scaled}}}} = \, {-} \, \left( {{\text{G }}{-}{\text{ H}}_{{{298}.{15}}} } \right)/T. $$

(39)

Isobaric molar specific heat capacity model of the DSO

The constant pressure (isobaric) molar heat capacity (in J mol⁻¹ K⁻¹) is evaluated from ${\text{C}}_{p} = \tfrac{{\partial \,{\text{H}}}}{\partial \,T}$^34,40. Substituting expression (35) into this equation gives

$$ {\text{C}}_{p} = \frac{5}{{2}}{\text{R}} + {\text{R}}\left\{ {T^{2} \left( {\frac{{{{{\rm Z}^{\prime\prime}}}_{{{\text{vib}}}} }}{{{\text{Z}}_{{{\text{vib}}}} }}} \right) - \left( {T\frac{{{{{\rm Z}^{\prime}}}_{{{\text{vib}}}} }}{{{\text{Z}}_{{{\text{vib}}}} }}} \right)^{2} } \right\} + \frac{{\text{R}}}{\upsilon }\left\{ {\frac{2T}{{\Theta_{{{\text{rot}}}} }} + \frac{8}{315}\left( {\frac{{\Theta_{{{\text{rot}}}} }}{T}} \right)^{2} } \right\}, $$

(40)

where ${{{\rm Z}^{\prime}}}_{{{\text{vib}}}}$ and ${{{\rm Z}^{\prime\prime}}}_{{{\text{vib}}}}$ are given by Eqs. (33) and (41), respectively

$$ \begin{aligned} T^{2} {{{\rm Z}^{\prime\prime}}}_{{{\text{vib}}}} & = \left\{ {\left( {\lambda_{0}^{2} - \omega + 1} \right)^{2} - \frac{{\lambda_{0} T}}{2\varsigma }\left( {\lambda_{0}^{2} - 2\omega + \frac{1}{2}} \right) - 1} \right\}{\text{A}}_{0} \, - \left\{ {\left( {\lambda_{1}^{2} - \omega + 1} \right)^{2} - \frac{{\lambda_{1} T}}{2\varsigma }\left( {\lambda_{1}^{2} - 2\omega + \frac{1}{2}} \right) - 1} \right\}{\text{A}}_{1} \\ & \;\;\;\, - \left\{ {\left( {\omega + \frac{1}{2}} \right)^{2} - \frac{1}{2}} \right\}{\text{A}}_{2} - \left\{ {\left( {\omega + 4\kappa \varsigma^{2} - \frac{1}{2}} \right)^{2} - \frac{1}{2}} \right\}{\text{A}}_{3} - \frac{T}{4\varsigma }\left( {\eta_{0} {\text{e}}^{{\eta_{0}^{2} }} - \eta_{1} {\text{e}}^{{\eta_{1}^{2} }} } \right){\text{e}}^{{ - \omega - 4\kappa \varsigma^{2} }} . \\ \end{aligned} $$

(41)

Results and discussion

In this section, the equation derived for the energy levels and thermochemical functions are applied to diatomic substances including ⁷Li₂ (2 ³Π_g), NaBr (X ¹Σ⁺), KBr (X ¹Σ⁺) and KRb (B ¹Π) molecules. The model parameters for these molecules are given in Table 1. The experimental values for D_e, r_e, ω_e and α_e were obtained from publications^77,78,79. The values of the potential parameters also listed in Table 1 were computed with Eqs. (3) and (6).

Table 1 Model parameters of the diatomic molecules investigated in this study.

To numerically affirm the accuracy of the model equations, the percentage average absolute deviation (PAAD) of the predicted results from the observed data for the molecule is employed as accuracy indicator. The PAAD values are interpreted according to Lippincott condition for the applicability of a model equation. The Lippincott criterion requires that the PAAD value of the predicted data from the observed data is at most 1% of the experimental results. The smaller the PAAD value, better the model equation. The PAAD value is written in compact form as⁸⁰

$$ {\text{PAAD}} = \frac{100}{{N_{{\text{p}}} }}\sum\limits_{n\, = \,1}^{{N_{{\text{p}}} }} {\left| {\frac{{{\text{X}} - {\text{Y}}}}{{\text{Z}}}} \right|_{n} } , $$

(42)

where N_p is the count of observed data, X, Y and Z are chosen in relation to the predicted and observed data.

Numerical results for potential energies

Utilizing the spectroscopic parameters in Table 1, Eq. (1) is used to generate numerical results for the potential energy U (≡ U_min, U_max) for different vales of r (≡ r_min, r_max). The results obtained are given in Tables 2, 3, 4 and 5. Available experimental Rydberg–Klein–Rees (RKR) data^77,79, and the multireference configuration interaction (MRCI) data⁷⁸ for the molecules are also included in the tables. The inclusion of the RKR and MRCI data is to allow for comparison of the predicted values of the potential energies with the observed data for the molecules. The variation in potential energy of the molecules as a function of interparticle separation is given in Figs. 1, 2, 3 and 4. The experimental RKR data are also plotted in the figures. The figures show that the computed potential energy of the molecules agree with the experimental data for the molecules.

Figure 1

Modeling of deformed Schiöberg potential with experimental RKR interparticle potential energy data for the ⁷Li₂ (2 ³Π_g) molecule.

Figure 2

Modeling of deformed Schiöberg potential with ab initio MRCI interparticle potential energy data for the NaBr (X ¹Σ⁺) molecule.

Figure 3

Modeling of deformed Schiöberg potential with ab initio MRCI interparticle potential energy data for the KBr (X ¹Σ⁺) molecule.

Figure 4

Modeling of deformed Schiöberg potential with experimental RKR interparticle potential energy data for the KRb (B ¹Π) molecule.

The accuracy of the DSO to model the experimental RKR data can be determined by letting X = RKR, Y = U and Z = D_e in Eq. (42). With the help of the resulting expression and the data in Tables 2, 3, 4 and 5, the PAAD values obtained are 1.3319%, 0.2108%, 0.2359% and 0.8841% for the ⁷Li₂ (2 ³Π_g), NaBr (X ¹Σ⁺), KBr (X ¹Σ⁺) and KRb (B ¹Π) molecules, respectively. Therefore, based on the Lippincott criterion, it can be inferred that the DSO could satisfactorily reproduce the experimental RKR and MRCI results for the selected diatomic molecules.

Table 2 Potential energies, bound-state energy eigenvalues and experimental Rydberg–Klein–Rees data for the ⁷Li₂ (2 ³Π_g) molecule.

Table 3 Potential energies, bound-state energy eigenvalues and experimental Rydberg–Klein–Rees (RKR) data for the NaBr (X ¹Σ⁺) molecule.

Table 4 Potential energies, bound-state energy eigenvalues and experimental Rydberg–Klein–Rees (RKR) data for the KBr (X ¹Σ⁺) molecule.

Table 5 Potential energies, bound-state energy eigenvalues and experimental Rydberg–Klein–Rees (RKR) data for the KRb (X ¹Σ⁺) molecule.

Applicability of the Pekeris approximation scheme to diatomic systems

To ascertain the significance of the Pekeris-type approximation model (11) suggested for the centrifugal barrier of the SE, the function F₁ = r⁻² is plotted as a function of interparticle separation. On the same scale and axes, the approximation function F₂ = d₁ + d₂ coth_q (αr) + d₃ coth_q² (αr) is also plotted. The graphical plots depicting F₁ and F₂ for the diatomic molecules are shown in Figs. 5, 6, 7 and 8. It is evident from the figures that for the range of r chosen for the interparticle separations, the Pekeris approximation F₂ is a good representation of the observed function F₁. The implication of the result is that based on the parameters of the diatomic molecules considered in this study, the Pekeris approximation model F₂ could be used to eliminate the function F₁ to analytically solve the SE (10).

Figure 5

Modeling of the Pekeris approximation scheme F2 to the function F₁ for the ⁷Li₂ (2 ³Π_g) molecule.

Figure 6

Modeling of the Pekeris approximation scheme F2 to the function F₁ for the NaBr (X ¹Σ⁺) molecule.

Figure 7

Modeling of the Pekeris approximation scheme F2 to the function F₁ for the KBr (X ¹Σ⁺) molecule.

Figure 8

Modeling of the Pekeris approximation scheme F2 to the function F₁ for the KRb (B ¹Π) molecule.

Numerical results for pure vibrational state energies

With the aid of Eq. (18), pure vibrational state energies are generated for the selected diatomic molecules. The computed results are summarized in Tables 2, 3, 4 and 5. To quantitatively compare the obtained bound-state energies with the experimental RKR results for the molecules, the parameters in Eq. (42) are adjusted so that X = Z = RKR and Y = E_n. The PAAD values obtained are 1.0956%, 0.2935%, 3.8667% and 1.4629% for the ⁷Li₂ (2 ³Π_g), NaBr (X ¹Σ⁺), KBr (X ¹Σ⁺) and KRb (B ¹Π) molecules, respectively. Therefore, based on the Lippincott requirement for the applicability of a model equation, the present formula for the pure vibrational state energies could satisfactorily predict the experimental data for the NaBr (X ¹Σ⁺) molecule, and marginally model the results for ⁷Li₂ (2 ³Π_g) and KRb (B ¹Π) molecules. The PAAD value obtained for the KRb (B ¹Π) molecule is relatively high (≈ 4% of the observed data), suggesting that the present energy levels equation could not satisfactorily predict the observed data for the KRb (B ¹Π) molecule.

Investigation of thermochemical properties of diatomic substances

In this section, the thermodynamic functions developed for the DSO are used to analyze the thermochemical properties of pure substances. To substantiate the accuracy of the model equations, numerical data are obtained analytically and the results are compared with the literature on gaseous substances. The experimental results were retrieved from the National Institute of Standards and Technology (NIST) database⁸¹. The NIST data is available for the gaseous NaBr and KBr molecules only. Thus, our discussions are restricted to these two molecules. PAAD values computed in the temperature range 300–6000 K are used to gauge the accuracy of the model equations.

Tables 6 and 7 summarize the data obtained using Eqs. (32), (36), (39) and (40). The NIST data for the molecules are also listed in the tables under the columns (S_NIST), (H_NIST), (G_NIST) and (C_pNIST). Graphical plots of the thermochemical equations versus temperature are represented in Figs. 9, 10, 11 and 12. The corresponding NIST data are also plotted in the figures. Due to the similarity in the figures for the NaBr and KBr molecules, only the plots for NaBr molecule are presented.

Table 6 Predicted and observed data on molar entropy (J mol⁻¹ K⁻¹), reduced molar enthalpy (kJ mol⁻¹), reduced molar Gibbs free energy (J mol⁻¹ K⁻¹) and constant pressure molar heat capacity (J mol⁻¹ K⁻¹) for the NaBr (X ¹Σ⁺) molecule.

Table 7 Predicted and observed data on molar entropy (J mol⁻¹ K⁻¹), reduced molar enthalpy (kJ mol⁻¹), reduced molar Gibbs free energy (J mol⁻¹ K⁻¹) and constant pressure molar heat capacity (J mol⁻¹ K⁻¹) for the KBr (X ¹Σ⁺) molecule.

Figure 9

Graphical representation of molar entropy versus temperature for the ground state NaBr molecule.

Figure 10

Graphical representation of scaled molar enthalpy versus temperature for the ground state NaBr molecule.

Figure 11

Graphical representation of scaled molar Gibbs free energy versus temperature for the ground state NaBr molecule.

Figure 12

Graphical representation of isobaric molar heat capacity versus temperature for the ground state NaBr molecule.

Figure 9 shows the graphical representation of the molar entropy against temperature. The figure shows that the predicted molar entropy agrees with the experimental results. To appraise the quality of the molar entropy model, the parameters X, Y, Z in Eq. (42) are chosen such that X = Z = S_NIST and Z = S. The PAAD values deduced are 0.5401% and 1.2919%, for NaBr and KBr molecules, respectively. The obtained PAAD values are within the Lippincott error limit. This means that molar entropy equation proposed for the DSO could satisfactorily predict the NIST data for the gaseous NaBr and KBr molecules.

In the plot shown in Fig. 10, scaled molar enthalpy is plotted as a function of temperature. The agreement between the observed and predicted data is evident in the figure. An estimate of the efficiency of the molar enthalpy model can be obtained by letting X = Z = H_NIST and Z = H_scaled in Eq. (42). Using the data in Tables 6 and 7, the computed PAAD values are 1.9428% and 1.5639% for the NaBr and KBr molecules, respectively. The PAAD values reveal that the DSO model for the scaled molar enthalpy could marginally predict the experimental results for the gaseous molecules. It is also noted from the tables that as the molecules are excited from moderate to high temperature region, the discrepancy between the predicted and observed data increases. The increased difference could be linked to lowest order approximation used to obtain expression (36). The absence of the quantum correction terms in the molar entropy equation is responsible for PAAD values exceeding 1%.

The variations in molar Gibbs free energy with temperature is graphically represented in Fig. 11. The figure show that the results obtained by analytical computations are in good agreement with the data reported in the NIST database for the gaseous substances. With the help of the data in Tables 6 and 7, the PAAD values obtained are 0.8164% and 1.5957% for the ground state NaBr and KBr molecules, respectively. The obtained PAAD values are deduced by setting X = Z = G_NIST, Y = G_scaled in (42). Based on the Lippincott condition, it can be inferred that the molar Gibbs free energy model for the DSO could satisfactorily predict the Gibbs free energy of the selected diatomic molecules.

In Fig. 12, the constant pressure molar heat capacity is plotted against the temperature of the molecules. From the figure, it is clear that in the low temperature range, the predicted isobaric molar heat capacity agrees with the experimental data for the molecules. However, in the moderate to high temperature domain, the predicted heat capacity results are smaller, and deviate significantly from the observed data. The reason for the relatively high deviation could be associated with the quantum corrections terms which are absent in Eq. (40).

Taking X = Z = C_pNIST, and Y = C_p, the PAAD values deduced for the molecules are 2.9770% and 2.4041% for the ground state NaBr and KBr, respectively. The results clearly suggest that the isobaric molar heat capacity could not accurately predict the experimental results for the NaBr and KBr molecules. Nevertheless, the results in the tables suggest that the model could be used to obtain the molar heat capacity of the molecules within the low temperature range.

Conclusions

In this work, the necessary conditions for a diatomic molecule oscillator were used to construct an improved version of the deformed Schiöberg oscillator (DSO). By employing the parametric Nikiforov-Uvarov solution recipe to solve the radial SE with the DSO, analytical expressions for energy spectra and canonical partition function were obtained. Using the obtained partition function, thermodynamic properties such as molar entropy, enthalpy, Gibbs free energy and isobaric heat capacity were developed for the DSO. The obtained equations were used to analyze the physical properties of diatomic substances including ⁷Li₂ (2 ³Π_g), NaBr (X ¹Σ⁺), KBr (X ¹Σ⁺) and KRb (B ¹Π) molecules. The percentage average absolute deviation (PAAD) of the predicted data from the experimental data of the molecules is used as the goodness-of-fit indicator. The PAAD values obtained with the DSO are 1.3319%, 0.2108%, 0.2359% and 0.8841% for the molecules. The equation of bound state energy levels gave PAAD of 1.0956%, 0.2935%, 3.8667% and 1.4629% from the experimental data of the ⁷Li₂ (2 ³Π_g), NaBr (X ¹Σ⁺), KBr (X ¹Σ⁺) and KRb (B ¹Π) molecules. PAAD values were also obtained using the expression for molar entropy, scaled molar enthalpy, scaled molar Gibbs free energy and constant pressure molar heat capacity models. The results obtained for NaBr (X ¹Σ⁺) molecule are 0.5401%, 1.9428%, 0.8164% and 2.9770%. The corresponding results for KBr (X ¹Σ⁺) are 1.2919%, 1.5639%, 1.5597% and 2.4041% from the NIST data. The results obtained are in good agreement with theoretic data reported in existing literature and available experimental data on diatomic systems. The results obtained in this study could have practical applications in the many fields of physics and engineering such as solid-state physics, chemical physics, chemical engineering and molecular physics.