Theoretical Prediction of Thermal Properties of K₂ Diatomic Molecule Using Generalized Mobius Square Potential

Abstract

In the present paper, thermal properties of K₂ diatomic molecule were theoretically investigated. To this goal, we have used the generalized Mobius square (GMS) potential and obtained the rotational–vibrational energy levels of the GMS potential analytically. The Schrödinger equation (SE) was solved by considering a Pekeris-type approximation framework and a suitable coordinate transformation. Using the calculated energy levels, we can analytically obtain the partition function and thereby thermal properties of the K₂ molecule such as mean energy, entropy, enthalpy and etc. Comparison of our results with experimental data shows a good agreement. Our results reveal that the GMS potential is a suitable candidate to predict the thermal properties of the K₂ molecule. We can use the model to predict thermal properties at the temperature ranges where there are no experimental results.

Appendix

We cannot obtain exact analytical relation for the energy levels due to the centrifugal potential ${U}_{\mathrm{cp}}\left(r\right)=\frac{l\left(l+1\right){\hbar }^{2}}{2\mu {r}^{2}}$ in Eq. 11. For this reason, we use a coordinate transformation z $=\frac{r-{r}_{\mathrm{e}}}{{r}_{\mathrm{e}}}$

$${U}_{\mathrm{cp}}\left(r\right)=\frac{l\left(l+1\right){\hbar }^{2}}{2\mu {r}^{2}}=\frac{l\left(l+1\right){\hbar }^{2}}{2\mu {r}_{\mathrm{e}}^{2}}\frac{1}{{\left(1+z\right)}^{2}}=\frac{l\left(l+1\right){\hbar }^{2}}{2\mu {r}_{\mathrm{e}}^{2}}\left(1-2z+3{z}^{2}+\cdots \right),$$

$$=\frac{l\left(l+1\right){\hbar }^{2}}{2\mu {r}_{\mathrm{e}}^{2}}\left({q}_{0}+\frac{{q}_{1}}{+{\mathrm{e}}^{\alpha r}}+\frac{{q}_{2}}{{\left(+{\mathrm{e}}^{\alpha r}\right)}^{2}}\right),$$

(20)

where the coefficients ${q}_{0}$, ${q}_{1}$ and ${q}_{2}$ are defined as [44]

$${q}_{0}=1+\frac{1}{{\alpha }^{2}{r}_{\mathrm{e}}^{2}}\left(3-3\alpha {r}_{\mathrm{e}}+6\eta{\mathrm{e}}^{-\alpha {r}_{\mathrm{e}}}+3\eta^{2}{\mathrm{e}}^{-2\eta\alpha {r}_{\mathrm{e}}}-2\eta\alpha {r}_{\mathrm{e}}{e}^{-\alpha {r}_{\mathrm{e}}}+\eta^{2}\alpha {r}_{\mathrm{e}}{\mathrm{e}}^{-2\alpha {r}_{\mathrm{e}}}\right),$$

(21)

$${q}_{1}=\frac{2}{{\alpha }^{2}{r}_{\mathrm{e}}^{2}}\left(-9\eta+3\eta\alpha {r}_{\mathrm{e}}-3{\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}+2\alpha {r}_{\mathrm{e}}{\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}-9\eta^{2}{\mathrm{e}}^{-\alpha {r}_{\mathrm{e}}}-3\eta^{3}{\mathrm{e}}^{-2\alpha {r}_{\mathrm{e}}}-\eta^{3}\alpha {r}_{\mathrm{e}}{\mathrm{e}}^{-2\alpha {r}_{\mathrm{e}}}\right),$$

(22)

$${q}_{2}=\frac{1}{{\alpha }^{2}{r}_{\mathrm{e}}^{2}}\left(18\eta^{2}+12\eta{\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}+3{\mathrm{e}}^{2\alpha {r}_{\mathrm{e}}}-2\eta\alpha {r}_{\mathrm{e}}{\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}-\alpha {r}_{\mathrm{e}}{\mathrm{e}}^{2\alpha {r}_{\mathrm{e}}}+12\eta^{3}{\mathrm{e}}^{-\alpha {r}_{\mathrm{e}}}+3\eta^{4}{\mathrm{e}}^{-2\alpha {r}_{\mathrm{e}}}+2\eta^{3}\alpha {r}_{\mathrm{e}}{\mathrm{e}}^{-\alpha {r}_{\mathrm{e}}}+\eta^{4}\alpha {r}_{\mathrm{e}}{\mathrm{e}}^{-2\alpha {r}_{\mathrm{e}}}\right).$$

(23)

Inserting Eq. 20 into Eq. 11 and using a transformation $S={\mathrm{e}}^{-\alpha {r}_{\mathrm{e}}}$, the SE can be written as

$${S}^{2}\frac{{\mathrm{d}}^{2}{R}_{nl}\left(S\right)}{\mathrm{d}{S}^{2}}+S\frac{\mathrm{d}{R}_{nl}\left(S\right)}{\mathrm{d}S}+\left(-{\varepsilon }_{nl}-\frac{G}{\eta-S}+\frac{H}{{\left(\eta-S\right)}^{2}}\right){R}_{nl}\left(S\right)=0,$$

(24)

where

$$-{\varepsilon }_{nl}=\frac{1}{{\hbar }^{2}{\alpha }^{2}}\left(2\mu {E}_{nl}-2\mu {D}_{\mathrm{e}}-\frac{l\left(l+1\right){\hbar }^{2}}{{r}_{\mathrm{e}}^{2}}{q}_{0}\right),$$

(25)

$$G=\frac{l\left(l+1\right){q}_{1}}{{\alpha }^{2}{r}_{\mathrm{e}}^{2}}-\frac{4\mu {D}_{\mathrm{e}}}{{\hbar }^{2}{\alpha }^{2}}\left(+{\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}\right),$$

(26)

$$H=\frac{2\mu {D}_{\mathrm{e}}}{{\hbar }^{2}{\alpha }^{2}}{\left(\eta+{\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}\right)}^{2}+\frac{l\left(l+1\right){q}_{2}}{{\alpha }^{2}{r}_{\mathrm{e}}^{2}}.$$

(27)

To solve Eq. 22, we apply the following relation

$${R}_{nl}\left(S\right)={S}^{\xi }{\left(\eta-S\right)}^{\delta }{F}_{nl}\left(S\right),$$

(28)

where

$$\xi =\pm \sqrt{{\varepsilon }_{nl}+\frac{G}{\eta}+\frac{H}{{\eta}^{2}}}; \quad\delta =\frac{1}{2}\left(1\pm \sqrt{1+\frac{4H}{{\eta}^{2}}}\right).$$

(29)

Inserting Eq. 26 into Eq. 22, we obtain

$$S\left(\eta-S\right)\frac{{\mathrm{d}}^{2}{F}_{nl}\left(S\right)}{\mathrm{d}{S}^{2}}+\left[\eta\left(1+2\xi \right)-S\left(1+2\xi +2\delta \right)\right]\frac{\mathrm{d}{F}_{nl}\left(S\right)}{\mathrm{d}S}-\left[\left(\xi +\delta -\sqrt{{\varepsilon }_{nl}}\right)\left(\xi +\delta +\sqrt{{\varepsilon }_{nl}}\right)\right]{F}_{nl}\left(S\right)=0.$$

(30)

According to above relation, we define the following parameters

$$a\equiv \xi +\delta -\sqrt{{\varepsilon }_{nl}}, \quad b\equiv \xi +\delta +\sqrt{{\varepsilon }_{nl}},\quad c\equiv \eta\left(1+2\xi \right).$$

(31)

The hypergeometric function becomes a polynomial of a certain degree when either $a$ or $b$ is equal to a negative integer ($-n$).

$$a=-n\quad n=0, 1, 2, 3, \ldots , {n}_{\mathrm{max}}.$$

(32)

Using Eqs. 23 to 25 and 27 and 29, we can obtain the energy levels as

$${E}_{nl}={D}_{\mathrm{e}}+\zeta {q}_{0}-\frac{{\hbar }^{2}{\alpha }^{2}}{2\mu }\left[\frac{2n+1\pm \sqrt{1+\frac{8\mu {D}_{\mathrm{e}}}{{\hbar }^{2}{\alpha }^{2}{\eta}^{2}}\left[{\left(\eta{+\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}\right)}^{2}+\zeta {q}_{2}\right]}}{4}-\frac{\frac{2\mu }{{\hbar }^{2}{\alpha }^{2}{\eta}^{2}}\left[{D}_{\mathrm{e}}\left({\mathrm{e}}^{2\alpha {r}_{\mathrm{e}}}-{\eta}^{2}\right)+\zeta {q}_{2}+\zeta {q}_{1}\eta\right]}{2n+1\pm \sqrt{1+\frac{8\mu {D}_{\mathrm{e}}}{{\hbar }^{2}{\alpha }^{2}{\eta}^{2}}\left[{\left(\eta{+\mathrm{e}}^{\alpha {r}_{\mathrm{e}}}\right)}^{2}+\zeta {q}_{2}\right]}}\right]$$