Thermal properties of anharmonic Eckart potential model using Euler–MacLaurin formula

Abstract

By employing the asymptotic iteration method (AIM), we solved the three-dimensional time-independent Schrödinger equation with the anharmonic Eckart potential model. The expression for the eigensolution of the anharmonic Eckart potential was obtained. With the help of the ro-vibrational energy spectra obtained, we derived the expressions for the ro-vibrational partition function and other thermodynamic functions, via the Euler MacLaurin formula. Effects of temperature and upper bound vibration quantum number on the thermodynamic functions of anharmonic Eckart potential were discussed for some diatomic molecular systems. It has been established that unique critical temperatures of ro-vibrational entropy and ro-vibrational specific heat capacity exist for the selected diatomic molecules.

Appendix A

1.1 A.1 Review of the asymptotic iteration method (AIM)

The AIM was proposed by Ciftci et al [65] to solve the homogeneous linear second-order differential equation of the form [66]

$$\begin{aligned} {{y}'}'(x)=\lambda _{0} (x){y}'(x)+s_{0} \left( x \right) y(x), \end{aligned}$$

(A.1)

where $\lambda _{0} \left( x \right) \ne 0$ and the prime denotes the derivative with respect to x. The functions, $s_{0} \left( x \right)$ and $\lambda _{0} \left( x \right)$ must be sufficiently differentiable. Differentiating eq. (A.1) with respect to x, we obtain

$$\begin{aligned} {y}'''\left( x \right) =\lambda _{1} \left( x \right) {y}'\left( x \right) +s_{1} \left( x \right) y\left( x \right) ,\quad \end{aligned}$$

(A.2)

where

$$\begin{aligned} \lambda _{1} \left( x \right)= & {} \lambda _{0}^{\prime }\left( x \right) +\lambda _{0}^{2} \left( x \right) +s_{0} \left( x \right) , \nonumber \\ s_{1} \left( x \right)= & {} s_{0}^{\prime }\left( x \right) +s_{0} \left( x \right) \lambda _{0} \left( x \right) . \end{aligned}$$

(A.3)

Taking the second derivative of eq. (A.1) yields

$$\begin{aligned} {{y}^{\prime \prime \prime \prime }} \left( x \right) =\lambda _{2} \left( x \right) {y}'\left( x \right) +s_{2} \left( x \right) y\left( x \right) ,\quad \end{aligned}$$

(A.4)

where

$$\begin{aligned} \lambda _{2} \left( x \right)= & {} \lambda _{1}^{\prime }\left( x \right) +\lambda _{0} \left( x \right) \lambda _{1} \left( x \right) +s_{1} \left( x \right) , \nonumber \\ s_{2} \left( x \right)= & {} s_{1}^{\prime }\left( x \right) +s_{0} \left( x \right) \lambda _{1} \left( x \right) . \end{aligned}$$

(A.5)

Again, by taking the $\left( {k+1} \right) \mathrm{th}$ and $\left( {k+2} \right) \mathrm{th}$-order derivative of eq. (A.1) for $k=1,2,3...$, we obtain the following differential equations:

$$\begin{aligned} y^{\left( {k+1} \right) }\left( x \right)= & {} \lambda _{k-1} \left( x \right) {y}'\left( x \right) +s_{k-1} \left( x \right) y\left( x \right) , \nonumber \\ y^{\left( {k+2} \right) }\left( x \right)= & {} \lambda _{k} \left( x \right) {y}'\left( x \right) +s_{k} \left( x \right) y\left( x \right) , \end{aligned}$$

(A.6)

where

$$\begin{aligned}&\lambda _{k-1} \left( x \right) ={\lambda }'_{k-2} \left( x \right) +\lambda _{0} \left( x \right) \lambda _{k-2} \left( x \right) +s_{k-2} \left( x \right) , \nonumber \\&s_{k-1} \left( x \right) =s_{0} \left( x \right) \lambda _{k-2} \left( x \right) +{s}'_{k-2} \left( x \right) , \nonumber \\&\lambda _{k} \left( x \right) ={\lambda }'_{k-1} \left( x \right) +\lambda _{0} \left( x \right) \lambda _{k-1} \left( x \right) +s_{k-1} \left( x \right) , \nonumber \\&s_{k} \left( x \right) =s_{0} \left( x \right) \lambda _{k-1} \left( x \right) +{s}'_{k-1} \left( x \right) . \end{aligned}$$

(A.7)

Solving eq. (A.6), we obtain the following relation:

$$\begin{aligned} \frac{y^{\left( {k+2} \right) }\left( x \right) }{y^{\left( {k+1} \right) }\left( x \right) }=\frac{\lambda _{k} \left( x \right) \left[ {{y}'\left( x \right) +\frac{s_{k} \left( x \right) }{\lambda _{k} \left( x \right) }y\left( x \right) } \right] }{\lambda _{k-1} \left( x \right) \left[ {{y}'\left( x \right) +\frac{s_{k-1} \left( x \right) }{\lambda _{k-1} \left( x \right) }y\left( x \right) } \right] }. \end{aligned}$$

(A.8)

For sufficiently large values of k, $\alpha \left( x \right)$ values are obtained as

$$\begin{aligned} \frac{s_{k} \left( x \right) }{\lambda _{k} \left( x \right) }=\frac{s_{k-1} \left( x \right) }{\lambda _{k-1} \left( x \right) }=\alpha \left( x \right) . \end{aligned}$$

(A.9)

This method consists of converting the Schrödinger-like equation into the form of eq. (A.1) for a given potential model. The corresponding energy eigenvalues are calculated by means of the quantisation condition [67]

$$\begin{aligned} \delta _{k} \left( x \right) {=}\,s_{k} \left( x \right) \lambda _{k-1} \left( x \right) {-}\lambda _{k} \left( x \right) s_{k-1} \left( x \right) ,k{=}1,2,3....\nonumber \\ \end{aligned}$$

(A.10)

The general solution of eq. (A.1) is obtained from eq. (A.8) as

$$\begin{aligned}&y\left( x \right) =\exp \left( {-\int ^{x} \alpha \left( {x_{1} } \right) \mathrm{d}x_{1} } \right) \nonumber \\&\quad \times \left[ \!{C_{2}{+}C_{1}\! \int ^{x} \exp \left( {\int ^{x} \left[ {\lambda _{0} \left( {x_{2} } \right) +2\alpha \left( {x_{2} } \right) } \right] \mathrm{d}x_{2} } \right) \mathrm{d}x_{1} }\!\right] , \end{aligned}$$

(A.11)

where $C_{1}$ and $C_{2}$ are integration constants. Also, the eigenfunction can be obtained by transforming the Schrödinger-like equation of the form

$$\begin{aligned} {y}''\left( x \right)= & {} 2\left( {\frac{ax^{N+1}}{1-bx^{N+2}}-\frac{t+1}{x}} \right) {y}'\left( x \right) \nonumber \\&-\frac{Wx^{N}}{1-bx^{N+2}}y\left( x \right) . \end{aligned}$$

(A.12)

The exact solutions for eq. (A.12) is given by

$$\begin{aligned} y\left( x \right)= & {} \left( {-1} \right) ^{2}C\left( {N+2} \right) \left( \sigma \right) _{n} {}_{2} F_{1}\nonumber \\&\left( {-n,\rho +n;\sigma ;bx^{N+2}} \right) , \end{aligned}$$

(A.13)

where

$$\begin{aligned}&\left( \sigma \right) _{n}=\frac{\Gamma \left( {\sigma +n} \right) }{\sigma },\sigma =\frac{2t+N+3}{N+2},\nonumber \\&\rho =\frac{\left( {2t+1} \right) b+2a}{\left( {N+2} \right) b}. \end{aligned}$$

(A.14)