$ Z_s =\displaystyle\frac{1}{ h ^{3 N } N !}\int\prod_i d^3 p_i \prod_i d^3 q_i e^{- \beta \sum_i( p_i ^2/2 m +V(q_i)) }$

Z_s =(1/( h ^(3* N )* N !))*@INT( @PROD(d^3 p_i , i )*@PROD( d^3 q_i , i )*exp(- beta *@SUM( p_i ^2/2m+ V(q_i) )

$ Z =\displaystyle\frac{1}{ h ^{3 N } N !}\left(\displaystyle\frac{2 \pi m }{ \beta }\right)^{3 N /2} V ^ N e^{- N \beta V_0 }$

Z =(1/ h ^(3* N )* N !)*(2* pi * m / beta )^3*( N /2)* V ^ N exp( - beta * V_0 * N )

$ Z =\displaystyle\frac{1}{ h ^{3 N }}\left(\displaystyle\frac{2 \pi }{\beta}\right)^{3 N }\left(\displaystyle\frac{ m a ^2}{ V_a }\right)^{3 N /2}e^{-3 N \beta V_0 }$

Z =(2* pi / beta )^(3* N )*( m * a ^2/ V_a )^(3* N /2)*e^(-3 * N * beta * V_0 )/( h ^(3* N ))

$ V = V_0 $

V = V_0

$ V(q_i,q_j) = V(q_i) \delta_{ij} $

V(q_i,q_j) = V(q_i) * delta_ij

$ V(q) = V_0 + \displaystyle\frac{1}{2} V_a \left(\displaystyle\frac{ q }{ a }\right)^2$

V = V_0 + V_a * ( q / a )^2/2

$ \Theta_s \equiv \displaystyle\frac{ \hbar }{ k_B }\sqrt{\displaystyle\frac{ V_a }{ m a ^2}}$

Theta_s =( hbar / k_B )*sqrt( V_a /( m * a ^2))

$ Z_s = \left(\displaystyle\frac{ T }{ \Theta_s }\right)^{3 N } e^{-3 N V_0 / k_B T }$

Z_s = ( T / Theta_s )^(3 * N )*exp(-3* N * V_0 /( k_B * T ))

$ \ln Z_s =- 3 N \ln\left(\displaystyle\frac{ \Theta_s }{ T }\right)-\displaystyle\frac{3 N V_0 }{ k_B T } $

ln( Z_s )=- 3* N *ln( Theta_s / T )- 3* N * V_0 /( k_B * T )

$ Z_s = \displaystyle\frac{1}{( \beta k_B \Theta_s )^{3 N }} e^{-3 N \beta V_0 }$

Z_s =exp(-3 * N * V_0 /( k_B * T ))/( beta * k_B * Theta_s )^(3* N )

$ \ln Z_s =- 3 N \ln( \beta k_B \Theta_s )- 3 N \beta V_0 $

ln( Z_s )=- 3* N *ln( beta * k_B * Theta_s )- 3* N * beta * V_0

$ \omega_s \equiv \sqrt{\displaystyle\frac{ V_a }{ m a ^2}}$

omega_s =sqrt( V_a /( m * a ^2))