Problem $1(50$ points$)$

The normal mode vibrations of the water molecule may be classified as symmetric bond stretching, asymmetric bond stretching and bond angle bending. The normal mode coordinates $(Q_j)$ for $H_{2}O$ may be represented by the following equations written in terms of displacements in internal coordinates.

$Q_1=0\mathrm{.}07+\frac{1}{\sqrt[\phantom{2}]{2}}(r_1+r_2),v_1=3800c{m}^{-1}$

$Q_2=-12\mathrm{.}2+\frac{1}{\sqrt[\phantom{2}]{2}}(r_1+r_2),v_2=1600c{m}^{-1}$

$Q_3=\frac{1}{\sqrt[\phantom{2}]{2}}(r_1-r_2),v_3=3670c{m}^{-1}$

where $Q,r_1$ and $r_2$ represent displacements in the $H-O-H$ bond angle and the lengths of the first and second $O-H$ bonds, respectively, from their equilibrium values. Note that lengths are given in units of $A,$ and the bond angle in radians. The frequencies associated with each normal mode are also given above.

Part a$.$ Evaluate the normal mode coordinates so that you can visualize them. Classify each mode $(Q_l,Q_2,Q_3)$ according to the type of vibration, i$.$e$.,$ either the symmetric stretch, the asymmetric stretch or the bond angle bending. You must show your work and clearly explain your answers to receive credit. No credit will be given for guessing.

Part b$.$ Calculate the total vibrational energy in units of $k\frac{J}{m}ol$ of a water molecule in vibrational state $(n_1,n_2,n_3)=(0,10,10),$ where $n_j$ represents the number of vibrational quanta in normal mode $j.$ Specify the reference energy used in your calculation, i$.$e$.,$ what is the zero of vibrational energy? Note that useful constants are given below.

Part c$.$ The potential energy function that governs the vibrational motion of the water molecule may be written as $V={\displaystyle \underset{?}{\overset{?}{}}}\frac{1}{2}{A}_j{Q}_j^2$ where $A_1=5530,A_2=1305$ and $A_3=1907k\frac{J}{m}ol{}^2.$ In this equation, the zero of $V$ is taken at the minimum of the electronic potential energy function. Calculate the maximum displacement in the bond angle associated with only mode $2$ when $n_2=10.$ Clearly explain your approach to receive credit.

Useful Information: $c=3{10}^{10}cm{s}^{-1},h=6\mathrm{.}626{10}^{-34}Js,v(s^{-1})=v(c{m}^{-1}{)}^{**}c(cm{s}^{-1})$

$R=8\mathrm{.}314\frac{J}{m}$olK,$1$mole$=6\mathrm{.}022{10}^{23}$ molecules