\documentclass[a4paper,12pt]{report}
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\usepackage{makeidx,shortvrb,latexsym}
\usepackage{amsthm,amssymb,amsmath}
\usepackage{subfigure}
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\usepackage{perpage}
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\begin{document}
\begin{gather*}
 {a_i}= \left\langle i \right|H\left| i \right\rangle  \\ 
  {b_{i + 1}} = \sqrt {\{ \left\langle i \right|(H-{a_i})-\left\langle {i-1} \right|{b_i}\} \{ (H-{a_i})\left| i \right\rangle -{b_i}\left| {i-1} \right\rangle \} }  \\ 
 \end{gather*}
 
 \begin{gather*}
\left\langle f \right|{H_{el-ph}}\left| i \right\rangle  = {\left( {\frac{{2\pi {c^2}N\hbar }}{{\omega V}}} \right)^{1/2}} \nonumber \\ 
 \times \left\{ {\begin{array}{*{20}{c}}
 {\left\langle {q,J,c} \right|{p_z}\left| {q',J',v} \right\rangle  \quad parallel\ polarization  } \\   {\left\langle {q,J,c} \right|{p_\varphi }\left| {q',J',v} \right\rangle \quad  perpendicular\ polarization}  \nonumber  \\  
\end{array}} \right. \\ 
\end{gather*}
 
 \begin{gather*}
{a_0} =  0.4581652932831429\\
{a_1} = 2.217058676663745\\
{a_2} = 0.7405551735357053\\
{a_3} = 0.01968227878617998\\
{b_1} = 1.0\\
{b_2} = 4.504130959426697\\
{b_3} = 1.110667363742916\\
{b_4} = 0.02359291751427506
\end{gather*}

 \begin{equation}
\begin{array}{l}
{a_0} =  0.4581652932831429\\
{a_1} = 2.217058676663745\\
{a_2} = 0.7405551735357053\\
{a_3} = 0.01968227878617998\\
{b_1} = 1.0\\
{b_2} = 4.504130959426697\\
{b_3} = 1.110667363742916\\
{b_4} = 0.02359291751427506
\end{array}
\end{equation}
\end{document}