设椭圆$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a>b>0\right)$的左、右焦点分别为$F_1,F_2$,点$P\left(a,b\right)$满足$\left|P{F}_2\right|=\left|F_1{F}_2\right|$．
（1）求椭圆的离心率$e$；

(2)设直线$P{F}_2$与椭圆${\left(x+1\right)}^2+{\left(y-\sqrt{3}\right)}^2=16$相交于$A,B$,两点若直线$P{F}_2$与圆相交于$M,N$两点,且$\left|MN\right|=\frac{5}{8}\left|AB\right|$,求椭圆的方程．