$Assumptions=$Assumptions && {a>0}
f[x_]=1/Cosh[a x]
ke=Integrate[f'[x]^2,{x,-Infinity,Infinity}]
        2 a
Out[3]= ---
         3

pe=Integrate[x^2 f[x]^2,{x,-Infinity,Infinity}]
          2
        Pi
Out[4]= ----
           3
        6 a

n=Integrate[f[x]^2,{x,-Infinity,Infinity}]
        2
Out[5]= -
        a

f[x_]=Exp[-a (x-2)^2]+Exp[-a (x+2)^2]

ke=Integrate[f'[x]^2,{x,-Infinity,Infinity}]
                             8 a
        Sqrt[a] (1 - 16 a + E   ) Sqrt[2 Pi]
Out[7]= ------------------------------------
                         8 a
                        E

pe=-2(1+Exp[-a 4^2])
direct=0, exchange=-Exp[-a 4^2], self=-1
n=Integrate[f[x]^2,{x,-Infinity,Infinity}]
              8 a
        (1 + E   ) Sqrt[2 Pi]
Out[9]= ---------------------
                     8 a
            Sqrt[a] E

(ke/2+pe)/n
Plot[%,{a,.1,2}]
FindMinimum[%%,{a,.5}]
Out[12]= {-0.340238, {a -> 0.490313}}

f[x_]=Exp[-a (x-2)^2]-Exp[-a (x+2)^2]
ke=Integrate[f'[x]^2,{x,-Infinity,Infinity}]
                               8 a
         Sqrt[a] (-1 + 16 a + E   ) Sqrt[2 Pi]
Out[15]= -------------------------------------
                          8 a
                         E
pe=-2(1-Exp[-a 4^2])
direct=0, exchange=+Exp[-a 4^2], self=-1
n=Integrate[f[x]^2,{x,-Infinity,Infinity}]
                8 a
         (-1 + E   ) Sqrt[2 Pi]
Out[17]= ----------------------
                       8 a
              Sqrt[a] E

(ke/2+pe)/n
Plot[%,{a,.1,2}]
FindMinimum[%%,{a,.75}]
Out[20]= {-0.306583, {a -> 0.766212}}

exact: -0.480369 & -0.517251
If provide exact as an option; RR gives exact

f1[x_]=Exp[-a (x-2)]
f0[x_]=Cosh[a x]/Cosh[a 2]
ke=Integrate[f0'[x]^2,{x,0,2}]+Integrate[f1'[x]^2,{x,2,Infinity}]
pe =-1
n=Integrate[f0[x]^2,{x,0,2}]+Integrate[f1[x]^2,{x,2,Infinity}]
(ke/2+pe)/n
Plot[%,{a,.1,2}]
FindMinimum[%%,{a,1}]
Out[33]= {-0.517251, {a -> 1.0171}}
Plot[Evaluate[f1[x] /. a -> 1.0171],{x,2,5}]
Plot[Evaluate[f0[x] /. a ->1.0171],{x,-2,2}]
Plot[Evaluate[f1[-x] /. a ->1.0171],{x,-5,-2}]
Show[%,%%,%%%,PlotRange->All]  
f[x_]=Exp[-a (x-2)^2]+Exp[-a (x+2)^2]
Plot[Evaluate[f[x] /.  a -> 0.490313],{x,-5,5}]
Show[%,%%%,PlotRange->All]

f1[x_]=Exp[-a (x-2)]
f0[x_]=Sinh[a x]/Sinh[a 2]
ke=Integrate[f0'[x]^2,{x,0,2}]+Integrate[f1'[x]^2,{x,2,Infinity}]
pe =-1
n=Integrate[f0[x]^2,{x,0,2}]+Integrate[f1[x]^2,{x,2,Infinity}]
(ke/2+pe)/n
Plot[%,{a,.1,2}]
FindMinimum[%%,{a,1}]
Out[41]= {-0.480369, {a -> 0.980173}}
Plot[Evaluate[f1[x] /. a -> 0.980173],{x,2,5}]
Plot[Evaluate[f0[x] /. a -> 0.980173],{x,-2,2}]
Plot[Evaluate[-f1[-x] /. a -> 0.980173],{x,-5,-2}]
Show[%,%%,%%%,PlotRange->All]  

f[x_]=Exp[-a (x-2)^2]-Exp[-a (x+2)^2]
Plot[Evaluate[f[x] /.  a -> 0.766212],{x,-5,5}]
Show[%,%%%,PlotRange->All]

GraphicsGrid[{{%70},{%55}}]
Export["delta2_RR.eps",%75] 



