Convolution of a data with multiple sigmas

I am trying to perform convolution on the attached data file (W_NAIF.mat). The input is shown in (spectrum.png) and the desired output in shown in (Exp.png).

Here,

Sigma = (-0.00789) + (0.06769)*E(i) + (0.021159)*(E(i)^2); [E(i) = bins at x-axis]

Can you please suggest what am I doing wrong with my code?

The code I am using for this is:

AAA=W_NAIF/sum(W_NAIF); %%I presume W_NAIF is imported data
numEbins = 1026;
E=linspace(0,10,numEbins);    % Energy bins
newSPECU(1:numEbins,1)=0;

for i=1:(numEbins-1)
    %new sigma for each energy
    sigmaFilter(i) = (-0.00789) + (0.06769)*E(i) + (0.021159)*(E(i)^2);
    % Gaussian function with new sigma for each energy and moving mean based on current energy          
    gaussFilter(i) = (1/sqrt(2*pi* (sigmaFilter(i)^2)))*exp(-(E(i)-((E(i)+E(i+1))/2)))/(2*(sigmaFilter(i)^2));
end
    gaussFilter(1,1026)=0;
    % a way to handle the edge cases is to just normalize the computed normal (Gaussian) distribution
    % (the edge cases being very low and very high energies)
    % but this may not be the best way to do that 
    gaussFilter = gaussFilter/sum(gaussFilter);
    newSPECU = (AAA').*gaussFilter; % ".*" is elemental multiplication
 
%%
figure(1);
stairs(E,newSPECU);
set(gca,'YScale','log');
xlabel('Energy Deposited (MeV)');
ylabel('Counts');
title('Geant4 Gamma energy spectra (NaI NEAR - Tap Water)');
grid minor;
grid on;

% THIS CODE IS NOT DOING A GOOD JOB AT THIS TASK.