Calculating 1/3 Octave Spectrum from FFT / DFT
Dr. Manuel Kuehner,
  You are close. You need to take the square root of the linear values squared.

$$P_{\mbox{total_linear}}=\sqrt{p_1^2+p_2^2+...}$$

$$P_{\mbox{total_dB}}=20 log_{10}\left( P_{\mbox{total_linear/20E-6}} \right)$$

FYI: I wrote a MATLAB function to do exactly as you request. It is here

Looking to read? See page 16 of this book:
https://www.amazon.com/Noise-Control-Engineers-Harold-Lord/dp/0070387389