Fitting a sum of three lorentzian based on a dataset

Question

as you can see below, I have developed a code that allows me to fit the present dataset via a Lorentzian. Now what I should do is to develop a sum of three Lorentzians based on this model so that this sum is the Lorentzian I have developed, exactly as it is clearly visible in the figure.

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ListPlot[data, PlotTheme -> "Detailed"]
model = a/((b - t)^2 + d) + c;
result = NonlinearModelFit[data, {model}, {a, {b, 4956}, c, d}, t]
Show[ListPlot[data], 
 Plot[result[t], {t, 4370, 5600}, PlotRange -> Full]]
max = NMaximize[result[t], t][[1]] 
min = NMinimize[result[t], t][[1]] 
minmax = data[[All, 1]] // MinMax
fwhm = t /. NSolve[{result[t] == (max + min)/2, 4370 < t < 5600}, t] //
    Differences // First
max1 = FindMaximum[result[t], t] // First
min1 = FindMinimum[{result[t], 4370 < t < 5600}, t] // First
middle = max1 + min1/2
t1t2 = NSolve[{result[t] == (max1 + min1)/2, 4370 < t < 5600}, t]
fwhm1 = t /. 
    NSolve[{result[t] == (max1 + min1)/2, 4370 < t < 5600}, t] // 
   Differences // First

To be clear, we are talking about a D2 transition of the cesium atom.

I have already tried several times to create a model with the sum of three Lorentzians of the type a1,a3,b1,b2,b3,c1,..., but I have not been able to obtain very interesting results.
Would any of you be able to help me?

Answer 1

Preferably you should have a more detailed model (e.g., amount of expected separation of the Lorentzians).

{tmin, tmax} = MinMax[data[[All, 1]]]

(* {4370, 5573.6} *)

bEst = Mean[MaximalBy[data, Last][[All, 1]]]

(* 4958.9 *)

model = Total[
  a[#]/((b[#] - t)^2 + d[#]) + c/3 & /@ Range[3]]

(* c + a[1]/((-t + b[1])^2 + d[1]) + a[2]/((-t + b[2])^2 + d[2]) + 
 a[3]/((-t + b[3])^2 + d[3]) *)

Let the Lorentzian be positioned such that tmin < b[1] <= b[2] <= b[3] < tmax

Manipulate[
 result = NonlinearModelFit[data,
   {model, tmin < b[1] <= b[2] <= b[3] < tmax,
    a[1] > 0, a[2] > 0, a[3] > 0,
    d[1] > 0, d[2] > 0, d[3] > 0},
   {{a[1], aEst}, {a[2], aEst}, {a[3], aEst}, {b[1], bEst - offset},
    {b[2], bEst}, {b[3], bEst + offset},
    d[1], d[2], d[3], c}, t];
 Column@{result // Normal,
   Show[
    ListPlot[data,
     PlotTheme -> "Detailed",
     PlotStyle -> Red],
    Plot[
     Evaluate@
      Flatten@
       {result[t], (a[#]/((b[#] - t)^2 + d[#]) + c/3 & /@ Range[3]) /. 
         result["BestFitParameters"]}, {t, tmin, tmax},
     PlotLegends -> {"result", "part 1", "part 2", "part 3"},
     PlotRange -> All],
    PlotRange -> All,
    ImageSize -> 300]},
 {{offset, 25}, 0, 50, 5, Appearance -> "Labeled"},
 {{aEst, 650}, 50, 1500, 50, Appearance -> "Labeled"},
 SynchronousUpdating -> False,
 TrackedSymbols :> All]