ClearAll[x1,y1,x2,y2,x3,y3,L,KK]

Assume[L>0]

rr1 = {0,0}

r1 = rr1+{x1,y1}

rr2 = {L,0}

r2 = rr2+{x2,y2}

rr3={L/2,Sqrt[3]*L/2}

r3=rr3+{x3,y3}

dr12 = r2-r1

dr23 = r3-r2

dr13 = r3-r1

dr12.dr12/.{x1->0,y1->0,x2->0,y2->0,x3->0,y3->0}

dr23.dr23/.{x1->0,y1->0,x2->0,y2->0,x3->0,y3->0}

dr13.dr13/.{x1->0,y1->0,x2->0,y2->0,x3->0,y3->0}

dr12.dr13/.{x1->0,y1->0,x2->0,y2->0,x3->0,y3->0}

dr13.dr23/.{x1->0,y1->0,x2->0,y2->0,x3->0,y3->0}

dr12.dr23/.{x1->0,y1->0,x2->0,y2->0,x3->0,y3->0}

U[d_] = (1/2)*(d-L)^2

U1=U[Sqrt[dr12.dr12]]

Expand[%]

Series[%,{L,∞,1}]//FullSimplify

U2=U[Sqrt[dr23.dr23]]

Expand[%]

Series[%,{L,∞,1}]//FullSimplify

U3=U[Sqrt[dr13.dr13]]

Expand[%]

Series[%,{L,∞,1}]//FullSimplify

Ebond=Series[U1+U2+U3,{L,∞,0}]//Normal//FullSimplify

Elaser = (1/2)KK r1.r1

Epot =Ebond+Elaser

D[Epot,x1]

D[Epot,y1]

D[Epot,x2]

D[Epot,y2]

D[Epot,x3]

D[Epot,y3]

varlist={x1,x2,x3,y1,y2,y3}

F=D[Epot+Elaser,#]&/@varlist

M=Table[D[F[[i]],#]&/@varlist,{i,1,6}]

M//MatrixForm

CharacteristicPolynomial[M,x]

4Map[Simplify,Series[%,{KK,0,2}]]

MM=M/.KK->0

MM//MatrixForm

Simplify[CharacteristicPolynomial[MM,x]]

{Evals,Evecs}=Eigensystem[MM]

Evals

N[Evals]

Evecs = Normalize /@ Evecs