(* Burkhard Luy and Steffen J. Glaser, Superposition of Scalar and Residual Dipolar Couplings: Analytical Transfer Functions for Three Spins 1/2 under Cylindrical Mixing Conditions Journal of Magnetic Resonance 148, 169-181 (2001). A Hamiltonian of the form H=2 pi j12l I1zI2z + 2 pi j12p (I1xI2x + I1yI2y) is assumed (expanded to 3 spins, of course). The longitudinal couplings j12l, j13l, j23l and the planar couplings j12p, j13p, j23p have to be defined right after the Clear command below. The variable transfer12 will give the polarization transfer function for I1z -> I2z under the above mentioned Hamiltonian. Correspondingly transfer13 or transfer11 corresponds to I1z -> I3z and I1z -> I1z, respectively. Transfer functions to zero quantum operators should be clear from the name given (e.g. transfer11y2z corresponds to the coherence transfer function I1z -> I1yI2z). Please cite the above mentioned reference whenever you use this program for scientific work. *) Clear[j12,j13,j23,x1,x2,x3,y1,y2,y3,z1,z2,z3,v1,v2,v3,w1,w2,w3, n1,n2,n3,c1,c3,lamb1,lamb2,lamb3,delta12,delta13,delta23, h, phi,s,p] j23p=11 j12p=-10 j13p=4.6 j23l=-22 j12l=20 j13l=-9.2 (* j23l=0 j12l=0 j13l=0 *) a12=(1/(2 n1^2 n2^2)) ((x1 x2-y1 y2)^2 - (z1 z2)^2) a13=(1/(2 n1^2 n3^2)) ((x1 x3-y1 y3)^2 - (z1 z3)^2) a23=(1/(2 n2^2 n3^2)) ((x2 x3-y2 y3)^2 - (z2 z3)^2) w12=x1 x2-y1 y2-z1 z2 w13=x1 x3-y1 y3-z1 z3 w23=x2 x3-y2 y3-z2 z3 v12=(1/(2 n1^2 n2^2)) v13=(1/(2 n1^2 n3^2)) v23=(1/(2 n2^2 n3^2)) n1=Sqrt[x1^2 + y1^2 + z1^2] n2=Sqrt[x2^2 + y2^2 + z2^2] n3=Sqrt[x3^2 + y3^2 + z3^2] (* x1=S2*j13p-j12p*j23p-j13p*(lamb1/Pi) x2=S2*j13p-j12p*j23p-j13p*(lamb2/Pi) x3=S2*j13p-j12p*j23p-j13p*(lamb3/Pi) y1=S1*j23p-j12p*j13p-j23p*(lamb1/Pi) y2=S1*j23p-j12p*j13p-j23p*(lamb2/Pi) y3=S1*j23p-j12p*j13p-j23p*(lamb3/Pi) z1=j12p^2-S1*S2+(S1+S2)*(lamb1/Pi)-(lamb1/Pi)^2 z2=j12p^2-S1*S2+(S1+S2)*(lamb2/Pi)-(lamb2/Pi)^2 z3=j12p^2-S1*S2+(S1+S2)*(lamb3/Pi)-(lamb3/Pi)^2 *) x1=j23p*(j23p-j12p-j13p)-S2*S3+S2*j13p+S3*j12p+(S2-j12p+S3-j13p)*lamb1/(Pi)-(lamb1/(Pi))^2 x2=j23p*(j23p-j12p-j13p)-S2*S3+S2*j13p+S3*j12p+(S2-j12p+S3-j13p)*lamb2/(Pi)-(lamb2/(Pi))^2 x3=j23p*(j23p-j12p-j13p)-S2*S3+S2*j13p+S3*j12p+(S2-j12p+S3-j13p)*lamb3/(Pi)-(lamb3/(Pi))^2 y1=j13p*(j13p-j12p-j23p)-S1*S3+S1*j23p+S3*j12p+(S1-j12p+S3-j23p)*lamb1/(Pi)-(lamb1/(Pi))^2 y2=j13p*(j13p-j12p-j23p)-S1*S3+S1*j23p+S3*j12p+(S1-j12p+S3-j23p)*lamb2/(Pi)-(lamb2/(Pi))^2 y3=j13p*(j13p-j12p-j23p)-S1*S3+S1*j23p+S3*j12p+(S1-j12p+S3-j23p)*lamb3/(Pi)-(lamb3/(Pi))^2 z1=j12p*(j12p-j13p-j23p)-S1*S2+S1*j23p+S2*j13p+(S1-j13p+S2-j23p)*lamb1/(Pi)-(lamb1/(Pi))^2 z2=j12p*(j12p-j13p-j23p)-S1*S2+S1*j23p+S2*j13p+(S1-j13p+S2-j23p)*lamb2/(Pi)-(lamb2/(Pi))^2 z3=j12p*(j12p-j13p-j23p)-S1*S2+S1*j23p+S2*j13p+(S1-j13p+S2-j23p)*lamb3/(Pi)-(lamb3/(Pi))^2 S1=(j23l-j12l-j13l)/2; S2=(j13l-j12l-j23l)/2; S3=(j12l-j13l-j23l)/2; no1={N[x1/n1],N[y1/n1],N[z1/n1]} no2={N[x2/n2],N[y2/n2],N[z2/n2]} no3={N[x3/n3],N[y3/n3],N[z3/n3]} delta12=lamb1-lamb2 delta13=lamb1-lamb3 delta23=lamb2-lamb3 lamb1=-aa/3+4 Pi h Cos[phi/3] lamb2=-aa/3-4 Pi h Cos[(Pi-phi)/3] lamb3=-aa/3-4 Pi h Cos[(Pi+phi)/3] phi=ArcCos[-q/(2*h^3)] h=Sqrt[Abs[p]/3] (* lamb1=-4 Pi h Cos[phi/3] lamb2=+4 Pi h Cos[(Pi-phi)/3] lamb3=+4 Pi h Cos[(Pi+phi)/3] phi=ArcCos[q/(2*h^3)] h=(Abs[q]/q) * Sqrt[Abs[p]/3] *) S11=(j23l-j12l-j13l)/4; S22=(j13l-j12l-j23l)/4; S33=(j12l-j13l-j23l)/4; p = -j12p^2/4 - j13p^2/4 - j23p^2/4 - S11^2/3 + (S11*S22)/3 - S22^2/3 + (S11* S33)/3 + (S22*S33)/3 - S33^2/3; q = -(j12p*j13p*j23p)/4 - (j12p^2*S11)/12 - (j13p^2*S11)/12 + (j23p^2*S11)/6 - (2*S11^3)/27 - (j12p^2*S22)/12 + (j13p^2*S22)/6 - (j23p^2*S22)/12 + (S11^2*S22)/ 9 + (S11*S22^2)/9 - (2*S22^3)/27 + (j12p^2*S33)/6 - (j13p^2*S33)/12 - (j23p^2* S33)/12 + (S11^2*S33)/9 - (4*S11*S22*S33)/9 + (S22^2*S33)/9 + (S11*S33^2)/9 + (S22*S33^2)/9 - (2*S33^3)/27; aa = Pi*(j12l + j13l + j23l)/2; dd=(p/3)^3 + (q/2)^2 com1=a12(1-Cos[delta12 t]) com2=a13(1-Cos[delta13 t]) com3=a23(1-Cos[delta23 t]) transfer11=N[1-(v12 w12^2 (1-Cos[delta12 t]) +v13 w13^2 (1-Cos[delta13 t]) +v23 w23^2 (1-Cos[delta23 t]))] transfer12= N[(a12(1-Cos[delta12 t]) +a13(1-Cos[delta13 t]) +a23(1-Cos[delta23 t]))] transfer13= N[v12 ((x1 x2-z1 z2)^2-(y1 y2)^2)(1-Cos[delta12 t]) +v13 ((x1 x3-z1 z3)^2-(y1 y3)^2)(1-Cos[delta13 t]) +v23 ((x2 x3-z2 z3)^2-(y2 y3)^2)(1-Cos[delta23 t])] transfer11x2y=N[v12 w12 (y1 x2-x1 y2) Sin[delta12 t] +v13 w13 (y1 x3-x1 y3) Sin[delta13 t] +v23 w23 (y2 x3-x2 y3) Sin[delta23 t]] transfer11x3y=N[v12 w12 (z1 x2-x1 z2) Sin[delta12 t] +v13 w13 (z1 x3-x1 z3) Sin[delta13 t] +v23 w23 (z2 x3-x2 z3) Sin[delta23 t]] transfer12x3y=N[v12 w12 (z1 y2-y1 z2) Sin[delta12 t] +v13 w13 (z1 y3-y1 z3) Sin[delta13 t] +v23 w23 (z2 y3-y2 z3) Sin[delta23 t]] transfer112323=N[v12 w12 (y1 z2+z1 y2) (1-Cos[delta12 t])/2 +v13 w13 (y1 z3+z1 y3) (1-Cos[delta13 t])/2 +v23 w23 (y2 z3+z2 y3) (1-Cos[delta23 t])/2] transfer121313=N[v12 w12 (x1 z2+z1 x2) (1-Cos[delta12 t])/2 +v13 w13 (x1 z3+z1 x3) (1-Cos[delta13 t])/2 +v23 w23 (x2 z3+z2 x3) (1-Cos[delta23 t])/2] transfer131212=N[v12 w12 (x1 y2+y1 x2) (1-Cos[delta12 t])/2 +v13 w13 (x1 y3+y1 x3) (1-Cos[delta13 t])/2 +v23 w23 (x2 y3+y2 x3) (1-Cos[delta23 t])/2]