Lecture 12: Strong coupling Strong coupling 71 = 72, <*>0,1 ~ ^0,2, ^1 ~ ^2 Secular approximation not applicable: = +"0,1^1Z + ^0,2^2^ + (2^lz^2z + 2^1x^2x + 2^ly^2y) j\z and Jiz do not commute with 2j?ixj?2X and 2-^iy^2y'- [j?1z,2j?1x^2x\ = lUlz, Slx\S2x = '^ly^2x [Slz,2slyS2y] = 2[^lz,^lyU2y = -'^lx^2y [j?2z,2^1x^2x\ = 2^1x[^2z, ^2x\ = '^\x^2y [S2z,2slyS2y] = 2^1y[^2z,^2y\ = -'^ly^x Effects Of 2j\xJ2x^ 2.J\yJ2y and ^\Zy ^2z cannot be analyzed separately in any order Strong coupling Hamiltonian not diagonal: Z+J 0 0 0 ) 0 A — J 2 J 0 0 2J -A-J 0 0 0 0 -E + J, ^ = (^0,1 + ^0,2)/tt A = (^0,1 - ^0,2)/tt f°l f°l 0 1 0 0 0 0 1 0 o o 1 NOT stationary states (eigenfunctions of jr) 7T je = — 2 New basis =>- diagonalized Hamiltonian Í1! 0 \ 0 o \ 0 \ A 1 -I__,_ 2 ^ 2VA2+4J2 1 2 A 2VA2+4J2 0 0 1 2 A 2\/A2+4J2 + A V 2y/A2+4J2 0 A = (^0,1 - ^0,2)/tt stationary states (eigenfunctions of jf') Diagonalized Hamiltonian 0 0 0 0 M2 + 4 J2 — J 0 0 ~ 2 0 0 -\/A2 + 4 J2 — J 0 0 0 ■£ + J J ' 1 0 0 0 ) 0 0 0 N (l 0 0 0N 2 0 10 0 0-1 0 0 ,"0,1 2 0 0 -1 0 0 0 1 0 + 7rJ 0 0 -1 0 0 0 -1 0 vo 0 0 -1 1° 0 0 -ij 1° 0 0 ij w0,l — w0,2 — ^ (^0,1 + ^0,2 + \ZOo,l - ^0,2)^ + 4tt2 J2 1 2 t2 - (^0,1 + ^0,2 - V(W0,1 - ^0,2) + 4?r J p and M+ in the new basis Ay + Ay = ^Oly + ^2y) + S^{2^lz^2y ~ ^ly^2z) Ax + Ax = + ^2x) + 8£(?SlzS<2x - 2^lx^2z) + A+ = c^lx + ^2x + i^ly + i^2y) Contribution —>■ -kJ-2,J?1z,J?2z —>• +c^cicJ - scsisJ +stc'1cj - Ccs'iSj Six 0 2^1y^2z 0 Signal of a strongly coupled pair Contribution —>• 7T'Jr-2^lz^2z —y +cecicJ - sis'isJ +^ Six 0 -cesicJ - sic'isJ 0 ~SCS1CJ ~ ctclsJ Contrib. Tr{jpr1(t)s{+} Six -clslcJ - cisic'lsJ ^sly^2z sfs^cj - C^S^Sj '} = ' J Signal of a strongly coupled pair / + 1- v 1+ V / + 14 V / + 1— J Af~f2h2B0 R2 + 4 J2, J 16kBT R% + (w - Qi - ttJ)2 Mi2h2B0 R2 M2 + 4 J2, J \ 16kBT R2 + (oo- - 7tj)2 j\fj2h2B0 R2 M2 + 4 J2 j J 16kBT #2 + - n'2 + ttJ)2 j\fj2h2B0 R2 M2 + 4 J2 j 16kBT #2 + - Q'2 + ttJ)2 1 ^2 = | (^i + Q2 - - Q2)2 + 4tt2J2 Spectrum of a strongly coupled pair 2k\J\ weak Spectrum of a strongly coupled pair n1-n2 = 8.0ttJ Spectrum of a strongly coupled pair Q-L - Q2 = 4.0ttJ Spectrum of a strongly coupled pair Q-L - Q2 = 2.0ttJ Spectrum of a strongly coupled pair Q-L - Q2 = 0.8ttJ Spectrum of a strongly coupled pair Q-L - Q2 = O.OttJ Magnetic equivalence • cjQi = cjq,2 molecular symmetry or accident • ^13 — ^23? Jl4 = J24, • • • Existence of a plane of symmetry is not sufficient, the plane must bisect the particular pair of nuclei: COOH COOH H (closer to OH) and H (closer to COOH) not equivalent Magnetic equivalence: eigenfunctions stationary states (eigenfunctions of j?') Magnetic equivalence: eigenvalues l2=[h + h) =lt + li + 2llxl2x + 2llvl2v + 2llzl2 — T2 T2 3?' = (wo + kJ)^1z + (w0 - kJ)^2z + 7rJ ■ 2^lz^2 z I2 f2T 1 J2 3h2/4 3h2/4 2hz 3h2/4 3h2/4 2h2 3h2/4 3h2/4 0 3h2/4 3h2/4 2h2 Eigenfunction \a) ® a) 1 | a) ® 1 1 a) ® \f>) ® |0> a) a) +h + 5 J o + f J o — H -UJQ + -J TOCSY (Totally Correlated Spectroscopy) 71 — 72> ^0,1 — ^0,2? — ^2 D ,HI_I 3C or15N a b c d e x/y 1 H ti yyyyyyyyyyyy illinium a b d t2 Simple example: J\2 CH CH «^23 x/y 1 H ti yyyyyyyyyyyy Iiiiiiiiiiii a b x/y 1 H ti yyyyyyyyyyyy Iiiiiiiiiiii a b d -i TOCSY x/y yyyyyyyyyyyy 1 H ti a b TOCSY pulse train applied with 90° (y) phases • ^iy, ^3y components of the density matrix intact • operators with jnx and jnz rotate "about" the jny "axis" • long rotation randomizes polarization in x and z • only the j?\y, j?2y, ^3y, "locked" in y, survive • only evolution of j?iy, j?2y, ^3y can give a signal yyyyyyyyyyyy iiiiiiiiiiii á ■y -COS(Qiíi) COS(7rJi2Íl)^l -COS(Q2*l) COS(-7rJi2Íl) COS(7rJ23ti)^2y -COS(Q3íi) COS(7rJ23ii)^3z/ TOCSY MIXING CH-CH-CH •^YOCSY = 7t^12(2^1x^2z + 2^iy^2y + 2^i^2z) All components of ^Yocsy commute their effects can be analyzed separately in any order TOCSY MIXING . . . but the analysis is not simple for > 2 nuclei Commutator relations provide insight: part of j?\y is lost • kly + ^2y,^YOCSY] = 2j7tJ23(^2ar^ ~ ^2z^3x) 0 the loss of j\y is not fully regained by ^2y • kly + ^2y + ^3y, ^TOCSyI = 0 =>• some j^3y must be created to keep j?\y + j^2y + j>-$y constant despite J13 = 0! x/y ti yyyyyyyyyyyy Iiiiiiiiiiii a b d) = - ^COS(Qiíi) COS(7rJi2íi)(aii^iy + ai2^2y + ai3^3y) — — COS(Q2íi) COS(7rJi2Íl) COS(7rJ23Íi) /í X 021^1 y + «22^2y + a23«^3y) - -COS(Q3íi) COS(7rJ23Čl)031^1y + «32^2y + «33^3y) TOCSY x/y yyyyyyyyyyyy 1 H ti a b t2 Evolution of p(t2) analyzed as usually: wallc-fl?t1 /e-i(ni-7rJi2)tl _|_ g-i(ni+7rJi2)tlj g-i?2t2 ^-K^i-irJi2)i2 _|_ g-i(«l+^12)*2j _|_ «°l2c-fl?t1 /e-i(fii-7rJi2)il _|_ e-i(ni+7rJi2)*lj g-#2*2 (q~K^2~^Jl2>*2 -|_ g-K^+T^12)*2j + ^13e-it2*l ^e-i(ni-7TJi2)tl _|_ e-i(ni+^12)*l^ e-^2*2 ^e-i(n3-7rJi2)t2 -|- e-'(«3+^12)t2j _|_ + ••• TOCSY spectrum DQF-COSY TOCSY TOCSY vs. COSY • different structural information • TOCSY: cross-peaks correlate all nuclei of a spin system (spin system = network of J-coupled nuclei) whole spin system in one spectrum • COSY: cross-peaks correlate only directly coupled nuclei who is whose neighbor HOMEWORK: Section 12.4.2 Strong coupling