Table analysis to cubic and higher terms

General objective: Extend the character table analysis to cubic and higher terms. Specific Objectives:  sum   Sn ir    0 n  n 0   NR 1. Apply character tables to parametric three-wave mixing. n  2   Sn  2    0 n   n 0   NR 2. Consider phase shifts in degenerate modes. n Note: Citations to the contents of these slides should reference the following textbook: Simpson, Garth J. (2017) Nonlinear Optical Polarization Analysis in Chemistry and Biology (Cambridge University Press, ISBN 978-0-521-51908-3). 1 Consider ir-vis SFG from an assembly with C4 molecular symmetry.  sum  Sn ir    0 n  n 0   NR z y x C4 E C4Character (z) C2(z) A 1 1 1 B 1 -1 E 1 1 i -i Lin. Quad. 1 z x2+y2,z2 1 1 - x2-y2,xy -1 -1 -i i (x,y) (xz,yz) XXX YXX ZXX XXY YXY ZXY XXZ YXZ ZXZ XYX YYX ZYX C2(z) C4(z) XYY YYY ZYY x  x y  y zz x y y  x zz XYZ YYZ ZYZ XZX YZX ZZX XZY YZY ZZY XZZ YZZ ZZZ 3C4(z) table 2 A more formal linear algebra approach exemplified for C4 :E-transitions Set of matrices corresponding to the C2(z) and C4(z) symmetry operations C4 E A 1 1 1 1 z x2+y2,z2 B 1 -1 1 1 - x2-y2,xy E 1 1 i -i -1 -1 -i i (x,y) (xz,yz) cos  2 / n   sin  2 / n  0    C  z    sin  2 / n  cos  2 / n  0   0 0 1  C4(z) Character C2(z) 3Ctable 4(z) Lin. Quad. 1 n 0 1 0    x   '  C41  z     1 0 0     y  0 0 1    z  i eigenvalues C41  z     i   1  1  eigenvec i    i  0  1 eigenvec  i    i   0   '  C41  z   C41  z     1 i  1 eigenvalues C4  z       i     1 0 0 0 0     1 0     0 0 eigenvec  i    0  ,  0       i   0  0 1      0   i  0 0     3 A more formal linear algebra approach exemplified for C4 : A & B -transitions Set of matrices corresponding to the C2(z) and C4(z) symmetry operations C4 E A 1 1 1 1 z x2+y2,z2 B 1 -1 1 1 - x2-y2,xy E 1 1 i -i -1 -1 -i i (x,y) (xz,yz) cos  2 / n   sin  2 / n  0    C  z    sin  2 / n  cos  2 / n  0   0 0 1  1 n 0  eigenvec 1  0  1  Quad.  '  C41  z   C41  z     1   0  0 0     0 0     0 0 eigenvec 1  1  , 0      0 0 0 0     1 0 0 i  0  1  eigenvalues C41  z           i     1 0 1 0    x   '  C41  z     1 0 0     y  0 0 1    z  i eigenvalues C41  z     i   1  C4(z) Character C2(z) 3Ctable 4(z) Lin. 1 0 0 1     0 0      0   1 eigenvec  1   1 ,  0      0 0 0 0     0 0 0 0     4 How about ? Parametric processes must transform as the totally symmetric group A.  '  C41  z   C41  z   C41  z      zzz  xxz   yyz  xzx   yzy  xzy    yzx  xyz    yxz  zxx   zyy  zxy    zyx 5 How do we account for these complex terms in construction of ? We don’t! Parametric processes must transform as the totally symmetric group A.  '  C41  z   C41  z   C41  z      zzz  xxz   yyz  xzx   yzy  xzy    yzx  xyz    yxz  zxx   zyy  zxy    zyx  00  Sn ir   0 n  n 0  NR n 0 1  i     0   0n n0 0 0 0 0     1 0     0 0  0 , 0       i   0  0 1      0   i  0 0     6 How do thing change for SHG resonant at the second harmonic frequency? Parametric processes must transform as the totally symmetric group A.  '  C41  z   C41  z   C41  z      zzz  xxz   yyz  xzx   yzy  xzy    yzx  xyz    yxz  zxx   zyy  zxy    zyx   Sn  2  n 0   0 n  NR 0n 1  i    0   n0 0 0 0 0     1 0     0 0  0 , 0       i   0  0 1      0   i  0 0     7