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Times-Bold" 14}{SCP_R 1 0 35{COM_R 2 0{TEXT 3 31 "Huckel Molecula
r Orbitals (HMO)"}}{COM_R 3 0{TEXT 1 91 "Chemistry: Organic (Quan
tum) Chemistry\015\015by Darren Redfern, Richard Pavelle and Tony
 Scott.\015"}}{COM_R 4 0{TEXT 4 10 "Reference:"}}{COM_R 5 0{TEXT 
1 347 "D. McLaughlin, \"Symbolic Computation in Chemical Educatio
n\", Applications of Computer Algebra, Ed. by R. Pavelle, Kluwer,
 Hingham, MA, (1985).\015\015Huckel Molecular Theory provides a s
imple approximation to the electronic structure of molecules, i.e
.: \015\0151) The eigenvalues approximate the energy spectrum. \0
152) The eigenvectors approximate the states.\015"}}{INP_R 6 0 ">
 "{TEXT 0 13 "with(linalg):"}}{COM_R 7 0{TEXT 1 69 "\015The HMO f
or Butadiene can be represented with the following matrix.\015"}}
{INP_R 8 0 "> "{TEXT 0 63 "matButadiene:=array([[a,b,0,0],[b,a,b,
0],[0,b,a,b],[0,0,b,a]]);"}}{OUT_R 9 0 8{DAG :3n6\`matButadiene`(
3n4\`MATRIX`,2[2,5[2,5n3\`a`n3\`b`j2x0000p13[2,5p11pFp11p13[2,5p1
3p11pFp11[2,5p13p13p11pF}}{COM_R 10 0{TEXT 1 71 "\015Calculate th
e eigenvalues, approximating Butadiene's energy spectrum.\015"}}
{INP_R 11 0 "> "{TEXT 0 25 "eigenvals(matButadiene); "}}{OUT_R 12
 0 11{DAG ,5+7n3\`a`j2x0001n3\`b`/3i2x0001j2x0002*5j2x0005/3p4pBp
6p4p10+7p2p4p6p8pDp8+7p2p4p6p10pDp10+7p2p4p6p10pDp8}}{COM_R 13 0
{TEXT 1 69 "Calculate the eigenvectors, approximating Butadiene's
 energy states.\015"}}{INP_R 14 0 "> "{TEXT 0 34 "eigenvects(matB
utadiene, radical);"}}{OUT_R 15 0 14{DAG ,5[2,4+7n3\`a`j2x0001n3\
`b`/3i2x0001j2x0002*5j2x0005/3p6pDp8p6p12p6%2,2(3n4\`VECTOR`,2[2,
5p6+3*5+5p8p12pFpAp6p8pBpBp24pB[2,4+7p4p6p8pApFpAp6%2,2(3p1C,2[2,
5p6+3*5+5p8p12pFp12p6p8pBpBp43pB[2,4+7p4p6p8p12pFp12p6%2,2(3p1C,2
[2,5p6+3*5+5p8pApFpAp6p8pBpBp61p6[2,4+7p4p6p8p12pFpAp6%2,2(3p1C,2
[2,5p6+3*5+5p8pApFp12p6p8pBpBp80p6}}{COM_R 16 0{TEXT 1 285 "\015A
s you can see, there is one eigenvector corresponding to each eig
envalue.  (The actual eigenvectors are the lists contained within
 the sets.) The eigenvectors are identical except for change(s) o
f sign(s). \015\015Moving on to the next example, the HMO for Nap
hthalene is represented thus."}}{COM_R 17 0{TEXT 1 0 ""}}{INP_R 
18 0 "> "{TEXT 0 23 "matNaphthalene:=array(["}}{INP_R 19 0 "> "
{TEXT 0 23 "[a,0,0,0,0,0,0,b,b,0], "}}{INP_R 20 0 "> "{TEXT 0 22 
"[0,a,b,0,0,0,0,0,b,0],"}}{INP_R 21 0 "> "{TEXT 0 23 "[0,b,a,b,0,
0,0,0,0,0], "}}{INP_R 22 0 "> "{TEXT 0 23 "[0,0,b,a,b,0,0,0,0,0],
 "}}{INP_R 23 0 "> "{TEXT 0 23 "[0,0,0,b,a,0,0,0,0,b], "}}{INP_R 
24 0 "> "{TEXT 0 23 "[0,0,0,0,0,a,b,0,0,b], "}}{INP_R 25 0 "> "
{TEXT 0 23 "[0,0,0,0,0,b,a,b,0,0], "}}{INP_R 26 0 "> "{TEXT 0 23 
"[b,0,0,0,0,0,b,a,0,0], "}}{INP_R 27 0 "> "{TEXT 0 23 "[b,b,0,0,0
,0,0,0,a,b], "}}{INP_R 28 0 "> "{TEXT 0 24 "[0,0,0,0,b,b,0,0,b,a]
]);"}}{OUT_R 29 0 28{DAG :3n6\`matNaphthalene`(3n4\`MATRIX`,2[2,B
[2,Bn3\`a`j2x0000p11p11p11p11p11n3\`b`p18p11[2,Bp11pFp18p11p11p11
p11p11p18p11[2,Bp11p18pFp18p11p11p11p11p11p11[2,Bp11p11p18pFp18p1
1p11p11p11p11[2,Bp11p11p11p18pFp11p11p11p11p18[2,Bp11p11p11p11p11
pFp18p11p11p18[2,Bp11p11p11p11p11p18pFp18p11p11[2,Bp18p11p11p11p1
1p11p18pFp11p11[2,Bp18p18p11p11p11p11p11p11pFp18[2,Bp11p11p11p11p
18p18p11p11p18pF}}{COM_R 30 0{TEXT 1 77 "\015Calculate the the ei
genvalues, approximating Naphthalene's energy spectrum.\015"}}
{INP_R 31 0 "> "{TEXT 0 26 "eigenvals(matNaphthalene);"}}{OUT_R 
32 0 31{DAG ,B+5n3\`a`j2x0001n3\`b`i2x0001+5p2p4p6p4+7p2p4p6/3p8j
2x0002*5j2x0005/3p4p15p6p4p1A+7p2p4p6p13p17p13+7p2p4p6p1A*5j2x001
3p1Ap6p4p1A+7p2p4p6p1Ap2Cp13+7p2p4p6p13p2Cp1A+7p2p4p6p13p2Cp13+7p
2p4p6p1Ap17p1A+7p2p4p6p1Ap17p13}}{COM_R 33 0{TEXT 1 172 "\015A sp
ecific eigenvector lambda can be computed by solving A x = lambda
 x or equivalently computing the  nullspace of the characteristic
 matrix  A - lambda I .  For example\015"}}{INP_R 34 0 "> "{TEXT 
0 44 "nullspace(charmat( matNaphthalene, (a+b) ));"}}{OUT_R 35 0 
34{DAG %2,2(3n4\`VECTOR`,2[2,Bj2x0000p9i2x0001pCp9p9pCpCj2x0001p1
3}}{INP_R 36 0 "> "{TEXT 0 0 ""}}}{END}
