Determinante Secular e Energia Molecular

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306 9MOLECULAR STRUCTURE
and hence the secular determinant is
RRRRRRRRRRRRR
αA − E βAB − SABE βAC − SACE
βBA − SBAE αB − E 0
βCA − SCAE 0 αC − E
RRRRRRRRRRRRR
(b) Substituting in the given values of the Coulomb integrals and the reso-
nance integrals gives
RRRRRRRRRRRRR
(−7.2 eV) − E (−1.0 eV) − SABE (−0.8 eV) − SACE
(−1.0 eV) − SBAE (−10.4 eV) − E 0
(−0.8 eV) − SCAE 0 (−8.4 eV) − E
RRRRRRRRRRRRR
(i) If the overlap integrals are set to zero the secular determinant ex-
pands to give
−E3 − (26 eV)E2 − (221.08 eV2)E − (613.936 eV3)
Setting this to zero and �nding the roots of the cubic (using mathe-
matical so�ware) gives the energies E1 = −10.7 eV , E2 = −8.71 eV
and E3 = −6.58 eV .
For the case where the overlap is zero the hamiltonian matrix is (ex-
pressed in units of eV)
⎛
⎜
⎝
−7.2 −1.0 −0.8
−1.0 −10.4 0
−0.8 0 −8.4
⎞
⎟
⎠
�e matrix of eigenvectors which diagonalizes this matrix is found
using mathematical so�ware as
⎛
⎜
⎝
0.294 0.349 0.890
0.950 −0.206 −0.233
0.102 0.914 −0.392
⎞
⎟
⎠
�e entries in each column of this matrix above give the coe�cients
of the atomic orbitals for the corresponding molecular orbital; the
columns correspond to the molecular orbitals in the same order as
the energies are quoted above. For example the molecular orbital
with E1 = −10.7 eV is Ψ1 = 0.294ψA + 0.950ψB + 0.102ψC.
(ii) If the overlap integrals SAB and SAC are set to 0.2, the secular deter-
minant becomes
RRRRRRRRRRRRR
(−7.2 eV) − E (−1.0 eV) − 0.2E (−0.8 eV) − 0.2E
(−1.0 eV) − 0.2E (−10.4 eV) − E 0
(−0.8 eV) − 0.2E 0 (−8.4 eV) − E
RRRRRRRRRRRRR
= (−0.92)E3 − (24.528 eV)E2 − (214.392 eV2)E − (613.936 eV3)
�e energies are found by setting the polynomial to zero and then
�nding the roots using mathematical so�ware. �e three roots give