# In the Hckel treatment of the cyclopropenyl radical, the basis functions are the three 2pz atomic..

In the Hckel treatment of the cyclopropenyl radical, the
basis functions are the three 2pz atomic orbitals, which we denote by f1,
f2 and f3

_ = c1 f1 + c2 f2
+ c3 f3.

The possible values of the orbital energy W are sought as
a function of the c coefficients by solving the three simultaneous equations

xc1 + c2 + c3 = 0,

c1 + xc2 + c3 = 0,

c1 + c2 + x c3 = 0,

where x = (_ _ W)/_ and where _ and
_ are certain integrals whose values are to be determined from empirical
data.

(a) The determinant of the c coefficients must be set
equal to zero in order for a nontrivial

In the Hckel treatment of the cyclopropenyl radical, the
basis functions are the three 2pz atomic orbitals, which we denote by f1,
f2 and f3

_ = c1 f1 + c2 f2
+ c3 f3.

The possible values of the orbital energy W are sought as
a function of the c coefficients by solving the three simultaneous equations

xc1 + c2 + c3 = 0,

c1 + xc2 + c3 = 0,

c1 + c2 + x c3 = 0,

where x = (_ _ W)/_ and where _ and
_ are certain integrals whose values are to be determined from empirical
data.

(a) The determinant of the c coefficients must be set
equal to zero in order for a nontrivial solution to exist. This is the secular
equation. Solve the secular equation and obtain the orbital energies.

(b) Solve the three simultaneous equations, once for each
value of x. Since there are only two independent equations, express c2
and c3 in terms of c1.

(c) Impose the normalization condition

to find the values of the c coefficients for each value
of W.

(d) Check your work by using Mathematical to find the
eigenvalues and eigenvectors of the matrix

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