Class Summary          11/29/04         

 

11/29               MO Descriptions of Biradicals

 

COVALENT wavefunctions

 

Singlet              [fA(1)fB(2) + fA(2)fB(1)][a(1)b(2) – b(1)a(2)]

Triplet              [fA(1)fB(2) + fA(2)fB(1)][a(1)b(2) – b(1)a(2)][ab + ba] (or) [aa] (or) [bb]

 

We can also consider IONIC wavefunctions

 

Singlet              [fA(1)fA(2)][ab - ba]   and  [fB(1)fB(2)][ab - ba]   (we can also combine these...)

 

 

CONSIDER p-benzyne as our model system, where the p-carbons are fA and fB

 

We can have MO’s made from out AO’s of                  y+ = fA + fB  and  y- = fA - fB

 

            y+2 (necessarily a singlet) = [fA(1)fA(2)] + [fB(1)fB(2)] + [fA(1)fB(2) + fA(2)fB(1)]

                                                            IONIC             IONIC             OPEN-SHELL

            y+1y-1 = [fA(1)fA(2)] - [fB(1)fB(2)] (+ a triplet term that zeroes out) IONIC

 

            y-2 = [fA(1)fA(2)] + [fB(1)fB(2)] - [fA(1)fB(2) + fA(2)fB(1)]                     

(IONIC components and OPEN-SHELL component, just like y+2)

 

TAKE-HOME LESSON      

We cannot make a wavefunction that is purely ionic or biradical from combining the AO’s.

We must use linear combinations of the MO’s:

 

e.g.      y+2 - y-2 = [fA(1)fB(2) + fA(2)fB(1)]            (purely biradical)

 

This leads to MULTI-CONFIGURATIONAL FUNCTIONS, where one can incorporate CONFIGURATION INTERACTION coefficients (CI):

 

            (1 – l)1/2 y+2 - ly-2                (0 £ l £ (1/2)1/2)

 

            OR, more simply          Y0 = c1y+2 – c2y-2

 

            A measure of biradical character is the RATIO of the coefficients, c12/c22           (c12 + c22 = 1)

 

                        As c12/c22 tends to 1, it becomes more biradical in character

                        As c12/c22 ® ∞, it becomes more closed-shell in character

 

EXAMPLES    Ethylene                       c12/c22 > 1010

                        cis-butadiene                c12/c22 ~ 80

                        o-benzyne                    c12/c22 = 11.1

                        m-benzyne                   c12/c22 = 4.3

                        p-benzyne                    c12/c22 = 0.6     (here, the ratio has actually inverted in favour of c2)

                                                                                    (the point is that it is quite biradical in nature)

 

RELEVANT REFERENCES

 

Nash, J. J., Squires, R. R.  Theoretical Studies of o-, m-, and p-Benzyne Negative Ions  J. Am. Chem. Soc. (1996), 118, 11872-11883.

 

Wierschke, S.G., Nash, J.J. and Squires, R.R.  A multiconfigurational SCF and correlation-consistent CI study of the structures, stabilities, and singlet-triplet splittings of o-, m-, and p-benzyne  J. Am. Chem. Soc. (1993) 115, 11958-11967.

 

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