The Overlap Integral

Later on we shall be concerned largely with ji-lTbond-ing, and it will be seen that the values of —irS j range between 0.20 to 0.27 over the usual range of carbon—carbon bond dis  

The graph of as a function of ry shows 2pr-n overlap to increase monotonically to unity as r j decreases. On the other hand for 2p.-o- overlap S j increases to a maximum, then goes to zero, and changes sign at 0. 7 A. Explain. 

If S j with i j is taken as zero, then the determinant simplifies as shown on the next page. Further progress at this point depends upon evaluation of the H integrals. 

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The last factor, the square of the overlap integral between the initial and final vibrational wavefunctions, is called the Franck-Condon factor for this transition. 

It can be observed that we can separate the overlap integral into the product of three independent spatial integrals... 

In the MNDO rnelluKi Ui e resonance integral, is proportion al Lo the overlap integral, S y ... 

S.-i is the overlap integral, and are ionisation potentials for the appropriate orbitals and /J. a is a parameter dependent upon both of the two atoms A and B. 

One restriction imposed by Huckel theory that is rather easy to release is that of zero overlap for nearest-neighbor interactions. One can retain a — as the diagonal elements in the secular matrix and replace p by p — EjS as nearest-neighbor elements where S is the overlap integral. Now,... 

Millikan has shown that the overlap integral for hydrogen-like p orbitals in linear hydrocarbons is about 0.27 (Millikan, 1949). 

What is the energy separation E2 — E of the bonding and antibonding orbitals in ethylene, assuming that the overlap integral S is 0.27 ... 

If the eonstituent atomie orbitals Xj have been orthonormalized as diseussed earlier in this ehapter, the overlap integrals [Pg.195]

The first term in this expansion, when substituted into the integral over the vibrational eoordinates, gives ifj(Re) , whieh has the form of the eleetronie transition dipole multiplied by the "overlap integral" between the initial and final vibrational wavefunetions. The if i(Rg) faetor was diseussed above it is the eleetronie El transition integral evaluated at the equilibrium geometry of the absorbing state. Symmetry ean often be used to determine whether this integral vanishes, as a result of whieh the El transition will be "forbidden". 

Computation of the overlap integrals in the atomic orbital basis set. 

The two-center one-electron integral Hj y, sometimes called the resonance integral, is approximated in MINDO/3 by using the overlap integral, Sj y, in a related but slightly different manner to... 

In the MNDO method the resonance integral, is proportional to the overlap integral,... 

Both the weak- and strong-coupling results (2.82a) and (2.86) could be formally obtained from multiplying Aq by the overlap integral (square root from the Franck-Condon factor) for the harmonic q oscillator,... 

If the overlap integral is neglected, the Heitler-London equation becomes... 

The Q s and i s are coulomb and exchange integrals defined as shown in Table 5-2. Notice that when A is infinitely separated from the B-C pair, Qg, Qc Jb, Jc are all zero, and Eq. (5-15) collapses to = Qa Ja, as it should. Similarly it gives the appropriate result for the other extreme cases. London did not derive Eq. (5-15), but it has since been derived and is known to apply only to s electrons moreover it neglects the overlap integrals. 

Sab is the overlap integral between atomic orbitals Isa and Isb, and the factor 1/V2(1 + Sab) is often called a normalization coefficient or the normalizing factor. It is introduced to make sure that... 

The overlap integrals (Table 8.6) were calculated from the master formulae of Mulliken et al. (1949). 

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