Integrals exchange

The initial values, a, , are derived by correlations with dipole moments of a series of conjugated systems. The exchange integrals are taken from Abraham and Hudson [38] and are considered as being independent of charge. The r-charges are then calculated from the orbital coefficients, c,j, of the HMO theory according to Eq. (14). 

The INDO meth od (In termediate N DO) corrects some of the worst problems with CNDO. Tor example, INDO exchange integrals between electrons on the same atom need not he eL tial, hut can depend on the orbitals involved. Though this introduces more parameters, additional compulation time is negligible. INDO and MINDO/11 (.Vlodilied INDO, version II) methods are different im piemen lalion s of the same approxim ation. ... 

The one-eenter exchange integrals that INDO adds to the CNDO schcmccan be related to th e-Slater-Condon param eters h", O. and F used to describe atomic spectra. In particular, for a set of s, p,. p,.. t, atom ie orbitals, all the on e-ecn ter in tegrals are given as ... 

The integral A, is the exchange integral corresponding to the linear combination... 

Coulomb integrals Jij describe the coulombic interaction of one charge density (( )i2 above) with another charge density (c )j2 above) exchange integrals Kij describe the interaction of an overlap charge density (i.e., a density of the form ( )i( )j) with itself ((l)i(l)j with ( )i( )j in the above). 

The three integrals shown above can be seen to be equal and to be of the exchange-integral form by expressing the integrals in terms of integrals over cartesian functions and recognizing identities due to the equivalence of the 2px, 2py, and 2pz orbitals. For example. 

A more complex set of functionals utilizes the electron density and its gradient. These are called gradient-corrected methods. There are also hybrid methods that combine functionals from other methods with pieces of a Hartree-Fock calculation, usually the exchange integrals. 

PM3, developed by James J.P. Stewart, is a reparameterization of AMI, which is based on the neglect of diatomic differential overlap (NDDO) approximation. NDDO retains all one-center differential overlap terms when Coulomb and exchange integrals are computed. PM3 differs from AMI only in the values of the parameters. The parameters for PM3 were derived by comparing a much larger number and wider variety of experimental versus computed molecular properties. Typically, non-bonded interactions are less repulsive in PM3 than in AMI. PM3 is primarily used for organic molecules, but is also parameterized for many main group elements. 

Note Do not use CNDO on any problem where electron-spin is critically important. Its complete neglect of atomic exchange integrals makes it incapable of dealing with these problems. 

The INDO (Intermediate Neglect of Differential Overlap) differs from CNDO in the treatment of one-center exchange integrals. The CNDO (Complete Neglect of Differential Overlap) treatment retains only the two-electron integrals (p.p. vv) = The Yj y are... 

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. 

Table 5-2. [dentification of Couiombic and Exchange Integrals for the Three-Electron System ... 

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