Configuration interaction mathematical methods

The above expansion of the full N-electron wavefunction is termed a "configuration-interaction" (Cl) expansion. It is, in principle, a mathematically rigorous approach to expressing P because the set of all determinants that can be formed from a complete set of spin-orbitals can be shown to be complete. In practice, one is limited to the number of orbitals that can be used and in the number of CSFs that can be included in the Cl expansion. Nevertheless, the Cl expansion method forms the basis of the most commonly used techniques in quantum chemistry. 

The configuration interaction (Cl) treatment of electron correlation [83,95] is based on the simple idea that one can improve on the HF wavefunction, and hence energy, by adding on to the HF wavefunction terms that represent promotion of electrons from occupied to virtual MOs. The HF term and the additional terms each represent a particular electronic configuration, and the actual wavefunction and electronic structure of the system can be conceptualized as the result of the interaction of these configurations. This electron promotion, which makes it easier for electrons to avoid one another, is as we saw (Section 5.4.2) also the physical idea behind the Mpller-Plesset method the MP and Cl methods differ in their mathematical approaches. 

The question for a more systematic inclusion of electronic correlation brings us back to the realm of molecular quantum chemistry [51,182]. Recall that (see Section 2.11.3) the exact solution (configuration interaction. Cl) is found on the basis of the self-consistent Hartree-Fock wave function, namely by the excitation of the electrons into the virtual, unoccupied molecular orbitals. Unfortunately, the ultimate goal oi full Cl is obtainable for very small systems only, and restricted Cl is size-inconsistent the amount of electron correlation depends on the size of the system (Section 2.11.3). Thus, size-consistent but perturbative approaches (Moller-Plesset theory) are often used, and the simplest practical procedure (of second order, thus dubbed MP2 [129]) already scales with the fifth order of the system s size N, in contrast to Hartree-Fock theory ( N ). The accuracy of these methods may be systematically improved by going up to higher orders but this makes the calculations even more expensive and slow (MP3 N, MP4 N ). Fortunately, restricted Cl can be mathematically rephrased in the form of the so-called coupled clus-... 

Other examples of optimizing functions that depend quadraticaUy of the parameters include the energy of Hartree-Fock (HF) and configuration interaction (Cl) wave functions. Minimization of the energy with respect to the MO or Cl coefficients leads to a set of linear equations. In the HF case, the Xy coefficients depend on the parameters Ui, and must therefore be solved iteratively. In the Cl case, the number of parameters is typically 10 -10 and a direct solution of the linear equations is therefore prohibitive, and special iterative methods are used instead. The use of iterative techniques for solving the Cl equations is not due to the mathematical nature of the problem, but due to computational efficiency considerations. 

The separation of polymer-polymer interactions into these two types is a division into qualitatively different kinds of interactions. Type a interactions involve only a few monomers at a time and therefore represent a few-body problem. The configurational statistics of polymers which have only short-range interactions can usually be treated exactly by using, say, the mathematical methods of the one-dimensional Ising model, or. 

The mathematical formalism employed in the theory of orbital interactions has been elaborated both in general terms [8-12] and at different levels of the MO theory approximation, including the Hiickel [13] and extended Hiickel molecular orbital [14] as well as the SCF MO [15] and configuration interaction [16, 17] methods. In what follows, only the principal general relationships and conclusions will be dealt with. 

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