Correlation energy interpretation

The set of molecules for which the relationship of Eq. (3.3) has been tested with the reported accuracy, has certain sinq>lifying features in common They all have standard bonding coordinations around each atom and the shortfall of the SCF energy is entirely due to dynamic correlations. Modifications are to be expected for systems where these premises are not satisfied. Even with these limitations, however, the molecules in Table 2 represent a variety of atom and bond combinations. It is therefore remarkable that, for all of them, the correlation energy can be recovered by a sinq>le system-independent formula that allows for a physically meaningful interpretation. 

In a 7r electron system the orbitals of an aromatic positive ion are similar to the corresponding orbitals of the neutral molecule. In contrast, in small molecules electronic rearrangement following excitation is often sufficiently important that changes in nuclear geometry, correlation energy, etc., are all essential to the correct interpretation of the excitation phenomenon. Because of the similarity in the orbital systems of neutral and positive ion aromatic compounds, we shall assume that it is possible to describe the photoionization of an aromatic molecule within the framework of a one-electron model. Given that the n and a electrons are describ-able by a set of separable equations of motion, we need consider only the initial and final orbitals of the most weakly bound electron to determine the ionization cross section near to the threshold of ionization. 

H. A. Physical interpretation of Kohn-Sham theory exchange-correlation energy functional and its derivative... 

For the details and derivation of the physical interpretation we refer the reader to the original literature14,15. Since the Coulomb self-energy component of the KS electron-interaction energy functional and its derivative, the Hartree potential, are known functionals of the density, we provide in Section HA the expressions governing the interpretation of the KS exchange-correlation energy... 

H. A. PHYSICAL INTERPRETATION OF KOHN-SHAM THEORY EXCHANGE-CORRELATION ENERGY FUNCTIONAL AND ITS DERIVATIVE... 

While the excellence of the agreement of the relative energies of the methylene group in the cyclic molecules with the measured strain energies may be to some extent due to the fortuitous cancellation of errors in the contributions not specifically considered, namely the correlation energy, the zero-point energy, and A AHf) between 0 and 298 K, the nature of the results leaves no doubt as to the correctness of the interpretation that has been given, that the atoms of theory recover the experimentally measured properties of atoms in molecules. 

Recalling the quantum-mechanical interpretation of the wave function as a probability-amplitude, we see that a product form of the many-body wave function corresponds to treating the probability amplitude of the many-electron system as a product of the probability amplitudes of individual electrons (the orbitals). Mathematically, the probability of a composed event is only equal to the probability of the individual events if the individual events are independent (i.e., uncorrelated). Physically, this means that the electrons described by the product wave function are independent. Such wave functions thus neglect the fact that, as a consequence of the Coulomb interaction, the electrons try to avoid each other. The correlation energy is the additional energy lowering obtained in a real system due to the mutual avoidance of the interacting electrons. 

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