Nonrelativistic energy

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

With sufficiently large basis set, the Hartree-Fock (HF) method is able to account for 99% of the total energy of the chemical systems. However, the remaining 1% is often very important for describing chemical reaction. The electron correlation energy is responsible for the same. It is defined as the difference between the exact nonrelativistic energy of the system ( 0) and Hartree-Fock energy (E0) obtained in the limit that the basis set approaches completeness [36] ... 

Also, the nonrelativistic energies of Wolniewicz are presented for comparison. All quantities in atomic units. 

The effort in the first stage of the calculations has been focused on generating very accurate variational wave functions and energies for the rotationless vibrational states of the HD+ ion. As mentioned, this system has been studied by many researchers and very accurate, virtually exact nonrelativistic energies have been published in the literature [112]. This includes the energy for the highest vibrational v = 22 state, which is only about 0.4309 cm below the D - - H+ dissociation limit. 

This section will be broken into a number of discussions. The first will be on a naive 5(7(2) x 5(7(2) extended standard model, followed by a more general chiral theory and a discussion on the lack of Lagrangian dynamics associated with the B3 field. This will be followed by an examination of non-Abelian QED at nonrelativistic energies and then at relativistic energies. It will conclude with a discussion of a putative 5(9(10) gauge unification that includes the strong interactions. 

The term "electron correlation energy" is usually defined as the difference between the exact nonrelativistic energy and the energy provided by the simplest MO wave function, the mono-determinantal Hartree-Fock wave function. This latter model is based on the "independent particle" approximation, according to which each electron moves in an average potential provided by the other electrons [14]. Within this definition, it is customary to distinguish between non dynamical and dynamical electron correlation. 

Further Hylleraas-type calculations with basis sets of increasing size and sophistication, culminating with the work of Pekeris and coworkers in the 1960 s (see Accad, Pekeris, and Schiff [26]) showed that nonrelativistic energies accurate to a few parts in 109 could be obtained by this method, at least for the low-lying states of helium and He-like ions. However, these calculations also revealed two serious numerical problems. First, it is difficult to improve upon this accuracy of a few parts in 109 without using extremely large basis sets where roundoff error and numerical linear dependence become a problem. Second, as... 

Correlation energy The difference between the Hartree-Fock energy calculated for a system and the exact nonrelativistic energy of that system. The correlation energy arises from the approximate representation of the electron-electron repulsions in the Hartree-Fock method. 

If the wave function is the exact wave function, we obtain for E the exact (nonrelativistic) energy. If is an approximate wave function, the variational principle (Levine, 1983) tells us that the lower the E the more closely il> resembles the exact wave function, so long as it satisfies certain conditions, the most important being the Pauli exclusion principle. If i ( contains parameters, then that choice of parameters giving the lowest E will give the best wave function in the sense of maximum overlap with the true wave function. 

When the single-configuration HF energy is subtracted, the energy (16a) represents 46% of the total electron correlation nonrelativistic energy of Be S, which is 1.19 eV. Nevertheless, it will be used in Section 9 for the calculation of excitation energies of low-lying excited states of Be, whose wavefunctions are also truncated appropriately. 

The reviews of the MCSCF method by Shepard [129] and Werner [130] are too technical to summarize briefly. We shall content ourselves with pointing out that the structure of our MCDHF(B) equations is consistent with the nonrela-tivistic theory, so that much of the nonrelativistic formalism can be adopted with little change. The nonrelativistic energy expression is of the form of (231) and can be regarded as a function of the CSF coefficients Cj and the MO expansion coefficients where... 

---