Exact energy

J. C. Simo, N. Tarnow, and K. K. Wang. Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Computer Methods in Applied Mechanics and Engineering, 100 63-116, 1994. 

Even though the problem of the hydrogen molecule H2 is mathematically more difficult than, it was the first molecular orbital calculation to appear in the literature (Heitler and London, 1927). In contrast to Hj, we no longer have an exact result to refer to, nor shall we have an exact energy for any problem to be encountered from this point on. We do, however, have many reliable results from experimental thermochemistry and spectroscopy. 

The expectation value of the Hamiltonian for any such function can be expressed in terms of its Cj coefficients and the exact energy levels Ej of H as follows ... 

Gaussian theory (Gl, G2, G3) a method for extrapolating from ah initio results to an estimation of the exact energy Gaussian-type orbital (GTO) mathematical function for describing the wave function of an electron in an atom... 

Another distinguishing aspect of MO methods is the extent to which they deal with electron correlation. The Hartree-Fock approximation does not deal with correlation between individual electrons, and the results are expected to be in error because of this, giving energies above the exact energy. MO methods that include electron correlation have been developed. The calculations are usually done using MoUer-Plesset perturbation theoiy and are designated MP calculations." ... 

Among the most widely used ab initio methods are those referred to as Gl" and 02." These methods incorporate large basis sets including d and / orbitals, called 6-311. The calculations also have extensive configuration interaction terms at the Moller-Plesset fourth order (MP4) and fiirther terms referred to as quadratic configuration interaction (QCISD). ° Finally, there are systematically applied correction terms calibrated by exact energies from small molecules. 

By equating Soft to die exact energy, this expression may be taken as the definition of... 

The Variational Principle states that an approximate wave function has an energy which is above or equal to the exact energy. The equality holds only if the wave function is exact. The proof is as follows. 

This proof shows that any approximate wave function will have an energy above or equal to the exact ground-state energy. There is a related theorem, known as MacDonald s Theorem, which states that the nth root of a set of secular equations (e.g. a Cl matrix) is an upper limit to the n — l)th excited exact state, within the given symmetry subclass. In other words, the lowest root obtained by diagonalizing a Cl matrix is an upper limit to the lowest exact wave functions, the 2nd root is an upper limit to the exact energy of the first excited state, the 3rd root is an upper limit to the exact second excited state and so on. 

The variation principle tells us that EHF is always larger than the exact energy, and this implies that the total correlation energy is always a negative quantity. From Eq. 11.70 it follows then that 7 corr is always positive, whereas VC0TT is negative according to the formulas ... 

According to Eq. 11.67, the correlation energy is simply defined as the difference between the exact energy and the energy of the Hartree-Fock approximation. Let us repeat this definition in a more precise form ... 

Kromhout, R. A., Phys. Rev. 107, 215, Exact energy self-consistent field. ... 

No known equation provides the exact energies of an atom that has more than one electron. Nevertheless, each electron in an atom can be assigned a value of n that is a positive integer and that correlates with the energy of the electron. The most stable energy for an atomic electron corresponds to = 1, and each successively higher value of tl describes a less stable energy state. 

As the amount of NO oxidised to NO2 in this unit has not been estimated, it is not possible to make an exact energy balance over the unit. However, the maximum possible quantity of steam generated can be estimated by assuming that all the NO is oxidised and the minimum quantity by assuming that none is. The plant steam pressure would be typically 150 to 200 psig 11 bar, saturation temperature 184°C. Taking the approach temperature of the outlet gases (difference between gas and steam temperature) to be 50°C, the gas outlet temperature will be = 184 + 50 = 234°C (507 K). 

The advantage over the HF scheme is that whereas in conventional ah initio theory we must resort to costly perturbation theory or configuration interaction expansions, in DFT electron correlation is already included explicitly in the exchange-correlation functional. The key problem is instead to find an appropriate expression for xc. As stated above, when we have the correct functional we should be able to extract the exact energy, the exact ground state density, and all properties for our system. 

The exact energy carried by an emitted negatron will depend upon the angle between its path and that of the antineutrino. As the angle can vary from atom to atom, so will the distribution of energy between the particles. Negatron spectra (Figure 10.3) thus do not have sharp peaks. 

---