Hartree-Fock approximation ground state energy

One of the simplest treatments for estimating the IP is Koopmans theorem [31], which indicates that the occupied orbital energy with an opposite sign is a reasonable approximation for the IP. Here, the IP is estimated by the difference between the Hartree-Fock (HF) ground-state and ionized-state energies. 

Figure 6. Correlation energy for ground states of two-electron atoms as a function of = 1/D. Values for real atoms Li" ", may be read off at = 1/3 (indicated by arrow). For nuclear charge Z >2, the maximum deviations from linearity are about 1%. (The curve for Z = 1 terminates at = 1/2 because the hydride ion is unbound in the Hartree-Fock approximation, and correlation energies therefore are not well-defined for D > 2.)...

An alternative approach to conventional methods is the density functional theory (DFT). This theory is based on the fact that the ground state energy of a system can be expressed as a functional of the electron density of that system. This theory can be applied to chemical systems through the Kohn-Sham approximation, which is based, as the Hartree-Fock approximation, on an independent electron model. However, the electron correlation is included as a functional of the density. The exact form of this functional is not known, so that several functionals have been developed. 

In quantum chemistry, the correlation energy Ecorr is defined as Econ = exact HF- In Order to Calculate the correlation energy of our system, we show how to calculate the ground state using the Hartree-Fock approximation. The main idea is to expand the exact wavefunction in the form of a configuration interaction picture. The first term of this expansion corresponds to the Hartree-Fock wavefunction. As a first step we calculate the spin-traced one-particle density matrix [5] (IPDM) y ... 

Table 1 shows analogous equations for po for the ground states of higher isoelec-tronic series, derived in the crude approximation where only one many-electron Sturmian basis function is used. Figure 1 shows the dementi s values [10] for the Hartree-Fock ground state energies of the 6-electron isoelectronic series... 

Figure 1 This figure shows the ground-state energies of the 6-electron iso-electronic series of atoms and ions, C, iV, 0 +, etc., as a function of the atomic number, Z. The energies in Hartrees, calculated in the crudest approximation, with only one 6-electron Sturmian basis function (as in Table 1), are represented by the smooth curve, while dementi s Hartree-Fock values [10] are indicated by dots. 

Density functional theory (DFT) is based on the Hohenberg-Kohn theorem, which states that there is a functional which gives the exact ground-state energy for the exact electron density. DFT models have become very popular because they are not more costly than Hartree-Fock models. The reason is that in the HF, Cl and MP models, a wavefunction for an N-electron system depends on 3N coordinates, whereas in the DFT approach, the electron density depends on only three coordinates, irrespective of the number of electrons. The problem is that the exact functional would be the Schrodinger equation itself Several approximate functionals have been developed by many authors (Becke, Parr, Perdew, and others) and different forms of the functional can yield slightly different results. Some of the most common DFT models are ... 

Photoelectron spectral measurements have prompted high-accuracy near-Hartree-Fock calculations on the Is hole states of 02. 261 Calculations were reported at Re for molecular O2. The frozen-orbital approximation evaluated the energy of Oj from the RHF calculations of Schaefer250 reported above. Then the IP are the difference between the O2 ground-state energy and the Ot energy. The IP obtained was 563.5 eV. Direct hole-state calculations for the relevant states of OJ, with the MO constrained to be of g or a symmetry, were also carried out. For the orbital occupancy (16), the computed IP was 554.4 eV. Finally, the restriction to g and u... 

The calculation of Mitroy started by calculating the Hartree—Fock approximation to the ground state 3s where we denote the states by the orbitals of the two active electrons in the configuration with the largest coefficient, in addition to the symmetry notation. The calculation used the analytic method with the basis set of Clementi and Roetti (1974) augmented by further Slater-type orbitals in order to give flexibility for the description of unoccupied orbitals. The total energy calculated by this method was —199.614 61, which should be compared with the result of a numerical Hartree—Fock calculation, —199.614 64. 

In practical calculations, it is never possible to obtain the exact solution to eqn (1). Instead, approximate methods are used that are based on either the Hartree-Fock approximation or the density-functional formalism in the Kohn-Sham formulation. In the case of the Hartree-Fock approximation one may add correlation elfects, which, however, is beyond the scope of the present discussion (for details, see, e.g., ref. 1). Then, in both cases the problem of calculating the best approximation to the ground-state electronic energy is transformed into that of solving a set of singleparticle equations,... 

It has already been pointed out several times that electron repulsion terms play a major part in the discussion of electronic excitation energies. Within the Hartree-Fock approximation, electron interaction in closed-shell ground states can be taken care of in a reasonable way using SCF methods. In a treatment of excited states, however, configuration interaction usually has to be taken into account. (Cf. Section 1.2.4.) This can be achieved either by semiempirical methods, especially in those cases where the jr approximation is sufficient for a discussion of light absorption, or, by ab initio methods in the case of small molecules. 

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