Hartree-Fock configuration

Having established the most important concepts for MQS, the next step is to actually compute the numerical values associated with the quantum similarity measures. Electron densities can naturally be obtained from many quantum chemical methods such as DFT, Hartree-Fock, configuration interaction, and many more, even from experiment. 

Thus, the ionic component has decreased while the covalent component has increased in Pmo-ci, relative to the Hartree—Fock configuration o-o-. ... 

MHz ( 2.8 MHz) can only be obtained if triple excitations are taken into account (45 MHz). A SD-CI gives a value of 30 MHz, i.e. an error of more than 30 %. As already discussed in the previous section, a Cl which includes only single excitations with respect to the Hartree-Fock configuration (S-CI) gives much better agreement (41 MHz) than the SD-CI calculations. 

Fig. 5.36. Electron-density distributions in borates — deformation density maps calculated for BlOHlj-. (a) static model map calculated from pseudoatom model (b) theoretical deformation density map from ab initio Hartree-Fock configuration-interaction calculation (after Gajhede et al., 1986 reproduced with the publisher s permission).

Fig. 5 Photo-ionization energies of the OSO4 molecule from experiment, from V-Xa calculations and Hartree Fock-Configuration Interaction (HF-CI) calculations.

It is sensible to choose the basis orbitals a) that form the basis configurations in an optimal way. The symmetry configuration that most-closely approximates the lowest-energy state of the Xj manifold is the Hartree—Fock configuration ro), in which the lowest-energy orbitals are occupied. The basis orbitals include ones that are unoccupied in ro). They may be calculated as eigenstates of the Hartree—Fock equation (5.26). 

In practice it is necessary to truncate the configuration basis rfe) to a dimension M < Mp. Satisfactory results are achieved by including all single and double excitations to unoccupied orbitals a) from the highest-energy major shell in the Hartree—Fock configuration. Examples of LS-coupled Hartree—Fock configurations po) are... 

If the perturbation V is reduced to zero then Ei is the same as the Hartree—Fock energy Ei. This is used to define the label i for a Hartree-Fock configuration pi). It is the configuration of the same symmetry as i) whose energy Ei is nearest to The residual electron—electron potential V splits the Hartree—Fock energy levels so that there is more than one atomic state for every Hartree—Fock state. 

Fig. 8.7. Energy level diagram of the H" ion showing states where Hartree—Fock configurations have two s orbitals.

Between the 2s2s and 2soos states there is a sequence of resonances with Hartree—Fock configurations 2sns, n = 3,oo. They occur just below the n=2 threshold at 10.20 eV in Eq and condense to this energy. A similar sequence of resonances occurs just below each inelastic threshold. Similar sequences occur in the other symmetry manifolds with Hartree—Fock configurations consisting of different orbitals. 

A simple example is the 2s state of the helium ion. It has a small overlap with the Is Hartree—Fock orbital of helium, since the Hartree—Fock potential of helium is not the same as the Coulomb potential of the helium ion. However, it has a large overlap with helium configurations that contain a 2s orbital. The 2s orbital is not occupied in the Hartree—Fock configuration. 

A. Hartree-Fock, Configuration-interaction Self-consistent Field Calculations... 

Calculation of the electronic structure of both SF6 and SFg is complicated by the fact that the single Hartree-Fock determinant may be an inadequate zeroth-order reference function, as may be seen from the values of C0 (the weight of the Hartree-Fock configuration) based on the MP2 results (UMP2 for the ion), which are collected in Table 1. 

The Cl and INO calculations discussed above yield very unwieldy wave-functions which do not provide any simple qualitative picture of the bonding. An alternative approach is to pick a reasonably small number of configurations and to try to find, not only the best coefficients CK in the Cl wavefunction of (15), but also to vary the molecular orbitals so as to obtain the optimum orbitals for the chosen form of the wavefunction. This method is known as the multi-configuration SCF method (MC-SCF). The configurations chosen will normally include the Hartree-Fock configuration, plus those additional configurations which add the most important types of... 

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