Orbital occupancy

Although HF orbitals are, by definition, the best possible for a singleconfiguration wavefunction, it is actually possible to find a better set of orbitals, called natural orbitals,31 to describe the correlated p(r). The natural orbitals are maximum-occupancy orbitals, determined from itself and guaranteed to give fastest possible convergence to p(r), i.e., consistently higher occupancies n, than HF orbitals inEq. (1.15). For a HF wavefunction the natural orbitals and HF orbitals are equivalent, but for more accurate wavefunctions the natural orbitals allow us to... 

For copper ions, the account of the electron correlation at the MP2 level decreases the charges about 1.5 times or on 32 % for the Cul and on 36.4% for the Cu2. The change of copper charges for the transition in calculation methods ROHF —> UHF is not such essential as for the transition UHF — MP2. The account ofthe electron correlation causes also the essential change of the occupation orbital numbers of the 4s and 3d copper shells. The 4s copper electrons are involved in the covalent bonding with the 2p oxygen electrons. 

On the next level of chemical reactivity, aiming to evaluate of the orbital chemical hardness (for an open chemical state and with unoccupied orbitals in neutral state) the derivative of the orbital electronegativity relation (3.192) respecting the occupancy orbital numbers will be considered while taking into account the total energy expression (3.189), so obtaining ... 

To find maximum-occupancy orbitals for atom A, one first finds the local eigenvectors of the one-center atomic block... 

After the maximum-occupancy orbitals have been... 

Comparing uncorrelated Hartree-Fock (RHF) with correlated multi-configurational (CASSCF) - complete valence-shell CASSCF(I2,I0), 13860 configurations - second-order Mpller-Plesset (MP2), and hybrid density functional (B3LYP) results, all at 6-311G basis level and RHF optimized geometry. NBO entries include the hybrid type (/lA. /named atom (lOOc- ), and occupancy (orbital population, in e). 

The co-ordination number in ionic compounds is determined by the radius ratio - a measure of the necessity to minimize cationic contacts. More subtle effects are the Jahn-Teller effect (distortions due to incomplete occupancy of degenerate orbitals) and metal-metal bonding. 

Because of the indistingiiishability of the electrons, the antisynnnetric component of any such orbital product must be fonned to obtain the proper mean-field wavefunction. To do so, one applies the so-called antisynnnetrizer operator [24] A= Y.p -lf p, where the pemuitation operator mns over all A pemuitations of the N electrons. Application of 4 to a product fiinction does not alter the occupancy of the fiinctions ( ). ] in it simply scrambles the order which the electrons occupy the ( ). ] and it causes the resultant fiinction... 

Here n,. = 0, 1 or 2 is the occupation number of tlie orbital ((),. in the state being studied. The kinetic energy... 

Flead and Silva used occupation numbers obtained from a periodic FIF density matrix for the substrate to define localized orbitals in the chemisorption region, which then defines a cluster subspace on which to carry out FIF calculations [181]. Contributions from the surroundings also only come from the bare slab, as in the Green s matrix approach. Increases in computational power and improvements in minimization teclmiques have made it easier to obtain the electronic properties of adsorbates by supercell slab teclmiques, leading to the Green s fiinction methods becommg less popular [182]. 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Figure 7-21. The MOs and energy levels given by HMO theory for 1,3-butadiene. The occupation of the orbitals is shown for the neutral molecule.

For Woodward-Hoffm an allowed thermal reactions (such as the con rotatory ring opening of cyclobulan e), orbital symmetry is conserved and there is no change in orbital occupancy. Hven though bonds are made and broken, you can use the RHFwave fun etion. 

For Iran sition metals th c splittin g of th c d orbitals in a ligand field is most readily done using HHT. In all other sem i-ctn pirical meth -ods, the orbital energies depend on the electron occupation. HyperCh em s m oiccii lar orbital calcii latiori s give orbital cri ergy spacings that differ from simple crystal field theory prediction s. The total molecular wavcfunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals arc the residue of SCT calculations, in that they are deemed least suitable to describe the molecular wavefunction, ... 

Atoms, linear molecules, and non-linear molecules have orbitals which can be labeled either according to the symmetry appropriate for that isolated species or for the species in an environment which produces lower symmetry. These orbitals should be viewed as regions of space in which electrons can move, with, of course, at most two electrons (of opposite spin) in each orbital. Specification of a particular occupancy of the set of orbitals available to the system gives an electronic configuration. For example,... 

In summary, an atom or molecule has many orbitals (core, bonding, non-bonding, Rydberg, and antibonding) available to it occupancy of these orbitals in a particular manner gives rise to a configuration. If some orbitals are partially occupied in this configuration. 

The essence of this analysis involves being able to write each wavefunction as a combination of determinants each of which involves occupancy of particular spin-orbitals. Because different spin-orbitals interact differently with, for example, a colliding molecule, the various determinants will interact differently. These differences thus give rise to different interaction potential energy surfaces. 

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