Electronic problem

At this point, it is appropriate to make some conmrents on the construction of approximate wavefiinctions for the many-electron problems associated with atoms and molecules. The Hamiltonian operator for a molecule is given by the general fonn... 

There can be subtle but important non-adiabatic effects [14, ll], due to the non-exactness of the separability of the nuclei and electrons. These are treated elsewhere in this Encyclopedia.) The potential fiinction V(R) is detennined by repeatedly solving the quantum mechanical electronic problem at different values of R. Physically, the variation of V(R) is due to the fact that the electronic cloud adjusts to different values of the intemuclear separation in a subtle interplay of mutual particle attractions and repulsions electron-electron repulsions, nuclear-nuclear repulsions and electron-nuclear attractions. 

The prototype system for all four electron problem is the H4 system, discussed in Section n. 

Future trends will include studies of grain-dependent surface adsorption phenomena, such as gas-solid reactions and surface segregation. More frequent use of the element-specific CEELS version of REELM to complement SAM in probing the conduction-band density of states should occur. As commercially available SAM instruments improve their spot sizes, especially at low Eq with field emission sources, REELM will be possible at lateral resolutions approaching 10 nm without back scattered electron problems. 

Reproducing the exact solution for the relevant n-electron problem a method ought to yield the same results as the exact solution to the Schrodinger equation to the greatest extent possible. What this means specifically depends on the theory underlying the method. Thus, Hartree-Fock theory should be (and is) able to reproduce the exact solution to the one electron problem, meaning it should be able to treat cases like HeH ... 

Higher order methods similarly ought to reproduce the exact solution to their corresponding problem. Methods including double excitations (see Appendix A) ought to reproduce the exact solution to the 2-electron problem, methods including triple excitations, like QCISD(T), ought to reproduce the exact solution to the three-electron problem, and so on. 

From this point on, we will focus entirely on the electronic problem. We will omit the superscripts on all operators and functions. 

But we can carry forward the knowledge of the Bom-Oppenheimer approximation gained from Chapter 2 and focus attention on the electronic problem. Thus... 

If we want to calculate the potential energy curve, then we have to change the intemuclear separation and rework the electronic problem at the new A-B distance, as in the H2 calculation. Once again, should we be so interested, the nuclear problem can be studied by solving the appropriate nuclear Schrodinger equation. This is a full quantum-mechanical equation, not to be confused with the MM treatment. 

The x s can individually be Is, 2s, 2p,. .. atomic orbitals. The lowest-energy solution will be when the x s correspond to Is orbitals on each of the two hydrogen atoms, the next-highest-energy solution will be when one of the ( s is a Is, the other a 2s atomic orbital, and so on. Possible solutions of the electronic problem, with the two H atoms at infinity, are shown in Table 4.1. 

In the Hiickel TT-electron model, ethene is a two-electron problem. 1 have numbered the carbon atoms C and Cj, and X is centred on Ci with X2 on Cj. The HF matrix becomes... 

A characteristic feature of the two-electron problem is that the total wave function may be factorized into a space part and a spin part ... 

Since our Hamiltonian involves a sum of hett(i), which are only functions of the coordinates and momenta of a single electron, we can use separation of variables and reduce the problem to m identical one-electron problems... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Unfortunately, in the molecular systems the theoretical predictions for the already formidable electronic problem carmot be checked fairly against the experimental data, since the nuclear motions may play major effects. From here the need to check these methods in calculations on atomic systems, where accurate theoretical and comparable experimental reference data are already available. 

A completely different route to the A-electron problem is provided by DPT. On an operational level it can be thought of as an attempt to improve on the HE method by including correlation effects into the self-consistent field procedure. 

Foyt et al. [137] interpreted the quadrupole-splitting parameters of low-spin ruthenium(II) complexes in terms of a crystal field model in the strong-field approximation with the configuration treated as an equivalent one-electron problem. They have shown that, starting from pure octahedral symmetry with zero quadrupole splitting, A q increases as the ratio of the axial distortion to the spin-orbit coupling increases. 

The attractive potential exerted on the electrons due to the nuclei - the expectation value of the second operator VNe in equation (1-4) - is also often termed the external potential, Vext, in density functional theory, even though the external potential is not necessarily limited to the nuclear field but may include external magnetic or electric fields etc. From now on we will only consider the electronic problem of equations (1 -4) - (1 -6) and the subscript elec will be dropped. 

There are several important conclusions to be drawn from the Hj example. As expected, separation of the electronic wave equation permits exact solution of the one-electron problem. The result provides a benchmark... 

In terms of this formulation the problem separates into several one-electron problems 10... 

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