Zero-electron pseudopotentials

In this section we report some theoretical work on the molecular spectroscopy in rare gas matrices. Due to the difficulty in modelling the environment, theoretical studies for a long time compared the results obtained from gas phase calculations with experimental data in matrices. However, it has been demonstrated that the effect of the matrix is not negligible, and may strongly depend on the rare gas, as discussed recently by Li et al [186]. Even if relativistic pseudopotentials lead to a substantial reduction of computational costs, one still needs to treat explicitly a certain number of valence electrons, for instance eight electrons for each rare gas (Rg) atom. A further simplification is to consider a rare gas atom as a zero-electron system with all its electrons in the core represented by the so-called zero-electron pseudopotential or e-Rg pseudopotential (for further details see Grofi and Spiegelmaim [187,188] and references therein). As 

GroB and Spiegelmann applied this technique to the spectroscopy of the NO molecule trapped in an argon matrix [188]. They simplified the problem, describing the NO molecule as a system of one electron interacting with a NO+ core. The molecular NO orbitals are obtained self-consistently fi-om the Fock operator corresponding to the NO+ ion in the argon matrix 

The calculated adiabatic transition energies for the trapped molecules reported in Table 10 are in excellent agreement with experiment, especially when the CPP is added. It should be noticed that the interaction with the matrix leads to an increase in the transition energies. However, the matrix relaxation effects were not taken into account in this calculation. 

This example demonstrates that such a spectroscopic problem is reducible to a quasi one-electron effective Hamiltonian. Whenever spin-orbit coupling is important, one can take advantage of the one-electron character to easily calculate the fine-structure of molecular excited states, by just adding the spin-orbit 

Adiabatic transition energies (in eV) of the NO molecule isolated and trapped in argon matrix, without (Te) or with (T ° ) CPP coirection, taken from Ref. [188]. 

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Relativistic effects may be also considered by other methods than pseudopotentials. It is possible to carry out relativistic all-electron quantum chemical calculations of molecules. This is achieved by various approximations to the Dirac equation, which is the relativistic analogue to the nonrelativistic Schrodinger equation. We do not want to discuss the mathematical details of this rather complicated topic, which is an area where much progress has been made in recent years and where the development of new methods is a field of active research. Interested readers may consult published reviews . A method which has gained some popularity in recent years is the so-called Zero-Order Regular Approximation (ZORA) which gives rather accurate results ". It is probably fair to say that... 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

In order to solve the Kohn Sham equations, an expansion of the one-electron wave functions on a basis set is performed. Both localized basis sets and plane wave ones are currently used. Localized basis sets have the advantage of their small size. However they are attached to the atomic positions, which yields non-zero Pulay forces in geometry optimization and molecular dynamics. Plane waves, on the other hand, provide a uniform sampling of space, whatever the specific conformation of the system they are independent of the atomic positions, but they require the use of pseudopotentials to mimick core electrons and a very large number of vectors is necessary in standard surface calculations. 

The electron affinity, which is very small for the Fe atom (0.15 eV), has so far not been reliably calculated. However, even the essentially zero affinity obtained is a tremendous improvement from the uncorrelated value of -2.36 eV. One of the reasons for the small remaining errors is that only simple trial functions were used. In particular, the determinants were constructed from Hartree-Fock orbitals. It is known that the Hartree-Fock wavefunction is usually more accurate for the neutral atom than for negative ion, and we conjecture that the unequal quality of the nodes could have created a bias on the order of the electron affinity, especially when the valence correlation energy is more than 20 eV. One can expect more accurate calculations with improved trial functions, algorithms, and pseudopotentials. 

The zero order contribution to the dielectric matrix imposes that Gi = G2. One then immediately recognizes the energy-wave number characteristic in the first term of this expression for the dynamical matrix. Furthermore, in the framework of the local ionic pseudopotential, the induced electron density is given by ... 

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