Frozen-nuclei approximation

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

Control of n-Electron Rotation in a Chiral Aromatic Molecule Within a Frozen-Nuclei Approximation... 

In this section, a pulse-design scheme to induce and control jt-electron rotation in a chiral aromatic molecule is provided within a frozen-nuclei approximation. We perform electron WP simulations and show that the initial direction of rr-electron rotation in a chiral aromatic molecule depends on the polarization direction of a linearly polarized laser pulse. A pump-dump method for performing unidirectional rotation of n electrons is also presented [15]. An ansa (planar-chiral) aromatic molecule with a six-membered ring, 2,5-dichloro[n](3,6)pyrazinophane (DCPH Fig. 6.1), was chosen. 

In the previous section, we treated rr-electron rotation within a frozen-nuclei approximation. However, the effects of nonadiabatic coupling should not be ignored when the duration of n-electron rotations becomes close to the period of molecular vibrations. Therefore, in this section, we explicitly take into account vibrational degrees of freedom and perform nuclear WP simulations in a model chiral aromatic molecule irradiated by a linearly polarized laser pulse. The potentials of the vibrational modes were determined by ab initio MO methods [12]. For reducing computational time, while maintaining properties of jt-electronic structures, we used 2,5-dichloropyrazine (DCP, Fig. 6.4) instead of 2,5-dichloro[n](3,6)pyrazinophane (DCPH), in which the ansa group is replaced by hydrogen atoms. 

The Born-Oppenheimer approximation [96], created to simpUfy the electronic calculus for frozen nuclei approximation, breaks down when computing, for instance, the magnetic dipole moment and its derivative with respect to the nuclear velocities or momenta for assessing the molecular properties of surfaces [97]. 

A reasonable assumption for any chemist is that molecules are made up of fixed cores (the nucleus and the inner electrons) and of chemically active valence electrons moving in the field of these fixed cores (frozen-core approximation). Obviously for accurate investigations in spectroscopy this assumption would fail, for example for alkali or alkaline-rich elements on the left of the periodic table, which possess highly polarizable cores. In the following only fixed atomic cores will be considered. 

Recent theoretical studies and accurate experimental determinations of electron densities in molecules have confirmed that the majority of electrons do indeed form a concentrated core near the nucleus which appears very atomic like [120]. The electTOTi density is very monotonic as the radial distance from the nucleus increases. Chemists sometimes wrongly consider that the electron density in the cores of atoms has outer maxima corresponding to the shell structures. The inner electron cores are almost transferable entities and cmisequently endorse the valence-care partition proposed by Lewis and Kossel, and this property is utilised in frozen core approximations [138]... 

While plane waves are a good representation of delocalized Kohn-Sham orbitals in metals, a huge number of them would be required in the expansion (O Eq. 7.67) to obtain a good approximation of atomic orbitals, in particular near the nucleus where they oscillate rapidly. Therefore, in order to reduce the size of the basis set, only the valence electrons are treated explicitly, while the core electrons (i.e., the inner shells) are taken into account implicitly through pseudopotentials combining their effect on the valence electrons with the nuclear Coulomb potential. This frozen core approximation is justified as typically only the valence electrons participate in chemical interactions. To minimize the number of basis functions the pseudopotentials are constructed in such a way as to produce nodeless atomic valence wavefunctions. Beyond a specified cutoff distance from the nucleus, Ecut the nodeless pseudo-wavefunctions are required to be identical to the reference all-electron wavefunctions. 

Electron spin resonance studies of silver(II) pyridine complexes have proved to be extremely useful in determining the nature of the spedes in solution. Since natural silver has two isotopes, 107Ag and 109Ag, in approximately the same abundance, both of spin / = J, and since their nuclear magnetic moments differ by less than 15%, interpretation of spectra is often considered in terms of a single nucleus. The forms of the hyperfine splitting patterns for IN, cis and trans 2N, 3N and 4N, would be expected to be quite different and hence the number of pyridines can be readily assessed from well-resolved spectra. Spin Hamilton parameters obtained from both solid and frozen solution spectra are collected in Table 64.497 499 501-510... 

There is a direcf correspondence (offen, approximate proportionality) between the polarization a nucleus picks up in the radical pairs and its h)/perfine coupling constant. The so-called polarization pattern " —the relative polarization intensities of the nuclei in a product—may thus be regarded as a frozen EPR spectrum of fhe intermediates, although the obtained values of the coupling constants are only relative ones, i.e. scaled by a constant factor. Hence, the paramagnetic intermediates can be identified and characferized by photo-CIDNP specfroscopy in a... 

Fig. 7.6. Failure of the Born-Oppenheimer approximation for HD. The shaded circle on the left represents the deuterium nucleus, the shaded circle on the right the proton. The frozen electron distribution is indicated by an ellipse.

However, if the valence spinors are modified to create pseudospinors—and this applies to both model potentials and pseudopotentials— the expectation of the valence wave function over the property operator is no longer the same as in the all-electron or the frozen-core case. For external fields, where the bulk of the contribution comes from the valence region of space, the deviation of the pseudospinor property integrals from their unmodified counterparts will be small. However, for nuclear fields, the approximation to the core part of the spinors will have serious consequences for the property. Pseudospinors used with pseudopotentials are designed to have vanishing amplitude at the nucleus, and therefore the property integrals will be much smaller than they should be. This means that properties such as NMR shielding constants calculated for the pseudopotential center will inevitably be erroneous. 

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