Electron correlation core-valence

Figure 8.7 Convergence of the CCSD(T) correlation energy for outer-core electrons using core-valence correlation consistent basis sets. Results are given relative to estimated CBS limits...

Whereas the core—core correlation energy arises from the internal correlation of the electrons in the core region, the core-valence correlation energy arises from the correlation between valence and core electrons. The core-valence correlation energy therefore represents the part of the core correlation energy that is most sensitive to the geometry of the molecule and the chief differential effects of core correlation can thus usually be recovered by including only the core—valence correlation in the calculation (in addition to the valence correlation). 

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

The MP2 and CCSD(T) values in Tables 11.2 and 11.3 are for correlation of the valence electrons only, i.e. the frozen core approximation. In order to asses the effect of core-electron correlation, the basis set needs to be augmented with tight polarization functions. The corresponding MP2 results are shown in Table 11.4, where the A values refer to the change relative to the valence only MP2 with the same basis set. Essentially identical changes are found at the CCSD(T) level. 

The magnitude of the core correlation can be evaluated by including the oxygen Is-electrons and using the cc-pCVXZ basis sets the results are shown in Table 11.9. The extrapolated CCSD(T) correlation energy is —0.370 a.u. Assuming that the CCSD(T) method provides 99.7% of the full Cl value, as indicated by Table 11.7, the extrapolated correlation energy becomes —0.371 a.u., well within the error limits on the estimated experimental value. The core (and core-valence) electron correlation is thus 0.063 a.u.. 

The effect of core-electron correlation is small, as shown in Table 11.16. It should be noted that the valence and core correlation energy per electron pair is of the same magnitude, however, the core correlation is almost constant over the whole energy surface and consequently contributes very little to properties depending on relative energies, like vibrational frequencies. It should be noted that relativistic corrections for the frequencies are expected to be of the order of 1 cm" or less. ... 

The active space used for both systems in these calculations is sufficiently large to incorporate important core-core, core-valence, and valence-valence electron correlation, and hence should be capable of providing a reliable estimate of Wj- In addition to the P,T-odd interaction constant Wd, we also compute ground to excited state transition energies, the ionization potential, dipole moment (pe), ground state equilibrium bond length and vibrational frequency (ov) for the YbF and pe for the BaF molecule. 

The quality of quantum-chemical calculations depends not only on the chosen n-electron model but also critically on the flexibility of the one-electron basis set in terms of which the MOs are expanded. Obviously, it is possible to choose basis sets in many different ways. For highly accurate, systematic studies of molecular systems, it becomes important to have a well-defined procedure for generating a sequence of basis sets of increasing flexibility. A popular hierarchy of basis functions are the correlation-consistent basis sets of Dunning and coworkers [15-17], We shall use two varieties of these sets the cc-pVXZ (correlation-consistent polarized-valence X-tuple-zeta) and cc-pCVXZ (correlation-consistent polarized core-valence X-tuple-zeta) basis sets see Table 1.1. 

As can be seen from the table, the number of AOs increases rapidly with the cardinal number X. Thus, with each increment in the cardinal number, a new shell of valence AOs is added to the cc-pVXZ set since the number of AOs added in each step is proportional to X2, the total number (Nbas) of AOs in a correlation-consistent basis set is proportional to X3. The core-valence sets cc-pCVXZ contain additional AOs for the correlation of the core electrons. As we shall see later, the hierarchy of correlation-consistent basis sets provides a very systematic description of molecular electronic systems, enabling us to develop a useful extrapolation technique for molecular energies. 

Optimization of augmenting functions for the description of electron affinities, weak interactions, or core-valence correlation effects. 

Obtained via a 2-point /n extrapolation of the VQZ-PP and V5Z-PP total energies cc-pwCVTZ-PP (Y) and cc-pCVTZ (C) core-valence basis sets with valence-only (7) and all-electrons (17) correlated. 

Relativistic Quantum Defect Orbital (RQDO) calculations, with and without explicit account for core-valence correlation, have been performed on several electronic transitions in halogen atoms, for which transition probability data are particularly scarce. For the atomic species iodine, we supply the only available oscillator strengths at the moment. In our calculations of /-values we have followed either the LS or I coupling schemes. 

As was understood in calculations of YbF [92], a fortuitous cancellation of effects of different types took place in the above calculations. Accounting for the polarization of the 5s, 5p-shells leads to an enhancement of the contributions of the valence shells to the Ay, A and PNC constants on the level of 50% of magnitude. A contribution of comparable magnitude but of opposite sign takes place when the relaxation-correlation effects of the 5d-shell are taken into account. This was confirmed in [120] when accounting for electron correlation both in the valence and core regions of HgF as compared to the YbF case. 

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