Energy frozen-core valence

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

The interaction energy and its many-body partition for Bejv and Lii r N = 2 to 4) were calculated in by the SCF method and by the M/ller-Plesset perturbation theory up to the fourth order (MP4), in the frozen core approximation. The calculations were carried out using the triply split valence basis set [6-311+G(3df)]. 

Example 4. Calculation of CBS-Q Energy for CH4 The geometry is first optimized at the HF/6-31G(d ) level and the HF/6-31G(d ) vibrational frequencies are calculated. The 6-31G(d ) basis set combines the sp functions of 6-31G with the polarization exponents of 6-311G(d,p). A scale factor of 0.91844 is applied to the vibrational frequencies that are used to calculate the zero-point energies and the thermal correction to 298 K. Next the MP2(FC)/6-31G(d ) optimization is performed and this geometry is used in all subsequent single-point energy calculations. In a frozen-core (FC) calculation, only valence electrons are correlated. 

The structural parameters and vibrational frequencies of three selected examples, namely, H2O, O2F2, and B2H6, are summarized in Tables 5.6.1 to 5.6.3, respectively. Experimental results are also included for easy comparison. In each table, the structural parameters are optimized at ten theoretical levels, ranging from the fairly routine HF/6-31G(d) to the relatively sophisticated QCISD(T)/6-31G(d). In passing, it is noted that, in the last six correlation methods employed, CISD(FC), CCSD(FC),..., QCISD(T)(FC), FC denotes the frozen core approximation. In this approximation, only the correlation energy associated with the valence electrons is calculated. In other words, excitations out of the inner shell (core) orbitals of the molecule are not considered. The basis of this approximation is that the most significant chemical changes occur in the valence orbitals and the core orbitals remain essentially intact. On... 

In electron correlation treatments, it is a common procedure to divide the orbital space into various subspaces orbitals with large binding energy (core), occupied orbitals with low-binding energy (valence), and unoccupied orbitals (virtual). One of the reasons for this subdivision is the possibility to freeze the core (i.e., to restrict excitations to the valence and virtual spaces). Consequently, all determinants in a configuration interaction (Cl) expansion share a set of frozen-core orbitals. For this approximation to be valid, one has to assume that excitation energies are not affected by correlation contributions of the inner shells. It is then sufficient to describe the interaction between core and valence electrons by some kind of mean-field expression. 

Table 8 Energies for various states and wave functions of CH4. These are valence only calculations with a C Is frozen core.

The geometry is at the MP2(full)/6-31G(d) level, and the zero-point energy is from the scaled (0.8929) HF/6-31G(d) harmonic oscillator frequencies. Other calculations are carried out on valence electrons only (the frozen core approximation). A typical script is... 

Relative Hartree-Fock (HF) energies (eV) of LS-states of Ce and Ce with respect to the 4f 5d 6s G ground state. Frozen-core errors (eV) in these relative energies are given for 4,12 and 30 valence electron systems. The core was taken from the neutral Ce atom in its ground state [92]. 

Besides the reduction of frozen-core errors when going from large-core to medium-core or small-core potentials also the valence correlation energies obtained in pseudopotential calculations become more accurate since the radial nodal structure is partially restored [97,98]. Clearly the accuracy of small-core potentials is traded against the low computational cost of the large-core po-... 

Table 3 lists the errors in excitation energies of Cs calculated with a relativistic ab initio one-valence electron PP and various forms of the cutoff-factor as well as with addition of a local potential. Clearly, for a given CPP the PP could be adjusted to reproduce (essentially exactly) the experimental energy levels, but then the PP without CPP does not model the frozen-core DHF AE case any more. 

The prefactor = —2ec is more or less arbitrary in atomic calculations, but one should note that only with a prefactor - oo is an AIMP calculation really equivalent to a frozen-core all-electron one, in the molecular case. One should also keep in mind that formally unoccupied core orbitals at finite energy in the (virtual) valence spectrum may lead to unphysical excitations, in ab initio Cl calculations, and should be removed beforehand. 

The lowest approximation to the removal energy is seen to be — e , where e is the eigenvalue of the frozen-core Hartree-Fock equation. It should be emphasized that the valence orbital is not treated self consistently. The orbitals of the closed-shell core are determined self-consistently, then the valence electron HF equation is solved in the frozen potential of the core. From Eq. (145) it follows that there is no first-order correction to the removal energy in the frozen-core HF potential. 

For consistency with RCI calculations, we also use DKS potentials for screened QED calculations. DKS potentials have been shown in Ref. [77] to give very good QED results for high-Z Li-like and Be-like ions and they appear to work just as well for Na-like to Si-like uranium [81]. Typically, QED corrections to transition energies are carried out in a frozen-core approximation where contributions from the valence electrons are considered but not those from the core electrons which cancel exactly between the initial and final states. In [77] and [81], however, it was found that core-relaxation effects are important and that they can be accounted for by summing the differences in QED energies of the core electrons as calculated with two different DKS potentials specific to the electronic configurations of the initial and final states of the transition. 

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