First-order shell corrections

One can rewrite the first-order shell-correction term given by Eqn (7b) by using the self-consistent energiese, and the above decomposition ... 

This process is repeated until self-consistency is reached, the final values p and n, being noted p and n, Using these values the first-order shell correction to the energy can be written ... 

In Section 3 we have formulated Strutinsky s shell-correction method in the framework of the analytic HFR scheme, for single open-shell atoms and molecules in their ground state. The consideration of two or many open-shell systems could be performed following the same pattern. Both the averaged part of the energy, E,jp, and its first-order shell-correction part, 8,E pr, have been derived in analytic form, and the self-consistent process for determining them has been described. 

Abstract. Calculations of the first-order shell corrections of the ionization potential, 6il, electron affinity, 5 A, electronegativity, ix, and chemical hardness. Sir] are performed for elements from B to Ca, using the previously described Strutinsky averaging procedure in the frame of the extended Kohn-Sham scheme. A good agreement with the experimental results is obtained, and the discrepancies appearing are discussed in terms of the approximations made. 

In Refs [10, If] we have shown that Eqn (30) is an expression for the first-order shell correction term in the EKS-DFT frame. As we pointed it out, the extended version [26,27] of the Kohn-Sham scheme [46] is appropriate because it allows fractional occupation numbers, thus permitting the... 

In Fig. 1 there is presented the first-order shell correction of the total binding energy of atoms, cations, and anions from beryllium to calcium. As expected, the oscillating part of the energy displays minima for the atoms... 

Figure 1. The first-order shell correction SiE of the total binding energy of atoms (squares), cations (circles) and airions (triangles) from beryllium to calcium, as a function of the atomic number Z.

The corresponding first-order Breit correction to the energy of a closed-shell atom is... 

As the reactants and products are all closed-shell molecules, there are no first-order spin-orbit corrections - the only first-order relativistic corrections are the mass-velocity and Darwin corrections (15.8.5), which are similar in magnitude to the anharmonic corrections but of opposite sign. The non-Born-Oppenheimer corrections may be assumed to be small for the reactions considered here. 

P/h can be interpreted as an effective spin density of this open shell system. Similarly to the electron binding exjvession there is no first order contribution in the correlation potential, that is, = 0, so that 5 is correct through second order. However, the second order correction in the electron correction for... 

The scalar relativistic contribution is computed as the first-order Darwin and mass-velocity corrections from the ACPF/MTsmall wave function, including inner-shell correlation. 

Equations (7) can be viewed as a formal Taylor-series expansion, around the averaged part of the one-particle density matrix, of the HF energy functional E[p] [16, 18], this defining a shell-correction series . In Eqn (13) the first-order term of this expansion is expressed in terms of the single-particle energies e,. 

In previous papers [10,11] we have formulated a procedure for splitting the ground-state energy of a multifermionic system into an averaged, structure-less part, E, and a residual, shell-structure part, 8E. The latter originates from quantum interference effects of the one-particle motion in the confining potential [12] and has the form of a shell-correction expansion 5E = It was also shown [11] that the first-order corrective term,... 

The second approach to calculating MCD starts from its definition in terms of the real part of first-order correction to the frequency-dependent polarizability in the presence of a magnetic field (Section II.A.6). This definition can be used to consider all types of MCD linear in the magnetic field (9). Our current implementation is restricted to systems with a closed-shell ground state. We shall therefore only consider the calculation of A and terms by this method. 

We calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

State average orbitals are not optimized for a specific electronic state. Normally, this is not a problem and a subsequent CASPT2 calculation will correct for most of it because the first order wave function contains CFs that are singly excited with respect to the CASSCF reference function. However, if the MOs in the different excited states are very different it may be needed to extend the active space such that it can describe the differences. A typical example is the double shell effect that appears for the late first row transition metals as described above. 

The CASSCF wavefiinction is used as reference function in a second-order estimate of the remaining dynamical correlation effects. All valence electrons were correlated in this step and also the 3s and 3p shells on copper. Relativistic corrections (the Darwin and mass-velocity terms) were added to all CASPT2 energies. They were obtained at the CASSCF level using first-order perturbation theory. A level-shift (typically 0.3 Hartree) was added to the zeroth order Hamiltonian in order to remove intruder states [30]. Transition moments were conputed with the CAS state-interaction method [31] at the CASSCF level. They were... 

Another important interaction which contributes to broadening of resonance peaks is the quadurpolar interaction. When the nuclear spin, / > 1, the nuclei possess electric quadurpole moment, eQ. The quadurpole moment 0 is a measure of the distortion of spherical symmetry of the nuclear charge distribution. Quadurpole moment interacts with the electric field gradient, EFG, present at the nucleus. These are the fields caused by the electrons in the atom. Spherical distribution of electrons in closed shells do not produce any EFG at the nucleus but the bonding electrons do. Therefore the quadurpolar interaction can be used to obtain information on chemical bonding. Quadurpolar interaction shifts the Zeeman levels. The first order correction due to quadurpolar interaction to the Zeeman splitting between levels m and (w-1) is given by. 

A truncation of the expansion (3.5) defines the zero- and first-order regular approximation (ZORA, FORA) (van Lenthe et al. 1993). A particular noteworthy feature of ZORA is that even in the zeroth order there is an efficient relativistic correction for the region close to the nucleus, where the main relativistic effects come from. Excellent agreement of orbital energies and other valence-shell properties with the results from the Dirac equation is obtained in this zero-order approximation, in particular in the scaled ZORA variant (van Lenthe et al. 1994), which takes the renormalization to the transformed large component approximately into account, using... 

The fully self-consistent handling is compared with a perturbative evaluation of only the beyond-Breit terms and a perturbative treatment of the complete Ej. Even for the heaviest atoms the perturbative evaluation of the retardation corrections to the Breit term seems to be sufficient. On the other hand, use of first-order perturbation theory for the complete Ej leads to errors of the order of 1 eV for heavy atoms. An accurate description of inner shell transitions in these systems requires the inclusion of second-order Breit corrections. 

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. 

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