Core polarization model

The 10% difference is probably due to core penetration in the 4snf states, which is not taken into account, and, according to the theoretical work of Vaidyanathan and Shorer12 should be of the same approximate size as the discrepancy. In any case, it is clear that the nonadiabatic core polarization model reproduces the observed intervals quite well, while the adiabatic model is substantially in error. 

While the agreement of the measured and calculated Ba+ quadrupole polarizabilities is not very good, compared to an analysis based on adiabatic core polarization the agreement in Table 17.3 is superb. The adiabatic core polarization model leads to ad = 146flo and aq = —5800. The ground state of Ba+ cannot have a negative quadrupole polarizability. Taken together, the Ca and Ba experiments show clearly that the nonadiabatic effects in core polarization in the alkaline earth atoms are important and may be calculated with some accuracy. 

Since the accuracy of the asymptotic expansion rapidly gets even better with increasing L, there is clearly no need to perform numerical solutions to the Schrodinger equation for L > 7. The entire singly excited spectrum of helium is covered by a combination of high precision variational solutions for small n and L, quantum defect extrapolations for high n, and asymptotic expansions based on the core polarization model for high L. The complete asymptotic expansion for helium up to (r-10) is [36,29]... 

In alkaline earth atoms, on the other hand, the static core polarization model clearly does not reproduce the energies of the atomic levels, because of the magnitude of the nonadiabatic effects/ Several theories have been formulated for the nonadiabatic effects. For example, the validity of the approach of Eissa and Opik has recently been verified by Vaidyanathan and Shorer by comparing calculated and measured quantum defects ofCa. 

Since the quantum defects of low angular momentum states are well known from optical spectroscopy and the high / quantum defects can be easily derived from the core polarization model, it is useful to present all the results together. In Figure 11 we show a plot of the alkali quantum defects vs. / on a logarithmic scale modified to include the s and p states. 

It was recently suggested by Nicklass and Peterson [60] that the use of core polarization potentials (CPPs) [61] could be an inexpensive and effective way to account, for the effects of inner shell correlation. The great potential advantage of this indeed rather inexpensive method over the MSFT bond-equivalent model is that it does not depend on... 

Abstract. We present a quantum-classieal determination of stable isomers of Na Arii clusters with an electronically excited sodium atom in 3p P states. The excited states of Na perturbed by the argon atoms are obtained as the eigenfunctions of a single-electron operator describing the electron in the field of a Na Arn core, the Na and Ar atoms being substituted by pseudo-potentials. These pseudo-potentials include core-polarization operators to account for polarization and correlation of the inert part with the excited electron . The geometry optimization of the excited states is carried out via the basin-hopping method of Wales et al. The present study confirms the trend for small Na Arn clusters in 3p states to form planar structures, as proposed earlier by Tutein and Mayne within the framework of a first order perturbation theory on a "Diatomics in Molecules" type model. 

Up to now,we always assumed a U coupled pair distribution to give a good description of both the proton 2p-2h excitation and of the neutron distribution over the valence model space.The quadrupole component of the proton-neutron force will induce 0+- 2+ pair breaking for both protons and neutrons which we call the core polarization effect.Thus,0 ground state and intruder wave functions... 

Shell model calculations predict a quasi-shell closure at 96Zr. Therefore, it is of interest to measure g-factors of states in 97Zr and test whether they can be described by simple shell model configurations. The 1264.4 keV level has a half-life of 102 nsec, and its g-factor was measured by the time-differential PAC method at TRISTAN [BER85a]. The result, g-0.39(4), is consistent with the Schmidt value of 0.43, which assumes no core polarization and the free value for the neutron g factor, g g free. This indicates that the 1264.4 keV level is a very pure single-particle state, thus confirming the shell model prediction of a quasi-shell closure at 96Zr. 

At present, the low-lying states of Na2 are better characterized computationally than experimentally, although multiphoton ionization experiments may change that picture eventually. We find reasonably close agreement between the results of our all-electron computations, pseudopotential (10), and model potential (11) computations. The latter two kinds of computations may give more accurate results than our ab initio computations since they may account for at least certain core polarization effects. 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et... 

The parameter Ka — 0.24 — 0.33 is determined from nuclear model calculations [49]. These two interactions can be treated together using (104) with Ka K = Ka — 7C2(k — 1/2)1 K. The resulting spin-dependent correction was evaluated in the Dirac-Fock approximation including weak core-polarization corrections. Combining that calculation with the previous spin-independent result, we obtain... 

Bond lengths R (A), binding energies D. (eV) and vibrational constants a>e (cm ) of the homonuclear halogen dimers from dl-electron (AE) Douglas-Kroll-HeB (DKH) and valence-only energy-consistent pseudopotential (EC-PP) Hartree-Fock self-consistent field (SCF) calculations. The effects of static and dynamic core-polarization at the valence-only level are modelled by a core-polarization potential (CPP). 

---