An exactly separable version of the Bohr Hamiltonian is developed using a potential of the form u(β) + u(γ)/β 2 , with the Davidson potential u(β) = β 2 + β 4 0 /β 2 (where β 0 is the position of the minimum) and a stiff harmonic oscillator for u(γ) centered at γ = 0 • . In the resulting solution, called exactly separable Davidson (ES-D), the ground state band, γ band and 0 + 2 band are all treated on an equal footing. The bandheads, energy spacings within bands, and a number of interband and intraband B(E2) transition rates are well reproduced for almost all well-deformed rare earth and actinide nuclei using two parameters (β 0 , γ stiffness). Insights regarding the recently found correlation between γ stiffness and the γ-bandhead energy, as well as the long standing problem of producing a level scheme with Interacting Boson Approximation SU(3) degeneracies from the Bohr Hamiltonian, are also obtained.

A collective Hamiltonian for the rotation-vibration motion of nuclei is considered, in which the axial quadrupole and octupole degrees of freedom are coupled through the centrifugal interaction. The potential of the system depends on the two deformation variables β 2 and β 3 . The system is considered to oscillate between positive and negative β 3 -values, by rounding an infinite potential core in the (β 2 , β 3 )-plane with β 2 > 0. By assuming a coherent contribution of the quadrupole and octupole oscillation modes in the collective motion, the energy spectrum is derived in an explicit analytic form, providing specific parity shift effects. On this basis several possible ways in the evolution of quadrupole-octupole collectivity are outlined. A particular application of the model to the energy levels and electric transition probabilities in alternating parity spectra of the nuclei 150 Nd, 152 Sm, 154 Gd and 156 Dy is presented.

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An analytic collective model in which the relative presence of the quadrupole and octupole deformations is determined by a parameter (φ 0 ), while axial symmetry is obeyed, is developed. The model [to be called the analytic quadrupole octupole axially symmetric model (AQOA)] involves an infinite well potential, provides predictions for energy and B(EL) ratios, which depend only on φ 0 , draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well 226 Th and 226 Ra, for which experimental energy data are shown to suggest that they lie close to this border. The similarity of the AQOA results with φ 0 = 45 • for ground-state band spectra and B(E2) transition rates to the predictions of the X(5) model is pointed out. Analytic solutions are also obtained for Davidson potentials of the form β 2 + β 4 0 /β 2 , leading to the AQOA spectrum through a variational procedure.

We propose a collective model formalism which describes the strong parity shift observed in low-lying spectra of nuclei with octupole deformations together with the fine rotational band structure developed at higher angular momenta. The parity effect is obtained by the Schrödinger equation for oscillations of the reflection asymmetric (octupole) shape between two opposite orientations in an angular momentum dependent double-well potential. The rotational structure is obtained by a collective quadrupole-octupole rotation Hamiltonian. The model scheme reproduces the complicated beat staggering patterns observed in the octupole bands of light actinide nuclei. It explains the angular momentum evolution of octupole spectra as the interplay between the octupole shape oscillation (parity shift) mode and the stable quadrupole-octupole rotation mode.

Strutinsky's standard averaging method is formulated in the framework of the extended Kohn-Sham scheme and a two-step procedure permitting the application of the method is proposed. A Taylor-series expansion of the groundstate energy-function of the occupation numbers is derived, which involves the averaged energy as the leading term and shell corrections as smaller terms. Numerical applications for atoms and ions from Be through Ar are presented and discussed.

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An analytic collective model in which the relative presence of the quadrupole and octupole deformations is determined by a parameter (φ 0 ), while axial symmetry is obeyed, is developed. The model [to be called the analytic quadrupole octupole axially symmetric model (AQOA)] involves an infinite well potential, provides predictions for energy and B(EL) ratios which depend only on φ 0 , draws the border between the regions of octupole deformation and octupole vibrations in an essentially parameter-independent way, and describes well 226 Th and 226 Ra, for which experimental energy data are shown to suggest that they lie close to this border. The similarity of the AQOA results with φ 0 = 45 o for ground state band spectra and B(E2) transition rates to the predictions of the X(5) model is pointed out. Analytic solutions are also obtained for Davidson potentials of the form β 2 + β 4 0 /β 2 , leading to the AQOA spectrum through a variational procedure.