Mixed-parity MBPT

In mixed-parity MBPT, we modify Hq by adding the weak-interaction h v to the HF potential. This approach leads to a generalization of the single-particle states in which each state acquires an opposite-parity admixture, 

Valence removal energies for cesium from an all-order calculation (see Ref [44]). Units a.u. 

Hyperfine constants (MHz) for Cs with 7 =, p/ = 0 7377208 and reduced dipole matrix elements (a.u.) for cesium from an all-order calculation (see Ref. [44]). 

for example, each Si/2 orbital will pick up a small pi/2 state admixture. Treating the weak interaction to lowest order (which is certainly justified), the induced correction satisfies the equation 

The approximation made in Eq. (84) ignores the dependence of Vhf on the core orbitals, which themselves acquire small opposite-parity admixtures. If we take this dependence into account, then the HF potential is also modified 

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We now proceed to carry out a consistent implementation of MBPT based on parity-mixed single-particle states. The above result (88) is the lowest-order result in such a perturbation theory, corresponding to the parity-mixed HF . To evaluate the second-order corrections, we linearize the second-order amplitude from Eq. (69) in the weak interaction. As discussed in the previous section, we use RPA amplitudes, rather than lowest-order amplitudes on the RHS of Eq. (69). Evaluating the resulting expression, one obtains the correction... 

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