PHYSICAL REVIE%' A VOLUME 34, NUMBER 2 Weak-interaction effects in heavy atomic systems. II AUGUST 1986 %. R. Johnson, D. S. Guo„M. Idrees, and J. Sapirstein University of Notre Dame, Noire Dame, Indiana 46556 (Received 14 January 1986) Evaluations of parity-violating and charge-parity — violating effects in heavy one-valence-electron atoms, employing the Hartree-Fock potential and several model potentials, are extended to include first-order electron-electron Coulomb corrections using many-body perturbation theory. Parity- conserving quantities, including valence energies, hyperfine splittings, and oscillator strengths, are also calculated and compared with experiment to determine the reliability of the weak-interaction calculations. It is found that the spread between calculations carried out in first-order perturbation theory starting from different potentials is of the same order of magnitude as the spread between the corresponding lowest-order evaluations. It is concluded that second-order many-body perturbation theory must give significant contributions. Some technical problems associated with going to second order are discussed. I. INTRODUCTION In a previous paper, ' referred to in the following as I, we evaluated weak-interaction effects along with various standard atomic properties in the heavy one-valence- electron atoms Rb, Cs, Au, and Tl. The weak-interaction effects were, first, the induction of a nonvanishing electric dipole matrix element between states of the same nominal parity due to exchange of a Zo boson and, second, the in- duction of an enhanced atomic electric dipole moment (EDM) by a charge-parity (CP) — violating electron EDM. We employed several different potentials in I to estimate the reliability of the calculations. Our principal result was that different potentials gave predictions for parity- violating and CP-violating dipole matrix elements having a spread of up to 20%, which we used as a measure of the accuracy of our predictions. While this accuracy is suffi- cient to establish qualitatively the existence of neutral weak-current effects, greater accuracy is clearly desirable. In particular, recent measurements of parity violation in Cs have reached the 8% level, and if atomic theory can achieve the same accuracy, information about the Wein- berg angle and one-loop radiative corrections to weak in- teractions in a low-energy regime, complementary to high-energy probes of the weak interactions can be ob- tained. Furthermore, if an atomic EDM were to be discovereda20, % theoretical calculation of the enhance- ment factor would permit a determination of the electron EDM only at the 20% level, so improvements in theoreti- cal EDM predictions are also desirable. While our pro- gram is directed at weak-interaction effects, we note that any techniques developed to predict these effects to within a few percent should also be applicable to accurate studies of other atomic properties such as hyperfine splittings, valence energies, and oscillator strengths, which are of considerable interest in their own right. %'hile other approaches could be applied to study heavy atomic systems, for example, multiconfiguration Hartree- Fock methods, we have chosen to employ many-body perturbation theory. Our hope is that, if we make a physically sensible lowest-order approximation, two or three orders of perturbation theory will suffice to achieve a few percent accuracy. In this paper we present the re- sults of calculations carried out to first order in the e)ectron-electron interaction. As described in I, our plan is to carry out calculations starting from several different potentials and to use the spread in values between the cal- culations in a given order of perturbation theory to mea- sure the reliability of the corresponding calculations. Presumably, this spread will vanish as higher-and-higher- order perturbations are included. For this purpose we em- ploy three model potentials (described in the Appendix) together with the Hartree-Fock potential. The principal result of our first-order calculations is that there is a great deal of sensitivity to core polarization, so that the spread in values between quantities calculated in first order start- ing from different potentials ranges up to 20%, compar- able to the spread found in lowest order. While somewhat better results were obtained for excited-state properties, due to the diminished effect of core polarization, it is clear that predictions of ground-state atomic properties at a level well under 10% will require the use of second- order perturbation theory and perhaps some form of in- finite summation. During the past decade, various many-body calculations of parity violation in heavy atoms have appeared, several of which go beyond the present calculations and include second-order correlation corrections. Closest to the present approach is the calculation of Martensson- Pendrill, which is a complete first-order calculation of the parity violation in Cs starting from a Hartree-Fock potential. The first-order parity-violating matrix element in Cs based on the Hartree-Pock potential in the present paper agrees very well with the result of Ref. 7. Indeed, such agreement is expected since the principal difference between the present calculation and that of Martensson- Pendrill concerns the way in which perturbation theory is implemented. %e also mention the elegant work of Dzu- ba et aI. on Cs which also starts with a Hartree-Fock po- tential and includes both first- and second-order correla- 34 1043 Qc1986 The American Physical Society