Atomic Hamiltonian parity

Parity of Atomic States. Consider the atomic Hamiltonian (11.1). We showed in Section 7.5 that the parity operator O commutes with the kinetic-energy operator. The quantity 1/r/ in (11.1) is = (jc + Replacement of each coordinate by its... 

The presence of a neutral weak current interaction between the electrons and nucleons in an atom gives rise to a parity non-conserving part of the atomic Hamiltonian which can be written as... 

The Hamiltonian considered above, which connmites with E, involves the electromagnetic forces between the nuclei and electrons. However, there is another force between particles, the weak interaction force, that is not invariant to inversion. The weak charged current mteraction force is responsible for the beta decay of nuclei, and the related weak neutral current interaction force has an effect in atomic and molecular systems. If we include this force between the nuclei and electrons in the molecular Hamiltonian (as we should because of electroweak unification) then the Hamiltonian will not conuuiite with , and states of opposite parity will be mixed. However, the effect of the weak neutral current interaction force is mcredibly small (and it is a very short range force), although its effect has been detected in extremely precise experiments on atoms (see, for... 

In conventional quantum mechanics, a wavefunction d ribing the ground or excited states of a many-particle system must be a simultaneous eigenfunction of the set of operators that commute with the Hamiltonian. Thus, for example, for an adequate description of an atom, one must introduce the angular momentum and spin operators L, S, L, and the parity operator H, in addition to the Hamiltonian operator. 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

P and S. Note that states with different values of /. or. S have different energies, partly because of the electron-electron interaction term in the electronic Hamiltonian. A further symmetry classification that should be mentioned is the parity of an atomic state which depends on the behaviour of the total wave function under space-fixed inversion. This is either even (g) or odd (u) and is determined by, summed over all the electrons in the atom. 

Here, k = 4, 7 = 7/2 and K2 = —0.05 for the valence proton of Cs. Additionally, parity violation in the nucleus leads to to a parity-violating nuclear moment, the anapole moment mentioned above, that couples elec-tromagnetically to the atomic electrons. The anapole-electron interaction is described by a Hamiltonian similar to (103),... 

Let us consider how independent /i(i ) 2 effects contribute to the v E) for the hydrogen halides, HX (X = I, Br, and Cl). The curves shown on Fig. 7.6 correspond to relativistic adiabatic potential energy curves (respectively 0 dotted, 0+ dashed, 1 and 2 solid) for HI obtained after diagonalization of the electronic plus spin-orbit Hamiltonians (see Section 3.1.2.2). The strong R-dependence of the electronic transition moment reflects the independence of the relative contributions of the case(a) A-S-Q basis states to each relativistic adiabatic II state. The independent experimental photodissociation cross sections are plotted as solid curves in Fig. 7.7 for HI and HBr. Note that, in addition to the independent variations in the A — S characters of each fl-state caused by All = 0 spin-orbit interactions, all transitions from the X1E+ state to states that dissociate to the X(2P) + H(2S) limit are forbidden in the separated atom limit because they are at best (2Pi/2 <— 2P3/2) parity forbidden electric dipole transitions on the X atom. In the case of the continuum region of an attractive potential, the energy dependence of the dissociation cross section exhibits continuity in the Franck-Condon factor density (see Fig. 7.18 Allison and Dalgarno, 1971 Smith, 1971 Allison and Stwalley, 1973). 

The potentials (94) and (95) are already quite similar to the leading effective Hamiltonians that have been used so far in one- and four-component calculations of molecular parity violating eflFects. We have assumed above that the fermions are elementary particles. The effective potentials may, however, also be applied for the description of low energy weak neutral scattering events, in which heavy non-elementary fermions like the proton and the neutron or even entire atomic nuclei are involved, provided that properly adjusted vector and axial coupling coefficients py and for non-elementary fermions are used. 

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

The parity nonconservation (PNC) terms of the molecular Hamiltonian originating in electroweak electron-nucleon interactions and their effect on NMR parameters have been discussed. Owing to the short-range nature of such PNC interactions, they occur only when the electron is inside the atomic nucleus. Terms linear and bilinear in the nuclear spins were obtained and their contributions to the J tensor were calculated employing RSPT. However, no numerical estimates of their importance were reported. 

D = I, for which / = 0 and / = 1 correspond to eigenstates of even and odd parity. A key theorem for S states of any N-body system is demonstrated for the N = Z case the D-dimensional Hamiltonian can be cast in the same form as D = Z, with the addition of a scalar centrifugal potential that contains the sole dependence on D as a quadratic polynomial. For two-electron atoms, the D —y oo limit and the first-order correction in 1/D are discussed for both the complete Hamiltonian and the Hartree-Fock approximation. 

To evaluate the influence of an electric field on hydrogen, therefore, we must consider at least the nearest state of opposite parity, which is the 2p state. These two states are separated in energy by an amount E s2p- Considering only these two states, and ignoring any spin structure, the atom-plus-field Hamiltonian is represented by a simple 2x2 matrix ... 

For 3 + lanthanide and actinide ions, almost all transitions within the f shell are electric dipole in nature. These transitions are formally parity (Laporte) forbidden. That such transitions are observable is attributed to non-centro-symmetric terms in the crystal-field Hamiltonian. Such terms have the effect of mixing higher-lying, opposite-pairty d and g states into the f shell. As Judd (1988) noted in a review of atomic theory and optical spectroscopy of rare earths No doubt that we shall eventually be able to calculate much of what we want with a high degree of accuracy. That day has not yet arrived. . 

As a simplest example, we shall again invoke the linear three atomic molecule XMX. Suppose the two states denoted by s> and > are nearly degenerate. If these states have different parity, say s> being symmetric and > antisymmetric, then they will couple via the perturbation term i odd <2odd- In the static approximation, we regard the normal coordinate Oodd as a parameter. Then we get a secular matrix of the Hamiltonian in the form... 

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