Hamiltonian parity violating

The traditional treatment of molecules relies upon a molecular Hamiltonian that is invariant under inversion of all particle coordinates through the center of mass. For such a molecular Hamiltonian, the energy levels possess a well-defined parity. Time-dependent states conserve their parity in time provided that the parity is well defined initially. Such states cannot be chiral. Nevertheless, chiral states can be defined as time-dependent states that change so slowly, owing to tunneling processes, that they are stationary on the time scale of normal chemical events. [22] The discovery of parity violation in weak nuclear interactions drastically changes this simple picture, [14, 23-28] For a recent review, see Bouchiat and Bouchiat. [29]... 

In the "nonrigid symmetric-top rotors" (such as NH ), the second-order Stark effect is observed under normal circumstances. Indeed, field strengths of the order of 1 600 000 [V/m] are required to bring the interaction into the first-order regime in this case [18]. In contrast, very weak interactions suffice to make the mixed-parity states and appropriate for the description of optically active systems. Parity-violating neutral currents have been proposed as the interaction missing from the molecular Hamiltonian [Eq.(1)] that is responsible for the existence of enantiomers [14,19]. At present, this hypothesis is still awaiting experimental verification. 

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),... 

Within the electroweak model, which unifies the electromagnetic and the weak interaction, the electroweak Hamiltonian Hew does not commute with the parity operator. Hew can be split into a parity conserving term Hpc behaving even under parity and a parity violating term Hpv that behaves... 

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. 

If we put all ingredients together, we arrive at the following effective Hamiltonian for the parity violating interaction between the n... 

We have finally arrived at those effective Hamiltonians, that have been employed in calculations of molecular parity violating effects either within a one-, two- or four-component scheme. In the following section I will outline the various strategies to include these Hamiltonians in perturbative computation of parity nonconservation effects in molecular systems. 

In this section I will outline the different methods that have been used and are currently used for the computation of parity violating effects in molecular systems. First one-component methods will be presented, then four-component schemes and finally two-component approaches. The term one-component shall imply herein that the orbitals employed for the zeroth-order description of the electronic wavefunction are either pure spin-up spin-orbitals or pure spin-down spin-orbitals and that the zeroth-order Hamiltonian does not cause couplings between the two different sets ( spin-free Hamiltonian). The two-component approaches use Pauli bispinors as basic objects for the description of the electronic wavefunction, while the four-component schemes employ Dirac four-spinors which contain an upper (or large) component and a lower (or small) component with each component being a Pauli bispinor. 

Since the one-component approaches employ the efi ective Hamiltonian (113) or various further approximations to it, the expectation value of this Hamiltonian for a system in a closed shell singlet state vanishes. This is due to the scalar product between either the spin and the momentum operator of the electron or the scalar product of the electron momentum with the nuclear spin. In the absence of a coupling mechanism between spin and coordinate space, the scalar product must therefore vanish. For the parity violating energy difference between enantiomers the main coupling contribution is expected to be due to spin-orbit coupling. The corresponding... 

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... 

First attempts to calculate molecular parity violating potentials within a two-component framework have been undertaken by Kikuchi and coworkers [168,169]. They have added the Breit-Pauli spin-orbit coupling operator Hso to the usual non-relativistic Hamiltonian Hq... 

R. Harris, L. Stodolski, The effect of the parity violating electron-nucleus interaction on the spin-spin coupling Hamiltonian of chiral molecules, J. Chem. Phys. 73 (1980) 3862-3863. 

Note that since under a space reflection R L the fact that ur ni can only arise from the parity-violating parts of the Hamiltonian. Our task is to relate nR,riL to the weak interaction Hamiltonian. 

Now look at octahedral complexes, or those with any other environment possessing a centre of symmetry e.g. square-planar). These present a further problem. The process of violating the parity rule is no longer available, for orbitals of different parity do not mix under a Hamiltonian for a centrosymmetric molecule. Here the nuclear arrangement requires the labelling of d functions as g and of p functions as m in centrosymmetric complexes, d orbitals do not mix with p orbitals. And yet d-d transitions are observed in octahedral chromophores. We must turn to another mechanism. Actually this mechanism is operative for all chromophores, whether centrosymmetric or not. As we shall see, however, it is less effective than that described above and so wasn t mentioned there. For centrosymmetric systems it s the only game in town. 

The conclusion of these works is that the parity (P) invariance and, separately, the charge conjugation (C) invariance are violated in P decay, while the time reversal (T) or combined CP invariance is not. The parity non-invariance (i.e., non-invariance of the Hamiltonian of the weak interaction under space reflection) can be expressed alternativelyby saying that the parity is not conserved. This formulation is a consequence of the fact that the parity P is an observable quantity. The presence of two-pion decay mode in the K° kaon decay implies, however, that even the CP invariance is violated in the weak interaction (Christenson et al. 1964). 

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