Heavy-atom molecules Hamiltonians

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. 

The most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

Generalized RECP When core electrons of a heavy-atom molecule do not play an active role, the effective Hamiltonian with RECP can be presented in the form... 

This approach was employed by Peralta et al. [649,661] and explored in detail by Thar and Kirchner [668] for the low-order DKH method. We refer to it as the diagonal local approximation to the Hamiltonian (DLH). It is clear that the DLH approximation can be applied to all relativistic exact-decoupling approaches. Obviously, the DLH approximation will work best at large interatomic distances. For example, it is a good approximation for heavy-atom molecules in solution for which it was conceived in Ref. [668]. Then, A represents the group of atoms that form one of the solute and/or solvent molecules. 

The following conclusions apply to organic molecules of about 25 heavy atoms (- 60 atoms total), assuming use of the MNDO or AMI Hamiltonian ... 

From a formal point of view, four-component correlation calculations [5, 6] based on the Dirac-Coulomb-Breit (DCB) Hamiltonian (see [7, 8, 9, 10, 11] and references therein) can provide with very high accuracy the physical and chemical properties of molecules containing heavy atoms. However, such calculations were not widely used for such systems during last decade because of the following theoretical and technical complications [12] ... 

HgH.—Das and Wahl have carried out a calculation on the HgH molecule which has many points of interest for the practical implementation of pseudopotentials on heavy-atomic molecular systems. As the nuclear charge increases so does the importance of the relativistic terms in the hamiltonian, and their influence is not only confined to the core orbitals (e.g. the Hg Is) where the kinetic energy of the electron is comparable with its rest mass, but even afiects the valence (Hg 6s) orbitals (Grant °) and the binding energy of Hgj (Grant and Pyper ),... 

Most applications to molecules that do not contain heavy atoms (beyond Br in the periodic table) have been based on the Breit-Pauli Hamiltonian which consists of six terms and operates on ordinary nonrelativistic wave functions. In the Breit-Pauli approximation, the total Hamiltonian is defined as... 

For heavy atoms and molecules, many-electron theory can be made to start with relativistic equations. Though the exact relativistic Hamiltonian is not known it seems a good approximation to base the theory on the relativistic Hartree-Fock Hamiltonian corrected by the non-relativistic 1/r,-, terms. 

For polyatomic molecules the operator 5 for the square of the total electronic spin angular momentum commutes with the electronic Hamiltonian, and, as for diatomic molecules, the electronic terms of polyatomic molecules are classified as singlets, doublets, triplets, and so on, according to the value of 25 + 1. (The commutation of 5 and H holds provided spin-orbit interaction is omitted from the Hamiltonian for molecules containing heavy atoms, spin-orbit interaction is considerable, and 5 is not a good quantum number.)... 

The treatment of molecules containing heavy atoms poses special problems for Cl methods because of the need to consider relativistic effects in the electronic Hamiltonian. If attention is restricted to low-lying states, it is quite useful to employ an accurate representation of the inner-shell... 

We will discuss the case where the motion of heavy atoms is confined to two dimensions, while the motion of light atoms can be either two- or three-dimensional. It will be shown that the Hamiltonian 10.76 with Ues in 10.44 supports the first-order quantum gas-crystal transition at T = 0 [68], This phase transition resembles the one for the flux lattice melting in superconductors, where the flux lines are mapped onto a system of bosons interacting via a two-dimensional Yukawa potential [73]. In this case Monte Carlo studies [74,75] identified the first-order liquid-crystal transition at zero and finite temperatures. Aside from the difference in the interaction potentials, a distinguished feature of our system is related to its stability. The molecules can undergo collisional relaxation into deeply bound states, or form weakly bound trimers. Another subtle question is how dilute the system should be to enable the use of the binary approximation for the molecule-molecule interaction, leading to Equations 10.76 and 10.44. 

Relativity adds a new dimension to quantum chemistry, which is the choice of the Hamiltonian operator. While the Hamiltonian of a molecule is exactly known in nonrelativistic quantum mechanics (if one focuses on the dominating electrostatic monopole interactions to be considered as being transmitted instantaneously), this is no longer the case for the relativistic formulation. Numerical results obtained by many researchers over the past decades have shown how Hamiltonians which capture most of the (numerical) effect of relativity on physical observables can be derived. Relativistic quantum chemistry therefore comes in various flavors, which are more or less well rooted in fundamental physical theory and whose relation to one another will be described in detail in this book. The new dimension of relativistic Hamiltonians makes the presentation of the relativistic many-electron theory very complicated, and the degree of complexity is far greater than for nonrelativistic quantum chemistry. However, the relativistic theory provides the consistent approach toward the description of nature molecular structures containing heavy atoms can only be treated correctly within a relativistic framework. Prominent examples known to everyone are the color of gold and the liquid state of mercury at room temperature. Moreover, it must be understood that relativistic quantum chemistry provides universal theoretical means that are applicable to any element from the periodic table or to any molecule — not only to heavy-element compounds. 

In recent years, there has been an increasing interest in the inclusion of relativistic effects for molecules containing heavy atoms. One of the most practical yet reliable methods is to use relativistically derived effective core potentials. Major relativistic effects such as the Darwin and mass-velocity effects are easily taken into account in the form of a spin-free (SF) one-electron operator. The spin-orbit (SO) interaction is in general too strong to be considered as a small perturbation, and therefore should be treated explicitly as a part of the total Hamiltonian. 

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