Atomic systems Hamiltonian parameters

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

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 presence of the collective parameters Ti2 and ft 2 introduces off-diagonal terms in the Hamiltonian II and in the dissipative part of the master equation. This suggests that in the presence of the interaction between the atoms the bare atomic states are no longer the eigenstates of the two-atom system. We can diagonalize the Hamiltonian (32) with respect to the dipole-dipole interaction and find collective states of the two-atom system. 

There are many experimental techniques for the determination of the Spin-Hamiltonian parameters g, Ux, J. D, E. Often applied are Electron Paramagnetic or Spin Resonance (EPR, ESR), Electron Nuclear Double Resonance (ENDOR) or Triple Resonance, Electron-Electron Double Resonance (ELDOR), Nuclear Magnetic Resonance (NMR), occasionally utilizing effects of Chemically Induced Dynamic Nuclear Polarization (CIDNP), Optical Detections of Magnetic Resonance (ODMR) or Microwave Optical Double Resonance (MODR), Laser Magnetic Resonance (LMR), Atomic Beam Spectroscopy, and Muon Spin Rotation (/iSR). The extraction of data from the spectra varies with the methods, the system studied and the physical state of the sample (gas, liquid, unordered or ordered solid). For these procedures the reader is referred to the monographs (D). Further, effective magnetic moments of free radicals are often obtained from static... 

The spin Hamiltonian parameters of a complex or a model system can in principle be determined from a quantum chemical calculation of the electronic structure. The Density Functional Theory (DFT) [13] method is at present popular for this purpose, since it allows fairly large systems ( 200 atoms) to be investigated. By comparing the calculated and experimental spin Hamiltonian parameters it is often possible to distinguish between different proposed models and to gain further insight into the electronic and geometric structure of the sample. 

Inner electrons in atoms and ions of high nuclear charge, Z, move with velocities comparable to the velocity of light Consequently, many-body calculations of structures and transitions for high Z atomic systems must account for the effects of relativity. Treating relativistic effects perturbatively is not satisfactory f or heavy systems since the relevant expansion parameter, aZ (a=e Mirhc -- 1/137), may be close to one. For such system approaches based on the approximate Dirac-Coulomb hamiltonian... 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The application of quantum-mechanical methods to the prediction of electronic structure has yielded much detailed information about atomic and molecular properties.13 Particularly in the past few years, the availability of high-speed computers with large storage capacities has made it possible to examine both atomic and molecular systems using an ab initio variational approach wherein no empirical parameters are employed.14 Variational calculations for molecules employ a Hamiltonian based on the nonrelativistic electrostatic nuclei-electron interaction and a wave function formed by antisymmetrizing a suitable many-electron function of spatial and spin coordinates. For most applications it is also necessary that the wave function represent a particular spin eigenstate and that it have appropriate geometric symmetry. 

---