Atomic systems system Hamiltonians

The evaluation of the action of the Hamiltonian matrix on a vector is the central computational bottleneck. (The action of the absorption matrix, A, is generally a simple diagonal damping operation near the relevant grid edges.) Section IIIA discusses a useful representation for four-atom systems. Section IIIB outlines one aspect of how the action of the kinetic energy operator is evaluated that may prove of general interest and also is of relevance for problems that require parallelization. Section IIIC discusses initial conditions and hnal state analysis and Section HID outlines some relevant equations for the construction of cross sections and rate constants for four-atom problems of the type AB + CD ABC + D. 

The simplest atomic system that we can consider is the hydrogen atom. To obtain the Hamiltonian operator for this three-dimensional system, we must replace the operator d2/dx2 by the partial differential operator... 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

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 ... 

Exciting new developments, not discussed in the review are the extension of time-dependent wavepacket reactive scattering theory to full dimensional four-atom systems [137,199-201], the adaptation of the codes to use the power of parallel computers [202], and the development of new computational techniques for acting with the Hamiltonian operator on the wavepacket [138]. 

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. 

Consider an atomic system with a singlet ground state a and doublet excited states b, the members of which are denoted by by and b2. The Hamiltonian for the states (ils b2) is assumed to be... 

Beginning therefore with an element (atomic number Za), and with one-and two-particle densities p (r) and pf (r,r )) as appropriate observables for macroscopic systems, the Hamiltonian for a neutral ensemble of nuclei (a = n) and electrons (a = e) established in a volume V is, in three-dimensions,... 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

The non relativistic Hamiltonian of an atomic system under a plasma environment is given by... 

We shall show how the Bom-Oppenheimer potential energy for the Hj ion can be calculated exactly using series expansion methods, even though an exact analytical solution cannot be obtained. Figure 6.29 shows the coordinate system used for an electron moving in the field of two clamped nuclei. In atomic units the Hamiltonian is... 

The formalism for treating light atom systems begins with the Breit equation. The atomic spin-orbit Hamiltonian is given by (5)... 

The chemical master equation (CME) for a given system invokes the same rate constants as the associated deterministic kinetic model. Yet the CME is more fundamental than the deterministic kinetic view. Just as Schrodinger s equation is the fundamental equation for modeling motions of atomic and subatomic particle systems, the CME is the fundamental equation for reaction systems. Remember that Schrodinger s equation is not a model for a specific mechanical system. Rather, it is a theoretical framework upon which models for particular systems can be developed. In order to write down a model for an atomic system based on Schrodinger s equation, one needs to know how to write down the Hamiltonian a priori. Similarly, the CME is not a model for a specific biochemical reaction system it is a theoretical framework. To determine the CME model for a reaction system, one must know what are the possible elementary reactions and the associated rate constants. 

The geometrical and electronic structure for molecular systems in general will depend on the balance between the different terms in the Hamiltonian i.e. electron-nucleus, electron-electron and nucleus-nucleus interaction including the valence as well as the core electrons of the constituent atoms. The full Hamiltonian for the molecular system is normally separated into a Hamiltonian Hn for the nuclei and another one Hgi for the electrons with fixed positions for the nuclei according to Born Oppenheimer approximation [31]. 

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 ),... 

The variational-principle energy for a system with Hamiltonian (12.14) approximated by a wavefunction (12.17) works out to a sum analogous to Eq (9.4) for atoms ... 

The second class of atomic systems studied in the search for manifestations of chaos consists of time-dependent Hamiltonian systems such as one-electron atoms in an oscillating field. The hydrogen atom in a microwave or laser field is the standard physical example and has been a focus of attention since the ionization of highly excited hydrogen atoms by intense microwave fields was first observed by Bayfield and Koch in 1974 [10]. 

A rigorous derivation of the form of the hybrid potentials treated in this chapter requires the construction an effective Hamiltonian, Her, for the system. This Hamiltonian can then be used as the Hamiltonian for the solution of the time-independent Schrodinger equation for the wavefunc-tion of the electrons on the QM atoms, P, and for the potential energy of the system, qm/mm- If R are the coordinates of the MM atoms, the Schrodinger equation (equation 1) becomes ... 

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