Many-atom systems

Two methods for time-dependent quantum simulations of many-atom systems are examined in this article the CSP-based and the Cl-CSP-based algorithms. The CSP method begins with a separable approximation for... 

Another important limitation of current MD methods is that of the conservation of zero point energy (see Ref. [21] and references therein). In a many-atom system the total quantum mechanical zero point energy can be quite high, and, in classical mechanics, this energy is available to the system. Additional sources of reviews of current methodologies are Refs. [22-24],... 

Two central problems remain. One is that one needs the potential which governs the motion. In many-atom systems, even if the motion is confined to the ground electronic state, this potential is a function of the spatial configuration of all the atoms. It is therefore a function of many variables, so its analytical form is far from obvious, nor do we necessarily want to know it everywhere. Indeed, we really only want it at each point along the actual trajectory of the system (so that the forces can be computed and thereby the next point to which the system will move to can be determined). Such an approach has been implemented [25] and applied to many-atom systems, and an extension to a multi-electronic state dynamics will be important... 

Many atomic systems (e.g. molecules, clusters and crystals) are systems with interacting electrons and nuclei and can be thus described by Schrodinger equation... 

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

Figure 15.3 Secondary KIEs are associated with normal modes other than the reaction coordinate, one of which is shown in this diagram. The heavy and light vibrational frequencies both change on going from the reactant (R) to the TS structure ( ) because in this example the mode is tighter in the TS structure, the difference between the heavy and light ZPVEs increases, and this causes the potential energy of activation to be larger for the light isotopomer than the heavy one (an example of an inverse secondary KIE). In a real many-atom system there are potentially a large number of modes that will contribute to the secondary KIE. some in a normal fashion and some in an inverse fashion...

A quantum mechanical theory is in principle needed to describe molecular phenomena in both few-atom and many-atom systems. In some cases a single electronic state is involved, and it is possible to gain valuable insight using only classical molecular dynamics, which can be relatively easy to apply even for a system of many atoms. A quantum mechanical description of molecular phenomena is however clearly needed for electronic states, insofar these have pronounced wavemechanical properties. The need for a quantum description of nuclear motions in molecular dynamics is less apparent, but it is required in some important situations. If we consider a generic interaction between two species A(a) and B(j3) leading to formation of two others, C(7) and D(6), all of them in the specified quantum states, so that... 

Working with the density operator is a convenient alternative to using wavefunctions when dealing with a few-atom, isolated molecular system, insofar it suggests more efficient computational procedures or more consistent approximations, but it is not stricktly needed. The density operator is however essential in treatments of a many-atom system, when this interacts with a medium which constrains thermodynamical properties such as temperature or pressure, because the density operator incorporates statistical averages which would not be included in a treatment based on wavefunctions. 

A many-atom system may contain hundreds of atoms, as in clusters, or macroscopic amounts of matter, as in the cases of condensed matter solutions or solid surface phenomena. Mesoscopic systems and nanostructures fall in between those two extremes. These objects may be embedded in a medium in thermodynamical equilibrium, which imposes constrains of temperature, pressure, or chemical potentials. The medium may alternatively be excited and near equilibrium, or even far from it, in which cases it may strongly affect the time evolution of the object of interest. A unified treatment of these situations can be done with the density operator and its L-vN equation of motion. 

To consider next the partitioning approach for many-atom systems, we start again with the full density operator f(t), describing both electrons and nuclei. We describe a procedure useful in a first principles dynamics where the nuclear motions are described in an eikonal approximation with Q = Q(t), so that the electronic part of the problem involves the calculation of an electronic density operator r[Q(f), Q t), f] = pei t). This satisfies an electronic L-vN equation... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

This overview has also considered the advantages and disadvantages of a description using first principles dynamics. Its application to many-atom systems undergoing electronic transitions is a very active and challenging subject of molecular quantum dynamics. 

D. A. Micha and B. Thorndyke. Dissipative dynamics in many-atom systems a density matrix treatment. Intern. J. Quantum Chem., vv ppp, 2001. submitted. 

Reduced Density Matrix Equations for Combined Instantaneous and Delayed Dissipation in Many-Atom Systems, and their Numerical Treatment... 

A many-atom system excited by light or by collisions, such as occurs in the photo-excitation of a molecule adsorbed on a surface or in photosynthesis and vision, leads to energy dissipation on different time scales. A fast dissipation typically occurs due to electronic energy relaxation in the medium, while a slow (delayed) dissipation arises from vibrational energy relaxation. Here we concentrate on localized phenomena where a relatively small number... 

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

In their study of the decomposition of nitromethane, Rice and Thompson [94] introduced a new approach for constructing potential energy surfaces for many-atoms systems that react via multiple pathways. The basic idea of the approach is to construct potentials that accurately describe the various equilibrium regions, e.g., reactants and products, and then write the overall global potential as Vtotal=E SjV where j denotes the various stable species, the Vi are the analytical potentials for those species, and the Sj are weighting functions that effect a switching between the potentials... 

EXAMPLES OF INTERACTION OF THEORY AND EXPERIMENT 1.2.1 High-resolution overtone spectroscopy of many-atom systems... 

The potential between the reactive atoms that we used is one that has been used before in studies of many-atom systems,eind which we have examined to insure that it recovers the essential chemical facts that are known to us. (However, one surely does not yet know all that there is to know about the high energy chemistry of the N/O system.) The potential has a functional form that allows for a weakening of a bond between a pair of atoms when one or more other reactive atoms are nearby. In other words, the potential has many-atom terms. It also Includes a long reinge van der Waals attraction which describes the packing in the cold cluster. For more details, see Ref. 95. 

A strength of direct dynamics is that it allows one to determine the classical dynamics for a particular level of electronic structure theory, without the need for an intermediate analytic PES fit. In addition, since it is difficult to develop analytic PES s for many-atom systems, direct dynamics greatly expands the range of unimolecular systems whose dynamics may be investigated by classical trajectories. However, a limitation of direct dynamics, is the large computer time needed to calculate the trajecto-... 

R.B. Gerber and M.A. Ratner, Self-consistent field methods for vibrational excitations in polyatomic molecules, Adv. Chem. Phys., 70 (1988) 97. P. Jungwirth and R.B. Gerber, Quantum dynamics of large polyatomic systems using a classically based separable potential method, J. Chem. Phys., 102 (1995) 6046 Quantum dynamics of many atom systems by classically based separable potential (CSP) method Calculations for T (Ar),2 in full dimensionality, J. Chem. Phys., 102 (1995) 8855. 

The present version of SDIBEX is designed to use both QM and QCT calculations. However, only one QCT conij)utational engine (ABCtraj [26] for three atom systems) is fully operative while other two ( enus[27] for more than three atom systems and DL POLY [28] for many atom systems) are still in the process of being implemented. 

The first term (kinetic energy) is a summation over all the particles in the molecule. The second term (potential enctgy) uses Coulomb s law to calculate the interaction between every pair of particles in the molecule, where e, and cj ate the charges on particles i and j. For electrons, the charge -e, while the charge for a nucleus is Ze, where Z Is the atomic number. The summation nutation ipairwise interaction terms in the summation (e.g., eg / = e e, and should only appear in the potential energy term once). The denominator r in the. second term is the distance between particles i anil j. J. i is understood to be the electronic wave function for a many-atom system. 

The previous examples were concerned with atomic targets, however, one of the advantages of the END formulation is the ability to deal with many-atom systems. In this section we present a review of molecular stopping cross sections a la END. 

This represents a formidable practical problem, as one is very unlikely to find isolated atoms with two nonorthogonal dipole moments and quantum states close in energy. Consider, for example, a V-type atom with the upper states 11), 3) and the ground state 2). The evaluation of the dipole matrix elements produces the following selection rules in terms of the angular momentum quantum numbers J — J2 = 1,0, J3 — J2 = 1,0, and Mi — M2 = M3 — M2 = 1,0. Since Mi / M3, in many atomic systems, p12 is perpendicular to p32 and the atomic transitions are independent. Xia et al. [62] have found transitions with parallel and antiparallel dipole moments in sodium molecules (dimers) and have demonstrated experimentally the effect of quantum interference on the fluorescence intensity. We discuss the experiment in more details in the next section. Here, we point out that the transitions with parallel and antiparallel dipole moments in the sodium dimers result from a mixing of the molecular states due to the spin-orbit coupling. 

The purpose of this book, edited by Bemd A. Hess, is twofold. On the one hand, the different theoretical tools for including relativistic effects in quantum-theoretical calculations (at very many different levels of theory and sophistication) developed recently are presented. And on the other hand, the results obtained with these methods for both simple and complicated (i.e. many-atom) systems are presented. 

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