Theoretical and computational details

In a very recent publication [1], we have presented a new model for the rotation-vibration motion of pyramidal XY3 molecules, based on the Hougen-Bunker-Johns (henceforth HBJ) approach [2] (see also Chapter 15, in particular Section 15.2, of Ref. [3]). In this model, inversion is treated as a large-amplitude motion in the HBJ sense, while the other vibrations are assumed to be of small amplitude they are described by linearized stretching and bending coordinates. The rotation-vibration Schrddinger equation is solved variationally to determine rotation-vibration energies. The reader is referred to Ref. [1] for a complete description of the theoretical and computational details. 

In order to complete our description of the theoretical and computational details behind the CIM-CCSD, CIM-CR-CC(2,3), and CIM-CCSD(T) approaches, we must provide information about the actual design of the occupied and unoccupied LMOs defining the CIM subsystems (P). This is done in the next subsection. Obviously, if we do not introduce any approximations and there is only one subsystem or orbital domain that corresponds to all orbitals of the system, Eqs. (23)-(38) become equivalent to the exact expressions, Eqs. (9) and (11) for the CCSD correlation energy and Eqs. (13) and (14) for the triples correction to CCSD. In this case, the CIM and canonical CC calculations yield identical results, i.e., and = 3 (2.3) -phe key idea of the CIM-CC... 

The first dynamical simulation of a protein based on a detailed atomic model was reported in 1977. Since then, the uses of various theoretical and computational approaches have contributed tremendously to our understanding of complex biomolecular systems such as proteins, nucleic acids, and bilayer membranes. By providing detailed information on biomolecular systems that is often experimentally inaccessible, computational approaches based on detailed atomic models can help in the current efforts to understand the relationship of the strucmre of biomolecules to their function. For that reason, they are now considered to be an integrated and essential component of research in modern biology, biochemistry, and biophysics. 

The Brookhart laboratory has contributed much of the knowledge of the polymerization mechanism for the late transition metal a-diimine catalysts. The review by Ittel provides a concise summary of the mechanistic understanding as of the year 2000 [26]. Some of the early findings will be reviewed here and additional insights reported afterward will be presented. In addition to the experimental work, many theoretical and computational studies worthy of discussion have also been carried out. These efforts have been most important in providing insight into the mechanistic details of the highly reactive nickel system, which is often difficult to study experimentally. 

The I2 system has been investigated experimentally, theoretically, and computationally by several groups, as a prototype for the study of dissociation and recombination dynamics influenced by the interactions with a surrounding solvent or cluster of solvent molecules[9],[36]-[41]. The system can be effectively modelled by two VB states[9],[41], which allows a focus on several key aspects of the implementation of the theory, without being hindered by the complexity of a multistate calculation. The implementation steps are conveniently collected in the flow chart in Table 1, to which the reader is referred to for a comprehensive overview of our strategy. All the details of the calculation are reported in BH-II. The effective wave function for the I2 reaction system can be written as... 

The elucidation of the structure, dynamics and self assembly of biopolymers has been the subject of many experimental, theoretical and computational studies over the last several decades. [1, 2] More recently, powerful singlemolecule (SM) techniques have emerged which make it possible to explore those questions with an unprecedented level of detail. [3-55] SM fluorescence resonance energy transfer (FRET), [56-60] in particular, has been established as a unique probe of conformational structure and dynamics. [26-55] In those SM-FRET experiments, one measures the efficiency of energy transfer between a donor dye molecule and an acceptor dye molecule, which label specific sites of a macromolecule. The rate constant for FRET from donor to acceptor is assumed to be given by the Forster theory, namely [59,61-64]... 

The model of a dipole in a spherical cavity can only provide qualitative insights into the behaviour of real molecules moreover, it cannot explain the effect of electrostatic interactions in the case of apolar molecules. More accurate predictions require a more detailed representation of the molecular charge distribution and of the cavity shape this is enabled by the theoretical and computational tools nowadays available. In the following, the application of these tools to anisotropic liquids will be presented. First, the theoretical background will be briefly recalled, stressing those issues which are peculiar to anisotropic fluids. Since most of the developments for liquid crystals have been worked out in the classical context, explicit reference to classical methods will be made however, translation into the quantum mechanical framework can easily be performed. Then, the main results obtained for nematics will be summarized, with some illustrative... 

Detailed modelling, or numerical simulation, provides a method we can use to study complex reactive flow processes (1). Predictions about the behavior of a physical system are obtained by solving numerically the multi-fluid conservation equations for mass, momentum, and energy. Since the success of detailed modelling is coupled to one s ability to handle an abundance of theoretical and numerical detail, this field has matured in parallel with the increase in size and speed of computers and sophistication of numerical techniques. 

Theoretical and computational methodologies are treated in detail elsewhere in this book. Experimental techniques for studying isolated molecules rely on their observation in the gas phase, where molecules can be studied free of interactions. This is different from single molecule studies in which molecules can interact with their environment, but are studied one by one [2], In the gas phase one may study a large ensemble of molecules or clusters, but each one of those is isolated and does not interact with its environment. Clusters represent a transition area between gas phase and bulk by allowing infra-cluster interactions, while being isolated from inter-cluster interactions. 

Owing to recent developments in theoretical and computational methods, the quantum mechanical approach to the polymer electronic structure problem has begun to associate very fruitfully with experimental research in this field. Combination of the methods of molecular quantum theory with the ideas of theoretical solid-state physics has provided a really efficient tool, not only for the interpretation of experimental results, but also for investigation of fine details in the electronic structure which would be only barely accessible in experiments. 

Because this chapter is a follow-up of previous work in the field it is not necessary to repeat the basics of ab initio methods. This has been done in detail by Basch and Hoz, who also discuss the most important atomic properties of Ge, Sn and Pb. We also recommend the theoretical section in the chapter by Apeloig about organosilicon compounds in this series who gave an excellent overview about the most important aspects of ab initio, semiempirical and force-field methods. The reader will find there an explanation of the most common standard methods which will be mentioned in this review without further explanation. We will focus in the following on those theoretical and computational aspects of methods which are particularly important for heavy-atom molecules that have been advanced in the last decade, i.e. ECPs and DFT. We also briefly discuss relativistic effects. We point out that semiempirical methods" and force field parameters are available for the elements Ge, Sn and Pb. However, the application of the two methods has not gained much popularity and not many papers have been published in the field. Most reports are restricted to special problems. ... 

In this final section, we examine in detail a number of recent research efforts in coupled cluster theory. This review is far from exhaustive, and because of space considerations, we choose to focus primarily on two specific areas in which the present authors have made contributions. We will then discuss some of the most important theoretical and computational advances expected in the near future. We also recommend Refs. 78 and 79 for a discussion of other recent work. 

The role of particle size in catalysis and electrocatalysis is a subject of longstanding interest. It is not our intention here to discuss in detail the available experimental and theoretical literature. Extensive reviews on particle-size effects in gas-phase catalysis and electrocatalysis can be found in the papers of Henry [25] and Kinoshita [26], respectively. Also, several monographs, reviews, and conference proceedings discuss particle-size effects from experimental, theoretical, and computational points of view [9,27,28]. 

The sources for all listed properties are either experimental observations or theoretical computations. We do not include results from semi-empirical calculations. For experimental and computational details of literature values the reader must check the original publications. Details for calculations done by us for this chapter with methods of Spartan-02 and -04 (Wave-function, Inc., Irvine, CA, USA) are given in Appendix 17.2. 

We note that, even if we start here from the same truncated basis B = B, B2,. . . , Bm as in the EOM method, the results are not necessarily the same, since (2.16) is a single-commutator secular equation whereas (1.50) is a double-commutator secular equation. It should be observed, however, that the column vectors d obtained by solving (2.15) are optimal in the sense of the variation principle, whereas this is not necessarily true for the vectors obtained by solving (1.49). In the following analysis, we will discuss the connection between these two approaches in somewhat greater detail. Since the variation principle (2.10) would provide an optimal approximation, the essential question is whether the theoretical and computational resources available today would permit the proper evaluation of the single-commutator matrix elements defined by (2.13) for a real many-particle system this remains to be seen. 

The present review has been very selective, stressing the rationale behind density-functional methods above their applications and excluding many important topics (both theoretical and computational). The interested reader may refer to anyone of the many books [91-93] or review articles [94-101] on density-functional theory for more details. Of special importance is the extension of density-functional theory to time-dependent external potentials [102-105], as this enables the dynamical behavior of molecules, including electronic excitation, to be addressed in the context of DFT [106-108]. As they are particularly relevant to the present discussion, we cite several articles related to the formal foundations of density-functional theory [85,100,109-111], linear-scaling methods [63,112-116], exchange-correlation energy functionals [25, 117-122], and qualitative tools for describing chemical reactions [123-126,126-132]. 

Continuing with the muguet theme, I then use examples to explain how a chemist might go about his search for novel materials. The more traditional approaches, such as the analysis of natural products, serendipity and lead optimisation, are illustrated with only one or two examples, since these techniques have been discussed in more detail in Chapters 3 and 12. Instead, I have focused on techniques that have the potential to lead to the discovery of new active compounds by design rather than chance. Over the past 20 years, such approaches have been made easier by the rapid advances in theoretical and computational chemistry and by the introduction of more powerful computers. 

Before discussing in detail the numerical results of our computational work, we describe the theoretical and computational context of the present calculations apart from deficiencies of models employed in the analysis of experimental data, we must be aware of the limitations of both theoretical models and the computational aspects. Regarding theory, even a single helium atom is unpredictable [14] purely mathematically from an initial point of two electrons, two neutrons and two protons. Accepting a narrower point of view neglecting internal nuclear structure, we have applied for our purpose well established software, specifically Dalton in a recent release 2.0 [9], that implements numerical calculations to solve approximately Schrodinger s temporally independent equation, thus involving wave mechanics rather than quantum... 

In this chapter, we will discuss theoretical and computational developments in intramolecular dynamics and vibrational spectroscopy which have occurred since the early 1980s. Emphasis will be placed upon the results arising from a long standing collaboration between research groups at The University of Texas at Austin and the University of Paris-Orsay (and more recently, at the University of Montpellier). The computational methods that were used and developed have resulted in the most detailed studies ever to be reported on the spectra and dynamics of moderate-sized molecules. Because of the very large number of available quantum states, frequently >10h, a number of new methods were developed, tested, and then incorporated into the production codes. 

In this chapter, we review some recent developments in the theoretical and computational aspects of acetylenes. There are several detailed reviews covering various aspects of the early work [1, 2]. It will become self-evident in this review that modern experimental and computational studies of acetylene constitute a paradigm for the rivalry and interplay between theory and experiment. As the theoretical treatments become increasingly sophisticated, and as the experimental design becomes more and more ingenious and precise, the better is our understanding. 

On the other hand, extreme theoretical and computational difficulties may also be caused in cases of resonance states whose fundamental cause is the very weak binding and the concomitant enhancement of the significance of the details of interelectronic interactions, regardless of the number of the system selectrons. In such cases, it is difficult to identify and to compute with accuracy the correct T oS. For example, investigations of such cases in resonances of He have been reported in Refs. [127,128]. 

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