The earliest theoretical studies

The assumption that a pentagonal dodecahedral cage structure was required to explain the special stability of Ti8C 2 and other met-cars had a double impact on the 

Ti-Ti = 3.06 A) and for silicon (Si-Si = 3.03 A). Carbon-carbon bond lengths (1.36 A C-C 1.44 A) are characteristic of an important n bonding interaction. The question of the stability of met-cars in their dodecahedral structure is discussed. Large stability is indeed found relative to either isolated atoms (6.67 eV/atom for TisC 6.90 eV/atom for ZrgC 6.80 eV/atom for VgC ) or cluster fragments Mg and 6C2 ( 58 eV for Ti, V, and Zr), but for the three metals able to form solid metal carbides (Ti, Zr, Si), the dodecahedral MgCn clusters are unstable relative to the reaction  

It is also worth noting the very small HOMO-LUMO gap especially for neutral TigCi2 (0.16 eV), a quasi-degeneracy that will be found in all DFT and other calculations involving a single-particle Hamiltonian. 

Next in chronological order comes the communication by I. Dance (submitted July 3, 1992). This report based on DFT-LDA calculations was the first to propose an alternative to the dodecahedral structure it has since been gaining both theoretical and experimental credibility. This important contribution, together with related theoretical work making similar structural assumptions, and their consequences, will be discussed in detail in Section 5.9.6. 

Despite general agreement on the high binding energy per atom computed for dodecahedral TigCi2 and related systems, a sense of discomfort emerged from those 

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The earliest theoretical studies of phosphonium ylides were EHT studies of methyl-enephosphorane62 and cyclopropylidenephosphorane63 by Hoffmann and coworkers. 

Possibly the earliest theoretical study of diffusion of aromatics in zeolites was published in 1987 by Nowak et al. (89), who considered diffusion of benzene and toluene in the pores of silicalite and theta-1. Theta-1 (90) has a unidimensional medium-sized pore opening bounded by 10-rings. In this study, only the straight channel of silicalite was considered, making the... 

One of the earliest theoretical studies in this area was work by Lord Kelvin [8,47] who derived an expression for the frequency of a propagating... 

In the earliest theoretical study of the SCD process Emslie al (14) obtained an analytical solution given by. 

The earliest theoretical studies of the behavior of an electrified interface were made by Helmholtz (1879). He discussed the adsorption of ions at a fixed double layer and he believed that this double layer formed the equivalent of a parallel-plate condenser. But this double layer model is an inadequate description of particles in electrolyte-containing systems. 

Analytical expressions for the primary echo modulation function of an B = 1 /2, / = 1/2 system were worked out for some of the earliest ESEEM studies that appeared in the literature. Perhaps the most general of these theoretical treatments is that of Mims where the two-pulse ESEEM function is given by... 

The earliest theoretical calculations of cocaine hydrolysis focused on the first step of the hydrolysis of the benzoyl ester [57,58]. hi these computational studies [57,58], MNDO, AMI, PM3, and SM3 semiempirical molecular orbital methods, as well as ab initio procedure at the HF/3-21G level of theory, were employed to optimize geometries of the transition states for the first step of the hydrolysis of cocaine and model esters, including methyl acetate [59,60] for which experimental activation energy in aqueous solu-... 

The studies of Vermilyea and Engell provided some of the earliest theoretical bases for the dissolution of iron oxides. Vermilyea expressed the rate of oxide... 

As is known, the piezoelectric effect was discovered in 1880 by the brothers Jacques and Pierre Curie, and ever since that discovery the effect has been intensively studied in various materials. The first investigations of the piezoelectric properties of polymers, such as cellulose, were made by Brain [137], and the earliest theoretical discussion of the piezoelectric properties of various polymeric materials can be found in the works of Rez [138] and Gubkin et al. [139]. (For recapitulation of the literature on this subject, tee, for example, Refs. 2a, 2b, 3. V 118, and 119]). 

We ve included several papers in the References section which perform theoretical and experimental studies of the IR and Raman spectra for these compounds. These compounds were among the earliest ab initio frequency studies of such systems. In addition, in the case of propellane, theoretical predictions of its energy and structure preceded its synthesis. 

The theoretical work that exploited the advantages of the multidimensional separation format appears to have been developed much later than the original experimental work. One of the earliest studies was conducted by Connors (1974), who assumed that the distribution of spots on a two-dimensional thin-layer chromatography (2DTLC) plate could be modeled using a Poisson distribution of data on each retention axis. He then constructed equations that related the number of chromatographic systems needed to resolve a specific number of compounds. One... 

The earliest method used at PicArsn was in 1933 It consisted of treating ethylenediamine with diethyl carbon ate to obtain ethyleneurea, followed by nitration to obtain Dinitroediyleneurea, which on hydrolysis gave Ethylenedinitramine. This process suffered from two disadvantages the overall yield was only 42% of the theoretical amt and the initial step, the reaction between ethylenediamine and diethyl carbon a re, required the use of high pressures and temperatures over an extended period of time. However, the expl props of EDNA were of sufficient in-, terest ro justify further study (Ref 4). See also Aaronson (Ref 5)... 

Low-dimensional Solids. - 2.10.1 Introduction. The magnetic properties of one and two-dimensional arrays of localized spins coupled by Heisenberg exchange interactions have been studied as a rather specialized branch of theoretical physics since the earliest days of quantum mechanics. However, recent advances in theory, and the preparation of real materials that are a good approximation to the theoretical models, have made low-dimensional systems much more central to condensed-matter science. There is enormous scope for synthetic chemistry in this area and, as will be seen later in this section, many new materials have been discovered recently. 

These studies suggested a planar ylidic carbon and definite participation of the P d orbitals in the description of the HOMO. The earliest ab initio study of methylenephosphorane found a very small rotational barrier [0.003 kcal mol-1 (1 kcal = 4.184 kJ)] about the P=C double bond64. These three points, the degree of pyramidalization at the ylidic carbon, the role of the P d orbitals and the P=C rotational barrier, remain the focal points of all theoretical studies of the ylides. A summary of the theoretical structures determined using ab initio techniques is given in Table 7. 

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