ECP method

The ECP method dates back to 1960, when Phillips and Kleinman suggested an approximation scheme for discarding core orbitals in band calculations [1]. They replaced the full Fock-operator with the following operator ... 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

The ECP method which will be discussed henceforth is derived from Huzinaga and Bonifacic s equations, and the full nodal structure of the valence orbitals is always kept. In the early ECP application on first row transition metals the only orbitals which were variationally determined were 3d and 4s[6]. However, experience showed that in certain cases it was important also to include the 3s and the 3p orbitals in the valence space[7-9], and ECP s with these characteristics were accordingly developed[10]. 

One of the advantages of the relativistic ECP methods is their ability to include spin-orbit effects simultaneously to correlation effects at a reasonable cost. Recently Wiillen [121] the ZORA method to coinage metal diatomics and other others H, F, Cl diatomics. These results for the gold atom are shown in Table 2 where we add other DFT calculations. The agreement of the DFT methods with the experimental values is excellent. The dissociation energy of AuH has also been calculated extensively using several methods. Results are shown in Table 3. 

The dynamic methods are based on direct chromatography and are popular because they are faster and easier to automate. Four direct chromatographic methods that are available for determination of adsorption isotherms are frontal analysis (FA) [13, 109] frontal analysis by characteristic points (FACP) [109], elution by characteristic points (ECP) [109] and the perturbation peak (PP) method [118-121], The FACP and ECP methods have... 

The FACP and ECP methods cannot be used to determine adsorption isotherm parameters from multicomponent mixtures. By contrast FA can be used to determine multi-component adsorption data but it is a complex and time-consuming process [124, 125],... 

The method of generating N, (moles of adsorbed gas or vapor per unit gram of polymer) versus p (partial pressure of gas or vapor) by IGC is based on the method described by Mohlin and Gray (16). This technique is the same as Elution by Characteristic Point (ECP) Method described by Conder and Young (18). 

A possibility to reduce the influence of column efficiency on the results obtained by the ECP method is to detect the position of the peak maximum only, which is called the peak-maximum or retention-time method. Graphs like Fig. 6.23 are then achieved by a series of pulse injections with different sample concentrations. The concentration and position of the maximum is strongly influenced by the adsorption equilibrium due to the compressive nature of either the front or the rear of the peak (Chapter 2.2.3). Thus, the obtained values are less sensitive to kinetic effects than in the case of the ECP method. The isotherm parameters can be evaluated in the same way as described in Section 6.5.7.6, but the same limitations have to be kept in mind. For some isotherm equations, analytical solutions of the ideal model can be used to replace the concentration at the maximum (Golshan-Shirazi and Guiochon, 1989 and Guiochon et al., 1994b). Thus, only retention times must be considered and detector calibration can be omitted in these cases. 

The principle of the effective core potential (ECP) method is to separate the electronic system into core and valence electrons. The core electrons are replaced with an effective core potential, and only valence electrons are treated explicitly in the quanmm chemical calculation. The ECP method assumes that the core electrons are chemically inert, that the atomic orbitals of core electrons do not change from the free atom for which they were derived to an atom in a molecule, and that valence electrons dictate the... 

Again, the primary disadvantage of this method is that the amoimt adsorbed, q C), is calculated from an equation derived from the ideal model, assuming that the coliunn efficiency is infinite. Therefore, the ECP method should be used only with highly efficient columns, where the contribution to band broadening is neg-... 

With the FACP and ECP methods, each point of the rear profile gives one point of the isotherm. With modem systems of data acquisition several himdred data points of an elution profile can be conveniently recorded and stored, so many points of an isotherm can be acquired rapidly. Therefore, these methods are more precise than the FA method (although they are less accurate see below). A significant advantage of FACP and ECP over FA is that they are approximately 25 times faster and require amoimts of pme compoimds and of mobile phase that are at least one order of magnitude less. The amount of wasted compoimds and solvents generated by these methods is comparatively much smaller than for FA. 

Figure 3.47 Dependence of the isotherm determined by ECP (or FACP) on the column efficiency. The ECP method is based on the ideal model profile (cf Eq. 7.4). A Langmuir isotherm (solid line, b) is used to calculate the band profiles obtained with columns of different efficiencies ( L/ = 10%). The profiles (a) are used to derive the isotherm following the ECP method. The isotherms differ from the initial Langmuir isotherm. The best fit of the data to a Langmuir model generates significant model errors, with deviations of the order of 1% between the initial and the measured coefficients of the isotherm for N = 5000 plates, larger at lower efficiencies and loading factors.

The experimental band profiles in Figures 7.15a and 7.15b are overlaid with the profiles calculated as solutions of the ideal model, using the isotherms determined previously by frontal analysis on the same column [46]. Note that using isotherms determined by the ECP method would lead us into a circular argiunent. 

Single component isotherm of T and P were measmed using the ECP method and fitted to single component Langmuir isotherms in order to determine the column saturation capacity of each solute, qs,T arid qg,p. The binary Langmuir isotherms were written as ... 

Extensive introductions to the effective core potential method may be found in Ref. [8-19]. The theoretical foundation of ECP is the so-called Phillips-Kleinman transformation proposed in 1959 [20] and later generalized by Weeks and Rice [21]. In this method, for each valence orbital (pv there is a pseudo-valence orbital Xv that contains components from the core orbitals and the strong orthogonality constraint is realized by applying the projection operator on both the valence hamiltonian and pseudo-valence wave function (pseudo-valence orbitals). In the generalized Phillips-Kleinman formalism [21], the effect of the projection operator can be absorbed in the valence Pock operator and the core-valence interaction (Coulomb and exchange) plus the effect of the projection operator forms the core potential in ECP method. 

Despite these disadvantages, there is one great advantage of using the nodeless ECP orbitals the primitive functions describing the undulation of the valence orbitals in the vicinity of the nuclei are not needed and the computations are more economic. For more details of the recent developments and applications of the relativistic ECP method, one may refer to the references [14,16,23,32-52]. 

Barnett et al. examined the bond between Au+ and ethene using both the ECP method and (in MCSCF calculations) the spdsMCPs for gold [259]. Subsequently, a series of five platinum(II) complexes of the form (N"N"N)PtCl were studied (where N N N represents the tridentate monoanionic ligands) using the time-dependent density functional theory [260]. 

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