Effective-core potentials, ECP

Basis sets for atoms beyond the third row of the periodic table are handled somewhat differently. For these very large nuclei, electrons near the nucleus are treated in an approximate way, via effective core potentials (ECPs). This treatment includes some relativistic effects, which are important in these atoms. The LANL2DZ basis set is the best known of these. 

Optimize these three molecules at the Hartree-Fock level, using the LANL2DZ basis set, LANL2DZ is a double-zeta basis set containing effective core potential (ECP) representations of electrons near the nuclei for post-third row atoms. Compare the Cr(CO)5 results with those we obtained in Chapter 3. Then compare the structures of the three systems to one another, and characterize the effect of changing the central atom on the overall molecular structure. 

Effective core potentials (ECP) replace the atomic core electrons in valence-only ab initio calculations, and they are often used when dealing with compounds containing elements from the second row of the periodic table and above. 

Valdes and Sordo have studied the interaction of PH3 and NH3 with BrCl using either all-electron or effective core potential (ECP) MP2 methods, and several different basis sets [178]. They found that ECPs could give results nearly equivalent to their all-electron counterparts. PH3 was found to form... 

The relativistic effects (Rl) and (R2) can be simulated by adjusting the sizes of basis functions used in a standard variational treatment. This adjustment is usually combined with an effective-core-potential [ECP] approximation in which inner-shell electrons are replaced by an effective [pseudo] potential of chosen radius. The calculations of this chapter were carried out with such ECP basis sets in order to achieve approximate incorporation of the leading relativistic effects.)... 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

The electronic structure calculations were carried out using the hybrid density functional method B3LYP [15] as implemented in the GAUSSIAN-94 package [16], in conjunction with the Stevens-Basch-Krauss (SBK) [17] effective core potential (ECP) (a relativistic ECP for Zr atom) and the standard 4-31G, CEP-31 and (8s8p6d/4s4p3d) basis sets for the H, (C, P and N), and Zr atoms, respectively. 

Calculations of ionization energies and electron affinities were performed with a modified development version of Gaussian 99 [48], Pople and Effective Core Potential (ECP) basis sets are provided in this software [49], Dunning and Atomic Natural Orbital (ANO) basis sets were obtained from the EMSL Gaussian Basis Set Library [50],... 

Initially, the level of theory that provides accurate geometries and bond energies of TM compounds, yet allows calculations on medium-sized molecules to be performed with reasonable time and CPU resources, had to be determined. Systematic investigations of effective core potentials (ECPs) with different valence basis sets led us to propose a standard level of theory for calculations on TM elements, namely ECPs with valence basis sets of a DZP quality [9, 10]. The small-core ECPs by Hay and Wadt [11] has been chosen, where the original valence basis sets (55/5/N) were decontracted to (441/2111/N-11) withN = 5,4, and 3, for the first-, second-, and third-row TM elements, respectively. The ECPs of the second and third TM rows include scalar relativistic effects while the first-row ECPs are nonrelativistic [11], For main-group elements, either 6-31G(d) [12-16] all electron basis set or, for the heavier elements, ECPs with equivalent (31/31/1) valence basis sets [17] have been employed. This combination has become our standard basis set II, which is used in a majority of our calculations [18]. 

Slovenia), using the DFT implementation in the Gaussian03 code. Revision C.02 (8). The orbitals were described by a mixed basis set. A fully uncontracted basis set from LANL2DZ was used for the valence electrons of Re (9), augmented by two / functions Q =1.14 and 0.40) in the full optimization. Re core electrons were treated by the Hay-Wadt relativistic effective core potential (ECP) given by the standard LANL2 parameter set (electron-electron and nucleus-electron). The 6-3IG basis set was used to describe the rest of the system. The B3PW91 density functional was used in all calculations. 

It was Hellmann (1935) who first proposed a rather radical solution to this problem -replace the electrons widi analytical functions that would reasonably accurately, and much more efficiently, represent the combined nuclear-electronic core to the remaining electrons. Such functions are referred to as effective core potentials (ECPs). In a sense, we have already seen ECPs in a very crude form in semiempirical MO theory, where, since only valence elections are tieated, the ECP is a nuclear point charge reduced in magnitude by the number of core elections. 

An even simpler approach to relativity, for heavy elements, is to use effective core potentials (ECPs) to represent the core electrons, taking the potentials from various compilations in the literature that explicitly include relativistic effects in the generation of the ECPs. References to such ECPs are given by Dyall et al. [103]. These relativistic ECPs (RECPs) allow the inclusion of some relativistic effects into a nonrelativistic calculation. Since ECPs will be treated in detail elsewhere, we will not pursue this approach further here. We may note, however, that recent comparisons with Dirac-Fock calculations suggest that the main weakness in the RECPs is not the treatment of relativity but the quality of the ECPs themselves [103]. Different RECPs gave spectroscopic constants with a noticeable scatter, compared to Dirac-Fock, but the relativistic corrections (difference between an RECP and the corresponding ECP value) were fairly consistent with one another. 

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. 

Results for first-row transition metal compounds were obtained from all-electron restricted and unrestricted Kohn-Sham calculations. Only for atoms of fifth or higher periods of the periodic table of the elements we have applied those effective core potentials (ECPs) of the Stuttgart group, which are the standard ECPs in TURBOMOLE for the core electrons. These ECPs account very well for the scalar relativistic effect in heavy-element atoms (126,127). 

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