Relativistic Effective Core Potentials RECP

Unlike semiempirical methods that are formulated to completely neglect the core electrons, ah initio methods must represent all the electrons in some manner. However, for heavy atoms it is desirable to reduce the amount of computation necessary. This is done by replacing the core electrons and their basis functions in the wave function by a potential term in the Hamiltonian. These are called core potentials, elfective core potentials (ECP), or relativistic effective core potentials (RECP). Core potentials must be used along with a valence basis set that was created to accompany them. As well as reducing the computation time, core potentials can include the effects of the relativistic mass defect and spin coupling terms that are significant near the nuclei of heavy atoms. This is often the method of choice for heavy atoms, Rb and up. 

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

Relativistic effects in heavy atoms are most important for inner-shell electrons. In ab initio and DFT calculations these electrons are often treated through relativistic effective core potentials (RECP), also known as pseudopotentials. This approach is sometimes called quasirelativistic, because it accounts for relativity effects in a rather simplified scalar way. The use of pseudopotentials not only takes into account a significant part of the relativistic corrections, but also diminishes the computational cost. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

An efficient way to solve a many-electron problem is to apply relativistic effective core potentials (RECP). According to this approximation, frozen inner shells are omitted and replaced in the Hamiltonian hnt by an additional term, a pseudopotential (UREP)... 

Effective core potentials address the aforementioned problems that arise when using theoretical methods to study heavy-element systems. First, ECPs decrease the number of electrons involved in the calculation, reducing the computational effort, while also facilitating the use of larger basis sets for an improved description of the valence electrons. In addition, ECPs indirectly address electron correlation because ECPs may be used within DFT, or because fewer valence electrons may allow implementation of post-HF, electron correlation methods. Finally, ECPs account for relativistic effects by first replacing the electrons that are most affected by relativity, with ECPs derived from atomic calculations that explicitly include relativistic effects via Dirac-Fock calculations. Because ECPs incorporate relativistic effects, they may also be termed relativistic effective core potentials (RECPs). 

For the heavier elements, relativistic effects due to the core may become important. To account for this in the simplest way, the electrons in the core can be replaced by a potential that produces the same valence electron distribution as an all-electron relativistic computation. This also reduces the computer time needed as well, since the number of functions is reduced. Another hazard of doing all-electron calculations with small basis sets on lower-row elements is that the bond lengths have large error. The relativistic effective core potential (RECP) that we employed was CEP-121G (12). For this RECP, the geometry was optimized at the MP2 level of theory, and a single-point energy was computed at the CCSD(T) level of theory (13). 

In Table 6.3, the values of De for RfCU are compared with those obtained within various approximations using relativistic effective core potentials (RECP) Kramers-restricted Hartree-Fock (KRHF) (Han et al 1999), averaged RECP including second-order M0ller-Plesset perturbation theory (AREP-MP2) for the correlation part (Han et al. 1999), RECP coupled-cluster single double (triple) [CCSD(T)] excitations (Han et al. 1999), and a Dirac-Fock-Breit (DFB) method (Malli and Styszynski 1998). The AREP-MP2 calculation of De gives 20.4 eV, while the RECP-CCSD(T) method with correlation leads to 18.8 eV. Our value of De of 19.5 eV is just between these calculated values. 

High-level theoretical calculations included ab initio relativistic effective core potentials (RECP) with MP2, MP4 and coupled-cluster methods, which emphasised the importance of elecfronic correlation in the correct description of fhe compounds (Di Bella ef al., 1996). This initial study was further refined by ab initio mefhods (Hong ef al., 1999) and relativistic DPT calculations (Dolg, 2001 Hong et al., 1999,2000 Lu and Li, 1999) The most recent calculations seem to favour an s d rafher fhan fhe f —> d promotion scheme in the metal-ligand bonding of fhe zero-valenf complexes. 

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

Relativistic effective core potential (RECP) methods, also called relativistic pseudopotential (PP) methods, are probably the most successful approximate methods for the various properties of molecules containing heavy atoms. 

An ab initio effective core potential method derived from the relativistic all-electron Dirac-Fock solution of the atom, which we call the relativistic effective core potential (RECP) method, has been widely used by several investigators to study the electronic structure of polyatomics including the lanthanide- and actinide-containing molecules. This RECP method was formulated by Christiansen et al. (1979). It differs from the conventional Phillips-Kleinman method in the representation of the nodeless pseudo-orbital in the inner region. The one-electron valence equation in an effective potential of the core electron can be written as... 

In preceding sections we have discussed several different relativistic methods four-component Dirac—Fock with and without correlation energy, the second-order Douglas—Kroll method, and perturbation methods including the mass—velocity and Darwin terms. The relativistic effective core potential (RECP) method is another well-established means of accounting for certain relativistic effects in quantum chemical calculations. This method is thoroughly described elsewhere - anJ is basically not different in the relativistic and... 

While accurate relativistic (both four- and two-component) calculations of simple heavy-atom molecules can be performed on modern computers the relativistic calculations of periodic systems are made mainly using relativistic effective core potential (RECP). We consider these potentials in the next section. 

Inasmuch as the values of the enthalpy of atomization of lanthanum monofluoride measured by other experimental methods are not available, the values obtained in this work can only be compared to the results of theoretical calculations. In particular, Chen and Shang (2003a) calculated AatH°(LaF, 0) by various methods and obtained 649.0 (B3LYP), 642.5 (CCSD(T)), and 680.9 (MP2) kj/mol. According to these authors, the first two values are more accruate than the last one. In addition, Cao and Dolg (2005) determined AatH°(LaF,0) = 649.4 kj/mol by the relativistic effective core potential (RECP) method. 

We also want to compare the amount of work in a spin-free modified Dirac calculation with that in a nonrelativistic calculation. One of the reasons given for the development of relativistic effective core potentials (RECPs) was the expense of four-component Dirac-Fock calculations. RECPs are often used in spin-free form, but the comparison is made with the spin-dependent unmodified Dirac approach. The comparison of the cost of a spin-free all-electron calculation with a nonrelativistic calculation will give a more realistic indication of the relative cost of the incorporation of spin-free relativistic effects. 

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