Orbital pseudo-valence

Reductions in the basis set used to represent the valence orbital can be only achieved if by admixture of core orbitals radial nodes are eliminated and the shape of the resulting pseudo ip) valence orbital core region (pseudo-valence orbital transformation)... 

Np denotes a normalization factor depending on the coefficients G)c- The origins valence orbital with the full nodal structure in terms of the pseudo valence orbital with the simplified nodal structure... 

Figure 10. Radial part of orbital products 2s 2p for gC entering in the 2s-2p exchange integral. Whereas the product formed with nodeless pseudo-valence orbitals generated by a 4-valence electron ([jHe] core) pseudopotential (PP) is always positive, the one formed with valence orbitals from all-electron (AE) calculations has a negative contribution in the core region due to the 2s radial node.

The origin of shape-consistent pseudopotentials [131,160] lies in the insight that the admixture of only core orbitals to valence orbitals in order to remove the radial nodes leads to too contracted pseudo valence orbitals and finally as a consequence to poor molecular results, e.g., to too short bond distances. It has been recognized about 20 years ago that it is indispensable to have the same shape of the pseudo valence orbital and the original valence orbital in the spatial valence region, where chemical bonding occurs. Formally this requires also an admixture of virtual orbitals in Eq. 37. Since these are usually not obtained in finite difference atomic calculations, another approach was developed. Starting point... 

Having a nodeless and smooth pseudo valence orbital (p. and the corresponding orbital energy e ij at hand, the corresponding radial rock equation... 

It may be asked how accurate energy-consistent pseudopotentials will reproduce the shape of the valence orbitals/spinors and their energies. Often radial expectation values < r > are used as a convenient measure for the radial shape of orbitals/spinors. Due to the pseudo-valence orbital transformation and the simplified nodal structure it is clear that values n < 0 are not suitable, since the resulting operator samples the orbitals mainly in the core region. Table 2 lists orbital energies, < r > and < > expectation values for the Db [Rn] 5f 6d ... 

The performance of energy-consistent quasirelativistic 7-valence electron PPs for all halogen elements has been investigated in a study of the monohydrides and homonuclear dimers [242]. Special attention was also paid to the accuracy of valence correlation energies obtained with pseudo valence orbitals [97,98]. Some of the results for the halogen dimers is presented in Tables 13... 

As a next step, a pseudopotential has to be adapted to the pseudo-valence orbitals defined in equation (13). The idea is to look for a one-electron operator Vpp which, when inserted into the valence Hamiltonian Hpp, equation... 

Within the molecular quantum-chemical regime, various strategies on how to generate sets of pseudopotentials have been followed, such as the shape-consistent pseudopotentials for which the pseudo-valence orbital is identical... 

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. 

The scalar-relativistic effects can be easily absorbed into the effective potential by taking the all-electron (AE) calculation results of the same order of relativistic approximation as the references to parametrize the potentials. Taking the two-component (or even four-component) form of the pseudo-valence orbitals, the spin-orbit coupling effect can also be absorbed into the ECR Because the pseudovalence orbitals are energetically the lowest-eigenvalue eigenvectors of the Fock... 

Different ECP approaches can be classified according to various criteria. If the original radial-node structure of the atomic valence orbitals is preserved, a model potential is produced [808-813]. If the nodal structure is not conserved, the ECP is called pseudo potential [814-817], While shape-consistent pseudo potentials [818-821] are optimized to obtain a maximum resemblance in the shape of pseudo-valence orbitals and original valence orbitals, energy-consistent pseudo potentials [822-829] reproduce the experimental atomic spectrum very accurately. 

In the MCP, or more advanced AIMP, approximations [72, 73], is represented by an adjustable local potential and a projection operator. This potential is constructed so that the inner nodal stmcture of the pseudo-valence orbitals is conserved, thus closely approximating all-electron valence AOs. Scalar relativistic effects are directly taken into account by relativistic operators such as Douglas-Kroll (DK) one. SO effects can be included with the use of the SO operator,... 

In the effective core potential (ECP) approximation, is represented by a semi-local potential [74]. Unlike in the MCP methods, there are no core functions and the pseudo-valence orbitals are nodeless for the radial part, which is an essential approximation. The semi-local ansatz gives rise to rather complicated integrals over the Gaussian functions compared to the MCP methods, though efficient algorithms were developed for their solution. Relativistic and SO effects are treated by relativistic one-electron PPs (RPP) [76]... 

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