Valence spinors

If we then partition the valence spinor in a manner analogous to Eq. (3)... 

Figure 8. Radial densities of the 4f, 5d and 6s valence spinors of jgCe in the 4f 5d 6s ...

Figure 13. Valence spinors of the Db atom in the 6d 7s ground state configuration from average-level all-electron (AE, dashed lines) multiconfiguration Dirac-Hartree-Fock calculations and corresponding valence-only calculations using a relativistic energy-consistent 13-valence-electron pseudopotential (PP, solid lines). A logarithmic scale for the distance r from the (point) nucleus is us in order to resolve the nodal structure of the all-electron spinors. The innermost parts have been truncated.

Figure 21. Valence spinors of 53I from average level multi-configuration calculations using the AE Dirac-Coulomb-Hamiltonian (solid lines) and a PP valence-only model Hamiltonian (dashed lines) [193],...

Accurate treatment of core spinors and of the valence spinors in the core region by a large... 

Similarly as in Section 1.2, one starts from atomic AE reference calculations at the independent-particle level (some kind of quasi-relativistic HF or fully relativistic DHF). The first step now in setting up pseudopotentials consists in a smoothing procedure for valence orbitals/spinors ( pseudo-orbital transformation ). In the DHF case, to be specific, the radial part ( )/ of the large component of the energetically lowest valence spinors for each //-combination is transformed according to... 

Some other versions of the DFT method like the Beijing Density Functional method (BDF) (see the chapter of C. van Wuellen in this issue) were also used for small compounds of the heaviest elements like 111 and 114 [115-117]. There, four-component numerical atomic spinors obtained by finite-difference atomic calculations are used for cores, while basis sets for valence spinors are a combination of numerical atomic spinors and kinetically balanced Slater-type functions. The non-relativistic GGA for F is used there. 

To summarise, the pure valence correlation (double excitations) tends to lower the spin-orbit splitting while the valence spinors relaxation (single excitations) tends to increase the splitting. Concerning the role of the core orbitals, the whole core polarisation, core-core and core-valence correlations taken into account via a semi-empirical CPP tend to enhance this splitting. ... 

The solution of the Dirac-F ock equation is a set of four-component spinors. If the spinors are partitioned as core and valence spinors, then one can write the overall many-electron relativistic wavefunction for a single configuration as... 

Fig, 5. Radial parts of the Dirac-Hartree-Fock valence spinors of Ce in the 4f 5d 6s ground state configuration, Calculated with the program GRASP (Dyall et al. 1989),... 

In plane-wave calculations of solids and in molecular dynamics, the separable pseudopotentials [93,492,515] are more popular now because they provide linear scaling of computational effort with the basis-set size in contrast to the radially local RECPs. Moreover, the nonlocal Huzinaga-type ab-initio model potentials [521-523] conserving the nodal structure for the valence spinors are often applied. Contrary to the... 

The difference between the models is not large, as shown by the data in table 7.3 for hydrogen-like mercury. The effect of the finite radius is approximately 2 h, but the difference between the models is only 0.01 E. For the valence spinors of many-electron atoms, the difference between the models is well below the limits of chemical... 

To formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

Once the projection operators have been introduced we may remove the requirement that the valence spinors should be orthogonal to the core spinors From the properties of determinants, we know that we ean always add a linear combination of the core spinors to the valence spinors without ehanging the total wave function. The resulting spinor we term a pseudospinor,... 

If 1 , is a valence spinor, we may insert the valence projection operator as defined by (20.8) into (20.17) to obtain a projected equation that is satisfied for any valence spinor. 

The sum on the right is restricted to the valence spinors because the terms involving the core spinors vanish. Now that we have introduced the projection operators, the equation is satisfied by the valence pseudospinors (t) of (20.12) as well. This is the SCF equation we would get from the projected frozen-core Hamiltonian. 

If there is only one valence spinor, and we substitute the pseudospinor for the spinor (20.20) simplifies to... 

We would also like to partition the Fock operator into a valence and a core part this we will do later. In the rest of this section, we explore the development of the generalized Philips-Kleinman pseudopotential for many valence electrons and many valence spinors. 

The generalized Philips-Kleinman pseudopotential depends on the eigenvalue of the spinor v, unlike the frozen-core pseudopotential, which depends only on the core spinors. The appearance of the term in the pseudopotential came about because we transferred a term from right-hand side of (20.21). This means that we have changed the metric, which has implications for orthogonality that we pursue later. The new operator makes the core spinors degenerate with the valence spinor ... 

The formation of a pseudospinor follows naturally we can always mix degenerate spinors. There is a danger here as well. If a reasonable representation of a core spinor could be made from the basis set, it too would have an eigenvalue equal to the valence spinor. Moreover, neither the overlap nor the one-electron matrix element between the core spinor and the valence pseudospinor would be zero. This would complicate matters in a real system, and it is therefore imperative to keep core-like functions out of the basis set. 

An atom in which we can only have one spinor in the valence region is not very interesting. We need a theory in which there are a number of valence spinors, and perhaps more than one of any given symmetry. Furthermore, we are interested not only in the occupied spinors but also in the virtual spinors or some combination of them, since it is these that will be used to provide the needed basis set flexibility in a molecular calculation. 

For spinors of different symmetries there is no problem we simply construct the pseudospinors in the same way for each symmetry. For an atomic shell the radial factorization and spinor degeneracy enable us to construct a single radial pseudopotential for the entire shell. And because there is rarely more than one symmetry in what we consider to be the valence spinors, this permits us to construct valence pseudopotentials for most atoms in the periodic table. 

For the main group elements, the lowest spinor is usually one of the valence spinors, which means that at least the valence pseudospinors for the atom will not be approximated, and these are the most important spinors in a molecular calculation. However, for the transition metals, it is often important to have both the ns and the (n -I- l)s spinors in the valence space. The pseudopotential must then be chosen for the ns pseudospinor, and this choice will affect the (n -I-1 )s pseudospinor. 

The presence and form of this last term has some important consequences. The first is that the definition of the pseudopotential is determined to some extent by the definition of the pseudospinors. This means that the pseudospinors and the pseudopotentials are not independent. The only dependence on the valence spinors in yOPK comes from the spinor eigenvalue, but the remainder terms and the term that came from the renormalization contain the valence pseudospinors, and therefore the detailed shape of the potential is dependent on the shape of the valence pseudospinors. 

The negative sign for the coefficient of rlr, implies a reduction in the amplitude of the pseudospinor in the valence region. So, if / only contains core and valence spinors, we are in the same situation as for the Philips-Kleinman pseudospinor, with a reduction in the valence density outside the chosen radius and the resultant problems with the pseudopotential. This means that fv must contain virtual spinors as well as occupied spinors, and the sum of virtual spinors must compensate for the reduction in the valenee spinor outside the radius of the core spinors (which is not necessarily the same as rc). 

The equation that is used to ensure that the core spinors are projected out is the Huzinaga-Cantu equation (Huzinaga and Cantu 1971, Huzinaga et al. 1973), which we will now derive. Instead of writing the Fock equations for the valence spinors in terms of Lagrange multipliers and imposing the criterion that the overlaps must be zero at convergence, we can introduce projection operators into the Fock operator ... 

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