The One-Step Solution

The unitary transformation required for the block diagonalization of the relativistic Fock operator can be obtained in one step if a matrix representation of the Fock operator is available this is achieved by the so-called eXact-2-Component (X2C) approach [725-728,731-734]. An important characteristic of the X2C approach is its noniterative construction of the key operator X of Eq. (11.2). In this noniterative construction scheme, the matrix operator X is obtained from the electronic eigenvectors of the relativistic (modified) Roothaan Eq. (14.13), 

Once the coefficients and have been obtained, Eq. (11.2) can simply be 

Multiplication from the right by then yields for the matrix represen- 

Since the X matrix is directly evaluated from the electronic solutions of the four-component Fock operator, which must therefore be diagonalized, the same pathologies regarding the negative-energy states discussed in chapters 8 and 10 pose a caveat. However, if a four-component calculation must be carried out before the two-dimensional operator can be evaluated (as in the X2C case), the projection to electronic states by elimination of the small component in an exact two-component approach has no valid formal advantage (as the four-component variational solution for the 4-spinors already required (implicit) projection to the electronic solutions). 

the evaluation of the X-operator in matrix form requires the diago-nalization of the modified Dirac-Roothaan equation, but its diagonalization posed the pitfalls that had led to the derivation of exact-decoupling methods in order to avoid pathologies originating from the negative-energy states in the first place. However, if this step can be accomplished with an approximate potential V — V, which does not include the full electron-electron interaction, then it can be a very efficient procedure. 

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The ocean power stations operate on the basis of the temperature difference between the upper and lower water layers. Taking advantage of the one-step solution of ozone in depth, the COD is decreased which prevents the befouling in the piping systems found in the upper layers. The ozone-enriched water from the lower layers is supplied to a cyclic filter and is directly injected into upper seawater for control of the sea microorganisms growth [33]. 

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