ZORA equation

It was previously shown [7] that one can, starting from the Dirac-Fock equation, derive the two component scaled ZORA equation... 

Using only the one component part of the operator we arrive at the scalar scaled ZORA equation... 

The scalar ZORA method has been implemented in the standard non relativistic Ab Initio electronic structure program GAMESS-UK [8]. The technical details of this implementation will be given in the following section. Comparing the Schrodinger equation with the ZORA equation (7) one sees that application of the ZORA method has resulted in a potential dependent correction on the kinetic energy term. 

It is interesting to note that the Coulomb matrix and the matrix of the nuclear potential present in Vc are opposite in sign. This means that an underestimation, or complete neglect, of the Coulomb matrix will lead to a larger Vc and thus to an overestimation of the relativistic effect. If Vc is negligable compared to 2c the ZORA equation reduces to the non relativistic Schrodinger equation. 

The zeroth-order regular approximation (ZORA) Hamiltonian can be derived from the upper part of the transformed Dirac equation (20). By using the ZORA ansatz for the small component (5) and assuming that the upper and the lower components are equal, the final ZORA equation for the upper component becomes... 

The assumption that the upper and the lower components were equal can actually be used to define the lower half of the ZORA equation. The lower ZORA... 

E. van Lenthe, The ZORA equation. Thesis, University of Amsterdam, 1996. 

The lORA equation corresponds to the ZORA equation with a modified metric operator. The numerical results with the lORA method show a considerable improvement over ZORA and for a many-electron system superior performance to FORA. While one of disadvantages of the ZORA and lORA methods is an incorrect dependence of energy eigenvalues on the choice of gauge in the electrostatic potential, the RA approach has an advantage of easier implementation in the MO calculation than other relativistic approximate approaches as will introduce in the following session. 

A truncation of this expansion for u) defines the zeroth- and first-order regular approximation abbreviated as ZORA and FORA, respectively [702]. The ZORA equation is then obtained from... 

This is the ZORA equation for an atomic Coulomb potential with = r, and... 

In the derivation of the ZORA equation, we made the assumption that < 2mc so that we could do the expansion of the inverse operator. However, the ZORA equation has a valid spectrum for all E. It is therefore not necessarily the case that the use of a truncated expansion is invalid outside the strict radius of convergence. We could... 

There are several ways in which we can develop a perturbation series for the ZORA equation. The first is simply to ignore the normalization—a perfectly valid procedure since the wave function is only defined up to a multiplicative constant. This we will do later in the present section. The second is to follow the same procedure as in the development of the Pauli Hamiltonian in chapter 17, and the third is to start from the Foldy-Wouthuysen transformation, as in the development of chapter 16. The last two of these both explicitly involve the normalization. We will commence here with the procedure used in chapter 17. 

To evaluate the integral, we may make use of the scaled coordinate system that was used to relate the Dirac and the ZORA equations in the previous section to show that... 

What is the relation between the lORA energies and the ZORA and Dirac energies There is a correspondence at =0 and we expect that the correspondence continues in the vicinity of this point. Unlike the ZORA equation, we cannot perform a scaling to obtain a relation with the Dirac ESC equation, and therefore we cannot obtain a direct relation with the Dirac eigenvalues. What we can do is to make use of the Rayleigh quotient for (18.37) to obtain a relation between the ZORA and lORA eigenvalues, since ZORA and lORA have the same Hamiltonian but a different metric. For an arbitrary wave function r] . 

Substitution of Xq for X in (19.14) gives the ZORA equation in the modified Dirac representation... 

As can be seen, the spin-orbit contribution (term 2) in equation (12.3) is present even at this lowest order of the expansion. Equation (12.3) shows the separation of the ZORA operator into a scalar (spin-free) part and the SO operator (the electron spin-dependent term with a). With )C 1 - - Vnuc/(2c ), the ZORA SO operator becomes equivalent to the Breit-Pauli one-electron counterpart in order However, the scalar part of ZORA misses some contributions in order c . The ZORA equation can also be written as,... 

Sun Q, Xiao Y and Liu W 2012 Exact two-component relativistic theory for NMR parameters General formulation and pilot application. J. Chem. Phys. 137(17), 174105—20. van Lenthe E 1996 The ZORA Equation PhD thesis Vrije Universiteit Amsterdam, Netherlands. He6 BA, Marian CM, Wahlgren U and Gropen O 1996 A mean-field spin-orbit method applicable to correlated wavefunctions. Chem. Phys. Lett. 251, 365—371. 

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