Second-order regular approximation

SOCI spin-orbit configuration interaction SOFT second-order perturbation theory SOREP spin-orbit relativistic effective potential TD-DFT time dependent density functional theory ZORA zero-order regular approximation... 

For transition metals from the second and third rows (like Mo and W), the inner electrons start to move at speeds approaching that of light. Consequently, relativistic effects may become sizeable. There are many ways to treat these effects, but the two most common ones for enzyme models are effective core potentials (ECP) or the zeroth-order regular approximation (ZORA). ECP means that the inner electrons are replaced by special one-electron potentials, which actually make the calculations faster. ZORA calculations, on the other hand, require special basis sets and are more demanding than non-relativistic calculations. However, in our test calculations, the two approaches have given similar results, with differences of 0-5 kj mol". ... 

The operator in the second bracket is the ZORA Hamiltonian, and it is sandwiched by normalization operators. If we expand these operators as we did above, we get the FORA Hamiltonian as the first term. The higher terms differ, however, because the final energy in the previous series must be the Dirac energy, whereas here it is the energy for the approximate Hamiltonian. Inclusion of the normalization terms corresponds to a resummation of certain parts of the ZORA perturbation series to infinite order, and the name coined by Dyall and van Lenthe (1999) is lORA—infinite-order regular approximation. 

Computational Studies. Of the three significant theory reports, two dealt with chemical reactivity while the second focused on eleetronie structure. The electronic and molecular structures of select bis-r -arene lanthanides and actinides were investigated at the scalar-relativistic level with an all-eleetron DFT method using the zero-order regular approximation Hamiltonian and with... 

The second term can be thought of as an effective kinetic energy operator that goes to the non-relativistic one when V 0. Proper renormalization gives the Infinite Order Regular Approximation (lORA) [17], often approximated by scaled ZORA [16], which improves on ZORA. 

Observe that provides a second-order approximation at the regular nodes A w — LaU = O(h ), while A m — L,yU = 0(1) at the irregular ones. 

A second-order approximation provided by the difference operator Aq, on a regular pattern... 

Of course, this second branch also accommodates DPT methods with approximate contemporary functionals. Again, approximate relativistic Hamiltonians such as the DKH or the ZORA Hamiltonian will do a good job. This holds true for the second-order DKH2 Hamiltonian, but one may always choose the fourth-order DKH4 Hamiltonian, which is still a uniquely defined operator (i.e., it is independent of the parametrization chosen for the unitary transformation). The fact that the lowest-order Hamiltonians in the regular approximation and in the DKH scheme work so well is also the reason why ZORA and DKH2 Hamiltonians yield similarly good results [1161]. Both Hamiltonians are now well established and heavily used in actual calculations. Needless to say, these all-electron methods are mandatory when effects of the core electrons become decisive (as, for instance, in MoRbauer spectroscopy). 

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