Picture-change error

The inclusion of (nonrelativistic) property operators, in combination with relativistic approximation schemes, bears some complications known as the picture-change error (PCE) [67,190,191] as it completely neglects the unitary transformation of that property operator from the original Dirac to the Schrodinger picture. Such PCEs are especially large for properties where the inner (core) part of the valence orbital is probed, for example, nuclear electric field gradients (EEG), which are an important... 

Figure 2. Quadrupolar point charge arrangement for the picture-change-error-free determination of electric field gradients. The six charges are placed close to the nucleus under consideration (10 —10 a.u. around...

Figure 3. Picture-change error, relativistic and correlation contributions for the coinage metal EFGs in their diatomic chloride compounds. Units are given in a.u.

One of the first molecular DHF EFG studies was done by Visscher et al. [162] for the hydrogen halides at the MP2, CCSD and CCSD(T) level. Here picture-change errors are absent and instead of the PCNQM model the common expression for the EFG operator can be applied. In a relativistic single-determinantal formalism the molecular electronic spinor i(r) consists of four components... 

The authors in [162] now associate with the picture change error the assumption that the operators Xi and Qzz commute leading to the expression... 

By applying the spin-free DHF formalism by Dyall [164] the spin-orbit effects can be separated from the scalar-relativistic effects and also allow for a determination of the PCE. The SO-effect is much smaller (-0.052 a.u. for the I field gradient in HI) than the picture-change error (-0.974 a.u. according to [163] but only -0.330 a.u. according to [162]). The differences in the calculated PCE can be attributed to the different approaches used. 

The neglect of this mandatory transition has been referred to as the picture-change effect or picture-change error (PCE) [762,763]. It should be emphasized that this phrase can easily cause misconceptions and has been a source of confusion over the years. As a matter of fact, the PCE is not a physical effect at all but a pure artifact due to an incomplete and therefore incorrect formulation of the property expression under consideration within the two-component framework. 

A formal analysis of the size of picture-change errors is, however, by no means simple. Although one could immediately try to write down (for electric... 

Hence, for the Ft complex, the scalar-relativistic DKH Hamiltonian and the spin-orbit ZORA-Hamiltonian recover relativistic effects on the electron density of the valence region found in four-component calculations. This is particularly remarkable since the DKH density has not been corrected for the picture-change error in these calculations. The difference density maps for the Ft complex clearly show the importance of the scalar-relativistic contraction of the inner shells, while spin-orbit effects can be neglected for this closed-shell complex (otherwise, the DKH results would deviate much more from the fourspin-orbit-coupling-induced splitting of the d shell is small but noticeable. 

L. Bucinsky, S. Biskupic, D. Jayatilaka. Picture change error correction of radon atom electron density. /. Chem. Phys., 133 (2010) 174125. 

Bucinsky L, Kuckova L, Malcek M, KoJisek J, BiskupiiS S et al (2014) Picture change error in quasirelativistic electron/spin density, Laplacian and bond critical points. Chem Phys 438 37 7... 

Turning now to the 2-component relativistic Hamiltonians we observe that the eXact 2-Component (X2C) Hamiltonians reproduces the DC orbital energies quite well, although there is a difference of about 10 Ei, for the lsi/2 level. This discrepancy stems entirely from picture change errors in the two-electron operator since the one-electron X2C Hamiltonian by construction reproduces exactly the positive-energy spectrum of the parent 4-component Hamiltonian. In the present calculations the one-electron X2C Hamiltonian... 

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