Orbital London

Ward, N.I. and J.M. Savage. 1994. Elemental status of grazing animals located adjacent to the London Orbital (M25) motorway. Sci. Total Environ. 146/147 185-189. 

Before one can compare different correlated methods and their results for the rotational g tensor, one should discuss the quality of the employed basis set. Therefore, I have performed also calculations with rotational London orbitals for all the molecules at the SCF and the level of theory and compare... 

Using perturbation dependent atomic orbitals (rotational London orbitals [61]) as basis functions. 

The differences between the results from the vD + D and daug-cc-pVTZ basis sets are smaller in rotational London orbital calculations than in conventional calculations, as one might expect [61]. Nevertheless, the differences between the vD + D and daug-cc-pVTZ results are not larger than 0.5% for all methods used here even without the usage of rotational London orbitals. Furthermore, the differences between the rotational London orbital results and conventional results for the vD + D basis set, 0.2% for CH4 and 0.1% otherwise, are smaller than for the daug-cc-pVTZ basis set and often smaller than the differences between rotational London orbital results from the two basis sets. 

I conclude therefore that the vD + D basis is closer to the basis set limit than the daug-cc-pVTZ basis set and that no important improvement could have been obtained by using rotational London orbitals in all calculations. 

Finally, I can compare my SCF and CAS results with corresponding rotational London orbital calculations by Ruud and co-workers [32-36,61]. My SCF results for HF using the vD + D basis set and conventional or rotational London orbitals are almost identical to a rotational London orbital result 0.7627 using the daug-cc-pVQZ basis set [34,61] or a result 0.7626 obtained with a large ANO basis set [34,35]. Similarly the 1000 54220 iooO(. 5633i jjp... 

Using perturbation dependent atomic orbitals (rotational London orbitals [61]) as basis functions. Experimental values for the u = 0 state gx =0.5654 + 0.0007 and gn =0.5024 + 0.0005 [74] minus a zero-point-vibrational correction Agx = —0.0135 and Agn = —0.0062 calculated with a... 

On the other hand, I observe larger dilferences between my SCF rotational London orbital results for HF, H2O and NH3 using both basis sets and the corresponding results in another study by Ruud et al. [32]. Similar larger deviations are found between the pjjjg jg partly due to slightly... 

It is found that the basis sets employed are well converged even though no rotational London orbitals are used in the calculations. The correlation effects are rather small, in the range of 1.5-3.5%, and for all but methane the correlation corrections are negative. 

SCF and MSCF electric field dependence of the magnetizability and nuclear magnetic shielding have been studied by Rizzo et al (17-19) within GIAO basis sets. The use of London orbitals guarantees invariance of theoretical estimates in a change of coordinate system, which is a basic requirement in the computation of magnetic response properties. 

London orbitals were introduced by Fritz London, who in 1937 used Hiickel theory to calculate the contribution to the magnetizability from the ring currents in the 7r-orbital backbone of some aromatic molecules [13]. The great virtue of London s approach is that each individual AO—the building blocks of molecular wave functions—has been harnessed to respond correctly (to first order at least) to the application of an external magnetic field, irrespective of the choice of the external gauge origin. Moreover, since, in London s approach, only the atomic orbitals are modified, this method is fully transparent to the treatment of the electronic structure otherwise. 

To illustrate the remarkable improvements in basis-set convergence that are obtained through the use of London orbitals, we have in Figure 1 plotted the isotropic magnetizability for the phosphorous trifluoride molecule (PF3) calculated with and without the use of London orbitals [14]. In these... 

Figure 1 The isotropic magnetizability (in ppm cgs) of PF3 calculated with and without the use of London orbitals...

Such a basis set combines well with coupled-cluster wave functions to tend to converge in a consistent and predictable manner towards limits of the basis set and the theory. Calculation of the rotational g tensor and magnetizability involved use of rotational London orbitals [10]. Optimization, first order in derivatives of energy with respect to internuclear distances, yielded all reported geometric stmctures of... 

Atomic orbitals are used in quantum chemistry as the building blocks of many-electron functions (cf. p. 423). Where to center the orbitals sometimes presents a serious problem. On top of this, in the case of a magnetic field, there is additionally the abovementioned arbitrariness of choice of the vector potential origin. A remedy to the second problem was found by Fritz London in the form of atomic orbitals that depend explicitly on the vector potential applied. Each atomic orbital xif c) centered on nucleus C (with position shown by vector Rc) and describing an electron pointed by vector r, is replaced by the London orbital in the following way ... 

Despite this property, the London orbitals are also known as gauge-invariant atomic orbitals (GIAOs). 

Let us calculate the overlap integral S between two London orbitals centered at points C and D. After shifting the origin of the coordinate system in Eq. (G.14) by vector R, we get... 

This means that when we use the London orbitals, the results do not depend on the choice of vector potential origin. 

The most important feature of London orbitals is that all the integrals appearing in calculations are invariant with respect to the origin of the vector potential. This is why results obtained using London orbitals are also independent of that choice. 

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