Correlation energy, plot

Figure 3. Basis set errors for the HF energy and CISD correlation energy with the final cc-pVnZ-PP basis sets for the ( , ) 5s 4d, (O, ) 5s4(f, and (A, A) 4

Figure 4. Contributions of correlating functions, as well as s, p, and d functions (inset), to the CISD correlation energy of the 5 d state of mercury. The absolute values of the incremental correlation energy lowerings, AEcon are plotted in mEh against the number offunctions in the expansions for spdf... functions. The solid lines are exponential fits.

For both the DZ and TZ sets a contracted function was included for the 6p orbital, but this was deleted in the QZ and 5Z sets due to near linear dependence. The contraction was also deleted from the 5Z set for the same reason. Figure 7 plots the correlation energies for both nonrelativistic and DK-relativistic CISD calculations. The CBS limits using a extrapolation of the QZ and 5Z correlation energies are -391.8 and -418.0 m /, for NR and DK, respectively. 

Figures 2 and 3 plot the valence correlation energies of Ne and FH obtained by various combinations of the CC or CC-R12 methods (using Ten-no s Slater-type correlation function) and basis sets (see ref. 35 for details). These figures are the stunning illustration of the extremely rapid convergence of correlation energies...

An even better correlation can be obtained by combining the steric radical diameter term with the Taft cr, constant. The use of linear free energy plots of the Hammett type to correlate orientation is discussed in detail in Section 8. 

Figure 9. Plot of the correlation energy, given by the [2/1] PadS approximant to the third-order energy, against basis set size for beryllium-like ions.

The inappropriate scaling of the RLDA with Z, and thus also with j6, becomes particularly obvious for fixed electron number. In Fig. 5.5 the percentage deviations of the RLDA for and Ej are shown for the Ne isoelectronic series. The error for the correlation energy in the RLDA shows little tendency to approach zero with increasing Z, indicating that the relativistic correction factor plotted in Fig. 4.4 is inadequate for electronic structure calculations. 

There was a slight rate acceleration with an inductive removal of electrons from the a-carbon. A correlation of the chemical shift of the a-proton in CCI4 with Taft (7 constants was made, and a fairly good linear free energy plot of the relative rate coefficients was obtained with a. This indicates a modest increase in rate by electron withdrawing substituents. 

Nelsen and coworkers [562] detected conformational equilibria in eq, eq- and ax, eq-N, A -disubstituted cyclic hydrazines from their oxidation potentials. The anodic oxidation reactions of trans- and cw-l,3-diisopropy-l-2,4-bis(diisopropylamino)-cyclodiphospha(III)azanes are quite different [563] The trans isomer is reversibly oxidized at 0.53 V (SCE) forming a stable cation radical the c/5-isomer undergoes a completely irreversible oxidation at a more positive potential because an unstable radical cation is formed. Evans and coworkers studied structural changes associated with electron transfer reactions of W(// -C5(CH3)5)(CH3)4 and related compounds [564,565]. Yoshida and coworkers found a linear correlation on plotting the oxidation potentials of a-silylated ethers, where the rotation around the C-0 bond is restricted, against the HOMO energy-torsion angle (Si-C-O-C) curve estimated by MO calculation [566]. 

Alternative definitions parallel mechanisms can be diagnosed when the observed rate on one side of the break-point in a free energy plot is greater than that calculated from the correlation on the other side. Parallel mechanisms are diagnosed if the non-linear free energy correlation exhibits a concave upwards curvature. 

Figure 6-2. Left Correlations between photo-induced hole transport rate constants and structural distances in synthetic capped hair-pin double-strand oligonucleotides for charge separation (open symbols) and charge recombination (filled symbols). Right Free energy plots for charge separation (filled symbols) and recombination (open symbols) Circles and triangles represent different sensitizer molecular units. Reprinted from ref. 52 with permission.

Figure 3 Plot of correlation energy (Ecor, obtained from a CASSCF calculation see text) versus the M d contribution (in terms of percentage) in the bonding eg orbital in a series of octahedral MLe complexes, with L = HjO, NH3, F, CP, Br, and P, and M a transition metal with a formal (a) or c/ (b) occupation number. Solid lines connect metals with a formal charge (+3) dashed lines connect metals with a formal charge (-I-4). For simplicity, the formal charges on the metals have been omitted from the plots.

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