Atomization, 163 correlation energy

Ec = E c - Ex have been employed. On the one hand, LDA and GGA type correlation functionals have been used [14], However, the success of the LDA (and, to a lesser extent, also the GGA) partially depends on an error cancellation between the exchange and correlation contributions, which is lost as soon as the exact Ex is used. On the other hand, the semiempirical orbital-dependent Colle-Salvetti functional [22] has been investigated [15]. Although the corresponding atomic correlation energies compare well [15] with the exact data extracted from experiment [23], the Colle-Salvetti correlation potential deviates substantially from the exact t)c = 8Ecl5n [24] in the case of closed subshell atoms [25]. 

Kutzelnigg, W., Morgan, J.D. III. Rates of convergence of the partial-wave expansions of atomic correlation energies. J. Chem. Phys. 1992, 96, 4484-508. 

Hamiltonian [38]) one finds the parameters of Table 1 for the PW91 correlation GGA [28]. Again, compared with the RLDA or the nonrelativistic GGA atomic correlation energies are clearly improved by this RGGA. 

Table 5.6 demonstrates once more the well known fact that the nonrelativistic LDA overestimates the exact atomic correlation energies by about a factor of 2. Here, however, not the accuracy of the complete functional (4.10) is of interest, but rather the relativistic corrections AE and Ej, as the correction scheme (4.10) could be combined with more accurate nonrelativistic c[n] like GGAs. Table 5.6 shows that both AEc and Ej are much smaller than their x-only counterparts. On the other hand, AE and Ej add up constructively so that the total correction AE + Ej is somewhat closer to AE + El than the individual components For Hg one obtains AEl + Ej = — 0.49 hartree within MBPT2 compared with the exact AE + El of about 2.19 hartree. Nevertheless, in absolute values the relativistic corrections to [n] are clearly less important than those to ,t[n]. 

The next obvious step was to obtain reliable estimate of the atomic correlation energy, but to do this 1 was in need of computing a reasonable accuracy the relativistic correction at least for the non-relativistic atoms from Z = 1 to Z = 36, the back bone of the chemical world the task was accomplished quickly [44]. At this point, using the laboratory available atomic ionization potentials (or suitable extrapolations obtained using the atomic data), it was simple to estimate rather accurately (within 0.001 hartrees) the atomic correlation correction, but more important, it was possible to learn of the notable regularity of the correction [45-47] and from this to extrapolate from atoms to simple molecules [48]. The correlation energy data were revised a few times but it took about 20 years before someone would produce a relatively final set of data [49]. 

Becke s 1995 correlation functional (B95) [168] was constructed to satisfy the following set of conditions (a) the correct uniform density limit (b) separation of the correlation energy into parallel-spin and opposite-spin components (c) zero correlation energy for one-electron systems (d) good fit to the atomic correlation energies. These requirements are met by the following analytic form... 

Brueckner-Goldstone (BG) MBPT [91,92] was used for the determination of atomic correlation energies and polarizabilities and soon applied... 

Figure 3. Comparison of total atomic correlation energies from several sources difference between full and HF subhamiltonians ( ) difference between total ionization energies and accurate HF calculations with relativistic corrections [15,16] (x) linear extrapolation of the latter values [20] (--) second-order M0ller-Plesset calculations [17,18]...

Becke s B95 meta-GGA correlation functional, containing two parameters whose values were fitted to atomic correlation energy data, is often used. The TPSS (Tan, Perdew, Staroverov, Scuseria) meta-GGA functional has given good results for many properties. A reparametrization of TPSS gave the oTPSS functional, where o is for optimized [L. Goerigk and S. Grimme, J. Chem. Theory CompuL, 6,107 (2010)]. 

Figure 1. Atomic correlation energi as a function of atomic number Z, Circles show the results of Clementi [18], essentially as extracted from experiment. Upper dashed curve shows results based on local density use of homogeneous electron liquid correlation energy. Lowest dashed curves show the results using the He-atom based result of CoUe and Salvetti [19]. Plainly atoms are not described quantitatively by local density electron liquids. (Redrawn from March and Wind [17])...

A more sensitive test for correlation functionals than total atomic correlation energies is provided by atomic EAs. In Table 2.11 the EAs for obtained with various functionals are listed [62]. In all cases the exact exchange is used, only the correlation part of varies. As to be expected, the x-only calculation predicts H to be unbound, while LDA correlation... 

M. Klobukowski, Atomic correlation energies from effective-core-potential and model-potential calculations, Chem. Phys. Lett., 172, 361-366 (1990). 

MP2 correlation energy calculations may increase the computational lime because a tw o-electron integral Iran sfonnalion from atomic orbitals (.40 s) to molecular orbitals (MO s) is ret]uired. HyperClicrn rnayalso need additional main memory arul/orcxtra disk space to store the two-eleetron integrals of the MO s. 

A more useful quantity for comparison with experiment is the heat of formation, which is defined as the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states. The heat of formation can thus be calculated by subtracting the heats of atomisation of the elements and the atomic ionisation energies from the total energy. Unfortunately, ab initio calculations that do not include electron correlation (which we will discuss in Chapter 3) provide uniformly poor estimates of heats of formation w ith errors in bond dissociation energies of 25-40 kcal/mol, even at the Hartree-Fock limit for diatomic molecules. 

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