Frozen-core approximation correlation

HyperChem supports MP2 (second order Mdllcr-l Icsset) correlation energy calcu latiou s u sin g any available basis set. lu order to save main memory and disk space, the HyperChem MP2 electron correlation calculation normally uses a so called frozen-core approximation, i.e. th e in n er sh el I (core) orbitals are omitted. A sett in g in CHHM.IX I allows excitation s from th e core orbitals to be include if necessary (melted core). Only the single poin t calcula-tion is available for this option. 

The MP2 and CCSD(T) values in Tables 11.2 and 11.3 are for correlation of the valence electrons only, i.e. the frozen core approximation. In order to asses the effect of core-electron correlation, the basis set needs to be augmented with tight polarization functions. The corresponding MP2 results are shown in Table 11.4, where the A values refer to the change relative to the valence only MP2 with the same basis set. Essentially identical changes are found at the CCSD(T) level. 

The structural parameters and vibrational frequencies of three selected examples, namely, H2O, O2F2, and B2H6, are summarized in Tables 5.6.1 to 5.6.3, respectively. Experimental results are also included for easy comparison. In each table, the structural parameters are optimized at ten theoretical levels, ranging from the fairly routine HF/6-31G(d) to the relatively sophisticated QCISD(T)/6-31G(d). In passing, it is noted that, in the last six correlation methods employed, CISD(FC), CCSD(FC),..., QCISD(T)(FC), FC denotes the frozen core approximation. In this approximation, only the correlation energy associated with the valence electrons is calculated. In other words, excitations out of the inner shell (core) orbitals of the molecule are not considered. The basis of this approximation is that the most significant chemical changes occur in the valence orbitals and the core orbitals remain essentially intact. On... 

Intramolecular nucleophilic substitution to form thiiranes was studied by means of ab initio MO computations based on the 6-31G basis set <1997JCC1773>. Systems studied included the anions SCH2CH2F and CH2C(=S)CH2F which would afford thiirane and 2-methylenethiirane, respectively (Equations Z and 3). It was important to include electron correlation which was done with the frozen-core approximation at the second-order Moller-Plesset perturbation level. Optimized structures were confirmed by means of vibrational frequency calculations. The main conclusions were that electron correlation is important in lowering AG and AG°, that the displacements are enthalpy controlled, and that reaction energies are strongly dependent on reactant stabilities. 

It is quite common in correlated methods (including many-body perturbation theory, coupled-cluster, etc., as well as configuration interaction) to invoke the frozen core approximation, whereby the lowest-lying molecular orbitals, occupied by the inner-shell electrons, are constrained to remain doubly-occupied in all configurations. The frozen core for atoms lithium to neon typically consists of the Is atomic orbital, while that for atoms sodium to argon consists of the atomic orbitals Is, 2s, 2px, 2py and 2pz. The frozen molecular orbitals are those made primarily from these inner-shell atomic orbitals. 

A justification for this approximation is that the inner-shell electrons of an atom are less sensitive to their environment than the valence electrons. Thus the error introduced by freezing the core orbitals is nearly constant for molecules containing the same types of atoms. In fact, it is often preferable to employ the frozen core approximation as a general rule because most of the basis sets commonly used in ab initio quantum chemistry do not provide sufficient flexibility in the core region to accurately describe the correlation of the core electrons. Recently, Woon and Dunning have attempted to alleviate this problem by publishing correlation consistent core-valence basis sets.125... 

Table 3. Basis set convergence of first hyperpolarizabilities (Sj (in a.u.) at the (unrelaxed) CCSD level. For FH. CO, and HjO results are given for the static limit /3 (0), while for HCl the results are for 6 (—ft), ft), 0) with i = 0.072003 a.u. (632.8 nm). In all calculations a frozen core approximation with the electrons in the ks shells of non-hydrogen atoms kept inactive in the correlation and response calculations is employed...

Table 17. Convergence of the anisotropy of the static hypermagnetizability An(0) of neon with the correlation treatment (from [39] finite field orbital-relaxed results (unless specified) for d-aug-cc-pVDZ, frozen core approximation for Is shell, numbers in atomic units.)...

Although the frozen-core approximation underlies all ECP schemes discussed so far, both static (polarization of the core at the Hartree-Fock level) and dynamic (core-valence correlation) polarization of the core may accurately and efficiently be accounted for by a core polarization potential (CPP). The CPP approach was originally used by Meyer and co-workers (MUller etal. 1984) for all-electron calculations and adapted by the Stuttgart group (Fuentealba et al. 1982) for PP calculations. The... 

The frozen-core approximation is one basic assumption underlying all ECP schemes described so far. Especially for main group elements, where a large-core ECP approximation works fairly well if not too high accuracy is desired, the polarizability of the cores (Fig. 14) has nonnegligible effects for elements from the lower part of the periodic table. Within the ECP approach it is indeed possible to account for both static (polarization of the core at the HF level) and dynamic (core-valence correlation) polarization of the cores in an efficient way. Meyer and coworkers [202] proposed in the framework of AE calculations the... 

The above correlation consistent prescription has since been used essentially unchanged for all the 2nd-row main group elements Al-Ar and 3rd-row elements Ga-Kr [10-14]. While the sizes of these basis sets generally range from n = D to n = 5 or 6, selected elements have been covered up to as large as cc-pVlOZ [7]. In the case of the post-3d elements, the HF set also included d-type functions, however these cc-pVnZ basis sets defined the 3d electrons to lie within the frozen core approximation. Hence the pattern of valence correlating functions is identical in these cases to the 1st and 2nd row p-block atoms. 

In post-HF calculations, the core electrons (e.g., Is for C, ls 2s 2p for Si) are usually left at the HF level, i.e., they are not correlated. This is called the frozen-core approximation. When desired, the contribution from core correlation is generally computed directly, although core-polarization potentials are effective [71,72] and parameterized estimation schemes are available [73]. 

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