Frozen-core basis sets

Table 8 Reduced one-bond metal-ligand spin-spin coupling constants for some transition metal complexes, in 1019T2J 1. DFT computations (BP non-hybrid functional) employing all-electron Slater-type basis sets for the 3d metal M-C and M-0 couplings, Slater-type frozen-core basis sets for the metals in all other cases. Data taken from Khandogin Ziegler, Ref. 119...

HypcrChcrn sup )orLs MP2 (second order Mollcr-Plessct) correlation cn crgy calculation s tisin g ah initio rn cth ods with an y ava liable basis set. In order lo save mam memory and disk space, the Hyper-Chern MP2 electron correlation calculation normally uses a so called frozen -core" appro.xiniatioii, i.e. the in n er sh ell (core) orbitals are om it ted,. A sett in g in CHKM. INI allows excitation s from the core orbitals lo be included if necessary (melted core). Only the single poin t calcii lation is available for this option. ... 

ADF uses a STO basis set along with STO fit functions to improve the efficiency of calculating multicenter integrals. It uses a fragment orbital approach. This is, in essence, a set of localized orbitals that have been symmetry-adapted. This approach is designed to make it possible to analyze molecular properties in terms of functional groups. Frozen core calculations can also be performed. 

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 localized basis function for the set 0 (Is) are usual frozen-core valence-shell Cl states all the bound states involved in the present calculations are also described at this level. 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

In G3(MP2) theory, the MP2(fu)/G2Large calculation of G3 is replaced with a frozen core calculation with the G3MP2Large basis set [23] that does not contain the core polarization functions of the G3Large basis set. 

As in the recent QCCD study by Head-Gordon et al. (28, 128), we tested the ECCSD, LECCSD, and QECCSD methods, based on eqs (52)-(59), using the minimum basis set STO-3G (145) model of N2. In all correlated calculations, the lowest two core orbitals were kept frozen. As in the earlier section, our discussion of the results focuses on the bond breaking region, where the standard CCSD approach displays, using a phrase borrowed from ref 128, a colossal failure (see Table II and Figure 2). 

The interaction energy and its many-body partition for Bejv and Lii r N = 2 to 4) were calculated in by the SCF method and by the M/ller-Plesset perturbation theory up to the fourth order (MP4), in the frozen core approximation. The calculations were carried out using the triply split valence basis set [6-311+G(3df)]. 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

We use Slater type basis sets that are of triple- quality in the valence region. These basis sets are augmented by two (all elements in ZORA calculations elements H - Kr in QR calculations ADF standard basis V) or one (all other elements in QR calculations ADF standard basis IV) sets of polarization functions. For Pauli (QR) calculations, we use the frozen core approach (27). The (frozen) core orbitals are... 

For the HgH system numerical wavefunctions were obtained for Hg using both relativistic (Desclaux programme87 was used) and non-relativistic hamiltonians. The orbitals were separated into three groups an inner core (Is up to 3d), an outer core (4s—4/), and the valence orbitals (5s—6s, 6p). The latter two sets were then fitted by Slater-type basis functions. This definition of two core regions enabled them to hold the inner set constant ( frozen core ) whilst making corrections to the outer set, at the end of the calculation, to allow some degree of core polarizability. The correction to the outer core was done approximately via first-order perturbation theory, and the authors concluded that in this case core distortion effects were negligible. 

The splitting factor of the d-polarization functions for the 3df basis set extension is 3 rather than the factor of 4 used for first- and second-row atoms. The 3d core orbitals and Is virtual orbitals are frozen in the single-point correlation calculations. 

Example 4. Calculation of CBS-Q Energy for CH4 The geometry is first optimized at the HF/6-31G(d ) level and the HF/6-31G(d ) vibrational frequencies are calculated. The 6-31G(d ) basis set combines the sp functions of 6-31G with the polarization exponents of 6-311G(d,p). A scale factor of 0.91844 is applied to the vibrational frequencies that are used to calculate the zero-point energies and the thermal correction to 298 K. Next the MP2(FC)/6-31G(d ) optimization is performed and this geometry is used in all subsequent single-point energy calculations. In a frozen-core (FC) calculation, only valence electrons are correlated. 

Another series of composite computational methods, Weizmann-n (Wn), with n = 1-4, have been recently proposed by Martin and co-workers W1 and W2 in 1999 and W3 and W4 in 2004. These models are particularly accurate for thermochemical calculations and they aim at approximating the CBS limit at the CCSD(T) level of theory. In all Wn methods, the core-valence correlations, spin-orbit couplings, and relativistic effects are explicitly included. Note that in G2, for instance, the single-points are performed with the frozen core (FC) approximation, which was discussed in the previous section. In other words, there is no core-valence effect in the G2 theory. Meanwhile, in G3, the corevalence correlation is calculated at the MP2 level with a valence basis set. In the Wn methods, the core-valence correlation is done at the more advanced CCSD(T) level with a specially designed core-valence basis set. 

Seven different results are given for each basis set, and in all of them the C Is orbital is doubly occupied in a frozen core. They are coded as follows ... 

Except for occasional discussions of the basis set dependence of the results, the numerical implementation issues such as grid integration techniques, electron-density fitting, frozen-cores, pseudopotentials, and linear-scaling techniques, are omitted. 

Table 4-1. CPU time for the perturbation selection. Cyan Fluorescent Protein, C H l C (Crsymmetry), with DZP level basis sets. The Is core and corresponding virtual orbitals were frozen. Total number of active space is 290 (51 occ. 239 unocc.)...

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