Contraction coefficients

To overcome the primary weakness of GTO fimetions (i.e. their radial derivatives vanish at the nucleus whereas the derivatives of STOs are non-zero), it is coimnon to combine two, tliree, or more GTOs, with combination coefficients which are fixed and not treated as LCAO-MO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs are multiplied by contraction coefficients and suimned to produce a CGTO, which approximates the proper cusp at the nuclear centre. 

This section gives a listing of some basis sets and some notes on when each is used. The number of primitives is listed as a simplistic measure of basis set accuracy (bigger is always slower and usually more accurate). The contraction scheme is also important since it determines the basis set flexibility. Even two basis sets with the same number of primitives and the same contraction scheme are not completely equivalent since the numerical values of the exponents and contraction coefficients determine how well the basis describes the wave function. 

The contracted basis set created from the procedure above is listed in Figure 28.3. Note that the contraction coefficients are not normalized. This is not usually a problem since nearly all software packages will renormalize the coefficients automatically. The atom calculation rerun with contracted orbitals is expected to run much faster and have a slightly higher energy. 

Usually, contractions are determined from atomic SCFcalculations. In these calculations one uses a relatively large basis of uncontracted Gaussians, optimizes all exponents, and determines the SCF coefficients of each of the derived atomic orbitals. The optimized exponents and SCF coefficients can then be used to derive suitable contraction exponents and contraction coefficients for a smaller basis set to be used in subsequent molecular calculations. 

As in the STO-LG basis, the 2s and 2p functions share the exponents for computational efficiency. The contraction coefficients djs, d2s , d2s , d2p , and d2p and the contraction exponents aisp , and a2sp were explicitly varied until the energy of an atomic SCFcalculation reached a minimum. Unlike the STO-NG basis. 

For two-phase flow, the phase contraction coefficients Cqc. nd Col relate the area of each phase Ac and A at the vena contracta to the known area of the orifice Ay. Thus ... 

In a segmented contraction each primitive as a rule is only used in one contracted function. In some cases it may be necessary to duplicate one or two PGTOs in two adjacent CGTOs. The contraction coefficients can be determined by a variational optimization, for example from an atomic HF calculation. 

In a general contraction primitives (on a given atom) and of a given angular momentum enter all the contracted functions having that angular momentum, but with different contraction coefficients. 

One popular way of obtaining general contraction coefficients is from Atomic Natural Orbitals (ANO), to be discussed in section 5.4.4. The difference between segmented and general contraction may be illustrated as shown in Figure 5.2. 

The basis functions are normally the same as used in wave mechanics for expanding the HF orbitals, see Chapter 5 for details. Although there is no guarantee that the exponents and contraction coefficients determined by the variational procedure for wave functions are also optimum for DFT orbitals, the difference is presumably small since the electron densities derived by both methods are very similar. ... 

Regarding current ab initio calculations it is probably fair to say that they are not really ab initio in every respect since they incorporate many empirical parameters. For example, a standard HF/6-31G calculation would generally be called "ab initio", but all the exponents and contraction coefficients in the basis set are selected by fitting to experimental data. Some say that this feature is one of the main reasons for the success of the Pople basis sets. Because they have been fit to real data these basis sets, not surprisingly, are good at reproducing real data. This is said to occur because the basis set incorporates systematical errors that to a large extent cancel the systematical errors in the Hartree-Fock approach. These features are of course not limited to the Pople sets. Any basis set with fixed exponent and/or contraction coefficients have at some point been adjusted to fit some data. Clearly it becomes rather difficult to demarcate sharply between so-called ab initio and semi-empirical methods.4... 

To some other experts the meaning of the term ab initio is rather clear cut. Their response is that "ab initio" simply means that all atomic/molecular integrals are computed analytically, without recourse to empirical parametrization. They insist that it does not mean that the method is exact nor that the basis set contraction coefficients were obtained without recourse to parametrization. Yet others point out that even the integrals need not be evaluated exactly for a method to be called ab initio, given that, for instance, Gaussian employs several asymptotic and other cutoffs to approximate integral evaluation. 

I Meanwhile others object to the suggestion that the optimization of basis sets are carried out by reference to experimental data. While accepting that the exponents and contraction coefficients are generally optimized in atomic calculations, they insist that these optimizations are in themselves ab initio. 

The atomic basis consists in a double-zeta set expanded with polarization functions (DZP) and augmented by diffuse functions (DZPR). Exponents and contraction coefficient are from McLean and Chandler 1980 [18] diffuse functions, centered on the heavy atoms with exponents of 0.023 for the s orbitals and 0.021 for the p orbitals are from Dunning and Hay 1977 [34]. Extension of the DZP basis set with two sets of diffuse s (0.0437, 0.0184) and p (0.0399, 0.0168) functions (DZPRR) has also been tested. 

The original motivation for contracting was that the contraction coefficients dST can be chosen in a way that the CGF resembles as much as possible a single STO function. In... 

Different structural materials have different thermal contraction coefficients, meaning that accommodations should be made for their different dimensions at cryogenic temperatures. If not, problems associated with safety (e.g., leaks) may arise. Generally, the contraction of most metals from room temperature (300 K) to a temperature close to the liquefaction temperature of hydrogen (20 K) is <1%, whereas the contraction for most common structural plastics is from 1% to 2.5% [23]. 

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