Unpolarized basis sets

One should mention however that our conclusions have been very recently questionned by Axe and Marynick (42) who carried out calculations on the reaction (3) with various basis sets ranging from split valence to double zeta quality, with and without polarization functions on C, O and H atoms. They found a marked increase in the endothermicity value on going from the unpolarized basis sets ( values ranging between 8.7 and 15.2 kcal/mol) to the polarized basis s.ets (with values between 19.5 and 25.2 kcal/mol, i.e. close to our SD-CI values). We have now carried out calculations adding to our original split valence basis set polarization functions on C, O and H. One polarized set includes the two sets of polarization functions ( = 0.920... 

Calculations of a similar nature have demonstrated that replacement of both hydrogens of water, yielding dimethyl ether, also has only a minor effect upon the nature of the H-bond in the water dimer. With their polarized basis set, and with inclusion of corrections for BSSE, dispersion, and intramolecular correlation effects, these authors found the first methyl substitution raises the binding energy by 0.5 kcal/mol and the second by 0.6. The authors cautioned that an unpolarized basis set would fail to pick up these small effects, which they attribute to Coulomb and dispersion components of the interaction. 

Data computed for the intramolecular vibrational modes of HF and its dimer are reported in Table 3.43. Taking the values in the last row, computed with a very extended basis set, as a benchmark, it is immediately apparent that frequencies computed with small unpolarized basis sets are several hundred cm too small. 3-21G is probably the worst offender in this regard. Once polarization functions have been added, even a single set, the frequencies are more in line with those of the better basis sets. The same patterns are observed in the intensities which are significantly underestimated with the small unpolarized basis sets. 

The water dimer adds a number of new dimensions to the problem since each water molecule contains three vibrational frequencies instead of one. The two stretching modes are labeled V, and Vji Vj refers to the symmetric bending motion. The frequencies computed for the water monomer are reported in the first three columns of Table 3.45, followed by the corresponding frequencies in the dimer. As in the case of (HF)2, the unpolarized basis sets strongly underestimate the stretching frequencies in the monomer. On the other hand, the bending frequency is computed reasonably well with all of the sets, albeit the small unpolarized sets do yield a bit of an overestimate. Rather similar patterns are evident in the dimer as well. The unpolarized sets underestimate v and Vj and yield a small overestimate, by less than 100 cm of the frequency for V2. 

A comprehensive comparison of various basis sets for the homodimers of HF and H2O offers hope that calculation of vibrational frequencies can be meaningful, even when restricted to the SCF level and with no account of anharmonicity. The frequencies are less demanding of basis set quality than are the intensities. Minimal basis sets are to be avoided in most cases, as are small split-valence sets such as 3-21G. In some cases, one can compute reasonable estimates of dimerization-induced frequency shifts with basis sets of 4-3IG type however, results with unpolarized basis sets can be deceptive. Polarization functions are strongly recommended for uniform quality of results, particularly if one is interested primarily in spectral changes induced by H-bond formation. Intensity calculations without polarization functions can be expected to yield only the crudest of estimates. Reasonable results can be achieved with only one set of such functions on each atom. In some cases, it may be useful to include diffuse + functions as well. 

The main difference between the G2 models is tlie way in which tlie electron correlation beyond MP2 is estimated. The G2 method itself performs a series of MP4 and QCISD(T) calculations, G2(MP2) only does a single QCISD(T) calculation with tlie 6-311G(d,p) basis, while G2(MP2, SVP) (SVP stands for Split Valence Polarization) reduces the basis set to only 6-31 G(d). An even more pruned version, G2(MP2,SV), uses the unpolarized 6-31 G basis for the QCISD(T) part, which increases the Mean Absolute Deviation (MAD) to 2.1 kcal/mol. That it is possible to achieve such good performance with tliis small a basis set for QCISD(T) partly reflects the importance of the large basis MP2 calculation and partly the absorption of errors in the empirical correction. 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

The results confirm the inadequacy of the unpolarized 4-3IG basis and the energy-optimized singly polarized DZP basis set, while DZP is seen to give results close to the ones obtained by the DZPP basis. Table V shows a large BSSE for the DZP basis because the polarization function exponents are not energy-optimized. One therefore has to correct for this BSSE in order to get meaningful interaction energies. 

Key to the optimized basis sets is shown in Table 10 the symbol B0 is used to denote the unpolarized sp basis set. In the ROHF and UHF calculations for SFg, the polarization sets used were the ones determined for the ion in the DF1 calculations. 

Minimal basis sets can be recommended only with reservations. Distortions of various components to the interaction energy are difficult to avoid with such small sets, and basis set superposition error is generally rather large. Of the various types of minimal sets, MlNI-1 is superior to STO-nG. In connection with slightly larger sets, 4-3IG or 6-3IG seem to yield much better results than 3-21G. In any case, there is little point in attempts to add correlation to either minimal or unpolarized double-i basis set calculations. 

The pseudo-atomic basis functions (p j) are obtained by solving the Kohn-Sham equation for a spherical symmetric spin-unpolarized neutral atom selfconsistently. From this procedure, we obtain for each atom type optimized atomic basis sets and atomic densities p , which are used to... 

Table 8 The equilibrium geometry of the water molecule optimized with unpolarized and polarized basis sets at the Hartree-Fock level...

The first point to note about the correlation-consistent basis sets in Table 8.16 is that the convergence is in all cases uniform and systematic - for the energies, for the bond distances, and for the bond angle. Scrutiny of the table reveals that, with each increment in the cardinal number, all errors are reduced by a factor of at least 3 or 4. Clearly, the correlation-consistent basis sets provide a convenient framework for the quantitative study of molecular systems at the Hartree-Fock level. We also note that the results for the cc-pVXZ and cc-pCVXZ basis sets are very similar. Apparently, the molecular core orbitals are quite atom-like and unpolarized by chemical bonding. In Hartree-Fock calculations, therefore, the use of the smaller valence cc-pVXZ sets is recommended. 

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