Valence polarization functions

Mulliken (41) distinguishes two kinds of polarization. He calls Coulomb polarization what we are concerned with in this paper the polarization produced by an electric field, and he calls valence polarization a kind of polarization du to quantum-mechanical valence forces. In order to correctly describe the chemical bond in H2, it is necessary to include the valence polarization function as soon as one calculates energy with the unperturbed function (i.e. the 2p orbital). 

The largest basis set, denoted as G3Large [21] includes some core polarization functions as well as multiple sets of valence polarization functions. It should be noted that MP2 calculation with the largest basis set in Eq. (3.4) is carried out at the MP2(fu) level. 

While steps (7) and (8) are very similar to the corresponding steps in G2 theory, the last step is somewhat different. The basis set used, denoted as G3Large [32], is significantly larger than the corresponding basis set used in G2 theory and includes some core polarization functions as well as multiple sets of valence polarization functions. In addition, it should be noted that the MP2 calculation in step (9) is carried out at the MP2(full) level. This is done to take account of core-related correlation contributions to total energies. 

Functions with higher / values and with sizes like those of lower-/ valence orbitals are also used to introduce additional angular correlation by pemiitting angularly polarized orbital pairs to be fomied. Optunal polarization functions for first- and second-row atoms have been tabulated and are included in the PNNL Gaussian orbital web site data base [45]. 

Basis sets can be extended indefinitely. The highest MOs in anions and weakly bound lone pairs, for instance, are very diffuse maybe more so than the most diffuse basis functions in a spht valence basis set. In this case, extra diffuse functions must be added to give a diffuse augmented basis set. An early example of such a basis set is 6-31+G [26]. Basis sets may also be split more than once and have many sets of polarization functions. 

An older, but still used, notation specihes how many contractions are present. For example, the acronym TZV stands for triple-zeta valence, meaning that there are three valence contractions, such as in a 6—311G basis. The acronyms SZ and DZ stand for single zeta and double zeta, respectively. A P in this notation indicates the use of polarization functions. Since this notation has been used for describing a number of basis sets, the name of the set creator is usually included in the basis set name (i.e., Ahlrichs VDZ). If the author s name is not included, either the Dunning-Hay set is implied or the set that came with the software package being used is implied. 

Even larger basis sets are now practical for many systems. Such basis sets add multiple polarization functions per atom to the triple zeta basis set. For example, the 6-31G(2d) basis set adds two d functions per heavy atom instead of just one, while the 6-311++G(3df,3pd) basis set contains three sets of valence region functions, diffuse functions on both heavy atoms and hydrogens, and multiple polarization functions 3 d functions and 1 f function on heavy atoms and 3 p functions and 1 d function on hydrogen atoms. Such basis sets are useful for describing the interactions between... 

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. 

In the case of water-soluble polymers, there is another factor that has to be taken into account when considering solubility, namely the possibility of hydrophobic interactions. If we consider a polymer, even one that is soluble in water, we notice that it is made up of two types of chemical species, the polar functional groups and the non-polar backbone. Typically, polymers have an organic backbone that consists of C—C chains with the majority of valence sites on the carbon atoms occupied by hydrogen atoms. In other words, this kind of polymer partially exhibits the nature of a hydrocarbon, and as such resists dissolution in water. 

The calculations were performed using a double-zeta basis set with addition of a polarization function and lead to the results reported in Table 5. The notation used for each state is of typical hole-particle form, an asterisc being added to an orbital (or shell) containing a hole, a number (1) to one into which an electron is promoted. In the same Table we show also the frequently used Tetter symbolism in which K indicates an inner-shell hole, L a hole in the valence shell, and e represents an excited electron. The more commonly observed ionization processes in the Auger spectra of N2 are of the type K—LL (a normal process, core-hole state <-> double-hole state ) ... 

G minimal 6G (nc = 6) core and triple-zeta 3G (nv = 3), lG(nv" = 1) valence sp sets, with polarization functions on heavy atoms... 

G (nv= 1) valence sp sets, with diffuse and polarization functions on all atoms... 

Dunning has developed a series of correlation-consistent polarized valence n-zeta basis sets (denoted cc-pVnZ ) in which polarization functions are systematically added to all atoms with each increase in n. (Corresponding diffuse sets are also added for each n if the prefix aug- is included.) These sets are optimized for use in correlated calculations and are chosen to insure a smooth and rapid (exponential-like) convergence pattern with increasing n. For example, the keyword label aug-cc-pVDZ denotes a valence double-zeta set with polarization and diffuse functions on all atoms (approximately equivalent to the 6-311++G set), whereas aug-cc-pVQZ is the corresponding quadruple-zeta basis which includes (3d2flg,2pld) polarization sets. 

Related basis sets in common usage include the original Dunning full and valence double-zeta sets, denoted D95 and D95V, respectively (built from nine s-type and five p-type primitives). These sets may be augmented in the usual way with diffuse and/or polarization functions, as in the example D95++ (diffuse and first-polarization sets on all atoms). 

We should note that inner polarization is strictly an SCF-level effect while, for instance, switching from an A VDZ to an A,VDZ+2d basis set affects the computed atomization energy of SO3 by as much as 40 kcal/mol ( ), almost all of this effect is seen in the SCF component of the TAE [28], In fact, we have recently found [29] that the effect persists if the (1, v, 2s, 2p) orbitals on the second-row atom are all replaced by a pseudopotential. What is really getting polarized here is the inner part of the valence orbitals, which requires polarizations functions that are much tighter (higher-exponent) than those required for the outer part of the valence orbital. The fact that these inner polarization functions are in the same exponent range as the d and / functions required for correlation out of the (2s, 2p) orbitals is merely coincidental the inner polarization effect has nothing to do with correlation, let alone with inner-shell correlation. 

A speculative proposal was made thirty years ago by Schmid and Krenmayr77, namely that a nitrosyl ion solvated, but not covalently bonded, by a water molecule may be involved in these systems. This hypothesis was investigated theoretically in 1984 by Nguyen and Hegarty78 who carried out ab initio SCF calculations of structure and properties employing the minimal STO-3G basis set, a split-valence basis set plus polarization functions. Optimized geometries of six planar and two nonplanar forms were studied for the nitrosoacidium ion. The lowest minimum of molecular electrostatic potential... 

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