Angular momentum higher

The cc-pVDZ basis contains two AOs for each occupied orbital in the valence region. In addition, there is a set of polarization functions - that is, AOs of angular momentum higher than that of the occupied orbitals in the atoms. As demonstrated by the systematic investigations of basis sets and wave functions in Chapter 15, the cc-pVDZ basis provides only a crude description of electron correlation - it is inadequate for high-precision work but sufficient for semi-quantitative investigations. In particular, since it provides a uniformly accurate representation of the whole energy surface of the water molecule, the cc-pVDZ basis is sufficient for our purposes - that is. 

In addition to the fundamental eore and valenee basis deseribed above, one usually adds a set of so-ealled polarization functions to the basis. Polarization funetions are funetions of one higher angular momentum than appears in the atom s valenee orbital spaee (e.g, d-funetions for C, N, and O and p-funetions for H). These polarization funetions have exponents ( or a) whieh eause their radial sizes to be similar to the sizes of the primary valenee orbitals... 

Polarization functions are functions of a higher angular momentum than the occupied orbitals, such as adding d orbitals to carbon or / orbitals to iron. These orbitals help the wave function better span the function space. This results in little additional energy, but more accurate geometries and vibrational frequencies. 

The parameters cr, (3,7 and 6 are optimized for each atom. The exponents are tire same for all types of angular momentum functions, and s-, p- and d-functions (and higher angular momentum) consequently have the same radial part. 

The derivative of the core operator h is a one-electron operator similar to the nucleus-electron attraction required for the energy itself (eq. (3.55)). The two-electron part yields zero, and the V n term is independent of the electronic wave function. The remaining terms in eqs. (10.89), (10.90) and (10.95) all involve derivatives of the basis functions. When these are Gaussian functions (as is usually the case) the derivative can be written in terms of two other Gaussian functions, having one lower and one higher angular momentum. 

Now we refer to the analysis of a functional relationship between the times of orientational and rotational (angular momentum) relaxation that are rg/ and tj, respectively. To lowest order in Jf/, this relationship is given by the Hubbard relation (2.28). It is universal in the sense that it does not depend on the mechanisms of rotational relaxation. However, this relation does not hold when rg/ is calculated to higher order in Jf/. Corrections to the Hubbard relation are expressed in terms of higher correlation moments of co,(t) whose dependence on tj is specific for different mechanisms. Let us demonstrate this, taking the impact theory as an example. In principle it distinguishes correlated behaviour of the... 

The original semiclassical version of the centrifugal sudden approximation (SCS) developed by Strekalov [198, 199] consistently takes into account adiabatic corrections to IOS. Since the orbital angular momentum transfer is supposed to be small, scattering occurs in the collision plane. The body-fixed correspondence principle method (BFCP) [200] was used to write the S-matrix for f — jf Massey parameter a>xc. At low quantum numbers, when 0)zc —> 0, it reduces to the usual non-adiabatic expression, which is valid for any Though more complicated, this method is the necessary extension of the previous one adapted to account for adiabatic corrections at higher excitation... 

In addition, it is possible that differences in the BSR may arise due to choice of the s-basis from which the higher angular momentum basis functions are constructed. However, on comparison of the Bethe sum rule for basis C, with the sum rule generated by the same method described above, from another energy optimized, frequently used basis (that of van Duijneveldt [16]), we find only small differences in the BSR, and only at large q values. 

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