Static Dipole Hyperpolarizabilities

We now turn our attention to the higher order polarizabilities (y etc.) in Eq. (9), which were named hyperpolarizabilities by Coulson et a/.[61]. The development of the theory 

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We will divide the survey into three parts (3.1) static dipole polarizabilities, (3.2) static dipole hyperpolarizabilities, and (3.3) dynamic dipole polarizabilities and hyperpolarizabilities. Within each part there will be sub-sections dealing with the three isoelectronic series He, Ne, and Ar. For (3.2) and (3.3) the hydrogen atom will also be included. 

The static dipole hyperpolarizability j8(0 0,0) can be obtained evaluating the third derivative of the energy. When we consider dynamic fields, the processes of particular interest are ... 

Table 43 The static dipole moment, the average polarizability, and the average first hyperpolarizability for different tungsten-carbon complexes. The second entry gives the conditions, and the third gives the method. All results are from ref. 100... 

Density functional static dipole polarizability and first-hyperpolarizability calculations of Na (w = 2,4,6,8) clusters using an approximate CPKS method and its comparison with MP2 calculations ... 

Soldan, P., Fee, E.P.F., and Wright, T.G. (2001) Static dipole polarizabilities (a) and static second hyperpolarizabilities (y) of the rare gas atoms. Phys. Chem. Chem. Phys., 3, 4661-4666. 

In other cases [166, 167] literature values for (supposedly) fc(w) taken from quantum chemical calculations were used to separate the Ag and Aj contributions. While this is in principle preferred over the first approach, the use of computational data leads to questions concerning their reliability, as low-level data might easily deteriorate the accuracy of the quadrupole moments obtained in this way. In addition, there has been some confusion in the earlier literature concerning the hyperpolarizability correction term / ( >). In the original papers by Buckingham and coworkers [134, 168] this term was called B and was referred to as a quadrupole hyperpolarizability. This was apparently misunderstood and led to the incorrect use of the averaged static dipole-dipole-quadrupole hyperpolarizability correction term instead of in the determination of the quadrupole moment [166, 167]. 

Maroulis et al.122 have applied their static polarizability, finite field technique to a study of the 22 electron diatomics CP , BC1, CC1+ and PO+. The vibrational contribution to the ground state polarizability has also been calculated. The dipole polarizability and other properties of YbF have been investigate in the unrestricted Dirac-Fock approximation by Parpia123 and the static second hyperpolarizability of the Cu2 dimer has been calculated in a correlation corrected UHF study by Shigemoto et al.124... 

Haskopoulos and MarouUs [10] studied the interaction electric properties of H20 Rg (Rg = He, Ne, Ar, Kr, Xe). Correlation effects have been taken into account by employing M0Uer-Plesset (MP2, MP4) and coupled-cluster theories (CCSD, CCSD(T)) in connection with flexible, carefully designed basis sets. Bara-nowska et al. [11] computed the interaction-induced axial static dipole moments, polarizabilities and first hyperpolarizabilities of HCHO (HF)n (n= 1,2). They employed a series of methods (e.g. MP2, CCSD(T)) in connection with various basis sets. 

The properties that are routinely computed and discussed are the mean (or average) static dipole polarizability (a), the anisotropy (Act) of the polarizability tensor, the vector component of the first hyperpolarizabUity tensor in the direction of the ground state permanent dipole moment (/3 ), the total first-order hyperpolarizabUity tot, and the scalar component of the second hyperpolarizability tensor y. Those quantities are related to the experiment and in terms of the Cartesian components are defined as... 

The CC2 model performes very different for static hyperpolarizabilities and for their dispersion. For methane, CC2 overestimates 70 by a similar amount as it is underestimated by CCS, thus giving no improvement in accuracy relative to the uncorrelated methods CCS and SCF. In contrast to this, the CC2 dispersion coefficients listed in Table 3 are by a factor of 3 - 8 closer to the CCSD values than the respective CCS results. The dispersion coefficients should be sensitive to the lowest dipole-allowed excitation energy, which determines the position of the first pole in the dispersion curve. The substantial improvements in accuracy for the dispersion coefficients are thus consistent with the good performance of CC2 for excitation energies [35,37,50]. 

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