Finite-difference methods, polarizability

The polarizability is the second derivative of the interaction energy by the external field The calculation formulas can be also obtained using the finite-difference method like for the dipole moment. As a result, for example, the 3-point finite difference approximation (with errors of order gives... 

We have performed a series of semiempirical quantum-mechanical calculations of the molecular hyperpolarzabilities using two different schemes the finite-field (FF), and the sum-over-state (SOS) methods. Under the FF method, the molecular ground state dipole moment fJ.g is calculated in the presence of a static electric field E. The tensor components of the molecular polarizability a and hyperpolarizability / are subsequently calculated by taking the appropriate first and second (finite-difference) derivatives of the ground state dipole moment with respect to the static field and using... 

A decade ago Laaksonen et al. published a paper giving an outline of the finite difference (FD) (or numerical) Hartree-Fock (HF) method for diatomic molecules and several examples of its application to a series of molecules (1). A summary of the FD HF calculations performed until 1987 can be found in (2). The work of Laaksonen et al. can be considered a second attempt to solve numerically the HF equations for diatomic molecules exactly. The earlier attempt was due to McCullough who in the mid 1970s tried to tackle the problem using the partial wave expansion method (3). This approach had been extended to study correlation effects, polarizabilities and hyper-fine constants and was extensively used by McCullough and his coworkers (4-6). Heinemann et al. (7-9) and Sundholm et al. (10,11) have shown that the finite element method could also be used to solve numerically the HF equations for diatomic molecules. 

An alternative to the use of finite differences or finite elements to discretize the differential operator is to use boundary element methods (BEM). " One of the most popular of these is the polarizable continuum model (PCM) developed originally by the Pisa group of Tomasi and co-workers. The main aspect of PCM is to reduce the electrostatic Poisson equation (1) into a boundary element problem with apparent charges (ASCs) on the solute cavity surface. 

The polarizability and first hyperpolarizability of p-nitroaniline and its methyl-substituted derivatives have been calculated using a non-iterative approximation to the coupled-perturbed Kohn-Sham equation where the first-order derivatives of the field-dependent Kohn-Sham matrix are estimated using the finite field method" . This approximation turns out to be reliable with differences with respect to the fully coupled-perturbed Kohn-Sham values smaller than 1% and 5% for a and p, respectively. The agreement with the MP2 results is also good, which enables to employ this simplified method to deduce structure-property relationships. 

The Tables 5.4, 5.5, 5.6 show the calculation results of some electrical characteristics for H2,02, N2, CO2, CO, CN, HCl, HCN, NaCl, OH, NaH"", CH4, and H2O molecules, which are important for astrophysical and atmospheric problems. In the work [88] the calculations were carried out using the finite-field method at the (R) CCSD(T) level of theory with different aVXZ basis sets (X = Q, 5). For these cases, the amplitudes of the applied fields have been chosen as follows F = 0.0025 a.u., = 0.0001 a.u., FajSy = 0.00,001 a.u. and Fg,pys = 0.000001 a.u. Multipole moments up to 4th order are presented in Table 5.4. For comparison, in Table 5.4 the other literature data are also given. Table 5.5 presents the calculated and measured values (we have chosen the more reliable ones) of multipole polarizabilities. 

A defining feature of the models discussed in the previous section, regardless of whether they are implemented via matrix inversion, iterative techniques, or predictive methods, is that they all treat the polarization response in each polarizable center using point dipoles. An alternative approach is to model the polarizable centers using dipoles of finite length, represented by a pair of point charges. A variety of different models of polarizability have used this approach, but especially noteworthy are the shell models frequently used in simulations of solid-state ionic materials. 

Since the intensity calculation can be reduced to calculating a and for deformed molecules, the availability of quantum chemical methods immediately led to attempts to employ them to determine vibrational intensities (Segal and Klein, 1967). The polarizability is the proportionality factor between the induced dipole moment and the inducing electric field. It is therefore necessary to use a perturbation treatment which takes the electric field into account. Two different approaches were explored the Finite... 

All the models above are finite in size and rely on explicit molecnlar additions subject to the same computational method as the species studied. Another way to take into account and model the real surrounding in an electrolyte is to use various continuum methods to implicitly mimic the effect of, e.g., the dielectric constant of the electrolyte. Popular since many years are different variants of the polarizable continuum methods (PCM) applicable to both ab initio and DFT methods and where parameters for a variety of different solvents exist, and with possibilities to tailor for special electrolytes. The use of continuum methods has demonstrated the importance of simulating solvent effects - especially the difference between the gas phase electronic energies and the free energies of solvation (AG) via PCM. The use of continuum methods can also be tweaked in various ways, e.g., in TD cycles to treat different dielectric constants for different parts of the cycle. 

The polarizability of T1 and element 113 has been calculated using the fully relativistic ab initio Dirac-Coulomb Fock-space coupled-cluster method and the finite field procedure. For Tl, the theoretical value is in good agreement with experiment. In group 13, the atomic polarizability increases from A1 to Ga, attains a maximum for In and then decreases towards Tl and furthermore towards element 113. So, element 113 presents the smallest polarizability, which results from the large relativistic contraction and stabilization of the 7pi/2 orbital. These values have then been used to estimate the adsorption enthalpies of Tl and element 113 on polyethylene and teflon surfaces and have shown that the difference of enthalpy attains 6 kJ/mol, which should be enough to separate and identify them. 

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