Polarizabilities finite-field methods

The variational theorem which has been initially proved in 1907 (24), before the birthday of the Quantum Mechanics, has given rise to a method widely employed in Qnantnm calculations. The finite-field method, developed by Cohen andRoothan (25), is coimected to this method. The Stark Hamiltonian —fi.S explicitly appears in the Fock monoelectronic operator. The polarizability is derived from the second derivative of the energy with respect to the electric field. The finite-field method has been developed at the SCF and Cl levels but the difficulty of such a method is the well known loss in the numerical precision in the limit of small or strong fields. The latter case poses several interconnected problems in the calculation of polarizability at a given order, n ... 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

Finite-field methods were first used to calculate dipole polarizabilities by Cohen and Roothaan [66]. For a fixed field strength V, the Hamiltonian potential energy term for the interaction between the electric field and ith electron is just The induced dipole moment with the applied field can be calculated from the Hartree-Fock wavefunction by integrating the dipole moment operator with the one-electron density since this satisfies the Hellmann-Feyman theorem. With the usual dipole moment expansion. 

In another study of the polarizability and hyperpolarizability of the Si atom Maroulis and Pouchan6 used the finite field method with correlation effects estimated through Moeller-Plesset perturbation theory. Correlation effects are found to be small. 

Atom-Atom Interactions. - The methods applied, usually to interactions in the inert gases, are a natural extension of diatomic molecule calculations. From the interaction potentials observable quantities, especially the virial coefficients can be calculated. Maroulis et al.31 have applied the ab initio finite field method to calculate the interaction polarizability of two xenon atoms. A sequence of new basis sets for Xe, especially designed for interaction studies have been employed. It has been verified that values obtained from a standard DFT method are qualitatively correct in describing the interaction polarizability curves. Haskopoulos et al.32 have applied similar methods to calculate the interaction polarizability of the Kr-Xe pair. The second virial coefficients of neon gas have been computed by Hattig et al.,33 using an accurate CCSD(T) potential for the Ne-Ne van der Waals potential and interaction-induced electric dipole polarizabilities and hyperpolarizabilities also obtained by CCSD calculations. The refractivity, electric-field induced SHG coefficients and the virial coefficients were evaluated. The authors claim that the results are expected to be more reliable than current experimental data. 

Maroulis117 has applied the finite field method to a study of HC1. In a systematic analysis with large basis sets, MBPT and CC techniques, the dipole, quadrupole, octupole and hexadecapole moments have been calculated at the experimental internuclear distance. The polarizability and several orders of hyperpolarizability have been calculated and the mean a and -values for the 18-electron systems HC1, HOOH, HOF, A, F2, H2S are compared. Fernandez et a/.118 have calculated the frequency dependent a, / and tensors for HC1 and HBr using the Multiple Configuration Self Consistent Field method (MCSCF), including the effect of molecular vibration. The results show good agreement with available experimental and theoretical data. 

The finite field method is the simplest method for obtaining nonlinear optical properties of molecules. This method was first used by Cohen and Roothaan to calculate atomic polarizabilities at the Hartree-Foclc level. The basic idea is to truncate the expansion of the energy (Eq. [6]) and solve for the desired coefficients by numerical differentiation. For example, if the expression is truncated after the quadratic term, the result is E(P) = E[0) — — iot yF,Fy. 

Analogically to the representation of the wave-function in structural terms, there is a way to separate (hyper)polarizabilities into the individual contributions from individual atoms. A method for such separation was developed by Bredas [15, 16] and is called the real-space finite-field method. The approach can be easily implemented for a post-Hartree-Fock method in the r-electron approximation due to the simplicity of e calculation of the one-electron reduced density matrix (RDMl) elements. In our calculations we are using a simple munerical-derivative two-points formula for RDMl matrix elements (Z ) [88] (see also [48]) ... 

The above calculation represents an example of the application to an atom of what is called the finite field method. In this method we solve the Schrodinger equation for the system in a given homogeneous (weak) electric field. Say, we are interested in the approximate values of Uqq/ for a molecule. First, we choose a coordinate system, fix the positions of the nuelei in space (the Born-Oppenheimer approximation) and ealeulate the number of electrons in the molecule. These are the data needed for the input into the reliable method we choose to calculate E S). Then, using eqs. (12.38) and (12.24) we calculate the permanent dipole moment, the dipole polarizability, the dipole hyperpolarizabilities, etc. by approximating E(S) by a power series of Sq A. 

Polarization of an atom or molecule can be calculated by using the finite field method described on p. 639. Let us apply this method to the hydrogen atom. Its polarizability was already calculated using a simple version of perturbation theory (p. 636). This time we will use the variational method. 

The finite field method, e.g., a variational approach in which the interaction with a weak homogeneous electric field is included in the Hamiltonian. The components of the polarizability are computed as the second derivatives of the energy with respect to the corresponding field components (the derivatives are calculated at the zero field). In practical calculations within the LCAO MO approximation, we often use the Sadlej relation that connects the shift of a Gaussian atomic orbital with its exponent and the electric field intensity. 

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. 

The finite-field method appears to be very simple and has been often applied, but it has some drawbacks. First, the values of (F) depend not only on the dipole moment, but also on the polarizability and hyperpolarizabUities, and high accuracy of the computed values of (F) is required to perform the numerical differentiation accurately. Moreover, the presence of the finite electric field may lower the symmetry of the system, making the calculations much more time-consuming. 

Finite field method for second-order properties (polarizabilities). 

The conventional approach to calculate the polarizability of metal clusters is to solve the Kohn-Sham equations using suitable approximate forms for the exchange correlation functionals and a finite field method. We have recently carried out a systematic all electron DFT-based calculations for the polarizability and binding energy of sodium as well as lithium metal clusters [51,52]. It has been shown that the effect of electron correlation plays a significant role in determining the polarizability of metal clusters, although the effect is less pronounced for lithium clusters. Electron... 

In principle, density functional theory calculations should be able to give answers that are more reliable than Hartree-Fock but at similar cost. Static a and can be calculated by finite field methods or by coupled perturbed Kohn-Sham theory (CPKS) and give answers that are broadly comparable with MP2. In 1986 Sennatore and Subbaswamy did some calculations of the dynamic polarizability and second hyperpolarizability of rare gas atoms, but there have been no calculations of frequency dependent polarizabilities or hyperpolarizabilities of molecules until very recently. 

Extended basis set SCF and configuration interaction (CISD) calculations for CO have been carried out by Amos [359] with die aim to assess the effect of electron correlation on calculated molecular polarizability and polarizability derivatives. The basis set employed has been of the type (5s4p2d) contracted Gaussian function on each atom. The configuration interaction function consisted of single and double excitations of the valence shell orbitals. Polarizabilities have been calculated by the finite field method [179]. Polarizability derivatives are evaluated by numerical differentiation. The results are given in Table 10.5. 

Here, the polarizability components (a x, -yy and are computed using the finite-field methods ... 

The molecular property function of the molecular hyperpolarizability is + 1 1 + Pi) where = Y.j iPajj + Pjaj + Pjja) and the subscript a represents v, y, or z. The hyper-polarizability components are computed using the finite-field methods ... 

In this section we present the nuclear relaxation (NR) contributions to the vibrational (hyper)polarizabilities of Li C6o and [Li C6o]. As previously stated our treatment requires a geometry optimization in the presence of a finite field. A problem can arise when there are multiple minima on the PES separated by low energy barriers. The finite field method works satisfactorily in that event as long as the field-dependent optimized structure corresponds to the same minimum as the field-free optimized structure. This was the case in previous work on ammonia [42], which has a double minimum potential. However, it is sometimes not the case for the endohedral fullerenes considered here, especially Li C6o- In fact, we were unable to determine the NR contribution in the x direction, i.e. perpendicular to the symmetry plane, for that molecule. It was possible to obtain based on the alternative analytical formulation [32-34], utilizing field-free dipole (first) derivatives and the Hessian. The analytical polarizability components in the other two directions were, then, used to confirm the values of the corresponding finite field method for those properties. 

Ohnishi, S., Gu, F.L., Naka, K., Imamura, A., Kirtman, B., Aoki, Y Calculation of static (hyper)polarizabilities for tr-conjugated donor/acceptor molecules and block copolymers by the elongation finite-field method. J. Rhys. Chem. A 108, 8478-8484 (2004)... 

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