Hyperpolarizabilities finite-field method

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. 

Maroulis126 has also investigated the static hyperpolarizability tensor (y) by the finite field method. The molecular geometries and levels of correlated calculation are as in reference 125, although in this case some very large basis... 

Spassova et al. have studied the possible structures of squaric acid and the monohydrogenosquarate anion at the MP2 level with 6-3IIG basis sets supplemented by a set of diffuse p and d functions. They use the finite field method to calculate the and y-hyperpolarizabilities and find that, compared with the acids, the former is reduced by 40% in the ion and the latter increased by a factor of more than two. 

Schiff base ligands and calculated the static fi using the finite field method with AML The effect of various groups as dentates on the hyperpolarizability is discussed. 

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. 

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 first hyperpolarizability of configurationally locked trienes (CLT) has been calculated using the finite field method with the aim of addressing the relationship between the molecular and crystal second-order NLO responses. In particular, the high performance of the 2-(5-methyl-3-(4-(pyrrolidin-l-yl)styryl)cyclohex-2-enylidene)malononitrile (MH2) (Fig. 1,6) species has been attributed to optimal orientation of the dominant tensor components with respect to the crystal polar axis for phase matching. 

The finite-field method can thus be applied at any level of approximation or correlation and even to approximations or methods for which a wavefunction or a ground-state energy is not defined. The latter approach was used for example for the calculation of the static second hyperpolarizability 7(0 0,0,0) of Li as second derivative of a(0 0) at the SOPPA(CCSD) level (Sauer, 1997). 

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. 

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 ... 

Ab initio methods have thus far only been used to calculate the static third-order hyperpolarizabilities of ligated organometallic fragments. The CFIF finite field (FF) method was utilized with MP2, MP3, MP4(SDTQ), QCISD(T), and CCSD(T) electron correlation corrections applied. The reader is directed elsewhere for a more detailed description of these electron correlation corrections.10... 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

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... 

During the past decade, theoretical calculations of hyperpolarizabilities have been performed to help synthetic chemists design optimum NLO structures. Although extremely accurate calculations are still out of reach, it is now possible to predict the influence of structural changes on the NLO coefficients. In the case of photochromes, theoretical calculations may be useful for predicting 3 values of thermally unstable colored forms. The theoretical methods generally employed to calculate molecular hyperpolarizabilities are of two types those in which the electric field is explicitly included in the Haniiltonian, frequently labeled as Finite Field (FF) and those which use standard time dependent perturbation theory, labeled Sum Over State (SOS) method. 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

Most of the NLO experiments performed with subphthalocyanines could not be interpreted without theoretical calculations. Thus, it is crucial to possess the knowledge of the minimum requirements for obtaining reliable theoretical values that may be compared to the experimental ones in the case of SubPcs. The molecular hyperpolarizabilities are generally calculated employing two parallel theoretical approaches (i) the finite field (FF) method and (ii) the sum over states (SOS) formalism as stated earlier. 

An attempt at an ab initio calculation of the pNA hyperpolarizability by Sim et a/.38 using double zeta quality basis sets with diffuse polarizing functions within the finite field/HF/MP2 regime gave values which are even lower than those obtained by the semi-empirical methods. These calculations are at zero frequency but have been corrected to 1064 nm using the usual two state model. Bartlett and Sekino90,91 have discussed the relationship between hyperpolarizabilities calculated by ab initio methods and by SOS formalisms. They conclude that only the most powerful recent methods for correlated calculations can be expected to produce accuracy of about 10% for gas phase hyperpolarizabilities, while ab initio calculations interpreted in terms of the SOS expansion give very poor results when the equivalent number of states is small. 

Kucharski et al.161 have calculated the static / -hyperpolarizability of new sulphonamide amphiphiles using finite field SCF and INDO/S methods. In the latter case a solvent correction (SCRF option) was also included. The ab initio and INDO/S results for the isolated molecule were similar while the inclusion of the solvent correction increased the values by about 55-65%. Kassimi and Lin 168 have calculated the dipole moment and static polarizability of aza-substituted thiophene derivatives within the Hartree-Fock approximation. For a representative sub-set, correlation up to the MP4(SDQ4) level has been included. The results are expected to be accurate to within a few percent. 

Figure 4 Results of finite field calculation on the second hyperpolarizability (7) of biphenyl obtained with the MNDO semiempirical method.

Lee et al. have investigated the two photon absorption cross-section and the y-hyperpolarizability of a series of quadrupolar molecules using ab initio finite field and SOS methods. 

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