Dipole polarizability finite-field methods

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. 

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 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 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. 

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. 

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... 

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. 

Hohm et al. have calculated the static dipole polarizability of P4 clusters using ab initio finite-field MP and coupled-cluster methods. The results have been compared with frequency-dependent measurements obtained fi-om the gas phase refi active index. 

All the calculations reported in this work were done on a DEC 20-60 using a modified version of the GAUSSIAN 80 series of program (6). Standard ST0-3G minimal basis set (7) was considered. Polarizabilities were calculated by the finite-field SCF method of Cohen and Roothaan (8) which is virtually equivalent to the analytic Coupled Hartree-Fock scheme. A term yf, describing the interaction between the electric field, E, and the molecule is added to the unperturbed molecular Hamiltonian, H y is the total dipole moment of the molecule. At the Hartree-Fock level, the electric field appears explicitly in the one-electron part of the modified Fock operator, F( ),... 

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... 

The many body perturbation theory has been applied to obtain higher accuracy in ab initio calculations of molecular properties. Pople etal. [181] have developed analytic derivative methods at second order perturbation theory level (MBPT(2)). Simandiras et al. [184] have derived specific expressions for analytic determination of dipole moment derivatives at MBPT(2). Dierksen and Sadlej [229] have shown by applying finite field MBPT in studying dipole and quadruple polarizabilities of the CO molecule that fourth and even higher level of MBPT is required to achieve satisfactory results. 

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... 

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