Method, finite field

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. 

If a uniform electric field is assumed to be aligned along the x direction only, this reduces to E(FJ - E(0) - Th s expressions can be solved 

If the energy is then calculated with a field of known strength oriented along the +x direction and the -x direction, the equations above give and A similar procedure can be repeated for other field orientations to get all the components of p and a. 

For the calculation of nonlinear optical properties, Eq. [6] is usually truncated beyond the F term and more complicated expressions result. Bartlett and Purvis advocated this procedure in 1979 and Kurtz et al. implemented it in the MOPAC program. The resulting equations that give NLO properties for the fewest energy calculations using this approach are as follows  

Formulas for the polarizability and hyperpolarizability components can likewise be derived from the dipole moment expression in Eq. [4]. The resulting equations are  

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. 

How do we put the moleeule in an electric field For example, at a long distance from the moleeule we loeate two point-like electric charges qx and qy on x and y axes, respeetively. Henee, the total external field at the origin (where the centre  

Negleeting the eubic and higher terms for a very small field (approximation) we obtain an equation for axx, cixy and ayy, because the polarizability tensor is symmetrie. Note that 

The results of the above procedure depend very much on the configuration of 

Example 2. Hydrogen atom in electric field-variationsi approach 

---

As an example, here is an output from Gaussian 98 on CH3F (Figure 17.2). I forced the finite field method by choice of Polar = Enonly (Polar = Energy only) in the route. The geometry was first optimized and stored in a checkpoint file. 

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 ... 

An alternative approach is to apply stronger fields and only use energies calculated for positive field strengths in generating the polynomial fit. In this case the energy is a function of both odd and even powers in the polynomial fit. We will show that the dipole moments derived from our non-BO calculations with the procedure that uses only positive fields and polynomial fits with both even and odd powers match very well the experimental results. Thus in the present work we will show results obtained using interpolations with even- and odd-power polynomials. Methods other than the finite field method exist where the noise level in the numerical derivatives is smaller (such as the Romberg method), but such methods still do not allow calculation of odd-ordered properties in the non-BO model. 

Much information of interest for atomic and molecular systems involves properties other than energy, usually observed via the energy shifts generated by coupling to some external field. The desired property is then the derivative of the energy with respect to the external field, which may be obtained by two different approaches. The finite-field method solves the Schrodinger equation in the presence of the external field, yielding... 

Barone and coworkers250 also determined EPR hyperfine splittings nN of the radical 40 at the UMP2/DZ + P level of theory using the Fermi contact operator and a finite field method with an increment size of 0.001 a.u. Expectation values of aN, < aN >, at higher temperatures T were calculated by assuming a Boltzmann population of vibrational levels according to equation 23 ... 

The approaches to this problem follow along two general lines. In the first approach, one computes derivatives of the dipole moment with respect to the applied field and relates them to the terms in the polarization expansion of equation 8. Inspection of equation 8 suggests that the second derivative of the dipole moment with respect to the field gives p. The choice of the exact form of the Hamiltonian, which incorporates the optical field and the atomic basis set, determines the accuracy of this procedure. In one popular version of this approach, the finite field method, the time dependence of the Hamiltonian is ignored for purposes of simplification and the effects of dispersion on p, therefore, cannot be accounted for. 

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,... 

There are both numerical and analytical ways of carrying out this procedure. The first is the easiest to understand and was first applied to the Hartree-Fock (HF) method by Cohen and Roothaan[23]. One simply takes various values of F (usually of the order of 0.001 au), finds the corresponding EF and makes a fit to Eq. (9). This is called the finite field method and it may be applied within the framework of any of the standard methods which determine energies, e.g., HF, MP2, MP4, coupled cluster (CC), MCSCF. 

John, 1. G., G. B. Bacskay, and N. S. Hush (1980). Finite field method calculations. VI. Raman scattering activities, infrared absorption intensities and higher order moments SCF and Cl calculations for the isotopic derivatives of H,0 and SCF calculations for CH4. Chem. Phys. 51, 49-60. 

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. 

Vibrational contributions to the a and (1 response functions of NaF and NaCl have been calculated by Andrade et al 5 at HF, MP and CC levels. The results obtained from perturbation theory are in agreement with those from the finite field method and demonstrate that the inclusion of vibrational effects is essential to get reliable electric response functions in these molecules. 

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. 

In principle, the differentiation is either done numerically in the so-called finite-field methods, or in an analytical scheme, or a combination of both. Numerical finite-field calculations are limited to derivatives with respect to static fields. Since SFG is an optical process that involves dynamic oscillating fields, it becomes necessary to use an analytical approach, such as the time-dependent Hartree Fock (TDHF) method. 

Finite-field methods can not handle time-dependent perturbations. 

---