The Finite-Field Method

The property P of which derivatives can be taken need not be a static property, but can also be a frequency-dependent polarizability a —u Lv), as e.g. done by Jaszunski (1987). Finite-field calculations on a —Lv cu) facilitate calculation of / (—a uj, 0), OJ, 0, 0) and so forth. 

The finite-field method is by far the easiest method to implement as long as the perturbations are real. Any program for the calculation of the property P can be used, as long as it is possible to include additional one-electron operators in the Hamiltonian. 

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). 

A disadvantage of the finite-field method lies in the nature of numerical differentiation. Care must be taken in choosing the field strength, in our example S, which must not be too high, and in the number of different field strengths for which the property P is evaluated. For higher-order properties or multiple perturbations the method becomes cumbersome since the number of calculations to be performed increases rapidly. Secondly, adding the field to the Hamiltonian lowers the symmetry and therefore increases the computational cost of these calculations compared to the calculations without field. Finally, the method can obviously not be used for time-dependent perturbations and therefore for frequency-dependent properties. 

A variation of this method is the finite point charge method, used by Maroulis and Thakkar (1988), in which the external electric field or field gradient is simulated by an appropriate arrangement of point charges. This method is even simpler to implement, since it only requires the option to include centres with a charge but no basis functions, rather than a modified one-electron Hamiltonian. 

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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... 

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. 

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. 

There is a distinction between two groups of methods. The first is the finite field technique.35 In this case finite perturbations representing the external fields are added to the molecular hamiltonian and the calculation of the ground state wavefunction and energy is carried out as for the unperturbed molecule. The finite field method can be applied in conjunction with any quantum mechanical method that is available for molecular calculations. There are two principal subdivisions of the finite field method. In one of these terms of the form tff-Fi... 

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... 

The Fermi-contact contribution, usually the dominant one, can be relar tively easily calculated by means of the finite field method. This approach, which does not require extensive programming, has been used by several groups for calculating the spin-spin coupling constants at the DFT level, also for the hydrogen-bond-transmitted couplings. ... 

Another DFT calculations of the interresidual coupling constants in DNA have been reported in the paper by Barfield et al . Here the model has been extended to the DNA triplets T-A-T and C" Gr-C, but the calculations have been restricted to the FC terms obtained by means of the finite field method. The DFT calculations for several different base pairs separated by various interresidual distances have led to a good correlation between J(NN ) coupling and confirmed by the experimental 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. 

One may solve the Schrbdinger equation, including the term —//, f, in the Hamiltonian. The solution is valid, then, for this particular . This procedure is known as the finite field method. 

Is it possible to obtain an even better result with the variational function Yes, it is. If we use the finite field method (with the electric field equalling — 0.01 a.u.), we will obtain the minimum of E of Eq. (V.3) as corresponding to opt = 0.797224. If we insert = opt into Eq. (V.4), we will obtain 4.475 a.u., which is only 0.5% off the exact result This nearly perfect result is computed with a single correction function. ... 

The most dramatic form of the problMii would appear if the finite field method were combined with the numerical solution of the Schrodinger or Fock equafim. 

The finite field method requires a large quantity of atomic orbitals with small exponents (they describe the lion s share of the electron cloud deformation), although, being diffuse, they do not contribute much to the minimized energy (and lowering the energy is the only indicator that tells us whether aparticular function is important or not). 

It seems that the SOS method will gradually fall out of favor. The finite field method (in the electric field responses) will become more and more important due to its simplicity. It remains, however, to solve the jxoblem of how to process fhe infnmation that we get from such computations and translate it into the abovementioned local characteristics of fhe molecule. 

However, the theory for the interaction of matter with the electromagnetic field has to be coherent. The finite field method, so gloriously successful in electric field effects, is in the stone age stage for magnetic field effects. The propagator methods look the most promising, these allow for easier calculation of NMR parameters than the sum-over-states methods. 

N(C2H2)bNH2 with respect to the number of units n. The DFT (LC-BOP, BOP, and B3LYP) results were obtained by the coupled-pertuibed Kohn-Sham method (see Sect. 4.7), the HF result was given by the coupled-perturbed Hartree-Fock method, and the ab initio results were provided by the finite-field method (see Sect. 4.7). The aug-cc-pVDZ basis functions tire used. See Kamiya et al. (2005)... 

The finite field method works well as long as the field-dependent optimum structure corresponds to the same minimum as the field-free optimized structure. For the endohedral fullerenes considered in Ref. [60], especially for Li C6o, it appears that there were several minima lying nearby on the potential energy surface, which are separated by low energy barriers. As a consequence it was not possible to determine the NR contribution perpendicular the symmetry plane of the Cs symmetry structure using the finite field method. However, otxx" could be determined using alternative analytical formulae [24, 25, 27], which do not require field-optimized structures, but only dipole derivatives and the Hessian of the field-free equilibrium structure. 

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