Finite field approach

The discussion of the pros and cons of the finite-field approach made it clear that we need an analytical formulation of the derivatives of the molecular energy with respect to the external fields in order to maintain computational efficiency and numerical stability. The phrase analytical derivative approaches is often used to denote methods where closed-form expressions have been derived for the part of tire molecular properties that regards the motions of the electrons, i.e. the part related to the first term in Eq. (99). 

Figure 4. Static hypermagnetizability anisotropy. Atj(O), computed with the d-aug-cc-pV5Z basis set (Neon) and d-aug-cc-pVQZ basis set (Argon). Orbital-relaxed results obtained with a finite field approach from analytically evaluated magnetizabilities are compared to those obtained from orbital-unrelaxed quadratic and cubic response functions...

In the finite field approach the FICs are generated simply by evaluating numerical derivatives of the change in equilibrium geometry induced by a finite field with respect to the magnitude of that field [35]. However, the FICs may also be determined analytically. 

The non-relativistic PolMe (9) and quasirelativistic NpPolMe (10) basis sets were used in calculations reported in this paper. The size of the [uncontractd/contracted] sets for B, Cu, Ag, and Au is [10.6.4./5.3.2], [16.12.6.4/9.7.3.2], [19.15.9.4/11.9.5.2], and [21.17.11.9/13.11.7.4], respectively. The PolMe basis sets were systematically generated for use in non-relativistic SCF and correlated calculations of electric properties (10, 21). They also proved to be successful in calculations of IP s and EA s (8, 22). Nonrelativistic PolMe basis sets can be used in quasirelativistic calculations in which the Mass-Velocity and Darwin (MVD) terms are considered (23). This follows from the fact that in the MVD approximation one uses the approximate relativistic hamiltonian as an external perturbation with the nonrelativistic wave function as a reference. At the SCF and CASSCF levels one can obtain the MVD quasi-relativistic correction as an expectation value of the MVD operator. In perturbative CASPT2 and CC methods one needs to use the MVD operator as an external perturbation either within the finite field approach or by the analytical derivative schems. The first approach leads to certain numerical accuracy problems. 

If effects from electron correlation on parity violating potentials shall be accounted for in a four-component framework, the situation becomes more complicated than in the Dirac Hartree-Fock case. This is related to the fact, that in four-component many body perturbation theory (MBPT) or in a four-component coupled cluster (CC) scheme the reduced density matrices on the respective computational level are required in order to determine the parity violating potentials. Since these densities were not available in analytic form, Thyssen, Laerdahl and Schwerdtfeger [153] used a finite field approach to compute parity violating potentials in a four-component framework on a correlated level. This amounts to adding the parity violating operator with different scaling factors A to the... 

For hyperpolarizabilities. Rice and Handy actually used equations developed for an arbitrary P(—whereas Aiga and Itoh used only a numerical derivative (finite field) approach to calculate P(-o) a),0) and -yl—(jd (jd,0,0). ... 

The nuclear relaxation (NR) contributions were computed using a finite field approach [73,74]. In this approach one first optimizes the geometry in the presence of a static electric field, maintaining the Eckart conditions. The difference in the static electric properties induced by the field can then be expanded as a power series in the field. Each coefficient in this series is the sum of a static electronic (hyper) polarizability at the equilibrium geometry and a nuclear relaxation term. The terms evaluated in Ref. [61] were the change of the dipole moment up to the third power of the field, and that of the linear polarizability up to the first power ... 

The derivatives of energy in the above equations can be obtained by numerical differentiation (Finite Field approach, FF). For instance, to calculate the second- and fourth-order derivatives which correspond to polarizability and 2nd hyperpolarizability the following seven point formulaes can be used ... 

This becomes an increasingly better approximation as the size of 8 is diminished. Of course, the numerical accuracy of the small energy difference in the numerator limits how small 8 should get. Another example is the finite field approach to finding molecular electrical properties. The energy of a molecule can be calculated in the absence of a field and then recomputed with a particular field, perhaps 0.0005 a.u. (2.6 X 10 V/cm) in the z-dirertion, entered into the Hamiltonian. From the energy difference and the size of the field used, an approximate value can be obtained for the first derivative of the energy with... 

Polarizability calculations were carried out using Eq. (3) for four values of the external field, -0.002,-0.001,+0.001,+0.002 a.u. The Richardson extrapolation procedure (9) was applied to check the numerical stability of the results, a point which remains difficult in the finite field approach. 

The MP2/aug-cc-pVTZ method has been applied within the finite field approach to calculate the polarizability and first hyperpolarizability of the Li2F and LigF systems. These systems present an alkalide character, i.e. some of its alkali metal atoms bear a negative charge, which is loosely bound and is therefore at the origin of large (hyper)polarizabilities. Using unrestricted MP2 calculations, the first hyperpolarizability has been shown... 

Basis set quality is very important in the computations for polarizabilities. Extends basis sets are es.sential for the accurate description of the small changes in energy and in charge density needed in the finite-field approach. Guan et al. showed that the S-VWN functional with a basis set that includes field-induced polarization (FIP) functions provides predictions for small molecules (a and p) which are in significantly better agreement with experiment, than without FIP functions, particularly for p. 

Gu, F.L., Aoki, Y, Imamura, A., Bishop, D.M., Kirtman, B. Application of the elongation method to nonlinear optical properties finite field approach for calculating static electric (hyper)polarizabilities. Mol. Rhys. 101,1487-1494 (2003)... 

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