Finite-field calculations

In practice the finite-field calculation is not so simple because the higher-order terms in the induced dipole and the interaction energy are not negligible. Normally we use a number of applied fields along each axis, typically multiples of 10 " a.u., and use the standard techniques of numerical analysis to extract the required data. Such calculations are not particularly accurate, because they use numerical methods to find differentials. 

Convergence continued) finite-field calculation. 327 geometry optimization, 40-50, 141, 183,... 

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]. 

In any finite field calculation, the choice of F is important because of the conflicting requirements of a small Fto make truncation valid and a large Fto minimize the rounding errors in the differences. The optimum choice clearly depends on the relative values of K0), a, / , etc. for the molecule concerned. 

In (II) the reaction field of the dipole is included in the molecular Hamiltonian, so that the QM calculation, at whatever level, is modified to give a new molecular wavefunction for one molecule at the centre of a cavity. This calculation can be carried out in the absence of an applied macroscopic field and would give the unperturbed properties (dipole moment, energy states etc) of a solvated molecule. The macroscopic field has then usually been applied in a finite field calculation of the hyperpolarizability. One source of uncertainty in this procedure arises from the fact that when the reaction field is introduced into the hamiltonian it appears in a specific form,... 

The perturbed total energies or other properties of the system can be written as an expansion in terms of moment and polarizability components (see Section I). If different values of the field strength or charge positions are used, a system of simultaneous equations can be written from the truncated series, and these equations are solved to find the unknown polarizabilities. The system of equations must be chosen sufficiently large to ensure that the truncation error is minimized, but sometimes it is not practical to carry out the number of finite-field calculations that this might call for. 

A number of calculations allow for internal consistency checks. For example, if either or B x.xx desired, one can use either total dipole moment or quadrupole moment expansions. These properties could not be calculated by uniform field computations alone. The finite-charge approach is easily carried out at all electronic structure levels (e.g., SCF, Cl, and MBPT) because it amounts to adding an extra nucleus, one with a negative or positive charge. It does not seem to be as widely employed as finite-field calculations. Examples include calculations on methane [77], Lij and ions [78], and HF [79]. 

Molecular electric properties give the response of a molecule to the presence of an applied field E. Dynamic properties are defined for time-oscillating fields, whereas static properties are obtained if the electric field is time-independent. The electronic contribution to the response properties can be calculated using finite field calculations , which are based upon the expansion of the energy in a Taylor series in powers of the field strength. If the molecular properties are defined from Taylor series of the dipole moment /x, the linear response is given by the polarizability a, and the nonlinear terms of the series are given by the nth-order hyperpolarizabilities ()6 and y). 

Coupled cluster response calculaAons are usually based on the HF-SCF wave-function of the unperturbed system as reference state, i.e. they correspond to so-called orbital-unrelaxed derivatives. In the static limit this becomes equivalent to finite field calculations where Aie perturbation is added to the Hamiltonian after the HF-SCF step, while in the orbital-relaxed approach the perturbation is included already in the HF-SCF calculation. For frequency-dependent properties the orbital-relaxed approach leads to artificial poles in the correlated results whenever one of the involved frequencies becomes equal to an HF-SCF excitation energy. However, in Aie static limit both unrelaxed and relaxed coupled cluster calculations can be used and for boAi approaches the hierarchy CCS (HF-SCF), CC2, CCSD, CC3,... converges in the limit of a complete cluster expansion to the Full CI result. Thus, the question arises, whether for second hyperpolarizabilities one... 

Finite field calculation in the Sadlej POL basis, [115]. Response calculation in the Sadlej POL basis, [121]. 

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. 

Shi and Garito recommend conventions I and II, because they are the ones most used to report values in organic nonlinear optics. As pointed out previously, convention IV is widely used in finite field calculations, and experimentally it is also used to report values on gas phase atoms or molecules. 

SOS and FF stand for sum over states and finite field calculation methods, respectively. 

The difference in jp and Dp is the incorporation of orbital relaxation for the reference determinant, so that the results for a dipole moment computed with the relaxed density matrix gives precisely the same results as if it were done by differentiating the CC energy in a finite-field calculation,... 

Fullerenes. - Jonsson et al.234 have carried out analytical Hartree-Fock calculations, expected to be near the basis set limit, of a, and the magnetiz-ability for the C70 and C84 fullerenes. The results are compared with earlier calculations on ) and the electronic structures of the molecules discussed. Moore et al.235 have made semi-empirical AMI finite field calculations of the static y-hyperpolarizability of Qo, C70, five isomers of C78 and two isomers of C84. The results are interpreted in terms of bonding and structural features. 

A breakthrough in the combination of PCE-free EFG calculations and the application of the efficient DK method was achieved by Malkin et al. [144] now explicitly performing the DK transformation of the EFG operator. In contrast to the numerical finite field calculations this method is now free from possible numerical errors and it was implemented as a scalar-relativistic version up to second order. The transformed EFG operator q is hereby split in the first and second-order contributions... 

The finite field procedure is the most often used procedure because of two main advantages (1) it is very easy to implement, and (2) it can be applied to a wide range of quantum mechanical methods. To calculate the energy in the presence of a uniform electric field of strength F, an F-r term needs to be added to the one-electron Hamiltonian. This interaction term can be constructed from just the dipole moment integrals over the basis set. Any ab initio or semiem-pirical method can then be used to solve the problem, with or without electron correlation. It is the ability to obtain properties from highly correlated methods that makes finite field calculations usually the most accurate available. 

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

Interaction-induced electric properties from finite-field calculations ... 

Recently, Bezchastnov et al. reported all-electron relativistic calculations of the interaction-induced dipole polarizability of the Xe dimer for the internuclear separations 2basis sets of [9s9p8d4f] and [10sl0p9d5f] size. The BSSE corrected electric polarizability values were extracted from finite-field calculations on Xc2 at the SCF, MP2, CCSD and CCSD(T) levels of theory. The results pertain to a range of internuclear separations. 

Baranowska et reported recently an interesting study of the interaction-induced electric properties in model (HCN)n (n = 2 ) chains. Their results were obtained via finite-field calculations at the SCF, MP2, CCSD and CCSD(T) levels of theory with LPol-n (n = ds, fs, dl, fl) basis sets. At the CCSD(T)/LPol-fl level of theory they obtained the following... 

From our experience to date with the transformations, we can immediately foresee a problem. If the transformation is some complicated function of the momentum, we might not be able to separate out the perturbation from the zeroth-order Hamiltonian. This would be unfortunate, because magnetic operators break Kramers symmetry and we would be forced to perform calculations without spin (or time-reversal) symmetry. We might also be forced to perform finite-field calculations. We will address this problem as it arises. 

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