Second-order properties, calculation

Table 3 Second Order Properties Calculated by EOM-CCSD Methods...

Analogous considerations apply, for example, to the calculation of second-order properties, for which a very similar computational problem must be addressed. For typical applications this step constitutes 95-100% of the total computational effort, and a successful parallelization will therefore reflect directly on overall performance. 

Recently, the assignment of the band at 980 cm to 28 has been doubted based on new calculations (this band is shifted to 976 cm if 28 is generated from 1,4-diiodobenzene (37), which is not unusual in the presence of iodine atoms. This shift may also be attributable to the change of the matrix host from argon to neon). ° On the other hand, ab initio calculations of the IR spectrum of 28 are complicated by the existence of orbital instabilities, the effect of which may (often) be negligible for first order properties (such as geometry and energy), but can result in severe deviations for second-order properties (vibrational frequencies, IR intensities). 

If W were known exactly, the value of a first-order property calculated from equation (12) would be exact. In practice, only an approximation to W is known, and it is important to know how the expectation value differs from the exact value. Since errors in calculated dipole moments due to the breakdown of the Bom-Oppenheimer approximation are likely to be small8 (typically 0.002 a.u.), and for most molecules relativistic effects can be ignored,6 there are two separate remaining problems in practice. The first concerns the likely accuracy when the wavefunction is at the Hartree-Fock limit, the second the effect of using a truncated basis set to obtain a wavefunction away from the Hartree-Fock limit. 

The inclusion of correlation effects in the calculation of second-order properties, such as the polarizability, has been examined by Bartlett and co-workers, and an application to H2 reported.73 A very accurate Cl wavefunction due to Liu has been used to calculate the Compton profile for molecular H2, with results in good agreement with experiment.74... 

Extensive numerical calculations for small systems (H2 (Ref. 79,87), H2 (Ref. 89,90) and He2 Ref. (81) show that the convergence properties of the SRS theory are excellent. Since in most cases the polarization expansion is divergent, one can expect that for many-electron monomers the SRS expansion will not be strictly convergent. However, the experience gained thus far for large many-electron systems suggests that a second-order SRS calculation correctly accounts for all major polarization... 

We have established that, for a fully variational wave function, we may calculate the first-order properties from the zero-order response of the wave function (i.e., from the unperturbed wave function) and the second-order properties from the first-order response of the wave function. In general, the 2n -f 1 rule is obeyed For fully variational wave functions, the derivatives (i.e., responses) of the wave function to order n determine the derivatives of the energy to order 2n+ 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wave function to first order, but that the calculation of the energy to fourth order requires a knowledge of the wave-function response to second order. 

Having set up the Hamiltonian, we may calculate the first- and second-order properties in the eigenvector representation. For the permanent electric and magnetic dipole moments, we obtain... 

Helgaker et alP presented a fully analytical implementation of spin-spin coupling constants at the DFT level. They used the standard procedure for linear response theory to evaluate second-order properties of PSO, FC and SD mechanisms. Their calculation involves all four contributions of the nonrelativistic Ramsey theory. They tested three different XC functionals -LDA (local density approximation), BLYP (Becke-Lee-Yang-Parr), " and B3LYP (hybrid BLYP). All three levels of theory represent a... 

CALCULATION OF SECOND-ORDER PROPERTIES IN KOHN-SHAM DENSITY FUNCTIONAL THEORY... 

The preceding remarks concerning the effect of correlation on second-order properties are best illustrated by the calculation of indirect nuclear spin-spin coupling constants Jnn> from molecular wave functions. The main contributions to such observables (at least for H—H coupling) comes from the contact terms associated with each nucleus and is proportional to the second-order quantity... 

Although evaluations of harmonic force constants [d E dq,dqj), elearic polarizabilities d EIdeide ), and dipole moment derivatives (d E/d ,dqj) are perhaps the most common applications of second-order properties (or, equivalently, second derivatives), other areas of interest to chemists can be treated with these techniques. One such field of application that holds great promise for the future is the calculation of nuclear magnetic resonance chemical shifts. 

At the moment of writing very few implementations of the theory of molecular properties at the 4-component relativistic molecular level, beyond expectation values at the closed-shell Hartree-Fock level, have been reported. The first implementation of the linear response function at the RPA level in a molecular code appears to be to MO-based module reported by Visscher et al. [97]. Quiney and co-workers [98] have reported the calculation of second-order properties at the uncoupled Hartree-Fock level (see section 5.3 for terminology). Saue and Jensen [99] have reported an AO-driven implementation of the linear response function at the RPA level and this work has been extended to quadratic response functions by Norman and Jensen [100]. Linear response functions at the DFT/LDA-level have been reported by Saue and Helgaker [101]. In this section we will review the calculation of linear and quadratic response functions at the closed-shell 4-component relativistic Hartree-Fock level. We will follow the approach of Saue and Jensen [99] where the reader is referred for further details. 

For the calculation of a second-order property the wave function P(k2) should be determined first by utilising a Hamiltonian of the form... 

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