Second-order polarization propagator approximation

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

The key step is to bring the Mpller-Plesset perturbation theory wavefunction, Eq. (9.64), into a form, that resembles a MCSCF wavefunction, i.e. 

The time-dependent Mpller Plesset wavefunction o (t)) is then written as 

One should note that single excitations and de-excitations are not included in the time-dependent MP state-transfer operator S(t) because they are already included 

Exercise 11.7 Derive the SOPPA Hessian and overlap matrices using the Ehrenfest theorem Eq. (11.55). 

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We have employed the second-order polarization propagator approximation (SOPPA) in this study, a method which was mainly developed by Jens Oddershede and his co-workers [3,4,20,51-56]. Barone et al. [32] have recently shown that SOPPA reproduces the vicinal F-F couplings reasonably well in 1,2-difluoroethene. 

The second-order polarization propagator approximation (see Section VII. D). 

The resulting second-order polarization propagator approximation (SOPPA) was first described in its present form by Nielsen et al. (1980). Splitting h4 up into 2p-2h and 2h-2p excitation operators as was done for h2 in Eqs (95)-(97)... 

The second-order polarization propagator approximation is closely related to the equation-of-motion (EOM) method (McCurdy et al., 1977). The equations that determine the excitation energies are the same up through the... 

For a polyyne chain142 the static al and dynamic a(— co)L polarizabilities have been computed using non-linear sequence transformations for the extrapolation and besides RPA the SOPPA (correlated second order polarization propagator approximation) method. In this way the authors have obtained for a C2 iH2 (polyyne) chain quite stable extrapolated values for both quantities. 

There do exist recent quantum chemical techniques which are size consistent. Among them, the Random Phase Approximation (RPA), its variants such as the Second-Order Polarization Propagator Approximation (SOPPA) [10], and the Coupled Cluster Approximation (CCA) [11] axe the most prominent and being widely used. In the SOPPA method, electron correlation effects are included in the two-particle polarization propagator to second order. The coupled cluster method uses an exponential ansatz through which higher-order exci-... 

Empirical equilibrium coupling constants can be compared as a benchmark with calculated equilibrium coupling constants obtained with various methods. A comparison of these empirical equilibrium constants with calculated equilibrium constants suggested that the restricted-active-space self-consistent field (RASSCF) method is the best approach for calculating the indirect nuclear spin-spin coupling constants of small molecules, and that the second-order polarization propagator approximation (SOPPA) and DFT are similar in performance. 

Second-Order Polarization Propagator Approximation Calculations. -... 

The paramagnetic contribution is more demanding to calculate and is a bit more sensitive to electron correlation, although not a great deal. For instance, for ammonia, the RPA (effectively uncorrelated) value is 38.45 ppm a.u., while the Second Order Polarization Propagator Approximation (correlated) yields 38.15 ppm a.u. 

J.E. Del Bene, I. Alkorta, I.J. Elguero, A systematic comparison of second-order polarization propagator approximation (SOPPA) and equation-of-motion coupled cluster singes and doubles (EOM — CCSD) spin — spin couphng constants for selected singly bonded molecules, and the hydrides NH3, H2O, and HF and their protonated and deprotonated ions and hydrogen-bonded complexes, J. Chem. Theor. Comput. 4 (2008) 967-973. 

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