Lagrangian multiplier expansion

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

Inserting the perturbation and Fourier expansion of the cluster amplitudes and the Lagrangian multipliers,... 

To derive working expressions for the dispersion coefficients Dabcd we need the power series expansion of the first-order and second-order responses of the cluster amplitudes and the Lagrangian multipliers in their frequency arguments. In Refs. [22,29] we have introduced the coupled cluster Cauchy vectors ... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

Clearly, approximation 1 leads to an Eq. [85] that is linear in the Lagrangian multipliers. Not surprisingly, its solution by the TB method is found to be inaccurate.- The reason for the inaccuracy is obvious in light of the steps of the matrix method the solution in Eq. [89] is just a linearization first estimate of the true solution, and no further iterations, using at least the lowest nonlinear term in the expansion, are carried out to refine this first estimate, unlike the procedure followed in the matrix method. To deal with this problem, Tobias and Brooks decouple the constraints and iterate over them until convergence is reached to within a certain tolerance. ... 

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