Clusters amplitudes

This choice produces asymmetric superoperator matrices. A simplified final form for the self-energy matrix that does not require optimization of cluster amplitudes is sought for large molecules the approximation... 

The Brueckner-reference method discussed in Section 5.2 and the cc-pvqz basis set without g functions were applied to the vertical ionization energies of ozone [27]. Errors in the results of Table IV lie between 0.07 and 0.17 eV pole strengths (P) displayed beside the ionization energies are approximately equal to 0.9. Examination of cluster amplitudes amd elements of U vectors for each ionization energy reveals the reasons for the success of the present calculations. The cluster operator amplitude for the double excitation to 2bj from la is approximately 0.19. For each final state, the most important operator pertains to an occupied spin-orbital in the reference determinant, but there are significant coefficients for 2h-p operators. For the A2 case, a balanced description of ground state correlation requires inclusion of a 2p-h operator as well. The 2bi orbital s creation or annihilation operator is present in each of the 2h-p and 2p-h operators listed in Table IV. Pole strengths are approximately equal to the square of the principal h operator coefiScient and contributions by other h operators are relatively small. 

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... 

These terms are present in the wave function T(,) defining variant b of the CR-CCSD(TQ) theory (the CRCCSD(TQ),b approach of ref 14). In consequence, the CR-CCSD(TQ),b results for multiple bond breaking are considerably better than the CRCCSD[T] and CR-CCSD(T) results (14). The Tj and other bilinear terms in cluster amplitudes, such as Jso... 

The first approach, taking the advantage of the BCH formula, was initiated hy Jeziorski and Monkhorst [23] and, so far, it has been intensively developed within Paldus s group [5,51-55] who formulated an orthogonally spin-adapted Hilbert space MR CC method for a special case of a two-dimensional model space spanned by closed-shell-type reference configurations. The unknown cluster amplitudes are obtained by the solution of the Bloch equation [45-49]... 

At the same time, Meissner, Kucharski and others [56,57] developed the quadratic MR CCSD method in a spin-orbital form which does not exploit the BCH formula. The unknown cluster amplitudes are calculated from the so-called generalized Bloch equation [45-47,49,64,65] (or in our language the Bloch equation in the Rayleigh-Schrodinger form)... 

Turning now back to the single-root MR BWCC approach, we derive the basic equations for the effective Hamiltonian and cluster amplitudes in the spin-orbital form without the use of the BCH formula. We limit ourselves to a complete model space which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. In our derivation we shall work with the Hamiltonian in the normal-ordered-product form, i.e. 

As concerns cluster amplitudes, if we employ the exact Hamiltonian in the normal-ordered-product form (31) with the /i-th configuration as a Fermi vacuum, the basic equation for the single-root wave operator (25) takes the form... 

For the sake of completeness, we recall that the idea of the single-root formalism exploiting the Hilbert space approach was also proposed by Banerjee and Simons [31] and Laidig and Bartlett [34,35]. In both approaches they start from the complete active space MC SCF wave function, however, in order to eliminate redundant cluster amplitudes they approximate the wave operator by... 

In this section, we derive basic equations for the monoexcited and biexcited cluster amplitudes at the CCSD level of approximation, i.e. with the cluster operators 7 being approximated by their singly and doubly excited cluster components... 

Now, the external cluster amplitudes are again given by the usual MR SU CCSD equations [55], namely... 

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