The coupled-cluster Lagrangian

In Section 12.2 it will be discussed that this approach for the calculation of expectation values is called the unrelaxed method, because the conditions for the molecular orbital coefficients were not included as additional constraints in the coupled cluster Lagrangian given in Eq. (9.95) or Eq. (9.98). A coupled cluster Lagrangian including orbital relaxation takes the following form... 

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. 

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 ... 

Presently, we are able to determine the coupled cluster energy based on the variational Lagrangian and expectation values for real operators... 

For the optimization of the coupled cluster wave function in the presence of the classical subsystem we write the CC/MM Lagrangian as [24]... 

Lagrangian incorporating the equations of motion and varying the quasienergy is used to allow variational calculations to be carried out on the interacting system within the coupled-cluster framework (which, when applied straightforwardly, is non-variational).56 58... 

Frequency-dependent higher-order properties can now be obtained as derivatives of the real part of the time-average of the quasi-energy W j- with respect to the field strengths of the external perturbations. To derive computational efficient expressions for the derivatives of the coupled cluster quasi-energy, which obey the 2n-(-1- and 2n-(-2-rules of variational perturbation theory [44, 45, 93], the (quasi-) energy is combined with the cluster equations to a Lagrangian ... 

The first derivative the excitation energies Lagrangian formulation [13, 14], to avoids the evaluation of the first derivative of the coupled-cluster ground state T and A amplitudes. 

The transition expectation value Eq , or coupled cluster Lagrangian Lq, is thus stationary with respect to the configuration or determinant coefficients and therefore satisfies partially the Hellmann-Feynman theorem... 

For reasons of computational practicality and efficiency, molecular electronic coupled-cluster energies are determined using a nonvariational projection technique. The chief deficiency of this approach is not so much the loss of boundedness (since the coupled-cluster energy is nevertheless rather accurate), but the difficulties that it creates for the calculation of properties as the conditions for the Hellmann-Feynman theorem are not satisfied - even in the limit of a eomplete one-electron basis. Fortunately, as discussed in Section 4.2.8, this situation may be remedied by the construction of a variational Lagrangian [14]. In this formulation, the conditions of the Hellmann-Feynman theorem are fulfilled and molecular properties may be calculated by a proeedure that is essentially the same as for variational wave functions. The Lagrangian formulation of the energy is also related to a variational treatment of coupled-cluster theory applicable to excited states, as discussed in Section 13.6. 

The calculation of the zero-order multipliers and Lagrangian requires the solution of one set of linear equations. The resulting expression for the coupled-cluster energy is then variational with respect to the amplitudes as well as their multipliers. Note that the linear equations for the multipliers (13.5.6) are similar in structure to the Newton equations for the amplitudes (13.4.5), both containing the coupled-cluster Jacobian (13.4.4). 

As discussed in Section 13.5.2, for the calculation of first-order properties, it is in general more useful to consider the Lagrangian formulation of the coupled-cluster energy. Following (13.5.11), we may then calculate the energy according to the expression... 

The reasons for not invoking the variation principle in the optimization of the wave function are given in Chapter 13, which provides a detailed account of coupled-cluster theory. We here only note that the loss of the variational property characteristic of the exact wave function is unfortunate, but only mildly so. Thus, even though the coupled-cluster method does not provide an upper bound to the FCI energy, the energy is usually so accurate that the absence of an upper bound does not matter anyway. Also, because of the Lagrangian method of Section 4.2.8, the complications that arise in connection with the evaluation of molecular properties for the nonvariational coupled-cluster model are of little practical consequence. 

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed the Unked and unlinked equations of coupled-clustCT theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter. 

To obtain an optimized coupled cluster state, we require that the Lagrangian, Lcc/MM(tf, t), is stationary with respect to both the t and t parameters. It is advantageous to define the following one-electron interaction operator, Tg, as... 

The same problem with the pole structure appears also for coupled cluster response functions, if one defines them as derivatives of a time-average quasi-energy Lagrangian including orbital relaxation, ft is therefore preferable also in the analytical derivative approach like in Section 11.4 to derive coupled cluster response functions as derivatives of a time-dependent quasi-energy Lagrangian without orbital relaxation... 

In terms of the Lagrangian densities, we may calculate coupled-cluster first-order properties in the same way as for variational wave functions, contracting the density-matrix elements with the molecular integrals [17]. The Lagrangian density matrices are also known as the variational or relaxed density matrices. For a closed-shell CCSD wave function, an expression for the one-electron variational density matrix is derived in Exercise 13.5. 

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