Coupled-cluster energy expression

This equation is not restricted to the CCSD approximation, however. Since higher excitation cluster operators such as T3 and T4 cannot produce fully contracted terms with the Hamiltonian, their contribution to the coupled cluster energy expression is zero. Therefore, Eq. [134] also holds for more complicated methods such as CCSDT and CCSDTQ. Higher excitation cluster operators can contribute to the energy indirectly, however, through the equations used to determine the amplitudes, and t-h, which are needed in the energy equation above. 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

Using the truncated Hausdorff expansion, we may obtain analytic expressions for the commutators in Eq. [52] and insert these into the coupled cluster energy and amplitude equations (Eqs. [50] and [51], respectively). However, this is only the first step in obtaining expressions that may be efficiently implemented on the computer. We must next choose a truncation of T and then derive expressions containing only one- and two-electron integrals and cluster amplitudes. This is a formidable task to which we will return in later sections. 

In this chapter we wiU finally follow the third approach, which means that we abandon the perturbation-theory approach all together and go back to the definitions of the properties as derivatives of the energy in the presence of the perturbation. We will illustrate with a few examples how this approach can be appfied to approximate expressions for the energy in the presence of both static as well as time-dependent perturbations. However, the presentation will be very brief and restricted to Mpller-Plesset perturbation theory and coupled cluster energies as nothing new is obtained for variational methods compared to the response theory approaches in Chapters 10 and 11. 

Inserting the Hamiltonian (13.2.42) into the expressions for the energy (13.2.22) and the nonlinear equations (13.2.23), we obtain the following expressions for the coupled-cluster energy and the amplitude equations ... 

We consider the variational reformulation of the coupled-cluster energy in the presence of a perturbation aV. The coupled-cluster energy is obtained from the standard expression... 

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... 

As discussed in Section 13.2.3, the full (untruncated) coupled-cluster energy may be calculated from the expression... 

Rather than calculating the energy from the general RSPT expression (14.3.32), we shall here use the equivalent form obtained from the coupled-cluster energy (14.3.13) (see the discussion at the end of this subsection), writing the energy correction as... 

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

Most of these extensions have involved electron correlation methods based on variational approaches (DFT, MCSCF, CI,VB). These methods can be easily formulated by optimizing the free energy functional (1.117), expressed as a function of the appropriate variational parameters, as in the case of the HF approximation. In contrast, for nonva-riational methods such as the Moller-Plesset theory or Coupled-Cluster, the parallel extension to solvation model is less straightforward. 

T. Korona, B. Jeziorski, One-electron properties and electrostatic interaction energies from the expectation value expression and wave function of singles and doubles coupled cluster theory. 

The projective techniques described above for solving the coupled cluster equations represent a particularly convenient way of obtaining the amplitudes that define the coupled cluster wavefunction, e o However, the asymmetric energy formula shown in Eq. [50] does not conform to any variational conditions in which the energy is determined from an expectation value equation. As a result, the computed energy will not be an upper bound to the exact energy in the event that the cluster operator, T, is truncated. But the exponential ansatz does not require that we solve the coupled cluster equations in this manner. We could, instead, construct a variational solution by requiring that the amplitudes minimize the expression ... 

The coupled cluster Schrodinger equation, which leads to the energy and amplitude expressions given in Eqs. [50] and [51], may be written as... 

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