Coupled-cluster Schrodinger equation

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

Inserting the coupled cluster wavefunction, Eq. (9.72), in the Schrodinger equation gives the coupled cluster Schrodinger equation... 

The equation for the T amplitudes has a clear physical meaning it corresponds to projection in the manifold spanned by all the orthonormal excitations to HF) of the coupled-cluster Schrodinger equation for the molecular solute,... 

The projected coupled-cluster Schrodinger equation (13.2.32) therefore yields at most quartic equations in the cluster amplitudes - even for the full cluster expansion. The BCH expansion terminates because of the special structure of the cluster operators, which are linear combinations of commuting excitation operators of the form (13.2.6) and (13.2.7). 

The unlinked coupled-cluster Schrodinger equation is given by... 

In this section we will introduce some wavefunction-based methods to calculate photoabsorption spectra. The Hartree-Fock method itself is a wavefunction-based approach to solve the static Schrodinger equation. For excited states one has to account for time-dependent phenomena as in the density-based approaches. Therefore, we will start with a short review of time-dependent Hartree-Fock. Several more advanced methods are available as well, e.g. configuration interaction (Cl), multireference configuration interaction (MRCI), multireference Moller-Plesset (MRMP), or complete active space self-consistent field (CASSCF), to name only a few. Also flavours of the coupled-cluster approach (equations-of-motion CC and linear-response CQ are used to calculate excited states. However, all these methods are applicable only to fairly small molecules due to their high computational costs. These approaches are therefore discussed only in a more phenomenological way here, and many post-Hartree-Fock methods are explicitly not included. 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

With the triples correction added, the error relative to experiment is still as large as 15 kJ/mol. More importantly, we are now above experiment and it is reasonable to assume that the inclusion of higher-order excitations (in particular quadruples) would increase this discrepancy even further, perhaps by a few kJ/mol (judging from the differences between the doubles and triples corrections). Extending the coupled-cluster expansion to infinite order, we would eventually reach the exact solution to the nonrelativistic clamped-nuclei electronic Schrodinger equation, with an error of a little more than 15 kJ/mol. Clearly, for agreement with experiment, we must also take into account the effects of nuclear motion and relativity. 

In order to use wave-function-based methods to converge to the true solution of the Schrodinger equation, it is necessary to simultaneously use a high level of theory and a large basis set. Unfortunately, this approach is only feasible for calculations involving relatively small numbers of atoms because the computational expense associated with these calculations increases rapidly with the level of theory and the number of basis functions. For a basis set with N functions, for example, the computational expense of a conventional HF calculation typically requires N4 operations, while a conventional coupled-cluster calculation requires N7 operations. Advances have been made that improve the scaling of both FIF and post-HF calculations. Even with these improvements, however you can appreciate the problem with... 

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... 

MR-MBPT methods, however, is not straightforward. The main problem here is similar With the coupled cluster wave function (4.46) the Schrodinger equation becomes... 

Since its introduction into quantum chemistry in the late 1960s by Qzek and Paldus, " coupled cluster theory has emerged as perhaps the most reliable, yet computationally affordable method for the approximate solution of the electronic Schrodinger equation and the prediction of molecular properties. The purpose of this chapter is to provide computational chemists who seek a deeper knowledge of coupled cluster theory with the background necessary to understand the extensive literature on this important ab initio technique. 

The exponential ansatz described above is essential to coupled cluster theory, but we do not yet have a recipe for determining the so-called cluster amplitudes (tf. If-- , etc.) that parameterize the power series expansion implicit in Eq. [31]. Naturally, the starting point for this analysis is the electronic Schrodinger equation,... 

As discussed earlier, the cluster amplitudes that parameterize the coupled cluster wavefunction may be determined from the projective Schrodinger equation given in Eq. [51]. In the CCSD approximation, the single-excitation amplitudes, t- , may be determined from... 

In the above the coupled cluster equations have been derived by multiplying the Schrodinger equation with ( o, ( 1 ( 1- alternative way of deriving the... 

Certain Schrodinger equation based methods, such as coupled cluster theory, are not based on a variational principle. They fall outside schemes that use the energy expectation value as a optimization function for simulated annealing, although these methods could be implemented within a simulated annealing molecular dynamics scheme with alternative optimization function. 

The truncated many-particle wave function in the coupled-cluster method is required to satisfy the Schrodinger equation... 

For the coupled cluster methods, which are non-variational, the initial values of the A s are nonzero, and 0) does not correspond to the unperturbed reference state but, in most applications, to the Hartree-Fock state. Tire initial values of the parameters are found in an iterative optimization of the coupled cluster state, and the time-dependent values of the parameters were determined from the coupled-cluster time-dependent Schrodinger equation by Koch and Jprgensen [35], The coupled cluster state is not norm conserving, but the inno roduct of the coupled cluster state vector CC(f)) and a constructed dual vector (CC(f) remains a constant of time... 

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