The 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,... 

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 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... 

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. 

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... 

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... 

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... 

The improvements within the hierarchy of A -electron models is probably a more complicated task. There are a couple of quantum-chemistry theories that allow us to approach the exact Schrodinger equation systematically. Among them the coupled-cluster (CC) method represents probably the most successful approach. It can be applied to relatively large systems and the theory is both size-extensive and size-consistent. So far the only way to approach the exact Schrodinger equation, within the hierarchy of the A -electron models (following the horizontal axis on Figure 1), is the systematic extension of the excitation level. In CC theory there is a series of models that refer to the way the cluster operator is truncated (CCS, CCSD, CCSDT, CCSDTQ and so on). In the limit of the untruncated cluster operator the CC wave function becomes equivalent with full Cl, which is the exact solution of the Schrodinger equation within a particular basis set. The truncation level indicates, in some sense, the accuracy of the model which is almost always limited by the available computational resources. 

The hierarchy of coupled-cluster models provides a clear route towards the exact solution of the Schrodinger equation, but the slow basis-set convergence limits the accuracy sometimes even for small molecules. The way to overcome this problem is to combine the coupled-cluster model with the explicitly correlated approach. It can be done, in principle, for any model within the coupled-cluster hierarchy. The main task of this work is, however, the implementation of explicitly correlated CCSD model, hence the discussion will be focused on this particular model. 

For actual applications of the cluster ansatz even in its truncated form it is necessary that no infinite sums as in Eq. (8.238) occur. Fortunately, we can benefit from the properties of the creation and annihilation operators. In order to understand how, we write the electronic Schrodinger equation for the coupled-cluster wave function. 

Contrary to response theory for exact states, in Section 3.11, or for coupled cluster wavefunctions, in Section 11.4, in MCSCF response theory the time dependence of the wavefunction is not determined directly from the time-dependent Schrodinger equation in the presence of the perturbation H t), Eq. (3.74). Instead, one applies the Ehrenfest theorem, Eq. (3.58), to the operators, which determine the time dependence of the MCSCF wavefunction, i.e. the operators hj ... 

Projecting the similarity-transformed Schrodinger equation (13.2.20) against the same determinants as in (13.2.18) and (13.2.19), we arrive at the following set of equations for the coupled-cluster amplitudes and energy ... 

For the study of excited states and molecular properties, the EOM-CC model constitutes a conceptually simple approach closely related to the Cl model, with the emphasis shifted away fiom excitations from determinants towards excitations from a more general state. However, the EOM-CC approach is somewhat restricted in the sense that it can be applied only to the standard coupled-cluster mcxlels such as CCSD and CCSDT. For the calculation of molecular properties and excitation energies, the response-fuiKtion approach - based on the solution of the time-dependent Schrodinger equation - constitutes a more general framework, also encompassing, for example, the quadratic Cl model of Section 13.8.2 and the iterative hybrid CC2 and CC3 models of Section 14.6.2 [20,21]. 

Let us consider the simultaneous solution of the Schrodinger and spin equations for high-spin open-shell states. The coupled-cluster wave function may be expressed in terms of a singledeterminant reference function with the excitation operators chosen to yield excitations from this reference state. The full coupled-cluster wave function, being equivalent to the FCI wave function in the same orbital basis, is a simultaneous eigenfunction of the Schrodinger equation and the spin equations ... 

We solve the Schrodinger equation (14.1.29) by a projection technique, in a manner reminiscent of that employed for the coupled-cluster energy and wave function in Section 13.2. Thus, the original equation (14.1.29) is equivalent to the following two equations, obtained by projection against the zero-order state and its orthogonal complement ... 

With the coupled cluster wave function (4.46) the Schrodinger equation becomes... 

There is nowadays agreement that the most systematic approach to the solution of the electronic Schrodinger equation is provided by coupled-cluster theory. The coupled-cluster singles and doubles scheme with noniterative treatment of triply excited clusters, CCSD(T), " is sometimes addressed as the gold standard in quantum chemistry, and recent efforts to facilitate calculations with higher-order methods have enabled the solution of the electronic Schrodinger equation with unprecedented accuracy. 

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