Coupled cluster wavefunction

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) ... 

To understand the structure of the coupled-cluster wavefunction, let us Taylor expand the exponential in Eq. (2.1). Sorting the resulting expansion according to the level of excitation, we obtain... 

Again, the coefficients describing the time dependence of the approximate coupled cluster wavefunctions [Pg.190]

The form of the wave operators need not be defined, but, in principle, they can describe any type of wavefunction, for example, Hartree-Fock or coupled-cluster wavefunctions. However, at their core, they always consist of strings of creation operators. We define the supermolecular wavefunction as... 

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 ... 

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... 

It is perhaps not immediately clear how one may go about solving the Tj and T2 amplitude equations given in Eqs. [152] and [153] for the individual amplitudes, tf and Iff- . A simple rearrangement of the equations, however, provides a more palatable form of these expressions that leads to a simple iterative approach for determining the coupled cluster wavefunction amplitudes. For example, the first few terms of Eq. [152] may be written as... 

It should be noted, however, that the use of a spin-symmetry-adapted determinant such as the ROHF wavefunction as a reference in a coupled cluster calculation does produce a spin-pure energy, but does not imply that the correlated wavefunction itself is an eigenfunction of as well. For the spin-orbital definition of T described here, spin contamination can still enter into the coupled cluster wavefunction through the nonlinear contributions of cluster operators to the amplitude equations, though the importance of this... 

The ROHF-CCSD energy is indeed completely spin-projected, as discussed in Refs. 35,27, and 37, but is still different from that computed using a spin-adapted coupled cluster wavefunction. 

T. D. Crawford, T. J. Lee, and H. F. Schaefer, /. Chem, Phys, 107, 9980 (1997). Spin-Restricted Brueckner Orbitals for Coupled-Cluster Wavefunctions. 

The time-dependent ground-state coupled cluster wavefunction for such a system is conveniently parameterized in a form, where the oscillating phase factor caused by the so-called level-shift [43 5, 88, 89] or time-dependent quasi-energy W(r, e) (vide infra) is explicitly isolated [42 6, 90, 91] ... 

R. A. Chiles and C. E. Dykstra, J. Chem. Phys., 74,4544 (1981). An Electron Pair Operator Approach to Coupled-Cluster Wavefunctions. Applications to He2, Bc2, and Mg2 and Comparison with CEPA Methods. 

Analytical first derivatives of Cl wavefunaions were first achieved by Pople and co-workers and Schaefer and co-workers " in 1980. After that, there was an explosion of effort to continue the differentiation of Cl energies to higher order [see, for example, 55-57]. For MBPT, methods for up through third derivatives of second-order wavefunctions have been developed. And, analytic differentiation of coupled cluster wavefunctions has even been achieved recently. " ... 

B. Kirtman and C. E. Eiykstra,/. Chem. Phys., 85,2791 (1986). Local Space Approximation for Configuration Interaction and Coupled Cluster Wavefunctions. 

In the PCM-EOM-CC/SACCI approximation the excited electronic states are represented by a linear (Cl-like) expansion build-up on the coupled-cluster wavefunction for the ground state [6-8]. We use the PTE couple-cluster wavefunction, computed in the presence of the frozen Hartree-Fock reaction field, as it leads to a more simpler and physically transparent PCM-EOM theory. The EOM-CC theory leads to a non-Hermitian eigenvalue problem with right and left eigenvalues. 

The basic premise of CC theory is that the coupled-cluster wavefunction Pec be expressed as a power series. 

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

In the case of the coupled cluster wavefunction the equations for the wavefunction parameters, i.e. for the coupled cluster amplitudes are simply the equations for the coupled cluster vector function in Eq. (9.81). The constraints are then = 0 and the coupled cluster Langrangian (Christiansen et ai, 1995a, 19986) is given as... 

However, other attempts have been made to improve on the treatment of electron correlation in SOPPA. Three SOPPA-like methods have thus been presented. All are based on the fact that a coupled cluster wavefunction gives a better description than the Mpller-Plesset first- and second-order wavefunctions, Eqs. (9.66) and (9.70). In the second-order polarization propagator with coupled cluster singles and doubles amplitudes-SOPPA(CCSD)-method (Sauer, 1997), the reference state in Eqs. (3.160) to (3.163) is approximated by a linearized CCSD wavefunction... 

In this chapter we will follow now the second approach, which means that we will apply time-independent and time-dependent perturbation theory from Chapter 3 to approximate solutions of the unperturbed molecular Hamiltonian. In particular, we will illustrate this in the following for Hartree-Fock, MCSCF and coupled cluster wavefunctions. 

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 ... 

Eq. (5.75) in Tables 13.4 to 13.6 as examples. The emphasis is here on the comparison of some of the methods introduced in Chapters 10 to 12 and in particular on the effect of electron correlation, meaning the difference in the results obtained with methods based on the Hartree-Fock wavefunction, like SCF linear response (section 11.2) or RPA (section 10.3) and CHF (section 11.1) on one side and with methods based on multiconfigurational (sections 10.4 and 11.2), Mpller-Plesset (sections 10.3 and 12.2) or coupled cluster wavefunctions (sections 10.3, 11.4 and 12.2) on the other side. Only results for small molecules are discussed here. 

To define the PCM free energy functional we introduce the bra coupled-cluster wavefunction ... 

We note that arguments of the expectation values (O Eqs. 28.12 and O 28.13) denote an explicit functional dependence on the T and A amplitudes. These amplitudes, which define completely the coupled-cluster wavefunction of the molecular solute, can be determined imposing the stationary conditions on the energy functional Gcc(A, T). 

The last term of AG ° in O Eq. 28.52 introduces the constraint for the ground state coupled-cluster wavefunction, and contains the de-excitation operator Z given by... 

Specification of the coupled-cluster wavefunction and the corresponding energy requires determination of the so far unknown amplitudes tf, tf, ... which are used to parametrize the exponential ansatz in equation (1). The usual procedure for this - often referred to as the standard coupled-cluster approach - involves the following steps... 

The problem with conventional configuration interaction or coupled-cluster wavefunctions is that the cusp conditions cannot be fulfilled, or more precisely that the overall shape of the wavefunction in the vicinity of electron coalescence, the Coulomb hole, converges extremely slowly with the size of the underlying basis set expansion. Figure 1 demonstrates that the expansion in orbital products does not provide terms which are linear in While at large separation, the wavefunction shape is well described even by the shortest Cl expansion, even the largest Cl expansion is far off at rj2 0. 

The coupled-cluster wavefunction is generated by an exponential wave operator acting on a reference determinant fPo),... 

With this, the general coupled-cluster wavefunction ansatz in F12 theory can be written as... 

The above equations might imply that only minimal changes have been made to the conventional coupled-cluster theory. Flowever, the non-linear character of the coupled-cluster wavefunction gives rise to a rather large number of additional terms. Originally, the theory was implemented within the standard approximation" which eliminates many terms, for a discussion see Ref. 46. These early implementations used linear correlation factors (CCSD-R12), the analogous implementation with Slater-type correlation factors (and the standard approximation) was reported more recently. ... 

Of the 33 invited speakers and the seven who contributed talks, 17 accepted our invitation to contribute a chapter to this book. These chapters are complemented by three additional chapters from individuals to help develop a more cohesive book as well as an overview chapter. Approximately half of the chapters are focused on the development of ab initio electronic structure methods and consideration of specific challenging molecular systems using electronic structure theory. Some of these chapters document the dramatic developments in the range of applicability of the coupled-cluster method, including enhancements to coupled-cluster wavefunctions based on additional small multireference configuration interaction (MR-CISD) calculations, the method of moments, the similarity transformed equation of motion (STEOM) method, a state-specific multireference coupled-cluster method, and a computationally efficient approximation to variational coupled-cluster theory. The concentration on the coupled-cluster approach is balanced by an approximately equal number of chapters discussing other aspects of modem electronic stracture theory. In particular, other methods appropriate for the description of excited electronic states, such as multireference... 

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