Completion The Coupled-Cluster Expansion

Since spin-orbit coupling is not present in the nonrelativistic limit and the spin-orbit-coupling-free Hamiltonian commutes with the spin operators, the N KS spinors can be constructed from spatial 2-spinors ff tensorially multiplied with spin eigenfunctions a such that four-component spin orbitals ipi,x = cp1 OL and = pf S respectively, are obtained. Then, the spin-orbit-coupling-free (SOfree) z-component of the magnetization resembles the nonrelativistic spin density. 

So far, we have considered variational wave functions. An important approach to approximate N-electron wave functions is the coupled-cluster (CC) 

The Aj are operators that contain the Cl coefficients and generate all excitations, i.e., singles. 

Ti generates all single excitations, T2 all double excitations and so forth. They are explicitly defined as 

In principle, the coupled-cluster ansatz for the wave function is exact if the excitation operator in Eq. (8.234) is not truncated. But this defines an FQ approach, which is unfeasible in actual calculations on general many-electron systems. A truncation of the CC expansion at a predefined order in the excitation operator T is necessary from the point of view of computational practice. Truncation after the single and double excitations, for instance, defines the CCSD scheme. However, in contrast with the linear Cl ansatz, a truncated CC wave function is still size consistent, because all disconnected cluster amplitudes which can be constructed from a truncated set of connected ones are kept [407]. The maximum excitation in T determines the maximum connected 

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MBPT starts with the partition of the Hamiltonian into H = H0 + V. The basic idea is to use the known eigenstates of H0 as the starting point to find the eigenstates of H. The most advanced solutions to this problem, such as the coupled-cluster method, are iterative well-defined classes of contributions are iterated until convergence, meaning that the perturbation is treated to all orders. Iterative MBPT methods have many advantages. First, they are economical and still capable of high accuracy. Only a few selected states are treated and the size of a calculation scales thus modestly with the basis set used to carry out the perturbation expansion. Radial basis sets that are complete in some discretized space can be used [112, 120, 121], and the basis... 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

Two relevant topics have been ignored completely in this short chapter the treatment of electron correlation with more sophisticated methods than DFT (that remains unsatisfactory from many points of view) and the related subject of excited states. Wave function-based methods for the calculation of electron correlation, like the perturbative Moller-Plesset (MP) expansion or the coupled cluster approximation, have registered an impressive advancement in the molecular context. The computational cost increases with the molecular size (as the fifth power in the most favorable cases), especially for molecules with low symmetry. That increase was the main disadvantage of these electron correlation methods, and it limited their application to tiny molecules. This scaling problem has been improved dramatically by modern reformulation of the theory by localized molecular orbitals, and now a much more favorable scaling is possible with the appropriate approximations. Linear scaling with such low prefactors has been achieved with MP schemes that the... 

Before proceeding to open-shell theory, it is worth noting that CCSD properly treats the nondynamical effects that are missing in a single-determinant reference function, which were discussed in section 12.1. This is because the coupled-cluster wave function is an infinite-order expansion to the given excitation level the coefficients of the determinants that complete the reference expansion and all the excitations from these are included and optimized in the coupled-cluster wave function. Also, the presence of single excitations accounts for the orbital relaxation that would correct the distortion of the reference determinant. 

Coupled cluster response calculaAons are usually based on the HF-SCF wave-function of the unperturbed system as reference state, i.e. they correspond to so-called orbital-unrelaxed derivatives. In the static limit this becomes equivalent to finite field calculations where Aie perturbation is added to the Hamiltonian after the HF-SCF step, while in the orbital-relaxed approach the perturbation is included already in the HF-SCF calculation. For frequency-dependent properties the orbital-relaxed approach leads to artificial poles in the correlated results whenever one of the involved frequencies becomes equal to an HF-SCF excitation energy. However, in Aie static limit both unrelaxed and relaxed coupled cluster calculations can be used and for boAi approaches the hierarchy CCS (HF-SCF), CC2, CCSD, CC3,... converges in the limit of a complete cluster expansion to the Full CI result. Thus, the question arises, whether for second hyperpolarizabilities one... 

Using the same kind of augmented correlation-consistent basis sets employed in the above coupled cluster methods, but working with the complete active subspace (CASSCF and CASSCF( +1, +2)) approximations to the Cl expansion, Lawson and Harrison101 have investigated the variation with interatomic distance and spatial distribution of the quadrupole moments of P2, S2 and CI2. The a and it contributions to the quadrupole are resolved and the poor results obtained at the SCF level are attributed to the inadequate representation of the it system in the SCF approximation for P2 and S2. 

The simplest way to realize an exponential expansion is to employ the exponential ansdtz of Jeziorski and Monkhorst [87] which exploits a complete model space. This is the approach that we followed in Section 4.2.2.2 in developing a multi-root multireference Brillouin-Wigner coupled cluster theory. The Jeziorski and Monkhorst exponential ansatz may be written... 

The preceding step to both MP2 and coupled-cluster calculations is to solve the Hartree-Fock equations. The standard approach is, of course, to solve the equations in a basis set expansion (Roothaan-Hall method), using atom-centered basis functions. This set of basis functions is used to expand the molecular orbitals and we will call it orbital basis set (OBS). It spans the computational (finite) orbital space. Occupied spin orbitals will be denoted (pi and virtual (unoccupied) spin orbitals pa- In order to address the terms that miss in a finite OBS expansion, the set of virtual spin orbitals in a formally complete space is introduced, pa- If we exclude from this space all those orbitals which can be represented by the OBS, we obtain the complementary space, with orbitals denoted cp i. The subdivision of the orbital space and the index conventions are summarized in the left part of Fig. 2. 

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