Coupled-cluster theory coefficients

Hence, if we are to make bE = 0, we can avoid these terms. But to do so we have to have E optimum with respect to the location of the atomic basis functions, t (R) the MO coefficients, c(R) and the Cl coefficients, C(R). The first cannot be satisfied unless the atomic orbital basis set is floated off the atomic centers to an optimum location [105], while the second requires optimum MO coefficients, and the third optimum Cl coefficients. In practice, we will introduce atomic orbital derivatives explicitly, so the AOs can follow their atoms. Now focusing only on the MO and Cl coefficients, in SCF we have optimum MOs and no Cl term. In MCSCF, both terms would vanish, whUe in Cl, the MO derivatives would remain, but the Cl coefficients contribution would vanish. In the non-variational coupled-cluster theory, neither will vanish and this means that CC theory forces us into some new considerations for analytical forces. 

In coupled cluster theory it is customary to use the term amplitudes for the expansion coefficients t, which are equivalent to the coefficients in eq. (4.1). 

It is important to note that, at each level of coupled-cluster theory, we include through the exponential parameterization of Eq. (28) all possible determinants that can be generated within a given orbital basis, that is, all determinants that enter the FCI wave function in the same orbital basis. Thus, the improvement in the sequence CCSD, CCSDT, and so on does not occur because more determinants are included in the description but from an improved representation of their expansion coefficients. For example, in CCS theory, the doubly-excited determinants are represented by ]HF), whereas the same determinants are represented by (T2 + Tf) HF) in CCSD theory. Thus, in CCSD theory, the weight of each doubly-excited determinant is obtained as the sum of a connected doubles contribution from T2 and a disconnected singles contribution from Tf/2. This parameterization of the wave function is not only more compact than the linear parameterization of configuration-interaction (Cl) theory, but it also ensures size-extensivity of the calculated electronic energy. 

A time-dependent coupled cluster theory with unrestricted electron spins and full treatment of orbital rotation has been implemented to calculate the polarizabilities and dispersion coefficients. Illustration calculations on Li, Ar, HCl, CO, N2, O2, and H2O at the coupled cluster singles and doubles level have demonstrated the reliabihty of the method. Comparisons with HF and MP2 results have further shown the importance of high-order electron correlation effects whereas basis sets of the aug-cc-pVXZ family have been compared. 

Dispersion coefficients for second hyperpolarizabilities using coupled cluster cubic response theory... 

Consequently, it is possible to write reference states for the electron propagator approach as expansion coefficients of the perturbation theory or as converged T amplitudes from the solution of the coupled-cluster equations. 

Second derivatives for configuration-interaction, coupled-cluster and many-body perturbation theory require first derivatives of those coefficients determined variationally and the second derivatives of those coefficients not... 

PHF methods can, in turn, be classified as the variational and nonvariational ones. In the former gronp of methods the coefficients in linear combination of Slater determinants and in some cases LCAO coefficients in HF MOs are optimized in the PHF calculations, in the latter such an optimization is absent. To the former group of PHF methods one refers different versions of the configuration interaction (Cl) method, the multi-configuration self-consistent field (MCSCF) method, the variational coupled cluster (CC) approach and the rarely used valence bond (VB) and generaUzed VB methods. The nonvariational PHF methods inclnde the majority of CC reaUza-tions and many-body perturbation theory (MBPT), called in its molecular realization the MoUer-Plessett (MP) method. In MP calculations not only RHF but UHF MOs are also used [107]. 

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. 

Hattig, C., Jorgensen, P. (1998). Dispersion coefficients for first hyperpolarizabilities using coupled cluster quadratic response theory. Theoretical Chemistry Accounts, 100, 230. 

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