Coupled-cluster analytic second derivative

Kallay, M., Gauss, J. Analytic second derivatives for general coupled-cluster and configuration-interaction models. J. Chem. Phys. 2004, 120, 6841-8. 

J. F. Stanton and J. Gauss, in Recent Advances in Coupled-Cluster Methods, R. J. Bartlett, Ed., World Scientific Publishing, Singapore, 1997, pp. 49-79. Analytic Evaluation of Second Derivatives of the Energy Computational Strategies for the CCSD and CCSD(T) Approximations. 

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. 

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. " ... 

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