Perturbation-dependent basis sets

For perturbation-dependent basis sets (e.g. geometry derivatives) the (first-order) CPHF equations involve (first) derivatives of the one- and two-electron integrals with respect to the perturbation. For basis functions which are independent of the... 

Furthermore, one is faced with the gauge-origin problem, as discussed in Section 5.10. A popular solution to this problem is to work with perturbation-dependent basis sets. In the case of an external magnetic field as perturbation such basis functions... 

Olsen, J., Bak, K. L., Ruud, K., Helgaker, T., Jorgensen, P (1995). Orbital connections for perturbation-dependent basis sets. Theoretica Chimica Acta, 90, 421-439. 

If the perturbation is spin—orbit coupling then a field-independent basis set is used in the calculation of interest (yv( 1)=0) and the t/1) coefficients are all that is needed to calculate the perturbed integrals. If the perturbation is a magnetic field then a field-independent basis set can be used. However, it is often desirable to utilize a field-dependent basis set such as GIAOs (66-69) that reduces the origin dependence of the results obtained. With such a basis set the evaluation of perturbed integrals is somewhat more involved as the Xv(1) terms are no longer zero. 

Using the complete form of the LR-CCSD (linear response CC formalism with single and double excitations) theory in combination with the ZPolC perturbation-tailored basis set Kowalski et have provided accurate estimates for the static and dynamic electronic polarizabilities of Ceo and have showed that the T2-dependent terms not included in CC2 play in important role in estimating these properties since they lead to a reduction of the polarizability by 12 13%. In the static limit, their estimate amounts to 82.23 in comparison with the 76.5 8 measured value. Note also that in the static limit the vibrational contribution is not necessary negligible. For a wavelength of 1064 nm, their estimate amounts to 83.62 A in comparison with the experimental value of 79 4 A. ... 

Spin-Rotation Interaction As explained above, the nuclear spin-rotation tensor consists of an electronic and a nuclear part, with the former computed as the second derivative of the electronic energy with respect to the rotational angular momentum and the appropriate nuclear spin as perturbations [38]. This is efficiently done using analytic second-derivative techniques [65], but, while calculations carried out using standard basis functions suffer from a slow basis set convergence, the use of perturbation-dependent basis functions significantly accelerates the basis set convergence [38]. 

As concerns the computational requirements, we note that electron correlation effects are important and that the CCSD(T) approach appears to be the method of choice for the corresponding calculations [e.g., 79]. With respect to the basis set convergence, even if a significant speed-up is already obtained by using perturbation-dependent basis functions, it should be noted that the basis set requirements are rather demanding In general, basis sets of at least quadruple-zeta quality are required to obtain converged values. Also important are in some cases consideration of additional diffuse functions [38, 39] and the inclusion of vibrational effects [79, 80]. 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

If the pertiubation is a change in geometry, the basis set is clearly perturbation dependent since the functions move along with the nuclei. Standard perturbation theory is therefore not suitable for calculating molecular gradients. 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

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