Definition of Basis Sets

A choice of basis set implies a partitioning of the Hamiltonian, H = Hel -I- Hso + Tn(R) + Hrot, into two parts a part, H ° which is fully diagonal in the selected basis set, and a residual part, H(1b The basis sets associated with the various Hund s cases reflect different choices of the parts of H that are included in fP°). Although in principle the eigenvalues of H are unaffected by the choice of basis, as long as this basis set forms a complete set of functions, one basis set is usually more convenient to use or better suited than the others for a particular problem. Convenience is a function of both the nature of the computational method and the relative sizes of electronic, spin-orbit, vibrational, and rotational energies. The angular momentum basis sets, from which Hund s cases (a)-(e) bases derive, are 

Some of the molecular frame R-axis quantum numbers (Sr,Ir,sr and Jr) correspond to quantum numbers C = Ir and S = Sr that have been in widespread use in the literature. There is, however, an ambiguity concerning onto which molecule frame rotation axis (J+,N+,or R), the projection quantum number is defined. The subscript- notation eliminates this ambiguity and we therefore recommend its use. Also, the lower-case forms of Ir and sr conform to the convention that single-electron quantum numbers appear as lower-case letters. 

The quantum numbers that are listed in any basis set must be eigenvalues of operators that form a set of mutually commuting operators. Watson (1999) analyzes the commutation rules among the magnitude, A2, and molecule frame component, Aa, angular momentum operators, where A = N, N+, 1, and explains why 

The Ir quantum number is a label that conveys information about the projection of 1 on the axis of nuclear rotation, R. In the limit of high R (or N), when R, N, and N+ all point in approximately the same direction, Ir does in fact specify the integer valued projection of 1 on R. However, Ir can always be regarded as an index that exactly specifies a difference in the magnitude quantum numbers, 

However, the expectation values of the operators derived by vector analysis to 

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Gutowski, M., Szczesniak, M. M., and Chalasinski, G., Comment on "A possible definition of basis set superposition error , Chem. Phys. Lett. 241, 140-145 (1995). 

The definition of basis sets involves the selection of a set of functions for each angular momentum of the atom. In non-relativistic calculations, is a good quantum number, but in relativistic calculations it is j or k which is the good quantum number. However for the lighter elements where the effects of relativity are small, is an approximately good quantum number, and basis sets for the spin-orbit components of a non-relativistic subshell can share exponents. Basis sets that are optimized with the same exponents for the two spin-orbit components are called -optimized. Similarly, basis... 

Electron correlation studies demand basis sets that are capable of very high accuracy, and the 6-31IG set I used for the examples above is not truly adequate. A number of basis sets have been carefully designed for correlation studies, for example the correlation consistent basis sets of Dunning. These go by the acronyms cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z and cc-pV6Z (double, triple, quadruple, quintuple and sextuple-zeta respectively). They include polarization functions by definition, and (for example) the cc-pV6Z set consists of 8. 6p, 4d, 3f, 2g and Ih basis functions. 

We have seen in the previous section that the definition of a set of OjS is intimately bound up with some choice of function space. The reader is cautioned, however, that not all function spaces can be used to define OjjS appropriate for a given point group. For example, the functions cosXi, sinx cosx and sinx, do not forma basis for a representation of the (symmetric tripod) point group xl and xt are the coordinates introduced before (see Fig. 5-2.2). 

Figure 10 Comparison of individual (left) and average (right) density matrices in the case of a non-mutual exchange. (A) Definition of spin set or spin system (see Figure 9). (B) Basis functions (lines) and intramolecular single quantum coherences (arrows) defined by the spin set or spin system. (C) Elements corresponding to the intramolecular single quantum coherences in the density matrix.

More effort has recently gone into including electron correlation by MCSCF or SCF-CI methods, and the first definitive study was by Schaefer and Bender in 1971,483 who used a variety of basis sets and the INO procedure. An important conclusion from these calculations was that minimal basis set plus full Cl essentially duplicates the correlation effects observed with larger basis sets and less than full Cl, as was found... 

With much more powerful quantum mechanical computations available (i.e., Gaussian 98), the method was applied to a variety of photochemical reactions (note Scheme 1.12). The expression in Equation 1.12 for the delta-density matrix elements includes overlap integrals to take care of basis set definitions. Weinhold NHOs (i.e., hybrids) were used in order to permit easy analysis in terms of basis orbital pair bonds comprising orbital pairs. Note A refers to a reactant and B refers to the corresponding excited state in this study. 

Finally, some spectroscopic applications for pseudopotentials within SOCI methods are presented in section 3. We focus our attention on applications related to relativistic averaged and spin-orbit pseudopotentials (other effective core potentials applications are presented in chapters 6 and 7 in this book). Due to the large number of theoretical studies carried out so far, we have chosen to illustrate the different SOCI methods and discuss a few results, rather than to present an extensive review of the whole set of pseudopotential spectroscopic applications which would be less informative. Concerning the works not reported here, we refer to the exhaustive and up-to-date bibliography on relativistic molecular studies by Pyykko [21-24]. The choice of an application is made on the basis of its ability to illustrate the performances on both the pseudopotential and the SOCI methods. One has to keep in mind that it is not easy to compare objectively different pseudopotentials in use since this would require the same conditions in calculations (core definition, atomic basis set, SOCI method). The applications are separated into gas phase (section 3.1) and embedded (section 3.2) molecular applications. Even if the main purpose of this chapter is to deal with applications to molecular spectroscopy, it is of great interest to underline the importance of the spin-orbit coupling on the ground state reactivity of open-shell systems. A case study is presented in section 3.1.4. 

The definition of the gas-phase acidity through reaction (7.3) implies that this quantity is a thermodynamic state function. Thus, one could use quantum chemical approaches to obtain gas-phase acidities from the theoretically computed enthalpies of the species involved. However, two points must be noted before one proceeds A chemical bond is being broken and an anion is being formed. Thus, one may anticipate the need for a proper treatment of electronic correlation effects and also of basis sets flexible enough to allow the description of these effects and also of the diffuse character of the anionic species, what immediately rules out the semi-empirical approaches. Hence, our discussion will only consider ab initio (Hartree-Fock and post-Hartree-Fock) and DFT (density functional theory) calculations. 

The Chemical Hamiltonian Approach uses the atomic orbital basis set for the definition of an atomic partition of the Hamiltonian operator. An atomic subsystem of a molecule consists of a nucleus and the set of basis functions centered on it. Accordingly, the partition of the finite basis set corresponds to the physical partition of a molecule into atomic subsystems. This raises the problem of non-orthogonality of the basis functions belonging to the different subsystems (atoms, in the present case) and also the problem of basis set superposition error (BSSE), which is a consequence of the finiteness of the basis set [46]. 

This new proposal is elaborated in detail elsewhere [25]. Clearly much more numerical corroboration is needed before claiming a definitive usefulness of LDMs in evaluating and comparing the quality of basis sets and/or levels of theory. 

The counterpoise correction typically overestimates the BSSE since the monomer basis set is enhanced not only by empty orbitals of the other fragment, but also by orbitals occupied by electrons of the other monomer molecule which are excluded by the Pauli principle. Thus, if CP-corrected and uncorrected interaction energies are plotted as function of basis set size, they approach from above and below, respectively, the true interaction energy at the complete basis set (CBS) limit. CP corrections are mandatory for all double-zeta calculations and with MP2 or CCSD(T) also for triple-zeta basis treatments. In triple-zeta basis set (e.g., cc-pVTZ or TZVPP) DPT calculations, the BSSE is typically less than 5-10% of the interaction energy which makes the laborious CP correction unnecessary. If sets of valence quadruple-zeta are used, it seems as if the error of the CP procedure is often similar to the (uncorrected) BSSE, but this is system-dependent and more definite conclusions about this issue requires further work. 

Here we indicate the effect of basis-set variation at the closed-shell HF level, for a single 1-electron perturbation, in a form similar to that given above but with small changes in the definitions. Thus, the matrix elements of h will suffer a first-order change arising partly from the change in the operator and partly from the change of the basis. When we write h = ho + -H. .. the first-order term will consequently be... 

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