Basis sets multiple electronic states

Figure 1 Energy spectrum of the low-lying states of four electrons confined in a quasi-one-dimensional Gaussian potential with (D, a>z,a>xy) = (4.0, 0.1, 20.0) for different-size basis sets. Energy levels of different spin multiplicities are indicated by different colors (See the caption to Figure 2). The number in the round brackets specifies the total number of basis functions and the parameter v p specifies the extended polyad quantum number (See the text for details). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this book.)...

We have a geometry and a basis set, and wish to do an SCF calculation on HHe+ with both electrons in the lowest MO, i/q, i.e. on the singlet ground state. In general, SCF calculations proceed from specification of geometry, basis set, charge and multiplicity. The multiplicity is a way of specifying the number of unpaired electrons ... 

Specify a geometry, basis set, and orbital occupancy (this latter is done by specifying the charge and multiplicity, with an electronic ground state being the default). 

We can thus conclude that states of different spin multiplicity (singlets, doublets, triplets, quartets, etc.) of very diverse jr-electron systems (Kekule or non-Kekule, alternant or nonalternant, aromatic, nonaromatic or antiaromatic) can be satisfactorily described by the PPP-VB method with a severely truncated set of covalent or maximally covalent structures using the same simple OEAO basis set hi. In contrast, the MO description requires a different handling of closed and open shell cases and the amount of correlation recovered in states of different multiplicity may be rather unbalanced. 

A basis set may be employed that is of the same form throughout the space of the system, or one in which the orbitals are expanded in different types of basis functions in different parts of space. Such partitioned bases are often used in solid-state calculations in which one must describe an overall wave function that is rapidly varying near the nuclei and slowly varying and free-electron-like when far from the nuclei. Such partitioned bases will be considered further in our discussion of band-theoretical calculations and the multiple-scattering Xa molecular-orbital method. 

The availability of numerical solutions of HF equations is still restricted to at most two-center (or linear) systems. Nonetheless, the so-called analytic approach, using suitable basis sets, enabled the computation of SCF solutions within the Roothaan linear combination of atomic orbitals (LCAO) SCF formalism [19]. Generation of such solutions, even for systems with several hundreds of electrons, is nowadays routine, although the handling of general open-shell states can still be frustrating at times, due to the possible multiplicity of various SCF solutions. 

Similar inputs are needed for other molecules. The initial geometry is to some extent arbitrary, and therefore in fact it cannot be considered as real input data. The only true information is the number and charge (kind) of the nuclei, the total molecular charge (i.e., we know how many electrons are in the system), and the multiplicity of the electronic state to be computed. The basis set issue (STO-3G) is purely technical and gives information about the quality of the results. 

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