Particle basis functions

It is obvious that the projection operators for the different species have different numbers of terms in them. The HON species have 12 terms (3 x 2 ) while the A2B-type species have four terms, and the HDT+ isotopomer has only two terms. This results in different sizes of the spin-projected basis sets, and for this reason the properties obtained in this work are not precisely comparable between the A3, A2B, and ABC systems, although a very good idea of the trends may be obtained from the data in Table XVI. While all of the above are given in terms of the original particles, it should be noted that the permutations used in the internal particle basis functions are pseudo -permutations induced by the permutations on real particles. 

If the TV-particle basis were a complete set of JV-electron functions, the use of the variational approach would introduce no error, because the true wave function could be expanded exactly in such a basis. However, such a basis would be of infinite dimension, creating practical difficulties. In practice, therefore, we must work with incomplete IV-particle basis sets. This is one of our major practical approximations. In addition, we have not addressed the question of how to construct the W-particle basis. There are no doubt many physically motivated possibilities, including functions that explicitly involve the interelectronic coordinates. However, any useful choice of function must allow for practical evaluation of the JV-electron integrals of Eq. 1.7 (and Eq. 1.8 if the functions are nonorthogonal). This rules out many of the physically motivated choices that are known, as well as many other possibilities. Almost universally, the iV-particle basis functions are taken as linear combinations of products of one-electron functions — orbitals. Such linear combinations are usually antisymmetrized to account for the permutational symmetry of the wave function, and may be spin- and symmetry-adapted, as discussed elsewhere ... 

The majority of quantum-chemistry calculations have been carried out by employing the independent particle model in the framework of the HF method. In the most widely used approach molecular orbitals are expanded in predefined one-particle basis functions which results in recasting the integro-differential HF equations into their algebraic equivalents. In practice, however, the basis set used is never complete and very often far too limited to describe essential features of HF orbitals, for example, their behaviour in the vicinity of nuclei. That is why such calculations always suffer from the so called basis set truncation error . This error is difficult to estimate and often leads to low credibility of the results. 

The one-particle basis functions x are often referred to as atomic orbitals (AOs). The MO coefficients C are obtained by solving an electronic structure problem simpler than that of Eq. (2), such as the independent particle (Hartree-Fock) approximation, or using a multiconfigurational Hartree-Fock approach. This has the advantage that these approximations generally... 

An important development in FCI methodology was Handy s observation that if determinants were used as the n-particle basis functions, rather than CSFs, the Hamiltonian matrix element formulas could be obtained very simply. Specifically, it was possible to create an ordering of the determinants such that for each determinant, a list of all other determinants with which it had a non-zero matrix element could easily be determined. Further, it was easy to evaluate the coupling coefficients A and B in Eq. (11) (the only non-zero values being + 1) and therefore to compute the matrix elements. While this greatly expanded the range of FCI calculations that could be carried out, the algorithm is essentially scalar in... 

The generalization of the HF methods toward degenerate or nearly degenerate systems is known as multiconfigurational self-consistent field (MCSCF). In this method, the MCSCF wavefunction is initially expanded in a set of many particle basis functions (Slater determinants or CSFs) [65],... 

In this context, the idea of discrete numerical basis sets, introduced by Sa-lomonson and Oster (129) for the bound-state problem and combined with the complex-rotation method by Lindroth (30), is very interesting. One-particle basis functions are defined on a discrete grid inside a spherical box containing the system under cosideration. The functions are evaluated by diagonalizing the discretized one-particle complex-rotated Hamiltonian. Such basis sets are then used to compute autoionizing state parameters by means of bound-state methods (30,31,66). 

The CCSD(F12) model was first introduced as CCSD(R12) approximation [41] to the full CCSD-R12 approach [40]. Here, we indicate by writing R12 that in this initial work, explicitly correlated two-particle basis functions of the form... 

A practical advantage of the finite-nucleus model is that extremely high exponents of the one-particle basis functions are avoided. Since for quantities of chemical interest it is not very important which nuclear model is actually used, the Gaussian charge distribution is often applied, being the most convenient choice. 

In approximate Cl methods, the set of many-particle basis functions is restricted and not infinitely large, i.e., it is not complete. Then, the many-particle basis is usually constructed systematically from a given reference basis function. (such as the Slater determinant, which is constructed to approximate the ground state of a many-electron system in (Dirac-)Hartree-Fock theory). 

In this chapter, the case of general molecules — meaning molecules of arbitrary structure and thus arbitrary external nuclear potential — is considered. We will understand how the numerical solution of mean-field equations for many-electron atoms helps us to solve the molecular problem. The key element is the introduction of analytically known one-particle basis functions rather than an elaborated numerical treatment on a three-dimensional spatial grid, which is feasible but not desirable (a fact that will become most evident in section 10.5). 

Figure 5 Illustration of the spaces spanned by the one-particle basis functions i, ],.. = occupied spin-orbitals, a,b,.. . = non-occupied spin-orbitals contained in the finite basis, a, p,... = complete set of virtuals, p,q,... = arbitrary spin-orbitals contained in the finite basis, 1C, A,... = complete basis set...

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