Basis finite

Caleulations that employ the linear variational prineiple ean be viewed as those that obtain the exaet solution to an approximate problem. The problem is approximate beeause the basis neeessarily ehosen for praetieal ealeulations is not suffieiently flexible to deseribe the exaet states of the quantnm-meehanieal system. Nevertheless, within this finite basis, the problem is indeed solved exaetly the variational prineiple provides a reeipe to obtain the best possible solution in the space spanned by the basis functions. In this seetion, a somewhat different approaeh is taken for obtaining approximate solutions to the Selirodinger equation. 

Head-Gordon M and Pople J A 1988 Optimization of wavefunotion and geometry in the finite basis Hartree-Fock method J. Phys. Chem. 92 3063... 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

Fig. 5. Total energy curve with finite basis correction (due to Francis and Payne [18]). (lhartree = 627kcal/mol or 2624kJ/mol).

Since the first formulation of the MO-LCAO finite basis approach to molecular Ilartree-Pock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — arc calculated and stored on external storage. Th e second step then con sists of the iterative solution of the Roothaan equations, where the integrals from the first step arc read once for every iteration. 

HyperChem s ab initio calculations solve the Roothaan equations (59) on page 225 without any further approximation apart from the use of a specific finite basis set. Therefore, ab initio calculations are generally more accurate than semi-empirical calculations. They certainly involve a more fundamental approach to solving the Schrodinger equation than do semi-empirical methods. 

The CBS models use the known asymptotic convergence of pair natural orbital expansions to extrapolate from calculations using a finite basis set to the estimated complete basis set limit. See Appendix A for more details on this technique. 

In developing perturbation theory it was assumed that the solutions to the unpermrbed problem formed a complete set. This is general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogously to the Cl method. When a finite basis set is employed it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is therefore truncated. 

Approximating a one-electron wave function (orbital) by an expansion in a finite basis set. 

G.P. Francis and M.C. Payne, Finite basis set corrections to total energy pseudopotential calculations, J. 

In the case of a finite system described by a finite basis set the spectrum of G E) and WIE) are discrete andG (E) has isolated real poles (31,99). As a result, the solution for the propagator consists in the diagonalization of the WfE) matrix... 

As discussed in (4), the K-matrix has a pole at energies near a resonance and this yields a convenient method for the analysis of the narrow autoionizing states. The matrix representation of equation [2] upon a finite basis may be in fact recast in the form (4)... 

If the basis set is mathematically complete, then the equation holds precisely. In practice, one has to work with an incomplete finite basis set and hence the equality is only approximate. Results close to the basis set limit (the exact HF solutions) can nowadays be found, but for all practical intents and purposes, one needs to live with a basis set incompleteness error that must be investigated numerically for specific applications. 

One specific problem becomes very acute in wavefunction based methods the basis set problem. The introduction of a finite basis set is not highly problematic in HE theory since the results converge quickly to the basis set limit. This is, unfortunately, not true in post-HE theory where the results converge very slowly with basis set size - which is another reason why the methods become computationally intractable for more than a few heavy atoms (heavy being defined as nonhydrogen in this context). These problems are now understood and appropriate approaches have been defined to overcome the basis set problem but a detailed description is not appropriate here. 

The matrix form for (5) expressed in a finite basis set is easily shown to be... 

Pople JA, Gill PMW, Johnson BG (1992) Kohn-Sham density-functional theory within a finite basis set. Chem Phys Lett 199 557... 

So far, the only approximation in our description of the FMS method has been the use of a finite basis set. When we test for numerical convergence (small model systems and empirical PESs), we often do not make any other approximations but for large systems and/or ab //i/Y/o-determined PESs (AIMS), additional approximations have to be made. These approximations are discussed in this subsection in chronological order (i.e., we begin with the initial basis set and proceed with propagation and analysis of the results). 

Similarly, improvement in the accuracy of the nuclear dynamics would be fruitful. While in this review we have shown that, in the absence of any approximations beyond the use of a finite basis set, the multiple spawning treatment of the nuclear dynamics can border on numerically exact for model systems with up to 24 degrees of freedom, we certainly do not claim this for the ab initio applications presented here. In principle, we can carry out sequences of calculations with larger and larger nuclear basis sets in order to demonstrate that experimentally observable quantities have converged. In the context of AIMS, the cost of the electronic structure calculations precludes systematic studies of this convergence behavior for molecules with more than a few atoms. A similar situation obtains in time-independent quantum chemistry—the only reliable way to determine the accuracy of a particular calculation is to perform a sequence of... 

In the present approach, the KS orbitals are expanded in a set of functions related to atomic orbitals (Linear Combination of Atomic Orbitals, LCAO). These functions usually are optimized in atomic calculations. In our implementation a basis set of contracted Gaussians VF/ is used. The basis set is in general a truncated (finite) basis set reasonably selected . 

---