Basis Set Problem

Compounding the difficulty of accounting for electron correlation effects properly, accurate computations of noncovalent interactions also require very large basis sets. This is not surprising because London dispersion interactions can be expressed in terms of the polarizabilities of the weakly interacting molecules, and polarizability computations are known to have large basis set requirements. In many weakly bound complexes, the dispersion terms can be the dominant ones. 

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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. 

When performing variable-cell AIMD simulations with plane-wave basis sets, problems originate from the fact that the basis set is not complete with respect to the cell vectors.71 This incompleteness can introduce fictitious forces (Pulay forces) into asys and lead to artificial dynamics. To overcome this problem, one must ensure that asys is well converged with respect to the basis set size. In general, it is found that one needs to employ a plane-wave kinetic... 

Suggest a computational procedure with which you believe the problem can be solved. Consider the basis set problem, the wave function, and the method you would like to use to reach the transition state geometry. This is a rather large computational problem. Try to set up a procedure that fits into the computer you are using. Try to estimate the computer time you would need to use on your installation. 

This is not associated with a particular method, like HF or Cl, but rather is a basis set problem. Consider what happens when we compare the energy of the hydrogen-bonded water dimer with that of two noninteracting water molecules. Here is the result of an MP2(fc)/6-31G calculation both structures were geometry-optimized, and the energies are corrected for ZPE ... 

In the first part of this section, the relationship between the solution of the Schrodinger equation and the hamiltonian in the space generated by a given basis set is discussed in some detail. Since basis set limitations appear to be one of the largest sources of error in most present day molecular calculations, the concept of a universal even-tempered basis set is discussed in the second part of this section. This concept represents an attempt to overcome the incomplete basis set problem, at least for diatomic molecules. Further aspects of the basis set truncation problem are discussed in the final part of this section. 

As a foreword it must be said, perhaps constructing a too late homage to the brilliant contribution of professor Boys to Quantum Chemistry, that the first description of cartesian exponential type orbitals (CETO) was made thirty years ago by Boys and Cook [1], One can probably think this fact as a consequence of the evolution of Boys s thought on the basis set problem and to the incipient ETO-GTO dilemma, which Boys has himself stated ten years earlier [2a]. 

One solution to this basis set problem in more recent classical VB-style approaches, such as some types of VBSCF (VB self-consistent-field) wavefunctions and the BOVB ( breathing orbital VB) method,is to use variational hybrid atomic orbitals (HAOs) expanded in terms of the basis functions on a single centre only. 

The optimization of trial functions for many-body systems is time consuming, particularly for complex trial functions. The dimension of the parameter space increases rapidly with the complexity of the system and the optimization can become very cumbersome since it is, in general, a nonlinear optimization problem. Here we are not speaking of the computer time, but of the human time to decide which terms to add, to program them and their derivatives in the VMC code. This allows an element of human bias into VMC the VMC optimization is more likely to be stopped when the expected result is obtained. The basis set problem is still plaguing quantum chemistry even at the SCF level where one only has 1-body orbitals. VMC shares this difficulty with basis sets as the problems get more complex. 

Gaussian functions are a very practical solution to the basis set problem. They differ from Slater orbitals but a fitted combination of gaussians [contraction, eq. (16)] can be used to simulate a Slater function. 

And what about the future of CC theory With these developments, much more effort can be devoted to multi-reference CC, Fock space CC, XCC, UCC, variational CC, methods like those discussed below for large molecules, and perhaps the ultimate current method, R12-CC [102]—where besides the correlation problem, we have the best current solution to the basis set problem—or a wealth of other methods that do not fit into the basic structure of the CC functional, discussed next, as that is the basis for the automated generation. 

The difficulty with VMC is exactly identical in spirit to all the problems of traditional methods the basis-set problem. Although the wavefunction is vastly improved in VMC, it is difficult to know when the wavefunction form is sufficiently flexible, and therefore it is always necessary to show that the basis-set limit of a given class of trial function has been reached. Moreover, the accuracy of energy in no way implies accuracy of other properties. One can assume that many of the variational errors cancel out in going from one system to another, but it is not very hard to find counterexamples. With the current class of wavefunctions it seems that we are far from getting chemical accuracies from VMC when applied to systems more complex than the electron gas or a single atom. In addition, in VMC one can waste a lot of time trying new forms rather than have the computer do the work. This problem is solved in a different way in the next two methods we discuss. 

B. The Basis Set Problem in Many-body Perturbation Theory 468... 

For a one-centre basis set, in contrast to multi-centre basis sets, problems arising from overcompleteness can usually be controlled if not avoided. The one-centre approach provides control over the convergence of a calculation with respect to the size of the basis set and control over computational linear dependence. Furthermore, because of the ea% with which integrals involving one-centre functions can be handled, the method can be used to explore the use of alternative types of basis function. The one-centre method is ideally suited to the calculation of energy derivatives with respect to the nuclear coordinates. 

Early work on the finite basis set problem in relativistic calculations has been reviewed by Kutzelnigg. Spurious unphysical solutions of the Dirac equation or the Dirac-Hartree-Fock equations are observed with too small a kinetic energy, leading to an overestimation of the binding energy. Furthermore, these solutions are found neither to tend to the solutions of the Schrodinger equation in the limit c- co nor to vary systematically with increasing size of basis set. 

To trace the origin of the finite basis set problem, let us observe that the Dirac equation for a hydrogenic system within the algebraic approximation may be written as two simultaneous equations. Thus Eq. (93) gives... 

K. McDowell, /. Chem. Phys., 68, 4151 (1978). The Incomplete Basis Set Problem. I. Perturbative Correaions to Hartree-Fock Energies. 

In addition to the basis-set problem, it should be noted that band structure results (as well as results derived from these, e.g., transport data) ate usually obtained by considering tubes of infinite length. The infinite-tube results, however, are quite different from those obtained for tubes of finite length. [5, 6, 7] Taking into account that realistic calculations have to be performed for tubes of finite length and that ab initio methods are not easily applicable to systems consisting of a moderate number of atoms ( 1000 atoms are necessary for simulations of finite tubes), it can be seen that a practical calculation would require use of semi-empirical methods such as the TB. 

The cluster energies are obtained from the solution of the nonrelativistic Schrodinger equation for each system. The expansion of the trial many-electron wave function delineates the level of theory (description of electron correlation), whereas the description of the constituent orbitals is associated with the choice of the orbital basis set. A recent review (Dunning 2000) outlines a path, which is based on hierarchical approaches in this double expansion in order to ensure convergence of both the correlation and basis set problems. It also describes the application of these hierarchical approaches to various chemical systems that are associated with very diverse bonding characteristics, such as covalent bonds, hydrogen bonds and weakly bound clusters. 

However, to achieve this goal, we shall probably have to find different solutions to the basis-set problem so as to reduce the size of the basis needed for convergence to the basis-set limit. The obvious candidates here are the ejq>licitly correlated methods, elements of which must be introduced into the standard models in a black-box manner, to enable the user to approach the basis-set limit of, for example, the CCSD(T) model at a cost smaller than presently possible. Again, there is enough work going on in this direction for the prospects to appear bright and exciting. 

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