Basis set truncation

Basis Set Truncation.—Basis set truncation does appear to be one of the main sources of error in the majority of calculations on small molecules. The efficiency of algorithms based on the diagrammatic perturbation expansion will allow increasingly large basis sets to be employed in molecular studies during the next few years. 

Clearly, this approach can also be used in the case of Slater basis sets and, moreover, in the case of universal Slater basis sets. Ruedenberg and co-workers104 105 have shown that, within the molecular orbital approximation, this systematic approach gives a series of energy values which smoothly approach the Hartree-Fock limit. Similarly smooth convergence is to be expected in the calculation of correlated wave functions and expectation values, and will be the subject of future studies in this area.106 

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The Complete Basis Set (CBS) methods were developed by George Petersson and several collaborators. The family name reflects the fundamental observation underlying these methods the largest errors in ab initio thermochemical calculations result from basis set truncation. 

However, as definitive values for E (CBS) became available from the MP2-R12 calculations of Klopper [48], it became clear that Eq. (6.1) seriously underestimates the magnitude of the basis set truncation error. Wilson and Dunning therefore examined [53] a wide variety of extrapolations (24 variations) based on generalizations of Eq. (1.1). They obtained RMS deviations from Klopper s results of less than 1 mEh using several different extrapolation schemes. We arrived at comparable results (Table 4.4) using just two points, E (lmax2) and E(2)(Zmaxi), so... 

The higher-order contributions to the correlation energy [such as CCSD(T)-MP2] are more than an order of magnitude smaller than their second-order counterparts. However, the basis set convergence to the CCSD(T)-R12 limit does not follow the simple linear behavior found for the second-order correlation energy. This is a consequence of the interference effect described in Eq. (2.2). The full Cl or CCSD(T) basis set truncation error is attenuated by the interference factor (Fig. 4.9). The CBS correction to the higher-order components of the correlation energy is thus the difference between the left-hand sides of Eqs. (2.2) and... 

Figure 4-9 The interference effect in Eq. (2.2) gives a quantitative description of the relationship between the MP2 and CCSD(T) basis set truncation errors. These calculations used [5s4p3d2f/4s3p2d] basis sets for the species Be, H2, C2H2, CH4, HCN, NH3, N2i H2CO, CO, H20, C02, HF, F2, and Oe+.

The ability to use precisely the same basis set parameters in the relativistic and non-relativistic calculations means that the basis set truncation error in either calculation cancels, to an excellent approximation, when we calculate the relativistic energy correction by taking the difference. The cancellation is not exact, because the relativistic calculation contains additional symmetry-types in the small component basis set, but the small-component overlap density of molecular spinors involving basis functions whose origin of coordinates are located at different centres is so small as to be negligible. The non-relativistic molecular structure calculation is, for all practical purposes, a precise counterpoise correction to the four-component relativistic molecular... 

As it is well known, the Basis Set Superposition Error (BSSE) affects calculations involving hydrogen bonds [1] and, more generally, intermolecular interaction investigations [2,3], This issue of consistency, as first pointed out in 1968 [4], arises from the use of an incomplete basis set but it does not correspond to the basis set truncation error and it is due to the use of diffuse functions on neighbouring interacting particles, which leads to a non physical contribution to the interaction energy within the complex. 

Table 8 reports equilibrium binding energies relative to the geometries described above. The differences between MCSCF-MI or SCF-MI energies and the SCF ones are obviously to be ascribed to the different basis set truncation effect and to BSSE. 

It is also clear from Table 5 that the absolute basis-set truncation error in Ne is about three times bigger for the antiparallel-spin correlation energy than for parallel. Thus the proposed spin-analysis hybrid of Ref. [35] may yet have some (limited) utility. 

Btiilding on atomic studies using even-tempered basis sets, universal basis sets and systematic sequences of even-tempered basis sets, recent work has shown that molecular basis sets can be systematically developed until the error associated with basis set truncation is less that some required tolerance. The approach has been applied first to diatomic molecules within the Hartree-Fock formalism[12] [13] [14] [15] [16] [17] where finite difference[18] [19] [20] [21] and finite element[22] [23] [24] [25] calculations provide benchmarks against which the results of finite basis set studies can be measured and then to polyatomic molecules and in calculations which take account of electron correlation effects by means of second order perturbation theory. The basis sets employed in these calculations are even-tempered and distributed, that is they contain functions centred not only on the atomic nuclei but also on the midpoints of the line segments between these nuclei and at other points. Functions centred on the bond centres were found to be very effective in approaching the Hartree-Fock limit but somewhat less effective in recovering correlation effects. 

Previous attempts to calculate bond energies in tin compounds employed levels of theory that were inadequate to provide accurate results. As discussed above, accurate bond energies require the use of either composite ah initio methods or methods employing a high level of electron correlation coupled with isogyric reactions to minimize basis set truncation and other systematic errors. Consequently, the results reported by Basch [46,96], which use a number of imcorrected ah initio methods or with very simple corrections (i.e., across-the-board energy corrections by finite amounts), are unhkely to be particularly accurate. 

Modern many-body methods have become sufficiently refined that the major source of error in most ab initio calculations of molecular properties is today associated with truncation of one-particle basis sets e.g. [1]- [4]) that is, with the accuracy with which the algebraic approximation is implemented. The importance of generating systematic sequences of basis sets capable of controlling basis set truncation error has been emphasized repeatedly in the literature (see [4] and references therein). The study of the convergence of atomic and molecular structure calculations with respect to basis set extension is highly desirable. It allows examination of the convergence of calculations with respect to basis set size and the estimation of the results that would be obtained from complete basis set calculations. 

All na-AIMD calculations reported in this chapter have been performed using the CPMD package [1] employing the BLYP exchange-correlation functional [3,48] and a plane-wave basis set truncated at 70 Ry in conjunction with Troullier-Martins pseudopotentials [93], For further details we refer the reader to the respective original articles. 

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. 

Space does not allow anything more than a brief overview of the many publications in which MP2 theory is used to investigate basis set truncation errors in molecular calculations during the period covered by this report. 

For most semiempirical MO methods or uncorrelated ab initio methods, the demand for basis sets is drastically increased, while the accuracy is usually reduced when going from light- to heavy-element systems. However, as we will show, the gradient-corrected DFT methods seem to have smaller basis set truncation error, so they are quite reliable for systems as heavy as those of actinides, and the results seem, at least qualitatively, to be less dependent on the choice of basis sets with high-angular-momentum components (40). 

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. 

There have been continuous attempts to develop schemes to circumvent the limitations imposed by the finite basis sets, e.g. the counterpoise method of Boys and Bernardi and its variants (28,29) or the complete basis set approach (30). It seems that the availibility of exact solutions of the Hartree-Fock equations for diatomic molecules could be of help in devising such methods by allowing the dependence of the basis set truncation and superposition errors on internuclear separation to be monitored (16). 

This approach is not very convenient when the potential energy curve has to be determined since for each internuclear separation the numerical basis set has to be constructed afresh. However, it could be of use for the study of weakly interacting systems for which basis set truncation errors make the algebraic approach too difficult to use. 

Conventional KS method <— Even basis set truncation Divide-and-conquer method <— Uneven basis set truncation... 

Finally, the Fermi energy controls directly the charge transfers between subsystems. It determines which subsystem orbitals to be utilized and also regulates their weights in both p ( r) and fi (r). The projection weights pick up subsystem densities and energy densities according to their profiles in real space. The Fermi function is the only approximation beyond the basis set truncation error in this construction. 

The final operational equations from both constructions are the same because the approximations in eqs.(24), (28), and (29) are equivalent to the basis set truncation error in eqs.(19) and (20). Yang s construction deals with the Hamiltonian while Zhou s construction manipulates basis sets. 

The Fermi function approximation of eqs.(l 1) and (12) is the only approximation beyond basis set truncation approximation. Figure 3 shows how the Fermi function approaches the step function, the function in the exact formula. The Fermi function is almost a step function when /3 = 200. The largest difference between Fermi and Heaviside functions comes from the neighborhood near the Fermi energy. 

Universal Basis Sets and Direct ccMBPT. - Early many-body perturbation theory calculations carried out within the algebraic approximation quickly led to the realization that basis set truncation is the dominant source of error in correlation studies seeking high precision when carried out with respect to an apprpriately chosen reference function. In more recent years, the importance of basis set truncation error control has been more widely recognized. We have described the concept of the universal basis set in Section 2.4.4 which provides a general approach to basis set truncation error reduction. 

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