Basis-set truncation error

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. 

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. 

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. 

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. 

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. 

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. 

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. 

The gas phase matrix Hartree-Fock energies of earli of the molecules N, CO, BF and 1VO+ at their resjjective equilibrium geometries were calculated for each of the basis sets designated STO-3G , 3-21G , 6-21G and 6-31G [30]-[39j. The results are presented in Table 1. The basis set truncation errors associated with the total energies presented in Table 1 can be determined by comparison with the exact Hartree-Fock energies which have been determined[26] for the four systems studies here using finite difference techniques[75]-[80]. For the N, CO,BF and NO ssretems the exact Hartree-Fock energies are —108.993826, —112.790907, —124.168779 and... 

The convergence behaviour observed in the present study for continuum models of solvated species using systematic sequences of even-tempered basis sets of Gaussian primitive functions of s- and p- symmetry mirrors that observed for the gas phase molecular species. Gas phase matrix Hartree-Fock calculations for the isoelectronic molecules studied in the present work which approach a suh-pHartreelevel of accuracy for the total energy have been reported elsewhere[18]-[25]. The present work demonstrates that this approach can be applied to molecules in solution by making use of continuum models. The basis set truncation error would then be 3 orders of magnitude smaller than the AE values. 

Let us now have a look at the usefulness of our theoretical AEC-formulae in calculations of accurate SAP energies in radially saturated basis sets including orbitals defined by quantum numbers I < Zmax/ which are followed by extrapolations to the CBS limit. This extrapolation is accomplished by taking into account the basis set truncation error, SEi, due to omission of functions of I > Imax- For /-contributions... 

Almost all computational methods expand the MOs in a basis set of one-electron functions. The basis set has a finite number of members and hence is incomplete. The incompleteness of the basis set produces the basis-set truncation error. In Cl treatments of correlation, one usually includes only CSFs (configuration state functions) with single and double excitations the failure to go to full Cl produces error (1). 

To do an MP electron-correlation calculation, one first chooses a basis set and carries out an SCF calculation to obtain o, hf> and virtual orbitals. One then evaluates EP (and perhaps higher corrections) by evaluating the integrals over spin-orbitals in (15.87) in terms of integrals over the basis functions. One ought to use a complete set of basis functions to expand the spin-orbitals. The SCF calculation will then produce the exact Hartree-Fock energy and will yield an infinite number of virtual orbitals. The first two sums in (15.87) will then contain an infinite number of terms. Of course, one always uses a finite, incomplete basis set, which yields a finite number of virtual orbitals, and the sums in (15.87) contain only a finite number of terms. One thus has a basis-set truncation error in addition to the error due to truncation of the MP perturbation energy at E or E or whatever. 

To apply the CC method, two approximations are made. First, instead of using a complete, and hence infinite, set of basis functions, one uses a finite basis set to express the spin-orbitals in the SCF wave function. One thus has available only a finite number of virtual orbitals to use in forming excited determinants. As usual, we have a basis-set truncation error. Second, instead of including all the operators fj, 72,..., r , one approximates the operator f by including only some of these operators. Theory shows (Wilson, p. 222) that the most important contribution to T is made by r2.The approximation T T 2 gives... 

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