Systematic truncation

Figure 5. Running fourier transforms of the linear cross-correlations taken for a series of laser pulses systematically truncated in the time domain. The entire pulse (a) is truncated in 500-fs intervals so that approximately half the pulse remains in panel (e). The center of a transform limited pulse was used to define the center (t = 0) of the optimized pulse.

In this paper, the systematic truncation of a distributed universal even-tempered basis set capable of supporting high precision in both mauix Haitree-Fock and second order many-body perturbation theory calculations is explored using the ground state of the boron fluoride molecule as a prototype. The truncation procedure adopted is based on the magnitude of the orbital expansion coefficients associated with a given basis function in each of the occupied orbitals. 

Systematic Truncation of a Distributed Universal Even-Tempered Basis Set of Gaussian Functions... 

Systematic truncation of a distributed universal even-tempered basis set of 323 Gaussian functions an application to the ground state of the BF molecule... 

These elements in turn couple to even higher-order many-particle correlations. A closed set of equations is derived by systematically truncating the correlations with more than two phonons. 

For the description of modulated roll patterns away from threshold (possibly only slightly) the ampUtude B must be included, now also transformed to real space. A uniform scaling in is then no more possible and coupling terms like AQyB appear in Eq. 36 as well as derivative terms in the cubic non-linearities. The gradient expansion starting from Eqs. 32 and 33 is systematically truncated in such a way that all contributions to the growth rate (Eq. 34) are included [24, 83]. The additional equation for B is of the form... 

These functions allow- the nonbonded potential energy Lo turn off smoothly and systematically, removing artifacts caused by a truncated potential. With an appropriate switching function, the potential function is unaffected except m the region of the switch. 

In order to systematically remedy the previous drawbacks, we recently proposed to perform a perturbation treatment, not on a wavefunction built iteratively, but on a wavefunction that already contains every components needed to properly account for the the chemistry of the problem under investigation [34], In that point of view, we mean that this zeroth-order wavefunction has to be at least qualitatively correct the quantitative aspects of the problem are expected to be recovered at the perturbation level that will include the remaining correlation effects that were not taken into account in the variational process any unbalanced error compensations or non-compensations between the correlation recovered for different states is thus avoided contrary to what might happen when using any truncated CIs. In this contribution, we will report the strategy developed along these lines for the determination of accurate electronic spectra and illustrate this process on the formaldehyde molecule H2CO taken as a benchmark. 

When the non-electrostatic terms are semiempirical, they also make up in an average way for systematic deficiencies in the treatment of electrostatics, e.g., for the truncation of the distributed multipole representation of the solute charge density at the monopole term on each center. 

Another contribution is represented by an investigation of a cubic thallium cluster phase of the Bergmann type Na13(TlA.Cdi A.)27 (0.24 < x <0.33) (Li and Corbett 2004). For this phase too the body centred cubic structure (space group Im 3, a = 1587-1599 pm) may be described in terms of multiple endo-hedral concentric shells of atoms around the cell positions 0, 0, 0, and 14,14,14. The subsequent shells in every unit are an icosahedron (formed by mixed Cd-Tl atoms), a pentagonal dodecahedron (20 Na atoms), a larger icosahedron (12 Cd atoms) these are surrounded by a truncated icosahedron (60 mixed Cd-Tl atoms) and then by a 24 vertices Na polyhedron. Every atom in the last two shells is shared with those of like shells in adjacent units. A view of the unit cell is shown in Fig. 4.38. According to Li and Corbett (2004), it may be described as an electron-poor Zintl phase. A systematic description of condensed metal clusters was reported by Simon (1981). 

Possibly the increase in the factor at high molar masses contains a hidden systematic error. Inspection of the curves in Fig. 34 reveals that the fit with the truncated Eq. (89) holds only for values of X<2 and sometimes up to X=4. At larger concentrations a fit becomes possible only when a further term in Eq. (89) is added ... 

Two further comments about formula 8 and its parameters are noteworthy. The number of terms with coefficients l i in the double sum is formally doubly infinite so that formula 8 can represent spectral terms involving arbitrarily large values of V and 7 with data of finite extent the sums become truncated in a systematic and consistent manner conforming to parameters of a minimum number required according to that intermediate model. Instead of exact equalities in formulae 10 and 11, the approximations arise because each term coefficient constitutes a sum of contributions [5],... 

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. 

---