The Truncation Procedure

In the simplest case for k = 1, we obtain the one-determinant wave function of the Hartree- Fock theory. The energy difference with respect to the FCI reference. 

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The truncation procedure for fiill-valence-space and N-electrons-in-N-orbitals SDTQ MCSCF waveflmctions is based on choosing split-localized molecular orbitals as configuration generators since they lead to the greatest number of deadwood configurations that can be deleted. A quite accurate estimation method of identifying the latter has been developed so that the truncation can be performed a priori. The method has been shown to be effective in applications to the molecules HNO, OCO and NCCN where, for instance, the energies of the full SDTQ[N/N] calculations are recovered to better than 1 mh by truncated expansions that require only 11.8%, 10.9% and 6.3%, respectively, of the number of determinants in the full calculations. Similar trends are observed for the FORS 1 model. 

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. 

The truncation procedure explored in the present smdy is described in detail in section 2. An analysis of the orbital expansion coefficients for the ground state of the BF molecule is presented in section 3, where the truncated basis sets employed in the present study are defined. The results of both matrix Hartree-Fock calculations and second-order many-body perturbation theory studies are given in section 4 together with a discussion of the properties of the truncated basis sets. The final section, section 5, contains a discussion of the results and conclusions are given. 

Figu re 3.24 (a, b) TEM images and illustrations showing the truncation procedure with time (c) UV-visible absorption spectra of sol with time, showing the blue-shift of the in-plane dipole, consistent with truncation of the sharp nanoprisms. Reprinted with permission from Ref [126] 2007 Wiley-VCH Verlag GmbH, Co. KGaA. 

However, this procedure depends on the existence of the matrix G(R) (or of any pure gauge) that predicates the expansion in Eq. (90) for a full electronic set. Operationally, this means the preselection of a full electionic set in Eq. (129). When the preselection is only to a partial, truncated electronic set, then the relaxation to the truncated nuclear set in Eq. (128) will not be complete. Instead, the now tmncated set in Eq. (128) will be subject to a YM force F. It is not our concern to fully describe the dynamics of the truncated set under a YM field, except to say (as we have already done above) that it is the expression of the residual interaction of the electronic system on the nuclear motion. 

This is an eigenvalue problem of the form of Eq. III.45 referring to the truncated basis only, and the influence of the remainder set is seen by the additional term in the energy matrix. The relation III.48 corresponds to a solution of the secular equation by means of a modified perturbation theory,19 and the problem is complicated by the fact that the extra term in Eq. III.48 contains the energy parameter E, which leads to an iteration procedure. So far no one has investigated the remainder problem in detail, but Eq. III.48 certainly provides a good starting point. 

The frequency dependence is taken into accoimt through a mixed time-dependent method which introduces a dipole-moment factor (i.e. a polynomial of first degree in the electronic coordinates ) in a SCF-CI (Self Consistent Field with Configuration Interaction) method (3). The dipolar factor, ensuring the gauge invariance, partly simulates the molecular basis set effects and the influence of the continuum states. A part of these effects is explicitly taken into account in an extrapolation procedure which permits to circumvent the sequels of the truncation of the infinite sum-over- states. 

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

A procedure based on the truncated Taylor series is named after Euler. With increment h = Ax, successive values of y are yi+i = Yi + h f (Xj, yj)... 

After following the procedure for random chimera production, the resulting chimeric molecules will have exchanges delimited by the truncated cDNA in step b)... 

The first step in our truncation procedure is the ordering of the determinants according to the importance of their contribution to the Cl expansion. The second step will be the deletion of all determinants after a certain truncation determinant. As mentioned above, a Cl waveftinction can be arranged in terms... 

Since the triply and quadruply excited configurations are truncated separately, it is usually possible to shorten the Cl expansion somewhat further, if this should be desirable, by applying the procedure described in Section 2.3 to the total calculated truncated SDTQ-CI wavefunction obtained as described in this section. Thus, for the SDTQ[ 18/18] wavefunction of NCCN, an accuracy of 1 mh is achieved with 49,033 determinants by the estimation method, and with 43,038 determinants by the just mentioned additional improvement, reducing the truncation from 6.32% to 5.54% of the 776,316 determinants of the full SDTQ-CI expansion. 

Figure 3 is the absorbance spectrum of a sample of the ambient laboratory air drawn into the cell. Here, in accord with the usual procedure, the initially determined spectrum was first corrected for radiation that had reached the detector without having passed through the sample (room temperature background radiation entering the optical path via imperfect optical components and nonoverlap of the source and detector pupils and fields), ratioed against a zero-sample spectrum, and converted to absorbance. Trace (A) shows the spectrum from 3600-600 cm l. The massive absorbances seen here truncated at 1% transmission are due to water vapor and to carbon dioxide. 

We want to mention here that the application of the near-nuclear corrections could have been performed with the complete relativistic functional, and we have utilized the semi-relativistic expressions just for simplicity and for testing them. For not large Z, the remaining errors above mentioned should be addressed to limitations of the semiclassical approach and of the procedure utilized for the near-nuclear corrections, rather than to the truncation of the expansion in powers of... 

This expression includes the curvature corrections to the Gaussian function, which play an important role in the averaging procedure they ensure the smoothed spectrum g(E) to be approximated, through the above definition, by its own truncated Taylor expansion. This smoothing procedure of the one-electron spectrum is an application of the method of moments, also used in other systems [23]. 

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