AnL 1 smoothing spline algorithm with cross validation

We propose an algorithm for the computation ofL ₁ (LAD) smoothing splines in the spacesW _M (D), with . We assume one is given data of the formy _i =(f(t _i ) + _i , i=1,...,N with {itt_i} _i=1 ^{N D} , the _i are errors withE( _i )=0, andf is assumed to be inW _M . The LAD smoothing spline, for fixed smoothing parameter0, is defined as the solution,s , of the optimization problem (1/N) _i=1 ^N ¦y _i –g(t _i ¦+J _M (g), whereJ _M (g) is the seminorm consisting of the sum of the squaredL ₂ norms of theMth partial derivatives ofg. Such an LAD smoothing spline,s , would be expected to give robust smoothed estimates off in situations where the _i are from a distribution with heavy tails. The solution to such a problem is a thin plate spline of known form. An algorithm for computings is given which is based on considering a sequence of quadratic programming problems whose structure is guided by the optimality conditions for the above convex minimization problem, and which are solved readily, if a good initial point is available. The data driven selection of the smoothing parameter is achieved by minimizing aCV() score of the form .The combined LAD-CV smoothing spline algorithm is a continuation scheme in 0 taken on the above SQPs parametrized in, with the optimal smoothing parameter taken to be that value of at which theCV() score first begins to increase. The feasibility of constructing the LAD-CV smoothing spline is illustrated by an application to a problem in environment data interpretation.     

The theoretical basis of column chromatography in multicomponent separations. Part 2: Peak capacity. Column systems with good selectivity

The selectivity of a column system, S, defined by equation 28, includes a variation coefficient (β) of plate number with capacity ratio which has a large influence on the peak capacity, as shown in Fig. 18. Some typical chromatograms are given. In order to predict S for a column system from Kovat's Index, equations 40 and 41 are given to calculate the constant of the carbon number rule for squalane at different temperatures or for different stationary phases. The specific retention value of heptane on squalane at different temperatures can be calculated from equation 42. The nonpolarity index, defined by equation 43, was used to calculate the retention value of heptane on various stationary phases. In liquid chromatography, the order of elution may be reversed by changing the composition of the eluent on the same chemically bonded silica (manufactured in China). The linear relations between the log retention values of different kinds of solutes or of a single solute on the silicas of different surface areas when using the same eluent are given.     

An experimental and theoretical study of vibrational spectra of picolinamide, nicotinamide, and isonicotinamide

The molecular structures and vibrational spectra of the three isomers of pyridinecarboxamide (picolinamide, nicotinamide, isonicotinamide) were calculated with the Density Functional Theory (DFT) method using the B3LYP function and the 6-31++G(d,p), Z2PolX, Z3PolX basis sets. The calculations were performed by using the Gaussian98W packet program set. The total energy distributions (TED) of the vibrational modes of these molecules were calculated by using the Scale 2.0 program and the vibrational modes of the molecules were determined. The Scaled Quantum Mechanical (SQM) method was used in the scaling procedure. In the experimental part of the study, the solid phase FT-IR and Micro Raman spectra of the three isomers of pyridinecarboxamide have been recorded in the range of 4000-650 and 1200-100 cm⁻¹, respectively. The calculated wavenumbers were compared to the corresponding experimental values. As a result, the observed bands of the three isomers of pyridinecarboxamide were assigned with good accuracy.     

Molecular integrals over Gaussian-type geminal basis functions

We present formulas for the evaluation of molecular integrals over basis functions with an explicit Gaussian dependence on interelectronic coordinates. These formulas use expansions in Hermite Gaussian functions and represent an extension to the work of McMurchie and Davidson to two-electron basis functions. Integrals that depend on the coordinates of up to four electrons are discussed explicitly. A key feature of this approach is that it allows full exploitation of the shell structure of the orbital part of the basis. Received: 24 February 1997 / Accepted: 4 March 1997     

Electronic structure of BeH2

Nonorthogonal single‐configuration and multiconfiguration ab initio calculations are carried out on the BeH₂ molecule in its equilibrium (D_∞h) geometry, using a highly optimized even‐tempered Slater‐type orbital (STO) basis set. The results are used as a basis for a discussion of the electronic structure of the molecule in modern‐VB terms. A value of R_e = 1.329 Å is obtained. Values are also obtained for the symmetric‐stretch harmonic frequency (2053 cm^?1), electric quadrupole moment (?5.60 Buckingham), and nuclear quadrupole coupling constants (?3.59 MHz for ⁹Be and 0.0915 MHz for ²H). © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Polarized basis sets of Slater-type orbitals: H to Ne atoms

We present three Slater-type atomic orbital (STO) valence basis (VB) sets for the first and second row atoms, referred to as the VB1, VB2, and VB3 bases. The smallest VB1 basis has the following structure: [3, 1] for the H and He atoms, [5, 1] for Li and Be, and [5, 3, 1] for the B to Ne series. For the VB2 and VB3 bases, both the number of shells and the number of functions per shell are successively increased by one with respect to VB1. With the exception of the H and Li atoms, the exponents for the VB1 bases were obtained by minimizing the sum of the Hartree-Fock (HF) and frozen-core singles and doubles configuration interaction (CISD FC) energies of the respective atoms in their ground state. For H and Li, we minimized the sum of the HF and CISD FC energies of the corresponding diatoms (i.e., of H(2) or Li(2)) plus the ground-state energy of the atom. In the case of the VB2 basis sets, the sum that was minimized also included the energies of the positive and negative ions, and for the VB3 bases, the energies of a few lowest lying excited states of the atom. To account for the core correlations, the VBx (x = 1, 2, and 3) basis sets for the Li to Ne series were enlarged by one function per shell. The exponents of these extended (core-valence, CV) basis sets, referred to, respectively, as the CVBx (x = 1, 2, and 3) bases, were optimized by relying on the same criteria as in the case of the VBx (x = 1, 2, and 3) bases, except that the full CISD rather than CISD FC energies were employed. We show that these polarized STO basis sets provide good HF and CI energies for the ground and excited states of the atoms considered, as well as for the corresponding ions.     

Reduced-size polarized basis sets for calculations of molecular electric properties. IV. First-row transition metals

Recent studies of the perturbation-dependent basis sets have indicated the possibility of a significant reduction of the size of the usual CGTO sets without considerable loss of accuracy in calculations of molecular electric properties. The resulting (ZPolX) basis sets have been developed for several atoms of the first and second row of the Periodic Table. The same method of the ZPolX basis set generation is extended for the first-row transition metals and the corresponding contracted ZPolX basis sets of the size [6s5p3d1f] are determined for both nonrelativistic and scalar relativistic calculations. The performance of the ZPolX basis sets is verified in calculations on the first-row transition metal oxides at the level of the ROHF, ROHF/CASPT2, and ROHF/CCSD(T) approximations. Also the study of the dipole polarizability of TiCl₄ confirms the excellent features of these very compact basis sets. The ZPolX basis sets for nonrelativistic and relativistic calculations of molecular electric properties are available on the web page http://www.chem.uni.torun.pl/zchk/basis-sets.html.     

The convergence of the Rayleigh-Ritz Method in quantum chemistry

Two sufficient criteria for the convergence of the Rayleigh-Ritz Method (RRM) with respect to the eigenvalues (E-convergence) of non-relativistic electronic Hamiltonians of molecules are discussed and compared. Moreover, a necessary and sufficient criterion is given. By example (Sect. 9) it is shown that the L ²-completeness of the basis is not sufficient to guarantee E-convergence. The convergence of the wave functions in different norms (-convergence) is also investigated. In particular, sufficient conditions for the one-particle basis functions (orbitals) are given, such that a CI calculation in this basis is both E- and -convergent.     

An approximate frozen core approach to valence only molecular calculations

A simple valence electron-only theory based on an approximate frozen core approach and an exact core-valence strong orthogonality condition is developed for atomic and molecular systems. A unique reduced basis is introduced in which both core and valence orbitals are expanded. The core representation is roughly approximated, and the valence orbital overlap with the corresponding all-electron reference functions is nearly exact. The size of the reduced basis in terms of primitive functions is practically the same as that adopted by effective core potential methods in which the valence orbitals have the correct nodal properties. Results obtained with the present approach are presented for LiO, BeO and CaO molecules, and compared with the corresponding all-electron frozen core calculations. In addition, a detailed investigation on Li _n Be clusters (n=1,..., 6) is carried out.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Medium-size Gaussian basis sets for hydrogen through argon

Summary Medium-sized Gaussian basis sets are reoptimized for the ground states of the atoms from hydrogen through argon. The composition of these basis sets is (4s), (5s), and (6s) for H and He, (9s5p) and (12s7p) for the atoms Li to Ne, and (12s8p) and (12s9p) for the atoms Na to Ar. Basis sets for the² P states of Li and Na, and the³ P states of Be and Mg are also constructed since they are useful in molecular calculations. In all cases, our energies are lower than those obtained previously with Gaussian basis sets of the same size.