Basis sets 524 Subject

A new period in theoretical work on arenediazonium ions began with Vincent and Radom s paper in 1978. This was the first ab initio study of the methane- and benzenediazonium ions, and was carried out with a minimal (STO-3G) basis set, subject only to some (specified) symmetry constraints and a fixed CH bond length (108.3 pm). The optimized structure of the benzenediazonium ion is given in Figure 4-2. 

Fundamental support for the d correction was expressed by Morokuma et who observed that in energy-partitioned SCF calculations the term became excessive for basis sets subject to large BSSE. They argued that a virtuals-only scheme is the proper one to correct for this. It was noted that the virtuals of the ghost to be included should really be those of the counterpoise-corrected instead of the isolated ghost. This recipe is difficult to implement. However, in their example the difference from the simple recipe was extremely small. Similarly. Hayes and Stone state that, while only the virtuals of the partner are available in their non-orthogonal second-order perturbation theory, the occupied orbitals may become available in higher orders. 

Supplementary to this subject, a recently published ab-initio calculation of silicondicarbonyl 0 = C = Si = C = 0 should be mentioned [195]. With an extended basis set (DZP functions) a bent C2 structure with C—Si—C 80.0°, Si—C 1.871, C —O 1.126 A is calculated the bent conformation is energetically more stable by 76.7 kcal/mol than a linear Dxh structure [196], These results also allow a reinterpretation of the experimental data (IR spectroscopy) for silicondicarbonyl [197]. 

The choice of basis set in ab initio calculations has been the subject of numerous theoretical studies. Early SCF calculations utilized mainly spht-va-lence basis sets such as 3-21G and 4-31G. The importance of inclusion of d polarization functions on sulfur atoms has been stressed by several authors. For instance, Suleimenov and Ha found that the omission of d polarization functions leads to a substantially lower barrier for the internal rotation ( 16 kj mol for the central bond of H2S4) and produces an unreahstically large S-S bond length for the most stable rotamer [4]. In general, the use of... 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

In molecular DFT calculations, it is natural to include all electrons in the calculations and hence no further subtleties than the ones described arise in the calculation of the isomer shift. However, there are situations where other approaches are advantageous. The most prominent situation is met in the case of solids. Here, it is difficult to capture the effects of an infinite system with a finite size cluster model and one should resort to dedicated solid state techniques. It appears that very efficient solid state DFT implementations are possible on the basis of plane wave basis sets. However, it is difficult to describe the core region with plane wave basis sets. Hence, the core electrons need to be replaced by pseudopotentials, which precludes a direct calculation of the electron density at the Mossbauer absorber atom. However, there are workarounds and the subtleties involved in this subject are discussed in a complementary chapter by Blaha (see CD-ROM, Part HI). 

The information obtainable upon solution of the eigenvalue problem includes the orbital energies eK and the corresponding wave function as a linear combination of the atomic basis set xi- The wave functions can then be subjected to a Mulliken population analysis<88) to provide the overlap populations Ptj ... 

A number of studies of lexitropsins and Hoechst agents make use of ab initio calculations to complement the experimental and molecular modeling results. Due to the large size of most of these molecules, it is necessary to perform fragment-wise calculations if a larger basis set is to be used. For smaller systems, the whole molecule can be subjected to geometry optimization. 

The molecules were subjected to complete geometry optimization via ab initio calculations, using the 6-31G basis set. The energies of the molecules, as well as the net atomic charges, were calculated. 

The complex formed by thymine with the guanidinium ion (Figs. 6.7, 6.8) and the complex formed by thymine with the aminopyrolidinium ion (Figs. 6.9,6.10) were subjected to ab initio (Hartree-Fock) calculations using the 6-31G find the 3-21G basis sets, in order to... 

An—at least, theoretically—simple example is the S = 1 system in weak-field subject to a dominant zero-field interaction and a weakly perturbing electronic Zeeman interaction (similar to the S = 2 case treated above). The initial basis set is... 

Highly energetic compounds with potential use in explosive devices must be characterized completely and safely, particularly as the explosive character may be linked directly to vibrational modes in the molecular structure, hence the application of computational methods to complement experimental observations. ANTA 5 has been the subject of various studies and, as an adjunct to one of these and to confirm the results of an inelastic neutron scattering experiment, an isolated molecule calculation was carried out using the 6-311G basis set <2005CPL(403)329>. 

For H at T in Si, Katayama-Yoshida and Shindo (1983, 1985) used a Green s function method to carry out spin-density-functional calculations. They found a reduction of the spin density by a factor 0.41. However, their results are subject to some uncertainty because they obtained an erroneous result for the position of the defect state in the band gap, probably due to an insufficiently converged LCAO basis set. 

Tomas et al. [281] have calculated the tautomeric equilibrium of 1,2,3-benzotriazole in the gas phase and compared their results to experimental data [282] derived from ultraviolet spectroscopy. Experiment suggests that 35 is about 4 kcal/mol more stable than 34 this result is consistent with calculations [281] at the MP2/6-31G level, which predict 35 to be 2.5 kcal/mol more stable than 34. The same level of theory predicts 33 to be 5.0 kcal/mol more stable than 32 in the parent triazole system. Although experimental data are available indicating 35 to be the dominant tautomer in CDCf and d6-dimethyl sulfoxide solutions [279, 283], this equilibrium does not appear to have been the subject of any modeling, continuum or otherwise. It may prove to be somewhat challenging, however. Tomas et al. point out that correlation effects favor 35 by about 5 kcal/mol at the MP2 level AMI, PM3, and HF calculations with moderate basis sets all predict... 

The second approach used in first-principles tribological simulations focuses on the behavior of the sheared fluid. That is, the walls are not considered and the system is treated as bulk fluid, as discussed. These simulations are typically performed using ab initio molecular dynamics (AIMD) with DFT and plane-wave basis sets. A general tribological AIMD simulation would be run as follows. A system representing the fluid would be placed in a simulation cell repeated periodically in all three directions. Shear or load is applied to the system using schemes such as that of Parrinello and Rahman, which was discussed above. In this approach, one defines a (potentially time-dependent) reference stress tensor aref and alters the nuclear and cell dynamics, such that the internal stress tensor crsys is equal to aref. When crsys = aref, the internal and external forces on the cell vectors balance, and the system is subject to the desired shear or load. 

This quantity was the subject of several indirect attempts at calculation based on physical properties as well as ab initio computations. A direct measurement was finally achieved by ion cyclotron experiments and gave a value of 163kJmol (McMahon and Larson, 1982a) for A [HF(g)+ F (g) -> FHF (g)]. Shortly after, ab initio calculations with the basis set [(14s) (9p) (2d)/(10s) (lp)] fell in line with a value of 169 kJ mol" (Emsley et ai, 1983). Prior to this, a wide range of values for the bond energy had been canvassed, often with theoretical computations providing support (see Table 8). 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

Hartree-Fock theory makes the fundamental approximation that each electron moves in the static electric field created by all of die other electrons, and then proceeds to optimize orbitals for all of the electrons in a self-consistent fashion subject to a variational constraint. The resulting wave function, when operated upon by the Hamiltonian, delivers as its expectation value the lowest possible energy for a single-detenninantal wave function formed from the chosen basis set. 

Of course, there is a key difference between HF theory and DFT - as we have derived it so far, DFT contains no approximations it is exact. All we need to know is xc as a function of p. .. Alas, while Hohenberg and Kohn proved diat a functional of the density must exist. their proofs provide no guidance whatsoever as to its fonn. As a result, considerable research effort has gone into dnding functions of die density diat may be expected to reasonably approximate xc, and a discussion of diese is die subject of the next section. We close here by emphasizing that the key contrast between HF and DFT (in the limit of an infinite basis set) is that HF is a deliberately approximate theory, whose development was in part motivated by an ability to solve die relevant equations exactly, while DFT is an exact theory, but the relevant equations must be solved approximately because a key operator has unknown form. 

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