Choice of basis for calculations

The correct choice of the basis for a calculation will often determine whether the calculation proves to be simple or complex. As with the choice of system boundaries, no all-embracing rules or procedures can be given for the selection of the right basis for any problem. The selection depends on judgement gained by experience. Some guide rules that will help in the choice are  

Time choose the time basis in which the results are to be presented for example kg/h, tonne/y. 

Choose as the mass basis the stream flow for which most information is given. 

It is often easier to work in mols, rather than weight, even when no reaction is involved. 

For gases, if the compositions are given by volume, use a volume basis, remembering that volume fractions are equivalent to mol fractions up to moderate pressures. 

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The form of the approximate wavefunctions suggests another choice of basis for this problem, namely one comprising some harmonic oscillator functions centred about one minimum and additional harmonic oscillator functions centred about the other minimum. The only minor difficulty in this calculation is that the basis set is not orthogonal (which should be clear simply by inspecting the overlap of the ground-state harmonic oscillator wavefunctions centred at the two points) and an orthonormalization based on equation... 

One now has a choice of how to construct a basis set in which to set up the Hamiltonian, and the best basis is the one which most nearly diagonalizes the Hamiltonian matrix. However, nature does not care which basis we choose and a full calculation in any basis yields the same results which basis is chosen is purely a matter of convenience. In diatomic molecules the choices of basis for the inclusion of spin most normally used were first considered by Hund, and are called Hund s cases a and b (see Figure 5 for explanation). 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

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... 

However, the values in Tables 26-30 are directly calculated BDEs, according to equation 1. Taking only those molecules for which there are reliable experimental information (Table 26), and excluding the problematic (S 2) cases (CCH-, COOH-, NCO-, CN-, NC-, CHCH2- and N3 ) and the decomposed radical (CH3CO2 ), the average MP2 error for CH3—Y is 2.0 kcalmol-1 (21 cases) and for MP4 is 3.8 kcal mol-1 (19 cases). It thus seems that the particular choice of basis set and level of calculation used here gives the best cancellation of errors for the MP2 method. 

The set of components used in a geochemical model is the calculation s basis. The basis is the coordinate system chosen to describe composition of the overall system of interest, as well as the individual species and phases that make up the system (e.g., Greenwood, 1975). There is no single basis that describes a given system. Rather, the basis is chosen for convenience from among an infinite number of possibilities (e.g., Morel, 1983). Any useful basis can be selected, and the basis may be changed at any point in a calculation to a more convenient one. We discuss the choice of basis species in the next section. 

At this point we should mention that we encountered instability problems in the linear response calculations for some of the MCSCF wavefunctions at intemuclear distances larger than R—S a.u. We believe those instabilities to be artifacts of the calculations because their existence or position depends on the choice of basis set, active space or number of electrons allowed in the RAS3 space. This implies that even though it might not be possible to generate... 

The methods for doing so are described in Chapter 9. The basic principles remain unchanged—the primary difference is the choice of a consistent basis for calculation, such as a solvent-free basis. Graphic techniques based on triangular coordinates provide approximate answers, but modern computational techniques are to be preferred. 

By appropriate choice of the type (or combination) of the organic solvent(s), selective polar dipole-dipole, proton-donor, or proton-acceptor interactions can be either enhanced or suppressed and the selectivity of separation adjusted [42]. Over a limited concentration range of methanol-water and acetonitrile-water mobile phases useful for gradient elution, semiempirical retention equation (Equation 5.7), originally introduced in thin-layer chromatography by Soczewinski and Wachtmeister [43], is used most frequently as the basis for calculations of gradient-elution data [4-11,29,30] ... 

Calculated CX stretching frequencies for these compounds (repeating the data in Appendix A7) are provided in Table 7-3 and compared to measured values. As expected, limiting (6-311+G basis set) Hartree-Fock frequencies are all larger than experimental values. In fact, with the sole exception of methyl chloride at the 3-2IG level, Hartree-Fock frequencies are always larger than experimental frequencies, irrespective of choice of basis set. 

Explosive Performance, Comparison of Two Methods for Its Evaluation. Until, the middle of 1950 s, the choice of expls for blasting a particular rock was made on the basis of the following methods a) Trial and error gained from previous experience in similar rocks b) Calculation of meaningful performance parameters of expls from their chemical compns by means of laws of thermochemistry or thermodynamics or c) Laboratory determination of expl characteristics... 

These systems provide a useful example because the calculations often work, but occasionally fail, either by distortion from planarity or by failure to locate a stable minimum for one of the tautomers. Thus, the students learn to consider their results critically with a healthy dose of skepticism, to analyze the success or failure of the calculation, to consider the influence of the choice of method (semi-empirical or ab initio), to consider the influence of the choice of basis set, and to determine the answer to the research question posed. 

In Fenske and Hall s parameter-free SCF calculations (80-84), the He1t 1-electron operator is substituted by a model 1-electron operator that has a kinetic energy and potential energy term for each atomic center in the complex. This approach assumes that the electron density may be assigned to appropriate centers. The partitioning of electron density is done through Mulliken population analyses (163) until self-consistency is obtained. The Hamiltonian elements are evaluated numerically, and the energies of the MO s depend only on the choice of basis functions and the intemuclear distance. 

The dimension of the secular determinant for a given molecule depends on the choice of basis set. EHT adopts two critical conventions. First, all core electrons are ignored. It is assumed that core electrons are sufficiently invariant to differing chemical environments that changes in their orbitals as a function of environment are of no chemical consequence, energetic or otherwise. All modern semiempirical methodologies make this approximation. In EHT calculations, if an atom has occupied d orbitals, typically the highest occupied level of d orbitals is considered to contribute to the set of valence orbitals. 

Having discussed ways to reduce the scope of the MCSCF problem, it is appropriate to consider the other limiting case. What if we carry out a CASSCF calculation for all electrons including all orbitals in the complete active space Such a calculation is called full configuration interaction or full CF. Witliin the choice of basis set, it is the best possible calculation that can be done, because it considers the contribution of every possible CSF. Thus, a full CI with an infinite basis set is an exact solution of the (non-relativistic, Bom-Oppenheimer, time-independent) Schrodinger equation. 

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