Methods and basis sets

Quantum-mechanical approximation methods can be classified into three generic types (1) variational, (2) perturbative, and (3) density functional. The first two can be systematically improved toward exactness, but a systematic correction procedure is generally lacking in the third case. 

Variational approximation methods are identified by the form of the variational trial function, particularly by the number and types of Slater determinants. 

The simplest approximation corresponds to a single-determinant wave function. The best possible approximation of this type is the Hartree-Fock (HF) molecular-orbital determinant. The HF wavefunction is constructed from the minimal number of occupied MOs (i.e., NI2 for an V-eleclron closed-shell system), each approximated as a variational linear combination of the chosen set of basis functions (vide infra). 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

More accurate multi-determinant configuration-interaction (Cl) wavefunctions are described by specifying the types of substitutions ( excitations ) from the starting HF 

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For many projects, a basis set cannot be chosen based purely on the general rules of thumb listed above. There are a number of places to obtain a much more quantitative comparison of basis sets. The paper in which a basis set is published often contains the results of test calculations that give an indication of the accuracy of results. Several books, listed in the references below, contain extensive tabulations of results for various methods and basis sets. Every year, a bibliography of all computational chemistry papers published in the previous... 

Table 12.1 gives the partial charges for the atoms in acetic acid computed with a number of dilferent methods and basis sets. All calculations use the molecular... 

Modeling the lighter main group inorganic compounds is similar to modeling organic compounds. Thus, the choice of method and basis set is nearly identical. The second-row compounds (i.e., sulfur) do have unfilled d orbitals, making it often necessary to use basis sets with d functions. 

The route section of a Gaussian input file specifies the kind of job you want to run as well as the specific theoretical method and basis set which should be used. All of these items are specified via keywords. Recall that the first line of the route section begins with a sign (or T to request terse output). 

Predict the zero point or thermal energy by running a frequency job at the optimized geometry, using the same method and basis set. (Note that these two steps maybe run via a single Gaussian job via the Opt Freq keyword.)... 

Seiufion Here are the results for a variety of theoretical methods and basis sets ... 

In the last two chapters, we discussed the ways that computational accuracy varies theoretical method and basis set. We ve examined both the successes and failures o variety of model chemistries. In this chapter, we turn our attention to mod designed for modeling the energies of molecular processes very accurately. 

Atoms defined in this way can be treated as quantum-mechanically distinct systems, and their properties may be computed by integrating over these atomic basins. The resulting properties are well-defined and are based on physical observables. This approach also contrasts with traditional methods for population analysis in that it is independent of calculation method and basis set. 

The Parameterized Configuration Interaction (PCI-X) method simply takes the correlation energy and scales it by a constant factor X (typical value 1.2), i.e. it is assumed that the given combination of method and basis set recovers a constant fraction of the correlation energy. 

In this chapter we will illustrate some of the methods described in the previous sections. It is of course impossible to cover all types of bonding and geometries, but for highlighting the features we will look at the H2O molecule. This is small enough that we can employ the full spectrum of methods and basis sets. 

Further details of the underlying computational methods and basis sets to determine the wavefunction and density are described in Appendix A. Unless otherwise noted, all numerical examples of this book employ the B3LYP/6-311++G level of theory. 

A brief introduction to the cryptic notation designating standard methods and basis sets of modern ab initio and density-functional calculations is given in Appendix A. Such designations will be used without further comment throughout this book. 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

This indicates that the deviations are due to systematic errors, for example deficiencies of the applied methods and basis sets. DFT-based methods, such as GIAO/DFT calculations are known to overestimate paramagnetic contributions to the chemical shielding. This results, for critical cases with small HOMO/LUMO separations, in overly deshielded competed chemical shifts. Notorious examples for these deficiencies are 29Si or 13C NMR chemical shift computations of silylenes, silylium ions or dienyl cation .(5/-54) Taking into account the deficiencies of the applied method, and bearing in mind very reasonable correlations shown in Figures 4 and 5, the computational results do support the structural characterization of the synthesized vinyl cations. 

Calculated NMR Chemical Shifts for Silylium Ions Using Different Methods and Basis Sets... 

The isotropic magnetic properties computed for TEMPO (4 in Figure 2.1) by different methods and basis sets are compared in Table 2.1 with the corresponding experimental values. 

Isomerization mechanisms of isolated trans- to cis-diazene were studied and transition states for two possible interconversion routes were found to be more than 200 kJ/mol unstable than traras-diazene (98). The relative stability of trans- to cis-diazene was calculated to be 21 to 29 kJ/mol using different quantum chemical methods and basis sets [see Ref. (98) and references cited therein]. The NNH2 isomer is about 87 kJ/mol (almost independent of the density functional) higher in energy than traras-diazene. 

Table 20-9. Adiabatic electron affinities (AEAs) of T, C, U, G and AT base pair in solution using different methods and basis set...

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