Basis Set Methods

The Complete Basis Set (CBS) methods were developed by George Petersson and several collaborators. The family name reflects the fundamental observation underlying these methods the largest errors in ab initio thermochemical calculations result from basis set truncation. 

Exploring Chemistry with Electronic Structure Methods 

As in G2 theory, the total energy is computed from the results of a series of calculations. The component calculations are defined on the basis of the following principles and observations  

CBS models typically include a Hartree-Fock calculation with a very large basis set, an MP2 calculation with a medium-sized basis set (and this is also the level where the CBS extrapolation is performed), and one or more higher-level calculations with a medium-to-modest basis set. The following table outlines the components of the CBS-4 and CBS-Q model chemistries  

Additional empirical corrections 1 and 2-electron higher-order corrections (size-consistent), spin contamination 2-electron higher-order correction (size-consistent), spin contamination, core correlation for sodium 

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The first quantum mechanical improvement to MNDO was made by Thiel and Voityuk [19] when they introduced the formalism for adding d-orbitals to the basis set in MNDO/d. This formalism has since been used to add d-orbitals to PM3 to give PM3-tm and to PM3 and AMI to give PM3(d) and AMl(d), respectively (aU three are available commercially but have not been published at the time of writing). Voityuk and Rosch have published parameters for molybdenum for AMl(d) [20] and AMI has been extended to use d-orbitals for Si, P, S and Q. in AMI [21]. Although PM3, for instance, was parameterized with special emphasis on hypervalent compounds but with only an s,p-basis set, methods such as MNDO/d or AMI, that use d-orbitals for the elements Si-Cl are generally more reliable. 

Two-dimensional semiclassical studies described in section 4 and applied to some concrete problems in section 6 show that, when no additional assumptions (such as moving along a certain predetermined path) are made, and when the fluctuations around the extremal path are taken into account, the two-dimensional instanton theory is as accurate as the one-dimensional one, and for the tunneling problem in most cases its answer is very close to the exact numerical solution. Once the main difficulty of going from one dimension to two is circumvented, there seems to be no serious difficulty in extending the algorithm to more dimensions that becomes necessary when the usual basis-set methods fail because of the exponentially increasing number of basis functions with the dimension. 

The main feature of the CBS (complete basis set) methods (e.g. CBS-Q [15] and CBS-QB3 [20]) is extrapolation to the complete basis set limit at the UMP2 level. Additional calculations [UMP4 and UQ-CISD(T) or UCCSD(T)] are performed to estimate higher-order effects. A scaled ZPVE, together with a size-consistent empirical correction and a spin-contamination correction, are added to yield the total CBS energy of the molecule. 

Valeev, E.F., Janssen, C.L. Second-order Moller-Plesset theory with linear R12 terms (MP2-R12) revisited auxiliary basis set method and massively parallel implementation. J. Chem. Phys. 2004, 121, 1214-27. 

One conceptually simple remedy for the shortcomings of DFT regarding dispersion forces is to simply add a dispersion-like contribution to the total energy between each pair of atoms in a material. This idea has been developed within localized basis set methods as the so-called DFT-D method. In DFT-D calculations, the total energy of a collection of atoms as calculated with DFT, dft> is augmented as follows ... 

Using a basis set method, Peternelj et al. [1987] calculated the spectrum of coupled rotation for a potential of a more general form than (7.44) ... 

Complete Basis Set Methods Petersson et al.61-63 developed a series of methods, referred to as complete basis set (CBS) methods, for the evaluation of accurate energies of molecular systems. The central idea in the CBS methods is an extrapolation procedure to determine the projected second-order (MP2) energy in the limit of a complete basis set. This extrapolation is performed pair by pair for all the valence electrons and is based on the asymptotic convergence properties of pair correlation energies for two-electron systems in a natural orbital expansion. As in G2 theory, the higher order correlation contributions are evaluated by a sequence of calculations with a variety of basis sets. 

Complete basis set methods [113] involve essentially seven or eight steps ... 

Atoms beyond the first row of the periodic table and molecules containing such atoms are commonly treated in QMC by an approach that avoids explicit consideration of core electrons see, for example [3,27]. The reason is the very considerable increase in computer time imposed by core electrons. For many properties of chemical interest, core electrons play a relatively minor role. The origin of the increase of computer time has been analyzed in various ways, but a major factor is differing time scales for valence and core electrons [28-30]. Just as pseudopotentials can serve to reduce the computational effort in basis set methods, they play an equivalent, if not more after important, role in QMC because of the heavy computational demands of sampling core electrons relative to valence electrons. 

Fliflet, A.W. and McKoy, V. (1978). Discrete basis set method for electron-molecule continuum wave functions, Phys. Rev. A 18,2107-2114. 

In addition to the Gaussian-n methods, there is a variety of CBS (complete basis set) methods, including CBS-Q, CBS-q, and CBS-4, which have also enjoyed some popularity among computational chemists. In these methods, special procedures are designed to estimate the complete-basis-set limit energy by extrapolation. Similar to the Gn methods, single-point calculations are also... 

Recently calculations on lithium have also reached impressive levels of precision on the order of 9 to 10 significant figures[3]. In the past, such accuracy would only have been associated with the most elaborate calculations on helium. Beyond lithium, this level of accuracy has not been achieved, as the use of correlated basis set methods becomes very cumbersome. At some point, large scale calculations based on simpler methods begin to surpass those produced with necessarily smaller, correlated basis sets. At present, this crossover occurs somewhere in the four to five electron range. Examples of such calculations on beryllium as well as some simple molecules will be presented. 

With the exception of the EC basis, the basis sets discussed to this point have utilized only one or two nonlinear parameters, relying on the flexibility available in large numbers of high- and low-order terms to simultaneously describe particle coalescences and the asymptotic behavior of the wave function. This same flexibility can be obtained with fewer terms by using multiple nonlinear parameters in the basis. Groups of terms with identical nonlinear parameters can be considered distinct basis sets, and the method can be described as a multiple basis set method. The first extensive use of such a double-basis set... 

An interesting mixed-basis-set method for use in SCF calculations has been described by Billingsley and Trindle,52 with application to LiC>2. One-centre and most two-centre integrals are evaluated analytically, whilst less tractable integrals are approximated by a gaussian expansion of the STO s. Examination of portions of the... 

Also as in the case of helium, asymptotic expansion methods can be applied to the Rydberg states of lithium and compared with high precision measurements [73,74]. This case is more difficult because the Li+ core is a nonhydrogenic two-electron ion for which the multipole moments cannot be calculated analytically, and variational basis set methods must be used instead. However, the method is in principle capable of the same high accuracy as for helium. 

In a second step, in order to determine the influence of the anharmonicity in the exact potential we will expand the term up to higher powers of the components of r and treat them as small perturbations to the harmonic approximation of the Hamiltonian by means of first order perturbation theory. These perturbative calculations offer insight into the effects of the anharmonic parts of the potential onto the energies and the form of the wave functions. For a discussion of the basis set method and the computational techniques used for the numerical calculation of the exact eigenenergies and eigenfunctions in the outer potential well we refer the reader to [7]. In the following we discuss the results of these numerical calculations of the exact eigenenergies and wave functions and... 

Second-order Moller-Plesset theory with linear R12 terms (MP2-R12) revisited Auxiliary basis set method and massively parallel implementation101... 

There have been numerous studies of the effects of different methods and different basis sets. General studies covering a wide range of molecules of special note relevant to this study are those by B. Johnson et al. [1] and B. Ma et al. [2], although the former does not include one of the methods used extensively in this study. Many papers study these effects for single molecules individually or for single types of molecule. The effects of basis sets/methods on small hydride molecules were considered many years ago and displayed in a graphical manner... 

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