Basis set approximation

The smallest basis sets are called minimal basis sets. The most popular minimal basis set is the STO—3G set. This notation indicates that the basis set approximates the shape of a STO orbital by using a single contraction of three GTO orbitals. One such contraction would then be used for each orbital, which is the dehnition of a minimal basis. Minimal basis sets are used for very large molecules, qualitative results, and in certain cases quantitative results. There are STO—nG basis sets for n — 2—6. Another popular minimal basis set is the MINI set described below. 

Minimal basis sets use fixed-size atomic-type orbitals. The STO-3G basis set is a minimal basis set (although it is not the smallest possible basis set). It uses three gaussian primitives per basis function, which accounts for the 3G in its name. STO stands for Slater-type orbitals, and the STO-3G basis set approximates Slater orbitals with gaussian functions. ... 

In practice, if the lower state energy and the corresponding wave function are known accurately then the coupling matrix element (4>o H i) is small. Experience shows that, because the finite basis set approximation is more restrictive for than it is for o, the calculated excited state energy lies above the corresponding exact value. 

The central tenet of CC theory is that the full-CI wave function (i.e., the exact one within the basis set approximation) can be described as... 

In the minimal-basis-set approximation, the lowest four ax MOs are linear combinations of g, through g4 (the coefficients being found from a 4x4 secular determinant), the 1 bx MO is g5, and the 1 b2 and 2b2 MOs are linear combinations of g6 and g7. In the ground electronic state, five of these seven MOs are occupied. (Results of SCF calculations on H20 are discussed in Levine, Section 15.4.)... 

The above discussion refers to full or complete CC theory. At this point, no approximations have been made, beyond any that are made in forming the underlying MOs, namely the one-particle basis set approximation. The results of full CC are equivalent to those of full Cl (FCI) with the same basis set. In practice, for other than model systems, approximations must be made in order for CC calculations to be tractable. Several directions may be followed. First, one may truncate the cluster operator at different levels of excitation. Setting... 

To this point, this is an exact formalism (apart from the basis set approximation in 0 >) and gives the same results (and costs as much) as FCI. In practice, of course, approximations must be made in order to have computationally tractable methods, just as is necessary for the ground state CC treatment. Conceptually, the most straightforward set of approximations run parallel to the approximations for the ground state ... 

All of the results to be presented in this section were obtained with the program deMon [16], using the Gaussian DZVPP basis set (approximately equivalent to 6-31G ). The exchange and correlation functionals were expressed in terms of the generalized gradient approximations (GGA) [17, 18]. The energy values that do not include zero-point corrections are indicated by an asterisk. 

Now the DK Hamiltonian may be calculated to the desired level of accuracy. Within our finite basis set approximation the multiple integral expressions occurring at the evaluation of the momentum space operators are reduced to simple matrix multiplications, which are computationally not very demanding. As soon as /foKHn has been evaluated within the chosen p -representation, it can be transformed back to the usual configuration space representation by applying the inverse transformation This Hamiltonian is then available for every variational procedure without any further modifications. 

Equation 3 still presents problems. First, it is an operator equation. Most of the expertise that has been developed in electronic structure calculations has centered on equations involving matrix elements of operators, rather than the operators themselves. Second, and also important, the vast majority of the solutions of (3) are ones in which we have no interest. Within the limited orbital basis set approximation, there are only a finite number n of linearly independent -electron states, or configuration functions, that can be formed. Within this basis, the exact -electron energies Eq, Ef,..., E i and the corresponding exact -electron states, 0>,..., I n — 1), are, respectively, the eigenvalues and eigenvectors of the n X n Hamiltonian matrix (i.e., the solutions of the complete Cl problem for... 

Consider the example of the boron atom and the making of comparisons of the Herman-Skillman numerical radial functions for 1 s, 2s and 2p with the Slater functions and possible Gaussian basis set approximations. 

The advantage of this approach is its simplicity, but one has to use relatively large orbital basis sets to ensure that the approximated unity operators in Eq. (68) are well represented [22]. This limitation quickly becomes a bottleneck and apparently such approach gives reliable results only for relatively small systems (treated with large orbital basis sets). A natural idea is to introduce an additional basis set (different than the orbital one) that is used to represent the resolution-of-the-identity operators. There are two (so far) closely related models, the auxiliary basis set approximation (ABS) [47] and the complementary auxiliary basis set approximation (CABS) [20]. In the implementation in Turbomole only the latter one was considered. Within this approach the whole basis (singly primed), used for the representation of the unity operators, is the union of the orbital and some auxiliary basis (doubly primed)... 

N. Rom, E. Engdahl, and N. Moiseyev,/. Chem. Phys., 93, 3413—3419 (1990). Tunneling Rates in Bound Systems Using Smooth Exterior Complex Scaling within the Framework of the Finite Basis Set Approximation. 

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