Basis set notation

The superaiatrix notation emphasizes the structure of the problem. Each diagonal operator drives a wavepaclcet, just as in the adiabatic case of Eq. (10), but here the motion of the wavepackets in different adiabatic states is mixed by the off-diagonal non-adiabatic operators. In practice, a single matrix is built for the operator, and a single vector for the wavepacket. The operator matrix elements in the basis set <() are... 

Quantum chemists have devised efficient short-hand notation schemes to denote the basis set aseti in an ab initio calculation, although this does mean that a proliferation of abbrevia-liijii.s and acronyms are introduced. However, the codes are usually quite simple to under-sland. We shall concentrate on the notation used by Pople and co-workers in their Gaussian aerie-, of programs (see also the appendix to this chapter). 

Most calculations today are done by choosing an existing segmented GTO basis set. These basis sets are identihed by one of a number of notation schemes. These abbreviations are often used as the designator for the basis set in the input to ah initio computational chemistry programs. The following is a look at the notation for identifying some commonly available contracted GTO basis sets. 

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. 

Another family of basis sets, commonly referred to as the Pople basis sets, are indicated by the notation 6—31G. This notation means that each core orbital is described by a single contraction of six GTO primitives and each valence shell orbital is described by two contractions, one with three primitives and the other with one primitive. These basis sets are very popular, particularly for organic molecules. Other Pople basis sets in this set are 3—21G, 4—31G, 4—22G, 6-21G, 6-31IG, and 7-41G. 

As the Pople basis sets have further expanded to include several sets of polarization functions, / functions and so on, there has been a need for a new notation. In recent years, the types of functions being added have been indicated in parentheses. An example of this notation is 6—31G(dp,p) which means that extra sets of p and d functions have been added to nonhydrogens and an extra set of p functions have been added to hydrogens. Thus, this example is synonymous with 6—31+G. ... 

In order to describe the number of primitives and contractions more directly, the notation (6s,5p) (ls,3p) or (6s,5p)/(ls,3p) is sometimes used. This example indicates that six s primitives and hve p primitives are contracted into one s contraction and three p contractions. Thus, this might be a description of the 6—311G basis set. However, this notation is not precise enough to tell whether the three p contractions consist of three, one, and one primitives or two, two, and one primitives. The notation (6,311) or (6,221) is used to distinguish these cases. Some authors use round parentheses ( ) to denote the number of primitives and square brackets [ ] to denote the number of contractions. 

An older, but still used, notation specihes how many contractions are present. For example, the acronym TZV stands for triple-zeta valence, meaning that there are three valence contractions, such as in a 6—311G basis. The acronyms SZ and DZ stand for single zeta and double zeta, respectively. A P in this notation indicates the use of polarization functions. Since this notation has been used for describing a number of basis sets, the name of the set creator is usually included in the basis set name (i.e., Ahlrichs VDZ). If the author s name is not included, either the Dunning-Hay set is implied or the set that came with the software package being used is implied. 

Sometimes it turns out that we need to include a number of polarization functions, not just one of each type. The notation 4-31G(3d, 2p) indicates a standard 4-31G basis set augmented with three d-type primitive Cartesian Gaussians per centre and two p-type primitives on every hydrogen atom. Again, details of the... 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

The calculations were performed using a double-zeta basis set with addition of a polarization function and lead to the results reported in Table 5. The notation used for each state is of typical hole-particle form, an asterisc being added to an orbital (or shell) containing a hole, a number (1) to one into which an electron is promoted. In the same Table we show also the frequently used Tetter symbolism in which K indicates an inner-shell hole, L a hole in the valence shell, and e represents an excited electron. The more commonly observed ionization processes in the Auger spectra of N2 are of the type K—LL (a normal process, core-hole state <-> double-hole state ) ... 

The formal vector cp (K) denotes the set of atomic orbital basis functions with centers at the original nuclear locations of the macromolecular nuclear configuration K, where the components cp(r, K) of vector q(K) are the individual AO basis functions. The macromolecular overlap matrix corresponding to this set cp (K) of AO s is denoted by S(K). The new macromolecular basis set obtained by moving the appropriate local basis functions to be centered at the new nuclear locations is denoted by cfcK ), where the notation cp(r, K ) is used for the individual components of this new basis set overlap matrix is denoted by S(K ). 

For compactness and clarity, Eq. (2.11) is written in matrix notation. It is similar to the more familiar case of a time-independent basis set expansion but with two important differences The AIMS basis is time-dependent and nonorthogonal. As a consequence, the proper propagation of the coefficients requires the inverse of the (time-dependent) nuclear overlap matrix... 

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. 

A series of single-point energy calculations is carried out at higher levels of theory. The first higher-level calculation is the complete fourth-order Mpller-Plesset perturbation theory [13] with the 6-31G(d) basis set, i.e. MP4/6-31G(d). For convenience of notation, we represent this as MP4/d. This energy is then modified by a series of corrections from additional calculations ... 

The following short notation for basis sets will be applied throughout in this contribution (a, /S, y/A, //) indicates that a s type, / p-ty-pe, and y d-type orbitals are applied for second-row atoms and A s-type and / p-type function for H-atoms. Contractions are given in square brackets (a, ft, y j/-, fi ). Basis sets C, D, and E are almost identical to or exactly the same as the ones used in Ref. 109-111) or 88> respectively. 

GTO basis sets are used unless stated otherwise, for the notation of the basis see Table 11. s) All lengths in (A), all energies in (kcal/mole)... 

That study at the MP2 correlated level for the dimers investigated are summarized in Table 7. In Table 7, we use the following notations SM denotes the values obtained in the super-molecule, while CP denotes those obtained in the counter-poise corrected calculations. The results clearly show, that the E(intra/net) values are close to each other (using any basis sets) for the monomers in both of SM and CP systems. This suggests that the SMO-LMBPT scheme takes into account the benefit effect of the basis set superposition. The deviations found in the intra terms between the SM and CP systems are explained recently (Kapuy etal, 1998) in detail. 

By Eq. (6) the sum on the right-hand side of the above equation is equal to the energy E, and from Eq. (2) we realize that the sums on the left-hand side are just Hamiltonian operators in the second-quantized notation. Hence, when the 2-RDM corresponds to an A -particle wavefunction i//, Eq. (12) implies Eq. (13), and the proof of Nakatsuji s theorem is accomplished. Because the Hamiltonian is dehned in second quantization, the proof of Nakatsuji s theorem is also valid when the one-particle basis set is incomplete. Recall that the SE with a second-quantized Hamiltonian corresponds to a Hamiltonian eigenvalue equation with the given one-particle basis. Unlike the SE, the CSE only requires the 2- and 4-RDMs in the given one-particle basis rather than the full A -particle wavefunction. While Nakatsuji s theorem holds for the 2,4-CSE, it is not valid for the 1,3-CSE. This foreshadows the advantage of reconstructing from the 2-RDM instead of the 1-RDM, which we will discuss in the context of Rosina s theorem. 

We conclude this section with a comment concerning notation. To identify a computational approach, it is customary to write procedure/basis set , e.g. MP2-FC/6-311G. If the geometry is obtained by procedure 1 and basis set 1, and that geometry is then used in calculating a particular property with procedure 2 and basis set 2, this is designated by procedure 2/basis set 2//procedure 1/basis set 1 . 

To specify the symbology we remember the familiar notation for the integrals over the basis set orbitals ... 

Hartree-Fock value of 1.1 kcaFmol using the 3-21G basis set. In this notation, MOVB(3)/3-21G, the number in parentheses specifies the number of configurations employed in the MOVE calculation. 

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