Basis sets of atomic orbitals

The choice of the basis set functions x (the incomplete set) is one of the most important practical (numerical) problems of quantum chemistry. Yet, because it is of a technical character, we will just limit ourselves to a few remarks. 

Although atomic functions do not need to be atomic orbitals (e.g., they may be placed in-between nuclei), in most cases they are centred directly on the nuclei of the atoms belonging to the molecule under consideration. If M is small (in the less precise calculations), the Slater atomic orbitals discussed above are often used as the expansion functions Xs, for larger M (in more accurate calculations), the 

Atomic Smicmre Calculations. I. Hanree—Fock Energy Results for the Elements Hthrou Lr , Report LA-3690 (Los Alamos National Laboratory, 1967). 

Electronic Motion in the Mean Field Atoms and Molecules 

The simplest level of the nonempirical (ab initio) and semiempirical all-valence calculations is the use of a minimal basis set of AO s where each AO in the expansion of Eq. (2.3) is represented by one function, for example, by a Slater-type orbital (STO)  

A considerable improvement in the accuracy of calculations can be achieved by making use of the DZ basis set where each valence orbital corresponds to two functions of the same type [Eq. (2.19)] but with different values of the Slater exponents  

Any basis set of the Slater functions that exceeds the DZ-type basis is regarded [2,6] as an extended basis set. Important is the case of a basis in which the functions of the DZ set are supplemented with the AO functions possessing a higher quantum number /, for example, with the d oribitals for the second period atoms and sometimes with the p orbitals for the hydrogen atoms. These additions make it possible to take into account the polarization of the orbitals of the atomic ground state, so when the d orbitals and the p-AO s are mixed we have 

The AO s having higher orbital quantum numbers / are termed the polarization (p) functions and the DZ basis set with the polarization functions is designated as DZ + P. 

When the basis of the STO functions are employed, by far the greater part of the computer time is spent on calculating the integrals of Eqs. (2.7) and (2.8). As one goes to the Gaussian-type (GTO) basis sets, this time consumption is drastically reduced. In this case, Eqs. (2.19) are approximated by a linear combination of several Cartesian Gaussian functions [6-8,13]. 

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Even with the minimal basis set of atomic orbitals used m most sem i-empirical calculatitm s. the n urn ber of molecii lar orbitals resulting from an SCFcalciilation exceeds the num ber of occupied molecular orbitals by a factor of about two. The n um ber of virtual orbitals in an ah initio calculation depends on the basis set used in this calculation. 

In addition most of the more tractable approaches in density functional theory also involve a return to the use of atomic orbitals in carrying out quantum mechanical calculations since there is no known means of directly obtaining the functional that captures electron density exactly. The work almost invariably falls back on using basis sets of atomic orbitals which means that conceptually we are back to square one and that the promise of density functional methods to work with observable electron density, has not materialized. 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

Basis Sets of Atomic Orbitals and the LCAO-MO Approximation 970... 

Construct a bonding model of molecular orbitals from a basis set of atomic orbitals. 

Based on first principles. Used for rigorous quantum chemistry, i. e., for MO calculations based on Slater determinants. Generally, the Schrodinger equation (Hy/ = Ey/) is solved in the BO approximation (see Born-Oppenheimer approximation) with a large but finite basis set of atomic orbitals (for example, STO-3G, Hartree-Fock with configuration interaction). 

The character tables of Tables 7.6 and 7.7 are best explained by example. For instance, consider the bent molecule NO2, which belongs to point group Czv, and choose a minimum basis-set of atomic orbitals centered on the three atoms (Fig. 7.7). To exploit the molecular symmetry, it is wise to orient the molecule with the z axis bisecting the ONObond angle and with the x axis normal to the N02 molecular plane. Consider what will happen to the column vector representing 2px orbitals centered on the three atoms 2px(N), 2px 0A), 2px(0B) ... 

Therefore, the m.o.s obtained (as linear combinations of a.o.s) which enable an approximate wavefunction to be constructed (a Slater determinant for a closed-shell system) must lead to the minimum energy attainable for the basis set of atomic orbitals used, within the constraints of the method being apphed. The larger the basis set, that is, the more flexible the wave-function, the lower the calculated energy and, by implication, the better the wavefunction. Thus, m.o.s based on an infinite set of a.o.s would lead to the best wavefunction (within the eventual constraints of the hamiltonian used). 

Until quite recently the role of a—n correlation effects was ignored in the. theoretical treatment of electronic transitions. Even now, nearly all ab initio calculations of excitation phenomena are based on independent-particle models using a minimal basis set of atomic orbitals, or involve a configuration interaction limited to the sr-electron system. In order to go far enough beyond the o—n separation, two improvements have to be simultaneously considered ... 

Another difficulty arises from the necessity of summing over all the excited states of the molecule, including the continuum. In general little is known about such states for most molecules. In addition, Ramsey s formulation produces screening data which depend upon the choice of origin, i.e. are gauge dependent, unless a complete basis set of atomic orbitals is included in the molecular orbital description. This is rarely possible, even for diatomic molecules, without the use of large amounts of computer time. 

Sponer, J. and Hobza, P. (2000) Interaction energies of hydrogen-bonded formamide dimer, formamidine dimer, and selected DNA base pairs obtained with large basis sets of atomic orbitals, J. Phys. Chem. A 104, 4592-4597. 

While semiempirical models which can be applied to molecules the size of 1 and 2 are necessarily only approximate, we were searching for trends rather than absolute values. In concept, the design of semiempirical quantum mechanical models of molecular electronic structure requires the definition of the electronic wavefunction space by a basis set of atomic orbitals representing the valence shells of the atoms which constitute the molecule. A specification of quantum mechanical operators in this function space is provided by means of parameterized matrices. Specification of the number of electrons in the system completes the information necessary for a calculation of electronic energies and wavefunctions if the molecular geometry is known. The selection of the appropriate functional forms for the parameterization of matrices is based on physical intuition and analogy to exact quantum mechanics. The numerical values of the parameters are obtained by fitting to selected experimental results, typically atomic properties. 

Within the LCAO approximation, when the molecular orbitals are expanded over the basis set of atomic orbitals xM)... 

Energy bands for the transition metals are constructed, using a minimal basis set of atomic orbitals. The eleven parameters required are reduced to two, the J-band width H, and its position , relative to the. s-band minimum, using Muffin-Tin Orbital theory. Relations giving Wj and all interatomic matrix elements in terms of a J-state radius r, and the internuclear distance are listed in the Solid State Table, along with values of r, and for all of the transition elements this makes possible elementary calculations of the bands for any transition metal, at any atomic volume. 

The LCAO approximation allows us to build up the MOs of a molecule as linear combinations of a basis set of atomic orbitals assigned to each atom. The approximation provides a simple interpretative framework for the nature of MOs and also has important computational advantages. 

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