Minimal-basis-set wave function

Several papers have dealt with the evaluation of wave functions including correlation in various ways. Bimstock34 has calculated the 13C shielding constants in CH4 and several other small molecules using an approximate form of uncoupled Hartree-Fock theory and the minimal basis set wave functions of Palke and Lipscomb.35 The results were similar to those obtained earlier by Ditchfield et al.33... 

The examples quoted in Section IV all refer to minimal basis set wave functions composed of Slater-type orbitals (best atom zetas82>) for molecules I-XIV, and of Gaussian orbitals for the others. Wave functions for compounds XV-XXI refer to a (7 s 3p 13 s) basis contracted to [2s 1 p 11 s] proposed by Clementi, Clementi and Davis33) (CD basis), while for compound XXII we have used another Gaussian basis (4 s 2p 3s) contracted to [2s 1 p 12s] proposed by M ly and Pullman (MP basis)34). 

It may be as well to emphasize here that the values in Table 3 refer to minimal basis set wave functions, and that the conclusions we have drawn are to be considered as provisional until equivalent analyses in terms of more flexible wave functions become available. To the extent that these results are satisfactory, we can define, a set of mean values for the expansion coefficients, since the spread of the values is sufficiently limited. — These mean values, up to the octopole terms, are reported in Table 4 and can be used in Eq. (17) for approximate calculations of the electrostatic potential. 

A double-zeta (DZ) basis set is obtained by replacing each STO of a minimal basis set by two STOs that differ in their orbital exponents ((zeta). (Recall that a single STO is not an accurate representation of an AO use of two STOs gives substantial improvement.) For example, for QHa a double-zeta set consists of two Is STOs on each H, two Is STOs, two 2s STOs, two 2p two 2py, and two 2p STOs on each carbon, for a total of 24 basis functions this is a (4s /2s) basis set. (Recall that we did a double-zeta SCF calculation on He in Section 13.16.) Since each basis function Xr in < i = 2, CriXr has its own independently determined variational coefficient c , the number of variational parameters in a double-zeta-basis-set wave function is twice that in a minimal-basis-set wave function. A triple-zeta (TZ) basis set replaces each STO of a minimal basis set by three STOs that differ in their orbital exponents. 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater Is orbital with a Gaussian function. That is, we replace the Is radial wave function... 

Semiempirical calculations are set up with the same general structure as a HF calculation in that they have a Hamiltonian and a wave function. Within this framework, certain pieces of information are approximated or completely omitted. Usually, the core electrons are not included in the calculation and only a minimal basis set is used. Also, some of the two-electron integrals are omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized. Parameters to estimate the omitted values are obtained by fitting the results to experimental data or ah initio calculations. Often, these parameters replace some of the integrals that are excluded. 

The second approach typically involves expanding the wave functions in terms of atomic or atomic-like orbitals. Frequently s- and p-symmetry functions suffice for silicon and impurities up through the third period. A minimal basis set for silicon would consist of four basis functions, one 5-function and three p-functions on each atom. Some approaches supplement the minimal basis either with more atomic-like functions or with additional types of functions, such as plane waves. Some calculations use only plane waves for the basis. 

It is not possible to use normal AO basis sets in relativistic calculations The relativistic contraction of the inner shells makes it necessary to design new basis sets to account for this effect. Specially designed basis sets have therefore been constructed using the DKH Flamiltonian. These basis sets are of the atomic natural orbital (ANO) type and are constructed such that semi-core electrons can also be correlated. They have been given the name ANO-RCC (relativistic with core correlation) and cover all atoms of the Periodic Table.36-38 They have been used in most applications presented in this review. ANO-RCC are all-electron basis sets. Deep core orbitals are described by a minimal basis set and are kept frozen in the wave function calculations. The extra cost compared with using effective core potentials (ECPs) is therefore limited. ECPs, however, have been used in some studies, and more details will be given in connection with the specific application. The ANO-RCC basis sets can be downloaded from the home page of the MOLCAS quantum chemistry software (http //www.teokem.lu.se/molcas). 

A second example is the minimal-basis-set (MBS) Hartree-Fock wave function for the diatomic molecule hydrogen fluoride, HF (Ransil 1960). The basis orbitals are six Slater-type (i.e., single exponential) functions, one for each inner and valence shell orbital of the two atoms. They are the Is function on the hydrogen atom, and the Is, 2s, 2per, and two 2pn functions on the fluorine atom. The 2sF function is an exponential function to which a term is added that introduces the radial node, and ensures orthogonality with the Is function on fluorine. To indicate the orthogonality, it is labeled 2s F. The orbital is described by... 

The electronic overlap populations in all three cases were calculated from the one electron Extended Hiickel MO s. For the photocyclizations of 1,2-difuryl ethylenes very similar results were obtained also from minimal basis set ab-initio wavefunctions l The possibility of obtaining useful reactivity analyses from wave-functions which are easily available even for large systems could prove to be an important practical consideration for further applications of this method. The dependence on Sri sj ill (5) ensures that electronic overlap populations show the desirable physical characteristics for their use as reactivity measures strong falling-off with increasing interatomic distance and proper directional dependence. This last point is of particular significance for bond formation in polyenes. Thus for two C 2 p atomic... 

The calculations in this illustration were not done with a minimal basis set, since, if such were used, they would not show the correct behavior, even qualitatively. This happens because we must represent both F and F in the same wave function. Clearly one set of AOs cannot represent both states of F. Li does not present such a difficult problem, since, to a first approximation, it has either one orbital or none. The calculations of Fig. 8.4 were done with wave functions of 1886 standard tableaux functions. These support 1020 s mimetry functions. We will discuss the arrangement of bases more fully in Chapter 9. 

For the minimal basis set l.sa and ls, group theory has given us the precise form (except for the orbital exponents, which must be found by a variational calculation) of the two lowest H2 MOs 1 ag and 1 au. [Since a a MO is unchanged upon reflection in a plane containing the molecular axis, any o one-electron function must be a a+ function hence there is no need to use the superscript for a MOs. We can, however, have 2+ and 2 many-electron electronic wave functions, and here the superscript must be included.]... 

At the lowest level of sophistication of quantum treatments, the tight-binding method and the semi-empirical HF method reduce the complexity of the interacting electron system to the diagonalization of an effective one-electron Hamiltonian matrix, whose elements contain empirical parameters. The electronic wave functions are expanded on a minimal basis set of atomic or Slater orbitals centered on the atoms and usually restricted to valence orbitals. The matrix elements are self-consistently determined or not, depending upon the method. 

The set of atomic orbitals used for LGAO (the basis set) can be either a minimal basis set or an extended basis set. In the first case only ground-state orbitals of the atoms are included in the combination, i.e. one atomic orbital is used for each independent occupied orbital in the component atom. SCF wavefunctions require the use of basis sets with more atomic orbitals (e.xtended basis sets). Still, useful wave-functions for ground states (but not excited states) can be obtained with minimal basis sets. Following the common usage these functions are also loosely described as SCF wavefunctions. 

Fig. 41. Comparison of the values of minima of electrostatic potential in some three-membered ring molecules according to STO and GTO minimal basis set SCF wave functions. The values are labeled as follows 1, aziridine (C—C), 2, cyclopropene (C=C), 3, cyclopropene (C—C), 4, cyclopropane (C—C), 5, oxaziridine (O), 6, oxazir-idine (O), 7, oxirane (O), 8, oxaziridine (N), 9, trans-diaziridine (N), 10, cis-diazir-idine (N), 11, aziridine (N)...

The provisional conclusion reached on the basis of checks performed so far is that, for molecules built up with atoms of the first and second row of the periodic table, a minimal basis set SCF wave function is... 

Fig. 42. Comparison of the values of minima of electrostatic potential in some three-membered ring molecules according to GTO minimal basis set SCF wave functions and to CNDO semiempirical calculations. The values are labeled as in the preceding figure...

Similar results have been obtained for other molecules (NH343), H2O43), H2NC017a>, etc.) on minimal basis set functions. By removing the constraint of contraction factors in the CD basis one obtains equivalent results. No controls have yet been performed on wave functions... 

The equality holds only when the exact wave function is used. Therefore the variational principle allows us to choose a basis set of functions gj such that a linear combination of them will tend to the exact solution. Thus the problem is reduced to finding the best set of coefficients that will minimize the right-hand side of eq. (9). This implies the calculation of a large number of integrals of the form,... 

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