Orbitals double-zeta

Split-valence or double zeta. Orbitals represented by 3 Gaussians near the nucleus and 2 away from the nucleus. Hydrogen atoms represented by one Gaussian. This is a good basis for conformational analysis of moderate-sized molecules... 

We are only interested in the localized wave function of the electron, for example, the highest occupied molecular orbital (HOMO), of the transition state complex with the electron transferred. Hence, we may use the HOMO wave function of the transition state complex after the electron transfer, including only the nearby surface metal atoms that contribute significantly to this HOMO [40]. This wave function at the cluster is calculated using the EHMO method together with the parameters of VSIP and double-zeta orbitals given in [31] ... 

T>y(rt, l, rri) is the Slater-type j th atomic orbital of the th atom multiphed by its associated spherical harmonics is the double-zeta orbital exponent a0 is the Bohr radius... 

The double zeta basis sets, such as the Dunning-Huzinaga basis set (D95), form all molecular orbitals from linear combinations of two sizes of functions for each atomic orbital. Similarly, triple split valence basis sets, like 6-3IIG, use three sizes of contracted functions for each orbital-type. 

We refer to such a basis set as a double zeta basis set. Where the minimal basis set for atomic lithium had a 1 s exponent of 2.6906, the double zeta basis set has two Is orbitals with exponents 2.4331 and 4.5177 (the outer and inner orbitals). 

The chemical bonding occurs between valence orbitals. Doubling the 1 s-functions in for example carbon allows for a better description of the 1 s-electrons. However, the Is-orbital is essentially independent of the chemical environment, being very close to the atomic case. A variation of the DZ type basis only doubles the number of valence orbitals, producing a split valence basis. In actual calculations a doubling of tire core orbitals would rarely be considered, and the term DZ basis is also used for split valence basis sets (or sometimes VDZ, for valence double zeta). 

The starting point to obtain a PP and basis set for sulphur was an accurate double-zeta STO atomic calculation4. A 24 GTO and 16 GTO expansion for core s and p orbitals, respectively, was used. For the valence functions, the STO combination resulting from the atomic calculation was contracted and re-expanded to 3G. The radial PP representation was then calculated and fitted to six gaussians, serving both for s and p valence electrons, although in principle the two expansions should be different. Table 3 gives the numerical details of all these functions. 

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 ) ... 

In a previous work [1,2], we were interested in the calculation of second order hyperpolarizabilities of eonjugated systems including substituted benzenes, pyridine N-oxydes and vinyl oligomers, in relation with non linear optical activity [3]. We showed that MNDO ealeulations were in good agreement with SCF ab initio results obtained using a double zeta basis set plus polarization and diffuse orbitals. 

The atomic basis consists in a double-zeta set expanded with polarization functions (DZP) and augmented by diffuse functions (DZPR). Exponents and contraction coefficient are from McLean and Chandler 1980 [18] diffuse functions, centered on the heavy atoms with exponents of 0.023 for the s orbitals and 0.021 for the p orbitals are from Dunning and Hay 1977 [34]. Extension of the DZP basis set with two sets of diffuse s (0.0437, 0.0184) and p (0.0399, 0.0168) functions (DZPRR) has also been tested. 

Ab initio calculations usually begin with a solution of the Hartree-Fock equations, which assumes the electronic wavefunction can be written as a single determinant of molecular orbitals. The orbitals are described in terms of a basis set of atomic functions and the reliability of the calculation depends on the quality of the basis set being used. Basis sets have been developed over the years to produce reliable results with a minimum of computational cost. For example, double zeta valence basis sets such as 3-21G [15] 4-31G [16] and 6-31G [17] describe each atom in the molecule with a single core Is function and two functions for the valence s and p functions. Such basis sets are commonly used, as there appears to be a cancellation of errors, which fortuitously allows them to predict quite accurate results. 

Robin et al.162 investigated the photoelectron spectrum of unsubstituted cyclo-propenone and interpreted its results with the aid of Gaussian-type orbital calculations of double-zeta quality for the electronic ground state using the experimentally established133 geometry of cyclopropenone. 

Each CGTO can be considered as an approximation to a single Slater-type orbital (STO) with effective nuclear charge f (zeta). The composition of the basis set can therefore be described in terms of the number of such effective zeta values (or STOs) for each electron. A double-zeta (DZ) basis includes twice as many effective STOs per electron as a single-zeta minimal basis (MB) set, a triple-zeta (TZ) basis three times as many, and so forth. A popular choice, of so-called split-valence type, is to describe core electrons with a minimal set and valence electrons with a more flexible DZ (or higher) set. 

For the related [CpIr(PH3)(CH3)]+ system, four basis sets were used. Basis set one (BS1) is the same as the ones described above for Ir and P, but the C and H are described as D95. Basis set two (BS2) is the Stuttgart relativistic, small core ECP basis set (49) augmented with a polarization function for Ir, and Dunning s correlation consistent double-zeta basis set with polarization function (50) for P, C and H. Basis set three (BS3) is the same as BS1 except the d-orbital of Ir was described by further splitting into triple-zeta (111) from a previous double-zeta (21) description and augmented with a f-polarization function (51). Basis set four (BS4) is the same as BS2 for Ir, P, and most of the C and H, but the C and H atoms involved in the oxidative addition were described with Dunning s correlation consistent triple-zeta basis set with polarization. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

In minimal basis sets, each atom is represented by a single orbital of each type. For example, oxygen is represented by Is, 2s, 2p, 2py, and 2p orbitals only. In double zeta basis sets, twice the functions in the minimum basis sets are used. Extended basis sets generally refer to sets that make use of functions that are more than the minimum basis set. 

A double-zeta (DZ) basis in which twice as many STOs or CGTOs are used as there are core and valence atomic orbitals. The use of more basis functions is motivated by a desire to provide additional variational flexibility to the LCAO-MO process. This flexibility allows the LCAO-MO process to generate molecular orbitals of variable diffuseness as the local electronegativity of the atom varies. Typically, double-zeta bases include pairs of functions with one member of each pair having a smaller exponent (C, or a value) than in the minimal basis and the other member having a larger exponent. 

The number of basis functions (defined by the chosen basis sets) used to construct the molecular orbitals also strongly affects the effort/accuracy ratio. The use of minimal basis sets yielded wrong results (56), whereas reasonable agreement with experiment is obtained when double zeta plus polarization basis sets are applied. Correlated methods require larger basis sets to include as much electron correlation as possible. This implies that in addition to the increased computational demand of such methods, a further increase of the computational cost results due to the requirement of using larger basis sets. 

Practically all of the calculation studies we discuss here were performed within the HF or DFT formalism, and most employed acid site cluster models that may contain anywhere between one and five Si and A1 atoms. Basis sets used to represent the electrons of the system were usually of double-zeta quality or higher i.e., each filled orbital of an atom has been represented by two separate exponential functions. In addition, extra functions have been added—so-called polarization functions—to represent orbitals that are empty. These allow the orbital more flexibility and result in better theoretical predictions. 

At the SCF or MCSCF level, the basis set requirements are fairly simple. We can imagine that the occupied molecular orbitals are given as a simple linear combination of atomic orbitals this corresponds to a minimal basis set. The results so obtained are fairly crude, but by admitting extra functions to represent the atomic orbitals more flexibly (split-valence, double zeta, etc) we can obtain a much better description. However, some effects require going beyond the occupied atomic orbitals ... 

Dunning-type contractions are characterized by considerable flexibility in the valence part of the primitive space. Typically, the outermost primitive functions are not contracted at all, contraction being reserved for the inner parts of the valence orbitals and the core orbitals. The commonest contracted set of this type is probably the [4s 2p] contraction of the (9s 5p) set. Unfortunately, there are at least two such double zeta contraction schemes in use, as well as an erroneous one. Some care may be required to reproduce results asserted to be obtained with a Huzinaga-Dunning [4s 2p] basis . Because of the relatively flexible contraction scheme these basis sets usually perform well, especially when large primitive sets such as van Duijneveldt s (13s 8p) sets are used. However, it should be noted that such primitive sets are difficult to contract this way without significant loss of accuracy at the atomic SCF level, unless very large contracted sets are used. 

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