Orbital, atomic wavefunction

Alternative methods are based on the pioneering work of Hylleraas ([1928], [1964]). In these cases orbitals do not form the starting point, not even in zero order. Instead, the troublesome inter-electronic terms appear explicitly in the expression for the atomic wavefunction. However the Hylleraas methods become mathematically very cumbersome as the number of electrons in the atom increases, and they have not been very successfully applied in atoms beyond beryllium, which has only four electrons. Interestingly, one recent survey of ab initio calculations on the beryllium atom showed that the Hylleraas method in fact produced the closest agreement with the experimentally determined ground state atomic energy (Froese-Fischer [1977]). 

The location of an electron in an atom is described by a wavefunction known as an atomic orbital atomic orbitals are designated by the quantum numbers , l, and mi and fall into shells and subshells as summarized in Fig. 1.30. 

The angular momenta of atoms are described by the quantum numbers L, S or J. When spin-orbit coupling is important, it is the total angular momentum J which is a constant of the system. A group of atomic wavefunctions with a common J value - akin to a term, as described in Section 3.6 - comprise (27 -i- 1) members with Mj... 

An electron transfer relay may also operate via the electronic coupling term, HDA. Mixing between the electron-donor and acceptor orbitals provides the physical basis for electron coupling. In view of the exponential decay of atomic wavefunctions and their corresponding linear combinations, it is not surprising that Hda exhibits an exponential dependence on the distance between the electron-acceptor (A) and donor (D) [197],... 

The functions 4> are the spatial molecular (or atomic) orbitals or wavefunctions that (along with the spin functions) make up the overall or total molecular (or atomic) wavefunction ijj, which can be written as a Slater determinant (Eq. 5.12). Concerning the energies from the fact that... 

To make the picture of 7T-electrons more intelligible the model of linear combinations of single electron atomic orbitals to molecular orbitals is helpful (Fig. 14). In this model one concentrates only on the outermost electrons or valence orbitals. Starting from the atomic wavefunctions the s, px and py atomic orbitals are combined in the (x,y)-plane to sp and sp2 orbitals. These sp and sp2 orbitals of the different atoms combine to molecular orbitals, building the molecular structure framework in the (x,y)-plane. The electrons in these molecular orbitals are called a-electrons and their wavefunctions are symmetric perpendicular to the (x,y)-plane extending only over two neighboring atoms. 

Orbital (Atomic or Molecular) A wavefunction which depends explicitly on the spatial coordinates of only one electron. 

The effect of relativity on the bonding in this molecule is so large that even the qualitative features of the bonding cannot be correctly described by non-relativistic theory which (i) fails to predict any 5d-6s hybridization in the localized bonding orbital (ii) seriously underestimates 5d-6s hybridization in one of the nonbonding orbitals (NBOs) (iii) predicts incorrectly that one of the orbitals with mj 1/2 is entirely tt in character and has pure spin and (iv) erroneously predicts that the bond (in AuH) is formed solely from the interaction between the gold 6s and the hydrogen Is atomic orbitals because the non-relativistic molecular orbital (MO) wavefunction constructed from these two atomic orbitals predicts the AuH molecule to be unbound by 0.19 eV. 

Consider an orbit-generating wavefunction, g [ x, ] such as the one given by Eq. (73), constructed from the atomic orbitals (.x,) which are products of radial functions times spherical harmonics. We assume that the radial functions are expanded as in Eq. (106) in terms of Slater type orbitals (Eq. (107)). The energy as a functional of the density and of the orbit-generating wavefunction is given by... 

R. S. Mulliken, / Chem. Phys., 23, 1833 (1955). Electronic Population Analysis on LCAO-MO [Linear Combination of Atomic Orbital-Molecular Orbital] Molecular Wavefunctions. 

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