Hydrogen molecule spin orbitals

The original VB wave function was introduced in the treatment of the hydrogen molecule by Heitler and London in 1932. This treatment considered only the one Is orbital on each hydrogen atom and assumed that the best wave function for a system of two electrons on two different atoms is a product of the two atomic Is orbitals i/ — XisXis- This wave function needs to be modified, however, to accommodate the antisymmetry of the wave function and to take into account the spin of the two electrons. 

Up to this point, we have considered the nonrelativistic Schrodinger equation. However, to calculate AEs to an accuracy of a few kJ/mol, it is necessary to account for relativistic effects, even for molecules containing only hydrogen and first-row atoms. Fortunately, the major relativistic contributions to the AEs of such molecules - the mass-velocity (MV), one-electron Darwin (ID), and first-order spin-orbit (SO) terms - are easily obtained [58]. 

The GUGA-Cl wavefunctions are spatial and spin symmetry-adapted, thus the projections of total orbital angular momentum and total spin of a hydrogen molecule in a particular electronic state are conserved for all the values of R. Therefore, the term remains constant for an electronic state, and it causes a... 

For a spin restricted two-electron system such as the hydrogen molecule the occupied orbitals can explicitly be expressed in the electron density as Kohn-Sham equations then yields an... 

The spin of the radical is characterized by two spin quantum numbers, the total spin S and the component of the total spin along the z-axis M. The simplest type of radical has one unpaired electron, and hence S = and M = , where the sign of M indicates the orientation of the electron spin in the z-direction. The dissociated singlet molecule, described by the N + l)-electron wavefunction, consists of the radical and a hydrogen atom in orbital (j) at infinity, ... 

The formal results of the quantum-mechanical treatment of valence, developed by Heitler, London, Born, Weyl, Slater, and other investigators, can be given the following simple statement an atom can form an electron-pair bond for each stabh orbital, the bond being of the type described for the hydrogen molecule and owing its stability to the same resonance phenomenon. In other words, for the formation of an eleo-tron-pair bond two electrons with opposed spins and a stable orbital of oach of the two bonded atoms are needed. 

The symbol for the normal hydrogen molecule is and means the following (/) there is no net orbital angular momentum around the axis of the molecule (/ /) the two electron spins are paired (as they have to be if the Pauli Exclusion Principle is not to be violated) (iii) the sum of the f s is an even number (zero in this case) and leads to the subscript g, which means that the wave function does not change sign by inversion through the center of symmetry (otherwise the symbol u would be used). 

According to Heitler and London [1], the electronic states of the hydrogen molecule can be formed by distributing the two electrons between the atomic orbitals (AOs), Ls and 1 sb (abbreviated by a and b), on the two hydrogen atoms. The singlet wave function characterized by quantum numbers S=0 and M=0 (M denotes the z-component of total spin S) can be described using the form... 

Corrections for Improper HF Asymptotic Behaviour.—There are two techniques which may be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single determinants] approximation goes to the incorrect asymptotic limit. These techniques are (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one) 137 and (ii) to employ a single determinantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3P and 1Z> states with a hydrogen molecule,138 and we illustrate them through a discussion of this work. 

It will be recalled that the approach of molecular orbital (MO) theory starts, on the other hand, from an independent-particle model (IPM) in which both electrons occupy the same bonding MO , 1 = Xa + Xb, similar to the one used [4] for the hydrogen molecule ion, Hj. The bonding MO is in fact the approximate wavefunction for a single electron in the field of the two nuclei and allocating two electrons to this same MO, with opposite spins, yields the 2-electron wavefunction... 

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