Wavefunction, hydrogen molecule

Calculating the Energy from the Wavefunction the Hydrogen Molecule... 

Ale now substitute the hydrogen molecule wavefunction into Equation (2.73) to provide the ollowing ... 

In our hydrogen molecule calculation in Section 2.4.1 the molecular orbitals were provided as input, but in most electronic structure calculations we are usually trying to calculate the molecular orbitals. How do we go about this We must remember that for many-body problems there is no correct solution we therefore require some means to decide whether one proposed wavefunction is better than another. Fortunately, the variation theorem provides us with a mechanism for answering this question. The theorem states that the... 

One widely used valence bond theory is the generalised valence bond (GVB) method of Goddard and co-workers [Bobrowicz and Goddard 1977]. In the simple Heitler-London treatment of the hydrogen molecule the two orbitals are the non-orthogonal atomic orbitals on the two hydrogen atoms. In the GVB theory the analogous wavefunction is written ... 

Imagine a model hydrogen molecule with non-interacting electrons, such that their Coulomb repulsion is zero. Each electron in our model still has kinetic energy and is still attracted to both nuclei, but the electron motions are completely independent of each other because the electron-electron interaction term is zero. We would, therefore, expect that the electronic wavefunction for the pair of electrons would be a product of the wavefunctions for two independent electrons in H2+ (Figure 4.1), which I will write X(rO and F(r2). Thus X(ri) and T(r2) are molecular orbitals which describe independently the two electrons in our non-interacting electron model. 

In the case of the hydrogen molecule-ion H2" ", we defined certain integrals Saa, Taa, Tab, Labra- The electronic part of the energy appropriate to the Heitler-London (singlet) ground-state wavefunction, after doing the integrations... 

In Chapter 3, I showed you how to write a simple LCAO wavefunction for the electronic ground state of the hydrogen molecule-ion, H2 ... 

In order to calculate the total probability (which comes to 1), we have to integrate over both space dr and spin ds. In the case of the hydrogen molecule-ion, we would write LCAO wavefunctions... 

The low-lying excited states of the hydrogen molecule conhned in the harmonic potential were studied using the configuration interaction method and large basis sets. Axially symmetric harmonic oscillator potentials were used. The effect of the confinement on the geometry and spectroscopic constants was analyzed. Detailed analysis of the effect of confinement on the composition of the wavefunction was performed. 

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... 

Table 3.2.1 summarizes the results of various approximate wavefunctions for the hydrogen molecule. This list is by no means complete, but it does show that, as the level of sophistication of the trial function increases, the calculated dissociation energy and bond distance approach closer to the experimental values. In 1968, W. Kolos and L. Wolniewicz used a 100-term function to obtain results essentially identical to the experimental data. So the variational treatment of the hydrogen molecule is now a closed topic. 

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. 

Nuclear Spin Effects on Rotation. There is an interesting effect on the rotational partition function, even for the hydrogen molecule, due to nuclear spin statistics. The Fermi postulate mandates that the overall wavefunction (including all sources of spin) be antisymmetric to all two-particle interchanges. A simple molecule like (1H1)2, made of two electrons (S = 1/2) and two protons (spin 7=1/2), will have two kinds of molecule ... 

To introduce the concepts, we must of course start from the classic 1927 paper of Heitler and London [1], referred to by Pauling himself [2] as the greatest single contribution to the clarification of the chemist s conception of valence. .. since G.N.Lewis s suggestion that the chemical bond between two atoms consists of a pair of electrons held jointly by the two . The electron-pair bond in the hydrogen molecule was described using a wavefunction of the form... 

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... 

When a population analysis is performed for the various wavefunctions used above, for the hydrogen molecule, it is found that the overlap populations are always positive (indicating a flow of electron density into the overlap region between the nuclei) but that the predicted enhancement of the density in the bond region depends on which function is used for example, the MO and VB functions give bond populations... 

In 1927, Heitler and London carried out a calculation for the hydrogen molecule using what has become known as valence bond theory.12 Each electron of the pair could be assigned to nuclei corresponding to wavefunctions of the type... 

In the first, electron 1 is in the Is atomic orbital < >a and electron 2 is in < >b. In the second, the electrons are reversed. The two wave function when considered separately lead to the same energy. If we wish to form the hydrogen molecule, it is necessary to take linear combinations of the above, and the two new wavefunctions become (neglecting the normalization constant) ... 

The basic idea of the Heitler-London model for the hydrogen molecule can be extended to chemical bonds between any two atoms. The orbital function (10.8) must be associated with the singlet spin function cro,o(l > 2) in order that the overall wavefunction be antisymmetric [cf. Eq (8.14)]. This is a quantum-mechanical realization of the concept of an electron-pair bond, first proposed by G. N. Lewis in 1916. It is also now explained why the electron spins must be paired, i.e., antiparallel. It is also permissible to combine an antisymmetric orbital function with a triplet spin function, but this will, in most cases, give a repulsive state, such as the one shown in red in Fig. 10.2. 

The first quantum-mechanical account of chemical bonding is due to Heitler and London in 1927, only one year after the Schrddinger equation was proposed. They reasoned that since the hydrogen molecule H2 was formed from a combination of hydrogen atoms A and B, a first approximation to its electronic wavefunction might be... 

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