Hydrogen molecule valence bond method

Unfortunately, the Schrodinger equation can be solved exactly only for one-electron systems such as the hydrogen atom. If it could be solved exactly for molecules containing two or more electrons,3 we would have a precise picture of the shape of the orbitals available to each electron (especially for the important ground state) and the energy for each orbital. Since exact solutions are not available, drastic approximations must be made. There are two chief general methods of approximation the molecular-orbital method and the valence-bond method. 

The second method of discussing the electronic structure of molecules, usually called the valence-bond method, involves the use of a wave function of such a nature that the two electrons of the electron-pair bond between, two atoms tend to remain on the two different atoms. The prototype of this method is the Heitler-London treatment of the hydrogen a olecule, which we shall now discuss. 

If we prefer to describe the bonding of a polyatomic molecule using localized two-center, two-electron (2c-2e) bonds, we can turn to the hybridization theory, which is an integral part of the valence bond method. In this model, for AX systems, we linearly combine the atomic orbitals on atom A in such a way that the resultant combinations (called hybrid orbitals) point toward the X atoms. For our BeH2 molecule in hand, two equivalent, colinear hybrid orbitals are constructed from the 2s and 2pz orbitals on Be, which can overlap with the two Is hydrogen orbitals to form two Be-H single bonds. (The 2p and 2py... 

In 1927, Burrau calculated the energy of Hj and Heitler and London treated the hydrogen molecule. In 1928, the Heitler-London or valence bond method was applied to many electron systems, and simultaneously Hund and Mulliken started the development of the molecular orbital theory. In 1931, Slater expressed the v/avefunctions of complex molecules in terms of Slater determinants made up of linear combinations of atomic orbitals. Thus, the Golden Age was born. 

The ground-state wave function, and the energy eigenvalue, g, of this system are expressed by Eq. (1.149) in a way analogous to the valence bond method described for the hydrogen molecule, where k is the force constant of the nucleus motion. 

The first quantum-mechanical treatment of the hydrogen molecule was by Heitler and London in 1927. Their ideas have been extended to give a general theory of chemical bonding, known as the valence-bond (VB) theory. The valence-bond method is more closely related to the chemist s idea of molecules as consisting of atoms held together by localized bonds than is the molecular-orbital method. The VB method views molecules as composed of atomic cores (nuclei plus inner-shell electrons) and bonding valence electrons. For H2, both electrons are valence electrons. 

D11.1 Our comparison of the two theories will focus on the manner of construction of the trial wavefunctions for the hydrogen molecule in the simplest versions of both theories. In the valence bond method, the trial function is a linear combination of two simple product wavefunctions, in which one electron resides totally in an atomic orbital on atom A. and the other totally in an orbital on atom B. See eqns I l.l and 11.2, as well as Fig. 11.2. There is no contribution to the wavefunction from products in which both electrons reside on either atom A or B. So the valence bond approach undervalues, by totally neglecting, any ionic contribution to the trial function. It is a totally covalent function. The molecular oibital function for the hydrogen molecule is a product of two functions of the form of eqn 11.8, one for each electron, that is. 

Now we want to write a wave equation that will mathematically describe the electron distribution in the hydrogen molecule. The valence bond method initially used by Heitler and London described one possible wave function as °... 

The generalised valence bond (GVB) method, developed by Goddard in 1970, is one of the simplest and oldest valence bond methods that use flexible orbitals in a general way. The generalised Coulson-Fischer theory for the hydrogen molecule mentioned above is used to describe every electron pair in a molecule. The orbitals for each electron pair are expanded in terms of the full basis set and are non-orthogonal. Orbitals from different pairs are forced to be orthogonal. This condition simplifies the calculations but may lead to some difficulties [160,161],... 

To obtain an additional comparison between the simple LCAOMO method and the valence-bond method for the hydrogen molecule, we obtain the one-electron probability... 

One powerful approach to the problem of bond breaking is the valence bond method. To understand the basic idea of the valence bond picture, let us consider the electronic structure of the ground, singlet state of the hydrogen molecule. The form of the valence bond (or Heitler-London) wavefunction is... 

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 ... 

We will use the valence bond approach extensively m our discussion of organic molecules and expand on it shortly First though let s introduce the molecular orbital method to see how it uses the Is orbitals of two hydrogen atoms to generate the orbitals of an H2 molecule... 

To calculate the molecular orbitals of the hydrogen molecule, the orbital equations of the two atoms are combined. When the orbital equations are added together, the result is a bonding molecular orbital that extends over both atoms. Subtracting the orbital equations of the atoms produces an antibonding molecular orbital. This process is called the linear combination of atomic orbitals or LCAO. It gives a more sophisticated and accurate approximation of how electrons behave in a molecule than the valence bond approach. It is also more difficult to use, so chemists choose the method that is adequate for their particular purpose. 

The valence bond (VB) theory grew directly out of the ideas of electron pairing by Lewis and others. In 1927 W. Heiller and F. London proposed a quantum-mechanical treatment of the hydrogen molecule. Their method has come to be known as the valence bond approach and was developed extensively by men such as Linus Pauling... 

Clearly, the most satisfactory way to decide between conflicting concepts of the structure and nature of the hydrogen bond is to treat quantum-mechani-cally a hydrogen-bonded complex as a single large molecule entity with no truncation and to compare the results obtained for this supermolecule to those obtained for the separated molecules treated in the same approximation. This mode of approach is now possible, and a number of such computations using both all-valence electrons methods and the SCF MO non-empirical procedure have recently appeared. The references pertinent to biochemistry have been listed in Tables I and II. These concern only various hydrogen-bonded amides and the base pairs of the nucleic acids. 

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