Hydrogen atom orbital normalization

Mulliken proposed a different trial function than that used in the valence bond approach. In general, this function is a normalized sum of hydrogen-atom orbitals and is written in the form... 

Fig. 39. Atomic orbitals in hydrogen fluoride. The fluorine 2p and 2p atomic orbitals, normal to the axis, are omitted for clarity.

Table 1.1 The radial and angular components of the hydrogenic atomic orbitals with distinct normalization constants for the radial and angular functions. The parameter, p = extends the application of the functions in the table entries for non-hydrogen one-electron atomic species. Remember that the solutions to the angular equation in are exp(-f / — im0) and the real forms given are obtained by taking the sums and differences of the expansions of the complex exponentials and then applying equations 1.1 to 1.3 to these results. The column headed -I-/- indicates the particular choices of sum when relevant. ...

The normalization integral for the hydrogen atom orbitals can be factored in spherical polar coordinates ... 

The valence theory (4) includes both types of three-center bonds shown as well as normal two-center, B—B and B—H, bonds. For example, one resonance stmcture of pentaborane(9) is given in projection in Figure 6. An octet of electrons about each boron atom is attained only if three-center bonds are used in addition to two-center bonds. In many cases involving boron hydrides the valence stmcture can be deduced. First, the total number of orbitals and valence electrons available for bonding are determined. Next, the B—H and B—H—B bonds are accounted for. Finally, the remaining orbitals and valence electrons are used in framework bonding. Alternative placements of hydrogen atoms require different valence stmctures. 

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

Excited states of the hydrogen molecule may be formed from a normal hydrogen atom and a hydrogen atom in various excited states.2 For these the interelectronic interaction will be small, and the Burrau eigenfunction will represent the molecule in part with considerable accuracy. The properties of the molecule, in particular the equilibrium distance, should then approximate those of the molecule-ion for the molecule will be essentially a molecule-ion with an added electron in an outer orbit. This is observed in general the equilibrium distances for all known excited states but one (the second state in table 1) deviate by less than 10 per cent from that for the molecule-ion. It is hence probable that states 3,4, 5, and 6 are formed from a normal and an excited atom with n = 2, and that higher states are similarly formed. 

Using the simplest picture (and neglecting the effect of overlap on the normalization), this doubly occupied og spatial molecular orbital can be thought of as being the symmetric linear combination of the two Is atomic orbitals on the left and right hydrogens, HL and Hr... 

Wave functions for the orbitals of molecules are calculated by linear combinations of all wave functions of all atoms involved. The total number of orbitals remains unaltered, i.e. the total number of contributing atomic orbitals must be equal to the number of molecular orbitals. Furthermore, certain conditions have to be obeyed in the calculation these include linear independence of the molecular orbital functions and normalization. In the following we will designate wave functions of atoms by % and wave functions of molecules by y/. We obtain the wave functions of an H2 molecule by linear combination of the Is functions X and of the two hydrogen atoms ... 

Toropov A, Toropova A (2004) Nearest neighboring code and hydrogen bond index in labeled hydrogen-filled graph and graph of atomic orbitals Application to model of normal boiling points of haloalkanes. J. Mol. Struct. (Theochem) 711 173-183. 

The formation of bonding molecular orbitals by an overlap of atomic orbitals applies not only to the Is orbitals of hydrogen, but also to other atomic orbitals. When the atomic orbitals overlap along the axis of the bond, a covalent bond, called a sigma (a) bond, results. This is normally referred to as end-on overlap. Some examples of the formation of a bonds from overlapping atomic orbitals are shown in the diagrams. 

Let the atoms in the chain be numbered 0, 1,. . . , iV, and let the foreign atom be denoted by X (Fig. 1). Associated with each atom we introduce an atomic orbital < (r, m). These orbitals are divided into two sets. One set (m = X) contains only one member, which is the orbital on the foreign atom the other set (m = 0, 1,. . . , A) consists of the orbitals on the crystal atoms. Thus, we have the problem of the interaction of a hydrogen-like atom with a crystal whose normal electronic structure consists of just one band of states. 

Problem 8-15. A normalized hydrogenic Is orbital for a one-electron atom or ion of nuclear charge Z has the form ... 

To get some idea of the use of trial wave functions and the variation principle, evaluate the expectation value of the energy using the Hydrogen atom Hamiltonian, and normalized Is orbitals with variable Z. That is, evaluate ... 

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