Is atomic orbitals

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... 

Thus for H and He, the basis set consists of one orbital, a Is atomic orbital. For atomsLi to Ne the 2 inner-shell electrons are combined with the nucleus and the basis set consists of 4 orbitals, the 2s, 2p, ... 

Figure 7.30 (a) In-phase and (b) out-of-phase Is atomic orbitals on the hydrogen atoms of linear... 

The lowest energy molecular orbital of singlet methylene looks like a Is atomic orbital on carbon. The electrons occupying this orbital restrict their motion to the immediate region of the carbon nucleus and do not significantly affect bonding. Because of this restriction, and because the orbital s energy is very low (-11 au), this orbital is referred to as a core orbital and its electrons are referred to as core electrons. 

The wave functions nlm) for the hydrogen-like atom are often called atomic orbitals. It is customary to indicate the values 0, 1, 2, 3, 4, 5, 6, 7,. .. of the azimuthal quantum number / by the letters s, p, d, f, g, h, i, k,. .., respectively. Thus, the ground-state wave function 100) is called the Is atomic orbital, 200) is called the 2s orbital, 210), 211), and 21 —1) are called 2p orbitals, and so forth. The first four letters, standing for sharp, principal, diffuse, and... 

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... 

As the ethylene molecule contains a total of 16 electrons, there are but two that are available to occupy the n system. Two pairs of electrons are assumed to Ell the two Is atomic orbitals of the carbon atoms. Five pairs of electrons contribute to the a orbitals that represent single bonds in Fig. 5. Thus, the two... 

Figure 1.8 The overlapping of two hydrogen Is atomic orbitals with the same phase sign (indicated by their identical color) to form a bonding molecular orbital.

Figure 3.4 shows a more correctly scaled energy level diagram that results for the hydrogen molecule. Note that the energy for the Is atomic orbital of a hydrogen atom is at — 1312 kJ moT1 because the... 

For example, each H atom has a Is atomic orbital, but in the H2 molecule, the 0 s molecular orbital belongs to the entire H2 molecule. 

This shows (as found for the actual NBO) that hA remains practically a pure Is atomic orbital at all distances, with no appreciable tendency to hybridize. Thus, we incur practically no error by picturing the spin-polarized NBO as a simple normalized linear combination of Is AOs... 

Before introducing the form of the Mulliken operator, w u11, it is appropriate to return to the concepts of early days of quantum mechanics. Heitler and London wrote down the singlet wave function for H2 in terms of the hydrogen Is atomic orbitals on both hydrogen atoms A and B [4] ... 

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

Molecular orbitals are generated by combining atomic orbitals. The number of molecular orbitals formed is always equal to the number of atomic orbitals that combine. So, if two atomic orbitals combine, then two molecular orbitals will be formed. This is the case when two hydrogen Is atomic orbitals combine to produce two molecular orbitals in a hydrogen molecule (H ). 

For a hydrogen atom, the lowest energy solution of the wave equation describes a spherical region about the nucleus, a Is atomic orbital. When the wave equation is solved to provide the next higher energy level, we also get a spherical region of high probability, but this 2s orbital is further away from the nucleus than the Is orbital. It also contains a node, or point of zero probability within the sphere... 

The H7+ molecule-ion, which consists of two protons and one electron, represents an even simpler case of a covalent bond, in which only one electron is shared between the two nuclei. Even so, it represents a quantum mechanical three-body problem, which means that solutions of the wave equation must be obtained by iterative methods. The molecular orbitals derived from the combination of two Is atomic orbitals serve to describe the electronic configurations of the four species H2+, H2, He2+ and He2. 

The equations for the bonding and anti-bonding molecular orbital wave functions for the linear combinations of the two Is atomic orbitals of the two hydrogen atoms may now be written in the forms ... 

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