The Hydrogen Orbitals

We saw earlier that the hydrogen atom can absorb energy to transfer the electron to a higher energy state (an excited state). 

In terms of the obsolete Bohr model, this meant the electron was transferred to an orbit with a larger radius. In the wave mechanical model, these higher energy states correspond to different kinds of orbitals with different shapes. 

The three-dimensional region in which there is a high prohahility of finding an electron in an atom 

The first four principal energy levels in the hydrogen atom. Each level is assigned an integer, n. 

An illustration of how principal levels can be divided into sublevels 

The probability map, or orbital, that describes the hydrogen electron in its lowest possible energy state. The more intense the color of a given dot, the more likely it is that the electron will be found at that point. We have no information about when the electron will be at a particular point or about how it moves. Note that the probability of the electron s presence is highest closest to the positive nucleus (located at the center of this diagram), as might be expected. 

Principal level 2 shown divided into the 2s and 2p sublevels. 

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The appropriate valence atomic orbitals which must be considered are 2s, 2pj., 2p, on carbon, and the Is orbitals lsH and Ish. The orbital 2py is clearly nonbonding (n) relative to the carbon-hydrogen interactions and need not be considered further. The hydrogen orbitals can be combined into a (lsH + Ish) combination of a symmetry and a (lsH — ls ) combination of 7r symmetry. (Fig. 5). Although there are three available basic cr orbitals, only two of these belong to the C1H2 group proper. We can first eliminate the ( out )... 

The hydrogen orbitals do not form a combination of symmetry type (iii) and so leave the oxygen orbital as nonbonding. 

Finally, note that no combination of ligand a orbitals interacts with members of the metal t2g set. The vanishing overlap between any ligand a orbital and, say, the dxy orbital is illustrated in Fig. 6-8. Overall, therefore, the metal t2g orbitals are nonbonding in this scheme. Recall how the 2p orbital of oxygen is similarly nonbonding to the hydrogen orbitals in water. 

This follows from the principle that bonds are formed only by overlap of orbitals of the same sign. Since this is a concerted reaction, the hydrogen orbital in the transition state... 

Although structures involving methyl groups bonded simultaneously to two carbon atoms by means of an overlap between the hydrogen orbitals and the />-orbitals of the carbon atoms may be readily enough assimilated, the state of structural theory is such that most of the cyclic intermediate or transition state structures are dubbed non-classical. In many cases they are best depicted by molecular orbitals, usually by diagramming the component atomic orbitals in the best position for overlap. Since maximum overlap of the component atomic orbitals imposes certain geometric requirements, pre-... 

The CHO radical is a o-radical and the main qualitative feature of the comparison between Tables 2 and 3 is the tendency for the a GHOs to be less contracted in CHO than is found in CH20. In particular, the sp2 hybrid on carbon which nominally contains the unpaired electron is considerably expanded (exponent 1.5923 compared to 1.8660 in CH20). The hydrogen orbital and the aoc orbital are also noticeably expanded while the lone pairs on oxygen are largely unaffected. 

In structure (a) the hydrogen orbital overlaps suprafacially with the terminal p orbitals of the n system while in structure (b) the overlap is antarafacially. Therefore the geometry of the two transition systems becomes different. While the suprafacial overlap has a plane of symmetry, the antarafacial migration has two fold axis. 

Each B-H-B three-center two-electron bridge bond corresponds to a filled three-center localized bonding orbital requiring the hydrogen orbital and one hybrid orbital from each boron atom. 

The linear molecule BeH, will serve as our first example of a triatomic species. The molecular orbitals for this molecule are constructed from the Is orbitals on the hydrogen atoms (labeled H and H ) and the 2j and one of the 2p orbitals of beryllium (the one directed along the H—Be—H bond axis). The remaining two Ip orbitals of beryllium cannot enter into the bonding because they are perpendicular to the molecular axis and thus have zero net overlap with the hydrogen orbitals. 

Figure 6-22 Generalized valence-bond orbitals calculated for ethene by the ab initio method. The nuclei are located in the x,y plane of the coordinate system at the positions indicated by crosses. The long dashes correspond to locations of change of phase. The dotted lines are contour lines of electron amplitude of opposite phase to the solid lines. Top shows both m-bonding carbon orbitals (almost sp2), middle-left is the carbon orbital and middle-right the hydrogen orbital of one of the C-H bonds, and bottom represents a side view of the ir orbitals in perpendicular section to the x,y plane. (Drawings furnished by Dr. W, A. Goddard, III.)...

In closing this section, a discussion of basis sets is in order. Almost all ab initio and DFT calculations require the use of a basis set, a set of functions in terms of which the molecular orbitals are constructed. In almost all cases, the functions chosen are atom-centered functions designed to mimic the shape of atomic orbitals. It is known that orbitals for many-electron atoms resemble the hydrogenic orbitals, and it is also observed that molecular orbitals can be expanded very efficiently in terms of atomic orbitals. One might think that a relatively small set of functions, essentially the optimized occupied atomic orbitals of the atoms making up the system, could be used... 

Furthermore, the wavefunction describing the bond direction for quantum numbers j and ni is identical to the angular portion of the hydrogen orbitals for quantum numbers / and in/. So the ground state j = m j = 0 is spherically symmetric, just like an s orbital the j = 1 states look like p orbitals, and so on. 

For hydrogenic orbitals with n — > 1, the most probable distance rmax may be defined as the distance of the principal maximum, namely the outermost and most prominent of the maxima of the > / function from the nucleus. The rmax values of the hydrogenic orbitals are listed in the fifth column of Table 2.1.5. It is seen that the size ratios for the Is, 2p, 3d, and 4f orbitals, which satisfy the relation = n — 1, are exactly the same as those of the Bohr orbits, which vary as n2. Furthermore, for orbitals in the same shell, rmax decreases as t increases, as in the case of r. The data in Table 2.1.5 show that orbital size ratios based on 7 are smaller than those based on rmax. 

Although ethylene has similar 0 orbitals, these are not only appreciably different in energy from the n orbital of ethylene, but the overlap between the hydrogen orbitals and the ethylene n orbitals is smaller upon bending both effects cause much less mixing of these orbitals upon deplanarization of ethylene. 

Conceptually, the STO basis is straightforward as it mimics the exact solution for the single electron atom. The exact orbitals for carbon, for example, are not hydrogenic orbitals, but are similar to the hydrogenic orbitals. Unfortunately, with STOs, many of the integrals that need to be evaluated to construct the Fock matrix can only be solved using an infinite series. Truncation of this infinite series results in errors, which can be significant. 

The hydrogenic orbitals (4.24) are linear combinations of Slater-type orbitals /,/(r) whose normalised form is parametrised in terms of positive integers n,/ and exponents Ci/- It is... 

These orbitals are often used as the basis for a more-general problem, since only a relatively-small number of them are needed to represent the low-lying states well and the integrals (4.37) are analytic, making the computation of the Hamiltonian matrix elements fast. The hydrogenic orbitals themselves constitute a better basis since they are orthonormal. 

The s and orbitals of nitrogen both have Aj symmetry, and the pair p, Py has E symmetry, exactly the same as the representations of the hydrogen I5 orbitals. Therefore, all orbitals of nitrogen are capable of combining with the hydrogen orbitals. As in water, the orbitals are grouped by symmetry and then combined. 

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