Hydrogen orbitals representations

It is seen from their orbital structures that hydrogen and fluorine both need to share 1 electron to complete their outer shells. Therefore the orbital representation of HF molecule is ... 

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

Hyperspherical harmonics are now explicitly considered as expansion basis sets for atomic and molecular orbitals. In this treatment the key role is played by a generalization of the famous Fock projection [5] for hydrogen atom in momentum space, leading to the connection between hydrogenic orbitals and four-dimensional harmonics, as we have seen in the previous section. It is well known that the hyperspherical harmonics are a basis for the irreducible representations of the rotational group on the four-dimensional hypersphere from this viewpoint hydrogenoid orbitals can be looked at as representations of the four-dimensional hyperspherical symmetry [14]. 

The simplest possible atomic orbital representation is termed a minimal basis set. This comprises only those functions required to accommodate all of the electrons of the atom, while still maintaining its overall spherical symmetry. In practice, this involves a single (Is) function for hydrogen and helium, a set of five functions (Is, 2s, 2px, 2py, 2pz) for lithium to neon and a set of nine functions (Is, 2s, 2px,... 

Fig. 9 may be viewed, also, as a localized molecular orbital representation of, e.g., a hydrocarbon (cf. Fig. 13, ref. 7). Thus, replacement of (i) the domains of the Si4+ cations (the atomic cores of silicon atoms) by the domains of C4+ cations (the atomic cores of carbon atoms r = 0.15 A 2>), (ii) the domains of the bridging (i.e., bonding) oxide ions by the domains of the electron-pairs of aliphatic carbon-carbon single bonds (r 0.6e A 40)), and (iii) the domains of the non-bridging oxide ions by the domains of the protonated electron-pairs of carbon-hydrogen bonds... 

The numbers in the table, the characters, detail the effect of the symmetry operation at the top of the colurrm on each representation labelled at the front of the row. The mirror plane that contains the H2O molecule, a (xz), leaves an orbital of bi symmetry unchanged while a Ci operation on the same basis changes the sign of the wavefimction (orbital representations are always written in the lower case). An orbital is said to span an irreducible representation when its response upon operation by each symmetry element reproduces the same characters in the row for that irreducible representation. For atoms that fall on the central point of the point group, the character table lists the atomic orbital subscripts (e.g. x, y, z as p , Pj, p ) at the end of the row of the irreducible representation that the orbital spans. A central s orbital always spans the totally synunetric representation (aU characters = 1). For the central oxygen atom in H2O, the 2s orbital spans ai and the 2px, 2py, and 2p span the bi, b2, and ai representations, respectively (see (25)). If two or more atoms are synunetry equivalent such as the H atoms in H2O, the orbitals must be combined to form symmetry adapted hnear combinations (SALCs) before mixing with fimctions from other atoms. A handy mathematical tool, the projection operator, derives the functions that form the SALCs for the hydrogen atoms. 

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. 

The chemical shifts (2.4 ppm) of acetylenic hydrogens (RCHC-H) are considerably more toward higher magnetic fields than those of alkene hydrogens (4.6 to 6.9 ppm). Show how this shielding effect might be explained in terms of the atomic orbital representation of acetylenes. 

Fig. 4.23. Various graphical representations of the hydrogen orbital (coordinates in panels a-c are expressed in a.u.).

Draw a three-dimensional orbital representation for each of the following molecules, indicate whether each bond in it is a cr or tt bond, and provide the hybridization for each non-hydrogen atom. 

The molecular orbital representation of covalent bond formation between two hydrogen atoms. 

Earlier, we eonsidered in some detail how the three Ish orbitals on the hydrogen atoms transform. Repeating this analysis using the short-eut rule just deseribed, the traees (eharaeters) of the 3 x 3 representation matriees are eomputed by allowing E, 2C3, and... 

Orbital Surfaces. Molecular orbitals provide important clues about chemical reactivity, but before we can use this information we first need to understand what molecular orbitals look like. The following figure shows two representations, a drawing and a computer-generated picture, of a relatively high-energy, unoccupied molecular orbital of hydrogen molecule, H2. 

In this representation there is no need to consider the next higher energy level cluster—the 2s, 2p orbitals. For hydrogen and helium these are much higher in energy and can give rise only to extremely weak attractions. 

In either representation, (22) or (23), we see that there is residual bonding capacity remaining in the species OH. In (22) the third 2p orbital has a single electron but a capacity for two. This means more bonding can occur. In (23) a census of the electrons near the oxygen atom indicates there are only seven. The oxygen atom would be more stable if it could add one more electron. With either representation, we conclude that OH should be able to react with another hydrogen atom. See representations (24), (25). 

In NH and NFS, three p orbitals are involved in the bonding [see representation (30)]. Figure 16-10 shows the spatial arrangement implied by assuming persistence of the hydrogen atom orbitals after bonding. We expect, then, that ant-... 

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