Phases in atomic and molecular orbitals

Phases in Atomic and Molecular Orbitals 363 The Ideal-Gas Equation 405 The Clausius—Clapeyron Equation 444 X-ray Diffraction 468 Hydrates 518... 

Phases in Atomic and Molecular Orbitals 379 The Ideal-Gas Equation 421 The Clausius-Clapeyron Equation 463 X-ray Diffraction 486... 

The boundary surfaces in Figure 3.11 are for the electron density, the probability of an electron being at any point in space. The electron density is given by the square of the wavefunction. Points with the same electron density will have the same numerical value for the wavefunction, but the wavefunction may be positive or negative. For example, if the probability of finding an electron at a particular point was one-quarter, 0.25, then the wavefunction at that point would have the value plus one half, + 0.5, or minus one half, -0.5, since both (0.5)2 and (-0.5)2 are equal to 0.25. The sign of the wavefunction gives its phase. To represent the wavefunction itself we can use the same contours as for electron density, but we also need to indicate the phase of the wavefunction. In atomic and molecular orbital representations, we shall use colour to show differences in phase. Is orbitals are all one phase and so are shown in one colour. 2p orbitals have two lobes, which are out... 

So far as AH+ is concerned, it would seem reasonable to suppose that the favoured pathway for a particular reaction would be that I one in which the greatest degree of residual bonding is maintained in the T.S. Maintenance of bonding implies maintenance of orbital overlap, and it is therefore necessary to establish the conditions that i ensure the maintenance of such overlap. To do this we have to ( consider a property of atomic and molecular orbitals not yet referred to, namely phase. 

The wave functions of atomic and molecular orbitals are used by chemists to understand many aspects of chemical bonding, spectroscopy, and reactivity. If you take a course in organic chemistry, you will probably see orbitals drawn to show the phases as in this box. 

Schematic pictures of atomic and molecular orbitals of the six states in the A and A" symmetry are drawn for the perpendicular approach in the Fig. 8(a) and (b). The orbitals in the figures are the three 2p orbitals of the F atom and the two s, 4

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2, but in somewhat different ways. Both assume that electron waves behave like more familiar waves, such as sound and light waves. One important property of waves is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other (in phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2.2). Recall from Section 1.1 that electron waves in atoms are characterized by then- wave function, which is the same as an orbital. For an electron in the most stable state of a hydrogen atom, for example, this state is defined by the I5 wave function and is often called the I5 orbital. The valence bond model bases the connection between two atoms on the overlap between half-filled orbitals of the two atoms. The molecular orbital model assembles a set of molecular- orbitals by combining the atomic orbitals of all of the atoms in the molecule. 

In some n molecular orbitals the phase at a given carbon atom may be zero and hence the p orbital in such a case is left unmarked. 

The EPR spectra of electrolytically produced anion radicals of Q -aminoanthraquin-ones were measured in DME and DMSO. The isotropic hyperfine coupling constants were assigned by comparison with the EPS spectra of dihydroxy-substituted antraquinones and molecular-orbital calculations. Isomerically pure phenylcarbene anion (PhCH ) has been generated in the gas phase by dissociative electron ionization of phenyldiazirine. PhCH has strong base and nucleophilic character. It abstracts an S atom from and OCS, an N atom from N2O, and an H atom from... 

Absorption of UV/VIS radiation in the solid state is different from UV/VIS absorption in the liquid or gaseous phase with respect to photophysical processes taking place in the crystal lattice and to the metallic, semiconductor (SC) or insulator properties of the absorbing solid (Bottcher, 1991). In crystals, multiple atomic or molecular orbitals are combined to form broad energy bands, i.e. a valence band (vb) fully occupied by electrons and a conduction band (cb) unoccupied or only partly occupied by electrons. Conduction bands and valence bands have different energetic positions relative to one another depending on the specific substrate. In a SC cluster, electronic transitions between the valence band and the conduction... 

The idea of distributed dipole moments has also been transferred to the dynamic domain and we shall discuss recent work from our laboratory in this section in more detail. With the help of maximally localized Wannier functions local dipoles and charges on atoms can be derived. The Wannier functions are obtained by Boys localization scheme [217]. Thus, Wannier orbitals [218] are the condensed phase analogs of localized molecular orbitals known from quantum chemistry. Access to the electronic structure during a CPMD simulation allows the calculation of electronic properties. Through an appropriate unitary transformation U of the canonical Kohn-Sham orbitals maximally localized Wannier functions (MLWFs)... 

Two atomic orbitals can combine to give two molecular orbitals - one bonding molecular orbital (lower in energy than the atomic orbitals) and one antibonding molecular orbital (higher in energy than the atomic orbitals) (see Section 1.4). Orbitals that combine in-phase form a bonding molecular orbital, and for the best orbital overlap, the orbitals should be of the same size. 

Let s review how the tt molecular orbitals of ethene are constructed. An MO description of ethene is shown in Figure 7.8. The twop orbitals can be either in-phase or out-of-phase. (The different phases are indicated by different colors.) Notice that the number of orbitals is conserved—the number of molecular orbitals equals the number of atomic orbitals that produced them. Thus, the two atomic p orbitals of ethene overlap to produce two molecular orbitals. Side-to-side overlap of in-phase p orbitals (lobes of the same color) produces a bonding molecular orbital designated i/ i (the Greek letter psi). The bonding molecular orbital is lower in energy than the p atomic orbitals, and it encompasses both carbons. In other words, each electron in the bonding molecular orbital spreads over both carbon atoms. 

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