Molecular orbitals atomic orbital representation

Fig. A. Molecular-orbital representation of the 1-centrc F-Xe-F bond, (a) The possible combinations of colinear p, atomic orbitals, and (b) the energies of the resulting MOs (schematic).

Canonical transformations from the tight-binding (atomic orbitals) representation to the eigenstate (molecular orbitals) representation play an important role, and we consider it in detail. Assume, that we find two unitary matrices SR and SR, such that the Hamiltonians of the left part Hi and of the right part Hi can be diagonalized by the canonical transformations... 

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... 

As noted in [4] and [5], and in the discussion above, the electronic structure of Ti " " has been addressed for both Ti20s [14], and the hydrated ion Ti(H20)6 complex [11]. However it is important understand these two applications are different. The d description applies exactly to each hydrated Ti ion. In contrast, the d designation for the Ti-atoms in Ti203 is based on a SALC molecular orbital representation of the Ti203 electronic structure, in particular on overall and local charge neutrality. 

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... 

MOLECULAR ORBITAL and valence bond calculations of the IT—electron energies of unsaturated molecules customarily start with models in which appropriate atomic orbitals are assigned to each nucleus to provide a framework for -notions of the binding electrons. Atomic orbital representations of organic molecules are now very commonly used in the teaching of elementary organic chemistry, although there often seems to be confusion between atomic orbital and molecular orbital representations. 

At its essence, a conjugated system involves at least one atom with ap orbital adjacent to at least one tt bond. The adjacent atom with the p orbital can be part of another ir bond, as in 1,3-butadiene, or a radical, cationic, or anionic reaction intermediate. If an example derives specifically from a propenyl group, the common name for this group is allyl. In general when we are considering a radical, cation, or anion that is adjacent to one or more TT bonds in a molecule other than propene, the adjacent position is called allylic. Below we show the formula for butadiene, resonance hybrids for the allyl radical and an allylic carbocation, and molecular orbital representations for each one. 

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

The molecular orbitals representation (2.110), following the orbital phases (+ or - in front of the p atomic orbitals from the primordial bases), is shown in Figure 2.24. 

In addition to the specific changes noted above, we have also changed much of the artwork throughout the textbook. In particular, all of the atomic and molecular orbital representations have been modified to be consistent across all chapters. We have redone all of the electrostatic potential maps (EPMs) to have the same potential energy color scale unless noted in the textbook. 

Boranes are typical species with electron-deficient bonds, where a chemical bond has more centers than electrons. The smallest molecule showing this property is diborane. Each of the two B-H-B bonds (shown in Figure 2-60a) contains only two electrons, while the molecular orbital extends over three atoms. A correct representation has to represent the delocalization of the two electrons over three atom centers as shown in Figure 2-60b. Figure 2-60c shows another type of electron-deficient bond. In boron cage compounds, boron-boron bonds share their electron pair with the unoccupied atom orbital of a third boron atom [86]. These types of bonds cannot be accommodated in a single VB model of two-electron/ two-centered bonds. 

Ferrocene (Figure 2-61a) has already been mentioned as a prime example of multi-haptic bonds, i.c, the electrons tlrat coordinate tire cyclopcntadicnyl rings with the iron atom are contained in a molecular orbital delocalized over all 11 atom centers [811, for w hich representation by a connection table having bonds between the iron atom and the five carbon atoms of cither cyclopcntadicnyl ring is totally inadequate. 

Drawing-, text-, and structure-input tools are provided that enable easy generation of flow charts, textual annotations or labels, structures, or reaction schemes. It is also possible to select different representation styles for bond types, ring sizes, molecular orbitals, and reaction arrows. The structure diagrams can be verified according to free valences or atom labels. Properties such as molecular... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

Fig. 7. Graphical representations of (a) the Highest Occupied Molecular Orbital (HOMO) and (b) the Lowest Unoccupied Molecular Orbital (LUMO) for ranitidine. It is possible, in the ordinarily visible color-coded data not shown here, to distinguish the strong localization (a) of the HOMO to the sulfur atom and adjacent nitroethyleneamine fragment and the contrasting localization (b) of the LUMO to the nitroethylenearnine fragment. Neither the LUMO not HOMO appear to have contributions from the dimethylaminomethyl-suhstitiited furan.

Fig. 1.17. Representation of the molecular orbitals of carbon monoxide. Energies are given in atomic units (1 a.u. 27.21 eV). (From W L. Jorgensen and L. Salem, 77i Organic Chemist s Book of Orbitals, Academic Press, New York, 1973. Reproduced wifli permission.)...

Now let us investigate the oxygen molecule, which experiment tells us has the molecular formula O2. We might begin by considering formation of a single bond between two oxygen atoms, as represented by the orbital representation... 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

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