Bond wave functions

In order to calculate the orbitals for a methane molecule, the four Lv functions of the four hydrogen atoms and the functions 2s, 2px, 2py and 2pz of the carbon atom are combined to give eight wave functions, four of which are bonding and four of which are antibonding. The four bonding wave functions are ... 

Using these relations we can construct the bond wave functions (2) and express the matching conditions (1) in terms of the vectors (f)1,11 only. The resulting quantization conditions can be expressed in terms of the matrix,... 

The equations for the bonding and anti-bonding wave functions used in Figure 3.3 are ... 

When the ionic terms are neglected, one gets the so-called valence-bond (VB) description. The (unnormalized) valence-bond wave function can also be written in the form... 

P. C. Hiberty, C. Leforestier, J. Am. Chem. Soc. 100, 2012 (1978). Expansion of Molecular Orbital Wave Functions into Valence Bond Wave Functions. A Simplified Procedure. 

WRITING AND REPRESENTING VALENCE BOND WAVE FUNCTIONS... 

For convenience and to avoid confusion, we will symbolize a purely covalent bond between A and B centers as A — B, while the notation A—B will be employed for a composite bond wave function like the one displayed in Equation 3.4. In other words, A—B refers to the real bond while A — B designates its covalent component. 

Valence Bond Wave Functions with Semilocalized AOs... 

Writing Valence Bond Wave Functions Beyond the 2e/2c Case... 

Denoting the tt AOs of the C1-C4 carbons by a,b,c, and d, respectively, the fully covalent VB wave function for the tt system of butadiene displays two singlet couplings one between a and b, and one between c and d. It follows that the wave function can be expressed in the form of Equation 3.8, as a product of the bond wave functions. 

The product of bond wave functions in Equation 3.8, involves so-called perfect pairing, whereby we take the Lewis structure of the molecule, represent each bond by a HL bond, and finally express the full wave function as a product of all these pair-bond wave functions. As a rule, such a perfectly paired polyelectronic VB wave function having n bond pairs will be described by 2" determinants, displaying all the possible 2x2 spin permutations between the orbitals that are singlet coupled. 

Pictorial Representation of Valence Bond Wave Functions by Bond Diagrams... 

The Relationship between Molecular Orbital and Valence Bond Wave Functions... 

A.1 NORMALIZATION CONSTANTS, ENERGIES, OVERLAPS, AND MATRIX ELEMENTS OF VALENCE BOND WAVE FUNCTIONS... 

G. A. Gallup, J. M. Norbeck, Chem. Phys. Lett. 21, 495 (1973). Population Analyses of Valence-Bond Wave Functions and BeH2. 

MAPPING A MOLECULAR ORBITAL CONFIGURATION INTERACTION WAVE FUNCTION INTO A VALENCE BOND WAVE FUNCTION... 

The wave functions fMo and vb can be compared with each other, by a simple expansion of Pvb in terms of elementary determinants as we did in Chapter 3 for the covalent and ionic parts of a bond wave function ... 

Each one of these wave functions is obtained, as explained in Chapter 3, as a product of the corresponding bond wave functions. Thus, in R the bonds are a-b and c-d, while in P these are b-c and a-d. 

Thus, as we did for the allyl radical case, here too the bonding characteristics of the two covalent states of benzene can be deduced from the respective wave functions. As discussed in Chapter 5, each Kekule structure can be generated by a product of the corresponding bond wave functions, each having a spin factor a(l)(3(2)- 3(l)a(2). Each Kekule structure possesses the two spin alternant QC determinants, which are related to each other by a cyclic permutation of the spins over the ring, and are shown in Fig. 7.4d. To illustrate clearly the building blocks of the two Kekule structures, we express them as a sum of all the permutations of the QC determinants, as follows ... 

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