Molecular orbitals from atomic orbital wave functions

A FIGURE 9.A0 Molecular orbitals from atomic orbital wave functions. 

In setting up molecular orbitals from atoms whose wave functions have already been determined, the following principle has been found to be both general and useful If the electron clouds of iwo atoms are likely to overlap when the atoms approach, then the molecular wave function for the combination can he obtained by a linear combination of the individual wave functions. (A linear combination of two quantities is obtained by multiplying each by a constant and adding.) Stated mathematically ... 

We wish to compare the valence band density of states (DOS) of f.c.c. and h.c.p. metals with and without stacking faults. We therefore adopt a mixture of the f.c.c. and h.c.p. structures as a representative of the stacking fault structure of either of these structures. To calculate the DOS we summed up the squares of the coefficients of molecular orbital wave functions and convoluted the summed squares with the Gaussian of full width 0.5 eV at half maximum. For these DOS calculations we chose the metals Mg, Ti, Co, Cu and Zn. The model clusters employed here for both the f.c.c. and the h.c.p. structures were made of 13 atoms i.e., a central atom and 12 equidistant neighbor atoms. These structures are shown in Fig. 1. We reproduced the typical electronic structures in bulk materials by extracting the molecular orbitals localized only on the central atom from all the molecular orbitals which contributed - those localized on ligand atoms as well as on the central atom. To perform calculations we take the symmetry of the cluster as C3, and the number... 

A molecular orbital wave function for the bond between H and Cl in HCl, constructed, assuming that the bond is formed from the Is electron of H atom and a 3p electron of Cl atom ... 

Qa Qb are the charges on atoms A and B, obtained from the Mulliken population analysis [28] of the molecular orbital wave function R is the A-B bond length in A. The Vmid values for the C-NO2 bonds in nitroaromatic molecules are taken to be indicative of the relative instabilities of the respective bonds indeed eq. (2) is closely related to a bond energy expression that has been developed by Fliszar [29,30]. 

Edgecombe, K. E.,Boyd,R. J. (1987). Atomic orbital populations and atomic charges from self-consistent field molecular orbital wave functions. J. Chem. Soc., Faraday Trans. 

An approximate molecular orbital wave function for the LiH molecule was constructed using hybrid orbitals called sp hybrids, which are a linear combination of 5 and p atomic orbitals on the same nucleus. A bonding molecular orbital made from a 2sp hybrid on the Li nucleus and a U orbital on the H nucleus provides an adequate description of the bonding in LiH. 

Molecular orbital (MO) theory describes covalent bond formation as arising from a mathematical combination of atomic orbitals (wave functions) on different atoms to form molecular orbitals, so called because they belong to the entire molecule rather than to an individual atom. Just as an atomic orbital, whether un hybridized or hybridized, describes a region of space around an atom where an electron is likely to be found, so a molecular orbital describes a region of space in a molecule where electrons are most likely to be found. 

Molecular orbital (MO) theory (Section 1.11) A description of covalent bond formation as resulting from a mathematical combination of atomic orbitals (wave functions) to form molecular orbitals. 

Mulliken introduced the term "orbital" distinct from "orbital wave function" in 1932 in the second of fourteen papers carrying the general title, "Electronic Structures of Polyatomic Molecules and Valence." Mulliken defined atomic orbitals (AOs) and molecular orbitals (MOs) as something like the... 

Mulliken, Life, 90. On the "orbital," Mulliken wrote in 1932 "From here on, one-electron orbital wave functions will be referred to for brevity as orbitals. The method followed here will be to describe unshared electrons always in terms of atomic orbitals but to use molecular orbitals for shared electrons." In Robert Mulliken, "Electronic Structures of Polyatomic Molecules and Valence," Physical Review 41 (1932) 4971, on 50. 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

It is convenient to have a word for the space-dependent part of a wave function for one electron, to distinguish it from the whole wave function which includes reference to the spin. Orbital is the customary word. Such a function as the Is wave function, obtained by solving Schrodinger s equation for an atom, is called an atomic orbital. Such functions as the wave functions used in the last chapter for the hydrogen-molecule ion are called molecular orbitals. The foregoing method of approximation in a many-electron problem is therefore often called the atomic orbital method. 

The lower-energy MO of H2 concentrates electron density between Ihe two hydrogen nuclei and is called Ihe bonding molecular orbital. This sausageshaped MO results from summing the two atomic orbitals so that Ihe atomic orbital wave functions enhance each other in the bond region. Because an electron in this MO is strongly attracted to both nuclei, the electron is more stable... 

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