Molecular orbitals, also

Hartree-FockWavefunction. The simplest quantum-mechanically correct representation of the many-electron wavefimction. Electrons are treated as independent particles and are assigned in pairs to functions termed Molecular Orbitals. Also known as Single-Determinant Wavefunction. 

The number of basis functions (defined by the chosen basis sets) used to construct the molecular orbitals also strongly affects the effort/accuracy ratio. The use of minimal basis sets yielded wrong results (56), whereas reasonable agreement with experiment is obtained when double zeta plus polarization basis sets are applied. Correlated methods require larger basis sets to include as much electron correlation as possible. This implies that in addition to the increased computational demand of such methods, a further increase of the computational cost results due to the requirement of using larger basis sets. 

At present there is consensus on the fact that the observed nephelauxetic effect in the spectra of lanthanide compounds is analogous to the phenomenon observed in the spectra of d-transition metal complexes. The nephelauxetic effect cannot be quantitatively interpreted by excluding the covalent interaction of lanthanide ions with surrounding ligands [34]. Jorgensen has proposed [38] two possible mechanisms of interaction for the observed nephelauxetic effect, namely (i) direct participation of lanthanide 4f orbitals in the formation of molecular orbitals also known as symmetry restricted covalency , (ii) transfer of some part of the ligand electron density to the unfilled 6s and 6p orbitals of the lanthanide also known as central field covalency . 

However the orientations of the atomic orbitals in space do not fit the directions predicted by the VSEPR theory according to Table 2- 1 on page 70. For this reason other orbitals than the atomic orbitals must be present in the molecules and composite ions in order to give the right bond directions according to the VSEPR theory. These orbitals are a type of molecular orbitals (also mentioned in section 2.2.2 Molecular orbital theory) which are called hybrid orbitals. These hybrid orbitals thus host the valence electrons which constitutes the chemical bond between the atoms. 

Accordingly, the molecular orbitals defined by the eigenvalue equation H = e

adiabatic states. The atomic orbitals are obtained from HQ = where Hq stands for the Hamiltonians Ha = T+ and... 

Finally, U, are the solutions of the coupled Hartree-Fock equations, which enter in the expression for the derivative, because the molecular orbitals also depend on field. Inserting the expression for U, from Eq. (11.25) the third term can be written more explicitly as... 

Delocalized Electrons An Explanation Based on Molecular Orbital Theory—Bonding in the benzene molecule, C H, is partly based on the concept of delocalized molecular orbitals. These are regions of high electron density that extend over several atoms in a molecule (Fig. 11-31). Delocalized molecular orbitals also provide an alternative to the concept of resonance in other molecules and ions. 

A more elaborate theoretical approach develops the concept of surface molecular orbitals and proceeds to evaluate various overlap integrals [119]. Calculations for hydrogen on Pt( 111) planes were consistent with flash desorption and LEED data. In general, the greatly increased availability of LEED structures for chemisorbed films has allowed correspondingly detailed theoretical interpretations, as, for example, of the commonly observed (C2 x 2) structure [120] (note also Ref. 121). 

The majority of photochemistry of course deals with nondegenerate states, and here vibronic coupling effects aie also found. A classic example of non-Jahn-Teller vibronic coupling is found in the photoelection spectrum of butatiiene, formed by ejection of electrons from the electronic eigenfunctions [approximately the molecular orbitals). Bands due to the ground and first... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

The Woodward-Hoffmann method [52], which assumes conservation of orbital symmetry, is another variant of the same idea. In it, the emphasis is put on the symmetries of molecular orbitals. Longuet-Higgins and Abramson [53] noted the necessity of state-to-state correlation, rather than the orbital correlation, which is not rigorously justified (see also, [30,44]). However, the orbital symmetry conservation rules appear to be very useful for most themial reactions. 

Depending on the application, models of molecular surfaces arc used to express molecular orbitals, clcaronic densities, van dor Waals radii, or other forms of display. An important definition of a molecular surface was laid down by Richards [182] with the solvent-accessible envelope. Normally the representation is a cloud of points, reticules (meshes or chicken-wire), or solid envelopes. The transparency of solid surfaces may also be indicated (Figure 2-116). 

Isovalue-based surfaces are also often used for the representation of molecular orbitals. 

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... 

The JME Editor is a Java program which allows one to draw, edit, and display molecules and reactions directly within a web page and may also be used as an application in a stand-alone mode. The editor was originally developed for use in an in-house web-based chemoinformatics system but because of many requests it was released to the public. The JME currently is probably the most popular molecule entry system written in Java. Internet sites that use the JME applet include several structure databases, property prediction services, various chemoinformatics tools (such as for generation of 3D structures or molecular orbital visualization), and interactive sites focused on chemistry education [209]. 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

HOMO and LLMO, also known as Frontier orbitals, are important in in tcrprcLitig results of a calculation (see Frontier Molecular Orbitals on page 141). You can use these m olecular orbiLals to comptiLe the lowest excited electronic singlet state of molecules and the ground states of radicals. 

The half-electron method can also apply to triplet states. For this calciilatioit. IlyperCheni populates selected molecular orbitals with pairs of half electron s. The final energy is computed by assigning the proper spins. 

Lei us consider how we might solve the Roothaan-Hall equations and thereby obtain the molecular orbitals. The first point we must note is that the elements of the Fock matrix, u liich appear on the left-hand side of Equation (2.162), depend on the molecular orbital oetficients which also appear on the right-hand side of the equation. Thus an iterative pi oeedure is required to find a solution. 

I he electron density distribution of individual molecular orbitals may also be determined and plotted. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are often of particular interest as these are the orbitals most cimimonly involved in chemical reactions. As an illustration, the HOMO and LUMO for Jonnamide are displayed in Figures 2.12 and 2.13 (colour plate section) as surface pictures. 

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