Orbital, molecular

MOLECULAR ORBITAL, ORBITAL SYMMETRY AIVO FROIXITIER MOLECULAR ORBITALS 

Molecular structure is best represented in terms of quantum mechanics. Quantum mechanical calculations are quite difficult. Therefore, approximation methods have been evolved which are result of mathematical simplifications. Molecular orbitals are centered around all the nuclei present in the molecule. Relative stabilities of molecules depend upon how electrons are distributed in them. In order to understand molecular symmetry it is essential to understand wave equations, phases of waves originated by the movement of electrons if we consider them as waves and also what are bonding and antibonding molecular orbitals. 

Phase Electrons not only behave as particles but as waves as well, i.e., they have dual nature. A stationary wave can be represened as follows. 

The vertical displacement of wave is known as amplitude which increases in one direction to the maximum, then decreases to zero and then again increases in opposite direction. The points of zero amplitude are known as nodes which lie is nodel plane perpendicular to the plane of paper. Upward and downward displacements are opposite phases, to distinguish between them we assign them positive and negative signs respectively. 

Wave nature of electrons can be expressed in terms of wave equation which describes amplitude, ( ), as a function of distance (x). 

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Molecules. The electronic configurations of molecules can be built up by direct addition of atomic orbitals (LCAO method) or by considering molecular orbitals which occupy all of the space around the atoms of the molecule (molecular orbital method). 

LCAO method A method of calculation of molecular orbitals based upon the concept that the molecular orbital can be expressed as a linear combination of the atomic orbitals. 

The carbon atom has a share in eight electrons (Ne structure) whilst each hydrogen atom has a share in two electrons (He structure). This is a gross simplification of covalent bonding, since the actual electrons are present in molecular orbitals which occupy the whole space around the five atoms of the molecule. 

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). 

Fig. XVIII-18. Interaction of the a and n molecular orbitals with the Pt d band Ef is the Fermi level. (From Ref. 172.)...

Sequences such as the above allow the formulation of rate laws but do not reveal molecular details such as the nature of the transition states involved. Molecular orbital analyses can help, as in Ref. 270 it is expected, for example, that increased strength of the metal—CO bond means decreased C=0 bond strength, which should facilitate process XVIII-55. The complexity of the situation is indicated in Fig. XVIII-24, however, which shows catalytic activity to go through a maximum with increasing heat of chemisorption of CO. Temperature-programmed reaction studies show the presence of more than one kind of site [99,1(K),283], and ESDIAD data show both the location and the orientation of adsorbed CO (on Pt) to vary with coverage [284]. 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

A superb treatment of applied molecular orbital theory and its application to organic, inorganic and solid state chemistry. Perhaps the best source for appreciating the power of the independent-particle approximation and its remarkable ability to account for qualitative behaviour in chemical systems. 

Salem L 1966 Molecular Orbital Theory of Conjugated Systems (Reading, MA Benjamin)... 

The relative acidities in the gas phase can be detennined from ab initio or molecular orbital calculations while differences in the free energies of hydration of the acids and the cations are obtained from FEP sunulations in which FIA and A are mutated into FIB and B A respectively. 

Ultraviolet photoelectron spectroscopy (UPS) results have provided detailed infomiation about CO adsorption on many surfaces. Figure A3.10.24 shows UPS results for CO adsorption on Pd(l 10) [58] that are representative of molecular CO adsorption on platinum surfaces. The difference result in (c) between the clean surface and the CO-covered surface shows a strong negative feature just below the Femii level ( p), and two positive features at 8 and 11 eV below E. The negative feature is due to suppression of emission from the metal d states as a result of an anti-resonance phenomenon. The positive features can be attributed to the 4a molecular orbital of CO and the overlap of tire 5a and 1 k molecular orbitals. The observation of features due to CO molecular orbitals clearly indicates that CO molecularly adsorbs. The overlap of the 5a and 1 ti levels is caused by a stabilization of the 5 a molecular orbital as a consequence of fomiing the surface-CO chemisorption bond. 

Figure Bl.6.12 Ionization-energy spectrum of carbonyl sulphide obtained by dipole (e, 2e) spectroscopy [18], The incident-electron energy was 3.5 keV, the scattered incident electron was detected in the forward direction and the ejected (ionized) electron detected in coincidence at 54.7° (angular anisotropies cancel at this magic angle ). The energy of the two outgoing electrons was scaimed keeping the net energy loss fixed at 40 eV so that the spectrum is essentially identical to the 40 eV photoabsorption spectrum. Peaks are identified with ionization of valence electrons from the indicated molecular orbitals.

B3.1.5.2 THE LINEAR COMBINATIONS OF ATOMIC ORBITALS TO FORM MOLECULAR ORBITALS EXPANSION OF THE SPIN ORBITALS... 

Flehre W J, Ditchfieid R and Popie J A 1972 Self-consistent molecular-orbital methods XII. Further extension of Gaussian-type basis sets for use in molecular orbital studies of organic molecules J. Chem. Phys. 56 2257-61 Flariharan P C and Popie J A 1973 The influence of polarization functions on molecular orbital hydrogenation energies Theoret. Chim. Acta. 28 213-22... 

Krishnan R, Binkley J S, Seeger R and Popie J A 1980 Self-consistent molecular orbital methods XX. A basis set for correlated wave functions J. Chem. Phys. 72 650-4... 

Stewart J J P 1991 Semiempirical molecular orbital methods Reviews in Computationai Chemistry vo 1, ed K B Lipkowitz and D B Boyd (New York VCH) pp 45-81... 

Thiel W 1996 Perspectives on semiempirical molecular orbital theory New Methods in Computationai Quantum Meohanios (Adv. Chem. Phys. XCiti) ed I Prigogine I and S A Rice (New York Wiley) pp 703-57 Earlier texts dealing with semi-empirical methods include ... 

Pisani C 1978 Approach to the embedding problem in chemisorption in a self-consistent-field-molecular-orbital formalism Phys. Rev. B 17 3143... 

Shephard M J and Paddon-Row M N 1996 Conformational analysis of Cgg ball and chain molecules a molecular orbital study Aust. J. Chem. 49 395-403... 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

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