Polyatomic molecules AB

A group-theoretical treatment of this symmetry contraint leads to the requirement that an MO must belong to an irreducible representation of the point group. A representation is a set of matrices - one for each symmetry operation - which constitutes a group isomorphous with the group of symmetry operations and can be used to represent the symmetry group. When we say that a function belongs to (or transforms as , or forms a basis for ) a particular representation, we mean that the matrices which constitute the representation act as operators which transform the function in the same way as the symmetry operations of the molecule. (The reader who knows little about matrices and their application as transformation operators can skip over such remarks.) An irreducible representation is one whose matrices cannot be simplified to sets of lower order. 

Two functions that belong to different irreducible representations are necessarily orthogonal to each other. If they belong to the same irreducible representation, they may not be orthogonal, and must be combined to produce a pair of orthogonal linear combinations. Thus the application of group-theoretical principles and the exploitation of molecular symmetry help to fulfil essential quantum-mechanical requirements in the construction of MOs. We now illustrate these principles by looking at the MOs of the H20 molecule. 

The first step is to specify a coordinate system. This is shown 

The z axis is a twofold proper axis of symmetry, designated C2. A molecule possesses a C axis of symmetry if the operation of rotation by 

Note that the effect of applying ov(jcz) and then applying ov(yz) is to transform (jc, yu z,) into (—JCj, —yu z,), which is the same as C2. Relationships like this endow the set of symmetry operations with group properties, and restrict the number of possible combinations of symmetry operations to those specified by the point groups. The point group to which the H20 molecule belongs is designated C2v. 

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Consider a di- or a polyatomic molecule AB in the gas phase, at T = 0. By means of an electron or a photon, this molecule can be ionized and excited to a state AB+, which subsequently decomposes into the fragments A+ and B ... 

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

More complex polyatomic molecules AB are tackled in essentially the same way as H20. In a qualitative treatment, the most difficult part is working out the group orbitals. In the simple case of HzO, this was done by inspection. Group-theoretical techniques are available in cases where this would not be practicable. The resulting MO energy diagram, after feeding in the appropriate number of electrons, will usually confirm or... 

Now we turn to polyatomic molecules, which we shall treat as quasidia-tomic. Remember that in partitioning a polyatomic molecule AB into two quasiatomic fragments A + B, the values of QA and QB usually are not known from experiment but are calculated with inevitable uncertainty. Furthermore, the master BOC equation [Eq. (8)] can be explicitly written for a polyatomic AB only in a rather crude fashion when we have to use some group terms [such as xA or jcb in Eq. (8b)], which further blurs the... 

Interaction in Polyatomic Molecules Ab Initio Computations with Gaussian Orbitals. 

Platzman in his theory of the initial processes in the radiolysis of chemicals assumed that when a polyatomic molecule AB is receiving energy of radiation, this energy may be utilized in two ways (a) in a direct ionization (equation 1) ... 

For a polyatomic molecule AB with n equal bonds, the energy of formation Eab of the corresponding hypothetical purely covalent gaseous molecule from gaseous atoms can be calculated in the same way as for a diatomic molecule. Analogous to Eq. (1) one obtains... 

From / > 1 and Qmoi <100 it follows that Qo < QmoB thus the actual ionic character Qo of a bond A—B in a polyatomic molecule AB is smaller than the ionic character Qmoi of the molecule calculated according to Eqs. (9) and (18). This is a consequence of the accumulation of charge at the central atom. For diatomic molecules Qo is identical with Qmoi-... 

Concerning completeness compare footnote on page 383 of Vol. II/6. In polyatomic molecules, AB corrections and pseudo-quadrupole coupling are normally not considered. However, see for example [53Bur]. 

Pulay P 1969 Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. I. Theory/Wo/. Phys. 17 197... 

Ab Initio Calculation of Force Constants and Equilibrium Geometries in Polyatomic Molecules. I. Theory P. Pulay... 

R.S. Mulliken and W.C. Ermler, Polyatomic Molecules. Results of ab initio calculations. Academic Press, New York, 1981. 

For polyatomic molecules, the dissociation energy can be measured directly only for the weakest bond, and even then the value may only be approximate because the energies of the other bonds in the molecule generally change when one bond is broken. To obtain the energies of other bonds, some assumptions must be made. For molecules of the type AB with only one type of bond, the enthalpy of atomization, that is, the enthalpy change for the reaction... 

Despite the obvious limitation of the LCAO procedure as revealed by the Hj and H2 problems it still is the most popular scheme used in the theoretical study of polyatomic molecules. There is a bewildering number of approximate methods, commonly distinguished in terms of cryptic acronyms, designated as either ab initio or semi-empirical, but all of them based on the LCAO construction of molecular orbitals. The precise details can be found in many books and reviews. The present summary uses the discussion of Richards and Cooper [92] as a guide. 

The adiabatic ionization energy of any molecule AB (mono-, di-, or polyatomic), represented by ) (AB), is the minimum energy required to remove an electron from the isolated molecule at 0 K ... 

Figure 1. Schematic representation of resonances in the continuum of a polyatomic molecule ABC(X) dissociating into products AB(X, 0) and C. The left-hand side shows an absorption-type cross section <7abs( ) with a rich resonance pattern. The term p(E) is the density of states at the energy E and N r(E) is the number of states at the TS, orthogonal to the dissociation path, that are accessible at energy E. Several experimental schemes for a spectroscopic analysis of resonances are also indicated. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)...

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