Diatomic molecules molecular orbital methods

A promising semi-rigorous molecular orbital method, PRDDO, partial retention of diatomic differential overlap has been reported in the interim [186]. Also, the method of diatomics-in-molecules has been revived and the derivations extended to include p orbitals appropriately [187]. 

Mention should be made here of a series of recent papers by Pople and coworkers on self-consistent molecular-orbital methods in which they use mainly small optimized sets of Gaussian orbitals as fits to STOs or energy-optimized atomic orbitals to study a wide variety of molecules, including triatomics [123-130]. (One of the papers [124] also discusses a variant of a semirigorous technique, PDDO (projection of diatomic differential overlap).)1... 

A wide range of theoretical methods has been applied to the study of the structure of small metal clusters. The extremes are represented on the one hand by semi-empirical molecular orbital (Extended Huckel) (8 ) and valence bond methods (Diatomics-In-Molecules) ( ) and on the other hand by rigorous initio calculations with large basis sets and extensive configuration interaction (Cl) (10). A number of approaches lying between these two extremes have been employed Including the X-a method (11), approximate molecular orbital methods such as CNDO (12) and PRDDO (13) and Hartree-Fock initio molecular orbital theory with moderate Cl. 

In principle the molecular-orbital method could be applied to any molecule whether simple or complex and whether finite (as in benzene) or infinite (as in diamond). In practice the mathematical difficulties are insuperable and we must adopt a compromise solution. For all but the simplest diatomic molecules, therefore, we first form the atomic orbitals of the individual atoms in the usual way and then, by a linear combination of the relevant bonding orbitals, we deduce the molecular orbitals of the molecule or crystal. We may illustrate this point by considering the molecules of ethylene, H2C=CH2, acetylene, HC=CH, and benzene, C6H6. 

Figure 3.6 shows the LCAO method for generating molecular orbitals of diatomic molecules such as H2. In real molecules, the atomic orbitals of elemental carbon are not really transformed into the molecular orbitals found in methane (CH4). Figure 3.6 represents a mathematical model that mixes atomic orbitals to predict molecular orbitals. Molecular orbitals exist in real molecules and the LCAO model attempts to use known atomic orbitals for atoms to predict the orbitals in the molecule. Molecular orbitals and atomic orbitals are very different in shape and energy, so it is not surprising that the model used for diatomic hydrogen fails for molecules containing other than s-orbitals. 

In the molecular orbital method [8] the bonding is described in terms of linear combinations of atomic orbitals and the localised description of the chemical bond is lost, but as Mulliken [9] showed, it is possible to partition the electron density and thereby get an estimate of the spatial distribution and bonding character of an orbital. The procedure can be illustrated by the simple case of a diatomic molecule with one basis function per atom containing N electrons in a molecular orbital. Let the molecular orbital be written as ... 

Describe the bonding in the boron nitride (BN) molecule using both the molecular orbital method and the valence-bond method. Compare the BN molecule with diatomic carbon. 

In the case of ethylene the a framework is formed by the carbon sp -orbitals and the rr-bond is formed by the sideways overlap of the remaining two p-orbitals. The two 7r-orbitals have the same symmetry as the ir 2p and 7T 2p orbitals of a homonuclear diatomic molecule (Fig. 1.6), and the sequence of energy levels of these two orbitals is the same (Fig. 1.7). We need to know how such information may be deduced for ethylene and larger conjugated hydrocarbons. In most cases the information required does not provide a searching test of a molecular orbital approximation. Indeed for 7r-orbitals the information can usually be provided by the simple Huckel (1931) molecular orbital method (HMO) which uses the linear combination of atomic orbitals (LCAO), or even by the free electron model (FEM). These methods and the results they give are outlined in the remainder of this chapter. 

The se the orie s are inevitablj based upem analyse s of the interactions and transformations of molecular orbitals, and consequently the accurate construction and re presentation of molecular orbitals has become essential, furthermore, although the forms of molecular orbitals in diatomics and of delocalized tt orbitals in conjugated systems are familiar, a general, non-computational method for determining the qualitative nature of or and t orbitals in arbitrary molecules has been lacking. 

The other approach, proposed slightly later by Hund[9] and further developed by Mulliken[10] is usually called the molecular orbital (MO) method. Basically, it views a molecule, particularly a diatomic molecule, in terms of its united atom limit . That is, H2 is a He atom (not a real one with neutrons in the nucleus) in which the two positive charges are moved from coinciding to the correct distance for the molecule. HF could be viewed as a Ne atom with one proton moved from the nucleus out to the molecular distance, etc. As in the VB case, further adjustments and corrections may be applied to improve accuracy. Although the imited atom limit is not often mentioned in work today, its heritage exists in that MOs are universally... 

The molecular orbitals (MOs) are formed by the linear combination of atomic orbitals (LCAO-MO method). For diatomic molecules, the component of the angular momentum (A) in the direction of the bond axis is now important. The energy states are expressed by the symbol... 

There are three different schemes for building up the electronic states of diatomic molecules (a) from separated atoms, (b) from the united atom, and (c) from the molecular orbitals of the diatomic molecule itself. It is the correlation between the electronic states of the diatomic molecule as built up from the separated atoms and as determined from the molecular orbitals of the diatomic which is most valuable for any general consideration of reactions and excited states. The correlation of molecular states obtained by these two methods is not limited solely to diatomic molecules but also forms a valid approach for polyatomic molecular systems. The correlation of separated atoms with the hypothetical united atom has value for diatomics and has been applied to simple polyatomic molecules, especially those with a heavy atom or two and a number of hydrogen atoms. However, it is conceptually less appealing even for simple polyatomic molecules and completely inapplicable for complex polyatomic molecules. 

Since the discussion of molecular orbitals for the intermediate species involved in collisions will be used extensively in this chapter, it seems appropriate to describe now the various LCAO-MO methods in most common usage for calculating molecular orbitals. This presentation will be general and is equally applicable to diatomic and polyatomic molecules. This description of MO methods is not intended as a detailed comprehensive review of quantum chemical calculational techniques, but rather as an indication of the different methods available. 

The above discussion of the LCAO-MO method and the terms of the electronic configurations is not restricted to diatomic molecules. It is general and completely applicable to polyatomic molecules hence, the emphasis in this chapter on the correlation between the reaction intermediates arising from states of the separated atoms (or in the next section on polyatomics from the separated molecular fragments and atoms) and arising from the molecular orbitals of the intermediates. 

From the practical standpoint, the first attempt to solve the self-consistent TF equation for a diatomic molecule was made by Hund.82 Following this, the density method was applied to the benzene molecule and compared with both the molecular orbital prediction for the density and with relevant experiments.88 Various other early molecular calculations are discussed in ref. 16 we refer here to the recent studies of Dreizler and his co-workers.84 The importance of such self-consistent calculations will be emphasized below, even though we shall not use them in any detail in the ensuing discussion. 

Molecular orbital theory originated from the theoretical work of German physicist Friederich Hund (1896-1997) and its apphcation to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. MuUiken (1896-1986) (Hund, 1926, 1927a, b Mulliken, 1926, 1928a, b, 1932). Inspired by the success of Heitler and London s approach, Finklestein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894-1954) later suggested that only valence electrons need be treated as delocalized inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones, 1929). 

The electronic structures of Group lA and IB metal clusters have been determined using two theoretical methods ah initio molecular orbital theory and the semi-empirical diatomics-in-molecules (DIM)... 

Relativistic molecular orbital calculations have been performed for the study of the atomic-number dependence of the relativistic effects on chemical bonding by examining the hexafluorides XFg (X=S, Se, Mo, Ru, Rh, Te, W, Re, Os, hr, Pt, U, Np, Pu) and diatomic molecules (CuH, AgH, AuH), using the discrete-variational Dirac-Slater and Hartree-Fock-Slater methods. The conclusions obtained in the present work are sununarized. 

Most molecules, of course, have far more than two atoms, and the method used for diatomic species is not sufficient to devise an appropriate molecular orbital picture. However, the assumptions fundamental to diatomics still apply orbitals interact if their lobes are oriented appropriately with respect to each other, and interactions are stronger if the orbitals interacting are closer in energy. 

As an alternative to ab initio methods, the semi-empirical quantum-chemical methods are fast and applicable for the calculation of molecular descriptors of long series of structurally complex and large molecules. Most of these methods have been developed within the mathematical framework of the molecular orbital theory (SCF MO), but use a number of simplifications and approximations in the computational procedure that reduce dramatically the computer time [6]. The most popular semi-empirical methods are Austin Model 1 (AMI) [7] and Parametric Model 3 (PM3) [8]. The results produced by different semi-empirical methods are generally not comparable, but they often do reproduce similar trends. For example, the electronic net charges calculated by the AMI, MNDO (modified neglect of diatomic overlap), and INDO (intermediate neglect of diatomic overlap) methods were found to be quite different in their absolute values, but were consistent in their trends. Intermediate between the ab initio and semi-empirical methods in terms of the demand in computational resources are algorithms based on density functional theory (DFT) [9]. 

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