Separated molecular orbitals

The method which produces the separated molecular orbitals (SMOs) was first published in ref. (Kozmutza et al., 1994) and then later applied successfully for dimer molecules (Kozmuta etal., 1994 Kozmutza etal., 1995) and also for several interacting systems (Kozmutza etal., 1996 Kozmutza etal., 1994). 

In the present work we summarize and continue a systematic study on van der Waals (vdW) interacting molecules using separated molecular orbitals (SMOs). Since the avoidance of basis set superposition error (BSSE) ([1] and reference therein) turned out to be one of the main problems in the study of vdW systems, we pay special attention to this question. It is known that the counter-poise (CP) method [2] is often used in order to correct the BSSE. 

The Separated Molecular Orbitals (SMO) method has been described earlier in details in refs. [16, 17]. At the Hartree-Fock SCF level the closed shell total energy has the following form ... 

The results presented in this work show that in the linear structured water dimer the partitioned energy terms calculated for the proton donor and acceptor molecules are significantly different (except the kinetic energy). The electron structure of the proton donor molecule was found more compact than that of the acceptor subsystem, when compared their (partitioned) total energy EM values. This result is in an excellent agreement with our pre-vious results obtained on the separated molecular orbital energies [17]. 

F 2p character than F 2s character and is also bonding with respect to the FI orbital. This set of orbitals (2cr, 3a) illustrates a central feature of the MO approach. Whereas a simple Lewis structure or valence picture would draw a localized electron pair interaction between two orbitals, the MO picture attributes some bonding character to two separate molecular orbitals. This simple MO diagram illustrates the difficulty of determining a meaningful definition for bond order in a polyatomic molecule. No single MO completely represents the bonding between two atoms. 

It has also been shown that using localized representation by the so-called LMBPT method [41], the interaction energy in van der Waals (vdW) systems can be treated in a straightforward manner [40,42-52]. A large variety of atomic and molecular, dimer and more components systems has been studied in the framework of the LMBPT procedure. It is known that the treatment of vdW systems reveals the introduction of counter-poise (CP) calculations due to the basis set superposition error [53]. The CP calculations, however, seriously increase the computational work (see, e.g. [54-56] and [57] and references therein). The introduction of separated molecular orbitals (SMOs) allows that the basis set superposition error could be taken into account without any additional (CP) calculation. The energy terms as compared at the HF level for several systems — resulted in the SMO as well as in the CP calculations — also affirmed this conclusion [43,49]. 

As mentioned before, distonic ions are distinguished from normal molecular ions by the occurrence of the positive charge and the radical cation in separate molecular orbitals. Several examples of intermediate distonic ions have been discussed above, and these distonic ions arise usually by hydrogen migration, for example during the initial step of the McLafferty rearrangement. Other modes for the... 

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. 

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. 

Frori tier Orbital theory supplies an additional asstim piion to ih is calculation. It considers on ly the interactions between the h ighest occupied molecular orbital (HOMO) and the lowest unoccupied rn olecular orbital (I.UMO). These orbitals h ave th e sin a 1 lest energy separation, lead in g to a sin all den oin in a tor in th e Klopinan -.Salem ct uation, fhe Hronticr orbitals are generally diffuse, so the numerator in the equation has large terms. 

In summary, we have made three assumptions 1) the Bom-Oppenheimer approximation, 2) the independent particle assumption governing molecular orbitals, and 3) the assumption of n-molecular orbital theory, but the third is unique to the Huckel molecular orbital method. 

These absorptions are ascribed to n-n transitions, that is, transitions of an electron from the highest occupied n molecular orbital (HOMO) to the lowest unoccupied n molecular orbital (LUMO). One can decide which orbitals are the HOMO and LUMO by filling electrons into the molecular energy level diagram from the bottom up, two electrons to each molecular orbital. The number of electrons is the number of sp carbon atoms contributing to the n system of a neuhal polyalkene, two for each double bond. In ethylene, there is only one occupied MO and one unoccupied MO. The occupied orbital in ethylene is p below the energy level represented by ot, and the unoccupied orbital is p above it. The separation between the only possibilities for the HOMO and LUMO is 2.00p. 

AEis the interaction energy q and qg are charges on atoms A and B, separated by Rab, on different molecules, r and s are molecular orbitals on the two different molecules, p and v label the atomic orbitals that contribute to these molecular orbitals, with coefficients Cjjj. and Cyg. Hj y is the matrix element between atomic orbitals p and V, which is a measure of the energy of their interaction, roughly proportional to their overlap. The energies of the molecular orbitals are and e. ... 

Figure 7.14 Molecular orbital energy level diagram for first-row homonuclear diatomic molecules. The 2p, 2py, 2p atomic orbitals are degenerate in an atom and have been separated for convenience. (In O2 and F2 the order of

The radical is much more stable if both stmctures exist. Quantum mechanical theory implies that the radical exists in both states separated by a small potential. Moreover, both molecular orbital theory and resonance theory show that the allyl carbocation is relatively stable. 

Fig. 8. Schematic illustration of the tunnelling in a CNT-based device (a) under no bias voltage, there are no orbitals available for conduction, (b) with small bias voltage, only one molecular orbital of a CNT contributes to the carrier transport and (c) when the next molecular orbital enters the bias window, current increases stepwise. Gate voltage can shift all the orbitals upward or downward. AE indicates the energy separation of molecular orbitals.

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