Mulliken overlap populations for

The sums are over the occupied MOs, since = 0 for the virtual MOs. The number nr is the Mulliken net population in the basis function Mulliken overlap population for the pair of basis functions s. The net population summed over all r plus the overlap population summed over all rls pairs equals the total number of electrons in the molecule. 

TABLE II Mulliken Overlap Populations for Selected Bonds in 7-Amino-3-acetoxy-methyl-3-cephem as a Function of OH" Approach Obtained from CNDO/2D MOs... 

For yq the overlap population 2clcyS12 is positive, and the electron is bonding for yr2 it is negative, and the electron is antibonding. The sum of the values 2clc2Sl2 of all occupied orbitals of the molecule, the Mulliken overlap population, is a measure of the bond strength or bond order (b.o.) ... 

The photocyclodehydrogenation of thienyl ethylenes is well-defined when both thiophene rings are bound via a C(2) atom to the ethylenic bond as in (70). In other cases, however, more cyclization products are possible. To predict the photocyclization mode for heterohelicenes the F s rule fails in many cases, because correction factors for the hetero atoms in the Huckel MO calculation have to be introduced and the systems are not well comparable with carbocyclic diaryl ethylenes. A better reaction parameter in these cases is the Mulliken overlap population (nrs)51), introduced by Muszkat52) for these cases. The overlap populations of the atoms r and s in ground and excited state (nIiS and n s), are calculated using the extended Huckel method. Cyclizations should not occur when nr>s and An s (= nr>s — n s) have negative values. (This method can also be used for diaryl olefins, but in these cases calculation of F s is more simple.). 

In Table 27 the simple Hiickel indices, Nr (Dewar number), Fr (free valence number) and Lr (localization energy) are given, together with the Mulliken overlap population pr, calculated by the extended Hiickel method and using the geometry of hexahelicene as determined by X-ray analysis141) to account for the non-planarity. 

Under mild conditions nitration and acetylation of hexahelicene give the 5-nitro-and 5-acetyl substitution product as the main product in about 50% yield. In both cases another monosubstitution product is formed, which was identified tentatively by NMR as the corresponding 8-substituted hexahelicene. From the relative rates of detritiation (krel) or the partial rate factors (f) given in Table 27, it seems more probable, however, that the 7-isomers are formed as the side product, as the positional reactivity order of detritiation is C(5) >C(7) >C(8) >C(1) >C-(4) >C(6) >C(2) > C-(3). The preferred reactivity at C(5), found in electrophilic substitutions, is predicted by all the simple Hiickel parameters, whereas the next two positions are correctly predicted by Nr and Lr. Judging from Nr-, Fr- and Lr-values the C-(l) position does not experience much steric hindrance in the H-exchange. Relative to some other positions (C(4), C(6)) its reactivity is higher than expected. The Mulliken overlap population predicts, however, the highest reactivity for C(l) and leaves room for the supposition that this position is considerably masked. 

At the end of Section 10.1 the Mulliken overlap population is mentioned as a quantity related to the bond order. A corresponding quantity for solids was introduced by R. Hoffmann the crystal orbital overlap population (COOP). It is a function that specifies the bond strength in a crystal, all states being taken into account by the Mulliken overlap populations Its calculation requires a powerful computer however, it can... 

Results of the calculations for the TiLn (L=H, Cl, CH3) neutral structures are displayed in Tables 2-4, which provide dissociation energies, bond lengths, Mulliken charges and overlap populations for these compounds. 

The quantities nr and nr s are used to calculate atom charges and bond orders. The Mulliken gross population in the basis function (fir is defined as the Mulliken net population nr (Eq. (5.211)) plus one half of all those Mulliken overlap populations nrh (Eq. (5.212)) which involve separated atoms Srs is very small) ... 

Molecular orbital calculations have also provided theoretical justification for these stereoelectronic effects in tetracovalent and pentacovalent phosphorus species (2-7). As has been shown in molecular orbital calculations on the X -P-X2 (X = 0,N) structural fragments, the X.-P bond is strengthened (as indicated by an increase in the Mulliken overlap population) while the P-X3 bond is weakened when the X atom lone pair is app to the P-X3 bond. Thus, in the g,t conformation of dimethyl phosphate (Structure ll the overlap population for the trans P-0 bond is. 017 electron lower than the overlap population for the gauche P-0 bond. As shown for g,t dimethyl phosphate one lone pair (shaded in 1) on the gauche bond oxygen is app to the trans bond, while no lone pairs on the trans bond oxygen are app to the gauche bond. Thus, the weakest X.-P bond has one app lone pair and no lone pairs on X. app to the P=X2 bond. 1... 

A Mulliken overlap population analysis [11] was employed to analyze the atomic orbital components in the energy levels caused by the impurities and to estimate the net charges of ions. The net charge AQx, for example, for the atom X with an atomic number Z was defined by the following equations ... 

In contrast to the nice, neat two-center-two-electron bond model, it is not so easy to determine the overall bonding character from MO orbital drawings alone. We need another measure. This comes from the Mulliken overlap population which is a numerical indicator of bonding (positive) and antibonding (negative) character between a pair of atoms within a molecule. For N2 in an approximate calculation the overall overlap populations are +0.68 for the three a filled MOs and +0.54 each for the it MOs. If one considers each it interaction of bond order one then the overall bond order is clearly three. 

Tables 1.2-1.6 contain the results of an approximate MO calculation (Fenske-Hall) on the BF molecule. From this output (a) construct a MO diagram showing MO energy levels and qualitative AO compositions in MO drawings (b) examine the HOMO and LUMO relative to Lewis acid/base behavior and compare it with CO. Would BF be suitable for coordination to, e.g., a Cr center (c) use the Mulliken charges to predict the direction of the dipole moment (d) examine the Mulliken overlap populations and decide whether it is proper to describe the B-F bond as a single, double or triple bond. 

Three-dimensional Hiickel molecular orbital calculations have been performed on a series of molecules made from the atoms H, C, N, O, F, and CL It has been found that the sum of the Mulliken overlap populations is closely related to the energy of atomization. For 40 compounds not containing carbon, the observed energies of atomization can be reproduced with a mean deviation of 11.1 kcal./mole by a simple empirical equation which includes a term to account for the extra stability of polar molecules. A slightly more complicated expression is needed for compounds of carbon, and the fit to the observed data is not as good. 

Free C2H2 molecule. The free acetylene molecule was optimized at the BLYP/6-31G level. Then, vibrational frequencies for the optimised free C2H2 molecule have been calculated. The computed geometry, C-C Mulliken overlap population and vibrational frequencies are listed in Table 2 along with the corresponding experimental values [41, 42], Both of these results show that the acetylene molecule in its ground state is linear with a Dmi, symmetry. 

TABLE 3. Total energies of the system C2H2/Cuii, optimized geometries and Mulliken overlap populations (M.O.P.) of the CC bond for C2H2 adsorbed on the diagonal fourfold hollow, aligned fourfold... 

TABLE 6. Adsorption energies (kJ.mof), optimized geometries, C-C and C-Pt Mulliken overlap populations (M.O.P.) and total adsorbate eharges, Q, for C2HJ adsorbed on the di-a and top sites of the ... 

First, we examine three kinds of basis sets to clarify the role of the sulfur 3d orbital. The basis sets we test here are Is Sp, ls 3d and ls 4s for S, and we fix F basis set to ls 2p. Table I shows the overlap populations obtained by the Mulliken population analysis for SF2, they clearly indicate the 3d orbital contribution through the change in the sulfur basis set The total overlap population for the ls- 3p basis set is particularly small, while the change in the total overlap population from the ls 3d basis set to the ls 4s basis set is not so large. This result indicates that a Is dp basis set is not suitable for sulfur. In addition, the overlap populations for each atomic orbital pair for the Is 3d and ls 4s basis sets are almost the same, and those involving 4s are very small. We have confirmed that the sulfur 3d orbital is essential for the understanding of its chemical bonding. 

MULLIKEN t-OVERLAP POPULATIONS FOR THE CARBONYL HALIDES [1857aa]... 

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