Hiickel theory determinant

In an early investigation (66T539) the two highest occupied and the two lowest unoccupied orbitals were calculated on the basis of an extended Hiickel theory to determine the electron transition responsible for the long wavelength UV absorption. An Ai- Bi, [Pg.197]

To illustrate the latter point, consider the butadiene radical cation (BD+ ). On the basis of Hiickel theory (or any single-determinant Hartree-Fock model) one would expect this cation to show two closely spaced absorption bands of very similar intensity, due to 7i i -> ji2 and ji2 —> JI3 excitation (denoted by subscripts a and v in Figure 28), which are associated with transition moments /xa and /xv of similar magnitude and orientation. Using the approximation fiwm) —3 eV288 the expected spacing amounts to about 0.7 eV. 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive to revisit one-electron effective Hamiltonian methods like the Hiickel model described in Section 4.4. Such models tend to involve the most drastic approximations, but as a result their rationale is tied closely to experimental concepts and they tend to be intuitive. One such model that continues to see extensive use today is the so-called extended Hiickel theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is the formation of the secular determinant for the secular equation... 

To obtain the pH, it is necessary to evaluate the activity coefficient of the chloride ion. So the acidity function is determined for at least three different molalities mci of added alkali chloride. In a subsequent step, the value of the acidity function at zero chloride molality, lg(flHyci)°, is determined by linear extrapolation. The activity of chloride is immeasurable. The activity coefficient of the chloride ion at zero chloride molality, yci, is calculated using the Bates-Guggenheim convention (Eq. 5) which is based on the Debye-Hiickel theory. The convention assumes that the product of constant B and ion size parameter a are equal to 1.5 (kg mol1)1/2 in a temperature range 5 to 50 °C and in all selected buffers at low ionic strength (I < 0.1 mol kg-1). 

The activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

The quantity given in equation (V-58) and expressing the average distance between the nearest ions in the solution (i. e. equalling the total of radii of both ions with opposite charges and being within the range of 3—5 x 10-8 cm) determines the specific influence of the electrolytes on the activity coefficient. Because this quantity iH not directly measurable, verification of the validity of the Debye - Hiickel theory is carried out in such a manner that a value is substituted for which conforms best with the values of y+c obtained by experiments. 

In the Hiickel theory the three occupied MO, which determine the it bonds in benzene, PMO = 11, where 1 stands for the unit row matrix, read... 

In the simplest application of Hiickel theory to the n electrons of planar conjugated hydrocarbons, a is taken to be the same for all C atoms, and ft to be the same for all bonded pairs of C atoms it is then customary to write the Hiickel secular determinant in terms of the dimensionless parameter x. 

In the previous two sections we have discussed the semiempirical extended Hiickel theory and the SCF-Xa-SW method. We have detailed the advantages and disadvantages of these methods for surface structure determination. 

It was necessary to measure the dielectric constant and density of each solvent mixture studied. Densities were determined in a constant-temperature bath maintained to within 0.02°C. Gay-Lussac pycnometers with a capacity of 25 mL were used for density measurements. Dielectric constants were determined with a Balsbaugh Model 2TN50 conductivity cell having a cell constant of 0.001. A Janz-Mclntyre a-c bridge (17) was used. The dielectric constants and densities of the solvents are listed in Table I, along with the constants A and B of the Debye-Hiickel theory. 

Having defined various activities and activity coefficients in solutions made up from strong electrolytes, we now turn to the determination of y . For this purpose we briefly discuss some aspects of the Debye-Hiickel Theory. 

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