Energy levels variational calculation

Because the approximation described by Eq. (3.47) fails if the inversion and vibrational wave functions are strongly mixed, the Coriolis operator defined by Eq. (5.8) cannot be treated by the numerical methods described in Sections 5.1 and 5.2. Instead of the perturbation treatment described in Section 5.1, we must use a variational approach in which the energy levels are calculated as eigenvalues of an energy matrix the off-diagonal elements of this matrix are the matrix elements of the Coriolis operator ) 2,4 ... 

The energy-level variation with molecular field has been calculated by Wang and Cooper (1970) for a Ce " ion in a cubic field and was used to explain the anomalous x(T) curves of CeSb near Tn. 

It is not possible to solve this equation analytically, and two different calculations based on the linear variational principle are used here to obtain the approximate energy levels for this system. In the first,... 

Recent theoretical studies have demonstrated that it is possible to calculate accurately adsorbate stmcture and energy levels, to explain trends with variations in metal composition, and to interpret and predict the influence of promoters and poisons on the adsorption of reactants. Additional efforts along these lines will contribute greatly to understanding how catalyst stmcture and composition influence catalyst-adsorbate interactions and the reactions of adsorbed species on a catalyst surface. With sufficient development of theoretical methods, it should be possible to predict the desired catalyst composition and stmcture to catalyze specific reactions prior to formulation and testing of new catalysts. 

The variation method is usually employed to determine an approximate value of the lowest eneigy state (the ground state) of a given atomic or molecular system. It can, furthermore, be extended to the calculation of energy levels of excited stales. It forms the basis of molecular orbital theory and that which is often referred to (incorrectly) as theoretical chemistry". 

Variational Calculation of the Bending Energy Levels of HF Trimer and DF Trimer. 

Tables 1 and 2 show the lowest torsional energy levels of hydrogen peroxide and deuterium peroxide which have been determined variationally using as basis functions the rigid rotor solutions. Experimental data are from Camy-Peiret et al [15]. The first set of leval data are from Camy-Peiret et al [15]. The first set of levels (SET I) has been calculated without including the pseudopotential V = 0). The levels corresponding to the other sets (SET II, SET III and SET IV) were obtained including pseudopotentials calculated with different numerical and analytical algorithms. Finally, the zero point vibration energy correction was introduced in the SET V [14],...

In the case of porphyrins different substitution patterns induce relevant electronic variations the energy level of the two HOMOs of the porphyrin n system (a/u and a2u) are in fact reversed in going from a meso-substituted porphyrin, such as tetraphenylporphyrin, to a -substituted one, such as octaethylporphyrin [44]. Electronic variations may also derive from conformational changes Smith and coworkers have reported hybrid compounds substituted both at meso and P positions, the Zn2+ complexes of 2,3,7,8,12,13,17,18-octaethyl- and 2,3,7,8,12,13,17,18-octamethyl-5,10,15,20-tetraphenylporphyrin [46] (see Sect. 3.3). These complexes have a nonplanar conformation caused by the steric interactions of the substituents. Theoretical calculations indicate that such conformational changes cause a differential shift in the energies of the HOMO and LUMO and so modulate the electronic structure of the macrocycles. 

Carter, S. and Handy, N.C. (1986). The variational method for the calculation of ro-vibrational energy levels, Computer Physics Reports 5, 115-172. 

Following a variational solution of the ground (Is) and first excited state (2p) of the electron in this potential well, various other polarization terms are added and a variety of characteristics for the solvated electron (optical transition energy, heat of solution, etc.) can be calculated (101,105). For illustrative purposes, we shall utilize this simple model because of its obvious transparency in relating certain (macroscopic) features of solvent properties to the energy levels and wave functions for the solvated electron in polar solvents. 

---