Function, wave

Now that we have expressions for the four energy levels, we can examine the wave functions ipi —04 corresponding to each energy level. For states (1) and (4) we have seen that the basis functions 0, do not mix, hence i/fj = 0( and 04 = 04. However, ip2 and are linear combinations (mixtures) of f 2 and 03, the extent of mixing depending on the ratio (vA — Vb)/Jab as shown in the following expressions  

The calculation of the wave functions ip from the Hamiltonian matrix uses standard matrix manipulations, which are not reproduced here. We shall discuss the spectrum arising from an AB system in detail in Section 6.8. 

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The wave function T i oo ( = 11 / = 0, w = 0) corresponds to a spherical electronic distribution around the nucleus and is an example of an s orbital. Solutions of other wave functions may be described in terms of p and d orbitals, atomic radii Half the closest distance of approach of atoms in the structure of the elements. This is easily defined for regular structures, e.g. close-packed metals, but is less easy to define in elements with irregular structures, e.g. As. The values may differ between allo-tropes (e.g. C-C 1 -54 A in diamond and 1 -42 A in planes of graphite). Atomic radii are very different from ionic and covalent radii. 

Only certain energy values ( ) will lead to solutions of this equation. The corresponding values of the wave functions are called Eigen functions or characteristic wave functions. 

The approximation of Fresnel is scalar approximation. Let u(, r],0-0) be the scalar wave function of the laser beam falling onto the optical element, and u( X,y,Cl) will the be scalar wave function in the plane Z = Cl. Then [3,4]... 

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

Small metal clusters are also of interest because of their importance in catalysis. Despite the fact that small clusters should consist of mostly surface atoms, measurement of the photon ionization threshold for Hg clusters suggest that a transition from van der Waals to metallic properties occurs in the range of 20-70 atoms per cluster [88] and near-bulk magnetic properties are expected for Ni, Pd, and Pt clusters of only 13 atoms [89] Theoretical calculations on Sin and other semiconductors predict that the stmcture reflects the bulk lattice for 1000 atoms but the bulk electronic wave functions are not obtained [90]. Bartell and co-workers [91] study beams of molecular clusters with electron dirfraction and molecular dynamics simulations and find new phases not observed in the bulk. Bulk models appear to be valid for their clusters of several thousand atoms (see Section IX-3). 

Gordon R G 1969 Constructing wave functions for bound states and scattering J. Chem. Phys. 51 14-25... 

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Luce T A and Bennemann K H 1998 Nonlinear optical response of noble metals determined from first-principles electronic structures and wave functions calculation of transition matrix elements P/rys. Rev. B 58 15 821-6... 

Krishnan R, Binkley J S, Seeger R and Popie J A 1980 Self-consistent molecular orbital methods XX. A basis set for correlated wave functions J. Chem. Phys. 72 650-4... 

Calculation of wave function components in Ursell-type expansion using... 

Olsen J and J0rgensen P 1995 Time-dependent response theory with applications to self-consistent field and multiconfigurational self-consistent field wave functions Modern Electronic Structure Theory vo 2, ed D R Yarkony (Singapore World Scientific) pp 857-990... 

Helgaker T, Gauss J, J0rgensen P and Olsen J 1997 The prediction of molecular equilibrium structures by the standard electronic wave functions J. Chem. Phys. 106 6430-40... 

Moiseyev N, Friesner R A and Wyatt R E 1986 Natural expansion of vibrational wave functions RRGM with residue algebra J. Chem. Phys. 85 331... 

Kroes G J and Neuhauser D 1996 Performance of a time-independent scattering wave packet technique using real operators and wave functions J. Chem. Phys. 105 8690... 

Herman M F, Kluk E and Davis H L 1986 Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited enharmonic oscillator J. Chem. Phys. 84 326... 

Mead C A and Truhlar D G 1979 On the determination of Born-Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei J. Chem. Phys. 70 2284... 

In tills weakly coupled regime, ET in an encounter complex can be described approximately using a two-level system model [23]. As such, tlie time-dependent wave function is... 

Comparison between the first and last lines of the table shows that the sign of the ground-state wave function has been reversed, which implies the existence of a conical intersection somewhere inside the loop described by the table. 

While the presence of sign changes in the adiabatic eigenstates at a conical intersection was well known in the early Jahn-Teller literature, much of the discussion centered on solutions of the coupled equations arising from non-adiabatic coupling between the two or mom nuclear components of the wave function in a spectroscopic context. Mead and Truhlar [10] were the first to... 

The radial factor in the total wave function must therefore be an eigenstate of... 

The first stoand of Ham s argument [11] is that V(c()) supports continuous bands of Floquet states, with wave functions of the form... 

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