Energy levels approximate

Here v is a quantum number characterizing the vibrational energy levels. Approximating the total wave function for /ftot of eq. (3-1) by... 

This calculation shows that in a mole of substance the C13-C12 bond will always lie at an energy level approximately thirty calories lower than the C12-C12 bond, and any reaction which involves the breaking of the carbon-hydrogen bond will require thirty calories more for the heavy carbon than for the light carbon. Applying this calculated difference to kinetics and substituting into equation (1) at 27°,... 

The vibrational energy of the nuclei is that which they possess by virtue of their oscillation about the point of minimum potential. This energy is quantized, and since near the minimum the potential well is approximately parabolic, the lower vibrational energy levels approximate to those of the harmonic oscillator. As a consequence of the uncertainty principle, the lowest allowed energy value cannot... 

The binding of an adsorbate to the surface of a solid by forces whose energy levels approximate those of a chemical bond. Contrast with physisorption. 

It should be mentioned that the single-particle Flamiltonians in general have an infinite number of solutions, so that an uncountable number of wavefiinctions [/ can be generated from them. Very often, interest is focused on the ground state of many-particle systems. Within the independent-particle approximation, this state can be represented by simply assigning each particle to the lowest-lying energy level. If a calculation is... 

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,... 

The variation of tlie frequency can be approximated by a series in the number of quanta, so the energy levels are given by... 

Often, it is a fair approximation to tnmcate the series at the quadratic tenn with The energy levels are then approximated as... 

The simplest case is a transition in a linear molecule. In this case there is no orbital or spin angular momentum. The total angular momentum, represented by tire quantum number J, is entirely rotational angular momentum. The rotational energy levels of each state approximately fit a simple fomuila ... 

The simplest example is that of tire shallow P donor in Si. Four of its five valence electrons participate in tire covalent bonding to its four Si nearest neighbours at tire substitutional site. The energy of tire fiftli electron which, at 0 K, is in an energy level just below tire minimum of tire CB, is approximated by rrt /2wCplus tire screened Coulomb attraction to tire ion, e /sr, where is tire dielectric constant or the frequency-dependent dielectric function. The Sclirodinger equation for tliis electron reduces to tliat of tlie hydrogen atom, but m replaces tlie electronic mass and screens the Coulomb attraction. 

Altogether, the three different models discussed so far are interconnected as sketched in Fig. 2. Now, we can by-pass the problems connected to caustics For e being small enough QCMD is justified as an approximation of QD if we exclude energy level crossings and discontinuities of the spectral decomposition. 

Hagedorn, G. A. Electron energy level crossing in the time-dependent Born-Oppenheimer approximation. Theor. Chim. Acta 67 (1990) 163-190... 

The complexity of molecular systems precludes exact solution for the properties of their orbitals, including their energy levels, except in the very simplest cases. We can, however, approximate the energies of molecular orbitals by the variational method that finds their least upper bounds in the ground state as Eq. (6-16)... 

This simple partiele-in-a-box model does not yield orbital energies that relate to ionization energies unless the potential inside the box is speeified. Choosing the value of this potential Vq sueh that Vq + ti2 h2/2m [ 52/E2] is equal to minus the lowest ionization energy of the 1,3,5,7-nonatetraene radieal, gives energy levels (as E = Vq + ti2 h2/2m [ n2/E2]) whieh then are approximations to ionization energies. 

Variational methods, in particular the linear variational method, are the most widely used approximation techniques in quantum chemistry. To implement such a method one needs to know the Hamiltonian H whose energy levels are sought and one needs to construct a trial wavefunction in which some flexibility exists (e.g., as in the linear variational method where the Cj coefficients can be varied). In Section 6 this tool will be used to develop several of the most commonly used and powerful molecular orbital methods in chemistry. 

The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the Yj m (0,(1)) as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule (Re), the energy levels are ... 

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