Effective Energy Function , potential

The characteristic structure of the actinide spectra is mainly determined by the special properties of the 5 f wave functions these, in turn, arise from the special nature of the 5 f effective potential-energy function Ueff (5f). Uetf describes the effect on the 5 f electrons of the attraction by the nucleus as well as of all the interactions with other electrons. It contains also a centrifugal term, 1(1 -I- l)/2 r, of particular importance for the properties of 5f wave functions, the effect of which will be illustrated later in more detail. 

Maciejczyk M, Rudnicki WR, Lesyng BA (2000) mezoscopic model of nucleic acids, part 2. an effective potential energy function for DNA, J. Mol. Struct, and Dynamics 17 1109-1115... 

Correlate the shape of the effective potential energy function between the nuclei with trends in the bond length and bond energies of diatomic molecules (Section 3.5, Problems 17-20). 

We calculate the electronic bonding energies for every value of Rar and add to that the nuclear-nuclear repulsion energy to generate the effective potential energy function Veff(l AB) that governs the nuclear motion. 

Some most impressive computer simulations have been made in efforts to model the structure of liquid water. Yet, because these calculations usually are based on pair additivity of the potentials for the H-bonded water molecules, the possibility exists that subtle effects may escape the theoretician, as no means are provided to incorporate the possibility of extensive cooperativity—an aspect that Henry Frank (1972) has so eloquently stressed. Very likely, this is the crux of the problem of interfacially modified water if nothing else, the thermal anomalies (discussed below) in the properties of vicinal water strongly implicate cooperativity on a large scale—a collective behavior of water molecules that no existing potential function is able to reproduce. The cooperativity reflects nonpair additivity, and it does not seem plausible that effective potential energy functions can be devised that will remedy the specific lack of a detailed understanding of many-body interactions in water. Attempts to allow for cooperativity have been made by Finney, Barnes, and co-workers, notably Quinn and Nicholas (see Barnes et al., 1979). 

Figure 1.7 Vibrational state-specific effective potential energy functions for the skeletal contortion vibration [86], Transitions at 754, 743.3 cm" forTRN(OH) isolated in a Ne matrix are assigned as the spectral doublet, and Aj, is the upper state splitting, ofthe v,...

From this there is subtracted the estimated contribution of the fcth electron, leaving the effective potential-energy function for this electron. 

Fi is an effective Hamiltonian function for the tth electron, and Vi the effective potential-energy function for the tth electron due to its interaction with the other electrons in the atom. Using the symbol Fi, we obtain as the condition that E be stationary with respect to variation in uf(t) the expression (Eq. 32-6)... 

Table 1. Effective Potential Energy Functions for Water" ...

Another concept often invoked for qualitative discussions and for calculations when the adiabatic approximation breaks down is that of diabatic potential energy surfaces. There are several nonequivalent ways of defining such surfaces, each of which may be useful under some circumstances. The simplest way is that already illustrated above in conjunction with the NaCl example namely, a diabatic state is the effective potential energy function for nuclear motion when the electronic state is artificially constrained to a state of prespecified pure valency. 

For each electronic state, there is an effective potential energy function that depends on the geometry. The actual energy necessary to move from one state to another will depend not only on what states are involved but also on the geometry of the molecule during the transition. 

Section 5.2 describes the effective potential energy function, which becomes the potential energy term in our vibrational Schrodinger equation in this... 

An elementary discussion of the neglected terms is given in [1.32]. Equation (1.8) is an equation for a wave function of the ions alone. The essential point is that for the ionic motion, an effective potential energy function... 

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