Electronic energy wave functions

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

Assume now that two different external potentials (which may be from nuclei), Vext and Vgjjj, result in the same electron density, p. Two different potentials imply that the two Hamilton operators are different, H and H, and the corresponding lowest energy wave functions are different, and Taking as an approximate wave function for H and using the variational principle yields... 

The electronically adiabatic wave functions v(/f ad(r q ) are defined as eigenfunctions of the electronic Hamiltonian Hel with electronically adiabatic potential energies ad(qjJ as their eigenvalues ... 

In order to obtain one-electron radial wave functions from the energy expression by the variational method, it is assumed that a set of coefficients i exists such that... 

The Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in Ti gives the kinetic energy of each electron in by computing the second derivative of the electron s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each parr of electrons, and (c) the repulsion between each pair of nuclei. 

In equation (A.8), is the wave function which describes the distribution of particles in the system. It may be the exact wave function [the solution to equation (A.l)] or a reasonable approximate wave function. For most molecules, the ground electronic state wave function is real, and in writing the expectation value in the form of equation (A.8), we have made this simplifying (though not necessary) assumption. The electronic energy is an observable of the system, and the corresponding operator is the Hamiltonian operator. Therefore, one may obtain an estimate for the energy even if one does not know the exact wave function but only an approximate one, P, that is,... 

These discrepancies result (a) from the harmonic approximation used in all calculations [to,- (theory) > v, (exp)], (b) the known deficiencies of minimal and DZ basis sets to describe three-membered rings [polarization functions are needed to describe small CCC bond angles a>,(DZ + P) > w,(DZ) > to,(minimal basis)] and (c) the need of electron correlated wave functions to correctly describe the curvature of the potential energy surface at a minimum energy point [ [Pg.102]

Product wave functions can clearly be constructed for any number of electrons. Early wave functions, were constructed on this basis together with the empirical rule that not more than two electrons could be assigned to a single orbital, one of each. spin. Further, electrons tend to occupy the orbitals with, lowest possible energy in the absence of other factors. 

As an example, suppose for an A-electron system that energy E is approximated by an orbital functional [ , ], which depends on one-electron orbital wave functions , and on occupation numbers n, through a variational A-electron trial wave function A momentum displacement is generated by U = cxp(- 7r D), where D = r - In (he momentum representation of the orbital wave functions,... 

Analyzing the semiempirical QC methods in relation to their suitability for developing the hybrid QM/MM methods reveals certain problems. Using the HFR form of the electron trial wave function together with the ZDO type of parametrization results in the decomposition of the total energy of a molecular system into a sum of mono-and diatomic increments ... 

Fig. 1. The QED contributions of order a/it) to the bound-electron gj factor depicted as Feynman diagrams. Double lines indicate bound fermions, wavy bnes indicate photons. The interaction with the magnetic field is denoted by a triangle. Diagram (a) is also termed SE, ve (self-energy vertex correction), diagrams (c) and (e) SE, wf (self-energy wave-function correction), diagram (b) VP, pot (vacuum-polarization potential correction), and diagrams (d) and (f) VP, wf (vacuum-polarization wave-function correction)...

The practical implication is the fact that in the CP MD simulation the molecular system does not evolve right on the Born-Oppenheimer PES, but stays close to it. A measure of deviations from the BO PES is the fictitious kinetic energy (wave-function temperature). Figure 4-2 demonstrates that this deviation is minor, as the electronic (fictitious) temperature is relatively low. The wave function stays cold (compared to the hot nuclei) in the MD terminology the term cold electrons is often used in this context. 

It is important to note that the energy of the electron in rest in vacuum, vac> is larger than the total energy of the impinging electron. Nevertheless, wave functions... 

H = the Hamiltonian operator E = energy of the electron = the wave function... 

Resonance hybrids Various possible chemical structures of molecules, each with identical atomic connectivity, but differing in the disposition of electrons. The wave function of the molecule is approximately represented by mixing the wave functions of the contributing structures. The energy calculated for such a mixture is lower, presumably because the representation is more nearly correct than it would be if formally represented by a single structure. 

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