Wave function difference

The biradical resonance structure for ozone requires two singly occupied MOs, and it is clear that an RHF type wave function, which requires all orbitals to be doubly occupied, cannot describe this. A UHF type wave function, however, allows the a and /3 orbitals to be spatially different, and can to a certain extent incorporate both resonance structures. Systems with biradical character will often have a (singlet) UHF wave function different from an RHF. 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

The radial wave functions i j(r) for = 1, 2, and 3 and 1 = 0 and 1 are shown plotted in Figure 21-2. The abscissas represent values of p hence the horizontal scale should be increased by the factor n in order to show R r) as functions of the electron-nucleus distance r. It will be noticed that only for s states (with 1 = 0) is the wave function different from zero at r = 0. The wave function crosses the p axis n — l — 1 times in the region between p = 0 and p = oo. 

The methods which we have used in Section 23 to obtain the first-order perturbation energy are not applicable when the energy level of the unperturbed system is degenerate, for the reason that in carrying out the treatment we assumed that the perturbed wave function differs only slightly from one function which is the solution of the unperturbed wave equation for a given energy value whereas now there are several such functions, all... 

The actual values of these wave functions differ, however, in one respect. If particles 1 and 2 are interchanged, i/t retains its identical value, and is therefore called the symmetrical combination, while if the corresponding exchange is made changes sign, since... 

Note that the form of a CIS wave function differs from that of an ordinary Cl wave function. In an ordinary Cl wave function, the reference function (the SCF wave function for the state of interest) makes the largest contribution. In the CIS method for an excited state, the reference function is the SCF wave function for the ground state, and this reference function does not appear in the CIS wave function. (This makes the CIS wave function orthogonal to the ground-state wave function, which is desirable, so as to avoid having the variational calculation collapse to the ground state.) The CIS wave function includes only a modest amount of electron correlation. 

It is. therefore, clear that the Heitler-London wave function differs considerably and that this huge difference does not vanish when R oo. 

Thus, the Heitler-London wave function differs from this difference is huge and does not vanish, when i 00. 

The ionization cross sections of the individual L subshells may provide a sensitive test for details of the ionization process as has been demonstrated for ion impact ionization. The binding energies of the three L subshells are comparable, but the corresponding wave functions differ significantly. Photoionization occurs dominantly by electric dipole transitions (A/ = 1), so that only a few final states are occupied. For ionization of the individual... 

Note that the form of a CIS wave function differs from that of an ordinary Cl wave function. In an ordinary Cl wave function, the reference function (the SCF wave function... 

If we include single excitations, other mechanisms become possible as well. With each excitation mechanism, we associate a characteristic probability amplitude. The final electronic state (5.7.1) is thus a linear combination of excited configurations, each with a total weight equal to the combined probabilities of all mechanisms leading to this particular configuration. In this respect, the coupled-cluster wave function differs fundamentally from the Cl wave fimction, where we parametrize the total weight of each excited configuration individually. 

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