Wavefunction multielectron

In the Bom interpretation (Section 4.2.6) the square of a one-electron wavefunction ij/ at any point X is the probability density (with units of volume-1) for the wavefunction at that point, and j/ 2dxdydz is the probability (a pure number) at any moment of finding the electron in an infinitesimal volume dxdydz around the point (the probability of finding the electron at a mathematical point is zero). For a multielectron wavefunction T the relationship between the wavefunction T and the electron density p is more complicated, being the number of electrons in the molecule times the sum over all their spins of the integral of the square of the molecular wavefunction integrated over the coordinates of all but one of the electrons (Section 5.5.4.5, AIM discussion). It can be shown [9] that p(x, y, z) is related to the component one-electron spatial wavefunctions ij/t (the molecular orbitals) of a single-determinant wavefunction T (recall from Section 5.2.3.1 that the Hartree-Fock T can be approximated as a Slater determinant of spin orbitals i/qoc and i// /i) by... 

Cf. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, p 422, equation (13.130) cf. p 624, problem 15.67, for the Kohn-Sham orbitals. The multielectron wavefunction is treated on pp 421 -23... 

The multielectron wavefunction T is a function of the spatial and spin coordinates of all the electrons. Physicists say that T for any system tells us all that can be known about the system. Do you think the electron density function p tells us everything that can be known about a system Why or why not ... 

The intensity of a spectral transition is calculated from matrix elements involving the initial and final state wavefunctions and the transition operator. This leads to selection rules, as for multielectron atoms, in terms of restrictions upon the values of S, L and / in the transition (Table 1). Considering the selection rules for vibrational spectra, the k=0 phonons of the initial and terminal states transform as T, and Tf, respectively, of the point group... 

The model that is outlined above is generated from a one-electron Hamiltonian and is only an approximation to the true wavefunction for a multielectron system. As suggested earlier, other components may be added as a linear combination to the wavefunction that has Just been derived. There are many techniques used to alter the original trial wavefunction. One of these is frequently used to improve wavefunctions for many types of quantum mechanical systems. Typically a small amount of an excited-state wavefunction is included with the minimal basis trial function. This process is called configuration interaction (Cl) because the new trial function is a combination of two molecular electron configurations. For example, in the H2+ system a new trial function can take the form... 

Complexity is used in very different fields (dynamical systems, time series, quantum wavefunctions in disordered systems, spatial patterns, language, analysis of multielectronic systems, cellular automata, neuronal networks, self-organization, molecular or DNA analyses, social sciences, etc.) [25-27]. Although there is no general agreement about the definition of what complexity is, its quantitative characterization is a very important subject of research in nature and has received considerable attention over the past years [28,29]. 

Elementary particles of the same kind are indistinguishable. The remaining symmetries do not refer to space or time but to the permutational symmetry of a set of particles. All electrons are the same, and thus the Hamiltonian does not change when we permute electron labels. This symmetry becomes apparent only when multielectronic wavefunctions are considered, and these will be treated in the next chapter. Likewise, identical nuclei can be permuted without changing the Hamiltonian. This is reflected in the potential energy terms, which consist of sums over all pairwise interactions. Permutations of particle labels change only the order of the terms in these summations. 

This last term generates a non linear perturbation. The general treatment of such a case has been considered already [17). It consists in expanding the wavefunction in terms of the multielectronic eigenfunctions of the unperturbed operator, to successive orders of perturbation and replacing F by this expansion in (23). 

To date, there is no known analytic solution to the second-order differential Schrodinger equation for the helium atom. This does not mean that there is no solution, or that wavefunctions do not exist. It simply means that we know of no mathematical function that satisfies the differential equation. In fact, for atoms and molecules that have more than one electron, the lack of separability leads directly to the fact that there are no known analytical solutions to any atom larger than hydrogen. Again, this does not mean that the wavefunctions do not exist. It simply means that we must use other methods to understand the behavior of the electrons in such systems. (It has been proven mathematically that there is no analytic solution to the so-called three-body problem, as the He atom can be described. Therefore, we must approach multielectron systems differently.)... 

We begin with the assumption that the electrons in a multielectron atom can in fact be assigned to approximate hydrogen-like orbitals, and that the wavefunction of the complete atom is the product of the wavefunctions of each occupied orbital. These orbitals can be labeled with the quantum number labels Is, 2s, 2p, 3s, 3p, and so on. Each s,p,d,f,... subshell can also be labeled by an quantum number, where ranges from — to T (2T + 1 possible values). But it can also be labeled with a spin quantum number m either -f or —The spin part of the wavefunction is labeled with either a or p, depending on the value of for each electron. Therefore, there are several simple possibilities for the approximate wavefunction for, say, the lowest-energy state (the ground state) of the helium atom ... 

FIGURE 12.5 Plots of 47rF Pp for the 3d and 4s wavefunctions. Note that the plots have the same x-axis, and that the 4s electron has some probability of being rather close to the nucleus. For multielectron atoms, the penetration of the 4s electron combined with the shielding effect of the other electrons serves to make the 4s orbital the next one occupied by electrons, rather than the 3d. 

In a previous section, we presumed that the wavefunctions of multielectron atoms can be approximated as products of hydrogen-like orbitals ... 

Spin orbitals are products of spatial and spin wave-functions, but correct antisymmetric forms of wavefunctions for multielectron atoms are sums and differences of spatial wavefunctions. Explain why acceptable antisymmetric wave-functions are sums and differences (that is, combinations) instead of products of spatial wavefunctions. 

The above example and selection rules are also applicable to hydrogen-like ions, which have a single electron. However, such systems are in the vast minority of atomic species whose spectra need to be understood. Recall that one of the final failings of classical mechanics was the inability to explain spectra. Although quantum mechanics does not provide analytic solutions for wavefunctions of multielectron systems, it does provide tools for understanding it. 

Note that the complete wavefunction as written in Eq. (2.47) changes sign if the labels of the electrons (1 and 2) are interchanged. W. PauU pointed out that the wavefunctions of all multielectronic systems have this property. The overall wavefunction invariably is antisymmetric for an interchange of the coordinates (both positional and spin) of any two electrons. This assertion rests on experimental measurements of atomic and molecular absorption spectra absorption bands predicted on the basis of antisymmetric electrOTiic wavefunctirais are seen experimentally, whereas bands predicted on the basis of symmetric electronic wave-functions are not observed. Its most important implication is the Pauli exclusion principle, which says that a given spatial wavefunction can hold no more than two electrons. This follows if an electron can be described completely by specifying its spatial and spin wavefunctions and electrons have only two possible spin wave-functions (a and fi). 

Molecules are multielectron systems. Approximate electronic wavefunctions of a molecule can be written as products of one-electron wavefunctions, each consisting of an orbital and a spin part ... 

The i are appropriate molecular orbitals (MO) and ij is one of the two possible spin eigenfunctions, a or p. The orbital part of this multielectron wavefunction defines the electronic configuration. 

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