Wave Function for Many Electrons

A wave function for many (N) electrons (or other particles) is interpreted in the same way. The wave function is a function of position and spin of each of the individual 

The absolute square of gives the probability density that electron 1 is at r, with spin 1, electron 2 at 2 with spin i 2, etc. Such a function can be described in many different ways. A correct although not very useful way would be to make a table of P in possible positions and spins of N electrons. We would need to do this for a great number of points for each electron, let us say for 100 values of r,. For each choice of r, 100 values of i2 are necessary and for each pair (ri, 2), 100 values of Fj, etc. In the case of the iron atom with N = 26, we need to give T in 10 different positions of the electrons Of course, this is impossible even using all the compact disks and USB sticks in the world. 

A simpler description is to assume that the electrons move independently of each other. The electronic motion is separable and the wave function is a product of orbitals (Section 1.5.4). Therefore, we try to approximate as an orbital product  

To describe the orbital product in Equation 1.83, we may plot 100 orbital values for the positions of each of the N = 26 electrons, altogether 2600 points in the case of the iron atom. Alternatively, one may use a linear expansion of the orbital, in which case only the coefficients have to be listed. 

Expansion in an orbital product is thus necessary, at least as a first approximation. The problem is that such an expansion cannot be exact in general. Equation 1.83 distributes electron 1 according to xy, irrespective of the coordinates of the other electrons. This is an approximation, since the probability amplitude for electron 1 depends on the coordinates of the other electrons. If electron 2 moves a bit, this should influence the way electron 1 is distributed. The positions are, in fact, correlated, and this correlation cannot be correctly represented by just a product of orbitals. 

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For systems that contain only one electron there is no difference in the molecular-orbital and the total electronic wave function. For many-electron systems, however, there is a considerable difference. 

Our earlier discussion of electronic wave functions for many-electron atoms drew attention to the main inadequacy of the Hartree-Fock single determinant treatment it does not take account of the correlation between the motions of electrons with opposite spins. In molecules this can even lead to qualitative deficiencies in the description of electronic structure, such as the failure to describe dissociation correctly. For example, the correct wave function for the singlet state of the hydrogen molecule at large... 

A necessary and reasonable approximation in the generation of wave functions for many-electron systems is to represent the many-electron wave function using products of one-electron wave functions or orbitals, c >,. To ensure compliance with the Pauli exclusion principle, the wave... 

Commentary on Independent Assessments of the Accuracy of Correlated Wave Functions for Many-Electron Systems, S. F. Boys, Symp. Faraday Soc., 1968, 2... 

Independent Assessments of the Accuracy of Correlated Wave Functions for Many-Electron Systems... 

RADIAL WAVE FUNCTIONS FOR MANY-ELECTRON ATOMS... 

Other prescriptions for the exponents, have been advanced over the years. Clementi and Raimondi (8) proposed in 1963 that the best exponents should be based on the criterion that the atomic energy should be minimized. Clementi, too, (9,10) and others (11) have investigated the use of more than one Slater function to obtain a better representation of the radial wave functions for many-electron atoms. 

The orbital concept and the Pauli exclusion principle allow us to understand the periodic table of the elements. An orbital is a one-electron spatial wave function. We have used orbiteils to obteiin approximate wave functions for many-electron atoms, writing the wave function as a Slater determinant of one-electron spin-orbitals. In the crudest approximation, we neglect all interelectronic repulsions and obtain hydrogenlike orbitals. The best possible orbitals are the Heu tree-Fock SCF functions. We build up the periodic table by feeding electrons into these orbitals, each of which can hold a pair of electrons with opposite spin. 

In this section we are concerned with the nomenclature, the conventions, and the procedure for writing down the wave functions that we use to describe many-electron systems. We will only consider many-electron wave functions that are either a single Slater determinant or a linear combination of Slater determinants. Sometimes, for very small systems, special functional forms are used for the wave function, but in most cases quantum chemists use Slater determinants. Before considering wave functions for many electrons, however, it is necessary to discuss wave functions for a single electron. 

For systems that contain only one electron there is no difference in the molecular-orbital and the total electronic wave function. For many-electron systems, however, there is a considerable difference. It should be noted that for many-electron systems it is only the symmetry of the total wave function which has physical (and chemical ) significance. This quantity is the only observable quantity. ... 

Boys, S. F., Proc. Roy. Soc. London) A207, 181, Electronic wave functions. IV. Some general theorems for the calculation of Schrodinger integrals between complicated vector-coupled functions for many-electron atoms."... 

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

All this is very straightforward if we are dealing with a system which contains only one electron such as the hydrogen atom or the hydrogen molecule-ion Ha+. But when we consider a many-eiectron molecule we are faced with the problem of combining the orbitals for the individual electrons into a total wave function for the whole system. Suppose we are dealing with two electrons which occupy space orbitals xpt and p2. The simplest compound wave function for both electrons is the product... 

The method of CFP is an elegant tool for the construction of wave functions of many-electron systems and the establishment of expressions for matrix elements of operators corresponding to physical quantities. Its major drawback is the need for numerical tables of CFP, normally computed by the recurrence method, and the presence in the matrix elements of multiple sums with respect to quantum numbers of states that are not involved directly in the physical problem under consideration. An essential breakthrough in this respect may be finding algebraic expressions for the CFP and for the matrix elements of the operators of physical quantities. For the latter, in a number of special cases, this can be done using the eigenvalues of the Casimir operators [90], however, it would be better to have sufficiently simple but universal formulas for the CFP themselves. 

The submatrix elements that enter into the left sides of these equations can be expressed in terms of complex CFP [112]. Using these coefficients the many-shell wave function for N electrons (N = Ni + N2 +...+ Nu) is composed of the antisymmetric wave functions of (N — 1) electrons... 

As we have seen in Chapter 4, relativistic operators of the Ek-transitions in the general case have several forms and are dependent on the gauge condition of the electromagnetic field potential. These forms are equivalent and do not depend on gauge for exact wave functions. Unfortunately, we are always dealing with the more or less approximate wave functions of many-electron systems, therefore we need general expressions for the appropriate matrix elements. 

Although 4>(r, t) is the wave function for one-electron systems only, it may provide p(r, t) and (r, t) as above, for many-electron systems as well. A generalized nonlinear Schrodinger equation (GNLSE) may be obtained [17,47] as follows, as the backbone of QFDFT, by combining equations (9a) and (9b),... 

Fig. 3 shows a qualitative graphical representation of hydrogen-like wave functions for one-electron atoms which have to be replaced for many-electron atoms at least by Slater-type 107) analytical wave functions ifnlm (1) which are approximate as they contain no nodes in the radial part R ,. 

It is convenient to have a word for the space-dependent part of a wave function for one electron, to distinguish it from the whole wave function which includes reference to the spin. Orbital is the customary word. Such a function as the Is wave function, obtained by solving Schrodinger s equation for an atom, is called an atomic orbital. Such functions as the wave functions used in the last chapter for the hydrogen-molecule ion are called molecular orbitals. The foregoing method of approximation in a many-electron problem is therefore often called the atomic orbital method. 

To see how electron localisation for large D would effect the partial wave expansions for many-electron atoms, we may for simplicity consider the helium atom, which is described by three internal coordinates, ri, T2, and. The localisation in the variables ri and V2 can easily be accommodated by using basis functions with flexible length scales, but the localisation in 6 would be very slowly described by truncated partial wave expansions, which contain no internal angular scaling parameter. Thus as D increases, the partial wave expansion has less trouble describing the electron-electron cusp, but more and more trouble describing the localisation of electrons. 

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