Wave Functions for Many-Electron Systems

Determinantal Wave Functions for Many-Electron Systems 

You can see that as the number of electrons increases the determinantal wave function will become difficult to actually write on a page. However there is a simple pattern there. 

Again N is the normalization constant. On first encounter, one may ask, just what is p The operation denoted by P is a permutation of the order of the orbitals in the basis set performed in the mind of the reader. There is no calculus operator P —it is just the operation you use in your mind when you write down aU the possible permutations that evolve from expanding 

Suppose we expand the determinant for three electrons in a Li atom using a basis of only l a, li 3, and IsoL. 

Here the (1, 2, 3) order has been maintained and the alternating order of determinant expansion has been used. 

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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. 

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... 

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. ... 

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. 

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),... 

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. 

In this chapter, we shall now come back to the question how physical observables are associated with proper operator descriptions, which has already been addressed in section 4.3. All preceding chapters dealt with the proper construction of Hamiltonians for the calculation of energies and wave functions of many electron systems. Here, we shall now transfer this knowledge to the construction of relativistic expressions for first-principles calculations of molecular properties for many-electron systems. The basic guideline for this is the fact that all molecular properties can be expressed as total electronic energy derivatives. 

Natural orbitals were defined by P. O. Lbwdin as the eigenfunctions of the first-order reduced density matrix. They provide a simple factorization of wave functions for two-electron systems which brings them into a standard, easily interpreted form. For many-electron systems, they provide a basis for constructing Slater determinants so that the importance of the first few terms is maximized. Used in this way, they were the basis for the construction of some of the first accurate wave functions for molecules. With modern computers, the number of Slater determinants involved in the wave function is no longer so much of an issue and natural orbitals are now mainly used to reduce the wave function and density matrix to a reasonably compact form that facilitates interpretation. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

The Schrodinger equations for many-electron systems are generahzations of the one-electron problem. The wave function is a function of the space and spin coordinates of all the electrons ... 

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... 

Unfortunately, the stationary Schrodinger equation (1.13) can be solved exactly only for a small number of quantum mechanical systems (hydrogen atom or hydrogen-like ions, etc.). For many-electron systems (which we shall be dealing with, as a rule, in this book) one has to utilize approximate methods, allowing one to find more or less accurate wave functions. Usually these methods are based on various versions of perturbation theory, which reduces the many-body problem to a single-particle one, in fact, to some effective one-electron atom. 

The concepts of hybridisation and resonance are the cornerstones of VB theory. Unfortunately, they are often misunderstood and have consequently suffered from much unjust criticism. Hybridisation is not a phenomenon, nor a physical process. It is essentially a mathematical manipulation of atomic wave functions which is often necessary if we are to describe electron-pair bonds in terms of orbital overlap. This manipulation is justified by a theorem of quantum mechanics which states that, given a set of n respectable wave functions for a chemical system which turn out to be inconvenient or unsuitable, it is permissible to transform these into a new set of n functions which are linear combinations of the old ones, subject to the constraint that the functions are all mutually orthogonal, i.e. the overlap integral J p/ip dT between any pair of functions ip, and op, (i = j) is always zero. This theorem is exploited in a great many theoretical arguments it forms the basis for the construction of molecular orbitals as linear combinations of atomic orbitals (see below and Section 7.1). 

The physical content of the VSCF approximation is simple within this approximation, each vibrational mode is described as moving in the mean field of the other vibrational modes. The mean fields, and the wave functions of the different modes are determined self-consistently, so that the approximation is analogous to the Hartree method for many-electron systems. The total wave function within this simplest level of VSCF is thus... 

The cusp values of the wave function are also the necessary conditions of the exact wave function. Kato [34] rigorously derived the cusp conditions for many-electron systems as... 

Through adiabatic approximation, motions of the electrons and the nuclei are handled separately. The crucial problem is that the many-electron wave functions are difficult to express and calculate. When a solid system s electron numbers exceed 1000 (the solid with volume 1 cm contains 10 electrons), the multi-electron wave function will be an unreasonable scientific concept [2, 3]. For many-electron system, Eq. (6.2) still cannot be solved strictly due to the electron-electron repulsion operator -3-y in terms of potential energy. To solve the... 

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