One-electron function

The simplest many-electron wave function that satisfies the Exclusion Principle is a product of N different one-electron functions that have been antisymmetrized, or written as a determinant. Here, N is the number of electrons (or valence electrons) in the molecule. HyperChem uses this form of the wave function for most semi-empirical and ab initio calculations. Exceptions involve using the Configuration Interaction option (see page 119). HyperChem computes one-electron functions, termed molecular spin orbitals, by relatively simple integration and summation calculations. The many-electron wave function, which has N terms (the number of terms in the determinant), never needs to be evaluated. 

Any set of one-electron functions can be a basis set in the LCAO approximation. However, a well-defined basis set will predict electronic properties using fewer terms than a poorly-defined basis set. So, choosing a proper basis set in ab initio calculations is critical to the reliability and accuracy of the calculated results. 

The next approximation involves expressing the jiiolecular orhiiah as linear combinations of a pre-defined set of one-electron functions kjiown as basis functions. These basis functions are usually centered on the atomic nuclei and so bear some resemblance to atomic orbitals. However, the actual mathematical treatment is more general than this, and any set of appropriately defined functions may be u.sed. 

I am assuming that this particular electronic state is the lowest-energy one of that given spatial symmetry, and that the i/f s are orthonormal. The first assumption is a vital one, the second just makes the algebra a little easier. The aim of HF theory is to find the best form of the one-electron functions i/ a,. .., and to do this we minimize the variational energy... 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

Since the exact density matrix is not known, the (approximate) density is written in terms of a set of auxiliary one-electron functions, orbitals, as... 

Before proving this theorem, we will make some general remarks about the nature of the one-electron functions ipk(x) or spin orbitals. For the two values of the spin coordinate f — 1, such a function y)k(r, f) has two space components... 

Let us now consider a system of N electrons, where N+ electrons occupy spin orbitals of a character or plus spin, and N electrons occupy spin orbitals of character or minus spin. By using the separation of the one-electron functions y>k x) into two groups having different spins, we may write the fundamental invariant (Eq. 11.41) in the form... 

The method of superposition of configurations described in the first paper (1928) implies that, after choosing a complete basic set of one-electron functions rpk rx) one can develop the space function in Eq. III. 1 in the form ... 

Every normalizable antisymmetric wave function can be expressed as the sum of a series of Slater determinants built up from a complete basic set of one-electron functions. 

The results in Table VI are of considerable interest also for atoms with more than two electrons, since they show the possibilities and limitations of the method of superposition of configurations/ when the latter are built up from one-electron functions... 

We note that it is possible to combine the method with correlation factor with the method using superposition of configurations to obtain any accuracy desired by means of comparatively simple wave functions. For a very general class of functions g(r12), one can develop the quotient (r r2)lg(r12) according to Eq. III.2 into products of one-electron functions y>k(r), which leads to the expansion... 

One-electron function, 228, 261 One particle scheme, 210 Ordered configuration, 261 Ortho-iodo-phenol, radiation resistance, 196... 

The total wavefunction, , is an antisymmetrized product of the one-electron functions i/q (a Slater determinant). The i/tj are called one-electron functions since they depend on the coordinates of only one electron this approximation is embedded in all MO methods. The effects that are missing when this approximation is used go under the general name of electron correlation. 

The same expression can be used with the appropriate restrictions to obtain matrix elements over Slater determinants made from non-orthogonal one-electron functions. The logical Kronecker delta expression, appearing in equation (15) as defined in (16)] must he substituted by a product of overlap integrals between the involved spinorbitals. 

A simple product 5 = XiC i) X2( 2) XiC i) 3Cj( j)-"Xn( n) 15 not acceptable as a model wave function for fermions because it assigns a particular one-electron function to a particular electron (for example X to X]) and hence violates the fact that electrons are indistinguishable. In addition,... 

Such a "general form of wave function is easily written explicitly for each set of values of N, S, and MS- Any appropriate form of approximate wave functions, like determinantal functions composed of one-electron functions ( molecular spin orbitals ), the "bond eigenfunctions" used in the valence bond approach, and so on, is shown to fulfil this requirement. 

The eigenfunctions of H are determinants formed from one-electron functions which are eigenfunctions of h9 say,... 

The most simple approach is the Hartree-Fock (HF) self-consistent field (SCF) approximation, in which the electronic wave function is expressed as an antisymmetrized product of one-electron functions. In this way, each electron is assumed to move in the average field of all other electrons. The one-electron functions, or spin orbitals, are taken as a product of a spatial function (molecular orbital) and a spin function. Molecular orbitals are constructed as a linear combination of atomic basis functions. The coefficients of this linear combination are obtained by solving iteratively the Roothaan equations. 

The HMO model further assumes that the wave function is a product of one-electron functions and that Hn is the sum of one-electron operators ... 

This equation shows the relationship between the one-electron function, yx(r r,), and the main part of the energy functional—the rest of the functional, which is the electron-electron repulsion, depends on y2(ri, r2). The last term is also a functional of the one-electron density. In the new reformulation of DFT, the methodology is... 

An important characteristic of ab initio computational methodology is the ability to approach the exact description - that is, the focal point [11] - of the molecular electronic structure in a systematic manner. In the standard approach, approximate wavefunctions are constructed as linear combinations of antisymmetrized products (determinants) of one-electron functions, the molecular orbitals (MOs). The quality of the description then depends on the basis of atomic orbitals (AOs) in terms of which the MOs are expanded (the one-electron space), and on how linear combinations of determinants of these MOs are formed (the n-electron space). Within the one- and n-electron spaces, hierarchies exist of increasing flexibility and accuracy. To understand the requirements for accurate calculations of thermochemical data, we shall in this section consider the one- and n-electron hierarchies in some detail [12]. 

Whereas the one-electron exponential form Eq. (5.5) is easily implemented for orbital-based wavefunctions, the explicit inclusion in the wavefunction of the interelectronic distance Eq. (5.6) goes beyond the orbital approximation (the determinant expansion) of standard quantum chemistry since ri2 does not factorize into one-electron functions. Still, the inclusion of a term in the wavefunction containing ri2 linearly has a dramatic impact on the ability of the wavefunction to model the electronic structure as two electrons approach each other closely. 

Hartree-Fock wavefunction as a sin e Slater determinant of one-electron functions and then considered the evolution operator, which generates the time variation of these functions. For the present chapter, we believe it is more transparent to deal with the one-electron functions themselves, rather than the c-operators or the evolution operator. Consequently, we work entirely in the Schrodinger representation. 

The Gaussian orbitals are very important in practical applications. In spite of their wrong asymptotic behaviour at both r —> 0 and r —> °o, nearly all molecular electronic structure calculation programs have been constructed using Gaussian sets of one-electron functions. In this example the Gaussian basis has been selected as... 

Formulae for calculating transition probabilities in both the LS and Jcl coupling schemes, within the context of the Relativistic Quantum Defect Orbital (RQDO) formalism, which yields one-electron functions, are given and applied to the complex atomic system Ar 11. The application of a given coupling scheme to the different energy levels dealt with is justified. 

Molecular orbital an initio calculations. These calcnlations represent a treatment of electron distribution and electron motion which implies that individual electrons are one-electron functions containing a product of spatial functions called molecular orbitals hi(x,y,z), 4/2(3 ,y,z), and so on. In the simplest version of this theory, a single assignment of electrons to orbitals is made. In turn, the orbitals form a many-electron wave function, 4/, which is the simplest molecular orbital approximation to solve Schrodinger s equation. In practice, the molecular orbitals, 4 1, 4/2,- -are taken as a linear combination of N known one-electron functions 4>i(x,y,z), 4>2(3,y,z) ... 

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