Independent-electron wavefunction

This form of Dp implies that Ap = 0 for each p > 1, a reflection of the fact that an independent-electron wavefunction consists of one-electron subsystems coupled only by exchange. 

Let the ground state of helium be our exanple. We take the ordinary independent-electron wavefunction as our initial approximation ... 

Imagine a model hydrogen molecule with non-interacting electrons, such that their Coulomb repulsion is zero. Each electron in our model still has kinetic energy and is still attracted to both nuclei, but the electron motions are completely independent of each other because the electron-electron interaction term is zero. We would, therefore, expect that the electronic wavefunction for the pair of electrons would be a product of the wavefunctions for two independent electrons in H2+ (Figure 4.1), which I will write X(rO and F(r2). Thus X(ri) and T(r2) are molecular orbitals which describe independently the two electrons in our non-interacting electron model. 

I ain going to leave you to prove for yourself that the wavefunction corresponding to this infinite-distance H2 problem is a product of two hydrogen atom wavefunc-tions. Physically, you might have expected this the two atoms are independent so the electronic wavefunctions multiply to give the molecular electronic... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

Unfortunately, the many-electron wavefunction itself does not necessarily provide insight into the chemistry of complex molecules as it describes the electronic distribution over the whole system. It is therefore assumed that the true many-electron wavefunction. can be represented as the product of a series of independent, one-electron wavefunctions ... 

Although in many cases, particularly in PE spectroscopy, single configurations or Slater determinants 2d> (M+ ) were shown to yield heuristically useful descriptions of the corresponding spectroscopic states 2 f i(M+ ), this is not generally true because the independent particle approximation (which implies that a many-electron wavefunction can be approximated by a single product of one-electron wavefunctions, i.e. MOs 4>, as represented by a Slater determinant 2 j) may break down in some cases. As this becomes particularly evident in polyene radical cations, it seems appropriate to briefly elaborate on methods which allow one to overcome the limitations of single-determinant models. 

The basic problem is to solve the time-independent electronic Schrodinger equation. Since the mass of the electrons is so small compared to that of the nuclei, the dynamics of nuclei and electrons can normally be decoupled, and so in the Born-Oppenheimer approximation the many-electron wavefunction P and corresponding energy may be obtained by solving the time-independent Schrodinger equation in which the nuclear positions are fixed. We thus solve... 

This kind of wavefunction is called a Hartree Product, and it is not physically realistic. In the first place, it is an independent-electron model, and we know electrons repel each other. Secondly, it does not satisfy the antisymmetry principle due to Pauli which states that the sign of the wavefunction must be inverted under the operation of switching the coordinates of any two electrons, or... 

Physically, the wavefunction of Eq. (19.6a) corresponds to adding one electron to Ba2+ to make the Ba+ n C states and then adding the second n electron to form the neutral Ba n Cni states. On each state of Ba+ there is built an entire system of Rydberg states and continua, as shown in Fig. 19.1 for the three lowest lying states of Ba+. Note that the Rydberg states converging to the 5d and 6p states of Ba+ are above the Ba+ 6s state. With the independent electron Hamiltonian H0 these states can only decay radiatively. They are not coupled to the degenerate continua above the Ba+ 6s state. 

The correlated wavefunction which incorporates ISCI follows in analogy to equ. (1.25a) as an expansion into independent-particle wavefunctions for the ground state and contributions from virtual two-electron excitations ... 

Inserting (2.29) into the time-independent Schrodinger equation (2.5), multiplying with (5%l from the left, and exploiting the orthogonality of the electronic wavefunctions for each nuclear configuration Q readily yields a set of coupled equations for the nuclear wavefunctions (Koppel, Domcke, and Cederbaum 1984),... 

Within the Born-Oppenheimer approximation the time-independent molecular wavefunctions for the various electronic states are written as... 

Using a plane wave representation for the electron wavefunction with 163 grid points and approximately 800 independent electronic and molecular configurations from the path integral molecular dynamics trajectories, we have also computed the density of states for the electron under different supercritical conditions of the solvents and the corresponding steady-state optical absorption spectra. The latter were computed within the dipolar approximation from the following expression within the Frank-Condon approximation ... 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

It will be recalled that the approach of molecular orbital (MO) theory starts, on the other hand, from an independent-particle model (IPM) in which both electrons occupy the same bonding MO , 1 = Xa + Xb, similar to the one used [4] for the hydrogen molecule ion, Hj. The bonding MO is in fact the approximate wavefunction for a single electron in the field of the two nuclei and allocating two electrons to this same MO, with opposite spins, yields the 2-electron wavefunction... 

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