Orbital wavefunctions solving Schrodinger equation

In 1926 the physicist Llewellyn Thomas proposed treating the electrons in an atom by analogy to a statistical gas of particles. No electron-shells are envisaged in this model which was independently rediscovered by Italian physicist Enrico Fermi two years later, and is now called the Thomas-Fermi method. For many years it was regarded as a mathematical curiosity without much hope of application since the results it yielded were inferior to those obtained by the method based on electron orbitals. The Thomas-Fermi method treats the electrons around the nucleus as a perfectly homogeneous electron gas. The mathematical solution for the Thomas-Fermi model is universal , which means that it can be solved once and for all. This should represent an improvement over the method that seeks to solve Schrodinger equation for every atom separately. Gradually the Thomas-Fermi method, or density functional theories, as its modem descendants are known, have become as powerful as methods based on orbitals and wavefunctions and in many cases can outstrip the wavefunction approaches in terms of computational accuracy. 

Here, we look at the atomic orbitals (AOs) that constitute the partly filled subshell we are dealing for the moment with free atoms/ions, as observed in the gas phase. An AO is a function of the coordinates of just one electron, and is the product of two parts the radial part is a function of r, the distance of the electron from the nncleus and thns has spherical syimnetry the angular part is a function of the x, y, and z axes and conveys the directional properties of the orbital. The notation nd indicates an AO whose / qnantum number is 2 we have five nd orbitals corresponding to m/ = 2,1, 0, —1, and —2. Solving the Schrodinger equation, we obtain the angular wavefunctions as equations (15). 

Undoubtedly, the methods most widely used to solve the Schrodinger equation are those based on the approach originally proposed by Hartree [1] and Fock [2]. Hartree-Fock (HF) theory is the simplest of the ab initio or "first principles" quantum chemical theories, which are obtained directly from the Schrodinger equation without incorporating any empirical considerations. In the HF approximation, the n-electron wavefunction is built from a set of n independent one-electron spin orbitals which contain both spatial and spin components. The HF trial wavefunction is taken as a single Slater determinant of spin orbitals. 

Although we have introduced references to bond formation with hybrid orbitals, we have not yet really tackled how to describe molecules using orbitals. The starting point for molecules, as for atoms, is the Schrodinger equation, and we can solve this to obtain electron wavefunctions or molecular orbitals. However, for molecules the electrons are attracted to all the nuclei in the molecule, not just one, and we have to include the repulsion between the nuclei in the energy. There are several methods and many programs available to calculate molecular orbitals. Nearly all employ two approximations. 

When the Schrodinger equation is solved in the Hartree-Fock— Roothaan procedure, the coefficients c,> are obtained and the wavefunction is at hand.2 Unfortunately, all the chemical information is contained in this wave-function, and it is expressed as a (very) long list of coefficients. As an example, a restricted Hartree-Fock calculation of benzene using the 6-31G basis set will have 102 atomic orbitals and 21 doubly occupied MOs for a total of 2142 coefficients. For the chemist, the interesting and pertinent data are entangled in a series of numbers, and the question becomes how to extract the chemical concepts from these numbers. 

Ab initio calculations rest on solving the Schrodinger equation the nature of the necessary approximations determine the level of the calculation. In the simplest approach, the HF method, the total molecular wavefunction F is approximated as a Slater determinant composed of occupied spin orbitals (each spin orbital is a product of a conventional... 

In terms of obtaining wavefunctions and energies for the atomic orbitals of He, it has not been possible to solve the Schrodinger equation exactly and only approximate solutions are available. For atoms containing more than two electrons, it is even more difficult to obtain accurate solutions to the wave equation. 

The procedure is called the linear combination of atomic orbitals (LC AO) approximation and can be used for molecules of any size. H2 is a special case in that a wavefunction can be found that will solve the Schrodinger equation exactly, yet the MO approach will be used so that molecular orbitals can be derived. The simplest trial function for the H2+ system is written ... 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

The next step is to develop a wavefunction. We will restrict our discussion to closed-shell atoms and molecules and to the most common approach that chemists take in solving Schrodinger s equation. That is, the wavefunction j is assumed to be a function of -electron coordinates with the nuclear coordinates frozen and is approximated by n one-electron functions referred to as orbitals. We will refer to these one-electron functions with the symbol x> > or i i, depending on the particular circumstance that we are discussing. But more about that later. We restrict our discussion to atoms for the moment. 

One approximate method for obtaining Eq. 7.182 in a central field form wa.s introduced by Hartree and named the self-consistent field approach. This method regards each electron in a many-electron atom as moving in the temporarily fixed field of the remaining electrons. The system can now be described in terms of one-electron wavefunctions (or orbitals) j(rj). The non-Coulomb potential energy for the jth electron is then V y(ry) atid this contains the other electronic coordinates only as parameters. Vjirj) can be chosen to be spherically symmetric. The computational procedure is to solve the Schrodinger equation for every electron in its own central field and then to make the wavefunctions, so found, self-consistent with their potential fields. The complete wavefunction for the system is a product of the one-electron functions. 

The modern theory of chemical bonding begins with the article The Atom and the Molecule published by the American chemist G. N. Lewis in 1916 [1], In this article, which is still well worth reading, Lewis for the first time associates a single chemical bond with one pair of electrons held in common by the two atoms "After a brief review of Lewis model we turn to a quantum-mechanical description of the simplest of all molecules, viz. the hydrogen molecule ion H J. Since this molecule contains only one electron, the Schrodinger equation can be solved exactly once the distance between the nuclei has been fixed. We shall not write down these solutions since they require the use of a rather exotic coordinate system. Instead we shall show how approximate wavefunctions can be written as linear combinations of atomic orbitals of the two atoms. Finally we shall discuss so-called molecular orbital calculations on the simplest two-electron atom, viz. the hydrogen molecule. 

Potentials in atoms are complicated as are the resulting wavefunctions of the valence electrons. However, these orbitals have some features in common with those of electrons in some simple potentials. The orbitals in a cubic flat-box potential (the potential is considered to be constant over the entire volume of the box) turn out to be sine functions if the Schrodinger equation is solved, like the eigenfunctions of a vibrating string. Each different potential imposes a different form on the wavefunctions, which are similar in shape to the sine functions of the flat-bottomed box with nodal planes that increase in number with the quantum number n. 

To go beyond the Hartree-Fock limit and obtain the full solution to the Schrodinger equation (in the non-relativistic and Bom-Oppenheimer limit), one would have to combine various solutions of the product type. In any calculation one obtains more molecular orbitals than needed to accommodate all the electrons in the system. In a system with 2n electrons, the n molecular orbitals with the lowest molecular orbital energies are used in the Hartree-Fock solution for the ground state (this assumes a closed shell system, where two electrons are paired up in each molecular orbital). The rest of the molecular orbitals obtained will be excited molecular orbitals. Of course, other possible wavefunctions of the product type can be formed by using excited molecular orbitals in the product. The set of all such possible products can be used as a basis set to solve the full Schrodinger equation. The solution now looks like ... 

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