Wave function orbitals

HyperClicm cati plot orbital wave fuuctious resulting fmni serni-cmpirical and ah i/iitw quan tii m m ecli an ica I calculations. It is ill tercstiu g to view both tli c u tidal properties an d th e relative sizes of the wave functions. Orbital wave functiou s can provide dietni-cal in sigh is. 

Section 1 1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron con figurations of atoms Neutral atoms have as many electrons as the num ber of protons m the nucleus These electrons occupy orbitals m order of increasing energy with no more than two electrons m any one orbital The most frequently encountered atomic orbitals m this text are s orbitals (spherically symmetrical) and p orbitals ( dumbbell shaped)... 

HyperChem can plot orbital wave functions resulting from semi-empirical and ab initio quantum mechanical calculations. It is interesting to view both the nodal properties and the relative sizes of the wave functions. Orbital wave functions can provide chemical insights. 

Approximating a one-electron wave function (orbital) by an expansion in a finite basis set. 

The quantum numbers n and i. Multi-electron atoms can be characterized by a set of principal and orbital quantum numbers n, t which labels one-electron wave functions (orbitals). 

Clearly, however, electrons exist. And they must exist somewhere. To describe where that somewhere is, scientists used an idea from a branch of mathematics called statistics. Although you cannot talk about electrons in terms of certainties, you can talk about them in terms of probabilities. Schrodinger used a type of equation called a wave equation to define the probability of finding an atom s electrons at a particular point within the atom. There are many solutions to this wave equation, and each solution represents a particular wave function. Each wave function gives information about an electron s energy and location witbin an atom. Chemists call these wave functions orbitals. 

In further studies of chemistry and physics, you will learn that the wave functions that are solutions to the Schrodinger equation have no direct, physical meaning. They are mathematical ideas. However, the square of a wave function does have a physical meaning. It is a quantity that describes the probability that an electron is at a particular point within the atom at a particular time. The square of each wave function (orbital) can be used to plot three-dimensional probability distribution graphs for that orbital. These plots help chemists visualize the space in which electrons are most likely to be found around atoms. These plots are... 

For a multi-Slater determinant wave function, orbitals which satisfy Eq. (3.6), and therefore Eq. (3.7), can still be defined. For these orbitals, referred to as the natural spin orbitals, the coefficients nt are not necessarily integers, but have the boundaries 0 n, 1. 

The minimum requirements for a many-electron wave function, namely, antisymmetry with respect to interchange of electrons and indistinguishability of electrons, are satisfied by an antisymmetrized sum of products of one-electron wave functions (orbitals), ( 1),... 

In summary, to obtain a many-electron wave function of the single determinantal form [equation (A.12)] which will give the lowest electronic energy [equation (A.14) or (A.27)], one must use one-electron wave functions (orbitals) which are eigenfunctions of the one-electron Fock operator according to equation (A.42). There are many, possibly an infinite number of, solutions to equation (A.42). We need the lowest Ne of them, one for each electron, for equation (A. 12) [or (A.27)]. When the Ne MOs of lowest energy satisfy equation (A.42), then Eq=Ehf [equation (A.27)] and o= hf [equation (A.12)]. 

Wave functions (orbitals) filling Electron configuration comprising ... 

As we have seen, when we solve the Schrodinger equation for the hydrogen atom, we find many wave functions (orbitals) that satisfy it. Each of these orbitals is characterized by a set of quantum numbers that arise when the boundary conditions are applied. Now we will systematically describe these quantum numbers in terms of the values they can assume and their physical meanings. 

In the quantum mechanical model the electron is described as a wave. This representation leads to a series of wave functions (orbitals) that describe the possible energies and spatial distributions available to the electron. 

Figure 7.1 Overlap of the atomic orbitals of hydrogen. Positive phase for wave functions (orbitals) shown red, negative as white...

The diagonal elements of this matrix give the probability density for the electrons in a point r with the spin s. A matrix representation of the density matrix can be obtained by an expansion in the basis of SOs used to construct the total wave function (orbitals which... 

For practical reasons, various graphical representations of atomic orbitals are used. The most useful are boundary surfaces, such as those shown in Figure 2. These enclose regions of space where the electron described by the corresponding wave function (orbital) can be fotmd with high probability (e.g., 99%) s orbitals are spherical, p orbitals are dumb-bell shaped, d orbitals have a four-leaf-clover shape, while/orbitals have complex shapes. 

List the most important ideas of the quantum mechanical model of the atom. Include in your discussion the terms or names wave function, orbital, Heisenberg uncertainty principle, de Broglie, Schrodinger, and probability distribution. 

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