Orbital Webisite Help Doc

Wave Function (ψ)

All particles (including electron) have wave-like property, and there is a wave function associated with the particle. In quantum mechanics, the wave function Ψ gives a complete description of the wavelike property of the electron. A stationary wave function Ψ is most naturally expressed in spherical coordinates (r, ѳ, φ). Ψ can be written as a product of the radial part (R) and the angular part (Y).

Ψ (r, ѳ, φ) = R(r) ·Y (ѳ, φ)            (hydrogen wave function)

Ψ of the electron in a hydrogenic atom can be found by solving the Schrödinger equation. The solutions to the Schrödinger equation are a set of possible wave functions (ψ), corresponding to a set of orbitals.

 HΨ = Eψ  (Schrödinger equation)

Probability Density (|ψ|²)

In general, the wavefunction ψ of an electron turns out to be complex function and does not have direct physical interpretation. Instead, we consider the real quantity |ψ|² , which when evaluated at a specific point is the probability of finding the electron at that point in space.  |ψ|² represents probability density - the probability of finding the electron in unit volume. An orbital is usually illustrated by a contour surface enclosing 90% of the electron probability.

Nodes

An orbital node, or more formally ‘nodal surface’, is a 2-dimensional surface where the value of the orbital, as well as the probability density (square of the wave function) vanishes, not including r=0 or r=∞.

Quantum Numbers

The solutions to the Schrödinger equation are a set of possible wave functions (ψ), corresponding to a set of orbitals. Each wave function is specified by a set of three interrelated integers - quantum numbers.

• The principal quantum number (n) determines the overall size and energy of an orbital for the hydrogen atom. The allowed values are n = 1, 2, 3, ……

• The angular momentum quantum number (l) determines the shape of the orbital. The allowed values are l = 0, 1, 2, … (n-1). For example, when n = 1, the possible value of l is 0; when n = 2, l = 0, 1. By convention, the values of l are represented by letters s, p, d, f, g, …… Orbitals with l = 0 are called s orbitals, orbitals with l = 1 are called p orbitals, and so on.
• The magnetic quantum number (m_l) determines the orientation of the orbital. The allowed values are: m_l = -l, … 0, … l. For example, when l = 0, m_l = 0; when l = 1, m_l has three possible values: -1, 0, 1. In other words, there are 2l +1 possible orbitals at the sublevel l. The wavefunctions with certain m_l values are complex (containing imaginary number), so we examine the linear combinations of them instead. The real orbitals resulted by taking linear combination of the complex wave functions are labeled by its orientation in the Cartesian coordinate, e.g. p_x , d_x2-y2.

Element

Select “Hydrogen” to view orbitals up to n=8. Select any other element in the first six periods to view the occupied atomic orbitals, whose radial wavefunction is approximated by hydrogenenic wavefunction with element- and orbital-dependent effective nuclear charge (Zeff).

Radial Wave Function: R(r)

Radial wave function R(r) is a separable part of the Ψ (r, ѳ, φ) that only depends on the distance r from the nucleus. It describes the wave-like property of the electron along r.

Radial Probability Density: R²(r)

The quantity of R(r)·R(r) gives the radial probability density of the electron --the probability for the electron to be found at a point with a distance from the nucleus is r. 

Radial Distribution Function (RDF): r²·R(r)²

Radial probability, also known as radial distribution function (RDF), is the probability of finding the electron within an infinitely thin shell of sphere with radius of r and nucleus as the center. The area of the sphere with radius r is 4πr², so the probability within that spherical shell is 4πr² · |ψ|². As the value of RDF only depends on r, it can be found by only evaluating the radial part of the wave function. RDF is given by r2·|R(r)|². Please note that 4π is dropped here so that RDF integrates to 1 over all distance r. 

Integral of RDF over r : ∫r²·|R(r)|²dr

The electron must be somewhere around the nuclear, so the integration of the radial distribution function (RDF) from the nuclear to infinity should be 1, which means:

.