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 (ml)
Wavefunctions with m valued other than 0 are complex (contain eimφ part). In chemistry we examine the real orbitals, which are the linear combinations of the complex orbitals. The real orbitals are labeled by its orientation in Cartesian coordinate. For example, φ211 (m =1) and φ21-1 (m =-1) orbitals are complex, but (1/√2)(φ211 + φ21-1 ) is real and is called 2px orbital as it orients along x-axis when plotted in Cartesian coordinate. In the drop-down menu, “2px [m=+-1(+)]” means: real orbital 2px is the normalized sum of 2p orbitals with m=1 and -1.
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.
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).
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:
.
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: |R(r)|2
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.
