The Orbitals Erwin Schrödinger (1933 Noble laureate) was the first to successfully apply the concept of the wave nature of matter to electronic structure. Erwin Schrödinger (1887 – 1961) He developed an equation that can be solved to give wave functions and energy levels for electrons trapped in them. Wave functions for electrons in atoms are called orbitals.

The Quantum Numbers The Spin Quantum Number . There are four different quantum numbers are associated with each electron in an atom, they are The Principal Quantum Number n The Angular Momentum (Azimuthal) Quantum Number l The Magnetic Quantum Number ml The Spin Quantum Number ms

The Principal Quantum Number (n) Put it simply the principal quantum numbers ‘n’ represent the different energy levels in an atom, with n = 1 at the lowest energy level. The principal quantum numbers are also called as the main shells. Higher numbers of ‘n’ indicates higher energy levels. The ‘n’ values are whole numbers, the possible values are n = 1, 2, 3 … Among the known elements the ‘n’ values do not exceed the number ‘7’.

The Angular Momentum Quantum Number (l ) The Angular Momentum(Azimuthal) Quantum Number (l ) is an integer that determines the shape of the orbital.

The Angular Momentum Quantum Number (l ) The Angular Momentum(Azimuthal) Quantum Number (l ) is an integer that determines the shape of the orbital. The Angular Momentum Quantum Number (l ) can be seen as the subdivision of the principal quantum number or simply subshells. The possible values of ‘l’ are 0, 1, 2 .. and the values are found by using the formula,. n – 1

Example – 1 Find the values of l for n = 1 1 – 1 = 0 Example – 2 Find the values of l for n = 2 2 – 1 = 1 Warning: This answer does not mean that ‘l ’ has a value of one. It means that it includes number 1 and anything below one including 0 (zero). Therefore, we have two subshells, with the Angular Momentum Quantum Numbers 0 and 1.

Example – 3 Find the values of l for n = 4 4 – 1 = 3 Warning again: This answer does not mean that ‘l ’ has a value of three. It means that it includes number 3 and anything below including 0 (zero). Therefore, we have four subshells, with the Angular Momentum Quantum Numbers 0, 1, 2 and 3.

Value of l Letter Designation l = 0 s l = 1 p l = 2 d l = 3 f The subshells are usually represented by letters, s, p, d and f Value of l Letter Designation l = 0 s l = 1 p l = 2 d l = 3 f .

The Magnetic Quantum Number (ml) The Magnetic Quantum Number (ml) is an integer that specifies the orientation of the orbital. The magnetic quantum number which essentially says how many types of subshells are there and how are they orientated. To find the magnetic quantum number (ml) first we have to know the value of angular momentum quantum number. The values magnetic quantum number (ml) range from – l to + l, Example – 1 Find the values of ml for l = 0 (s orbital). This zero says that, there is only one type of s orbital in any given energy level.

The Magnetic Quantum Number (ml) The Magnetic Quantum Number (ml) is an integer that specifies the orientation of the orbital. The magnetic quantum number which essentially says how many types of subshells are there and how are they orientated. To find the magnetic quantum number (ml) first we have to know the value of angular momentum quantum number. The values magnetic quantum number (ml) range from – l to + l, Example – 1 Find the values of ml for l = 0 (s orbital). This zero says that, there is only one type of s orbital in any given energy level.

Example – 2 Find the values of ml for l = 1 (p orbital). – 1, 0, + 1 This answer says that there are three types of p orbitals in any given energy level. Find the values of ml for an where l = 2 (d orbital). Example – 3 – 2, – 1, 0, + 1, +2 This answer says that there are five types of d orbitals in any given energy level.

Example – 4 Find the values of ml for l = 3 (f’ orbital). – 3, – 2, – 1, 0, + 1, +2, +3 This answer says that there are seven types of f orbitals in any given energy levels. Example – 5 Find the values of ml for l = 4 – 4, – 3, – 2, – 1, 0, + 1, +2, +3, +4 This answer says that there are nine types of orbitals corresponding to l = 4, but no known element contains these type of orbitals.

The figure below shows the total probability of finding the electron within a thin shell at a distance r from the nucleus. The figure also shows that the probability finding an electron at the nucleus is zero, also, the probability of finding the 1s electron at higher values r is zero as well.

The s orbital (l = 0) Each principal level will have an s orbital.

The s orbital (l = 0) Each principal level will have an s orbital. Middle Image: Plot of surface density (radial distribution plot) = Probability of finding an electron in a thin shell at a given distance from the nucleus.

The p orbitals (l = 1) px py pz Each principal level with n = 2 or higher will have p orbitals. There are no p orbitals in n = 1 level. There are three p orbitals (ml = – 1, 0, +1), these three are called, px, py and pz based on their orientation. px py pz

The d orbitals (l = 2) Each principal level with n = 3 or higher will have d orbitals. There are no d orbitals in n = 1 and n = 2 levels.

The d orbitals (l = 2) Each principal level with n = 3 or higher will have d orbitals. There are no d orbitals in n = 1 and n = 2 levels.

The f orbitals (l = 3) Each principal level with n = 4 or higher will have d orbitals. There are no f orbitals in n = 1, n = 2, and n = 3 levels.

The Spin Quantum Number Electrons within atoms interact with a magnet field in one of two ways: Electrons can spin in either direction in the presence of an external magnetic field. This gives rise to the spin quantum number, ms with allowed values of + ½ (spin “up”) or – ½ (spin “down”). Electron spin is important in determining electronic structure

Aufbau principle 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 7s

1 2 Quantum numbers and electron configuration . Principle quantum number (n) Azimuthal quantum number l. Subshell (sublevel) design nation Magnetic Quantum number ml Number of Orbitals in Subshell Number of electrons in subshell Nunber of electrons in Prnciple quantum 1 2 .

3 Quantum numbers and electron configuration . Principle quantum number (n) Azimuthal quantum number l. Subshell (sublevel) design nation Magnetic Quantum number ml Number of Orbitals in Subshell Number of electrons in subshell Nunber of electrons in Prnciple quantum 3 (3dxy, 3dxz, 3dx2-y2, 3dyz, 3dz2) .

4 Quantum numbers and electron configuration . Principle quantum number (n) Azimuthal quantum number l. Subshell (sublevel) design nation Magnetic Quantum number ml Number of Orbitals in Subshell Number of electrons in subshell Nunber of electrons in Prnciple quantum 4 (3dxy, 3dxz, 3dx2-y2, 3dyz, 3dz2) .

Write the complete set of quantum numbers for each electron in the 3p sublevel in the table below: ml ms .

A complete description of an electron in an atom must have four quantum numbers: n, l, ml, ms

Depending on the arrangement of the electrons in an atom, the atom may be paramagnetic or diamagnetic. Paramagnetic atoms tend to be attracted to an external magnetic field. As we will see, these atoms have one or more “unpaired” electrons in the atom. Diamagnetic atoms are slightly repelled by an external magnetic field.

There are three ways of representing the electrons in an atom. 1.Electron configuration notation notation (spdf notation) given by: 1s22s22p2, etc. The leading numbers are the n numbers, the letters are the l numbers and the superscripted number gives the total number of electrons within that suborbital. 2. Noble gas configuration given by: [Ne]3s23p4 Only the higher energy electrons are explicitly given. The core electrons are represented by the noble gas symbol in brackets. Orbital box diagrams. Given by: Where the arrows denote ms

Two other important rules to keep in mind besides the Pauli exclusion principle: The Aufbau (“building up”) principle: Electrons should be filled in using the lowest energy state possible. Hund’s Rule: When filling in electrons in a sublevel (such as the p sublevel), one electron should be placed in each orbital with parallel spin orientations before pairing them up.

Example: Give the proper ECN, condensed ECN and orbital notations for the following elements: Oxygen, Gallium and Calcium Predict whether each is expected to be paramagnetic or diamagnetic

Exceptions to the Aufbau principle Note Cr and Cu

Electron Configurations of Ions When cations are formed, electrons are removed from the highest n number first. Ex. The ground state electron configuration for zinc is 1s22s22p63s23p63d104s2. When it becomes a 2+ ion, the electrons are removed from the 4s sublevel to become 1s22s22p63s23p63d10. All common transition metal cations have electron configurations of the general type [noble gas core](n-1)dx. In the process of ionization, the ns electrons are lost first.

Example: Depict the electron configurations for V2+, V3+, and Co3+. Use orbital notation and noble gas notation. Are any of the ions paramagnetic? If so, give the number of unpaired electrons.