Atomic Orbitals Glenn V. Lo Department of Physical Sciences Nicholls State University

Quantum Theory Quantum Theory proposes a “wave mechanical model” for atoms. developed by Schrodinger Postulate: there is a function (a “wavefunction”),  ), which give us all accessible information about the a system

Example True or False. If we know  for an electron in an atom, then at any given time, we can use  to precisely determine where it is, how fast it is moving, and in which direction it is headed.

Atomic Orbitals Orbital = mathematical function that describes an electron. An orbital does NOT describe how an electron moves or where the electron is; it has no physical significance. We perform mathematical operations on an orbital to obtain information about the electron. Information available includes probability of finding electron in a specified region. Customary to refer to an electron as being “in an orbital” or “occupying an orbital”

Quantum Numbers Quantum numbers: ID numbers for orbitals “Quantum” means “restricted” Restrictions: Principal quantum number: n = positive integer Orbital (angular momentum) quantum number: l (“ell”) = non-negative integer less than n Magnetic quantum number: m or m l (“em sub ell”) = integer from –l to +l

Example Which of the following is an allowed value for the quantum number n? A. 0 B. 0.5 C. 2 D. -3 Rules: n=1,2,3…; l=0,1,..,n-1; m=-l to +l

Example Which of the following is an allowed value for the quantum number l if n=2? A. -1 B. 1.2 C. 0 D. 2 Rules: n=1,2,3…; l=0,1,..,n-1; m=-l to +l

Example Which of the following is an allowed value for the quantum number m if n=3 and l=2? A. -3 B. 1.5 C. 2, D. 3 Rules: n=1,2,3…; l=0,1,..,n-1; m=-l to +l

Shells and subshells Orbitals with same n are said to belong to the same shell or level Orbitals with the same n and l are said to belong to the same subshell or sublevel. “Spectroscopic Notation” - more common way of referring to subshells and orbitals; use code letter for the quantum number l “s” for l=0, “p” for l=1, “d” for l=2, “f” for l=3 Example: What is spectroscopic notation for the subshell with n=2 and l=1?

Orbitals in a subshell An orbital is specified by specifying n, l, and m. Example: (n=2, l=1, m=+1) or 2p +1 Since m= -l to +l. There are 2l +1 orbitals in a subshell. Example: how many orbitals are there in a “d” subshell?

Orbitals in a subshell In general, the number of orbitals in An s subshell is …. A p subshell is …. A d subshell is … An f subshell is … Note: an expression like “2p” could means the 2p subshell or any one of the orbitals in the 2p subshell.

Example How many orbitals belong to the n=2 shell? List them.

Test Yourself Which of the following are allowed values of the quantum numbers n and l respectively ? A. 1.5 and 0, B. 3 and 3, C. 2 and 1, D. 0 and 1

Test Yourself What is the value of the orbital quantum number (l ) for an electron in a 3p orbital? A. 0, B. 1, C. 2, D. 3

Test Yourself Which of the following is not an allowed value of the magnetic quantum number (m) for an electron in a 3d orbital? A. 3, B. 2, C. 0, D. -1

Test Yourself Which of the following is not a valid name for an atomic orbital? A. 1s, B. 2s, C. 2d, D. 3p

Information from orbitals Probability of finding an electron in a particular region Energy needed to remove an electron from the atom (ionization energy) Photons that can be absorbed or released How the electron would react to an external magnetic field

Radial distribution function Tells us how the probability of finding the electron depends on distance. Example: Most probable distance of electron in 1s orbital is 52.9 pm; this is called the “Bohr radius” a o = 52.9 pm 1 Bohr = 52.9 pm For which orbital is the electron, on average, closer to the nucleus: 1s or 2s ?

Average distance from nucleus For electron in an H atom, or an ion with only one electron Z=nuclear charge (in atomic units). Approximation for atoms or ions with more than one electron: replace Z by Z eff where Z eff = effective nuclear charge; an electron won’t feel the full attractive force of nucleus due to repulsions from the other electrons

Example For which of these orbitals is the distance of the electron, on average, farthest from the nucleus? A. 2s of H B. 2p of H C. 2s of He +

Average distance from nucleus In general, electron is farther (on average) from the nucleus in orbitals with larger n  the most probable region is in a larger “shell” 1s (H) closer to nucleus than 2s and 2p Within a given shell (same n), larger l  closer to nucleus on average (if only one electron): 2p (H) closer to nucleus than 2s (H) Identify radial distribution functions for 1s, 2s, and 2p in the diagram:

Average distance from nucleus The function associated with an orbital name is not the same in different atoms. Same orbital, higher Z  closer to nucleus Which orbital describes an electron that is closer to the nucleus (on average): A. 1s orbital of He + B. 1s orbital of H

3-D Representation of Orbitals Use electron cloud to represent probability; cloud is thicker in regions where probability of finding electron is higher Example: 3p orbital

3-D Representation of Orbitals Boundary surface resembles the shape of the electron cloud tells us where regions of high electron density are found The s orbitals have a spherical shape. The real p orbitals generally have two lobes

Real Orbitals Orbitals with nonzero m have real and imaginary parts (“complex numbers”). The i in the expression for  (2p +1 ) below is the square root of –1; an imaginary number. For convenience, we can combine pair of orbitals with same |m| to get an equivalent pair of orbitals that have the no imaginary parts. Example: 2p x = (1/  2) (2p p -1 ) 2p y = (i/  2) (2p p -1 )

Orbital energies and ionization energy In H and ions with only one electron: energy of electron depends only on n, E = -R H (Z/n) 2 R H = 13.6 eV or 2.18x J (Rydberg constant) What does a negative energy mean? It’s the energy compared to the case where the electron is at infinite distance from the nucleus. By definition: E = 0 when electron is at infinite distance from the nucleus. Ionization energy, the minimum energy needed to remove an electron = - E If more energy is provided, it becomes kinetic energy of electron.

Example In a hydrogen atom, which orbital describes an electron with higher energy: 3s or 3p?

Example What is the minimum energy needed to remove an electron from the 2s orbital of H? If this electron were to encounter a photon with energy of 5.0 eV, what would be its kinetic energy once it leaves the atom? Answer: 3.4 eV, 1.6 eV

Example From which orbital would it be hardest to remove an electron? A. 1s of H B. 1s of He + C. 2s of He +

Photon energy absorbed or released Allowed photon energy = difference between energies of two orbitals What is the energy of the photon released when an electron in the 3p orbital of H loses energy and ends up in the 2s orbital? What color light is produced? Answer: 1.89 eV or 3.03x J, red light (656 nm)

Orbital energies In H and ions with only one electron: energy of electron depends only on n, E = -R H (Z/n) 2 where Z=nuclear charge. For atoms and ions with more than one electron, an electron will not “feel” the full effect of nucleus because it is repelled by other electrons. Approx. E by replacing Z by Z eff (“effective nuclear charge”). Z eff will depend on n and l.

Magnetic Properties Motion of electron around the nucleus (“orbital motion”) generates a magnetic field. Imagine electron moving as shown below Red arrow points to “south pole” of magnetic force generated.

Magnetic Properties Each m corresponds to one way that the magnetic field generated by an electron can be oriented. Exception: motion of electrons in s orbitals do not generate a magnetic field. The North-South line is NOT fixed.It is precessing about a “cone of uncertainty” Figure below shows 5 possible orientations of magnetic field generated by electron in a d orbital

Electron Spin Postulate: Each electron has intrinsic magnetism, which oriented two possible ways. Since we associate magnetism is associated with some kind of curved motion of a charged particle, we imagine that the intrinsic magnetism to be due to “spin.” Complete description of electron: specify 4 quantum numbers n, l, m, m s Specify orbital (n, l, m) and spin. “spin up”; m s =+1/2 “spin down”; m s =-1/2”

Example How many ways can the intrinsic magnetism of an electron in a 3d orbital be oriented? A. 2, B. 3, C. 5, D. 10

Pauli’s Exclusion Principle Pauli’s Exclusion Principle: (postulate) no more than two electrons per orbital two electrons in an orbital must have opposite spins. no two electrons can have the same set of 4 quantum numbers

Test Yourself How many ways can the magnetism generated by the orbital motion of an electron in a 3p orbital be oriented? A. 1, B. 3, C. 5, D. 7

Test Yourself How many ways can the intrinsic magnetism of an electron in a 4d orbital be oriented? A. 2, B. 3, C. 5, D. 10

Example How many electrons can be assigned to a 4p subshell?

Example How many electrons can be assigned to a 4p orbital?

Example How many electrons can be assigned to the n=3 shell?

Test Yourself What is the maximum number of electrons that can be assigned to a 2p orbital of an atom? A. 2, B. 3, C. 6, D. 10

Test Yourself How many sets of quantum numbers can an electron in a 4p subshell have? A. 2, B. 3, C. 6, D. 10