The Quantum (Wave) Mechanics Model In 1924, a French physicist named Louis de Broglie suggested that, like light, electrons could act as both particles and waves. De Broglie's hypothesis was soon confirmed in experiments that showed electron beams could be diffracted or bent as they passed through a slit much like light could. The waves produced by an electron confined in its orbit about the nucleus sets up a standing wave of specific wavelength, energy and frequency (i.e., Bohr's energy levels) much like a guitar string sets up a standing wave when plucked. De Broglie's vision of Bohr's atom      

Quantum (Wave) Mechanics Quantum mechanics, or wave mechanics, is the treatment of atomic structure through the wavelike properties of the electron Erwin Schrödinger developed an equation to describe the hydrogen atom A wave function is a solution to the Schrödinger equation and represents an energy state of the atom

Wave Mechanics = Probability Wave mechanics provides a probability of where an electron will be in certain regions of an atom This region of space where there’s a high probability of finding an electron is called an orbital Wave mechanics led to the idea of a “cloud of electron density” rather than a discrete location

Quantum Numbers and Atomic Orbitals A wave function with a given set of these three quantum numbers is called an atomic orbital In quantum mechanics the atomic orbitals require three integer quantum numbers to completely describe the energy and the shape of the 3-D space occupied by the electron (n, l, and ml)

Principal Quantum Number (n) Is independent of the other two quantum numbers Can only be a positive integer indicates the size of an orbital (distance from the nucleus) and its electron energy n can be 1, 2, 3, 4, … These orbitals are the volume around the atom that the electrons occupy 90-95% of the time. 1 2 3

Orbital Angular Momentum Quantum Number (l) (aka Azimuthal quantum number) Determines the shape of the orbital: s, p, d, f , which corresponds to values l values of: 0, 1, 2, 3 Possible values of l: 0 to n – 1; e.g. if n = 2, l can only be 0 or 1 Each of these orbitals is in a different region of space and has a different shape All the ‘l’ quantum values represent different sublevels or subshells When n = 1, there is only one “l” value meaning there is only one sublevel in the first energy level; when n= 2; there are two values for ‘l’ indicating two sublevels in the second energy level

Summary n l subshell name 1 s ( sharp) 2 p (principal) 3 d (diffuse) 4 s ( sharp) 2 p (principal) 3 d (diffuse) 4 f (fundamental)

Magnetic Quantum Number (ml) Determines the orientation in space of the orbital; so named because in a magnetic field, these different orientations have different energies Possible values: –l to +l; e.g., if l = 2, ml can be –2, –1, 0, 1, 2 The magnetic quantum number, ml, defines the number of orbitals in a sublevel. E.g. in the l = 0 sublevel, there is only one ml value, therefore there is only orbital in this sublevel; when l=1; there are 3 possible ml values (-1, 0, +1) 3 orbitals in this sublevel

Quantum Numbers Summary Taken together the three quantum numbers specific the orbital the electron occupies. Namely: the energy of the orbital, the shape of the orbital, and the orientation of the orbital .

writing 3 quantum numbers to indicate every possible orbital an electron can occupy is cumbersome; instead do we do the following: retain the numeric value of the principal quantum number and use a letter to indicate the azimuthal quantum number: l = 0  s; l = 1 p; l = 2  d; l = 3  d - When combined, they indicate an a specific orbital e.g. 1s orbital; 2s orbital; 2p orbital

Radial Distributions Electrons are most likely to reside nearest the nucleus because of electrostatic attraction Probability of finding an electron decreases as distance (radius) from the nucleus increases

Electron Probabilities and the 1s Orbital The 1s orbital looks very much like a fuzzy ball, that is, the orbital has spherical symmetry (the probability of finding an electron is the same in direction) The electrons are more concentrated near the center

Electron Probabilities and the 2s Orbital The 2s orbital has two regions of high electron probability, both being spherical The region near the nucleus is separated from the outer region by a spherical node - a spherical shell in which the electron probability is zero EOS

The Three p Orbitals -There are three p orbital; each orbital is cylindrically symmetrical with respect to rotation around one of the 3 axes, x, y, or z Each ‘p’ orbital has two lobes of high probability density separated by a node (region of zero probability)

The Five d Orbitals The first d orbitals appear in the n=3 shell. The five d orbitals have two different shapes ( 4 are clover and 1 is … complex) There are 5 d orbitals per n level They have an l=2 value, therefore ml= -2, -1, 0, +1 , +2

The f orbitals The first f orbital appears in the the n=4 shell. There are seven f orbitals per n levels They have an l=3 value ml= -3,-2,-1,0, +1, +2, +3

Electron Spin (ms) The electron spin quantum number explains some of the finer features of atomic emission spectra The spin refers to a magnetic field induced by the moving electric charge of the electron as it spins Only possible values = –1/2 to +1/2 EOS

Pauli’s Exclusion Principle “ No two electrons in an atom can have the same set of 4 quantum numbers”