9.02 Quantum Mechanics and The Schrodinger Equation

05.2015 1 9.2 Quantum Mechanics and The Schrodinger Equation

9.02 Quantum Mechanics and The Schrodinger Equation

Dr. Fred Omega Garces Chemistry 152 Miramar College


Wave Treatment of the Atom What is the Schrodinger Wave Equation?

How does the solution of the Schrodinger Wave equation lead to a model of the atom ?

Representation of Orbitals

Quantum Numbers / Restrictionsn - Energy; Principal Quantum #, Shell, n=1,2,3 ...

l - orbital shape; Azimuthal Quantum #, l = 0,1,2,3 ...n-1

ml - orbital orientation, Magnetic Quantum #, m l = - l ...0...+ l ms - electron spin, Direction of electron spin, ms = +1/2, -1/2

Arrangement of electrons Pictorial - illustration of shells in an atom Energy Diagram- Energy level depicting the relative energies Mnemonic - Upside down “Hotel del Orbital” Orbital Block Diagram- electron box illustrating e- config. e- Configuration - Nomenclature of e- configuration.

Electron configuration Notation of the electron address


Schrodinger Wave Equation

H-Hamiltonian Operator (Math function) i.e. ex, ln, yx, !, E

H Ψ = EΨ

E-Eigen Value Total energy of the atom Sum of P.E.; attraction of p+, e- and K.E. of moving e-.

Ψ -Psi - wave function: wave properties Ψ2 -probability distribution

Properties of the Schrodinger equation provides information about the electronic arrangement of each atom.


Significance of the Schrodinger Equation Properties of the Schrodinger is mathematically sophisticated, but it has four important features that can be appreciated without understanding how to solve the equation.

1. H - Hamiltonian is different for every atom, ion or molecule.

2. Ψ - Wave function has wave properties and has spatial variables: Ψ = ψ(x,y,z)

• Ψ - is the math function that gives information of the electron at any point in space; shell and orbital.

• Ψ2 - is the probability of finding e- at any point in space (Probability distribution).

3. Schrodinger equation has solution only for specific value of Energy. This is the quantum condition.

4. Schrodinger equation has solution for any atom or molecule and has an infinite number of solution. Molecules or atoms have infinite number of discrete energies En and each of these correspond to a different Ψn


Electron Address and Ψ

Ψ - wave function for e - at ground state. Ψ2 - probability of e- distribution about atom E - energy of electron

e e

When e- is promoted to new orbital there is a new: Ψ - wave function for e - at ground state. Ψ2 - probability of e- distribution about atom E - Energy of electron

Many Wavefunction Ψ are acceptable with each Ψ having unique set of quantum numbers.

Quantum number - Address of each electron; yields size, shape, orientation and spin of e-

It is essential to realize that an atomic orbital bears no resemblance whatsoever to an “orbital” in the Bohr model: the orbital is a mathematical function with no independent physical reality.


Representation of Orbitals

1s

3s

Classic Orbit: Orbital (3D) Probability distribution


Ψ, Ψ2 and Atomic shells

Ψ - Schrodinger equation. Math formula describing e- with particle-wave duality character.

Orbital density picture

Ψ2 - Probability of e- distribution from center of nucleus

Electron density plots, electron cloud depictions, and radial probability distribution plots for three s orbitals. Information for each of the s orbital is shown as a plot of electron density (top), a cloud representation of the electron density (middle), in which shading coincides with peaks in the plot above, and a radial probability distribution (bottom) that shows where the electrons spends its time


Quantum numbers 4 Parameters (quantum number) which provide address of each electron of an atom. Quantum numbers give probability of electron location in 3-dimensional space. Description Quantum Name Value

Energy n-Shell Principal Quantum# n = 1,2,3,...∞ shell - Probability Vol info. of e- location from nucleus. .

Orbital l - subshell Azimuthal Quantum # l =0,1,2,3,...n-1 Shape Angular momentum l =0, s-suborbital

info. of shape of orbital l =1, p-suborbital

Orbital ml Magnetic Quantum # ml = - l , - l +1,.. 0, ... l +1, l Orientation info. on the orbital Total = 2 l +1 state

orientation

Electron ms Electron spin Quantum # ms = +1/2 or -1/2 spin info. on direction of

e- spin.


Relationship of Quantum Numbers Quantum numbers are dependent on each other

Shell Subshell Orbital

n = 3

n = 2

n = 1

l = 3 d

l = 1 p

l = 1 p

l = 0 s

l = 0 s

l = 0 s

+2 +1 0 -1 -20+1

-10+1

-10+1

00

0

0

n ml l


Quantum Numbers and electron Address

Your address is defined by

Your name Your street Your City Your State Zip Code

Similarly, in an atom each electron has a unique .

The shell (Energy level-Size)

n- Principal Quantum Number. The Orbital Orientation (Shape)

l -azimuthal Quantum number. The suborbital probability Volume

ml - suborbital orientation (Orientation)

Spin of electron

ms- Magnetic Spin

Address Code:

n = 1, 2, 3, ...∞

l = 0, 1, 2, ... n-1 → s, p, d,... ml = -l... 0 ... l → (p) px, py, pz

→ (d) dxy, dyz, dxz, dx2-y2 , dx2

ms → +1/2, -1/2


Wavefunction Ψ and Quantum number (n)

n Principle Quantum Number Size of shell which determines the energy. n = 1,2,3,4,...

n =1

n =2

n =3

n =4


Wavefunction Ψ and Quantum number (l) l Azimuthal Quantum Number (Angular momentum) Determines the Shape of the orbital.

l = 0, 1, 2, ... n-1

l = 0 l = 1 l =2 l = 3 ...

s p d f


Wavefunction Ψ and Quantum number ( ml ) ml Magnetic Quantum Number Orientation of Orbital (Sub Orbital) ml = 0, 1, 2, ... n-1

l = 0 l = 1 l =2 l = 3 ... ml = 0 ml= -1 ,0 ,1, ml =2, -1 ,0 ,1, 2 ml = -3, -2, -1, 0, 1, 2, 3

s p d f

Lobes and nodes?


Wavefunction Ψ and Quantum number ( ms )

ms Electron Spin Quantum Number

Spin orientation of electron.

ms = - 1/2 , +1/2


Atomic Orbitals Shown below are orbitals density plots and Ψ2 vs. r plots for the 1s, 2s, 2p and 3p orbitals

1s

2s

2p

3p


Ψ2 and Atomic Orbitals Electron density plots for specific

orbitals Ψ2 for 1s, 2s and 3s orbitals

Electron density plots for the 1s, 2s, and 3s atomic orbitals of the hydrogen atom. The vertical lines indicate the value of r where the 90% contour surface would be located. Notice that this value is about four times as large for 2s as for 1s and about nine times as large for 3s as far 1s.

Ψ2

r distance from the nucleus

1s

2s

3s


Ψ2 and Atomic Orbitals

1s 2s

2p 3p

3s

3d

Electron density plots for 1s, 2s, 2p, 3s, 3p, and 3d orbitals for the hydrogen atom. All orbitals belonging to the same principle quantum number, n has their maximum electron density occurring at about the same distance from the nucleus. In other words, all orbitals with the same principle quantum number are about the same size.

1s 2s 2p

3s 3p 3d


Shells and Orbitals and Atomic Structure

Shells of an atom contain a number of stacked orbitals

1

2

3

4

s p d f


Relative Energies for Shells and Orbitals Not only do shells have different energies (n levels), but different suborbitals have different energies Note that the d and f-orbitals have energies which overlap energy level of the s-orbital.

1

2

3

4

567∞8

s p d f

Relative Energies of the orbitals


Summary The electron’s wave function (ϕ, atomic orbital) is a mathematical description of the electron's wavelike motion in an atom. Each wave function is associated with one of the atom’s allowed energy states. The probability of finding the electron at a particular location is represented by ϕ2. An electron density diagram and a radial probability distribution plot show how the electron occupies the space near the nucleus for a particular energy level. Three features of the atomic orbital are described by quantum numbers: size (n), shape (l), and orientation (ml). Orbitals are part of sublevels (defined by n and l), which are part of an energy level (defined by n). A sublevel with l =0 has a spherical (s) orbital (no nodes); one with l =1 has two-lobed (p) orbitals (one node); and one with l =2 has four lobed (d) orbitals (two nodes). For the H atom, the energy levels depend only on the n value.

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9.02 Quantum Mechanics and The Schrodinger Equation