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Home , Azimuthal quantum number, Quantum number

9.02 and The Schrodinger Equation

Dr. Fred Omega Garces 152 Miramar College

1 9.2 and The Schrodinger Equation 05.2015 Wave Treatment of the

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 / Restrictions n - 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 - , Direction of electron spin, ms = +1/2, -1/2 Arrangement of Pictorial - illustration of shells in an atom Energy Diagram- depicting the relative energies Mnemonic - Upside down “Hotel del Orbital” Orbital Diagram- electron box illustrating e- config. e- Configuration - Nomenclature of e- configuration. Notation of the electron address

2 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Schrodinger Wave Equation

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

H Ψ = EΨ

H-Hamiltonian E-Eigen Value (Math function) Total energy of the atom i.e. ex, ln, yx, !, E Sum of P.E.; attraction of p+, e- and K.E. of moving e-.

Ψ -Psi - : wave properties Ψ2 -probability distribution

3 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 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(,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 have

infinite number of discrete energies En and each of these

correspond to a different Ψn

4 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Electron Address and Ψ

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

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

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

5 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Representation of Orbitals

Classic Orbit:

Orbital (3D)

Probability

distribution 3s

1s

6 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Ψ, Ψ2 and Atomic shells

Ψ - Schrodinger equation. Electron density plots, Math formula electron cloud depictions, - describing e and radial probability with particle- distribution plots for three wave duality s orbitals. Information for character. each of the s orbital is shown as a plot of electron density (top), a cloud representation of the electron density (middle), in Orbital which shading coincides with density peaks in the plot above, and picture a radial probability distribution (bottom) that shows where the electrons Ψ2 - Probability of spends its time e- distribution from center of nucleus

7 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 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 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.

8 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Relationship of Quantum Numbers

Quantum numbers are dependent on each other

Shell Subshell Orbital

l = 3 d +2 +1 0 -1 -2

n = 3 l = 1 p +1 0 -1

l = 0 s 0

l = 1 p +1 0 -1 n = 2

l = 0 s 0

n = 1 l = 0 s 0

n l ml

9 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Quantum Numbers and electron Address

Your address is defined by Your name Your street Similarly, in an atom each Your City electron has a unique . Your State The shell (Energy level-Size) Zip Code n- . The Orbital Orientation (Shape) l -. The suborbital probability Volume Address Code: ml - suborbital orientation n = 1, 2, 3, ... (Orientation) ∞ Spin of electron l = 0, 1, 2, ... n-1 → s, p, d,... ms- Magnetic Spin ml = -l... 0 ... l

→ (p) px, py, pz

→ (d) dxy, dyz, dxz, dx2-y2 , dx2

ms → +1/2, -1/2

10 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Wavefunction Ψ and Quantum number (n)

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

n =4 n =3 n =2 n =1

11 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 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

12 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Wavefunction Ψ and Quantum number ( ml )

ml Orientation of Orbital (Sub Orbital) ml = 0, 1, 2, ... n-1 l = 0 l = 1 l =2 l = 3 ... m = 0 m = -1 ,0 ,1, m =2, -1 ,0 ,1, 2 m = -3, -2, -1, 0, 1, 2, 3 l l l l s p d f

Lobes and nodes?

13 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Wavefunction Ψ and Quantum number ( ms )

ms Electron Spin orientation of electron.

ms = - 1/2 , +1/2

14 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 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

15 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Ψ2 and Atomic Orbitals

Electron density plots for specific orbitals Electron density plots Ψ2 for 1s, 2s and 3s orbitals for the 1s, 2s, and 3s atomic orbitals of the 1s atom. The vertical lines indicate 2s the value of r where the 90% contour surface would be located. 3s Notice that this value is about four times as large for 2s as for 1s 2 Ψ and about nine times as large for 3s as far 1s. r distance from the nucleus

16 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Ψ2 and Atomic Orbitals

Electron density plots for 1s, 2s, 2p, 3s, 3p, and 3d orbitals for the . 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 3d quantum number are about the same size.

2p 3p

1s 2s 3s

3s 3p 3d

1s 2s 2p

17 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Shells and Orbitals and Atomic Structure f d s p

Shells of an atom contain a number of stacked orbitals 4

3

2

1

18 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Relative Energies for Shells and Orbitals p ∞ s d f 8 Not only do shells 7 have different 6 energies (n levels), 5 but different suborbitals have 4 different energies Note that the d and f-orbitals have 3 energies which overlap energy level of the s-orbital.

2

Relative Energies of the 1 orbitals 19 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015 Summary

The electron’s wave function (ϕ, atomic orbital) is a mathematical description of the electron's wavelike 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.

20 9.2 Quantum Mechanics and The Schrodinger Equation 05.2015