X y z (1s + 1s) σ.

x y z (1s + 1s) σ

(1s - 1s) sσ*

(2pz + pz) 2pσ

(pz - pz) 2pσ*

(2px + 2px) 2pπ

(2px - 2px) 2pπ*

(2py + 2py) 2pπ

(2py - 2py) π*

1sσ* 1sσ (1sσ)2 (1sσ*)2

H2+ 1sσ* ↑ 1sσ (1sσ)2 (1sσ*)2

H2 1sσ* ↑ ↓ 1sσ (1sσ)2 (1sσ*)2

He2+ 1sσ* ↑ ↑ ↓ 1sσ (1sσ)2 (1sσ*)2

He2+ H2– 1sσ* ↑ ↑ ↓ 1sσ (1sσ)2 (1sσ*)2

He2 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2

2sσ 1sσ* 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

2sσ* 2sσ 1sσ* 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

2p 2pσ 2sσ* 2sσ 1sσ* 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

2pσ* 2p 2pσ 2sσ* 2sσ 1sσ* 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

2pσ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

O2 2pσ* 2pπ* 2pπ 2pσ 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
(1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

(1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2
↑ ↑ ↑ ↓ 2pσ ↑ ↓ ↑ ↓ 2pπ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

Li2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

Be2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

B2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

C2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↑ ↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

N2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

CO 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

O2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↑ 2pπ ↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

Cl2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ ↑ ↓ 2pπ ↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

Ne2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ 2pσ* 2pπ* ↑ ↓ ↑ ↓ 2pπ ↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

O2 2pσ* 2pπ* 2pσ 2pπ 2sσ* 2sσ 1sσ* 1sσ)

F2 2pσ* 2pπ* 2pσ 2pπ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ ↑ ↓ 2pπ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

O2 2pσ 2sσ* 2sσ 1sσ* 1sσ (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

O2 2pσ* 2pπ* 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ

O2 2pσ* 2pπ* 2pπ 2pσ 2pπ 2pσ 2sσ* 2sσ 1sσ* 1sσ)

↑ ↑ 2pπ ↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

NO 2pσ* 2pπ* 2pσ 2pπ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↑ ↓ 2pσ ↑ ↓ ↑ ↓ 2pπ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

F2 2pσ* 2pπ* 2pσ 2pπ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ ↑ ↓ 2pπ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

(Ne2) 2pσ* 2pπ* 2pσ 2pπ 2sσ* 2sσ 1sσ* 1sσ)
↑ ↓ 2pσ* 2pπ* ↑ ↓ ↑ ↓ ↑ ↓ 2pσ ↑ ↓ ↑ ↓ 2pπ ↑ ↓ 2sσ* 2sσ ↑ ↓ 1sσ* ↑ ↓ ↑ ↓ 1sσ) (1sσ)2 (1sσ*)2 (2sσ)2 (2pσ)2 (2pπ)4 (2pπ*)4 (2pσ*)2 (2sσ*)2

1σg 1σu 2σg 2σu 3σg 1πu 1πg 1s (AO) 2s (AO) 2p (AO)

1σg 1σu 2σg 2σu 3σg 1πu 1πg 1s (AO) 2s (AO) 2p (AO) H2 Li2 B2 C2 N2 O2 F2 1σg 1σu 2σg 2σu 3σg 1πu 1πg 1s (AO) -0.5 2s (AO) 2p (AO)

(1sσ)2(1sσ*)2(2sσ)2(2pσ)2(2pπ)4(2pπ*)4 (2pσ*)2(2sσ*)2
↑ ↓ 2sσ* 2pσ* 2pπ* ↑ ↓ ↑ ↓ ↑ ↓ ● ↑ ↓ ↑ ↓ 2pπ 2pσ 2sσ ↑ ↓ ↑ ↓ 1sσ* 1sσ) ↑ ↓ ↑ ↓ (1sσ)2(1sσ*)2(2sσ)2(2pσ)2(2pπ)4(2pπ*)4 (2pσ*)2(2sσ*)2

↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓

(p - p) 2pσ*

Hamiltonian for the H atom

Hψ = E ψ H = H1 + H2 Separated H atoms

H12 extra term due to interaction as the atoms get close
H = H1 + H2 + H12 H12 extra term due to interaction as the atoms get close

LCAO-MO Approximation
Linear Combination of Atomic Orbitals – Molecular Orbital Approximation

Ψ = Σi ciφi

Ψ = Σi ciφi Ψ molecular wavefunction

In this simple case: Ψ = c1φ1 + c2φ2

Ψ = Σi ciφi Ψ molecular wavefunction Σi summation operator

Ψ = Σi ciφi Ψ molecular wavefunction Σi summation operator ci orbital coefficient

Ψ = Σi ciφi Ψ molecular wavefunction Σi summation operator ci orbital coefficient φi atomic orbital

Linear Combination of Atomic Orbitals
H = H1 + H2 + H12 Hψ = E ψ Ψ = c1φ1 + c2φ2 LCAO MO Approximation Linear Combination of Atomic Orbitals

Electron Density is given by
Ψ2 or Ψ*Ψ

φ φ2 Ψ = c1φ1 + c2φ2 c12 = c22

φ φ2 Ψ = c1φ1 + c2φ2 c12 = c22 ∫φ12 dτ = ∫φ22 dτ = 1

∫φ12 dτ = ∫φ22 dτ = 1 ∫c12φ12 dτ = ∫c22φ22 dτ φ1 φ2 Ψ = c1φ1 + c2φ2
φ φ2 Ψ = c1φ1 + c2φ2 c12 = c22 ∫φ12 dτ = ∫φ22 dτ = 1 ∫c12φ12 dτ = ∫c22φ22 dτ

c12 = c22

c12 = c22 c1 = ±?c2

c12 = c22 c1 = ±c2

Electron Wavefunction Ψ

Bonding Orbital Electron Density Ψ2

Antibonding Orbital Electron Density Ψ2

H = H1 + H2 + H12

Ψ = φ1 + φ2

Ψ = φ1 + φ2 + - Ψ = φ1 - φ2

+ 1sa + 1sb σbonding σ - + 1sa - 1sb σantibonding σ*

The + and – signs are not charge signs they are phase indicators
- +

- + + Protons pulled towards each other by the build-up of –ve charge in the centre

- + + Protons pulled towards each other by the build-up of –ve charge in the centre Protons repelled as little negative charge build up in the centre + +

Note - Now we are discussing the charges
+ + Protons pulled towards each other by the build-up of –ve charge in the centre Protons repelled as little negative charge build up in the centre + + Note - Now we are discussing the charges

Separate atoms ↑ ↑

At the bond separation the bonding and antibonding orbitals split apart in energy
↑ ↑

The electrons pair up in the lower level – energy is gained - relative to the separate atoms and a stable molecule is formed σ* ↑ ↑ ↓↑ σ

σ* E = α - β ↓↑ σ E = α + β

Note β the stabilisation energy is –ve
↓↑ E = α + β Note β the stabilisation energy is –ve

↑ σ* ↑ σ

↓↑ σ* σ

H2+ σ* ↑ σ

Fig. 6-2. A contour map of the electron density distribution (or the molecular charge distribution) for H2 in a plane containing the nuclei. Also shown is a profile of the density distribution along the internuclear axis. The internuclear separation is 1.4 au. The values of the contours increase in magnitude from the outermost one inwards towards the nuclei. The values of the contours in this and all succeeding diagrams are given in au; 1 au = e/ao3 = e/Å3.

Hydrogen. The two electrons in the hydrogen molecule may both be accommodated in the 1sg orbital if their spins are paired and the molecular orbital configuration for H2 is 1sg2. Since the 1sg orbital is the only occupied orbital in the ground state of H2, the density distribution shown previously in Fig. 6-2 for H2 is also the density distribution for the 1sg orbital when occupied by two electrons. The remarks made previously regarding the binding of the nuclei in H2 by the molecular charge distribution apply directly to the properties of the 1sg charge density. Because it concentrates charge in the binding region and exerts an attractive force on the nuclei the 1sg orbital is classified as a bonding orbital.

Hydrogen. II Excited electronic configurations for molecules may be described and predicted with the same ease within the framework of molecular orbital theory as are the excited configurations of atoms in the corresponding atomic orbital theory. For example, an electron in H2 may be excited to any of the vacant orbitals of higher energy indicated in the energy level diagram. The excited molecule may return to its ground configuration with the emission of a photon. The energy of the photon will be given approximately by the difference in the energies of the excited orbital and the 1sg ground state orbital. Thus molecules as well as atoms will exhibit a line spectrum. The electronic line spectrum obtained from a molecule is, however, complicated by the appearance of many accompanying side bands. These have their origin in changes in the vibrational energy of the molecule which accompany the change in electronic energy.

Electron Density Ψ*Ψ

The nucleus is the very dense region consisting of nucleons (protons and neutrons) at the center of an atom. Almost all of the mass in an atom is made up from the protons and neutrons in the nucleus, with a very small contribution from the orbiting electrons. The diameter of the nucleus is in the range of 1.6 fm (1.6 × 10−15 m) (for a proton in light hydrogen) to about 15 fm (for the heaviest atoms, such as uranium). These dimensions are much smaller than the diameter of the atom itself, by a factor of about 23,000 (uranium) to about 145,000 (hydrogen). The branch of physics concerned with studying and understanding the atomic nucleus, including its composition and the forces which bind it together, is called nuclear physics.

1 2 a b T + V → H V = -1/ra1 - 1/rb2 H12 = 1/rab + 1/r12 - 1/ra2 + 1/rb1 Setting charge e2 = 1

The wave function is usually represented by ψ

The electron density is given by ψ2

probability = ψ2 Electron Density Radius r

- jpg - classweb.gmu.edu/.../graphics/H2-orbitals.jpg
Image may be subject to copyright.

X y z (1s + 1s) σ.

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X y z (1s + 1s) σ.

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