Normal vibrational frequencies of water molecule

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Question 1

Normal modes and vibrational frequencies of water molecule

HF/3-21* optimised geometry of the water molecule

H bond length


HOH bond angle


(ii) Energy of the HF/3-21G optimised water molecule = -75.58596 au



Max. Grad.

Max. Dist.













Frequency (cm-1)

Relative Motion

Stretch or Bend


Symmetry (S or A)
















75.58596 au


Bond angle


Bond length


Vibrational frequencies


H moves faster than D



D moves quickly whereas H moves slightly



H moves quickly whereas D moves slightly - asymmetric

Normal modes and vibrational frequencies of the water dimmer (H2O)2

Hydrogen-Bond Acceptor

Hydrogen bond

Hydrogen Bond Donor

Hydrogen bond length (H''''O) = 1.808

Hydrogen bond angle (O-H''''O) = 174.9˚

Energy of the F/3.12G optimised water dimer = -151.18902 au

(a) Potential energy calculation:

ΔE = E(dimer) - 2xE(H2O)

= (-396 871.2KJ/mol) - 2x(-198 413.2KJ/mol)

= (-396 871.2) - (-396 826.3)

= - 44.9 KJmol-1

(b)As seen from the surface diagram for H2O, the oxygen has negative charge (δ-) whereas the hydrogens are positively charged (δ+).

In the water dimer molecule, the hydrogen atoms (on the H-bond donor oxygen) are δ+/blue region. The oxygen atom that is bonded to the hydrogen that is the H-bond acceptor has δ- charge/red region. Between in the H-bond, the positive(H) and negative(O) charges combine/green region.

The hydrogen bond is formed between one of the H atoms and one O, instead between the two oxygens, because the two oxygen atoms are negatively charged, and have δ-, and therefore repulsive interactions are formed between them. So, one H reacts with the O, which donates one of its lone pairs to form the H-bond.

In the structure of the molecule, the H''''O bond is almost linear, very close to 180˚ but it is distorted so it is about 175˚. Also, the distortion causes the bond H'''O to become longer.

(c) For the water molecule:

H bond length = 0.967

For the water dimer:

H bond length of H-bond donor = 0.965

H bond length of H-bond acceptor = 0.966, 0.974 (H of H-bond)

The H bond length of the hydrogen of the H-bond is bigger than the other O-H bonds in the molecule. This is because this H is bonded to the oxygen through the H-bond, and it is pulled towards the oxygen, causing its bond with the other oxygen to become a bit longer.

Question 2

The water dimer consists of two fragments, the H-bond acceptor (top OH2 group) and the H-bond donor (bottom OH2 group). When a vibration causes both fragments and H-bond to move, then it is considered to be the inter-monomer because it is a vibration between the two molecules. If only one of the fragments vibrates, then the vibration is only in one of the molecules (it is internal) and it is considered to be an intra-monomer.

The vibrational frequencies of the water dimer are the following:

Frequency = 81 cm-1

Type = A'

Bending Mode

Top part of the molecule moving slightly up and down, while the two bottom hydrogens move up and down as well

Inter-monomer: The vibration affects both molecules connected through the hydrogen bond.

Frequency = 133

Type = A''

Bending mode

Top part and bottom part moving right and left.

Inter monomer

Frequency = 172

Type = A''

Bending mode

Middle hydrogen moving right and left and two bottom H atoms moving up and down symmetrically (when one is up, other is down)

Inter Monomer

Frequency = 242

Type = A'

Stretching Mode

Inter monomer

Frequency = 425

Type = A'

Bending Mode

The H-bond acceptor fragment moves to the front and then back, and the H-bond donor fragment moves up and down as well.


Frequency = 826

Type = A''

Bending mode

The H of the H-bond (middle H) is moving to the right and left, causing the rest of the molecule to move in that way as well


Frequency = 1782

Type = A'

Bending Mode

The hydrogen atoms on the H-bond donor fragment move up and down to the sides going further away and then coming closer.


Frequency = 1854

Type = A'

Bending Mode

The hydrogen atoms on the H-bond acceptor fragment separate and go further away and then come closer together again.


Frequency = 3724

Type = A'

Stretching mode

The hydrogen forming the H-bond moves closer to the oxygen of the H-bond and then further from it, causing the O-H bond to come smaller and the H'''''O bond to become bigger, and the opposite.


Frequency = 3849

Type = A'

Stretching mode

The hydrogen atoms move symmetrically so that their bonds with the O of the H-bond donor are becoming bigger (stretch out) and then smaller.


Frequency = 3907

Type = A'

Stretching mode

The O-H bond of the H not involved in the H-bond acceptor fragment is stretching out, causing the bond to become longer, while the bond of the oxygen with the other H, which is involved in the H-bond, becomes shorter.


Frequency = 3982

Type = A''

Stretching mode

It is an unsymmetrical movement, where one O-H bond in the H-bond donor fragment becomes shorter and the other longer.


Question 3

Isotopic substitution in the water dimer

Free Energy (H-TS) = 37.8

ΔΗ Total = 127.5

Free Energy (H-TS) = 39.7

ΔΗ Total = 126.5

ΔG = G(B) - G(A) = 39.7KJmol-1 - 37.8KJmol-1 = 1.9 KJ/mol

K = e(-ΔG/RT) = exp(-1.9x10-3Jmol-1/8.314JK-1mol-1x298K) = 1.00000077

Deuterium prefers the position shown in B (connected to the oxygen of the H-bond acceptor fragment, but doesn't take part directly in the H-bond) because the molecule has higher free energy for this arrangement.

Question 4

Interconversion of water dimer structures

Frequency = i302

Type = B1

Frequency = 105

Type = B2

Frequency = 208

Type = A1

Frequency = 225

Type = B1

Frequency = 256

Type = A2

Frequency = 591

Type = B2

Frequency = 1785

Type = A1

Frequency = 1831

Type = A1

Frequency = 3829

Type = A1

Frequency = 3862

Type = A1

Frequency = 3952

Type = B1

Frequency = 3961

Type = B2

Acyclic water dimer Cyclic water dimer

  • The acyclic water molecule energy is 3.969x10-5 KJmol-1 whereas the energy of the cyclic one is ----. The cyclic molecule is less stable than the acyclic one because its ability to move around is effectively reduced compared to the acyclic one, due to the two bonds formed between the oxygen of one molecule and the two H of the other molecule.
  • The imaginary frequency has the value of i306.9. One of the middle hydrogens moves up while the other moves down, in an unsymmetrical movement as shown in the pictures above.
  • For the acyclic water dimer there are no imaginary frequencies and it corresponds to the valley. This shows that it is very stable and this structure is preferred.
  • The cyclic molecule contains one vibrational frequency and this suggests that it is not as stable as the acyclic one. It corresponds to the hilltop of molecule-mountain.
  • If a molecule has more than one vibrational frequency it corresponds to the mountain passes and it is a very unstable and unfavoured structure for the molecule to be at, which most probably does not exist.
  • The cyclic structure is not very stable, and therefore it is not preferred over the acyclic one.

Question 5


No imaginary frequencies à Valley à stable structure, highly favoured

Boat cyclohexane:

ne imaginary frequency à Hilltop à fairly unstable, exists but not preferred

All-syn cyclohexane:

More than one imaginary frequencies à Mountain Pass à does not exist, very unstable