Rep:Mod:Seagull2

Introduction

Aluminium trichloride (AlCl₃) in the solid state (at room temperature) has a layered structure with octahedrally coordinated Al atoms^[1]. In the vapor phase, the compound exists as Al₂Cl₆ dimers and at high temperatures they dissociate to AlCl₃ monomers. In the following investigation, four hypothetical dimers of AlCl₂Br are investigated in terms of energies, bonding and molecular vibrations. The dimers are shown on the figure on the right - there are 5 possible dimers and two of them are enantiomers of each other (2 and ent-2). All properties investigated in this text are expected to be identical for both enantiomers, so computations are performed only for dimer 2.

Structure optimizations and energy comparison

As an input for structure optimization, all 4 dimers were drawn with appropriate symmetry group imposed on them. The following bond lengths and angles were set:

bond lengths: terminal Al-Cl: 2.24000 A, terminal Al-Br: 2.39000 A, all bridging Al-X bonds: 2.2400 A

bond angles: between terminal bonds: ca. 121 degree, between the bridging bonds at Al: ca. 80 degree

For all isomers, B3LYP DFT method was used with 6-31G(d,p) basis set on Al and Cl atoms and LanL2DZ psuedopotential basis set on Br atoms. Frequency analysis (shown in the next section) showed that energy minima were achieved for all structures (all calculated frequencies were positive).

Isomer 1

Output file: https://wiki.ch.ic.ac.uk/wiki/images/e/ee/SB_1_OPT.LOG

The energy converged, as can be seen in the "Item" table of the output file:

         Item               Value     Threshold  Converged?
 Maximum Force            0.000041     0.000450     YES
 RMS     Force            0.000016     0.000300     YES
 Maximum Displacement     0.000594     0.001800     YES
 RMS     Displacement     0.000224     0.001200     YES
 Predicted change in Energy=-2.278200D-08
 Optimization completed.
    -- Stationary point found.

Isomer 2

Output file: https://wiki.ch.ic.ac.uk/wiki/images/6/60/SB_2_OPT.LOG

The energy converged, as can be seen in the "Item" table of the output file:

         Item               Value     Threshold  Converged?
 Maximum Force            0.000095     0.000450     YES
 RMS     Force            0.000040     0.000300     YES
 Maximum Displacement     0.001268     0.001800     YES
 RMS     Displacement     0.000435     0.001200     YES
 Predicted change in Energy=-2.318735D-07
 Optimization completed.
    -- Stationary point found.

Isomer 3

Output file: https://wiki.ch.ic.ac.uk/wiki/images/d/d0/SB_3_OPT.LOG

The energy converged, as can be seen in the "Item" table of the output file:

         Item               Value     Threshold  Converged?
 Maximum Force            0.000094     0.000450     YES
 RMS     Force            0.000043     0.000300     YES
 Maximum Displacement     0.001586     0.001800     YES
 RMS     Displacement     0.000704     0.001200     YES
 Predicted change in Energy=-2.371649D-07
 Optimization completed.
    -- Stationary point found.

Isomer 4

Output file: https://wiki.ch.ic.ac.uk/wiki/images/8/87/SB_4_OPT.LOG

The energy converged, as can be seen in the "Item" table of the output file:

         Item               Value     Threshold  Converged?
 Maximum Force            0.000091     0.000450     YES
 RMS     Force            0.000042     0.000300     YES
 Maximum Displacement     0.001284     0.001800     YES
 RMS     Displacement     0.000406     0.001200     YES
 Predicted change in Energy=-2.151388D-07
 Optimization completed.
    -- Stationary point found.

Comparison

Summary tables of all four isomers generated in Gaussview 5.0 software are shown side-by side below.

Isomer	1	2	3	4
File Type	.log	.log	.log	.log
Calculation Type	FOPT	FOPT	FOPT	FOPT
Calculation Method	RB3LYP	RB3LYP	RB3LYP	RB3LYP
Basis Set	Gen	Gen	Gen	Gen
Charge	0	0	0	0
Spin	Singlet	Singlet	Singlet	Singlet
E(RB3LYP) [a.u.]	-2352.40630793	-2352.41109909	-2352.41626653	-2352.41628792
RMS Gradient Norm [a.u.]	0.00001689	0.00007099	0.00005297	0.00005086
Imaginary Freq
Dipole Moment [Debye]	0.0000	0.1402	0.1670	0.0000
Point Group	D2H	C1	C2V	C2H
Job cpu time	0 days 0 hours 11 minutes 42.0 seconds	0 days 0 hours 5 minutes 24.0 seconds	0 days 0 hours 8 minutes 54.0 seconds	0 days 0 hours 9 minutes 26.0

From the above computations isomer 4 turned out to be most stable, having energy very close to 3. Energies of isomers relative to isomer 4 are shown below:

Isomer	1	2	3	4
Energy [a.u.]	-2352.40630793	-2352.41109909	-2352.41626653	-2352.41628792
Energy relative to 4 [a.u]	0.00997999	0.00518883	0.00002139	0.00000000
Energy relative to 4 [kJ/mol], rounded to 1 dp	26.2	13.6	0.1	0.0

The expected uncertainty of computed energy values is ca. 10 kJ/mol. It has various sources, including:

method of numerical computation
basis set used
pseudopotential used

Considering this uncertainty, we can't state which isomer (3 or 4) is the most stable one. Very close energy values of 3 and 4 are predictable: they are cis-/trans- isomers with the differing fragments being far apart (on the opposite sides of the dimer). Because they are far, one can't expect any major steric clash between opposite terminal atoms is or any electronic/bonding effects (the opposite terminal atoms are 4 bonds apart). The highest energy isomer (1) has two bridging bromine atoms (in total: 4 Al-Br bonds, 4 Al-Cl bonds) while the second highest energy isomer (2) has one bridging bromine atom (in total: 3 Al-Br bonds, 5 Al-Cl bonds). This is in accordance with higher magnitude of Al-Cl bond energy determined in gaseous AlCl and AlBr molecules^[2]. A larger bond energy for Al-Cl bond can probably be exaplained by better overlap (size match) between p orbitals of Cl and Al comparing to overlap between Br and Al orbitals (Al and Cl are in the same period).

Dissociation energy computation

In order to calculate the dissociation energy of dimer 4 (assumed to be the most stable dimer), a monomer molecule (AlCl₂Br) was optimized using exactly the same method and basis sets as for dimer optimization. The following bond lengths and angles were set:

bond lengths: Al-Cl: 2.24000 A, Al-Br: 2.39000

bond angles: all 120 degree

Output file: https://wiki.ch.ic.ac.uk/wiki/images/3/3d/SB_ALCL2BR_OPT.LOG

The energy converged, as can be seen in the "Item" table of the output file:

         Item               Value     Threshold  Converged?
 Maximum Force            0.000136     0.000450     YES
 RMS     Force            0.000073     0.000300     YES
 Maximum Displacement     0.000760     0.001800     YES
 RMS     Displacement     0.000497     0.001200     YES
 Predicted change in Energy=-7.984418D-08
 Optimization completed.
    -- Stationary point found.

Summary table:

AlCl2Br optimization
File Name = SB_ALCL2BR_OPT
File Type = .log
Calculation Type = FOPT
Calculation Method = RB3LYP
Basis Set = Gen
Charge = 0
Spin = Singlet
E(RB3LYP) = -1176.19013679 a.u.
RMS Gradient Norm = 0.00004196 a.u.
Imaginary Freq =
Dipole Moment = 0.1075 Debye
Point Group = C2V
Job cpu time: 0 days 0 hours 1 minutes 53.0 seconds.

Frequency analysis (using the same method and basis set showed that the energy minimum was reached (all calculated frequencies positive) - output file: https://wiki.ch.ic.ac.uk/wiki/images/4/4f/SB_ALCL2BR_FREQ.LOG.

From the energies of optimized monomer and dimer 4 it is possible to calculate the energy of dissociation:

$Δ E = 2 E_{m o n} - E_{4}$

$E_{4} = - 2352.41628792 a . u .$

$E_{m o n} = - 1176.19013679 a . u .$

$Δ E = 0.03601434 a . u . \approx 94.6 k J / m o l$

The positive value means that the dimer is more stable than two monomers (i.e. dissociation is an endothermic process). What is actually observed is dissociation of Al₂Cl₆ in higher temperatures^[1]) but the onset of reaction at certain temperature is influenced by activation energy and here we only compared the energies of initial and final state.

Frequency analysis

The optimized dimer molecules were used for vibration frequency analysis using the same method and basis set.

Isomer 1

Output file: https://wiki.ch.ic.ac.uk/wiki/images/e/e1/SB_1_FREQ.LOG

Mass-weighted force constants:

 Full mass-weighted force constant matrix:
Low frequencies --- -5.2347 -4.9339 -3.1387 -0.0046 -0.0039 -0.0035
Low frequencies --- 14.8221 63.2842 86.0928

Isomer 2

Output file: https://wiki.ch.ic.ac.uk/wiki/images/3/30/SB_2_FREQ.LOG

Mass-weighted force constants:

 Full mass-weighted force constant matrix:
Low frequencies --- -2.8375 -0.6433 0.0016 0.0030 0.0033 3.0142
Low frequencies --- 17.1556 55.9429 80.0969

Isomer 3

Output file: https://wiki.ch.ic.ac.uk/wiki/images/6/64/SB_3_FREQ.LOG

Mass-weighted force constants:

 Full mass-weighted force constant matrix:
Low frequencies --- -3.3737 -1.6429 -0.0047 -0.0038 -0.0033 1.3738
Low frequencies --- 17.2119 50.9383 78.5889

Isomer 4

Output file: https://wiki.ch.ic.ac.uk/wiki/images/8/89/SB_4_FREQ.LOG

Mass-weighted force constants:

 Full mass-weighted force constant matrix:
Low frequencies --- -3.2770 -1.4933 0.0039 0.0045 0.0046 1.0767
Low frequencies --- 17.7778 48.9870 72.9547

Comparison

The computed vibrational spectra of dimers, together with the number of IR-active modes are shown below:

Isomer	spectrum, IR activity of modes
1	8 modes IR-active
2	18 modes IR-active
3	15 modes IR-active
4	9 modes IR-active

Every dimer has 18 vibrational modes in total (in general, N-atomic non-linear molecule has 3N-6 normal modes). In order for each mode to be excited by absorption of a photon (called "IR-active", because the energy difference falls within the infrared region of the electromagnetic spectrum), the dipole moment of the molecule must change during the vibrational mode. The transition dipole associated with $ν^{'} - ν^{″}$ transition ( $μ_{ν^{'} ν^{″}}$ ) can be expressed^[3] as integral:

$μ_{ν^{'} ν^{″}} = \int ψ_{ν^{'}}^{*} (Q_{a}) μ ψ_{ν^{″}} (Q_{a}) d Q_{a}$

where $ψ$ is vibrational eigenfunction, $Q_{a}$ is normal coordinate of given mode, $μ$ is the dipole moment. This transition dipole must be non-zero for a transition to be active. Using some results of group theory, one can prove^[3], that given mode is IR active if this mode belongs to the same species (symmetry label) of molecule's point group as one of the components of the dipole moment vector. The implication in our case is that only for isomer 2 all 18 modes are IR active because they all change the dipole moment while for other isomers various modes of vibration don't.

Bridging bond stretching

For each isomer, 4 modes were found which consist of stretching of bridging bonds and bending of terminal bonds. These modes are presented in the table below, together with computerd frequencies and absorption intensities:


mode (shown: isomer 1)	1		2		3		4
	frequency [cm^-1]	intensity	frequency [cm^-1]	intensity	frequency [cm^-1]	intensity	frequency [cm^-1]	intensity
a)	197	0.0	211	21.2	264	0.0	264	0.0
b)	241	99.7	424	275.1	279	25.6	280	28.9
c)	247	0.0	257	9.6	309	2.2	308	0.0
d)	341	160.7	385	153	413	149	413	149

As mentioned earlier (Mod:Seagull#GaBr3_and_comparison_with_BBr3), the frequency (wavenumber) of transition for harmonic oscillator is:

$ν = \frac{1}{2 π c} \sqrt{\frac{k}{m}}$

In case of polyatomic molecule, k is the "effective" force constant of given mode - it's influenced by rigidity of bonds oscillating in that mode. In the same way, m is the "effective" mass in the mode - a quantity influenced by the masses of atoms oscillating in that mode. The effective force constant (k) value of these modes is dominated by force constants of the bonds being stretched (in general, k value of bond bending is noticably lower than k value of bond stretching).

In the table above one can notice that in general the frequencies of modes in 3 and 4 (which have 2 bridging Cl atoms) are higher than in 2 (which has 1 bridging Br atom) and those in 2 are higher than in 1 (which has 2 bridging Br atoms). This is not surprizing, since:

a) Al-Br bonds are weaker than Al-Cl (conclusion of the energy analyses - see earlier section of this text), hence k is smaller

b) Br atoms are heavier than Cl atoms, hence effective mass (m) is can be larger. But it may be not, because the terminal atoms are also oscillating to some extent in these modes (especially in b) and c)) and so effective mass of the mode can be decreased by increasing the number of (ligther) terminal chlorine atoms.

One exception is mode b) which has the highest energy in isomer 2 (instead of 3 or 4). Mode b) in this isomer is quite different than in others - it's very asymmetric (only one Al atom oscillates significantly out of entire molecule) and much stretching of terminal bonds is involved, which explains the different frequency.

Terminal bond stretching

In a similar way, 2 modes were found which consist of terminal bond stretching and bridging bond bending. They are shown below together with frequencies and absorption intensities:


mode (shown: isomer 1)	1		2		3		4
	frequency [cm^-1]	intensity	frequency [cm^-1]	intensity	frequency [cm^-1]	intensity	frequency [cm^-1]	intensity
e)	608	0.0	574	121.7	570	32.3	574	0.0
f)	616	331.8	614	197.1	582	277.6	579	316.1

The frequency trend is the same as in the bridging bond stretching frequencies - isomer 1 (having 4 Al-Cl terminal bonds stretching) has the highest frequency, followed by 2 (3 Al-Cl terminal bonds), 3 and 4 (2 Al-Cl terminal bonds).

Computations of molecular orbitals

Optimized molecule of dimer 4 was used as an input for energy computation (using the same method and basis sets) in order to plot molecular orbitals.

Output files:

Checkpoint file: https://wiki.ch.ic.ac.uk/wiki/images/c/cd/SB_4_MO_3.chk

Log file: https://wiki.ch.ic.ac.uk/wiki/images/7/79/SB_4_MO_3.LOG

Five computed molecular orbitals (boundary surfaces, isovalue=0.02) together with their energies and descriptions are shown below:


MO number	energy [Hartree]	description
55 (LUMO)	-0.06	The contributing atomic orbitals are 3p of all Cl atoms, 4p of Br atoms and 3s of Al atoms. The 3s Al orbitals have the largest contribution to this molecular orbital (visible as large red boundary surface). There are antibonding interactions along both terminal bonds (halogen p and Al s form σ* orbitals) and bridging bonds, however there is some overlap (constructive interference) between the outer lobes of orbitals of bridging Cl atoms and Al s orbital.
54 (HOMO)	-0.32	The Al orbitals have no contribution to this MO. All atomic orbitals are 3p Cl and 4p Br. There are two kinds of weakly bonding interactions: between the terminal halogens on the opposite molecule ends and between the bridging Cl atoms. On the other hand there are weakly antibonding interactions: between terminal atoms on each end of the dimer and between terminal and bridging halogens. The are all weak, as can be seen by the distance between orbitals' boundary surfaces.
42	-0.41	The largest contribution is from the 3p oritals of bridging Cl atoms. They combine "side-on" with Al 3p orbitals to form a π orbital with a nodal plane perpendicular to the "bridging ring".
40	-0.43	The contributing orbitals are: Cl 3s, Al 3p and Br 4p. The p orbitals of Al atoms and bridging Cl atoms all combine "side-on" to form a delocalized π orbital with a nodal surface. The same p Al orbitals overlap with p orbitals of all terminal halogens to form σ terminal bonds.
31 (lowest energy valence MO)	-0.91	Two 3s Cl orbitals combine to form a strongly bonding sigma orbital. No contributions from orbitals of other atoms, except some very small p orbital contribution from terminal Cl atoms.

References

↑ ^1.0 ^1.1 C. E. Housecroft and A. G. Sharpe, in Inorganic Chemistry, Pearson Education, 4th edn., 2012, pp. 390–392.
↑ R. F. Barrow, Trans. Faraday Soc., 1960, 56, 952–958 DOI:10.1039/TF9605600952
↑ ^3.0 ^3.1 K. Nakamoto, in Infrared and Raman Spectra of Inorganic and Coordination Compounds, Wiley, 1st edn., 1997, pp. 53-57.

[Housecroft_AlCl3-1] 1.0 ^1.1 C. E. Housecroft and A. G. Sharpe, in Inorganic Chemistry, Pearson Education, 4th edn., 2012, pp. 390–392.

[Harrow_bonden-2] R. F. Barrow, Trans. Faraday Soc., 1960, 56, 952–958 DOI:10.1039/TF9605600952

[Nakamoto_dipole-3] 3.0 ^3.1 K. Nakamoto, in Infrared and Raman Spectra of Inorganic and Coordination Compounds, Wiley, 1st edn., 1997, pp. 53-57.

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