Point Group D5d

Results for Point Group D_5d

**Characters of representations for molecular motions**
Motion	E	2C₅	2(C₅)²	5C'₂	i	2(S₁₀)³	2S₁₀	5σ_d
Cartesian 3N	63	1.618	-0.618	-1	-3	-1.618	0.618	5
Translation (x,y,z)	3	1.618	-0.618	-1	-3	-1.618	0.618	1
Rotation (R_x,R_y,R_z)	3	1.618	-0.618	-1	3	1.618	-0.618	-1
Vibration	57	-1.618	0.618	1	-3	-1.618	0.618	5

**Decomposition to irreducible representations**
Motion	A_1g	A_2g	E_1g	E_2g	A_1u	A_2u	E_1u	E_2u	Total
Cartesian 3N	4	2	6	6	2	5	7	6	38
Translation (x,y,z)	0	0	0	0	0	1	1	0	2
Rotation (R_x,R_y,R_z)	0	1	1	0	0	0	0	0	2
Vibration	4	1	5	6	2	4	6	6	34

**Molecular parameter**
Number of Atoms (N)	21
Number of internal coordinates	57
Number of independant internal coordinates	4
Number of vibrational modes	34

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A_1g	A_2g	E_1g	E_2g	A_1u	A_2u	E_1u	E_2u	Total
Linear (IR)	4	1	5	6	2	4	6	6	10 / 24
Quadratic (Raman)	4	1	5	6	2	4	6	6	15 / 19
IR + Raman	- - - -	1	- - - -	- - - -	2	- - - -	- - - -	6	0^* / 9

* Parity Mutual Exclusion Principle

Characters of force fields
(Symmetric powers of vibration representation)
Force field	E	2C₅	2(C₅)²	5C'₂	i	2(S₁₀)³	2S₁₀	5σ_d
linear	57	-1.618	0.618	1	-3	-1.618	0.618	5
quadratic	1.653	1.618	-0.618	29	33	1.618	-0.618	41
cubic	32.509	-1.000	-1.000	29	-91	-1.000	-1.000	165
quartic	487.635	-0.000	0.000	435	555	-0.000	-0.000	811
quintic	5.949.147	12.000	12.000	435	-1.425	-0.000	-0.000	2.791
sextic	61.474.519	-19.416	7.416	4.495	6.355	0.000	0.000	10.571

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field	A_1g	A_2g	E_1g	E_2g	A_1u	A_2u	E_1u	E_2u
linear	4	1	5	6	2	4	6	6
quadratic	102	67	169	168	78	84	162	162
cubic	1.669	1.572	3.242	3.242	1.596	1.664	3.260	3.260
quartic	24.721	24.098	48.819	48.819	24.260	24.448	48.708	48.708
quintic	298.195	296.582	594.771	594.771	296.942	298.120	595.056	595.056
sextic	3.077.809	3.070.276	6.148.085	6.148.091	3.071.888	3.074.926	6.146.814	6.146.820

Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of D_5d

Subtotal: <Number of nonvanishing force constants in subsection> / <number of nonzero irrep combinations in subsection> / <number of irrep combinations in subsection>
Total: <Number of nonvanishing force constants in force field> / <number of nonzero irrep combinations in force field> / <number of irrep combinations in force field>

**Contributions to nonvanishing quadratic force field constants**
Irrep combinations (i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..10.	A_1gA_1g.	..1.	A_2gA_2g.	..15.	E_1gE_1g.	..21.	E_2gE_2g.	..3.	A_1uA_1u.	..10.	A_2uA_2u.	..21.	E_1uE_1u.	..21.	E_2uE_2u.
Subtotal: 102 / 8 / 8
Irrep combinations (i,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
Subtotal: 0 / 0 / 28
Total: 102 / 8 / 36

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..20.	A_1gA_1gA_1g.
Subtotal: 20 / 1 / 8
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..90.	E_1gE_1gE_2g.	..4.	A_1gA_2gA_2g.	..60.	A_1gE_1gE_1g.	..84.	A_1gE_2gE_2g.	..12.	A_1gA_1uA_1u.	..40.	A_1gA_2uA_2u.	..84.	A_1gE_1uE_1u.	..84.	A_1gE_2uE_2u.	..10.	A_2gE_1gE_1g.	..15.	A_2gE_2gE_2g.
..15.	A_2gE_1uE_1u.	..15.	A_2gE_2uE_2u.	..105.	E_1gE_2gE_2g.	..105.	E_1gE_2uE_2u.	..126.	E_2gE_1uE_1u.
Subtotal: 849 / 15 / 56
Irrep combinations (i,j,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_2u)
..8.	A_2gA_1uA_2u.	..60.	E_1gA_1uE_1u.	..120.	E_1gA_2uE_1u.	..180.	E_1gE_1uE_2u.	..72.	E_2gA_1uE_2u.	..144.	E_2gA_2uE_2u.	..216.	E_2gE_1uE_2u.
Subtotal: 800 / 7 / 56
Total: 1.669 / 23 / 120

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A_1g) ≤ i ≤ pos(E_2u)
..35.	A_1gA_1gA_1gA_1g.	..1.	A_2gA_2gA_2gA_2g.	..120.	E_1gE_1gE_1gE_1g.	..231.	E_2gE_2gE_2gE_2g.	..5.	A_1uA_1uA_1uA_1u.	..35.	A_2uA_2uA_2uA_2u.	..231.	E_1uE_1uE_1uE_1u.	..231.	E_2uE_2uE_2uE_2u.
Subtotal: 889 / 8 / 8
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..210.	E_1gE_1gE_1gE_2g.	..336.	E_1uE_1uE_1uE_2u.	..280.	E_1gE_2gE_2gE_2g.	..336.	E_1uE_2uE_2uE_2u.
Subtotal: 1.162 / 4 / 56
Irrep combinations (i,i,j,j) with indices: pos(A_1g) ≤ i ≤ j ≤ pos(E_2u)
..10.	A_1gA_1gA_2gA_2g.	..150.	A_1gA_1gE_1gE_1g.	..210.	A_1gA_1gE_2gE_2g.	..30.	A_1gA_1gA_1uA_1u.	..100.	A_1gA_1gA_2uA_2u.	..210.	A_1gA_1gE_1uE_1u.	..210.	A_1gA_1gE_2uE_2u.	..15.	A_2gA_2gE_1gE_1g.	..21.	A_2gA_2gE_2gE_2g.	..3.	A_2gA_2gA_1uA_1u.
..10.	A_2gA_2gA_2uA_2u.	..21.	A_2gA_2gE_1uE_1u.	..21.	A_2gA_2gE_2uE_2u.	..465.	E_1gE_1gE_2gE_2g.	..45.	E_1gE_1gA_1uA_1u.	..150.	E_1gE_1gA_2uA_2u.	..780.	E_1gE_1gE_1uE_1u.	..465.	E_1gE_1gE_2uE_2u.	..63.	E_2gE_2gA_1uA_1u.	..210.	E_2gE_2gA_2uA_2u.
..666.	E_2gE_2gE_1uE_1u.	..1.107.	E_2gE_2gE_2uE_2u.	..30.	A_1uA_1uA_2uA_2u.	..63.	A_1uA_1uE_1uE_1u.	..63.	A_1uA_1uE_2uE_2u.	..210.	A_2uA_2uE_1uE_1u.	..210.	A_2uA_2uE_2uE_2u.	..666.	E_1uE_1uE_2uE_2u.
Subtotal: 6.204 / 28 / 28
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ pos(E_2u)
..80.	E_1gE_1gA_1uA_2u.	..180.	E_1gE_1gA_1uE_2u.	..360.	E_1gE_1gA_2uE_2u.	..540.	E_1gE_1gE_1uE_2u.	..120.	E_2gE_2gA_1uA_2u.	..252.	E_2gE_2gA_1uE_1u.	..504.	E_2gE_2gA_2uE_1u.	..756.	E_2gE_2gE_1uE_2u.	..360.	A_1gE_1gE_1gE_2g.	..90.	A_2gE_1gE_1gE_2g.
..252.	A_1uE_1uE_1uE_2u.	..504.	A_2uE_1uE_1uE_2u.	..40.	A_1gA_2gE_1gE_1g.	..60.	A_1gA_2gE_2gE_2g.	..60.	A_1gA_2gE_1uE_1u.	..60.	A_1gA_2gE_2uE_2u.	..420.	A_1gE_1gE_2gE_2g.	..420.	A_1gE_1gE_2uE_2u.	..504.	A_1gE_2gE_1uE_1u.	..105.	A_2gE_1gE_2gE_2g.
..105.	A_2gE_1gE_2uE_2u.	..126.	A_2gE_2gE_1uE_1u.	..630.	E_1gE_2gE_1uE_1u.	..630.	E_1gE_2gE_2uE_2u.	..120.	A_1uA_2uE_1uE_1u.	..120.	A_1uA_2uE_2uE_2u.	..252.	A_1uE_1uE_2uE_2u.	..504.	A_2uE_1uE_2uE_2u.
Subtotal: 8.154 / 28 / 168
Irrep combinations (i,j,k,l) with indices: pos(A_1g) ≤ i ≤ j ≤ k ≤ l ≤ pos(E_2u)
..32.	A_1gA_2gA_1uA_2u.	..240.	A_1gE_1gA_1uE_1u.	..480.	A_1gE_1gA_2uE_1u.	..720.	A_1gE_1gE_1uE_2u.	..288.	A_1gE_2gA_1uE_2u.	..576.	A_1gE_2gA_2uE_2u.	..864.	A_1gE_2gE_1uE_2u.	..60.	A_2gE_1gA_1uE_1u.	..120.	A_2gE_1gA_2uE_1u.	..180.	A_2gE_1gE_1uE_2u.
..72.	A_2gE_2gA_1uE_2u.	..144.	A_2gE_2gA_2uE_2u.	..216.	A_2gE_2gE_1uE_2u.	..360.	E_1gE_2gA_1uE_1u.	..360.	E_1gE_2gA_1uE_2u.	..720.	E_1gE_2gA_2uE_1u.	..720.	E_1gE_2gA_2uE_2u.	..2.160.	E_1gE_2gE_1uE_2u.
Subtotal: 8.312 / 18 / 70
Total: 24.721 / 86 / 330

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Results for Point Group D5d

Force field analysis

Further Reading

Contributions to nonvanishing force field constants

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Results for Point Group D_5d