Point Group D6d

Results for Point Group D_6d

**Characters of representations for molecular motions**
Motion	E	2S₁₂	2C₆	2S₄	2(S₁₂)⁵	C₂	6C'₂	6σ_d
Cartesian 3N	75	0.732	2	-1	-2.732	-1	-1	5
Translation (x,y,z)	3	0.732	2	-1	-2.732	-1	-1	1
Rotation (R_x,R_y,R_z)	3	-0.732	2	1	2.732	-1	-1	-1
Vibration	69	0.732	-2	-1	-2.732	1	1	5

**Decomposition to irreducible representations**
Motion	A₁	A₂	B₁	B₂	E₁	E₂	E₃	E₄	E₅	Total
Cartesian 3N	4	2	2	5	7	6	6	6	6	44
Translation (x,y,z)	0	0	0	1	1	0	0	0	0	2
Rotation (R_x,R_y,R_z)	0	1	0	0	0	0	0	0	1	2
Vibration	4	1	2	4	6	6	6	6	5	40

**Molecular parameter**
Number of Atoms (N)	25
Number of internal coordinates	69
Number of independant internal coordinates	4
Number of vibrational modes	40

Force field analysis

**Allowed / forbidden vibronational transitions**
Operator	A₁	A₂	B₁	B₂	E₁	E₂	E₃	E₄	E₅	Total
Linear (IR)	4	1	2	4	6	6	6	6	5	10 / 30
Quadratic (Raman)	4	1	2	4	6	6	6	6	5	15 / 25
IR + Raman	- - - -	1	2	- - - -	- - - -	- - - -	6	6	- - - -	0 / 15

Characters of force fields
(Symmetric powers of vibration representation)
Force field	E	2S₁₂	2C₆	2S₄	2C₃	2(S₁₂)⁵	C₂	6C'₂	6σ_d
linear	69	0.732	-2	-1	0	-2.732	1	1	5
quadratic	2.415	-0.732	2	1	0	2.732	35	35	47
cubic	57.155	-1.000	-1	-1	23	-1.000	35	35	195
quartic	1.028.790	0.000	0	18	0	0.000	630	630	1.078
quintic	15.020.334	-0.000	0	-18	0	-0.000	630	630	3.886
sextic	185.250.786	-0.000	12	18	276	0.000	7.770	7.770	16.354

Decomposition to irreducible representations
Column with number of nonvanshing force constants highlighted
Force field	A₁	A₂	B₁	B₂	E₁	E₂	E₃	E₄	E₅
linear	4	1	2	4	6	6	6	6	5
quadratic	123	82	99	105	198	204	198	204	199
cubic	2.442	2.327	2.345	2.425	4.758	4.764	4.764	4.764	4.758
quartic	43.321	42.467	42.779	43.003	85.680	85.782	85.680	85.788	85.680
quintic	627.001	624.743	625.061	626.689	1.251.642	1.251.750	1.251.642	1.251.744	1.251.642
sextic	7.725.163	7.713.101	7.716.983	7.721.275	15.436.896	15.438.186	15.436.962	15.438.192	15.436.896

Contributions to nonvanishing force field constants

pos(X) : Position of irreducible representation (irrep) X in character table of D_6d

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(A₁) ≤ i ≤ pos(E₅)
..10.	A₁A₁.	..1.	A₂A₂.	..3.	B₁B₁.	..10.	B₂B₂.	..21.	E₁E₁.	..21.	E₂E₂.	..21.	E₃E₃.	..21.	E₄E₄.	..15.	E₅E₅.
Subtotal: 123 / 9 / 9
Irrep combinations (i,j) with indices: pos(A₁) ≤ i ≤ j ≤ pos(E₅)
Subtotal: 0 / 0 / 36
Total: 123 / 9 / 45

**Contributions to nonvanishing cubic force field constants**
Irrep combinations (i,i,i) with indices: pos(A₁) ≤ i ≤ pos(E₅)
..20.	A₁A₁A₁.	..56.	E₄E₄E₄.
Subtotal: 76 / 2 / 9
Irrep combinations (i,i,j) (i,j,j) with indices: pos(A₁) ≤ i ≤ j ≤ pos(E₅)
..126.	E₁E₁E₂.	..126.	E₂E₂E₄.	..4.	A₁A₂A₂.	..12.	A₁B₁B₁.	..40.	A₁B₂B₂.	..84.	A₁E₁E₁.	..84.	A₁E₂E₂.	..84.	A₁E₃E₃.	..84.	A₁E₄E₄.	..60.	A₁E₅E₅.
..15.	A₂E₁E₁.	..15.	A₂E₂E₂.	..15.	A₂E₃E₃.	..15.	A₂E₄E₄.	..10.	A₂E₅E₅.	..42.	B₁E₃E₃.	..84.	B₂E₃E₃.	..90.	E₂E₅E₅.
Subtotal: 990 / 18 / 72
Irrep combinations (i,j,k) with indices: pos(A₁) ≤ i ≤ j ≤ k ≤ pos(E₅)
..8.	A₂B₁B₂.	..60.	B₁E₁E₅.	..72.	B₁E₂E₄.	..120.	B₂E₁E₅.	..144.	B₂E₂E₄.	..216.	E₁E₂E₃.	..216.	E₁E₃E₄.	..180.	E₁E₄E₅.	..180.	E₂E₃E₅.	..180.	E₃E₄E₅.
Subtotal: 1.376 / 10 / 84
Total: 2.442 / 30 / 165

**Contributions to nonvanishing quartic force field constants**
Irrep combinations (i,i,i,i) with indices: pos(A₁) ≤ i ≤ pos(E₅)
..35.	A₁A₁A₁A₁.	..1.	A₂A₂A₂A₂.	..5.	B₁B₁B₁B₁.	..35.	B₂B₂B₂B₂.	..231.	E₁E₁E₁E₁.	..231.	E₂E₂E₂E₂.	..357.	E₃E₃E₃E₃.	..231.	E₄E₄E₄E₄.	..120.	E₅E₅E₅E₅.
Subtotal: 1.246 / 9 / 9
Irrep combinations (i,i,i,j) (i,j,j,j) with indices: pos(A₁) ≤ i ≤ j ≤ pos(E₅)
..336.	E₁E₁E₁E₃.	..224.	A₁E₄E₄E₄.	..56.	A₂E₄E₄E₄.	..112.	B₁E₂E₂E₂.	..224.	B₂E₂E₂E₂.	..210.	E₃E₅E₅E₅.
Subtotal: 1.162 / 6 / 72
Irrep combinations (i,i,j,j) with indices: pos(A₁) ≤ i ≤ j ≤ pos(E₅)
..10.	A₁A₁A₂A₂.	..30.	A₁A₁B₁B₁.	..100.	A₁A₁B₂B₂.	..210.	A₁A₁E₁E₁.	..210.	A₁A₁E₂E₂.	..210.	A₁A₁E₃E₃.	..210.	A₁A₁E₄E₄.	..150.	A₁A₁E₅E₅.	..3.	A₂A₂B₁B₁.	..10.	A₂A₂B₂B₂.
..21.	A₂A₂E₁E₁.	..21.	A₂A₂E₂E₂.	..21.	A₂A₂E₃E₃.	..21.	A₂A₂E₄E₄.	..15.	A₂A₂E₅E₅.	..30.	B₁B₁B₂B₂.	..63.	B₁B₁E₁E₁.	..63.	B₁B₁E₂E₂.	..63.	B₁B₁E₃E₃.	..63.	B₁B₁E₄E₄.
..45.	B₁B₁E₅E₅.	..210.	B₂B₂E₁E₁.	..210.	B₂B₂E₂E₂.	..210.	B₂B₂E₃E₃.	..210.	B₂B₂E₄E₄.	..150.	B₂B₂E₅E₅.	..666.	E₁E₁E₂E₂.	..666.	E₁E₁E₃E₃.	..666.	E₁E₁E₄E₄.	..780.	E₁E₁E₅E₅.
..666.	E₂E₂E₃E₃.	..1.107.	E₂E₂E₄E₄.	..465.	E₂E₂E₅E₅.	..666.	E₃E₃E₄E₄.	..465.	E₃E₃E₅E₅.	..465.	E₄E₄E₅E₅.
Subtotal: 9.171 / 36 / 36
Irrep combinations (i,i,j,k) (i,j,j,k) (i,j,k,k) with indices: pos(A₁) ≤ i ≤ j ≤ k ≤ pos(E₅)
..756.	E₁E₁E₂E₄.	..630.	E₁E₁E₃E₅.	..630.	E₂E₂E₃E₅.	..504.	A₁E₁E₁E₂.	..504.	A₁E₂E₂E₄.	..126.	A₂E₁E₁E₂.	..126.	A₂E₂E₂E₄.	..252.	B₁E₁E₁E₄.	..504.	B₂E₁E₁E₄.	..756.	E₁E₂E₂E₃.
..630.	E₁E₂E₂E₅.	..1.260.	E₁E₃E₃E₅.	..630.	E₁E₄E₄E₅.	..1.512.	E₂E₃E₃E₄.	..630.	E₃E₄E₄E₅.	..60.	A₁A₂E₁E₁.	..60.	A₁A₂E₂E₂.	..60.	A₁A₂E₃E₃.	..60.	A₁A₂E₄E₄.	..40.	A₁A₂E₅E₅.
..168.	A₁B₁E₃E₃.	..336.	A₁B₂E₃E₃.	..360.	A₁E₂E₅E₅.	..42.	A₂B₁E₃E₃.	..84.	A₂B₂E₃E₃.	..90.	A₂E₂E₅E₅.	..120.	B₁B₂E₁E₁.	..120.	B₁B₂E₂E₂.	..120.	B₁B₂E₃E₃.	..120.	B₁B₂E₄E₄.
..80.	B₁B₂E₅E₅.	..252.	B₁E₂E₄E₄.	..180.	B₁E₄E₅E₅.	..504.	B₂E₂E₄E₄.	..360.	B₂E₄E₅E₅.	..756.	E₁E₃E₄E₄.	..540.	E₁E₃E₅E₅.	..540.	E₂E₄E₅E₅.
Subtotal: 14.502 / 38 / 252
Irrep combinations (i,j,k,l) with indices: pos(A₁) ≤ i ≤ j ≤ k ≤ l ≤ pos(E₅)
..32.	A₁A₂B₁B₂.	..240.	A₁B₁E₁E₅.	..288.	A₁B₁E₂E₄.	..480.	A₁B₂E₁E₅.	..576.	A₁B₂E₂E₄.	..864.	A₁E₁E₂E₃.	..864.	A₁E₁E₃E₄.	..720.	A₁E₁E₄E₅.	..720.	A₁E₂E₃E₅.	..720.	A₁E₃E₄E₅.
..60.	A₂B₁E₁E₅.	..72.	A₂B₁E₂E₄.	..120.	A₂B₂E₁E₅.	..144.	A₂B₂E₂E₄.	..216.	A₂E₁E₂E₃.	..216.	A₂E₁E₃E₄.	..180.	A₂E₁E₄E₅.	..180.	A₂E₂E₃E₅.	..180.	A₂E₃E₄E₅.	..432.	B₁E₁E₂E₃.
..360.	B₁E₁E₂E₅.	..432.	B₁E₁E₃E₄.	..360.	B₁E₂E₃E₅.	..360.	B₁E₃E₄E₅.	..864.	B₂E₁E₂E₃.	..720.	B₂E₁E₂E₅.	..864.	B₂E₁E₃E₄.	..720.	B₂E₂E₃E₅.	..720.	B₂E₃E₄E₅.	..1.296.	E₁E₂E₃E₄.
..2.160.	E₁E₂E₄E₅.	..1.080.	E₂E₃E₄E₅.
Subtotal: 17.240 / 32 / 126
Total: 43.321 / 121 / 495

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Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement

Results for Point Group D6d

Force field analysis

Further Reading

Contributions to nonvanishing force field constants

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Results for Point Group D_6d