Group Theory CSIR NET GATE Chemistry Mock Test

Question 1

A molecule has a 2-fold axis and a mirror plane perpendicular to that. The point group must have a:

Accepted Answer

व्युत्क्रमण केंद्र (Centre of inversion, $i$)

Answer

$C_2$ अक्ष ($C_2$ axis)

Answer

$\sigma_h$ तल ($\sigma_h$ plane)

Answer

$\sigma_v$ तल ($\sigma_v$ plane)

Question 2

The symmetry elements that are present in $BF_3$ are:

Accepted Answer

$C_3, \sigma_v, \sigma_h, 3C_2$

Answer

$C_3, 3C_2, S_2, \sigma_v$

Answer

$C_3, 3C_2, \sigma_h, S_2$

Answer

$C_3, \sigma_h, \sigma_v, i$

Question 3

The $S_2$ operation on a molecule with the axis of rotation as the z axis, moves a nucleus at $(x, y, z)$ to:

Accepted Answer

$(-x, -y, -z)$

Answer

$(-x, -y, z)$

Answer

$(x, -y, -z)$

Answer

$(-x, y, -z)$

Question 4

The point group of $NSF_3$ is:

Accepted Answer

$C_{3v}$

Answer

$D_{3d}$

Answer

$C_{3h}$

Answer

$D_{3h}$

Question 5

The $T_d$ point group has 24 elements and 5 classes. Given that it has two 3-dimensional irreducible representations, the number of one dimensional irreducible representations is:

Accepted Answer

2

Answer

1

Answer

6

Answer

3

Question 6

The point group symmetry of the given planar shape is:

Accepted Answer

$D_{3h}$

Answer

$C_3$

Answer

$C_{3h}$

Answer

$C_{3v}$

Question 7

Statement for Linked Answer Questions 7, 8 and 9: Trans 1,2 difluoroethylene molecule has a 2-fold rotational axis, a symmetry plane perpendicular to the rotational axis and an inversion centre.

(7) The number of distinct symmetry operations that can be performed on the molecule is:

Accepted Answer

4

Answer

2

Answer

6

Answer

8

Question 8

(8) The number of irreducible representations of the point group of the molecule is:

Accepted Answer

4

Answer

1

Answer

2

Answer

3

Question 9

(9) When two H atoms of the above molecule are also replaced by F atoms, the point group of the resultant molecule will be:

Accepted Answer

$D_{2h}$

Answer

$C_i$

Answer

$C_{2h}$

Answer

$C_{2v}$

Question 10

Linked Answer Questions (10 and 11): Character table for the point group $C_{2v}$ is given below:
$C_{2v}$$E$$C_2$$\sigma_v(xz)$$\sigma_v(yz)$LinearQuadratic$A_1$1111$z$$x^2, y^2, z^2$$A_2$11-1-1$R_z$$xy$$B_1$1-11-1$x, R_y$$xz$$B_2$1-1-11$y, R_x$$yz$
(10) The reducible representation corresponding to the three translational degrees of freedom, $	au_	ext{tr}$:

Accepted Answer

$3, -1, 1, 1$

Answer

$3, 1, 1, 1$

Answer

$3, -1, -1, -1$

Answer

$3, 1, -1, -1$

Question 11

(11) The asymmetric stretching mode of the $H_2O$ is shown below. The molecular plane is yz and the symmetry axis of $H_2O$ is z:

O
  
  H
  
  H

This vibration transforms as the irreducible representation:

Accepted Answer

$B_2$

Answer

$A_1$

Answer

$B_1$

Answer

$A_2$

Question 12

$[CoCl_4]^{2-}$ a blue coloured complex. Controlled-treatment of this complex with water generates two Isomeric light pink coloured complexes, of composition $[Co(H_2O)_4Cl_2]$. Identify the correct point groups for $[CoCl_4]^{2-}$ and two isomeric complexes $[Co(H_2O)_4Cl_2]$:

Accepted Answer

$T_d$ and ($C_{2v}$ & $D_{4h}$)

Answer

$D_{4h}$ and ($C_{2v}$ & $C_{2h}$)

Answer

$D_{4h}$ and ($C_{2v}$ & $D_{4h}$)

Answer

$T_d$ and ($C_{2v}$ & $C_{4v}$)

Question 13

The point group of $ClF_3$ molecule and its corresponding number of irreducible representations are respectively:

Accepted Answer

$C_{2v}$ & 4

Answer

$C_{3v}$ & 4

Answer

$C_{3v}$ & 3

Answer

$C_{2v}$ & 3

Question 14

Symmetry operations of the four $C_2$ axes perpendicular to the principal axis belong to the same class in the point group(s):

Accepted Answer

$D_{4h}$

Answer

$D_4$

Answer

$D_{4d}$

Answer

$D_{4h}$ and $D_{4d}$

Question 15

The point group symmetry of $CH_2=C=CH_2$ is:

Accepted Answer

$D_{2d}$

Answer

$D_{2h}$

Answer

$C_{2h}$

Answer

$C_{2v}$

Question 16

Conversion of boron trifluoride to tetrafluoroborate accompanies:

Accepted Answer

सममिति में वृद्धि तथा आबंध लंबाई में वृद्धि (increase in symmetry and bond elongation)

Answer

सममिति में वृद्धि तथा आबंध लंबाई में संकुचन (increase in symmetry and bond contraction)

Answer

सममिति में कमी तथा आबंध लंबाई में संकुचन (decrease in symmetry and bond contraction)

Answer

सममिति में कमी तथा आबंध लंबाई में वृद्धि (decrease in symmetry and bond elongation)

Question 17

The number of $C_2$ axes in $CCl_4$ is:

Accepted Answer

तीन (Three)

Answer

दो (Two)

Answer

चार (Four)

Answer

पांच (Five)

Question 18

The point group of $IF_7$ is:

Accepted Answer

$D_{5h}$

Answer

$D_{6h}$

Answer

$C_{6v}$

Answer

$C_{5v}$

Question 19

Character table of point group $D_8$ is given below:
$D_8$$E$$2C_8$$2C_4$$2C_8^3$$C_2$$4C_2'$$4C_2''$$A_1$$a$111111$A_2$$b$1111$h$$i$$B_1$$c$-11-111$j$$B_2$$d$-11-11-11$E_1$$e$$\sqrt{2}$0$-\sqrt{2}$-200$E_2$$f$0-20$k$00$E_3$$g$$-\sqrt{2}$0$\sqrt{2}$-200
Value of $(a + b + c + d + e + f + g + h + i + j + k)$ is equal to:

Accepted Answer

9

Answer

7

Answer

16

Answer

0

Question 20

The character table for a pyramidal $AB_3$ molecule of $C_{3v}$ point group is given below:
$C_{3v}$$E$$2C_3$$3\sigma_v$LinearQuadratic$A_1$111$z$$x^2+y^2, z^2$$A_2$11-1$R_z$-$E$2-10$(x, y), (R_x, R_y)$$(x^2-y^2, xy), (xz, yz)$
The reducible representation of pyramidal $AB_3$ is:
$C_{3v}$$E$$2C_3$$3\sigma_v$$\Gamma$1202
The CORRECT option representing all the normal Raman active modes of pyramidal $AB_3$ is:

Accepted Answer

$2A_1 + 2E$

Answer

$A_1 + A_2 + 2E$

Answer

$3E$

Answer

$3A_1 + A_2 + E$

Question 21

The characters of $E, C_2, \sigma_v$, and $\sigma_v'$ symmetry operations, in this order, for valid irreducible representation(s) of the $C_{2v}$ point group is/are:

Accepted Answer

$1, 1, 1, 1$

Answer

$-1, 1, 1, -1$

Answer

$1, -1, 1, -1$

Answer

$1, -1, -1, -1$

Question 22

The point group of $SF_6$ is:

Accepted Answer

$O_h$

Answer

$D_{6h}$

Answer

$D_{6d}$

Answer

$C_{6v}$

Question 23

A point originally at $(1, 3, 5)$ was subjected to a symmetry operation ($\hat{O}_1$) that shifted the point to $(-1, -3, 5)$. Subsequently, the point at $(-1, -3, 5)$ was subjected to another symmetry operation ($\hat{O}_2$) that shifted this point to $(-1, -3, -5)$. The symmetry operators $\hat{O}_1$ and $\hat{O}_2$ are, respectively:

Accepted Answer

$\hat{C}_2(z)$ and $\hat{\sigma}(xy)$

Answer

$\hat{C}_2(x)$ and $\hat{\sigma}(xy)$

Answer

$\hat{\sigma}(xy)$ and $\hat{C}_2(z)$

Answer

$\hat{S}_1$ and $\hat{S}_2$

Question 24

The symmetry point group of ethane in its staggered conformation is:

Accepted Answer

$D_{3d}$

Answer

$C_{3v}$

Answer

$D_{3h}$

Answer

$S_6$

Question 25

Among the following, the CORRECT statement is:

Accepted Answer

अनपचनीय निरूपणों (irreducible representations) की संख्या सममिति संक्रियाओं के वर्गों (classes) की संख्या के बराबर होती है।

Answer

अनपचनीय निरूपणों की संख्या सममिति बिन्दु समूह की कोटि (order) के बराबर होती है।

Answer

किसी भी बिन्दु समूह में निहित अनपचनीय निरूपण हमेशा एकविमीय (one dimensional) होते हैं।

Answer

एक सममिति बिन्दु समूह में पूर्णतः सममित अनपचनीय निरूपण (totally symmetric irreducible representation) उपस्थित नहीं हो सकता है।

Question 26

The molecule that has an $S_6$ symmetry element is:

Accepted Answer

$SF_6$

Answer

$B_2H_6$

Answer

$CH_4$

Answer

$PH_5$

Question 27

A molecule contains the following symmetry operations: $E, 2C_6, 2C_3, C_2, 3\sigma_d$ & $3\sigma_v$. The number of classes and order of the symmetry point group is:

Accepted Answer

6, 12

Answer

3, 12

Answer

5, 12

Answer

6, 6

Question 28

The symmetry point group of propyne is:

Accepted Answer

$C_{3v}$

Answer

$C_3$

Answer

$D_3$

Answer

$D_{3d}$

Question 29

The point group symmetries for $trans	ext{-}[Cr(en)_2F_2]^+$ and $[TiCl_6]^{3-}$, respectively are:

Accepted Answer

$D_{4h}$ and $D_{3h}$

Answer

$D_{4d}$ and $D_{3d}$

Answer

$D_{3d}$ and $D_{4d}$

Answer

$D_{3h}$ and $D_{4h}$

Question 30

The product $C_2^x \sigma_{xy}$ ($C_2^x$ is the two-fold rotation axis around the x-axis and $\sigma_{xy}$ is the xy mirror plane) is:

Accepted Answer

$\sigma_{xz}$

Answer

$\sigma_{yz}$

Answer

$C_2^y$

Answer

$C_2^z$

Question 31

The symmetry point group of the most stable geometry of the following molecule $Cl(H)C=C=C(H)Cl$ is:

Accepted Answer

$C_2$

Answer

$C_1$

Answer

$C_{2v}$

Answer

$C_{2h}$

Question 32

The molecule that possesses $S_4$ symmetry element is:

Accepted Answer

एलीन (allene)

Answer

एथिलीन (ethylene)

Answer

बेंजीन (benzene)

Answer

1,3-ब्यूटाडाईन (1,3-butadiene)

Question 33

The molecule with a $C_2$ axis of symmetry among the following is:

Accepted Answer

$BH_2Cl$

Answer

$CH_3Cl$

Answer

$NH_2Cl$

Answer

$HOCl$

Question 34

$N_2O$ molecule belongs to the point group:

Accepted Answer

$C_{\infty v}$

Answer

$D_{\infty h}$

Answer

$C_{2v}$

Answer

$S_2$

Question 35

The total number of symmetry elements in diborane molecule is:

Accepted Answer

8

Answer

2

Answer

4

Answer

6

Question 36

A cube does not have the symmetry element:

Accepted Answer

$C_6$

Answer

$C_2$

Answer

$C_3$

Answer

$C_4$

Question 37

The following is the character table for the $C_{2v}$ point group:
$C_{2v}$$E$$C_2$$\sigma_v$$\sigma_v'$$A_1$1111$A_2$11-1-1$B_1$1-11-1$B_2$1-1-11
There are two functions $f_1$ and $f_2$ belonging to $A_2$ and $B_1$ representation, respectively. The correct option for the product of two functions $f_1$ and $f_2$ and the integral $\int f_1 f_2 d	au$ is:

Accepted Answer

गुणनफल $B_2$ निरूपण से संबंधित है और समाकल शून्य (zero) है।

Answer

गुणनफल $A_2$ निरूपण से संबंधित है और समाकल अशून्य (nonzero) है।

Answer

गुणनफल $A_1$ निरूपण से संबंधित है और समाकल शून्य (zero) है।

Answer

गुणनफल $B_1$ निरूपण से संबंधित है और समाकल अशून्य (nonzero) है।

Question 38

The point groups of cis- and trans-1,2-dichloroethylene are respectively:

Accepted Answer

$C_{2v}$ and $C_{2h}$

Answer

$C_{2v}$ and $C_{2v}$

Answer

$C_{2h}$ and $C_{2v}$

Answer

$C_{2h}$ and $C_{2h}$

Question 39

The combination of two reflections, $\sigma_v' \sigma_v''$ about an intersecting mirror plane is equivalent to:

Accepted Answer

$C_n$

Answer

$S_n$

Answer

$\sigma_h$

Answer

$i$

Question 40

Given the character table of the point group $C_{3v}$:
$C_{3v}$$E$$2C_3$$3\sigma_v$Linear / RotQuadratic$A_1$111$z$$x^2+y^2, z^2$$A_2$11-1$R_z$-$E$2-10$(x, y), (R_x, R_y)$$(x^2-y^2, xy), (xz, yz)$
Consider the reducible representation, $	au$:
$E$$2C_3$$3\sigma_v$$	au$630
Its irreducible components are:

Accepted Answer

$E + 2A_1 + 2A_2$

Answer

$2E + A_1 + A_2$

Answer

$3A_1 + 3A_2$

Answer

$E^2 + 2A_1$

Question 41

Character table of $C_{2v}$ point group is given below:
$C_{2v}$$E$$C_2$$\sigma_v$$\sigma_v'$Linear$A_1$1111$z$$A_2$11-1-1-$B_1$1-11-1$x$$B_2$1-1-11$y$
If the initial and final states belong to $A_1$ and $B_1$ irreducible representation respectively, the allowed electronic transition from $A_1$ to $B_1$ is:

Accepted Answer

$x$-ध्रुवित ($x$-polarized)

Answer

$z$-ध्रुवित ($z$-polarized)

Answer

$y$-ध्रुवित ($y$-polarized)

Answer

$x, z$-ध्रुवित ($x, z$-polarized)

Question 42

Identify the Mulliken notation for the following irreducible representation:
$E$$C_n$$nC_2$$i$$\sigma_h$11-1-1-1

Accepted Answer

$A_{2u}''$

Answer

$A_{1u}'$

Answer

$B_{2u}'$

Answer

$A_{2u}'$

Question 43

Identify the point group symmetry of the following molecule (all C−C bond lengths are equal):

X
  
  X
  
  X
  
  X

Accepted Answer

$D_{2d}$

Answer

$C_{2v}$

Answer

$S_4$

Answer

$D_{4d}$

Question 44

The transition that is allowed by x-polarized light in trans-butadiene is (The character table for $C_{2h}$ is given below):
$C_{2h}$$E$$C_2$$i$$\sigma_h$Linear / Quad$A_g$1111$R_z, x^2, y^2, z^2, xy$$B_g$1-11-1$R_x, R_y, xz, yz$$A_u$11-1-1$z$$B_u$1-1-11$x, y$

Accepted Answer

$^1A_u 	o ^1B_g$

Answer

$^1A_u 	o ^1A_u$

Answer

$^1B_u 	o ^1B_g$

Answer

$^1B_g 	o ^1A_g$

Question 45

The low temperature ($-98^\circ C$) $^{19}F$ NMR spectrum of $SF_4$ shows doublet of triplets. It is consistent with the point group symmetry:

Accepted Answer

$C_{2v}$

Answer

$C_{3v}$

Answer

$C_{4v}$

Answer

$T_d$

Question 46

The result of the product $C_2(x) C_2(y)$ is:

Accepted Answer

$C_2(z)$

Answer

$E$

Answer

$\sigma_{xy}$

Answer

$i$

Question 47

The $E \otimes E$ direct product in $D_3$ point group contains the irreducible representations:
$D_3$$E$$2C_3$$3C_2$$A_1$111$A_2$11-1$E$2-10

Accepted Answer

$A_1 + A_2 + E$

Answer

$2A_1 + E$

Answer

$2A_2 + E$

Answer

$2A_1 + 2A_2$

Question 48

The product $\sigma_{xy} \cdot S_4^z$ ($S_4^z$ is the four fold improper axis of rotation around the z axis, and $\sigma_{xy}$ is the reflection in the xy plane) is:

Accepted Answer

$C_4^z$

Answer

$C_4^z \cdot i$

Answer

$C_4^y$

Answer

$C_2^z$

Question 49

IR active normal modes of methane belong to the irreducible representation (The character table for $T_d$ is given below):
$T_d$$E$$8C_3$$3C_2$$6S_4$$6\sigma_d$Linear / Quad$A_1$11111$x^2+y^2+z^2$$A_2$111-1-1-$E$2-1200$2z^2-x^2-y^2, x^2-y^2$$T_1$30-11-1$R_x, R_y, R_z$$T_2$30-1-11$x, y, z, xy, yz, zx$

Accepted Answer

$T_2$

Answer

$E + A_1$

Answer

$E + A_2$

Answer

$T_1$

Question 50

The molecule diborane belongs to the symmetry point group:

Accepted Answer

$D_{2h}$

Answer

$C_{2v}$

Answer

$C_{2h}$

Answer

$D_{2d}$

Question 51

The irreducible representations of $C_{2h}$ are $A_g, B_g, A_u$, and $B_u$. The Raman active modes of trans-1,3-butadiene belong to the irreducible representations:

Accepted Answer

$A_g$ and $B_g$

Answer

$A_g$ and $A_u$

Answer

$A_u$ and $B_g$

Answer

$B_g$ and $B_u$

Question 52

The point group obtained by adding symmetry operation $\sigma_h$ to the point group $C_4$ is:

Accepted Answer

$C_{4h}$

Answer

$S_4$

Answer

$D_{2h}$

Answer

$D_4$

Question 53

The character table of $C_{3v}$ point group is provided below, along with an additional reducible representation $\Gamma$:
$C_{3v}$$E$$2C_3$$3\sigma_v$$A_1$111$A_2$11-1$E$2-10$\Gamma$602
$\Gamma$ is given as:

Accepted Answer

$2A_1 + 2E$

Answer

$A_1 + A_2 + 2E$

Answer

$2A_2 + 2E$

Answer

$2A_1 + 2A_2 + E$

Question 54

The character table for the $D_3$ point group is provided below:
$D_3$$E$$2C_3$$3C_2$Linear / Quad$A_1$111$x^2+y^2, z^2$$A_2$11-1$z, R_z$$E$2-10$(x, y), (R_x, R_y), (x^2-y^2, xy), (xz, yz)$
For this point group, the correct statement among the following is:

Accepted Answer

यह संभव है कि IR सक्रिय प्रसामान्य विधाओं का एक युग्म अपभ्रष्ट (degenerate) हो।

Answer

यह संभव है कि कंपन की एक पूर्णतः सममित प्रसामान्य विधा IR सक्रिय हो।

Answer

सभी IR सक्रिय प्रसामान्य विधाएं आवश्यक रूप से रमन निष्क्रिय होती हैं।

Answer

सभी रमन सक्रिय प्रसामान्य विधाएं आवश्यक रूप से IR सक्रिय होती हैं।

Question 55

For $H_2O$ molecule, the electronic transition from the ground state to an excited state of $B_1$ symmetry is:
$C_{2v}$$E$$C_2$$\sigma_v$$\sigma_v'$Linear / Quad$A_1$1111$z, z^2, x^2, y^2$$A_2$11-1-1$xy$$B_1$1-11-1$x, xz$$B_2$1-1-11$y, yz$

Accepted Answer

$x$-ध्रुवण के साथ अनुमत (allowed with $x$ polarisation)

Answer

अनुमत नहीं (not allowed)

Answer

$y$-ध्रुवण के साथ अनुमत (allowed with $y$ polarisation)

Answer

$z$-ध्रुवण के साथ अनुमत (allowed with $z$ polarization)

Question 56

The pair of symmetry points groups that are associated with only polar molecules is:

Accepted Answer

$C_{2v}, C_{\infty v}$

Answer

$C_{2v}, D_\infty$

Answer

$C_{3v}, C_{2h}$

Answer

$D_{2h}, C_{2h}$

Question 57

The compound $N_2F_2$ has two isomers. Choose the correct option from the following:

Accepted Answer

एक समावयवी में $\sigma_h$ तल होता है जबकि दूसरे में $\sigma_v$ तल होता है।

Answer

दोनों समावयवियों में $\sigma_v$ तल उपस्थित होता है।

Answer

दोनों समावयवियों में $\sigma_h$ तल उपस्थित होता है।

Answer

उनमें से किसी में भी $\sigma_h$ तल नहीं होता है।

Question 58

The correct relation involving symmetry operations:

Accepted Answer

$S_6^3 = S_2$

Answer

$S_4^2 = S_2$

Answer

$\sigma(xy)\sigma(yz) = C_2(x)$

Answer

$S_4^3 = C_4^3$

Question 59

For a point group, an incomplete character table is given below with one irreducible representation missing:
$E$$2C_3$$3\sigma_v$$A_1$111????$E$2-10
The Mulliken symbol and characters of the missing representation are:

Accepted Answer

$A_2 \quad 1 \quad 1 \quad -1$

Answer

$A_1' \quad 1 \quad -1 \quad 1$

Answer

$B_1 \quad 1 \quad -1 \quad -1$

Answer

$B_2 \quad 1 \quad -1 \quad 1$

Question 60

Given below is a specific vibrational mode of $BCl_3$ with $\oplus$ and $\ominus$ denoting movements of the respective atoms above and below the plane of the molecule respectively:

B
  
  +
  
  Cl
  
  −
  
  Cl
  
  −
  
  Cl
  
  −

The irreducible representation of the vibrational mode and its IR / Raman activity are:
$D_{3h}$$E$$2C_3$$3C_2$$\sigma_h$$2S_3$$3\sigma_v$Linear / Quad$A_1'$111111$x^2+y^2, z^2$$A_2'$11-111-1$R_z$$E'$2-102-10$(x, y), (x^2-y^2, xy)$$A_1''$111-1-1-1-$A_2''$11-1-1-11$z$$E''$2-10-210$(R_x, R_y), (xz, yz)$

Accepted Answer

$A_2''$; IR सक्रिय

Answer

$A_2'$; न तो IR और न ही Raman सक्रिय

Answer

$E'$; IR तथा Raman दोनों सक्रिय

Answer

$A_1'$; Raman सक्रिय

Question 61

A part of the character table of a point group (of order 4) is given below:
$E$$X_1$$X_2$$X_3$$\Gamma_1$1111$\Gamma_2$1-11-1$\Gamma_3$1-1-11$\Gamma_4$????
The four characters of $\Gamma_4$ are, respectively:

Accepted Answer

$1, 1, -1, -1$

Answer

$2, 0, 0, 2$

Answer

$1, i, i, 1$

Answer

$1, -i, i, -1$

Question 62

Two opposite sides (in the y-direction) of a square box of side L are slightly stretched. Consider the following four statements:
A. The point group changes from $D_{4h}$ to $D_{2h}$.
B. The (1,2) and (2,1) energy levels remain doubly degenerate.
C. Both the energy levels are lowered & the energy of the (1,2) level is higher than that of the (2,1) level.
D. Both the energy levels are lowered & the energy of the (1,2) level is lower than that of the (2,1) level.
The two correct statements are:

Accepted Answer

A and D

Answer

A and B

Answer

A and C

Answer

B and C

Question 63

A molecule $AB_2$ shows the following IR and Raman spectra:
$\bar{
u} (	ext{cm}^{-1})$IRRaman2215 vsvs, PRs, depol1250 vsvs, PRvs, pol.560 ss, PQR-
The structure of the molecule is:

Accepted Answer

रैखिक असममित ($C_{\infty v}$)

Answer

रैखिक सममित ($D_{\infty h}$)

Answer

कोणीय सममित ($C_{2v}$)

Answer

कोणीय असममित ($C_s$)

Question 64

Consider Table-1:
$D_{2h}$$E$$C_2(z)$$C_2(y)$$C_2(x)$$i$$\sigma(xy)$$\sigma(xz)$$\sigma(yz)$Linear / Quad$A_g$11111111$x^2, y^2, z^2$$B_{1g}$11-1-111-1-1$R_z, xy$$B_{2g}$1-11-11-11-1$R_y, xz$$B_{3g}$1-1-111-1-11$R_x, yz$$A_u$1111-1-1-1-1-$B_{1u}$11-1-1-1-111$x$$B_{2u}$1-11-1-11-11$y$$B_{3u}$1-1-11-111-1$z$
The $\pi_u$-orbital of ethylene, when placed in the xy-plane with the C=C bond aligned to the x-axis, transforms according to the irreducible representation:

Accepted Answer

$b_{2u}$

Answer

$a_u$

Answer

$b_{1u}$

Answer

$b_{3u}$

Molecule / System	Symmetry Elements Present	Point Group	Center of Inversion (i)?
Water (H₂O), SO₂	E, C₂, 2 σ_v	C_2v (h = 4)	No
Ammonia (NH₃), PCl₃	E, 2 C₃, 3 σ_v	C_3v (h = 6)	No
Boron Trifluoride (BF₃)	E, 2 C₃, 3 C₂, σ_h, 2 S₃, 3 σ_v	D_3h (h = 12)	No
Allene (CH₂=C=CH₂)	E, C₂, 2 C‘₂, 2 σ_d, 2 S₄	D_2d (h = 8)	No
Ethylene (C₂H₄)	E, 3 C₂, 3 σ, i	D_2h (h = 8)	Yes (Mutual Exclusion applies)
Ferrocene (Staggered conformer)	E, 2 C₅, 2 C₅², 5 C₂, i, 2 S₁₀, 5 σ_d	D_5d (h = 20)	Yes
Ferrocene (Eclipsed conformer)	E, 2 C₅, 2 C₅², 5 C₂, σ_h, 2 S₅, 5 σ_v	D_5h (h = 20)	No
Hydrogen Peroxide (Non-planar open-book)	E, C₂	C₂ (h = 2)	No

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