“Molecular symmetry”的意思、由来-开放百科全书

The predominant framework for the study of molecular symmetry is group theory. Symmetry is useful in the study of molecular orbitals, with applications such as the Hückel method, ligand field theory, and the Woodward-Hoffmann rules. Another framework on a larger scale is the use of crystal systems to describe crystallographic symmetry in bulk materials.

Many techniques for the practical assessment of molecular symmetry exist, including X-ray crystallography and various forms of spectroscopy. Spectroscopic notation is based on symmetry considerations.

Symmetry concepts

Elements

The point group symmetry of a molecule can be described by 5 types of symmetry element.

Operations

The five symmetry elements have associated with them five types of symmetry operation, which leave the molecule in a state indistinguishable from the starting state. They are sometimes distinguished from symmetry elements by a caret or circumflex. Thus, Ĉ_n is the rotation of a molecule around an axis and Ê is the identity operation. A symmetry element can have more than one symmetry operation associated with it. For example, the C₄ axis of the square xenon tetrafluoride (XeF₄) molecule is associated with two Ĉ₄ rotations (90°) in opposite directions and a Ĉ₂ rotation (180°). Since Ĉ₁ is equivalent to Ê, Ŝ₁ to σ and Ŝ₂ to î, all symmetry operations can be classified as either proper or improper rotations.

Symmetry groups

Groups

The symmetry operations of a molecule (or other object) form a group, which is a mathematical structure usually denoted in the form (G,*) consisting of a set G and a binary combination operation say '*' satisfying certain properties listed below.

In a symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of a C₄ rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C₄. By convention the order of operations is from right to left.

For every pair of elements x and y in G, the product x*y is also in G.
( in symbols, for every two elements x, y∈G, x*y is also in G ).
This means that the group is closed so that combining two elements produces no new elements. Symmetry operations have this property because a sequence of two operations will produce a third state indistinguishable from the second and therefore from the first, so that the net effect on the molecule is still a symmetry operation.

For every x and y and z in G, both (x*y)*z and x*(y*z) result with the same element in G.

There must be an element ( say e ) in G such that product any element of G with e make no change to the element.

For each element ( x ) in G, there must be an element y in G such that product of x and y is the identity element e.

( in symbols, for each x∈G there is a y ∈ G such that x*y=y*x= e for every x∈G )

The order of a group is the number of elements in the group. For groups of small orders, the group properties can be easily verified by considering its composition table, a table whose rows and columns correspond to elements of the group and whose entries correspond to their products.

Point groups and permutation-inversion groups

The successive application (or composition) of one or more symmetry operations of a molecule has an effect equivalent to that of some single symmetry operation of the molecule. For example, a C₂ rotation followed by a σ_v reflection is seen to be a σ_v' symmetry operation: σ_v*C₂ = σ_v'. (Note that "Operation A followed by B to form C" is written BA = C).^[8] Moreover, the set of all symmetry operations (including this composition operation) obeys all the properties of a group, given above. So (S,*) is a group, where S is the set of all symmetry operations of some molecule, and * denotes the composition (repeated application) of symmetry operations.

This group is called the point group of that molecule, because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarizes all symmetry operations that all molecules in that category have.^[8] The symmetry of a crystal, by contrast, is described by a space group of symmetry operations, which includes translations in space.

One can determine the symmetry operations of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one USES a point group, the operations in it are not to be interpreted in the same way. Instead the operations are interpreted as rotating and/or reflecting the vibronic (vibration-electronic) coordinates and these operations commute with the vibronic Hamiltonian. They are "symmetry operations" for that vibronic Hamiltonian. The point group is used to classify by symmetry the vibronic eigenstates. The symmetry classification of the rotational levels, the eigenstates of the full (rovibronic nuclear spin) Hamiltonian, requires the use of the appropriate permutation-inversion group as introduced by Longuet-Higgins^[9]. The relation between point groups and permutation-inversion groups is explained in this pdf file [https://chemphys.ca/pbunker/PointGroupspdffile.html Link] .

Examples of point groups

Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl₃, POF₃, XeO₃, and NH₃ all share identical symmetry operations.^[10] They all can undergo the identity operation E, two different C₃ rotation operations, and three different σ_v plane reflections without altering their identities, so they are placed in one point group, C_3v, with order 6.^[11] Similarly, water (H₂O) and hydrogen sulfide (H₂S) also share identical symmetry operations. They both undergo the identity operation E, one C₂ rotation, and two σ_v reflections without altering their identities, so they are both placed in one point group, C_2v, with order 4.^[12] This classification system helps scientists to study molecules more efficiently, since chemically related molecules in the same point group tend to exhibit similar bonding schemes, molecular bonding diagrams, and spectroscopic properties.^[8]

Common point groups

The following table contains a list of point groups labelled using the Schoenflies notation, which is common in chemistry and molecular spectroscopy. The description of structure includes common shapes of molecules, which can be explained by the VSEPR model.

Representations

The symmetry operations can be represented in many ways. A convenient representation is by matrices. For any vector representing a point in Cartesian coordinates, left-multiplying it gives the new location of the point transformed by the symmetry operation. Composition of operations corresponds to matrix multiplication. Within a point group, a multiplication of the matrices of two symmetry operations leads to a matrix of another symmetry operation in the same point group.^[8] For example, in the C_2v example this is:

Although an infinite number of such representations exist, the irreducible representations (or "irreps") of the group are commonly used, as all other representations of the group can be described as a linear combination of the irreducible representations.

Character tables

For each point group, a character table summarizes information on its symmetry operations and on its irreducible representations. As there are always equal numbers of irreducible representations and classes of symmetry operations, the tables are square.

The table itself consists of characters that represent how a particular irreducible representation transforms when a particular symmetry operation is applied. Any symmetry operation in a molecule's point group acting on the molecule itself will leave it unchanged. But, for acting on a general entity, such as a vector or an orbital, this need not be the case. The vector could change sign or direction, and the orbital could change type. For simple point groups, the values are either 1 or −1: 1 means that the sign or phase (of the vector or orbital) is unchanged by the symmetry operation (symmetric) and −1 denotes a sign change (asymmetric).

The tables also capture information about how the Cartesian basis vectors, rotations about them, and quadratic functions of them transform by the symmetry operations of the group, by noting which irreducible representation transforms in the same way. These indications are conventionally on the righthand side of the tables. This information is useful because chemically important orbitals (in particular p and d orbitals) have the same symmetries as these entities.

Consider the example of water (H₂O), which has the C_2v symmetry described above. The 2p_x orbital of oxygen has B₁ symmetry as in the fourth row of the character table above, with x in the sixth column). It is oriented perpendicular to the plane of the molecule and switches sign with a C₂ and a σ_v'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B₁ irreducible representation. Likewise, the 2p_z orbital is seen to have the symmetry of the A₁ irreducible representation (i.e.: none of the symmetry operations change it), 2p_y B₂, and the 3d_xy orbital A₂. These assignments and others are noted in the rightmost two columns of the table.

Historical background

Molecular nonrigidity

Point groups are useful for describing rigid molecules which undergo only small oscillations about a single equilibrium geometry, and for which the distorting effects of molecular rotation can be ignored, so that the symmetry operations all correspond to simple geometrical operations. However Longuet-Higgins has introduced a more general type of symmetry group suitable not only for rigid molecules but also for non-rigid molecules that tunnel between equivalent geometries (called versions) and which can also allow for the distorting effects of molecular rotation.^[9]^[16] These groups are known as permutation-inversion groups, because the symmetry operations in them are energetically feasible permutations of identical nuclei, or inversion with respect to the center of mass, or a combination of the two.

For example, ethane (C₂H₆) has three equivalent staggered conformations. Tunneling between the conformations occurs at ordinary temperatures by internal rotation of one methyl group relative to the other. This is not a rotation of the entire molecule about the C₃ axis. Although each conformation has D_3d symmetry, as in the table above, description of the internal rotation and associated quantum states and energy levels requires the more complete permutation-inversion group G₃₆.

Similarly, ammonia (NH₃) has two equivalent pyramidal (C_3v) conformations which are interconverted by the process known as nitrogen inversion. This is not an inversion in the sense used for point group symmetry operations of rigid molecules (i.e., the inversion of vibrational displacements and electronic coordinates in the center of mass) since NH₃ has no inversion center. Rather it the inversion of all nuclei and electrons in the center of mass (close to the nitrogen atom), which happens to be energetically feasible for this molecule. The appropriate permutation-inversion group to be used in this situation is D_3h(M) which is isomorphic with the point group D_3h.

Additionally, as examples, the methane (CH₄) and H₃⁺ molecules have highly symmetric equilibrium structures with T_d and D_3h point group symmetries respectively; they lack permanent electric dipole moments but they do have very weak pure rotation spectra because of rotational

centrifugal distortion.^[17]^[18] The permutation-inversion groups required for the complete study of CH₄ and H₃⁺ are T_d(M) and D_3h(M), respectively.

A second and less general approach to the symmetry of nonrigid molecules is due to Altmann.^[19]^[20] In this approach the symmetry groups are known as Schrödinger supergroups and consist of two types of operations (and their combinations): (1) the geometric symmetry operations (rotations, reflections, inversions) of rigid molecules, and (2) isodynamic operations, which take a nonrigid molecule into an energetically equivalent form by a physically reasonable process such as rotation about a single bond (as in ethane) or a molecular inversion (as in ammonia).^[20]

See also

References

1. ^Quantum Chemistry, Third Edition John P. Lowe, Kirk Peterson {{ISBN|0-12-457551-X}}
2. ^Physical Chemistry: A Molecular Approach by Donald A. McQuarrie, John D. Simon {{ISBN|0-935702-99-7}}
3. ^The chemical bond 2nd Ed. J.N. Murrell, S.F.A. Kettle, J.M. Tedder {{ISBN|0-471-90760-X}}
4. ^Physical Chemistry P.W. Atkins and J. de Paula (8th ed., W.H. Freeman 2006) {{ISBN|0-7167-8759-8}}, chap.12
5. ^G. L. Miessler and D. A. Tarr Inorganic Chemistry (2nd ed., Pearson/Prentice Hall 1998) {{ISBN|0-13-841891-8}}, chap.4.
6. ^{{cite web |url=http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/CharacterTables.html |title=Symmetry Operations and Character Tables |last= |first= |date=2001 |website=University of Exeter |access-date=29 May 2018 }}
7. ^LEO Ergebnisse für "einheit"
8. ^¹²³⁴{{Cite book|title=Principles of Inorganic Chemistry|last=Pfenning|first=Brian|publisher=John Wiley & Sons|year=2015|isbn=9781118859025|location=|pages=}}
9. ^¹{{cite journal | last1 = Longuet-Higgins | first1 = H.C. | year = 1963 | title = The symmetry groups of non-rigid molecules | url = | journal = Molecular Physics | volume = 6 | issue = 5| pages = 445–460 | doi = 10.1080/00268976300100501 | bibcode = 1963MolPh...6..445L }}
10. ^{{cite book|last1=Pfennig|first1=Brian|title=Principles of Inorganic Chemistry|publisher=Wiley|isbn=978-1-118-85910-0|pages=191}}
11. ^{{cite book|last1=pfennig|first1=Brian|title=Principles of Inorganic Chemistry|publisher=Wiley|isbn=978-1-118-85910-0}}
12. ^{{cite book|last1=Miessler|first1=Gary|title=Inorganic Chemistry|publisher=Pearson|isbn=9780321811059}}
13. ^Group Theory and its application to the quantum mechanics of atomic spectra, E. P. Wigner, Academic Press Inc. (1959)
14. ^Correcting Two Long-Standing Errors in Point Group Symmetry Character Tables Randall B. Shirts J. Chem. Educ. 2007, 84, 1882. Abstract
15. ^{{cite journal | last1 = Rosenthal | first1 = Jenny E. | last2 = Murphy | first2 = G. M. | year = 1936 | title = Group Theory and the Vibrations of Polyatomic Molecules | url = | journal = Rev. Mod. Phys. | volume = 8 | issue = | pages = 317–346 | doi = 10.1103/RevModPhys.8.317 | bibcode=1936RvMP....8..317R}}
16. ^Philip R. Bunker and Per Jensen (2005), Fundamentals of Molecular Symmetry (Institute of Physics Publishing) {{ISBN|0-7503-0941-5}}
17. ^{{cite journal | last1 = Watson | first1 = J.K.G | year = 1971 | title = Forbidden rotational spectra of polyatomic molecules | url = | journal = Journal of Molecular Spectroscopy | volume = 40 | issue = 3| pages = 546-544 | doi = 10.1016/0022-2852(71)90255-4 | bibcode = 1971JMoSp..40..536W }}
18. ^{{cite journal | last = Oldani | first = M.|display-authors=etal | year = 1985 | title =Pure rotational spectra of methane and methane-d4 in the vibrational ground state observed by microwave Fourier transform spectroscopy| url = | journal = Journal of Molecular Spectroscopy | volume = 110 | issue = 1| pages = 93-105 | doi = 10.1016/0022-2852(85)90215-2 | bibcode = 1985JMoSp.110...93O}}
19. ^Altmann S.L. (1977) Induced Representations in Crystals and Molecules, Academic Press
20. ^¹Flurry, R.L. (1980) Symmetry Groups, Prentice-Hall, {{ISBN|0-13-880013-8}}, pp.115-127