Group Theory and Chemistry (Dover Books on Chemistry)

Group Theory and Chemistry

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Bishop , Paperback, Reprint. Symmetry, symmetry operations, point groups, matrices, matrix representations, equivalent and reducible representations, irreducible representations and character tables, representations and quantum mechanics, molecular vibrations, molecular orbital theory, hybrid orbitals, and transition metal chemistry. Relevant math after each chapter. Group theoretical principles are an integral part of modern chemistry. Not only do they help account for a wide variety of chemical phenomena, they simplify quantum chemical calculations. Indeed, knowledge of their application to chemical problems is essential for students of chemistry.

There is a fruitful relation between infinite abstract groups and topological groups: A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups profinite groups: During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups , and other related groups.

One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups , which may be viewed as dealing with " continuous symmetry ", is strongly influenced by the associated Weyl groups.

These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry. Saying that a group G acts on a set X means that every element of G defines a bijective map on the set X in a way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.

For example, if G is finite, it is known that V above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole V via Schur's lemma. Given a group G , representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: The totality of representations is governed by the group's characters.

For example, Fourier polynomials can be interpreted as the characters of U 1 , the group of complex numbers of absolute value 1 , acting on the L 2 -space of periodic functions.

A Lie group is a group that is also a differentiable manifold , with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie , who laid the foundations of the theory of continuous transformation groups.

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Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures , which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations differential Galois theory , in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations.

An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. Groups can be described in different ways. A more compact way of defining a group is by generators and relations , also called the presentation of a group. The kernel of this map is called the subgroup of relations, generated by some subset D. Combinatorial group theory studies groups from the perspective of generators and relations.

The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element.

By relating the problem to Turing machines , one can show that there is in general no algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the group isomorphism problem , which asks whether two groups given by different presentations are actually isomorphic.

Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X , for example a compact manifold , then G is quasi-isometric i. Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure.

This occurs in many cases, for example. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: The identity keeping the object fixed is always a symmetry of an object.

Dover Books on Chemistry: Group Theory and Chemistry by David M. Bishop (1993, Paperback, Reprint)

Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object.

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The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms , and the symmetry group is the automorphism group of the object in question. Applications of group theory abound.

Almost all structures in abstract algebra are special cases of groups. Rings , for example, can be viewed as abelian groups corresponding to addition together with a second operation corresponding to multiplication. Therefore, group theoretic arguments underlie large parts of the theory of those entities.

Group theory and chemistry / [by] David M. Bishop - Details - Trove

Galois theory uses groups to describe the symmetries of the roots of a polynomial or more precisely the automorphisms of the algebras generated by these roots. The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory.

It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S 5 , the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory. Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in.

By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. I am advising a student on her senior math thesis.

She would like to involve chemistry somehow her other major. I know no chemistry. I have read in several places that representation theory features prominently in some areas of chemistry. When I try to read anything online I am not sure how much chemistry one needs to assume. So, if my student is in her senior year of a chemistry major, are there any places where she can start to look for a project in group theory and chemistry? However there are numerous books available.

Again chemwiki has a good documentation on group theory. Several answers and comments have suggested general texts that have good foundation information; however if a student is writing a thesis, these sources will be insufficient since they are geared towards the teaching of group theory rather than the application of group theory to generate new knowledge. Even at an undergraduate level, there should be an attempt to generate new knowledge in a thesis.

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