GROUP THEORY TUTORIAL

What's new in GTPack 1.3

GTPack version 1.3 was released December, 2020.

Angular Momentum Package

GTPack 1.3 comes with a new package for angular momentum operators and representations.

GTJxgives the x component of the total angular momentum operator
GTJygives the y component of the total angular momentum operator
GTJzgives the z component of the total angular momentum operator

The components of the total angular momentum operator in terms of matrix representations acting on a finite sub space indexed by the total angular momentum quantum number J.

Components of the total angular momentum operator
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Verify commutation relations.
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GTJplusgives the raising operator
GTJminusgives the lowering operator

The raising operator acts as J+|j; m> = |j; m+1>

The lowering operator acts as J-|j; m> = |j; m-1>

J+ and J- are related to Jx and Jy:

J+= Jx + i Jy

J-= Jx - i Jy

Matrix representations
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Define a state vector
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Verify action of J+ and normalization.
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Verify action of J+ as a raising operator:
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GTJTransformapplies a symmetry transformation to the basis functions of an irreducible representation of O(3)
GTJMatrixgives the representation matrix of a symmetry element for an irreducible representation of O(3)
GTJTransform and GTJMatrix are closely related. While GTJTransform gives the action of a symmetry element on one specific basis function, GTJMatrix gives the transformation matrix of the entire subspace.
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GTAngularMomentumRepapplies a symmetry transformation to the basis functions of an irreducible representation of O(3)
GTAngularMomentumCharsgives the representation matrix of a symmetry element for an irreducible representation of O(3)

For the implementation of irreducible representations of O(3), SO(3) and SU(2) we follow [1].

[1] Altman, S. L., Rotations, quaternions, and double groups. Chapter 14. Clarendon, 1986

Install the point group Oh (GTInstallGroup)
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Calculate the character Table (GTCharacterTable)
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Calculate the character system of a single s, p, and d electron
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Calculate the qualitative splitting in a cubic crystal field
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Calculate a matrix representation for d-electrons
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Further modifications

GTPack 1.3 contains a new implementation of the modules GTLeftCosets and GTRightCosets. Furthermore, cosets are now ordered by the order of elements. The reordering has also been implemented in GTSGLeftCosets and GTSGRightCosets, respectively. GTGetIreps now outputs names of irreducible representations, e.g., if Mulliken notation is chosen. A new implementation of GTSubGroupQ and the option GOFast has been added.