Tensor of rank

The spin operator S is an irredueible tensor of rank one with the following transformational properties... 

An important purpose of tensor analysis is to describe any physical or geometrical quantity in a form that remains invariant under a change of coordinate system. The simplest type of invariant is a scalar. The square of the line element ds of a space is an example of a scalar, or a tensor of rank zero. 

Tensors of higher rank are defined in the same way, for example, a mixed tensor of rank four is... 

This result shows that, by its transformation properties, Aljkl is equivalent to a covariant vector of rank two. This process of summing over a pair of contravariant and covariant indices is called contraction. It always reduces the rank of a mixed tensor by two and thus, when applied to a mixed tensor of rank two, the result is a scalar ... 

The spin interactions, dipole-dipole (D), nuclear electric quadrupole (Q) and chemical shielding (C.S), may be formally written in terms of irreducible tensors of rank l34 in Hz ... 

One defines2 a spherical tensor of rank k by the commutation relations [/z,rf] = K7f ... 

Two ingredients are needed to compute the intensities of transitions the wave functions of the initial and final states and the form of the transition operator (Ogilvie and Tipping, 1983). For infrared transitions the appropriate operator is the dipole operator, M(r, 0, (j>). This operator is a vector (tensor of rank 1) and thus can be written as... 

A computation of Raman intensities can be done precisely in the same way as for infrared intensities. One needs here, in addition to the wave functions of the initial and final state, the polarizability tensor a(r,0, < )). This is a symmetric tensor of rank 2 that in Cartesian coordinates can be written as... 

The result (2.167) is particularly important, since it is used to analyze experimental data. It is merely a consequence of the fact that the quadrupole operator is a tensor of rank 2. Sj is just the square of the Clebsch-Gordan coefficients in... 

The transition operator, or electric multipole operator, is a tensor of rank k and it is given the symbol We, thus, have... 

BTAs(m, 0) collects simple operations on tensors of rank m, e.g. copying. btas(1,1) is equivalent to blasI, btas(2, 1) is equivalent to blas2, and btas(2, 2) to blas3. 

Let us briefly recall a few of the basics of the algebra of tensors. An nth rank tensor in m-dimensional space is an object with n indices and rrf components. For a general tensor a distinction is made between contravari-ant (upper) indices and covariant (lower) indices. A tensor of rank mi + m2 may have mi contravariant indices and m2 covariant indices. The order of the indices is significant. Tensors can be classified according to whether they are... 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

The factor C k of B(R) is often called the /th component of the Racah spherical tensor of rank k the three tensor components of rank 1 may be considered unit basis vectors spanning the (spherical) space. 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

In this book, vector quantities such as x and y above are normally column vectors. When necessary, row vectors are indicated by use of the transpose (e.g., r). If the components of x and y refer to coordinate axes [e.g., orthogonal coordinate axes ( i, 2, 3) aligned with a particular choice of right, forward, and up in a laboratory], the square matrix M is a rank-two tensor.9 In this book we denote tensors of rank two and higher using boldface symbols (i.e., M). If x is an applied force and y is the material response to the force (such as a flux), M is a rank-two material-property tensor. For example, the full anisotropic form of Ohm s law gives a charge flux Jq in terms of an applied electric field E as... 

By computing the commutators of the components of the quasispin operator with electron creation and annihilation operators, we can directly see that the latter behave as the components of a tensor of rank q = 1/2 in quasispin space and obey the relationship of the type (14.2)... 

Since the wave functions with N > v can be found from the wave functions with N = v using (16.1), in the second-quantization representation it is necessary to construct in an explicit form only wave functions with the number of electrons minimal for given v, i.e. IolQLSMq = —Q). But even such wave functions cannot be found by generalizing directly relation (15.4) if operator cp is still defined so that it would be an irreducible tensor in quasispin space, then the wave function it produces in the general case will not be characterized by some value of quantum number Q v). This is because the vacuum state 0) in quasispin space of one shell is not a scalar, but a component of a tensor of rank Q = l + 1/2... 

The rank of the operator q> in quasispin space will only be determined by the number of electrons N and can be ignored as a characteristic. In particular, operator - a tensor of rank one in quasispin space - acting on the vacuum state gives rise to the two-electron wave function with v = 0... 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

We shall now introduce the electron creation operators a and b and the electron annihilation operators a = (—b = (—1 y mb for the electrons in the subshells nihjNl and n2hjNl, respectively. They are irreducible tensors of rank t = 1 /2... 

A quantity T that is invariant under all proper and improper rotations (that is, under all orthogonal transformations) so that T = T, is a scalar, or tensor of rank 0, written 7(0). If T is invariant under proper rotations but changes sign on inversion, then it is a pseudoscalar. 

Pseudoscalars with the property T T (where the positive sign applies to proper rotations and the negative sign applies to improper rotations) are also called axial tensors of rank 0, 7 (0)ax. A quantity T with three components 7 7 2 7) that transform like the coordinates x x2 x3 of a point P, that is like the components of the position vector r, so that... 

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