Tensor first rank

First rank (linear after pseudospin) Zeeman splitting tensorgap (i.e., the conventional tensor), its main values, including the sign of the product gxgySz> and the main magnetic axes (Xm, Ym,Zm) in the initial coordinate frame. 

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. 

A vectorial product will be defined below by (5.14), and V as a tensor of first rank is defined by (2.12). Operator L may be defined also in a more general way by the commutation relations of its components. Such a definition is applicable to electron spin s, as well. Therefore, we can write the following commutation relations between components of arbitrary angular momentum j ... 

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions (2.13), as well as the Clebsch-Gordan and 3n -coefficients. Below we shall illustrate this procedure by the examples of operators (1.16) and (2.1). Formulas (1.15), (1.18)—(1.22) present concrete expressions for each term of Eq. (1.16). It is convenient to divide all operators (1.15), (1.18)—(1.22) into two groups. The first group is composed of one-electron operators (1.18), the first two... 

There is no paradox [112] in the use of e(3) as an operator as well as a unit vector. In the same sense [112], there is no paradox in the use of the scalar spherical harmonics as operators. The rotation operators in space are first-rank Toperators, which are irreducible tensor operators, and under rotations, transform into linear combinations of each other. The Toperators are directly proportional to the scalar spherical harmonic operators. The rotation operators, J, of the full rotation group are related to the T operators as follows... 

This may be considered a scalar product of two vector operators L1 and S1 (the first-rank tensors)... 

Since the first-rank (orbital) unit tensor yields a simple expression... 

The angular momentum operator is a first-rank tensor whose Cartesian and spherical components transform into each other as... 

The unit tensor operator of the first rank is as simple as ... 

The Cartesian operators may be expressed through the components (q = -1, 0, +1) of the first-rank spherical irreducible tensor Lnamely,... 

First-rank tensor operators (vector operators), K = 1 ... 

A scalar is a zero-rank tensor an ordinary vector is a first-rank tensor. A Cartesian tensor T consists of 3 quantities. If the index sets are denoted as... 

The first selection rule is a consequence of the fact that the transition dipole moment has negative parity. The second reflects that the quantization axis is parallel to the polarization of the electric field. The third follows from the fact that the transition dipole moment is a tensor of rank 1 (corresponding to an angular momentum with quantum number 1). 

S= kX, provided they point in the same direction. If not, the quotient S/X is of more complicated form. Like the quotient of two integers, which is not always another integer, the quotient of two non-aligned vectors is not necessarily a vector itself, but something else, called a tensor. The scalar k for co-aligned vectors is a tensor of rank zero. A tensor of the first rank has three components, e.g. T = ayTj and is equivalent to a vector. A second rank tensor has the form of a square matrix with nine elements in three dimensions e.g. Tij = Yjk,iaikajiTu, and so forth. 

The state of strain in a body is fully described by a second-rank tensor, a strain tensor , and the state of stress by a stress tensor, again of second rank. Therefore the relationships between the stress and strain tensors, i.e. the Young modulus or the compliance, are fourth-rank tensors. The relationship between the electric field and electric displacement, i.e. the permittivity, is a second-rank tensor. In general, a vector (formally regarded as a first-rank tensor) has three components, a second-rank tensor has nine components, a third-rank tensor has 27 components and a fourth-rank tensor has 81 components. 

A first-rank tensor operator 3 V) is also called a vector operator. It has three components, 2T and jH j. Operators of this type are the angular momentum operators, for instance. Relations between spherical and Cartesian components of first-rank tensor operators are given in Eqs. [36] and [37], Operating with the components of an arbitrary vector operator ( 11 on an eigenfunction u1fF) of the corresponding operators and 3 yields... 

As an example, consider the product of two arbitrary first-rank tensor operators 0 and It is nine-dimensional and can be reduced to a sum of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin-spin coupling Hamiltonians. In terms of spherical and Cartesian components of 0 and J2, the resulting irreducible tensors are given in Tables 8 and 9, respectively.70... 

Table 8 Irreducible Spherical Compound Tensor Operators Resulting from a Product of Two First-Rank Tensor Operators J T and. 9.T...

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

The first term is a tensor of rank zero involving only spin variables. It does not contribute to the multiplet splitting of an electronic state but yields only a (small) overall shift of the energy and is, henceforth, neglected. The operator 7 is a traceless (irreducible) second-rank tensor operator, the form of which in Cartesian components is... 

The condition j + j > 1 for a matrix element of a first rank tensor operator implies, e.g., that there is no first-order SOC of singlet wave functions. Two doublet spin wave functions may interact via SOC, but the selection rule /+ / > 2 for i (2)(Eq. [171]) tells us that electronic spin-spin interaction does not contribute to their fine-structure splitting in first order. 

For the special case of a first-rank tensor operator and m1 = j and m = /, analytical formulas for the symmetry related factors in Eq. [172] have been worked out by McWeeny70 and by Cooper and Musher.80 Note, however, that the formulas in both publications contain typos concerning a sign or a square root. 

The resolution of this discrepancy is closely related to another question How is an operator such as S, when combined with j , capable of coupling electronic states of different multiplicities while, according to Eqs. [149] and [150], S as a first rank tensor operator is only able to change the Ms quantum number of a state, but not its S value. 

Further, so may be expressed formally as = J2i where the s, are the usual one-electron spin operators and the first rank tensor operators denote the spatial part of >so related to electron i. (In the BP Hamiltonian (Eq. [104]), for example, 2 corresponds to the terms in braces.) One then obtains... 

Second-rank tensors such as transport properties relate two first-rank tensors, or vectors. Thus, a second-rank tensor representing a physical property has nine components (32), usually written in 3 x 3 matrixlike notation. Each component is associated with two axes one from the set of some reference frame and one from the material frame. Three equations, each containing three terms on the right-hand side, are needed to describe a second-rank tensor exactly. For a general... 

We will present the effective Hamiltonian terms which describe the interactions considered, sometimes using cartesian methods but mainly using spherical tensor methods for describing the components. These subjects are discussed extensively in chapters 5 and 7, and at this stage we merely quote important results without justification. We will use the symbol T to denote a spherical tensor, with the particular operator involved shown in brackets. The rank of the tensor is indicated as a post-superscript, and the component as a post-subscript. For example, the electron spin vector A is a first-rank tensor, T1 (A), and its three spherical components are related to cartesian components in the following way ... 

The spherical components of the new first-rank tensor in (1.57) are defined, in the molecule-fixed axes system, by... 

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

Although there is a one-to-one correspondence between the first-rank cartesian and spherical tensors, the same is not true for second- and higher-rank cartesian tensors. 

The reduced matrix element of a first-rank tensor is given by... 

The remaining reduced matrix element in the last line is developed further by noting the construction of the first-rank tensor,... 

The first-rank spherical spin tensor components in (8.472) may now be rewritten in cartesian form using the definitions... 

The third term in equation (8.510) describes only the electron-nuclear spin dipolar interaction, with the first-rank tensor T1 (.S. C2) being constructed so that... 

The reduced matrix element in (9.124) is evaluated by decomposing the second-rank tensor into its constituent first-rank tensors one finds that... 

Classically, Raman spectroscopy arises from an induced dipole in a molecule resulting from the interaction of an electromagnetic field with a vibrating molecule. In electromagnetic theory, an induced dipole is a first-rank tensor formed from the dot product of the molecular polarizability and the oscillating electric field of the photon, (jl = a-E. Assuming a harmonic potential for the molecular vibration, and that the polarizability does not deviate significantly from its equilibrium value (a0) as a result of the vibration... 

The induced polarization in a piezoelectric, Pj, is a first-rank tensor (vector), and mechanical stress, is a second-rank tensor (nine components), which is represented in a Cartesian coordinate system with axes x, y, and z, as ... 

Because stress and strain are vectors (first-rank tensors), the forms of Eqs. 10.5 and 10.6 state that the elastic constants that relate stress to strain must be fourth-rank tensors. In general, an wth-rank tensor property in p dimensional space requires p" coefficients. Thus, the elastic stiffness constant is comprised of 81 (3 ) elastic stiffness coefficients,... 

The first non-linear polarizability is a tensor of rank 3 and possesses non-vanishing elements only in the case of molecules without a centre of symmetry. This Born distortion tensor was resorted to by Piekara for explaining the non-linear dielectric behaviour of nitrobenzene. In the case of axially symmetric molecules, the tensor of the first non-linear distortion (hyperpolarizability ) can be written as follows ... 

The physical meaning of the first-rank tensor can be seen when these are related to the components of the mean angular momentum vector (/). With the help of the Wigner—Eckart theorem one can show that... 

Prior to discussing the properties of octopolar molecules, it is instructive to consider first some of the basic properties of tensors. In general, any tensor of rank n can be decomposed in a sum of so-called irreducible tensors that are invariant under three-dimensional rotation [99] ... 

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