Operators First-rank tensor

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

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 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... 

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

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

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... 

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 evaluation of the SSC matrix elements is analogous to the treatment of the SOC matrix elements with exception that the SSC Hamiltonian of (49) is a product of two irreducible second-rank tensors operators. Because SSC has nonvanishing contribution in the first order, it is usually sufficient to neglect contributions from states of different multiplicities. Application of the WET theorem to SSC Hamiltonian of (49) reads (S = S ) ... 

It should be noted that the simplified angular momentum operators (48) satisfy the commutation relations (49) but are not tensor operators in that sense. However, one can use them to construct angular momentum operators that are first-rank tensors (i.e., so, (3) vectors) in the following way ... 

Case 4. When the first-rank tensor is an angular momentum operator (Tk = l, k = 0, Tt = 7(72), / = 1) then its reduced matrix elements become... 

In the above formulae, spherical components of the irreducible tensor operators occur. In changing to Cartesian components one can use the transformations listed in Table 1.13 (Section 1.5.3) the following relationships hold true (a) three components of the first-rank tensor (vector) operator are... 

The compound irreducible tensor operators of the second rank are constructed from the first-rank tensors (spherical vectors) as follows... 

Add the tensorial rank of each cofactor and subtract 2 for the dot operation. Scalars are zeroth-rank tensors and vectors are first-rank tensors. Since the del operator is a vector and convective momentum flux is a second-rank tensor, V pvv is a vector. The fcth component of V pvv is... 

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... 

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,... 

In a rotating molecule containing one quadrupolar nucleus there is an interaction between the angular momentum J of the molecule and the nuclear spin momentum I. The operator of this interaction can be written as a scalar product of two irreducible tensor operators of second rank. The first tensor operator describes the nuclear quadrupole moment and the second describes the electrical field gradient at the position of the nucleus under investigation. 

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. 

At first order, it can be shown that only the symmetrized part of the interaction tensor contributes to the frequency shift. The majority of second-order contributions arise from large EFG, the EFG tensor being symmetric by definition. Thus, only symmetric second-rank tensors T can be considered, which can be decomposed into two contributions T = isol3 + AT with D3 the identity matrix. The first term is the isotropic part so = l/3Tr(T) that is invariant by any local symmetry operation. The second term is the anisotropic contribution AT, a symmetric second-rank traceless tensor, which depends then on five parameters the anisotropy 8 and the asymmetry parameter rj that measures the deviation from axial symmetry, and three angles to orient the principal axes system (PAS) in the crystal frame. The most common convention orders the eigenvalues of AT such that IAzzL and defines 8 — Xzz and rj — fzvv i )... 

The reduced matrix element of a vector operator (tensor operator of the first rank) is expressed in terms of the 67-symbol as follows... 

---